Algorithms for multiple attribute decision making with dual hesitant Pythagorean fuzzy information and their application to supplier selection

Abstract

In this paper, we investigate the multiple attribute decision making (MADM) problem based on the Hamacher aggregation operators and Choquet integral with dual Pythagorean hesitant fuzzy information. Then, motivated by the ideal of Hamacher operation and Choquet integral, we have developed some Hamacher correlated operators for aggregating dual hesitant Pythagorean fuzzy information. The prominent characteristic of these proposed operators is studied. Then, we have utilized these two operators to develop some approaches to solve the dual hesitant Pythagorean fuzzy MADM problems. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Keywords

Multiple attribute decision making (MADM)dual Pythagorean hesitant fuzzy numbers dual hesitant Pythagorean fuzzy Hamacher correlated average (DHPFHCA) operator dual hesitant Pythagorean fuzzy Hamacher correlated geometric (DHPFHCG) operator supplier selection supply chain management

1. Introduction

In practical decisionmaking environment, it’s difficult for decision makers (DMs) to give evaluate information with exact real numbers [1, 2, 3, 4]. To overcome this disadvantage, Zadeh [5] developed the fuzzy set (FS) theory which utilized the function of membership degree to express decision making information instead of crisp results between 0 and 1. Based on the studied of FS, Atanassov [6] introduced the concept of intuitionistic fuzzy set (IFS) characterized by a membership function and a non-membership function, which is a generalization of the concept of fuzzy set whose basic component is only a membership function. After that, more and more scholars have studied about the IFS in many multiple attribute decision making (MADM) problems [7, 8, 9, 10, 11, 12, 13, 13]. Xu [14] defined some intuitionistic fuzzy weighted average operators. Xu and Yager [15] proposed some intuitionistic fuzzy geometric mean operators. Wu et al. [16] defined the VIKOR method for financing risk assessment of rural tourism projects under interval-valued intuitionistic fuzzy environment. Wu et al. [17] gave some models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Lu and Wei [11] defined the TODIM method for performance appraisal on social-integration-based rural reconstruction with interval-valued intuitionistic fuzzy information. Besides, as an effective MADM tool, Pythagorean fuzzy set (PFS) [18, 19] has emerged to describe the indeterminacy and complexity of the evaluation information. Similar to IFS, The PFS is also consisted of the membership degree and non-membership degree, and the sum of squares of them is restricted to 1. Zhang and Xu [20] defined the Pythagorean fuzzy TOPSIS model to solve the MADM issues. Peng and Yang [21] primarily proposed two Pythagorean fuzzy operations including the division and subtraction operations to better understand PFSs. Reformat and Yager [22] handled the collaborative-based recommender system with PFSs. Garg [23] defined the Pythagorean fuzzy Einstein weighted averaging (PFEWA) operator, Pythagorean fuzzy Einstein ordered weighted averaging (PFEOWA) operator, generalized Pythagorean fuzzy Einstein weighted averaging (GPFEWA) operator and generalized Pythagorean fuzzy Einstein ordered weighted averaging (GPFEOWA) operator. Zeng et al. [24] used the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator to cope with Pythagorean fuzzy MADM issues. Wei [25] defined some Pythagorean fuzzy interaction aggregation operators. Wei and Wei [26] defined ten cosine similarity measures under PFSs. Liang et al. [27] presented Pythagorean fuzzy Bonferroni mean aggregation operators based on geometric averaging (GA) operations. Wu and Wei [28] proposed some Pythagorean fuzzy Hamacher aggregation operators to fuse Pythagorean fuzzy information. Gou et al. [29] studied some precious properties of continuous Pythagorean fuzzy information. Ren et al. [30] defined the Pythagorean fuzzy TODIM model. Wei and Lu [31] proposed some Pythagorean fuzzy power aggregation operators. Wei and Lu [32] developed some new Maclaurin symmetric mean (MSM) [33] operator based on PFSs. Liang et al. [34] investigated some Bonferroni mean operator with PFSs Combined the PFSs [18, 19] and dual hesitant fuzzy sets (DHFSs) [35, 36], Wei and Lu [37] introduced the definition of the dual hesitant Pythagorean fuzzy sets (DHPFSs) and proposed some dual hesitant Pythagorean fuzzy Hamacher aggregation operators. Obviously, the DHPFSs have the advantages of considering the hesitance of DMs and expressing fuzzy information more effectively and reasonably. Tang et al. [38] proposed some dual hesitant Pythagorean fuzzy Heronian mean aggregation operators such as: the dual hesitant Pythagorean fuzzy generalized weighted Heronian mean (DHPFGWHM) operator and the dual hesitant Pythagorean fuzzy generalized geometric weighted Heronian mean (DHPFGGWHM) operator. Wei et al. [39] defined some dual hesitant Pythagorean fuzzy Hamy mean aggregation operators such as: the dual hesitant Pythagorean fuzzy Hamy mean (DHPFHM) operator, the dual hesitant Pythagorean fuzzy weighted Hamy mean (DHPFWHM) operator, the dual hesitant Pythagorean fuzzy dual Hamy mean (DHPFDHM) operator and the dual hesitant Pythagorean fuzzy weighted dual Hamy mean (DHPFWDHM) operator. Tang and Wei [40] designed the dual hesitant Pythagorean fuzzy Bonferroni mean (DHPFBM) operator dual hesitant Pythagorean fuzzy geometric Bonferroni mean (DHPFGBM) operator, dual hesitant Pythagorean fuzzy generalized Bonferroni mean (DHPFGBM) operator dual hesitant Pythagorean fuzzy generalized geometric Bonferroni mean (DHPFGGBM) operator, dual hesitant Pythagorean fuzzy dual Bonferroni mean (DHPFDBM) operator and dual hesitant Pythagorean fuzzy dual geometric Bonferroni mean (DHPFDGBM) operator. Tang and Wei [41] built proposed the dual hesitant Pythagorean fuzzy weighted Bonferroni mean (DHPFWBM) operator, dual hesitant Pythagorean fuzzy weighted geometric Bonferroni mean(DHPFWGBM) operator, dual hesitant Pythagorean fuzzy generalized weighted Bonferroni mean (DHPFGWBM) operator, dual hesitant Pythagorean fuzzy generalized weighted geometric Bonferroni mean (DHPFGWGBM) operator, dual hesitant Pythagorean fuzzy dual weighted Bonferroni mean (DHPFDWBM) operator and dual hesitant Pythagorean fuzzy dual weighted geometric Bonferroni mean (DHPFDWGBM) operator. Lu et al. [42] proposed a bidirectional project method of DHPFS to handle the MADM problem under the dual hesitant Pythagorean fuzzy environment.

