EMGT 501 HW #1 3.1-4 3.1-11 3.4-16 Due Day: Sep 12

1. EMGT 501 HW #1 3.1-4 3.1-11 3.4-16 Due Day: Sep 12

2. 3.1-4 Use the graphical method to solve the problem:

3. 3.1-11 The Omega Manufacturing Manufacturing Company has discontinued the production of a certain unprofitable product line. This act created considerable excess production capacity. Management is considering devoting this excess capacity to one ore more of three products; call them products 1, 2, and 3. The available capacity on the machines that might limit output is summarized in the following table: Available Time (Machine Hours per Week) 500 350 150 Machine Type Milling machine Lathe Grinder

4. The number of machine hours required for each unit of the respective products is Productivity coefficient (in machine hours per unit) Machine Type Milling machine Lathe Grinder Product 1 9 5 3 Product 2 3 4 0 Product 3 5 0 2 The sales department indicates that the sales potential for products 1 and 2 exceeds the maximum production rate and that the sales potential for product 3 is 20 units per week. The unit profit would be $50, $20, and $25, respectively, on products 1, 2, and 3. The objective is to determine how much of each product Omega should produce to maximize profit. (a) Formulate a linear programming model for this problem. (b) Use a computer to solve this model by the simplex method.

5. 3.4-16 A cargo plane has three compartments for storing cargo: front, center, and back. These compartments have capacity limits on both weight and space, as summarized below: Weight Capacity (Tons) Space Capacity (Cubic Feet) Compartment Front Center Back 12 18 10 7,000 9,000 5,000 Further more, the weight of the cargo in the respective compartment must be the same proportion of that compartment’s weight capacity to maintain the balance of the airplane.

6. The following four cargoes have been offered for shipment on an upcoming flight as space is available: Weight (Tons) Volume (Cubic Feet/Ton) Profit ($/Ton) Cargo 20 16 25 13 1 2 3 4 500 700 600 400 320 400 360 290 Any portion of these cargoes can be accepted. The objective is to determine how much (if any) of each cargo should be accepted and how to distribute each among the compartments to maximize the total profit for the flight. (a) Formulate a linear programming model for this problem. (b) Solve this model by the simplex method to find one of its multiple optimal solutions.

7. EMGT 501 HW #1 Answer 3-1-4 3-1-11 3-4-16

8. Max s.t. 3-1-4 (1) (2) (3) (4) (3) and (2) (1) 10 8 (4) 6 (13,5) Optimal Solution with Z=31 4 2 2 4 6 8 10 12

9. 3-1-11 (a) Let

10. (b)

11. 3-4-16

13. (b)

14. EMGT 501 HW #2 4.4-6(b) (c) Due Day: Sep. 19

15. 4.4-6 Consider the following problem. (b) Work through the simplex method step by step in tabular form. (c) Use a software package based on the simplex method to solve the problem.

16. EMGT 501 HW #2 Answer 4.4-6 (b), (c)

17. 4.4-6 (b) Basis Z X X X X X X RHS 1 2 3 4 5 6 Z 1 -2 -4 -3 0 0 0 0 X 0 3 4 2 1 0 0 60 4 X 0 2 1 2 0 1 0 40 5 X 0 1 3 2 0 0 1 80 6 Z 1 1 0 -1 1 0 0 60 X 0 3/4 1 1/2 1/4 0 0 15 2 X 0 5/4 0 3/2 -1/4 1 0 25 5 X 0 -5/4 0 1/2 -3/4 0 1 35 6 Z 1 11/6 0 0 5/6 2/3 0 230/3 X 0 1/3 1 0 1/3 -1/3 2 = Optimal solution ( x *, x *, x *, x *, x *, x *) ( 0 , 20 / 3 , 50 / 3 , 0 , 0 , 80 / 3 ) 1 2 3 4 5 6 = with Z 230 / 3 . 0 20/3 X 0 5/6 0 1 -1/6 2/3 0 50/3 3 X 0 -5/3 0 0 -2/3 -1/3 1 80/3 6 (c)

18. EMGT 501 HW #3 6.1-4 6.1-5 Dual Formulation of DEA

19. 6.1-4 For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead. Explain. (a) Maximize subject to (b) Maximize subject to and and

20. 6.1-5 Consider the following problem. Maximize subject to and (a) Construct the dual problem. (b) Use duality theory to show that the optimal solution for the primal problem has

21. Formulate the Dual Form of the DEA Problem Primal Form of DEA Max s.t.

22. EMGT 501 HW#3 Answer 6.1-4 6.1-5 Dual Formulation of DEA

23. 6.1-4 (a) Dual formulation becomes Min s.t. # of constraints of Dual = 3 # of constraints of Primal = 5 So, Dual is better than Primal because the size of B-1 in Dual is smaller than that of Primal.

24. (b) Dual formulation becomes Min s.t. # of constraints of Dual = 5 # of constraints of Primal = 3 So, Primal is better than Dual because the size of B-1 in Primal is smaller than that of Dual.

25. 6.1-5 (a) Min s.t. (b) It is clear that Z*=0, y1*=0, y2*=0.

