Chapter 1 - Introduction True / False 1. The process of decision making is more limited than that of problem solving. a. True b. False ANSWER: True POINTS: 1 TOPICS: Problem solving and decision making 2. The terms 'stochastic' and 'deterministic' have the same meaning in quantitative analysis. a. True b. False ANSWER: False POINTS: 1 TOPICS: Model development 3. The volume that results in marginal revenue equaling marginal cost is called the break-even point. a. True b. False ANSWER: False POINTS: 1 TOPICS: Problem solving and decision making 4. Problem solving encompasses both the identification of a problem and the action to resolve it. a. True b. False ANSWER: True POINTS: 1 TOPICS: Problem solving and decision making 5. The decision making process includes implementation and evaluation of the decision. a. True b. False ANSWER: False POINTS: 1 TOPICS: Problem solving and decision making 6. The most successful quantitative analysis will separate the analyst from the managerial team until after the problem is fully structured. a. True b. False ANSWER: False POINTS: 1 TOPICS: Quantitative analysis 7. The value of any model is that it enables the user to make inferences about the real situation.
Chapter 1 - Introduction a. True b. False ANSWER: True POINTS: 1 TOPICS: Model development 8. Uncontrollable inputs are the decision variables for a model. a. True b. False ANSWER: False POINTS: 1 TOPICS: Model development 9. The feasible solution is the best solution possible for a mathematical model. a. True b. False ANSWER: False POINTS: 1 TOPICS: Model solution 10. A company seeks to maximize profit subject to limited availability of man-hours. Man-hours is a controllable input. a. True b. False ANSWER: False POINTS: 1 TOPICS: Model development 11. Frederick Taylor is credited with forming the first MS/OR interdisciplinary teams in the 1940's. a. True b. False ANSWER: False POINTS: 1 TOPICS: Introduction 12. To find the choice that provides the highest profit and the fewest employees, apply a single-criterion decision process. a. True b. False ANSWER: False POINTS: 1 TOPICS: Problem solving and decision making 13. The most critical component in determining the success or failure of any quantitative approach to decision making is problem definition. a. True b. False Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction ANSWER: True POINTS: 1 TOPICS: Quantitative analysis 14. The first step in the decision making process is to identify the problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Introduction 15. All uncontrollable inputs or data must be specified before we can analyze the model and recommend a decision or solution for the problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Quantitative analysis 16. In quantitative analysis, the optimal solution is the mathematically-best solution. a. True b. False ANSWER: True POINTS: 1 TOPICS: Quantitative analysis 17. If you are deciding to buy either machine A, B, or C with the objective of minimizing the sum of labor, material and utility costs, you are dealing with a single-criterion decision. a. True b. False ANSWER: True POINTS: 1 TOPICS: Problem solving and decision making 18. Model development should be left to quantitative analysts; the model user's involvement should begin at the implementation stage. a. True b. False ANSWER: False POINTS: 1 TOPICS: Problem solving and decision making 19. A feasible solution is one that satisfies at least one of the constraints in the problem. a. True b. False ANSWER: False Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction POINTS: 1 TOPICS: Model solution 20. A toy train layout designed to represent an actual railyard is an example of an analog model. a. True b. False ANSWER: False POINTS: 1 TOPICS: Model development Multiple Choice 21. The field of management science a. approaches decision making rationally, with techniques based on the scientific method. b. concentrates on the use of quantitative methods to assist in decision making. c. is another name for decision science and for operations research. d. each of these choices are true. ANSWER: d POINTS: 1 TOPICS: Introduction 22. Identification and definition of a problem a. is the final step of problem solving. b. cannot be done until alternatives are proposed. c. requires consideration of multiple criteria. d. is the first step of decision making. ANSWER: d POINTS: 1 TOPICS: Problem solving and decision making 23. Decision alternatives a. should be identified before decision criteria are established. b. are limited to quantitative solutions c. are evaluated as a part of the problem definition stage. d. are best generated by brain-storming. ANSWER: a POINTS: 1 TOPICS: Problem solving and decision making 24. Decision criteria a. are the ways to evaluate the choices faced by the decision maker. b. are the choices faced by the decision maker. c. must be unique for a problem. d. are the problems faced by the decision maker. ANSWER: a Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction POINTS: 1 TOPICS: Problem solving and decision making 25. In a multicriteria decision problem a. successive decisions must be made over time. b. it is impossible to select a single decision alternative. c. the decision maker must evaluate each alternative with respect to each criterion. d. each of these choices are true. ANSWER: c POINTS: 1 TOPICS: Problem solving and decision making 26. The quantitative analysis approach requires a. mathematical expressions for the relationships. b. the manager's prior experience with a similar problem. c. a relatively uncomplicated problem. ANSWER: a POINTS: 1 TOPICS: Quantitative analysis and decision making 27. A physical model that does not have the same physical appearance as the object being modeled is a. a qualitative model. b. a mathematical model. c. an analog model. d. an iconic model. ANSWER: c POINTS: 1 TOPICS: Model development 28. Inputs to a quantitative model a. must all be deterministic if the problem is to have a solution. b. are uncertain for a stochastic model. c. are a trivial part of the problem solving process. d. are uncontrollable for the decision variables. ANSWER: b POINTS: 1 TOPICS: Model development 29. When the value of the output cannot be determined even if the value of the controllable input is known, the model is a. deterministic. b. analog. c. stochastic. d. digital. ANSWER: c POINTS: 1 Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction TOPICS: Model development 30. The volume that results in total revenue being equal to total cost is the a. profit mix. b. marginal volume. c. marginal cost. d. break-even point. ANSWER: d POINTS: 1 TOPICS: Break-even analysis 31. Management science and operations research both involve a. operational management skills. b. quantitative approaches to decision making. c. scientific research as opposed to applications. d. qualitative managerial skills. ANSWER: b POINTS: 1 TOPICS: Introduction 32. George Dantzig is important in the history of management science because he developed a. the scientific management revolution. b. powerful digital computers. c. World War II operations research teams. d. the simplex method for linear programming. ANSWER: d POINTS: 1 TOPICS: Introduction 33. The first step in problem solving is a. definition of decision variables. b. the identification of a difference between the actual and desired state of affairs. c. determination of the correct analytical solution procedure. d. implementation. ANSWER: b POINTS: 1 TOPICS: Problem solving and decision making 34. Problem definition a. must involve the analyst and the user of the results. b. includes specific objectives and operating constraints. c. must occur prior to the quantitative analysis process. d. each of these choices are true. ANSWER: d POINTS: 1 Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction TOPICS: Quantitative analysis 35. A model that uses a system of symbols to represent a problem is called a. iconic. b. constrained. c. mathematical. d. analog. ANSWER: c POINTS: 1 TOPICS: Model development 36. Which of the following is not one of the commonly used names for the body of knowledge involving quantitative approaches to decision-making? a. efficiency studies b. management science c. business analytics d. operations research ANSWER: a POINTS: 1 TOPICS: Introduction Subjective Short Answer 37. A snack food manufacturer buys corn for tortilla chips from two cooperatives, one in Iowa and one in Illinois. The price per unit of the Iowa corn is $6.00 and the price per unit of the Illinois corn is $5.50. a. Define variables that would tell how many units to purchase from each source. b. Develop an objective function that would minimize the total cost. The manufacturer needs at least 12000 units of corn. The Iowa cooperative can supply up to c. 8000 units, and the Illinois cooperative can supply at least 6000 units. Develop constraints for these conditions. ANSWER: a. Let x1 = the number of units from Iowa Let x2 = the number of units from Illinois b. Min 6x1 + 5.5x2 c. x1 + x 2 ≥ 12000 x1 ≥ 8000 x1 ≥ 6000 POINTS: 1 TOPICS: Model development 38. The relationship d = 5000 − 25p describes what happens to demand (d) as price (p) varies. Here, price can vary between $10 and $50. a. How many units can be sold at the $10 price? How many can be sold at the $50 price? b. Model the expression for total revenue. Consider prices of $20, $30, and $40. Which of these three price alternative will maximize c. total revenue? What are the values for demand and revenue at this price? ANSWER: Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction a. b. c.
For p = 10, d = 4750 For p = 50, d = 3750 TR = p(5000 − 25p) For p = 20, d = 4500, TR = $90,000 For p = 30, d = 4250, TR = $127,500 For p = 40, d = 4000, TR = $160,000 (maximum total revenue)
POINTS: 1 TOPICS: Model development 39. There is a fixed cost of $50,000 to start a production process. Once the process has begun, the variable cost per unit is $25. The revenue per unit is projected to be $45. a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. Find the break-even point. ANSWER: a. C(x) = 50000 + 25x b. R(x) = 45x c. P(x) = 45x − (50000 + 25x) d. x = 2500 POINTS: 1 TOPICS: Break-even analysis 40. An author has received an advance against royalties of $10,000. The royalty rate is $1.00 for every book sold in the United States, and $1.35 for every book sold outside the United States. Define variables for this problem and write an expression that could be used to calculate the number of books to be sold to cover the advance. ANSWER: Let x1 = the number of books sold in the U.S. Let x2 = the number of books sold outside the U.S. 10000 = 1x1 + 1.35x2 POINTS: 1 TOPICS: Break-even analysis 41. A university schedules summer school courses based on anticipated enrollment. The cost for faculty compensation, laboratories, student services, and allocated overhead for a computer class is $8500. If students pay $920 to enroll in the course, how large would enrollment have to be for the university to break even? ANSWER: Enrollment would need to be 10 students. POINTS: 1 TOPICS: Break-even analysis 42. As part of their application for a loan to buy Lakeside Farm, a property they hope to develop as a bed-and-breakfast operation, the prospective owners have projected: Monthly fixed cost (loan payment, taxes, insurance, maintenance) Variable cost per occupied room per night Revenue per occupied room per night a. b. c.
$6000 $ 20 $ 75
Write the expression for total cost per month. Assume 30 days per month. Write the expression for total revenue per month. If there are 12 guest rooms available, can they break even? What percentage of rooms would need to be occupied, on average, to break even?
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Chapter 1 - Introduction ANSWER:
a. b. c.
C(x) = 6000 + 20(30)x (monthly) R(x) = 75(30)x (monthly) Break-even occupancy = 3.64 or 4 occupied rooms per night, so they have enough rooms to break even. This would be a 33% occupancy rate.
