Geometric consistency of triangular fuzzy multiplicative preference relation and its application to group decision making

Abstract

The triangular fuzzy multiplicative preference relation (TFMPR) has attracted the attention of many scholars. This paper investigates the geometric consistency of TFMPR and applies it to group decision making (GDM). Firstly, by introducing two parameters, a triangular fuzzy number is transformed into an interval. According to the geometric consistency of interval multiplicative preference relation (IMPR), the geometric consistency of TFMPR is defined. Then, two corresponding IMPRs are extracted from the TFMPR by programming models in the majority case and minority case, respectively. Using the constructed linear programming models, two interval priority weight vectors are obtained from the two extracted IMPRs, respectively. Combining two interval priority weight vectors, a linear programming model is established to derive the triangular fuzzy priority weights. Subsequently, the closeness degrees of alternatives by experts are defined to obtain the group utility indices and individual regret indices of alternatives. Then, the compromise indices of alternatives are calculated considering experts’ compromise attitude. By minimizing the compromise indices of alternatives, a multi-objective programming model is constructed to obtain experts’ weights. By aggregating the individual TFMPRs, the collective TFMPR is obtained to derive the triangular fuzzy priority weights. Using the arithmetic mean values, the ranking order of alternatives is generated. Therefore, a method is proposed to solve GDM with TFMPRs. Finally, a performance evaluation example of precise poverty alleviation is provided to illustrate the advantage of the proposed method.

1 Introduction

Decision making refers to the problem of selecting an alternative from a finite set of alternatives or ranking them based on the evaluation information of alternatives provided by several experts. The poverty alleviation in China has entered a stage of storming fortifications. With the development of national precision poverty alleviation, it is important to evaluate the precise poverty alleviation performance for measuring the alleviation work comprehensively and systematically. The evaluation of the precise poverty alleviation performance can be regarded as a kind of decision-making problems. The performance evaluation of precise poverty alleviation includes the factors on economic, living, social, educational and so on. It is difficult to incorporate all factors affecting precise poverty alleviation performance into the evaluation system.

To model such decision-making problems, preference relation, also known as pairwise comparison matrix, is a useful tool for expressing expert’s assessments on alternatives. Preference relation-based decision making has become an interesting topic in the decision theory. The fuzzy preference relation (FPR) [1] and multiplicative preference relation (MPR) [2] are two classical preference relations. Considering expert exhibits some hesitation and fuzziness for the assessments, the elements in preference relation are indicated by intervals, intuitionistic fuzzy values, triangular fuzzy numbers (TFNs) and so on. For the above preference relations, all possible membership degree, non-membership degree or hesitation degree of elements have the same importance in uncertain judgment. However, preference relation with TFNs considers the values with different membership degrees, which can characterize the uncertainty and fuzziness flexibly. Similar to classical preference relations, preference relations with TFNs also include two types. For convenience, the preference relations with TFNs in 0–1 scale and 1/9–9 scale are called triangular fuzzy MPR (TFMPR) and triangular fuzzy additive preference relation (TFAPR), respectively. Moreover, TFMPR and TFAPR can be transformed each other.

It is worth noting that the decision making under TFMPRs has attracted a great deal of attention. Chang [3] firstly introduced an approach for handling fuzzy analytic hierarchy process (AHP) with TFNs. Based on Chang [3], Wang et al. [4] proposed an extent analysis method on fuzzy AHP to obtain a crisp priority vector from TFMPR. Liu et al. [5] defined the consistent TFMPR and provided an algorithm of obtaining the ranking of alternatives. Wang [6] defined consistent TFMPR by a triangular fuzzy arithmetic based transitivity equation and established a logarithmic least square model. Dong et al. [7] constructed a consistent recovery method and proposed two integrated approaches to the multi-criteria GDM problem. Liu et al. [8] put forward the approximation consistency of TFMPR. Wang and Lin [9] proposed a geometric consistency index of TFMPR and developed a method for hierarchical multi-criteria decision making. Tang and Meng [10] defined the consistency for TFMPR by imaginary TFMPR and proposed a decision-making method with TFMPRs. Using the consistency of IMPR, Meng and Chen [11] defined the consistency of TFMPR and introduced an algorithm to derive the triangular fuzzy priority vector. Wang [12] defined the consistent TFMPR and developed a two-step approach for decision making. Although some works have done on TFMPR, there exist some limitations as follows:

• (1) Methods [3, 4] ignored the consistency of TFMPR. The consistency concepts of TFMPR in [5,6,7,8,9,10] are depended on the lower bounds, upper bounds and median values of TFNs. They may not sufficiently capture the original triangular fuzzy preference information. The possible values between the lower bound and the upper bound are neglected in the existing consistency definitions, which could possibly cause distortion of original information.

