Case of fuzzy technological coefficients

83 ⎪ ⎩ ⎪ ⎨ ⎧ + − + + = − = + + = − − = + + = = − = otherwise i T n n j and T K m K Km i T K m K Km i n j r a ij ij , 1 1 , 1 , 1 , , 1 , 1 , 1 1 , , 1 , 1 , , 1 , ⎩ ⎨ ⎧ + = = = otherwise mT i n j a ij , 1 , , 1 , 1 ⎩ ⎨ ⎧ + = = − = otherwise mT i n j a ij , 2 , , 1 , 1 ⎩ ⎨ ⎧ + + + = = + + − + = = − − − otherwise m mT mT i m K TK n K T n j p a K T n j ij , 2 , 3 , , 1 , , 1 1 , 1 In the next section we extended this result to fuzzy decisions theory.

5.5 Case of fuzzy technological coefficients and fuzzy right-hand side numbers

5.5.1 Case of fuzzy technological coefficients

In this section presents an approach to portfolio selection using fuzzy decisions theory. We consider the problem 5.34 – 5.36 with fuzzy technological coefficients Gasimov [57]. max X ϕ = ∑ = n j j j X c 1 5.37 subject to i mT n j j ij b X a ≤ ∑ + =1 ~ , 2 , 1 + + = m mT i , 5.38 ≥ j X , mT n j + = ,1 . 5.39 Assumption 5.1. ij a ~ is a fuzzy number for any i and j. In this case we consider the following membership functions: i 1. For 1 , , 1 , 1 − = + + = T K m K Km i and n j , 1 = ⎪ ⎩ ⎪ ⎨ ⎧ + − ≥ + − ≤ − − + − − = . , , 1 ij ij ij ij ij ij ij ij ji a d r t if d r t r if d t d r r t if t ij μ 84 2. For 1 , , 1 , 1 − = + + = T K m K Km i and j=n+Ti-Km-1+K+1 ⎪ ⎩ ⎪ ⎨ ⎧ + − ≥ + − ≤ − − + − − = , 1 , 1 1 1 1 1 ij ij ij ij a d t if d t if d t d t if t ij μ ii For 2 , 3 + + + = m mT mT i , m K , 1 = and TK n K T n j + + − + = , 1 1 ⎪ ⎩ ⎪ ⎨ ⎧ + ≥ + ≤ − + = − − − − − − − − − − − − − − − , 1 1 1 1 1 , , 1 ij ij ij ij K T n j a d p t if d p t p if d t d p p t if t K T n j K T n j K T n j K T n j ij μ where R t ∈ and ij d for all 2 , 1 + + = m mT i , 1 , − = T K and mT n j + = ,1 . For defuzzification of this problem, we first fuzzify the objective function. This is done by calculating the lower and upper bound of the optimal values first. The bounds of the optimal values l z and u z are obtained by solving the standard linear programming problems max 1 X z ϕ = 5.40 subject to i j mT n j ij b X a ≤ ∑ + = 1 , 2 , 1 + + = m mT i , 5.41 ≥ j X , mT n j + = ,1 , 5.42 and max 2 X z ϕ = 5.43 subject to i j mT n j ij b X a ≤ ∑ + = 1 ˆ , 2 , 1 + + = m mT i , 5.44 ≥ j X , mT n j + = ,1 , 5.45 where 85 ⎪ ⎩ ⎪ ⎨ ⎧ − = + − + + = + + = + − − = + + = = + − = otherwise d T K and i T n n j m Km Km i d T K and m K Km i n j d r a ij ij ij ij ij , 1 , , 1 1 , 1 , 1 , 1 , 1 1 , 1 , 1 , , 1 , ˆ ⎪⎩ ⎪ ⎨ ⎧ + = = + = otherwise d mT i n j d a ij ij ij , 1 , , , 1 , 1 ˆ ⎪⎩ ⎪ ⎨ ⎧ + = = + − = otherwise d mT i n j d a ij ij ij , 2 , , 1 , 1 ˆ ⎪⎩ ⎪ ⎨ ⎧ + + + = = + + − + = + = − − − otherwise d m mT mT i and m K TK n K T n j d p a ij ij K T n j ij , 2 , 3 , , 1 , , 1 1 , ˆ 1 The objective function takes values between 1 z and 2 z while technological coefficients vary between ij a and ij ij d a + . Let , min 2 1 z z z = l and , max 2 1 z z z = u . Then l z and u z are called the lower and upper bounds of the optimal values, respectively. Assumption 5.2. The linear crisp problems 5.40- 5..42 and 5.43-5.45 