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Introduction to Management Science, 10e (Taylor)

Chapter 1 Management Science

1) Management science involves the philosophy of approaching a problem in a subjective manner.

2) Management science techniques can be applied only to business and military organizations.

3) Once management scientist makes his or her decision and recommendation to management, then typically, his or her involvement with the problem is finished.

4) A variable is a value that is usually a coefficient of a parameter in an equation.

5) Parameters are known, constant values that are usually coefficients of variables in equations.

6) Data are pieces of information from the problem environment.

7) A model is a mathematical representation of a problem situation including variables, parameters, and equations.

8) A management science technique usually applies to a specific model type.

9) The first step of the management science process is to define the problem.

10) Management science modeling techniques provide results that are known with certainty.

11) The term sensitivity analysis refers to testing how a problem solution reacts to changes in one or more of the model parameters.

12) Fixed costs depend on the number of items produced.

13) Variable costs depend on the number of items produced.

14) Fixed cost is the difference between total cost and total variable cost.

15) The break-even point is the volume that equates total revenue with total cost.

16) In general, an increase in price increases the break even point if all costs are held constant.

17) If variable costs increase, but price and fixed costs are held constant, the break even point will decrease.

18) Managers utilize spreadsheets to conduct their own analyses in management science studies.

19) Management science techniques focus primarily on observation, model construction and implementation to find an appropriate solution to a problem.

20) Management science modeling techniques focus on model construction and problem solution.

21) Decision Support Systems (DSS) use computers to help decision makers address complex problems.

22) Enterprise Resource Planning (ERP) system is a data oriented decision support system that utilizes specific management science solution procedures to solve individual problems such as cost-volume analysis.

23) A __________ is a symbol used to represent an item that can take on any value.

24) __________ are known, constant values that are coefficients of variables in equations.

25) __________ are pieces of information from the problem environment.

26) A __________ is a functional relationship including variables, parameters, and equations.

27) Management science techniques include __________ techniques, models that are represented as diagrams, presenting a pictorial representation of the system being analyzed.

28) __________ techniques provide results that contain uncertainty, unlike mathematical programming techniques which are deterministic.

29) __________ costs are independent of the volume of goods produced and remain constant.

30) __________ depend on the number of items produced.

31) Total revenue minus total cost equals __________ .

32) The __________ is the volume that equates total revenue with total cost.

33) A __________ represents a limitation to achieving maximum profits due to limited resources.

34) A __________ programming technique refers to a predetermined set of mathematical steps used to solve a problem.

35) A __________ is a computer-based system that helps decision-makers address complex problems that involve different parts of an organization and operations.

36) The relationship d = 5000 – 25p describes what happens to demand (d) as price (p) varies. Price can vary between $10 and $50. How many units can be sold when the price is $10?

37) The supplier of cans for Coors Brewery, Valley Metal Container, uses a __________ to determine the weekly production schedule for cans in order to meet brewery demand.

38) A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the revenue per unit is projected to be $45. Write a mathematical expression for total cost.

39) A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the revenue per unit is projected to be $45. Write a mathematical expression for total revenue.

40) A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the revenue per unit is projected to be $45. Write a mathematical expression for total profit.

41) A production process requires a fixed cost of $50,000. The variable cost per unit is $25 and the revenue per unit is projected to be $45. Find the break-even point.

42) A production process requires a fixed cost of $50,000 and the variable cost per unit is $25. The revenue per unit was projected to be $45 but a recent marketing study shows that because of an emerging competitor, the revenue will be about 12% lower. How does this affect the break even point?

43) Administrators at a university will charge students $150 to attend a seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. How many students would have to register for the seminar for the university to break even?

44) Administrators at a university are planning to offer a summer seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. If we know that 20 people will attend, what price should be charged per person to break even?

45) Administrators at a university are planning to offer a summer seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. If 30 students attend the seminar, how much of a profit (or loss) will be incurred?

46) A newly opened bed-and-breakfast projects the following:

Monthly fixed costs $8000

Variable cost per occupied room per night $40

Revenue per occupied room per night $165

Write the expression for total cost per month.

47) A newly opened bed-and-breakfast projects the following:

Monthly fixed costs $8000

Variable cost per occupied room per night $40

Revenue per occupied room per night $165

Write the expression for total revenue per month.

48) A newly opened bed-and-breakfast projects the following:

Monthly fixed costs $8000

Variable cost per occupied room per night $40

Revenue per occupied room per night $165

How many rooms would have to be occupied per month in order to break even?

49) A script writer has received an advance against royalties of $10000. The royalty rate is $2 for every performance in the US, and $3 for every performance outside the US. Define variables for this problem.

50) A script writer has received an advance against royalties of $10,000. The royalty rate is $2 for every performance in the US, and $3 for every performance outside the US. Write an expression that could be used to compute the number of performances in order to cover the advance.

51) Students are organizing a “Battle of the Bands” contest. They know that at least 100 people will attend. The rental fee for the hall is $150 and the winning band will receive $500. In order to guarantee that they break even, how much should they charge for each ticket?

52) A manufacturer buys peas for vegetable pies from 2 cooperatives. The price per unit is $6 from cooperative A, and $5.50 per unit from cooperative B. Define variables that would tell how many units to purchase from each source.

53) A manufacturer buys peas for vegetable pies from 2 cooperatives. The price per unit is $6 from cooperative A, and $5.50 per unit from cooperative B. Develop an objective function that would minimize the total cost.

54) A manufacturer buys peas for vegetable pies from 2 cooperatives. The price per unit is $6 from cooperative A, and $5.50 per unit from cooperative B. The manufacturer needs at least 12000 units of peas. Cooperative A can supply up to 8000 units, and cooperative B can supply at least 6000 units. Develop constraints for these conditions.

55) A manager of the cereal bar at the college campus has determined that the profit made for each bowl of Morning Buzz cereal sold, x, is equal to: Z = $4x – 0.5x. Each bowl of Morning Buzz weighs 6 ounces, and the manager has 12 lbs (192 ounces) of cereal available each day, which can be written as the constraint, 6x ≤192. How much profit will be made from Morning Buzz if it is all sold in one day?

56) The College Coffee Café buys tea from 3 suppliers. The price per pound is $15.00 from supplier A, $17.50 from supplier B, and $21.00 from supplier C. They have budged $175 to purchase the tea. The café needs at least 12 pounds of tea, and supplier C can supply no more than 4 pounds. Develop constraints for these conditions.

57) The College Coffee Café receives a profit of $1.25 for each cup of house tea that they sell, $1.40 for each cup of the premium brand, and $1.50 for each cup of their special blend that they sell. Develop an objective that maximizes profit.

58) The steps of the management science process are:

A) problem definition, model construction, observation, model solution, implementation.

B) observation, problem definition, model construction, model solution, implementation.

C) model construction, problem definition, observation, model solution, implementation.

D) observation, implementation, problem definition, model construction, model solution.

59) A model is a functional relationship that includes:

A) variables

B) parameters

C) equations

D) all of the above

60) Which of the following is an equation or an inequality that expresses a resource restriction in a mathematical model?

A) a decision variable.

B) data

C) an objective function.

D) a constraint.

E) a parameter.

61) Which of the following is incorrect with respect to the use of models in decision making?

A) they improve understanding of the problem

B) they promote subjectivity in decision making

C) they are generally easy to use

D) they provide a systematic approach to problem solving

62) The field of management science

A) approaches decision making rationally with techniques based on the scientific method

B) is another name for decision science and for operations research

C) concentrates on the use of quantitative methods to assist managers in decision making

D) all of the above

63) The processes of problem observation

A) cannot be done until alternatives are proposed

B) requires consideration of multiple criteria

C) is the first step of decision making

D) is the final step of problem solving

64) The limits of the problem and the degree to which it pervades other units in the organization must be included during the __________ step of the management science process.

A) observation

B) definition

C) solution

D) implementation

65) __________ involves determining the functional relationship between variables, parameters and equations

A) Problem observation

B) Problem definition

C) Model construction

D) Model solution

E) Model implementation

66) Which steps of the management science process can either be a recommended decision or information that helps a manager make a decision?

A) model implementation

B) model construction

C) problem definition

D) model solution

E) problem formulation

67) The quantitative analysis approach requires

A) mathematical expressions for the relationship

B) uncomplicated problems

C) the manager to have prior experience with similar problems

D) all of the above

68) The result of an effective decision making process should be monitored in order to

A) reveal wrong assumptions

B) reveal errors in the implementation

C) insure the achievement of desired results

D) all of the above

69) The management science process does not include

A) problem definition

B) feedback

C) implementation

D) subjective preference

E) information

70) The indicator that results in total revenues being equal to total cost is called the

A) marginal cost

B) marginal volume

C) break-even point

D) profit mix

71) Variable cost

A) depends on the number of units produced

B) plus marginal cost equals fixed cost

C) is equal to total cost in deterministic models

D) is the same as average cost

72) The components of break-even analysis are

A) cost and profit

B) volume and cost

C) volume, cost and profit

D) volume and profit

73) __________ are generally independent of the volume of units produced and sold.

A) Fixed costs

B) Variable costs

C) Profits

D) average cost

74) The purpose of break-even analysis is to determine the number of units of a product to sell that will

A) appeal to the consumer

B) result in a profit

C) result in a loss

D) result in zero profit

75) Variable cost does not include

A) raw materials and resources

B) staff and management salaries

C) material handling and freight

D) direct labor and packaging

76) Which variable is not a component of break-even analysis?

A) fixed costs

B) variable costs

C) number of employees

D) total costs

E) number of customers

77) At the break-even point

A) total revenue equals total cost

B) profit is maximized

C) revenue is maximized

D) costs are minimized

78) If the price increases but fixed and variable costs do not change, the break even point

A) decreases

B) increases

C) remains the same

D) may increase or decrease, depending on sales

79) If the price decreases but fixed and variable costs do not change, the break even point

A) decreases

B) increases

C) remains the same

D) may increase or decrease, depending on sales

80) The term __________ refers to testing how a problem solution reacts to changes in one or more of the model parameters.

A) graphical solution

B) decision analysis

C) decision science

D) sensitivity analysis

E) break-even analysis

81) If fixed costs decrease, but variable cost and price remain the same, the break even point

A) decreases

B) increases

C) remains the same

D) may increase or decrease depending on sales

82) If fixed costs increase, but variable cost and price remain the same, the break even point

A) decreases

B) increases

C) remains the same

D) may increase or decrease depending on sales

83) EKA manufacturing company produces Part # 2206 for the aerospace industry. Each unit of part # 2206 is sold for $15. The unit production cost of part # 2206 is $3. The fixed monthly cost of operating the production facility is $3000. How many units of part # 2206 have to be sold in a month to break-even?

A) 166.67

B) 200

C) 250

D) 500

E) 1000

84) EKA manufacturing company produces Part # 2206 for the aerospace industry. The unit production cost of part # 2206 is $3. The fixed monthly cost of operating the production facility is $3000. Next month’s demand for part # 2206 is 200 units. How much should the company charge for each unit of part # 2206 to break-even?

A) 10

B) 12

C) 15

D) 18

E) 20

85) A bed and breakfast even every month if they book 30 rooms over the course of a month. Their fixed cost is $6000 per month and the revenue they receive from each booked room is $180. What their variable cost per occupied room?

A) $30

B) $40

C) $48

D) $62

86) Administrators at a university will charge students $150 to attend a seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. How many students would have to register for the seminar for the university to break even?

A) 16

B) 18

C) 20

D) 24

E) 30

87) A university is planning a seminar. It costs $3000 to reserve a room, hire an instructor, and bring in the equipment. Assume it costs $25 per student for the administrators to provide the course materials. If we know that 20 people will attend, what price should be charged per person to break even?

A) 100

B) 120

C) 150

D) 175

E) 200

88) It costs $50,000 to start a production process. Variable cost is $25 per unit and revenue is $$45 per unit. What is the break even point?

A) 1000 units

B) 1111 units

C) 2000 units

D) 2500 units

89) Which of the following statements is false?

A) Decision models selectively describe the managerial situation.

B) Decision models consider all factors from the real world.

C) Decision models designate performance measures that reflect objectives.

D) Decision models designate decision variables.

90) A difficult aspect of using spreadsheets to solve management science problems is

A) obtaining the solution to standard management science problems

B) data entry

C) performing sensitivity analysis

D) setting up a spreadsheet with complex models and formulas

91) A technique that assumes certainty in its solution is referred to as

A) indeterminate

B) probabilistic

C) deterministic

D) parametric

92) Classification of management science techniques does not recognize

A) linear mathematical programming

B) probabilistic techniques

C) network techniques

D) computer programming

93) Linear mathematical programming techniques assume that all parameters in the models are

A) known with certainty

B) unknown

C) predictable

D) unpredictable

94) Decision analysis is a __________ technique.

A) linear mathematical programming

B) probabilistic

C) network

D) simulation

E) non-linear programming technique

95) Which one of the following techniques is not a mathematical programming technique?

A) linear programming models

B) transportation models

C) analytical hierarchy process

D) goal programming

E) integer linear programming technique

96) Which one of the following management science methods is not a probabilistic technique?

A) assignment models

B) decision analysis

C) queuing analysis

D) statistical analysis

97) A baker uses organic flour from a local farmer in all of his baked goods. For each batch of bread (x1), he uses 4 lbs. For a batch of cookies (x2), he uses 3 pounds, and for a batch of muffins (x3) he uses 2 pounds. The local farmer can supply him with no more than 24 pounds per week. The constraint that represents this condition is:

A) x1 ≤ 8, x2 ≤ 8, x3 ≤ 8

B) x1+ x2 + x3 ≥ 24

C) x1 ≤ 6, x2 ≤ 8, x3 ≤ 12

D) x1+ x2 + x3 ≤ 24

E) 4×1+ 3×2 + 2×3 ≤ 24

98) An objective function

A) is a part of a model

B) represents the objective of the firm

C) can represent costs or profits

D) A and B only

E) all of the above

99) Larry’s Fish Market buys salmon (S) for $5 per pound and a local whitefish (W) for $3.50 per pound. Larry wants to minimize his cost, but he cannot spend more than $160. The objective function that minimizes these costs for Larry is:

A) 5S + 3.5W = 160

B) 5S + 3.5W ≤ 160

C) Min 5S + 3.5 W

D) Max 5S + 3.5 W

E) 5S + 3.5W ≥ 160

100) Taco Bell used which of the following management science techniques to help save over $53 million?

A) linear programming and network analysis

B) forecasting, queuing theory and inventory analysis

C) goal programming and network analysis

D) forecasting, simulation and integer programming

Introduction to Management Science, 10e (Taylor)

Chapter 2 Linear Programming: Model Formulation and Graphical Solution

1) Linear programming is a model consisting of linear relationships representing a firm’s decisions given an objective and resource constraints.

2) The objective function is a linear relationship reflecting the objective of an operation.

3) A constraint is a linear relationship representing a restriction on decision making.

4) A linear programming model consists of only decision variables and constraints.

5) A feasible solution violates at least one of the constraints.

6) Proportionality means the slope of a constraint is proportional to the slope of the objective function.

7) The terms in the objective function or constraints are additive.

8) The terms in the objective function or constraints are multiplicative.

9) The values of decision variables are continuous or divisible.

10) All model parameters are assumed to be known with certainty.

11) In linear programming models , objective functions can only be maximized.

12) All linear programming models exhibit a set of constraints.

13) Linear programming models exhibit linearity among all constraint relationships and the objective function.

14) The equation 8xy = 32 satisfies the proportionality property of linear programming.

15) Objective functions in linear programs always minimize costs.

16) The feasible solution area contains infinite solutions to the linear program.

17) There is exactly one optimal solution point to a linear program.

18) The following equation represents a resource constraint for a maximization problem: X + Y ≥ 20

19) A minimization model of a linear program contains only surplus variables.

20) In the graphical approach, simultaneous equations may be used to solve for the optimal solution point.

21) Slack variables are only associated with maximization problems.

22) Surplus variables are only associated with minimization problems.

23) If the objective function is parallel to a constraint, the constraint is infeasible.

24) Multiple optimal solutions occur when constraints are parallel to each other.

25) Graphical solutions to linear programming problems have an infinite number of possible objective function lines.

26) The first step in formulating a linear programming model is to define the objective function.

27) __________ are mathematical symbols representing levels of activity.

28) The __________ is a linear relationship reflecting the objective of an operation.

29) A __________ is a linear relationship representing a restriction on decision making.

30) If at least one constraint in a linear programming model is violated the solution is said to be __________.

31) A graphical solution is limited to solving linear programming problems with __________ decision variables

32) The __________ solution area is an area bounded by the constraint equations.

33) Multiple optimal solutions can occur when the objective function line is __________ to a constraint line.

34) When a maximization problem is __________, the objective function can increase indefinitely without reaching a maximum value.

35) A linear programming problem that results in a solution that is __________ usually indicates that the linear program has been incorrectly formulated.

Answer: infeasible

Diff: 2 Page Ref: 54

Main Heading: Graphical Solutions of Linear Programming Models

Key words: graphical solution, infeasible solution

36) In a constraint the __________ variable represents unused resources.

37) If the objective function is parallel to a constraint, the linear program could have __________.

38) Corner points on the boundary of the feasible solution area are called __________ points.

39) The __________ step in formulating a linear programming model is to define the decision variables.

40) The __________ property of linear programming models indicates that the values of all the model parameters are known and are assumed to be constant.

41) The __________ property of linear programming models indicates that the rate of change or slope of the objective function or a constraint is constant.

42) The __________ property of linear programming models indicates that the decision variables cannot be restricted to integer values and can take on any fractional value.

43) The constraint, 2X +XY violates the __________ property of linear programming.

44) Consider the following minimization problem:

Min z = x1 + 2×2

s.t. x1 + x2 ≥ 300

2×1 + x2 ≥ 400

2×1 + 5×2 ≤ 750

x1, x2 ≥ 0

What is the optimal solution?

45) Consider the following minimization problem:

Min z = x1 + 2×2

s.t. x1 + x2 ≥ 300

2×1 + x2 ≥ 400

2×1 + 5×2 ≤ 750

x1, x2 ≥ 0

Which constraints are binding at the optimal solution? (x1 =250, x2 = 50)

46) Solve the following graphically

Max z = 3×1 +4×2

s.t. x1 + 2×2 ≤ 16

2×1 + 3×2 ≤ 18

x1 ≥ 2

x2 ≤ 10

x1, x2 ≥ 0

What are the optimal values of x1, x2, and z?

47) Consider the following linear program:

MAX Z = 60A + 50B

s.t. 10A + 20B ≤ 200

8A + 5B ≤ 80

A ≥ 2

B ≥ 5

Solve this linear program graphically and determine the optimal quantities of A, B, and the value of Z.

48) Consider the following linear program:

MIN Z = 60A + 50B

s.t. 10A + 20B ≤ 200

8A + 5B ≤ 80

A ≥ 2

B ≥ 5

Solve this linear program graphically and determine the optimal quantities of A, B, and the value of Z.

49) A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.

If this is a maximization, which extreme point is the optimal solution?

50) A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.

If this is a minimization, which extreme point is the optimal solution?

51) A graphical representation of a linear program is shown below. The shaded area represents the feasible region, and the dashed line in the middle is the slope of the objective function.

What would the be the new slope of the objective function if multiple optimal solutions occurred along line segment AB?

52) Consider the following linear programming problem:

Max Z = $15x + $20y

Subject to: 8x + 5y ≤ 40

0.4x + y ≥ 4

x, y ≥ 0

Determine the values for x and y that will maximize revenue. Given this optimal revenue, what is the amount of slack associated with the first constraint?

53) Max Z = $3x + $9y

Subject to: 20x + 32y ≤ 1600

4x + 2y ≤ 240

y ≤ 40

x, y ≥ 0

Solve for the quantities of x and y which will maximize Z. What is the value of the slack variable associated with constraint 2?

54) Max Z = 5×1 + 3×2

Subject to: 6×1 + 2×2 ≤ 18

15×1 + 20×2 ≤ 60

x1 , x2 ≥ 0

Find the optimal profit and the values of x1 and x2 at the optimal solution.

55) Max Z = 3×1 + 3×2

Subject to: 10×1 + 4×2 ≤ 60

25×1 + 50×2 ≤ 200

x1 , x2 ≥ 0

Find the optimal profit and the values of x1 and x2 at the optimal solution.

56) Consider the following linear programming problem:

MIN Z = 10×1 + 20×2

Subject to: x1 + x2 ≥ 12

2×1 + 5×2 ≥ 40

x2 ≥ 13

x1 , x2 ≥ 0

What is minimum cost and the value of x1 and x2 at the optimal solution?

57) Consider the following linear programming problem:

MIN Z = 10×1 + 20×2

Subject to: x1 + x2 ≥ 12

2×1 + 5×2 ≥ 40

x2 ≥ 13

x1 , x2 ≥ 0

At the optimal solution, what is the value of surplus and slack associated with constraint 1 and constraint 3 respectively?

58) Consider the following linear programming problem:

MIN Z = 2×1 + 3×2

Subject to: x1 + 2×2 ≤ 20

5×1 + x2 ≤ 40

4×1 +6×2 ≤ 60

x1 , x2 ≥ 0

What is the optimal solution?

59) A company producing a standard line and a deluxe line of dishwashers has the following time requirements (in minutes) in departments where either model can be processed.

Standard Deluxe

Stamping 3 6

Motor installation 10 10

Wiring 10 15

The standard models contribute $20 each and the deluxe $30 each to profits. Because the company produces other items that share resources used to make the dishwashers, the stamping machine is available only 30 minutes per hour, on average. The motor installation production line has 60 minutes available each hour. There are two lines for wiring, so the time availability is 90 minutes per hour.

Let x = number of standard dishwashers produced per hour

y = number of deluxe dishwashers produced per hour

Write the formulation for this linear program:

60) In a linear programming problem, the binding constraints for the optimal solution are:

5×1 + 3×2 ≤ 30

2×1 + 5×2 ≤ 20

As long as the slope of the objective function stays between __________ and __________, the current optimal solution point will remain optimal.

61) In a linear programming problem, the binding constraints for the optimal solution are:

5×1 + 3×2 ≤ 30

2×1 + 5×2 ≤ 20

Which of these objective functions will lead to the same optimal solution?

a. 2×1 + 1×2

b. 7×1 + 8×2

c. 80×1 + 60×2

d. 25×1 + 15×2

62) Decision variables

A) measure the objective function

B) measure how much or how many items to produce, purchase, hire, etc.

C) always exist for each constraint

D) measure the values of each constraint

63) In a linear programming problem, a valid objective function can be represented as

A) Max Z = 5xy

B) Max Z 5×2 + 2y2

C) Max 3x + 3y + 1/3z

D) Min (x1 + x2) / x3

64) Which of the following could not be a linear programming problem constraint?

A) 1A + 2B ≠ 3

B) 1A + 2B = 3

C) 1A + 2B ≤ 3

D) 1A + 2B ≥ 3

65) A linear programming model consists of

A) decision variables

B) an objective function

C) constraints

D) all of the above

66) The minimization of cost or maximization of profit is the

A) objective of a business

B) constraint of operations management

C) goal of management science

D) objective of linear programming

E) both A and D

67) Which of the following could be a linear programming objective function?

A) Z = 1A + 2BC + 3D

B) Z = 1A + 2B + 3C + 4D

C) Z = 1A + 2B / C + 3D

D) Z = 1A + 2B2 + 3D

E) all of the above

68) The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet (D). Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the objective function?

A) MAX $2R + $4D

B) MAX $3R + $2D

C) MAX $3D + $2R

D) MAX $4D + $2R

E) MAX $4R + $2D

69) The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet(D). Two of the limited resources are production time (8 hours = 480 minutes per day) and syrup limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint?

A) 2R + 5D ≤ 480

B) 2D + 4R ≤ 480

C) 2R + 3D ≤ 480

D) 3R + 2D ≤ 480

E) 2R + 4D ≤ 480

70) Non-negativity constraints

A) restrict the decision variables to zero.

B) restrict the decision variables to positive values

C) restrict the decision variables to negative values

D) do not restrict the sign of the decision variable.

E) both A and B

71) Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the objective function?

A) MAX Z = $300B + $100 M

B) MAX Z = $300M + $150 B

C) MAX Z = $300B + $150 M

D) MAX Z = $300B + $500 M

E) MAX Z = $500B + $300 M

72) Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the storage space constraint?

A) 90B + 100M ≥ 18000

B) 90B + 100M ≤ 18000

C) 100B + 90M ≤ 18000

D) 500B + 300M ≤ 18000

E) 300B + 500M ≤ 18000

73) The __________ property of linear programming models indicates that the decision variables cannot be restricted to integer values and can take on any fractional value.

A) linearity

B) additive

C) divisibility

D) certainty

E) proportionality

74) The __________ property of linear programming models indicates that the rate of change or slope of the objective function or a constraint is constant.

A) additive

B) divisibility

C) certainty

D) proportionality

E) feasibility

75) The __________ property of linear programming models indicates that the values of all the model parameters are known and are assumed to be constant.

A) additive

B) divisibility

C) certainty

D) proportionality

76) The region which satisfies all of the constraints in a graphical linear programming problem is called the

A) region of optimality

B) feasible solution space

C) region of non-negativity

D) optimal solution space

77) Which of the following statements is not true?

A) An infeasible solution violates all constraints.

B) A feasible solution point does not have to lie on the boundary of the feasible solution.

C) A feasible solution satisfies all constraints.

D) An optimal solution satisfies all constraints.

78) Except satisfying the non-negativity constraint, a solution that satisfies all the other constraints of a linear programming problem is called

A) feasible

B) infeasible

C) semi-feasible

D) optimal

79) The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. Which of the following is not a feasible production combination?

A) 90R and 75D

B) 135R and 0D

C) 0R and 120D

D) 75R and 90D

E) 40R and 100D

80) The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What are the optimal daily production quantities of each product and the optimal daily profit?

A) R = 75, D = 90, Z = $405

B) R = 135, D = 0, Z = $405

C) R = 0, D= 120, Z = $360

D) R = 90, D = 75, Z = $420

E) R = 40, D= 100, Z = $320

81) __________ is used to analyze changes in model parameters.

A) Optimal solution

B) Feasible solution

C) Sensitivity analysis

D) none of the above

82) Cully furniture buys 2 products for resale: big shelves (B)and medium shelves (M).Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. Which of the following is not a feasible purchase combination?

A) 0 big shelves and 200 medium shelves

B) 100 big shelves and 82 medium shelves

C) 150 big shelves and 0 medium shelves

D) 100 big shelves and 100 medium shelves

E) 100 big shelves and 0 medium shelves

83) Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. What is the maximum profit?

A) $25000

B) $35000

C) $45000

D) $55000

E) $65000

84) Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. In order to maximize profit, how many big shelves (B) and how many medium shelves (M) should be purchased?

A) B = 90, M = 75

B) B = 135, M = 15

C) B = 150, M = 0

D) B = 0, M = 200

E) B = 100, M = 100

85) The theoretical limit on the number of constraints that can be handled by a linear programming problem is

A) 2

B) 3

C) 4

D) unlimited

86) Consider the following maximization problem.

MAX z = x + 2y

s.t. 2x + 3y ≤ 6

5x + 6y ≤ 30

y ≥ 1

The optimal solution

A) cannot be determined

B) occurs where x = 4.67 and y = 1.11

C) occurs where x = 0 and y = 5

D) occurs where x = 6 and y = 0

E) results in an objective function value of 12

The following is a graph of a linear programming problem. The feasible solution space is shaded, and the optimal solution is at the point labeled Z*.

87) This linear programming problem is a:

A) maximization problem

B) minimization problem

C) irregular problem

D) cannot tell from the information given

88) The equation for constraint DH is:

A) 4X + 8Y ≥ 32

B) 8X + 4Y ≥ 32

C) X + 2Y ≥ 8

D) 2X + Y ≥ 8

E) None of the above

89) Which of the following points are not feasible?

A) A

B) J

C) H

D) G

E) B

Answer: D

Diff: 1 Page Ref: 54

Main Heading: Graphical Solutions of Linear Programming Models

Key words: graphical solution, feasible point

90) Which line is represented by the equation 2X + Y ≥ 8?

A) BF

B) CG

C) DH

D) AJ

91) Which of the following constraints has a surplus greater than 0?

A) BF

B) CG

C) DH

D) AJ

92) The constraint AJ

A) Is not a binding constraint.

B) Has no surplus

C) Does not contain feasible points

D) A and B

E) B and C

93) Multiple optimal solutions can occur when the objective function is __________ a constraint line.

A) unequal to

B) equal to

C) perpendicular to

D) parallel to

94) A slack variable

A) is the amount by which the left side of a ≥ constraint is larger than the right side

B) is the amount by which the left side of a ≤ constraint is smaller than the right side

C) is the difference between the left and right side of a constraint

D) exists for each variable in a linear programming problem

95) The production manager for the Coory soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her limited resources are production time (8 hours = 480 minutes per day) and syrup (1 of her ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. For the production combination of 135 cases of regular and 0 cases of diet soft drink, which resources will not be completely used?

A) only time

B) only syrup

C) time and syrup

D) neither time nor syrup

96) Cully furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150. If the furniture company purchases no big shelves and 200 medium shelves, which of the two resources will be completely used (at capacity)?

A) investment money only

B) storage space only

C) investment money and storage space

D) neither investment money nor storage space

97) Consider the following linear program:

MAX z = 5x + 3y

s.t. x – y ≤ -1

x ≤ 1

The optimal solution

A) is infeasible

B) occurs where x = 1 and y = 0

C) occurs where x = 0 and y = 1

D) results in an objective function value of 11

98) The first step in solving a graphical linear programming model is

A) plot the model constraints as equations on the graph and indicate the feasible solution area

B) plot the objective function and move this line out from the origin to locate the optimal solution point

C) solve simultaneous equations at each corner point to find the solution values at each point

D) determine which constraints are binding

99) The optimal solution of a minimization problem is at the extreme point __________ the origin.

A) farthest from

B) closest to

C) exactly at

D) parallel to

100) Multiple optimal solutions provide __________ flexibility to the decision maker.

A) greater

B) less

C) greater or equal

D) less or equal

101) Which of the following special cases does not require reformulation of the problem in order to obtain a solution?

A) unboundedness

B) infeasibility

C) alternate optimality

D) each one of these cases requires reformulation

102) If the feasible region for a linear programming problem is unbounded, then the solution to the corresponding linear programming problem is __________ unbounded.

A) always

B) sometimes

C) never

D) there is not enough information to complete this statement

Introduction to Management Science, 10e (Taylor)

Chapter 3 Linear Programming: Computer Solution and Sensitivity Analysis

1) The reduced cost (shadow price) for a positive decision variable is 0.

2) When the right-hand sides of 2 constraints are both increased by 1 unit, the value of the objective function will be adjusted by the sum of the constraints’ prices.

3) When a linear programming problem is solved using a computer package decision variables will always be integer and therefore decision variable values never need to be rounded.

4) Sensitivity ranges can be computed only for the right hand sides of constraints.

5) Sensitivity analysis determines how a change in a parameter affects the optimal solution.

