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Hesitant fuzzy programming technique for multidimensional analysis of hesitant fuzzy preferences

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Abstract

The linear programming technique for multidimensional analysis of preferences (LINMAP) is the most representative method for handling the multiple criteria decision making (MCDM) problems with respect to the preference information over alternatives. This paper utilizes the main structure of LINMAP to develop a novel hesitant fuzzy mathematical programming technique to handle MCDM problems within the decision environment of hesitant fuzzy elements (HFEs). Considering the hesitancy of the decision maker, both the pair-wise comparison preference information over alternatives and the evaluation information of alternatives with criteria are represented by the HFEs. Based on the incomplete pair-wise preference judgments over alternatives, we propose the concepts of the hesitant fuzzy consistency and inconsistency indices. Furthermore, we construct a hesitant fuzzy mathematical programming model to derive the weights of criteria and the positive-ideal solution. In this hesitant fuzzy programming model both the objective function and partial constraints’ coefficients take the form of HFEs, and an effective approach based on the ranking method of HFEs is further developed to solve the new derived model. To address the incomplete and inconsistent preference structures of criteria weights, we introduce several deviation variables and establish the bi-objective nonlinear programming model. At length, we employ a green supplier selection problem to illustrate the feasibility and applicability of the proposed technique and conduct a comparison analysis to validate its effectiveness.

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Acknowledgments

The authors are very grateful to the anonymous reviewers and the editor for their insightful and constructive comments and suggestions that have led to an improved version of this paper. The work was supported by the National Natural Science Foundation of China (Nos. 61273209, 71263020629 and 71363020), and the Central University Basic Scientific Research Business 526 Expenses Project (No. skgt201501).

Author information

Authors and Affiliations

  1. The Collaborative Innovation Center, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    Xiaolu Zhang

  2. Business School, Sichuan University, Chengdu, 610065, Sichuan, China

    Zeshui Xu

  3. Institute of Industrial Economics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    Xiaoming Xing

Authors
  1. Xiaolu Zhang
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  2. Zeshui Xu
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  3. Xiaoming Xing
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Corresponding author

Correspondence to Zeshui Xu.

