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A Novel Preference Measure for Multi-Granularity Probabilistic Linguistic Term Sets and its Applications in Large-Scale Group Decision-Making

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Abstract

Comparing probabilistic linguistic term sets (PLTSs) is quite essential in solving PLTS-expressed multi-attribute group decision-making problems (PLTS-MAGDM). Researchers have designed various comparison measures to obtain the rank of PLTSs. However, most of the existing PLTS comparison measures need additional tedious adjustments before conducting a specific computation. Besides, these measures do not adequately consider the effects of the semantics of the basic linguistic term set and the probabilistic distributions. This paper proposes a new preference degree for g-granularity probabilistic term sets (g-GPLTSs) to overcome the two shortcomings simultaneously by integrating the effect from basic linguistic terms and probabilistic distributions without any adjustment. Moreover, the g-GPLTS preference degree also shows the extended adaptability for comparing PLTSs with unbalanced semantics. Based on the newly proposed preference degree, we construct a useful min-conflict model to solve PLTS-MAGDM with a large number of experts expressing the three-way primary grading. Finally, an illustrative example concerning software supplier selections, followed by the comparative analysis, is presented to verify the feasibility and effectiveness of the proposed method.

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Acknowledgements

We would like to thank the reviewers for their insightful suggestions. The authors also thank assistant professor Yarong Hu for her help in completing the theoretical proof. This work was supported by the National Natural Science Foundation of China (Nos. 61703363, 61876103), the Applied Basic Research Program of Shanxi Province (Nos. 201901D211462, 201801D121148), the Key R&D program of Shanxi Province (International Cooperation, 201903D421041), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (No. 2019L0864), and the Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018008).

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Appendices

Appendix A

Proof of the transitivity of the GLTS preference degree

Let \(p^{1}=(p_1^1,p_2^1,\ldots ,p_n^1),\)\(p^2=(p_1^2,p_2^2,\ldots ,p_n^2)\) and \(p^3=(p_1^3,p_2^3,\ldots ,p_n^3)\) be three probability distributions. n discrete numbers \(s_i\ (i=1,2,\ldots ,n)\) satisfy that \(0\le s_1<s_2<\ldots <s_n\le 1.\) Let

$$\begin{aligned} P^{uv}=\left( \begin{array}{cccc} p^{(uv)}_1p^{uv}_1 &{} p^{(uv)}_1p^{uv}_2 &{}\cdots &{} p^{(uv)}_1p^{uv}_n \\ p^{(uv)}_2p^{uv}_1 &{} p^{(uv)}_2p^{uv}_2 &{} \cdots &{} p^{(uv)}_2p^{uv}_n \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ p^{(uv)}_np^{uv}_1 &{} p^{(uv)}_np^{uv}_2 &{}\cdots &{} p^{(uv)}_np^{uv}_n \\ \end{array} \right) \end{aligned}$$

be the joint distribution of \(p^i\) and \(p^j.\) Let

$$\begin{aligned} S=\left( \begin{array}{cccc} s_1-s_1 &{} s_1-s_2 &{}\cdots &{} s_1-s_n \\ s_2-s_1 &{} s_2-s_2 &{}\cdots &{} s_2-s_n \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ s_n-s_1 &{} s_n-s_2 &{}\cdots &{} s_n-s_n \\ \end{array} \right) . \end{aligned}$$

Define

$$\begin{aligned} \parallel P^{uv}\otimes S\parallel =\sum \limits _{i=1}^n\sum \limits _{j=1}^n p_i^up_j^v(s_i-s_j). \end{aligned}$$

If \(\parallel P^{12}\otimes S\parallel \ge 0\) and \(\parallel P^{23}\otimes S\parallel \ge 0,\) then \(\parallel P^{13}\otimes S\parallel \ge 0.\)

