# Credibilistic Programming: An Introduction to Models and Applications (Uncertainty and Operations Research)

Language: English

Pages: 144

ISBN: 364236375X

Format: PDF / Kindle (mobi) / ePub

It provides fuzzy programming approach to solve real-life decision problems in fuzzy environment. Within the framework of credibility theory, it provides a self-contained, comprehensive and up-to-date presentation of fuzzy programming models, algorithms and applications in portfolio analysis.

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ak +bl ≥ai +bj bl >bj max ak +bl ai +bj It is easy to prove that n wi = m wij , j =1 μk ∧ νl ∧ 0.5 wj = wij i=1 μk ∧ νl ∧ 0.5. 50 3 Expected Value Model for all i = 1, 2, . . . , m, j = 1, 2, . . . , n. If {ai }, {bj } and {ai + bj } are sequences consisting of distinct elements, then m E[ξ ] = n m E[η] = a i wi , b j wj , j =1 i=1 n E[ξ + η] = (ai + bj )wij . i=1 j =1 Thus E[ξ + η] = E[ξ ] + E[η]. Otherwise, we may give them a

exp π(e − r)/ 6σ ≥ α √ = e − ln(1 − α) − ln α 6σ/π which is shown by Fig. 4.14. Theorem 4.8 (Liu 2004) The pessimistic value ξinf (α) is an increasing and leftcontinuous function of α. 4.2 Pessimistic Value 87 Fig. 4.13 Pessimistic value of an exponential fuzzy variable Fig. 4.14 Pessimistic value of a normal fuzzy variable Proof It is easy to prove that ξinf (α) is an increasing function with respect to α. Next, we prove the left-continuity. Let αi be an arbitrary sequence of positive

0.0001 43 1.8994 0.0003 19 1.8995 0.0003 44 1.8998 0.0001 20 1.8993 0.0004 45 1.8997 0.0002 21 1.8987 0.0007 46 1.8977 0.0012 22 1.8997 0.0001 47 1.9001 0.0001 23 1.9000 0.0000 48 1.8985 0.0008 24 1.8991 0.0005 49 1.8997 0.0002 25 1.8996 0.0002 50 1.8986 0.0007 Step 3. Calculate the minimum and maximum objective values a = min f (x, y i ) | 1 ≤ i ≤ N , Step 4. Step 5. Step 6. Step 7. b = max f (x, y i ) | 1 ≤ i ≤ N . Set r = (a + b)/2. If L(r) ≥ α,

Definition 7.1, the distance between ξ and η is d(ξ, η) = (a1 − c2 ) + 2(b1 − b2 ) + (c1 − a2 ) /4. In general, the distance between triangular fuzzy variables ξ and η is d(ξ, η) = |a1 − c2 | + 2|b1 − b2 | + |c1 − a2 | /4. Example 7.4 Suppose that ξ = (a1 , b1 , c1 , d1 ) and η = (a2 , b2 , c2 , d2 ) are two independent trapezoidal fuzzy variables such that (a1 , d1 ) ∩ (a2 , d2 ) = ∅ (see Fig. 7.2). It has been proved that ξ − η is also a trapezoidal fuzzy variable, denoted by (a1 − d2 , b1 − c2

a function from m to , then fuzzy variable f (ξ1 , ξ2 , . . . , ξm ) has the credibility function ⎧ min νi (xi ), if sup min νi (xi ) < 0.5 ⎪ ⎨ sup 1≤i≤m f (x )=z f (x )=z 1≤i≤m μ(z) = (1.29) ⎪ 1 − sup min ν (x ), if sup min νi (xi ) ≥ 0.5 i i ⎩ 1≤i≤m 1≤i≤m f (x )=z f (x )=z for all z ∈ , where νi is the credibility function of ξi for i = 1, 2, . . . , m. Proof Since fuzzy variables ξ1 , ξ2 , . . . , ξm are independent, the fuzzy vector (ξ1 , ξ2 , . . . , ξm ) has a joint credibility function