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# Chapter 3: Linear Programming Modeling Applications

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1. Chapter 3:Linear Programming Modeling Applications Jason C. H. Chen, Ph.D. Professor of MIS School of Business Administration Gonzaga University Spokane, WA 99223 chen@jepson.gonzaga.edu

2. Linear Programming (LP) Can Be Used for Many Managerial Decisions: • 1. Manufacturing applications • Product mix • Make-buy • 2. Marketing applications • Media selection • Marketing research • 3. Finance application • Portfolio selection • 4. Transportation application and others • Shipping & transportation • Multiperiod scheduling

3. For a particular application we begin with the problem scenario and data, then: • Define the decision variables • Formulate the LP model using the decision variables • Write the objective function equation • Write each of the constraint equations • Implement the model in Excel • Solve with Excel’s Solver

4. Manufacturing ApplicationsProduct Mix Problem: Fifth Avenue Industries • Produce 4 types of men's ties • Use 3 materials (limited resources) Decision: How many of each type of tie to make per month? Objective: Maximize profit

5. Resource Data Labor cost is \$0.75 per tie

6. Product Data

7. Material Requirements(yards per tie)

8. Decision Variables S = number of silk ties to make per month P = number of polyester ties to make per month B1 = number of poly-cotton blend 1 ties to make per month B2 = number of poly-cotton blend 2 ties to make per month

9. Profit Per Tie Calculation Profit per tie = (Selling price) – (material cost) –(labor cost) Silk Tie Profit = \$6.70 – (0.125 yds)(\$20/yd) - \$0.75 = \$3.45 per tie

10. Objective Function(in \$ of profit) Max 3.45S + 2.32P + 2.81B1 + 3.25B2 Subject to the constraints: Material Limitations(in yards) 0.125S < 1,000 (silk) 0.08P + 0.05B1 + 0.03B2< 2,000 (poly) 0.05B1 + 0.07B2< 1,250 (cotton)

11. Min and Max Number of Ties to Make 6,000 < S < 7,000 10,000 < P < 14,000 13,000 < B1 < 16,000 6,000 < B2 < 8,500 Finally nonnegativity S, P, B1, B2 > 0

12. LP Model for Product Mix Problem Max 3.45S + 2.32P + 2.81B1 + 3.25B2 Subject to the constraints: 0.125S < 1,000 (yards of silk) 0.08P + 0.05B1 + 0.03B2 < 2,000 (yards of poly) 0.05B1 + 0.07B2 < 1,250 (yards of cotton) 6,000 < S < 7,000 10,000 < P < 14,000 13,000 < B1 < 16,000 6,000 < B2 < 8,500 S, P, B1, B2 > 0 Go to file 3-1.xls

13. Go to file 3-1.xls

14. Marketing applicationsMedia Selection Problem: Win Big Gambling Club • Promote gambling trips to the Bahamas • Budget: \$8,000 per week for advertising • Use 4 types of advertising Decision: How many ads of each type? Objective: Maximize audience reached

15. Data

16. Other Restrictions • Have at least 5 radio spots per week • Spend no more than \$1800 on radio Decision Variables T = number of TV spots per week N = number of newspaper ads per week P = number of prime time radio spots per week A = number of afternoon radio spots per week

17. Objective Function (in num. audience reached) Max 5000T + 8500N + 2400P + 2800A Subject to the constraints: Budget is \$8000 800T + 925N + 290P + 380A < 8000 At Least 5 Radio Spots per Week P + A > 5

18. No More Than \$1800 per Week for Radio 290P + 380A < 1800 Max Number of Ads per Week T < 12 P < 25 N < 5 A < 20 Finally nonnegativity T, N, P, A > 0

19. LP Model for Media Selection Problem Objective Function Max 5000T + 8500N + 2400P + 2800A Subject to the constraints: 800T + 925N + 290P + 380A < 8000 P + A > 5 290P + 380A < 1800 T < 12 P < 25 N < 5 A < 20 T, N, P, A > 0 Go to file 3-3.xls

