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Chapter 02A Linear Programming using the Excel Solver True / False Questions 1. Linear programming is a single-objective model, meaning that it has a single objective function to be maximized or minimized. TRUE

Level: Easy

2. Linear programming is useless when resources are plentiful relative to demand. TRUE

Level: Medium

3. A common application of linear programming is in materials handling. TRUE

Level: Easy

4. The decision variables in a linear programming model must be non-negative. TRUE

Level: Easy

5. Linear programming is useful when resources are unlimited. FALSE

Level: Easy

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6. Each term in a linear program's objective function should be expressed in the same units. TRUE

Level: Medium

7. Excel Solver solutions include a sensitivity report. This report gives information on making changes in objective function coefficients. TRUE

Level: Easy

8. The objective function in a linear programming model can be nonlinear. FALSE

Level: Easy

9. In the conventional formulation of a linear programming model we will see all of the decision variables on the right-hand-side of a constraint and a constant value on the left-handside. FALSE

Level: Easy

10. In the formulation of a linear programming model we expect to see a requirement on all the decision variables to be either zero or some positive value. TRUE

Level: Easy

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11. Finding the optimal way to use aircraft and their operating crews to provide transportation services to customers to be moved between different locations is one kind of problem that can be solved by linear programming. TRUE

Level: Easy

12. Finding the optimal location of a new plant by evaluating shipping costs between alternative locations and supply and demand sources is one kind of problem that can be solved by linear programming. TRUE

Level: Easy

13. Minimizing the amount of scrap material generated by cutting steel, leather or fabric from a roll or sheet of stock material is one kind of problem that cannot be solved by linear programming. FALSE

Level: Easy

14. Finding the optimal combination of products to stock in a retail store cannot be solved using linear programming. FALSE

Level: Easy

15. Finding the optimal product mix where several products have different costs and resource requirements cannot be solved using linear programming. FALSE

Level: Easy

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16. Finding the optimal routing for a product that must be processed sequentially through several machine centers, with each machine in a center having its own cost and output characteristics cannot be solved using linear programming. FALSE

Level: Easy

17. If products and resources can not be subdivided into fractions, the condition of divisibility is violated. In these cases, a modification of linear programming called integral programming can be used. FALSE

Level: Medium

18. If products and resources can not be subdivided into fractions, the condition of divisibility is violated. In these cases, a modification of linear programming called integer programming can be used. TRUE

Level: Medium

19. Linear programming is gaining wide acceptance in many industries due to the availability of detailed operating information and the interest in optimizing processes to reduce cost. TRUE

Level: Easy

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Multiple Choice Questions 20. Which of the following is an essential condition in a situation for linear programming to be useful? A. Nonlinear constraints B. Bottlenecks in the objective function C. Homogeneity D. Uncertainty E. Competing objectives

Level: Easy

21. Which of the following is not an essential condition in a situation for linear programming to be useful? A. An explicit objective function B. Uncertainty C. Linearity D. Limited resources E. Divisibility

Level: Easy

22. Which of the following is not a common application of linear programming in operations and supply management? A. Waiting time analysis B. Product planning C. Product routing D. Process control E. Service productivity analysis

Level: Easy

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23. Which of the following is a common application of linear programming in operations and supply management? A. Cycle counting analysis B. Cost of quality studies C. Cost allocation studies D. Plant location studies E. Product design decisions

Level: Medium

24. There are other related mathematical programming techniques that can be used instead of linear programming if the problem has a unique characteristic. If the problem has multiple objectives we should use which of the following methodologies? A. Goal programming B. Orthogonal programming C. Integer programming D. Multiplex programming E. Dynamic programming

Level: Easy

25. There are other related mathematical programming techniques that can be used instead of linear programming if the problem has a unique characteristic. If the problem is best solved in stages or time frames we should use which of the following methodologies? A. Goal programming B. Temporal programming C. Integer programming D. Genetic programming E. Dynamic programming

