Hesitant Fuzzy Multiple Criteria Decision Analysis Based on LINMAP

Abstract

The LINMAP technique is one of the most representative methods for handling the MCDM or MCGDM problems with respect to the preference information over alternatives. This chapter utilizes the main structure of LINMAP to develop the hesitant fuzzy group LINMAP technique with interval programming model and the hesitant fuzzy programming model-based LINMAP technique. The former is mainly used to solve the MCGDM problems in which the ratings of alternatives with respect to criteria are taken as HFEs, and all pair-wise comparison judgments are represented by interval numbers. While the latter is mainly utilized to address the MCDM problems with incomplete weight information in which the ratings of alternatives with each criterion are taken as HFEs and the incomplete judgments on pair-wise comparisons of alternatives with hesitant degrees are also represented by HFEs. The main contributions of this chapter are that (1) the developed techniques not only can take sufficiently into account the experts’ hesitancy in expressing their assessment information for criteria values by using HFEs but also can simultaneously consider the uncertainty of preference information over alternatives by using interval numbers or HFEs; (2) the concept of hesitant fuzzy programming model in which both the objective function and the constraints’ coefficients take the form of HFEs has been proposed, and an effective technique to solve this sort of model is developed; (3) the bi-objective programming model has been constructed to address the issues of incomplete and inconsistent weights of the criteria.

Appendix

$$\begin{aligned} & \hbox{min} \;\left\{ \begin{aligned} & (0.05\theta + 0.75) \times z_{12}^{1} + (0.05\theta + 0.85) \times z_{34}^{1} + (0.05\theta + 0.85) \times z_{53}^{1} + (0.05\theta + 0.65) \times z_{13}^{2} + \\ & (0.05\theta + 0.85) \times z_{34}^{2} + (0.05\theta + 0.75) \times z_{42}^{2} + (0.05\theta + 0.75) \times z_{53}^{2} + (0.05\theta + 0.75) \times z_{21}^{3} + \\ & (0.05\theta + 0.55) \times z_{31}^{3} + (0.05\theta + 0.95) \times z_{45}^{3} \\ \end{aligned} \right\} \\ & s.t.\;\left\{ \begin{aligned} & 0.014w_{1} + 0.002w_{2} - 0.148w_{3} + \, 0.136w_{4} + 0.04w_{3} \times (3h_{31} + \, 2h_{32} + 3h_{33} - h_{34} h_{35} ) - 0.04w_{1} \times h_{12} - \\ & 0.04w_{2} \times (h_{22} + \, h_{23} - 4h_{24} - h_{25} ) - 0.08w_{4} \times (h_{41} + h_{42} \, + h_{43} ) + z_{12}^{1} \ge 0 \\ & 0.092w_{1} \, + \, 0.296w_{2} \, + \, 0.066w_{3} - 0.306w_{4} - 0.04w_{1} \times (2h_{13} - h_{12} - h_{11} + 3h_{14} + 3h_{15} ) + 0.04w_{4} \times (3h_{41} + 4h_{42} + \\ & 2h_{43} + 2h_{44} + 2h_{45} ) - 0.04w_{3} \times (4h_{32 \, } - h_{31} + 2h_{33} + 2h_{34} + 2h_{35} ) - 0.08w_{2} \times (h_{12} + h_{22} + 2h_{23} + 2h_{24} + 2h_{25} ) + z_{34}^{1} \ge 0 \\ & 0.382w_{3} - 0.088w_{2} - 0.034w_{1} + 0.168w_{4} + 0.04w_{4} \times (3h_{41} + 3h_{42} + 2h_{43} - h_{44} - h_{45} ) + 0.04w_{2} \times (h_{12} - \\ & h_{22} + 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (2h_{31} + 3h_{32} + 4h_{33} + 4h_{34} + 4h_{35} ) - 