Page - 167 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 35 andbudget.Moreover, thesupplychaincontrolof therootnodeof thesupplychain ismuchmore important thanthecontrolofanyof its successors ([6,8]). Considera tree-typegraphrepresentingthehierarchical structureofan industrial supplychain Define the “parent layer”, called also the “main layer”, as consisting of a single node, as follows: L0 = {n0}wheren0 is the single node of the parent layer. Called also the original equipment manufacturer (OEM), or the root node. OEM is afirm (or company) that creates an endproduct, for, instance,assemblesandcreatesanautomobile. Definea layer Ls (alsodenotedas layer s) as the setofnodeswhichareon thesamedistance s fromtherootnoden0 in theunderlyinggraphof theSC. Layer 1 (also called Tier 1) are the companies supplying components directly to the OEM that set up the chain. In a typical supply chain, companies in Tier 2 supply the companies in Tier 1; Tier 3 supplies Tier 2, and so on. Tiered supply chains are common in industries such as aerospaceorautomotivemanufacturingwherethefinalproductconsistsofmanycomplexcomponents andsub-assemblies. Definea“cut” (alsocalleda“cross-section”)Cs asaunionofall the layersL0,L1,L2, . . .Ls, from0 to s. It isevident thatC0=L0, Cs−1⊂Cs, Cs= { C(s−1),Ls } s=1,2, . . ., S. Assumethat, foreachnodeof theSC, the listof riskdriversF= {f1, f2, . . . fN} isknown,each beingasourceofdifferentadverseevents in thenodesof theSC.Forsimplicity,butwithout lossof generality,assumethatanyadverseevent iscausedbyasingleriskdriver (otherwise,onecansplit suchamulti-driveevent into several elementaryevents eachonebeingcausedbyasingledriver). HereNis the totalnumberofall thedrivers. Anotherbasicassumptiontobeusedinthisworkis that thedrivefactorsaremutuallydependent. Itmeans, forexample, thatanunfavorabletechnologicaldecisionoranenvironmentalpollution, that is, causedbya technology-baseddriver insomecomponentatTier2may lead toanadverseevent in supplyoperations toanodeatTier1.AtechnologicalmistakeatTier3maybeasourceofadelayed supply toTier2,andsoon. Ingeneral, anyfactors f happeningat tier smaybedependingonafactor f ′ at anearlier tier s+1, f =1, . . . ,N; f ′=1, . . . ,N; s=1, . . . ,S. Below, thedependencieswill be describedwith thehelpof theN×Nmatrixof relativeprobabilities. The followingMarkovian property is assumed to take place. Assume that the dependence betweenanyfactor intiers,ontheonehand,andthefactors inthelowertierss+1,s+2, . . . ,Sactually existsonlyfor thefactors inapairofneighboringtiers (s, s+1),where s=0,1, . . . ,S.Moreover,assume that thepitfallsandanydefectivedecisionsdonotflowdownwards, that is, anyrisk factor in tier s doesnotdependupontheriskdrivers in thenodesofhigher layers,numbered s−1, s−2, . . . , 1. Ineach layer s,whencomputing theprobabilityof riskdrivers f occurring in thenodesof the layer s, twotypesofadverseeventshaveaplace. First, thereare theevents (called“primaryevents” anddenotedbyAfprime(s)) thathavehappened in thenodesof layer sandwhicharecausedby the riskdriver f, f =1, . . . ,N. Second, thereare theevents (called“secondaryevents”anddenotedby Afsecond(s+1, s)) thathavehappenedin thenodesof thenext layer (s±1)buthavean indirect impact uponinverseevents ins, since therisk factorsaredependent.Moreprecisely,differentdrivers f ′ ins+1 have impactuponthedriver f in layer s, f =1, . . . ,N; f ′=1, . . . ,N; s=1,2, . . . ,S. The impact from f ′ to f is estimatedwith thehelpof the transitionprobabilitymatrixMwhich is definedbelowandcomputedfromthedata in theriskprotocols. DenotebyAj(s) the followingevents: Af(s)= {riskdriver f is thesourceofvariousadverseevents insupply toall thenodesof layer s}, f =1, . . . ,N, s=0,1, . . . ,S. Denotebypf(s)=Pr (Af(s)) theprobability that theriskdriver f is thesourceofdifferentadverse events insupply in layer s, pi(s)=Pr(Ai(s))=Pr {theriskdriver fi is thecauseofadverseeventonthe layer sonly} Denotebypfprime(s)=Pr (Af prime(s)) theprobability that theriskdriver f is thesourceofdifferent adverseevents in layer s, andwhicharecausedbytheriskdriver f. Theseprobabilitiesare termedas 167