Level: Easy

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26. There are other related mathematical programming techniques that can be used instead of linear programming if the problem has a unique characteristic. If the problem prevents divisibility of products or resources we should use which of the following methodologies? A. Goal programming B. Primary programming C. Integer programming D. Unit programming E. Dynamic programming

Level: Easy

27. A company wants to determine how many units of each of two products, A and B, they should produce. The profit on product A is $50 and the profit on product B is $45. Applying linear programming to this problem, which of the following is the objective function if the firm wants to make as much money as possible? A. Minimize Z = 50 A + 45 B B. Maximize Z = 50 A + 45 B C. Maximize Z = A + B D. Minimize Z = A + B E. Maximize Z = A/45B + B/50A

Level: Easy

28. An agribusiness company mixes and sells chicken feed to farmers. The costs of the chicken feed ingredients vary throughout the chicken feeding season but the selling price of chicken feed is independent of the ingredients. On August 1, management needs to know how many units of each of three grains (Q, R and S) should be included in their chicken feed in order to produce the product most economically. The cost of each grain is, for a unit of Q, $30; for a unit of R, $37; and for a unit of S, $78. Applying linear programming to this problem, which of the following is the objective function? A. Minimize Z = 30 Q + 37 R + 78 S B. Maximize Z = 30 Q + 37 R + 78 S C. Minimize Z = (Q x R x S)/3 D. Minimize Z = Q + R + S E. Maximize Z = Q + R + S

Level: Easy

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29. Apply linear programming to this problem. A firm wants to determine how many units of each of two products (products D and E) they should produce to make the most money. The profit in the manufacture of a unit of product D is $100 and the profit in the manufacture of a unit of product E is $87. The firm is limited by its total available labor hours and total available machine hours. The total labor hours per week are 4,000. Product D takes 5 hours per unit of labor and product E takes 7 hours per unit. The total machine hours are 5,000 per week. Product D takes 9 hours per unit of machine time and product E takes 3 hours per unit. Which of the following is one of the constraints for this linear program? A. 5 D + 7 E =< 5,000 B. 9 D + 3 E => 4,000 C. 5 D + 7 E = 4,000 D. 5 D + 9 E =< 5,000 E. 9 D + 3 E =< 5,000

Level: Medium

30. Apply linear programming to this problem. A firm wants to determine how many units of each of two products (products X and Y) they should produce in order to make the most money. The profit from making a unit of product X is $190 and the profit from making a unit of product Y is $112. The firm has a limited number of labor hours and machine hours to apply to these products. The total labor hours per week are 3,000. Product X takes 2 hours of labor per unit and Product Y takes 6 hours of labor per unit. The total machine hours available are 750 per week. Product X takes 1 machine hour per unit and Product Y takes 5 machine hours per unit. Which of the following is one of the constraints for this linear program? A. 1 X + 5 Y =< 750 B. 2 X + 6 Y => 750 C. 2 X + 5 Y = 3,000 D. 1 X + 3 Y =< 3,000 E. 2 X + 6 Y =>3,000

Level: Medium

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31. Apply linear programming to this problem. David and Harry operate a discount jewelry store. They want to determine the best mix of customers to serve each day. There are two types of customers for their store, retail (R) and wholesale (W). The cost to serve a retail customer is $70 and the cost to serve a wholesale customer is $89. The average profit from either kind of customer is the same. To meet headquarters' expectations, they must serve at least 8 retail customers and 12 wholesale customers daily. In addition, in order to cover their salaries, they must at least serve 30 customers each day. Which of the following is one of the constraints for this model? A. 1 R + 1 W =< 8 B. 1 R + 1 W => 30 C. 8 R + 12 W => 30 D. 1 R => 12 E. 20 x (R + W) =>30

Level: Medium

32. Apply linear programming to this problem. A one-airplane airline wants to determine the best mix of passengers to serve each day. Their airplane seats 25 people and flies 8 one-way segments per day. There are two types of passengers: first class (F) and coach (C). The cost to serve each first class passenger is $15 per segment and the cost to serve each coach passenger is $10 per segment. The marketing objectives of the airplane owner are to carry at least 13 first class passenger-segments and 67 coach passenger-segments each day. In addition, in order to break even, they must at least carry a minimum of 110 total passenger segments each day. Which of the following is one of the constraints for this linear program? A. 15 F + 10 C => 110 B. 1 F + 1 C => 80 C. 13 F + 67 C => 110 D. 1 F => 13 E. 13 F + 67 C =< (80/200)

