A Weighted possibilistic mean value approach

138

6.4 A Weighted possibilistic mean value approach

In this section introducing a weighting function measuring the importance of λ - level sets of fuzzy numbers we shall define the weighted lower possibilistic and upper possibilistic mean values, crisp possibilistic mean value of fuzzy numbers, which is consistent with the extension principle and with the well-known definitions of expectation in probability theory. We shall also show that the weighted interval-valued possibilistic mean is always a subset moreover a proper subset excluding some special cases of the interval-valued probabilistic mean for any weighting function. A trapezoidal fuzzy number , , , ~ 4 3 2 1 r r r r r = is a fuzzy set of the real line R with a normal, fuzzy convex and continuous membership function of bounded support. The family of fuzzy numbers will be denoted by F. A λ -level set of a fuzzy number , , , ~ 4 3 2 1 r r r r r = is defined by } , { ] ~ [ R ∈ ≥ = x x x r λ μ λ , then  ] , [ } , { ] ~ [ 4 1 4 1 2 1 r r r r r r x x x r − − − + = ∈ ≥ = λ λ λ μ λ R , if λ and } { ] ~ [ ≥ ∈ = x x cl r μ λ R the closure of the support of r~ if = λ . It is well-known that if r~ is a fuzzy number then λ ] ~ [ r is a compact subset of R for all ] 1 , [ ∈ λ . Definition 6.1 Majlender [110] Let  F ∈ r~ be fuzzy number with = λ ] ~ [ r ], , [ 2 1 λ λ a a ] 1 , [ ∈ λ . A function ] 1 , [ : w → R is said to be a weighting function if w is non-negative, monoton increasing and satisfies the following normalization condition ∫ = 1 1 λ λ d w . 6.33 We define the w-weighted possibilistic mean or expected value of fuzzy number r~ as ∫ + = 1 2 1 2 ~ ~ λ λ λ λ d w a a r E w . 6.34 It should be noted that if ] 1 , [ , 2 ∈ = λ λ λ w then ∫ + = 1 2 1 . ] [ ~ ~ λ λ λ λ d a a r E w 139 That is the w-weighted possibilistic mean value defined by 6.24 can be considered as a generalization of possibilistic mean value in Chen [32]. From the definition of a weighting function it can be seen that λ w might be zero for certain unimportant λ -level sets of r~ . So by introducing different weighting functions we can give different case-dependent important to λ -levels sets of fuzzy numbers. Let , , , ~ 2 1 β α r r r = be a fuzzy number of trapezoidal form and with peak ] , [ 2 1 r r , left- with α and right-with β and let ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − = − 1 1 1 2 2 1 γ λ γ λ w , where 1 ≥ γ . It’s clear that w is weighting function with = w and ∞ = − → lim 1 λ λ w . Then the w-weighted lower and upper possibilistic mean values of r~ are computed by [ ] ∫ − − − − − = − − 1 2 1 1 1 1 1 2 ] 1 [ ~ ~ λ λ γ α λ γ d r r E w 1 4 2 1 2 1 − − − = γ γ α r , and [ ] ∫ − − − − − = − + 1 2 1 1 1 1 1 2 ] 1 [ ~ ~ λ λ γ β λ γ d r r E w 1 4 2 1 2 2 − − + = γ γ β r and therefore ⎥⎦ ⎤ ⎢⎣ ⎡ − − + − − − = 1 4 2 1 2 , 1 4 2 1 2 ~ ~ 2 1 γ γ β γ γ α r r r E w 1 4 4 1 2 2 ~ ~ 2 1 − − − + + = γ α β γ r r r E w . 6.35 This observation along with Theorem 6.1 as in Section 6.3.3 leads to the following theorem. 140 Theorem 6.4 The mean VaR efficient portfolio model is ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ ∑ ∑ = = = ∈ n j j ji n j j ji w q i i R x x c x r E n 1 1 1 ~ ~ max λ 6.36 i n j i j ji b x r Pos to subject β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ = 1 ~ ~ , q i , 1 = , 6.37 ∑ = = n j j x 1 1 , 6.38 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.39 In the next theorem we extend Theorem 6.3 to the case weighted possibility mean value approach with a special weighted λ w . Theorem 6.5 Let ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − = − 1 1 1 2 2 1 γ λ γ λ w , 1 ≥ γ the weighted possibility mean of the trapezoidal fuzzy number 4 3 2 1 , , , ~ ji ji ji ji ji r r r r r = where ≤ 2 1 ji ji r r 4 3 ji ji r r and addition 4 3 2 1 , , , ~ i i i i i b b b b b = is a trapezoidal fuzzy number for VaR i level, q i , 1 = . Then the possibilistic mean VaR portfolio selection model is ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ n j j ji n j j ji n j j ji n j j ji n j j ji q i i R x x c x r x r x r x r n 1 1 3 1 4 1 2 1 1 1 1 4 4 1 2 2 max γ γ λ 6.40 to subject 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = , 6.41 ∑ = = n j j x 1 1 , 6.42 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.43 141 Proof : From the equation 6.16, we have 1 4 4 1 2 2 ~ ~ 1 3 1 4 1 2 1 1 1 − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − + + = ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ ∑ ∑ ∑ ∑ = = = = = γ γ n j j ji n j j ji n j j ji n j j ji n j j ji w x r x r x r x r x r E , 1 ≥ γ . From Lemma 6.1, we have that i n j i j ji b x r Pos β ≤ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∑ = 1 ~ ~ , q i , 1 = is equivalent with 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = . Furthermore, from 6.40-6.43 given by Theorem 6.4, we get the following form : ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ n j j ji n j j ji n j j ji n j j ji n j j ji q i i R x x c x r x r x r x r n 1 1 1 1 4 1 2 1 1 1 1 4 4 1 2 2 max γ γ λ 6.44 to subject 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = 6.45 ∑ = = n j j x 1 1 , 6.46 n j M x M j j j , 1 , 2 1 = ≤ ≤ . 6.47 This completes the proof. □ Problem 6.44-6.47 is a standard multi-objective linear programming problem. Also we can obtain an optimal solution by using some algorithms of multi-objective programming Kacprzyk and Yager [76] and Stanley and Li [177]. For ∞ → γ we see that = ∞ → ~ ~ lim r E w γ 8 2 4 3 2 1 r r r r − + + . Thus we get Corollary 6.1 For +∞ → γ , the weighted possibilistic mean VaR efficient portfolio selection model can be reduce to the following linear programming problem: 142 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − + + ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = ∈ n j j ji n j j ji n j j ji n j j ji n j j ji q i i R x x c x r x r x r x r n 1 1 1 1 4 1 2 1 1 1 8 2 max λ subject to 1 1 3 2 1 4 1 ≥ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − + ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − − ∑ ∑ = = n j i j ji i n j i j ji i b x r b x r β β , q i , 1 = , ∑ = = n j j x 1 1 , n j M x M j j j , 1 , 2 1 = ≤ ≤ .

6.5 A weighted possibilistic mean variance and covariance of fuzzy numbers