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The Aviation Technology Two-Sided Matching with the Expected Time Based on the Probabilistic Linguistic Preference Relations

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Abstract

The two-sided matching has been widely applied to the decision-making problems in the field of management. With the limited working experience, the two-sided agents usually cannot provide the preference order directly for the opposite agent, but rather to provide the preference relations in the form of linguistic information. The preference relations based on probabilistic linguistic term sets (PLTSs) not only allow agents to provide the evaluation with multiple linguistic terms, but also present the different preference degrees for linguistic terms. Considering the diversities of the agents, they may provide their preference relations in the form of the probabilistic linguistic preference relation (PLPR) or the probabilistic linguistic multiplicative preference relation (PLMPR). For two-sided matching with the expected time, we first provide the concept of the time satisfaction degree (TSD). Then, we transform the preference relations in different forms into the unified preference relations (u-PRs). The consistency index to measure the consistency of u-PRs is introduced. Besides, the acceptable consistent u-PRs are constructed, and an algorithm is proposed to modify the unacceptable consistent u-PRs. Furthermore, we present the whole two-sided matching decision-making process with the acceptable consistent u-PRs. Finally, a case about aviation technology suppliers and demanders matching is presented to exhibit the rationality and practicality of the proposed method. Some analyses and discussions are provided to further demonstrate the feasibility and effectiveness of the proposed method.

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Author information

Authors and Affiliations

  1. Institute for Disaster Management and Reconstruction, Sichuan University, Chengdu, 610207, China

    Bo Li

  2. Business School, State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu, 610064, China

    Yi-**n Zhang & Ze-Shui Xu

  3. School of Computer Science, University of Manchester, Manchester, M13 9PL, United Kingdom

    Yi-**n Zhang

Authors
  1. Bo Li
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  2. Yi-**n Zhang
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  3. Ze-Shui Xu
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Corresponding author

Correspondence to Ze-Shui Xu.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 71771155, 71571123), the scholarship under the UK-China Joint Research and Innovation Partnership Fund Ph.D. Placement Programme (No. 201806240416) and the Teacher-Student Joint Innovation Research Fund of Business School of Sichuan University (No. H2018016).

Appendix

Appendix

See Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20.

