1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.1 Table of Contents Chapter 4 (Linear Programming: Formulation and Applications) Super Grain Corp. Advertising-Mix Problem (Section 4.1)4.2–4.5 Resource Allocation Problems & Think-Big Capital Budgeting (Section 4.2)4.6–4.10 Cost-Benefit-Trade-Off Problems & Union Airways (Section 4.3)4.11–4.15 Distribution-Network Problems & Big M Co. (Section 4.4)4.16–4.20 Continuing the Super Grain Corp. Case Study (Section 4.5)4.21–4.24 Mixed Formulations & Save-It Solid Waste Reclamation (Section 4.6)4.25–4.30

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.2 Super Grain Corp. Advertising-Mix Problem Goal: Design the promotional campaign for Crunchy Start. The three most effective advertising media for this product are –Television commercials on Saturday morning programs for children. –Advertisements in food and family-oriented magazines. –Advertisements in Sunday supplements of major newspapers. The limited resources in the problem are –Advertising budget ($4 million). –Planning budget ($1 million). –TV commercial spots available (5). The objective will be measured in terms of the expected number of exposures. Question: At what level should they advertise Crunchy Start in each of the three media?

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.3 Cost and Exposure Data Costs Cost Category Each TV Commercial Each Magazine Ad Each Sunday Ad Ad Budget$300,000$150,000$100,000 Planning budget90,00030,00040,000 Expected number of exposures 1,300,000600,000500,000

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.4 Spreadsheet Formulation

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.5 Algebraic Formulation LetTV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending:300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost:90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots:TV ≤ 5 and TV ≥ 0, M ≥ 0, SS ≥ 0.

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.6 Think-Big Capital Budgeting Problem Think-Big Development Co. is a major investor in commercial real-estate development projects. They are considering three large construction projects –Construct a high-rise office building. –Construct a hotel. –Construct a shopping center. Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years. Question: At what fraction should Think-Big invest in each of the three projects?

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.7 Financial Data for the Projects Investment Capital Requirements YearOffice BuildingHotelShopping Center 0$40 million$80 million$90 million 160 million80 million50 million 290 million80 million20 million 310 million70 million60 million Net present value$45 million$70 million$50 million

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.8 Spreadsheet Formulation

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.9 Algebraic Formulation LetOB = Participation share in the office building, H = Participation share in the hotel, SC = Participation share in the shopping center. Maximize NPV = 45OB + 70H + 50SC subject to Total invested now:40OB + 80H + 90SC ≤ 25 ($million) Total invested within 1 year:100OB + 160H + 140SC ≤ 45 ($million) Total invested within 2 years:190OB + 240H + 160SC ≤ 65 ($million) Total invested within 3 years:200OB + 310H + 220SC ≤ 80 ($million) and OB ≥ 0, H ≥ 0, SC ≥ 0.

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.10 Summary of Formulation Procedure for Resource- Allocation Problems 1.Identify the activities for the problem at hand. 2.Identify an appropriate overall measure of performance (commonly profit). 3.For each activity, estimate the contribution per unit of the activity to the overall measure of performance. 4.Identify the resources that must be allocated. 5.For each resource, identify the amount available and then the amount used per unit of each activity. 6.Enter the data in steps 3 and 5 into data cells. 7.Designate changing cells for displaying the decisions. 8.In the row for each resource, use SUMPRODUCT to calculate the total amount used. Enter ≤ and the amount available in two adjacent cells. 9.Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.11 Union Airways Personnel Scheduling Union Airways is adding more flights to and from its hub airport and so needs to hire additional customer service agents. The five authorized eight-hour shifts are –Shift 1:6:00 AM to 2:00 PM –Shift 2:8:00 AM to 4:00 PM –Shift 3:Noon to 8:00 PM –Shift 4:4:00 PM to midnight –Shift 5:10:00 PM to 6:00 AM Question: How many agents should be assigned to each shift?

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.12 Schedule Data Time Periods Covered by Shift Time Period12345 Minimum Number of Agents Needed 6 AM to 8 AM√48 8 AM to 10 AM√√79 10 AM to noon√√65 Noon to 2 PM√√√87 2 PM to 4 PM√√64 4 PM to 6 PM√√73 6 PM to 8 PM√√82 8 PM to 10 PM√43 10 PM to midnight√√52 Midnight to 6 AM√15 Daily cost per agent$170$160$175$180$195

