Page - 133 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 54 Notations xij:Quantityofcommoncomponentmanufactured inplant i, andassembledatmarket j. pj: Sellingpriceoffinalassemblyatmarket j. Ctij: Costof transportationofcommoncomponent fromplant i tomarket j. (Ctii =0) Cmi: Costofmanufacturingthecommoncomponent inplant i Kmi: Capacity limitonthecommoncomponentmanufacturingatplant i Kaj: Capacity limitonthefinalassemblyatmarket j. Dj: Demandat themarket j, followingauniformdistribution,Dj~U[aj,bj] aj:Lowerboundontheuniformdemanddistribution inmarket j bj:Upperboundontheuniformdemanddistribution inmarket j Caj: Average cost per item incurred at market j due to various sources from which the items are transported vij: Valueaddition in termsofperunitprofitobtainedduetoacommoncomponent transportation fromplant iandsoldatmarket j Aj: Coefficientofx2 termintheEquation(2) forprofit function inmarket j Bj: Coefficientofx termintheEquation(2) forprofit function inmarket j Cj: Constant termintheEquation(2) forprofit function inmarket j βj: Boundedrationalityparameter formarket j πj: Profit inmarket j n: Totalnumberofcountries in themulti-marketnetwork. Plant, i=1,2 . . . n Market j=1,2 . . . n Toapplythenewsvendor’smodel toeachmarket,weneedtodeterminetheaveragecostper item ateachmarket. This is takenas theweightedaverageof thecommoncomponentmanufacturingcost per itemfromthevarioussourceplants fromwhichthe itemsare transported. Caj= a ∑ j=1 xij(Cmi+Ctij) a ∑ i=1 xij (3) Note here thatCtii =0, as there is no transportation cost fromplant i tomarket i. Thevalue addition ineachpath in termsofperunitprofitobtainedduetoacommoncomponent transportation fromplant iandsoldatmarket j isgivenby, Vij= pj−Cmi−Ctij Thus, from Equation (2), the profit in each market j, under uniform demand distribution Dj~U[aj,bj]wouldbegivenas, πj=Aj( a ∑ i=1 xij)2+Bj( a ∑ i=1 xij)+Cj (4) where, Aj= −pj 2(bj−aj), Bj= pjbj (bj−aj)−Caj,Cj= −pja2j 2(bj−aj) NoteherethatCaj isnotaconstantterm.ThustheequationsisexpandedandtakingBBj= pjbj (bj−aj), wehavetheexpressionfor the totalprofitπas, 133