6) The sensitivity range for an objective function coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

7) The sensitivity range for an objective function coefficient is the range of values over which the profit does not change.

8) The sensitivity range for a constraint quantity value is the range over which the shadow price is valid.

9) If we change the constraint quantity to a value outside the sensitivity range for that constraint quantity, the shadow price will change.

10) The sensitivity range for a constraint quantity value is the range over which the optimal values of the decision variables do not change.

11) Linear programming problems are restricted to decisions in a single time period.

12) A maximization problem may be characterized by all greater than or equal to constraints.

13) A change in the value of an objective function coefficient will always change the value of the optimal solution.

14) The terms reduced cost, shadow price, and dual price all mean the same thing.

15) Sensitivity analysis can be used to determine the effect on the solution for changing several parameters at once.

16) For a profit maximization problem, if the allowable increase for a coefficient in the objective function is infinite, then profits are unbounded.

17) The reduced cost (shadow price) for a positive decision variable is __________.

18) The sensitivity range for a __________ is the range of values over which the quantity values can change without changing the shadow price

19) __________ is the analysis of the effect of parameter changes on the optimal solution.

20) The sensitivity range for a constraint quantity value is also the range over which the __________ is valid.

21) The sensitivity range for an __________ coefficient is the range of values over which the current optimal solution point (product mix) will remain optimal.

Consider the following linear program, which maximizes profit for two products, regular (R), and super (S):

MAX 50R + 75S

s.t.

1.2R + 1.6 S ≤ 600 assembly (hours)

0.8R + 0.5 S ≤ 300 paint (hours)

.16R + 0.4 S ≤ 100 inspection (hours)

22) The optimal number of regular products to produce is __________, and the optimal number of super products to produce is __________, for total profits of __________.

23) If the company wanted to increase the available hours for one of their constraints (assembly, painting, or inspection ) by 2 hours, they should increase __________.

24) The profit on the super product could increase by __________ without affecting the product mix.

25) If downtime reduced the available capacity for painting by 40 hours (from 300 to 260 hours), profits would be reduced by __________.

26) A change in the market has increased the profit on the super product by $5. Total profit will increase by __________.

Tracksaws, Inc. makes tractors and lawn mowers. The firm makes a profit of $30 on each tractor and $30 on each lawn mower, and they sell all they can produce. The time requirements in the machine shop, fabrication, and tractor assembly are given in the table.

27) How many tractors and saws should be produced to maximize profit, and how much profit will they make?

28) Determine the sensitivity range for the profit for tractors.

29) What is the shadow price for assembly?

30) What is the shadow price for fabrication?

31) What is the maximum amount a manager would be willing to pay for one additional hour of machining time?

32) A breakdown in fabrication causes the available hours to drop from 120 to 90 hours. How will this impact the optimal number of tractors and mowers produced?

33) What is the range for the shadow price for assembly?

The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular (R) and diet (D). The company operates one “8 hour” shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case.

34) What is the optimal daily profit?

35) How many cases of regular and how many cases of diet soft drink should Whoppy produce to maximize daily profit?

36) What is the sensitivity range for the per case profit of a diet soft drink?

37) What is the sensitivity range of the production time?

38) if the company decides to increase the amount of syrup it uses during production of these soft drinks to 990 lbs. will the current product mix change? If show what is the impact on profit?

39) What is the optimal product mix and maximum profit?

40) Determine the sensitivity range for the profit on the big shelf.

41) If the Mallory Furniture is able to increase the profit per medium shelf to $200, would the company purchase medium shelves. If so, what would be the new product mix and the total profit?

The linear programming problem whose output follows is used to determine how many bottles of fire red nail polish (x1), bright red nail polish (x2), basil green nail polish(x3), and basic pink nail polish(x4) a beauty salon should stock. The objective function measures profit; it is assumed that every piece stocked will be sold. Constraint 1 measures display space in units, constraint 2 measures time to set up the display in minutes. Note that green nail polish does not require any time to prepare its display. Constraints 3 and 4 are marketing restrictions. Constraint 3 indicates that the maximum demand for fire red and green polish is 25 bottles, while constraint 4 specifies that the minimum demand for bright red, green and pink nail polish bottles combined is at least 50 bottles.

42) How much space will be left unused? How many minutes of idle time remaining for setting up the display?

43) a) To what value can the per bottle profit on fire red nail polish drop before the solution (product mix) would change?

b) By how much can the per bottle profit on green basil nail polish increase before the solution (product mix) would change?

44) a) By how much can the amount of space decrease before there is a change in the profit?

b) By how much can the amount of space decrease before there is a change in the product mix?

c) By how much can the amount of time available to setup the display can increase before the solution (product mix) would change?

d) What is the lowest value for the amount of time available to setup the display before the solution (product mix) would change?

45) You are offered the chance to obtain more space. The offer is for 15 units and the total price is $1500. What should you do? Why?

46) Max Z = 5×1 + 3×2

Subject to: 6×1 + 2×2 ≤ 18

15×1 + 20×2 ≤ 60

x1 + x2 ≥ 0

Determine the sensitivity range for each constraint.

47) Max Z = 5×1 + 3×2

Subject to: 6×1 + 2×2 ≤ 18

15×1 + 20×2 ≤ 60

x1 + x2 ≥ 0

Determine the sensitivity range for each objective function coefficient.

48) Max Z = 3×1 + 3×2

Subject to: 10×1 + 4×2 ≤ 60

25×1 + 50×2 ≤ 200

x1 , x2 ≥ 0

Determine the sensitivity range for each objective function coefficient.

49) For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the:

A) same product mix, different total profit

B) different product mix, same total profit as before

C) same product mix, same total profit

D) different product mix, different total profit

50) A plant manager is attempting to determine the production schedule of various products to maximize profit. Assume that a machine hour constraint is binding. If the original amount of machine hours available is 200 minutes., and the range of feasibility is from 130 minutes to 340 minutes, providing two additional machine hours will result in:

A) the same product mix, different total profit

B) a different product mix, same total profit as before

C) the same product mix, same total profit

D) a different product mix, different total profit

The production manager for Beer etc. produces 2 kinds of beer: light (L) and dark (D). Two resources used to produce beer are malt and wheat. He can obtain at most 4800 oz of malt per week and at most 3200 oz of wheat per week respectively. Each bottle of light beer requires 12 oz of malt and 4 oz of wheat, while a bottle of dark beer uses 8 oz of malt and 8 oz of wheat. Profits for light beer are $2 per bottle, and profits for dark beer are $1 per bottle.

51) If the production manager decides to produce of 0 bottles of light beer and 400 bottles of dark beer, it will result in slack of

A) malt only

B) wheat only

C) both malt and wheat

D) neither malt nor wheat

52) Which of the following is not a feasible solution?

A) 0 L and 0 D

B) 0 L and 400 D

C) 200 L and 300 D

D) 400 L and 400 D

53) What is the optimal weekly profit?

A) $1000

B) $900

C) $800

D) $700

E) $600

Mallory Furniture buys 2 products for resale: big shelves (B) and medium shelves (M). Each big shelf costs $500 and requires 100 cubic feet of storage space, and each medium shelf costs $300 and requires 90 cubic feet of storage space. The company has $75000 to invest in shelves this week, and the warehouse has 18000 cubic feet available for storage. Profit for each big shelf is $300 and for each medium shelf is $150.

54) Which of the following is not a feasible purchase combination?

A) 0 big shelves and 200 medium shelves

B) 0 big shelves and 0 medium shelves

C) 150 big shelves and 0 medium shelves

D) 100 big shelves and 100 medium shelves

55) If the Mallory Furniture company decides to purchase 150 big shelves and no medium shelves, which of the two resources will be left over?

A) investment money only

B) storage space only

C) investment money and storage space

D) neither investment money nor storage space

The production manager for the Whoppy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. The company operates one “8 hour” shift per day. Therefore, the production time is 480 minutes per day. During the production process, one of the main ingredients, syrup is limited to maximum production capacity of 675 gallons per day. Production of a regular case requires 2 minutes and 5 gallons of syrup, while production of a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case.

56) Which of the following is not a feasible production combination?

A) 90 R and 75 D

B) 135 R and 0 D

C) 0 R and 120 D

D) 75 R and 90 D

E) 50 R and 50 D

57) For the production combination of 135 regular cases and 0 diet cases, which resource is completely used up (at capacity)?

A) only time

B) only syrup

C) time and syrup

D) neither time nor syrup

58) The sensitivity range for the profit on a regular case of soda is

A) $2 to $3

B) $2 to $4

C) $1 to $3

D) $1 to $3.33

59) Which of the following could not be a linear programming problem constraint?

A) A + B ≤ -3

B) A – B ≤ -3

C) A – B ≤ 3

D) A + B ≥ -3

E) -A + B ≤ -3

60) Use the constraints given below and determine which of the following points is feasible.

(1) 14x + 6y ≤ 42

(2) x – y ≤ 3

A) x = 1; y = 5

B) x = 2; y = 2

C) x = 2; y = 8

D) x = 2; y = 4

E) x = 3; y = 0.5

61) For the constraints given below, which point is in the feasible region of this minimization problem?

(1) 14x + 6y ≤ 42

(2) x + 3y ≥ 6

A) x = 0; y = 4

B) x = 2; y = 5

C) x = 1; y = 2

D) x = 2; y = 1

E) x = 2; y = 3

62) What combination of x and y is a feasible solution that minimizes the value of the objective function ?

Min Z = 3x + 15y

(1) 2x + 4y ≥ 12

(2) 5x + 2y ≥10

A) x = 0; y = 3

B) x = 0; y = 5

C) x = 5; y = 0

D) x = 6; y = 0

E) x = 4; y = 1

63) A shadow price reflects which of the following in a maximization problem?

A) the marginal gain in the objective that would be realized by adding 1 unit of a resource

B) the marginal gain in the objective that would be realized by subtracting 1 unit of a resource

C) the marginal cost of adding additional resources

D) the marginal gain of selling one more unit

64) Given the following linear programming problem:

Max Z = 15x + 20 y

s.t.

8x + 5y ≤ 40

4x + y ≥ 4

What would be the values of x and y that will maximize revenue?

A) x = 5; y = 0

B) x = 0; y = 8

C) x = 0; y = 1

D) x = 1; y = 0

E) x = 3; y = 4

65) Given the following linear program that maximizes revenue:

Max Z = 15x + 20 y

s.t.

8x + 5y ≤ 40

4x + y ≥ 4

What is the maximum revenue at the optimal solution?

A) $120

B) $160

C) $185

D) $200

Given the following linear programming problem that minimizes cost.

Min Z = 2x + 8y

Subject to (1) 8x + 4y ≥ 64

(2) 2x + 4y ≥ 32

(3) y ≥ 2

66) Determine the optimum values for x and y.

A) x = 2; y = 6

B) x = 6; y = 2

C) x = 12; y = 2

D) x = 2; y = 2

E) x = 6; y = 5

67) At the optimal solution the minimum cost is:

A) $30

B) $40

C) $50

D) $52

E) $53.33

68) What is the sensitivity range for the cost of x?

A) 0 to 2

B) 4 to 6

C) 2 to 4

D) 0 to 4

Answer: D

Diff: 2 Page Ref: 81

Main Heading: Sensitivity Analysis and Computer Solution

Key words: sensitivity analysis/range for objective function coefficients

69) What is the sensitivity range for the third constraint, y ≥ 2?

A) 0 to 4

B) 2 to 5.33

C) 0 to 5.33

D) 4 to 6.33

70) For a maximization problem, the shadow price measures the __________ in the value of the optimal solution, per unit increase for a given __________.

A) decrease, resource

B) increase, parameter

C) improvement, resource

D) change, objective function coefficient

E) decrease, parameter

71) Sensitivity analysis is the analysis of the effect of __________ changes on the __________.

A) price, company

B) cost, production

C) parameter, optimal solution

D) none of the above

72) For a linear programming problem, assume that a given resource has not been fully used. We can conclude that the shadow price associated with that constraint:

A) will have a positive value

B) will have a negative value

C) will have a value of zero

D) could have a positive, negative or a value of zero. (no sign restrictions)

Answer: C

Diff: 3 Page Ref: 90

Main Heading: Sensitivity Analysis

Key words: sensitivity analysis, shadow price

73) For a resource constraint, either its slack value must be __________ or its shadow price must be __________.

A) negative, negative

B) negative, zero

C) zero, zero

D) zero, negative

Aunt Anastasia operates a small business: she produces seasonal ceramic objects to sell to tourists. For the spring, she is planning to make baskets, eggs, and rabbits. Based on your discussion with your aunt you construct the following table.

Your aunt also has committed to make 25 rabbits for a charitable organization. Based on the information in the table, you formulate the problem as a linear program.

74) Which additional resources would you recommend that Aunt Anastasia try to obtain?

A) mix/mold

B) kiln

C) paint and seal

D) demand

E) Cannot tell from the information provided

75) Suppose the charitable organization contacted Aunt Anastasia and told her that they had overestimated the amount of rabbits they needed. Instead of 25 rabbits, they need 35. How would this affect Aunt Anastasia’s profits?

A) Profits would increase by $5.

B) Profits would decrease by $5

C) Profits would increase by $2.50

D) Profits would decrease by $2.50

E) Cannot tell from the information provided.

76) Aunt Anastasia feels that her prices are too low, particularly for her eggs. How much would her profit have to increase on the eggs before it is profitable for her to make and sell eggs?

A) $0.50

B) $1.00

C) $1.50

D) $2.50

E) None of the above

77) Aunt Anastasia’s available hours for paint and seal have fallen from 80 hours to 60 hours because of other commitments. How will this affect her profits?

A) Profits will decrease by $30.

B) Profits will increase by $30.

C) Profits will decrease by $20.

D) Profits will increase by $20.

E) Profits will not change.

78) Aunt Anastasia can obtain an additional 10 hours of kiln capacity free of charge from a friend. If she did this, how would her profits be affected?

A) Profit would increase by $25.

B) Profits would decrease by $25.

C) Profits would increase by $6.25.

D) Profits would decrease by $6.25

E) Cannot tell from the information provided.

79) Aunt Anastasia is planning for next spring, and she is considering making only 2 products. Based on the results from the linear program, which two products would you recommend that she make?

A) baskets and eggs

B) baskets and rabbits

C) eggs and rabbits

D) She should continue to make all 3.

E) Cannot tell from the information provided.

Billy’s Blues sells 3 types of T-shirts: Astro, Bling, and Curious. Manufacturing Astros requires 2 minutes of machine time, 20 minutes of labor, and costs $10. Brand Bling requires 2..5 minutes of machine time, 30 minutes of labor, and costs $14 to produce. Brand Curious requires 3 minutes of machine time, 45 minutes of labor, and costs $18 to produce. There are 300 machining hours available per week, 3,750 labor hours, and he has a budget of $3,000. Brand Astro sells for $15, Brand Bling for $18, and Brand Curious for $25.

80) If Billy could acquire more of any resource, which would it be?

A) machining time

B) labor time

C) money

D) buyers

81) If one of Billy’s machines breaks down, it usually results in about 6 hours of downtime. When this happens, Billy’s profits are reduced by

A) $15

B) 18

C) $25

D) $35

82) Billy’s accountant made an error, and the budget has been reduced from $3000 to $2500. Billy’s profit will go down by

A) $0

B) $625

C) $1350

D) $1650

83) Billy has decided that he can raise the price on the Curious t-shirt by 10% without losing sales. If he raises the price, his profits will

A) increase by 10%

B) decrease by 10%

C) increase by $2.50

D) increase by $125

Introduction to Management Science, 10e (Taylor)

Chapter 4 Linear Programming: Modeling Examples

1) When formulating a linear programming problem constraint, strict inequality signs (i.e., less than < or, greater than >) are not allowed.

2) When formulating a linear programming model on a spreadsheet, the measure of performance is located in the target cell.

3) The standard form for the computer solution of a linear programming problem requires all variables to be to the right and all numerical values to be to the left of the inequality or equality sign

4) The standard form for the computer solution of a linear programming problem requires all variables to be on the left side, and all numerical values to be on the right side of the inequality or equality sign.

5) Fractional relationships between variables are not permitted in the standard form of a linear program.

6) A constraint for a linear programming problem can never have a zero as its right-hand-side value.

7) The right hand side of constraints cannot be negative.

8) A systematic approach to model formulation is to first define decision variables.

9) A systematic approach to model formulation is to first construct the objective function before determining the decision variables.

10) In a linear programming model, a resource constraint is a problem constraint with a greater-than-or-equal-to (≥) sign.

11) Determining the production quantities of different products manufactured by a company based on resource constraints is a product mix linear programming problem.

12) Product mix problems cannot have “greater than or equal to” (≥) constraints.

13) When using a linear programming model to solve the “diet” problem, the objective is generally to maximize profit.

14) When using a linear programming model to solve the “diet” problem, the objective is generally to maximize nutritional content.

15) In formulating a typical diet problem using a linear programming model, we would expect most of the constraints to be related to calories.

16) Solutions to diet problems in linear programming are always realistic.

17) Diet problems usually maximize nutritional value.

18) In most media selection decisions, the objective of the decision maker is to minimize cost.

19) In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the audience exposure.

20) Linear programming model of a media selection problem is used to determine the relative value of each advertising media.

21) In a media selection problem, maximization of audience exposure may not result in maximization of total profit.

22) In a balanced transportation model, supply equals demand such that all constraints can be treated as equalities.

23) In an unbalanced transportation model, supply does not equal demand and supply constraints have ≤ signs.

24) Transportation problems can have solution values that are non-integer and must be rounded.

25) In a transportation problem, the supply constraint represents the maximum amount of product available for shipment or distribution at a given source (plant, warehouse, mill).

26) In a transportation problem, a supply constraint (the maximum amount of product available for shipment or distribution at a given source) is a greater-than-or equal-to constraint (≥).

27) In a transportation problem, a demand constraint for a specific destination represents the amount of product demanded by a given destination (customer, retail outlet, store).

28) In a transportation problem, a demand constraint (the amount of product demanded at a given destination) is a less-than-or equal-to constraint (≤).

29) Blending problems usually require algebraic manipulation in order to write the LP in “standard form.”

30) Data Envelopment Analysis indicates which type of service unit makes the highest profit.

31) Data Envelopment Analysis indicates the the relative _________ of a service unit compared with others.

32) __________ types of linear programming problems often result in fractional relations between variables which must be eliminated.

33) When formulating a linear programming model on a spreadsheet, the measure of performance is located in the __________ cell.

34) When the __________ command is used in an Excel spreadsheet, all the values in a column (or row) are multiplied by the values in another column (or row) and then summed.

35) For product mix problems, the constraints are usually associated with __________.

36) The __________ for the computer solution of a linear programming problem requires all variables on the left side, and all numerical values on the right side of the inequality or equality sign.

37) The objective function of a diet problem is usually to __________ subject to nutritional requirements.

38) Investment problems maximize __________.

39) In a media selection problem, instead of having an objective of maximizing profit or minimizing cost, generally the objective is to maximize the __________.

40) In __________ problem, maximization of audience exposure may not result in maximization of total profit.

41) In a balanced transportation model, supply equals __________ .

42) In a __________ transportation problem, supply exceeds demand.

The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50.

43) What is the formulation for this problem?

44) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

45) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which resource is not completely used up and how much is remaining?

A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS (tablespoons) of almond paste. An almond- filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. The shop must produce at least 400 almond filled croissants due to customer demand. Bear claw profits are 20 cents each, and almond-filled croissant profits are 30 cents each.

46) This represents what type of linear programming application?

47) What is the formulation for this problem?

48) For the production combination of 600 bear claws and 800 almond filled croissants, how much flour and almond paste is remaining?

49) If Xij = the production of product i in period j, write an expression to indicate that the limit on production of the company’s 3 products in period 2 is equal to 400.

50) Small motors for garden equipment is produced at 4 manufacturing facilities and needs to be shipped to 3 plants that produce different garden items (lawn mowers, rototillers, leaf blowers). The company wants to minimize the cost of transporting items between the facilities, taking into account the demand at the 3 different plants, and the supply at each manufacturing site. The table below shows the cost to ship one unit between each manufacturing facility and each plant, as well as the demand at each plant and the supply at each manufacturing facility.

Write the formulation for this problem.

51) Quickbrush Paint Company makes a profit of $2 per gallon on its oil-base paint and $3 per gallon on its water-base paint. Both paints contain two ingredients, A and B. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. The company wishes to use linear programming to determine the appropriate mix of oil-base and water-base paint to produce to maximize its total profit. How much oil based and water based paint should the Quickbrush make?

Andy Tyre manages Tyre’s Wheels, Inc. Andy has received an order for 1000 standard wheels and 1200 deluxe wheels next month, and for 750 standard wheels and 1000 deluxe wheels the following months. He must fill all the orders. The cost of regular time production for standard wheels is $25 and for deluxe wheels, $40. Overtime production costs 50% more. For each of the next two months there are 1000 hours of regular time production and 500 hours of overtime production available. A standard wheel requires .5 hours of production time and a deluxe wheel, .6 hours. The cost of carrying a wheel from one month to the next is $2.

52) Define the decision variables and objective function for this problem.

53) Write the constraints for this problem.

Bullseye Shirt Company makes three types of shirts: Athletic, Varsity, and Surfer. The shirts are made from different combinations of cotton and rayon. The cost per yard of cotton is $5 and the cost for rayon is $7. Bullseye can receive up to 4,000 yards of cotton and 3,000 yards of rayon per week.

The table below shows relevant manufacturing information:

Shirt Total Yards of fabric per shirt Fabric requirement Minimum weekly contracts Maximum Demand Selling Price

Athletic 1.00 at least 60% cotton 500 600 $30

Varsity 1.20 no more than 30% rayon 650 850 $40

Surfer 0.90 As much as 80% cotton 300 700 $36

54) Assume that the decision variables are defined as follows:

A = total number of athletic shirts produced

V = total number of varsity shirts produced

S = total number of surfer shirts produced

C = yards of cotton purchased

R = yards of rayon purchased

Xij = yards of fabric i (C or R) blended into shirt J (A, V or S)

Write the objective function.

55) Write the constraints for the fabric requirements.

56) Write the constraints for the total number of shirts of each style produced.

57) Kitty Kennels provides overnight lodging for a variety of pets. An attractive feature is the quality of care the pets receive, including well balanced nutrition. The kennel’s cat food is made by mixing two types of cat food to obtain the “nutritionally balanced cat diet.” The data for the two cat foods are as follows:

58) A credit union wants to make investments in the following:

The firm will have $2,500,000 available for investment during the coming year. The following restrictions apply:

∙ Risk free securities may not exceed 30% of the total funds, but must comprise at least 5% of the total.

∙ Signature loans may not exceed 12% of the funds invested in all loans (vehicle, consumer, other secured loans, and signature loans)

∙ Consumer loans plus other secured loans may not exceed the vehicle loans

∙ Other secured loans plus signature loans may not exceed the funds invested in risk free securities. How should the $2,500,000 be allocated to each alternative to maximize annual return? What is the annual return?

59) When systematically formulating a linear program, the first step is

A) Construct the objective function

B) Formulate the constraints

C) Identify the decision variables

D) Identify the parameter values

E) Identify a feasible solution

60) The following types of constraints are ones that might be found in linear programming formulations:

1. ≤

2. =

3. >

A) 1 and 2

B) 2 and 3

C) 1 and 3

D) all of the above

61) Assume that x2, x7 and x8 are the dollars invested in three different common stocks from New York stock exchange. In order to diversify the investments, the investing company requires that no more than 60% of the dollars invested can be in “stock two”. The constraint for this requirement can be written as:

A) x2 ≥ .60

B) x2 ≥ .60 (x2 + x7 + x8)

C) .4×2 – .6×7 – .6×8 ≤ 0

D) .4×2 – .6×7 – .6×8 ≥ 0

E) -.4×2 + .6×7 + .6×8 ≤ 0

62) The owner of Black Angus Ranch is trying to determine the correct mix of two types of beef feed, A and B which cost 50 cents and 75 cents per pound, respectively. Five essential ingredients are contained in the feed, shown in the table below. The table also shows the minimum daily requirements of each ingredient.

Ingredient Percent per pound in Feed A Percent per pound in Feed B Minimum daily requirement (pounds)

1 20 24 30

2 30 10 50

3 0 30 20

4 24 15 60

5 10 20 40

The constraint for ingredient 3 is:

A) .5A + .75B = 20

B) .3B = 20

C) .3 B≤ 20

D) .3B ≥ 20

E) A + B = .3(20)

The owner of Chips etc. produces 2 kinds of chips: Lime (L) and Vinegar (V). He has a limited amount of the 3 ingredients used to produce these chips available for his next production run: 4800 ounces of salt, 9600 ounces of flour, and 2000 ounces of herbs. A bag of Lime chips requires 2 ounces of salt, 6 ounces of flour, and 1 ounce of herbs to produce; while a bag of Vinegar chips requires 3 ounces of salt, 8 ounces of flour, and 2 ounces of herbs. Profits for a bag of Lime chips are $0.40, and for a bag of Vinegar chips $0.50.

63) For the production combination of 800 bags of Lime and 600 bags of Vinegar, which of the three resources is (are) not completely used?

A) flour only

B) salt only

C) herbs only

D) salt and flour

E) salt and herbs

64) What is the constraint for salt?

A) 6L + 8V ≤ 4800

B) 1L + 2V ≤ 4800

C) 3L + 2V ≤ 4800

D) 2L + 3V ≤ 4800

E) 2L + 1V ≤ 4800

Answer: D

Diff: 2 Page Ref: 111-116

Main Heading: Product Mix Example

Key words: formulation, constraint

65) Which of the following is not a feasible production combination?

A) 0L and 0V

B) 0L and 1000V

C) 1000L and 0V

D) 0L and 1200V

66) If Xab = the production of product a in period b, then to indicate that the limit on production of the company’s “3” products in period 2 is 400,

A) X32 ≤ 400

B) X21 + X22 + X23 ≤ 400

C) X12 + X22 + X32 ≤ 400

D) X12 + X22 + X32 ≥ 400

E) X23 ≤ 400

67) Balanced transportation problems have the following type of constraints:

A) ≥

B) ≤

C) =

D) < E) None of the above 68) Compared to blending and product mix problems, transportation problems are unique because A) They maximize profit. B) The constraints are all equality constraints with no “≤” or “≥” constraints. C) They contain fewer variables. D) The solution values are always integers. E) All of the above are True. 69) The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are production time (8 hours = 480 minutes per day) and syrup (1 of the ingredients) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the time constraint? A) 2R + 4D ≤ 480 B) 2D + 4R ≤ 480 C) 2R + 3D ≤ 480 D) 3R + 2D ≤ 480 E) 3R + 4D ≤ 480 70) A croissant shop produces 2 products: bear claws (B) and almond filled croissants (C). Each bear claw requires 6 ounces of flour, 1 ounce of yeast, and 2 TS of almond paste. An almond filled croissant requires 3 ounces of flour, 1 ounce of yeast, and 4 TS of almond paste. The company has 6600 ounces of flour, 1400 ounces of yeast, and 4800 TS of almond paste available for today’s production run. Bear claw profits are 20 cents each, and almond filled croissant profits are 30 cents each. What is the optimal daily profit? A) $380 B) $400 C) $420 D) $440 E) $480 71) The production manager for the Softy soft drink company is considering the production of 2 kinds of soft drinks: regular and diet. Two of her resources are constraint production time (8 hours = 480 minutes per day) and syrup (1 of her ingredient) limited to 675 gallons per day. To produce a regular case requires 2 minutes and 5 gallons of syrup, while a diet case needs 4 minutes and 3 gallons of syrup. Profits for regular soft drink are $3.00 per case and profits for diet soft drink are $2.00 per case. What is the optimal daily profit? A) $220 B) $270 C) $320 D) $420 E) $520 72) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the supply constraint for component 1. A) x21 + x22 ≤ 8000 B) x12 + x22 ≥ 8000 C) x11 + x12 ≤ 8000 D) x21 + x22 ≥ 8000 E) x11 + x12 ≥ 8000 73) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the demand constraint for gasoline type 1. A) x21 + x22 = 11000 B) x12 + x22 = 11000 C) x11 + x21 ≤ 11000 D) x11 + x21 = 11000 E) x11 + x12 ≥ 11000 74) Let xij = gallons of component i used in gasoline j. Assume that we have two components and two types of gasoline. There are 8,000 gallons of component 1 available, and the demand gasoline types 1 and 2 are 11,000 and 14,000 gallons respectively. Write the constraint stating that the component 1 cannot account for more than 35% of the gasoline type 1. A) x11 + x12 (.35)(x11 + x21) B) x11 .35 (x11 + x21) C) x11 .35 (x11 + x12) D) -.65×11 + .35×21 ≤ 0 E) .65×11 – .35×21 ≤ 0 75) Quickbrush Paint Company is developing a linear program to determine the optimal quantities of ingredient A and ingredient B to blend together to make oil based and water based paint. The oil-base paint contains 90 percent A and 10 percent B, whereas the water-base paint contains 30 percent A and 70 percent B. Quickbrush currently has 10,000 gallons of ingredient A and 5,000 gallons of ingredient B in inventory and cannot obtain more at this time. Assuming that x represents the number of gallons of oil based paint, and y represents the gallons of water based paint, which constraint is correctly represents the constraint on ingredient A? A) .9A + .1B ≤ 10,000 B) .9x + .1y ≤ 10,000 C) .3x + .7y ≤ 10,000 D) .9x + .3y ≤ 10,000 E) .1x + .9y ≤ 10,000 76) A systematic approach to model formulation is to first A) construct the objective function B) develop each constraint separately C) define decision variables D) determine the right hand side of each constraint E) all of the above 77) Let: rj = regular production quantity for period j, oj =overtime production quantity in period j, ii = inventory quantity in period j, and di = demand quantity in period j Correct formulation of the demand constraint for a multi-period scheduling problem is: A) rj + oj + i2 – i1 ≥ di B) rj + oj + i1 – i2 ≥ di C) rj + oj + i1 – i2 ≤ di D) rj – oj – i1 + i2 ≥ di E) rj + oj + i2 – i1 ≤ di 78) In a multi-period scheduling problem the production constraint usually takes the form of: A) beginning inventory + demand – production = ending inventory B) beginning inventory – demand + production = ending inventory C) beginning inventory – ending inventory + demand = production D) beginning inventory – production – ending inventory = demand E) beginning inventory + demand + production = ending inventory 79) The type of linear program that compares services to indicate which one is less productive or inefficient is called A) product mix B) data envelopment analysis C) marketing D) blending E) multi period scheduling 80) The stockbroker suggests limiting the investments so that no more than $10,000 is invested in stock 2 or the total number of shares of stocks 2 and 3 does not exceed 350, whichever is more restrictive. How would this be formulate as a linear programming constraint? A) X2 ≤ 10000 X2 + X3 ≤350 B) 10,000 X2 ≤ 350X2 + 350X3 C) 47.25X2 ≤10,000 X2 + X3 ≤ 350 D) 47.25X2 ≤10,000 47.25 X2 + 110X3 ≤ 350 81) An appropriate part of the model would be A) 15X1 + 47.25X2 +110 X3 ≤ 50,000 B) MAX 15X1 + 47.25X2 + 110X3 C) X1 + X2 +X3 ≤ 50,000 D) MAX 50(15)X1 + 50 (47.25)X2 + 50 (110)X3 82) The expected returns on investment of the three stocks are 6%, 8%, and 11%. An appropriate objective function is A) MAX .06X1 +.08X2 +.11X3 B) MAX .06(15)X1 +.08(47.25)X2 +.11(110)X3 C) MAX 15X1 + 47.25X2 +.110X3 D) MAX (1/.06)X1 +.(1/08)X2 + (1/.11)X3 83) The investor stipulates that stock 1 must not account for more than 35% of the number of shares purchased. Which constraint is correct? A) X1 ≤ 0.35 B) X1 = 0.35 (50000) C) X1 ≤ 0.35(X1 + X2 +.X3) D) X1 = 0.35(X1 + X2 +.X3) Introduction to Management Science, 10e (Taylor) Chapter 5 Integer Programming 1) The 3 types of integer programming models are total, 0 – 1, and mixed. 2) In a total integer model, all decision variables have integer solution values. 3) In a 0 – 1 integer model, the solution values of the decision variables are 0 or 1. 4) In a mixed integer model, some solution values for decision variables are integer and others can be non-integer. 5) In a mixed integer model, all decision variables have integer solution values. 6) In a mixed integer model, the solution values of the decision variables are 0 or 1. 7) The branch and bound method can only be used for maximization integer programming problems. 8) The branch and bound solution method cannot be applied only to 0-1 integer programming problems. 9) In a problem involving capital budgeting applications, the 0-1 variables designate the acceptance or rejection of the different projects. 10) In a 0-1 integer programming problem involving a capital budgeting application (where xj = 1, if project j is selected, xj = 0, otherwise) the constraint x1 – x2 ≤ 0 implies that if project 2 is selected, project 1 can not be selected. 11) The divisibility assumption is violated by integer programming. 12) One type of constraint in an integer program is a multiple choice constraint. 13) If exactly 3 projects are to be selected from a set of 5 projects, this would be written as 3 separate constraints in an integer program. 14) A conditional constraint specifies the conditions under which variables are integers or real variables. 15) Rounding non-integer solution values up to the nearest integer value can result in an infeasible solution to an integer programming problem. 16) A feasible solution to an integer programming problem is ensured by rounding down non-integer solution values. 17) A feasible solution to an integer programming problem is ensured by rounding down integer solution values. 18) A rounded-down integer solution can result in a less than optimal solution to an integer programming problem. 19) Rounding down integer solution values ensures an infeasible solution to an integer linear programming problem. 20) Rounding non-integer solution values up to the nearest integer value will result in an infeasible solution to an integer linear programming problem. 21) Rounding non-integer solution values up to the nearest integer value will still result in a feasible solution. 22) The solution to the LP relaxation of a maximization integer linear program provides a lower bound for the value of the objective function. 23) The solution to the LP relaxation of a maximization integer linear program provides an upper bound for the value of the objective function. 24) The solution to the LP relaxation of a minimization integer linear program provides a lower bound for the value of the objective function. 25) The solution to the LP relaxation of a minimization integer linear program provides an upper bound for the value of the objective function. 26) If we perform sensitivity analysis for an integer linear programming problem, we can provide the same interpretation as we would have obtained from interpreting the corresponding linear programming problem. 27) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 + x3 ≤ 3 is a mutually exclusive constraint. 28) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a mutually exclusive constraint. 29) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a mutually exclusive constraint. 30) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a mutually exclusive constraint. 31) If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a conditional constraint. 32) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a conditional constraint. 33) In a __________ linear programming model, some of the solution values for the decision variables are required to assume integer values and others can be integer or noninteger. 34) In a __________ linear programming model, the solution values of the decision variables are zero or one. 35) If exactly one investment is to be selected from a set of five investment options, then the constrain is often called a __________ constraint. 36) If we graph the problem that requires x1 and x2 to be an integer, it has a feasible region consisting of __________. 37) __________ variables are best suited to be the decision variables when dealing with yes-or-no decisions. 38) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a __________ constraint. 39) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint. 40) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a __________ constraint. 41) If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________ constraint. 42) Because of their structure, __________ types of linear programming problems always result in integer solutions, even though integer solutions are not specified in the linear program. 43) If one location for a warehouse can be selected only if a specific location for a manufacturing facility is also selected, this decision can be represented by a __________ constraint. 44) In an integer program, if we were choosing between two locations to build a facility, this would be written as: __________. 45) In an integer program, if building one facility required the construction of another type of facility, this would be written as: __________. 46) Consider the following integer linear programming problem Max Z = 3×1 + 2×2 Subject to: 3×1 + 5×2 ≤ 30 4×1 + 2×2 ≤ 28 x1 ≤ 8 x1 ,x2 ≥ 0 and integer The solution to the Linear programming relaxation is: x1 = 5.714, x2 = 2.571. What is the upper bound for the value of the objective function? What is the value of the objective function for the rounded down solution? Is the rounded down solution feasible? 47) Consider the following integer linear programming problem Max Z = 3×1 + 2×2 Subject to: 3×1 + 5×2 ≤ 30 4×1 + 2×2 ≤ 28 x1 ≤ 8 x1 , x2 ≥ 0 and integer The solution to the Linear programming relaxation is: x1 = 5.714, x2 = 2.571. What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function. 48) Consider the following integer linear programming problem Max Z = 3×1 + 2×2 Subject to: 3×1 + 5×2 ≤ 30 5×1 + 2×2 ≤ 28 x1 ≤ 8 x1 ,x2 ≥ 0 and integer The solution to the Linear programming relaxation is: x1 = 5.714, x2= 2.571. What is the optimal solution to the integer linear programming problem? State the optimal values of decision variables and the value of the objective function. Consider a capital budgeting example with 5 projects from which to select. Let x1 = 1 if project a is selected, 0 if not, for a = 1, 2, 3, 4, 5. Projects cost $100, $200, $150, $75, and $300 respectively. The budget is $450. 49) Write the appropriate constraint for the following condition: Choose no fewer than 3 projects. 50) Write the appropriate constraint for the following condition: If project 3 is chosen, project 4 must be chosen. 51) Write the appropriate constraint for the following condition: If project 1 is chosen, project 5 must not be chosen. The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. 52) This problem requires 2 different kinds of decision variables. Clearly define each kind. 53) Write the objective function. 54) Write a constraint to ensure that if machine 4 is used, machine 1 will not be used. 55) Write a constraint that will ensure that Weithoff purchases exactly 2 machine.s 56) Max Z = x1 + 6×2 Subject to: 17×1 + 8×2 ≤ 136 3×1 + 4×2 ≤ 36 x1, x2 ≥ 0 and integer Find the optimal solution. 57) Max Z = 3×1 + 5×2 Subject to: 7×1 + 12×2 ≤ 136 3×1 + 5×2 ≤ 36 x1, x2 ≥ 0 and integer Find the optimal solution. 58) Solve the following integer linear program graphically. MAX Z = 5×1 + 8×2 s.t. x1 + x2 ≤ 6 5×1 + 9×2 ≤ 45 x1, x2 ≥ 0 and integer 59) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint for the first restriction. 60) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint(s) for the second restriction. 61) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint for the third restriction. Due to increased sales, a company is considering building 3 new distribution centers (DCs) to serve 4 regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. 62) Define the decision variables for this situation. 63) Write the objective function for this problem . 64) Write the constraints for the 3 distribution centers. 65) Types of integer programming models are __________. A) total B) 0 – 1 C) mixed D) all of the above 66) In a __________ integer model, some solution values for decision variables are integers and others can be non-integer. A) total B) 0 – 1 C) mixed D) all of the above 67) In a __________ integer model, all decision variables have integer solution values. A) total B) 0 – 1 C) mixed D) all of the above 68) In a __________ integer model, the solution values of the decision variables are 0 or 1. A) total B) 0 – 1 C) mixed D) all of the above 69) Binary variables are A) 0 or 1 only B) any integer value C) any continuous value D) any negative integer value 70) Which of the following is not an integer linear programming problem? A) pure integer B) mixed integer C) 0-1integer D) continuous 71) If the solution values of a linear program are rounded in order to obtain an integer solution, the solution is A) always optimal and feasible B) sometimes optimal and feasible C) always optimal D) always feasible E) never optimal and feasible 72) The branch and bound method of solving linear integer programming problems is __________. A) an integer method B) a relaxation method C) a graphical solution D) an enumeration method 73) If a maximization linear programming problem consist of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will __________ result in a(n) __________ solution to the integer linear programming problem. A) always, optimal B) always, non-optimal C) never, non-optimal D) sometimes, optimal E) never, optimal 74) If a maximization linear programming problem consist of all less-than-or-equal-to constraints with all positive coefficients and the objective function consists of all positive objective function coefficients, then rounding down the linear programming optimal solution values of the decision variables will __________ result in a feasible solution to the integer linear programming problem. A) always B) sometimes C) never 75) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 ≤ 1 is a __________ constraint. A) multiple choice B) mutually exclusive C) conditional D) corequisite E) none of the above 76) If we are solving a 0-1 integer programming problem, the constraint x1 + x2 = 1 is a __________ constraint. A) multiple choice B) mutually exclusive C) conditional D) corequisite E) none of the above 77) If we are solving a 0-1 integer programming problem, the constraint x1 ≤ x2 is a __________ constraint. A) multiple choice B) mutually exclusive C) conditional D) corequisite E) none of the above 78) If we are solving a 0-1 integer programming problem, the constraint x1 = x2 is a __________ constraint. A) multiple choice B) mutually exclusive C) conditional D) corequisite E) none of the above 79) For a maximization integer linear programming problem, feasible solution is ensured by rounding __________ non-integer solution values if all of the constraints are less-than -or equal-to type. A) up and down B) up C) down D) up or down 80) The implicit enumeration method A) generates an optimal integer solution when no new constraints can be added to the relaxed linear programming model B) eliminates obviously infeasible solutions and evaluates the remaining solutions to determine which one is optimal C) is used to solve a mixed integer linear programming model D) cannot be used to solve linear programming models with multiple infeasible solutions 81) The linear programming relaxation contains the objective function and the original constraints of the integer programming problem, but drops all __________. A) different variables B) slack values C) integer restrictions D) decision variables E) nonnegativity constraints 82) The solution to the linear programming relaxation of a minimization problem will always be __________ the value of the integer programming minimization problem. A) greater than or equal to B) less than or equal to C) equal to D) different than 83) If the optimal solution to the linear programming relaxation problem is integer, it is __________ to the integer linear programming problem. A) a real solution B) a degenerate solution C) an infeasible solution D) the optimal solution E) a feasible solution 84) In a capital budgeting problem, if either project 1 or project 2 is selected, then project 5 cannot be selected. Which of the alternatives listed below correctly models this situation? A) x1 + x2 + x5 ≤ 1 B) x1 + x2 + x5 ≥1 C) x1 + x5 ≤ 1, x2 + x5 ≤ 1 D) x1 – x5 ≤ 1, x2 – x5 ≤ 1 E) x1 – x5 = 0, x2 – x5 = 0 85) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, the constraint for the first restriction is A) S1 + S3 + S7 ≥ 1 B) S1 + S3 + S7 ≤1 C) S1 + S3 + S7 = 2 D) S1 + S3 + S7 ≤ 2 E) S1 + S3 + S7 = 3 Due to increased sales, a company is considering building 3 new distribution centers (DCs) to serve 4 regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. Assume that Xij = quantity shipped from distribution i to region j, i = 1,2,3; j = 1, 2, 3, 4. Assume that Yi = 0 or 1 where i = distribution center 1, 2 or 3. 86) The constraint for distribution center 1 is: A) X11 + X12 +X13 + X14- 500y1 ≤ 0 B) X11 + X12 +X13 + X14D + 500y1 ≤ 0 C) X11 + X12 +X13 + X14 ≤ 500 D) X11 + X12 +X13 + X14 ≥ 500 87) The objective function is A) B) C) D) 88) You have been asked to select at least 3 out of 7 possible sites for oil exploration. Designate each site as S1, S2, S3, S4, S5, S6, and S7. The restrictions are: Restriction 1. Evaluating sites S1 and S3 will prevent you from exploring site S7. Restriction 2. Evaluating sites S2 or S4 will prevent you from assessing site S5. Restriction 3. Of all the sites, at least 3 should be assessed. Assuming that Si is a binary variable, write the constraint(s) for the second restriction A) S2 +S5 ≤ 1 B) S4 +S5 ≤ 1 C) S2 +S5 + S4 +S5 ≤ 2 D) S2 +S5 ≤ 1, S4 +S5 ≤ 1 Future Plastics manufactures plastic products for industrial use worldwide. In order to meet demand, they are considering setting up a facility in each region in order to lower transportation cost and to possibly avoid duties that could be imposed if the product is imported from another region. The disadvantage of this approach is that plants are sized to meet local demand and may not fully exploit economies of scale. Therefore, Future Plastics is also interested in determining the appropriate size of facility to build in each location and are choosing between facilities with capacities of 5 or 10 million. The fixed costs of each facility as well as the cost of shipping between regions is shown in the table below. The decision variables are defined as follows: Xij = quantity shipped from supply region i to demand region j. i = 1,2, 3, 4 and j = 1, 2, 3, 4. Yik = 1 if facility k is selected for supply region i; 0 otherwise. where i = 1, 2, 3, 4 for each supply region; k = 1 (low capacity facility) or 2 (high capacity facility) Demand Region Production and Transportation Cost per 1,000,000 Units Low Capacity High Capacity Supply Region North America South America South Asia Europe Fixed Cost Capacity Fixed Cost Capacity North America 40 45 51 65 3,000 5 4,500 10 South America 51 57 47 60 3,200 5 4,800 10 South Asia 58 63 45 31 2,000 5 3,000 10 Europe 71 50 51 53 1,800 5 2,700 10 Demand 6 4 7 8 89) The constraint for the North American supply region is: A) X11 + X21 + X31 + X34 – 5Y11 – 10Y21 ≤ 0 B) X11 + X12 + X13 + X14 – 5Y11 – 10Y12 ≤ 0 C) X11 + X12 + X13 + X14 – 3200Y11 – 4800Y12 ≤ 0 D) X11 + X12 + X13 + X14 – 5Y11 – 10Y12 = 0 90) The constraint for the South Asia demand region is: A) X31 + X32 + X33 + X34 =7 B) X31 + X32 + X33 + X34 ≤ 7 C) X13 + X23 + X33 + X43 ≤ 7 D) X13 + X23 + X33 + X43 = 7 91) Which of these constraints will ensure that either a low capacity or a high facility capacity facility is built in the European supply region? A) Y41 + Y42 = 1 B) Y41 + Y42 ≤ 1 C) Y14 + Y24 = 1 D) Y14 + Y24 ≤ 1 92) Which of these constraints will ensure that a low capacity facility is NOT built in South America? A) Y12 + Y22 ≤ 1 B) Y12 + Y22 =1 C) Y12 + Y22 =0 D) Y21 + Y22 ≤ 1 93) Max Z = 5×1 + 6×2 Subject to: 17×1 + 8×2 ≤ 136 3×1 + 4×2 ≤ 36 x1, x2 ≥ 0 and integer What is the optimal solution? A) x1 = 6, x2 = 4, Z = 54 B) x1 = 3, x2 = 6, Z = 51 C) x1 = 2, x2 = 6, Z = 46 D) x1 = 4, x2 = 6, Z = 56 E) x1 = 0, x2 = 9 Z = 54 94) Assume that we are using 0-1 integer programming model to solve a capital budgeting problem and xj = 1 if project j is selected and xj = 0, otherwise. The constraint (x1 + x2 + x3 + x4 ≤ 2) means that __________ out of the 4 projects must be selected. A) exactly 1, 4 B) exactly 2, 4 C) at least 2, 4 D) at most 2, 4 95) In a 0-1 integer programming model, if the constraint x1-x2 = 0, it means when project 1 is selected, project 2 __________ be selected. A) can also B) can sometimes C) can never D) must also 96) In a 0-1 integer programming model, if the constraint x1-x2 ≤ 0, it means when project 2 is selected, project 1 __________ be selected. A) must always B) can sometimes C) can never D) A and B 97) In formulating a mixed integer programming problem, the constraint x1 + x2 ≤ 500y1 where y1 is a 0-1 variable and x1 and x2 are continuous variables, then x1 + x2 = 500 if y1 is __________. A) 0 B) 1 C) 0 or 1 D) none of the above The Wiethoff Company has a contract to produce 10000 garden hoses for a customer. Wiethoff has 4 different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. 98) Write a constraint to ensure that if machine 4 is used, machine 1 will not be used. A) Y1 + Y4 ≤ 0 B) Y1 + Y4 = 0 C) Y1 + Y4 ≤ 1 D) Y1 + Y4 ≥ 0 E) Y1 + Y4 ≥ 1 99) Write the constraint for machine 4. A) Y4 =11 B) Y4 ≤ 300 C) X4 ≤ 5000 D) X4 -300Y1 ≤ 0 E) 2X4 -300Y1≤ 5000 100) Write the constraint that indicates they can purchase no more than 3 machines. A) Y1 + Y2 + Y3+ Y4 ≤ 3 B) Y1 + Y2 + Y3+ Y4 = 3 C) Y1 + Y2 + Y3+ Y4 ≥3 D) Y1 + Y2 + Y3 = 3 Y2 + Y3+ Y4 = 3 E) none of the above Introduction to Management Science, 10e (Taylor) Chapter 6 Transportation, Transshipment, and Assignment Problems 1) In a transportation problem, items are allocated from sources to destinations at a minimum cost. 2) In a transportation problem, items are allocated from sources to destinations at a maximum value. 3) The linear programming model for a transportation problem has constraints for supply at each source and demand at each destination. 4) In a balanced transportation model where supply equals demand, all constraints are equalities. 5) In an unbalanced transportation model all constraints are equalities. 6) In order to model a “prohibited route” in a transportation or transshipment problem, the route should be omitted from the linear program. 7) In an unbalanced transportation problem, if demand exceeds supply, the optimal solution will be infeasible. 8) The transshipment model includes intermediate points between the sources and destinations. 9) In a transshipment problem, items may be transported from sources through transshipment points on to destinations. 10) In a transshipment problem, items may be transported from one source to another. 11) In a transshipment problem, items may be transported from one transshipment point to another. 12) In a transshipment problem, items may be transported from one destination to another. 13) In a transshipment problem, items may be transported directly from sources to destinations. 14) In a transshipment problem, items may be transported from destination to destination and from source to source. 15) An assignment problem is a special form of transportation problem where all supply and demand values equal 1. 16) An assignment problem is a special form of transportation problem. 17) Assignment linear programs always result in integer solutions. 18) Networks may be used to represent assignment problems. 19) Assignment problems are always balanced. 20) In a __________ problem, items are allocated from sources to destinations at a minimum cost. 21) In a __________ transportation model where supply equals demand, all constraints are equalities. 22) In an unbalanced transportation problem, if supply exceeds demand, the shadow price for at least one of the supply constraints will be equal to __________. 23) In order to model a “prohibited route” in a transportation or transshipment problem, the cost assigned to the route should be __________. 24) The __________ model is an extension of the transportation model in which intermediate points are added between the sources and destinations. 25) An example of a __________ point is a distribution center or warehouse located between plants and stores. 26) An appropriate choice of a model for analyzing the best shipping routes for a supply chain consisting of a manufacturer, warehouse, and retailer would be the __________ model. 27) A form of the transportation problem in which all supply and demand values equal 1 is the __________ problem. 28) A plant has 4 jobs to be assigned to 4 machines, and each machine has different manufacturing times for each product. The production manager wants to determine the optimal assignments of 4 jobs to 4 machines to minimize total manufacturing time. This problem can be most efficiently solved using the __________ model. 29) If the number of sources is greater than the number of destinations, then we have an __________ assignment problem. 30) In an assignment problem, all demand and supply values are equal to __________. A logistics specialist for Wiethoff Inc. must distribute cases of parts from 3 factories to 3 assembly plants. The monthly supplies and demands, along with the per-case transportation costs are: 31) If 120 units are shipped from Factory C to Assembly Plant 1, 60 units from Factory C to Assembly Plant 3, and 400 units from Factory B to Assembly Plant 2, what are the remaining shipments? 32) What is the objective function for the linear programming formulation of this problem? 33) What are the supply constraints for the factories? 34) What are the demand constraints for the assembly plants? 35) What are the total monthly transportation costs for the optimal solution? Consider the following transportation problem: 36) How many supply-side constraints are there? Write the supply-side constraints. 37) How many demand-side constraints are there? Write the demand-side constraints. 38) If the optimal solution includes x11 = 100 and x22 = 200, determine the remaining shipments that will result in a minimum cost of $1700. A large book publisher has five manuscripts that must be edited as soon as possible. Five editors are available for doing the work, however their working times on the various manuscripts will differ based on their backgrounds and interests. The publisher wants to use an assignment method to determine who does what manuscript. Estimates of editing times (in hours) for each manuscript by each editor is: 39) a) How many supply-side constraints are needed? b) How many demand-side constraints are needed? c) How many variables are involved in this assignment method? 40) If the optimal assignments include manuscript 1 to editor B, manuscript 2 to editor E and manuscript 3 to editor C with a total editing time of 47 minutes, what manuscripts are assigned to editors D and A? 41) What is the linear programming constraints for manuscript 1 and editor C? 42) How many supply-side constraints are there? What are the supply-side constraints? 43) How many demand-side constraints are there? What are the demand-side constraints? 4 demand side constraints 44) Write the assignment problem matrix as a network flow problem. Assume that the numbers in each cell represent the travel distance required between nodes. The dash indicates that there is not a route between nodes. A B C 1 4 6 – 2 – 2 1 3 3 5 9 Awards committees need to be formed to review potential award recipients. In the past, 3 people have been assigned to review each applicant. The only stipulation is that a reviewer cannot be assigned to an applicant if the applicant is a co-worker. The matrix below shows 9 reviewers, 3 candidates, and a matrix. If an entry in the matrix contains an “X”, then that specific reviewer is ineligible to review an applicant’s material. For example, Reviewer 1 cannot review materials submitted by candidate B. It is possible that some reviewers may not receive an assignment. Applicant Reviewer A B C 1 X 2 X 3 X 4 5 X 6 X 7 X 8 X 9 X 45) Formulate this as an assignment problem in which 2 reviewers are assigned to review each applicant’s material. 46) A partial solution to this problem is shown below, where the number 1 indicates when a reviewer is assigned to an applicant. Assign reviewers to Applicant B and 1 additional reviewer to Applicant C. Applicant Reviewer A B C 1 X 2 X 1 3 X 4 1 5 X 6 X 7 X 8 1 X 9 X Demand 2 2 2 Assigned 2 47) The committee would like to assign 3 reviewers to each applicant. A partial solution to this problem is shown below, where the number 1 indicates when a reviewer is assigned to an applicant. Assign reviewers to Applicant B and Applicant C. Applicant Reviewer A B C 1 X 2 X 3 1 X 4 1 5 X 6 X 7 X 8 X 9 1 X In setting up the an intermediate (transshipment) node constraint, assume that there are three sources, two intermediate nodes and two destinations, and travel is possible between all sources and the intermediate nodes and between all intermediate nodes and all destinations for a given transshipment problem. In addition, assume that no travel is possible between source nodes, between intermediate nodes and between destination nodes and no direct travel from source nodes to destination nodes. Let the source nodes be labeled as 1,2,3 and the intermediate nodes be labeled as 4 and 5, and the destination nodes be labeled as 6 and 7. 48) State the constraint for intermediate node 4. 49) If there are 300 units available at source 2, state the constraint for source node 2. 50) If there are 175 units demanded at destination 6, state the constraint for destination 6. Madlantic Devices designs and manufactures high end medical devices. The facilities in Madison and Atlanta serve as design and component manufacturing facilities. Components are then shipped to warehouses in Philadelphia or Knoxville where they are held until final assembly is completed at either Dayton, Bloomington or Albany. Manufacturing capacity in Madison and Atlanta is 1000 units. Demand at Dayton, Bloomington, and Albany is 450, 500, and 610 respectively. The network representing the shipping routs is shown below. The costs for shipping between each facility is shown below. A blank cell indicates that shipping between two facilities is not permitted. Warehouses Final Assembly Facilities Design and Component Manufacturing Philadelphia Knoxville Dayton Bloomington Albany Madison 7 8 Atlanta 4 7 Warehouses Philadelphia 3 25 6 17 Knoxville 3 29 8 5 51) What is the objective function for this problem? Use the notation Xij where i and j correspond to the node numbers indicated in the diagram. 52) What is the constraint for the transshipment node in Knoxville? 53) What is the constraint for Bloomington? 54) Due to increased sales, a company is considering building 3 new distribution centers (DCs) to serve 4 regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. Define the decision variables for this situation. 55) Due to increased sales, a company is considering building 3 new distribution centers (DCs) to serve 4 regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. Write the objective function for this problem . 56) Due to increased sales, a company is considering building 3 new distribution centers (DCs) to serve 4 regional sales areas. The annual cost to operate DC 1 is $500 (in thousands of dollars). The cost to operate DC 2 is $600 (in thousands of dollars.). The cost to operate DC 3 is $525 (thousands of dollars). Assume that the variable cost of operating at each location is the same, and therefore not a consideration in making the location decision. The table below shows the cost ($ per item) for shipping from each DC to each region. Region DC A B C D 1 1 3 3 2 2 2 4 1 3 3 3 2 2 3 The demand for region A is 70,000 units; for region B, 100,000 units; for region C, 50,000 units; and for region D, 80,000 units. Assume that the minimum capacity for the distribution center will be 500,000 units. Write the constraints for the 3 distribution centers. 57) In a balanced transportation model where supply equals demand, A) all constraints are equalities B) none of the constraints are equalities C) all constraints are inequalities D) none of the constraints are inequalities 58) In a transportation problem, items are allocated from sources to destinations A) at a maximum cost B) at a minimum cost C) at a minimum profit D) at a minimum revenue 59) The linear programming model for a transportation problem has constraints for supply at each __________ and __________ at each destination. A) destination, source B) source, destination C) demand, source D) source, demand 60) Which of the following assumptions is not an assumption of the transportation model? A) Shipping costs per unit are constant. B) There is one transportation route between each source and destination. C) There is one transportation mode between each source and destination. D) Actual total supply and actual total demand must be equal. 61) The problem that deals with the distribution of goods from several sources to several destinations is the A) network problem B) assignment problem C) transportation problem D) transshipment problem 62) In the linear programming formulation of a transportation network A) there is one variable for each arc B) there is one constraint for each node C) the sum of variables corresponding to arcs out of an origin node is constrained by the supply at that node D) all of the above 63) The assignment problem constraint x41+x42+x43+x44 ≤ 3 means A) agent 3 can be assigned to 4 tasks B) agent 4 can be assigned to 3 tasks C) a mixture of agents 1, 2, 3 and 4 will be assigned to tasks 1, 2 or 3 D) there is no feasible solution 64) Which of the following are assumptions or requirements of the transportation problem? A) There must be multiple sources. B) Goods are the same, regardless of source C) A and B D) none of the above The following table represents the cost to ship from Distribution Center 1, 2, or 3 to Customer A, B, or C. 65) The constraint that represents the quantity supplied by DC 1 is: A) 4X1A + 6X1B + 8X1C ≤ 500 B) 4X1A + 6X1B + 8X1C = 500 C) X1A + X1B + X1C ≤ 500 D) X1A + X1B + X1C =500 E) X1A + X1B + X1C ≥ 500 66) The constraint that represents the quantity demanded by Customer B is: A) 6X1B + 2X2B + 8X3B ≤ 350 B) 6X1B + 2X2B + 8X3B = 350 C) X1B + X2B + X3B ≤ 350 D) X1B + X2B + X3B = 350 E) X1B + X2B + X3B ≥ 350 67) In a transshipment problem, items may be transported A) from destination to destination B) from one transshipment point to another C) directly from sources to destinations D) all of the above 68) In a transshipment problem, items may not be transported A) from source to source B) from sources to destinations C) from sources to transshipment points D) from transshipment points to destinations E) none of the above Consider the following network representation of shipment routes between plants, a distribution center, and retail outlets. The numbers next to the arcs represent shipping costs. For example, the cost of shipping from plant 1 to distribution center 3 is equal to 2. Assume that Plant 1 can supply 400 units and Plant 2, 500 units. Demand at the retail outlets are: Outlet 4, 300 units; Outlet 5, 250 units; Outlet 6, 450 units. 69) Which constraint represents transshipment through the distribution center? A) 2X13+3X23 = 900 B) 2X13 + 3X23 + 5X34 + 4X35 + 3X36 = 0 C) X13 +X23 – X34 – X35 – X36 = 0 D) X13 + X23 – X34 – X35 – X36 ≥ 0 E) X34 + X35 + X36 =900 70) Which constraint represents the quantity shipped to retail outlet 6? A) X23 + X36 = 450 B) X23 + X36 + X26= 450 C) X36 + X26 ≤ 450 D) X36 + X26 = 450 E) 3X36 + 5X26= 450 71) In an assignment problem all supply and demand values equal A) 0 B) 1 C) 2 D) greater than 1 E) none of the above 72) In the process of evaluating location alternatives, the transportation model method minimizes the A) total demand B) total supply C) total shipping cost D) number of destinations 73) The assignment problem constraint x31+x32+x33+x34 ≤ 2 means A) agent 3 can be assigned to 2 tasks B) agent 3 can be assigned to no more than 2 tasks C) a mixture of agents 1, 2, 3 and 4 will be assigned to tasks D) agent 2 can be assigned to 3 tasks 74) In an assignment problem, A) one agent can do parts of several tasks B) one task can be done by only one agent C) each agent is assigned to its own best task D) B and C 75) The difference between the assignment and the transportation problem is that A) total supply must equal total demand in the assignment problem B) the number of origins must equal the number of destinations in the transportation problem C) each supply and demand value is 1 in the assignment problem D) A and B E) none of the above Madlantic Devices designs and manufactures high end medical devices. The facilities in Madison and Atlanta serve as design and component manufacturing facilities. Components are then shipped to warehouses in Philadelphia or Knoxville where they are held until final assembly is completed at either Dayton, Bloomington or Albany. Manufacturing capacity in Madison and Atlanta is 1000 units. Demand at Dayton, Bloomington, and Albany is 450, 500, and 610 respectively. The network representing the shipping routs is shown below. The costs for shipping between each facility is shown below. A blank cell indicates that shipping between two facilities is not permitted. Warehouses Final Assembly Facilities Design and Component Manufacturing Philadelphia Knoxville Dayton Bloomington Albany Madison 7 8 Atlanta 4 7 Warehouses Philadelphia 3 25 6 17 Knoxville 3 29 8 5 76) The transshipment locations are: A) Madison and Atlanta B) Philadelphia and Knoxville C) Madison, Atlanta, Philadelphia and Knoxville D) Dayton, Bloomington, and Albany 77) The constraint for the quantity shipped from Madison is: A) X13 + X 14 = 1000 B) X13 + X 14 ≤ 1000 C) X13 + X 14 ≥ 1000 D) X13 + X 14 – X34 = 1000 78) The constraint for Philadelphia is A) X13 + X23 – X35 – X36 – X37 = 0 B) X13 + X23 – X35 – X36 – X37 ≥ 0 C) X13 + X23 + X43 – X34 – X35 – X36 – X37 = 0 D) X13 + X23 + X43 – X34 – X35 – X36 – X37 ≥ 0 79) The objective function is: A) MIN 7X13 + 8X14 + 4X23 + 7X24 + 3X34 + 3X43 + 25X35 + 6X36 + 17X37 + 29X45 + 8X46 + 5X47 B) MIN 7X13 + 8X14 + 4X23 + 7X24 – 3X34 – 3X43 + 25X35 + 6X36 + 17X37 + 29X45 + 8X46 + 5X47 C) MIN 7X13 + 8X14 + 4X23 + 7X24 + 3X34 + 3X43 – 25X35 – 6X36 – 17X37 – 29X45 – 8X46 – 5X47 D) MIN 7X13 + 8X14 + 4X23 + 7X24 + 25X35 + 6X36 + 17X37 + 29X45 + 8X46 + 5X47 A professor needs help from 3 student helpers to complete 4 tasks. The first task is grading; the second is scanning; the third is copying, and the fourth is organizing student portfolios. The estimated time for each student to do each task is given in the matrix below. 80) How many tasks will be assigned to the students? A) 2 tasks B) 3 tasks C) 4 tasks D) none, because the students take too long 81) Which of the following constraints represents the assignment for student A? A) XA1 +XA2 + XA3 + XA4 = 0 B) XA1 +XA2 + XA3 + XA4 = 1 C) XA1 +XA2 + XA3 + XA4 ≤ 1 D) XA1 +XA2 + XA3 + XA4 ≥ 0 82) Which of the following constraints represents the assignment for task 2, scanning? A) X2A +X2B + X2C = 0 B) X2A +X2B + X2C = 1 C) X2A +X2B + X2C ≤ 1 D) XX2A +X2B + X2C ≥ 0 83) Based on the information in the table, which tasks are least likely to be assigned to a student? A) grading B) scanning C) copying D) organizing portfolios Introduction to Management Science, 10e (Taylor) Chapter 7 Network Flow Models 1) A network is an arrangement of paths connected at various points through which items move. 2) Networks are popular because they provide a picture of a system and because a large number of systems can be easily modeled as networks. 3) Nodes represent junction points connecting branches. 4) Branches connect nodes and show flow from one point to another. 5) The values assigned to branches typically represent distance, time, or cost. 6) Flows in a network can only be in one direction. 7) The shortest route problem is to find the shortest distance between an origin and various destination points. 8) The shipping company manager wants to determine the best routes for the trucks to take to reach their destinations. This problem can be solved using the minimal spanning tree. 9) The shortest route network problem could help identify the best route for pizza delivery drivers from the pizza parlor to a specific customer. 10) The minimal spanning tree problem is to connect all nodes in a network so that the total branch lengths are minimized. 11) The first step of the minimal spanning tree solution to compute the distance of any path through the network. Answer: FALSE Diff: 2 Page Ref: 291-293 Main Heading: The Minimal Spanning Tree Problem Key words: minimal spanning tree problem 12) The last step of the minimal spanning tree solution method is to make sure all nodes have joined the spanning tree. 13) In a minimal spanning tree, the source and destination nodes must be connected along a single path. 14) The choice of the initial node in the minimal spanning tree technique must be the first node. 15) The minimal spanning tree allows the visitation of each node without backtracking. 16) The shortest route network problem could help identify the best plan for running cables for televisions throughout a building. 17) The goal of the maximal flow problem is to maximize the amount of flow of items from an origin to a destination. 18) For a directed branch, flow is possible in only one direction. 19) To determine the maximum possible flow of railroad cars through the rail system they should first select the longest path from origin to destination and ship as much as possible on that path. 20) The shortest route problem requires that there be a branch from each destination to every other destination. 21) The maximal flow algorithm may end with capacity remaining at the source. 22) The source node is the input node in a maximal flow problem. 23) The direction of the flow is not critical in the maximal flow problem. 24) A traffic system could be represented as a network in order to determine bottlenecks using the maximal flow network algorithm. 