• (2) To rank alternatives, methods [3,4,5, 8, 11] summed up each row of consistent TFMPR and normalized the above row sums. Dong et al. [7] used the aggregation operator to obtain the synthetic extent indices in TFMPR, which could result in information loss. Methods [6, 9, 12] constructed some programming models to derive the triangular fuzzy priority weights (TFPWs), which only considers the lower bound, upper bound and median value of TFN.

• (3) Methods [3,4,5,6, 8, 10,11,12] are applied to solve individual decision-making problems. However, considering the increasingly complex decision-making problems, it is natural and reasonable to invite several experts to decision making. Thus, GDM is widely used to deal with such complex problems.

To overcome the limitations, this paper focuses on the geometric consistency of TFMPR and proposes a method for GDM with TFMPRs. The motivations of this paper mainly come from the following facts:

• (1) Considering the increasing complexity of the real decision problems, it is necessary to investigate the GDM with TFMPRs.

• (2) The prior consistent definitions of TFMPR only take the special values of TFMPR into account. It is reasonable and comprehensive to define the consistency of TFMPR considering all possible membership degrees in TFMPR.

• (3) The ranking order of alternatives should be generated by the priority weights. It is important to use the consistency of TFMPR derive the TFPWs.

Based on the above analysis, the contributions of this paper are summarized in the following.

• (1) The geometric consistency of TFMPR is defined according to the geometric consistency of IMPR, which could sufficiently capture the original uncertain information in TFMPR.

• (2) In the majority case and minority case, two linear programs or two adjusted programs are constructed to extract the geometrically consistent IMPRs from the TFMPR, respectively. Then, the two interval priority weight vectors are derived from the extracted IMPRs, respectively, which are applied to derive the TFPWs by a linear program.

• (3) The closeness degrees of alternatives by experts are calculated to obtain the group utility indices and individual regret indices of alternatives. The compromise indices of alternatives are defined considering experts’ compromise attitude. To obtain experts’ weights, a multi-objective programming model is constructed by minimizing the compromise indices of alternatives. Then, the collective TFMPR is obtained to derive the TFPWs for ranking alternatives. Therefore, a method is proposed to solve GDM with TFMPRs.

This paper is structured as follows. In Sect. 2, some definitions about TFN, TFMPR and IMPR are reviewed. In Sect. 3, the geometric consistency of TFMPR is analyzed. Section 4 presents a new method for GDM with TFMPRs. An example is provided to illustrate the effectiveness of the proposed method in Sect. 5. The main conclusions are presented in Sect. 6.

2 Preliminaries

In the section, some concepts about TFN, TFMPR and IMPR are reviewed.

Definition 1

[13] A TFN $\tilde{r}$ is expressed by an order triple $\tilde{r}=(l,m,u)$ such that $l\le m\le u$ with the following membership function:

$$\begin{aligned} \mu _{\tilde{r}}(x)=\left\{ \begin{array}{ll} \frac{x-l}{m-l},&{}\quad l\le x\le m\\ \frac{u-x}{u-m},&{}\quad m\le x\le u\\ 0,&{}\quad \hbox {otherwise} \end{array} \right. \end{aligned}$$

where l and u, respectively, denote the lower and upper bounds of $\tilde{r}$, and m is the median value. If $l\ge 0$, then $\tilde{r}$ is said to be a positive TFN.

For positive TFNs $\tilde{r}_i=(l_i,m_i,u_i)$ $(i=1,2)$, the operations are defined as $\tilde{r}_1\tilde{r}_2=(l_1l_2,m_1m_2,u_1u_2)$ and $\tilde{r}_1^c=(l_1^c,m_1^c,u_1^c)$ where $c>0$ [13].

