Multiple-attribute group decision-making method based on intuitionistic multiplicative linguistic information

Abstract

This paper considers a multiple-attribute group decision-making (MAGDM) problem, where the evaluation information and the weight of attribute both take the form of intuitionistic multiplicative linguistic variables (IMLVs). Some new operational laws for IMLVs are introduced, which can guarantee the closeness of operation. Then, several intuitionistic multiplicative linguistic aggregation operators are proposed, and some desirable properties are investigated. Considering the application of intuitionistic multiplicative linguistic preference relation (IMLPR), a new concept of consistent IMLPR is defined. We build a mathematical programming model to obtain the normalized intuitionistic multiplicative linguistic priority weight vector from an IMLPR. An automatic convergent algorithm is designed to repair the unacceptably consistent IMLPR to be of acceptable consistency. Moreover, a model is established to estimate the unknown values of an incomplete IMLPR. Finally, the novel method is applied to some practical problems and the comparison analysis is conducted. The new approach in this paper has two prominent advantages: (i) the constructed consistency-based model can deal with a group decision-making (GDM) problem under the intuitionistic multiplicative linguistic environment; (ii) the proposed aggregation operators can fuse the intuitionistic multiplicative linguistic information and solve the multiple-attribute decision-making (MADM) problem.

Appendix

Given an IMLPR $R = (r_{ij})_{n\times n}$ with $r_{ij} = (s_{\mu _{ij}}, s_{\nu _{ij}})$, if some values of R are unknown, then R is called an incomplete IMLPR. In this Appendix, an approach is provided to estimate the unknown values. From Definition 7 and Theorem 5, we can easily get the following theorem.

Theorem 9

An IMLPR $R = (r_{ij})_{n\times n}$ with $r_{ij} = (s_{\mu _{ij}}, s_{\nu _{ij}})$ is called consistent if

$$\begin{aligned}&\log _{q}I(s_{\mu _{ij}})+\log _{q}I(s_{\mu _{jk}})+\log _{q}I(s_{\mu _{ki}})\\&\quad = \log _{q}I(s_{\nu _{ij}})+\log _{q}I(s_{\nu _{jk}})+\log _{q}I(s_{\nu _{ki}}),~~ i,j,k = 1,2,\ldots ,n,~i<j<k. \end{aligned}$$

Assume

$$\begin{aligned} \zeta _{ijk}&= \log _{q}I(s_{\mu _{ij}})+\log _{q}I(s_{\mu _{jk}})+\log _{q}I(s_{\mu _{ki}})\\&\quad -\log _{q}I(s_{\nu _{ij}})-\log _{q}I(s_{\nu _{jk}})-\log _{q}I(s_{\nu _{ki}}),~~ i,j,k = 1,2,\ldots ,n,~i<j<k. \end{aligned}$$

Considering the property of consistency and from Theorem 9, the unknown values should make the absolute deviation $\vert \zeta _{ijk}\vert$ as small as possible. Let $U_{ij} = \{(i,j)\mid i<j~\mathrm{and}~s_{\mu _{ij}}~ \mathrm{is} ~\mathrm{unknown}\}$ and $V_{ij} = \{(i,j)\mid i<j~\mathrm{and}~s_{\nu _{ij}}~ \mathrm{is} ~\mathrm{unknown}\}$, we can build a Model 5 to estimate the missing values.

$$\begin{aligned}&{\textbf {Model}}~ {\textbf {5}} ~~~~~~~ \min ~z = \sum _{i = 1}^{n-2}\sum _{j = i+1}^{n-1}\sum _{k = j+1}^{n}\vert \zeta _{ijk}\vert \\&\begin{aligned}&s.t. {\left\{ \begin{array}{ll} \log _{q}I(s_{\mu _{ij}})+\log _{q}I(s_{\mu _{jk}})+\log _{q}I(s_{\mu _{ki}})-\log _{q}I(s_{\nu _{ij}})\\ -\log _{q}I(s_{\nu _{jk}})-\log _{q}I(s_{\nu _{ki}})-\zeta _{ijk} = 0,~~ 1\le i<j<k\le n,\\ I(s_{\mu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\in U_{ij}, ~(i,j)\notin V_{ij}, \\ I(s_{\nu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\notin U_{ij}, ~(i,j)\in V_{ij}, \\ I(s_{\mu _{ij}}),I(s_{\nu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\in U_{ij}, ~(i,j)\in V_{ij}. \end{array}\right. } \end{aligned} \end{aligned}$$

$$\begin{aligned}&{\textbf {Model}}~ {\textbf {6}} ~~~~~~~ \min ~z = \sum _{i = 1}^{n-2}\sum _{j = i+1}^{n-1}\sum _{k = j+1}^{n}(a_{ijk}^{+}+a_{ijk}^{-})\\&\begin{aligned}&s.t. {\left\{ \begin{array}{ll} \log _{q}I(s_{\mu _{ij}})+\log _{q}I(s_{\mu _{jk}})+\log _{q}I(s_{\mu _{ki}})-\log _{q}I(s_{\nu _{ij}})\\ -\log _{q}I(s_{\nu _{jk}})-\log _{q}I(s_{\nu _{ki}})-a_{ijk}^{+}+a_{ijk}^{-} = 0,~~ 1\le i<j<k\le n,\\ I(s_{\mu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\in U_{ij}, ~(i,j)\notin V_{ij}, \\ I(s_{\nu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\notin U_{ij}, ~(i,j)\in V_{ij}, \\ I(s_{\mu _{ij}}),I(s_{\nu _{ij}})\in [1/q,q],~ I(s_{\mu _{ij}})I(s_{\nu _{ij}})\le 1,~~(i,j)\in U_{ij}, ~(i,j)\in V_{ij},\\ a_{ijk}^{+}\ge 0,~a_{ijk}^{-}\ge 0,~~ 1\le i<j<k\le n. \end{array}\right. } \end{aligned} \end{aligned}$$

By deleting the absolute value symbols, Model 5 can be rewritten as a Model 6.

After solving Model 6, we can obtain some optimal solutions $I(s_{\mu _{ij}^{*}})$ ($(i,j)\in U_{ij}$) and $I(s_{\nu _{ij}^{*}})$ ($(i,j)\in V_{ij}$), then from Definition 3, a complete IMLPR $R = (r_{ij})_{n\times n}$ can be constructed.