Level: Medium

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33. An objective function in a linear program can be which of the following? A. A maximization function B. A nonlinear maximization function C. A quadratic maximization function D. An uncertain quantity E. A divisible additive function

Level: Easy

34. The number of constraints allowed in a linear program is which of the following? A. Less than 5 B. Less than 72 C. Less than 512 D. Less than 1,024 E. Unlimited

Level: Easy

35. The number of decision variables allowed in a linear program is which of the following? A. Less than 5 B. Less than 72 C. Less than 512 D. Less than 1,024 E. Unlimited

Level: Easy

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Short Answer Questions 36. Formulate and solve the following linear program. A firm wants to determine how many units of each of two products (products D and E) they should produce to make the most money. The profit in the manufacture of a unit of product D is $100 and the profit in the manufacture of a unit of product E is $87. Although the firm can readily sell any amount of either product, it is limited by its total labor hours and total machine hours available. The total labor hours per week are 4,000. Product D takes 5 hours of labor per unit and product E takes 7 hours of labor per unit. The total machine hours are 5,000 per week. Product D takes 9 hours of machine time per unit and product E takes 3 hours of machine time per unit. Write the constraints and the objective function for this problem, solve for the best mix of product D and E and report the maximum value of the objective function? Objective function: Maximize Z = $100 D + $87 E Subject to: 5 D + 7 E <= 4,000 9 D + 3 E <= 5,000 Solution: D = 479.17 E = 229.17 Z = $67,854

Level: Hard

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37. Formulate and solve the following linear program. A firm wants to determine how many units of each of two products (products X and Y) they should produce in order to make the most money. The profit from making a unit of product X is $100 and the profit from making a unit of product Y is $80. Although the firm can readily sell any amount of either product, it is limited by its total labor hours and total machine hours available. The total labor hours per week are 800. Product X takes 4 hours of labor per unit and Product Y takes 2 hours of labor per unit. The total machine hours available are 750 per week. Product X takes 1 machine hour per unit and Product Y takes 5 machine hours per unit. Write the constraints and the objective function for this problem, solve for the best mix of product X and Y and report the maximum value of the objective function? Objective function: Maximize Z = $100 X + $80 Y Subject to: 04 X + 2 <= 800 1 X + 5 Y <= 750 Solution: X = 138.89 Y = 122.22 Z = $23,666.67

Level: Hard

Essay Questions

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38. Two of the five essential conditions for linear programming to pertain to a real world problem are linearity and homogeneity. Discuss why these are difficult to achieve in the real world and how they might limit your confidence in the solution? These five essential conditions are described on page 38 of the text. Linearity is unlikely to be precise from one unit of resource or output to the next and, further, the trade-off between one output unit to another is unlikely to be identical over the entire range of outputs. Scale economies and learning effects are among the concepts that mitigate against pure linearity. Similarly, homogeneity, defined as resources and products being precisely alike is an illusion. The key here is not that these conditions pertain precisely, but that we can assume that they pertain so that a tractable answer can arise from the analysis. Putting that answer back into the real world requires that the analyst make some judgment that the assumptions of linearity and homogeneity are met nearly enough that the results are a reasonable guide to action.

Level: Medium

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39. What is a "shadow price"? A shadow price is the amount the value of the objective function (target cell) would change if one more or one less unit of resource were available up to the limits imposed by the "allowable increase" or the "allowable decrease". See text, page 48-49.

Level: Medium

40. What is the shadow price of a non-binding constraint? Why is this so? Zero. Having one more or one less unit of a resource that is not completely used in the optimal solution will have no effect on the solution and therefore will have a zero shadow price. This is from the discussion of a problem in the text on page 52.

Level: Easy

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