Algorithms for multiple attribute group decision making with intuitionistic 2-tuple linguistic information and its application

Abstract

Recently, the TODIM has been used to deal with multiple attribute decision making (MADM) problems. The intuitionistic 2-tuple linguistic sets (I2TLSs) are useful tools to depict uncertainty in MADM. This paper use TODIM method to solve MADM with the intuitionistic 2-tuple linguistic numbers (I2TLNs). Firstly, the concepts of I2TLNs are briefly presented, and classical TODIM for MADM are introduced. Then, the classical TODIM method is used to solve MADM with I2TLNs, and its important characteristic is to fully consider the decision makers’ bounded rationality which is a real action in MADM. And a comparative analysis is also given. Finally, a numerical example for service outsourcing providers selection is proposed.

Keywords

Multiple attribute decision making (MADM)intuitionistic 2-tuple linguistic set TODIM method service outsourcing providers selection

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1. Introduction

Zadeh [1] designed the fuzzy set (FS) and opened the door to a new field of fuzzy theory in 1965. As a very useful and effective tool, it is applicable to various fields [2]. It could be founded the footprint and research results from the engineering science and technology to the social and human sciences [3, 4, 5]. To extend Zadeh’s classical fuzzy set, the intuitionistic fuzzy set (IFS), which takes into account the membership degree, the non-membership degree and the hesitancy degree [6] at the same time, was developed and studied by Atanassov [7, 8] in 1986 and 1989. Bao et al. [9] defined an intuitionistic fuzzy decision method based on prospect theory and the evidential reasoning approach which aiming at analyzing MADM problems in which the criteria values are intuitionistic fuzzy numbers and the information of attributes weights is unknown. Chen and Wang [10] investigated the MADM problems for evaluating the performance of projects loaned by international financial organizations with the fuzzy number intuitionistic fuzzy numbers (FNIFNs). Damgaci et al. [11] defined the evaluation of appropriate renewable energy sources for Turkey considering technical, economic, environmental and social factors by using Intuitionistic Fuzzy TOPSIS Method. Das et a. [12] investigated the EBM operator with Atanassov’s intuitionistic fuzzy sets (AIFSs) by using additive generators of strict t-conorms in the operations of AIFSs. Liang et al. [13] discussed the decision principles of three-way decision rules based on the variation of loss functions with intuitionistic fuzzy sets (IFSs) and introduced the intuitionistic fuzzy point operator (IFPO) into DTRSs and explore three-way decisions. Liu et al. [14] proposed two prioritized aggregation (PA) operators for HIFLSs: prioritized weighted averaging operator for HIFLSs (HIFLPWA) and prioritized weighted geometric operator for HIFLSs (HIFLPWG). Liu et al. [15] proposed a novel approach for multi-attribute group decision making problem (MAGDM) with trapezoidal intuitionistic fuzzy numbers (TIFNs). Jiang et al. [16] proposed a new type of operator called an intuitionistic fuzzy entropy weighted power average aggregation operators. Li et al. [17] used dynamic intuitionistic fuzzy weighted geometric (DIFWG) operator to integrate different time periods of the intuitionistic fuzzy decision matrices. Xu [18] investigated the MADM problems, in which attribute weights is incomplete and the attribute values are depicted in intuitionistic fuzzy numbers (IFNs). Chen [19] proposed a useful model of relating optimism and pessimism to MCDM in the context of IFSs based on the unipolar bivariate model. Wan and Li [20] built a new intuitionistic fuzzy programming model to solve heterogeneous multi-attribute GDM problems with intuitionistic fuzzy truth degrees in which there are four types of attribute values such as IFSs, trapezoidal fuzzy numbers, intervals, and real numbers. Lu and Wei [21] designed the TODIM method for performance appraisal on social-integration-based rural reconstruction under IVIFSs. Wu et al. [22] proposed some interval-valued intuitionistic fuzzy Dombi Heronian mean operators for evaluating the ecological value of forest ecological tourism demonstration areas. Wu et al. [23] designed the algorithms for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Wu et al. [24] gave the VIKOR method for financing risk assessment of rural tourism projects under IVIFSs.

Although, intuitionistic fuzzy set (IFS) theory has been successfully applied in some areas, the IFS is also characterized by the membership degree and the non-membership degree, whose sum is less than or equal to 1. However, all the above approaches are unsuitable to describe the membership degree and the non-membership degree of an element to a set by linguistic variables on the basis of the given linguistic term sets, which can reflect the decision maker’s confidence level when they are making an evaluation. In order to overcome this limit, Wei [25] proposed the concept of 2-tuple intuitionistic linguistic sets (2TILSs) to solve this problem based on the IFSs and 2-tuple linguistic information processing model [26, 27, 28]. Beg and Rashid [29] defined the intuitionistic 2-tuple linguistic sets (I2LSs). Until now, there is no scholar to investigate the TODIM model under intuitionistic 2-tuple linguistic sets (I2TLSs). Thus, it is very necessary to keep a watchful eye on this novel and worthy issue. The purpose of our paper is to use the TODIM idea to cope with the MADM with the I2TLSs, to fill up this vacancy. For Section 2, we introduce the definition of IFSs and I2TLSs and classical TODIM method for MADM problems. We develop the TODIM model to solve the MADM under I2TLSs in Section 3. An illustrative example for service outsourcing providers selection is pointed out and a comparative analysis is given in Section 4. We give a conclusion in the Section 5.

2. Preliminaries

In the following, some basic concepts related to I2TLSs and IFSs are introduced.

