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DSCI 3870: Management Science Exam # 3, Version - Green Date December 12th,2013 Allotted Time 120 Min Name SID Please read this carefully The questions, which you attempt today, consist of True/False and multiplechoice questions worth 133 points total. Some of these questions are based on descriptive cases. There are a total of 37 questions. Please answer all the questions on the scantron sheet provided in the order that they appear. After you are done, please turn in the scantron and this question booklet. Please note that you have to enter your name and Student ID Number (SID#) in the above area and on the scantron. Failure to do so will result in a grade of zero. This is an open book, open notes exam. Laptop computers are also allowed. On the exam the following acronyms may have been used: LP - Linear Program/Programming IP – Integer Program/Programming ILP – Integer Linear Program/Programming (used synonymously with IP) NLP – Non-linear Program/ Programming MILP – Mixed Integer Linear Program/ Programming QP – Quadratic Program/Programming An “(s)” appended to these acronyms denotes the plural Be sure to allocate your time wisely between the multiple choice and T/F questions. Best of luck!! 1 1. In an all-integer linear program, A. all variables must be integer*. B. all objective function coefficients and right-hand side values must be integer. C. all objective function coefficients must be integer. D. all right-hand side values must be integer. 2. The objective of the transportation problem is to A. identify one origin that can satisfy total demand at the destinations and at the same time minimize total shipping cost. B. minimize the number of origins used to satisfy total demand at the destinations. C. minimize the cost of shipping products from several origins to several destinations*. D. minimize the number of shipments necessary to satisfy total demand at the destinations. 3. The assignment problem constraint x31 + x32 + x33 + x34 < 2 means A. agent 3 can be assigned to 2 tasks.* B. agent 2 can be assigned to 3 tasks. C. a mixture of agents 1, 2, 3, and 4 will be assigned to tasks. D. there is no feasible solution. 4. The measure of risk most often associated with the Markowitz portfolio model is the A. portfolio average return. B. portfolio minimum return. C. portfolio variance.* D. portfolio standard deviation. 5. We assume in the maximal flow problem that A. the flow out of a node is equal to the flow into the node.* B. the source and sink nodes are at opposite ends of the network. C. the number of arcs entering a node is equal to the number of arcs exiting the node. D. None of the alternatives is correct. 6. Dual or shadow prices, similar to those that we get for linear programming models, are mathematically impossible to derive for integer programming models. A. True* B. False 7. Whenever total supply is less than total demand in a transportation problem, the LP model does not determine how the unsatisfied demand is handled. A. True* B. False 2 8. In a model, x1 > 0 and integer, x2 > 0, and x3 = 0, 1. Which solution would not be feasible? A. x1 = 5, x2 = 3, x3 = 0 B. x1 = 4, x2 = .389, x3 = 1 C. x1 = 2, x2 = 3, x3 = .578* D. x1 = 0, x2 = 8, x3 = 0 The next two questions are based on the following case: Optimal Ovens, Inc. (OOI) makes home toaster ovens at plants in Alabama and Wisconsin. Completed ovens are shipped by rail to one of OOI’s warehouses in Memphis and Pittsburgh and then distributed to customer facilities in Fresno, Peoria and Newark. The two warehouses can also transfer small quantities between themselves using company trucks. The task is to plan OOI’s distribution over the next month. Each plant can ship up to 1000 units during this period and none are presently in Fresno, Peoria and Newark. Fresno, Peoria and Newark customers require 450, 500 and 610 ovens respectively. Transfers between warehouses are limited to 25 ovens but no cost is charged. The network flow diagram is given below. Wisconsin 1 Memphis 5 3 6 2 Fresno Peoria 4 7 Alabama Pittsburgh Newark Unit costs (cij) of the flow (Fij)on arc (i, j) are detailed in the following tables: From/To 3. Memphis 4. Pittsburgh 1. Wisconsin 