0.04w_{1} \times (2h_{11} + h_{12} - 3h_{14} - 5h_{15} ) + z_{53}^{1} \ge 0 \\ & 0.134w_{2} - 0.06w_{1} - 0.096w_{3} + 0.13w_{4} + 0.04w_{1} (h_{11} + 2h_{12} + h_{13} - 2h_{15} ) - 0.04w_{2} \times (2h_{22} + h_{23} + h_{24} + h_{25} ) - \\ & w_{4} \times (0.12h_{41} + 0.04h_{42} + 0.04h_{43} ) + 0.04w_{3} \times (2h_{31} + h_{32} + h_{33} + h_{34} + h_{35} ) + z_{13}^{2} \ge 0 \\ & 0.296w_{2} - 0.02w_{1} - 0.094w_{3} - 0.24w_{4} + 0.04w_{4} \times (3h_{41} + 3h_{42} + 2h_{43} + 2h_{44} ) + 0.04w_{1} \times (h_{11} + h_{12} - \\ & 2h_{13} - h_{14} + h_{15} ) - 0.08w_{2} \times (h_{12} + h_{22} + 2h_{23} + 2h_{24} + 2h_{25} ) + 0.04w_{3} \times (h_{31} + h_{32} + h_{33} + h_{34} + h_{35} ) + z_{34}^{2} \ge 0 \\ & 0.144w_{1} - 0.23w_{2} - 0.018w_{3} + 0.246w_{4} - 0.04w_{4} \times (2h_{41} + 4h_{42} + 3h_{43} + 2h_{44} ) + 0.04w_{3} \times h_{33} + \\ & 0.04w_{2} \times (2h_{12} + 3h_{22} + 4h_{23} + 3h_{24} - h_{25} ) - 0.12w_{1} \times (h_{11} + h_{12} ) + z_{42}^{2} \ge 0 \\ & 0.216w_{1} - 0.088w_{2} + 0.308w_{3} + 0.154w_{4} + 0.04w_{2} \times (h_{12} - h_{22} + 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (2h_{31} + 3h_{32} + \\ & 4h_{33} + 3h_{34} + 2h_{35} ) - 0.04w_{4} \times (3h_{41} + 3h_{42} - h_{43} ) - 0.04w_{1} \times (2h_{11} + h_{12} + h_{13} + 2h_{14} + 4h_{15} ) + z_{53}^{2} \ge 0 \\ & 0.234w_{3} - 0.088w_{2} - 0.148w_{1} - 0.1w_{4} + 0.04w_{1} \times (2h_{11} + 3h_{12} + 2h_{13} + h_{14} ) + 0.04w_{2} \times (h_{23} - h_{12} + \\ & 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (3h_{31} + 2h_{32} + 2h_{33} + 3h_{34} + 3h_{35} ) + 0.04w_{4} \times (2h_{41} + 2h_{44} + 2h_{45} ) + z_{21}^{3} \ge 0 \\ & 0.05w_{1} - 0.07w_{2} + 0.086w_{3} - 0.004w_{4} + 0.04w_{2} \times (h_{22} - h_{21} + h_{23} + h_{24} + h_{25} ) - w_{1} \times (0.04h_{14} + 0.08h_{15} ) + \\ & 0.04w_{4} \times (h_{41} - h_{42} - h_{43} + h_{45} ) - 0.04w_{3} \times (2h_{31} + h_{32} + h_{34} + h_{35} ) + z_{31}^{3} \ge 0 \\ & 0.138w_{4} - 0.208w_{2} - 0.448w_{3} - 0.058w_{1} - 0.04w_{4} \times (h_{42} + 3h_{44} + 3h_{45} ) + 0.04w_{1} (h_{11} + 2h_{13} - 2h_{15} ) + \, \\ & 0.04w_{2} \times (h_{12} + 3h_{22} + 4h_{23} + 2h_{24} + 2h_{25} ) + 0.04w_{3} \times (h_{31} + 7h_{32} + 6h_{33} + 6h_{34} + 6h_{35} ) + z_{45}^{3} \ge 0 \\ & 0.0258w_{2} - 0.145w_{1} - 0.1558w_{3} - 0.2352w_{4} + 0.028w_{1} h_{11} - 0.028w_{1} h_{12} + 0.004w_{1} h_{13} + \\ & 0.028w_{2} h_{12} + 0.092w_{1} h_{14} + 0.204w_{1} h_{15} + 0.06w_{2} h_{22} + 0.004w_{2} h_{23} - 0.212w_{2} h_{24} + 0.012w_{2} h_{25} + \\ & 0.016w_{3} h_{31} + 0.028w_{3} h_{32} - 0.024w_{3} h_{33} + 0.16w_{3} h_{34} + 0.128w_{3} h_{35} + 0.112w_{4} h_{41} + \\ & 0.236w_{4} h_{42} + 0.108w_{4} h_{43} - 0.028w_{4} h_{44} - 0.044w_{4} h_{45} \ge \underline{a} \\ & 0.0236w_{2} - 0.1352w_{1} - 0.1422w_{3} - 0.2191w_{4} + 0.026w_{1} h_{11} - 0.026w_{1} h_{12} + 0.004w_{1} h_{13} + \\ & 0.026w_{2} h_{12} \, + 0.086w_{1} h_{14} + 0.19w_{1} h_{15} + 0.056w_{2} h_{22} + 0.004w_{2} h_{23} - 0.198w_{2} h_{24} + \\ & 0.012w_{2} h_{25} + 0.014w_{3} h_{31} + 0.024w_{3} h_{32} - 0.024w_{3} h_{33} + 0.148w_{3} h_{34} + 0.118w_{3} h_{35} + \\ & 0.104w_{4} h_{41} + 