2.1 2-tuple linguistic term sets

Let $S=\{s_{i}|i=1,2,\cdots,t\}$ be a linguistic term set with odd cardinality. Any label, $s_{i}$ depicts a possible value for a linguistic variable, and it should meet the following characteristics [27]:

(1) The set is ordered: $s_{i}>s_{j}$ , if $i>j$ ; (2) Max operator: $\max(s_{i},s_{j})=s_{i}$ , if $s_{i}\geqslant s_{j}$ ; (3) Min operator: $\min(s_{i},s_{j})=s_{i}$ , if $s_{i}\leqslant s_{j}$ . For example, $S$ can be listed as

$\displaystyle S=\{s_{1}=\textit{extremely poor},s_{2}=\textit{very poor},s_{3}% =\textit{poor},s_{4}=\textit{medium},s_{5}=\textit{good},s_{6}=\textit{very % good},$ $\displaystyle\quad s_{7}=\textit{extremely good}\}$

Herrera and Martinez [26] defined the 2-tuple fuzzy linguistic information model based on the definition of symbolic translation. It is employed for depicting the linguistic information by means of a 2-tuple $(s_{i},\alpha_{i})$ , where $s_{i}$ is a linguistic label from $S$ and $\alpha_{i}$ is the value of symbolic translation, and $\alpha_{i}\in[-0.5,0.5)$ .

Definition 1 [26, 27]. Let $S=\{s_{1},s_{2},\cdots,s_{t}\}$ be a linguistic term set and $\beta\in[1,t]$ is a number value depicting the fused result of linguistic symbolic. Then the mathematical function $\Delta$ employed to derive the 2-tuple linguistic information equivalent to $\beta$ is designed as:

$\displaystyle\Delta[1,t]\to S\times[-0.5,0.5),$ (1) $\displaystyle\Delta(\beta)=\left\{\begin{array}[]{l}s_{i},i=\textit{round}(% \beta)\\ \alpha=\beta-i,\alpha\in[-0.5,0.5)\\ \end{array}\right.,$ (2)

where round(.) is the usual round operation, $s_{i}$ has the closest label to $\beta$ and $\alpha$ is the value of the symbolic translation.

Definition 2 [26, 27]. Let $S=\{s_{1},s_{2},\cdots,s_{t}\}$ be a linguistic term set and $(s_{i},\alpha_{i})$ be a 2-tuple. There is always a mathematical function $\Delta^{-1}$ can be designed, such that, from a 2-tuple $(s_{i},\alpha_{i})$ it return its equivalent value $\beta\in[1,t]\subset R$ , which is:

$\displaystyle\Delta^{-1}:S\times[-0.5,0.5)\to[1,t],$ (3) $\displaystyle\Delta^{-1}(s_{i},\alpha)=i+\alpha=\beta,$ (4)

From Definitions 1 and 2, we can conclude that the conversion of a linguistic term into a linguistic 2-tuple consists of giving a value 0 as symbolic translation:

$\displaystyle\Delta(s_{i})=(s_{i},0),$ (5)

2.2 Intuitionistic fuzzy set

Atanassov [7] extended the fuzzy set to the IFS, defined as follows:

Definition 3 [7]. An IFS $A$ in $X$ is given by

$\displaystyle A=\{\langle x,\mu_{A}(x),\nu_{A}(x)\rangle|x\in X\}$ (6)

Where $\mu_{A}:X\to[0,1]$ and $\nu_{A}:X\to[0,1]$ , with the condition $0\leqslant\mu_{A}(x)+\nu_{A}(x)\leqslant 1$ , $\forall x\in X$ . The $\mu_{A}(x)$ and $\nu_{A}(x)$ depict, respectively, the membership and non-membership of the element $x$ to the set $A$ .

Definition 4 [30]. Let $\tilde{a}=(\mu,\nu)$ be an IFN, a score function $S$ of an IFN can be defined:

$\displaystyle S(\tilde{a})=\mu-\nu,S(\tilde{a})\in[-1,1].$ (7)

Definition 5 [31]. Let $\tilde{a}=(\mu,\nu)$ be an IFN, an accuracy function $H$ of an IFN can be defined:

$\displaystyle H(\tilde{a})=\mu+\nu,H(\tilde{a})\in[0,1].$ (8)

to evaluate the degree of accuracy of $\tilde{a}=(\mu,\nu)$ . The larger the value of $H(\tilde{a})$ , the more the degree of accuracy of the $\tilde{a}$ is.

2.3 Intuitionistic 2-tuple linguistic sets

In the following, Beg and Rashid [29] defined the concepts of the intuitionistic 2-tuple linguistic sets (I2TLSs) on the basis of the IFSs and 2-tuple linguistic information model.

Definition 6 [29]. An I2TLSs $\phi$ in $X$ is given

$\displaystyle\phi=\{(s_{\theta(x)},\rho),(\mu_{\phi}(x),\nu_{\phi}(x)),x\in X\}$ (9)

where $s_{\theta(x)}\in S$ , $\rho\in[-0.5,0.5)$ , $u_{\phi}(x)\in[0,1]$ and $v_{\phi}(x)\in[0,1]$ , with the condition $0\leqslant u_{\phi}(x)+v_{\phi}(x)\leqslant 1$ , $\forall x\in X$ . The $\mu_{\phi}(x),\nu_{\phi}(x)$ depict, respectively, the membership and non-membership of the element $x$ to linguistic variable $(s_{\theta(x)},\rho)$ . For convenience, we call $\phi=\langle(s_{\theta},\rho),(u_{\phi},v_{\phi})\rangle$ an intuitionistic 2-tuple linguistic number (I2TLN), where $\mu_{\phi}\in[0,1],\nu_{\phi}\in[0,1]$ , $\mu_{\phi}+\nu_{\phi}\leqslant 1$ , $s_{\theta}\in S$ and $\rho\in[-0.5,0.5)$ .