Appendix

Appendix

$$\begin{aligned} \begin{array}{l} \min \;\left\{ {\begin{array}{l} \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{array}} \right\} \\ \mathrm{s.t.}\;\left\{ {\begin{array}{l} -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 ){\succeq }\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 \\ 0\le \vartheta _j^1 \le \vartheta _j^2 \le \vartheta _j^3 \le w_j \;(j=1,2,3,4) \\ \end{array}} \right. \\ \end{array}\nonumber \\ \end{aligned}$$
(5.1)
$$\begin{aligned} \begin{array}{l} \min \;\left\{ {\begin{array}{l} \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{array}} \right\} \\ \min \left\{ {\varpi _{141}^- +\varpi _{132}^- +\varpi _{223}^- +\varpi _{223}^+ +\varpi _{34321}^- +\varpi _{421}^- +\varpi _{54}^+ +\varpi _{54}^- } \right\} \\ \mathrm{s.t.}\;\left\{ {\begin{array}{l} -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 ){\succeq }\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 +\varpi _{141}^- \ge w_1 , w_3 +\varpi _{132}^- \ge w_2 ,w_2 -w_3 \;+\varpi _{223}^- \ge 0.05\text{, } w_2 -w_3 -\varpi _{223}^+ \le 0.3, \; \\ w_4 -w_3 -w_2 +w_1 +\varpi _{34321}^- \ge 0 \text{, } {w_2 } /{w_1 }+\varpi _{421}^- \ge 2, w_4 -\varpi _{54}^+ \le 0.5,\;w_4 +\varpi _{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) \\ \varpi _{141}^- ,\varpi _{132}^- ,\varpi _{223}^- ,\varpi _{223}^+ ,\varpi _{34321}^- ,\varpi _{421}^- ,\varpi _{54}^+ ,\varpi _{54}^- \ge 0 \\ \end{array}} \right. \\ \end{array} \end{aligned}$$
(5.2)
$$\begin{aligned} \begin{array}{l} \min \;\left\{ \chi \right\} \\ \mathrm{s.t.}\;\left\{ {\begin{array}{l} 0.6z_{12} +0.65z_{23} +0.85z_{24} +0.6z_{25} +0.5z_{31} +0.75z_{34} +0.8z_{45} \le \chi \\ \varpi _{141}^- +\varpi _{132}^- +\varpi _{223}^- +\varpi _{223}^+ +\varpi _{34321}^- +\varpi _{421}^- +\varpi _{54}^+ +\varpi _{54}^- \le \chi \\ -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 +\varpi _{141}^- \ge w_1 , w_3 +\varpi _{132}^- \ge w_2 ,w_2 -w_3 \;+\varpi _{223}^- \ge 0.05\text{, } w_2 -w_3 -\varpi _{223}^+ \le 0.3, \; \\ w_4 -w_3 -w_2 +w_1 +\varpi _{34321}^- \ge 0 \text{, } {w_2 } /{w_1 }+\varpi _{421}^- \ge 2, w_4 -\varpi _{54}^+ \le 0.5,\;w_4 +\varpi _{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) \\ 0\le \vartheta _j^1 \le \vartheta _j^2 \le \vartheta _j^3 \le w_j \;(j=1,2,3,4) \\ \varpi _{141}^- ,\varpi _{132}^- ,\varpi _{223}^- ,\varpi _{223}^+ ,\varpi _{34321}^- ,\varpi _{421}^- ,\varpi _{54}^+ ,\varpi _{54}^- \ge 0 \\ \end{array}} \right. \\ \end{array} \end{aligned}$$
(5.3)
$$\begin{aligned} \begin{array}{l} \min \;\left\{ {\left\langle {0.5,0.3} \right\rangle z_{12} +\left\langle {0.6,0.3} \right\rangle z_{23} +\left\langle {0.8,01} \right\rangle z_{24} +\left\langle {0.5,0.3} \right\rangle z_{25} +\left\langle {0.4,0.4} \right\rangle z_{31} +\left\langle {0.6,0.05} \right\rangle z_{34} +\left\langle {0.7,0.1} \right\rangle z_{45} } \right\} \\ \mathrm{s.t.}\;\left\{ {\begin{array}{l} 0.2u_2 -0.1u_1 +0.6u_4 -0.2v_1 +0.4v_2 -0.2v_4 +0.13w_1 -0.26w_2 -0.1w_4 +z_{12} \ge 0 \\ 0.2u_2 -0.2u_1 +0.1u_3 +0.5u_4 -0.4v_1 +0.1v_2 +0.2v_3 +0.2v_4 +0.28w_1 -0.17w_2 -0.14w_3 -0.13w_4 +z_{23} \ge 0 \\ 0.2u_1 +0.3u_2 -0.6u_3 +0.2u_4 +0.1v_1 -0.3v_2 +0.15v_3 -0.35v_4 -0.14w_1 -0.19w_2 -0.3675w_3 -0.02w_4 +z_{24} \ge 0 \\ 0.4u_2 -0.2u_1 +0.05u_4 +0.2v_1 -0.4v_2 -0.15v_3 +0.3v_4 -0.36w_1 -0.48w_2 +0.0725w_3 -0.595w_4 +z_{25} \ge 0 \\ 0.3u_1 -0.4u_2 -0.1u_3 +0.5u_4 +0.6v_1 -0.5v_2 -0.2v_3 -0.48w_1 +0.36w_2 +0.13w_3 +0.22w_4 +z_{31} \ge 0 \\ 0.4u_1 +0.1u_2 -0.7u_3 +0.1u_4 +0.5v_1 -0.4v_2 -0.05v_3 -0.55v_4 -0.48w_1 +0.07w_2 +0.4825w_3 +0.105w_4 +z_{34} \ge 0 \\ 0.4u_2 -0.2u_1 +0.05u_4 +0.2v_1 -0.4v_2 -0.15v_3 +0.3v_4 +0.04w_1 -0.08w_2 +0.0725w_3 -0.395w_4 +z_{45} \ge 0 \\ \left\langle {0.5,0.3} \right\rangle ( {0.2u_2 -0.1u_1 +0.6u_4 -0.2v_1 +0.4v_2 -0.2v_4 +0.13w_1 -0.26w_2 -0.1w_4 })\, + \\ \left\langle {0.6,0.3} \right\rangle (0.2u_2 -0.2u_1 + 0.1u_3 + 0.5u_4 -0.4v_1 +0.1v_2 +0.2v_3 +0.2v_4 + 0.28w_1 -0.17w_2 -0.14w_3 -0.13w_4 ) \,+ \\ \left\langle {0.8,01} \right\rangle ( {0.2u_1 + 0.3u_2 -0.6u_3 + 0.2u_4 +0.1v_1 -0.3v_2 +0.15v_3 -0.35v_4 -0.14w_1 -0.19w_2 -0.3675w_3 -0.02w_4 })\, + \\ \left\langle {0.5,0.3} \right\rangle (0.4u_2 -0.2u_1 + 0.05u_4 +0.2v_1 -0.4v_2 -0.15v_3 +0.3v_4 -0.36w_1 -0.48w_2 + 0.0725w_3 -0.595w_4 )\, + \\ \left\langle {0.4,0.4} \right\rangle (0.3u_1 -0.4u_2 -0.1u_3 +0.5u_4 +0.6v_1 -0.5v_2 -0.2v_3 -0.48w_1 +0.36w_2 + 0.13w_3 +0.22w_4 ) \,+ \\ \left\langle {0.6,0.05} \right\rangle (0.4u_1 + 0.1u_2 -0.7u_3 +0.1u_4 +0.5v_1 -0.4v_2 -0.05v_3 -0.55v_4 -0.48w_1 + 0.07w_2 + 0.4825w_3 \,+ \\ 0.105w_4 )+\left\langle {0.7,0.1} \right\rangle (0.4u_2 -0.2u_1 +0.05u_4 +0.2v_1 -0.4v_2 -0.15v_3 +\,0.3v_4\, + \\ 0.04w_1 -0.08w_2 +0.0725w_3 -0.395w_4 ){\succeq }\left\langle {0.01,0.99} \right\rangle \\ w_2 \ge 2w_1 ,\;0.05\le w_2 -w_2 \le 0.3,\;\;0.15\le w_4 \le 0.5 \\ w_1 +w_2 +w_3 +w_4 =1,\;0.05\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 \\ u_j ,v_j \ge 0,\;u_j +v_j \le w_j \quad (j=1,2,3,4) \\ \end{array}} \right. \\ \end{array}\nonumber \\ \end{aligned}$$
(5.4)

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Zhang, X., Xu, Z. & Xing, X. Hesitant fuzzy programming technique for multidimensional analysis of hesitant fuzzy preferences. OR Spectrum 38, 789–817 (2016). https://doi.org/10.1007/s00291-015-0420-0

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