P (Ei*) PI(Ei*).P2(Ei*) [ i~i,pl(ei)'p2(ei) ]-1 i=l [ Infi~i,Pl(Ei).P2(Ei) 1-1 [ Inf~ (Ei).P2(Ei)] -1

Transcription

1 APPENDIX Application of Formula (3.21) to Structures Defined by k-ppds. We have P,(Ei*) i=l P(Ei*) = ~PI(EI) "..." PI(Ei*) "..." PI(Ei) (3.21) and Sup Pj(Ei) = Uji PiES~. Inf Pj(Ei) = Lji PjES~. i=l,...,k; j=l,...,1 We designate: Ui*: = Sup~ P(Ei*) PjES~ j=l...,1 Li*: = Inf P.ES* J J 3=1,...,1 P (Ei*) and confine ourselves at first to the case 1 = 2. We start by calculating Ui*, for a fixed i*, if Pje S j, j = 1,2 : PI(Ei*).P2(Ei*) [ i~i,pl(ei)'p2(ei) ]-1 i=l [ Infi~i,Pl(Ei).P2(Ei) 1-1 [ Inf~ (Ei).P2(Ei)] -1 = Sup 1 + = Ui* = Sup ~ P,(Ei)'P2(Ei) P,(Ei*)'P2(Ei*) = 1 + = 1 + i;~i*pl Sup PI(Ei*) "P2 (Ei*) U1 i* " U2 i* Evidently the sum in the numerator reaches its smallest value, if Pl(Ei*)= Uli* and P2(Ei*) = U2i*. Therefore Ui* is calculated by solving a problem of Non-Linear Programming: the search for Infi~i,Pl(Ei ) "P2(Ei), under the restrictions defined by the two k-pris, combined with E Pt(Ei) = 1 - Ufi* and i~ i,p2(ei) = 1 - U2i* (*) i~i* It is easy to see, that the infimum cannot be found in the interior of the two sub-structures defined by (*); it must appear in a situation which may be described as a combination of two corners be- longing to both sub-structures defined by (*). Therefore the search for the infimum, which in this case is indeed a minimum, can be achieved by means of a computer program which constructs all pairs of corners belonging to the two sub-structures. The values of Li*, if Pj E Sj, j = 1,2 are found in an analogous way:

2 128 Li* = Inf PI(Ei*) "P2 (Ei*) E PI(Ei)'P2(Ei) + SuPi~i,Pl(Ei)'P2(Ei) ]-1 = 1 L1 i* ' L2 i* if Lli* L2i* ~ 0. If Lli* L2i* = 0, then obviously Li* = 0. This time we search for Sup i~i,pl(ei)"p2(ei) under the restriction defined by the k-pri combined with i~i,p,(ei) : 1 - Lli* ~nd i~i,r2(ei) : 1 - L21* (**) As this supremum must appear in a situation in which two corners of the sub-structures (**) are combined, it is indeed a maximum and may also be found by means of a computer program. Whilst the procedures described up to now are sufficient to apply (3.21) to a situation in which two k-pris are combined, the extension of these methods to situations in which more than two k-pris are involved, consists of a stepwise repetition of the same kind of analysis: Once the first two k-pris have been combined resulting in a single one, this can be combined with a third one, and so on. The final result does not depend upon the order of these operations because commutativity and associativity are granted. We shall demonstrate the application of Formula (3.21) to two k-pris with the following example. Example: Let k = 4 and Lll = 0.0 Ull = 0.2 L21 = 0.4 U~I = 0.5 L12 = 0.1 U12 = 0.4 L22 = 0.0 U22 = 0.2 L13 = 0.2 U13 = 0.5 L23 = 0. I U23 = 0.3 L14 = 0.3 U14 = 0.7 L24 = 0.2 U24 = The calculation of U1 requires the use of Inf Z Pi(Ei).P2(Ei) i=2 when 4 4 i~2pl(ei) = 0.8, i~21)2(ei) : 0.5 Controlling all pairs of corners, we find the minimum to be 0.12 (for instance if and P,(E2) = 0.3, PI(E3) = 0.2, P,(E4) = 0.3 P2(E2) = 0.0, Pu(E3) = 0.3, P2(E4) = 0.2 ). [ ]104 4 Ui= Therefore we can calculate: In the same way we derive U2 = , U3 = and U4 = Concerning the lower limits it is immediately seen that LI = 0 and L2 = 0.

3 129 The calculation of L3 uses Sup [ PI(E1)"P2(E1) + Pt(E2)"P2(E2) + PI(E4)"Pa(E4) ] with PI(E1) + PI(E2) + PI(E4) = 0.8 P2(E1) + P2(E2) + P2(E4) = 0.9 and results in: + ]-' = i-~ = La = [ : In the same manner we calculate L4 = so that our result of applying (3.21) to the combina- tion of the two 4-PRIs is: Li = 0.0 U1 = L2 = 0.0 U2 = L 3 = U3 = L U4 = []

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