Definition 7 [29]. Let $\phi_{1}=\langle(s_{\theta_{1}},\rho_{1}),(u_{\phi_{1}},v_{\phi_{1}})\rangle$ and $\phi_{2}=\langle(s_{\theta_{2}},\rho_{2}),(u_{\phi_{2}},v_{\phi_{2}})\rangle$ be two I2TLNs, $S(\phi_{1})=\Delta(\Delta^{-1}(s_{\theta_{1}},\rho_{1})\cdot\frac{1+\mu_{\phi_% {1}}-\nu_{\phi_{1}}}{2})$ and $S(\phi_{2})=\Delta(\Delta^{-1}(s_{\theta_{2}},\rho_{2})\cdot\frac{1+\mu_{\phi_% {2}}-\nu_{\phi_{2}}}{2})$ be the scores of $\phi_{1}$ and $\phi_{2}$ , respectively, and let $H(\phi_{1})=\Delta(\Delta^{-1}(s_{\theta_{1}},\rho_{1})\cdot\frac{\mu_{\phi_{1% }}+\nu_{\phi_{1}}}{2})$ and $H(\phi_{2})=\Delta(\Delta^{-1}(s_{\theta_{2}},\rho_{2})\cdot\frac{\mu_{\phi_{2% }}+\nu_{\phi_{2}}}{2})$ be the accuracy degrees of $\phi_{1}$ and $\phi_{2}$ , respectively, then if $S(\phi_{1})<S(\phi_{2})$ , then $\phi_{1}<\phi_{2}$ ; if $S(\phi_{1})=S(\phi_{2})$ , then (1) if $H(\phi_{1})=H(\phi_{2})$ , then $\phi_{1}=\phi_{2}$ ; (2) if $H(\phi_{1})<H(\phi_{2})$ , $\phi_{1}<\phi_{2}$ .

Definition 8 [29]. Let $\phi_{1}=\langle(s_{\theta_{1}},\rho_{1}),(u_{\phi_{1}},v_{\phi_{1}})\rangle$ and $\phi_{2}=\langle(s_{\theta_{2}},\rho_{2}),(u_{\phi_{2}},v_{\phi_{2}})\rangle$ be two I2TLNs, the normalized Hamming distance between $\phi_{1}$ and $\phi_{2}$ is:

$\displaystyle d(\phi_{1},\phi_{2})=\frac{1}{2T}\left[\left|\begin{array}[]{l}{% (1+\mu_{\phi_{1}}-\nu_{\phi_{1}})\cdot\Delta^{-1}(s_{\theta_{1}},\rho_{1})-}\\ {(1+\mu_{\phi_{2}}-\nu_{\phi_{2}})\cdot\Delta^{-1}(s_{\theta_{2}},\rho_{2})}% \end{array}\right|\right]$ (10)

2.4 The TODIM method

The TODIM approach [32], developed to consider the DM’s psychological behavior, can effectively solve the MADM problems. Based on the prospect theory, this approach depicts the dominance of each alternative over others by constructing a function of multi-attribute values [33, 34].

Let $G=\{G_{1},G_{2},\cdots,G_{n}\}$ be the set of attributes, $w=(w_{1},w_{2},\cdots,w_{n})$ be the weight vector of attributes $G_{j}$ , where $w_{j}\in[0,1]$ , $j=1,2,\cdots,n$ , $\sum\limits_{j=1}^{n}w_{j}=1$ . Let $A=\{A_{1},A_{2},\cdots,A_{m}\}$ be a discrete set of alternatives. Suppose that $A=(a_{ij})_{m\times n}$ be a decision matrix, where $a_{ij}$ is the attribute value, given by an expert, for the alternative $A_{i}\in A$ with respect to the attribute $G_{j}\in G$ , $i=1,2,\cdots,m$ , $j=1,2,\cdots,n$ . We define $w_{jr}=w_{j}/w_{r}(j,r=1,2,\cdots,n)$ are the relative weight of the attribute $G_{j}$ to $G_{r}$ , and $w_{r}=\max\{w_{j}|j=1,2,\cdots,n\}$ , and $0\leqslant w_{jr}\leqslant 1$ . Then the classical TODIM approach involves the following steps:

Step 1. Normalize the decision matrix $A=(a_{ij})_{m\times n}$ into the matrix $B=(b_{ij})_{m\times n}$ .

Step 2. Compute the dominance degree of $A_{i}$ over each alternative $A_{t}$ with respect to the attribute $G_{j}$ :

$\displaystyle\delta(A_{i},A_{t})=\sum\limits_{j=1}^{n}\phi_{j}(A_{i},A_{t}),(i% ,t=1,2,\cdots,m)$ (11)

where

$\displaystyle\phi_{j}(A_{i},A_{t})=\left\{\begin{array}[]{ll}\sqrt{{w_{jr}(b_{% ij}-b_{tj})\left/\sum\limits_{j=1}^{n}w_{jr}\right.}},&\text{if }b_{ij}-b_{tj}% >0\\ 0,&\text{if }b_{ij}-b_{tj}=0\\ -\frac{1}{\theta}\sqrt{{\left.\left(\sum\limits_{j=1}^{n}w_{jr}\right)(b_{tj}-% b_{ij})\right/w_{jr}}},&\text{if }b_{ij}-b_{tj}<0\\ \end{array}\right.$ (12)

and the parameter $\theta$ shows the attenuation factor of the losses. If $b_{ij}-b_{tj}>0$ , then $\phi_{j}(A_{i},A_{t})$ represents a gain; if $b_{ij}-b_{tj}<0$ , then $\phi_{j}(A_{i},A_{t})$ signifies a loss.

Step 3. Compute the overall value of the alternative $A_{i}$ by the following formula:

$\displaystyle\phi(A_{i})=\frac{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})-{% \mathop{\min}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})\right% \}}{{\mathop{\max}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_{t})% \right\}-{\mathop{\min}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\delta(A_{i},A_% {t})\right\}},i=1,2,\cdots,m.$ (13)

Step 4. Rank all alternatives and select the most desirable alternative by the overall values $\phi(A_{i})(i=1,2,\cdots,m)$ . The alternative with the alternative with the minimum value is the worst. Inversely, the maximum value is the most desirable one.

