1. 1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Slides by JOHN LOUCKS St. Edward’s University

2. 2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 4 Linear Programming Applications in Marketing, Finance and Operations PART 2: n Workforce Assignment n Product Mix Problems n Blending Problems

3. 3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment National Wing Company (NWC) is gearing up for National Wing Company (NWC) is gearing up for the new B-48 contract. NWC has agreed to produce 20 wings in April, 24 in May, and 30 in June. Currently, NWC has 100 fully qualified workers. A fully qualified worker can either be placed in production or can train new recruits. A new recruit can be trained to be an apprentice in one month. After another month, the apprentice becomes a qualified worker. Each trainer can train two recruits.

4. 4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment The production rate and salary per employee The production rate and salary per employee type is listed below. Type ofProduction Rate Wage Type ofProduction Rate Wage Employee (Wings/Month) Per Month Employee (Wings/Month) Per Month Production.6 $3,000 Production.6 $3,000 Trainer.3 $3,300 Trainer.3 $3,300 Apprentice.4 $2,600 Apprentice.4 $2,600 Recruit.05 $2,200 Recruit.05 $2,200 At the end of June, NWC wishes to have no recruits At the end of June, NWC wishes to have no recruits or apprentices, but have at least 140 full-time workers.

5. 5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Define the Decision Variables P i = number of producers in month i (where i = 1, 2, 3 for April, May, June) (where i = 1, 2, 3 for April, May, June) T i = number of trainers in month i (where i = 1, 2 for April, May) (where i = 1, 2 for April, May) A i = number of apprentices in month i (where i = 2, 3 for May, June) (where i = 2, 3 for May, June) R i = number of recruits in month i (where i = 1, 2 for April, May) (where i = 1, 2 for April, May)

6. 6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Define the Objective Function Minimize total wage cost for producers, trainers, apprentices, and recruits for April, May, and June: Min 3000 P 1 + 3300 T 1 + 2200 R 1 + 3000 P 2 + 3300 T 2 + 2600 A 2 +2200 R 2 + 3000 P 3 + 2600 A 3 + 2600 A 2 +2200 R 2 + 3000 P 3 + 2600 A 3

7. 7 7 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Define the Constraints Total production in Month 1 (April) must equal or exceed contract for Month 1: (1).6 P 1 +.3 T 1 +.05 R 1 > 20 Total production in Months 1-2 (April, May) must equal or exceed total contracts for Months 1-2: (2).6 P 1 +.3 T 1 +.05 R 1 +.6 P 2 +.3 T 2 +.4 A 2 +.05 R 2 > 44 Total production in Months 1-3 (April, May, June) must equal or exceed total contracts for Months 1-3: (3).6 P 1 +.3 T 1 +.05 R 1 +.6 P 2 +.3 T 2 +.4 A 2 +.05 R 2 +.6 P 3 +.4 A 3 > 74 +.6 P 3 +.4 A 3 > 74

8. 8 8 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Define the Constraints (continued) The number of producers and trainers in a month must equal the number of producers, trainers, and apprentices in the previous month: (4) P 1 - P 2 + T 1 - T 2 = 0 (5) P 2 - P 3 + T 2 + A 2 = 0 The number of apprentices in a month must equal the number of recruits in the previous month: (6) A 2 - R 1 = 0 (7) A 3 - R 2 = 0

9. 9 9 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Define the Constraints (continued) Each trainer can train two recruits: (8) 2 T 1 - R 1 > 0 (9) 2 T 2 - R 2 > 0 In April there are 100 employees that can be producers or trainers: (10) P 1 + T 1 = 100 At the end of June, there are to be at least 140 employees: (11) P 3 + A 3 > 140 Non-negativity: P 1, T 1, R 1, P 2, T 2, A 2, R 2, P 3, A 3 > 0 P 1, T 1, R 1, P 2, T 2, A 2, R 2, P 3, A 3 > 0