25) In a network flow problem, __________ represent junction points connecting branches. Answer: nodes Diff: 1 Page Ref: 281 Main Heading: Network Models Key words: network flow models, nodes 26) In a network flow problem, __________ connect nodes and show flow from one point to another. 27) In a network flow problem, the values assigned to __________ typically represent distance, time, or cost. 28) The shipping company manager wants to determine the best routes for the trucks to take to reach their destinations. This problem can be solved using the __________ solution technique. 29) The __________ connects all nodes in a network so that the total branch lengths are minimized. 30) The goal of the __________ problem is to maximize the amount of flow of items from an origin to a destination. 31) A __________ network model could be used to represent the capacity of a series of dams for flood control. 32) A company plans to use an automatic guided vehicle for delivering mail to ten departments. The vehicle will begin from its docking area, visit each department, and return to the docking area. Cost is proportional to distance traveled. The type of network model that best represent this situation is __________. 33) Determining where to build roads at the least cost within a park that reaches every popular sights represents a __________ network model. 34) Determining where to build one way roads at the least cost within a park that takes visitors to every popular sight and returns them to the entrance represents a __________ network model. 35) Determining where capacity needs to be added within a series of one way roads within a park represents a __________ model. Figure 1. Delivery Routes 36) Consider the network diagram given in Figure 1. Assume that the amount on each branch is the distance in miles between the respective nodes. What is the shortest route from the source node (node 1) to nodes 2, 3, and 4. Indicate the total distance for each route. 37) Consider the network diagram given in Figure 1. Assume that the amount on each branch is the distance in miles between the respective nodes. What is the shortest route from the source node (node 1) to nodes 5 and 6. Indicate the total distance for each route. 38) Consider the network diagram given in Figure 1. Assume that the amount on each branch is the distance in miles between the respective nodes. Also assume that it is not possible to travel from a node with a higher number to a node with a lower number. Write the constraint associated with the second node (node 2) for the 0-1 integer linear programming formulation of the shortest route problem. 39) Consider the network diagram given in Figure 1. Assume that the amount on each branch is the distance in miles between the respective nodes. Also assume that it is not possible to travel from a node with a higher number to a node with a lower number. Write the constraint associated with the fifth node (node 5) for the 0-1 integer linear programming formulation of the shortest route problem. 40) Consider the network diagram given in Figure 1. Assume that he numbers on the branches indicate the length of cable (in miles) six nodes on a telecommunication network. What is the minimum number of miles of cable to be used to connect all six nodes? Pro-Carpet company manufactures carpets in Northwest Indiana and delivers them to warehouses and retail outlets. The network diagram given in the Figure below shows the possible routes and travel times (in minutes) from the carpet plant to the various warehouses or retail outlets. V = Valparasio, P=Portage, G=Gary, Ha=Hammond, Hi=Highland, M = Merillville, L = Lansing 41) Determine the shortest route for a carpet delivery truck from the carpet plant in Valparaiso, Indiana to warehouses in Hammond – IN, Gary – IN and Merillville – IN. State the total completion time in minutes or each route 42) Determine the shortest route for a carpet delivery truck from the carpet plant in Valparaiso, Indiana to retail outlets in Portage – IN, Highland – IN, and Lansing, Illinois. State the total completion time in minutes or each route 43) Write the constraint associated with the Valparasio (source) node for the 0-1 integer linear programming formulation of the shortest route problem. 44) Write the constraint associated with the Lansing (destination) node for the 0-1 integer linear programming formulation of the shortest route problem. 45) Draw the network associated with the following constraints for a shortest route problem. X12 + X13 = 1 X12 – X24 = 0 X13 – X34 = 0 X24 + X34 – X45 = 0 X45 = 1 Consider the following network, which shows the location of various facilities within a youth camp and the distances (in tens of yards) between each facility. 46) Walking trails will be constructed to connect all the facilities. In order to preserve the natural beauty of the camp (and to minimize the construction time and cost), the directors want to determine which paths should be constructed. Use this network to determine which paths should be built. 47) The camp nurse is stationed at Facility B. What is the shortest route from B to C? Consider the following network, which shows the location of various facilities within a youth camp and the distances (in tens of yards) between each facility. There is a swampy area between facility A and E. 48) Walking trails will be constructed to connect all the facilities. In order to preserve the natural beauty of the camp (and to minimize the construction time and cost), the directors want to determine which paths should be constructed. Use this network to determine which paths should be built. 49) A clean up crew visits is stationed at facility D and wants to take the shortest route to each site. They usually clean up facilities C, E, A and F on the same day and therefore want the shortest route from D to each facility. . Recommend a route for the crew to leave from D, clean up each facility one after the other, and return to facility D. (Assume all paths are accessible.) 50) A clean up crew is stationed at facility F and wants to take the shortest route to each site. They usually clean up facilities B and A on the same day and therefore want the shortest route from F to each facility. Recommend a route for the crew to leave from F, clean up each facility A and B, and then return to facility D. (Assume all paths are accessible.) Refer to the figure below to answer the following questions. Figure 3 51) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. Determine the maximal flow from source node 1 to destination node 9. 52) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. Determine the maximal flow on the following path: node 1 to node 2 to node 7 to destination node 9. 53) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. What is the objective function for the 0-1 integer linear programming formulation of the maximal flow problem? 54) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. What is the input- output constraint associated with the first node of the network diagram for the 0-1 integer linear programming formulation of the maximal flow problem? 55) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. What is the input-output constraint associated with the ninth node of the network diagram for the 0-1 integer linear programming formulation of the maximal flow problem? 56) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. What is the capacity constraint associated with the branch from node 7 to node 9 of the network diagram for the 0-1 integer linear programming formulation of the maximal flow problem? 57) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. What is the input-output constraint associated with the fifth node of the network diagram for the 0-1 integer linear programming formulation of the maximal flow problem? 58) In a network flow model, a directed branch A) is a branch with a positive distance value B) has the same flow capacity in each direction C) is a branch in which flow is possible in only one direction D) is a branch on which the flow capacity is exhausted E) is a branch in which flow is not possible in either direction 59) In a network modeling problem, the linear programming decision variables are given by A) source node B) sink node C) network branches D) origin node E) network nodes 60) A branch where flow is permissible in either direction is a A) directed branch B) undirected branch C) labeled branch D) unlabeled branch E) unbounded branch 61) If we wanted to represent water resources as a network flow problem, which of the following would be represented as nodes? A) canals B) pumping stations C) rivers D) pipelines E) all of the above 62) If we wanted to represent an office layout as a network flow problem, which of the following would be represented as a branch? A) offices B) waiting areas C) heating and ventilation systems D) computer rooms E) none of the above 63) If we wanted to represent an urban transportation system as a network flow problem, which of the following would be represented as nodes? A) streets B) railway lines C) street intersections D) pedestrian right of ways E) none of the above 64) The shipping company manager wants to determine the best routes for the trucks to take to reach their destinations. This problem can be solved using the A) shortest route solution technique B) minimal spanning tree solution method C) maximal flow solution method D) minimal flow solution method 65) In the linear programming formulation of the shortest route problem, the constraint for each node represents A) capacity on each path B) conservation of flow C) capacity on each branch D) minimum flow 66) The first step in the shortest route solution method is to A) select the node with the shortest direct route from the origin B) determine all nodes directly connected to the permanent set nodes C) arbitrarily select any path in the network from origin to destination D) make sure that all nodes have joined the permanent set E) select any starting node Figure 2. 67) Consider the network diagram given in Figure 2. Assume that the amount on each branch is the distance in miles between the respective nodes. What is the distance for the shortest route from the source node (node 1) to nodes 4. A) 8 B) 9 C) 10 D) 11 E) 12 68) Consider the network diagram given in Figure 2. Assume that the amount on each branch is the distance in miles between the respective nodes. What is the distance for the shortest route from the source node (node 1) to nodes 5. A) 13 B) 14 C) 15 D) 16 E) 17 Pro-Carpet company manufactures carpets in Northwest Indiana and delivers them to warehouses and retail outlets. The network diagram given in the Figure below shows the possible routes and travel times (in minutes) from the carpet plant to the various warehouses or retail outlets. V = Valparasio, P=Portage, G=Gary, Ha=Hammond, Hi=Highland, M = Merillville, L = Lansing 69) What is the distance for the shortest route from the carpet plant in Valparaiso, Indiana to retail outlet in Lansing, Illinois. State the total completion time in minutes. A) 36 B) 37 C) 39 D) 41 E) 43 70) Determine the shortest route for a carpet delivery truck from the carpet plant in Valparaiso, Indiana to retail outlet in Hammond, Indiana. A) 26 B) 28 C) 30 D) 32 E) 34 71) The minimal spanning tree problem determines the A) minimum amount that should be transported along any one path B) maximum amount that can be transported along any one path C) shortest distance between a source node and a destination node D) minimum total branch lengths connecting all nodes in the network 72) The first step of the minimal spanning tree solution method is to A) select any starting node B) select the node closest to the starting node to join the spanning tree C) select the closest node not presently in the spanning tree D) . arbitrarily select any path in the network from origin to destination E) select the node with the shortest direct route from the origin 73) The local Internet provider wants to develop a network that will connect its server at its satellite center in Valparaiso with the main city computer centers in Northwest Indiana to improve the Internet service and to minimize the amount of cable used to connect network nodes. If we represent this problem with a network, A) the cities are branches and cables are nodes B) the cables are the branches and the cities are the nodes C) the length of cables in miles are the branches, and the cities are the nodes D) the cities are the branches and the length of cables in miles are the nodes E) none of the above Figure 2. 74) Consider the network diagram given in Figure 2. Assume that the numbers on the branches indicate the length of cable (in miles) six nodes on a telecommunication network. What is the minimum number of miles of cable to be used to connect all six nodes? A) 16 miles B) 17 miles C) 18 miles D) 19 miles E) 20 miles Consider the following network, which shows the location of various facilities within a youth camp and the distances (in tens of yards) between each facility. There is a swampy area between facility A and E. 75) Walking trails will be constructed to connect all the facilities. In order to preserve the natural beauty of the camp (and to minimize the construction time and cost), the directors want to determine which paths should be constructed. What is the minimum number of paths (in tens of yards) that must be built to connect each facility? A) 54 B) 56 C) 60 D) 65 E) 82 76) The objective of the maximal flow solution approach is to A) maximize resource allocation B) maximize the total amount of flow from an origin to a destination C) determine the longest distance between an originating point and one or more destination points D) determine the shortest distance between an originating point and one or more destination points 77) The first step of the maximal flow solution method is to A) arbitrarily select any path in the network from origin to destination B) select the node with the shortest direct route from the origin C) add the maximal flow along the path to the flow in the opposite direction at each node D) make sure there is no path with available flow capacity left E) select any starting node 78) The shortest route problem requires A) each destination to be visited only once. B) finding the quickest route from the source to each node. C) that there be a branch from each destination to every other destination D) that there be no two-way branches between nodes 79) The maximal flow algorithm A) does not require flow on every branch for the final solution B) may end with capacity remaining at the source C) may end with capacity at those nodes leading immediately to the sink D) all of the above Refer to the figure below to answer the following questions. Figure 3 80) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. Determine the maximal flow from source node 1 to destination node 9. A) 10 B) 11 C) 12 D) 13 E) 14 81) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. Determine the maximal flow on the following path: node 1 to node 4 to node 3 to node 5 to node 8 to destination node 9. A) 2 B) 3 C) 4 D) 5 E) 6 82) Consider the network diagram given in Figure 3 with the indicated flow capacities along each branch. Determine the maximal flow on the following path: node 1 to node 2 to node 7 to destination node 9. A) 2 B) 3 C) 4 D) 5 E) 6 Figure 4. 83) Determine the maximal flow through the network in Figure 4. Assume that all branches are directed branches. A) 10 B) 11 C) 13 D) 16 84) Determine the minimum distance required to connect all nodes in Figure 4. A) 22 B) 24 C) 26 D) 30 Introduction to Management Science, 10e (Taylor) Chapter 8 Project Management 1) Project teams are made up of individuals from various areas and departments within a company. 2) A statement of work is a written description of the goals, work, and time frame of a project. 3) The sequence of activities in a project is depicted by the precedence relationships. 4) A work breakdown structure breaks down a project into subcomponents, components, activities, and tasks. 5) An element of project planning is to compare the project schedule objectives. 6) Estimating project funds is an element of project planning. 7) Establishing precedence relationships is an element of project planning. 8) Making time estimates is an element of project planning. 9) Determining project completion time is an element of project planning. 10) Setting deadlines is an element of project planning. 11) One aspect of project control is team building. 12) The critical path is the shortest path through the project network. 13) The critical path is the longest path through the project network. 14) Slack is the amount of time an activity can be delayed without delaying the project. 15) In PERT, three time estimates (optimistic, most likely and pessimistic) provide an estimate of the mean and variance for each activity based on the normal distribution. 16) The critical path activities have no slack. 17) In PERT, the project variance is the sum of the variance of all project activities. 18) In PERT, the expected project completion time is assumed to be distributed according to Beta distribution. 19) The expected project completion time is assumed to be normally distributed. 20) Project crashing is a method for shortening the project completion time by reducing the completion time of the one or more critical activities at a cost. 21) An AOA network will never require dummy activities. 22) An AOA network will never require dummy activities. 23) An AON network will never require dummy activities. 24) Two concurrent activities in a project can be represented in an AOA network by starting and ending on the same node. 25) The early start for an activity on the critical path is always equal to the earliest finish of the preceding activity. 26) The early start for a non critical activity is always equal to the earliest finish of the preceding activity. 27) The early finish for an activity on the critical path is equal to the latest finish of the preceding activity. 28) Microsoft project constructs project planning networks with activities on the arrows. 29) A work breakdown structure is a useful tool because it addresses the timing of individual work elements. 30) The PERT pessimistic time estimate is an estimate of the minimum time an activity will require. 31) In PERT, the path with the fewest activities is referred to as the critical path. 32) The standard deviation of project duration is the average of the standard deviation of all activities on the critical path. 33) When earliest finish is subtracted from latest finish, we obtain the slack value for the activity. 34) Crashing cost is always higher than the normal cost. 35) Shortening the project’s duration by deleting unnecessary activities is called crashing. 36) The earliest finish time for the final activity on a project network is also the total completion of the project. 37) The early start of an activity that has only one predecessor is equal to the early finish of that predecessor 38) The early start of an activity that has more than one predecessor is the minimum of the earliest finishes of all predecessors. 39) In PERT, the actual project completion time will never exceed the expected duration of the critical path. 40) Project __________ a method for shortening the project completion time by reducing the completion time of the one or more critical activities at a cost. 41) The expected project completion time is assumed to be __________ distributed based on the central limit theorem. 42) Three time estimates (optimistic, most likely and pessimistic) provide an estimate of the mean and variance for each activity based on the __________ distribution. 43) The critical path activities have no __________. 44) __________ is the amount of time an activity can be delayed without delaying the project. 45) A(n) __________ chart is a graph or a bar chart for each activity that shows the project schedule and the passage of time 46) In an AOA network, a __________ activity is used to show a precedence relationship, but it does not represent passage of time. 47) When the earliest start time (ES) of an activity is subtracted from the latest start time (LS) of an activity, the value obtained is called __________. 48) A __________ shows who in the organization is responsible for doing the work in a project. 49) A __________ breaks down a project into modules: subcomponents, components, activities, and tasks. 50) When applying linear programming models to project crashing, the objective is to minimize __________. 51) When applying linear programming models to project planning, the objective is to minimize __________. Consider the following network. 52) The early start for activity 2 is __________. 53) The latest start for activity 3 is _________. Consider the following network: 54) The critical path for this network is _________ 55) The slack for activity 2 is equal to __________. 56) The slack for activity 3 is equal to ______. Refer to the information provided in the table below to answer the following questions. 57) What are the estimated expected completion (mean) times (in weeks) for activities A-H? 58) What is the critical path for this project network? 59) If the early start for activity A is equal to 23 months and the duration of activity A is equal to 5 months, what is the earliest finish for activity A? 60) If the earliest finish for activity C is equal to 10 months, its duration 3 months, what is its earliest start? Refer to the information provided in the table below to answer the following questions. 61) Determine the latest start for activity E. 62) What are the estimated slack times (in weeks) for activities A-H? 63) What are the estimated standard deviations (in weeks) in the times for activities A-H? 64) What is the estimated expected mean time (in weeks) for the project completion? 65) What is the estimated standard deviation (in weeks) for the critical path completion time? 66) What is the probability that the critical path for this project will be completed within 75 weeks? 67) What is the probability that the critical path for this project will be completed within 80 weeks? 68) Within what amount of time (in weeks) is there a 90% probability that the critical path for this project will be completed? 69) Within what amount of time (in weeks) is there a 99% probability that the critical path for this project will be completed? 70) Given the following information for a project, draw the AON (Activity-On -Node) project network. Activity Immediate Processor Activity Time (days) A – 6 B – 8 C A, B 5 D B 4 E C 7 F C, D 3 G D 6 H E, F, G 5 71) Given the following information for a project, draw the AOA (Activity-On -Arrow) project network. Activity Immediate Processor Activity Time (days) A – 6 B – 8 C A, B 5 D B 4 E C 7 F C, D 3 G D 6 H E, F, G 5 72) Given the following information for a project, identify the critical path and determine the total completion time of the project. Activity Immediate Processor Activity Time (days) A – 6 B – 8 C A, B 5 D B 4 E C 7 F C, D 3 G D 6 H E, F, G 5 73) Given the following information for a project, What are the estimated slack times (in weeks) for activities A-H? Activity Immediate Processor Activity Time (days) A – 6 B – 8 C A, B 5 D B 4 E C 7 F C, D 3 G D 6 H E, F, G 5 CP = Critical Path Consider the following network. 74) Identify all paths in the network. 75) Identify the critical path. 76) How long will it take to complete the project? 77) Assume that the variance for each activity in the network is equal to 1. Compute the probability that path A-B-E-F will be completed in 15 days. The diagram below shows the activities on the nodes, and the table shows the normal time, crash time and cost for each activity, in days. Activity Normal Time Crash Time Cost per day to crash A 6 6 — B 10 8 $100 C 5 4 $300 D 4 1 $700 E 9 7 $500 F 2 1 $650 78) Determine which activities should be crashed to shorten the project by 1 day. 79) Determine which activities should be crashed to shorten the project by 2 days. 80) Determine which activities should be crashed to shorten the project by 3 days. What is the cost? 81) Once a project is underway, the project manager is responsible for the A) people B) cost C) time D) all of the above 82) A work breakdown structure breaks down a project into A) subcomponents and components B) activities C) tasks D) all of the above 83) Elements of project planning are A) defining project objectives B) identifying activities C) establishing precedence relationships D) all of the above 84) Elements of project planning include A) making time estimates B) determining project completion time C) comparing project schedule objectives D) all of the above 85) If an activity cannot be delayed without affecting the entire project, then it is a __________ activity. A) completed B) critical C) crashed D) non-critical E) normal 86) A list of the tasks, broken down into modules, components, and individual tasks, is called: A) work breakdown structure(WBS) B) PERT C) planning matrix D) crashing E) critical path 87) Project control involves A) time management B) cost management C) performance management D) earned value analysis E) all of the above 88) On an AOA diagram, a __________ represents an activity of a project. A) route B) branch C) path D) node E) event 89) On an AOA diagram, a __________ represents the beginning and end of activities, referred to as events. A) path B) arc C) branch D) node E) route 90) The activities that must be completed prior to the start of an activity in question are called the immediate __________ of the activity in question. A) successors B) predecessors C) successors and predecessors D) followers E) none of the above 91) Project management differs from management of more traditional day-to-day activities because A) it has limited time frame B) its has an unlimited budget C) A and B D) none of the above 92) A dummy activity is used to show a precedence relationship, but it does not represent a __________. A) relationship between activities B) change in relationship C) real activity D) passage of time 93) The critical path is the __________ path through the network. A) longest B) shortest C) straightest D) none of the above 94) The advantage of a Gantt chart over other charts is its A) feasibility B) practicality C) simplicity D) linearity 95) In a CPM/PERT network the critical path is the A) lowest path through the network B) highest path through the network C) shortest path through the network D) longest path through the network 96) A GANTT chart represents mainly A) an immediate identification of predecessor task B) a record-keeping device used in scheduling activities C) a general identifier D) none of the above 97) If t is the expected completion time for a given activity, then A) LF = LS – t B) EF = ES – t C) 335EF = ES + t D) EF = LS – t E) none of the above 98) The LS and LF are calculated using the A) backward pass through the network B) forward pass through the network C) values for ES and EF D) backward and forward pass through the network E) none of the above Consider the following project. Activity Immediate Processor Activity Time (days) A – 6 B – 8 C A, B 5 D B 4 E C 7 F C, D 3 G D 6 H E, F, G 5 99) Determine the critical path. A) A-C-E-H B) B-C-E-H C) B-D-F-H D) B-D-G-H E) B-C-F-H 100) Determine the estimated completion time of the project. A) 23 days B) 19 days C) 22 days D) 25 days E) 20 days 101) Determine the slack for activity D. A) 0 days B) 2 days C) 4 days D) 6 days E) 7 days 102) Determine the slack for activity F. A) 2 days B) 5 days C) 6 days D) 9 days E) 13 days 103) How much can activity F be delayed without delaying the project completion? A) 0 days B) 1 day C) 2 days D) 3 days E) 4 days 104) A dummy activity in a PERT network allows us to A) specify the positive time and resources required to complete the activity B) maintain the precedence relationships C) add more starting nodes D) add more ending nodes Consider the following project. Activity Immediate Predecessor Activity Time (days) A — 3 B — 4 C A 4 D A 6 E B 5 F C,D 4 G E,F 7 105) What is the critical path? A) ACFG B) ADFG C) BEG D) ACDFG 106) What is the minimum possible time for completing this project? A) 14 days B) 15 days C) 16 days D) 18 days E) 20 days 107) Compute the slack time for activity C. A) 0 B) 7 C) 4 D) 2 E) 3 108) What is the latest possible time that Activity E can be started without delaying the completion of the project? A) 0 B) 3 C) 4 D) 7 E) 9 109) What is the latest finish time for activity F? A) 9 B) 7 C) 13 D) 16 E) 18 110) What is the latest time that activity B can start without delaying the project? A) 0 B) 8 C) 9 D) 4 E) 7 111) What is the latest time that activity B can finish without delaying any other activities? A) 0 B) 8 C) 9 D) 4 E) 7 112) Joe used a project management software package and has determined the following results for a given project. Expected completion time of the project = 22 days Variance of project completion time = 2.77 What is the probability of completing the project over 20 days? A) 0.3849 B) 0.8849 C) 0.1151 D) 0.7642 E) 0.2358 113) Joe used a project management software package and has determined the following results for a given project. Expected completion time of the project = 22 days Variance of project completion time = 2.77 What is the probability of completing the project within 20 days? A) 0.3849 B) 0.8849 C) 0.1151 D) 0.7642 E) 0.2358 114) A PERT/CPM activity has an optimistic time estimate of 3 days, a most likely time estimate of 8 days, and a pessimistic time estimate of 10 days. The expected time (in days) of this activity is A) 7.0 B) 7.5 C) 8.0 D) 8.5 E) 10.0 115) A PERT/CPM activity has an optimistic time estimate of 3 days, a most likely time estimate of 8 days, and a pessimistic time estimate of 10 days. The standard deviation of this activity is A) 7/9 B) 7/6 C) 1/3 D) 2/3 E) 8/9 116) A PERT/CPM activity has an optimistic time estimate of 4 days, a most likely time estimate of 6 days, and a pessimistic time estimate of 10 days. The expected time (in days) of this activity is A) 6.0 B) 6.33 C) 7.0 D) 7.5 E) 10.0 117) A PERT/CPM activity has an optimistic time estimate of 4 days, a most likely time estimate of 6 days, and a pessimistic time estimate of 10 days. The standard deviation of this activity is A) 7/9 B) 7/6 C) 1/3 D) 2/3 E) 1.0 118) The normal cost for an activity is $7,000 and the crash cost is $12,000. The normal time to complete this activity is 8 days and crash time is 4 days. If this activity is crashed by 2 days it will cost an additional __________. A) $1,000 B) $1,250 C) $2,000 D) $2,500 E) $5,000 119) Which of these statements regarding project crashing is true? A) Crashing is not possible unless there are multiple critical paths. B) Activities not on the critical path cannot become critical after crashing. C) Crashing shortens the project duration by assigning more resources to one or more of the critical tasks. D) Crashing a project often reduces the time it takes for lengthy or complex, but noncritical activities. E) 8/9 The diagram below shows the activities on the nodes, and the table shows the normal time, crash time and cost for each activity, in days. Activity Normal Time Crash Time Cost per day to crash A 6 6 — B 10 8 $100 C 5 4 $300 D 4 1 $700 E 9 7 $500 F 2 1 $650 120) Which activity should be crashed to reduce the project completion time by one day? A) B B) C C) D D) E E) F 121) Which activity should be crashed to reduce the project completion by 2 days? A) Crash activity B by 1 day and Activity C by 1 day. B) Crash activity E by 2 days. C) Crash activity C by 1 day and activity E by 1 day. D) Crash activity B by 2 days. E) Crash activity D by 2 days. Introduction to Management Science, 10e (Taylor) Chapter 9 Multicriteria Decision Making 1) The different objectives in a goal programming problem are referred to as goals. 2) All goal constraints are inequalities that include deviational variables. 3) A negative deviational variable is the amount by which a goal level is exceeded. 4) At least one or both deviational variables in a goal constraint must equal 0. 5) The objective function in a goal programming model seeks to minimize the deviation from goals in the order of the goal priorities. 6) In goal programming, terms are summed in the objective function in order to make consistent decisions. 7) Goal constraints can include all deviational variables. 8) In goal programming, problems cannot have two or more goals at the same priority level. 9) Goal programming provides a method for simultaneously striving to achieve several objectives. 10) A requirement for the solution procedure for the goal programming problem is that all goals must be achieved. 11) One goal is never achieved at the expense of another higher-priority goal. 12) A goal can be achieved at the expense of another lower-priority goal. 13) Goal programming solutions do not always achieve all goals. 14) In a goal programming model, the terms in the objective function are summed to determine the maximum profit or minimum cost. 15) Objective function terms are not summed in goal programming because not all goals are achievable. 16) Goal programming violates the divisibility property of linear programming. 17) In a pairwise comparison, 2 alternatives are compared according to a criterion and one is preferred. 18) A preference scale assigns numerical values to different levels of preference. 19) On a preference scale for pairwise comparisons, the number “1” indicates that two objects are equally preferred. 20) A pairwise comparison matrix summarizes the pairwise comparisons for a criterion. 21) In synthesization, decision alternatives are prioritized within each criterion. 22) A consistency index measures the degree of inconsistency in pairwise comparisons. 23) A preference scale assigns numerical values to different levels of preferences. 24) In a given AHP problem, the consistency index is .15, and the random index is .90. In this instance, there are probably serious inconsistencies and therefore AHP result may not be meaningful. 25) Scoring model is more subjective than the Analytical Hierarchy Process. 26) Scoring models use consistency indexes to measure the degree of consistency between choices. 27) In scoring models, the two alternatives with the highest scores are compared to make the final decision. 28) At least one or both deviational variables in a goal constraint must equal __________. 29) Consider the following constraint: 2×1 + 3×2 =60. Assume that we convert this constraint into a goal constraint and the value of x1 = 15 and the value of x2 = 15, then the values of d1+ and d1- are __________ and __________ respectively. 30) A(n) __________ variable is the amount by which a goal level is underachieved. 31) A(n) __________ variable is the amount by which a goal level is exceeded. 32) A(n) __________ assigns numerical values to different levels of preferences. 33) Deviation variables that occur in the objective function and the constraints of a goal programming model indicate the difference between all actual and __________ values. 34) A(n) __________ measures the degree of inconsistency in pairwise comparisons. 35) A __________ vector is the average of the values in each row of the normalized matrix in the AHP process. 36) If the CI/RI ration is more than __________ then there are probably serious inconsistencies in the AHP results. 37) The analytic hierarchy process, scoring models, and goal programming are all considered to be __________ decision making techniques because they incorporate decisions based on more than 1 objective. 38) A(n) __________ model is a similar to AHP, but simpler and easier to understand. 39) In a facility location problem, if location A receives a score of 45 for the criteria, “traffic congestion”, which has a weight of .30; a score of 50 for the criteria, “labor force”, which has a weight of .45; and a score of 60 for “utilities”, which has a weight of .25, the overall score for location A is: __________. 40) The objective function formulation for a goal programming model is as follows: Min P1 d1-, P2 d2-, P3 d1+, P4 d3- Which one of the variables would the program designers like to have as the second goal such that its deviation from zero is minimized? 41) The objective function formulation for a goal programming model is as follows: Min P1d1-, P2 d2-, (4 P4 d3- + 6 P4 d2+) At the priority level 4, which one of the deviational variables is more important? Ashley’s manufactures home furnishings for department stores. Planning is underway for the production of the following items during the next production period: Quilts (X1) Blinds (X2) Pillows (X3) Fabric required (yards) (d1) 7 4 9 Time required (hours) (d2) 1.5 2 0.5 Packaging material (ounces) (d3) 3 2 1 Profit (d4) 12 10 8 Ashley has 3000 yards of material in stock for this production period. Five hundred hours of production time are scheduled and 400 ounces of packaging material is available. Each of these quantities can be adjusted through overtime or extra purchases. Ashley’s highest priority is to achieve a profit of $3200. Her second priority is to avoid additional purchases of packaging material. Third, she wants to use all of the scheduled production hours and fourth, minimize any fabric remaining from the 3000 yards. Note that the deviational variables associated with each item is given in the table. 42) What is the objective function? 43) What is the fabric constraint? 44) What is the profit constraint? 45) What is the production time constraint? 46) What is the packaging material constraint? An investor has $80,000 to invest in three stocks, stock A costs $100, stock B costs$120 and stock C costs $80. Each stock A has a risk factor of 8, each stock B has a risk factor of 10 and each stock C has a risk factor of 7. The investor believes that the sum of the risk factors for all stocks purchase should not exceed 6,000. The projected annual growth rate for the three stocks are 9%, 13% and 8% respectively. The projected annual dividend income from these stocks are as follows: Stock A: $14/stock, Stock B: $15/stock and Stock C: $20/stock. The investor desires an annual dividend income of $10,000. The investor has established the following goals in order of their importance: (1) The investor believes that the budget cannot be exceeded. (d1) (2) The risk factor should not exceed the target amount of 6,000. (d2) (3) The average annual growth rate in stock prices must be at least 10% (d3) (4) The investor desires a dividend income of at least $10,000 (d4) 47) State the goal programming objective function. 48) Write the budget constraint 49) Write the risk factor constraint 50) Centerville City council is in the process of developing city tax rates. The annual tax base for real estate property is $750 million and for general sales, $80 million. Annual local gas consumption is estimated at 12 million gallons. They have 3 goals, listed in order of priority: 1. Tax revenues must be at least $25 million to meet the city’s financial commitments. 2. Sales tax cannot exceed 25% of all taxes collected. 3. Gasoline tax cannot exceed 8 cents per gallon. Formulate as a goal programming problem. A production of 300 units of Twiddle Bugs, an educational toy for children, must be completed within 1 week by BugU Manufacturing. Two production lines are available, each for 30 hours during the week. Production line 1 can produce five units per hour and production line 2 can produce 4 units an hour. Line 1 costs $50 per hour to operate and line 2 costs $55 per hour. Overtime is available for line 1 at $15 per hour and for line 2 at $12 per hour. Management goals in decreasing order are: P1: Produce 300 units. P2: Maximum allowable overtime of 6 hours for line 1. P3: Cost of overtime must not exceed $750. P4: Avoid the underutilization of either production line. Assign weights that are proportional to their production capability. P5: Producing more than 300 units is 1 1/2 times as undesirable as producing under 300 units. 51) What are the constraints for this problem? 52) What is the objective function? 53) Assume that a decision maker using analytical hierarchy process generated the following pairwise comparison matrix. 54) Using the normalized matrix given below, calculate the row averages. 55) Assume that a decision maker has to make a choice between three types of cars. Chevy, Honda, and Ford. Based on three criteria: comfort, MPG and style. The row averages for criteria is summarized in the following column vector: The preference matrix for the three types of cars is given as follows: Compute the overall score for each decision alternative (car). 56) Assume that a plant manager has to decide where to locate its warehouse. The decision has been narrowed down to choices among the following three cities: Detroit, Michigan, Cleveland, Ohio and St. Louis Missouri. The following pairwise comparison matrix summarizes the preferences of the plant manager. 57) Consider the following normalized matrix for Car Comfort that was computed using the analytical hierarchy process: Car A Car B Car C Car A 0.30 0.25 0.50 Car B 0.60 0.50 0.33 Car C 0.10 0.25 0.17 Which car is the most preferred based on comfort? 58) The analytical hierarchy process was used to determine which car to purchase. Four criteria were used: price, miles per gallon, comfort, and durability. Consider the following normalized matrix and the preference vector for durability. Multiplying the normalized matrix by the preference vector gives the following results: 2.17 2.26 1.69 Compute the consistency index. 59) Three job applicants were rated on 4 criteria, resulting in the following preference vectors: Applicant Criterion 1 Criterion 2 Criterion 3 Criterion 4 A .3085 .0792 .5986 .6192 B .2147 .4615 .0126 .3045 C .4768 .4593 .3888 .0763 The priorities of the criteria are: Criterion Priority 1 .2178 2 .1915 3 .4123 4 .1784 Rank the 3 applicants. 60) Assume that a plant manager has to decide where to locate its warehouse The decision has been narrowed down to choices among the following three cities: Detroit, Michigan, Cleveland, Ohio and St. Louis Missouri. The company has weighted each of these criteria in terms of its relative importance in the decision making process, and it has analyzed each potential warehouse location and graded them according to each criteria as shown in the following table. Decision Criteria Weight Grades for Detroit Grades for Cleveland Grades for St. Louis Proximity to suppliers .45 80 70 60 Proximity to customers .35 75 90 80 Land and construction costs .20 60 50 85 Calculate the scores for each location and state where the warehouse should be located? 61) A business is trying to decide which restaurant to hold its annual awards banquet. Use the information below to determine the best choice. Restaurant Criteria Weight A B C Appearance 0.15 40 65 60 Service 0.30 75 80 70 Atmosphere 0.15 60 40 70 Location 0.10 90 100 75 Quality of food 0.30 70 80 75 62) A positive deviational variable is the amount by which a goal level is A) underachieved B) exceeded C) kept short off D) none of the above 63) A __________ deviational variable is the amount by which a goal level is exceeded. A) negative B) positive C) positive or negative D) positive and negative 64) At least one or both deviational variables in a goal constraint must equal A) 1 B) -1 C) 0 D) 2 E) none of the above 65) A decision with more than 1 objective A) should be decomposed into a separate model for each objective B) depends on the probability of satisfying each objective C) requires the decision maker to put the objectives in some order of importance D) cannot have an optimal solution 66) Deviation variables A) represent the actual value of the function part of a goal constraint. B) are limited so that only one appears per constraint. C) represent the difference between the target and actual values D) must sum to one E) equal the difference between actual and predicted values of the decision variables 67) The objective function min P1d1- , P2d2+ A) attempts to avoid being below target for the priority 1 goal B) attempts to avoid being below target for the priority 2 goal C) will not have any feasible solutions D) will have a solution only if P1 > P2

