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Table of Contents Chapter 4 Linear Programming: Formulation and Applications

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Table of Contents Chapter 4 Linear Programming: Formulation and Applications

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    1. © 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 Applications of Linear Programming with Spreadsheets (UW Lecture) 4.31–4.50 These slides are based upon lectures to first-year MBA students at the University of Washington that discuss the application and formulation of linear programming models (as taught by one of the authors).

    2. © 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. © The McGraw-Hill Companies, Inc., 2003 4.3 Cost and Exposure Data Table 4.1 Cost and exposure data for the Super Grain advertising-mix problemTable 4.1 Cost and exposure data for the Super Grain advertising-mix problem

    4. © The McGraw-Hill Companies, Inc., 2003 4.4 Spreadsheet Formulation Figure 4.1 The spreadsheet model for the Super Grain problem (Section 4.1), including the target cell Total Exposures (H13), the changing cells Number of Ads (C13:E13), and the optimal solution obtained by Solver.Figure 4.1 The spreadsheet model for the Super Grain problem (Section 4.1), including the target cell Total Exposures (H13), the changing cells Number of Ads (C13:E13), and the optimal solution obtained by Solver.

    5. © The McGraw-Hill Companies, Inc., 2003 4.5 Algebraic Formulation

    6. © 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. © The McGraw-Hill Companies, Inc., 2003 4.7 Financial Data for the Projects Table 4.2 Financial data for the projects being considered for partial investment by the Think-Big Development Co.Table 4.2 Financial data for the projects being considered for partial investment by the Think-Big Development Co.

    8. © The McGraw-Hill Companies, Inc., 2003 4.8 Spreadsheet Formulation Figure 4.2 The spreadsheet model for the Think-Big problem (Section 4.1), including the target cell Total NPV (H16), the changing cells Participation Share (C16:E16), and the optimal solution obtained by Solver.Figure 4.2 The spreadsheet model for the Think-Big problem (Section 4.1), including the target cell Total NPV (H16), the changing cells Participation Share (C16:E16), and the optimal solution obtained by Solver.

    9. © The McGraw-Hill Companies, Inc., 2003 4.9 Algebraic Formulation

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

    11. © 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. © The McGraw-Hill Companies, Inc., 2003 4.12 Schedule Data Table 4.4 Data for the Union Airways personnel scheduling problemTable 4.4 Data for the Union Airways personnel scheduling problem

    13. © The McGraw-Hill Companies, Inc., 2003 4.13 Spreadsheet Formulation Figure 4.3 The spreadsheet model for the Union Airways problem, including the target cell Total Cost (J21), the changing cells Number Working (C21:G21), and the optimal solution as obtained by the Solver.Figure 4.3 The spreadsheet model for the Union Airways problem, including the target cell Total Cost (J21), the changing cells Number Working (C21:G21), and the optimal solution as obtained by the Solver.

    14. © The McGraw-Hill Companies, Inc., 2003 4.14 Algebraic Formulation

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

    16. © 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. © The McGraw-Hill Companies, Inc., 2003 4.17 Some Data Table 4.5 Some data for the Big M Company distribution-network problemTable 4.5 Some data for the Big M Company distribution-network problem

    18. © The McGraw-Hill Companies, Inc., 2003 4.18 The Distribution Network Figure 4.4 The distribution network for the Big M Company problem.Figure 4.4 The distribution network for the Big M Company problem.

    19. © The McGraw-Hill Companies, Inc., 2003 4.19 Spreadsheet Formulation Figure 4.5 The spreadsheet model for the Big M Company problem, including the target cell Total Cost (H15), the changing cells Units Shipped (C11:E12), and the optimal solution obtained by the Solver.Figure 4.5 The spreadsheet model for the Big M Company problem, including the target cell Total Cost (H15), the changing cells Units Shipped (C11:E12), and the optimal solution obtained by the Solver.

    20. © The McGraw-Hill Companies, Inc., 2003 4.20 Algebraic Formulation

    21. © 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. © The McGraw-Hill Companies, Inc., 2003 4.22 Benefit and Fixed-Requirement Data

    23. © The McGraw-Hill Companies, Inc., 2003 4.23 Spreadsheet Formulation Figure 4.6 The spreadsheet model for the revised Super Grain problem (Section 4.5), including the target cell Total Exposures (H19), the changing cells Number of Ads (C19:E19), and the optimal solution obtained by the Solver.Figure 4.6 The spreadsheet model for the revised Super Grain problem (Section 4.5), including the target cell Total Exposures (H19), the changing cells Number of Ads (C19:E19), and the optimal solution obtained by the Solver.