0.22w_{4} h_{42} + 0.1w_{4} h_{43} - 0.026w_{4} h_{44} - 0.04w_{4} h_{45} \ge \bar{a} \\ & z_{12}^{1} ,z_{34}^{1} ,z_{53}^{1} ,z_{13}^{2} ,z_{34}^{2} ,z_{42}^{2} ,z_{53}^{2} ,z_{21}^{3} ,z_{31}^{3} ,z_{45}^{3} \ge 0 \\ & 0 \le h_{11} \le h_{12} \le h_{13} \le h_{14} \le h_{15} \le 1,\quad 0 \le h_{21} \le h_{22} \le h_{23} \le h_{24} \le h_{25} \le 1 \\ & 0 \le h_{31} \le h_{32} \le h_{33} \le h_{34} \le h_{35} \le 1,\quad 0 \le h_{41} \le h_{42} \le h_{43} \le h_{44} \le h_{45} \le 1 \\ & 0 \le w_{j} \le 1(j = 1,2,3,4),\quad w_{1} + w_{2} + w_{3} + w_{4} = 1 \\ \end{aligned} \right. \\ \end{aligned}$$

(MOD-A.1)

$$\begin{aligned} & \hbox{min} \;\left\{ \begin{aligned} & (0.05\theta + 0.75) \times z_{12}^{1} + (0.05\theta + 0.85) \times z_{34}^{1} + (0.05\theta + 0.85) \times z_{53}^{1} + (0.05\theta + 0.65) \times z_{13}^{2} + \\ & (0.05\theta + 0.85) \times z_{34}^{2} + (0.05\theta + 0.75) \times z_{42}^{2} + (0.05\theta + 0.75) \times z_{53}^{2} + (0.05\theta + 0.75) \times z_{21}^{3} + \\ & (0.05\theta + 0.55) \times z_{31}^{3} + (0.05\theta + 0.95) \times z_{45}^{3} \\ \end{aligned} \right\} \\ & s.t.\;\left\{ \begin{aligned} & 0.014w_{1} + 0.002w_{2} - 0.148w_{3} + \, 0.136w_{4} + 0.04w_{3} \times (3h_{31} + \, 2h_{32} + 3h_{33} - h_{34} h_{35} ) - 0.04w_{1} \times h_{12} - \\ & 0.04w_{2} \times (h_{22} + \, h_{23} - 4h_{24} - h_{25} ) - 0.08w_{4} \times (h_{41} + h_{42} \, + h_{43} ) + z_{12}^{1} \ge 0 \\ & 0.092w_{1} \, + \, 0.296w_{2} \, + \, 0.066w_{3} - 0.306w_{4} - 0.04w_{1} \times (2h_{13} - h_{12} - h_{11} + 3h_{14} + 3h_{15} ) + 0.04w_{4} \times (3h_{41} + 4h_{42} + \\ & 2h_{43} + 2h_{44} + 2h_{45} ) - 0.04w_{3} \times (4h_{32 \, } - h_{31} + 2h_{33} + 2h_{34} + 2h_{35} ) - 0.08w_{2} \times (h_{12} + h_{22} + 2h_{23} + 2h_{24} + 2h_{25} ) + z_{34}^{1} \ge 0 \\ & 0.382w_{3} - 0.088w_{2} - 0.034w_{1} + 0.168w_{4} + 0.04w_{4} \times (3h_{41} + 3h_{42} + 2h_{43} - h_{44} - h_{45} ) + 0.04w_{2} \times (h_{12} - \\ & h_{22} + 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (2h_{31} + 3h_{32} + 4h_{33} + 4h_{34} + 4h_{35} ) - 0.04w_{1} \times (2h_{11} + h_{12} - 3h_{14} - 5h_{15} ) + z_{53}^{1} \ge 0 \\ & 0.134w_{2} - 0.06w_{1} - 0.096w_{3} + 0.13w_{4} + 0.04w_{1} (h_{11} + 2h_{12} + h_{13} - 2h_{15} ) - 0.04w_{2} \times (2h_{22} + h_{23} + h_{24} + h_{25} ) - \\ & w_{4} \times (0.12h_{41} + 0.04h_{42} + 0.04h_{43} ) + 0.04w_{3} \times (2h_{31} + h_{32} + h_{33} + h_{34} + h_{35} ) + z_{13}^{2} \ge 0 \\ & 0.296w_{2} - 0.02w_{1} - 0.094w_{3} - 0.24w_{4} + 0.04w_{4} \times (3h_{41} + 3h_{42} + 2h_{43} + 2h_{44} ) + 0.04w_{1} \times (h_{11} + h_{12} - \\ & 2h_{13} - h_{14} + h_{15} ) - 0.08w_{2} \times (h_{12} + h_{22} + 2h_{23} + 2h_{24} + 2h_{25} ) + 0.04w_{3} \times (h_{31} + h_{32} + h_{33} + h_{34} + h_{35} ) + z_{34}^{2} \ge 0 \\ & 0.144w_{1} - 0.23w_{2} - 0.018w_{3} + 0.246w_{4} - 0.04w_{4} \times (2h_{41} + 4h_{42} + 3h_{43} + 2h_{44} ) + 0.04w_{3} \times h_{33} + \\ & 0.04w_{2} \times (2h_{12} + 3h_{22} + 4h_{23} + 3h_{24} - h_{25} ) - 0.12w_{1} \times (h_{11} + h_{12} ) + z_{42}^{2} \ge 0 \\ & 0.216w_{1} - 0.088w_{2} + 0.308w_{3} + 0.154w_{4} + 0.04w_{2} \times (h_{12} - h_{22} + 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (2h_{31} + 3h_{32} + \\ & 4h_{33} + 3h_{34} + 2h_{35} ) - 0.04w_{4} \times (3h_{41} + 3h_{42} - h_{43} ) - 0.04w_{1} \times (2h_{11} + h_{12} + h_{13} + 2h_{14} + 4h_{15} ) + z_{53}^{2} \ge 0 \\ & 0.234w_{3} - 0.088w_{2} - 0.148w_{1} - 0.1w_{4} + 0.04w_{1} \times (2h_{11} + 3h_{12} + 2h_{13} + h_{14} ) + 0.04w_{2} \times (h_{23} - h_{12} + \\ & 2h_{24} + 2h_{25} ) - 0.04w_{3} \times (3h_{31} + 2h_{32} + 2h_{33} + 3h_{34} + 3h_{35} ) + 0.04w_{4} \times (2h_{41} + 2h_{44} + 2h_{45} ) + z_{21}^{3} \ge 0 \\ & 0.05w_{1} - 0.07w_{2} + 0.086w_{3} - 0.004w_{4} + 0.04w_{2} \times (h_{22} - h_{21} + h_{23} + h_{24} + h_{25} ) - w_{1} \times (0.04h_{14} + 0.08h_{15} ) + \\ & 0.04w_{4} \times (h_{41} - h_{42} - h_{43} + h_{45} ) - 0.04w_{3} \times (2h_{31} + h_{32} + h_{34} + h_{35} ) + z_{31}^{3} \ge 0 \\ & 0.138w_{4} - 0.208w_{2} - 0.448w_{3} - 0.058w_{1} - 0.04w_{4} \times (h_{42} + 3h_{44} + 3h_{45} ) + 0.04w_{1} (h_{11} + 2h_{13} - 2h_{15} ) + \, \\ & 0.04w_{2} \times (h_{12} + 3h_{22} + 4h_{23} + 2h_{24} + 2h_{25} ) + 0.04w_{3} \times (h_{31} + 7h_{32} + 6h_{33} + 6h_{34} + 6h_{35} ) + z_{45}^{3} \ge 0 \\ & 0.0258w_{2} - 0.145w_{1} - 0.1558w_{3} - 0.2352w_{4} + 0.028w_{1} h_{11} - 0.028w_{1} h_{12} + 0.004w_{1} h_{13} + \\ & 0.028w_{2} h_{12} + 0.092w_{1} h_{14} + 0.204w_{1} h_{15} + 0.06w_{2} h_{22} + 0.004w_{2} h_{23} - 0.212w_{2} h_{24} + 0.012w_{2} h_{25} + \\ & 0.016w_{3} h_{31} + 0.028w_{3} h_{32} - 0.024w_{3} h_{33} + 0.16w_{3} h_{34} + 0.128w_{3} h_{35} + 0.112w_{4} h_{41} + \\ & 0.236w_{4} h_{42} + 0.108w_{4} h_{43} - 0.028w_{4} h_{44} - 0.044w_{4} h_{45} \ge \underline{a} \\ & 0.0236w_{2} - 0.1352w_{1} - 0.1422w_{3} - 0.2191w_{4} + 0.026w_{1} h_{11} - 0.026w_{1} h_{12} + 0.004w_{1} h_{13} + \\ & 0.026w_{2} h_{12} \, + 0.086w_{1} h_{14} + 0.19w_{1} h_{15} + 0.056w_{2} h_{22} + 0.004w_{2} h_{23} - 0.198w_{2} h_{24} + \\ & 0.012w_{2} h_{25} + 0.014w_{3} h_{31} + 0.024w_{3} h_{32} - 0.024w_{3} h_{33} + 0.148w_{3} h_{34} + 0.118w_{3} h_{35} + \\ & 0.104w_{4} h_{41} + 0.22w_{4} h_{42} + 0.1w_{4} h_{43} - 0.026w_{4} h_{44} - 0.04w_{4} h_{45} \ge \bar{a} \\ & z_{12}^{1} ,z_{34}^{1} ,z_{53}^{1} ,z_{13}^{2} ,z_{34}^{2} ,z_{42}^{2} ,z_{53}^{2} ,z_{21}^{3} ,z_{31}^{3} ,z_{45}^{3} \ge 0 \\ & 0 \le h_{11} \le h_{12} \le h_{13} \le h_{14} \le h_{15} \le 1,\quad 0 \le h_{21} \le h_{22} \le h_{23} \le h_{24} \le h_{25} \le 1 \\ & 0 \le h_{31} \le h_{32} \le h_{33} \le h_{34} \le h_{35} \le 1,\quad 0 \le h_{41} \le h_{42} \le h_{43} \le h_{44} \le h_{45} \le 1 \\ & 0.15 \le w_{1} \le 0.2,0.16 \le w_{2} \le 0.18,0.3 \le w_{3} \le 0.35,0.3 \le w_{4} \le 0.45,w_{1} + w_{2} + w_{3} + w_{4} = 1 \\ \end{aligned} \right. \\ \end{aligned}$$

(MOD-A.2)

