Page - 136 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 54 multi-nomial logit choicemodel is applied and assumed that decisionmakers selects alterative j, withprobability. φj= e uj β ∑ j e uj β (6) Correspondingly, the following density function gives the logit choice probabilities over a continuousdomain: φ(y)= e u(y) β∫ y e u(y) β (7) Thus, theselectionofagents isarandomvariable. Thesuperioroptionsareselectedfrequently usingtheabove logit structure. Thebestalternativedependsonthemodeof theselectiondistribution. The magnitude of cognitive and computational restrictions suffered by the decision makers is elucidated by β parameter. This can be observedwhen selection distribution approximates the uniformdistributionover theoptions in β→∞ limit. In someutmost instances, decisionmakers randomize theoptionswithequalprobabilitieswhentheyfailed tomakeany informedchoices. In the otherscenario,when β→∞ thechoicedistribution in (1)completely focusedonutilitymaximizing options. The selectionofperfect rational decisionmaker correspondswith this choice. Therefore, wecanconsider themagnitudeofβas theextentofboundedrationality. The logit choice framework to thenewsvendorproblemisemployedhere. Theprofit function isasstated inEquation(1) is π(x)= pE(min(D,x))−cx This isuniquelymaximizedat, x∗=F−1(1−c/p) (8) where,F is thecumulativedistributionfunctionofdemandD.Allothersymbolsareasusedbefore. This solution is selectedwhenthedecisionmaker isperfectly rational, althoughthenewsvendor’s orderingamount isdependentonnoiseanditcalledthebehavioral solution inboundedrationality. Theboundedrationalnewsvendorcanorderanyamountofproductwithin therangeofminimum possibleandmaximumpossibledemandrealizations in thedomainS.Then, theEquation(9)gives the probabilitydensity functionassociatedwithbehavioral solution: φ(y)= e π(x) β∫ s e π(v) β dv = e pE(min(D,x))−cx β∫ s e pE(min(D,v))−cv β dv (9) Now, if thedemandD isuniformlydistributed in in the interval of [a, b], then thebehavioral solution would follow truncated normal distribution over [a, b] with mean μ and standard deviationσ [13]. μ= b− c p (b−a) (10) σ2= β b−a p (11) Theexpectedvalueof this truncatednormaldistributionwouldbe, E(x)=μ−σφ((b−μ)/σ)−φ((a−μ)/σ) ϕ((b−μ)/σ)−ϕ((a−μ)/σ) (12) where,φ(.) denotescumulativenormaldistributionandϕ(.) denotesnormaldensity function. 136