Page - 137 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 54 Asanexample in thedualmarketscenario,eachmarketdecisionmakerwouldbesubjectedto boundedrationality. Thus, theexpectedorderingquantities ineachmarket,withuniformlydistributed demandbetween [ai, bi] (i=1 .. . n)andwithboundedrationalityparametersasβ1 andβ2wouldbe, E(x1+x3)=μ1−σ1φ((b1−μ1)/σ1)−φ((a1−μ1)/σ1)ϕ((b1−μ1)/σ1)−ϕ((a1−μ1)/σ1) (13) where, μ1= b1− c1p1(b1−a1) σ 2 1 = β1− b1−a1 p1 and, E(x2+x4)=μ1−σ1φ((b1−μ1)/σ1)−φ((a1−μ1)/σ1)ϕ((b1−μ1)/σ1)−ϕ((a1−μ1)/σ1) (14) The concept of bounded rationality canbe reconciledwith original problemofmulti-market networkdesigninwaythat initiallyunderuniformdemanddistribution, theproductionallocation strategy isdecidedbythecompanyandthenorderingdecisionsaremadeunderboundedrationality, givingsuboptimal. Similarly, themodelcanbeappliedtomulti-marketscenariowithanynumberof plantsandmarkets. Thus,underboundedrationalconditions,wecanfindtheexpectedprofitasgiven in (3).Hence,acomprehensivecomparison ismadeamongst theprofitsundervariousconditions in the followingsection. 4.TestResultsandComparativeAnalysis Acomprehensiveanalytical studywasmadefor thedifferent testexamplesofvaryingproblem sizes. Eachcasedepictstheresults indifferentscenarios.Weillustratetheoptimalproductionallocation in thenetworkdesign for theuniformdemanddistribution. Finally,wemakeacomparisonof this withprofits in thescenarioofboundedrationaldecisionmaker. Alldatasetsusedfor the testexamplesareprovidedinAppendixA.These includethepricing details, themanufacturingandtransportationcostsandthedemanddistributionparameters. X:matrixproductionallocation,wherexij represents theoptimalproductionallocationbetween plant iandmarket j. 4.1. I.TestCase1 Problemsizen=2. 1. ScenarioofUniformDemandDistribution X: Theproductionallocationvalues for test case1aregiven inTable1. Table1.Productionallocationvalues for test case1. Market1 Market2 Plant1 1636.36 0 Plant2 0 1187.5 Objective functionvalue=108,678. Weobserve that thesolution indicatesamarket focusedstrategy for thecompanyasdepicted inFigure4. 137