However, all the above approaches are unsuitable to aggregate these dual hesitant Pythagorean fuzzy numbers on the basis of the Hamacher operations [43] and Choquet integral [44]. Thus, based on the Hamacher operations and Choquet integral, how to aggregate these dual hesitant Pythagorean fuzzy numbers is an interesting topic. To solve this issue, in this paper, we shall develop some dual hesitant Pythagorean fuzzy Hamacher aggregation operators on the basis of the traditional Hamacher operations and Choquet integral. In order to do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to PFSs, DHPFSs and their operational laws. In Section 3, we shall propose some dual hesitant Pythagorean fuzzy Hamacher correlated aggregation operators. In Section 4, based on these two operators, we shall propose some models for MADM problems with DHPFSs. In Section 5, we present a numerical example for supplier selection in supply chain management with DHPFSs in order to illustrate the method proposed in this paper. Section 6 concludes the paper with some remarks.

2. Preliminaries

2.1 Pythagorean fuzzy set

The basic concepts of PFSs [29, 30] are briefly reviewed in this section. Afterwards, novel score and accuracy functions for PFNs are proposed. Furthermore, a new comparison method for PFNs is developed.

Definition 1 [29, 30]. Let $X$ be a fix set. A PFS is an object having the form

$\displaystyle P=\{{\langle{x,({\mu_{P}(x),\nu_{P}(x)})}\rangle|{x\in X}.}\}$ (1)

where the function $\mu_{P}:X\to[{0,1}]$ defines the degree of membership and the function $\nu_{P}:X\to[{0,1}]$ defines the degree of nonmembership of the element $x\in X$ to $P$ , respectively, and, for every $x\in X$ , it holds that

$\displaystyle({\mu_{p}(x)})^{2}+({\nu_{p}(x)})^{2}\leqslant 1.$ (2)

Definition 2 [31]. Let $\tilde{a}_{1}=({\mu_{1},\nu_{1}})$ , $\tilde{a}_{2}=({\mu_{2},\nu_{2}})$ , and $\tilde{a}=({\mu,\nu})$ be three PFNs and some basic operations on them are defined as follows:

(1)

$\tilde{a}_{1}\oplus\tilde{a}_{2}=({\sqrt{({\mu_{1}})^{2}+({\mu_{2}})^{2}-({\mu% _{1}})^{2}({\mu_{2}})^{2}},\nu_{1}\nu_{2}})$ ;

(2)

$\tilde{a}_{1}\otimes\tilde{a}_{2}=({\mu_{1}\mu_{2},\sqrt{({\nu_{1}})^{2}+({\nu% _{2}})^{2}-({\nu_{1}})^{2}({\nu_{2}})^{2}}})$ ;

(3)

$\lambda\tilde{a}=({\sqrt{1-({1-\mu^{2}})^{\lambda}},\nu^{\lambda}}),\lambda>0$ ;

(4)

$(\tilde{a})^{\lambda}=({\mu^{\lambda},\sqrt{1-({1-\nu^{2}})^{\lambda}}}),% \lambda>0$ ;

(5)

$\tilde{a}^{c}=({\nu,\mu})$ .

2.2 Dual hesitant Pythagorean fuzzy set

Combined the PFSs [18, 19] and dual hesitant fuzzy sets (DHFSs) [35, 36], Wei and Lu [37] introduced the definition of the dual hesitant Pythagorean fuzzy sets (DHPFSs).

