Page - 132 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 54 methodbyRosen. Thus,a three-foldcomprehensivecomparisonismadefor themulti-marketnetwork design. Thecanonical settingat eachmarket includesanewsvendor setupwhohas toobtainhow muchofthecommonalitytoorder. Everyitemcostscbutcanbesoldatpricep,wherep>c. Foragiven demand, theprofit isgivenas theproductofpriceandminimumofdemandandorderingquantity minus theprocurementcost,which iscostper itemmultipliedbyorderingquantity. Supposewehave arandomdemand,Dhavingdensity f andcumulativedistributionF.Theexpectedprofit inanyof the sellingpoints in thenetworkundernewsvendormodelwithdemanduncertainty isgivenby: π(x)= pE(min(D,x))−cx (1) where, x:Numberof itemsordered p: Sellingpriceof the item c: Costof item,where c<p D:Demand E(y): Expectedvalueofy Consideringthat thedemandD isuniformlydistributedbetween [a,b]. Thentheprofit function inEquation(1)canbeexpressedas [13]: π(x)=Ax2+Bx+C (2) where, A= −p 2(b−a),B= pb b−a−c,C= −pa2 2(b−a) Now,we apply the abovemodel to themulti-market scenario. Each of themarkets can be consideredasanewsvendormodel.Consider themulti-marketmodelasdepicted inFigure2. Figure2.Generalizedmulti-marketdesign. Withreference toFigure2,weuse the followingnotations: 132