POINTS: 1 TOPICS: Break-even analysis 43. Organizers of an Internet training session will charge participants $150 to attend. It costs $3000 to reserve the room, hire the instructor, bring in the equipment, and advertise. Assume it costs $25 per student for the organizers to provide the course materials. a. How many students would have to attend for the company to break even? If the trainers think, realistically, that 20 people will attend, then what price should be b. charged per person for the organization to break even? ANSWER: a. C(x) = 3000 + 25x R(x) = 150x Break-even students = 24 b. Cost = 3000 + 25(20) Revenue = 20p Break-even price = 175 POINTS: 1 TOPICS: Break-even analysis 44. In this portion of an Excel spreadsheet, the user has given values for selling price, the costs, and a sample volume. Give the cell formula for a. cell E12, break-even volume. b. cell E16, total revenue. c. cell E17, total cost. d. cell E19, profit (loss). A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
B
C
D
E
Break-even calculation Selling price per unit
10
Costs Fix cost Variable cost per unit
8400 4.5
Break-even volume Sample calculation Volume Total revenue Total cost
2000
Profit (loss)
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Chapter 1 - Introduction ANSWER:
a. =E9/(E6-E10) b. =E15*E6 c. =E9+E10*E15 d. =E16-E17 POINTS: 1 TOPICS: Spreadsheets for management science 45. A furniture store has set aside 800 square feet to display its sofas and chairs. Each sofa utilizes 50 sq. ft. and each chair utilizes 30 sq. ft. At least five sofas and at least five chairs are to be displayed. a. Write a mathematical model representing the store's constraints. Suppose the profit on sofas is $200 and on chairs is $100. On a given day, the probability b. that a displayed sofa will be sold is .03 and that a displayed chair will be sold is .05. Mathematically model each of the following objectives: 1. Maximize the total pieces of furniture displayed. 2. Maximize the total expected number of daily sales. 3. Maximize the total expected daily profit. ANSWER: a. 50s + 30c ≤ 800 s≥5 c≥5 b. (1) Max s + c (2) Max .03s + .05c (3) Max 6s + 5c POINTS: 1 TOPICS: Model development 46. A manufacturer makes two products, doors and windows. Each must be processed through two work areas. Work area #1 has 60 hours of available production time per week. Work area #2 has 48 hours of available production time per week. Manufacturing of a door requires 4 hours in work area #1 and 2 hours in work area #2. Manufacturing of a window requires 2 hours in work area #1 and 4 hours in work area #2. Profit is $8 per door and $6 per window. a. Define decision variables that will tell how many units to build (doors and windows) per week. b. Develop an objective function that will maximize total profit per week. c. Develop production constraints for work area #1 and #2. ANSWER: a. Let D = the number of doors to build per week Let N = the number of windows to build per week b. Weekly Profit = 8D + 6W c. 4D + 2W ≤ 60 2D + 4W ≤ 48 POINTS: 1 TOPICS: Model development 47. A small firm builds galvanized swing sets. The investment in plant and equipment is $200,000. The variable cost per swing set is $500. The selling price of the swing set is $1000. How many swing sets would have to be sold for the firm to break even? ANSWER: 400 swing sets POINTS: 1 TOPICS: Break-even analysis Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction 48. A computer rework center has the capacity to rework 300 computers per day. The expected number of computers needing to be reworked per day is 225. The center is paid $26 for each computer reworked. The fixed cost of renting the reworking equipment is $250 per day. Work space rents for $150 per day. The cost of material is $18 per computer and labor costs $3 per computer. What is the break-even number of computers reworked per day? ANSWER: 80 computers POINTS: 1 TOPICS: Break-even analysis 49. To establish a driver education school, organizers must decide how many cars, instructors, and students to have. Costs are estimated as follows. Annual fixed costs to operate the school are $30,000. The annual cost per car is $3000. The annual cost per instructor is $11,000 and one instructor is needed for each car. Tuition for each student is $350. Let x be the number of cars and y be the number of students. a. Write an expression for total cost. b. Write an expression for total revenue. c. Write an expression for total profit. d. The school offers the course eight times each year. Each time the course is offered, there are two sessions. If they decide to operate five cars, and if four students can be assigned to each car, will they break even? ANSWER: a. C(x) = 30000 + 14000x b. R(y) = 350y c. P(x,y) = 350y − (30000 + 14000x) d. Each car/instructor can serve up to (4 students/session)(2 sessions/course)(8 courses/year) = 64 students annually. Five cars can serve 320 students. If the classes are filled, then profit for five cars is 350(320) − (30000 + 14000(5)) = 12000. So, the school can reach the break-even point. POINTS: 1 TOPICS: Break-even analysis 50. Zipco Printing operates a shop that has five printing machines. The machines differ in their capacities to perform various printing operations due to differences in the machines' designs and operator skill levels. At the start of the workday there are five printing jobs to schedule. The manager must decide what the job-machine assignments should be. a. How could a quantitative approach to decision making be used to solve this problem? b. What would be the uncontrollable inputs for which data must be collected? Define the decision variables, objective function, and constraints to appear in the c. mathematical model. d. Is the model deterministic or stochastic? e. Suggest some simplifying assumptions for this problem. ANSWER: A quantitative approach to decision making can provide a systematic way for deciding the a. job-machine pairings so that total job processing time is minimized. How long it takes to process each job on each machine, and any job-machine pairings that b. are unacceptable. Decision variables: one for each job-machine pairing, taking on a value of 1 if the pairing is c. used and 0 otherwise. Objective function: minimize total job processing time. Constraints: each job is assigned to exactly one machine, and each machine be assigned no more than one job. Stochastic: job processing times vary due to varying machine set-up times, variable operator d. performance, and more. e. Assume that processing times are deterministic (known/fixed). POINTS: 1 Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction TOPICS: Model development 51. Consider a department store that must make weekly shipments of a certain product from two different warehouses to four different stores. a. How could a quantitative approach to decision making be used to solve this problem? b. What would be the uncontrollable inputs for which data must be gathered? What would be the decision variables of the mathematical model? the objective function? the c. constraints? d. Is the model deterministic or stochastic? e. Suggest assumptions that could be made to simplify the model. ANSWER: A quantitative approach to decision making can provide a systematic way to determine a a. minimum shipping cost from the warehouses to the stores. Fixed costs and variable shipping costs; the demand each week at each store; the supplies b. each week at each warehouse. Decision variables--how much to ship from each warehouse to each store; objective c. function--minimize total shipping costs; constraints--meet the demand at the stores without exceeding the supplies at the warehouses. Stochastic--weekly demands fluctuate as do weekly supplies; transportation costs could vary d. depending upon the amount shipped, other goods sent with a shipment, etc. Make the model deterministic by assuming fixed shipping costs per item, demand is constant e. at each store each week, and weekly supplies in the warehouses are constant. POINTS: 1 TOPICS: Model development 52. Three production processes - A, B, and C - have the following cost structure: Process
Fixed Cost per Year
Variable Cost per Unit
A B C
$120,000 90,000 80,000
$3.00 4.00 4.50
a. What is the most economical process for a volume of 8,000 units? How many units per year must be sold with each process to have annual profits of $50,000 if the selling price is $6.95 b. per unit? c. What is the break-even volume for each process? ANSWER: a. C(x) = FC + VC(x) Process A: C(x) = $120,000 + $3.00(8,000) = $144,000 per year Process B: C(x) = $ 90,000 + $4.00(8,000) = $122,000 per year Process C: C(x) = $ 80,000 + $4.50(8,000) = $116,000 per year Process C has the lowest annual cost for a production volume of 8,000 units. b. Q = (profit + FC)/(price - VC) Process A: Q = ($50,000 + $120,000)/($6.95 - $3.00) = 43,038 units Process B: Q = ($50,000 + $ 90,000)/($6.95 - $4.00) = 47,458 units Process C: Q = ($50,000 + $ 80,000)/($6.95 - $4.50) = 53,062 units Process A requires the lowest production volume for an annual profit of $50,000. c. At breakeven, profit (the pretax profits per period) is equal to zero. Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction Q = FC/(price - VC) Process A: Q = $120,000/ ($6.95 - $3.00) = 30,380 units Process B: Q = $ 90,000/ ($6.95 - $4.00) = 30,509 units Process C: Q = $ 80,000/ ($6.95 - $4.50) = 32,654 units Process A has the lowest break-even quantity, while Process B’s is almost as low. POINTS: 1 TOPICS: Break-even analysis 53. Jane Persico, facility engineer at the El Paso plant of Computer Products Corporation (CPC), is studying a process selection decision at the plant. A new printer is to be manufactured and she must decide whether the printer will be autoassembled or manually assembled. The decision is complicated by the fact that annual production volume is expected to increase by almost 50% over three years. Jane has developed these estimates for two alternatives for the printer assembly process:
Annual fixed cost Variable cost per product Estimated annual production (in number of products):
Year 1 Year 2 Year 3
AutoAssembly Process
Manual Assembly Process
$690,000 $29.56
$269,000 $31.69
152,000 190,000 225,000
152,000 190,000 225,000
a. Which production process would be the least-cost alternative in Years 1, 2, and 3? b. How much would the variable cost per unit have to be in Year 2 for the auto-assembly process to justify the additional annual fixed cost for the auto-assembly process over the manual assembly process? ANSWER:
a. C(x) = fixed cost + variable cost(x) Year 1: CA = 690,000 + 29.56(152,000) = $5,183,120 CM = 269,000 + 31.69(152,000) = $5,085,880 (least-cost alternative) Year 2: CA = 690,000 + 29.56(190,000) = $6,306,400 CM = 269,000 + 31.69(190,000) = $6,290,100 (least-cost alternative) Year 3: CA = 690,000 + 29.56(225,000) = $7,341,000 (least-cost alternative) CM = 269,000 + 31.69(225,000) = $7,399,250 b. CA = CM FCA + vA(190,000) = FCM + vM(190,000) 690,000 + v(190,000) = 269,000 + 31.69(190,000) vA = (269,000 + 6,021,100 - 690,000)/190,000 vA = $29.47 (roughly a 0.3% reduction)
