EF–522 - Decision ScienceProblems for Assignment Problem 1 Consider the following LP problem: Maximize: Profit = 5X + 6Y Subject to: 2X + Y ≤ 120 2X + 3Y ≤ 240 (a) What is the optimal solution to this problem? Solve graphically as well as using Simplex Method. (b) If a technical breakthrough occurred that raised the profit per unit of X to $8, would this affect the optimal solution? (c) Instead of an increase in the profit per unit on X to $8, suppose that profit rate was overestimated and should only have been $3. Does this change the optimal solution? (d) Consider the original problem again. If the second constraint is changed to 2X + 3Y ≤ 240 what effect will this have on the optimal solution? Problem 2 The Heinlein and Krampf Brokerage firm has just been instructed by one of its clients to invest $250,000 of her money obtained recently through the sale of land holdings in Ohio. The client has a good deal of trust in the investment house, but she also has her own ideas about the distribution of the funds being invested. In particular, she requests that the firm select whatever stocks and bonds they believe are well rated, but within the following guidelines: (a) Municipal bonds should constitute at least 20% of the investment. (b) At least 40% of the funds should be placed in a combination of electronic firms, aerospace firms, and drug manufacturers. (c) No more than 50% of the amount invested in municipal bonds should be placed in a high- risk, high-yield nursing home stock. Subject to these restraints, the client’s goal is to maximize projected return on investments. The analysts at Heinlein and Krampf, aware of these guidelines, prepare a list of high-quality stocks and bonds and their corresponding rates of return: Investment Projected Rate of Return (%) Los Angeles municipal bonds 5.3 Thompson Electronics, Inc. 6.8 United Aerospace Corp. 4.9 Palmer Drugs 8.4 Happy Days Nursing Homes 11.8 (a) Formulate this portfolio selection problem using LP. (b) Solve this problem both graphically and numerically. Problem 3 Kathy Roniger, campus dietitian for a small Idaho college, is responsible for formulating a nutritious meal plan for students. For an evening meal, she feels that the following five meal- content requirements should be met: (1) between 900 and 1,500 calories; (2) at least 4 milligrams of iron; (3) no more than 50 grams of fat; (4) at least 26 grams of protein; and (5) no more than 50 grams of carbohydrates. On a particular day, Roniger’s food stock includes seven items that can be prepared and served for supper to meet these requirements. The cost per pound for each food item and the contribution to each of the five nutritional requirements are given in the table below. What combination and amounts of food items will provide the nutrition Roniger requires at the least total food cost? TABLE OF FOOD VALUES AND COSTS FOOD CALORIES/ IRON FAT PROTEIN CARBOHYDRATES COST/ ITEM LB (MG/LB) (GM/LB) (GM/LB) (GM/LB) LB ($) Milk 295 0.2 16 16 22 0.60 Ground Meat 1216 0.2 96 81 0 2.35 Chicken 394 4.3 9 74 0 1.15 Fish 358 3.2 0.5 83 0 2.25 Beans 128 3.2 0.8 7 28 0.58 Spinach 118 14.1 1.4 14 19 1.17 Potatoes 279 2.2 0.5 8 63 0.33 Formulate this problem as an LP problem and solve both the primal and the dual. What does the solution to each problem reveals? What is the cost per meal? Is this a well-balanced diet? Problem 4 The state of Missouri has three major power-generating companies (A, B, and C). During the months of peak demand, the Missouri Power Authority authorizes these companies to pool their excess supply and to distribute it to smaller independent power companies that do not have generators large enough to handle the demand. Excess supply is distributed on the basis of cost per kilowatt hour transmitted. The following table shows the demand and supply in millions of kilowatt hours and the cost per kilowatt hour of transmitting electric power to four small companies in cities W, X, Y, and Z: FROM TO W X Y Z EXCESS SUPPLY A 12¢ 4¢ 9¢ 5¢ 55 B 8¢ 1¢ 6¢ 6¢ 45 C 1¢ 12¢ 4¢ 7¢ 30 UNFILLED POWER DEMAND 40 20 50 20 Problem 5 The Patricia Garcia Company is producing seven new medical products. Each of Garcia’s eight plants can add one more product to its current line of medical devices. The unit manufacturing costs for producing the different parts at the eight plants are shown in the table above. How should Garcia assign the new products to the plants to minimize manufacturing costs? Electronic Plant Component 1 2 3 4 5 6 7 8 C53 $0.10 $0.12 $0.13 $0.11 $0.10 $0.06 $0.16 $0.12 C81 0.05 0.06 0.04 0.08 0.04 0.09 0.06 0.06 D5 0.32 0.40 0.31 0.30 0.42 0.35 0.36 0.49 D44 0.17 0.14 0.19 0.15 0.10 0.16 0.19 0.12 E2 0.06 0.07 0.10 0.05 0.08 0.10 0.11 0.05 E35 0.08 0.10 0.12 0.08 0.09 0.10 0.09 0.06 G99 0.55 0.62 0.61 0.70 0.62 0.63 0.65 0.59 NOTE Each student will need to submit an independent assignment. The last date for submission is 11 th July 2014. All assignments should be submitted to the Class C.R. on or before the deadline. I will NOT collect any assignments personally. The student’s name and seat number must be clearly mentioned on the top. Assignments without names or seat numbers will not earn any marks. Assignments must be submitted in proper format. M. Shahzad Mirza