3. TODIM method for MADM problems with I2TLNs

The following mathematical notations are used to depict the MADM problems with I2TLNs. Let $A=\{A_{1},A_{2},\cdots,A_{m}\}$ be a discrete set of given alternatives, and $G=\{G_{1},G_{2},\cdots,G_{n}\}$ be the set of predefined attributes. Let $w=(w_{1},w_{2},\cdots,w_{n})$ be the weight vector of attributes, where $w_{j}\in[0,1]$ , $\sum\limits_{j=1}^{n}w_{j}=1$ . Suppose that $R=(r_{ij})_{m\times n}=\langle(s_{ij},\rho_{ij}),(\mu_{ij},\nu_{ij})\rangle_{m% \times n}$ is the decision matrix, where $r_{ij}$ take the form of the I2TLNs, where $\mu_{ij}$ indicates the degree that the alternative $A_{i}$ satisfies the attribute $G_{j}$ , $\nu_{ij}$ indicates the degree that the alternative $A_{i}$ doesn’t satisfy the attribute $G_{j}$ , $\mu_{ij}\in[0,1]$ , $\eta_{ij}\in[0,1]\nu_{ij}\in[0,1]$ , $(\mu_{ij})^{2}+(\nu_{ij})^{2}\leqslant 1$ , $s_{ij}\in S$ , $\rho_{ij}\in[-0.5,0.5)$ , $i=1,2,\cdots,m$ , $j=1,2,\cdots,n$ .

To solve the MADM problem with I2TLNs, we shall define the intuitionistic 2-tuple linguistic TODIM (I2TL-TODIM) method, which is proposed based on the prospect theory and can depict the DMs’ behaviors under risk.

Firstly, we calculate the relative weight of each attribute $G_{j}$ as:

$\displaystyle w_{jr}=w_{j}/w_{r},j,r=1,2,\cdots,n.$ (14)

where $w_{j}$ is the weight of the of $G_{j}$ , $w_{r}=\max\{w_{j}|j=1,2,\cdots,n\}$ , and $0\leqslant w_{jr}\leqslant 1$ .

Based on the Eq. (15), we can derive the dominance degree of $A_{i}$ over each alternative $A_{t}$ for $G_{j}$ :

$\displaystyle\eta_{j}(A_{i},A_{t})=\left\{\begin{array}[]{ll}\sqrt{{w_{jr}d(r_% {ij},r_{tj})\left/\sum\limits_{j=1}^{n}w_{jr}\right.}},&\text{if }r_{ij}>r_{tj% }\\ 0,&\text{if }r_{ij}=r_{tj}\\ -\frac{1}{\theta}\sqrt{{\left.\left(\sum\limits_{j=1}^{n}w_{jr}\right)d(r_{ij}% ,r_{tj})\right/w_{jr}}},&\text{if }r_{ij}<r_{tj}\\ \end{array}\right.$ (15) $\displaystyle d(r_{ij},r_{tj})=\frac{1}{2T}\left[\left|\begin{array}[]{l}{(1+(% \mu_{ij})^{2}-(\nu_{ij})^{2})\cdot\Delta^{-1}(s_{ij},\rho_{ij})-}\\ {(1+(\mu_{tj})^{2}-(\nu_{tj})^{2})\cdot\Delta^{-1}(s_{tj},\rho_{tj})}\end{% array}\right|\right]$ (16)

where the parameter $\theta$ shows the attenuation factor of the losses, and $d(r_{ij},r_{tj})$ is to measure the distances between the I2TLNs $r_{ij}$ and $r_{tj}$ by Definition 8. If $r_{ij}>r_{tj}$ , then $\eta_{j}(A_{i},A_{t})$ represents a gain; if $r_{ij}<r_{tj}$ , then $\eta_{j}(A_{i},A_{t})$ signifies a loss.

In order to indicate the functions $\eta_{j}(A_{i},A_{t})$ clearly, we depict it in a dominance degree matrix with respect to $G_{j}$ as:

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\begin{array}[]{cccc}∼{% }∼{}∼{}∼{}A_{1}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{2}&\ldots&∼{}∼{}∼{}∼{}∼% {}∼{}A_{m}\\ \end{array}$ $\displaystyle\eta_{j}=[\eta_{j}(A_{i},A_{t})]_{m\times m}=\begin{array}[]{c}A_% {1}\\ A_{2}\\ {\vdots}\\ A_{m}\\ \end{array}\left[\begin{array}[]{cccc}{o}&{\eta_{j}(A_{1},A_{2})}&{\cdots}&{% \eta_{j}(A_{1},A_{m})}\\ {\eta_{j}(A_{2},A_{1})}&{0}&{\cdots}&{\eta_{j}(A_{2},A_{m})}\\ {\vdots}&{\vdots}&{\cdots}&{\vdots}\\ {\eta_{j}(A_{m},A_{1})}&{\eta_{j}(A_{m},A_{2})}&{\cdots}&{0}\\ \end{array}\right]$ $\displaystyle j=1,2,\cdots,n.$ (17)

based on which we could derive the overall dominance degree of the $A_{i}$ over each $A_{t}$ with Eq. (18).

