Page - 168 - in Algorithms for Scheduling Problems

Algorithms 2018,11, 35 “primary”.Next,denotebypfsecond(s)=Pr (Af second(s)) theprobability that theriskdriver f isasource ofdifferentadverseevents in layer swhich isaresultof the indirecteffectonthe f bytheriskdrivers f ′ thathavecausedtheadverseevents in layer s+1; theseprobabilitiesare termedas“secondary”. Introduce the followingnotation: p(1)i (s)=Pr ( Aprimei (s) ) =Pr ( A(1)i ) =Pr{theriskdriver fi is thecauseofadverseeventonthe layersonly} p(2)i (s)=Pr ( Asecondi (s) ) =Pr ( A(2)i ) =Pr {theriskdriver fi is thecauseofadverseeffectonthe layer sas theresultof theriskdriversonthe layers s+1}. Forsimplicity,andwithoutlossofgenerality,supposethatthelistofriskdriversF={f1, f2, . . . fN} is complete foreach layer. Thenthefollowingholds N ∑ i=1 pi(s)=1 for s=0,1,2, . . .. (1) Denotep(s)=(p1(s),p2(s), . . .pN(s)). It isobvious that Ai(s)=A (1) i (s)∪A(2)i (s)andA(1)i (s)∩A(2)i (s)=∅, j=1,2, . . . ,N. Therefore, p(Ai(s))= p ( A(1)i (s) ) +p ( A(2)i (s) ) or pi(s)= p (1) i (s)+p (2) i (s) i=1,2, . . . ,N. (2) Then thevectorof riskdriverprobabilitiesp(s) = (p1(s),p2(s), . . .pN(s)) canbedecomposed into twovectorsas p(s)=p(1)(s)+p(2)(s), (3) where p(1)(s) = ( p(1)1 (s),p (1) 2 (s), . . . ,p (1) N (s) ) is the vector of drivers’ primary probabilities and p(2)(s)= ( p(2)1 (s),p (2) 2 (s), . . . ,p (2) N (s) ) thevectorofdrivers’ secondaryprobabilities. Forany layer s,define the transitionmatrixM(2)(s)ofconditionalprobabilitiesof theriskdrivers on layers thatareobtainedas theresultof riskdriversexistingonlayer s+1 M(2)(s)= ( p(2)ij (s) ) N×N , s=0,1,2, . . . , with p(2)ij (s)=Pr ( A(2)j (s) ∣∣∣Ai(s+1))i, j=1,2, . . . ,N, s=0,1,2. . . (4) Next,definethematricesML(s)of theprimarydrivers’probabilitiesas ML(s)= ( qij(s) ) N×N, s=0,1,2, . . . , qij(s)= p (1) j (s), i, j=1,2, . . . ,N, s=0,1, . . . ML(s)= ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝ p(1)1 (s) p (1) 2 (s) . . p (1) N (s) p(1)1 (s) p (1) 2 (s) . . p (1) N (s) . . . . . . . . . . p(1)1 (s) p (1) 2 (s) . . p (1) N (s) ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠ (5) 168