Multi-criteria Outranking Methods with Hesitant Probabilistic Fuzzy Sets

Abstract

Due to the defects of hesitant fuzzy sets (HFSs) in the actual decision-making process, it is necessary to add the probabilities corresponding to decision maker’s preferences to the values in HFSs. Hesitant probabilistic fuzzy sets (HPFSs) are suitable for presenting this kind of information and contribute positively to the efficiency of depicting decision maker’s preferences in practice. However, some important issues in HPFSs utilization remain to be addressed. In this paper, the qualitative flexible multiple criteria method (QUALIFLEX) and the preference ranking organization method for enrichment evaluations II (PROMETHEE II) are extended to HPFSs. First, we provide a comparison method for hesitant probabilistic fuzzy elements (HPFEs). Second, we propose a novel possibility degree depicting the relations between two HPFEs, and then, employ the possibility degree to extend the QUALIFLEX and PROMETHEE II methods to hesitant probabilistic fuzzy environments based on the proposed possibility degree. Third, an information integration method is introduced to simplify the processing of HPFE evaluation information. Finally, we provide an example to demonstrate the usefulness of the proposed methods. An illustrative example in conjunction with comparative analyses is employed to demonstrate that our proposed methods are feasible for practical multi-criteria decision-making (MCDM) problems, and the final ranking results show that the proposed methods are more accurate than the compared methods in an actual decision-making processes. HPFSs are more practical than HFSs due to their efficiency in comprehensively representing uncertain, vague, and probabilistic information. The proposed methods are effective for solving hesitant probabilistic MCDM problems and are expected to contribute to the solution of MCDM problems involving uncertain or vague information.

Appendices

Appendix 1

Table 8 Table of acronyms

Appendix 2. Proof of Property 1

Proof

(1) can be easily obtained, and is omitted here.

Since (2), (3), and (4) are similar, we choose (4) to prove. If h(p _x ) < h(p _y ), then according to Definition 9, we obtain ${\gamma}_x^{\sigma (j)}\le {\gamma}_y^{\sigma (j)}$ (j = 1, 2,  ⋯ , l; l = min {m, n}), where at least one of the “≤” relations should be “<_.” In the following, we will show the proof in two different conditions:

• Case 1: m = n

$$ \begin{array}{c}\hfill \varphi \left( h\left({p}_x\right), h\left({p}_y\right)\right)=\sum_{i=1}^m\sum_{j=1}^n\frac{\gamma_{i x}}{\gamma_{i x}+{\gamma}_{j y}}{p}_{i x}{p}_{j y}=\sum_{i=1}^m\sum_{j=1}^m\frac{\gamma_{i x}}{\gamma_{i x}+{\gamma}_{j y}}{p}_{i x}{p}_{j y}\hfill \\ {}\hfill ={\sum}_{j=1}^m\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (j)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (j)}+\cdots +{\sum}_{j=1}^m\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (j)}}{p}_x^{\sigma (m)}{p}_y^{\sigma (j)}\hfill \\ {}\hfill ={\sum}_{i=1}^m\frac{\gamma_x^{\sigma (i)}}{\gamma_x^{\sigma (i)}+{\gamma}_y^{\sigma (i)}}{p}_x^{\sigma (i)}{p}_y^{\sigma (i)}+\left(\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (2)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (2)}+\frac{\gamma_x^{\sigma (2)}}{\gamma_x^{\sigma (2)}+{\gamma}_y^{\sigma (1)}}{p}_x^{\sigma (2)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (m)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma (1)}}{p}_x^{\sigma (m)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{\gamma_x^{\sigma \left( m-1\right)}}{\gamma_x^{\sigma \left( m-1\right)}+{\gamma}_y^{\sigma (m)}}{p}_x^{\sigma \left( m-1\right)}{p}_y^{\sigma (m)}+\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma \left( m-1\right)}}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m-1\right)}\right)\hfill \\ {}\hfill <\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (1)}+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma (m)}\right)+\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (2)}+\frac{1}{2}{p}_x^{\sigma (2)}{p}_y^{\sigma (1)}\right)+\cdots +\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{1}{2}{p}_x^{\sigma \left( m-1\right)}{p}_y^{\sigma (m)}+\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m-1\right)}\right)\hfill \\ {}\hfill =\frac{1}{2}\left({p}_x^{\sigma (1)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}\right)+\frac{1}{2}\left({p}_x^{\sigma (2)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (2)}{p}_y^{\sigma (m)}\right)+\cdots +\frac{1}{2}\left({p}_x^{\sigma (m)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (m)}{p}_y^{\sigma (m)}\right)\hfill \\ {}\hfill =\frac{1}{2}{p}_x^{\sigma (1)}\left({p}_y^{\sigma (1)}+\cdots +{p}_y^{\sigma (m)}\right)+\frac{1}{2}{p}_x^{\sigma (2)}\left({p}_y^{\sigma (1)}+\cdots +{p}_y^{\sigma (m)}\right)+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}\left({p}_y^{\sigma (1)}+\cdots +{p}_y^{\sigma (m)}\right)\hfill \\ {}\hfill =\frac{1}{2}{p}_x^{\sigma (1)}+\frac{1}{2}{p}_x^{\sigma (2)}+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}=\frac{1}{2}.\hfill \end{array} $$

