Understanding the Effect of Assignment of Importance Scores of Evaluation Criteria Randomly in the Application of DOE-TOPSIS in Decision Making

Abstract

In conventional applications of hybrid DoE- TOPSIS technique in decision making problems, full factorial design layouts are generally used because of their ability to measure the effects of all possible combinations for evaluation factors. In a typical application, for a design layout, a number of replications are generated by assigning different sets of relative importance scores for evaluation factors. A TOPSIS score is then obtained for each experiment and replication pair. Regression analysis is finally applied to obtain a relationship with inputs (values of evaluation factors) and outputs (alternatives’ TOPSIS meta-model scores). The key in conventional application of hybrid DoE-TOPSIS technique is generation of relative importance scores. Each set of scores can be assigned by a decision maker or generated randomly. This paper aims to determine whether using either of the two methods in determination of relative importance scores makes any difference in the ranking orders of alternatives.

Appendix 1. Illustration of the Application of the DoE-TOPSIS Model

A typical Do-E model uses five evaluation factors (denoted with A, B, C, D and E) as illustrated in Fig. 1. In the model, the factors can be assigned one of the two extreme values namely, low (−1) and high (+1) as an input. Five factors lead to development of a 25 full factorial experimental design plan (32 combinations). The experimental design plan also includes replication of the same experiment combination five times each with a different weight to provide measurement of the amount of variation/experiment error. The assigned weights for the five factors are normalized by dividing each weight with the sum of the weights.

Fig. 1.

The flowchart of the DoE approach

For the illustrative purposes, application of the DoE-TOPSIS model for the first replication is provided in Tables 7 and 8. Table 7 contains the decision matrix (R) whose rows are 32 different sets of randomly assigned values for the five factors (32 alternatives (called as experiments) form the rows and the five factors provide the columns). The randomly generated weight set for the five factors are normalized and provided in Table 1 also. The decision matrix is converted to a weighted normalized decision matrix V and a positive ideal solution and a negative ideal solution are calculated with the application of the TOPSIS approach as shown in Table 7. In Table 8, a ranking score (C_i*) for each company is calculated using its distance to the positive ideal solution (S_i*) and its distance to the negative ideal solution (S ⁻_i ). The ranking score, C_i*, takes values between 0 and 1.

Table 7. Application of the TOPSIS approach

Table 8. Calculation of ranking scores of the 32 companies

The DoE-TOPSIS model is applied four more times each with a different weight set to obtain a total of five replications. Using the TOPSIS results of the five replications the polynomial regression equation (TOPSIS meta-model) is obtained and provided in Eq. (10) [1]:

$$ \begin{aligned} Y & = \beta_{0} + \sum\limits_{i = 1}^{k} {\beta_{i} \,x_{i} } + \sum\limits_{i} {\sum\limits_{{\mathop j\limits_{i < j} }} {\beta_{ij} \,x_{i} \,x_{j} } } + \sum\limits_{i} {\sum\limits_{{\mathop j\limits_{i < j < k} }} {\sum\limits_{k} {\beta_{ijk} \,x_{i} \,x_{j} \,x_{k} } } } \\ & \quad + \sum\limits_{i} {\sum\limits_{{\mathop j\limits_{i < j < k < l} }} {\sum\limits_{k} {\sum\limits_{l} {\beta_{ijkl} \,x_{i} \,x_{j} \,x_{k} \,x_{l} } } } } + \beta_{12345} \,x_{1} \,x_{2} \,x_{3} \,x_{4} \,x_{5} + \varepsilon \\ \end{aligned} $$

(10)

In Eq. (1), Y is the TOPSIS score, x represents five input factors, β₀ the overall mean response or intercept coefficient, β_i is the main or first-order effect of factor i, β_ij is the two-factor interaction between factors i and j with i ≠ j, β_ijk is the three-factor interaction between factors i,j and k with i ≠ j≠k, βijkl is the four-factor interaction between factors i,j,k and l with i ≠ j≠k ≠ l,β₁₂₃₄₅ is the five-factor interaction between all factors, and ε is the error term [1, 5].

The polynomial regression equation can be obtained with the ANOVA procedure available in MINITAB software package. Table 9 provides a summary of the main effects and interactions. It can be observed that main factors A, B, C, D, and E are statistically significant (p < 0.05) whereas the other terms are statistically insignificant. In that case, the regression equation given in Eq. (10) is reduced to Eq. (11). Equation (11) can be used to calculate ranking scores of alternatives as illustrated in Table 10.

Table 9. Estimated effects and coefficients

Table 10. An example of obtaining ranking score for an alternative

$$ Y = 0.50 + 0.06683 \times A - 0.08994 \times B - 0.05824 \times C - 0.07413 \times D - 0.08661 \times E $$

(11)