E) none of the above

68) The majority of the goal constraints are equalities that include __________ variables.

A) only deviational

B) deviational and decision

C) only decision

D) neither deviational nor decision variables

E) undeviational

69) A __________ deviational variable is the amount by which a goal level is underachieved.

A) negative

B) positive

C) positive or negative

D) positive and negative

70) The objective function in a goal programming model seeks to __________ the deviation from goals in the order of the goal priorities.

A) free

B) even out

C) maximize

D) minimize

71) Goal constraints can include __________ deviational variables.

A) no

B) some

C) all

D) multiple

72) Deviational variables in a goal programming model constraint represent the

A) underachievement or overachievement of a goal level

B) resource constraint as defined by the linear model

C) probability of an optimal solution

D) probabilistic variables in the objective constraints

73) Two or more goals at the same priority level can be assigned weights to indicate their relative

A) meaning

B) slack

C) difference

D) importance

74) A company has 3 goals, listed in order of importance.

1. Achieve a total profit of at least $240 million.

2. Maintain the current employment level of 3,000 employees

3. Invest no more than $70 million in capital.

What is the objective function for a goal program for this situation?

A) P1d1- , P2d2- , P3d3+

B) P1d1- , P2d2+ , P2d2- , P3d3-

C) P1d1- , P2d2- , P2d2- , P3d3+

D) P1d1+ , P2d2- , P2d2- , P3d3+

E) None of the above

75) If the constraint 3×1 + 4×2 + (d1-) – (d1+) = 250 measures hours, then

A) 3×1 + 4×2 is equal to the actual time spent

B) if d1- is equal to 25, then 3×1 + 4×2 must equal 225.

C) overtime could be represented by d1+

D) all of the above are true.

76) A company has a goal to maintain the currently employment level, but places more importance on laying employees off than on hiring new employees. In other words, the company would rather hire than lay off employees. If the company assigns a “penalty weight” of 4 for the amount under the employment goal (goal 2) , and a weight of 2 for the amount over the employment goal, how would this be expressed in the objective function?

A) Min 4P2d2- + 2P2d2+

B) Min 2P2d2- + 4P2d2+

C) Max 4P2d2- + 2P2d2+

D) Min 4P2d2+ + 2P2d2-

E) None of the above

77) A company has a goal of calling on at least 400 customers a month (goal 1), but no fewer than 260 customers a month (goal 2). If X represents the number of customers contacted, which pair of constraints listed below is the appropriate representation in a goal programming problem?

A) X + d1- – d1+ = 400, X + d2- – d2+ = 260

B) X – d1- + d1+ = 400, X – d2- – d2+ = 260

C) X – d1- + d1+ = 400, X + d2- – d2+ = 260

D) X + d1- – d1+ = 400, X – d2- + d2+ = 260

78) Consider the following constraint: 2×1 + 3×2 =60. Assume that we convert this constraint into a goal constraint and the value of x1 = 15 and the value of x2 = 15, then the values of d1+ and d1- are __________ and __________ respectively.

A) 0, 15

B) 15, 0

C) 20, 0

D) 0, 20

E) 15, 15

79) One goal __________ achieved at the expense of another higher-priority goal.

A) is never

B) can sometimes be

C) is always

D) is under certain circumstances

80) The objective function in all goal programming models is to __________ from the goal constraint levels.

A) maximize alternatives

B) minimize alternatives

C) minimize deviation

D) maximize deviation

81) Goal programming solutions __________ achieve all goals.

A) always

B) don’t always

C) sometimes

D) never

Riverside Industries makes two products and each product is processed in three departments. The time requirements for each product in each department are given below. The profit for each product is also provided as well as the available hours in each department.

Product Department A Department B Department C Profit

1 3 4 2 1

2 2 1 2 2

Available Hours 600 400 400

Management wants to achieve 3 goals. The first two goals are equal in priority.

Priority 1: Produce at least 125 units of product 1.

Priority 1: Produce at least 80 units of product 2.

Priority 2: Achieve a profit of at least 300.

82) The goal programming model for this problem has how many constraints?

A) 3

B) 4

C) 6

D) 7

83) The goal programming constraint for the first goal is:

A) 3×1 + d1- – d1+ = 125

B) 3×1 – d1- + d1+ = 125

C) 1×1 + d1- – d1+ = 125

D) 1×1 – d1- + d1+ = 125

84) The objective function for this goal program is:

A) Min P1d1- + P1d2-, P2d3-

B) Min P1d1+ + P1d2-, P2d3-

C) Min P1d1- + P1d2+, P2d3-

D) Min P1d1- + P1d2-, P2d3+

85) If the second goal was to produce no more than 80 units of product two, the objective function would be:

A) Min P1d1- + P1d2-, P2d3-

B) Min P1d1+ + P1d2-, P2d3-

C) Min P1d1- + P1d2+, P2d3-

D) Min P1d1- + P1d2-, P2d3+

86) Goal programming solutions achieve the __________ satisfactory solution possible.

A) best or most

B) worst or least

C) only

D) somewhat

87) If choice A is strongly preferred to choice B and choice B is moderately preferred to choice C, and the decision maker says choice C is equally preferred to choice A, what conclusion can be drawn?

A) The decision maker is consistent.

B) The decision maker is inconsistent.

C) The pairwise comparison matrix is symmetric.

D) The decision maker has already synthesized.

88) In a pairwise comparison, __________ alternatives are compared according to a criterion and one is preferred.

A) 6

B) 4

C) 8

D) 2

89) In synthesization, decision alternatives are prioritized __________ criteria/ion.

A) within some

B) among some

C) among all

D) within each

90) Pairwise comparisons are made among

A) all alternatives for a particular criterion

B) all alternatives for all criteria

C) some alternatives for a particular criterion

D) some alternatives for all criteria

91) In a pairwise comparison matrix the diagonal values will __________ equal __________.

A) sometimes, 1

B) always, 0

C) sometimes, 0

D) always, 1

E) none of the above

92) The analytic hierarchy process

A) optimizes procedures with a single goal

B) requires no pairwise comparison

C) uses both qualitative and subjective assessment

D) does not require the input of a decision maker

93) In determining the pair wise comparison matrix, if the decision maker rates the option A compared to option B as “4”, then option B compared to option A __________.

A) would have to be also Answered by the decision maker

B) is also 4

C) 1/4

D) 2

E) would have to be computed by making various calculations based on other values in the pairwise comparison matrix

94) The analytical hierarchy process is a method for __________ decision alternatives.

A) creating

B) ranking

C) changing

D) deleting

95) Values with an acceptable consistency ratio are values

A) less than or equal to 0.1

B) greater than 0.5 but less than 0.6

C) less than 1 but greater than 0.5

D) greater than 0.2 but less than 0.5

96) Assume that a decision maker has to make a choice between three types of cars. Chevy, Honda, and Ford. Based on three criteria: comfort, MPG and style.

The row averages for criteria is summarized in the following column vector:

The preference matrix for the three types of cars is given as follows:

The first row of the following matrix represents Chevy, the second row represents Honda, and the third row represents Ford.

Compute the overall score for each decision alternative (car). Which choice does the decision maker prefer?

A) Chevy

B) Honda

C) Ford

D) Honda and Chevy are equally preferred

E) Ford and Chevy are equally preferred

97) Assume that a plant manager has to decide where to locate its warehouse The decision has been narrowed down to choices among the following three cities: Detroit, Michigan, Cleveland, Ohio and St. Louis Missouri. The following pairwise comparison matrix summarizes the preferences of the plant manager.

Detroit Cleveland St. Louis

Detroit 1 1/3 1/4

Cleveland 3 1 1/2

St. Louis 4 2 1

Which choice does the decision maker prefer?

A) Detroit

B) Cleveland

C) St. Louis

D) Cleveland and St. Louis are equally preferred

E) Detroit and St. Louis are equally preferred

98) Three fast food restaurants on a college campus have been subjected to pairwise comparisons on the quality of their food. The matrix is

Pizza Mama’s Papa’s Freddie’s

Mama’s 1 3 8

Papa’s 1 4

Freddie’s 1

Which choice does the decision maker prefer?

A) Mama’s

B) Papa’s

C) Freddie’s

D) Mama’s and Papa’s are equally preferred.

E) Cannot be determined from the information provided

99) In the __________ process, the decision maker determines how well each alternative scores on a criterion using pairwise comparisons.

A) linear programming

B) simplex tableau

C) goal programming

D) analytical hierarchy

100) In the analytical hierarchy process, a high consistency ratio is considered __________ consistent than a low consistency ratio.

A) less

B) more

C) none of the above

101) A required step in the analytic hierarchy process is to determine the

A) number of hierarchies to use

B) relative importance of a set of features based on a criterion

C) goals to be satisfied

D) expected value of the criteria

102) An art critic is evaluating four different interpretations “The Nutcracker”. The pairwise comparison matrix for the criterion “originality of choreography” is given below.

Choreographer A B C D

A 1 1/3 3 4

B 3 1 5 2

C 1/3 1/5 1 6

D 1/4 1/2 1/6 1

Determine the priorities of the four choreographers relative to “originality”.

A) A is preferred to B; B is preferred to C; and C is preferred to D.

B) B is preferred to A; A is preferred to D; and D is preferred to C.

C) B is preferred to A; A is preferred to C; and C is preferred to D.

D) C is preferred to A; A is preferred to D; and D is preferred to B.

103) In synthesization, dividing each value in each column of the pairwise comparison matrix by the corresponding column sum, we obtain the __________ matrix.

A) pairwise

B) normalized

C) preference

D) criteria

E) criteria preference

104) Assume that a plant manager has to decide where to locate its warehouse The decision has been narrowed down to choices among the following three cities: Detroit, Michigan, Cleveland, Ohio and St. Louis Missouri. The company has weighted each of these criteria in terms of its relative importance in the decision making process, and it has analyzed each potential warehouse location and graded them according to each criteria as shown in the following table.

Decision Criteria Weight Grades for Detroit Grades for Cleveland Grades for

St. Louis

Proximity to suppliers .45 80 70 60

Proximity to customers .35 75 90 80

Land and construction costs .20 60 50 85

Calculate the scores for each location and state where the warehouse should be located?

A) Detroit

B) Cleveland

C) St. Louis

D) Cleveland and St. Louis are equally preferred

E) Detroit and St. Louis are equally preferred

105) A business is trying to decide which restaurant to hold its annual awards banquet. Use the information below to determine the best choice.

Restaurant

Criteria Weight A B C

Appearance 0.15 40 65 60

Service 0.30 75 80 70

Atmosphere 0.15 60 40 70

Location 0.10 90 100 75

Quality of food 0.30 70 80 75

Where should the restaurant be located?

A) A

B) B

C) C

D) A and B equally preferred

E) B and C equally preferred

Introduction to Management Science, 10e (Taylor)

Chapter 10 Nonlinear Programming

1) The slope of a curve at any point is equal to the derivative of the curves function.

2) The slope of a curve at its highest point equals 1.

3) Decision variables cannot be multiplied by each other in the objective function of a nonlinear program.

4) More attention has been devoted to linear programs than to non linear programs because these types of problems are more realistic in business.

5) Both linear and nonlinear programming models are examples of constrained optimization models.

6) An optimal solution to a nonlinear programming problem will always occur at the boundary of the feasible solution space formed by the constraint.

7) The Lagrange multiplier is analogous to the dual variables in a linear programming problem.

8) The Lagrange multiplier at the optimum gives only the instantaneous rate of change in the objective value.

9) Both linear and nonlinear programming models have the general form of an objective function subject to more than 1 constraint.

10) The classic optimization is the use of calculus to determine the optimal value of a variable.

11) If a nonlinear program has been correctly formulated, procedures guarantee a solution.

12) In an unconstrained nonlinear programming problem, we have a single nonlinear objective function and no constraints.

13) Constraints for nonlinear programs are usually nonlinear.

14) In portfolio selection problems, risk is measured by the variance of the return on the portfolio.

15) In solving the facility location problem, the objective is to locate a centralized facility that serves customers or other facilities such that the distance traveled between the facility and customers or other facilities is minimized.

16) If a nonlinear programming model consists of a single nonlinear objective function and a single linear constraint, it is called a(n) __________ optimization problem.

17) If a nonlinear programming model consists of a single nonlinear objective function and no constraints, it is called a(n) __________ optimization problem.

18) The __________ reflects the approximate change in the objective function resulting from a unit change in the quantity (right-hand-side) value of the constraint.

19) Assume a nonlinear programming problem with a single constraint has been solved. The value of the Lagrange multiplier is $0.75 and the value of the optimal profit (Z) is $25. If the right-hand-side of the constraint is increased from 38 to 42, the new value of Z will be __________.

20) If a nonlinear programming problem results in profit (Z) of $50, and the Lagrange multiplier for a constraint is -2, the new profit will be __________ if the right hand side of the constraint is increased by 1 unit.

21) If a firm’s profit is Z = 100p -8p2 +16, then the maximum profit occurs where p = __________.

22) If a firm’s profit is z = 20p -2p2 + 40, then the optimal value of p yields a maximum profit of __________.

23) Assume price and demand are related by the following function: v = 200 – p. If fixed cost = $10,000 and variable cost = $8, then the expression for profit is __________.

24) Assume price and demand are related by the following function: v = 100 – 2.5p. If fixed cost = $5,000 and variable cost = $10, then the expression for profit is __________.

25) __________, a measure of correlation between returns on investment i and returns on investment j is used to reflect risk.

26) The __________ of the value of investment is a measure of risk.

27) The __________ measure of distance between two points on a set of X and Y coordinates is the hypotenuse of right triangle.

28) The objective of a facility location problem is to minimize __________.

The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1,000 – 8p.

29) What is the nonlinear profit function for the XYZ company? Simplify the terms as much as possible.

30) What is the derivative of the profit function for the XYZ company? Simplify the terms as much as possible.

31) What price for the ball bearings will maximize the profit?

32) What is the optimal production quantity?

33) What is the optimal profit?

34) Determine the quantity of soap and shampoo that should be produced to maximize profit.

35) Determine the profit for the optimal production quantities of soap and shampoo.

36) Lush Lawns, Inc. provides a lawn fertilizer and weed control service. They are adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A budget of $2000 is to be used on this promotional campaign over the next quarter. Based on past experience in promoting its other services, Lush Lawns has been able to obtain an estimate of the relationship between sales and the amount spent on promotion in these two media:

s = 2 X12 – 10 X22 – 2x1x2 + 18×1 + 34×2

s.t. x1 + x2 = 2

Solve.

37) Zoey’s Catnip Toys faces the following relationship between price and demand: v = 2000 – 200p. The fixed cost is $500 and variable cost is $1. Write an expression for the total profit.

38) Zoey’s Catnip Toys faces the following relationship between price and demand: v = 2000 – 200p. The fixed cost is $500 and variable cost is $1. What price should Zoey charge to maximize profit?

Sara’s Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:

x1 = 1800 – 150p1

x2 = 1500 – 300p2

The cost for a catnip ball is $2 and for the mouse, $3.

Customer demand requires that she make 2.5 times more balls than mice.

39) Write the formulation for this problem

40) Determine the prices that Sara should charge to maximize profit

41) Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. Write the new constraint.

42) Sara has found an unlimited source of catnip so that is no longer a constraint. However, customer demand dictates that she produce 2.5 times more catnip balls than mice. How will this impact the prices that she should charge to maximize profit?

43) A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20×2 – 10C2 + 40Cx + 120x – 200C + 600. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?

44) The slope of a curve at its highest point equals

A) 0

B) 1

C) 2

D) 3

45) A store has determined that the weekly sales of a product is related to the number of customers who visit the store and the square feet of shelf space, x, according to the following equation: -20×2 – 10C2 40Cx + 120x – 200. C represents the hundreds of customers who visit their store. If a store averages 200 customers per week, how many square feet of shelf space is required to maximize sales?

A) 3

B) 4

C) 5

D) 1 or 9

E) None of the above

46) If a firm’s profit is Z = 12x -6×2 + 30, and their minimum production level of x is equal to 0.5, then the level of x that maximizes profit is

A) .5

B) 1

C) 1.5

D) 2

E) None of the above

47) A customer molder produces 6-ounce juice glasses and 10-ounce cocktail glasses. The per unit contribution for the juice glasses (x1) is equal to 60 – 5×1, and the per unit contribution for the cocktail glasses (x2) is 80 – 4×2. An expression for the total contribution is:

A) 20 – 4×2 – 5×1

B) 60×1 – 5 x12 + 80×2 – 4 x22

C) 80×1 – 5 x12 + 60×2 – 4 x22

D) 20 – (4×2)(5×1)

E) None of the above

48) Classical optimization is the use of __________ to determine the optimal value of a variable.

A) calculus

B) linear programming

C) nonlinear programming

D) goal programming

49) The derivative of a function __________ the slope of the curve defined by that function.

A) is larger than

B) equals

C) is smaller than

D) is similar to

50) Both linear and nonlinear programming models are examples of

A) goal programming models

B) simplex tableaus

C) constrained likelihood models

D) constrained optimization models

51) The Lagrange multiplier reflects the appropriate change in the objective function resulting from a unit change in the __________ of the constraint equation.

A) coefficient

B) objective function

C) right hand side

D) shadow price

52) The Lagrange multiplier is __________ to the dual variables in a linear programming problem.

A) analogous

B) contradictory

C) inversely related

D) opposite

53) The Lagrange multiplier

A) is the shadow price for the objective function coefficients

B) is not valid over a range of changes in the RHS

C) is the rate of change in the objective value as the RHS of the constraint increases

D) all of the above

54) The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1,000 – 8p. What is the optimal price of the ball bearings that will maximize the profit?

A) 47.99

B) 53.99

C) 58.99

D) 63.99

E) 67.99

55) The XYZ manufacturing company produces ball bearings. The annual fixed cost is $20,000 and the variable cost per ball bearing is $3. The price is related to demand according to the following equation: 1,000 – 8p. What is optimal profit?

A) $5668

B) $6668

C) $7668

D) $8668

E) $9668

The Salt Creek soap company has determined the following nonlinear model to determine the optimal pounds of industrial soap (X1) and shampoo (X2) it should produce each day.

Maximize Z = X12 + 2 X22 – 8X1 – 12X2 + 34

Subject to: X1 + 2X2 = 4 lbs

56) What quantities of soap and shampoo maximize profit?

A) X1 = 2, X2 = 1

B) X1 = 1, X2 = 2

C) X1 = 3, X2 = 1

D) X1 = 2, X2 = 2

E) X1 = 2, X2 = 3

57) What is profit when the optimal values of soap and shampoo are produced?

A) Z = 10

B) Z = 11

C) Z = 12

D) Z = 13

E) Z = 14

Sara’s Sensible Critters makes two kinds of catnip toys: balls (x1) and mice (x2). The relationship between demand and price for balls and mice is:

x1 = 1800 – 150p1

x2 = 1500 – 300p2

The cost for a catnip ball is $2 and for the mouse, $3.

Customer demand requires that she make 2.5 times more balls than mice.

58) Write the appropriate expression for profit

A) Max Z = (p1 – 3)x1 + (p2 – 2)x2

B) Max Z = (p1 + 2)x1 + (p2 + 3)x2

C) Max Z = (p1 – 2)x1 + (p2 – 3)x2

D) Min Z = (p1 – 2)x1 + (p2 – 3)x2

E) None of the above are correct

59) Write the appropriate expression for the demand constraint.

A) 2.5×1 = x2

B) x1 – 2.5×2 ≥ 0

C) x1 + 2.5×2≤ 0

D) x1 = 2.5×2

E) None of the above are correct

Introduction to Management Science, 10e (Taylor)

Chapter 11 Probability and Statistics

1) Deterministic techniques assume that no uncertainty exists in model parameters.

2) Probabilistic techniques assume that no uncertainty exists in model parameters.

3) Objective probabilities that can be stated prior to the occurrence of an event are classical or a priori.

4) Objective probabilities that are stated after the outcomes of an event have been observed are relative frequencies.

5) Relative frequency is the more widely used definition of objective probability.

6) Subjective probability is an estimate based on personal belief, experience, or knowledge of a situation.

7) An experiment is an activity that results in one of several possible outcomes.

8) The events in an experiment are mutually exclusive if only one can occur at a time.

9) In a given experiment, the probabilities of all mutually exclusive events sum to one.

10) A set of events is collectively exhaustive when it includes all the events that can occur in an experiment.

11) A marginal probability is the probability of a single event occurring.

12) A Venn diagram visually displays mutually exclusive and non-mutually exclusive events.

13) A joint probability is the probability that two or more events that are mutually exclusive can occur simultaneously.

14) A conditional probability is the probability that an event occurs given that another event has already occurred.

15) Conditional probabilities are shown in Venn diagrams.

16) Probability trees are used only to compute conditional probabilities.

17) A succession of events that does not affect other events is independent.

18) A binomial probability distribution indicates the probability of r successes in n trials.

19) The chi-square test is a statistical test to determine if data that are squared exhibit bias.

20) A continuous random variable may assume only integer values within a given interval.

21) Seventy two percent of all observations fall within 1 standard deviation of the mean if the data is normally distributed.

22) Another name for the mean of a probability distribution is its expected value.

23) An inspector correctly identifies defective products 90% of the time. For the next 10 products, the probability that he makes fewer than 2 incorrect inspections is 0.736.

24) In Bayesian analysis, additional information is used to alter the conditional probability of the occurrence of an event.

25) Objective probabilities that can be stated prior to the occurrence of an event are __________.

26) __________ probability is an estimate based on a personal belief, experience, and knowledge of a situation.

27) The events in an experiment are __________ if only one can occur at a time.

28) A __________ organizes numerical data to describe the events of an experiment.

29) A __________ is the probability of occurrence of a single event.

30) __________ can enable one to improve marginal probabilities of the occurrence of an event by gathering additional information.

31) A succession of events that do not affect each other are __________.

32) A __________ probability is the probability that an event will occur given that another event has already occurred.

33) In a binomial distribution process, there are __________ possible outcomes.

34) One of the properties of the __________ distribution is that the probability of success remains constant over time.

35) Altered marginal probability of an event based on additional information is a __________ probability.

36) The __________ of a random variable is computed by multiplying each possible value of the variable by its probability and summing these products.

37) If events A and B are independent, then P(AB) = __________.

38) If events A and B are independent, then P(A|B) = __________.