Definition 3 [37]. Let $X$ be a fixed set, then a dual hesitant Pythagorean fuzzy set (DHPFS) on $X$ is described as:

$\displaystyle D=({\langle{x,h_{P}(x),g_{P}(x)}\rangle|{x\in X}})$ (3)

in which $h_{P}(x)$ and $g_{P}(x)$ are two sets of some values in $[{0,1}]$ , denoting the possible membership degrees and nonmembership degrees of the element $x\in X$ to the set $D$ respectively, with the conditions:

$\displaystyle\gamma^{2}+\eta^{2}\leqslant 1$

where $\gamma\in h_{P}(x),\eta\in g_{P}(x)$ for all $x\in X$ . For convenience, the pair $d(x)=({h_{P}(x),g_{P}(x)})$ is called a dual hesitant Pythagorean fuzzy number (DHPFN) denoted by $d=({h,g})$ , with the conditions: $\gamma\in h,\eta\in g$ , $0\leqslant\gamma,\eta\leqslant 1,0\leqslant\gamma^{2}+\eta^{2}\leqslant 1$ .

To compare the DHPFNs, in the following, Wei and Lu [37] gave the following comparison laws:

Definition 4 [37]. Let $d=({h,g})$ be a DHPFNs, $s(d)=\frac{1}{2}({1+\frac{1}{\#h}\sum\nolimits_{\gamma\in h}{\gamma^{2}}-\frac% {1}{\#g}\sum\nolimits_{\eta\in g}{\eta^{2}}})$ the score function of $d$ , and $p(d)=\frac{1}{\#h}\sum\nolimits_{\gamma\in h}{\gamma^{2}}+\frac{1}{\#g}\sum% \nolimits_{\eta\in g}{\eta^{2}}$ the accuracy function of $d$ , where $\#h$ and $\#g$ are the numbers of the elements in $h$ and $g$ respectively, then, Let $d_{i}=({h_{i},g_{i}})({i=1,2})$ be any two DHPFNs, we have the following comparison laws:

•

If $s({d_{1}})>s({d_{2}})$ , then $d_{1}$ is superior to $d_{2}$ , denoted by $d_{1}\succ d_{2}$ ;

•

If $s({d_{1}})=s({d_{2}})$ , then

(1)

If $p({d_{1}})=p({d_{2}})$ , then $d_{1}$ is equivalent to $d_{2}$ , denoted by $d_{1}\sim d_{2}$ ;

(2)

If $p({d_{1}})>p({d_{2}})$ , then $d_{1}$ is superior to $d_{2}$ , denoted by $d_{1}\succ d_{2}$ .

Then, Wei and Lu [37] defined some new operations on the DHPFNs $d$ , $d_{1}$ and $d_{2}$ :

(1)

$d^{\lambda}=\cup_{\gamma\in h,\eta\in g}\{{\{{\gamma^{\lambda}}\},\{{\sqrt{1-(% {1-\eta^{2}})^{\lambda}}}\}}\},\lambda>0$ ;

(2)

$\lambda d=\cup_{\gamma\in h,\eta\in g}\{{\{{\sqrt{1-({1-\gamma^{2}})^{\lambda}% }}\},\{{\eta^{\lambda}}\}}\},\lambda>0$ ;

(3)

$d_{1}\oplus d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}\in h_{2},\eta_{1}\in g_% {1},\eta_{2}\in g_{2}}\{{\{{\sqrt{({\gamma_{1}})^{2}+({\gamma_{2}})^{2}-({% \gamma_{1}})^{2}({\gamma_{2}})^{2}}}\},\{{\eta_{1}\eta_{2}}\}}\}$ ;

(4)

$d_{1}\otimes d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}\in h_{2},\eta_{1}\in g% _{1},\eta_{2}\in g_{2}}\{{\{{\gamma_{1}\gamma_{2}}\},\{{\sqrt{({\eta_{1}})^{2}% +({\eta_{2}})^{2}-({\eta_{1}})^{2}({\eta_{2}})^{2}}}\}}\}$ .

2.3 Hamacher operations of dual hesitant Pythagorean fuzzy set

Based on the traditional Hamacher operations [43], in the following, Wei and Lu [37] defined the Hamacher operations on the DHPFNs $d$ , $d_{1}$ and $d_{2}$ .