26. Primal Form Max s.t. Dual Form Min s.t.

27. EMGT 501 HW#4 10.3-1 10.4-5

28. 10.3-1 You and several friends are about to prepare a lasagna dinner. The tasks to be performed, their immediate predecessors, and their estimated durations are as follows: Tasks that Task Task Description Must Precede Time A Buy the mozzarella cheese* 30 minutes B Slice the mozzarella A 5 minutes C Beat 2 eggs 2 minutes D Mix eggs and ricotta cheese C 3 minutes E Cut up onions and mushrooms 7 minutes F Cook the tomato sauce E 25 minutes G Boil large quantity of water 15 minutes H Boil the lasagna noodles G 10 minutes I Drain the lasagna noodles H 2 minutes J Assemble all the ingredients I, F, D, B 10 minutes K Preheat the oven 15 minutes L Bake the lasagna J, K 30 minutes

29. * There is none in the refrigerator. (a) Construct the project network for preparing this dinner. (b) Find all the paths and path lengths through this project network. Which of these paths is a critical path? (c) Find the earliest start time and earliest finish time for each activity. (d) Find the latest start time and latest finish time for each activity. (e) Find the slack for each activity. Which of the paths is a critical path? (f) Because of a phone call, you were interrupted for 6 minutes when you should have been cutting the onions and mushrooms. By how much will the dinner be delayed? If you use your food processor, which reduces the cutting time from 7 to 2 minutes, will the dinner still be delayed?

30. 10.4-5 Sharon Lowe, vice president for marketing for the Electronic Toys Company, is about to begin a project to design an advertising campaign for a new line of toys. She wants the project completed within 57 days in time to launch the advertising campaign at the beginning of the Christmas season. Sharon has identified the six activities (labeled A, B, …, F) needed to execute this project. Considering the order in which these activities need to occur, she also has constructed the following project network. A C E F START FINISH B D

31. Using the PERT three-estimate approach, Sharon has obtained the following estimates of the duration of each activity. Optimistic Most Likely Pessimistic Activity Estimate Estimate Estimate A 12 days 12 days 12 days B 15 days 21 days 39 days C 12 days 15 days 18 days D 18 days 27 days 36 days E 12 days 18 days 24 days F 2 days 5 days 14 days (a) Find the estimate of the mean and variance of the duration of each activity. (b) Find the mean critical path.

32. (c) Use the mean critical path to find the approximate probability that the advertising campaign will be ready to launch within 57 days. (d) Now consider the other path through the project network. Find the approximate probability that this path will be completed within 57 days. (e) Since these paths do not overlap, a better estimate of the probability that the project will finish within 57 days can be obtained as follows. The project will finish within 57 days if both paths are completed within 57 days. Therefore, the approximate probability that the project will finish within 57 days is the product of the probabilities found in parts (c) and (d). Perform this calculation. What does this answer say about the accuracy of the standard procedure used in part (c)?

33. HW #4 Answer 10.3-1 10.4-5

34. Start 0 10.3-1 a) 7 2 30 15 15 A G K C E 10 5 3 25 B H D F 10 J 2 I L 30 Finish 0

36. c, d & e) ES 0 0 30 0 2 0 7 0 15 25 35 0 45 75 EF 0 30 35 2 5 7 32 15 25 27 45 15 75 75 LS 0 0 30 30 32 3 10 8 23 33 35 30 45 75 LF 0 30 35 32 35 10 35 23 33 35 45 45 75 75 Slack 0 0 0 30 30 3 3 8 8 8 0 30 0 0 Activity Start A B C D E F G H I J K L Finish Critical Path Yes Yes Yes No No No No No No No Yes No Yes Yes Critical Path: Start A B J L Finish

37. f ) Dinner will be delayed 3 minutes because of the phone call. If the food processor is used, then dinner will not be delayed because there was 3 minutes of slack and 5 minutes of cutting time saved.

38. 10.4 – 5 a) Activity A 12 0 B 23 16 C 15 1 D 27 9 E 18 4 F 6 4 b)

40. HW #5 14.5-2

41. 14.5-2 Consider the game having the following payoff table. (a) Use the approach described in Sec. 14.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as a linear programming problem. (b) Use the simplex method to find these optimal mixed strategies.

42. Answer HW #5 14.5-2

43. 14.5-2 a) b) Solve Automatically by the Simplex Method Optimal Solution Value of the Objective Function: Z = 2.368

44. HW #6 16.5 - 4 16.5 - 6 Due Day: Nov. 7

45. 16.5 - 4. The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst believes that brand switching can be modeled as a Markov chain using three states, with states A and B representing customers drinking beer produced from the aforementioned breweries and the analyst has constructed the following (one-step) transition matrix from past data. What are the steady-state market shares for the two major breweries?

46. 16.5 - 6. A soap company specializes in a luxury type of bath soap. The sales of this soap fluctuate between two levels - “Low” and “High” - depending upon two factors: (1) whether they advertise, and (2) the advertising and marketing of new products being done by competitors. The second factor is out of the company’s control, but it is trying to determine what its own advertising policy should be. For example, the marketing manager’s proposal is to advertise when sales are low but not to advertise when sales are high. Advertising in any quarter of a year has its primary impact on sales in the following quarter. Therefore, at the beginning of each quarter, the needed information is available to forecast accurately whether sales will be low or high that quarter and to decide whether to advertise that quarter.

47. The cost of advertising is $1 million for each quarter of a year in which it is done. When advertising is done during a quarter, the probability of having high sales the next quarter is 1/2 or 3/4, depending upon whether the current quarter’s sales are low or high. These probabilities go down to 1/4 or 1/2 when advertising is not done during the current quarter. The company’s quarterly profits (excluding advertising costs) are $4 million when sales are high but only $2 million when sales are low. (Hereafter, use units of millions of dollars.) (a) Construct the (one-step) transition matrix for each of the following advertising strategies: (i) never advertise, (ii) always advertise, (iii) follow the marketing manager’s proposal. (b) Determine the steady-state probabilities manually for each of the three cases in part (a). (c) Find the long-run expected average profit (including a deduction for advertising costs) per quarter for each of the three advertising strategies in part (a). Which of these strategies is best according to this measure of performance?

48. HW #6 ANSWER 16.5-4 16.5-6

49. 16.5-4

50. 16.5-6 0 1 0 1 0 1