39) If two events are not mutually exclusive, then P(A or B) = __________.

40) __________ is a measure of dispersion of random variable values about the expected value.

41) A continuous random variable can take on a(n) __________ number of values within a given interval.

42) The __________ test is a statistical test to see if an observed data fit a particular probability distribution.

43) The __________ normal distribution has a mean of 0 and a standard deviation of 1.

44) Almost all of the data from a normal distribution fall within __________ standard deviations of the mean.

45) The expected value of the standard normal distribution is equal to __________.

46) The standard deviation of the standard normal distribution is equal to __________.

Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y.

47) What is the probability that Jim will be accepted at both universities?

48) What is the probability that Jim will not be accepted at either university?

49) What is the probability that Jim will be accepted by at least one of the two universities?

Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

Male (M) Female (F)

Job

Administrative (AD) 110 10

Salaried staff (SS) 30 50

Hourly staff (HS) 60 40

50) If an employee is selected at random, what is the probability that the employee is male?

51) If an employee is selected at random, what is the probability that the employee is male and salaried staff?

52) If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member?

53) If an employee is selected at random, what is the probability that the employee is female or works as a member of the administration

An automotive center keeps tracks of customer complaints received each week. The probability distribution for complaints can be represented as a table or a graph, both shown below. The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.

xi 0 1 2 3 4 5 6

p(xi) .10 .15 .18 .20 .20 .10 .07

54) What is the probability that they receive less than 3 complaints in a week?

55) What is the average number of complaints received per week?

56) A fair die is rolled nine times. What is the probability that an odd number (1,3 or 5) will occur less than 3 times?

57) A fair die is rolled 8 times. What is the probability that an even number (2,4, 6) will occur between 2 and 4 times?

A company markets educational software products, and is ready to place three new products on the market. Past experience has shown that for this particular software, the chance of “success” is 80%. Assume that the probability of success is independent for each product.

58) Find the probability that exactly 1 of the 3 products is successful.

59) Find the probability that none of the 3 products is successful.

60) If X has the following probability distribution

X 1 2 3 4

P(X) .1 .5 .2 .2

Compute the expected value of X

61) If X has the following probability distribution

X 1 2 3 4

P(X) .1 .5 .2 .2

Compute the standard deviation of X.

62) If x is normally distributed with a mean of 10 and a standard deviation of 3, then P(x ≤ 6) is equal to P( Z ≤ __)?

63) For a standard normal distribution, what is the probability that z is greater than 1.75?

Two Psychology majors, in 2 different sections of Clinical Psychology, were comparing test scores. The following gives the students’ scores, class mean, and standard deviation for each section:

Section 1 Section 2

Student Score 84 75

Mean 75 60

Standard deviation 7 8

64) What is the z-score of the student from section 1 and what is the probability that a student in section 1 will score higher than 84?

65) What is the z-score of the student from section 2 and what is the probability that a student in section 2 will score higher than 75?

66) Which student scored better compared to the rest of the section?

67) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is larger than 21 oz?

68) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf of bread is larger than 23 oz?

69) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is less than 24 oz?

70) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is between 20.75 and 23.25 oz?

71) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is between 21.75 and 22.25 oz?

72) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is more than 24 oz?

73) A loaf of bread is normally distributed with a mean of 22 oz and a standard deviation of 0.5 oz. What is the probability that a loaf is more than 22.25 oz?

74) A life insurance company wants to estimate their annual payouts. Assume that the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 4 years. What proportion of the plan recipients would receive payments beyond age 75?

75) A life insurance company wants to estimate their annual payouts. Assume that the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 4 years. What proportion of the participants die before they reach the age of 65?

76) A life insurance company wants to estimate their annual payouts. Assume that the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 4 years. By what age have 80% of the plan participants pass away?

77) For the normal distribution, the mean plus and minus 1.96 standard deviations will include what percent of the observations?

78) What is the area under the normal curve for ≥ 1.79?

79) A study of a company’s practice regarding the payment of invoices revealed that on the average an invoice was paid 20 days after it was received. The standard deviation equaled 5 days. Assuming that the distribution is normal, what percent of the invoices is paid within 15 days of receipt?

80) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of the crab are normally distributed, what is the probability that a randomly selected crab will weigh more than 2 .2 pounds?

81) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of crab are normally distributed, what is the probability that a randomly selected crab will weigh between 1 and 2 pounds?

82) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. Assuming the weights of crab are normally distributed, the probability that a randomly selected crab will weigh less than 1.2 pounds is __________.

83) Assume that X is a normally distributed random variable with a mean of 50 and a standard deviation of 2. Find the probability that X is between 48 and 55.

84) A paint manufacturer’s production process is normally distributed with a mean of 100,000 gallons and a standard deviation of 10,000 gallons. Management wants to create an incentive bonus for the production crew when the daily production exceeds the 94th percentile of the distribution. At what level of production should management pay the incentive bonus?

An online sweepstakes has the following payoffs and probabilities. Each person is limited to one entry.

85) The probability that someone wins any money is ________.

86) The probability of winning at least $1,000.00 is ________.

87) __________ techniques assume that no uncertainty exists in model parameters.

A) Probability

B) Probabilistic

C) Deterministic

D) Distribution

88) __________ probability is an estimate based on personal belief, experience, or knowledge of a situation.

A) Binomial

B) Subjective

C) Marginal

D) Joint

89) Objective probabilities that can be stated prior to the occurrence of an event are

A) subjective

B) a priori

C) classical or a priori

D) none of the above

90) The events in an experiment are __________ if only one can occur at a time.

A) mutually exclusive

B) non-mutually exclusive

C) mutually inclusive

D) independent

91) In a given experiment the probabilities of mutually exclusive events sum to

A) 0

B) 0.5

C) 1

D) none of the above

92) A __________ probability is the probability of a single event occurring.

A) subjective

B) binomial

C) marginal

D) joint

93) A frequency distribution is an organization of __________ data about the events in an experiment.

A) quantitative

B) numerical

C) qualitative

D) A and B

94) P(A U B) is the probability that __________ will occur.

A) A

B) B

C) A and B

D) A or B or both

95) Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will be accepted at both universities?

A) .65

B) .25

C) .20

D) 09

E) .05

96) Jim is considering pursuing an MS in Information Systems degree. He has applied to two different universities. The acceptance rate for applicants with similar qualifications is 20% for University X and 45% for University Y. What is the probability that Jim will not be accepted at either university?

A) .20

B) .30

C) .36

D) .44

E) 56

97) Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

Male (M) Female (F)

Job

Administrative (AD) 110 10

Salaried staff (SS) 30 50

Hourly staff (HS) 60 40

If an employee is selected at random, what is the probability that the employee is female or works as a member of the administration

A) .1667

B) .60

C) .67

D) .70

E) .73

98) A __________ probability distribution indicates the probability of r successes in n trials.

A) joint

B) subjective

C) marginal

D) binomial

99) The probability of independent events occurring in succession is computed by __________ the probabilities of each event.

A) multiplying

B) adding

C) subtracting

D) dividing

100) A __________ probability is the probability that an event will occur given that another event has already occurred.

A) subjective

B) objective

C) conditional

D) binomial

101) In Bayesian analysis, additional information is used to alter the __________ probability of the occurrence of an event.

A) marginal

B) conditional

C) binomial

D) revised

102) Employees of a local company are classified according to gender and job type. The following table summarizes the number of people in each job category.

Male (M) Female (F)

Job

Administrative (AD) 110 10

Salaried staff (SS) 30 50

Hourly staff (HS) 60 40

If an employee is selected at random, what is the probability that the employee is female given that the employee is a salaried staff member.

A) .1667

B) .50

C) .60

D) .625

E) .70

103) A __________ probability is the altered marginal probability of an event based on additional information.

A) posterior

B) joint

C) marginal

D) conditional

E) A and B

104) Mutually exclusive events are

A) events with identical probabilities

B) events that have no outcomes in common

C) events that have no effect on each other

D) all of the above

105) Bayesian analysis involves a(n) __________ probability.

A) a priori

B) posterior

C) joint

D) relative frequency

106) In a __________ distribution, for each of n trials, the event always has the same probability of occurring.

A) binomial

B) joint

C) frequency

D) standard

107) Experiments with repeated independent trials will be described by the binomial distribution if

A) each trial result influences the next

B) each trial has exactly 2 outcomes whose probabilities do not change

C) the trials are continuous

D) the time between trials is constant

108) In a binomial distribution, for each of n trials, the event

A) time between trials is constant

B) always has the same probability of occurring

C) result of the first trial influence the next trial

D) trials are continuous

109) A fair die is rolled nine times. What is the probability that an odd number (1,3 or 5) will occur less than 3 times?

A) .0899

B) .2544

C) .7456

D) .9101

E) .9916

110) A fair die is rolled 8 times. What is the probability that an even number (2,4, 6) will occur between 2 and 4 times?

A) .1640

B) .2188

C) .4922

D) .6016

E) .8204

111) A company markets educational software products, and is ready to place three new products on the market. Past experience has shown that for this particular software, the chance of “success” is 80%. Assume that the probability of success is independent for each product. What is the probability that exactly 1 of the 3 products is successful?

A) 0.80

B) 2.4

C) 0.032

D) 0.24

E) 0.096

112) __________ is a measure of the dispersion of random variable values about the expected value or mean.

A) Standard deviation

B) Sample mean

C) Population mean

D) Variance

E) A and D

113) An automotive center keeps tracks of customer complaints received each week. The probability distribution for complaints can be represented as a table or a graph, both shown below. The random variable xi represents the number of complaints, and p(xi) is the probability of receiving xi complaints.

xi 0 1 2 3 4 5 6

p(xi) .10 .15 .18 .20 .20 .10 .07

What is the average number of complaints received per week?

A) 2.12

B) 3.32

C) 4.12

D) 2.83

E) None of the above

114) The expected value of the standard normal distribution is equal to

A) 0

B) 1

C) 1.5

D) 2

E) 2.5

115) The area under the normal curve represents probability, and the total area under the curve sums to

A) 0

B) 0.5

C) 1

D) 2

116) The __________ and variance are derived from a subset of the population data and are used to make inferences about the population.

A) Population variance

B) Population standard deviation

C) population mean

D) sample mean

E) sample range

117) Under the normal curve, the area between z=1 and z =-2 includes approximately __________ of the values.

A) 99%

B) 98%

C) 95%

D) 85%

E) 82%

118) For the normal distribution, the mean plus and minus 1.96 standard deviations will include what percent of the observations?

A) 80%

B) 84%

C) 90%

D) 95%

E) 97%

119) A jar of jelly is normally distributed with a mean of 16 oz and a standard deviation of 0.02 oz. What is the probability that a jar of jelly contains less than 16 oz?

A) .1915

B) .3085

C) .5000

D) .7257

E) .8413

120) A jar of jelly is normally distributed with a mean of 16 oz and a standard deviation of 0.02 oz. What is the probability that a jar of jelly contains more than 16.03 oz?

A) .0668

B) .1587

C) .3413

D) .4332

E) .9332

121) Under the normal curve, the area between z=2 and z =-2 includes __________ of the values.

A) 98%

B) 96%

C) 95%

D) 93%

E) 90%

122) The metropolitan airport commission is considering the establishment of limitations on noise pollution around a local airport. At the present time, the noise level per jet takeoff in one neighborhood near the airport is approximately normally distributed with a mean of 100 decibels and a standard deviation of 3 decibels. What is the probability that a randomly selected jet will generate a noise level of more than 105 decibels?

A) 0.0228

B) 0.0475

C) 0.0485

D) 0.0500

E) None of the above

123) For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.2910. The value of Z is:

A) 0.17

B) 0.81

C) 1.25

D) 1.65

124) For some value of Z, the probability that a standard normal variable is below Z is 0.3783. The value of Z is:

A) -0.81

B) -0.31

C) 0.82

D) 1.55

125) For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3554. The value of Z is:

A) 0.31

B) 0.36

C) 0.95

D) 1.06

126) If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.5 minutes and a standard deviation of 1 minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes.

A) 0.3551

B) 0.3085

C) 0.2674

D) 0.1915

127) Assume that it takes a college student an average of 5 minutes to find a parking spot in the main parking lot. Assume also that this time is normally distributed with a standard deviation of 2 minutes. Find the probability that a randomly selected college student will take between 2 and 6 minutes to find a parking spot in the main parking lot.

A) 0.1950

B) 0.4772

C) 0.4332

D) 0.6247

128) Assume that it takes a college student an average of 5 minutes to find a parking spot in the main parking lot. Assume also that this time is normally distributed with a standard deviation of 2 minutes. What time is exceeded by approximately 75% of the college students when trying to find a parking spot in the main parking lot?

A) 3.5 minutes

B) 5.75 minutes

C) 6.36 minutes

D) 9.2 minutes

129) The owner of a seafood market determined that the average weight for a crab is 1.6 pounds with a standard deviation of 0.4 pound. What weight is exceeded by 2% of all of the crabs? (Assume the weights are normally distributed.)

A) 0.78 pounds

B) 1.82 pounds

C) 2.42 pounds

D) 4.36 pounds

130) A professor would like to assign grades such that 5% of students receive As. If the exam average is 62 with a standard deviation of 13, what grade should be the cutoff for an A? (Round your answer.)

A) 80

B) 83

C) 90

D) 93

131) A professor would like to assign grades such that 7% of students receive Fs. If the exam average is 62 with a standard deviation of 13, what grade should be the cutoff for an F? (Round your answer.)

A) 43

B) 49

C) 50

D) 55

Introduction to Management Science, 10e (Taylor)

Chapter 12 Decision Analysis

1) A state of nature is an actual event that may occur in the future.

2) A payoff table is a means of organizing a decision situation, including the payoffs from different decisions given the various states of nature.

3) The maximax criterion results in the maximum of the maximum payoffs.

4) The maximin approach involves choosing the alternative with the highest payoff.

5) Regret is the difference between the payoff from the best decision and all other decision payoffs.

6) The minimax regret criterion minimizes the maximum regret.

7) The minimax regret criterion maximizes the maximum regret.

8) The Hurwicz criterion is a compromise between the maximax and maximin criteria.

9) The Hurwicz criterion is a compromise between the minimax and minimin criteria.

10) The coefficient of optimism is a measure of the decision maker’s optimism.

11) The Hurwicz criterion multiplies the best payoff by the coefficient of optimism.

12) The Hurwicz criterion multiplies the worst payoff by the coefficient of optimism.

13) A dominant decision is one that has better payoff than another decision under each state of nature.

14) The appropriate criterion is dependent on the risk personality and philosophy of the decision maker.

15) The maximax criterion is optimistic.

16) The maximin criterion maximizes the minimum regret.

17) The minimax criterion minimizes the maximum payoff.

Answer: FALSE

Diff: 1 Page Ref: 531

Main Heading: Decision Making without Probabilities

Key words: minimax criterion

18) Regret and opportunity loss mean the same thing.

19) The equal likelihood criterion assigns a probability of 0.5 to each state of nature.

20) Expected opportunity loss is the expected value of the regret for each decision.

21) When using decision trees, branches with the greatest expected value are selected.

22) A decision tree is a diagram consisting of circles decision nodes, square probability nodes, and branches.

23) When the __________ criterion is used, the maximum of the maximum payoffs is observed.

24) When the __________ criterion is used, the maximum of the minimum payoffs is observed

25) __________ is the difference between the payoff from the best decision and all other decision payoffs.

26) The __________ is a compromise between the maximax and the maximin criterion.

27) The __________ is a measure of the decision makers optimism.

28) A(n) __________ decision is one that has a better payoff than another decision under the state of nature.

29) A __________ structures decisions with series of nodes.

30) The __________ of sample information is the ratio of the expected value of sample information to the expected value of perfect information.

31) When the __________ criterion is used, the decision maker selects the decision alternative that minimizes the maximum regret.

32) A ________ decision tree illustrates a situation requiring a services of decisions.

33) ________ is a measure of personal satisfaction derived from money.

34) People who forgo a high expected value to avoid a disaster with a low probability are __________.

A group of friends are planning a recreational outing and have constructed the following payoff table to help them decide which activity to engage in. Assume that the payoffs represent their level of enjoyment for each activity under the various weather conditions.

Weather

Cold Warm Rainy

S1 S2 S3

Bike: A1 10 8 6

Hike: A2 14 15 2

Fish: A3 7 8 9

35) If the group is optimistic, what decision should they make?

36) If the group is conservative, what decision will they make?

37) If the group chooses to minimize their maximum regret, what activity will they choose?

38) If the probabilities of cold weather (S1), warm weather (S2), and rainy weather (S3) are 0.2, 0.4, and 0.4, respectively, then what decision should be made using the expected value criterion?

39) What is the EVPI for this situation?

An investor is consider 4 different opportunities, A, B, C, or D. The payoff for each opportunity will depend on the economic conditions, represented in the payoff table below.

Economic Condition

Poor Average Good Excellent

Investment (S1) (S2) (S3) (S4)

A 50 75 20 30

B 80 15 40 50

C -100 300 -50 10

D 25 25 25 25

40) What decision would be made under maximax?

41) What decision would be made under maximin?

42) What decision would be made under minimax regret?

43) If the probabilities of each economic condition are 0.5, 0.1, 0.35, and 0.05 respectively, what investment would be made using the expected value criterion?

44) What is the expected value of perfect information?

A manager has developed a payoff table that indicates the profits associated with a set of alternatives under 2 possible states of nature.

Alt S1 S2

1 10 2

2 -2 8

3 8 5

45) If the manager uses maximin as the decision criterion, which of the alternatives should she choose?

46) If the manager uses minimax regret as the decision criterion, which of the alternatives would she choose?

47) Use the expected value criterion to select the best alternative. Assume that the probability of S2 is equal to 0.4.

48) Compute the expected value of perfect information assuming that the probability of S2 is equal to 0.4.

The local operations manager for the IRS must decide whether to hire 1, 2, or 3 temporary workers. He estimates that net revenues will vary with how well taxpayers comply with the new tax code.

49) If he uses the maximin criterion, how many new workers will he hire?

50) If he uses the minimax regret criterion, how many new workers will he hire?

51) If he thinks the chances of low, medium, and high compliance are 20%, 30%, and 50% respectively, what are the expected net revenues for the number of workers he will decide to hire?

52) If he thinks the chances of low, medium, and high compliance are 20%, 30%, and 50% respectively, what is the expected value of perfect information?

A manufacturer must decide whether to build a small or a large plant at a new location. Demand at the location can be either small or large, with probabilities estimated to be 0.4 and 0.6 respectively. If a small plant is built, and demand is large, the production manager may choose to maintain the current size or to expand. The net present value of profits is $223,000 if the firm chooses not to expand. However, if the firm chooses to expand, there is a 50% chance that the net present value of the returns will be 330,000 and 50% chance the estimated net present value of profits will be $210,000. If a small facility is built and demand is small, there is no reason to expand and the net present value of the profits is $200,000. However, if a large facility is built and the demand and the demand turns out to be small, the choice is to do nothing with a net present value of $40,000 or to stimulate demand through local advertising. The response to advertising can be either modest with a probability of .3 or favorable with a probability of .7. If the response to advertising is modest the net present value of the profits is $20,000. However, if the response to advertising is favorable, then the net present value of the profits is$220,000. Finally, the when large plant is built and the demand happens to be high, the net present value of the profits $800,000.

53) Draw a decision tree.

54) Draw a decision tree and determine the payoff for each decision and event node. Which alternative should the manufacturer choose?

55) If a student attends every management science class, the probability of passing the course is 0.80; but if the student only attends randomly, then the probability of passing the course is 0.50. If a student fails, they can take a makeup exam where the probability of passing is 0.60 if the student has attended every class. This probability of passing the makeup exam drops to 0.10 if the student has attended at random.

Passing the course is worth 5 credits. Full time attendance “costs” 3 credits in terms of energy and time whereas random attendance “costs” only 1 credit.

Use a decision tree to decide which is the best attendance pattern to adopt. Assume that all failing students take the make up exam and that the payoff for failing is equal to 0.

The quality control manager for ENTA Inc. must decide whether to accept (a1), further analyze (a2) or reject (a3) a lot of incoming material. Assume the following payoff table is available. Historical data indicates that there is 30% chance that the lot is poor quality (s1), 50 % chance that the lot is fair quality (s2) and 20% chance that the lot is good quality (s3).

56) What action would you choose according to maximax criterion?

57) What action would you choose according to maximin criterion?

58) Construct the regret table

59) What action would you choose according to minimax regret criterion?

60) What action would you choose according to expected value criterion?

61) What is the maximum amount that you would be willing to pay for perfect information?

62) Lucky Lucy is playing the slots in Reno, Nevada, holding her last silver dollar. There are three possible payoffs if she wins: one cherry, $1.00; two cherries, $5.00; or three cherries, $50.00. Anything else on the slot machine loses.

Construct the payoff table for Lucky Lucy

63) Consider the following decision tree.

What is the expected value at node 4? 64) Consider the following decision tree.

What is the value associated with node 3?

Which decision, A or B, is best? What is the expected value of this decision?

66) The maximax criterion results in the

A) maximum of the minimum payoffs

B) maximum of the maximum payoffs

C) minimum of the maximum payoffs

D) minimum of the minimum payoffs

67) The maximin criterion results in the

A) minimum of the maximum payoffs

B) maximum of the maximum payoffs

C) maximum of the minimum payoffs

D) minimum of the minimum payoffs

68) Regret is the difference between the payoff from the

A) best decision and all other decision payoffs

B) worst decision and all other decision payoffs

C) best decision and the worst decision payoffs

D) none of the above

69) The __________ minimizes the maximum regret.

A) maximax regret criterion

B) minimax regret criterion

C) minimin regret criterion

D) maximin regret criterion

70) The minimax regret criterion

A) maximizes the minimum regret

B) minimizes the minimum regret

C) minimizes the maximum regret

D) maximizes the maximum regret

71) Determining the worst payoff for each alternative and choosing the alternative with the best worst is called

A) maximin

B) minimin

C) maximax

D) minimax

72) The maximin approach to decision making refers to

A) minimizing the maximum return

B) maximizing the minimum return

C) maximizing the maximum return

D) minimizing the minimum return

73) The term opportunity loss is most closely related to

A) maximin regret

B) maximax regret

C) minimax regret

D) minimin regret

74) The Hurwicz criterion is a compromise

A) for the maximin criterion

B) for the maximax criterion

C) between the maximax and maximin criteria

D) none of the above

75) The Hurwicz criterion multiplies the

A) best payoff by the coefficient of optimism

B) worst payoff by the coefficient of optimism

C) best payoff by the worst payoff

D) none of the above

76) The basic decision environment categories are

A) certainty

B) risk

C) uncertainty

D) all of the above

Answer: D

Diff: 2 Page Ref: 527

Main Heading: Decision Making without Probabilities

Key words: decision making

77) The basic decision environment categories are

A) certainty and risk

B) risk and uncertainty

C) certainty and uncertainty

D) certainty, uncertainty and risk

78) The Hurwicz criterion

A) multiplies the worst payoff by one minus the coefficient of optimism

B) multiplies the best payoff by the coefficient of optimism

C) is a compromise between the maximax and maximin criteria

D) all of the above

79) The appropriate criterion is dependent on

A) the risk personality of the decision maker

B) the philosophy of the decision maker

C) all of the above

D) none of the above

80) The __________ is a measure of the decision maker’s optimism.

A) equal likelihood criterion

B) dominant decision

C) coefficient of optimism

D) none of the above

81) The __________ multiplies the decision payoff for each state of nature by an equal weight.

A) dominant decision

B) coefficient of optimism

C) equal likelihood criterion

D) none of the above

82) A __________ is one that has better payoff than another decision under each state of nature.

A) coefficient of optimism

B) equal likelihood criterion

C) dominant decision

D) none of the above

83) A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

The maximax strategy is:

A) Buy

B) Rent

C) Lease

D) Brisk

E) Slow

84) A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

The maximin strategy is:

A) Buy

B) Rent

C) Lease

D) Brisk.

E) Slow

85) A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

The equal likelihood criterion strategy is:

A) Buy

B) Rent

C) Lease

D) High

E) Low

86) A business owner is trying to decide whether to buy, rent, or lease office space and has constructed the following payoff table based on whether business is brisk or slow.

If the probability of brisk business is .40 and for slow business is .60, the expected value of perfect information is:

A) 12

B) 55

C) 57

D) 69

E) 90

87) The __________ is computed by multiplying each decision outcome under each state of nature by the probability of its occurrence.

A) expected value

B) expected value of perfect information

C) expected opportunity loss

D) none of the above

88) The __________ is the expected value of the regret for each decision.

A) expected value

B) expected opportunity loss

C) expected value of perfect information

D) none of the above

89) A tabular presentation that shows the outcome for each decision alternative under the various possible states of nature is called a

A) decision tree

B) payoff table

C) feasible region

D) payback matrix

90) The __________ is the maximum amount a decision maker would pay for additional information.

A) expected opportunity loss

B) expected value

C) expected value of perfect information

D) none of the above

91) A decision tree is a diagram consisting of

A) square decision nodes

B) circle probability nodes

C) branches representing decision alternatives

D) all of the above

92) In __________ additional information is used to alter the marginal probability of the occurrence of an event.

A) Bayesian analysis

B) decision analysis

C) probability analysis

D) all of the above

93) A __________ probability is the probability that an event will occur given that another event has already occurred.

A) posterior

B) conditional

C) marginal

D) all of the above

94) A __________ probability is the altered marginal probability of an event based on additional information.

A) marginal

B) conditional

C) posterior

D) none of the above

95) The efficiency of sample information is the ratio of the expected value of sample information to the

A) expected value of perfect information

B) expected value

C) utilization rate

D) coefficient of optimism

E) expected opportunity loss

96) The expected value of sample information

A) is never more than EVPI

B) can be compared to the sample cost to judge whether to sample.

C) is never negative

D) all of the above are true

E) Only A and C are true.

97) People who forgo a high expected value to avoid a disaster with a low probability are

A) risk takers

B) risk averters

C) risk calculators

D) risk predictors

98) People who take a chance on a bonanza with a very low probability of occurrence in lieu of a sure thing are

A) risk takers

B) risk averters

C) risk calculators

D) risk predictors

99) Utiles are units of __________ measures of utility.

A) quantitative

B) objective

C) subjective

D) qualitative

A small entrepreneurial company is trying to decide between developing two different products that they believe they can sell to two potential companies, one large and one small. If they develop Product A, they have a 50% chance of selling it to the large company with annual purchases of about 20,000 units. If the large company won’t purchase it, then they think they have an 80% chance of placing it with a smaller company, with sales of 15,000 units. On the other hand if they develop Product B, they feel they have a 40% chance of selling it to the large company, resulting in annual sales of about 17,000 units. If the large company doesn’t buy it, they have a 50% chance of selling it to the small company with sales of 20,000 units.

100) What is the probability that Product Awill being purchased by the smaller company?

A) 0.8

B) 0.5

C) 0.4

D) 0.2

E) 0.1

101) What is the probability that Product B will being purchased by the smaller company?

A) 0.8

B) 0.5

C) 0.4

D) 0.3

E) 0.1

102) How many units of Product A can they expect to sell?

A) 20,000

B) 17,000

C) 15,500

D) 15,000

E) 13,100

103) How many units of Product A can they expect to sell?

A) 20,000

B) 17,000

C) 15,500

D) 15,000

E) 13,100

104) How many units can they expect to sell for the optimum alternative?

A) 20,000

B) 17,000

C) 15,500

D) 15,000

E) 13,100

Introduction to Management Science, 10e (Taylor)

Chapter 13 Queuing Analysis

1) Providing quick service is an important aspect of quality customer service.

2) Operating characteristics describe the methods used by the service process.

3) Waiting lines form because people or things arrive at the servicing function, or server, faster than they can be served.

4) Components of a waiting line system include arrivals, servers, and the calling population.

5) The most important factors to consider in analyzing a queuing system are queuing discipline, arrival and service rate, and the nature of the calling population.

6) The queue discipline is the order in which waiting customers are served.

7) The calling population is the source of customers.

8) Calling populations are always finite.

9) The arrival rate is the frequency at which customers arrive at a waiting line according to a probability distribution.

10) The arrival rate is most frequently described by negative exponential distribution.

11) The service rate is the average time it takes to serve a customer.

12) The service time can often be described by the Poisson distribution.

13) Queuing system operating statistics are constant over time.

14) As the level of service improves, the cost of service decreases.

15) Queue discipline describes customers’ behavior in the queue.

16) The basic single server queuing model assumes an infinite calling population.

17) Utilization of 100% is necessary for a queuing system to reach a steady state.

18) Queuing models provide optimal solutions to waiting line problems

19) All single-server queuing models require the utilization factor to be less than 1.

20) Queue discipline refers to the willingness of customers to wait in line for service.

21) The basic single server model assumes that arrival rates are normally distributed.

22) A car wash with two attendants who work together as a team would be an example of a multiple-server system.

23) In multiple server models, two or more servers work as a team to serve a single waiting line.

24) In systems with finite queue length, the service rate does not have to exceed toe arrival rate.

25) The basic single server model assumes that customers who arrive first are served first.

26) If it takes 5 minutes to serve a customer at a fast food restaurant the service rate is _________.

27) The __________ is the average number of customers who can be served during a given time period.

28) The service time can most often be described by the __________ distribution.

Answer: negative exponential

Diff: 2 Page Ref: 590

Main Heading: Queuing

Key words: service time, negative exponential distribution

29) The __________ is the frequency at which the customers arrive at a waiting line according to a probability distribution.

30) The arrival rate can generally be described by a(n) __________ distribution.

31) The __________ is the source of the customers or objects being simulated.

32) A system has one service facility that can service 10 customers per hour. The customers arrive at an average rate of 6 per hour. Utilization is __________.

33) A situation in which a mechanic is responsible for repairing a pool of fleet vehicles should be analyzed for a waiting line system with a __________ calling population

34) Customers arrive at a candy shop every 8 minutes on average. The arrival rate is __________.

35) If λ = 24 and μ = 30, then utilization is equal to __________.

36) If λ = 24 and μ = 30, then L is equal to __________.

37) If λ = 24 and μ = 30, then W is equal to __________ minutes.

38) If the __________ queuing system, which is a variation of the single phase single channel model, the service rate does not have to exceed the arrival rate.

39) A single channel queuing system has an average service time of 10 minutes and an average time between arrivals of 15 minutes. What is the arrival rate?

40) In a single server queuing system, if 10 customers arrive per hour, and 20 customers are served per hour, what is the probability that there are no customers in the system?

41) A single bay car wash with a Poisson arrival rate and an exponential service time has cars arriving an average of 15 minutes apart. It takes approximately 9 minutes to wash a car. What is the system utilization?

A crew of mechanics at the Department of Transportation garage make minor repairs to snowplows during the winter. The snowplows break down at an average rate of 4 vehicles per day and breakdowns are distributed according to the Poisson distribution. The mechanic can service an average of 7 vehicles per day with a repair time distribution that approximates a negative exponential distribution. Assume an 8 hour day.

42) What is the utilization rate for the mechanic?