$\displaystyle\pi(A_{i},A_{t})=\sum\limits_{j=1}^{n}\eta_{j}(A_{i},A_{t}),(i,t=% 1,2,\cdots,m)$ (18)

Therefore, by Eq. (18), the overall dominance degree matrix can be got as:

$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\begin{array}[]{cccc}∼{% }A_{1}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{2}&\ldots&∼{}∼{}∼{}∼{}∼{}A_{m}\\ \end{array}$ $\displaystyle\pi=[\pi(A_{i},A_{t})]_{m\times m}=\begin{array}[]{c}A_{1}\\ A_{2}\\ \vdots\\ A_{m}\\ \end{array}\left[\begin{array}[]{cccc}{o}&{\pi(A_{1},A_{2})}&{\cdots}&{\pi(A_{% 1},A_{m})}\\ {\pi(A_{2},A_{1})}&{0}&{\cdots}&{\pi(A_{2},A_{m})}\\ {\vdots}&{\vdots}&{\cdots}&{\vdots}\\ {\pi(A_{m},A_{1})}&{\pi(A_{m},A_{2})}&{\cdots}&{0}\\ \end{array}\right]$ (19)

Finally, the overall value of each $A_{i}$ can be obtained with the following equation:

$\displaystyle\pi(A_{i})=\frac{\sum\limits_{t=1}^{m}\pi(A_{i},A_{t})-{\mathop{% \min}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\pi(A_{i},A_{t})\right\}}{{% \mathop{\max}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\pi(A_{i},A_{t})\right\}-% {\mathop{\min}\limits_{i}}\left\{\sum\limits_{t=1}^{m}\pi(A_{i},A_{t})\right\}% },i=1,2,\cdots,m.$ (20)

And the order of each alternative can be ranked by the principle, that is, the greater the overall value $\pi(A_{i})(i=1,2,\cdots,m)$ , the better the alternative $A_{i}$ . In general, I2TL-TODIM method includes the following calculating steps:

Step 1. Identify the decision matrix $R=(r_{ij})_{m\times n}=\langle(s_{ij},\rho_{ij}),(\mu_{ij},\nu_{ij})\rangle_{m% \times n}$ given by the DM in the MADM, where $r_{ij}$ is a I2TLN.

Step 2. Calculate the relative weight of each attribute $G_{j}$ by employing Eq. (14).

Step 3. Calculate the dominance degree $\eta_{j}(A_{i},A_{t})$ of the $A_{i}$ over each $A_{t}$ with respect to the $G_{j}$ by Eq. (3).

Step 4. Calculate the overall dominance degree $\pi(A_{i},A_{t})$ of $A_{i}$ over each alternative $A_{t}$ using Eq. (18).

Step 5. Derive the overall value $\pi(A_{i})(i=1,2,\cdots,m)$ of each alternative $A_{i}$ by using Eq. (20).

Step 6. Determine the order of the alternatives in accordance with the overall values $\pi(A_{i})(i=1,2,\cdots,m)$ .

4. Numerical example and comparative analysis

4.1 Numerical example

Outsourcing has become a global business development trend, when enterprises face competitive challenges, and excellent outsourcing supply manufacturers, suppliers can use all the resources, so that enterprises can focus on their core functions and capabilities, improve production, profitability, reduce costs, increase enterprise competitive advantage. Thus, in this section we shall present a numerical example for selecting the service outsourcing provider with I2TLNs in order to illustrate the method proposed in this paper. Let us suppose there is a problem for selecting the service outsourcing provider which is classical MADM problems. There is a panel with five possible service outsourcing providers of communications industry $A_{i}(i=1,2,3,4,5)$ to select. The expert team selects four attributes to evaluate the five possible service outsourcing providers of communications industry: ⟀ G ${}_{1}$ is the business reputation; ⟁ G ${}_{2}$ is the technical ability; ⟂ G ${}_{3}$ is the management ability; ⟃ G ${}_{4}$ is the service quality. The five possible service outsourcing providers of communications industry $A_{i}(i=1,2,\cdots,5)$ are to be evaluated using the I2TLNs by the DMs under the above four attributes (whose weighting vector is $\omega=(0.2,0.1,0.3,0.4)^{T}$ ), and construct the following matrix $\tilde{R}=(\tilde{r}_{ij})_{5\times 4}$ as listed in Table 1.

Table 1
The decision matrix with I2TLNs

	G ${}_{1}$	G ${}_{2}$
A ${}_{1}$	${<}$ (S ${}_{5}$ , 0), (0.60, 0.30) ${>}$	${<}$ (S ${}_{4}$ , 0.2), (0.70, 0.20) ${>}$
A ${}_{2}$	${<}$ (S ${}_{2}$ , 0), (0.70, 0.40) ${>}$	${<}$ (S ${}_{3}$ , $-$ 0.3), (0.40, 0.50) ${>}$
A ${}_{3}$	${<}$ (S ${}_{4}$ , $-$ 0.2), (0.80, 0.10) ${>}$	${<}$ (S ${}_{4}$ , 0), (0.60, 0.20) ${>}$
A ${}_{4}$	${<}$ (S ${}_{6}$ , 0.4), (0.80, 0.20) ${>}$	${<}$ (S ${}_{7}$ , 0), (0.90, 0.10) ${>}$
A ${}_{5}$	${<}$ (S ${}_{3}$ , $-$ 0.3), (0.70, 0.20) ${>}$	${<}$ (S ${}_{2}$ , 0.1), (0.80, 0.10) ${>}$
	G ${}_{3}$	G ${}_{4}$
A ${}_{1}$	${<}$ (S ${}_{7}$ , 0), (0.70, 0.20) ${>}$	${<}$ (S ${}_{5}$ , 0), (0.70, 0.30) ${>}$
A ${}_{2}$	${<}$ (S ${}_{3}$ , 0), (0.60, 0.30) ${>}$	${<}$ (S ${}_{1}$ , 0), (0.80, 0.20) ${>}$
A ${}_{3}$	${<}$ (S ${}_{4}$ , 0.1), (0.60, 0.20) ${>}$	${<}$ (S ${}_{5}$ , $-$ 0.2), (0.70, 0.20) ${>}$
A ${}_{4}$	${<}$ (S ${}_{6}$ , $-$ 0.2), (0.60, 0.30) ${>}$	${<}$ (S ${}_{7}$ , 0.3), (0.80, 0.10) ${>}$
A ${}_{5}$	${<}$ (S ${}_{2}$ , 0.3), (0.70, 0.20) ${>}$	${<}$ (S ${}_{2}$ , 0), (0.60, 0.30) ${>}$

Then, we use the proposed I2TL-TODIM method to select the best service outsourcing providers of communications industry.

