Models for MADM with hesitant interval-valued fuzzy information under uncertain environment

Abstract

This paper is a research of interval-valued fuzzy and Muirhead Mean algorithms. We deduced new algorithms named as hesitant interval-valued fuzzy Muirhead Mean (HIVFMM) and hesitant interval-valued fuzzy Muirhead Mean (HIVFWMM) with Muirhead Mean algorithms based on Hesitant interval-valued fuzzy set (HIVFS). Firstly, we introduced some concepts and operation laws of HIVFS and the formula form of MM, then we combined them both and gave the proof process of properties and theorems, a mathematic model applying to MADM and a numerical example was given to illustrate the effectively and practically.

Keywords

Multiple attribute decision making (MADM)hesitant interval-valued fuzzy set (HIVFS)muirhead mean (MM) algorithm possibility degree

1. Introduction

Territory of Multiple attribute decision making (MADM) is a decision science of assessing multiple attributes or criteria before a final choice is selected [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Atanassov [21], in his paper, introduced the interval valued intuitionistic fuzzy sets (IVIFSs) and defined some definitions and concepts, and he also made some comparison of IVIFSs and intuitionistic fuzzy set simply. Atanassov [22] deduced some operators and properties of the interval valued intuitionistic fuzzy sets (IVIFSs). Wei et al. [23] defined some operation laws and score function of hesitant interval-valued fuzzy elements (HIVFEs), and also deduced some novel algorithms, such as hesitant interval-valued fuzzy aggregation operators: hesitant interval-valued fuzzy weighted averaging (HIVFWA) operator and hesitant interval-valued fuzzy ordered weighted geometric (HIVFOWG) operator and so on. Torra [24] developed hesitant fuzzy set and introduced some basic operations, they also improved the hesitant fuzzy set is a intuition fuzzy set. Wan [25] developed some novel geometric aggregation operators after the research of law and properties of 2-tuple linguistic fuzzy set, deduced some algorithms such as the 2-tuple hybrid linguistic weighted geometric average (T-HLWG) operator and the extended 2-tuple hybrid linguistic weighted geometric average (ET-HLWG) operator. Muirhead Mean algorithm which is used to solve the symmetric algebraic function is created by professor Muirhead [26] firstly. Then Qin et al. [27] import Muirhead Mean into 2-tuple linguistic fuzzy set and deduced some algorithms such as the 2-tuple linguistic Muirhead mean (2TLMM) operator and the 2-tuple linguistic dual Muirhead mean (2TLDMM) operator. We must introduce the Bonferroni Mean (BM) which is first introduced by Bonferroni [28] when it comes to MM, for the MM is a extent form of BM and they can inter-contain in some special cases.

In the beginning of this paper, we firstly introduced some basic concepts of Hesitant interval-valued fuzzy set (HIVFS) and interval-valued fuzzy set (IVFS) and their basic operation laws, secondly, the introducing of Muirhead Mean is also main task of this part. In the following, we combined the Hesitant interval-valued fuzzy set and Muirhead Mean algorithm and deduced the new form of Muirhead Mean under Hesitant interval-valued fuzzy set and proved some properties of it. After that we gave a decision-making model to solve Multiple attribute decision making problems and applied it to a numerical expel, compared its result with a exist algorithms to prove the effectively and practically of the deduced algorithms.

2. Hesitant interval-valued fuzzy set and Muirhead Mean algorithm

On this part, we will introduce some basic definitions of hesitant interval-valued fuzzy and concept of Muirhead Mean (MM) algorithm.

2.1 Hesitant interval-valued fuzzy set

Definition 1 [23]. Let $X$ be a fixed set, a hesitant interval-valued fuzzy set (HIVFS) on $X$ is in terms of a function that when applied to each $x$ in $X$ and returns a subset of interval values in $[0,1]$ . To be easily understood, we express the HIVFS by a mathematical symbol:

$\displaystyle E=\{{\langle{x,\tilde{h}_{E(x)}}\rangle|{x\in X}}\}$ (1)

where $\tilde{h}_{E(x)}$ is a set of some interval values in $[0,1]$ , denoting the possible membership degrees of the element $x\in X$ to the set $E$ .

In the following, $\tilde{h}_{E(x)}=\tilde{h}=[\gamma^{L},\gamma^{R}]$ can be called as hesitant interval-valued fuzzy element (HIVFE) and $\tilde{H}$ as the set of all HIVFEs.