For TFNs $\tilde{r}_i=(l_i,m_i,u_i)$ $(i=1,2)$, the Hamming distance between $\tilde{r}_1=(l_1,m_1,u_1)$ and $\tilde{r}_2=(l_2,m_2,u_2)$ is defined as [14]

$$\begin{aligned} d(\tilde{r}_1,\tilde{r}_2)=(|l_1-l_2|+|m_1-m_2|+|u_1-u_2|)/3. \end{aligned}$$

(1)

For two triangular fuzzy vectors $\tilde{{\varvec{{r}}}}_i=(\tilde{r}_{i1},\tilde{r}_{i2},\ldots ,\tilde{r}_{im})^\mathrm {T}$ $(i=1,2)$ with $\tilde{r}_{ij}=(l_{ij},m_{ij},u_{ij})$, the distance between $\tilde{{\varvec{{r}}}}_1$ and $\tilde{{\varvec{{r}}}}_2$ is defined as follows:

$$\begin{aligned} d(\tilde{{\varvec{{r}}}}_1,\tilde{{\varvec{{r}}}}_2)=\sum _{j=1}^md(\tilde{r}_{1j},\tilde{r}_{2j}). \end{aligned}$$

(2)

Plugging Eq. (1) into Eq. (2), the distance between $\tilde{{\varvec{{r}}}}_1$ and $\tilde{{\varvec{{r}}}}_2$ is further calculated as

$$\begin{aligned} d(\tilde{{\varvec{{r}}}}_1,\tilde{{\varvec{{r}}}}_2)=\frac{1}{3}\sum _{j=1}^m(|l_{1j}-l_{2j}|+|m_{1j}-m_{2j}|+|u_{1j}-u_{2j}|). \end{aligned}$$

(3)

By introducing the parameters $\alpha$ and $\beta$ where $0\le \alpha ,\beta \le 1$, TFN $\tilde{r}=(l,m,u)$ can be transformed into an interval

$$\begin{aligned} r=[l^\alpha m^{(1-\alpha )},m^{(1-\beta )}u^\beta ]. \end{aligned}$$

(4)

Definition 2

[3] A TFMPR ${\widetilde{R}}$ on the object set $X=\{x_1,x_2,\ldots ,x_n\}$ is denoted by a triangular fuzzy multiplicative judgment matrix ${\widetilde{R}}=(\tilde{r}_{ij})_{n\times n}$ $\subset X\times X$, where TFN $\tilde{r}_{ij}=(l_{ij},m_{ij},u_{ij})$ illustrates the fuzzy preference ratio of alternative $x_i$ to $x_j$. Furthermore, $l_{ij},m_{ij}$ and $u_{ij}$ fulfill $1/9\le l_{ij}\le m_{ij}\le u_{ij}\le 9$, $l_{ij}u_{ji}=m_{ij}m_{ji}=u_{ij}l_{ji}=1$, $l_{ii}=m_{ii}=u_{ii}=1$ $(i,j=1,2,\ldots ,n)$.

Definition 3

[15] An IMPR R on the object (or alternative) set $X=\{x_1,x_2, \ldots ,x_n\}$ is denoted by a judgment matrix $R=(r_{ij})_{n\times n}\subset X\times X$, where $r_{ij}=[r_{ij}^-,r_{ij}^+]$ is an interval and indicates that object $x_i$ is between $r_{ij}^-$ and $r_{ij}^+$ times as important as $x_j$ and satisfies the following conditions:

$$\begin{aligned} r_{ij}^+ \ge r_{ij}^-\ge 0,\quad r_{ij}^+r_{ji}^-{=}r_{ij}^-r_{ji}^+{=}1,\quad r_{ii}^-{=}r_{ii}^+{=}1\quad (i,j=1,2,\ldots ,n). \end{aligned}$$

Definition 4

[16] An IMPR $R=(r_{ij})_{n\times n}$ with $r_{ij}=[r_{ij}^-,r_{ij}^+]$ is geometrically consistent if it satisfies the geometric transitivity condition as:

$$\begin{aligned} r_{ij}^-r_{ij}^+=r_{ik}^-r_{ik}^+r_{kj}^-r_{kj}^+\ (i,j,k=1,2,\ldots ,n) \end{aligned}$$

(5)

Theorem 1

An IMPR $R=(r_{ij})_{n\times n}$ with $r_{ij}=[r_{ij}^-,r_{ij}^+]$ is geometrically consistent if it satisfies:

$$\begin{aligned} r_{ij}^-r_{ij}^+=r_{ik}^-r_{ik}^+r_{kj}^-r_{kj}^+\ (i,j,k=1,2,\ldots ,n; i<k<j). \end{aligned}$$