$$\begin{aligned} & \hbox{min} \;\left\{ \begin{aligned} & \left\{ {0.5,0.6,0.7} \right\} \otimes z_{12} + \left\{ {0.6,0.65,0.7} \right\} \otimes z_{23} + \left\{ {0.8,0.85,0.9} \right\} \otimes z_{24} + \left\{ {0.5,0.7} \right\} \otimes z_{25} \\ & + \left\{ {0.4,0.5,0.6} \right\} \otimes z_{31} + \left\{ {0.6,0.7,0.95} \right\} \otimes z_{34} + \left\{ {0.7,0.9} \right\} \otimes z_{45} \\ \end{aligned} \right\} \\ & s.t.\;\left\{ \begin{aligned} & - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + \, 0.0667\vartheta_{1}^{1} - 0.2\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} + \\ & 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{12} \ge 0 \\ & - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + \, 0.0933w_{4} + \, 0.133\vartheta_{1}^{1} + 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + \\ & 0.0667\vartheta_{2}^{3} - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} + z_{23} \ge 0 \\ & - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + \, 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} + 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} - \\ & 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{24} \ge 0 \\ & 0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} + 0.267\vartheta_{2}^{2} + \\ & 0.267\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{25} \ge 0 \\ & 0.14w_{1} - 0.02w_{2} - 0.608w_{3} + \, 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} + \\ & 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} + z_{31} \ge 0 \\ & 0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} - 0.0667\vartheta_{2}^{2} + \\ & 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{34} \ge 0 \\ & 0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} + \\ & 0.0667\vartheta_{2}^{3} + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} + z_{45} \ge 0 \\ & \left\{ {0.5,0.6,0.7} \right\} \otimes ( - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + \, 0.0667\vartheta_{1}^{1} - \\ & 0.2\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} + 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) + \\ & \left\{ {0.6,0.65,0.7} \right\} \otimes ( - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + \, 0.0933w_{4} + \, 0.133\vartheta_{1}^{1} + \\ & 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} ) + \\ & \left\{ {0.8,0.85,0.9} \right\} \otimes ( - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + \, 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} + \\ & 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} - 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) + \\ & \left\{ {0.5,0.7} \right\} \otimes (0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} + \\ & 0.267\vartheta_{2}^{2} + 0.267\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) + \\ & \left\{ {0.4,0.5,0.6} \right\} \otimes (0.14w_{1} - 0.02w_{2} - 0.608w_{3} + 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} - \\ & 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} ) + \\ & \left\{ {0.6,0.7,0.95} \right\} \otimes (0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} - \\ & 0.0667\vartheta_{2}^{2} + 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) + \\ & \left\{ {0.7,0.9} \right\} \otimes (0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} + \\ & 0.0667\vartheta_{2}^{3} + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} )\, \underline{ \succ } \, \left\{ {0.01} \right\} \\ & w_{4} \ge w_{1} ,\;0.05 \le w_{2} - w_{2} \le 0.3,\;\;w_{4} - w_{3} \ge w_{2} - w_{1} ,\;w_{2} \ge 2w_{1} ,\;0.15 \le w_{4} \le 0.5 \\ & w_{1} + w_{2} + w_{3} + w_{4} = 1,\;0.01 \le w_{j} \le 1\quad (j = 1,2,3,4) \\ & z_{12} ,z_{23} ,z_{24} ,z_{31} ,z_{34} ,z_{25} ,z_{45} \ge 0 \\ & 00 \le \vartheta_{j}^{1} \le \vartheta_{j}^{2} \le \vartheta_{j}^{3} \le w_{j} \;(j = 1,2,3,4) \\ \end{aligned} \right. \\ \end{aligned}$$