Suppose $\tilde{h}=[\gamma^{L},\gamma^{R}]$ , $\tilde{h}_{1}=[\gamma_{1}^{L},\gamma_{1}^{R}]$ and $\tilde{h}_{2}=[\gamma_{2}^{L},\gamma_{2}^{R}]$ are three HIVFEs and $\lambda>0$ , there are four operation laws as follow:

(1) (1)

$({\tilde{h}})^{\lambda}=\bigcup\nolimits_{\tilde{{\gamma}}\in\tilde{h}}{\{{[{(% {\gamma^{L}})^{\lambda}({\gamma^{R}})^{\lambda}}]}\}}$ ;

(2)

$\lambda({\tilde{h}})=\bigcup\nolimits_{\tilde{{\gamma}}\in\tilde{h}}\{[1-({1-% \gamma^{L}})^{\lambda},1-({1-\gamma^{R}})^{\lambda}]\}$ ;

(3)

$\tilde{h}_{1}\oplus\tilde{h}_{2}=\bigcup\nolimits_{\tilde{{\gamma}}_{1}\in% \tilde{h}_{1}\tilde{{\gamma}}_{2}\in\tilde{h}_{2}}\{[\gamma_{1}^{L}+\gamma_{2}% ^{L}-\gamma_{1}^{L}\gamma_{2}^{L},$ $\gamma_{1}^{R}+\gamma_{2}^{R}-\gamma_{1}^{R}\gamma_{2}^{R}]\}$ ;

(4)

$\tilde{h}_{1}\otimes\tilde{h}_{2}=\bigcup\nolimits_{\tilde{{\gamma}}_{1}\in% \tilde{h}_{1}\tilde{{\gamma}}_{2}\in\tilde{h}_{2}}\{{[{\gamma_{1}^{L}\gamma_{2% }^{L},\gamma_{1}^{R}\gamma_{2}^{R}}]}\}$ .

HIVFE, as we know, is a set interval-valued fuzzy numbers (IVFNs), In terms of combining several a HIVFE to be an interval-valued fuzzy umber, we’ll introduce the concept of IVFNs next.

Definition 2 [23]. Let $\tilde{a}=[a^{L},a^{R}]=\{x|a^{L}\leqslant x\leqslant a^{R}\}$ , then $\tilde{a}$ is a IVFN, if $0\leqslant a^{L}\leqslant a^{R}$ , the $\tilde{a}$ is named as positive IVFN.

Four operations also exit to calculate any two interval-valued number $\tilde{a}=[{a^{L},a^{R}}]$ and $\tilde{b}=[{b^{L},b^{R}}]$ :

(1) (1)

$\tilde{a}+\tilde{b}=[{a^{L}+b^{L},a^{R}+b^{R}}]$ ;

(2)

$\lambda\tilde{a}=[{\lambda a^{L},\lambda a^{R}}]$ ;

(3)

$\tilde{a}^{\lambda}=[{({a^{L}})^{\lambda},({a^{R}})^{\lambda}}]$ ;

(4)

$\tilde{a}\cdot\tilde{b}=[{a^{L}\cdot b^{L},a^{R}\cdot b^{R}}]$ .

To solve the problem of comparing two IVFNs, we can utilize the degree of possibility, so we’ll, in the following, give the approach of measuring the degree of possibility.

Definition 3 [23]. Let $\tilde{a}_{1}=[{a_{1}^{L},a_{1}^{R}}]$ and $\tilde{a}_{2}=[{a_{2}^{L},a_{2}^{R}}]$ be two interval fuzzy numbers, and let $\text{len}({\tilde{a}_{1}})=a_{1}^{R}-a_{1}^{L}$ , $\text{len}({\tilde{a}_{2}})=a_{2}^{R}-a_{2}^{L}$ , then the degree of possibility of $\tilde{a}_{1}\geqslant\tilde{a}_{2}$ is defined as:

$\displaystyle p({\tilde{a}_{1}\geqslant\tilde{a}_{2}})$ (2) $\displaystyle=\max\left\{{1-\max\left({\frac{a_{2}^{R}-a_{1}^{L}}{\text{len}({% \tilde{a}_{1}})+\text{len}({\tilde{a}_{2}})}0}\right),0}\right\}$

At the same times, the degree of possibility of $\tilde{a}_{2}\geqslant\tilde{a}_{1}$ is defined as:

$\displaystyle p({\tilde{a}_{2}\geqslant\tilde{a}_{1}})$ (3) $\displaystyle=\max\left\{{1-\max\left({\frac{a_{1}^{R}-a_{2}^{L}}{\text{len}({% \tilde{a}_{1}})+\text{len}({\tilde{a}_{2}})}0}\right),0}\right\}$

Under the condition of existing several alternatives, the comparisons of inter-alternative are too complicated to be done. For convenience, we can struct a matrix to simplify the program. Then we can utilize

$\displaystyle p({\tilde{a}_{i}\geqslant\tilde{a}_{j}})$ (4) $\displaystyle=\max\left\{{1-\max\left({\frac{a_{j}^{R}-a_{i}^{L}}{\text{len}({% \tilde{a}_{j}})+\text{len}({\tilde{a}_{i}})}0}\right),0}\right\}$

to note the comparison of any two alternatives. Both $\tilde{a}_{i}$ and $\tilde{a}_{j}$ are IVFNs with $i,j=1,2,\cdots,n$ . Suppose $p_{ij}=p({\tilde{a}_{i}\geqslant\tilde{a}_{j}})$ , then the constructed matrix is developed as follows:

$\displaystyle p=\begin{bmatrix}{p_{11}}&{p_{12}}&\cdots&{p_{1n}}\\ {p_{21}}&{p_{22}}&\cdots&{p_{2n}}\\ \vdots&\vdots&\ddots&\vdots\\ {p_{n1}}&{p_{n2}}&\cdots&{p_{nn}}\end{bmatrix}$

where $p_{ij}\geqslant 0$ , $p_{ij}+p_{ji}=1$ , $p_{ii}=0.5$ , $i,j=1,2,\cdots,n$ .

In order to compare alternatives, we can sum each line of matrix, the formula showed as follow:

$\displaystyle p_{i}=\sum_{j=1}^{n}{p_{ij}}\quad i,j=1,2,\cdots,n$ (5)

Then, we can acquire the rank through the results.

However, the precondition of definition 3 is the transformation from HIVFE to IVFN. So a demanding of score function is necessary.

Definition 4 [23]. Suppose $\tilde{a}$ is a HIVFE, the we can get a score by the function

$\displaystyle s({\tilde{a}})=\frac{1}{\#\tilde{a}}\sum{\tilde{a}}$ (6)

where $\#\tilde{a}$ is the number of IVFN in $\tilde{a}$ . For two HIVFEs $\tilde{a}_{1}$ and $\tilde{a}_{2}$ , if $s({\tilde{a}_{1}})>s({\tilde{a}_{2}})$ , then $\tilde{a}_{1}>\tilde{a}_{2}$ . After this step, we can use the Definition 3 go on.

Definition 4 [23]. If $\tilde{a}=[{\gamma^{L},\gamma^{R}}]$ is a interval valued number, then the expected value of $\tilde{a}$ is

$\displaystyle E({\tilde{a}})=\frac{1}{2}({\gamma^{L}+\gamma^{R}})$ (7)

2.2 Muirhead mean (MM)

The MM, deduced by Muirhead firstly, has the advanced advantage of capturing the overall interrelationships among the multiple input arguments.

Definition 5. Let $a_{i}({i=1,2,\cdots,n})$ be a set of crisp numbers and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ , then the Muirhead mean (MM) operator is defined as

$\displaystyle\text{MM}^{\lambda}({a_{1},a_{2},\cdots,a_{n}})$ $\displaystyle=\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{a_{% \vartheta(j)}^{\lambda_{j}}}}}\right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_% {j}}}}$ (8)

Meanwhile, there is also another form of weighted MM (WMM),

$\displaystyle\text{WMM}^{\lambda}({a_{1},a_{2},\cdots,a_{n}})$ $\displaystyle=\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{({% nw_{\vartheta(j)}a_{\vartheta(j)}})^{\lambda_{j}}}}}\right)^{\frac{1}{\sum% \limits_{j=1}^{n}{\lambda_{j}}}}$ (9)

where $\vartheta(j)({j=1,2,\cdots,n})$ is any permutation of $(1,2,\cdots,n)$ and $S_{n}$ is the set of all permutation of $({1,2,\cdots,n})$ .

3. Hesitant interval-valued fuzzy Muirhead Mean operator

In Section 2, we introduced the fundamental conception of HIVF and the form of MM algorithm. On this part, we will combine them both and deduce a new algorithm named as Hesitant interval-valued fuzzy Muirhead Mean (HIVFMM) operator and Hesitant interval-valued fuzzy dual Muirhead Mean (HIVFDMM) operator.

3.1 The deducing of HIVFMM

Definition 6. Let $\tilde{h}_{i}=[\gamma_{i}^{L},\gamma_{i}^{R}]$ be a set of HIVFE and $[\lambda]=(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in R$ be a vector of parameters, then the HIVFMM operator is defined as

$\displaystyle\text{HIVFMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\cdots,% \tilde{h}_{n}})$ $\displaystyle=\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{% \tilde{h}_{\vartheta(j)}^{\lambda_{j}}}}}\right)^{\frac{1}{\sum\limits_{j=1}^{% n}{\lambda_{j}}}}$ (10)

where $\vartheta_{j}(j=1,2,\cdots,n)$ is any permutation of $(1,2,$ $\cdots,n)$ , and $S_{n}$ is the collection of all permutations of $({1,2,\cdots,n})$ . The aggregated result of HIVFMM is also a HIVFE and

$\displaystyle\text{HIVFMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\cdots,% \tilde{h}_{n}})=$ $\displaystyle\left[\left(1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j% =1}^{n}{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}% \right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}},\right.$ $\displaystyle\left.\left(1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j% =1}^{n}{({\gamma_{i}^{R}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}% \right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\right]$ (11)