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.13 Spreadsheet Formulation

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.14 Algebraic Formulation LetS i = Number working shift i (for i = 1 to 5), Minimize Cost = $170S 1 + $160S 2 + $175S 3 + $180S 4 + $195S 5 subject to Total agents 6AM–8AM:S 1 ≥ 48 Total agents 8AM–10AM:S 1 + S 2 ≥ 79 Total agents 10AM–12PM:S 1 + S 2 ≥ 65 Total agents 12PM–2PM:S 1 + S 2 + S 3 ≥ 87 Total agents 2PM–4PM:S 2 + S 3 ≥ 64 Total agents 4PM–6PM:S 3 + S 4 ≥ 73 Total agents 6PM–8PM:S 3 + S 4 ≥ 82 Total agents 8PM–10PM:S 4 ≥ 43 Total agents 10PM–12AM:S 4 + S 5 ≥ 52 Total agents 12AM–6AM:S 5 ≥ 15 and S i ≥ 0 (for i = 1 to 5)

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.15 Summary of Formulation Procedure for Cost-Benefit-Tradeoff Problems 1.Identify the activities for the problem at hand. 2.Identify an appropriate overall measure of performance (commonly cost). 3.For each activity, estimate the contribution per unit of the activity to the overall measure of performance. 4.Identify the benefits that must be achieved. 5.For each benefit, identify the minimum acceptable level and then the contribution of each activity to that benefit. 6.Enter the data in steps 3 and 5 into data cells. 7.Designate changing cells for displaying the decisions. 8.In the row for each benefit, use SUMPRODUCT to calculate the level achieved. Enter ≤ and the minimum acceptable level in two adjacent cells. 9.Designate a target cell. Use SUMPRODUCT to calculate this measure of performance.

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.16 The Big M Distribution-Network Problem The Big M Company produces a variety of heavy duty machinery at two factories. One of its products is a large turret lathe. Orders have been received from three customers for the turret lathe. Question: How many lathes should be shipped from each factory to each customer?

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.17 Some Data Shipping Cost for Each Lathe ToCustomer 1Customer 2Customer 3 FromOutput Factory 1$700$900$80012 lathes Factory 280090070015 lathes Order Size10 lathes8 lathes9 lathes

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.18 The Distribution Network

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.19 Spreadsheet Formulation

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.20 Algebraic Formulation LetS ij = Number of lathes to ship from i to j (i = F1, F2; j = C1, C2, C3). Minimize Cost = $700S F1-C1 + $900S F1-C2 + $800S F1-C3 + $800S F2-C1 + $900S F2-C2 + $700S F2-C3 subject to Factory 1:S F1-C1 + S F1-C2 + S F1-C3 = 12 Factory 2:S F2-C1 + S F2-C2 + S F2-C3 = 15 Customer 1:S F1-C1 + S F2-C1 = 10 Customer 2:S F1-C2 + S F2-C2 = 8 Customer 3:S F1-C3 + S F2-C3 = 9 and S ij ≥ 0 (i = F1, F2; j = C1, C2, C3).

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.21 Continuing the Super Grain Case Study David and Claire conclude that the spreadsheet model needs to be expanded to incorporate some additional considerations. In particular, they feel that two audiences should be targeted — young children and parents of young children. Two new goals –The advertising should be seen by at least five million young children. –The advertising should be seen by at least five million parents of young children. Furthermore, exactly $1,490,000 should be allocated for cents-off coupons.

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.22 Benefit and Fixed-Requirement Data Number Reached in Target Category (millions) Each TV Commercial Each Magazine Ad Each Sunday Ad Minimum Acceptable Level Young children1.20.105 Parents of young children0.50.2 5 Contribution Toward Required Amount Each TV Commercial Each Magazine Ad Each Sunday Ad Required Amount Coupon redemption0$40,000$120,000$1,490,000

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.23 Spreadsheet Formulation

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.24 Algebraic Formulation LetTV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending:300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost:90TV + 30M + 30SS ≤ 1,000 ($thousand) Number of TV Spots:TV ≤ 5 Young children:1.2TV + 0.1M ≥ 5 (millions) Parents:0.5TV + 0.2M + 0.2SS ≥ 5 (millions) Coupons:40M + 120SS = 1,490 ($thousand) and TV ≥ 0, M ≥ 0, SS ≥ 0.

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.25 Types of Functional Constraints TypeForm*Typical InterpretationMain Usage Resource constraintLHS ≤ RHS For some resource, Amount used ≤ Amount available Resource-allocation problems and mixed problems Benefit constraintLHS ≥ RHS For some benefit, Level achieved ≥ Minimum Acceptable Cost-benefit-trade-off problems and mixed problems Fixed-requirement constraint LHS = RHS For some quantity, Amount provided = Required amount Distribution-network problems and mixed problems * LHS = Left-hand side (a SUMPRODUCT function). RHS = Right-hand side (a constant).