20. Go to file 3-3.xls

21. Finance applicationPortfolio Selection: International City Trust Has \$5 million to invest among 6 investments Decision: How much to invest in each of 6 investment options? Objective: Maximize interest earned

22. Data

23. Constraints • Invest up to \$ 5 million • No more than 25% into any one investment • At least 30% into precious metals • At least 45% into trade credits and corporate bonds • Limit overall risk to no more than 2.0

24. Decision Variables T = \$ invested in trade credit B = \$ invested in corporate bonds G = \$ invested gold stocks P = \$ invested in platinum stocks M = \$ invested in mortgage securities C = \$ invested in construction loans

25. Objective Function (in \$ of interest earned) Max 0.07T + 0.10B + 0.19G + 0.12P + 0.08M + 0.14C Subject to the constraints: Invest Up To \$5 Million T + B + G + P + M + C < 5,000,000

26. No More Than 25% Into Any One Investment T < 0.25 (T + B + G + P + M + C) B < 0.25 (T + B + G + P + M + C) G < 0.25 (T + B + G + P + M + C) P < 0.25 (T + B + G + P + M + C) M < 0.25 (T + B + G + P + M + C) C < 0.25 (T + B + G + P + M + C)

27. At Least 30% Into Precious Metals G + P > 0.30 (T + B + G + P + M + C) At Least 45% Into Trade Credits And Corporate Bonds T + B > 0.45 (T + B + G + P + M + C)

28. Limit Overall Risk To No More Than 2.0 Use a weighted average to calculate portfolio risk 1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C< 2.0 T + B + G + P + M + C OR 1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C < 2.0 (T + B + G + P + M + C) finally nonnegativity: T, B, G, P, M, C > 0

29. LP Model for Portfolio Selection Max 0.07T + 0.10B + 0.19G + 0.12P+ 0.08M + 0.14C Subject to the constraints: T + B + G + P + M + C < 5,000,000 (total funds) T < 0.25 (T + B + G + P + M + C) (Max trade credits) B < 0.25 (T + B + G + P + M + C) (Max corp bonds) G < 0.25 (T + B + G + P + M + C) (Max gold) P < 0.25 (T + B + G + P + M + C) (Max platinum) M < 0.25 (T + B + G + P + M + C) (Max mortgages) C < 0.25 (T + B + G + P + M + C) (Max const loans) 1.7T + 1.2B + 3.7G + 2.4P + 2.0M + 2.9C < 2.0 (T + B + G + P + M + C) (Risk score) G + P > 0.30 (T + B + G + P + M + C) (precious metal) T + B > 0.45 (T + B + G + P + M + C) (Trade credits & bonds) T, B, G, P, M, C > 0 Go to file 3-5.xls

30. Go to file 3-5.xls

31. Employee Staffing ApplicationLabor Planning: Hong Kong Bank Number of tellers needed varies by time of day Decision: How many tellers should begin work at various times of the day? Objective: Minimize personnel cost

32. Total minimum daily requirement is 112 hours

33. Full Time Tellers • Work from 9 AM – 5 PM • Take a 1 hour lunch break, half at 11, the other half at noon • Cost \$90 per day (salary & benefits) • Currently only 12 are available

34. Part Time Tellers • Work 4 consecutive hours (no lunch break) • Can begin work at 9, 10, 11, noon, or 1 • Are paid \$7 per hour (\$28 per day) • Part time teller hours cannot exceed 50% of the day’s minimum requirement • (50% of 112 hours = 56 hours)

35. Decision Variables F = num. of full time tellers (all work 9–5) P1 = num. of part time tellers who work 9–1 P2 = num. of part time tellers who work 10–2 P3 = num. of part time tellers who work 11–3 P4 = num. of part time tellers who work 12–4 P5 = num. of part time tellers who work 1–5