43) What is the average time that a snowplow is out of service?

44) On average, how long does a snow plow wait before the mechanic can begin his repair?

45) Approximately how many vehicles are in the garage, waiting for or being repaired?

46) Poultry Processing processes chickens for fast food restaurants. The chickens arrive from the farms on trucks, in cages, at a rate of 8 trucks per hour according to the Poisson distribution. The quality standards of Poultry Processing require that the chickens be processes within 30 minutes, which includes the time from when the trucks arrive until the chickens are finished processing. Determine the maximum average processing rate (in truckloads per hour) that must be designed for the machine, in order to ensure that the cages will be processed, on the average, in 30 minutes or less. Assume processing time is exponentially distributed.

Lenny, a graduate research assistant “moonlights” at the short order counter in the student union snack bar in the evenings. He is the only one on duty at the counter during the hours he works. Arrivals to the counter seem to follow the Poisson distribution with a mean of 8 per hour. Each customer is served one at a time and the service time follows an exponential distribution with a mean of 5 minutes.

47) How long will a student wait in line, on average?

48) The manager thinks that students will go elsewhere for lunch if they have to wait more than 5 minutes. Therefore he’s thinking of hiring another server to help Lenny, reducing the customer service time to 4 minutes. How long will students wait in line if Lenny gets help?

49) Instead of having another student helps Lenny the manager is thinking of having two lines instead. Customers will equally divide themselves between the two lines. How long will students wait in line if there’s a second server? (Assume that the service time is 5 minutes.)

50) A hotel is considering changing its waiting line system. In the current system hotel guests divide themselves equally between the lines that form in front of 4 hotel clerks. The manager is considering having hotel guests wait in one line and then proceeding to the next available clerk. If average service time is 10 minutes and the average number of arrivals per hour is 12 guests, determine which system results in the lowest customer waiting time.

51) The local grocery store consists of two cashiers. The customers arrive according to Poisson distribution and the service times are based on negative exponential distribution. The average customer service time is five minutes and the average time between the arrivals of successive customers is 3 minutes. What is the probability that there are no customers in the grocery store?

The local grocery store consists of two cashiers. The customers arrive at the checkout according to the Poisson distribution and the service times are based on negative exponential distribution. The average customer service time is 4 minutes and the average time between the arrivals of successive customers is 3 minutes. Assume that customers equally divide themselves between the two cashiers.

52) What is the average number of customers waiting in each line and being checked out?

53) How much time is a customer expected to spend waiting in line and being checked out?

54) On average, how much time will the customer spend in line waiting to be served?

55) On the average, how many customers are waiting in line to be served?

56) A queuing system has 3 crews with 2 members each. What is the number of servers?

57) A multiple channel queuing system with a Poisson arrival rate and exponential service time has an average arrival rate of 6 customers per hour and an average service time of 20 minutes per customer. What is the minimum number of servers required to avoid an overloaded system?

In a factory machines breakdown at an average of 6 machines per hour according to a Poisson distribution. The time a repair person takes to repair the machine is not defined by any probability distribution but has a mean of 8 minutes and a standard deviation of 3 minutes.

58) What is the probability that no machine is being fixed?

59) On the average how many machines are waiting in line to be fixed?

60) On the average, how many machines are in the system either being repaired or waiting in line to be repaired?

61) On the average, how long will a machine have to wait before it is fixed?

62) On the average, how long will a machine be down and out of service?

In a bank drive-through, there is a single service window and room only for 2 cars to line-up to wait for service. The mean time between arrivals for drive through customers is five minutes. The mean time to complete a customer transaction is 3 minutes. The number of arrivals is distributed according to Poisson distribution and the service times are exponentially distributed.

63) What is the probability that there are no vehicles in the system?

64) On the average how many cars are in the system?

65) What is the probability that the system is full and the customer must drive on?

66) On the average, how many customers are waiting to be served?

67) What is the average time a customer spends in the system?

Answer: W = 7.653 minutes

Diff: 3 Page Ref: 601

Main Heading: Finite Queue Length

Key words: the avg time customer spends in system, finite queue length

68) What is the average time a customer spends in the line waiting to be served?

69) Operating characteristics for a waiting line system include

A) queue discipline

B) the Poisson distribution

C) a waiting line structure

D) utilization

70) The most important factors to consider in analyzing a queuing system are

A) the queue discipline

B) the queue structure

C) the queue order

D) all of the above

71) Customers may be served

A) according to a number assigned to each item

B) on a first-come-first serve basis

C) on a last-come-first-serve basis

D) all of the above

72) In a single-server queuing model, L represents

A) the length of time a customer waits

B) the size of the queue

C) the average number of customers in the queuing system

D) the length of the line

73) Operating characteristics

A) are averages

B) are constant over time

C) represent the steady state

D) all of the above

74) Queuing discipline refers to the

A) reason waiting occurs in underloaded systems

B) willingness of customers to wait in line

C) order in which customers are processed

D) all of the above

75) The arrival rate is the

A) time between arrivals to the service facility

B) rate items arrive at the server after being in queue

C) rate of arrivals to the service facility

D) all of the above

76) A single server queuing system has average time between arrivals of 20 minutes and a service time of 10 minutes each. Assuming Poisson arrivals and exponential service times, the utilization factor is approximately

A) 0.25

B) 0.33

C) 0.50

D) 0.67

E) 2.0

77) What happens to the customer waiting time if system utilization increases?

A) decreases exponentially

B) decreases proportionally

C) increases proportionally

D) increases exponentially

78) What is not considered a measure of system performance in a queuing analysis?

A) average number in the system

B) system utilization

C) average number waiting in line

D) service time

79) Which of the following will not decrease system utilization?

A) increase in arrival rate

B) increase in service rate

C) increase in the number of servers

D) decrease in service time

A single server waiting line system has an arrival pattern characterized by a Poisson distribution with 3 customers per hour. The average service time is 12 minutes. The service times are distributed according to the negative exponential distribution.

80) The average time a customer can expect to wait in line is:

A) 18 minutes

B) 36 minutes

C) 30 minutes

D) 60 minutes

E) 72 minutes

81) The probability that the system is idle is:

A) 0

B) .20

C) .40

D) .60

E) .80

82) The expected number of customers in the system is:

A) 3.0

B) 1.5

C) 1.0

D) .90

E) .60

83) The expected number of customers in the waiting line is:

A) .6

B) .7

C) .8

D) .9

E) 1.0

84) A system has 5 servers. Customers arrive at a rate of 6 per hour and service time is 20 minutes. What is the system utilization?

A) .83

B) .40

C) .50

D) 20

A crew of mechanics at the Department of Transportation garage make minor repairs to snowplows during the winter. The snowplows break down at an average rate of 4 vehicles per day and breakdowns are distributed according to the Poisson distribution. The mechanic can service an average of 7 vehicles per day with a repair time distribution that approximates a negative exponential distribution. Assume an 8 hour day.

85) The utilization is

A) .30

B) .45

C) .57

D) .85

E) 1.00

86) Determine the average time that a snowplow is out of service.

A) .33 hours

B) 20 minutes

C) 2.64 hours

D) .33 days

E) 1.15 hours

87) On average, how long does a plow wait before the mechanic begins the repair?

A) 1 hour

B) 1.25 hours

C) 1.52 hours

D) 2 hours

E) 2.64 hours

88) What is the expected average number of snowplow in the garage (waiting for repair and being repaired)?

A) 1

B) 1.33

C) 2

D) 2.52

89) In a single server queuing system, if 12 customers arrive per hour, and 30 customers are served per hour, what is the probability that there are no customers in the system?

A) 0.75

B) 0.60

C) 0.40

D) 0.2 5

90) A manager is trying to improve a single-server queueing system through automation. The average service time is 20 minutes per customer, exponentially distributed, and the arrival rate is 16 customers per 8-hour day (Poisson arrivals). The automated system will have a constant service time of 16 minutes. The effect of this change will:

A) decrease utilization

B) increase waiting time

C) decrease waiting time

D) have no effect since the service time is unchanged

91) A multiple channel system has customers arriving at an average rate of 5 per hour and an average service time of 40 minutes. What minimum number of servers is required to ensure that the system is not overloaded?

A) 4

B) 5

C) 6

D) 3

92) A single channel queuing system has an average service time of 8 minutes and an average time between arrivals of 10 minutes. What is the hourly arrival rate?

A) 8

B) 6

C) 4

D) 2

93) A queuing system has 5 crews with 2 members each. What is the number of servers?

A) 2

B) 5

C) 10

D) none of the above

94) A multiple channel queuing system with a Poisson arrival rate and exponential service time has an average arrival rate of 4 customers per hour and an average service time of 18 minutes per customer. What is the minimum number of servers required to avoid an overloaded system?

A) 1

B) 2

C) 3

D) 4

95) Poultry Processing processes chickens for fast food restaurants. The chickens arrive from the farms on trucks, in cages, at a rate of 8 trucks per hour according to the Poisson distribution. The quality standards of Poultry Processing require that the chickens be processes within 30 minutes, which includes the time from when the trucks arrive until the chickens are finished processing. What is the maximum average processing rate (in truckloads per hour) that must be designed for the machine, in order to ensure that the cages will be processed, on the average, in 30 minutes or less. Assume processing time is exponentially distributed.

A) 4.5 trucks per hour

B) 9 trucks per hour

C) 10 trucks per hour

D) 18 trucks per hour

96) Constant service times occur with

A) machinery

B) well-trained employees

C) service processes

D) assembly processes

97) Customers arrive at a music store at an average of 1 per minute (Poisson arrivals). The service rate is 15 customers per hour (exponential service times). What is the minimum number of servers needed to keep the waiting time in the system under 5 minutes?

A) 4

B) 5

C) 6

D) 7

98) A single bay car wash with a Poisson arrival rate and an exponential service time has cars arriving on average of 10 minutes apart and an average service time of 4 minutes. What is the system utilization?

A) 0.2

B) 0.3

C) 0.4

D) 0.5

E) 0.6

99) Cars arrive at a single bay car wash on the average of 6 per hour according to the Poisson distribution. The wash time is a constant 4 minutes. What is the average number of cars in line?

A) .022

B) .133

C) .267

D) .667

100) Cars arrive at a single bay car wash on the average of 6 per hour according to the Poisson distribution. The wash time averages 4 minutes with a standard deviation of 1 minute, but the wash time is not defined by any distribution. What is the average number of cars in line?

A) .142

B) .267

C) .283

D) 2.83

E) 3.33

101) In a finite queue, the length of the queue is

A) limited

B) unlimited

C) limited or unlimited

D) limited and unlimited

102) Pickmeup is a drive through coffee house that has room for 3 cars in the driveway. The line cannot exceed 3 cars, and because they are exclusively drive through, customers may be turned away. In the morning, the arrival rate is 40 cars per hour (Poisson distributed) and with the servers working in teams, they can process 50 cars per hour (The service rate is exponential). What is the probability of turning customers away?

A) .173

B) .210

C) .339

D) .410

103) In multiple-server models, __________ independent servers in parallel serve a single waiting line.

A) 2 or more

B) 3 or more

C) 4 or more

D) 5 or more

The local grocery store consists of two cashiers. The customers arrive at the checkout according to the Poisson distribution and the service times are based on negative exponential distribution. The average customer service time is 4 minutes and the average time between the arrivals of successive customers is 3 minutes. Assume that customers equally divide themselves between the two cashiers.

104) How much time is a customer expected to spend in line at the checkout?

A) 11.2 minutes

B) 13.6 minutes

C) 14.4 minutes

D) 16.2 minutes

E) 18.2 minutes

105) On the average, how much time will the customer spend in line waiting to be served?

A) 11.2 minutes

B) 13.6 minutes

C) 14.4 minutes

D) 16.2 minutes

E) 18.2 minutes

106) What is the probability that there are no customers in the grocery store?

A) .0809

B) .0909

C) .10909

D) .1209

E) .1409

Introduction to Management Science, 10e (Taylor)

Chapter 14 Simulation

1) In computer mathematical simulation, a system is replicated with a mathematical model that is analyzed with the computer.

2) Monte Carlo is a technique for selecting numbers randomly from a probability distribution.

3) The Monte Carlo process is analogous to gambling devices.

4) In the Monte Carlo process, values for a random variable are generated by sampling from a probability distribution.

5) A long period of real time can be represented by a short period of simulated time.

6) Random numbers are equally likely to occur.

7) Simulation results will always equal analytical results if 30 trials of the simulation have been conducted.

8) It’s often difficult to validate that the results of a simulation truly replicate reality.

9) Simulation applies mathematical models to determine optimal solutions to business problems.

10) Validation of a simulation model occurs when the true steady state average results have been reached.

11) Random numbers generated by a mathematical process instead of a physical process are pseudorandom numbers.

12) Random numbers are typically generated on the computer using a numerical technique.

13) A table of random numbers must be normally distributed and efficiently generated.

14) When using Excel to simulate a system, it is necessary to compute only the probability (relative frequency) distribution.

15) Excel can only be used to simulate systems that can be represented by continuous random variables.

16) In Excel the “VLOOKUP” function is used to determine values for continuous random variables.

17) Developing the cumulative probability distribution helps to determine random number ranges.

18) Starting conditions have no impact on the validity of a simulation model.

19) Simulations should always be started with an empty system.

20) Manual simulation is limited because of the amount of real time required to simulate even one trial.

21) __________ is a technique for selecting numbers randomly from a probability distribution.

22) Developing the __________ probability distribution helps to determine random number ranges.

23) Random numbers of a mathematical process instead of a physical process are __________ numbers.

24) Simulation models must be __________ to make sure they are accurately replicating the system being simulated.

25) A table of random numbers must be __________ distributed, efficiently generated, and absent of patterns.

26) The __________ command is used in generating the random numbers with Excel.

27) In order to determine the value of discrete demand in a simulation model using Excel, the __________ function is used to associate a specific value of demand with a random number.

28) The __________, a 10 kilometer race held in Colorado, used simulation to help deal with queuing problems at the finish line.

29) __________ is a risk analysis and forecasting program that uses Monte Carlo simulation.

30) __________, __________, and __________ are common application of simulation.

31) Consider the following distribution and random numbers:

If a simulation begins with the first random number, what would the first simulation value would be __________.

The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

32) Determine the drying time for these 5 random numbers: 0.13; 0.09; 0.19; 0.81; and 0.12.

33) Determine the drying time for these 5 random numbers: 0.53; 0.95; 0.97; 0.96; and 0.07.

34) The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

What is the average drying time if you simulate the 10 trials using these random numbers: 0.13; 0.09; 0.19; 0.81; 0.12; 0.53; 0.95; 0.97; 0.96; and 0.07.

35)

Determine the random number ranges for the above data set (Start with 00).

36) Given the following random number ranges and the following random number sequence: 62, 13, 25, 40, 86, 93, determine the average demand for the following distribution of demand.

Demand Random

Number Ranges

5 00-14

6 15-44

7 45-69

8 70-84

9 85-99

37) If f(x) = 2x, what is the equation for generating x, given the random number r?

38) The drying rate in an industrial process is dependent on many factors and varies according to the following distribution.

Compute the mean drying time.

39) A normal distribution has a mean of 500 and a standard deviation of 50. A manager wants to simulate 2 values from this distribution, and has drawn these random numbers: -0.6 and 1.4. What are the 2 values respectively?

40) The number of cars arriving at Joe Kelly’s oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Determine the probability distribution of car arrivals.

41) The number of cars arriving at Joe Kelly’s oil change and tune-up place during the last 200 hours of operation is observed to be the following:

Based on the above frequencies, use two digit random numbers, start with random number 00 and determine the random number ranges for the data set given above.

42) Consider the following manual simulation of machine breakdowns and repair

Breakdowns Time between breakdowns, x (weeks) Repair time, y (days) Cost ($2,000 y)

1 4 1 $2000

2 5.7 2 $4000

3 5.5 2 $4000

4 2.5 1 $2000

5 5.2 2 $4000

6 5.8 2 $4000

7 1.6 1 $2000

8 2.3 2 $4000

9 1.2 2 $4000

10 3.3 2 $4000

What operating characteristics can be obtained from this simulation?

43) George Nanchoff owns a gas station. The cars arrive at the gas station according to the following inter-arrival time distribution. The time to service a car is given by the following service time distribution. Using the following random number sequence: 92, 44, 15, 97, 21, 80, 38, 64, 74, 08, estimate the average customer waiting time , average idle time of the assistant and the average time a car spends in the system.

Interarrival time (in minutes) P(X) Random Numbers Service Time (in minutes) P (X) Random Numbers

4 .35 00-34 2 .30 00-29

7 .25 35-59 4 .40 30-69

10 .30 60-89 6 .20 70-89

20 .10 90-99 8 .10 90-99

44) An answering service for a doctor’s office wants to evaluate the service by simulation calls. They used past data to determine the distributions for the time between calls and the time spent on the phone with patients.

Use the table below to manually simulate 8 calls. The time of the call and the service time have already been determined.

Customer

Number Time of

Call Time call answered Service

Time Time call ends Customer

Wait Time Operator

Wait Time

1 15 15 25 40 0 15

2 45 45 25 70 0 5

3 70 15

4 95 45

5 125 15 155

6 150 35

7 175 190 25 15 0

8 205 25 240 10 0

Calculate the average waiting time per customer.

Calculate the utilization of the operator.

A graduate research assistant “moonlights” at the short order counter in the student union snack bar in the evenings. He is considering asking for help taking orders, but needs to convince the management that they should hire another student. Because he is taking a simulation class, he thinks it may be the perfect way to convince management to hire more help if he can show that customers have to wait a long time. When a customer arrives, he takes their order and their payment, prepares the food, gives it to the customer, and then takes the order from the next person in line. If someone arrives while he’s cooking an order, they have to wait until he’s completed the current order. He is working on the simulation and a portion is shown below.

45) Complete the table below and determine the average customer wait time.

Customer

Number Time of

Arrival Time Order Taken Order Prep Time Customer Receives Food Customer Wait Time

1 2 2 4 6 0

2 14 14 7

3 19 6

4 28 8

5 33 4

46) Complete the table and determine the average customer waiting time and the utilization of the cook.

Customer Number Time of Arrival Time Order Taken Order Prep Time Customer Receives Food Customer Wait Time Cook “Idle” TIme

1 2 2 4 6 0 2

2 14 14 7

3 19 6

4 28 8

5 33 4

Complete the table and determine the average customer waiting time and the utilization of the cook.

47) Assume that order prep time is based on the following distribution:

Prep Time Probability

3 .10

4 .20

5 .25

6 .20

7 .15

8 .10

Complete the following table given that the random numbers for order prep time for customers 3, 4, and 5 are 62, 93 and 26, respectively.

Arrival

Number Time of

Arrival Time Order Taken Order Prep Time Customer Receives Food Customer Wait Time Cook “Idle” Time

1 1 1 4 5 0 1

2 5 5 7 12 0 0

3 7

4 10

5 18

What is the average customer waiting time and the graduate student’s utilization?

A newsboy sells newspapers and his goal is to maximize profit. He kept a record of his sales for 125 days with the following result:

Newspapers demand per day Number of days

15 10

16 20

17 42

18 31

19 12

20 10

Total 125

His ordering policy is to order an amount each day that is equal to the previous day’s demand.

A newspaper costs the carrier 50 centers and he sells it for $1.00. Unsold papers are returned and he receives 25 cents (for a loss of 25 cents).

48) Develop the cumulative distribution table and the corresponding random numbers.

Newspapers demanded per day Number of Days Probability Cumulative Probability

15 10

16 20

17 42

18 31

19 12

20 10

Total

49) Use the information and random numbers given in the table below to simulate the sale of newspapers for 10 days.

Day Demand Random Number Quantity Ordered Sales Unsatisfied Demand Unsold Papers

1 .78 18

2 .43

3 .93

4 .87

5 .48

6 .84

7 .87

8 .27

9 .20

10 .52

After completing the simulation, determine his total revenue for the ten days as well as monetary losses that result from unmet demand and unsold papers.

50) Analogue simulation replaces a physical system with an analogous physical system that is __________ to manipulate.

A) harder

B) easier

C) equally difficult

D) none of the above

51) __________ is a technique for selecting numbers randomly from a probability distribution.

A) Marseille

B) Monaco

C) Monte Carlo

D) Analogue simulation

E) all of the above

52) In the Monte Carlo process, values for a random variable are generated by __________ a probability distribution.

A) sampling from

B) running

C) integrating

D) implementing

53) The __________ process is analogous to gambling devices.

A) Simulation

B) Monte Carlo

C) Monaco

D) none of the above

54) __________ numbers are numbers derived from a mathematical process that appear to be random.

A) Random

B) Pseudorandom

C) Randomized

D) Semi-random

55) Pseudorandom numbers exhibit __________ in order to be considered truly random.

A) a limited number of possible outcomes

B) a uniform distribution

C) a detectable pattern

D) a detectable run of certain numbers

56) __________ is not part of a Monte Carlo simulation.

A) Analyzing results

B) Analyzing a real problem

C) Finding an optimal solution

D) Evaluating the results

57) A seed value is a(n)

A) steady state solution of a simulation experiment

B) number used to start a stream of random numbers

C) first run of a simulation model

D) analytic solution of a simulation experiment

58) In assigning random numbers to probabilistic events in a simulation,

A) several events are associated with the same random number

B) every random number is associated with a particular event

C) every event is associated with the same random number

D) all of the above

59) __________ are the values that express the state of the system being modeled at the beginning of the Monte Carlo simulation.

A) Outputs

B) Random events

C) Initial conditions

D) Random numbers

60) Simulation does not usually provide recommended decisions. Instead it provides

A) operating characteristics

B) optimal solutions

C) realistic results

D) system parameters

61) For the following frequency distribution of demand, the random number 0.23 would be interpreted as a demand of:

A) 0

B) 1

C) 2

D) 3

Consider the following frequency of demand and random numbers:

Random numbers: 0.13, 0.81, 0.53.

62) If the simulation begins with the first random number the simulated value for demand would be

A) 0

B) 1

C) 2

D) 3

63) If the simulation begins with the second random number the simulated value for demand would be

A) 1

B) 2

C) 3

D) 4

64) If the simulation begins with the third random number the simulated value for demand would be

A) 0

B) 1

C) 2

D) 3

E) 4

65) Which of the following would not be considered a limitation of simulation modeling?

A) Models are typically unstructured.

B) Validation of simulation models can be difficult.

C) The cost of building simulation can be prohibitive.

D) Simulation allows flexibility in analyzing systems.

Key words: simulation models, limitations

66) Simulations are normally done

A) manually

B) in a casino

C) by a spreadsheet

D) on the computer

67) Random numbers generated by a __________ process instead of a __________ process are pseudorandom numbers.

A) physical / physical

B) physical / mathematical

C) mathematical / physical

D) mathematical / mathematical

68) A table of random numbers must be

A) uniform

B) efficiently generated

C) absent of patterns

D) all of the above

69) Developing the cumulative probability distribution helps to determine

A) simulation numbers

B) data sets

C) random number ranges

D) all of the above

A graduate research assistant “moonlights” at the short order counter in the student union snack bar in the evenings. He is considering asking for help taking orders, but needs to convince the management that they should hire another student. Because he is taking a simulation class, he thinks it may be the perfect way to convince management to hire more help if he can show that customers have to wait a long time. When a customer arrives, he takes their order and their payment, prepares the food, gives it to the customer, and then takes the order from the next person in line. If someone arrives while he’s cooking an order, they have to wait until he’s completed the current order. He has simulated 5 orders.

Customer Number Time of Arrival Time Order Taken Order Prep Time Customer Receives Food Customer Wait Time

1 2 2 4 6 0

2 14 14 7

3 19 6

4 28 8

5 33 4

70) Average customer waiting time is:

A) 0 minutes

B) 1 minute

C) 2 minutes

D) 2.5 minutes

E) 3 minutes

71) Average utilization is:

A) 50%

B) 67%

C) 72.5%

D) 83.4%

E) 95%

72) __________ simulation is limited because of the amount of real time required to simulate even one trial.

A) Manual

B) Monte Carlo

C) Monaco

D) Any kind of

73) If the probability of an event is 0.36, what random number range specifies this properly?

A) 0.10 – 0.20

B) 0.20 – 0.30

C) 0.30 – 0.40

D) 0.40 – 0.50

74) Sometimes manual simulation of several trials is __________ way to validate a simulation.

A) a good

B) a bad

C) no

D) the only

75) Unlike optimization models, simulation provides

A) recommendations

B) operating characteristics

C) suggestions

D) solutions

76) A limitation of simulation is

A) models are typically unstructured and must be developed for problems that are also unstructured

B) it is often impossible to realistically validate simulation results

C) model building is costly and time-consuming

D) all of the above

The U.S. Department of Agriculture estimates that the yearly yield of limes per acre is distributed as follows:

Yield, bushels per acre Probability

350 .10

400 .18

450 .50

500 .22

The estimated average price per bushel is $16.80.

77) What is the expected yield of the crop?

A) 425

B) 442

C) 440

D) 475

78) Use the following random numbers to simulation crop yield for 10 years: 37, 23, 92, 01, 69, 50, 72, 12, 46, 81, 31, 89. What is the estimated crop yield from the simulation?

A) 425

B) 442

C) 440

D) 475

79) Use the following random numbers to simulation crop yield for 10 years: 37, 23, 92, 01, 69, 50, 72, 12, 46, 81, 31, 89. What is the estimated crop yield from the simulation?

A) 425

B) 442

C) 440

D) 475

A bakery is considering hiring another clerk to better serve customers. To help with this decision, records were kept to determine how many customers arrived in 10-minute intervals. Based on 100 ten-minute intervals, the following probability distribution and random number assignments developed.

Number of Arrivals Probability Random numbers

6 .1 .01 – .10

7 .3 .11 – .40

8 .3 .41 – .70

9 .2 .71 – .90

10 .1 .91 – .00

80) Suppose the next three random numbers were .18, .89 and .67. How many customers would have arrived during this 30-minute period?

A) 22

B) 23

C) 24

D) 25

E) none of the above

81) Suppose the next three random numbers were .08, .50 and .69. How many customers would have arrived during this 30-minute period?

A) 22

B) 23

C) 24

D) 25

E) none of the above

82) Suppose the next three random numbers were .11, .42 and .84. Use these values to simulate arrivals into the bakery and determine the average number of arrivals per 10-minute period based on these occurrences. The average number of arrivals per 10-minute period is:

A) 6

B) 7

C) 8

D) 9

E) none of the above

Two hundred simulation runs were completed using the probability of a machine breakdown from the table below. The average number of breakdowns from the simulation trials was 1.93 with a standard deviation of 0.20.

No. of breakdowns per week Probability Cumulative probability

0 .10 .10

1 .25 .35

2 .36 .71

3 .22 .93

4 .07 1.00

83) What is the probability of 2 or fewer breakdowns?

A) .10

B) .25

C) .35

D) .36

E) .71

84) What is the probability that there are more than 3 breakdowns?

A) .07

B) .10

C) .22

D) .25

E) .36

85) Construct a 95% confidence interval for the average number of machine breakdowns.

A) 1.896 to 1.938

B) 1.902 to 1.958

C) .1.877 to 1.943

D) 1.907 to 1.953

E) none of the above

86) The use of simulation to determine the impact of projects such as nuclear power plants, reservoirs and dams is known as

A) public service operation

B) environmental and resource analysis

C) cost benefit analysis

D) none of the above

87) The use of simulation to analyze airport operations, and fire or police department operations are known as

A) public service operation

B) environmental and resource analysis

C) cost benefit analysis

D) none of the above

Introduction to Management Science, 10e (Taylor)

Chapter 15 Forecasting

1) A trend is a gradual, long-term, up or down movement of demand.

2) A seasonal pattern is an up-and-down repetitive movement within a trend occurring periodically.

3) Random variations are movements that are not predictable and follow no pattern.

4) The basic types of forecasting methods include time series, regression, and qualitative methods.

5) Time series is a category of statistical techniques that uses historical data to predict future behavior.

6) Regression methods attempt to develop a mathematical relationship between the item being forecast and factors that cause it to behave the way it does.

7) Qualitative methods use management judgment, expertise, and opinion to make forecasts.

8) Qualitative methods are the least common type of forecasting method for the long-term strategic planning process.

9) A cycle is an up and down movement in demand that repeats itself in less than 1 year.

10) Seasonal patterns are observed only during the 4 seasons – winter, spring, summer and fall.

11) The Delphi develops a consensus forecast about what will occur in the future.

12) Technological forecasting helps determine the technological feasibility of new products by surveying large numbers of consumers.

13) Irregular variations exhibit no pattern.

14) Data cannot exhibit both trend and cyclical patterns.

15) Time series methods assume that what has occurred in the past will continue to occur in the future.

16) Moving averages are good for stable demand with no pronounced behavioral patterns.

17) Longer-period moving averages react more slowly to recent demand changes than do shorter-period moving averages.

18) Shorter-period moving averages react more slowly to recent demand changes than do longer-period moving averages.

19) In a weighted moving average, weights must sum to one.

20) Adjusted exponential smoothing is an exponential smoothing forecast adjusted for seasonality.

21) If average forecast error is positive, it indicates that the forecast is biased high.

22) __________ is a gradual long term upward or downward movement of demand.

23) A(n) __________ forecast typically encompass a period longer than one years.

24) __________ error is the sum of the forecast errors.

25) A(n) __________ forecast encompasses the immediate future, is concerned with daily activities of the firm and does not go beyond one or two months in to the future.

26) A(n) __________ is an up-and-down repetitive movement within a trend occurring periodically.

27) __________ relates demand to two or more independent variables.

28) One problem with multiple regression is __________, which is a measure of the amount of “overlapping” information about he dependent variable that’s provided by several independent variables.

29) Exponential smoothing forecasts are more sensitive or reactive to the changes in demand as the value of the smoothing constant, α __________.

30) The closer the value of α is to zero, the __________ will be the dampening or smoothing effect.

31) __________ measures the strength of the relationship between two variables.

32) The __________ is the percentage of variation in the dependent variable that results from the independent variable.

33) Longer period moving averages react more __________ to recent demand changes than do shorter period moving averages.

34) The __________ movements or variations in demand exhibit no pattern and occur on a random basis.

35) The __________ is the sum of the absolute value of forecasting errors divided by the number of periods in which a forecast was made.

36) The __________ is the square of the average of the sum of the squared errors.

37) MAPD measures __________.

38) __________ is a type of exponential smoothing that can also include trend.

Given the following data on the number of pints of ice cream sold at a local ice cream store for a 6-period time frame:

39) Compute a 3-period moving average for period 4.

40) Compute a 3-period moving average for period 6.

41) Compute a 3-period moving average for period 7.

42) Compute a 5-period moving average for period 6.

43) Compute a 5-period moving average for period 7.

44) Daily highs in Sacramento for the past week (from least to most recent) were: 95, 102, 101, 96, 95, 90 and 92. Develop a forecast for today using a 2 day moving average.

45) Daily highs in Sacramento for the past week (from least to most recent) were: 95, 102, 101, 96, 95, 90 and 92. Develop a forecast for today using a 3 day moving average.