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.26 Save-It Company Waste Reclamation The Save-It Company operates a reclamation center that collects four types of solid waste materials and then treats them so that they can be amalgamated into a salable product. Three different grades of product can be made: A, B, and C (depending on the mix of materials used). Question: What quantity of each of the three grades of product should be produced from what quantity of each of the four materials?

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.27 Product Data for the Save-It Company GradeSpecification Amalgamation Cost per Pound Selling Price per Pound A Material 1: Not more than 30% of total Material 2: Not less than 40% of total Material 3: Not more than 50% of total Material 4: Exactly 20% of total $3.00$8.50 B Material 1: Not more than 50% of total Material 2: Not less than 10% of the total Material 4: Exactly 10% of the total 2.507.00 CMaterial 1: Not more than 70% of the total2.005.50

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.28 Material Data for the Save-It Company Material Pounds/Week Available Treatment Cost per PoundAdditional Restrictions 13,000$3.001. For each material, at least half of the pounds/week available should be collected and treated. 2. $30,000 per week should be used to treat these materials. 22,0006.00 34,0004.00 41,0005.00

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.29 Spreadsheet Formulation

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.30 Algebraic Formulation Let x ij = Pounds of material j allocated to product i per week (i = A, B, C; j = 1, 2, 3, 4). Maximize Profit = 5.5(x A1 + x A2 + x A3 + x A4 ) + 4.5(x B1 + x B2 + x B3 + x B4 ) + 3.5(x C1 + x C2 + x C3 + x C4 ) subject toMixture Specifications:x A1 ≤ 0.3 (x A1 + x A2 + x A3 + x A4 ) x A2 ≥ 0.4 (x A1 + x A2 + x A3 + x A4 ) x A3 ≤ 0.5 (x A1 + x A2 + x A3 + x A4 ) x A4 = 0.2 (x A1 + x A2 + x A3 + x A4 ) x B1 ≤ 0.5 (x B1 + x B2 + x B3 + x B4 ) x B2 ≥ 0.1 (x B1 + x B2 + x B3 + x B4 ) x B4 = 0.1 (x B1 + x B2 + x B3 + x B4 ) x C1 ≤ 0.7 (x C1 + x C2 + x C3 + x C4 ) Availability of Materials:x A1 + x B1 + x C1 ≤ 3,000 x A2 + x B2 + x C2 ≤ 2,000 x A3 + x B3 + x C3 ≤ 4,000 x A4 + x B4 + x C4 ≤ 1,000 Restrictions on amount treated:x A1 + x B1 + x C1 ≥ 1,500 x A2 + x B2 + x C2 ≥ 1,000 x A3 + x B3 + x C3 ≥ 2,000 x A4 + x B4 + x C4 ≥ 500 Restriction on treatment cost:3(x A1 + x B1 + x C1 ) + 6(x A2 + x B2 + x C2 ) + 4(x A3 + x B3 + x C3 ) + 5(x A4 + x B4 + x C4 ) = 30,000 and x ij ≥ 0 (i = A, B, C; j = 1, 2, 3, 4).

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.31 Formulating an LP Spreadsheet Model Enter all of the data into the spreadsheet. Color code (blue). What decisions need to be made? Set aside a cell in the spreadsheet for each decision variable (changing cell). Color code (yellow with border). Write an equation for the objective in a cell. Color code (orange with heavy border). Put all three components (LHS, ≤/=/≥, RHS) of each constraint into three cells on the spreadsheet. Some Examples: –Production Planning –Diet / Blending –Workforce Scheduling –Transportation / Distribution –Assignment

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.32 George Dantzig’s Diet Stigler (1945) “The Cost of Subsistence” –heuristic solution. Cost = $39.93. Dantzig invents the simplex method (1947) –Stigler’s problem “solved” in 120 man days. Cost = $39.69. Dantzig goes on a diet (early 1950’s), applies diet model: –≤ 1,500 calories –objective: maximize (weight minus water content) –500 food types Initial solutions had problems –500 gallons of vinegar –200 bouillon cubes For more details, see July-Aug 1990 Interfaces article “The Diet Problem”, available for download at www.mhhe.com/hillier2e/articles

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 4.33 Least-Cost Menu Planning Models in Food Systems Management Used in many institutions with feeding programs: hospitals, nursing homes, schools, prisons, etc. Menu planning often extends to a sequence of meals or a cycle. Variety important (separation constraints). Preference ratings (related to service frequency). Side constraints (color, categories, etc.) Generally models have reduced cost about 10%, met nutritional requirements better, and increased customer satisfaction compared to traditional methods. USDA uses these models to plan food stamp allotment. For more details, see Sept-Oct 1992 Interfaces article “The Evolution of the Diet Model in Managing Food Systems”, available for download at www.mhhe.com/hillier2e/articles