Proof $\tilde{h}_{\vartheta(j)}^{\lambda_{j}}=[{({\gamma_{i}^{L}})^{\lambda_{j}},({% \gamma_{i}^{R}})^{\lambda_{j}}}]$

Then,

$\displaystyle\prod_{j=1}^{n}\tilde{h}_{\vartheta(j)}^{\lambda_{j}}=\left[{% \prod_{j=1}^{n}{({\gamma_{i}^{L}})^{\lambda_{j}}}\prod_{j=1}^{n}{({\gamma_{i}^% {R}})^{\lambda_{j}}}}\right]$ $\displaystyle\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{\tilde{h}_{\vartheta(j)% }^{\lambda_{j}}}}=\left[1-\prod_{\vartheta\in S_{n}}\left({1-\prod_{j=1}^{n}{(% {\gamma_{i}^{L}})^{\lambda_{j}}}}\right),\right.\left.1-\prod_{\vartheta\in S_% {n}}\left({1-\prod_{j=1}^{n}{({\gamma_{i}^{R}})^{\lambda_{j}}}}\right)\right]$

Thereafter,

$\displaystyle\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{\tilde{h}_{% \vartheta(j)}^{\lambda_{j}}}}=$ $\displaystyle\left[1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j=1}^{n% }{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}},\right.$ $\displaystyle\left.1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j=1}^{n% }{({\gamma_{i}^{R}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}\right]$

Therefore,

$\displaystyle\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{% \tilde{h}_{\vartheta(j)}^{\lambda_{j}}}}}\right)^{\frac{1}{\sum\limits_{j=1}^{% n}{\lambda_{j}}}}=$ $\displaystyle\left[\left(1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j% =1}^{n}{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}% \right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}},\right.$ $\displaystyle\left.\left(1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j% =1}^{n}{({\gamma_{i}^{R}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}% \right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\right]$

First of all, we know that $0\leqslant\gamma_{i}^{L},\gamma_{i}^{R}\leqslant 1$ and $\gamma_{i}^{L}<\gamma_{i}^{R}$ , then we can easily remind the conclusion of

$\displaystyle 0\leqslant\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-% \prod_{j=1}^{n}{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n% !}}}\right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\leqslant 1$

and

Secondly, we can easily prove that

$\displaystyle\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j=1}^{% n}{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}$ $\displaystyle\leqslant\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-% \prod_{j=1}^{n}{({\gamma_{i}^{R}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n% !}}}\right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}$

Obviously, this process has proved that the result calculated by HIVFMM is also a HIVFE.

In the following, we will prove that HIVFMM has three properties.

Theorem 1 (Idempotency). Suppose $\tilde{h}_{j}(j=1,2,\ldots,$ $n)$ is a set of HIVFEs, if all $\tilde{h}_{j}$ are equal, that’s to say $\tilde{h}_{j}=\tilde{h}$ for all $j=1,2,\ldots,n$ , then

$\displaystyle\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j=1}^{% n}{({\gamma_{i}^{L}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}$ $\displaystyle=\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-({\gamma^{L}% })^{\sum_{j=1}^{n}{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}$ $\displaystyle=\left({1-\left({1-({\gamma^{L}})^{\sum_{j=1}^{n}{\lambda_{j}}}}% \right)}\right)^{\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}=\gamma^{L}$

so can we get that

$\displaystyle\left({1-\left({\prod_{\vartheta\in S_{n}}{\left({1-\prod_{j=1}^{% n}{({\gamma^{R}})^{\lambda_{j}}}}\right)}}\right)^{\frac{1}{n!}}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}$ $\displaystyle=\gamma^{R}.$

So, the $\text{HIVFMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\ldots,\tilde{h}_{n}})=[{% \gamma^{L},\gamma^{R}}]=\tilde{h}$ , the process has finished.

Theorem 2 (Monotonicity). Suppose $\tilde{h}_{j}=[\gamma_{j}^{L},\gamma_{j}^{R}]$ $(j$ $=1,2,\cdots,n)$ and $\tilde{h}_{j^{\prime}}=[{\gamma_{j}^{L^{\prime}},\gamma_{j}^{R^{\prime}}}]∼{}(% {j=1,2,\cdots,n})$ are two sets of HIVFEs, where $T_{j}=\prod_{k=1}^{j-1}{E({s({\tilde{h}_{k}})})}$ , $T_{j}^{\prime}=\prod_{k=1}^{j-1}{E({s({\tilde{h}_{k}^{\prime}})})}$ , $T_{1}=T_{1}^{\prime}=1$ . $s({\tilde{h}_{k}})$ and $s({\tilde{h}_{k}^{\prime}})$ are the score values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ respectively. $E({s({\tilde{h}_{k}})})$ and $E({s({\tilde{h}_{k}^{\prime}})})$ are the expect values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ respectively. We easily know that the more bigger of $\gamma_{j}^{L^{\prime}},\gamma_{j}^{R^{\prime}}$ the more bigger the result calculated by algorithm HIVFMM, if $\tilde{h}_{j}\leqslant\tilde{h}_{j}^{\prime}$ , then