7 8 2. Alabama 4 7 From/To 5. Fresno 6. Peoria 7. Newark 3. Memphis 25 5 17 4. Pittsburgh 29 8 5 3 9. The overall flow constraint at Memphis is given as: A. - F13 - F23 - F43 + F34 + F35 + F36 + F37 = 0* B. +F13 + F23 + F43 – F34 – F35 – F36 – F37 <= 0 C. -F13 - F23 - F43 = F34 + F35 + F36 + F37 D. +F13 + F23 + F43 – F34 – F35 – F36 – F37 >= 0 10. Which of the following represents the objective function? A. Max. 7F31 + 8F41 + 4F32 + 7F42 + 25F53 + 5F63 + 17F73 + 29F54 + 8F64 + 5F74 B. Min. 7F31 + 8F41 + 4F32 + 7F42 + 25F53 + 5F63 + 17F73 + 29F54 + 8F64 + 5F74 C. Min. F13 + F14 + F23 + F24 + F35 + F36 + F37 + F45 + F46 + F47 D. Min. 7F13 + 8F14 + 4F23 + 7F24 + 25F35 + 5F36 + 17F37 + 29F45 + 8F46 + 5F47* Read the following case and answer the three questions that follow: Fix-It shop has received four luxury cars Aston Martin (A), Bentley (B), Carrera (C) and Daimler (D) to repair. Five workers with different talents are available to repair the cars. The fix-it shop owner has an estimate of the profit if a particular worker does a particular car. The profits in dollars per hour, which are shown in the table differ because the owner believes that each worker will differ in speed and skill on these quite varied jobs. Assume that the variables are Xij where i=A,B,C,D denote cars and j=1,2,3,4,5 denote workers, and assume that the owner wants to maximize his profits by repairing all the cars and making sure that each car is assigned to a single worker and each worker works on at most one car. Worker # 1 2 3 4 5 A 25 30 28 27 24 B 31 25 24 22 19 C 18 35 31 32 19 D 20 22 25 24 22 Car 11. The constraint for person 5 is given as: A. 25XA5 + 31XB5 + 18XC5 + 20XD5 =1 B. 25XA5 + 31XB5 + 18XC5 + 20XD5 ≤ 1 C. XA5 + XB5 + XC5 + XD5 ≥ 1 D. XA5 + XB5 + XC5 + XD5 ≤ 1* 4 12. The constraint for Aston Martin is given as: A. XA1 + XA2 + XA3 + XA4 =1 B. XA1 + XA2 + XA3 + XA4 ≤ 1 C. XA1 + XA2 + XA3 + XA4 + XA5 =1* D. 25XA1 + 30XA2 + 28XA3 + 27XA4 + 24XA5 ≥ 1 13. Using the Greedy Heuristic discussed in class, what would be the assignment of cars to workers? A. Aston Martin5, Bentley1, Carrera2, Daimler4 B. Aston Martin5, Bentley2, Carrera3, Daimler4 C. Aston Martin3, Bentley5, Carrera1, Daimler4 D. Aston Martin3, Bentley1, Carrera2, Daimler4* 14. In class we saw the Max Cover problem for the state of Ohio. Based on the map of the optimal solution to such a Max Cover problem shown below, how many counties, excluding the hubs, have been covered to achieve maximum coverage? A. B. C. D. 25 22 19* 3 5 The next three questions are based on the following case: Consider the Excel implementation and subsequent Sensitivity Analysis for a typical portfolio optimization model below. Let x, y and z represent the fraction invested in stocks AT&T, GM and USS respectively. The “Average Return” and the information in the “Covariance Matrix” represent basic statistical analysis performed on returns data provided to us for AT&T, GM and USS. The optimization model embedded in MS Excel appears below the covariance matrix. Average Return Covariance Matrix AT&T GM USS Decision: Stock % Requirements Expected Return -0.47% AT&T 0.0014 -0.0016 -0.0006 65.00% <=65% -0.30% 7.61% GM -0.0016 0.0355 0.0141 5.78% <=75% 0.44% 33.76% USS -0.0006 0.0141 0.0244 29.22% <=70% 9.86% Total 100% =100% 10% Portfolio Variance 0.0029 >=10.0% Sensitivity Analysis Adjustable Cells Cell $C$19 $D$19 $E$19 Name Decision: Stock % AT&T Decision: Stock % GM Decision: Stock % USS Final Value 65.00% 5.78% 29.22% Reduced Gradient -0.75% 0.00% 0.00% Name Decision: Stock % Total Expected Return Total Final Value 100% 10% Lagrange Multiplier 1% 2% Constraints Cell $F$19 $F$21 15. The appropriate objective function is given as: A. Max. 0.65x2 + 0.0578 y2 + 0.2922 z2 B. Min. 0.0014x2 +0.0355 y2 + 0.0244 z2 0.0032 x y 0.012 x z + 0.0282 y z* C. Min. 0.65x2 + 0.0578 y2 + 0.2922 z2 0.0016 x y 0.006 x z + 0.0141 y z D. Max. 0.0014x2 +0.0355 y2 + 0.0244 z2 0.0016 x y 0.006 x z + 0.0141 y z 16. One of the constraints in the problem would be: A. 0.65x + 0.0578 y + 0.2922 z = 1.0 B. 0.65x + 0.0578 y + 0.2922 z ≤ 0.75 C. 0.003 x + 0.0044 y + 0.0986z = 0.1 D. 0.0047 x + 0.0761 y + 0.3376 z ≥ 0.1* 6 17. If the investor demands one percent more return, then the additional risk that He/she will have to bear would be approximately: A. 0.0002* B. 1.0029% C. 0.0129 D. 0.02 Consider the shaded feasible regions shown below: Region A Region B 18. Which of the following items is correct? A. both regions are non-convex* B. both regions are convex C. Region A is convex and Region B is concave D. Region A is non-convex and Region B is convex Consider the network below. Consider the LP for finding the shortest-route path from node 1 to node 7. Bidirectional arrows () indicate that travel is possible both ways. Let Xij = 1 if the route from node i to node j is taken and 0 otherwise. 