Firstly, since $w_{4}=\max\{w_{1},w_{2},w_{3},w_{4}\}$ , then $G_{4}$ is the reference attribute and the reference weight is $w_{r}=0.4$ . Therefore, the relative weights of all the attributes $G_{j}(j=1,2,3,4)$ are $w_{1r}=0.50,w_{2r}=0.25,w_{3r}=0.75$ and $w_{4r}=1.00$ , respectively. Then, we need to calculate the dominance degree of the candidate $A_{i}$ over each candidate $A_{t}$ with respect to the attributes $G_{j}(j=1,2,3,4)$ . Here we take an example from the information given by the decision matrix. Let $\theta=2.5$ , then the dominance degree matrices with respect to the attributes $G_{j}(j=1,2,3,4)$ are:

$\displaystyle\qquad\qquad\qquad\begin{array}[]{ccccc}A_{1}&∼{}∼{}∼{}∼{}∼{}∼{}∼% {}∼{}∼{}∼{}A_{2}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{3}&∼{}∼{}∼{}∼{}∼{}∼{}A_{4}&∼{}∼% {}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{5}\\ \end{array}$ $\displaystyle\phi_{1}=\begin{array}[]{c}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\\ A_{5}\\ \end{array}\left[\begin{array}[]{rrrrr}0.0000&0.2252&-0.4055&0.2357&-0.3730\\ 0.2252&0.0000&0.2566&0.3260&0.1262\\ 0.0472&0.2202&0.0000&0.2404&-0.3609\\ -0.4715&-0.6520&-0.1387&0.0000&-0.6012\\ 0.1865&0.1262&0.2428&0.3006&0.0000\\ \end{array}\right]$ $\displaystyle\qquad\qquad\qquad\begin{array}[]{ccccc}A_{1}&∼{}∼{}∼{}∼{}∼{}∼{}∼% {}∼{}∼{}∼{}A_{2}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{3}&∼{}∼{}∼{}∼{}∼{}∼{}A_{4}&∼{}∼% {}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{5}\\ \end{array}$ $\displaystyle\phi_{2}=\begin{array}[]{c}{A_{1}}\\ {A_{2}}\\ {A_{3}}\\ {A_{4}}\\ {A_{5}}\\ \end{array}\left[\begin{array}[]{rrrrr}0.0000&0.1611&-0.4859&0.2156&-0.5521\\ 0.1611&0.0000&0.1702&0.2692&0.0831\\ 0.0761&0.1420&0.0000&0.2287&-0.4607\\ -0.8626&-1.0767&-0.4573&0.0000&-1.0241\\ 0.1380&0.0831&0.1582&0.2560&0.0000\\ \end{array}\right]$ $\displaystyle\qquad\qquad\quad\begin{array}[]{ccccc}∼{}∼{}A_{1}&∼{}∼{}∼{}∼{}∼{% }∼{}A_{2}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{3}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{4}&∼{% }∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{5}\\ \end{array}$ $\displaystyle\phi_{3}=\begin{array}[]{c}{A_{1}}\\ {A_{2}}\\ {A_{3}}\\ {A_{4}}\\ {A_{5}}\\ \end{array}\left[\begin{array}[]{rrrrr}0.0000&0.3686&-0.0460&-0.3257&-0.5095\\ 0.3686&0.0000&0.2708&0.2760&-0.1345\\ 0.3186&0.1853&0.0000&0.2046&-0.2813\\ 0.2442&0.2760&-0.2442&0.0000&-0.3919\\ 0.3821&0.1009&0.2797&0.2939&0.0000\\ \end{array}\right]$ $\displaystyle\qquad\qquad\qquad\begin{array}[]{ccccc}A_{1}&∼{}∼{}∼{}∼{}∼{}∼{}∼% {}∼{}∼{}∼{}A_{2}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{3}&∼{}∼{}∼{}∼{}∼{}∼{}A_{4}&∼{}∼% {}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{5}\\ \end{array}$ $\displaystyle\phi_{4}=\begin{array}[]{c}{A_{1}}\\ {A_{2}}\\ {A_{3}}\\ {A_{4}}\\ {A_{5}}\\ \end{array}\left[\begin{array}[]{rrrrr}0.0000&-0.4078&-0.2806&0.3577&-0.3734\\ 0.4078&0.0000&0.4156&0.5425&0.1639\\ 0.1146&-0.3913&0.0000&0.3757&-0.3554\\ -0.3577&-0.5425&-0.1499&0.0000&-0.5171\\ 0.3734&0.1639&-0.4012&0.5171&0.0000\\ \end{array}\right]$

Secondly, the overall dominance degree of the candidate $A_{i}$ over each candidate $A_{t}$ can be obtained by Eq. (18), and the overall dominance degree matrix is:

$\displaystyle\qquad\qquad\quad\begin{array}[]{ccccc}∼{}∼{}A_{1}&∼{}∼{}∼{}∼{}∼{% }∼{}∼{}∼{}∼{}∼{}A_{2}&∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{3}&∼{}∼{}∼{}∼{}∼{}∼{}A_{4}% &∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}∼{}A_{5}\\ \end{array}$ $\displaystyle\delta=\begin{array}[]{c}{A_{1}}\\ {A_{2}}\\ {A_{3}}\\ {A_{4}}\\ {A_{5}}\\ \end{array}\left[\begin{array}[]{rrrrr}0.0000&0.3471&-1.2181&0.4834&-1.8080\\ 1.1627&0.0000&1.1132&1.4137&0.2386\\ 0.5565&0.1561&0.0000&1.0494&-1.4582\\ -1.4475&-1.9951&-0.9901&0.0000&-2.5343\\ 1.0801&0.4740&1.0820&1.3676&0.0000\\ \end{array}\right]$