• Case 2: m < n

$$ \begin{array}{c}\hfill \varphi \left( h\left({p}_x\right), h\left({p}_y\right)\right)=\sum_{i=1}^m\sum_{j=1}^n\frac{\gamma_{i x}}{\gamma_{i x}+{\gamma}_{j y}}{p}_{i x}{p}_{j y}\hfill \\ {}\hfill ={\sum}_{j=1}^n\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (j)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (j)}+\cdots +{\sum}_{j=1}^n\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (j)}}{p}_x^{\sigma (m)}{p}_y^{\sigma (j)}\hfill \\ {}\hfill ={\sum}_{i=1}^m\frac{\gamma_x^{\sigma (i)}}{\gamma_x^{\sigma (i)}+{\gamma}_y^{\sigma (i)}}{p}_x^{\sigma (i)}{p}_y^{\sigma (i)}+\left(\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (2)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (2)}+\frac{\gamma_x^{\sigma (2)}}{\gamma_x^{\sigma (2)}+{\gamma}_y^{\sigma (1)}}{p}_x^{\sigma (2)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (m)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma (1)}}{p}_x^{\sigma (m)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{\gamma_x^{\sigma \left( m-1\right)}}{\gamma_x^{\sigma \left( m-1\right)}+{\gamma}_y^{\sigma (m)}}{p}_x^{\sigma \left( m-1\right)}{p}_y^{\sigma (m)}+\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma \left( m-1\right)}}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m-1\right)}\right)\hfill \\ {}\hfill +\left(\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma \left( m+1\right)}}{p}_x^{\sigma (1)}{p}_y^{\sigma \left( m+1\right)}+\cdots +\frac{\gamma_x^{\sigma (1)}}{\gamma_x^{\sigma (1)}+{\gamma}_y^{\sigma (n)}}{p}_x^{\sigma (1)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma \left( m+1\right)}}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m+1\right)}+\cdots +\frac{\gamma_x^{\sigma (m)}}{\gamma_x^{\sigma (m)}+{\gamma}_y^{\sigma (n)}}{p}_x^{\sigma (m)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill <\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (1)}+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma (m)}\right)+\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (2)}+\frac{1}{2}{p}_x^{\sigma (2)}{p}_y^{\sigma (1)}\right)+\cdots +\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma (1)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{1}{2}{p}_x^{\sigma \left( m-1\right)}{p}_y^{\sigma (m)}+\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m-1\right)}\right)+\left(\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma \left( m+1\right)}+\cdots +\frac{1}{2}{p}_x^{\sigma (1)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill +\cdots +\left(\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma \left( m+1\right)}+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill =\frac{1}{2}\left({p}_x^{\sigma (1)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\cdots +{p}_x^{\sigma (1)}{p}_y^{\sigma (n)}\right)+\frac{1}{2}\left({p}_x^{\sigma (2)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (1)}{p}_y^{\sigma (m)}+\cdots +{p}_x^{\sigma (1)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill +\cdots +\frac{1}{2}\left({p}_x^{\sigma (m)}{p}_y^{\sigma (1)}+\cdots +{p}_x^{\sigma (m)}{p}_y^{\sigma (m)}+\cdots +{p}_x^{\sigma (m)}{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill =\frac{1}{2}{p}_x^{\sigma (1)}\left({p}_y^{\sigma (1)}+\cdots +{p}_y^{\sigma (n)}\right)+\cdots +\frac{1}{2}{p}_x^{\sigma (m)}\left({p}_y^{\sigma (1)}+\cdots +{p}_y^{\sigma (n)}\right)\hfill \\ {}\hfill =\frac{1}{2}{p}_x^{\sigma (1)}+\cdots +\frac{1}{2}{p}_x^{\sigma (n)}=\frac{1}{2}.\hfill \end{array} $$

A similar proof can be obtained when m > n.

The details of proofs (5) and (6) are omitted to save space.

This completes the proof.