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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
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book Algorithms for Scheduling Problems"
Algorithms for Scheduling Problems
- Title
- Algorithms for Scheduling Problems
- Authors
- Frank Werner
- Larysa Burtseva
- Yuri Sotskov
- Editor
- MDPI
- Location
- Basel
- Date
- 2018
- Language
- English
- License
- CC BY 4.0
- ISBN
- 978-3-03897-120-7
- Size
- 17.0 x 24.4 cm
- Pages
- 212
- Keywords
- Scheduling Problems in Logistics, Transport, Timetabling, Sports, Healthcare, Engineering, Energy Management
- Categories
- Informatik
- Technik