$\displaystyle(1)\quad d^{\lambda}=\cup_{\gamma\in h,\eta\in g}\left\{\left\{{% \frac{\sqrt{\gamma}({\gamma_{1}})^{\lambda}}{\sqrt{({1+({\gamma-1})({1-({% \gamma_{1}})^{2}})})^{\lambda}+({\gamma-1})({\gamma_{1}})^{2\lambda}}}}\right% \},\right.$ $\displaystyle\quad\quad\left.{\left\{{\sqrt{\frac{({1+({\gamma-1})({\eta_{1}})% ^{2}})^{\lambda}-({1-({\eta_{1}})^{2}})^{\lambda}}{({1+({\gamma-1})({\eta_{1}}% )^{2}})^{\lambda}+({\gamma-1})({1-({\eta_{1}})^{2}})^{\lambda}}}}\right\}}% \right\},\lambda>0;$ $\displaystyle(2)\quad\lambda d=\cup_{\gamma\in h,\eta\in g}\left\{\left\{{% \sqrt{\frac{({1+({\gamma-1})({\gamma_{1}})^{2}})^{\lambda}-({1-({\gamma_{1}})^% {2}})^{\lambda}}{({1+({\gamma-1})({\gamma_{1}})^{2}})^{\lambda}+({\gamma-1})({% 1-({\gamma_{1}})^{2}})^{\lambda}}}}\right\},\right.$ $\displaystyle\quad\quad\left.{\left\{{\frac{\sqrt{\gamma}({\eta_{1}})^{\lambda% }}{\sqrt{({1+({\gamma-1})({1-({\eta_{1}})^{2}})})^{\lambda}+({\gamma-1})({\eta% _{1}})^{2\lambda}}}}\right\}}\right\},\lambda>0;$ $\displaystyle(3)\quad d_{1}\oplus d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}% \in h_{2},\eta_{1}\in g_{1},\eta_{2}\in g_{2}}\left\{\left\{{\sqrt{\frac{({% \gamma_{1}})^{2}+({\gamma_{2}})^{2}-({\gamma_{1}})^{2}({\gamma_{2}})^{2}-(1-% \gamma)({\gamma_{1}})^{2}({\gamma_{2}})^{2}}{1-(1-\gamma)({\gamma_{1}})^{2}({% \gamma_{2}})^{2}}}}\right\},\right.$ $\displaystyle\quad\quad\left.{\left\{{\frac{\eta_{1}\eta_{2}}{\sqrt{\gamma+(1-% \gamma)({({\eta_{1}})^{2}+({\eta_{2}})^{2}-({\eta_{1}})^{2}({\eta_{2}})^{2}})}% }}\right\}}\right\},\lambda>0$ $\displaystyle(4)\quad d_{1}\otimes d_{2}=\cup_{\gamma_{1}\in h_{1},\gamma_{2}% \in h_{2},\eta_{1}\in g_{1},\eta_{2}\in g_{2}}\left\{\left\{{\frac{\gamma_{1}% \gamma_{2}}{\sqrt{\gamma+(1-\gamma)({({\gamma_{1}})^{2}+({\gamma_{2}})^{2}-({% \gamma_{1}})^{2}({\gamma_{2}})^{2}})}}}\right\},\right.$ $\displaystyle\quad\quad\left.{\left\{{\sqrt{\frac{({\eta_{1}})^{2}+({\eta_{2}}% )^{2}-({\eta_{1}})^{2}({\eta_{2}})^{2}-(1-\gamma)({\eta_{1}})^{2}({\eta_{2}})^% {2}}{1-(1-\gamma)({\eta_{1}})^{2}({\eta_{2}})^{2}}}}\right\}}\right\},\lambda>0.$

3. Some Hamacher correlated aggregation operators with DPHFNs

For real decisionmaking issues, there is always some degree of inter-dependent characteristics between attributes. Usually, there is interaction among attributes of decision makers. However, this assumption is too strong to match decision behaviour in the real world. The independence axiom generally can’t be satisfied. Thus, it is necessary to consider this issue.

Definition 5 [44]. Let $f$ be a positive real-valued function on $X$ , and $\mu$ be a fuzzy measure on $X$ . The discrete Choquet integral of $f$ with respective to $\mu$ is defined by

$\displaystyle C_{\mu}(f)=\sum\limits_{i=1}^{n}{f_{\sigma(i)}}[{\mu({A_{\sigma(% i)}})-\mu({A_{\sigma({i-1})}})}]$ (4)

where $({\sigma(1),\sigma(2),\cdots,\sigma(n)})$ is a permutation of $({1,2,\cdots,n})$ , such that $f_{\sigma({i-1})}\geqslant f_{\sigma(i)}$ for all $j=2,\cdots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}.}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ .

Based on the aggregation principle for DPHFNs and Choquet integral [44], in the following, we shall develop some Hamacher correlated aggregation operators with DPHFNs.

