Multi-criteria PROMETHEE method based on possibility degree with Z-numbers under uncertain linguistic environment

Abstract

People make decisions based on their cognitive information about the objective world. Zadeh’s Z-number allows people to better express their cognition of the real world by considering the fuzzy restriction and reliability restriction of information. However, the Z-number is a complex construct, and some important issues must be discussed in its study. Here, a computationally simple method of ranking Z-numbers for multi-criteria decision-making (MCDM) problems is proposed, and a comprehensive possibility degree of Z-numbers is defined, as inspired by the possibility degree concept of interval numbers. The outranking relations of Z-numbers are also discussed on the basis of the proposed method. Then, a weight acquisition algorithm relative to the possibility degree of Z-numbers is presented. Finally, an extended Preference Ranking Organization Method for Enrichment Evaluation II (PROMETHEE II) based on the possibility degree of Z-numbers is developed for the MCDM problem under Z-evaluation, and a numerical example about the selection of travel plans is used to illustrate the validity of the proposed method. The applicability and superiority of the proposed method is demonstrated through sensitivity and comparative analyses along with other existing methods.

Appendices

Appendix A. Special computation of the possibility degree of triangular fuzzy numbers

Let $\tilde {a}=({a_1},{a_2},{a_3})$ and $\tilde {b}=({b_1},{b_2},{b_3})$ be any two TFNs. The possibility degree of $\tilde {a} \geq \tilde {b}$ can be computed as follows:

If ${a_1}={a_2}$, ${a_2}={a_3}$, ${b_1}={b_2}$ and ${b_2}={b_3}$, then

$$p(\tilde {a} \geq \tilde {b})=\left\{ {\begin{array}{*{20}{c}} 0&{{a_2}<{b_2}} \\ {0.5}&{{a_2}={b_2}} \\ 1&{{a_2}>{b_2}} \end{array}} \right.$$

Otherwise,

$p(\tilde {a} \geq \tilde {b})=\left\{ {\begin{array}{*{20}{c}} 0&{{a_2}<{b_2}\,{\text{and}}\,{a_3} \leq {b_1}} \\ {\frac{{{a_3} - {b_1}}}{{({b_3} - {b_1})+({a_3} - {a_1})}}+\frac{{{b_2} - {a_2}}}{{({b_3} - {b_1})+({a_3} - {a_1})}}\ln \left( {\frac{{{b_2} - {a_2}}}{{({b_2} - {b_1})+({a_3} - {a_1})}}} \right)}&{{a_2}<{b_2}\,{\text{and}}\,{a_3}>{b_1}} \\ {\frac{{{a_3} - {b_1}}}{{({b_3} - {b_1})+({a_3} - {a_1})}}}&{{a_2}={b_2}} \\ {\frac{{{a_3} - {b_1}}}{{({b_3} - {b_1})+({a_3} - {a_1})}}+\frac{{{a_2} - {b_2}}}{{({b_3} - {b_1})+({a_3} - {a_1})}}\ln \left( {\frac{{{a_2} - {b_2}}}{{({a_2} - {a_1})+({b_3} - {b_2})}}} \right)}&{{a_2}>{b_2}\,{\text{and}}\,{b_3}>{a_1}} \\ 1&{{a_2}<{b_2}\,{\text{and}}\,{b_3}>{a_1}} \end{array}} \right.$

Appendix B. Proof for the conclusion in Remark 1.

As shown in Fig. 2, two TFNs, denoted by ${A_i}$ and ($i<j$), exist.

If ${A_i}$ and ${A_j}$ are non-intersecting (e.g. ${A_3}$ and ${A_4}$), then ${p^\alpha }({A_i} \geq {A_j})=0,\,\forall \alpha \in [0,1]$ Consequently, $p({A_i} \geq {A_j})<0.5$ is satisfied according to Definition 9.

If ${A_i}$ and ${A_j}$ are partially intersecting (e.g. ${A_1}$ and ${A_2}$), whose cut sets under level $\alpha$ are $\left[ {A_{{i\alpha }}^{ - },A_{{i\alpha }}^{+}} \right]$and $\left[ {A_{{j\alpha }}^{ - },A_{{j\alpha }}^{+}} \right]$, then ${p^\alpha }({A_i} \geq {A_j})=\left\{ {\begin{array}{*{20}{l}} 0&{A_{{i\alpha }}^{+} - A_{{j\alpha }}^{ - } \leq 0} \\ {\frac{{A_{{i\alpha }}^{+} - A_{{j\alpha }}^{ - }}}{{(A_{{j\alpha }}^{+} - A_{{j\alpha }}^{ - })+(A_{{i\alpha }}^{+} - A_{{i\alpha }}^{ - })}}}&{A_{{i\alpha }}^{+} - A_{{j\alpha }}^{ - }>0} \end{array}} \right.$. Furthermore, the following is obtained:

$\frac{{A_{{i\alpha }}^{+} - A_{{i\alpha }}^{ - }}}{{(A_{{j\alpha }}^{+} - A_{{j\alpha }}^{ - })+(A_{{i\alpha }}^{+}+A_{{i\alpha }}^{ - })}} \leq \frac{{A_{{i\alpha }}^{+} - A_{{j\alpha }}^{ - }+\left[ {\frac{{A_{{j\alpha }}^{+}+A_{{j\alpha }}^{ - }}}{2} - \frac{{A_{{i\alpha }}^{+}+A_{{i\alpha }}^{ - }}}{2}} \right]}}{{(A_{{j\alpha }}^{+} - A_{{j\alpha }}^{+})+(A_{{i\alpha }}^{+} - A_{{i\alpha }}^{ - })}}=\frac{{\frac{1}{2}\left[ {(A_{{j\alpha }}^{+} - A_{{j\alpha }}^{ - })+(A_{{i\alpha }}^{+} - A_{{i\alpha }}^{ - })} \right]}}{{(A_{{j\alpha }}^{+} - A_{{j\alpha }}^{ - })+(A_{{i\alpha }}^{+} - A_{{i\alpha }}^{ - })}}=0.5.$Consequently, ${p^\alpha }({A_i} \geq {A_j}) \leq 0.5,\forall \alpha \in \left[ {0,1} \right]$ is always true. Therefore, $p({A_i} \geq {A_j})<0.5$ is satisfied according to Definition 9.

On the basis of the above definitions, the relevant conclusion can be obtained.