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An improved multi-criteria emergency decision-making method in environmental disasters

  • Soft computing in decision making and in modeling in economics
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Abstract

Multi-criteria decision-making (MCDM) methods deeply assist decision-makers in evaluating and analyzing input data as accurately as possible in the area of safety and reliability. The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) can be used to rank a set of alternatives based on the different types of criteria. However, this method still needs to be improved; fuzzy set theory and its corresponding extensions are bold in the line of TOPSIS improvement. In this study, the original form of TOPSIS method is extended using Pythagorean fuzzy-based Chebyshev distance measures. Pythagorean fuzzy set has better reflection compared to the conventional fuzzy set theory and Chebyshev distance measures can provide more accurate results, which are used to compute the separation measures in the TOPSIS method. Therefore, the novel improved TOPSIS methods have been developed underlying the ideal of Pythagorean fuzzy-based Chebyshev distance measures. The developed method is applied on a humanitarian case-study to show the efficiency of the approach in an emergency response situation. Results show that the proposed method is efficiently applicable in real-life decision-making situations.

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Fig. 1

Source: Web of Science, keywords search: (Title: TOPSIS AND Topic: Safety, AND Reliability, AND Risk Assessment)

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Source: Web of Science, keywords search: (Title: TOPSIS AND Topic: Safety, AND Reliability, AND Risk Assessment)

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Modified from an example of Engineering Mechanics, Chapter 5, Section 8, written by Meriam et al. (2009)

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Modified from an example of Engineering Mechanics, Chapter 5, Section 8, written by Meriam et al. (2009)

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Modified from an example of Engineering Mechanics, Chapter 5, Section 8, written by Meriam et al. (2009)

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Acknowledgements

This work was partially supported by National Natural Science Foundation of China under the contract number 71761030 and 51965051, Natural Science Foundation of Inner Mongolia under the contract number 2019LH07003.

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Guang-Jun Jiang, Hong-Xia Chen and Hong-Hua Sun contributed to methodology, software, and writing—original draft; ; Mohammad Yazdi contributed to conceptualization, methodology, software, validation, and writing—original draft; Arman Nedjati was involved in writing—original draft and investigation; Kehinde A. Adesina contributed to writing—original draft and investigation.

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Correspondence to Hong-Xia Chen.

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Appendices

Appendix 1

Proof of Theorem 5 (T.4.A)

According to the definition 3 and definition 7, we can simply means that \({\left({\fancyscript{u}}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({\fancyscript{v}}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({\dot{\pi }}_{\fancyscript{p}_{t }}\right)}^{2}\), \({\left({r}_{\fancyscript{p}_{t }}\right)}^{2}\), and \({d}_{\fancyscript{p}_{t}}\) fit into the interval range between one and zero [0,1] for all \(t\). The theorem shows that \({-1\le \left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \(\le {\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \(-1{\le \left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), \({1\le \left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }\le 1\), and \(-1\le {d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\le 1\). Therefore, the property of \({0\le D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)\le 1\) is acetated and T.4.A is valid.

Agree that the T.4.B and T.4.C are valid.□

Proof of Theorem 4 (T.4.D)

In order to meet the property of Separability, \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)=0\) is assumed, and the following circumstances must be fulfilled: \({\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\le \left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2}\), \({\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }={\left({r}_{\fancyscript{p}_{2 }}\right)}^{2}\), and \({d}_{\fancyscript{p}_{1 }}={d}_{\fancyscript{p}_{2}}\). T.4.D proves that, \({\fancyscript{u}}_{\fancyscript{p}_{1 }}={\fancyscript{u}}_{\fancyscript{p}_{2}}\), \({\fancyscript{v}}_{\fancyscript{p}_{1 }}={\fancyscript{v}}_{\fancyscript{p}_{2}}\), and \({r}_{\fancyscript{p}_{1 }}={r}_{\fancyscript{p}_{2}}\). One can say that once \({\fancyscript{p}}_{1 }={\fancyscript{p}}_{2}\), which is requirement of Separability, then \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)=0\). Therefore, T.4.D is viable and valid. □

Proof of Theorem 4 (T.4.E)

