Application of the Fuzzy Weighted Average in Strategic Portfolio Management*

Fuzzy weighted average

When the environment is vague, the rating criteria and the weights of their corresponding importance are often evaluated as a fuzzy number (Liou & Wang, 1992). In order to obtain the weighted sum of those criteria evaluated by fuzzy numbers in terms of rating and importance, we use the fuzzy weighted average for the calculation. There has been some research involved in the field of fuzzy weighted average (Vanegas & Labib, 2001; Buyukozkan & Feyzioglu, 2004). Dong and Wong (1987) addressed the computational aspect of the extension principle when the principle is applied to the weighted average operations in risk and decision analysis. Their computational algorithm is based on the α-cut representation of fuzzy sets and interval analysis. Lee and Park (1997) proposed an efficient algorithm, named the Efficient Fuzzy Weighted Average (EFWA), to compute a fuzzy weighted average, which was an improvement over the previous methods by reducing the number of comparisons and arithmetical operations to O(n log n). We adopted the EFWA algorithm (Lee & Park, 1997) to do the calculations in the current work, which is described as follows:

Suppose x^sub i^ and w^sub i^, i = 1, 2, . . . , n, have the corresponding intervals [a^sub i^, b^sub i^] and [c^sub i^, d^sub i^] with c^sub i^ ≤ 0, respectively.

1. Sort a's in non-decreasing order. Let (a^sub 1^, a^sub 2^, . . . , a^sub n^) be the resulting sequence. Let first =1 and last = n.

2. Let δ - threshold = [(first + last)/2] = (n + 1)/2. For each i = 1, 2, . . . , δ - threshold, let e^sub i^ = d^sub i^, and for each i = δ - threshold + 1, . . . n, let e^sub i^ = c^sub i^. For an n-tuple S = (e^sub 1^, e^sub 2^, . . . , e^sub n^), evaluate δ^sub S^sub δ-threshold^^ and δ^sub S^sub (δ-threshold+1)^^. δ^sub S^sub i^^ = (a^sub 1^ - a^sub i^)e^sub 1^ + (a^sub 2^ - a^sub i^)e^sub 2^ + . . . + (a^sub n^ - a^sub i^)e^sub n^/e^sub 1^ + e^sub 2^ + . . . + e^sub n^.

3. If δ^sub S^sub δ-threshold^^ > 0 and δ^sub S^sub (δ-threshold+1)^^ ≤ 0; then L = f^sub L^(e^sub 1^, e^sub 2^, . . . , e^sub n^) = a^sub 1^ + δ^sub S^sub 1^^ = a^sub 2^ + δ^sub S^sub 2^^ = . . . = a^sub n^ + δ^sub S^sub n^^ and go to Step 4; otherwise execute the following step.

3.1. If δ^sub S^sub δ-threshold^^ > 0, then first =δ - threshold + 1; otherwise last = δ threshold, and go to Step 2.

4. Sort b's in non-decreasing order. Let (b^sub 1^, b^sub 2^, . . . , b^sub n^) be the resulting sequence. Let first = 1 and last = n.

5. Let ξ - threshold = [(first + last)/2] = (n + 1)/2. For each i = 1, 2, . . . , ξ - threshold, let e^sub i^ = c, and for each i = ξ - threshold + 1, . . . , n, let e^sub i^ = d^sub i^. For an n-tuple S = (e^sub 1^, e^sub 2^, . . . , e^sub n^), evaluate ξ^sub S^sub ξ-threshold^^ and ξ^sub S^sub (ξ-threshold+1)^^.ξ^sub S^sub i^^ = (b^sub 1^ - b^sub i^)e^sub 1^ + (b^sub 2^ - b^sub i^)e^sub 2^ + . . . + (b^sub n^ - b^sub i^)e^sub n^/e^sub 1^ + e^sub 2^ + . . . + e^sub n^.

6. If ξ^sub S^sub ξ-threshold^^ > 0 and ξ^sub S^sub (ξ-threshold+1)^^ ≤ 0; then U = f^sub U^(e^sub 1^, e^sub 2^, . . . , e^sub n^) = b^sub 1^ + ξ^sub S^sub 1^^ = b^sub 2^ + ξ^sub S^sub 2^^ = . . . = b^sub n^ + ξ^sub S^sub n^^ and stop; otherwise execute the following step.