(6)

Set ${\varvec{{W}}}=\{{\varvec{{w}}}=(w_1,w_2,\ldots ,w_n)^\mathrm {T}|w_i=[w_i^-,w_i^+], 0\le w_i^-\le w_i^+\le 1, w_i^-+\sum _{j=1,j\ne i}^nw_j^+\ge 1, w_i^++\sum _{j=1,j\ne i}^nw_j^-\le 1 (i=1,2,\ldots ,n)\}$. Thus, ${\varvec{{w}}}\in {\varvec{{W}}}$ is a normalized interval weight vector [17, 18].

The IMPR $R=(r_{ij})_{n\times n}$ is geometrically consistent if there exists a normalized priority weight vector ${\varvec{{w}}}=(w_1,w_2,\ldots ,w_n)^\mathrm {T}$ such that [19]

$$\begin{aligned} r_{ij}=\left\{ \begin{array}{ll} [w_i^-/w_j^+,w_i^+/{w_j^-}], &{}\quad i=j\\ {[1,1]},&{}\quad i\ne j \end{array} \right. \end{aligned}$$

(7)

Suppose the parameters $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;i<j)$ satisfying $0\le \alpha _{ij},\beta _{ij}\le 1$. According to Eq. (7), TFMPR ${\widetilde{R}}$ can be transformed into IMPR $R=(r_{ij})_{n\times n}$ where

$$\begin{aligned} r_{ij}=[r_{ij}^-,r_{ij}^+]=\left\{ \begin{array}{ll} [l_{ij}^{\alpha _{ij}} m_{ij}^{(1-\alpha _{ij})},m_{ij}^{(1-\beta _{ij})}u_{ij}^{\beta _{ij}}],&{}\quad {i<j}\\ {[1,1]},&{}\quad {i=j}\\ {[}1-r_{ji}^+,1-r_{ji}^-{]},&{}\quad {i>j} \end{array}\right. \end{aligned}$$

(8)

3 Consistency analysis of TFMPR

In this section, the Geometric consistency of TFMPR is defined. Then, two corresponding IMPRs are extracted from the TFMPR by the majority case and minority case, respectively. The TFPWs are obtained from the TFMPR by some linear programming models.

3.1 Geometric consistency of TFMPR

According to Eq. (8), an TFMPR can be transformed into IMPR $R=(r_{ij})_{n\times n}$ with parameters $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n; i<j)$. Now, the consistency of IMPR has been analyzed maturely. Thus, it is reasonable and necessary to investigate the consistency of TFMPR by the extracted IMPR.

Definition 5

For TFMPR ${\widetilde{R}}=(\tilde{r}_{ij})_{n\times n}$ with $\tilde{r}_{ij}=(l_{ij},m_{ij},u_{ij})$, if there exists an geometrically consistent IMPR $R=(r_{ij})_{n\times n}$ extracted by Eq. (8), then TFMPR ${\widetilde{R}}$ is geometrically consistent.

Combining Eqs. (6) with (8), the geometrically consistent condition is converted into

$$\begin{aligned}&l_{ij}^{\alpha _{ij}} m_{ij}^{(1-\alpha _{ij})}m_{ij}^{(1-\beta _{ij})}u_{ij}^{\beta _{ij}}=l_{ik}^{\alpha _{ik}} m_{ik}^{(1-\alpha _{ik})}m_{ik}^{(1-\beta _{ik})}u_{ik}^{\beta _{ik}} l_{kj}^{\alpha _{kj}} m_{kj}^{(1-\alpha _{kj})}m_{kj}^{(1-\beta _{kj})}u_{kj}^{\beta _{kj}}\nonumber \\&\qquad (i,j,k=1,2,\ldots ,n; i<k<j) \end{aligned}$$

(9)

Using the logarithmic operation, Eq. (9) is transformed into a linear condition as

$$\begin{aligned}&\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}+(2-\alpha _{ik}-\beta _{ik})\ln m_{ik}\nonumber \\&\quad +\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}+(2-\alpha _{kj}-\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}\nonumber \\&\qquad (i,j,k=1,2,\ldots ,n; i<k<j) \end{aligned}$$

(10)

Based on Definition 5, more than one geometrically consistent IMPR can be extracted from a geometrically consistent TFMPR. Thus, two special cases are considered to extract the IMPR from TFMPR.