The triangular inequality can be represented in the form of different functions. Let use the membership function as an example, i.e. \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\). To prove the property of triangular inequality, four assumptions that must be taken into account: the relationships between three membership functions \({\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{2}}\), and \({\fancyscript{u}}_{\fancyscript{p}_{3}}\) as: (1) \({\fancyscript{u}}_{\fancyscript{p}_{1 }}\le \) \({\fancyscript{u}}_{\fancyscript{p}_{2}}{\le \fancyscript{u}}_{\fancyscript{p}_{3}}\), (2) \({\fancyscript{u}}_{\fancyscript{p}_{1 }}\ge {\fancyscript{u}}_{\fancyscript{p}_{2}}{\ge \fancyscript{u}}_{\fancyscript{p}_{3}}\), (3) \({\fancyscript{u}}_{\fancyscript{p}_{2 }}\le {\mathrm{min}}\{{\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{3 }}\}\), and (4) \({\fancyscript{u}}_{\fancyscript{p}_{2 }}\ge \mathrm{max}\left\{{\fancyscript{u}}_{\fancyscript{p}_{1 }},{\fancyscript{u}}_{\fancyscript{p}_{3 }}\right\}.\) An inference can be concluded based on assumption (1) and (2), \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), that according to the assumption (3), the two conditions as \({\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\ge 0\) and \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\) underlies the ideal of this assumption. In addition, the following results can also be concluded:

$$\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|-\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|=\langle \begin{array}{c}{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 } \mathrm{if}{ \fancyscript{u}}_{\fancyscript{p}_{1 }}\ge {\fancyscript{u}}_{\fancyscript{p}_{3 }} \\ {\left({\fancyscript{u}}_{\fancyscript{p}_{1}}\right)}^{2}-{\left({\fancyscript{u}}_{\fancyscript{p}_{2}}\right)}^{2}+{\left({\fancyscript{u}}_{\fancyscript{p}_{2}}\right)}^{2}-{\left({\fancyscript{u}}_{\fancyscript{p}_{3}}\right)}^{2}+{\left({\fancyscript{u}}_{\fancyscript{p}_{1}}\right)}^{2 }{- \left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 } \mathrm{if}{ \fancyscript{u}}_{\fancyscript{p}_{1 }}\le {\fancyscript{u}}_{\fancyscript{p}_{3 }}\end{array}=2\times \left[{\left({\mathrm{min}}\left\{{ \fancyscript{u}}_{\fancyscript{p}_{1 }},{ \fancyscript{u}}_{\fancyscript{p}_{3 }}\right\}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right]\ge 0.$$

Therefore, \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3}}\right)}^{2}\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\) underlies the ideal of assumption (3). In addition, assumption (4) denotes that \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }\ge 0\) and \({\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\ge 0\) and accordingly the following conclusion can be obtained:

$$\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{{}_{3 }}\right)}^{2 }\right|-\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|=\langle \begin{array}{c}{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 } \mathrm{if}{ \fancyscript{u}}_{\fancyscript{p}_{1 }}\ge {\fancyscript{u}}_{\fancyscript{p}_{3 }} \\ {\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }+{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 } \mathrm{if}{ \fancyscript{u}}_{\fancyscript{p}_{1 }}\le {\fancyscript{u}}_{\fancyscript{p}_{3 }}\end{array}=2\times \left[{{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-\left({\mathrm{min}}\left\{{ \fancyscript{u}}_{\fancyscript{p}_{1 }},{ \fancyscript{u}}_{\fancyscript{p}_{3 }}\right\}\right)}^{2 }\right]\ge 0.$$