$\displaystyle\text{HIVFMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\ldots,% \tilde{h}_{n}})$ $\displaystyle\leqslant\text{HIVFMM}^{\lambda}({\tilde{h}_{1}^{\prime},\tilde{h% }_{2}^{\prime},\ldots,\tilde{h}_{n}^{\prime}})$

Theorem 3 (Boundedness). Suppose $\tilde{h}_{j}=[{\gamma_{j}^{L},\gamma_{j}^{R}}]$ $({j=1,2,\ldots,n})$ is a set of HIVFEs, where $T_{j}=\prod_{k=1}^{j-1}$ ${E({s({\tilde{h}_{k}})})}$ , $T_{1}=1$ . $s({\tilde{h}_{k}})$ is the score values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ . $E({s({\tilde{h}_{k}})})$ is the expect values of $\tilde{h}_{j}$ . If $\tilde{h}^{-}=\min_{j}\tilde{h}_{j}$ and $\tilde{h}^{+}=\max_{j}\tilde{h}_{j}$ ,then considering the extant of the theorem 1 and 2, we can easily know that

$\displaystyle\tilde{h}^{-}\leqslant\text{HIVFMM}^{\lambda}({\tilde{h}_{1},% \tilde{h}_{2},\ldots,\tilde{h}_{n}})\leqslant\tilde{h}^{+}$

3.2 The deducing of HIVFWMM

$\displaystyle\text{HIVFWMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\cdots,% \tilde{h}_{n}})$ $\displaystyle=\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{({% nw_{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{\lambda_{j}}}}}\right)^{\frac{1}{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}$

$\displaystyle\text{HIVFWMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\cdots,% \tilde{h}_{n}})=$ (12) $\displaystyle\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{({nw% _{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{\lambda_{j}}}}}\right)^{\frac{1}{% \sum\limits_{j=1}^{n}{\lambda_{j}}}}=$ $\displaystyle\left[\begin{array}[]{l}\left(1-\left(\prod\limits_{\vartheta\in S% _{n}}\left(1-\right.\right.\right.\\ \left.\left.\left.\prod\limits_{j=1}^{n}(1-(1-\gamma_{i}^{L})^{nw_{\vartheta(j% )}})^{\lambda_{j}}\right)^{\frac{1}{n!}}\right)\right)^{\frac{1}{\sum\limits_{% j=1}^{n}{\lambda_{j}}}},\\ \left(1-\left(\prod\limits_{\vartheta\in S_{n}}\left(1-\right.\right.\right.\\ \left.\left.\left.\prod\limits_{j=1}^{n}(1-(1-\gamma_{i}^{R})^{nw_{\vartheta(j% )}})^{\lambda_{j}}\right)\right)^{\frac{1}{n!}}\right)^{\frac{1}{\sum\limits_{% j=1}^{n}{\lambda_{j}}}}\end{array}\right]$

Proof

$\displaystyle nw_{\vartheta(j)}\tilde{h}_{\vartheta(j)}=\left[1-({1-\gamma_{i}% ^{L}})^{nw_{\vartheta(j)}},\right.\left.1-({1-\gamma_{i}^{R}})^{nw_{\vartheta(% j)}}\right]$ $\displaystyle({nw_{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{\lambda_{j}}=\left% [({1-({1-\gamma_{i}^{L}})^{nw_{\vartheta(j)}}})^{\lambda_{j}},\right.\left.({1% -({1-\gamma_{i}^{R}})^{nw_{\vartheta(j)}}})^{\lambda_{j}}\right]$

Then,

$\displaystyle\prod_{j=1}^{n}{({nw_{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{% \lambda_{j}}}$ $\displaystyle=\left[\prod_{j=1}^{n}{({1-({1-\gamma_{i}^{L}})^{nw_{\vartheta(j)% }}})^{\lambda_{j}}},\right.$ $\displaystyle\left.\prod_{j=1}^{n}{({1-({1-\gamma_{i}^{R}})^{nw_{\vartheta(j)}% }})^{\lambda_{j}}}\right]$ $\displaystyle\sum_{\vartheta\in S_{n}}{\prod_{j=1}^{n}{({nw_{\vartheta(j)}% \tilde{h}_{\vartheta(j)}})^{\lambda_{j}}}}=$ $\displaystyle\left[1-\prod_{\vartheta\in S_{n}}\left(1-\prod_{j=1}^{n}(1-(1-% \gamma_{i}^{L})^{nw_{\vartheta(j)}})^{\lambda_{j}}\right),\right.$ $\displaystyle\left.1-\prod_{\vartheta\in S_{n}}\left(1-\prod_{j=1}^{n}{({1-({1% -\gamma_{i}^{R}})^{nw_{\vartheta(j)}}})^{\lambda_{j}}}\right)\right]$