36. Objective Function (in \$ of personnel cost) Min 90 F + 28 (P1 + P2 + P3 + P4 + P5) Subject to the constraints: Part Time Hours Cannot Exceed 56 Hours 4 (P1 + P2 + P3 + P4 + P5) < 56

37. Minimum Num. Tellers Needed By Hour Time of Day F + P1> 10 (9-10) F + P1 + P2> 12 (10-11) 0.5 F + P1 + P2 + P3> 14 (11-12) 0.5 F + P1 + P2 + P3+ P4 > 16 (12-1) F + P2 + P3+ P4 + P5> 18 (1-2) F + P3+ P4 + P5> 17 (2-3) F + P4 + P5> 15 (3-4) F + P5 > 10 (4-5)

38. Only 12 Full Time Tellers Available F < 12 finally nonnegativity: F, P1, P2, P3, P4, P5> 0

39. LP Model for Labor Planning Min 90 F + 28 (P1 + P2 + P3 + P4 + P5) Subject to the constraints: F + P1> 10 (9-10) F + P1 + P2> 12 (10-11) 0.5 F + P1 + P2 + P3> 14 (11-12) 0.5 F + P1 + P2 + P3+ P4 > 16 (12-1) F + P2 + P3+ P4 + P5> 18 (1-2) F + P3+ P4 + P5> 17 (2-3) F + P4 + P5> 15 (3-4) F + P5 > 10 (4-5) F < 12 4 (P1 + P2 + P3 + P4 + P5) < 56 F, P1, P2, P3, P4, P5> 0 Go to file 3-6.xls

40. Go to file 3-6.xls

41. Transportation application and othersVehicle Loading: Goodman Shipping How to load a truck subject to weight and volume limitations Decision: How much of each of 6 items to load onto a truck? Objective: Maximize the value shipped

42. Data

43. Decision Variables Wi = number of pounds of item i to load onto truck , (where i = 1,…,6) Truck Capacity • 15,000 pounds • 1,300 cubic feet

44. Objective Function (in \$ of load value) Max 3.10W1 + 3.20W2 + 3.45W3 + 4.15W4 + 3.25W5 + 2.75W6 Subject to the constraints: Weight Limit Of 15,000 Pounds W1 + W2 + W3 + W4 + W5 + W6< 15,000

45. Volume Limit Of 1300 Cubic Feet 0.125W1 + 0.064W2 + 0.144W3 + 0.448W4 + 0.048W5 + 0.018W6< 1300 Pounds of Each Item Available W1< 5000 W4< 3500 W2< 4500 W5< 4000 W3< 3000 W6< 3500 Finally nonnegativity: Wi > 0, i=1,…,6

46. LP Model for Vehicle Loading Objective Function Max 3.10W1 +3.20W2 +3.45W3 +4.15W4 +3.25W5+2.75W6 Subject to the constraints: W1 + W2 + W3 + W4 + W5 + W6< 15,000 (Weight Limit) 0.125W1 + 0.064W2 + 0.144W3 +0.448W4 + 0.048W5 + 0.018W6< 1300 (volume limit of truck) Pounds of Each Item Available W1< 5000 (item 1 availability) W2< 4500 (item 2 availability) W3< 3000 (item 3 availability) W4< 3500 (item 4 availability) W5< 4000 (item 5 availability) W6< 3500 (item 6 availability) Wi > 0, i=1,…,6 Go to file 3-7.xls

47. Go to file 3-7.xls

48. Blending Problem:Whole Food Nutrition Center Making a natural cereal that satisfies minimum daily nutritional requirements Decision: How much of each of 3 grains to include in the cereal? Objective: Minimize cost of a 2 ounce serving of cereal

49. Decision Variables A = pounds of grain A to use B = pounds of grain B to use C = pounds of grain C to use Note: grains will be blended to form a 2 ounce serving of cereal