Definition 6. Let $\tilde{d}_{j}({j=1,2,\cdots,n})$ be a collection of DHPFNs, and $\mu$ be a fuzzy measure on $X$ , then we define the dual hesitant Pythagorean fuzzy Hamacher correlated averaging (DHPFHCA) operator as follows:

$\displaystyle\text{DHPFHCA}_{\mu}({\tilde{d}_{1},\tilde{d}_{2},\cdots,\tilde{d% }_{n}})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A_{% \sigma({j-1})}})})\tilde{d}_{\sigma(j)}})$ $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{\left\{\right\},\right.$ (5) $\displaystyle\quad\left.{\left\{{\frac{\sqrt{\gamma}\prod\limits_{j=1}^{n}{({% \gamma_{\sigma(j)}})^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}}{% \sqrt{\prod\limits_{j=1}^{n}({1+({\gamma-1})({1-({\gamma_{\sigma(j)}})^{2}})})% ^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}+({\gamma-1})\prod\limits_% {j=1}^{n}({\gamma_{\sigma(j)}})^{2(\mu(A_{\sigma(j)})-\mu(A_{\sigma({j-1})}))}% }}}\right\}}\right\}$

where $({\sigma(1),\sigma(2),\cdots,\sigma(n)})$ is a permutation of $({1,2,\cdots,n})$ , such that $\tilde{d}_{\sigma({j-1})}\geqslant\tilde{d}_{\sigma(j)}$ for all $j=2,\cdots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}.}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ , $\gamma>0$ .

Now, we can discuss some special cases of the DHPFHCA operator with respect to the parameter $\gamma$ .

•

When $\gamma=1$ , DHPFHCA operator reduces to the dual hesitant Pythagorean fuzzy correlated average (DHFCA) operator as follows:

$\displaystyle\text{DHPFCA}_{\mu}({\tilde{d}_{1},\tilde{d}_{2},\cdots,\tilde{d}% _{n}})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A_{\sigma% ({j-1})}})})\tilde{d}_{\sigma(j)}})$ (6) $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{{\begin{array}[]{l}\left\{{\sqrt{1-\prod\limits_{j=1}% ^{n}{({1-({\gamma_{\sigma(j)}})^{2}})^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({% j-1})}})})}}}}\right\},\\ \left\{{\frac{\sqrt{2}\prod\limits_{j=1}^{n}{\gamma_{j}^{({\mu({A_{\sigma(j)}}% )-\mu({A_{\sigma({j-1})}})})}}}{\sqrt{\prod\limits_{j=1}^{n}{({2-({\gamma_{j}}% )^{2}})^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}+\prod\limits_{j=1% }^{n}{({\gamma_{j}})^{2({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}}}}% \right\}\\ \end{array}}\right\}$

•

When $\gamma=2$ , DHPFHCA operator reduces to the dual hesitant Pythagorean fuzzy Einstein correlated average (DHPFECWA) operator as follows:

$\displaystyle\text{DHPFECA}_{\mu}({\tilde{d}_{1},\tilde{d}_{2},\cdots,\tilde{d% }_{n}})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({A_{\sigma(j)}})-\mu({A_{% \sigma({j-1})}})})\tilde{d}_{\sigma(j)}})$ $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{\left\{\right\},\right.$ (7) $\displaystyle\quad\left.{\left\{{\frac{\sqrt{2}\prod\limits_{j=1}^{n}{({\eta_{% \sigma(j)}})^{({\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}}{\sqrt{\prod% \limits_{j=1}^{n}{({2-({\eta_{\sigma(j)}})^{2}})^{({\mu({A_{\sigma(j)}})-\mu({% A_{\sigma({j-1})}})})}}+\prod\limits_{j=1}^{n}{({\eta_{\sigma(j)}})^{2({\mu({A% _{\sigma(j)}})-\mu({A_{\sigma({j-1})}})})}}}}}\right\}}\right\}$

In the following, we shall develop the dual Pythagorean hesitant fuzzy Hamacher correlated geometric (DPHFHCG) operator based on the DPHFHCA operator and geometric mean [45, 46].

Definition 7. Let $\tilde{d}_{j}({j=1,2,\cdots,n})$ be a collection of DPHFNs on $X$ , and $\mu$ be a fuzzy measure on $X$ , then we call

$\displaystyle\text{DPHFHCG}_{\mu}({\tilde{p}_{1},\tilde{p}_{2},\cdots,\tilde{p% }_{n}})=\mathop{\otimes}\limits_{j=1}^{n}({p_{\sigma(j)}})^{\mu({A_{\sigma(j)}% })-\mu({A_{\sigma({j-1})}})}$ $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{\left\{\right\},\right.$ (8) $\displaystyle\quad\left.{\left\{{\sqrt{\frac{\prod\limits_{j=1}^{n}{({1+({% \gamma-1})({\eta_{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1% })}})}-\prod\limits_{j=1}^{n}{({1-({\eta_{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j% )}})-\mu({A_{\sigma({j-1})}})}}}}{\prod\limits_{j=1}^{n}{({1+({\gamma-1})({% \eta_{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}+({% \gamma-1})\prod\limits_{j=1}^{n}{({1-({\eta_{\sigma(j)}})^{2}})^{\mu({A_{% \sigma(j)}})-\mu({A_{\sigma({j-1})}})}}}}}}\right\}}\right\}$

the dual Pythagorean hesitant fuzzy Hamacher correlated geometric (DPHFHCG) operator, where $(\sigma(1),\sigma(2),\cdots,\linebreak\sigma(n))$ is a permutation of $({1,2,\cdots,n})$ , such that $\tilde{d}_{\sigma({j-1})}\geqslant\tilde{d}_{\sigma(j)}$ for all $j=2,\cdots,n$ , $A_{\sigma(k)}=\{{x_{\sigma(j)}|{j\leqslant k}}\}$ , for $k\geqslant 1$ , and $A_{\sigma(0)}=\phi$ , $\gamma>0$ .