Up to this point, by evaluating all possible four assumptions, the triangular inequality, \(\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), is fully satisfied. Analogously, the following relationships between the square difference of the non-membership function, the indeterminacy, and the commitment strength can be achieved: \(\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), \(\left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\), and \(\left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|\le \left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|+\left|{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2}\right|\). In addition, assuming that relationships between the difference of commitment direction exists, it is proposed that \(\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2} }\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3}}\right|\) satisfies \({d}_{\fancyscript{p}_{1 }}\le {d}_{\fancyscript{p}_{2 }}\le {d}_{\fancyscript{p}_{3}}\) and \({d}_{\fancyscript{p}_{1 }}\ge {d}_{\fancyscript{p}_{2 }}\ge {d}_{\fancyscript{p}_{3}}\). According to the aforementioned assumption, \({d}_{\fancyscript{p}_{2 }}\le {\mathrm{min}}\{{ d}_{\fancyscript{p}_{1 }}\,\mathrm{and}\,\,{ d}_{\fancyscript{p}_{3 }}\}\), the following can be concluded that:

$$\begin{aligned}\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|&\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3 }}\right|\\ &=\left\langle \begin{array}{l}{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}+{d}_{\fancyscript{p}_{3 }}-{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{1 }}+{d}_{\fancyscript{p}_{3 }}\,\mathrm{if}\,{ d}_{\fancyscript{p}_{1 }}\ge {d}_{\fancyscript{p}_{3 }} \\ {d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}+{d}_{\fancyscript{p}_{3 }}-{d}_{\fancyscript{p}_{2 }}+{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\,\mathrm{if}\,{ d}_{\fancyscript{p}_{1}}\ge {d}_{\fancyscript{p}_{3}}\end{array},\right.\\ & =2\times \left[{\left({\mathrm{min}}\left\{{d}_{\fancyscript{p}_{1 }},{d}_{3 }\right\}\right)}^{2 }-{d}_{\fancyscript{p}_{2 }}\right]\ge 0\end{aligned}$$
$$\begin{aligned}\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3 }}\right|&=\langle \begin{array}{c}{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{1 }}+{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3 }}-{d}_{\fancyscript{p}_{1 }}+{d}_{\fancyscript{p}_{3 }} \mathrm{if}{ d}_{\fancyscript{p}_{1 }}\ge {d}_{\fancyscript{p}_{3 }} \\ {d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{1 }}+{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3 }}+{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }} \mathrm{if}{ d}_{\fancyscript{p}_{1 }}\ge {d}_{\fancyscript{p}_{3}}\end{array}, \\ &=2\times \left[{d}_{\fancyscript{p}_{2 }}-{\left(\mathrm{max}\left\{{d}_{\fancyscript{p}_{1 }},{d}_{\fancyscript{p}_{3 } }\right\}\right)}^{2 }\right]\ge 0.\end{aligned}$$

Finally, the relationship \(\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\le \left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\right|+\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3}}\right|\) is fulfilled. Therefore, combination of the aforementioned outputs, with consideration of a maximization operator, the following result can be derived:

$$\mathrm{max}\left\{\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{3 }}\right|\right\}\le \mathrm{max}\left\{\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|,\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|,\left|{\left({\dot{\pi }}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|,\left|{\left({r}_{\fancyscript{p}_{1 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }\right|,\left|{d}_{\fancyscript{p}_{1 }}-{d}_{\fancyscript{p}_{2 }}\right|\right\}+\mathrm{max}\left\{\left|{\left({\fancyscript{u}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{u}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({\fancyscript{v}}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\fancyscript{v}}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({\dot{\pi }}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({\dot{\pi }}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{\left({r}_{\fancyscript{p}_{2 }}\right)}^{2 }-{\left({r}_{\fancyscript{p}_{3 }}\right)}^{2 }\right|,\left|{d}_{\fancyscript{p}_{2 }}-{d}_{\fancyscript{p}_{3 }}\right|\right\}.$$

According to the aforementioned above results, the property of triangular inequality \({D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{3 }\right)\le {D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{1 },{\fancyscript{p}}_{2 }\right)+{D}_{\mathrm{ Chebyshev }}^{ 5\text{-terms }}\left({\fancyscript{p}}_{2 },{\fancyscript{p}}_{3}\right)\) is proved. Therefore, T.4.E is viable and valid. □

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Jiang, GJ., Chen, HX., Sun, HH. et al. An improved multi-criteria emergency decision-making method in environmental disasters. Soft Comput 25, 10351–10379 (2021). https://doi.org/10.1007/s00500-021-05826-x

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