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Linear Programming: Two GoalsWhere These Skills Fit in the Abstraction ProcessSpreadsheet DesignSteps in Formulating a Linear Programming Problem: Take #1Formulating a Linear Programming Problem: Take #2Common Types of LPs (aka Beginning of Your “Toolbox”)Product Mix & Make vs. Buy Pbs#1: Product Mix Decisions (Like Howie’s Hot Tub Problem)#2: Make vs. Buy DecisionsMake vs. Buy DecisionsMake vs. Buy Textbook Example: Weedwacker CompanyGeneralize Make vs. Buy(Investment Portfolio) Allocation Pbs#3: Investment Portfolio AllocationAllocation Problems Beyond Investment Portfolios#6: Multi-period Planning ProblemsKey Idea Elaborated#6a: Using Inventory to Meet Time Varying DemandExamples#6b: Multi-period Financial PlanningExamples1Linear Programming:Two GoalsFormulate LP’sFormulating LP’s is an artMany problems are variants on a standard problem“Pattern match” your current problem to one of those classicsLike brainteasers: can be frustrating but also funSolve LP’s in ExcelThink ahead about layout/design of spreadsheetCan use text’s examples as templates2Where These Skills Fit in the Abstraction ProcessReal world “problem opportunity”big confusing messProfessional assignmentObject/scope defined; rest is murkyTextbook level natural language descriptionall and only the relevant info includedCareful natural language descriptionfairly unambiguous but not compactMathematical formulationformal, precise, unambiguous, compactTractable implementation usually on computer, e.g., in Excel3Spreadsheet DesignObjectivesClear (communicate results/intuition to others)Reliable (is error-free)Auditable (can understand & know it’s error-free)Modifiable (re-use code rather than reinventing)GuidelinesBuild model around display of dataDon’t bury constants in formulasLogically close quantities should be physically closeDesign so formulae can be copiedUse color, shading, borders, and protectionDocument with text boxes & cell notes4Steps in Formulating a Linear Programming Problem: Take #1Understand the problem(Be willing to “get your hand’s dirty”)Identify the Decision Maker (DM)Identify the decision variables (DV’s)(This is often the key step.)State objective function as a linear combination of the DV’sState constraints as a linear combination of the DV’sIdentify upper or lower bounds on DV’s5Formulating a Linear Programming Problem: Take #2Master a “toolbox” of standard problem typesUnderstand the particular problem at handBe willing to “get your hand’s dirty”Identify the Decision Maker (DM), DM’s goals, etc.Recognize what type of problem it isUsually will be a “base type” + “whistles and bells”Sometimes need to blend several base typesSometimes won’t match any & have to start from scratchImport notation for DV’s from standard problem typeWrite down objective, constraints, bounds, etc.6Common Types of LPs (aka Beginning of Your “Toolbox”)1) Product mix2) Make vs. buy3) Investment/Portfolio allocation4) Transportation (will do in Chapter 5)5) Assignment Problem (will do in Chapter 5)6) Blending (will do with Chapter 5)7) Multi-period planningproduction schedulesfinancial flows8) DEA (covered in 94-833)We did product mix already. (Howie’s Pb.)We’ll cover three more of these today.7Product Mix & Make vs. Buy Pbs8#1: Product Mix Decisions(Like Howie’s Hot Tub Problem)Decision: How many of each type of product should be made (offered)Decision variablesXi = amount of product i to make (offer)ConstraintsNonnegative productionProduction capacity constraints (e.g., limits on resources which are consumed in the process of producing products)Objective: Maximize profit (or min cost)9#2: Make vs. Buy DecisionsSeveral products, each can be made in house or purchased from vendorsDecision: How much to make and how much to buy (separate sources), so ...Decision variablesfor each product i:Mi = amount of product i to make in houseBi = amount of product i to purchase10Make vs. Buy DecisionsConstraintsMeet demandProduction capacity constraintsNonnegativityObjective: Minimize cost (or max profit)ExamplesCHAMPUSOutsourcing/privatizationStaffing courses with adjuncts11Make vs. Buy Textbook Example: Weedwacker CompanyMeet demand for two types of trimmers30,000 electric & 15,000 gasCheaper to make than to buy $55 vs. $67 and $85 vs. $95, respectivelyBut producing in-house constrained by limited time in three departments10,000 hours in production department15,000 hours in assembly department5,000 hours in packaging departmentObjective: Minimize cost12Generalize Make vs. BuyProducts can be obtained through more than two sourcesDecision variablesXij = amount of product i obtained from source jConstraintsSupply constraints on each source jDemand constraints on each product iProduction capacity constraints on own productionNon-negativity13(Investment Portfolio) Allocation Pbs14#3: Investment Portfolio AllocationAllocate a pool of resources (e.g., money or workers) across several available “instruments”Decision variables – one for each instrumentXi = amount invested in instrument iConstraintsAll resources allocatedDiversity constraints on amount invested in any one instrument or type of instrumentNon-negativityObjective: Maximize return/benefitClassic Example: Ragsdale’s Blacksburg National Bank Problem15Allocation ProblemsBeyond Investment PortfoliosOther examples that fit the problem structureDollars to development projectsDollars to public works projectsStaff/personnel to work projectsDollars to media outlets in marketing campaignsCase 3.4: Saving the Manatee16#6: Multi-period Planning ProblemsShow up in many contexts, e.g.production planning (over time) investing (over time)Workforce or demographic planning (over time)Key characteristicsdecisions about multiple time periodssome quantity is “conserved” over time, creating constraints that link different time periodsThis is the classic example used in most textbooks.This is the classic example used in most textbooks.Also useful; Wolverine Retirement Case is great example.Also useful; Wolverine Retirement Case is great example.Mention just to show MPP as a class of pbs,

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