have finite optimal values. In this case the fuzzy set of optimal values, G, which is subset of mT n R + , is defined as Klir and Yuan [84 ] ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≤ ≤ − − = ∑ ∑ ∑ ∑ = = = = n j j j n j j j n j j j n j j j G z X c if z X c z if z z z X c z X c if X 1 1 1 1 1 u u l l u l l μ 5.46 The fuzzy set of the ith constraint, i C , which is a subset of mT n R + , is defined by i 1. For m K Km i 1 , 1 + + = and 1 , − = T K 86 X i C μ = ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ + − ≥ + − ≤ − + − ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = n j j ij ji i n j n j j ij ij i j ij n j n j j ij j ij i n j j ji i X d r b X d r b X r X d X r b X r b 1 1 1 1 1 1 , 1 , , 5.47 2. For m K Km i 1 , 1 + + = and 1 , − = T K = X i C μ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ + − ≥ + − ≤ − + − ∑ ∑ ∑ ∑ ∑ ∑ + = + = + = + = = + = , 1 , 1 , 1 , 1 1 , 1 , 1 , 1 , 1 , , , K i n n j j ij i K i n n j K i n n j j ij i j K i n n j n j j ij j i K i n n j j i X d b X d b X X d X b X b 5.48 where ni,K=n+Ti-Km-1+K+1 ii For , 2 , 3 + + + = m mT mT i and m K , 1 = X i C μ = ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ + ≥ + ≤ − ∑ ∑ ∑ ∑ ∑ ∑ + − + = − − − + − + = + − + = − − − − − − + − + = + − + = − − − + − + = − − − TK n K T n j j ij K T n j i TK n K T n j TK n K T n j j ij K T n j i j K T n j TK n K T n j TK n K T n j j ij j K T n j i TK n K T n j j K T n j i X d p b X d p b X p X d X p b X p b 1 1 1 1 1 1 1 1 1 1 1 . , 1 , , , , 5.49 By using the definition of the fuzzy decisions proposed by Bellman and Zadeh [9], we have min , min X X X j C j G D μ μ μ = . i.e. min , min max max X X X j C j G X D X μ μ μ ≥ ≥ = 87 Consequently, the problem 5.37-5.39 can be written as λ max 5.50 , λ μ ≥ X G 5.51 2 , 1 , + + = ≥ m mT i X i C λ μ , 5.52 ≥ j X , 1 ≤ ≤ λ , mT j , 1 = . 5.53 By using 5.46 and 5.47-5.53, we obtain the following theorem. Theorem 5.1 The portfolio problem with fuzzy technological coefficient can be reduced to the following problem λ max 5.54 2 1 2 1 ≤ + − − ∑ = z X c z z n j j j λ , 5.55 ∑ + = ≤ − mT n j i j ij b X a 1 ˆ λ , 2 , 1 + + = m mT i , 5.56 ≥ j X , 1 ≤ ≤ λ , mT n j + = ,1 . 4.57 where ⎪ ⎩ ⎪ ⎨ ⎧ + − + = − = + + = + − − = + + = = + − = otherwise, , , 1 1 , 1 1 , , 1 , 1 , 1 , 1 , , 1 , 1 , , 1 , ˆ ij ij ij ij ij d i T n j and T K m K Km i d T K and m K Km i n j d r a λ λ λ λ ⎪⎩ ⎪ ⎨ ⎧ + = = + = , otherwise , , 1 , , 1 , 1 ˆ ij ij ij d mT i n j d a λ λ λ ⎪⎩ ⎪ ⎨ ⎧ + = = + − = otherwise, , , 2 , , 1 , 1 ˆ ij ij ij d mT i n j d a λ λ λ ⎪⎩ ⎪ ⎨ ⎧ + + + = + + − + = + = − − − . otherwise , 2 , 3 , , 1 1 , ˆ 1 ij ij K T n j ij d m mT mT i TK n K T n j d p a λ λ λ Notice that, the constraints in problem 5.54-5.57 containing the cross product term j X λ are not convex. Therefore the solution of this problem requires the special approach 88 adopted for solving general nonconvex optimization problem Rockafellar and Wets [153] and White [209].

5.5.2 Portfolio problems with fuzzy technological coefficients and fuzzy right-