(MOD-A.3)

$$\begin{aligned} & \hbox{min} \;\left\{ \begin{aligned} & H\left\{ {0.5,0.6,0.7} \right\} \otimes z_{12} + H\left\{ {0.6,0.65,0.7} \right\} \otimes z_{23} + H\left\{ {0.8,0.85,0.9} \right\} \otimes z_{24} + H\left\{ {0.5,0.7} \right\} \otimes z_{25} \\ & + H\left\{ {0.4,0.5,0.6} \right\} \otimes z_{31} + H\left\{ {0.6,0.7,0.95} \right\} \otimes z_{34} + H\left\{ {0.7,0.9} \right\} \otimes z_{45} \\ \end{aligned} \right\} \\ & \hbox{min} \left\{ {o_{141}^{ - } + o_{132}^{ - } + o_{223}^{ - } + o_{223}^{ + } + o_{34321}^{ - } + o_{421}^{ - } + o_{54}^{ + } + o_{54}^{ - } } \right\} \\ & s.t.\;\left\{ \begin{aligned} & - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + \, 0.0667\vartheta_{1}^{1} - 0.2\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} \\ & + 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{12} \ge 0 \\ & - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + \, 0.0933w_{4} + \, 0.133\vartheta_{1}^{1} + 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} \\ & - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} + z_{23} \ge 0 \\ & - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + \, 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} + 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} \\ & - 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{24} \ge 0 \\ & 0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} + 0.267\vartheta_{2}^{2} + 0.267\vartheta_{2}^{3} \\ & + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{25} \ge 0 \\ & 0.14w_{1} - 0.02w_{2} - 0.608w_{3} + \, 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} \\ & + 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} + z_{31} \ge 0 \\ & 0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} - 0.0667\vartheta_{2}^{2} \\ & + 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{34} \ge 0 \\ & 0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} \\ & + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} + z_{45} \ge 0 \\ & \left\{ {0.5,0.6,0.7} \right\} \otimes ( - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + \, 0.0667\vartheta_{1}^{1} - 0.2\vartheta_{1}^{2} \\ & - 0.0667\vartheta_{2}^{2} + 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) \\ & + \left\{ {0.6,0.65,0.7} \right\} \otimes ( - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + \, 0.0933w_{4} + \, 0.133\vartheta_{1}^{1} \\ & + 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} ) + \\ & \left\{ {0.8,0.85,0.9} \right\} \otimes ( - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + \, 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} \\ & + 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} - 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) \\ & + \left\{ {0.5,0.7} \right\} \otimes (0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} \\ & + 