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Chapter 3

Chapter 3. 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.

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Chapter 3

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  1. Chapter 3

  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 ($1 million). TV commercial spots available (5). The objective will be measured in terms of the expected number of exposures. Question: At what level should they advertise Crunchy Start in each of the three media?
  3. Cost and Exposure Data
  4. Spreadsheet Formulation
  5. Algebraic Formulation Let TV = Number of commercials for separate spots on televisionM = Number of advertisements in magazines.SS = Number of advertisements in Sunday supplements.Maximize Exposure = 1,300TV + 600M + 500SSsubject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost: 90TV + 30M + 40SS ≤ 1,000 ($thousand) Number of TV Spots: TV ≤ 5andTV ≥ 0, M ≥ 0, SS ≥ 0.
  6. The TBA Airlines Problem TBA Airlines is a small regional company that specializes in short flights in small airplanes. The company has been doing well and has decided to expand its operations. The basic issue facing management is whether to purchase more small airplanes to add some new short flights, or start moving into the national market by purchasing some large airplanes, or both. Question: How many airplanes of each type should be purchased to maximize their total net annual profit?
  7. Data for the TBA Airlines Problem
  8. Violates Divisibility Assumption of LP Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values. Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.
  9. Spreadsheet Model
  10. Integer Programming Formulation Let S = Number of small airplanes to purchaseL = Number of large airplanes to purchaseMaximize Profit = S + 5L ($millions)subject to Capital Available: 5S + 50L ≤ 100 ($millions) Max Small Planes: S ≤ 2and S ≥ 0, L ≥ 0S, L are integers.
  11. Think-Big Capital Budgeting Problem Think-Big Development Co. is a major investor in commercial real-estate development projects. They are considering three large construction projects Construct a high-rise office building. Construct a hotel. Construct a shopping center. Each project requires each partner to make four investments: a down payment now, and additional capital after one, two, and three years. Question: At what fraction should Think-Big invest in each of the three projects?
  12. Financial Data for the Projects
  13. Spreadsheet Formulation
  14. Algebraic Formulation Let OB = Participation share in the office building,H = Participation share in the hotel,SC = Participation share in the shopping center.Maximize NPV = 45OB + 70H + 50SCsubject to Total invested now: 40OB + 80H + 90SC ≤ 25 ($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) andOB ≥ 0, H ≥ 0, SC ≥ 0.
  15. Template for Resource-Allocation Problems
  16. 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.
  17. 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?
  18. Schedule Data
  19. Spreadsheet Formulation
  20. Algebraic Formulation Let Si = Number working shift i (for i = 1 to 5),Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5subject to Total agents 6AM–8AM: S1 ≥ 48 Total agents 8AM–10AM: S1 + S2 ≥ 79 Total agents 10AM–12PM: S1 + S2 ≥ 65 Total agents 12PM–2PM: S1 + S2 + S3 ≥ 87 Total agents 2PM–4PM: S2 + S3 ≥ 64 Total agents 4PM–6PM: S3 + S4 ≥ 73 Total agents 6PM–8PM: S3 + S4 ≥ 82 Total agents 8PM–10PM: S4 ≥ 43 Total agents 10PM–12AM: S4 + S5 ≥ 52 Total agents 12AM–6AM: S5 ≥ 15andSi ≥ 0 (for i = 1 to 5)
  21. Template for Cost-Benefit Tradoff Problems
  22. Summary of Formulation Procedure forCost-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.
  23. Types of Functional Constraints
  24. 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 $1,490,000 should be allocated for cents-off coupons.
  25. Benefit and Fixed-Requirement Data
  26. Spreadsheet Formulation
  27. Algebraic Formulation Let TV = Number of commercials for separate spots on televisionM = Number of advertisements in magazines.SS = Number of advertisements in Sunday supplements.Maximize Exposure = 1,300TV + 600M + 500SSsubject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($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) andTV ≥ 0, M ≥ 0, SS ≥ 0.
  28. Template for Mixed Problems
  29. 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. Question: What should the diet for each prisoner be?
  30. Algebraic Formulation Let x1 = gallons of milk per prisoner,x2 = cups of beans per prisoner,x3 = number of oranges per prisoner.Minimize Cost = $2.00x1 + $0.20x2 + $0.25x3subject to Niacin: 3.2x1 + 4.9x2 + 0.8x3 ≥ 13 mg Thiamin: 1.12x1 + 1.3x2 + 0.19x3 ≥ 1.5 mg Vitamin C: 32x1 + 93x3 ≥ 45 mgandx1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
  31. Spreadsheet Formulation
  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”
  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”
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