    24. © The McGraw-Hill Companies, Inc., 2003 4.24 Algebraic Formulation

    25. © The McGraw-Hill Companies, Inc., 2003 4.25 Types of Functional Constraints

    26. © 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. © The McGraw-Hill Companies, Inc., 2003 4.27 Product Data for the Save-It Company Table 4.9 Product data for the Save-It Company.Table 4.9 Product data for the Save-It Company.

    28. © The McGraw-Hill Companies, Inc., 2003 4.28 Material Data for the Save-It Company

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

    30. © The McGraw-Hill Companies, Inc., 2003 4.30 Algebraic Formulation

    31. © 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. © The McGraw-Hill Companies, Inc., 2003 4.32 LP Example #1 (Product Mix) The Quality Furniture Corporation produces benches and picnic tables. The firm has a limited supply of two resources: labor and wood. 1,600 labor hours are available during the next production period. The firm also has a stock of 9,000 pounds of wood available. Each bench requires 3 labor hours and 12 pounds of wood. Each table requires 6 labor hours and 38 pounds of wood. The profit margin on each bench is $8 and on each table is $18. Question: What product mix will maximize their total profit?

    33. © The McGraw-Hill Companies, Inc., 2003 4.33 Algebraic Formulation

    34. © The McGraw-Hill Companies, Inc., 2003 4.34 Spreadsheet Formulation

    35. © The McGraw-Hill Companies, Inc., 2003 4.35 LP Example #2 (Diet Problem) A prison is trying to decide what to feed its prisoners. They would like to offer some combination of milk, beans, and oranges. Their goal is to minimize cost, subject to meeting the minimum nutritional requirements imposed by law. The cost and nutritional contents of each food, along with the minimum nutritional requirements are shown below.

    36. © The McGraw-Hill Companies, Inc., 2003 4.36 Algebraic Formulation

    37. © The McGraw-Hill Companies, Inc., 2003 4.37 Spreadsheet Formulation

    38. © The McGraw-Hill Companies, Inc., 2003 4.38 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

    39. © The McGraw-Hill Companies, Inc., 2003 4.39 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

    40. © The McGraw-Hill Companies, Inc., 2003 4.40 LP Example #3 (Scheduling Problem) An airline reservations office is open to take reservations by telephone 24 hours per day, Monday through Friday. The number of reservation agents needed for each time period is shown below. A union contract requires that all employees work 8 consecutive hours.

    41. © The McGraw-Hill Companies, Inc., 2003 4.41 Algebraic Formulation

    42. © The McGraw-Hill Companies, Inc., 2003 4.42 Spreadsheet Formulation

    43. © The McGraw-Hill Companies, Inc., 2003 4.43 Workforce Scheduling at United Airlines United employs 5,000 reservation and customer service agents. Some part-time (2-8 hour shifts), some full-time (8-10 hour shifts). Workload varies greatly over day. Modeled problem as LP: Decision variables: how many employees of each shift length should begin at each potential start time (half-hour intervals). Constraints: minimum required employees for each half-hour. Objective: minimize cost. Saved United about $6 million annually, improved customer service, still in use today. For more details, see Jan-Feb 1986 Interfaces article “United Airlines Station Manpower Planning System”, available for download at www.mhhe.com/hillier2e/articles

    44. © The McGraw-Hill Companies, Inc., 2003 4.44 LP Example #4 (Transportation Problem) A company has two plants producing a certain product that is to be shipped to three distribution centers. The unit production costs are the same at the two plants, and the shipping cost per unit is shown below. Shipments are made once per week. During each week, each plant produces at most 60 units and each distribution center needs at least 40 units.

    45. © The McGraw-Hill Companies, Inc., 2003 4.45 Algebraic Formulation

    46. © The McGraw-Hill Companies, Inc., 2003 4.46 Spreadsheet Formulation

    47. © The McGraw-Hill Companies, Inc., 2003 4.47 Distribution System at Proctor and Gamble Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s. 50 product categories 60 plants 15 distribution centers 1000 customer zones Solved many transportation problems (one for each product category). Goal: find best distribution plan, which plants to keep open, etc. Closed many plants and distribution centers, and optimized their product sourcing and distribution location. Implemented in 1996. Saved $200 million per year. For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at www.mhhe.com/hillier2e/articles

    48. © The McGraw-Hill Companies, Inc., 2003 4.48 LP Example #5 (Assignment Problem) The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below.

    49. © The McGraw-Hill Companies, Inc., 2003 4.49 Algebraic Formulation

    50. © The McGraw-Hill Companies, Inc., 2003 4.50 Spreadsheet Formulation

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