Then, we derive the overall value $\delta(A_{i})(i=1,2,\cdots,m)$ of each alternative $A_{i}$ using Eq. (20):

$\displaystyle\delta(A_{1})=3.5520,\delta(A_{2})=2.9518,\delta(A_{3})=3.2063$ $\displaystyle{\delta(A_{4})=4.3024,\delta(A_{5})=1.8008}$

Finally, we determine the ranking of the alternatives according to the overall values $\delta(A_{i})(i=1,2,3,4,5):A_{4}\succ A_{1}\succ A_{3}\succ A_{2}\succ A_{5}$ , and thus the most desirable service outsourcing provider of communications industry is $A_{4}$ .

5. Conclusion

The MADM methods are widely applied in real-life decision-making situations, and the TODIM method which could fully consider DM’s bounded rationality. Additionally, the intuitionistic 2-tuple linguistic sets (I2TLSs) are more easily express uncertain or fuzzy information. Therefore, it is meaningful and valuable to research uncertain MADM problems with the I2TLNs which considering the behavior preferences of the DMs. This paper use TODIM method to solve MADM with the I2TLNs. Firstly, the definition, comparison and distance of I2TLNs are briefly introduced, and classical TODIM for MADM are introduced. Then, the intuitionistic 2-tuple linguistic TODIM (I2TL-TODIM) method is defined to deal with MADM problems, and its important characteristic is to fully consider the decision makers’ bounded rationality which is a real action in practical MADM. Finally, a numerical example for service outsourcing providers selection is proposed and a comparative analysis is also given. In our future researches, the proposed models and methods of I2TLNs needs to be extended in the MADM [35, 36, 37, 38, 39, 40], risk assessment [41, 42, 43] and some other uncertainty and fuzziness [44, 45, 46, 47, 48].

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. 14YJA630017).

References

Zadeh

. Fuzzy Sets. In: Information and Control, 1965, pp. 338–356.

Orlovsky

. Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems. 1978; 1: 155–167.

Fei

. A novel method for heterogeneous multi-attribute group decision making with preference deviation. Computers & Industrial Engineering. 2018; 124: 58–64.

Qiu

Zheng

. Some operators of intuitionistic uncertain 2-tuple linguistic variables and application to multi-attribute group decision making with heterogeneous relationship among attributes. Journal of Intelligent & Fuzzy Systems. 2018; 34: 599–611.

Wang

Madden

Ding

Tang

Sun

Zhang

Zhou

. Analyzing stock market trends using social media user moods and social influence. Journal of the Association for Information Science and Technology. 2019; 70: 1000–1013.

Szmidt

Kacprzyk

. Distances between intuitionistic fuzzy sets. Fuzzy Sets and Systems. 2000; 114: 505–518.

Atanassov

. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20: 87–96.

Atanassov

. More on intuitionistic fuzzy-sets. Fuzzy Sets and Systems. 1989; 33: 37–45.

Bao

Xie

Long

Wei

. MADM method based on prospect theory and evidential reasoning approach with unknown attribute weights under intuitionistic fuzzy environment. Expert Systems with Applications. 2017; 88: 305–317.

10.

Chen

Wang

. Research on performance evaluation of projects loaned by international financial organizations with fuzzy number intuitionistic fuzzy information. Journal of Intelligent & Fuzzy Systems. 2017; 33: 3505–3514.

11.

Damgaci

Boran

. Evaluation of turkey’s renewable energy using intuitionistic fuzzy TOPSIS method. Journal of Polytechnic-Politeknik Dergisi. 2017; 20: 629–637.

12.

Das

Guha

Mesiar

. Extended bonferroni mean under intuitionistic fuzzy environment based on a strict t-conorm. Ieee Transactions on Systems Man Cybernetics-Systems. 2017; 47: 2083–2099.

13.

Liang

Liu

. Three-way decisions with intuitionistic fuzzy decision-theoretic rough sets based on point operators. Information Sciences. 2017; 375: 183–201.

14.

Liu

Mahmood

Khan

. Multi-attribute decision-making based on prioritized aggregation operator under hesitant intuitionistic fuzzy linguistic environment. Symmetry-Basel. 2017; 9.

15.

Liu

Changyong

. Some Einstein aggregating operators for trapezoidal intuitionistic fuzzy MAGDM and application in investment evolution. Journal of Intelligent & Fuzzy Systems. 2017; 32: 63–74.

16.

Jiang

Wei

Liu

Zheng

. Intuitionistic fuzzy power aggregation operator based on entropy and its application in decision making. International Journal of Intelligent Systems. 2018; 33: 49–67.

17.

Chen

Yang

. A time-preference and VIKOR-based dynamic intuitionistic fuzzy decision making method. Filomat. 2018; 32: 1523–1533.

18.

. Models for multiple attribute decision making with intuitionistic fuzzy information. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems. 2007; 15: 285–297.

19.

Chen

. Bivariate models of optimism and pessimism in multi-criteria decision-making based on intuitionistic fuzzy sets. Information Sciences. 2011; 181: 2139–2165.

20.

Wan

. Atanassov’s intuitionistic fuzzy programming method for heterogeneous multiattribute group decision making with atanassov’s intuitionistic fuzzy truth degrees. Ieee Transactions on Fuzzy Systems. 2014; 22: 300–312.

21.

Wei

. TODIM method for performance appraisal on social-integration-based rural reconstruction with interval-valued intuitionistic fuzzy information. Journal of Intelligent & Fuzzy Systems. 2019; 37: 1731–1740.