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Chapter 1 - Introduction TOPICS: Cost and volume models Essay 54. Should the problem solving process be applied to all problems? ANSWER: Answer not provided. POINTS: 1 TOPICS: Problem solving and decision making 55. Explain the difference between quantitative and qualitative analysis from the manager's point of view. ANSWER: Answer not provided. POINTS: 1 TOPICS: Quantitative analysis and decision making 56. Explain the relationship among model development, model accuracy, and the ability to obtain a solution from a model. ANSWER: Answer not provided. POINTS: 1 TOPICS: Model solution 57. What are three of the management science techniques that practitioners use most frequently? How can the effectiveness of these applications be increased? ANSWER: Answer not provided. POINTS: 1 TOPICS: Methods used most frequently 58. What steps of the problem solving process are involved in decision making? ANSWER: Answer not provided. POINTS: 1 TOPICS: Introduction 59. Give three benefits of model development and an example of each. ANSWER: Answer not provided. POINTS: 1 TOPICS: Model development 60. Explain the relationship between information systems specialists and quantitative analysts in the solution of large mathematical problems. ANSWER: Answer not provided. POINTS: 1 TOPICS: Data preparation 61. Define and contrast the terms feasible solution, infeasible solution and optimal solution. ANSWER: Answer not provided. POINTS: 1 TOPICS: Model solution 62. Define three forms of models and provide an example of each. Cengage Learning Testing, Powered by Cognero
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Chapter 1 - Introduction ANSWER: Answer not provided. POINTS: 1 TOPICS: Model development 63. Explain the difference between controllable and uncontrollable inputs to a mathematical model and provide an example of each. ANSWER: Answer not provided. POINTS: 1 TOPICS: Model development
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Chapter 2 - An Introduction to Linear Programming True / False 1. Increasing the right-hand side of a nonbinding constraint will not cause a change in the optimal solution. a. True b. False ANSWER: False POINTS: 1 TOPICS: Introduction 2. In a linear programming problem, the objective function and the constraints must be linear functions of the decision variables. a. True b. False ANSWER: True POINTS: 1 TOPICS: Mathematical statement of the RMC Problem 3. In a feasible problem, an equal-to constraint cannot be nonbinding. a. True b. False ANSWER: True POINTS: 1 TOPICS: Graphical solution 4. Only binding constraints form the shape (boundaries) of the feasible region. a. True b. False ANSWER: False POINTS: 1 TOPICS: Graphical solution 5. The constraint 5x1 − 2x2 ≤ 0 passes through the point (20, 50). a. True b. False ANSWER: True POINTS: 1 TOPICS: Graphing lines 6. A redundant constraint is a binding constraint. a. True b. False ANSWER: False POINTS: 1 TOPICS: Slack variables
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Chapter 2 - An Introduction to Linear Programming 7. Because surplus variables represent the amount by which the solution exceeds a minimum target, they are given positive coefficients in the objective function. a. True b. False ANSWER: False POINTS: 1 TOPICS: Slack variables 8. Alternative optimal solutions occur when there is no feasible solution to the problem. a. True b. False ANSWER: False POINTS: 1 TOPICS: Alternative optimal solutions 9. A range of optimality is applicable only if the other coefficient remains at its original value. a. True b. False ANSWER: True POINTS: 1 TOPICS: Simultaneous changes 10. Because the dual price represents the improvement in the value of the optimal solution per unit increase in right-handside, a dual price cannot be negative. a. True b. False ANSWER: False POINTS: 1 TOPICS: Right-hand sides 11. Decision variables limit the degree to which the objective in a linear programming problem is satisfied. a. True b. False ANSWER: False POINTS: 1 TOPICS: Introduction 12. No matter what value it has, each objective function line is parallel to every other objective function line in a problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Graphical solution 13. The point (3, 2) is feasible for the constraint 2x1 + 6x2 ≤ 30. Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming a. True b. False ANSWER: True POINTS: 1 TOPICS: Graphical solution 14. The constraint 2x1 − x2 = 0 passes through the point (200,100). a. True b. False ANSWER: False POINTS: 1 TOPICS: A note on graphing lines 15. The standard form of a linear programming problem will have the same solution as the original problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Surplus variables 16. An optimal solution to a linear programming problem can be found at an extreme point of the feasible region for the problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Extreme points 17. An unbounded feasible region might not result in an unbounded solution for a minimization or maximization problem. a. True b. False ANSWER: True POINTS: 1 TOPICS: Special cases: unbounded 18. An infeasible problem is one in which the objective function can be increased to infinity. a. True b. False ANSWER: False POINTS: 1 TOPICS: Special cases: infeasibility 19. A linear programming problem can be both unbounded and infeasible. a. True b. False Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming ANSWER: False POINTS: 1 TOPICS: Special cases: infeasibility and unbounded 20. It is possible to have exactly two optimal solutions to a linear programming problem. a. True b. False ANSWER: False POINTS: 1 TOPICS: Special cases: alternative optimal solutions Multiple Choice 21. The maximization or minimization of a quantity is the a. goal of management science. b. decision for decision analysis. c. constraint of operations research. d. objective of linear programming. ANSWER: d POINTS: 1 TOPICS: Introduction 22. Decision variables a. tell how much or how many of something to produce, invest, purchase, hire, etc. b. represent the values of the constraints. c. measure the objective function. d. must exist for each constraint. ANSWER: a POINTS: 1 TOPICS: Objective function 23. Which of the following is a valid objective function for a linear programming problem? a. Max 5xy b. Min 4x + 3y + (2/3)z c. Max 5x2 + 6y2 d. Min (x1 + x2)/x3 ANSWER: b POINTS: 1 TOPICS: Objective function 24. Which of the following statements is NOT true? a. A feasible solution satisfies all constraints. b. An optimal solution satisfies all constraints. c. An infeasible solution violates all constraints. Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming d. A feasible solution point does not have to lie on the boundary of the feasible region. ANSWER: c POINTS: 1 TOPICS: Graphical solution 25. A solution that satisfies all the constraints of a linear programming problem except the nonnegativity constraints is called a. optimal. b. feasible. c. infeasible. d. semi-feasible. ANSWER: c POINTS: 1 TOPICS: Graphical solution 26. Slack a. is the difference between the left and right sides of a constraint. b. is the amount by which the left side of a ≤ constraint is smaller than the right side. c. is the amount by which the left side of a ≥ constraint is larger than the right side. d. exists for each variable in a linear programming problem. ANSWER: b POINTS: 1 TOPICS: Slack variables 27. To find the optimal solution to a linear programming problem using the graphical method a. find the feasible point that is the farthest away from the origin. b. find the feasible point that is at the highest location. c. find the feasible point that is closest to the origin. d. None of the alternatives is correct. ANSWER: d POINTS: 1 TOPICS: Extreme points 28. Which of the following special cases does not require reformulation of the problem in order to obtain a solution? a. alternate optimality b. infeasibility c. unboundedness d. each case requires a reformulation. ANSWER: a POINTS: 1 TOPICS: Special cases 29. The improvement in the value of the objective function per unit increase in a right-hand side is the a. sensitivity value. b. dual price. Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming c. constraint coefficient. d. slack value. ANSWER: b POINTS: 1 TOPICS: Right-hand sides 30. As long as the slope of the objective function stays between the slopes of the binding constraints a. the value of the objective function won't change. b. there will be alternative optimal solutions. c. the values of the dual variables won't change. d. there will be no slack in the solution. ANSWER: c POINTS: 1 TOPICS: Objective function 31. Infeasibility means that the number of solutions to the linear programming models that satisfies all constraints is a. at least 1. b. 0. c. an infinite number. d. at least 2. ANSWER: b POINTS: 1 TOPICS: Alternate optimal solutions 32. A constraint that does not affect the feasible region is a a. non-negativity constraint. b. redundant constraint. c. standard constraint. d. slack constraint. ANSWER: b POINTS: 1 TOPICS: Feasible regions 33. Whenever all the constraints in a linear program are expressed as equalities, the linear program is said to be written in a. standard form. b. bounded form. c. feasible form. d. alternative form. ANSWER: a POINTS: 1 TOPICS: Slack variables 34. All of the following statements about a redundant constraint are correct EXCEPT a. A redundant constraint does not affect the optimal solution. b. A redundant constraint does not affect the feasible region. Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming c. Recognizing a redundant constraint is easy with the graphical solution method. d. At the optimal solution, a redundant constraint will have zero slack. ANSWER: d POINTS: 1 TOPICS: Slack variables 35. All linear programming problems have all of the following properties EXCEPT a. a linear objective function that is to be maximized or minimized. b. a set of linear constraints. c. alternative optimal solutions. d. variables that are all restricted to nonnegative values. ANSWER: c POINTS: 1 TOPICS: Problem formulation 36. If there is a maximum of 4,000 hours of labor available per month and 300 ping-pong balls (x1) or 125 wiffle balls (x2) can be produced per hour of labor, which of the following constraints reflects this situation? a. 300x1 + 125x2 > 4,000 b. 300x1 + 125x2 < 4,000 c. 425(x1 + x2) < 4,000 d. 300x1 + 125x2 = 4,000 ANSWER: b POINTS: 1 37. In what part(s) of a linear programming formulation would the decision variables be stated? a. objective function and the left-hand side of each constraint b. objective function and the right-hand side of each constraint c. the left-hand side of each constraint only d. the objective function only ANSWER: a POINTS: 1 38. The three assumptions necessary for a linear programming model to be appropriate include all of the following except a. proportionality b. additivity c. divisibility d. normality ANSWER: d POINTS: 1 39. A redundant constraint results in a. no change in the optimal solution(s) b. an unbounded solution c. no feasible solution Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming d. alternative optimal solutions ANSWER: a POINTS: 1 40. A variable added to the left-hand side of a less-than-or-equal-to constraint to convert the constraint into an equality is a. a standard variable b. a slack variable c. a surplus variable d. a non-negative variable ANSWER: b POINTS: 1 Subjective Short Answer 41. Solve the following system of simultaneous equations. 6X + 2Y = 50 2X + 4Y = 20 ANSWER: X = 8, Y =1 POINTS: 1 TOPICS: Simultaneous equations 42. Solve the following system of simultaneous equations. 6X + 4Y = 40 2X + 3Y = 20 ANSWER: X = 4, Y = 4 POINTS: 1 TOPICS: Simultaneous equations 43. Consider the following linear programming problem Max s.t.