10. 10 Slide © 2008 Thomson South-Western. All Rights Reserved Workforce Assignment n Solution Summary P 1 = 100, T 1 = 0, R 1 = 0 P 2 = 80, T 2 = 20, A 2 = 0, R 2 = 40 P 3 = 100, A 3 = 40 Total Wage Cost = $1,098,000 ProducersTrainersApprenticesRecruits April May June July 100 80 100 140 0 20 0 0 0 20 0 0 0 0 40 0 0 0 40 0 0 40 0 0 0 40 0 0

11. 11 Slide © 2008 Thomson South-Western. All Rights Reserved Product Mix Floataway Tours has $420,000 that can be used to purchase new rental boats for hire during the summer. The boats can be purchased from two different manufacturers. Floataway Tours would Floataway Tours would like to purchase at least 50 boats and would like to purchase the same number from Sleekboat as from Racer to maintain goodwill. At the same time, Floataway Tours wishes to have a total seating capacity of at least 200.

12. 12 Slide © 2008 Thomson South-Western. All Rights Reserved Formulate this problem as a linear program. Maximum Expected Maximum Expected Boat Builder Cost Seating Daily Profit Boat Builder Cost Seating Daily Profit Speedhawk Sleekboat $6000 3 $ 70 Silverbird Sleekboat $7000 5 $ 80 Catman Racer $5000 2 $ 50 Classy Racer $9000 6 $110 Product Mix

13. 13 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the Decision Variables x 1 = number of Speedhawks ordered x 1 = number of Speedhawks ordered x 2 = number of Silverbirds ordered x 2 = number of Silverbirds ordered x 3 = number of Catmans ordered x 3 = number of Catmans ordered x 4 = number of Classys ordered x 4 = number of Classys ordered n Define the Objective Function Maximize total expected daily profit: Maximize total expected daily profit: Max (Expected daily profit per unit) Max (Expected daily profit per unit) x (Number of units) x (Number of units) Max 70 x 1 + 80 x 2 + 50 x 3 + 110 x 4 Product Mix

14. 14 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the constraints Spend no more than $420,000: Spend no more than $420,000: (1) 6000 x 1 + 7000 x 2 + 5000 x 3 + 9000 x 4 < 420,000 (1) 6000 x 1 + 7000 x 2 + 5000 x 3 + 9000 x 4 < 420,000 Purchase at least 50 boats: Purchase at least 50 boats: (2) x 1 + x 2 + x 3 + x 4 > 50 (2) x 1 + x 2 + x 3 + x 4 > 50 Number of boats from Sleekboat must equal Number of boats from Sleekboat must equal number of boats from Racer: number of boats from Racer: (3) x 1 + x 2 = x 3 + x 4 or x 1 + x 2 - x 3 - x 4 = 0 (3) x 1 + x 2 = x 3 + x 4 or x 1 + x 2 - x 3 - x 4 = 0 Product Mix

15. 15 Slide © 2008 Thomson South-Western. All Rights Reserved n Define the constraints (continued) Capacity at least 200: Capacity at least 200: (4) 3 x 1 + 5 x 2 + 2 x 3 + 6 x 4 > 200 (4) 3 x 1 + 5 x 2 + 2 x 3 + 6 x 4 > 200 Non-negativity of variables: Non-negativity of variables: x i > 0, for i = 1, 2, 3, 4 x i > 0, for i = 1, 2, 3, 4 Product Mix

16. 16 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Science Output Objective Function Value = 5040.000 Objective Function Value = 5040.000 Variable Value Reduced Cost Variable Value Reduced Cost x 1 28.000 0.000 x 1 28.000 0.000 x 2 0.000 2.000 x 2 0.000 2.000 x 3 0.000 12.000 x 3 0.000 12.000 x 4 28.000 0.000 x 4 28.000 0.000 Constraint Slack/Surplus Dual Price Constraint Slack/Surplus Dual Price 1 0.000 0.012 1 0.000 0.012 2 6.000 0.000 2 6.000 0.000 3 0.000 -2.000 3 0.000 -2.000 4 52.000 0.000 4 52.000 0.000 Product Mix

