A PROMETHEE II Approach Based on Probabilistic Hesitant Fuzzy Linguistic Information with Applications to Multi-Criteria Group Decision-Making (ICSSE 2020)

Abstract

Multi-criteria group decision-making (MCGDM) problems are very common in the real world. The complexity of the problem necessitates a solution method that is more in line with the decision-making habits of decision-makers (DMs). This paper introduces a novel type of integrated linguistic information, namely, the Probabilistic Hesitant Fuzzy Linguistic Sets (PHFLSs), which combines the concepts of Hesitant Fuzzy Linguistic Sets and Probabilistic Linguistic Term Sets for solving MCGDM problems. By integrating PHFLSs into the Preference Ranking Organization Method for Enrichment Evaluations II (PROMETHEE II), a group decision-making framework is constructed to effectively generate the best decision given various evaluations from multiple DMs. More specifically, the DMs’ assessments are first developed and normalized based on PHFLSs definitions. Then the Hausdorff distance is employed to compute the distances between different PHFLSs, from which the weights of criteria are derived and then fed into PROMETHEE II for the best group decision. To demonstrate the practicality and capability of the proposed decision framework, a case study on seeking the best open-source software project is presented and discussed.

Appendix

The proof of the basic operations are as follows.

For PHFLS ${\tilde{A}}_1(p)$, $k=1,2,\ldots ,u$;

For PHFLS ${\tilde{A}}_2(p)$, $k=1,2,\ldots ,v$.

