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Consistency measure of the WH-PLPR under the risk identification of PPP projects

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Abstract

Risk identification is the primary link and a significant basis for risk management. It is difficult to identify critical risk factors (CRFs) in terms of the uncertainty, diversity, and incompleteness of risk factors. The various preferences for different stakeholders could cause different identification results. Hence, we propose the weakened hedged probabilistic linguistic preference relation (WH-PLPR) to identify CRFs from the stakeholders’ preferences. For the WH-PLPR, checking and revising individual consistency is the basic part of the decision support model. Hence, the main contribution of this paper is studied in three parts: First, the concept of the WH-PLPR is given. Some consistency concepts, namely, weakened consistency, additive consistency, and satisfactory consistency of the WH-PLPR are defined. After that, the algorithms for improving the consistency of the WH-PLPR are studied. Then, we identify CRFs from stakeholders’ perspectives with the WH-PLPR information. A case study of a PPP project illustrates the utility and effectiveness of the proposed model. A sensitivity analysis of the WH-PLPR is introduced to illustrate the focus on consistency in the WH-PLPR, as well as the comparison of the consistency of the WH-PLPR with the linguistic hedged preference relations (LHPR) and the probabilistic linguistic preference relations (PLPR), which illustrates that weak consistency is the basis for satisfactory consistency. Moreover, the ranking results of CRFs show robustness for WH-PLPRs reaching satisfactory consistency.

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Acknowledgements

This work was supported by the Natural Science Foundation of China (No. 72071135) and the Scientific Research Foundation of Graduate School of Southeast University (No. YBPY2034).

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Correspondence to Zeshui Xu.

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A Preference matrix

A Preference matrix

(1) WH-PLPRs from Expert 1

$$\begin{aligned}&P_1^1\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_6}} \right\rangle ,0.9} \right\} ,\\ \left\{ {\left\langle {{h_0},{s_2}} \right\rangle ,0.1} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.63} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.37} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_2}} \right\rangle ,0.1} \right\} ,\\ \left\{ {\left\langle {{h_0},{s_6}} \right\rangle ,0.9} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_6}} \right\rangle ,0.65} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.35} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.37} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.63} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_2}} \right\rangle ,0.35} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.65} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_2^1\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_7}} \right\rangle ,0.7} \right\} ,\\ \left\{ {\left\langle {{h_4},{s_1}} \right\rangle ,0.3} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_2}} \right\rangle ,0.7} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_2}} \right\rangle ,0.3} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_1}} \right\rangle ,0.3} \right\} ,\\ \left\{ {\left\langle {{h_4},{s_7}} \right\rangle ,0.7} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.43} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.57} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_6}} \right\rangle ,0.3} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_6}} \right\rangle ,0.7} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.57} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.43} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_3^1\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.3} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_2}} \right\rangle ,0.7} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.52} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.48} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.7} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_6}} \right\rangle ,0.3} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{h_4}} \right\rangle ,0.18} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.82} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.48} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.52} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.82} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.18} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad {\begin{array}{{c}} P_{4}^{1} \end{array}}\\&\quad = \left[ \begin{array}{cccc} s_{4} &{} {\left( {\left\{ {\left\langle {h_{3},s_{5}} \right\rangle ,0.55} \right\} ,\left\{ {\left\langle {h_{1},s_{3}} \right\rangle ,0.45} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{3},s_{3}} \right\rangle ,0.45} \right\} ,\left\{ {\left\langle {h_{1},s_{5}} \right\rangle ,0.55} \right\} } \right) } &{} s_{4}\\ {\left( {\left\{ {\left\langle {h_{3},s_{4}} \right\rangle ,0.63} \right\} ,\left\{ {\left\langle {h_{1},s_{4}} \right\rangle ,0.37} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{4},s_{5}} \right\rangle ,0.56} \right\} ,\left\{ {\left\langle {h_{0},s_{5}} \right\rangle ,0.44} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{1},s_{1}} \right\rangle ,0.4} \right\} ,\left\{ {\left\langle {h_{3},s_{7}} \right\rangle ,0.6} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{0},s_{5}} \right\rangle ,0.1} \right\} ,\left\{ {\left\langle {h_{0},s_{3}} \right\rangle ,0.9} \right\} } \right) } \end{array}\right. \\&\left. \begin{array}{*{20}{c}c} {\left( {\left\{ {\left\langle {h_{3},s_{4}} \right\rangle ,0.37} \right\} ,\left\{ {\left\langle {h_{1},s_{4}} \right\rangle ,0.63} \right\} } \right) }&{} {\left( {\left\{ {\left\langle {h_{1},s_{7}} \right\rangle ,0.6} \right\} ,\left\{ {\left\langle {h_{3},s_{1}} \right\rangle ,0.4} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{4},s_{3}} \right\rangle ,0.44} \right\} ,\left\{ {\left\langle {h_{0},s_{3}} \right\rangle ,0.56} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{0},s_{3}} \right\rangle ,0.9} \right\} ,\left\{ {\left\langle {h_{0},s_{5}} \right\rangle ,0.1} \right\} } \right) }\\ s_{4} &{} {\left( {\left\{ {\left\langle {h_{3},s_{2}} \right\rangle ,0.44} \right\} ,\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.56} \right\} } \right) } \\ {\left( {\left\{ {\left\langle {h_{3},s_{6}} \right\rangle ,0.56} \right\} ,\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.44} \right\} } \right) } &{} s_{4}\\ \end{array}\right] \\&\quad P_5^1\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.12} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.88} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.59} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.41} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.88} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.12} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.77} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_7}} \right\rangle ,0.23} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.41} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.59} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.23} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_1}} \right\rangle ,0.77} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad {P_6^{1}} = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( {\left\{ {\left\langle {h_{1},s_{5}} \right\rangle ,0.35} \right\} ,\left\{ {\left\langle {h_{3},s_{3}} \right\rangle ,0.65} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{1},s_{3}} \right\rangle ,0.65} \right\} ,\left\{ {\left\langle {h_{3},s_{5}} \right\rangle ,0.35} \right\} } \right) }&{}{{s_4}} \end{array}} \right] \end{aligned}$$