17. 17 Slide © 2008 Thomson South-Western. All Rights Reserved n Solution Summary Purchase 28 Speedhawks from Sleekboat. Purchase 28 Speedhawks from Sleekboat. Purchase 28 Classy’s from Racer. Purchase 28 Classy’s from Racer. Total expected daily profit is $5,040.00. Total expected daily profit is $5,040.00. The minimum number of boats was exceeded by 6 (surplus for constraint #2). The minimum number of boats was exceeded by 6 (surplus for constraint #2). The minimum seating capacity was exceeded by 52 (surplus for constraint #4). The minimum seating capacity was exceeded by 52 (surplus for constraint #4). Product Mix

18. 18 Slide © 2008 Thomson South-Western. All Rights Reserved Blending Problem Ferdinand Feed Company receives four raw grains from which it blends its dry pet food. The pet food advertises that each 8-ounce packet meets the minimum daily requirements for vitamin C, protein and iron. The cost of each raw grain as well as the vitamin C, protein, and iron units per pound of each grain are summarized on the next slide.

19. 19 Slide © 2008 Thomson South-Western. All Rights Reserved Blending Problem Vitamin C Protein Iron Vitamin C Protein Iron Grain Units/lb Units/lb Units/lb Cost/lb Grain Units/lb Units/lb Units/lb Cost/lb 1 9 12 0.75 1 9 12 0.75 2 16 10 14.90 2 16 10 14.90 3 8 10 15.80 3 8 10 15.80 4 10 8 7.70 Ferdinand is interested in producing the 8-ounce mixture at minimum cost while meeting the minimum daily requirements of 6 units of vitamin C, 5 units of protein, and 5 units of iron.

20. 20 Slide © 2008 Thomson South-Western. All Rights Reserved Blending Problem n Define the decision variables x j = the pounds of grain j ( j = 1,2,3,4) x j = the pounds of grain j ( j = 1,2,3,4) used in the 8-ounce mixture used in the 8-ounce mixture n Define the objective function Minimize the total cost for an 8-ounce mixture: Minimize the total cost for an 8-ounce mixture: MIN.75 x 1 +.90 x 2 +.80 x 3 +.70 x 4 MIN.75 x 1 +.90 x 2 +.80 x 3 +.70 x 4

21. 21 Slide © 2008 Thomson South-Western. All Rights Reserved Blending Problem Define the constraints Define the constraints Total weight of the mix is 8-ounces (.5 pounds): (1) x 1 + x 2 + x 3 + x 4 =.5 (1) x 1 + x 2 + x 3 + x 4 =.5 Total amount of Vitamin C in the mix is at least 6 units: (2) 9 x 1 + 16 x 2 + 8 x 3 + 10 x 4 > 6 (2) 9 x 1 + 16 x 2 + 8 x 3 + 10 x 4 > 6 Total amount of protein in the mix is at least 5 units: (3) 12 x 1 + 10 x 2 + 10 x 3 + 8 x 4 > 5 (3) 12 x 1 + 10 x 2 + 10 x 3 + 8 x 4 > 5 Total amount of iron in the mix is at least 5 units: (4) 14 x 2 + 15 x 3 + 7 x 4 > 5 (4) 14 x 2 + 15 x 3 + 7 x 4 > 5 Non-negativity of variables: x j > 0 for all j

22. 22 Slide © 2008 Thomson South-Western. All Rights Reserved n The Management Scientist Output OBJECTIVE FUNCTION VALUE = 0.406 OBJECTIVE FUNCTION VALUE = 0.406 VARIABLE VALUE REDUCED COSTS VARIABLE VALUE REDUCED COSTS X1 0.099 0.000 X1 0.099 0.000 X2 0.213 0.000 X2 0.213 0.000 X3 0.088 0.000 X3 0.088 0.000 X4 0.099 0.000 X4 0.099 0.000 Thus, the optimal blend is about.10 lb. of grain 1,.21 lb. of grain 2,.09 lb. of grain 3, and.10 lb. of grain 4. The mixture costs Frederick’s 40.6 cents. Blending Problem

23. 23 Slide © 2008 Thomson South-Western. All Rights Reserved End of Chapter 4