• 1.
  $$\begin{aligned}&{\tilde{A}}_1(p)\oplus {\tilde{A}}_2(p) \\&\quad =\langle \{[A_1^{(1)}(p_1^{(1)})+ A_2^{(1)}(p_2^{(1)})+A_1^{(1)}(p_1^{(1)})\\&\qquad + A_2^{(2)}(p_2^{(2)})+\cdots +A_1^{(1)}(p_1^{(1)})\\&\qquad + A_2^{(v)}(p_2^{(v)})]+[A_1^{(2)}(p_1^{(2)})+ A_2^{(1)}(p_2^{(1)})\\&\qquad +A_1^{(2)}(p_1^{(2)})+ A_2^{(2)}(p_2^{(2)})+\cdots +A_1^{(2)}(p_1^{(2)})+ A_2^{(v)}(p_2^{(v)})]+\cdots \\&\qquad +[A_1^{(u)}(p_1^{(u)})+ A_2^{(1)}(p_2^{(1)})+A_1^{(u)}(p_1^{(u)})+A_2^{(2)}(p_2^{(2)})\\&\qquad +\cdots +A_1^{(u)}(p_1^{(u)})+ A_2^{(v)}(p_2^{(v)})]\},\\&\qquad \{[(\gamma_1^{(1)}+\gamma_2^{(1)}-\gamma_1^{(1)}\gamma_2^{(1)})+(\gamma_1^{(1)}+\gamma_2^{(2)}-\gamma_1^{(1)}\gamma_2^{(2)})\\&\qquad +\cdots +(\gamma_1^{(1)}+\gamma_2^{(v)}- \gamma_1^{(1)}\gamma_2^{(v)})],\\&\qquad [(\gamma_1^{(2)}+\gamma_2^{(1)}-\gamma_1^{(2)}\gamma_2^{(1)})+(\gamma_1^{(2)}+\gamma_2^{(2)}-\gamma_1^{(2)}\gamma_2^{(2)})+\cdots \\&\qquad +(\gamma_1^{(2)}+\gamma_2^{(v)}-\gamma_1^{(2)}\gamma_2^{(v)})],\ldots ,[(\gamma_1^{(u)}+\gamma_2^{(1)}\\&\qquad -\gamma_1^{(u)}\gamma_2^{(1)})+(\gamma_1^{(u)}+\gamma_2^{(2)}-\gamma_1^{(u)}\gamma_2^{(2)})+\cdots \\&\qquad +(\gamma_1^{(u)}+\gamma_2^{(v)}-\gamma_1^{(u)}\gamma_2^{(v)})]\}\rangle \\ &= \left \langle \bigcup_{A_1^{(k)}\in {\tilde{A}}_1(p),A_2^{(k)}\in {\tilde{A}}_2(p)}(A_1^{(k)}p_1^{(k)}+ A_2^{(k)}p_2^{(k)}), \right.\\&\left.\qquad \bigcup_{\gamma_1\in h_1,\gamma_2 \in h_2}\{\gamma_1+\gamma_2-\gamma_1\gamma_2\}\right.\rangle. \end{aligned}$$
• 2.
  $$\begin{aligned}&{\tilde{A}}_1(p)\otimes {\tilde{A}}_2(p)\\&\quad =\langle \{[{A_1^{(1)}}^{p_1^{(1)}}\cdot {A_2^{(1)}}^{p_2^{(1)}}+{A_1^{(1)}}^{p_1^{(1)}}\cdot {A_2^{(2)}}^{p_2^{(2)}}+\cdots \\&\qquad +{A_1^{(1)}}^{p_1^{(1)}}\cdot {A_2^{(v)}}^{p_2^{(v)}}]+[{A_1^{(2)}}^{p_1^{(2)}}\cdot {A_2^{(1)}}^{p_2^{(1)}}\\&\qquad +{A_1^{(2)}}^{p_1^{(2)}}\cdot {A_2^{(2)}}^{p_2^{(2)}} + \cdots \\&\qquad +{A_1^{(2)}}^{p_1^{(2)}}\cdot {A_2^{(v)}}^{p_2^{(v)}}]+\cdots \\&\qquad +[{A_1^{(u)}}^{p_1^{(u)}}\cdot {A_2^{(1)}}^{p_2^{(1)}}+{A_1^{(u)}}^{p_1^{(u)}}\cdot {A_2^{(2)}}^{p_2^{(2)}}+\cdots \\&\qquad +{A_1^{(u)}}^{p_1^{(1)}}\cdot {A_2^{(v)}}^{p_2^{(v)}}]\},\\&\qquad \{[(\gamma_1^{(1)}\gamma_2^{(1)})+(\gamma_1^{(1)}\gamma_2^{(2)})+\cdots +(\gamma_1^{(1)}\gamma_2^{(v)})],\\&\qquad [(\gamma_1^{(2)}\gamma_2^{(1)})+(\gamma_1^{(2)}\gamma_2^{(2)})+\cdots +(\gamma_1^{(2)}\gamma_2^{(v)})],\ldots ,\\&\qquad [(\gamma_1^{(u)}\gamma_2^{(1)})+(\gamma_1^{(u)}\gamma_2^{(2)})\\&\qquad +\cdots +(\gamma_1^{(u)}\gamma_2^{(v)})]\}\rangle \\&\quad = \left\langle \bigcup_{A_1^{(k)}\in {\tilde{A}}_1(p),A_2^{(k)}\in {\tilde{A}}_2(p)}({A_1^{(k)}}^{p_1^{(k)}}\cdot {A_2^{(k)}}^{p_2^{(k)}}),\bigcup_{\gamma_1\in h_1,\gamma_2 \in h_2}\{\gamma_1\gamma_2\}\right\rangle. \end{aligned}$$
• 3.
  $$\begin{aligned}&{\tilde{A}}(p)^\lambda =\langle \{({A^{(1)}}^{\lambda p^{(1)}})+({A^{(2)}}^{\lambda p^{(2)}})+\cdots \\&\qquad +({A^{(u)}}^{\lambda p^{(u)}})\},\{(\gamma ^{(1)})^\lambda +(\gamma ^{(2)})^\lambda +\cdots \\&\qquad +(\gamma ^{(u)})^\lambda \}\rangle \\&\quad =\langle \bigcup_{A^{(k)}\in {\tilde{A}}(p)}({A^{(k)}}^{\lambda p^{(k)}}),\bigcup_{\gamma \in h}\{\gamma ^\lambda \}\rangle , \quad \lambda >0. \end{aligned}$$
• 4.
  $$\begin{aligned}&\lambda {\tilde{A}}(p)=\langle \{\lambda p^{(1)}A^{(1)}+\lambda p^{(2)}A^{(2)}+\cdots \\&\qquad +\lambda p^{(u)}A^{(u)}\},\{[1-(1-\gamma ^{(1)})^\lambda ]+[1-(1-\gamma ^{(2)})^\lambda ]+\cdots \\&\qquad +[1-(1-\gamma ^{(u)})^\lambda ]\}\rangle \\&\quad = \langle \bigcup_{A^{(k)}\in {\tilde{A}}(p)}(\lambda p^{(k)}{A^{(k)}}),\bigcup_{\gamma \in h}\{1-(1-\gamma )^\lambda \}\rangle , \,\,\,\, \lambda >0. \end{aligned}$$

Table 17 Directed Hausdorff distance for criterion 1

Table 18 Directed Hausdorff distance for criterion 2

Table 19 Directed Hausdorff distance for criterion 3

Table 20 Directed Hausdorff distance for criterion 4

Table 21 Weights $w_j$ for criteria $C_j$

Table 22 Multi-criteria preference index $\varPi (A_i,A_k)$

Table 23 Positive, negative flows