8 4 2 6 10 12 3 1 5 12 4 3 7 3 7 9 7 6 19. Which of the following represents the constraint for Node 5? A. X25 X35 X45 X75 + X52 + X53 + X57 = 0* B. X25 X35 X45 X75 + X52 + X53 + X57 = 1 C. X25 7X35 3X45 + 8X52 + 7X53 + 3X54 + 4X57 = 0 D. X25 X35 X75 + X52 + X53 + X54 + X57 = 0 Read the following case and answer the three questions that follow: A summer camp recreation director is trying to choose activities for a rainy day. Information about possible choices is given in the table below. A higher numeric popularity rating is considered better. Category Time (minutes) Activity 1 - Painting 30 2 - Drawing 20 3 - Nature craft 30 Music 4 - Rhythm band 20 Sports 5 - Relay races 45 6 - Basketball 60 Computer 7 - Internet 45 8 - Writing 30 9 - Games 40 Let xi = 1 if activity i is chosen, 0 if not, for i = 1, …, 9 Art Popularity with Campers Popularity with Counselors 4 5 3 5 2 1 1 4 1 2 2 1 5 1 3 1 3 2 20. If the objective is to keep the campers happy, what should the objective function be? A. Max 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9* B. Max 30x1 + 20x2 + 30x3 + 20x4 + 45x5 + 60x6 + 45x7 + 30x8 + 40x9 C. Max 2x1 + 2x2 + x3 + 5x4 + 1x5 + 3x6 + 1x7 + 3x8 + 2x9 D. Min 4x1 + 5x2 + 3x3 + 5x4 + 2x5 + 1x6 + 1x7 + 4x8 + 1x9 21. Which of the following represents the constraint : “at most one art activity can be done.”? A. 4x1 + 5x2 +3 x3 1 B. 2x1 + 2x2 + x3 1 C. x1 + x2 + x3 1* D. 30x1 + 20x2 + 30x3 1 22. Which of the following represents the constraint : “if basketball is chosen, then music must be chosen” A. x6 5x4 B. 60x6 20x4 C. 5x6 x4 D. x6 x4* 8 Max. 54X1 - 9X12 + 78X2 -13 X22 St. X1 < 4 2 X2 < 12 3 X1 + 2 X2 < 18 Xi > 0 i 6 Z = 189 4 2 Z = 162 Z = 117 Feasible Region 2 4 The next two questions are based upon the following case: Consider the picture above/on the earlier page which shows a “Feasible Region” and several objective function contours. Answer the next two questions based on this picture. 23. The optimal solution will likely result in: A. 1 binding constraint B. 2 binding constraints C. no binding constraints* D. 3 binding constraints 24. The optimal solution will likely occur: A. in the interior of the feasible region* B. at a corner point C. at two corner points D. on the boundary of the feasible region 9 The next three questions are based upon the following case: The Westfall Company has a contract to produce 10,000 yards of garden hoses for a large discount chain. Westfall has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same. Fixed Cost to Variable Capacity Set Up Cost (yards) Production Run Per Hose 1 750 1.25 6000 2 500 1.50 7500 3 1000 1.00 4000 4 300 2.00 5000 Let Pi = the number of hoses produced on machine i and Ui = 1 if machine i is used, = 0 otherwise Machine 25. A. B. C. D. If the company wants to minimize total cost what will be the objective function? Min 1.25P1 + 1.5P2 + P3 + 2P4 Min750U1 + 500U2 + 1000U3 + 300U4 Min750U1 + 500U2 + 1000U3 + 300U4 + 1.25P1 + 1.5P2 + P3 + 2P4* Min500U2 + 300U4 + 1.25P1 + 1.5P2 + P3 + 2P4 26. A. B. C. D. What does the constraint “P1 + P2 + P3 + P4 ≥ 10000” capture? Total amount of production Minimum amount of production Maximum amount of production Minimum required contracted amount* 27. If we add the constraint of “U1 + U4 ≤ 1”, what does the constraint capture? A. Machine 1 and machine 4 should be used. B. If machine 4 is used, machine 1 cannot be used and vice versa.