Page - 169 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 35 Definethecomplete transitionmatricesas Mˆ(1)(s)= ( pˆ(1)ij (s) ) N×N , s=0,1,2, . . . , with pˆ(1)ij (s)= p (1) j (s)+p (2) ij (s), i, j=1,2, . . . ,N, s=0,1,2, . . . (6) From(5)and(6) it followsthat Mˆ(1)(s)=ML(s)+M(2)(s), s=0,1,2. . . (7) p(2)j ( A(2)j (s) ) = N ∑ i=1 pi(s+1) ·p(2)ij (s), j=1,2, . . . ,N, s=0,1, . . . or p(2)j ( A(2)j (s) ) = N ∑ i=1 pi(s+1) ·p(2)ij (s), j=1,2, . . . ,N, s=0,1, . . . (8) In thematrix form,Equation(8)canberewrittenas p(2)(s)=p(s+1) ·M(2)(s), s=0,1, . . . (9) Thefollowingclaimis true. Claim. The followingrelationholds: N ∑ i=1 ( pi(s+1) · ( N ∑ j=1 p(2)ij (s) )) =1− N ∑ j=1 p(1)j (s), s=0,1, . . . (10) Theproof is straightforwardandskippedhere. 3. InformationEntropyasaMeasureofSupplyChainComplexity InformationentropyisdefinedbyShannonas follows[20].GivenasetofeventsE={e1, . . . , en} with apriori probabilities of event occurrenceP= {p1, . . . , pn}, pi≥ 0, such that pi + .. . +pn =1, theentropyfunctionH isdefinedby H=−∑I pi logpi. (11) Inorder toyieldallnecessary informationontheSCcomplexity issues, thispaperuses thedata recordingofall adverseeventsoccurred. Theenterprise is tocollect andstore thedataaboutmain adverseevents thatoccurredandledtoeconomic losses in theenterprise, compensationcost, aswell as the statistical analysisof the recordeddata. Such requirementalsoapplies to the registrationof informationaboutcontrolofcomplianceof targetandactualenvironmental characteristics. Similar to [12], foreachnodeu, consideran informationdatabasecalleda ‘riskprotocol’. This isa registration listofmost importantevents thathaveoccurred in thenodeduringapre-specifiedtime period. Theprotocolprovidesus the informationwhetherornot theeventsareundesirableand, in the lattercase,whatare its riskdriversandpossible losses. Thisdata is recordedintablesTBLu, representing listsofevents ineachnodeuduringacertain timeperiodT (e.g.,month,oryear). Eachrowinthetablecorrespondstoanindividualeventoccurring inagivennodeatacertain timemoment (forexample, aday).Weusesymbol f asan indexof risk drivers,Fas the totalnumberof riskdrivers,and rasan indexof theevent (row). Thevaluezrf at the intersectionofcolumn f androw r is equal to1 if the risk factor f is a sourceof theadverseevent r, and0—otherwise. The last column,F+1, ineachrow(r) contains themagnitudeofeconomic loss causedbythecorrespondingevent r. 169