2 Super Grain Corp. Advertising-Mix Problem Goal: Design the promotional campaign for Crunchy Start.The three most effective advertising media for this product areTelevision 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 areAdvertising 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 Costs Cost Category Each TV Commercial Each Magazine AdEach Sunday AdAd Budget$300,000$150,000$100,000Planning budget90,00030,00040,000Expected number of exposures1,300,000600,000500,000Table 3.1 Cost and exposure data for the Super Grain advertising-mix problem
4 Spreadsheet Formulation Figure 3.1 The spreadsheet model for the Super Grain problem (Section 3.1), including the target cell Total Exposures (H13), the changing cells Number of Ads (C13:E13), and the optimal solution obtained by Solver.
5 Algebraic Formulation Let TV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($thousand) Planning Cost: 90TV + 30M + 40SS ≤ 1,000 ($thousand) Number of TV Spots: TV ≤ 5 and TV ≥ 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 Small AirplaneLarge AirplaneCapital AvailableNet annual profit per airplane$1 million$5 millionPurchase cost per airplane5 million50 million$100 millionMaximum purchase quantity2—Table 3.2 Data for the TBA Airlines problem.
8 Violates Divisibility Assumption of LP Divisibility Assumption of Linear Programming: Decision variables in a linear programming model are allowed to have any values, including fractional values, that satisfy the functional and nonnegativity constraints. Thus, these variables are not restricted to just integer values.Since the number of airplanes purchased by TBA must have an integer value, the divisibility assumption is violated.
9 Spreadsheet ModelFigure 3.2 A spreadsheet model for the TBA Airlines integer programming problem where the changing cells, Units Produced (C12:D12), show the optimal airplane purchases obtained by the Solver and the target cell, Total Profit (G12), gives the resulting total profit in millions of dollars.
10 Integer Programming Formulation Let S = Number of small airplanes to purchase L = Number of large airplanes to purchase Maximize Profit = S + 5L ($millions) subject to Capital Available: 5S + 50L ≤ 100 ($millions) Max Small Planes: S ≤ 2 andS ≥ 0, L ≥ 0 S, 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 projectsConstruct 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 Investment Capital RequirementsYearOffice BuildingHotelShopping Center$40 million$80 million$90 million160 million80 million50 million290 million20 million310 million70 millionNet present value$45 million$70 million$50 millionTable 3.3 Financial data for the projects being considered for partial investment by the Think-Big Development Co.
13 Spreadsheet Formulation Figure 3.3 The spreadsheet model for the Think-Big problem (Section 3.2), including the target cell Total NPV (H16), the changing cells Participation Share (C16:E16), and the optimal solution obtained by Solver.
14 Algebraic Formulation Let OB = Participation share in the office building, H = Participation share in the hotel, SC = Participation share in the shopping center. Maximize NPV = 45OB + 70H + 50SC subject to Total invested now: 40OB + 80H + 90SC ≤ 25 ($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.
15 Template for Resource-Allocation Problems Figure 3.4 A template of a spreadsheet model for pure 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 areShift 1: 6:00 AM to 2:00 PMShift 2: 8:00 AM to 4:00 PMShift 3: Noon to 8:00 PMShift 4: 4:00 PM to midnightShift 5: 10:00 PM to 6:00 AMQuestion: How many agents should be assigned to each shift?
18 Time Periods Covered by Shift Minimum Number of Agents Needed Schedule DataTime Periods Covered by ShiftTime Period12345Minimum Number of Agents Needed6 AM to 8 AM√488 AM to 10 AM7910 AM to noon65Noon to 2 PM872 PM to 4 PM644 PM to 6 PM736 PM to 8 PM828 PM to 10 PM4310 PM to midnight52Midnight to 6 AM15Daily cost per agent$170$160$175$180$195Table 3.5 Data for the Union Airways personnel scheduling problem
19 Spreadsheet Formulation Figure 3.5 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.
20 Algebraic Formulation Let Si = Number working shift i (for i = 1 to 5), Minimize Cost = $170S1 + $160S2 + $175S3 + $180S4 + $195S5 subject 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 ≥ 15 and Si ≥ 0 (for i = 1 to 5)
21 Template for Cost-Benefit Tradoff Problems Figure 3.6 A template of a spreadsheet model for pure cost-benefit-trade-off problems.
22 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.
23 Types of Functional Constraints Form*Typical InterpretationMain UsageResource constraintLHS ≤ RHSFor some resource, Amount used ≤ Amount availableResource-allocation problems and mixed problemsBenefit constraintLHS ≥ RHSFor some benefit, Level achieved ≥ Minimum AcceptableCost-benefit-trade-off problems and mixed problemsFixed-requirement constraintLHS = RHSFor some quantity, Amount provided = Required amountTransportation problems and mixed problems* LHS = Left-hand side (a SUMPRODUCT function). RHS = Right-hand side (a constant).Table 3.6 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 goalsThe 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 Number Reached in Target Category (millions)Each TV CommercialEach Magazine AdEach Sunday AdMinimum Acceptable LevelYoung children1.20.15Parents of young children0.50.2Contribution Toward Required AmountRequired AmountCoupon redemption$40,000$120,000$1,490,000Table 3.7 and Table 3.8 The benefit data and fixed-requirement data for the Revised Super Grain Corp. Advertising Mix Problem.
26 Spreadsheet Formulation Figure 3.7 The spreadsheet model for the revised Super Grain problem (Section 3.4), including the target cell Total Exposures (H19), the changing cells Number of Ads (C19:E19), and the optimal solution obtained by the Solver.
27 Algebraic Formulation Let TV = Number of commercials for separate spots on television M = Number of advertisements in magazines. SS = Number of advertisements in Sunday supplements. Maximize Exposure = 1,300TV + 600M + 500SS subject to Ad Spending: 300TV + 150M + 100SS ≤ 4,000 ($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.
28 Template for Mixed Problems Figure 3.8 A template of a spreadsheet model for pure cost-benefit-trade-off 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.Milk (gallons)Navy Beans (cups)Oranges (large Calif. Valencia)Minimum Daily RequirementNiacin (mg)22.214.171.1243.0Thiamin (mg)126.96.36.199.5Vitamin C (mg)329345Cost ($)2.000.200.25Question: 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 + $0.25x3 subject to Niacin: 3.2x x x3 ≥ 13 mg Thiamin: 1.12x x x3 ≥ 1.5 mg Vitamin C: 32x1 + 93x3 ≥ 45 mg and x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
32 George Dantzig’s Diet heuristic solution. Cost = $39.93. 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 caloriesobjective: maximize (weight minus water content)500 food typesInitial solutions had problems500 gallons of vinegar200 bouillon cubesFor 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”