Table 16 The \( \phi_{1} \), \( \phi_{2} \) and \( \phi_{3} \)
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Table 17 The funding provided by the demanders (unit: ten thousand yuan)
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Table 18 The matching results and \( Z \) with different values of \( \lambda \) and \( \varepsilon_{1} = \varepsilon_{2} = 0.5 \)
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Table 19 The matching results and \( Z \) with different values of \( \varepsilon \) and \( \lambda_{1} = \lambda_{2} = 0.5 \)
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Table 20 The results for the three methods
Full size table
$$ U_{{s_{1} }} = \left[ {\begin{array}{*{20}c} {\{ 0.50(1)\} } & {\{ 0.66(0.25),0.5(0.75)\} } & {\{ 0.5(0.44),0.66(0.56)\} } & {\{ 0.34(0.2),0.5(0.5),0.66(0.3)\} } & {\{ 0.5(0.4),0.66(0.6)\} } \\ {\{ 0.34(0.25),0.25(0.75)\} } & {\{ 0.5(1)\} } & {\{ 0.5(0.63),0.66(0.38)\} } & {\{ 0.5(0.44),0.75(0.56)\} } & {\{ 0.5(0.11),0.66(0.89)\} } \\ {\{ 0.50(0.44),0.34(0.56)\} } & {\{ 0.5(0.63),0.34(0.37)\} } & {\{ 0.5(1)\} } & {\{ 0.34(0.3),0.5(0.7)\} } & {\{ 0.5(0.5),0.66(0.5)\} } \\ {\{ 0.66(0.2),0.50(0.5),0.34(0.3)} & {\{ 0.5(0.44),0.25(0.56)\} } & {\{ 0.66(0.3),0.5(0.7)\} } & {\{ 0.5(1)\} } & {\{ 0.75(1)\} } \\ {0.5(0.4),0.34(0.6)} & {\{ 0.5(0.11),0.34(0.89)\} } & {\{ 0.5(0.5),0.34(0.5)\} } & {\{ 0.25(1)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{s_{2} }} = \left[ {\begin{array}{*{20}c} {\{ 0.50(1)\} } & {\{ 0.83(0.14),1(0.86)\} } & {\{ 0.67(0.5),0.83(0.5)\} } & {\{ 0.33(0.25),0.83(0.75)\} } & {\{ 0.5(0.4),1(0.6)\} } \\ {\{ 0.17(0.14),0(0.86)\} } & {\{ 0.5(1)\} } & {\{ 0.17(0.5),0.67(0.5)\} } & {\{ 0.67(0.5),1(0.5)\} } & {\{ 0.67(0.2),0.83(0.8)\} } \\ {\{ 0.33(0.5),0.17(0.5)\} } & {\{ 0.83(0.5),0.33(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.25),0.67(0.75)\} } & {\{ 0.67(0.4),0.83(0.6)\} } \\ {\{ 0.67(0.25),0.17(0.75)\} } & {\{ 0.83(0.5),0(0.5)\} } & {\{ 0.67(0.25),0.33(0.75)\} } & {\{ 0.5(1)\} } & {\{ 0.83(1)\} } \\ {0.5(0.4),0(0.6)} & {\{ 0.33(0.2),0.17(0.8)\} } & {\{ 0.33(0.4),0.17(0.6)\} } & {\{ 0.17(1)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{s_{3} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.33(0.25),0.67(0.75)\} } & {\{ 0.67(0.2),0.83(0.8)\} } & {\{ 0.5(0.5),0.67(0.5)\} } & {\{ 0.33(0.3),0.83(0.7)\} } \\ {\{ 0.67(0.25),0.33(0.75)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.5),0.83(0.5)\} } & {\{ 0.33(0.4),0.83(0.6)\} } & {\{ 0.67(0.2),0.83(0.8)\} } \\ {\{ 0.33(0.2),0.17(0.8)\} } & {\{ 0.67(0.5),0.17(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.3),0.5(0.7)\} } & {\{ 0.67(0.5),0.83(0.5)\} } \\ {\{ 0.5(0.5),0.33(0.5)\} } & {\{ 0.67(0.4),0.17(0.6)\} } & {\{ 0.67(0.3),0.5(0.7)\} } & {\{ 0.5(1)\} } & {\{ 0.83(1)\} } \\ {\{ 0.67(0.3),0.17(0.7)\} } & {\{ 0.33(0.2),0.17(0.8)\} } & {\{ 0.33(0.5),0.17(0.5)\} } & {\{ 0.17(1)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{s_{4} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.5(0.4),0.67(0.6)\} } & {\{ 0.67(1)\} } & {\{ 0.33(0.5),0.67(0.5)\} } & {\{ 0(0.5),0.83(0.5)\} } \\ {\{ 0.5(0.4),0.33(0.6)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.5),0.83(0.5)\} } & {\{ 0.33(0.4),1(0.6)\} } & {\{ 0.33(0.2),0.83(0.8)\} } \\ {\{ 0.33(1)\} } & {\{ 0.67(0.5),0.17(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.3),0.5(0.7)\} } & {\{ 0.17(0.5),0.83(0.5)\} } \\ {\{ 0.67(0.5),0.33(0.5)\} } & {\{ 0.67(0.44),0(0.56)\} } & {\{ 0.67(0.3),0.5(0.7)\} } & {\{ 0.5(1)\} } & {\{ 0.5(1)\} } \\ {\{ 1(0.5),0.17(0.5)\} } & {\{ 0.67(0.2),0.17(0.8)\} } & {\{ 0.83(0.5),0.17(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{d_{1} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.5(0.33),0.83(0.67)\} } & {\{ 0.33(0.44),0.83(0.56)\} } & {\{ 0.33(0.4),0.67(0.6)\} } \\ {\{ 0.5(0.33),0.17(0.67)\} } & {\{ 0.5(1)\} } & {\{ 0.5(0.5),0.83(0.5)\} } & {\{ 0.33(0.4),0.83(0.6)\} } \\ {\{ 0.67(0.44),0.17(0.56)\} } & {\{ 0.5(0.5),0.17(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.67(0.5),1(0.5)\} } \\ {\{ 0.67(0.4),0.33(0.6)\} } & {\{ 0.67(0.4),0.17(0.6)\} } & {\{ 0.33(0.5),0(0.5)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{d_{2} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.5(0.25),0.83(0.75)\} } & {\{ 0.33(0.4),0.5(0.6)\} } & {\{ 0.5(1)\} } \\ {\{ 0.5(0.25),0.17(0.75)\} } & {\{ 0.5(1)\} } & {\{ 0.5(0.5),0.67(0.5)\} } & {\{ 0.33(0.4),0.5(0.6)\} } \\ {\{ 0.67(0.4),0.5(0.6)\} } & {\{ 0.5(0.5),0.33(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.5(0.5),0.67(0.5)\} } \\ {\{ 0.5(1)\} } & {\{ 0.67(0.4),0.5(0.6)\} } & {\{ 0.5(0.5),0.33(0.5)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{d_{3} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.67(1)\} } & {\{ 0.33(0.3),0.17(0.1),0.5(0.6)\} } & {\{ 0.5(1)\} } \\ {\{ 0.33(1)\} } & {\{ 0.5(1)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.4),0.83(0.6)\} } \\ {\{ 0.67(0.3),0.83(0.1),0.5(0.6)\} } & {\{ 0.5(1)\} } & {\{ 0.5(1)\} } & {\{ 0.67(1)\} } \\ {\{ 0.5(1)\} } & {\{ 0.67(0.4),0.17(0.6)\} } & {\{ 0.33(1)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right], $$
$$ U_{{d_{4} }} = \left[ {\begin{array}{*{20}l} {\{ 0.5(1)\} } \hfill & {\{ 0.33(0.4)\} } \hfill & {\{ 0.5(1)\} } \hfill & {\{ 0.67(1)\} } \hfill \\ {\{ 0.67(0.4),0.17(0.6)\} } \hfill & {\{ 0.5(1)\} } \hfill & {\{ 0.5(0.5),0.67(0.5)\} } \hfill & {\{ 0.33(0.56),0.83(0.44)\} } \hfill \\ {\{ 0.5(1)\} } \hfill & {\{ 0.5(0.5),0.33(0.5)\} } \hfill & {\{ 0.5(1)\} } \hfill & {\{ 0.67(0.5),0.83(0.5)\} } \hfill \\ {\{ 0.33(1)\} } \hfill & {\{ 0.67(0.56),0.17(0.44)\} } \hfill & {\{ 0.33(0.5),0.17(0.5)\} } \hfill & {\{ 0.5(1)\} } \hfill \\ \end{array} } \right], $$
$$ U_{{d_{5} }} = \left[ {\begin{array}{*{20}c} {\{ 0.5(1)\} } & {\{ 0.5(0.2),0.83(0.8)\} } & {\{ 0.33(0.4),0.5(0.6)\} } & {\{ 0.67(1)\} } \\ {\{ 0.5(0.2),0.17(0.8)\} } & {\{ 0.5(1)\} } & {\{ 0.33(0.5),0.17(0.5)\} } & {\{ 0.33(0.14),0.5(0.86)\} } \\ {\{ 0.67(0.4),0.5(0.6)\} } & {\{ 0.67(0.5),0.83(0.5)\} } & {\{ 0.5(1)\} } & {\{ 0.67(0.4),0.83(0.6)\} } \\ {\{ 0.33(1)\} } & {\{ 0.67(0.14),0.5(0.86)\} } & {\{ 0.33(0.4),0.17(0.6)\} } & {\{ 0.5(1)\} } \\ \end{array} } \right]. $$
$$\begin{aligned} & \hat{E}^{{s_{1} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.57} & \hfill {0.57} & \hfill {0.52} & \hfill {0.65} \\ \hfill {0.43} & \hfill {0.50} & \hfill {0.56} & \hfill {0.51} & \hfill {0.64} \\ \hfill {0.43} & \hfill {0.44} & \hfill {0.50} & \hfill {0.45} & \hfill {0.58} \\ \hfill {0.48} & \hfill {0.36} & \hfill {0.55} & \hfill {0.50} & \hfill {0.63} \\ \hfill {0.35} & \hfill {0.36} & \hfill {0.42} & \hfill {0.37} & \hfill {0.50} \\ \end{array} } \right], \quad \hat{E}^{{s_{2} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.71} & \hfill {0.63} & \hfill {0.71} & \hfill {0.89} \\ \hfill {0.29} & \hfill {0.50} & \hfill {0.42} & \hfill {0.50} & \hfill {0.68} \\ \hfill {0.37} & \hfill {0.58} & \hfill {0.50} & \hfill {0.58} & \hfill {0.75} \\ \hfill {0.29} & \hfill {0.50} & \hfill {0.42} & \hfill {0.50} & \hfill {0.68} \\ \hfill {0.11} & \hfill {0.32} & \hfill {0.25} & \hfill {0.32} & \hfill {0.50} \\ \end{array} } \right],\\ & \quad \hat{E}^{{s_{3} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.55} & \hfill {0.63} & \hfill {0.58} & \hfill {0.82} \\ \hfill {0.45} & \hfill {0.50} & \hfill {0.58} & \hfill {0.53} & \hfill {0.77} \\ \hfill {0.37} & \hfill {0.42} & \hfill {0.50} & \hfill {0.45} & \hfill {0.69} \\ \hfill {0.42} & \hfill {0.47} & \hfill {0.55} & \hfill {0.50} & \hfill {0.74} \\ \hfill {0.18} & \hfill {0.24} & \hfill {0.32} & \hfill {0.26} & \hfill {0.50} \\ \end{array} } \right], \end{aligned} $$
$$\begin{aligned} & \hat{E}^{{s_{4} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.21} & \hfill {0.39} & \hfill {0.41} & \hfill {0.64} \\ \hfill {0.75} & \hfill {0.50} & \hfill {0.70} & \hfill {0.70} & \hfill {0.70} \\ \hfill {0.61} & \hfill {0.30} & \hfill {0.50} & \hfill {0.50} & \hfill {0.50} \\ \hfill {0.51} & \hfill {0.30} & \hfill {0.50} & \hfill {0.50} & \hfill {0.50} \\ \hfill {0.36} & \hfill {0.30} & \hfill {0.50} & \hfill {0.50} & \hfill {0.50} \\ \end{array} } \right],\quad \hat{E}^{{d_{1} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.72} & \hfill {0.61} & \hfill {0.78} \\ \hfill {0.28} & \hfill {0.50} & \hfill {0.40} & \hfill {0.63} \\ \hfill {0.39} & \hfill {0.59} & \hfill {0.50} & \hfill {0.75} \\ \hfill {0.20} & \hfill {0.37} & \hfill {0.21} & \hfill {0.50} \\ \end{array} } \right],\\ & \quad \hat{E}^{{d_{2} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.59} & \hfill {0.43} & \hfill {0.50} \\ \hfill {0.41} & \hfill {0.50} & \hfill {0.38} & \hfill {0.43} \\ \hfill {0.57} & \hfill {0.61} & \hfill {0.50} & \hfill {0.57} \\ \hfill {0.50} & \hfill {0.57} & \hfill {0.43} & \hfill {0.50} \\ \end{array} } \right],\end{aligned} $$
$$\begin{aligned} & \hat{E}^{{d_{3} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.48} & \hfill {0.42} & \hfill {0.58} \\ \hfill {0.51} & \hfill {0.50} & \hfill {0.46} & \hfill {0.63} \\ \hfill {0.58} & \hfill {0.54} & \hfill {0.50} & \hfill {0.67} \\ \hfill {0.41} & \hfill {0.37} & \hfill {0.33} & \hfill {0.50} \\ \end{array} } \right],\quad \hat{E}^{{d_{4} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.60} & \hfill {0.50} & \hfill {0.66} \\ \hfill {0.40} & \hfill {0.50} & \hfill {0.39} & \hfill {0.56} \\ \hfill {0.50} & \hfill {0.60} & \hfill {0.50} & \hfill {0.66} \\ \hfill {0.34} & \hfill {0.44} & \hfill {0.34} & \hfill {0.50} \\ \end{array} } \right],\\ & \quad \hat{E}^{{d_{5} }} = \left[ {\begin{array}{*{20}r} \hfill {0.50} & \hfill {0.70} & \hfill {0.43} & \hfill {0.67} \\ \hfill {0.30} & \hfill {0.50} & \hfill {0.25} & \hfill {0.47} \\ \hfill {0.57} & \hfill {0.75} & \hfill {0.50} & \hfill {0.71} \\ \hfill {0.33} & \hfill {0.53} & \hfill {0.17} & \hfill {0.50} \\ \end{array} } \right].\end{aligned} $$

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Li, B., Zhang, YX. & Xu, ZS. The Aviation Technology Two-Sided Matching with the Expected Time Based on the Probabilistic Linguistic Preference Relations. J. Oper. Res. Soc. China 8, 45–77 (2020). https://doi.org/10.1007/s40305-019-00274-9

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