• (1) majority case In the majority case, a multi-objective program is constructed as follows:

  $$\begin{aligned} \begin{aligned}&\max \ \{\alpha _{ij}+\beta _{ij}\}\ (i,j=1,2,\ldots ,n;i<j)\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}\\ &{}\quad +(2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}+(2-\alpha _{kj}\\ &{}\quad -\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}\ (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j) \end{array} \right. \end{aligned} \end{aligned}$$

  (11)

  Since there is no any preference among objectives $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;i<j)$, Eq. (11) is transformed into a single-objective linear program by linear sum of equal weights as follows:

  $$\begin{aligned} \begin{aligned}&\max \ \sum _{i=1}^n\sum _{j=i+1}^n(\alpha _{ij}+\beta _{ij})\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}\\ &{}\quad +(2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}+(2-\alpha _{kj}\\ &{}\quad -\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}\ (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j) \end{array}\right. \end{aligned} \end{aligned}$$

  (12)

  By maximizing the values of $\alpha _{ij}$ and $\beta _{ij}$, we extend the search space of $r_{ij}$ in $\tilde{r}_{ij}$ as much as possible in the above programming model. The larger the values of $\alpha _{ij}$ and $\beta _{ij}$, the closer the search space of $r_{ij}$ to interval $[l_{ij},u_{ij}]$. In this case, the majority possibility of the value of $r_{ij}$ is considered. Thus, it can be regarded as the majority case. Solving Eq. (12) to yield the optimal solutions $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;$ $i<j)$. Then using Eq. (8), the corresponding IMPR $R=(r_{ij})_{n\times n}$, denoted as $\dot{R}=(\dot{r}_{ij})_{n\times n}$ with $\dot{r}_{ij}=[\dot{r}_{ij}^-,\dot{r}_{ij}^+]$ is extracted from the TFMPR ${\widetilde{R}}$.

• (2) minority case In the majority case, a multi-objective program is constructed as follows:

  $$\begin{aligned} \begin{aligned}&\min \ \{\alpha _{ij}+\beta _{ij}\}\ (i,j=1,2,\ldots ,n;i<j)\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}+\\ &{}\quad (2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}+(2-\alpha _{kj}\\ &{}\quad -\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}\ (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j) \end{array}\right. \end{aligned} \end{aligned}$$

  (13)

  Similar to the majority case, Eq. (13) is transformed into

  $$\begin{aligned} \begin{aligned}&\min \ \sum _{i=1}^n\sum _{j=i+1}^n(\alpha _{ij}+\beta _{ij})\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}+\\ &{}\quad (2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}+(2-\alpha _{kj}\\ &{}\quad -\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}\ (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j) \end{array}\right. \end{aligned} \end{aligned}$$

  (14)

  By minimizing the values of $\alpha _{ij}$ and $\beta _{ij}$, we increase the search space of $r_{ij}$ in $\tilde{r}_{ij}$ as much as possible in the above programming model. The smaller the values of $\alpha _{ij}$ and $\beta _{ij}$, the closer the search space of $r_{ij}$ to numerical $m_{ij}$. In this case, the minority possibility of the value of $r_{ij}$ is considered. Thus, it can be regarded as the minority case. Solving Eq. (14) to yield the optimal solutions $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;$ $i<j)$. Then using Eq. (8), the corresponding IMPR $R=(r_{ij})_{n\times n}$, denoted as $\ddot{R}=(\ddot{r}_{ij})_{n\times n}$ with $\ddot{r}_{ij}=[\ddot{r}_{ij}^-,\ddot{r}_{ij}^+]$ is extracted from the TFMPR ${\widetilde{R}}$. However, if the TFMPR ${\widetilde{R}}=(\tilde{r}_{ij})_{n\times n}$ is not geometrically consistent, then it can not obtain the corresponding IMPR with geometric consistency. Furthermore, the above linear programming models in the majority and minority cases, i.e., Eqs. (12) and (14) have the empty feasible region. To overcome the limitation, some deviations $d_{ikj}^-$ and $d_{ikj}^+$ $(i,j,k=1,2,\ldots ,n; i<k<j)$ are introduced to extend the feasible region. To obtain the IMPRs which are as consistent as possible, the adjusted linear programming models are established in the majority and minority cases by considering minimizing the deviations $d_{ikj}^-$ and $d_{ikj}^+$ $(i,j,k=1,2,\ldots ,n; i<k<j)$ as follows:

  $$\begin{aligned}&\begin{aligned}&\max \ \sum _{i=1}^n\sum _{j=i+1}^n(\alpha _{ij}+\beta _{ij})-\sum _{i=1}^n\sum _{k=i+1}^n\sum _{j=k+1}^n(d_{ikj}^++d_{ikj}^-)\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}\\ &{}\quad +(2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}\\ &{}\quad +(2-\alpha _{kj}-\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}+d_{ikj}^+-d_{ikj}^-\\ &{}\qquad (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j)\\ &{}d_{ikj}^-,d_{ikj}^+\ge 0, d_{ikj}^-d_{ikj}^+=0\ (i,j,k=1,2,\ldots ,n;i<k<j)\\ \end{array}\right. \end{aligned} \end{aligned}$$

  (15)
  $$\begin{aligned}&\begin{aligned}&\min \ \sum _{i=1}^n\sum _{j=i+1}^n(\alpha _{ij}+\beta _{ij})+\sum _{i=1}^n\sum _{k=i+1}^n\sum _{j=k+1}^n(d_{ikj}^++d_{ikj}^-)\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\alpha _{ij}\ln l_{ij}+(2-\alpha _{ij}-\beta _{ij})\ln m_{ij}+\beta _{ij}\ln u_{ij}=\alpha _{ik}\ln l_{ik}\\ &{}\quad +(2-\alpha _{ik}-\beta _{ik})\ln m_{ik}+\beta _{ik}\ln u_{ik}+\alpha _{kj}\ln l_{kj}\\ &{}\quad +(2-\alpha _{kj}-\beta _{kj})\ln m_{kj}+\beta _{kj}\ln u_{kj}+d_{ikj}^+-d_{ikj}^-\\ &{}\qquad (i,j,k=1,2,\ldots ,n; i<k<j)\\ &{}0\le \alpha _{ij},\beta _{ij}\le 1\ (i,j=1,2,\ldots ,n;i<j)\\ &{}d_{ikj}^-,d_{ikj}^+\ge 0, d_{ikj}^-d_{ikj}^+=0\ (i,j,k=1,2,\ldots ,n;i<k<j)\\ \end{array}\right. \end{aligned} \end{aligned}$$

  (16)

  If the feasible regions of Eqs. (12) and (14) are empty, the optimal solutions $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;$ $i<j)$ are obtained by solving Eqs. (15) and (16), respectively. Then, using Eq. (8), the corresponding IMPRs $\dot{R}=(\dot{r}_{ij})_{n\times n}$ and $\ddot{R}=(\ddot{r}_{ij})_{n\times n}$ are extracted from the TFMPR ${\widetilde{R}}$. An example is provided to illustrate the above method.

Example 1

Consider the TFMPR provided by Example 2 in [9].

$$\begin{aligned} {\widetilde{R}}=\left( \begin{array}{cccc} (1,1,1)&{}\quad (1/2,1,2)&{}\quad (8/15,1,15/8)&{}\quad (5/9,1,9/5)\\ (1/2,1,1/2)&{}\quad (1,1,1)&{}\quad (3/5,1,5/3)&{}\quad (5/8,1,8/5)\\ (8/15,1,15/8)&{}\quad (3/5,1,5/3)&{}\quad (1,1,1)&{}\quad (2/3,1,3/2)\\ (5/9,1,9/5)&{}\quad (5/8,1,8/5)&{}\quad (2/3,1,3/2)&{}\quad (1,1,1)\\ \end{array}\right) \end{aligned}$$

For the TFMPR ${\widetilde{R}}$, the feasible regions of Eqs. (12) and (14) are not empty. The TFMPR ${\widetilde{R}}$ is geometrically consistent. Thus solving Eqs. (12) and (14), the optimal solutions are derived, respectively, and applied to generate the corresponding IMPRs $\dot{R}$ and $\ddot{R}$ using Eq. (8).