8X + 7Y 15X + 5Y ≤ 75 10X + 6Y ≤ 60 X+ Y≤8 X, Y ≥ 0
a.
Use a graph to show each constraint and the feasible region. Identify the optimal solution point on your graph. What are the values of X and Y at the b. optimal solution? c. What is the optimal value of the objective function? ANSWER:
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Chapter 2 - An Introduction to Linear Programming
a.
b.
The optimal solution occurs at the intersection of constraints 2 and 3. The point is X = 3, Y = 5. The value of the objective function is 59.
c. POINTS: 1 TOPICS: Graphical solution
44. For the following linear programming problem, determine the optimal solution by the graphical solution method −X + 2Y 6X − 2Y ≤ 3 −2X + 3Y ≤ 6 X+ Y≤3 X, Y ≥ 0 ANSWER: X = 0.6 and Y = 2.4 Max s.t.
POINTS: 1 TOPICS: Graphical solution Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming 45. Use this graph to answer the questions.
Max s.t.
20X + 10Y 12X + 15Y ≤ 180 15X + 10Y ≤ 150 3X − 8Y ≤ 0 X,Y≥0
a. Which area (I, II, III, IV, or V) forms the feasible region? b. Which point (A, B, C, D, or E) is optimal? c. Which constraints are binding? d. Which slack variables are zero? ANSWER: a. Area III is the feasible region b. Point D is optimal c. Constraints 2 and 3 are binding d. S2 and S3 are equal to 0 POINTS: 1 TOPICS: Graphical solution 46. Find the complete optimal solution to this linear programming problem. Min s.t.
5X + 6Y 3X + Y ≥ 15 X + 2Y ≥ 12 3X + 2Y ≥ 24 X,Y≥0 ANSWER:
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Chapter 2 - An Introduction to Linear Programming
The complete optimal solution is POINTS: 1 TOPICS: Graphical solution
X = 6, Y = 3, Z = 48, S1 = 6, S2 = 0, S3 = 0
47. Find the complete optimal solution to this linear programming problem. Max s.t.
5X + 3Y 2X + 3Y ≤ 30 2X + 5Y ≤ 40 6X − 5Y ≤ 0 X,Y≥ 0 ANSWER:
The complete optimal solution is Cengage Learning Testing, Powered by Cognero
X = 15, Y = 0, Z = 75, S1 = 0, S2 = 10, S3 = 90 Page 11
Chapter 2 - An Introduction to Linear Programming POINTS: 1 TOPICS: Graphical solution 48. Find the complete optimal solution to this linear programming problem. Max s.t.
2X + 3Y 4X + 9Y ≤ 72 10X + 11Y ≤ 110 17X + 9Y ≤ 153 X,Y≥0 ANSWER:
The complete optimal solution is POINTS: 1 TOPICS: Graphical solution
X = 4.304, Y = 6.087, Z = 26.87, S1 = 0, S2 = 0, S3 = 25.043
49. Find the complete optimal solution to this linear programming problem. Min s.t.
3X + 3Y 12X + 4Y ≥ 48 10X + 5Y ≥ 50 4X + 8Y ≥ 32 X,Y≥0 ANSWER:
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Chapter 2 - An Introduction to Linear Programming
The complete optimal solution is POINTS: 1 TOPICS: Graphical solution
X = 4, Y = 2, Z = 18, S1 = 8, S2 = 0, S3 = 0
50. For the following linear programming problem, determine the optimal solution by the graphical solution method. Are any of the constraints redundant? If yes, then identify the constraint that is redundant. Max s.t.
X + 2Y X+ Y≤3 X − 2Y ≥ 0 Y≤1 X, Y ≥ 0 ANSWER: X = 2, and Y = 1 Yes, there is a redundant constraint; Y ≤ 1
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Chapter 2 - An Introduction to Linear Programming 51. Maxwell Manufacturing makes two models of felt tip marking pens. Requirements for each lot of pens are given below. Fliptop Model 3 5 5
Plastic Ink Assembly Molding Time
Tiptop Model 4 4 2
Available 36 40 30
The profit for either model is $1000 per lot. a. What is the linear programming model for this problem? b. Find the optimal solution. c. Will there be excess capacity in any resource? ANSWER: a. Let F = the number of lots of Fliptop pens to produce Let T = the number of lots of Tiptop pens to produce Max s.t.
1000F + 1000T 3F + 4T ≤ 36 5F + 4T ≤ 40 5F + 2T ≤ 30 F,T≥0
b.
The complete optimal solution is F = 2, T = 7.5, Z = 9500, S1 = 0, S2 = 0, S3 = 5 There is an excess of 5 units of molding time available.
c. POINTS: 1 TOPICS: Modeling and graphical solution
52. The Sanders Garden Shop mixes two types of grass seed into a blend. Each type of grass has been rated (per pound) according to its shade tolerance, ability to stand up to traffic, and drought resistance, as shown in the table. Type A seed costs $1 and Type B seed costs $2. If the blend needs to score at least 300 points for shade tolerance, 400 points for traffic resistance, and 750 points for drought resistance, how many pounds of each seed should be in the blend? Which targets will be exceeded? How much will the blend cost? Shade Tolerance Traffic Resistance
Type A 1 2
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Type B 1 1 Page 14
Chapter 2 - An Introduction to Linear Programming Drought Resistance 2 5 ANSWER: Let A = the pounds of Type A seed in the blend Let B = the pounds of Type B seed in the blend Min s.t.
1A + 2B 1A + 1B ≥ 300 2A + 1B ≥ 400 2A + 5B ≥ 750 A, B ≥ 0
The optimal solution is at A = 250, B = 50. Constraint 2 has a surplus value of 150. The cost is 350. POINTS: 1 TOPICS: Modeling and graphical solution 53. Muir Manufacturing produces two popular grades of commercial carpeting among its many other products. In the coming production period, Muir needs to decide how many rolls of each grade should be produced in order to maximize profit. Each roll of Grade X carpet uses 50 units of synthetic fiber, requires 25 hours of production time, and needs 20 units of foam backing. Each roll of Grade Y carpet uses 40 units of synthetic fiber, requires 28 hours of production time, and needs 15 units of foam backing. The profit per roll of Grade X carpet is $200 and the profit per roll of Grade Y carpet is $160. In the coming production period, Muir has 3000 units of synthetic fiber available for use. Workers have been scheduled to provide at least 1800 hours of production time (overtime is a possibility). The company has 1500 units of foam backing available for use. Develop and solve a linear programming model for this problem. ANSWER: Let X = the number of rolls of Grade X carpet to make Let Y = the number of rolls of Grade Y carpet to make Max
200X + 160Y
s.t.
50X + 40Y ≤ 3000 25X + 28Y ≥ 1800 20X + 15Y ≤ 1500 X,Y≥0
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Chapter 2 - An Introduction to Linear Programming
The complete optimal solution is X = 30, Y = 37.5, Z = 12000, S1 = 0, S2 = 0, S3 = 337.5 POINTS: 1 TOPICS: Modeling and graphical solution 54. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain. Min s.t.
1X + 1Y 5X + 3Y ≤ 30 3X + 4Y ≥ 36 Y≤7 X,Y≥0 ANSWER: The problem is infeasible.
POINTS: 1 TOPICS: Special cases 55. Does the following linear programming problem exhibit infeasibility, unboundedness, or alternate optimal solutions? Explain. Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming Min s.t.
3X + 3Y 1X + 2Y ≤ 16 1X + 1Y ≤ 10 5X + 3Y ≤ 45 X,Y≥0 ANSWER: The problem has alternate optimal solutions.
POINTS: 1 TOPICS: Special cases 56. A businessman is considering opening a small specialized trucking firm. To make the firm profitable, it is estimated that it must have a daily trucking capacity of at least 84,000 cu. ft. Two types of trucks are appropriate for the specialized operation. Their characteristics and costs are summarized in the table below. Note that truck 2 requires 3 drivers for long haul trips. There are 41 potential drivers available and there are facilities for at most 40 trucks. The businessman's objective is to minimize the total cost outlay for trucks. Truck Small Large
Cost $18,000 $45,000
Capacity (Cu. Ft.) 2,400 6,000
Drivers Needed 1 3
Solve the problem graphically and note there are alternate optimal solutions. Which optimal solution: a. uses only one type of truck? b. utilizes the minimum total number of trucks? c. uses the same number of small and large trucks? ANSWER: a. 35 small, 0 large b. 5 small, 12 large c. 10 small, 10 large POINTS: 1 TOPICS: Alternative optimal solutions 57. Consider the following linear program: Max s.t.
60X + 43Y X + 3Y ≥ 9
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Chapter 2 - An Introduction to Linear Programming 6X − 2Y = 12 X + 2Y ≤ 10 X, Y ≥ 0 a. b.
Write the problem in standard form. What is the feasible region for the problem? Show that regardless of the values of the actual objective function coefficients, the optimal c. solution will occur at one of two points. Solve for these points and then determine which one maximizes the current objective function. ANSWER: a. Max 60X + 43Y s.t. X + 3Y − S1 = 9 6X − 2Y = 12 X + 2Y + S3 = 10 X, Y, S1, S3 ≥ 0 b. Line segment of 6X − 2Y = 12 between (22/7,24/7) and (27/10,21/10). c. Extreme points: (22/7,24/7) and (27/10,21/10). First one is optimal, giving Z = 336. POINTS: 1 TOPICS: Standard form and extreme points 58. Solve the following linear program graphically. Max s.t.
5X + 7Y X ≤6 2X + 3Y ≤ 19 X+ Y≤8 X, Y ≥ 0 ANSWER: From the graph below we see that the optimal solution occurs at X = 5, Y = 3, and Z = 46.
POINTS: 1 TOPICS: Graphical solution procedure 59. Given the following linear program: Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming Min s.t.
150X + 210Y 3.8X + 1.2Y ≥ 22.8 Y≥6 Y ≤ 15 45X + 30Y = 630 X, Y ≥ 0
Solve the problem graphically. How many extreme points exist for this problem? ANSWER: Two extreme points exist (Points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (Point B).
POINTS: 1 TOPICS: Graphical solution procedure 60. Solve the following linear program by the graphical method. Max s.t.
