Page - 142 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 54 5.ConclusionsandFutureWork The decision regarding themanufacturing of the common component has been analytically modeledandsolved for amulti-market scenario. For the caseofdeterministicdemand, the linear formulationwassolvedfor the3distinct scenarios. Then, thecaseofuncertaintyhasbeenanalyzed, where the demand is uniformly distributed. This was formulated as a quadratic problem and solvedusingaproposed integratedmethodologyexploiting theFrank-Wolfe linearization technique, alongwithBenders’decompositionmethod. Toestablish the superiorityof theproposedsolution methodology, in termsofconvergence,wecompareditwithRosen’sgradientsearchtechnique. Incase of uniformdemandwith bounded rationality, the decisionmaker ordering a specific number of units for selling in the respectivemarket cannotmake theoptimal choice, buthis choice isguided byaprobabilistic logit choicemodel,withhigherprobabilityofmaking the right choice. Wehave demonstrated(asexpected)thatboundedrationaldecisionsarelessprofitablethantherationaldecision makingandgetreinforcedwhenthe lackofclarityofdecisionmakersbecomesmoreprominent. Scope for futurestudiescaninclude,but isnot limitedto, incorporatingsomeotherbehavioralcharacteristics alongwithboundedrationality. Author Contributions: Manoj Kumar Tiwari and Narasimha Kamath contributed to the overall idea and manuscript drafting. D.G.Mogale andGeet Lahoti defined the problemand formulated themathematical model. ShashiBhushanJhaandManishShuklawritten theoverallmanuscript. Finally,all theauthorshaveread andapprovedthefinalmanuscript. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. AppendixA Thenotationsusedhavefollowing implications: n Numberofcountrieswhere themarketsarepresent. p Vectorwherepj is thesellingpriceoffinalassemblyatmarket j. Ct Matrix for the transportationcostwhere,Ctij is thecostof transportationofcommon component fromplant i tomarket j. (Ctii=0). Cm VectorwhereCmi is costofmanufacturingthecommoncomponent inplant i. Km VectorwhereKmi is thecapacity limitonthecommoncomponentmanufacturingatplant i. Ka VectorwhereKaj is thecapacity limitonthefinalassemblyatmarket j. a Vectorwhereai is lowerboundontheuniformdemanddistribution. b Vectorwherebi isupperboundontheuniformdemanddistribution. I. TestCase1 1. n=2 2. p 3. Ct 4. Cm 5. Km 142