Seite - 166 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 35 Similar tomanyotherresearchers (see,e.g., [11–13]), inorder toaccuratelymeasure therisksof pitfalls andcasesof failed supplies, thispaper tends todetermine theprobabilities ofundesirable eventsandtheirnegative impactsonmaterial,financial, andinformationflows. Theeconomic losses arecausedbyfailures tosupply, shipmentsof inappropriateproducts, incompletedeliveries,delays in deliveries,etc., andtheharmfuleffectof theadverseeventsmaybeexpressedinmonetaryformby relevantpenalties. The main differences of the present work in comparison with close earlier papers (see, e.g., [7,12,14,15])alsoexploitingtheentropyapproachfor theSCcomplexityanalysisare thefollowing: • This paperdevelops anewgraph-theoreticmodel as a tool for selecting themost vulnerable disruptionrisks inside theSCwhich, in turn, canessentiallydecrease thesizeof the initial SC modelwithoutsacrificingessentialknowledgeabout therisks; • ThispaperintroducestheconditionalentropyasatoolforintegratedanalysisoftheSCcomplexity underuncertainty,providesmorepreciseestimationof thesupplychaincomplexity taking into account linksbetweenthenodesofdifferent layers; • Thispapersuggestsanewfastentropy-basedalgorithmforminimizingtheSCsize. Thispaper is structuredas follows. Therelateddefinitions fromgraphtheoryarepresented in the next section. Thedefinitionof informationentropyandthedetailedproblemdescriptionaregiven inSection3. Section4describes theentropy-basedalgorithmpermitting toreduce theSCmodelsize withouta lossofessential information. Section5describes thenumericalexample. Section6concludes thepaper. 2.BasicDefinitions Wishing to avoidanyambiguity in furtherdiscussions,webeginwithkeydefinitionsof risk andambiguity in relationshipsbetween themanufacturerand its supplier. There isawidespecter ofdifferentdefinitionsof riskanduncertainty. In this study,we followKnight’s [16]viewandhis numerous followers. Theuncertainty is theabsenceofcertainty inourknowledge,or, inotherwords, a situationwherein it is impossible topreciselydescribe futureoutcomes. In theKnightian sense, therisk ismeasurableuncertainty,which ispossible tocalculate. Similar tomanyotherriskevaluators,weassumethat thenotionof riskcanbedescribedas the expectedvalueofanundesirableoutcome, that is, theproductof twocharacteristics, theprobabilityof anundesirableevent (that is,anegativedeviationof thedelayedsupplyor failure toreachtheplanned supplytarget),andtheimpactorseverity, that is,anexpectedloss inthecaseof thedisruptionaffecting thesupplyofproductsacrossorganizations inasupplynetwork. Inasituationwithseveralpossible accidents,weadmit that thetotal risk is thesumoftherisks fordifferentaccidents (see,e.g., [11,17,18]). In themodelconsideredbelow,an“event” is theobservablediscretechange in thestateof theSC or itscomponents.A“riskdriver” isa factor,adrivingforce thatmaybeacauseof theundesirable unforeseen event, suchasdisruptions, breakdowns, defects,mistakes in thedesignandplanning, shortagesofmaterial insupplyintheSC,etc. Inthispaper,westudythesituationswithan“observable uncertainty”where there is anobjectiveopportunity to register, for apre-specifiedperiodof time, theadverse events in the relationshipsbetween the components in theSC. Sucha registration list, calleda“riskprotocol”,providesus the informationwhetherornot theeventsareundesirableand, in the case if the event isundesirablewhat are its riskdrivers andpossible loss (see [12,19]). Such statistics in theriskprotocolspermits thedecisionmaker toquantitativelyevaluateacontributionof eachdriverandthe total (entropy-based)observable information in theSC. Thereexistsawidediversityof risk types, riskdrivers, andoptions for theirmitigation in theSC. Their taxonomyliesbeyondthescopeof thispaper.Manyauthorsnoticedthat ifa researcher tries to analyzepotential failures/disruptionsofall thesuppliers inaSCor theirabsolutemajority,he/she encountersasimply impracticalandunrealisticproblemdemandinganastronomicamountof time 166