4X + 5Y X + 3Y ≤ 22 −X + Y ≤ 4 Y≤6 2X − 5Y ≤ 0 X, Y ≥ 0 ANSWER: Two extreme points exist (Points A and B below). The optimal solution is X = 10, Y = 6, and Z = 2760 (Point B).
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Chapter 2 - An Introduction to Linear Programming
POINTS: 1 TOPICS: Graphical solution procedure Essay 61. Explain the difference between profit and contribution in an objective function. Why is it important for the decision maker to know which of these the objective function coefficients represent? ANSWER: Answer not provided. POINTS: 1 TOPICS: Objective function 62. Explain how to graph the line x1 − 2x2 ≥ 0. ANSWER: Answer not provided. POINTS: 1 TOPICS: Graphing lines 63. Create a linear programming problem with two decision variables and three constraints that will include both a slack and a surplus variable in standard form. Write your problem in standard form. ANSWER: Answer not provided. POINTS: 1 TOPICS: Standard form 64. Explain what to look for in problems that are infeasible or unbounded. ANSWER: Answer not provided. POINTS: 1 TOPICS: Special cases 65. Use a graph to illustrate why a change in an objective function coefficient does not necessarily lead to a change in the optimal values of the decision variables, but a change in the right-hand sides of a binding constraint does lead to new values. ANSWER: Answer not provided. POINTS: 1 TOPICS: Graphical sensitivity analysis Cengage Learning Testing, Powered by Cognero
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Chapter 2 - An Introduction to Linear Programming 66. Explain the concepts of proportionality, additivity, and divisibility. ANSWER: Answer not provided. POINTS: 1 TOPICS: Notes and comments 67. Explain the steps necessary to put a linear program in standard form. ANSWER: Answer not provided. POINTS: 1 TOPICS: Surplus variables 68. Explain the steps of the graphical solution procedure for a minimization problem. ANSWER: Answer not provided. POINTS: 1 TOPICS: Graphical solution procedure for minimization problems
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution True / False 1. Classical sensitivity analysis provides no information about changes resulting from a change in the coefficient of a variable in a constraint. a. True b. False ANSWER: True POINTS: 1 TOPICS: Changes in constraint coefficients 2. The reduced cost for a positive decision variable is 0. a. True b. False ANSWER: True POINTS: 1 TOPICS: Reduced cost 3. When the right-hand sides of two constraints are each increased by one unit, the objective function value will be adjusted by the sum of the constraints' dual prices. a. True b. False ANSWER: False POINTS: 1 TOPICS: Simultaneous changes 4. If the range of feasibility indicates that the original amount of a resource, which was 20, can increase by 5, then the amount of the resource can increase to 25. a. True b. False ANSWER: True POINTS: 1 TOPICS: Range of feasibility 5. The 100% Rule does not imply that the optimal solution will necessarily change if the percentage exceeds 100%. a. True b. False ANSWER: True POINTS: 1 TOPICS: Simultaneous changes 6. For any constraint, either its slack/surplus value must be zero or its dual price must be zero. a. True b. False ANSWER: True POINTS: 1 TOPICS: Dual price Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 7. A negative dual price indicates that increasing the right-hand side of the associated constraint would be detrimental to the objective. a. True b. False ANSWER: True POINTS: 1 TOPICS: Dual price 8. In order to tell the impact of a change in a constraint coefficient, the change must be made and then the model resolved. a. True b. False ANSWER: True POINTS: 1 TOPICS: Changes in constraint coefficients 9. Decreasing the objective function coefficient of a variable to its lower limit will create a revised problem that is unbounded. a. True b. False ANSWER: False POINTS: 1 TOPICS: Range of optimality 10. The dual price for a percentage constraint provides a direct answer to questions about the effect of increases or decreases in that percentage. a. True b. False ANSWER: False POINTS: 1 TOPICS: Dual price 11. The dual price associated with a constraint is the change in the value of the solution per unit decrease in the right-hand side of the constraint. a. True b. False ANSWER: False POINTS: 1 TOPICS: Interpretation of computer output 12. For a minimization problem, a positive dual price indicates the value of the objective function will increase. a. True b. False ANSWER: False POINTS: 1 TOPICS: Interpretation of computer output--a second example Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 13. There is a dual price for every decision variable in a model. a. True b. False ANSWER: False POINTS: 1 TOPICS: Interpretation of computer output 14. The amount of a sunk cost will vary depending on the values of the decision variables. a. True b. False ANSWER: False POINTS: 1 TOPICS: Cautionary note on the interpretation of dual prices 15. If the optimal value of a decision variable is zero and its reduced cost is zero, this indicates that alternative optimal solutions exist. a. True b. False ANSWER: True POINTS: 1 TOPICS: Interpretation of computer output 16. Any change to the objective function coefficient of a variable that is positive in the optimal solution will change the optimal solution. a. True b. False ANSWER: False POINTS: 1 TOPICS: Range of optimality 17. Relevant costs should be reflected in the objective function, but sunk costs should not. a. True b. False ANSWER: True POINTS: 1 TOPICS: Cautionary note on the interpretation of dual prices 18. If the range of feasibility for b1 is between 16 and 37, then if b1 = 22 the optimal solution will not change from the original optimal solution. a. True b. False ANSWER: False POINTS: 1 TOPICS: Right-hand sides Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 19. The 100 percent rule can be applied to changes in both objective function coefficients and right-hand sides at the same time. a. True b. False ANSWER: False POINTS: 1 TOPICS: Simultaneous changes 20. If the dual price for the right-hand side of a ≤ constraint is zero, there is no upper limit on its range of feasibility. a. True b. False ANSWER: True POINTS: 1 TOPICS: Right-hand sides Multiple Choice 21. To solve a linear programming problem with thousands of variables and constraints a. a personal computer can be used. b. a mainframe computer is required. c. the problem must be partitioned into subparts. d. unique software would need to be developed. ANSWER: a POINTS: 1 TOPICS: Computer solution 22. A negative dual price for a constraint in a minimization problem means a. as the right-hand side increases, the objective function value will increase. b. as the right-hand side decreases, the objective function value will increase. c. as the right-hand side increases, the objective function value will decrease. d. as the right-hand side decreases, the objective function value will decrease. ANSWER: a POINTS: 1 TOPICS: Dual price 23. If a decision variable is not positive in the optimal solution, its reduced cost is a. what its objective function value would need to be before it could become positive. b. the amount its objective function value would need to improve before it could become positive. c. zero. d. its dual price. ANSWER: b POINTS: 1 TOPICS: Reduced cost 24. A constraint with a positive slack value Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution a. will have a positive dual price. b. will have a negative dual price. c. will have a dual price of zero. d. has no restrictions for its dual price. ANSWER: c POINTS: 1 TOPICS: Slack and dual price 25. The amount by which an objective function coefficient can change before a different set of values for the decision variables becomes optimal is the a. optimal solution. b. dual solution. c. range of optimality. d. range of feasibility. ANSWER: c POINTS: 1 TOPICS: Range of optimality 26. The range of feasibility measures a. the right-hand-side values for which the objective function value will not change. b. the right-hand-side values for which the values of the decision variables will not change. c. the right-hand-side values for which the dual prices will not change. d. each of these choices are true. ANSWER: c POINTS: 1 TOPICS: Range of feasibility 27. The 100% Rule compares a. proposed changes to allowed changes. b. new values to original values. c. objective function changes to right-hand side changes. d. dual prices to reduced costs. ANSWER: a POINTS: 1 TOPICS: Simultaneous changes 28. An objective function reflects the relevant cost of labor hours used in production rather than treating them as a sunk cost. The correct interpretation of the dual price associated with the labor hours constraint is a. the maximum premium (say for overtime) over the normal price that the company would be willing to pay. b. the upper limit on the total hourly wage the company would pay. c. the reduction in hours that could be sustained before the solution would change. d. the number of hours by which the right-hand side can change before there is a change in the solution point. ANSWER: a POINTS: 1 TOPICS: Dual price Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 29. A section of output from The Management Scientist is shown here. Variable 1
Lower Limit 60
Current Value 100
Upper Limit 120
What will happen to the solution if the objective function coefficient for variable 1 decreases by 20? a. Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same. b. The value of the objective function will change, but the values of the decision variables and the dual prices will remain the same. c. The same decision variables will be positive, but their values, the objective function value, and the dual prices will change. d. The problem will need to be resolved to find the new optimal solution and dual price. ANSWER: b POINTS: 1 TOPICS: Range of optimality 30. A section of output from The Management Scientist is shown here. Constraint 2
Lower Limit 240
Current Value 300
Upper Limit 420
What will happen if the right-hand-side for constraint 2 increases by 200? a. Nothing. The values of the decision variables, the dual prices, and the objective function will all remain the same. b. The value of the objective function will change, but the values of the decision variables and the dual prices will remain the same. c. The same decision variables will be positive, but their values, the objective function value, and the dual prices will change. d. The problem will need to be resolved to find the new optimal solution and dual price. ANSWER: d POINTS: 1 TOPICS: Range of feasibility 31. The amount the objective function coefficient of a decision variable would have to improve before that variable would have a positive value in the solution is the a. dual price. b. surplus variable. c. reduced cost. d. upper limit. ANSWER: c POINTS: 1 TOPICS: Interpretation of computer output 32. The dual price measures, per unit increase in the right hand side of the constraint, a. the increase in the value of the optimal solution. b. the decrease in the value of the optimal solution. Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution c. the improvement in the value of the optimal solution. d. the change in the value of the optimal solution. ANSWER: d POINTS: 1 TOPICS: Interpretation of computer output 33. Sensitivity analysis information in computer output is based on the assumption of a. no coefficient changes. b. one coefficient changes. c. two coefficients change. d. all coefficients change. ANSWER: b POINTS: 1 TOPICS: Simultaneous changes 34. When the cost of a resource is sunk, then the dual price can be interpreted as the a. minimum amount the firm should be willing to pay for one additional unit of the resource. b. maximum amount the firm should be willing to pay for one additional unit of the resource. c. minimum amount the firm should be willing to pay for multiple additional units of the resource. d. maximum amount the firm should be willing to pay for multiple additional units of the resource. ANSWER: b POINTS: 1 TOPICS: Dual price 35. Which of the following is not a question answered by standard sensitivity analysis information? a. If the right-hand side value of a constraint changes, will the objective function value change? b. Over what range can a constraint's right-hand side value without the constraint's dual price possibly changing? c. By how much will the objective function value change if the right-hand side value of a constraint changes beyond the range of feasibility? d. By how much will the objective function value change if a decision variable's coefficient in the objective function changes within the range of optimality? ANSWER: c POINTS: 1 TOPICS: Interpretation of computer output 36. The cost that varies depending on the values of the decision variables is a a. reduced cost. b. relevant cost. c. sunk cost. d. dual cost. ANSWER: b POINTS: 1 TOPICS: Sunk and relevant costs 37. A cost that is incurred no matter what values the decision variables assume is Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution a. a reduced cost. b. an optimal cost. c. a sunk cost. d. a dual cost. ANSWER: c POINTS: 1 TOPICS: Sunk and relevant costs 38. Sensitivity analysis is often referred to as a. feasibility testing. b. duality analysis. c. alternative analysis. d. postoptimality analysis. ANSWER: d POINTS: 1 TOPICS: Introduction 39. Sensitivity analysis is concerned with how certain changes affect a. the feasible solution. b. the unconstrained solution. c. the optimal solution. d. the degenerative solution. ANSWER: c POINTS: 1 TOPICS: Introduction 40. The dual price for a < constraint a. will always be < 0. b. will always be > 0. c. will be < 0 in a minimization problem and > 0 in a maximization problem. d. will always equal 0. ANSWER: b POINTS: 1 TOPICS: Dual price Subjective Short Answer 41. In a linear programming problem, the binding constraints for the optimal solution are 5X + 3Y ≤ 30 2X + 5Y ≤ 20 a.