22.

Wei

. Some interval-valued intuitionistic fuzzy dombi heronian mean operators and their application for evaluating the ecological value of forest ecological tourism demonstration areas. International Journal of Environmental Research and Public Health. 2020; 17: 829.

23.

Wang

Gao

. Models for competiveness evaluation of tourist destination with some interval-valued intuitionistic fuzzy Hamy mean operators. Journal of Intelligent & Fuzzy Systems. 2019; 36: 5693–5709.

24.

Gao

Wei

. VIKOR method for financing risk assessment of rural tourism projects under interval-valued intuitionistic fuzzy environment. Journal of Intelligent & Fuzzy Systems. 2019; 37: 2001–2008.

25.

Wei

. 2-tuple intuitionistic fuzzy linguistic aggregation operators in multiple attribute decision making. Iranian Journal of Fuzzy Systems. 2019; 16: 159–174.

26.

Herrera

Martinez

. A 2-tuple fuzzy linguistic representation model for computing with words. Ieee Transactions on Fuzzy Systems. 2000; 8: 746–752.

27.

Herrera

Martinez

. An approach for combining linguistic and numerical information based on the 2-tuple fuzzy linguistic representation model in decision-making. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems. 2000; 8: 539–562.

28.

Herrera

Martinez

. The 2-tuple linguistic computational model. Advantages of its linguistic description, accuracy and consistency. International Journal of Uncertainty Fuzziness and Knowledge-Based Systems. 2001; 9: 33–48.

29.

Beg

Rashid

. An intuitionistic 2-tuple linguistic information model and aggregation operators. International Journal of Intelligent Systems. 2016; 31: 569–592.

30.

Chen

Tan

. Handling multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Systems. 1994; 67: 163–172.

31.

Hong

Choi

. Multicriteria fuzzy decision-making problems based on vague set theory. Fuzzy Sets Systems. 2000; 114: 103–113.

32.

Gomes

Rangel

LAD

. An application of the TODIM method to the multicriteria rental evaluation of residential properties. European Journal of Operational Research. 2009; 193: 204–211.

33.

Wang

Wei

. TODIM method for multiple attribute group decision making under 2-tuple linguistic neutrosophic environment. Symmetry-Basel. 2018; 10: 486.

34.

Wei

. Cumulative prospect theory: Performance evaluation of government purchases of home-based elderly-care services using the pythagorean 2-tuple linguistic TODIM method. International Journal of Environmental Research and Public Health. 2020; 17: 1939.

35.

Wang

Wei

. CODAS method for multiple attribute group decision making under 2-tuple linguistic neutrosophic environment. Informatica. 2020; 31: 161–184.

36.

Wei

Lei

Wei

Guo

. Green supplier selection with an uncertain probabilistic linguistic MABAC method. Journal of Intelligent & Fuzzy Systems. 2020; 39: 3125–3136.

37.

Xiao

Zhang

Wei

Guo

Wei

. Green supplier selection in steel industry with intuitionistic fuzzy Taxonomy method. Journal of Intelligent & Fuzzy Systems. 2020; 39: 7247–7258.

38.

Keshavarz Ghorabaee

Zavadskas

Amiri

Esmaeili

. Multi-criteria evaluation of green suppliers using an extended WASPAS method with interval type-2 fuzzy sets. Journal of Cleaner Production. 2016; 137: 213–229.

39.

Turskis

Daniunas

Zavadskas

Medzvieckas

. Multicriteria evaluation of building foundation alternatives. Computer-Aided Civil and Infrastructure Engineering. 2016; 31: 717–729.

40.

Yazdani

Zavadskas

Ignatius

Abad

. Sensitivity analysis in MADM methods: Application of material selection. Inzinerine Ekonomika-Engineering Economics. 2016; 27: 382–391.

41.

Wei

Liang

Zhang

Wei

. Can CBOE gold and silver implied volatility help to forecast gold futures volatility in China? Evidence based on HAR and Ridge regression models, Finance Research Letters. 2019. doi: 10.1016/j.frl.2019.1009.1002.

42.

Wei

Bai

Yang

Wei

. Are industryâ€level indicators more helpful to forecast industrial stock volatility? Evidence from Chinese manufacturing purchasing managers index. Journal of Forecasting. 2020; 39. doi: 10.1002/for.2696.

43.

Wei

Nan

Wei

. The impact of employee welfare on innovation performance: Evidence from China’s manufacturing corporations. International Journal of Production Economics. 2020; 228: 107753.

44.

Lei

Wei

Guo

. QUALIFLEX method for MAGDM with probabilistic uncertain linguistic information and its application to green supplier selection. Journal of Intelligent & Fuzzy Systems. 2020; 39: 6819–6831.

45.

Wang

Wei

. Maximizing deviation method for multiple attribute decision making under q-rung orthopair fuzzy environment. Defence Technology. 2020; 16: 1073–1087.

46.

Liang

Cao

. q-Rung orthopair fuzzy sets-based decision-theoretic rough sets for three-way decisions under group decision making. International Journal of Intelligent Systems. 2019; 34: 3139–3167.

47.

Liu

Peng

Liu

. The distance measures between q-rung orthopair hesitant fuzzy sets and their application in multiple criteria decision making. International Journal of Intelligent Systems. 2019; 34: 2104–2121.

48.

Wang

Liu

. Multi-attribute group decision-making methods based on q-rung orthopair fuzzy linguistic sets. International Journal of Intelligent Systems. 2019; 34: 1129–1157.

Algorithms for multiple attribute group decision making with intuitionistic 2-tuple linguistic information and its application

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 2-tuple linguistic term sets

4.1 Numerical example

Table 1 The decision matrix with I2TLNs

Footnotes

Acknowledgments

References

Table 1
The decision matrix with I2TLNs