Thereafter,

$\displaystyle\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{n}{({nw% _{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{\lambda_{j}}}}=$ $\displaystyle\left[1-\left(\prod\limits_{\vartheta\in S_{n}}\left(1\!-\!\prod% \limits_{j=1}^{n}{(1\!-\!(1\!-\!\gamma_{i}^{L})^{nw_{\vartheta(j)}})^{\lambda_% {j}}}\right)^{\frac{1}{n!}}\right),\right.$ $\displaystyle\left.1-\left(\prod\limits_{\vartheta\in S_{n}}\left(1\!-\!\prod% \limits_{j=1}^{n}(1\!-\!(1\!-\!\gamma_{i}^{R})^{nw_{\vartheta(j)}})^{\lambda_{% j}}\right)\right)^{\frac{1}{n!}}\right]$

Therefore,

$\displaystyle\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod\limits_{j=1}^{% n}{({nw_{\vartheta(j)}\tilde{h}_{\vartheta(j)}})^{\lambda_{j}}}}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}=$ $\displaystyle\left[\left(1-\left(\prod\limits_{\vartheta\in S_{n}}\left(1-% \right.\right.\right.\right.$ $\displaystyle\left.\left.\left.\left.\prod\limits_{j=1}^{n}(1-(1-\gamma_{i}^{L% })^{nw_{\vartheta(j)}})^{\lambda_{j}}\right)^{\frac{1}{n!}}\right)\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\right.,$ $\displaystyle\left.\left(1-\left(\prod\limits_{\vartheta\in S_{n}}\left(1-% \right.\right.\right.\right.$ $\displaystyle\left.\left.\left.\left.\prod\limits_{j=1}^{n}(1-(1-\gamma_{i}^{R% })^{nw_{\vartheta(j)}})^{\lambda_{j}}\right)\right)^{\frac{1}{n!}}\right)^{% \frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\right]$

Similar with the process of proof of HIVFMM, we omitted the proof of HIVFDMM. We can also get that the result calculated by HIVFDMM is also an HIVFN and have two properties.

Theorem 4 (Monotonicity). Suppose $\tilde{h}_{j}=[{\gamma_{j}^{L},\gamma_{j}^{R}}]$ $(j$ $=1,2,\ldots,n)$ and $\tilde{h}_{j}^{\prime}=[{\gamma_{j}^{L^{\prime}},\gamma_{j}^{R^{\prime}}}]∼{}(% {j=1,2,\cdots,n})$ are two sets of HIVFEs, where $T_{j}=\prod_{k=1}^{j-1}{E({s({\tilde{h}_{k}})})}$ , $T_{j}^{\prime}=\prod_{k=1}^{j-1}{E({s({\tilde{h}_{k}^{\prime}})})}$ , $T_{1}=T_{1}^{\prime}=1$ . $s({\tilde{h}_{k}})$ and $s({\tilde{h}_{k}^{\prime}})$ are the score values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ respectively. $E({s({\tilde{h}_{k}})})$ and $E({s({\tilde{h}_{k}^{\prime}})})$ are the expect values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ respectively. We easily know that the more bigger of $\gamma_{j}^{L^{\prime}},\gamma_{j}^{R^{\prime}}$ the more bigger the result calculated by algorithm HIVFWMM, if $\tilde{h}_{j}\leqslant\tilde{h}_{j}^{\prime}$ , then

$\displaystyle\text{HIVFWMM}^{\lambda}({\tilde{h}_{1},\tilde{h}_{2},\ldots,% \tilde{h}_{n}})$ $\displaystyle\leqslant\text{HIVFWMM}^{\lambda}({\tilde{h}_{1}^{\prime},\tilde{% h}_{2}^{\prime},\ldots,\tilde{h}_{n}^{\prime}})$

Theorem 5 (Boundedness). Suppose $\tilde{h}_{j}=[{\gamma_{j}^{L},\gamma_{j}^{R}}]$ $(j$ $=1,2,\ldots,n)$ is a set of HIVFEs, where $T_{j}=\prod_{k=1}^{j-1}{E({s({\tilde{h}_{k}})})}$ , $T_{1}=1$ . $s({\tilde{h}_{k}})$ is the score values of $\tilde{h}_{j}$ and $\tilde{h}_{j}^{\prime}$ . $E({s({\tilde{h}_{k}})})$ is the expect values of $\tilde{h}_{j}$ . If $\tilde{h}^{-}=\min_{j}\tilde{h}_{j}$ and $\tilde{h}^{+}=\max_{j}\tilde{h}_{j}$ , then considering the extant of the theorem 3, there is

$\displaystyle\tilde{h}^{-}\leqslant\text{HIVFMM}^{\lambda}({\tilde{h}_{1},% \tilde{h}_{2},\ldots,\tilde{h}_{n}})\leqslant\tilde{h}^{+}$

4. An approach to multiple attribute decision making with hesitant interval-valued fuzzy information

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 hesitant interval-valued fuzzy element $\tilde{h}_{ij}$ . Suppose that the decision matrix $\tilde{H}=({\tilde{h}_{ij}})_{m\times n}$ is the hesitant interval-valued fuzzy decision matrix, where $\tilde{h}_{ij}({i=1,2,\cdots,m,j=1,2,\cdots,n})$ are in the form of HIVFEs.