* C. Machine 1 and machine 4 should be used together. D. Neither machine 1 nor machine 4 should be used. The next four questions are based upon the following case: Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year. The projects, the number of engineers and the number of support personnel required for each project, and the expected profits for each project are summarized in the following table: 10 Project Engineers Required Support Personnel Required Profit ($1,000,000s) 1 2 3 4 5 6 20 55 47 38 90 63 15 45 50 40 70 70 1.0 1.8 2.0 1.5 3.6 2.2 28. Which of the following captures the constraint of “If either project 6 or project 4 is undertaken, then both must be undertaken”? A. P4 + P6 ≥1 B. P4+P6 = 0 C. P4 P6 ≥ 0 D. P4 P6 = 0* 29. What is the meaning of the constraint “P1 P2 ≥ 0”? A. Project 2 can be undertaken only if project 1 is undertaken.* B. If either project 2 or project 1 is undertaken, both must be undertaken C. If project 2 is undertaken, then project 1 cannot be undertaken. D. If project 1 is undertaken, then project 2 cannot be undertaken. 30. What is the meaning of the constraint “P3 + P5 ≥ 1”? A. If either project 3 or project 5 is undertaken, both must be undertaken B. Either project 3 or 5 must be undertaken.* C. If project 5 is undertaken, project 3 must not be undertaken and vice versa D. If project 3 is undertaken, then project 5 cannot be undertaken. 31. Which of the following captures the constraint: “project 2 cannot be undertaken unless both projects 6 and 4 are undertaken, however, when projects 6 and 4 are undertaken, then project 2 must be undertaken” A. P4 + P6 ≥ P2 and P4 ≥ P2, P6 ≥ P2 B. P4 ≥ P2, P6 ≥ P2 and P4 + P6 – 1 ≤ P2* C. P4 + P6≤ P2 and P4 ≥ P2, P6 ≥ P2 D. P4 ≥ P2, P6 ≥ P2 and P4 + P6 ≤ 2 P2 11 32. Consider a map-based solution of the sales-rep example discussed in class, which was somewhat different from the Max Cover or Set Cover problems. The map (where shaded counties are part of the optimal solution), suggests that we are optimally locating ___ sales reps. A. 10 B. 5* C. 3 D. 2 12 The next two questions are based on the following case: Consider the following integer programming problem and partial graphical solution which follows. Answer the next two questions based on this information. Payoff: 3.0 x + 5.0 y = -20.4 y 6 5 4 3 Constraint 1: 2.0 x + 3.0 y = 16.0 Constraint 3: 2.0 x + 0.0 y = 3.0 2 1 Constraint 2: 1.0 x + 4.0 y = 6.0 0 0 1 2 3 4 5 6 7 8 Optimal Decisions(x,y): ( 2.0, 4.0) Constraint 1: 2.0x + 3.0y <= 16.0 Constraint 2: 1.0x + 4.0y >= 6.0 Constraint 3: 2.0x + 0.0y >= 3.0 Objective function: 3 x + 5 y Subject to: 2 x > 3 2 x + 3 y < 16 x + 4y > 6 x > 0, y > 0 x - Integer, y - Integer 33. If the objective function were to be minimized, then the linear programming relaxation of the problem would yield an objective function value: A. slightly more than 11 B. exactly the same as the integer program C. slightly less than 11* D. slightly less than 18 34. If the objective function were to be maximized, then the optimal objective function value for the integer program would be. A. 24 B. 25 C. 26* D. 26.1 13 x 35. Consider below a screen shot of an exercise that we did in class. This represents the ______. A. Traveling Salesman Game* B. Transportation Game C. Shortest Path Game D. Max Flow Game 36. Consider below a picture that represents an optimal solution to a network model similar to one we discussed in class and as mentioned, this is one of the few models that can be solved to optimality using simply a “greedy” approach. We call this a _________. A. Shortest Path Model B. Multicommodity Network Flow Model C. Traveling Salesman Model D. Spanning Tree Model* 14 37. Consider below a picture that represents an optimal solution to the “Share of Choices” model similar to the one that we discussed in class. The optimal pizza covers __ customers. A. 5 B. 4 C. 3* D. 2 Salem Foods Consumer 1 2 3 4 5 6 7 8 Crust Cheese Sauce Thin Thick Mozzarella Blend Smooth Chunky 11 2 6 7 3 17 11 7 15 17 16 26 7 5 8 14 16 7 13 20 20 17 17 14 2 8 6 11 30 20 12 17 11 9 2 30 9 19 12 16 16 25 5 9 4 14 23 16 Design Variables L11 0 L21 1 L12 0 L22 1 L13 0 Mild L23 1 L14 0 Max Share Constraints LHS RHS 1 51 42 80 44 68 4 10 1 1 1 1 > > > > > > > > = = = = 1 1 1 1 1 1 1 1 1 1 1 1 26 14 29 25 15 22 30 16 Consumers Preferring (Y's) 1 0 0 0 0 0 1 1 15 Sausage Medium 27 1 16 29 5 12 23 30 L24 1 Hot 8 10 19 10 12 20 19 3 L34 0 Current Favorite 52 88 66 83 58 70 79 59