McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Chapter 4 (Linear Programming: Formulation and Applications) Advertising-Mix Problem (Section 4.1) – Super Grain Corp | 4.2–4.5 Resource Allocation Problems (Section 4.2) –Think-Big Capital Budgeting | 4.6–4.10 Cost-Benefit-Trade-Off Problems (Section 4.3) –Union Airways | 4.11–4.15 Distribution-Network Problems (Section 4.4) –Big M Co. | 4.16–4.20 Student Exercises

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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?

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Spreadsheet Formulation

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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?

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Spreadsheet Formulation

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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?

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Spreadsheet Formulation

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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)

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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?

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Some Data Shipping Cost for Each Lathe ToCustomer 1Customer 2Customer 3 FromOutput Factory 1$700$900$80012 lathes Factory lathes Order Size10 lathes8 lathes9 lathes

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., The Distribution Network

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Spreadsheet Formulation

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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).

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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).

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., 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

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Product Mix Exercise Blue Ridge Hot Tubs manufactures and sells two models of hot tubs: the Aqua- Spa and the Hydro-Lux. Howie Jones, the owner and manager of the company needs to decide how many of each type of hot tub to produce during his next production cycle. Howie buys prefabricated fiberglass hot tub shells from a local supplier and adds the pump and tubing to the shells to create his hot tubs. (The supplier has the capacity to deliver as many hot tub shells as Howie needs.) Howie installs the same type of pump into both hot tubs. He will have only 200 pumps available during his next production cycle. From a manufacturing standpoint, the main difference between the two models of hot tubs is the amount of tubing and labor required. Each Aqua-Spa requires 9 hours of labor and 12 feet of tubing. Each Hydro-Lux requires 6 hours of labor and 16 feet of tubing. Howie expects to have 1,566 production labor hours and 2,880 feet of tubing available during the next production cycle. Howie earns a profit of $350 on each Aqua-Spa he sells and $300 on each Hydro-Lux he sells. He is confident that he can sell all the hot tubs he produces. The question is, how many Aqua-Spas and Hydro-Luxes should Howie produce if he wants to maximize his profits during the next production cycle? Source: Ragsdale, Spreadsheet Modeling and Decision Analysis.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Elements Common to Every Problem Decision variables: number of aqua-spas (A) to produce and number of hydro-luxes (H) to produce. Objective function: Max: Profit = 350 A H Constraints: Pump 1A + 1H <= 200 Labor9A + 6H <= 1566 Tubing12A + 16H <= 2880

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Graphical Solution Using Graphic LP Optimizer

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Graphical Solution Using Graphic LP Optimizer

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Graphical Solution Using Graphic LP Optimizer Constraints

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Graphical Solution Using Graphic LP Optimizer Feasible Solution Space

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Graphical Solution Using Graphic LP Optimizer Optimal Solution

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Organizing Spreadsheet & Entering Formulas D6. =SUMPRODUCT($B$5:$C$5,B6:C6) D9. =SUMPRODUCT($B$5:$C$5,B9:C9) D10. =SUMPRODUCT($B$5:$C$5,B10:C10) D11. =SUMPRODUCT($B$5:$C$5,B11:C11) Decision Variable Cells Decision Variable Coefficients Constraint Coefficients Constraint RHS Formulas Constraint RHS Limits

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Tools | Solver

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., Sensitivity Analysis

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., A Classic Problem See In class handout. First, identify the decision variables, objective function and constraints. Second, think about spreadsheet layout. Third, implement and solve model.

McGraw-Hill/Irwin Modified for Quan 6610 by Dr. Jim Grayson Optimization© The McGraw-Hill Companies, Inc., In Class Exercise End of chapter problem 4.6