46) Daily highs in Sacramento for the past week (from least to most recent) were: 95, 102, 101, 96, 95, 90 and 92. Develop a forecast for today using a weighted moving average, with a weights of .6, .3 and .1, where the highest weights are applied to the most recent data.

Given the following data on hotel check-ins for a 6-month period:

July: 70 rooms

August: 105 rooms

September: 90 rooms

October: 120 rooms

November: 110 rooms

December: 115 rooms

47) What is the 3-month moving average forecast for January?

48) With alpha = 0.2, what is the simple exponential smoothing forecast for October? Assume the forecast for July was 12 rooms.

49) Using a 3-month moving average, how many check-ins can be forecasted for January?

50) Using the exponential smoothing factor 0.3, how many check-ins can be forecasted for January? Assume the forecast for December was 22 rooms.

51) The following data summarizes the historical demand for a product.

Month Actual Demand

March 20

April 25

May 40

June 35

July 30

August 45

Use exponential smoothing with α = .2 and the smoothed forecast for July is 32 and determine August and September’s smoothed forecasts.

52) If the forecast is 33 and the actual value is 44, then the error this period is

53) If the forecast is 25 and the actual value is 25, then the error this period is

54) The following sales data are available for 2003-2008 :

Determine a 4-year moving average forecast for 2008 and 2009.

55) The following sales data are available for 2003-2008 :

Year Sales Forecast

2003 7 9

2004 12 10

2005 14 15

2006 20 22

2007 16 18

2008 25 21

Calculate the average error.

56) If the forecast is 14 and the actual value is 15, then the error this period is

57) If the absolute errors were calculated for 5 periods and the sum of the absolute deviation is 60, what is the value of the MAD?

58) Assume that the forecasted demand for 2006 is 15, use the following data set and exponential smoothing with α = 0.4 and determine the forecasted demand for 2009.

59) Given the following data, compute the MAD for the forecast.

Year Demand Forecast

2001 16 18

2002 20 19

2003 18 24

60) The following sales data are available for 2003-2008.

Determine a 4-year weighted moving average forecast for 2009, where weights are W1 = .1, W2 = .2, W3 = .2 and W4 = .5.

61) Quarterly sales is given for the past 3 years, determine the seasonal factors for each quarter.

Winter Spring Summer Fall

Year 1 4800 4500 4100 5500

Year 2 570 3800 4500 6000

Year 3 6000 4600 4900 6500

Month Actual Demand

February 20

March 22

April 33

May 35

June 31

July 48

August 41

62) Determine the forecasted demand for April and May based on adjusted exponential smoothing with α = .2, β = .3.

63) The forecasted demand for May, June, July August and September are 25, 30, 33, 38, 40 respectively. Determine the MSE and MAD.

64) Forecasted demand for May, June, July August and September are 25, 30, 33, 38, 40 respectively. Determine the MAPD

65) Base on a three month weighted moving average with weights w1 = .1, w2 = .4, and w3 = .5, determine the forecasted demand for August and September. What is the forecast error in August?

66) Use simple exponential smoothing with alpha = .4 and determine the forecasted demand for August and September. Assume that the smoothed forecast for July is 38.

67) Simple exponential smoothing is being used to forecast demand. The previous forecast of 66 turned out to be four units less than actual demand. If the next forecast is 66.6, what is the value of the smoothing constant, α ?

68) Robert wants to know if there is a relation between money spent on gambling and winnings.

What is the coefficient of determination?

Robert has the following accounts on money spent on gambling and winnings:

69) Develop a regression equation that relates the money Robert spends and the money he wins.

70) Determine the correlation coefficient and the coefficient of determination.

Sally has been running the following number of adds in the local newspaper to help attract customers into her store. She has also been keeping track of customers who have come into the store as a result of the ads, as well as the amount of money they spend.

This is the data from the last 4 weeks:

71) Determine the equation that relates ads and increased sales.

72) If Sally runs 15 ads, how much will sales increase?

73) If Sally runs 10 ads, how much will sales increase?

Consider the following annual sales data for 2001-2008.

Year Sales

2001 2

2002 4

2003 10

2004 8

2005 14

2006 18

2007 17

2008 20

74) Use the linear regression method and determine the estimated sales equation.

75) Calculate the coefficient of determination.

76) Calculate the correlation coefficient .

The following data summarizes the historical demand for a product

Month Actual Demand

March 20

April 25

May 40

June 35

July 30

August 45

77) Use a four period moving average to determine the forecasted demand for July, August and September.

78) If the forecasted demand for June, July and August is 32, 38 and 42, respectively, what is MAD?

79) If the forecasted demand for June, July and August is 32, 38 and 42, respectively, what is MSE?

80) If the forecasted demand for June, July and August is 32, 38 and 42, respectively, what is MAPD?

81) A manager uses the following equation to predict monthly receipts: Yt = 4,000 + 30t. What is the forecast for July of next year if t = 0 in April of this year?

82) A local gym has discovered that their demand for personal trainers (measured in hours) is related not only to their own advertising expenditures in the prior month, but also to the demand for doughnuts and swim gear in the prior month at neighboring stores. The gym has developed the following regression model to forecast demand for personal trainers:

Demand = 185 + (0.15*advertising expenditures) – (0.05*doughnuts) + (0.23*swim gear sales)

What is the forecast for October, given advertising expenditures of 1000, doughnut sales of 2450, and swim gear sales of 782 in September?

83) Managers use __________ in forecasting.

A) judgment

B) opinion

C) past experience

D) all of the above

84) The applicability of forecasting methods depends on

A) the time frame of the forecast

B) the existence of patterns in the forecast

C) the number of variables to which the forecast is related

D) all of the above

85) __________ is a gradual, long-term, up or down movement of demand.

A) Seasonal pattern

B) Cycle

C) Trend

D) Prediction

86) A __________ is an up-and-down repetitive movement that repeats itself over a time span of more than 1 year.

A) prediction

B) seasonal pattern

C) trend

D) cyclical pattern

87) __________ methods are the most common type of forecasting method for the long-term strategic planning process.

A) Regression

B) Qualitative

C) Time series

D) all of the above

88) __________ is a category of statistical techniques that uses historical data to predict future behavior.

A) Qualitative methods

B) Regression

C) Time series

D) Quantitative methods

89) __________ use management judgment, expertise, and opinion to make forecasts.

A) Qualitative methods

B) Regression

C) Time series

D) Quantitative methods

90) The __________ is a procedure for developing a consensus forecast about what will occur in the future.

A) Delphi method

B) quantitative method

C) regression equation

D) time series forecasting method

91) __________ has become increasingly crucial to compete in the modern international business environment.

A) The Delphi method

B) Technological forecasting

C) Prediction

D) all of the above

92) Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

A) Trend only

B) Trend plus seasonal

C) Trend plus random

D) Cyclical only

E) None of the above

93) Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

A) Trend only

B) Trend plus seasonal

C) Trend plus random

D) Seasonal only

E) None of the above

94) Consider the following graph of sales.

Which of the following characteristics is exhibited by the data?

A) Trend only

B) Trend plus seasonal

C) Trend plus irregular

D) Seasonal

E) None of the above

95) __________ moving averages react more slowly to recent demand changes than do __________ moving averages.

A) Longer-period, shorter-period

B) Shorter-period, longer-period

C) Longer-period, longer-period

D) Shorter-period, shorter-period

96) __________ is good for stable demand with no pronounced behavioral patterns.

A) Longer-period moving average

B) Shorter-period moving average

C) Moving average

D) Weighted moving average

97) __________ methods assume that what has occurred in the past will continue to occur in the future.

A) Time series

B) Regression

C) Quantitative

D) Qualitative

98) In exponential smoothing, the closer alpha is to __________, the greater the reaction to the most recent demand.

A) -1

B) 0

C) 1

D) -1 or 1

99) In adjusted exponential smoothing, the closer beta is to __________, the stronger a trend is reflected.

A) -1 / 1

B) -1

C) 0

D) 1

100) __________ is a linear regression model relating demand to time.

A) Linear trend

B) Linear regression

C) Forecast demand

D) Linear equation

101) Which of the following possible values of alpha would cause exponential smoothing to respond the most slowly to sudden changes in forecast errors?

A) .01

B) .10

C) .20

D) .50

E) .90

102) Given an actual demand of 59, a previous forecast of 64, and an alpha of .3, what would the forecast for the next period be using simple exponential smoothing?

A) 36.9

B) 57.5

C) 60.5

D) 62.5

E) 65.5

103) __________ is the difference between the forecast and actual demand.

A) Forecast mistake

B) Forecast error

C) MAD

D) Forecast accuracy

104) __________ is absolute error as a percentage of demand.

A) Cumulative error

B) MAD

C) MAPD

D) Average error

105) __________ indicates a forecast is biased low.

A) Large :

B) Large –

C) Large +

D) Large x

106) __________ indicates a forecast is biased high.

A) Large +

B) Large –

C) Large :

D) Large x

107) __________ is a measure of the strength of the relationship between independent and dependent variables.

A) Correlation

B) Linear regression

C) Coefficient of determination

D) Regression

108) __________ is the percentage of the variation in the dependent variable that results from the independent variable.

A) Regression

B) Coefficient of determination

C) Correlation

D) Linear regression

109) Coefficient of determination is the percentage of the variation in the __________ variable that results from the __________ variable.

A) dependent, dependent

B) independent, dependent

C) dependent, independent

D) independent, independent

110) Consider the following demand and forecast.

Period Demand Forecast

1 7 10

2 12 15

3 18 20

4 22

If MAD = 2, what is the forecast for period 4?

A) 19

B) 20

C) 21

D) 22

E) none of the above

Given the following data on the number of pints of ice cream sold at a local ice cream store for a 6-period time frame:

111) Use a 2-period moving average to forecast demand for period 7.

A) 227.5

B) 275

C) 280

D) 290

E) none of the above

112) Use a 3-period moving average to forecast demand for period 7.

A) 283.33

B) 280

C) 290

D) 310

E) none of the above

113) If the forecast for period 5 is equal to 275, use exponential smoothing to compute a forecast for period 7 if α = .40.

A) 273

B) 277

C) 267.8

D) 286.2

E) none of the above

The following data represents quarterly sales of lawnmowers.

Year Quarter 1 Quarter 2 Quarter 3 Quarter 4

1 150 140 190 165

2 160 148 210 175

114) What is the seasonal index for the third quarter? (Round to the nearest hundredth.)

A) .20

B) .22

C) .26

D) .30

E) none of the above

115) What is the seasonal index for the fourth quarter? (Round to the nearest hundredth.)

A) .20

B) .23

C) .25

D) .30

E) none of the above

116) The manager of “Skis 4 U” is preparing a forecast for February of 2011. Demand exhibits both trend and seasonality. The trend equation for monthly demand is y = 4375 + 80t, where t = 1 for January 2009. The seasonal index for February is 1.25. The forecast for February is

A) 4575

B) 4583

C) 5719

D) 5729

E) none of the above

117) Given forecast errors of 6, 4, 0 and -2, what is the mean absolute deviation?

A) 2

B) 3

C) 4

D) 2.67

E) none of the above

118) Given forecast errors of 6, 4, 0 and -2, what is the mean squared error?

A) 14

B) 18.67

C) 16

D) 12

E) none of the above

Introduction to Management Science, 10e (Taylor)

Chapter 16 Inventory Management

1) Independent demand items are used internally to produce a final product.

2) Dependent demand items are final products demanded by an external customer.

3) Inventory costs include carrying, ordering, and shortage costs.

4) The purpose of inventory management is to determine how much and when to order.

5) In a continuous inventory system, a constant amount is ordered when inventory declines to a predetermined level.

6) In a periodic inventory system, a constant amount is ordered when inventory declines to a predetermined level.

7) The EOQ is the optimal order quantity that will minimize total carrying costs.

8) Assumptions of the basic EOQ model include constant demand, no shortages, constant lead time, and gradual usage.

9) The non-instantaneous receipt model applies only to manufacturing.

10) The EOQ model with shortages does not allow backorders.

11) The basic EOQ model plays no role in determining order sizes in the presence of quantity discounts.

12) Quantity discounts are always evaluated with carrying cost as a percentage of price.

13) The service level is the probability that the inventory available during lead time will meet demand.

14) If service level is 50%, then safety stock is equal to 50% of lead time demand.

15) The reorder point is the date when a new order should be placed.

16) If lead time and demand are constant, safety stock is equal to demand multiplied by lead time.

17) Periodic inventory systems normally require smaller safety stock than a continuous inventory system.

18) Techniques for inventory analysis are widely used to analyze other types of problems.

19) Carrying costs include storage cost, interest and depreciation.

20) Ordering costs include transportation, shipping and inspection.

21) Shortage costs include loss of customer goodwill.

22) If a business frequently runs out of inventory their service levels are negative.

23) __________ demand items are generally final products demanded by customers.

24) The purpose of inventory management is to determine __________ and __________ to order.

25) __________ demand items are used internally to produce a final product.

26) Inventory __________ costs include storage cost and the cost of capital.

27) Inventory __________ costs include transportation and inspection.

28) A __________ occurs when customer demand cannot be met because of insufficient inventory.

29) In a __________ inventory system, a constant amount is ordered when inventory declines to a predetermined level.

30) In a __________ inventory system an order is placed for a variable amount after a fixed passage of time

31) __________ is the optimal order quantity that will minimize the total inventory costs.

32) A __________ occurs when an item is out of stock and is sold to the customer when a shipment arrives.

33) Costs involved in a typical inventory model include __________ and __________.

34) In the basic EOQ model, as the size of the order increases, the annual __________ cost decreases.

35) In the basic EOQ model, as the size of the order increases, the annual __________ cost increases.

36) In the quantity discounts model, the __________ must be included in the total cost calculation. This variable is not included in the basic EOQ model.

37) If all the variables are held constant, the total inventory cost in a non-instantaneous receipts model is __________ than the total cost in the basic EOQ model.

38) The __________ determines when an order should be placed for a continuous review inventory system.

39) __________ is inventory that is used to help protect against stockouts.

40) __________ is the probability that the inventory available during the lead time will meet demand.

41) If lead time and demand are constant then __________ is zero.

42) For the __________ inventory system, Q, the quantity ordered, can vary.

43) The EOQ model is __________ , or resistant to errors in the cost estimates and demand.

44) __________ costs and _________ costs react inversely to each other in response to an increase in order size.

45) The basic EOQ model assumes that __________ is known with certainty and is relatively constant over time.

46) In the basic EOQ model, if D=80 per month, Co=$13, and Cc=$11 per unit per month, what is the EOQ?

47) In the basic EOQ model, if D=40 per month, Co=$9, and Cc=$8 per unit per month, what is the EOQ?

48) In the basic EOQ model, if D=100 per month, Co=$20, and Cc=$15 per unit per month, what is the EOQ?

49) In the basic EOQ model, if annual demand is 50 units, carrying cost is $2 per unit per year, and ordering cost is $15, what is the EOQ?

A company produces item Y, and uses the basic EOQ model for managing its inventory. Lead time to obtain item Y is two weeks. Demand is normally distributed with a mean of 400 units per week and a standard deviation of 40 units per week. The desired service level is 98.5%. The ordering cost is $20, and carrying cost is 20% of the items cost, which is $10.

50) Determine the order quantity for product Y. (Assume 52 weeks of operation per year.)

51) Determine the annual setup cost and the annual carrying cost for product Y. (Assume 52 weeks of operation per year.)

52) Determine the total annual inventory cost for product Y. Include the item cost in your calculations. (Assume 52 weeks of operation per year.)

53) Determine the reorder point for product Y.

The injection molding department of Alver Inc. uses an average of 40 pounds of a special powder per day. The plant operates 250 days per year. The daily usage of the powder is normally distributed with a standard deviation of 5 pounds per day. The lead time to obtain the powder from a supplier is 9 days. The annual holding cost is $2. per unit and the cost of ordering the powder is $50.

54) How many units should Alver Inc. order in order to minimize annual ordering and carrying cost?

55) How many orders will be placed each year?

56) Determine the reorder point for a service level of 97%.

The daily sales of a peanut butter at Power’s Grocery are normally distributed, with a mean of 12 jars and a standard deviation of 4. The manager checks the inventories on shelves and places an order every three days. Delivery lead time is two days.

57) How much safety stock of peanut butter should they have for a 99% service level?

58) If there are 4 jars on the shelf when an order is placed, how much should the store order?

59) The daily demand for a product is normally distributed with a mean of 80 and a standard deviation of 8. Constant lead time is 4 days. The cost of placing an order is $20. The item costs $8 and the carrying rate per year is 10% of the item cost. Determine the economic order quantity.

60) In a non-instantaneous receipt model, daily demand is 55 units and daily production is 120 units, Co=$70 and Cc=$4 per unit/year. The production facility operates 300 days per year. What is the maximum inventory level?

61) In a non-instantaneous receipt model, daily demand is 55 units and daily production is 120 units, Co=$70 and Cc=$4 per unit/year. The production facility operates 300 days per year. What is the optimal order quantity?

62) A product has an annual demand of 3600 units. Unit cost for this product is $3. Set up cost is $20 and the inventory carrying rate as a percent of the unit cost is 25%. The product is produced in-house where the daily production rate is 50 units. Assume 360 working days per year and determine the economic production quantity.

63) A product has an annual demand of 3600 units. Unit cost for this product is $3. Set up cost is $20 and the inventory carrying rate as a percent of the unit cost is 25%. The product is produced in-house where the daily production rate is 50 units. Assume 360 working days per year and determine the annual ordering cost and carrying cost.

64) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the total annual shortage cost?

65) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the optimal order quantity?

66) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the annual ordering cost?

67) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the optimal stock out level?

68) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the annual carrying cost?

69) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and CS =$25, what is the total minimum annual inventory cost?

A company distributes repair parts for high end appliances. The annual demand is 81,000 and the company operates 300 days per year. The annual carrying cost is 20% of the item cost, which is $500. The ordering cost is estimated at $60 and the shortage cost is $150.

70) Determine the optimal order quantity.

71) Determine the optimal shortage level.

72) How many order per year will they place?

73) What is the maximum inventory level?

74) Determine the annual shortage cost.

75) If the daily demand is 30 and the lead time in days is 3, what is the reorder point?

76) If the daily demand is 67 and the lead time in days is 2, what is the reorder point?

77) If the daily demand is 25 and the lead time in days is 4, what is the reorder point?

78) Sonny Lawler’s law office uses EOQ models to manage their office supplies. They’ve been ordering ink refills for their printers in quantities of 60 units. The firm estimates carrying cost at 40% of the $10 unit cost and that annual demand is about 240 units per year. The assumptions of the basic EOQ model are thought to apply. For what value of ordering cost would its action be optimal?

79) Dana Swor’s Dream Store sells weight loss products. Her best-selling item, an energy booster and fat burning pill, has an annual demand of 400 units. Ordering cost is $40 and carrying cost is $5 per unit year. How many units should she order to minimize total inventory costs?

80) Annual demand for a paperback dictionary at the bookstore is 1200 units. Ordering costs are 350, carrying costs are $6 per unit per year, and the lead time is 9 days. The bookstore is open for 300 days of the year. What is the reorder point ?

81) If average demand for an inventory item is 200 units per day, lead time is 3 days, and safety stock is 100 units, what is the reorder point?

82) If average demand for an inventory item is 180 units per day, lead time is 5 days, and safety stock is 90 units, what is the reorder point?

83) Annual demand for notecards at Suzie’s Stationery shop is 10,000 units. Deliveries take about 5 working days and Suzie operates 300 days per year. Calculate the reorder point for the notecards that she stocks.

A bakery uses an average of 60 ounces of organic orange juice daily. Demand is normally distributed with a standard deviation of 15 ounces. The bakery places orders every seven days. The lead time for delivery of the juice is three days.

84) Compute the safety stock required to achieve a 98% service level

85) If the bakery has 190 ounces at the time an order is placed, how much should be ordered?

86) The daily demand for a product is normally distributed with a mean of 80 and a standard deviation of 8. Constant lead time is 4 days. The cost of placing an order is $20. The item costs $8 and the carrying rate per year is 10% of the item cost. Determine the reorder point to satisfy 90% of the orders.

87) A bakery’s use of corn syrup is normally distributed with a mean of 50 gallons per day and a standard deviation of 5 gallons per day. Lead time for delivery of the syrup is normal with a mean of 4 days and a standard deviation of 2 days. The manager wants a service level of 99 percent. Calculate the reorder point.

88) A bakery’s use of corn syrup is normally distributed with a mean of 50 gallons per day and a standard deviation of 5 gallons per day. Lead time for delivery of the syrup is normal with a mean of 4 days and a standard deviation of 2 days. The manager reorders when his inventory drops to 300 gallons. What cycle service level is implied by this policy?

89) A manager has just received a revised price schedule from a vendor. What order quantity should the manager use in order to minimize total costs? Annual Demand is 120 units, ordering cost is $10, and annual carrying cost is $1 per unit.

Quantity Unit Price

1-59 $15

60-99 $14

100 or more $13

90) An office manager uses 400 boxes of file folders per year. The price is $8.50 per box for an order size of 199 boxes or less, $8.00 per box for orders of 200 to 799 boxes, and $7.50 per box for an order of 800 or more boxes. Carrying cost is 20 percent of the price of the product and ordering costs are $80. What order quantity minimizes total annual cost?

91) __________ demand items are used internally to produce a final product.

A) Independent

B) Dependent

C) Assumed

D) Internal

E) Integrated

92) __________ demand items are final products demanded by an external customer.

A) Assumed

B) Dependent

C) Independent

D) External

E) Integrated

93) Inventory costs include

A) carrying

B) ordering

C) shortage costs

D) all of the above

94) The purpose of inventory management is to determine

A) timing and cost of orders

B) quantity and cost of orders

C) timing and quantity of orders

D) ordering and carrying costs

95) A keyboard costs $1,000, and the annual holding cost is 25%. Annual demand is 10,000 units, and the order cost is $150 per order. What is the approximate economic order quantity?

A) 16

B) 70

C) 110

D) 183

96) In a(n) __________ inventory system a constant amount is ordered when inventory declines to a predetermined level.

A) optimal

B) economic

C) periodic

D) continuous

97) In a(n) __________ inventory system, an order is placed for a variable amount after a fixed passage of time.

A) periodic

B) continuous

C) optimal

D) economic

98) EOQ is a(n) __________ inventory system.

A) periodic

B) continuous

C) optimal

D) economic

99) EOQ is the optimal order quantity that will __________ total inventory costs.

A) maximize

B) minimize

C) steady

D) maintain

E) improve

100) Assumptions of the EOQ model include

A) constant demand and no shortages

B) constant lead time

C) instantaneous order receipt

D) all of the above

101) In the basic EOQ model, if lead time increases from 5 to 10 days , the EOQ will

A) double

B) increase, but not by double the amount

C) remain the same

D) decrease

102) The economic production quantity is 500 units (units are delivered to the user department as they come off the production line). If the firm decides to buy this item from an outside supplier rather than producing it, the economic purchase quantity would probably be (assume that inventory costs of production and purchasing an item are the same):

A) less than 500 units

B) more than 500 units

C) 500 units

D) the direction of change in quantity cannot be determined without additional information

103) If order quantity is increased, annual holding cost __________, annual order cost __________, and change in annual total cost __________.

A) decreases, increases, is positive

B) decreases, increases, can not be determined

C) decreases, decreases, can not be determined

D) increases, decreases, is negative

E) increases, decreases, can not be determined

104) The EOQ minimizes total __________ cost.

A) inventory

B) purchase

C) ordering

D) marketing

E) carrying

105) In an EOQ model, as the carrying cost increases, the order quantity :

A) increases.

B) decreases.

C) remains the same

D) cannot be determined

106) In the basic EOQ model, if D=60 per month, S=$12, and H=$10 per unit per month, what is the EOQ?

A) 11

B) 12

C) 13

D) 14

107) In the basic EOQ model, if annual demand is 50, carrying cost is $2 per unity per year, and ordering cost is $15, what is the EOQ?

A) 27.39

B) 26.39

C) 25.39

D) 24.39

E) 22.39

108) The daily sales of a peanut butter at Power’s Grocery are normally distributed, with a mean of 12 jars and a standard deviation of 4. The manager checks the inventories on shelves and places an order every three days. Delivery lead time is two days and they carry 21 jars for safety stock. If there are 4 jars on the shelf when an order is placed, how much should the store order?

A) 77

B) 81

C) 32

D) 36

E) cannot be determined from the information provided

109) In a non instantaneous receipt model, daily demand is 55 units and daily production is 120 units, Co=$70 and Cc=$4 per unit per year. What is the maximum inventory level? (Assume that the facility is open 365 days per year)

A) 616.9

B) 618.4

C) 620.3

D) 622.9

E) 625.5

110) A firm is presently purchasing an itme for inventory using the basic EOQ model. They plan on making the product themselves and will be using the EOQ model based on non-instantaneous receipt of inventory. If everything else stays the same, what changes should the firm expect?

A) EOQ decreases

B) Total relevant inventory (annual setup and annual carrying) cost increase

C) Average inventory level decreases

D) Reorder point increases

E) Safety stock increases

111) When using the EOQ Formula with non-instantaneous production, as the demand rate (D) increases more than the rate of production, the EOQ:

A) increases

B) decreases

C) remains the same

D) cannot be determined

112) The diagram above represents which type of inventory model?

A) Economic Order Quantity (EOQ)

B) EOQ with Noninstantaneous Receipt

C) Economic Production Quantity

D) Fixed Period Model

E) none of the above

113) The slope of the line labeled “B” in the diagram above is:

A) Order rate

B) Rate of inventory demand

C) Production rate

D) Shipping rate

E) Production rate minus rate of inventory demand

114) The slope of the line labeled “A” in the diagram below is:

A) Order rate

B) Rate of inventory demand

C) Production rate

D) Shipping rate

E) Production rate minus rate of inventory demand

115) The slope of the line labeled “C” in the diagram below is:

A) Order rate

B) Rate of inventory demand

C) Production rate

D) Shipping rate

E) Production rate minus rate of inventory demand

116) The interval labeled “E” in the diagram below is:

A) Production cycle

B) Production run length

C) Shipping lead time

D) Inventory fill rate

117) The interval labeled “D” in the diagram below is:

A) Production cycle

B) Production run length

C) Shipping lead time

D) Inventory fill rate

118) A product has demand during lead time of 100 units, with a standard deviation of 25 units. What safety stock (approximately) provides a 95% service level?

A) 41

B) 55

C) 95

D) 140

119) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and Cs =$25, what is the total annual shortage cost?

A) 294.72

B) 296.51

C) 298.53

D) 299.17

E) 285.91

120) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and Cs =$25, what is the order quantity?

A) 394.72

B) 285.91

C) 495.74

D) 296.51

E) 456.34

121) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and Cs =$25, what is the optimal stock out level?

A) 96.44

B) 102.36

C) 108.44

D) 114.64

E) 121.43

122) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and Cs =$25, what is the annual carrying cost?

A) 2711.09

B) 1059.03

C) 1355.55

D) 296.51

E) 495.74

123) Given an EOQ model with shortages in which annual demand is 4200 units, Co=$160, Cc=$7 per unit per year and Cs =$25, what is the annual ordering cost?

A) 2711.09

B) 1059.03

C) 1355.55

D) 296.51

E) 495.74

The manager of the Quick Stop Corner Convenience Store (which is open 360 days per year) sells four cases of Stein soda each day (1,440 cases per year). Order costs are $8.00 per order. The lead time for an order is three days. Annual holding costs are equal to $57.60 per case.

124) If the manager orders 16 cases each time she placed an order, what is the average inventory level?

A) 4 cases

B) 8 cases

C) 12 cases

D) 20 cases

E) 16 cases

125) If the manager orders 16 cases each time she placed an order, how many orders would she place in a year?

A) 10

B) 22.5

C) 50

D) 72

E) 90

126) What is the optimal order quantity for Stein soda?

A) 4 cases

B) 8 cases

C) 12 cases

D) 20 cases

E) 16 cases

127) What is the reorder point for Stein soda?

A) 4 cases

B) 8 cases

C) 12 cases

D) 20 cases

E) 16 cases

128) If the daily demand is 50 and the lead time in days is 4, what is the reorder point?

A) 200

B) 220

C) 240

D) 260

E) 300

129) If the daily demand is 40 and the level time in days is 4, what is the reorder point?

A) 120

B) 140

C) 160

D) 180

E) 200

130) If the daily demand is 10 and the level time in days is 8, what is the reorder point?

A) 50

B) 60

C) 70

D) 80

E) 90

131) If average demand for an inventory item is 200 units per day, lead time is 3 days, and safety stock is 100 units, what is the reorder point?

A) 300

B) 500

C) 600

D) 700

E) 800

132) Ruby owns a small café and uses a linen supplier for her tablecloths. Whenever she needs more tablecloths, she calls the supplier. She uses an average of 12 tablecloths a day with a standard deviation of 3 tablecloths. Lead time is a constant 2 days. If Ruby is willing to accept a 5% stockout risk, what is the reorder point, rounded to the nearest tablecloth? Assume demand is normally distributed.

A) 28

B) 31

C) 34

D) 42

E) none of the above

133) The service level is the probability that

A) the inventory will meet demand

B) the inventory available during lead time will meet demand

C) the inventory available during lead time will not meet demand

D) the inventory will not meet demand

134) What service level results in zero safety stock in reorder point calculations?

A) 0%

B) 33%

C) 50%

D) 100%

E) 80%

135) The __________ is the probability that the inventory available during lead time will meet demand.

A) service level

B) inventory level

C) review period

D) reorder point

E) maximum inventory level

136) If annual demand equals 1000 units, the number of working days per year is 250, Co=$300 per order and Cc=$3 per unit, how many days are between orders in the basic EOQ model?

A) 105

B) 110

C) 112

D) 115

E) 120

137) Annual demand for a paperback dictionary at a bookstore is 1200 units. The order cost is $350 and the carrying cost is $6 per unit per year. The number of working days per year is 365. What is the reorder point? Assume that the basic EOQ model is applicable.

A) 29.2

B) 29.4

C) 29.6

D) 29.8

E) 30.1

138) As service level increases, expected number of stock outs __________ and safety stock __________.

A) increase, increases

B) decrease, decreases

C) decrease, increases

D) increase, decreases

139) Adding 1.5 standard deviations of safety stock to the average demand during lead time will result in a service level of approximately:

A) 50%

B) 68.4%

C) 84.1%

D) 93.3%

E) 97.7%

140) A periodic inventory system

A) uses fixed order sizes at variable time intervals

B) normally requires a larger safety stock

C) cannot be used if demand is variable

D) is used to periodically manage inventory

141) Demand for a product is constant but lead time is variable. The demand during the lead time is normally distributed with a mean of 40 and a standard deviation of 4. They computed a reorder point of 45 units Approximately what service level is being used?

A) 85%

B) 90%

C) 95%

D) 98%

E) none of the above

142) The economic order quantity is especially sensitive to which of the following?

A) ordering cost

B) carrying cost

C) annual demand

D) lead time

E) none of the above

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