(2) WH-PLPRs from Expert 2

$$\begin{aligned}&P_1^2\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.74} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_1}} \right\rangle ,0.26} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_2}} \right\rangle ,0.28} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_1}} \right\rangle ,0.72} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.26} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_7}} \right\rangle ,0.74} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_7}} \right\rangle ,0.3} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.7} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_6}} \right\rangle ,0.72} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_7}} \right\rangle ,0.28} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_1}} \right\rangle ,0.7} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.3} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_2^2 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_1}} \right\rangle ,0.39} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.61} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_0}} \right\rangle ,0.49} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.51} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_7}} \right\rangle ,0.61} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.39} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_7}} \right\rangle ,0.64} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.36} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_8}} \right\rangle ,0.51} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.49} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_1}} \right\rangle ,0.36} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.64} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_3^2 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_5}} \right\rangle ,0.54} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.46} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_6}} \right\rangle ,0.62} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.38} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_3}} \right\rangle ,0.46} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.54} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_7}} \right\rangle ,0.35} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.65} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_2}} \right\rangle ,0.38} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.62} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_1}} \right\rangle ,0.65} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.35} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad {\begin{array}{{c}} P_{4}^{2} \end{array}}= \left[ \begin{array}{cccc} s_{4} &{} {\left( {\left\{ {\left\langle {h_{2},s_{3}} \right\rangle ,0.46} \right\} ,\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.54} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{2},s_{5}} \right\rangle ,0.54} \right\} ,\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.46} \right\} } \right) } &{} s_{4}\\ {\left( {\left\{ {\left\langle {h_{0},s_{2}} \right\rangle ,0.55} \right\} ,\left\{ {\left\langle {h_{4},s_{5}} \right\rangle ,0.45} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{4},s_{1}} \right\rangle ,0.4} \right\} ,\left\{ {\left\langle {h_{0},s_{3}} \right\rangle ,0.6} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{2},s_{2}} \right\rangle ,0.48} \right\} ,\left\{ {\left\langle {h_{2},s_{6}} \right\rangle ,0.52} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{3},s_{5}} \right\rangle ,0.5} \right\} ,\left\{ {\left\langle {h_{0},s_{3}} \right\rangle ,0.5} \right\} } \right) } \end{array}\right. \\ \left. \begin{array}{cc} {\left( {\left\{ {\left\langle {h_{0},s_{6}} \right\rangle ,0.45} \right\} ,\left\{ {\left\langle {h_{4},s_{3}} \right\rangle ,0.55} \right\} } \right) }&{} {\left( {\left\{ {\left\langle {h_{2},s_{6}} \right\rangle ,0.52} \right\} ,\left\{ {\left\langle {h_{2},s_{2}} \right\rangle ,0.48} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{4},s_{7}} \right\rangle ,0.6} \right\} ,\left\{ {\left\langle {h_{0},s_{5}} \right\rangle ,0.4} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{3},s_{3}} \right\rangle ,0.5} \right\} ,\left\{ {\left\langle {h_{0},s_{5}} \right\rangle ,0.5} \right\} } \right) }\\ s_{4} &{} {\left( {\left\{ {\left\langle {h_{2},s_{1}} \right\rangle ,0.53} \right\} ,\left\{ {\left\langle {h_{1},s_{5}} \right\rangle ,0.47} \right\} } \right) } \\ {\left( {\left\{ {\left\langle {h_{2},s_{7}} \right\rangle ,0.47} \right\} ,\left\{ {\left\langle {h_{1},s_{3}} \right\rangle ,0.53} \right\} } \right) } &{} s_{4}\\ \end{array}\right] \\&\quad P_5^2 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_6}} \right\rangle ,0.52} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_5}} \right\rangle ,0.48} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.63} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.37} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_4},{s_2}} \right\rangle ,0.48} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_3}} \right\rangle ,0.52} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_3}} \right\rangle ,0.4} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.6} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.37} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.63} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_5}} \right\rangle ,0.6} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.4} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_6^2 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( {\left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.46} \right\} ,\left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.54} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.54} \right\} ,\left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.46} \right\} } \right) }&{}{{s_4}} \end{array}} \right] \end{aligned}$$