b.
Fill in the blanks in the following sentence: As long as the slope of the objective function stays between _______ and _______, the current optimal solution point will remain optimal. Which of these objective functions will lead to the same optimal solution? 1) 2X + 1Y 2) 7X + 8Y 3) 80X + 60Y 4) 25X + 35Y
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution ANSWER:
a. −5/3 and −2/5 b. Objective functions 2), 3), and 4) POINTS: 1 TOPICS: Graphical sensitivity analysis 42. The optimal solution of the linear programming problem is at the intersection of constraints 1 and 2. Max s.t.
2x1 + x2 4x1 + 1x2 ≤ 400 4x1 + 3x2 ≤ 600 1x1 + 2x2 ≤ 300 x1 , x2 ≥ 0
Over what range can the coefficient of x1 vary before the current solution is no longer optimal? b. Over what range can the coefficient of x2 vary before the current solution is no longer optimal? c. Compute the dual prices for the three constraints. ANSWER: a. 1.33 ≤ c1 ≤ 4 b. .5 ≤ c2 ≤ 1.5 c. Dual prices are .25, .25, 0 POINTS: 1 TOPICS: Graphical sensitivity analysis a.
43. The binding constraints for this problem are the first and second. Min
x1 + 2x2
s.t.
x1 + x2 ≥ 300 2x1 + x2 ≥ 400 2x1 + 5x2 ≤ 750 x1 , x2 ≥ 0
Keeping c2 fixed at 2, over what range can c1 vary before there is a change in the optimal solution point? b. Keeping c1 fixed at 1, over what range can c2 vary before there is a change in the optimal solution point? c. If the objective function becomes Min 1.5x1 + 2x2, what will be the optimal values of x1, x2, and the objective function? d. If the objective function becomes Min 7x1 + 6x2, what constraints will be binding? e. Find the dual price for each constraint in the original problem. ANSWER: a. .8 ≤ c1 ≤ 2 b. 1 ≤ c2 ≤ 2.5 c. x1 = 250, x2 = 50, z = 475 d. Constraints 1 and 2 will be binding. a.
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution e. Dual prices are .33, 0, .33 (The first and third values are negative.) POINTS: 1 TOPICS: Graphical sensitivity analysis 44. Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a maximization objective function and all ≤ constraints. Input Section Objective Function Coefficients X Y 4 6 Constraints #1 #2 #3
3 3 1
5 2 1
Avail. 60 48 20
Variables Profit
13.333333 53.333333
4 24
77.333333
Constraint #1 #2 #3
Usage 60 48 17.333333
Slack 1.789E-11 -2.69E-11 2.6666667
Output Section
a. Give the original linear programming problem. b. Give the complete optimal solution. ANSWER: a. Max 4X + 6Y 3X + 5Y ≤ 60 3X + 2Y ≤ 48 1X + 1Y ≤ 20 X,Y≥0 The complete optimal solution is X = 13.333, Y = 4, Z = 73.333, S1 = 0, S2 = 0, S3 = 2.667 s.t.
b.
POINTS: 1 TOPICS: Spreadsheet solution of LPs 45. Excel's Solver tool has been used in the spreadsheet below to solve a linear programming problem with a minimization objective function and all ≥ constraints. Input Section Objective Function Coefficients X 5 Cengage Learning Testing, Powered by Cognero
Y 4 Page 10
Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution Constraints #1 #2 #3
4 2 9
3 5 8
Req'd 60 50 144
Variables Profit
9.6 48
7.2 28.8
76.8
Constraint #1 #2 #3
Usage 60 55.2 144
Slack 1.35E-11 -5.2 -2.62E-11
Output Section
a. Give the original linear programming problem. b. Give the complete optimal solution. ANSWER: a. Min 5X + 4Y 4X + 3Y ≥ 60 2X + 5Y ≥ 50 9X + 8Y ≥ 144 X,Y≥0 The complete optimal solution is X = 9.6, Y = 7.2, Z = 76.8, S1 = 0, S2 = 5.2, S3 = 0 s.t.
b.
POINTS: 1 TOPICS: Spreadsheet solution of LPs 46. Use the spreadsheet and Solver sensitivity report to answer these questions. a. What is the cell formula for B12? b. What is the cell formula for C12? c. What is the cell formula for D12? d. What is the cell formula for B15? e. What is the cell formula for B16? f. What is the cell formula for B17? g. What is the optimal value for x1? h. What is the optimal value for x2? i. Would you pay $.50 each for up to 60 more units of resource 1? Is it possible to figure the new objective function value if the profit on product 1 increases by j. a dollar, or do you have to rerun Solver? A 1 2 3 4 5 6 7
B
C
D
E
Var. 1 2 3 1
Var. 2 5 1 1
(type) < < >
Avail. 40 30 12
Input Information Constraint 1 Constraint 2 Constraint 3
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 8 Profit 9 10 Output Information 11 Variables 12 Profit 13 14 Resources 15 Constraint 1 16 Constraint 2 17 Constraint 3 18 19
5
4
= Total Used
Slack/Surplus
Name Variable 1 Variable 2
Final Reduced Value Cost 8.461538462 0 4.615384615 0
Objective Coefficient 5 4
Final Value
Shadow Price
Constraint Allowable R.H. Side Increase
40
0.538461538 40
110
7
30
1.307692308 30
30
4.666666667
Sensitivity Report Changing Cells Cell $B$12 $C$12
Allowable Increase 7 8.5
Allowable Decrease 3.4 2.333333333
Constraints Cell $B$15 $B$16 $B$17
Name constraint 1 Used constraint 2 Used constraint 3 Used
13.07692308 0
12
Allowable Decrease
1.076923077 1E+30
ANSWER:
a. =B8*B11 b. =C8*C11 c. =B12+C12 d. =B4*B11+C4*C11 e. =B5*B11+C5*C11 f. =B6*B11+C6*C11 g. 8.46 h. 4.61 i. yes j. no POINTS: 1 TOPICS: Spreadsheet solution of LPs 47. Use the following Management Scientist output to answer the questions. LINEAR PROGRAMMING PROBLEM MAX 31X1+35X2+32X3 Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution S.T. 1) 3X1+5X2+2X3>90 2) 6X1+7X2+8X3<150 3) 5X1+3X2+3X3<120 OPTIMAL SOLUTION Objective Function Value = 763.333 Variable X1 X2 X3
Value 13.333 10.000 0.000
Reduced Cost 0.000 0.000 10.889
Constraint 1 2 3
Slack/Surplus 0.000 0.000 23.333
Dual Price −0.778 5.556 0.000
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3
Lower Limit 30.000 No Lower Limit No Lower Limit
Current Value 31.000 35.000 32.000
Upper Limit No Upper Limit 36.167 42.889
Current Value 90.000 150.000 120.000
Upper Limit 107.143 163.125 No Upper Limit
RIGHT HAND SIDE RANGES Constraint 1 2 3
Lower Limit 77.647 126.000 96.667
a. Give the solution to the problem. b. Which constraints are binding? c. What would happen if the coefficient of x1 increased by 3? d. What would happen if the right-hand side of constraint 1 increased by 10? ANSWER: a. x1 = 13.33, x2 = 10, x3 = 0, s1 = 0, s2 = 0, s3 = 23.33, z = 763.33 b. Constraints 1 and 2 are binding. c. The value of the objective function would increase by 40. d. The value of the objective function would decrease by 7.78. POINTS: 1 TOPICS: Interpretation of Management Scientist output 48. Use the following Management Scientist output to answer the questions. MIN 4X1+5X2+6X3 S.T. 1) X1+X2+X3<85 2) 3X1+4X2+2X3>280 3) 2X1+4X2+4X3>320 Objective Function Value = 400.000 Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution Variable X1 X2 X3
Value 0.000 80.000 0.000
Reduced Cost 1.500 0.000 1.000
Constraint 1 2 3
Slack/Surplus 5.000 40.000 0.000
Dual Price 0.000 0.000 −1.250
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3
Lower Limit 2.500 0.000 5.000
Current Value 4.000 5.000 6.000
Upper Limit No Upper Limit 6.000 No Upper Limit
Current Value 85.000 280.000 320.000
Upper Limit No Upper Limit 320.000 340.000
RIGHT HAND SIDE RANGES Constraint 1 2 3
Lower Limit 80.000 No Lower Limit 280.000
a. What is the optimal solution, and what is the value of the profit contribution? b. Which constraints are binding? c. What are the dual prices for each resource? Interpret. d. Compute and interpret the ranges of optimality. e. Compute and interpret the ranges of feasibility. ANSWER: a. x1 = 0, x2 = 80, x3 = 0, s1 = 5, s2 = 40, s3 = 0, Z = 400 b. Constraint 3 is binding. c. Dual prices are 0, 0, and −1.25. They measure the improvement in Z per unit increase in each right-hand side. d. 2.5 ≤ c1 < ∞ 0 ≤ c2 ≤ 6 5 ≤ c3 < ∞
e.
