Page - 170 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 35 As far as the tables TBLu, for all the nodes belonging to a certain arbitrary SC layer, says, arederived,all the tablesaregatheredinto theCut_TableCTs for theentireSCcut. LetRs(u)denote the totalnumberofobservedadverse (critical)events inanodeuofcutCsduringacertainplanning period. If suchcutcontainsn(s)nodes, the totalnumberofcriticalevents in it isNs=∑u=1, . . . ,n(s)Rs(u). Assumethat thereareF riskdrivers. Foreachriskdriver f (f =1, . . . ,F),wecancompute thenumber Ns(u, f)of criticaleventscausedbydriver f in thenodeuandthe totalnumberNs(f)of criticalevents inallnodesofcutCs, as registered in theriskprotocols. Therelative frequencyps(f)of thatdriver f is thesourceofdifferent critical events innodesof sandcanbetreatedas theestimationof thecorrespondingprobability. Thenwecompute the latter probabilityas ps(f)=Ns(f)/Ns (12) Then∑f ps(f)=1. Forthesakeofsimplicityof furtheranalysis,ourmodelapplies tothecasewhenthecriticalevents are independentwithin thesametierandthe lossesareadditive (theseassumptionswillberelaxed in our futureresearch). Foranynodeu from s,wecandefinecorrespondingprobabilitiesps(u, f)of the event thatadriver f is thesourceofadverseevents innodeu ps(u, f)=Ns(u, f)/Rs(u). (13) Thispaper treats theps(u, f)valuesdefinedbyEquation(3)asprobabilitiesofeventsparticipating incalculationof theentropyfunction inEquation(1). Themain ideaof thesuggestedentropicapproach is that the informationentropyinthisstudy estimatestheaverageamountof informationcontainedinastreamofcriticaleventsof theriskprotocol. Thus theentropycharacterizesouruncertainty,or theabsenceofknowledge,about therisks. The idea here is that the less theentropy is, themore informationandknowledgeabout risks isavailable for the decisionmakers. Theentropyvaluecanbecomputed iteratively foreachcutof theSC.Assumethat thenodesofa cutCs−1, aredefinedatstep(iteration) s−1.DenotebyTsall supplier-nodes in thesupply layer sof thegiventree. LetLs(Ts)denote the total lossesdefinedbytheriskprotocolandsummedupforall nodesofcutCs in tiersTs:Ls(Ts)=∑u∈Ts cs(u). Further, letLTdenote the total losses forallnodesof theentirechain. Thus,Ls(Ts) arecontributionsof thesuppliersofcutCs into the total lossesLT. ThenLs(Ts)/LTdefinetherelative losses inthe s-truncatedsupplychain. Therelativecontribution of lower tiers, that is,of thosewith larger svalues,are, respectively, (LT−Ls(Ts))/LT.Onecanobserve that the larger is the share (LT−Ls(Ts))/LT in comparisonwithLs(Ts)/LT, the less is theavailable informationabout the losses in thecutCs. Forexample, if theratioLs(Ts)/LT=0.2, thiscasegiveus less informationabout the losses incutCs in comparisonwith theopposite caseofLs(Ts)/LT=0.8. This argumentmotivatesus to take the ratios (LT−Ls(Ts))/LT as the coefficients (weights) of the entropy(or,ofourunawareness)about theeconomic losses incurredbyadverseeventsaffectingthe environmentalquality. Inotherwords, the lattercoefficientsweighthe lackofourknowledgeabout the losses;as faras thenumber sgrows, thesecoefficientsbecomelessandlesssignificant. Thenthe totalentropyofallnodesu includedinto thecut swillbedefinedas H(s)=∑uH(u) (14) where theweightedentropyineachnodeu is computedas H(u)=−Us∑f ps(u, f) logps(u, f), (15) Us=(LT−Ls(Ts))/LT, (16) Letxu=1ifnodeu fromTs is includedinto s, andxu=0,otherwise. 170