Now, we can discuss some special cases of the DHPFHCG operator with respect to the parameter $\gamma$ .

•

When $\gamma=1$ , DHPFHCG operator reduces to the dual hesitant Pythagorean fuzzy correlated geometric (DHPFCG) operator as follows:

$\displaystyle\text{DHPFCG}_{\mu}({\tilde{d}_{1},\tilde{d}_{2},\cdots,\tilde{d}% _{n}})=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{d}_{\sigma(j)}})^{\mu({A_{% \sigma(j)}})-\mu({A_{\sigma({j-1})}})}$ (9) $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{{\begin{array}[]{l}\left\{{\prod\limits_{j=1}^{n}{({% \gamma_{\sigma(j)}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}}\right\}% ,\\ \left\{{\sqrt{1-\prod\limits_{j=1}^{n}{({1-({\eta_{\sigma(j)}})^{2}})^{\mu({A_% {\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}}}\right\}\\ \end{array}}\right\}$

•

When $\gamma=2$ , DHPFHCG operator reduces to the dual hesitant Pythagorean fuzzy Einstein correlated geometric (DHPFECG) operator as follows:

$\displaystyle\text{DHPFECG}_{\mu}({\tilde{d}_{1},\tilde{d}_{2},\cdots,\tilde{d% }_{n}})=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{d}_{\sigma(j)}})^{\mu({A_{% \sigma(j)}})-\mu({A_{\sigma({j-1})}})}$ $\displaystyle\quad=\cup_{\gamma_{\sigma(j)}\in h_{\sigma(j)},\eta_{\sigma(j)}% \in g_{\sigma(j)}}\left\{\left\{\right\},\right.$ (10) $\displaystyle\quad\left.{\left\{{\sqrt{\frac{\prod\limits_{j=1}^{n}{({1+({\eta% _{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}-\prod% \limits_{j=1}^{n}{({1-({\eta_{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j)}})-\mu({A_% {\sigma({j-1})}})}}}}{\prod\limits_{j=1}^{n}{({1+({\eta_{\sigma(j)}})^{2}})^{% \mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}+\prod\limits_{j=1}^{n}{({1-({% \eta_{\sigma(j)}})^{2}})^{\mu({A_{\sigma(j)}})-\mu({A_{\sigma({j-1})}})}}}}}}% \right\}}\right\}$

4. An approach to MADM with DPHFNs

In this section, we shall utilize the dual hesitant aggregation operators to MADM with DPHFNs Let $A=\{{A_{1},A_{2},\cdots,A_{m}}\}$ be a discrete set of alternatives, and $G=\{{G_{1},G_{2},\cdots,G_{n}}\}$ be the state of nature. If the decision makers provide several values for the alternative $A_{i}$ under the state of nature $G_{j}$ with anonymity, these values can be considered as a DPHFN $\tilde{d}_{ij}=({h_{ij},g_{ij}})$ . In such case where two decision makers provide the same value, then the value emerges only once in $\tilde{d}_{ij}$ . Suppose that the decision matrix $\tilde{D}=({\tilde{d}_{ij}})_{m\times n}$ is the decision matrix, where $\tilde{d}_{ij}({i=1,2,\cdots,m,j=1,2,\cdots,n})$ are in the form of DHPFNs.

In the following, we apply the DHPFHCA (or DHPFHCG) operator to the MADM problems for supplier selection in supply chain management with DHPFNs.

Step 1. We utilize the decision information given in matrix $\tilde{D}$ , and the DHPFHCA operator

$\displaystyle\tilde{d}_{i}=\text{DHPFHCA}({\tilde{d}_{i1},\tilde{d}_{i2},% \cdots,\tilde{d}_{in}})=\mathop{\oplus}\limits_{j=1}^{n}({({\mu({G_{\sigma(j)}% })-\mu({G_{\sigma({j-1})}})})\tilde{d}_{ij}})$ (11) $\displaystyle\quad=\cup_{\gamma_{ij}\in h_{ij},\eta_{ij}\in h_{ij}}\left\{% \left\{\right\},\right.$ (12) $\displaystyle\quad\left.\left\{\!\!\frac{\sqrt{\gamma}\prod\limits_{j=1}^{n}{(% {\eta_{\sigma({ij})}})^{({\mu({G_{\sigma(j)}})-\mu({G_{\sigma({j-1})}})})}}}{% \sqrt{\prod\limits_{j=1}^{n}{({1+({\gamma-1})({1-({\eta_{\sigma({ij})}})^{2}})% })^{({\mu({G_{\sigma(j)}})-\mu({G_{\sigma({j-1})}})})}}+({\gamma-1})\prod% \limits_{j=1}^{n}({\eta_{\sigma({ij})}})^{2({\mu({G_{\sigma(j)}})-\mu({G_{% \sigma({j-1})}})})}}}\!\!\right\}\right\}$