As long as the objective function coefficient stays within its range, the current optimal solution point will not change, although Z could. 80 ≤ b1 < ∞ −∞ < b2 ≤ 320 280 ≤ b3 ≤ 340 As long as the right-hand side value stays within its range, the currently binding constraints will remain so, although the values of the decision variables could change. The dual variable values will remain the same.
POINTS: 1 TOPICS: Interpretation of Management Scientist output 49. The following linear programming problem has been solved by The Management Scientist. Use the output to answer the questions. Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution LINEAR PROGRAMMING PROBLEM MAX 25X1+30X2+15X3 S.T. 1) 4X1+5X2+8X3<1200 2) 9X1+15X2+3X3<1500 OPTIMAL SOLUTION Objective Function Value = 4700.000 Variable X1 X2 X3
Value 140.000 0.000 80.000
Reduced Cost 0.000 10.000 0.000
Constraint 1 2
Slack/Surplus 0.000 0.000
Dual Price 1.000 2.333
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3
Lower Limit 19.286 No Lower Limit 8.333
Current Value 25.000 30.000 15.000
Upper Limit 45.000 40.000 50.000
Current Value 1200.000 1500.000
Upper Limit 4000.000 2700.000
RIGHT HAND SIDE RANGES Constraint 1 2 a. b. c. d. e.
Lower Limit 666.667 450.000
Give the complete optimal solution. Which constraints are binding? What is the dual price for the second constraint? What interpretation does this have? Over what range can the objective function coefficient of x2 vary before a new solution point becomes optimal? By how much can the amount of resource 2 decrease before the dual price will change?
What would happen if the first constraint's right-hand side increased by 700 and the second's decreased by 350? ANSWER: a. x1 = 140, x2 = 0, x3 = 80, s1 = 0, s2 = 0, z = 4700 b. Constraints 1 and 2 are binding. Dual price 2 = 2.33. A unit increase in the right-hand side of constraint 2 will increase the c. value of the objective function by 2.33. d. As long as c2 ≤ 40, the solution will be unchanged. e. 1050 The sum of percentage changes is 700/2800 + (−350)/(−1050) < 1 so the solution will not f. change. POINTS: 1 f.
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution TOPICS: Interpretation of Management Scientist output 50. LINDO output is given for the following linear programming problem. MIN 12 X1 + 10 X2 + 9 X3 SUBJECT TO 2) 5 X1 + 8 X2 + 5 X3 >= 60 3) 8 X1 + 10 X2 + 5 X3 >= 80 END LP OPTIMUM FOUND AT STEP 1 OBJECTIVE FUNCTION VALUE 1) 80.000000 VARIABLE X1 X2 X3 ROW 2) 3)
VALUE .000000 8.000000 .000000
REDUCED COST 4.000000 .000000 4.000000
SLACK OR SURPLUS 4.000000 .000000
DUAL PRICE .000000 −1.000000
NO. ITERATIONS= 1 RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE X1 X2 X3
ROW 2 3
CURRENT COEFFICIENT 12.000000 10.000000 9.000000 CURRENT RHS 60.000000 80.000000
OBJ. COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE INFINITY 4.000000 5.000000 10.000000 INFINITY 4.000000
RIGHTHAND SIDE RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 4.000000 INFINITY INFINITY 5.000000
a. What is the solution to the problem? b. Which constraints are binding? c. Interpret the reduced cost for x1. d. Interpret the dual price for constraint 2. e. What would happen if the cost of x1 dropped to 10 and the cost of x2 increased to 12? ANSWER: a. x1 = 0, x2 = 8, x3 = 0, s1 = 4, s2 = 0, z = 80 b. Constraint 2 is binding. c. c1 would have to decrease by 4 or more for x1 to become positive. d. Increasing the right-hand side by 1 will cause a negative improvement, or increase, of 1 in Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution this minimization objective function. The sum of the percentage changes is (−2)/(−4) + 2/5 < 1 so the solution would not change.
e. POINTS: 1 TOPICS: Interpretation of LINDO output
51. The LP problem whose output follows determines how many necklaces, bracelets, rings, and earrings a jewelry store 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. Constraints 3 and 4 are marketing restrictions. LINEAR PROGRAMMING PROBLEM MAX 100X1+120X2+150X3+125X4 S.T. 1) X1+2X2+2X3+2X4<108 2) 3X1+5X2+X4<120 3) X1+X3<25 4) X2+X3+X4>50 OPTIMAL SOLUTION Objective Function Value = 7475.000 Variable X1 X2 X3 X4
Value 8.000 0.000 17.000 33.000
Reduced Cost 0.000 5.000 0.000 0.000
Constraint 1 2 3 4
Slack/Surplus 0.000 63.000 0.000 0.000
Dual Price 75.000 0.000 25.000 −25.000
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3 X4
Lower Limit 87.500 No Lower Limit 125.000 120.000
Current Value 100.000 120.000 150.000 125.000
Upper Limit No Upper Limit 125.000 162.500 150.000
Current Value 108.000 120.000 25.000 50.000
Upper Limit 123.750 No Upper Limit 58.000 54.000
RIGHT HAND SIDE RANGES Constraint 1 2 3 4
Lower Limit 100.000 57.000 8.000 41.500
Use the output to answer the questions. a.
How many necklaces should be stocked?
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution b. c. d. e. f. g. h. i. j. k.
Now many bracelets should be stocked? How many rings should be stocked? How many earrings should be stocked? How much space will be left unused? How much time will be used? By how much will the second marketing restriction be exceeded? What is the profit? To what value can the profit on necklaces drop before the solution would change? By how much can the profit on rings increase before the solution would change? By how much can the amount of space decrease before there is a change in the profit? You are offered the chance to obtain more space. The offer is for 15 units and the total price l. is 1500. What should you do? ANSWER: a. 8 b. 0 c. 17 d. 33 e. 0 f. 57 g. 0 h. 7475 i. 87.5 j 12.5 k. 0 l. Say no. Although 15 units can be evaluated, their value (1125) is less than the cost (1500). POINTS: 1 TOPICS: Interpretation of Management Scientist output 52. The decision variables represent the amounts of ingredients 1, 2, and 3 to put into a blend. The objective function represents profit. The first three constraints measure the usage and availability of resources A, B, and C. The fourth constraint is a minimum requirement for ingredient 3. Use the output to answer these questions. a. How much of ingredient 1 will be put into the blend? b. How much of ingredient 2 will be put into the blend? c. How much of ingredient 3 will be put into the blend? d. How much resource A is used? e. How much resource B will be left unused? f. What will the profit be? g. What will happen to the solution if the profit from ingredient 2 drops to 4? h. What will happen to the solution if the profit from ingredient 3 increases by 1? i. What will happen to the solution if the amount of resource C increases by 2? What will happen to the solution if the minimum requirement for ingredient 3 increases to j. 15? LINEAR PROGRAMMING PROBLEM MAX 4X1+6X2+7X3 S.T. 1) 3X1+2X2+5X3<120 2) 1X1+3X2+3X3<80 3) 5X1+5X2+8X3<160 4) +1X3>10 Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution OPTIMAL SOLUTION Objective Function Value = 166.000 Variable X1 X2 X3
Value 0.000 16.000 10.000
Reduced Cost 2.000 0.000 0.000
Constraint 1 2 3 4
Slack/Surplus 38.000 2.000 0.000 0.000
Dual Price 0.000 0.000 1.200 −2.600
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3
Lower Limit No Lower Limit 4.375 No Lower Limit
Current Value 4.000 6.000 7.000
Upper Limit 6.000 No Upper Limit 9.600
Current Value 120.000 80.000 160.000 10.000
Upper Limit No Upper Limit No Upper Limit 163.333 20.000
RIGHT HAND SIDE RANGES Constraint 1 2 3 4 ANSWER:
Lower Limit 82.000 78.000 80.000 8.889
a. 0 b. 16 c. 10 d. 44 e. 2 f. 166 g. rerun h. Z = 176 i. Z = 168.4 j. Z = 153 POINTS: 1 TOPICS: Interpretation of Management Scientist output 53. The LP model and LINDO output represent a problem whose solution will tell a specialty retailer how many of four different styles of umbrellas to stock in order to maximize profit. It is assumed that every one stocked will be sold. The variables measure the number of women's, golf, men's, and folding umbrellas, respectively. The constraints measure storage space in units, special display racks, demand, and a marketing restriction, respectively. MAX 4 X1 + 6 X2 + 5 X3 + 3.5 X4 SUBJECT TO 2) 2 X1 + 3 X2 + 3 X3 + X4 <= 120 3) 1.5 X1 + 2 X2 <= 54 4) 2 X2 + X3 + X4 <= 72 Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution 5) X2 + X3 >= 12 END OBJECTIVE FUNCTION VALUE 1) 318.00000 VARIABLE X1 X2 X3 X4 ROW 2) 3) 4) 5)
VALUE 12.000000 .000000 12.000000 60.000000
REDUCED COST .000000 .500000 .000000 .000000
SLACK OR SURPLUS .000000 36.000000 .000000 .000000
DUAL PRICE 2.000000 .000000 1.500000 −2.500000
RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE X1 X2 X3 X4
ROW 2 3 4 5
CURRENT COEFFICIENT 4.000000 6.000000 5.000000 3.500000 CURRENT RHS 120.000000 54.000000 72.000000 12.000000
OBJ. COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 1.000000 2.500000 .500000 INFINITY 2.500000 .500000 INFINITY .500000
RIGHTHAND SIDE RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 48.000000 24.000000 INFINITY 36.000000 24.000000 48.000000 12.000000 12.000000
Use the output to answer the questions. a. b. c. d. e. f. g. h.