0.267\vartheta_{2}^{2} + 0.267\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) \\ & + \left\{ {0.4,0.5,0.6} \right\} \otimes (0.14w_{1} - 0.02w_{2} - 0.608w_{3} + 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} \\ & - 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} ) \\ & + \left\{ {0.6,0.7,0.95} \right\} \otimes (0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} \\ & - 0.0667\vartheta_{2}^{2} + 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) \\ & + \left\{ {0.7,0.9} \right\} \otimes (0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} \\ & + 0.0667\vartheta_{2}^{3} + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} )\, \underline{ \succ } \, H\left\{ {0.01} \right\} \\ & z_{12} ,z_{23} ,z_{24} ,z_{31} ,z_{34} ,z_{25} ,z_{45} \ge 0 \\ & 0 \le \vartheta_{j}^{1} \le \vartheta_{j}^{2} \le \vartheta_{j}^{3} \le w_{j} \;(j = 1,2,3,4) \\ & w_{4} + o_{141}^{ - } \ge w_{1} ,w_{3} + o_{132}^{ - } \ge w_{2} ,w_{2} - w_{3} \; + o_{223}^{ - } \ge 0.05 , { }w_{2} - w_{3} - o_{223}^{ + } \le 0.3 ,\; \\ & w_{4} - w_{3} - w_{2} + w_{1} + o_{34321}^{ - } \ge 0{ , }{{w_{2} } \mathord{\left/ {\vphantom {{w_{2} } {w_{1} }}} \right. \kern-0pt} {w_{1} }} + o_{421}^{ - } \ge 2, \, w_{4} - o_{54}^{ + } \le 0.5,\;w_{4} + o_{54}^{ - } \ge 0.15\; \, \\ & w_{1} + w_{2} + w_{3} + w_{4} = 1,\;\;0.01 \le w_{j} \le 1\quad (j = 1,2,3,4) \\ & o_{141}^{ - } ,o_{132}^{ - } ,o_{223}^{ - } ,o_{223}^{ + } ,o_{34321}^{ - } ,o_{421}^{ - } ,o_{54}^{ + } ,o_{54}^{ - } \ge 0 \\ \end{aligned} \right. \\ \end{aligned}$$

(MOD-A.4)

$$\begin{aligned} & \hbox{min} \;\left\{ y \right\} \\ & s.t.\;\left\{ \begin{aligned} & 0.6z_{12} + 0.65z_{23} + 0.85z_{24} + 0.6z_{25} + 0.5z_{31} + 0.75z_{34} + 0.8z_{45} \le y \\ & o_{141}^{ - } + o_{132}^{ - } + o_{223}^{ - } + o_{223}^{ + } + o_{34321}^{ - } + o_{421}^{ - } + o_{54}^{ + } + o_{54}^{ - } \le y \\ & - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + 0.0667\vartheta_{1}^{1} - 0.2\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} \\ & + 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{12} \ge 0 \\ & - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + 0.0933w_{4} + 0.133\vartheta_{1}^{1} + 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} \\ & - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} + z_{23} \ge 0 \\ & - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} + 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} \\ & - 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{24} \ge 0 \\ & 0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} + 0.267\vartheta_{2}^{2} + 0.267\vartheta_{2}^{3} \\ & + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} + z_{25} \ge 0 \\ & 0.14w_{1} - 0.02w_{2} - 0.608w_{3} + 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} - 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} \\ & + 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} + z_{31} \ge 0 \\ & 0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} - 0.0667\vartheta_{2}^{2} \\ & + 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} + z_{34} \ge 0 \\ & 0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} \\ & + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} + z_{45} \ge 0 \\ & 0.6 \times ( - 0.00667w_{1} + 0.157w_{2} + 0.0175w_{3} - 0.321w_{4} + \, 0.0667\vartheta_{1}^{1} - 0.2\vartheta_{1}^{2} \\ & - 0.0667\vartheta_{2}^{2} + 0.0333\vartheta_{2}^{3} + 0.133\vartheta_{4}^{1} + 0.167\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) \\ & + 0.65 \times ( - 0.133w_{1} - 0.137w_{2} + 0.0433w_{3} + \, 0.0933w_{4} + \, 0.133\vartheta_{1}^{1} \\ & + 0.133\vartheta_{1}^{2} + 0.133\vartheta_{2}^{2} + 0.0667\vartheta_{2}^{3} - 0.0667\vartheta_{3}^{1} - 0.0667\vartheta_{4}^{1} - 0.0667\vartheta_{4}^{2} ) + \\ & 0.85 \times ( - 0.09w_{1} - 0.39w_{2} + 0.522w_{3} - 0.345w_{4} + \, 0.333\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} \\ & + 0.2\vartheta_{2}^{2} + 0.2\vartheta_{2}^{3} - 0.2\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.267\vartheta_{4}^{1} + 0.233\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) \\ & + 0.6 \times (0.203w_{1} - 0.48w_{2} - 0.0983w_{3} - 0.095w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} - 0.133\vartheta_{1}^{3} \\ & + 0.267\vartheta_{2}^{2} + 0.267\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.1\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.0333\vartheta_{4}^{1} + 0.1\vartheta_{4}^{2} + 0.133\vartheta_{4}^{3} ) \\ & + 0.5 \times (0.14w_{1} - 0.02w_{2} - 0.608w_{3} + 0.288w_{4} - 0.2\vartheta_{1}^{1} - 0.0667\vartheta_{1}^{2} \\ & - 0.0667\vartheta_{2}^{2} - 0.0667\vartheta_{2}^{3} + 0.0667\vartheta_{3}^{1} + 0.0333\vartheta_{3}^{2} - 0.0667\vartheta_{4}^{1} - 0.1\vartheta_{4}^{2} - 0.133\vartheta_{4}^{3} ) \\ & + 0.75 \times (0.0433w_{1} - 0.253w_{2} + 0.478w_{3} - 0.438w_{4} - 0.133\vartheta_{1}^{1} + 0.2\vartheta_{1}^{2} + 0.0667\vartheta_{1}^{3} \\ & - 0.0667\vartheta_{2}^{2} + 0.133\vartheta_{2}^{3} - 0.133\vartheta_{3}^{1} - 0.233\vartheta_{3}^{2} - 0.3\vartheta_{3}^{3} + 0.333\vartheta_{4}^{1} + 0.3\vartheta_{4}^{2} + 0.3\vartheta_{4}^{3} ) \\ & + 0.8 \times (0.293w_{1} - 0.09w_{2} - 0.63w_{3} + 0.25w_{4} - 0.133\vartheta_{1}^{1} - 0.133\vartheta_{1}^{2} + 0.2\vartheta_{1}^{3} + 0.0667\vartheta_{2}^{2} \\ & + 0.0667\vartheta_{2}^{3} + 0.267\vartheta_{3}^{1} + 0.333\vartheta_{3}^{2} + 0.0333\vartheta_{3}^{3} - 0.3\vartheta_{4}^{1} - 0.133\vartheta_{4}^{2} - 0.167\vartheta_{4}^{3} ) \ge 0.01 \\ & z_{12} ,z_{23} ,z_{24} ,z_{31} ,z_{34} ,z_{25} ,z_{45} \ge 0 \\ & w_{4} + o_{141}^{ - } \ge w_{1} ,w_{3} + o_{132}^{ - } \ge w_{2} ,w_{2} - w_{3} \; + o_{223}^{ - } \ge 0.05 , { }w_{2} - w_{3} - o_{223}^{ + } \le 0.3 ,\; \\ & w_{4} - w_{3} - w_{2} + w_{1} + o_{34321}^{ - } \ge 0{ , }{{w_{2} } \mathord{\left/ {\vphantom {{w_{2} } {w_{1} }}} \right. \kern-0pt} {w_{1} }} + o_{421}^{ - } \ge 2, \, w_{4} - o_{54}^{ + } \le 0.5,\;w_{4} + o_{54}^{ - } \ge 0.15\; \, \\ & w_{1} + w_{2} + w_{3} + w_{4} = 1,\;\;0 \le w_{j} \le 1\quad (j = 1,2,3,4) \\ & 0 \le \vartheta_{j}^{1} \le \vartheta_{j}^{2} \le \vartheta_{j}^{3} \le w_{j} \;(j = 1,2,3,4) \\ & o_{141}^{ - } ,o_{132}^{ - } ,o_{223}^{ - } ,o_{223}^{ + } ,o_{34321}^{ - } ,o_{421}^{ - } ,o_{54}^{ + } ,o_{54}^{ - } \ge 0 \\ \end{aligned} \right. \\ \end{aligned}$$

(MOD-A.5)