In the following, we apply the Hesitant interval-valued fuzzy Muirhead Mean (HIVFMM) operator and Hesitant interval-valued fuzzy weighted Muirhead Mean (HIVFWMM) operator to the MADM problems with hesitant interval-valued fuzzy information.

Step 1: Step 1:
Utilize the decision information given in matrix $\tilde{R}$ , and the HIVFWMM operator

$\displaystyle r_{t}=\text{HIVFWMM}^{\lambda}({\tilde{h}_{t1},\tilde{h}_{t2},% \cdots,\tilde{h}_{tn}})=\left({\frac{1}{n!}\sum_{\vartheta\in S_{n}}{\prod_{j=% 1}^{n}{({nw_{\vartheta(j)}\tilde{h}_{t\vartheta(j)}})^{\lambda_{j}}}}}\right)^% {\frac{1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}}=\left[\begin{array}[]{@{}l@{}}% \displaystyle\left(1-\left(\prod_{\vartheta\in S_{n}}\left(1\!-\!\right.\right% .\right.\\ \displaystyle\left.\left.\left.\prod_{j=1}^{n}{(1\!-\!(1\!-\!\gamma_{ti}^{L})^% {nw_{\vartheta(j)}})^{\lambda_{j}}}\right)^{\frac{1}{n!}}\right)\right)^{\frac% {1}{\sum\limits_{j=1}^{n}{\lambda_{j}}}},\\ \displaystyle\left(1-\left(\prod_{\vartheta\in S_{n}}\left(1\!-\!\right.\right% .\right.\\ \displaystyle\left.\left.\left.\prod_{j=1}^{n}(1\!-\!(1\!-\!\gamma_{ti}^{R})^{% nw_{\vartheta(j)}})^{\lambda_{j}}\right)\right)^{\frac{1}{n!}}\right)^{\frac{1% }{\sum\limits_{j=1}^{n}{\lambda_{j}}}}\end{array}\right],t=1,2,\cdots,m$ (13)
Step 2:
Calculate the scores $S({\tilde{h}_{i}})({i=1,2,\cdots,m})$ of the overall hesitant interval-valued fuzzy preference values $\tilde{h}_{i}({i=1,2,\cdots,m})$ .
Step 3:
To rank these overall hesitant interval-valued fuzzy preference values $S({\tilde{h}_{i}})$ $(i=1,2,$ $\cdots,$ $m)$ , then we develop a complementary matrix as $P=({p_{ij}})_{m\times m}$ .

Summing all the elements in each line of matrix $P$ , we have

$\displaystyle p_{i}=\sum_{j=1}^{m}{p_{ij}},i=1,2,\cdots,m$
Step 4:
Rank all the alternatives $A_{i}(i=1,2,\cdots,$ $m)$ and select the best one(s) in accordance with $p_{i}(i=1,2,$ $\cdots,m)$ .

5. Numerical example

Thus, in this section we shall present a numerical example for evaluating software quality with hesitant interval-valued fuzzy information in order to illustrate the method proposed in this paper. There is a panel with five possible software systems $A_{i}({i=1,2,3,4,5})$ to select. The team of experts must take a decision according to the following four attributes: $G_{1}$ is the functionality and reliability; $G_{2}$ is the efficiency; $G_{3}$ is the easiness to use; $G_{4}$ is the maintainability and transferability, their weight vector is $W=({0.2,0.5,0.3,0.2})$ . In order to avoid influence each other, the decision makers are required to evaluate the five possible software systems $A_{i}({i=1,2,3,4,5})$ under the above four attributes in anonymity and the decision matrix $\tilde{H}=({\tilde{h}_{ij}})_{5\times 4}$ is presented in Table 1, where $\tilde{h}_{ij}({i=1,2,3,4,5,j=1,2,3,4})$ are in the form of HIVFEs. Let $\lambda=({0.2,0.20.3,0.3})$

In the following, we utilize the approach developed to evaluate the software quality with hesitant interval-valued fuzzy information.