Or the dual hesitant Pythagorean fuzzy correlated geometric (DHPFHCG) operator:

$\displaystyle\tilde{d}_{i}=\text{DHPFHCG}({\tilde{d}_{i1},\tilde{d}_{i2},% \cdots,\tilde{d}_{in}})=\mathop{\otimes}\limits_{j=1}^{n}({\tilde{d}_{\sigma({% ij})}})^{({\mu({G_{\sigma(j)}})-\mu({G_{\sigma({j-1})}})})}$ $\displaystyle\quad=\cup_{\gamma_{ij}\in h_{ij},\eta_{ij}\in h_{ij}}\left\{% \left\{\right\},\right.$ (13) $\displaystyle\quad\left.{\left\{{\sqrt{}}\right\}\right\}}$

to derive the overall preference values $\tilde{d}_{i}({i=1,2,\cdots,m})$ of the alternative $A_{i}$ .

Step 2. Calculate the scores $S({\tilde{d}_{i}})({i=1,2,\cdots,m})$ of the overall dual hesitant Pythagorean fuzzy preference values $\tilde{d}_{i}({i=1,2,\cdots,m})$ . if there is no difference between two scores $S({\tilde{d}_{i}})$ and $S({\tilde{d}_{j}})$ , then we need to calculate the accuracy degrees $p({\tilde{d}_{i}})$ and $p({\tilde{d}_{j}})$ of the collective overall preference values $\tilde{d}_{i}$ and $\tilde{d}_{j}$ , respectively, and then rank the alternatives $A_{i}$ and $A_{j}$ in accordance with the accuracy degrees $p({\tilde{d}_{i}})$ and $p({\tilde{d}_{j}})$ .

Step 3. Rank all the alternatives $A_{i}({i=1,2,\cdots,m})$ and select the best one(s) in accordance with the scores $S({\tilde{d}_{i}})({i=1,2,\cdots,m})$ .

Step 4. End.

5. Numerical example

Since the beginning of the 21 st century, the social economy has continued to grow at an unprecedented rate. However, as human beings continue to pursue economic growth, they have neglected environmental pollution, resulting in global warming, acid rain and other evils, and the environment on which humans depend for survival is facing a serious threat. Especially in recent years, with the progress of science and technology, various electronic products have emerged in an endless stream, the life cycle has been shortened, and the number of electronic products that have been eliminated has increased dramatically, and 80% of electronic waste has not been properly recovered, resulting in serious Environmental pollution. With the increasing awareness of human environmental protection and the increasing number of environmental regulations introduced by countries around the world, environmental protection has become the focus of attention. In the face of deteriorating natural environment and increasingly strict environmental protection policies, it is imperative for China’s electronics manufacturing companies to implement green supply chain management. To effectively promote the construction of a green supply chain, it is necessary to first strengthen the evaluation and selection of green suppliers. It is necessary to consider the environmental benefits when evaluating suppliers. Thus, in this section we shall present a numerical example for supplier selection in supply chain management with DHPFNs in order to illustrate the method proposed in this paper. Let us suppose there is a problem to deal with the supplier selection in supply chain management which is classical multiple attribute decision making problems. There are five prospect suppliers $A_{i}({i=1,2,3,4,5})$ for four attributes $G_{j}({j=1,2,3,4})$ . The four attributes include product quality $({G_{1}})$ , service $({G_{2}})$ , delivery $({G_{3}})$ and environmental protection $({G_{4}})$ , respectively. In order to avoid influence each other, the decision makers are required to evaluate the five suppliers $A_{i}({i=1,2,3,4,5})$ under the above four attributes in anonymity and the decision matrix $\tilde{D}=({\tilde{d}_{ij}})_{5\times 4}$ is presented in Table 1, where $\tilde{d}_{ij}({i=1,2,3,4,5,j=1,2,3,4})$ are in the form of DHPFNs.

Table 1
Dual hesitant Pythagorean fuzzy decision matrix

	G ${}_{1}$	G ${}_{2}$	G ${}_{3}$	G ${}_{4}$
A ${}_{1}$	{{0.2, 0.3, 0.4}, {0.6}}	{{0.5, 0.6, 0.7}, {0.3}}	{{0.5, 0.7}, {0.2}}	{{0.2}, {0.7, 0.8}}
A ${}_{2}$	{{0.1, 0.2}, {0.3}}	{{0.1}, {0.6, 07, 0.8}}	{{0.7}, {0.3})}	{{0.6, 0.7, 0.8}, {0.2}}
A ${}_{3}$	{{0.4, 0.5}, {0.2}}	{{0.2, 0.3, 0.4}, {0.5}}	{{0.6, 0.7}, {0.2}}	{{0.2, 0.3, 0.4}, {0.5}}
A ${}_{4}$	{{0.2, 0.3}, {0.7)}	{{0.4, 0.5}, {0.5}}	{{0.3, 0.4}, {0.6}}	{{0.4, 0.5}, {0.3, 0.4)}
A ${}_{5}$	{{0.2}, {0.6, 07, 0.8}}	{{0.5), {0.4, 0.5)}	{{0.6}, {0.4}}	{{0.2, 0.4, 0.5}, {0.4}}