Proof

Compute that

$$\begin{aligned} \begin{array}{l} \parallel P^{12}\otimes S\parallel \\ =\sum \limits _{i=1}^n\sum \limits _{j=1}^n p_i^1p_j^2(s_i-s_j) \\ =p_1^1p_1^2(s_1-s_1) +p_1^1p_2^2(s_1-s_2)+\cdots +p_1^1p_n^2(s_1-s_n)+ \\ \ \ \ \ p_2^1p_1^2(s_2-s_1) +p_2^1p_2^2(s_2-s_2)+\cdots +p_2^1p_n^2(s_2-s_n)+\\ \ \ \ \ \cdots \\ \ \ \ \ p_n^1p_1^2(s_n-s_1) +p_n^1p_2^2(s_n-s_2)+\cdots +p_n^1p_n^2(s_n-s_n)\\ = p_1^2(p_1^1(s_1-s_1)+p_2^1(s_2-s_1)+p_n^1(s_n-s_1))+\\ \ \ \ \ p_2^2(p_1^1(s_1-s_2)+p_2^1(s_2-s_2)+p_n^1(s_n-s_2))+\\ \ \ \ \ \cdots \\ \ \ \ \ p_n^2(p_1^1(s_1-s_n)+p_2^1(s_2-s_n)+p_n^1(s_n-s_n))\\ = p_1^2(\sum \limits _{i=1}^np_i^1s_i-s_1\sum \limits _{i=1}^n p_i^1)+p_2^2(\sum \limits _{i=1}^np_i^1s_i-s_2\sum \limits _{i=1}^n p_i^1)+\\ \ \ \ \ \cdots \ + p_1^n(\sum \limits _{i=1}^np_i^1s_i-s_n\sum \limits _{i=1}^n p_i^1)\\ =(\sum \limits _{i=1}^np_i^2)(\sum \limits _{i=1}^np_i^1s_i)-\sum \limits _{i=1}^n p_i^2s_i\\ =\sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^2s_i. \\ \end{array} \end{aligned}$$

Similarly, we have

$$\begin{aligned} \parallel P^{23}\otimes S\parallel =\sum \limits _{i=1}^np_i^2s_i-\sum \limits _{i=1}^n p_i^3s_i \end{aligned}$$

and

$$\begin{aligned} \parallel P^{13}\otimes S\parallel =\sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^3s_i. \end{aligned}$$

Since \(\parallel P^{12}\otimes S\parallel \ge 0\) and \(\parallel P^{23}\otimes S\parallel \ge 0,\) we can easily obtain that

$$\begin{aligned} \begin{array}{l} \parallel P^{13}\otimes S\parallel =\sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^3s_i \\ = \sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^3s_i\\ = \sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^3s_i-\sum \limits _{i=1}^n p_i^2s_i+\sum \limits _{i=1}^n p_i^2s_i\\ =(\sum \limits _{i=1}^np_i^1s_i-\sum \limits _{i=1}^n p_i^2s_i)+(\sum \limits _{i=1}^n p_i^2s_i-\sum \limits _{i=1}^n p_i^3s_i)\\ =\parallel P^{12}\otimes S\parallel +\parallel P^{23}\otimes S\parallel \\ \ge 0+0\\ =0. \end{array} \end{aligned}$$

Therefore, we have \(\parallel P^{13}\otimes S\parallel \ge 0.\)\(\square \)

Appendix B

Definition of Hadamard product

Let \(A=\big (a_{ij}\big )_{m\times n}\) and \(B=\big (b_{ij}\big )_{m\times n}\) be two matrices.

$$\begin{aligned} A\circ B= \left( \begin{array}{cccc} a_{11}\times b_{11} &{} a_{12}\times b_{12} &{} \cdots &{} a_{1n}\times b_{1n} \\ a_{21}\times b_{21} &{} a_{22}\times b_{22} &{} \cdots &{} a_{2n}\times b_{2n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{m1}\times b_{m1} &{} a_{m2}\times b_{m2} &{} \cdots &{}a_{mn}\times b_{mn} \\ \end{array} \right) \end{aligned}$$

is called the Hadamard Product of A and B.

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Wang, B., Liang, J. A Novel Preference Measure for Multi-Granularity Probabilistic Linguistic Term Sets and its Applications in Large-Scale Group Decision-Making. Int. J. Fuzzy Syst. 22, 2350–2368 (2020). https://doi.org/10.1007/s40815-020-00887-w

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