Table 1
Hesitant interval-valued fuzzy decision matrix

	C1	C2	C3	C4
A1	([0.4, 0.5], [0.5, 0.6])	([0.2, 0.3])	([0.3, 0.4])	([0.2, 0.4], [0.3, 0.5])
A2	([0.7, 0.9])	([0.3, 0.4], [0.4, 0.5])	([0.2, 0.3], [0.6, 0.8])	([0.2, 0.5], [0.4, 0.6])
A3	([0.6, 0.7], [0.7, 0.8])	([0.7, 0.8])	([0.5, 0.6], [0.6, 0.8])	([0.6, 0.7])
A4	([0.1, 0.3], [0.5, 0.7])	([0.7, 0.8])	([0.6, 0.7], [0.7, 0.9])	([0.4, 0.6])
A5	([0.6, 0.7], [0.7, 0.8])	([0.5, 0.6])	([0.5, 0.6], [0.6, 0.8])	([0.6, 0.7])

Table 2

Preference values of HIVFEs

	HIVFWA	HIVFWMM
$A_{1}$	[0.3275, 0.4580]	[0.3236, 0.3407]
$A_{2}$	[0.5011, 0.6984]	[0.4587, 0.4819]
$A_{3}$	[0.7100, 0.8172]	[0.6526, 0.6722]
$A_{4}$	[0.6669, 0.8085]	[0.4544, 0.4766]
$A_{5}$	[0.6256, 0.7415]	[0.6158, 0.6358]

Table 3

The ranking results

HIVFWA	$A_{3}>$ $A_{4}>$ $A_{5}>$ A ${}_{2}>$ A ${}_{1}$
HIVFWMM	$A_{3}>$ A ${}_{5}>$ A ${}_{2}>$ A ${}_{4}>$ A ${}_{1}$

Step 1: Step 1:

Utilize the decision information given in matrix $\tilde{R}$ , and the HIVFMM operator, such the case of $A_{1}$ :

$\displaystyle r_{1}=\text{HIVFWMM}^{\lambda}(([0.4,0.5],[0.5,0.6]),([0.2,0.3])% ,([0.3,0.4]),([0.2,0.4],[0.3,0.5]))=\{[{0.2990,0.3155}],[{0.3250,0.3419}],[{0.% 3211,0.3384}],[{0.3491,0.3668}]\}$

Step 2:

Calculate the scores $S({\tilde{h}_{i}})({i=1,2,\cdots,m})$ of the overall hesitant interval-valued fuzzy preference values $\tilde{h}_{i}({i=1,2,\cdots,m})$ . The results are showed as Table 2. In comparison of the HIVFDMM, we also use the exist algorithm of HIVFWA to calculate the data in Table 1.

Step 3:

To rank these overall hesitant interval-valued fuzzy preference values $S({\tilde{h}_{i}})$ $(i=1,2,$ $\cdots,$ $m)$ , then we develop a complementary matrix as $P=({p_{ij}})_{m\times m}$

$\displaystyle P_{\text{HIVFWMM}}$ $\displaystyle=\begin{bmatrix}0.5&0&0&0&0\\ 1&0.5&0&0.5059&0\\ 1&1&0.5&1&1\\ 1&0.3941&0&0.5&0\\ 1&1&0&1&0.5\\ \end{bmatrix}$ $\displaystyle P_{\text{HIVFWA}}$ $\displaystyle=\begin{bmatrix}0.5&0&0&0&0\\ 1&0.5&0&0.0928&0.2325\\ 1&1&0.5&0.6040&0.8586\\ 1&0.9072&0.3960&0.5&0.7104\\ 1&0.7675&0.1414&0.2896&0.5\\ \end{bmatrix}$

Step 4:

Rank all the alternatives $A_{i}(i=1,2,\cdots,$ $m)$ and select the best one(s) in accordance with $p_{i}(i=1,2,$ $\cdots,m)$ . And we, then, acquired the best alternative is $A_{3}$ , even though the results of HIVFWA and HIVFWMM are not exactly same.

6. Conclusion

In this paper, we deduced HIVFMM and HIVFWMM algorithms under hesitant inter-valued fuzzy information and gave a numerical example to investigate the multiple attribute decision making (MADM) problems. However, we, obviously, know that the HIVFWMM is more one factor of weight information than HIVFWMM, because the weight information is a due importance in MADM, so we use the HIVFWMM of illustrating numerical example is reasonable. The results giving by HIVFMM and HIVFWMM algorithms demonstrate the novel approach we deduced is effectively and practically. In the future, the extension and application of the proposed models and methods with PFNs needs to be investigated into other uncertain and fuzzy decision making [29, 30, 31, 32] and other uncertain and fuzzy environment [33, 34, 35, 36, 37, 38, 39, 40].

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. 15XJA630006).

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Models for MADM with hesitant interval-valued fuzzy information under uncertain environment

Abstract

Keywords

1. Introduction

2. Hesitant interval-valued fuzzy set and Muirhead Mean algorithm

2.1 Hesitant interval-valued fuzzy set

3.1 The deducing of HIVFMM

Table 1 Hesitant interval-valued fuzzy decision matrix

Footnotes

Acknowledgments

References

Table 1
Hesitant interval-valued fuzzy decision matrix