Assume that the fuzzy measure of attribute of $G_{j}({j=1,2,\cdots,n})$ and attribute sets of $G$ as follows:

$\displaystyle\mu({G_{1}})=0.35,\mu({G_{2}})=0.30,\mu({G_{3}})=0.25,\mu({G_{4}}% )=0.20,\mu({G_{1},G_{2}})=0.60,$ $\displaystyle\mu({G_{1},G_{3}})=0.55,\mu({G_{1},G_{4}})=0.50,\mu({G_{2},G_{3}}% )=0.65,\mu({G_{2},G_{4}})=0.50$

In the following, we utilize the approach developed for supplier selection in supply chain management with DHPFNs.

Step 1. We utilize the decision information given in matrix $\tilde{D}$ , and the DHPFHCA operator to obtain the overall preference values $\tilde{d}_{i}$ of the supplier in supply chain management $A_{i}({i=1,2,3,4,5})$ and calculate the scores $s({\tilde{d}_{i}})({i=1,2,3,4,5})$ of the overall DHPFNs $\tilde{d}_{i}({i=1,2,3,4,5})$ :

$\displaystyle s({\tilde{d}_{1}})=0.3831,s({\tilde{d}_{2}})=0.4565,s({\tilde{d}% _{3}})=0.5013$ $\displaystyle s({\tilde{d}_{4}})=0.4786,s({\tilde{d}_{5}})=0.6185$

Step 2. Rank all the suppliers $A_{i}({i=1,2,3,4,5})$ in accordance with the scores $s({\tilde{d}_{i}})({i=1,2,3,4,5})$ of the overall DHPFNs: $A_{5}\succ A_{3}\succ A_{4}\succ A_{2}\succ A_{1}$ , and thus the most desirable supplier is $A_{5}$ .

Based on the DHPFHCG operator, then, in order to select the most desirable supplier, we can develop an approach to multiple attribute decision making problems with DHPFNs, which can be described as following:

Step 1’. Aggregate all DHPFNs $\tilde{h}_{ij}({j=1,2,3,4})$ by using the DHPFHCG operator to derive the overall DHPFNs $\tilde{d}_{i}({i=1,2,\cdots,5})$ of the supplier and calculate the scores $s({\tilde{d}_{i}})({i=1,2,3,4,5})$ of the overall DHPFNs $\tilde{d}_{i}({i=1,2,3,4,5})$ of the supplier $A_{i}$ :

$\displaystyle s({\tilde{d}_{1}})=0.4879,s({\tilde{d}_{2}})=0.4058,s({\tilde{d}% _{3}})=0.4023$ $\displaystyle s({\tilde{d}_{4}})=0.5287,s({\tilde{d}_{5}})=0.5566$

Step 2.’ Rank all the suppliers in supply chain management $A_{i}({i=1,2,3,4,5})$ in accordance with the scores $s({\tilde{d}_{i}})({i=1,2,3,4,5})$ of the overall DHPFNs $\tilde{d}_{i}({i=1,2,\cdots,5})$ by using definition 5: $A_{5}\succ A_{4}\succ A_{1}\succ A_{2}\succ A_{3}$ and thus the most desirable supplier in supply chain management is $A_{5}$ .

From the above analysis, it is easily seen that although the overall rating values of the alternatives are slightly different by using two operators respectively. However, the most desirable supplier in supply chain management is $A_{5}$ .

6. Conclusion

In this paper, we investigate the MADM problem based on the Hamacher aggregation operators and Choquet integral with DHPFNs. Then, motivated by the ideal of Hamacher operation and Choquet integral, we have developed some Hamacher correlated operators for aggregating the DHPFNs: dual hesitant Pythagorean fuzzy Hamacher correlated average (DHPFHCA) operator and dual hesitant Pythagorean fuzzy Hamacher correlated geometric (DHPFHCG) operator. The prominent characteristic of these proposed operators is studied. Then, we have utilized these two operators to develop some approaches to solve the MADM problems with DHPFNs. Finally, a practical example for supplier selection in supply chain management is given to verify the developed approach and to demonstrate its practicality and effectiveness. In the future, we shall continue working in the extension and application of the developed operators to other uncertain domains [47, 48, 49, 50, 51, 52].

Footnotes

Acknowledgments

The work was supported by the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 14YJCZH082).

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Algorithms for multiple attribute decision making with dual hesitant Pythagorean fuzzy information and their application to supplier selection

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Pythagorean fuzzy set

3. Some Hamacher correlated aggregation operators with DPHFNs

Table 1 Dual hesitant Pythagorean fuzzy decision matrix

Footnotes

Acknowledgments

References

Table 1
Dual hesitant Pythagorean fuzzy decision matrix