How many women's umbrellas should be stocked? How many golf umbrellas should be stocked? How many men's umbrellas should be stocked? How many folding umbrellas should be stocked? How much space is left unused? How many racks are used? By how much is the marketing restriction exceeded? What is the total profit? By how much can the profit on women's umbrellas increase before the solution would i. change? j. To what value can the profit on golf umbrellas increase before the solution would change? k. By how much can the amount of space increase before there is a change in the dual price? You are offered an advertisement that should increase the demand constraint from 72 to 86 l. for a total cost of $20. Would you say yes or no? ANSWER: a. 12 Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution b. c. d. e. f. g. h. i. j. k. l.
0 12 60 0 18 0 318 1 6.5 48 Yes. The dual price is 1.5 for 24 additional units. The value of the ad (14)(1.5)=21 exceeds the cost of 20.
POINTS: 1 TOPICS: Interpretation of solution 54. Eight of the entries have been deleted from the LINDO output that follows. Use what you know about linear programming to find values for the blanks. MIN 6 X1 + 7.5 X2 + 10 X3 SUBJECT TO 2) 25 X1 + 35 X2 + 30 X3 >= 2400 3) 2 X1 + 4 X2 + 8 X3 >= 400 END LP OPTIMUM FOUND AT STEP 2 OBJECTIVE FUNCTION VALUE 1) 612.50000 VARIABLE X1 X2 X3 ROW 2) 3)
VALUE ________ ________ 27.500000
REDUCED COST 1.312500 ________ ________
SLACK OR SURPLUS ________ ________
DUAL PRICE −.125000 −.781250
NO. ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE X1 X2 X3
CURRENT COEFFICIENT 6.000000 7.500000 10.000000
OBJ. COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE _________ _________ 1.500000 2.500000 5.000000 3.571429 RIGHTHAND SIDE RANGES
ROW
CURRENT
ALLOWABLE
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ALLOWABLE Page 21
Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution RHS INCREASE DECREASE 2 2400.000000 1100.000000 900.000000 3 400.000000 240.000000 125.714300 ANSWER: It is easiest to calculate the values in this order. x1 = 0, x2 = 45, reduced cost 2 = 0, reduced cost 3 = 0, row 2 slack = 0, row 3 slack = 0, c1 allowable decrease = 1.3125, allowable increase = infinity POINTS: 1 TOPICS: Interpretation of solution 55. Portions of a Management Scientist output are shown below. Use what you know about the solution of linear programs to fill in the ten blanks. LINEAR PROGRAMMING PROBLEM MAX 12X1+9X2+7X3 S.T. 1) 3X1+5X2+4X3<150 2) 2X1+1X2+1X3<64 3) 1X1+2X2+1X3<80 4) 2X1+4X2+3X3>116 OPTIMAL SOLUTION Objective Function Value = 336.000 Variable X1 X2 X3
Value ______ 24.000 ______
Reduced Cost 0.000 ______ 3.500
Constraint 1 2 3 4
Slack/Surplus 0.000 ______ ______ 0.000
Dual Price 15.000 0.000 0.000 ______
OBJECTIVE COEFFICIENT RANGES Variable X1 X2 X3
Lower Limit 5.400 2.000 No Lower Limit
Current Value 12.000 9.000 7.000
Upper Limit No Upper Limit 20.000 10.500
RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value 1 145.000 150.000 2 ______ ______ 3 ______ ______ 4 110.286 116.000 ANSWER: x3 = 0 because the reduced cost is positive.
Upper Limit 156.667 64.000 80.000 120.000
x1 = 24 after plugging into the objective function Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution The second reduced cost is 0. s2 = 20 and s3 = 22 from plugging into the constraints. The fourth dual price is −16.5 from plugging into the dual objective function, which your students might not understand fully until Chapter 6. The lower limit for constraint 2 is 44 and for constraint 3 is 58, from the amount of slack in each constraint. There are no upper limits for these constraints. POINTS: 1 TOPICS: Interpretation of solution 56. A large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: the adult Open Trail, the adult Cityscape, the girl's Sea Sprite, and the boy's Trail Blazer. It is assumed that every bike ordered will be sold, and their profits, respectively, are 30, 25, 22, and 20. The LP model should maximize profit. There are several conditions that the store needs to worry about. One of these is space to hold the inventory. An adult's bike needs two feet, but a child's bike needs only one foot. The store has 500 feet of space. There are 1200 hours of assembly time available. The child's bike need 4 hours of assembly time; the Open Trail needs 5 hours and the Cityscape needs 6 hours. The store would like to place an order for at least 275 bikes. a. Formulate a model for this problem. b. Solve your model with any computer package available to you. c. How many of each kind of bike should be ordered and what will the profit be? d. What would the profit be if the store had 100 more feet of storage space? e. If the profit on the Cityscape increases to $35, will any of the Cityscape bikes be ordered? f. Over what range of assembly hours is the dual price applicable? If we require 5 more bikes in inventory, what will happen to the value of the optimal g. solution? h. Which resource should the company work to increase, inventory space or assembly time? ANSWER: NOTE TO INSTRUCTOR: The problem is suitable for a take-home or lab exam. The student must formulate the model, solve the problem with a computer package, and then interpret the solution to answer the questions. a.
MAX 30 X1 + 25 X2 + 22 X3 + 20 X4 SUBJECT TO 2) 2 X1 + 2 X2 + X3 + X4 <= 500 3) 5 X1 + 6 X2 + 4 X3 + 4 X4 <= 1200 4) X1 + X2 + X3 + X4 >= 275
b.
OBJECTIVE FUNCTION VALUE 1) 6850.0000 VARIABLE X1 X2 X3 X4
VALUE 100.000000 .000000 175.000000 .000000
REDUCED COST .000000 13.000000 .000000 2.000000
ROW 2) 3) 4)
SLACK OR SURPLUS 125.000000 .000000 .000000
DUAL PRICE .000000 8.000000 −10.000000
NO. ITERATIONS= 2 RANGES IN WHICH THE BASIS IS UNCHANGED: Cengage Learning Testing, Powered by Cognero
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution
X1 X2 X3 X4
CURRENT COEFFICIENT 30.000000 25.000000 22.000000 20.000000
OBJ. COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE INFINITY 2.500000 13.000000 INFINITY 2.000000 2.000000 2.000000 INFINITY
ROW 2 3 4
CURRENT RHS 500.000000 1200.000000 275.000000
RIGHTHAND SIDE RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE INFINITY 125.000000 125.000000 100.000000 25.000000 35.000000
VARIABLE
c.
Order 100 Open Trails, 0 Cityscapes, 175 Sea Sprites, and 0 Trail Blazers. Profit will be 6850. 6850 No. The $10 increase is below the reduced cost. 1100 to 1325 It will decrease by 50. Assembly time.
d. e. f. g. h. POINTS: 1 TOPICS: Formulation and computer solution
57. A company produces two products made from aluminum and copper. The table below gives the unit requirements, the unit production man-hours required, the unit profit and the availability of the resources (in tons). Product 1 Product 2 Available
Aluminum 1 1 10
Copper 0 1 6
Man-hours 2 3 24
Unit Profit 50 60
The Management Scientist provided the following solution output: Objective Function Value = 540.000 VARIABLE X1 X2 CONSTRAINT 1 2 3
VALUE 6.000 4.000
REDUCED COST 0.000 0.000
SLACK/SURPLUS .000 2.000 0.000
DUAL PRICE 30.000 0.000 10.000
RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE X1 X2
CURRENT COEFFICIENT 50.000 60.000
OBJ. COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 10.000 10.000 15.000 10.000 RIGHTHAND SIDE RANGES
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution CONSTRAINT 1 2 3
CURRENT RHS 10.000 6.000 24.000
ALLOWABLE INCREASE 2.000 INFINITY 2.000
ALLOWABLE DECREASE 1.000 2.000 4.000
a.
What is the optimal production schedule? Within what range for the profit on product 2 will the solution in (a) remain optimal? What is b. the optimal profit when c2 = 70? c. Suppose that simultaneously the unit profits on x1 and x2 changed from 50 to 55 and 60 to 65 respectively. Would the optimal solution change? Explain the meaning of the "DUAL PRICES" column. Given the optimal solution, why d. should the dual price for copper be 0? e. What is the increase in the value of the objective function for an extra unit of aluminum? Man-hours were not figured into the unit profit as it must pay three workers for eight hours f. of work regardless of the number of man-hours used. What is the dual price for man-hours? Interpret. On the other hand, aluminum and copper are resources that are ordered as needed. The unit profit coefficients were determined by: (selling price per unit) - (cost of the resources per g. unit). The 10 units of aluminum cost the company $100. What is the most the company should be willing to pay for extra aluminum? ANSWER: a. 6 product 1, 4 product 2, Profit = $540 b. Between $50 and $75; at $70 the profit is $580 c. No; total % change is 83 1/3% < 100% Dual prices are the shadow prices for the resources; since there was unused copper (because d. S2 = 2), extra copper is worth $0 e. $30 f. $10; this is the amount extra man-hours are worth The shadow price is the "premium" for aluminum -- would be willing to pay up to $10 + $30 g. = $40 for extra aluminum POINTS: 1 TOPICS: Interpretation of solution 58. Given the following linear program: MAX
5x1 + 7x2
s.t.
x1 ≤ 6 2x1 + 3x2 ≤ 19 x1 + x2 ≤ 8 x1 , x2 ≥ 0
The graphical solution to the problem is shown below. From the graph we see that the optimal solution occurs at x1 = 5, x2 = 3, and z = 46.
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Chapter 3 - Linear Programming: Sensitivity Analysis and Interpretation of Solution
a. Calculate the range of optimality for each objective function coefficient. b. Calculate the dual price for each resource. ANSWER: a. Ranges of optimality: 14/3 ≤ c1 ≤ 7 and 5 ≤ c2 ≤ 15/2 Summarizing, the dual price for the first resource is 0, for the second resource is 2, and for b. the third is 1 POINTS: 1 TOPICS: Introduction to sensitivity analysis 59. Consider the following linear program: MAX
3x1 + 4x2 ($ Profit)
s.t.
x1 + 3x2 ≤ 12 2x1 + x 2 ≤ 8 x1 ≤ 3 x1 , x2 ≥ 0
The Management Scientist provided the following solution output: OPTIMAL SOLUTION Objective Function Value = 20.000 Variable X1 X2
Value 2.400 3.200
Reduced Cost 0.000 0.000
Constraint
Slack/Surplus
Dual Price
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