Application of the Fuzzy Weighted Average in Strategic Portfolio Management*

In Step 8, the three managers decided the optimism levels that were used to transfer fuzzy numbers into crisp numbers in Step 9. In Step 10, facilitators used the average method to aggregate the evaluation results from the three managers. With the aggregation results of the weighted scores, the financial data collected from Steps 11 and 12, and the importance ratings of evaluation constructs obtained from Step 13, a corresponding ILP model was set up in Step 14. The ILP model maximizes 8r^sub 11^ + 6r^sub 12^ + 6r^sub 13^ + 7r^sub 14^ + 5r^sub 21^ + 6r^sub 22^ + 4r^sub 31^ + 5r^sub 32^ + 4r^sub 33^ + 7r^sub 41^ + 6r^sub 42^ and satisfies several constraints. Solving the ILP model using LINDO (Schrage, 1991), a commercial mathematical programming software, we obtained the optimal solution: r^sub 14^ = r^sub 22^ = r^sub 31^ = r^sub 42^ = 1, all other r^sub ij^' s = 0. The results imply that if SBU1 adopts its fourth strategic plan, SBU2 adopts its second strategic plan, SBU3 adopts its first strategic plan, and SBU4 adopts its second strategic plan, then the entire company can maximize its returns with acceptable cost resulting from implementing these strategic plans. The maximum profit that could thus be generated is estimated to be $30 million.

DISCUSSION AND ANALYSIS

In the above example, to simplify the arithmetical process of EFWA, the three managers decided to use confidence level α = 0 to calculate the weighted scores. However, it should be noted that the confidence levels of different managers toward a strategic plan may not be the same (Hsu & Chen, 1996). Thus, we asked the three managers to provide their own confidence levels in terms of SBUs, strategic plans, and the importance of evaluation factors (Table 4). We also take Manager 1 evaluating Strategic Plan 2 submitted by SBU3 on the feasibility factors as an example to show the computational procedure of EFWA by using different confidence levels (Example 2 of the Appendix). Accordingly, the interval for confidence level α = 0.5 is [5.073, 8.167] (α = 1 is 5.432). Then, we used the other two managers' confidence levels to recalculate the weighted score results. Consequently, different evaluation results were obtained. The optimal solution is r^sub 13^ = r^sub 22^ = r^sub 32^ = r^sub 42^ = 1, all other r'^sub ij^ s = 0 and the maximum profit is $28 million. Hence, the two evaluation results, the former obtained through using the same confidence level α = 0, and the latter obtained through using the three managers' own confidence levels (Table 4), are different. In practice, top managers will encounter plan selection situations in which they have different confidence levels in terms of SBUs, strategic plans, and the importance of evaluation factors. The proposed integrated approach containing EFWA can consider the diversity of confidence levels of managers instead of treating their confidence levels as equal when conducting evaluations.

After calculating the weighted sum of scores by using EFWA, we derived the results presented in triangular fuzzy numbers. The next step is to decide the optimism levels to use when we transfer the triangular fuzzy numbers into crisp numbers. The optimism level is an attitude a decision maker possesses regarding personal expectations of how things will proceed. The managerial meaning of optimism level has been mentioned above. Therefore, just like the role of the confidence level, the optimism level can be viewed as a factor expressing the diversity of the managers' attitudes toward the whole situation instead of seeing all of the managers' perspectives as equal. In this case, the three managers had the same medium optimism level. In addition, during the step of using integer linear programming to select the most suitable strategic plans, the different proportion of the values w^sub A^, w^sub I^, and w^sub F^ in the objective function will also affect the final selection results. The values w^sub A^, w^sub I^, and w^sub F^ presenting, respectively, the importance of constructs A, I, and F on the selection of strategic plans, will vary somewhat from industry to industry, or firm to firm, according to the evaluator's determination. In this case, the three managers assigned w^sub A^, w^sub I^, and w^sub F^ as .3, .3, and .4, respectively.