(3) WH-PLPRs from Expert 3

$$\begin{aligned}&P_1^3 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_2}} \right\rangle ,0.45} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_1}} \right\rangle ,0.55} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.43} \right\} ,\\ \left\{ {\left\langle {{h_4},{s_6}} \right\rangle ,0.57} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_0},{s_6}} \right\rangle ,0.55} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_7}} \right\rangle ,0.45} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_0}} \right\rangle ,0.51} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.49} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_4}} \right\rangle ,0.57} \right\} ,\\ \left\{ {\left\langle {{h_4},{s_2}} \right\rangle ,0.43} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_8}} \right\rangle ,0.49} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.51} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_2^3\mathrm{{ = }}\left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_3}} \right\rangle ,0.56} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_1}} \right\rangle ,0.44} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_1}} \right\rangle ,0.33} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_1}} \right\rangle ,0.67} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_5}} \right\rangle ,0.44} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_7}} \right\rangle ,0.56} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_1}} \right\rangle ,0.54} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.46} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_2},{s_7}} \right\rangle ,0.67} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_7}} \right\rangle ,0.33} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_7}} \right\rangle ,0.46} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.54} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_3^3 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.55} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_1}} \right\rangle ,0.45} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.52} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_2}} \right\rangle ,0.48} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.45} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_7}} \right\rangle ,0.55} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_6}} \right\rangle ,0.44} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_3}} \right\rangle ,0.56} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_4}} \right\rangle ,0.48} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_6}} \right\rangle ,0.52} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_2}} \right\rangle ,0.56} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_5}} \right\rangle ,0.42} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad {\begin{array}{{c}} P_{4}^{3} \end{array}}=\left[ \begin{array}{cccc} s_{4} &{} {\left( {\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.47} \right\} ,\left\{ {\left\langle {h_{2},s_{1}} \right\rangle ,0.53} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.53} \right\} ,\left\{ {\left\langle {h_{2},s_{7}} \right\rangle ,0.47} \right\} } \right) } &{} s_{4}\\ {\left( {\left\{ {\left\langle {h_{1},s_{4}} \right\rangle ,0.54} \right\} ,\left\{ {\left\langle {h_{3},s_{6}} \right\rangle ,0.46} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{2},s_{4}} \right\rangle ,0.35} \right\} ,\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.65} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{1},s_{7}} \right\rangle ,0.38} \right\} ,\left\{ {\left\langle {h_{2},s_{3}} \right\rangle ,0.62} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.54} \right\} ,\left\{ {\left\langle {h_{3},s_{6}} \right\rangle ,0.46} \right\} } \right) } \end{array}\right. \\&\left. \begin{array}{cc} {\left( {\left\{ {\left\langle {h_{1},s_{4}} \right\rangle ,0.46} \right\} ,\left\{ {\left\langle {h_{3},s_{2}} \right\rangle ,0.54} \right\} } \right) }&{} {\left( {\left\{ {\left\langle {h_{1},s_{1}} \right\rangle ,0.62} \right\} ,\left\{ {\left\langle {h_{2},s_{5}} \right\rangle ,0.38} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {h_{2},s_{4}} \right\rangle ,0.65} \right\} ,\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.35} \right\} } \right) } &{} {\left( {\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.46} \right\} ,\left\{ {\left\langle {h_{3},s_{2}} \right\rangle ,0.54} \right\} } \right) }\\ s_{4} &{} {\left( {\left\{ {\left\langle {h_{1},s_{3}} \right\rangle ,0.44} \right\} ,\left\{ {\left\langle {h_{1},s_{6}} \right\rangle ,0.56} \right\} } \right) } \\ {\left( {\left\{ {\left\langle {h_{1},s_{5}} \right\rangle ,0.56} \right\} ,\left\{ {\left\langle {h_{1},s_{2}} \right\rangle ,0.44} \right\} } \right) } &{} s_{4}\\ \end{array}\right] \\&\quad P_5^3 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.37} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.63} \right\} \end{array} \right) }{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.63} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_6}} \right\rangle ,0.37} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.63} \right\} ,\\ \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.37} \right\} \end{array} \right) }&{}{{s_4}}&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_2}} \right\rangle ,0.52} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.48} \right\} \end{array} \right) }\\ {\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.37} \right\} ,\\ \left\{ {\left\langle {{h_1},{s_2}} \right\rangle ,0.63} \right\} \end{array} \right) }&{}{\left( \begin{array}{l} \left\{ {\left\langle {{h_3},{s_6}} \right\rangle ,0.48} \right\} ,\\ \left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.52} \right\} \end{array} \right) }&{{s_4}} \end{array}} \right] \\&\quad P_6^3 = \left[ {\begin{array}{*{20}{c}} {{s_4}}&{}{\left( {\left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.39} \right\} ,\left\{ {\left\langle {{h_1},{s_5}} \right\rangle ,0.61} \right\} } \right) }\\ {\left( {\left\{ {\left\langle {{h_2},{s_4}} \right\rangle ,0.61} \right\} ,\left\{ {\left\langle {{h_1},{s_3}} \right\rangle ,0.39} \right\} } \right) }&{}{{s_4}} \end{array}} \right] \end{aligned}$$

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Wang, L., Xu, Z. & Hao, Z. Consistency measure of the WH-PLPR under the risk identification of PPP projects. Int. J. Mach. Learn. & Cyber. 13, 3441–3461 (2022). https://doi.org/10.1007/s13042-022-01606-7

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