$$\begin{aligned} \dot{R}= & {} \left( \begin{array}{cccc} [1,1]&{}\quad [0.50,2.00]&{}\quad [0.53,1.88]&{}\quad [0.56,1.80]\\ {[}0.50,2.00]&{}\quad [1,1]&{}\quad [0.60,1.67]&{}\quad [0.63,1.60]\\ {[}0.53,1.88]&{}\quad [0.60,1.67]&{}\quad [1,1]&{}\quad [0.67,1.50]\\ {[}0.56,1.80]&{}\quad [0.63,1.60]&{}\quad [0.67,1.50]&{}\quad [1,1]\\ \end{array}\right) ,\\ \ddot{R}= & {} \left( \begin{array}{cccc} {[}1,1]&{}\quad [1,1]&{}\quad [1,1]&{}\quad [1,1]\\ {[}1,1]&{}\quad [1,1]&{}\quad [1,1]&{}\quad [1,1]\\ {[}1,1]&{}\quad [1,1]&{}\quad [1,1]&{}\quad [1,1]\\ {[}1,1]&{}\quad [1,1]&{}\quad [1,1]&{}\quad [1,1]\\ \end{array}\right) \end{aligned}$$

3.2 Determination of triangular fuzzy priority weights

Based on Eq. (7), the geometrically consistent IMPR $R=(r_{ij})_{n\times n}$ with $r_{ij}=[r_{ij}^-,r_{ij}^+]$ satisfies

$$\begin{aligned} r_{ij}^-=w_i^-/w_j^+,\quad r_{ij}^+=w_i^+/w_j^-\quad (i,j=1,2,\ldots ,n;i<j). \end{aligned}$$

(17)

Equation (17) can be transformed into

$$\begin{aligned} w_i^--r_{ij}^-w_j^+=0,\quad w_i^+-r_{ij}^+w_j^-=0\quad (i,j=1,2,\ldots ,n;i<j). \end{aligned}$$

(18)

To derive the interval priority weights $w_i (i=1,2,\ldots ,n)$ from IMPR $R=(r_{ij})_{n\times n}$ with $r_{ij}=[r_{ij}^-,r_{ij}^+]$, a good enough solution is to find the interval priority weights that satisfies Eq. (18) as well as possible. Thus by minimizing the deviations $|w_i^--r_{ij}^-w_j^+|$ and $|w_i^+-r_{ij}^+w_j^-|$ $(i,j=1,2,\ldots ,n;i<j)$, a goal programming model is established as follows:

$$\begin{aligned} \begin{aligned}&\min \ \sum _{i=1}^n\sum _{j=i+1}^n(|w_i^--r_{ij}^-w_j^+|+|w_i^+-r_{ij}^+w_j^-|)\\&\hbox {s.t.}\quad {\varvec{{w}}}\in {\varvec{{W}}} \end{aligned} \end{aligned}$$

(19)

Set the parameters $\dot{e}_{ij}^+$, $\dot{e}_{ij}^-$, $\ddot{e}_{ij}^+$ and $\ddot{e}_{ij}^-$ $(i,j=1,2,\ldots ,n;i<j)$ where

$$\begin{aligned}&\dot{e}_{ij}^+=(|w_i^--r_{ij}^-w_j^+|+w_i^--r_{ij}^-w_j^+)/2,\\&\dot{e}_{ij}^-=(|w_i^--r_{ij}^-w_j^+|-w_i^-+r_{ij}^-w_j^+)/2,\\&\ddot{e}_{ij}^+=(|w_i^+-r_{ij}^+w_j^-|+w_i^+-r_{ij}^+w_j^-)/2,\\&\ddot{e}_{ij}^-=(|w_i^+-r_{ij}^+w_j^-|-w_i^++r_{ij}^+w_j^-)/2. \end{aligned}$$

The above goal programming model is transformed into

$$\begin{aligned} \begin{aligned}&\min \ \sum _{i=1}^n\sum _{j=i+1}^n(\dot{e}_{ij}^++\dot{e}_{ij}^-+\ddot{e}_{ij}^++\ddot{e}_{ij}^-)\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\dot{e}_{ij}^+-\dot{e}_{ij}^-=w_i^--r_{ij}^-w_j^+\ (i,j=1,2,\ldots ,n;i<j)\\ &{}\ddot{e}_{ij}^+-\ddot{e}_{ij}^-=w_i^+-r_{ij}^+w_j^-\ (i,j=1,2,\ldots ,n;i<j)\\ &{}{\varvec{{w}}}\in {\varvec{{W}}}, \dot{e}_{ij}^+,\dot{e}_{ij}^-,\ddot{e}_{ij}^+,\ddot{e}_{ij}^-\ge 0\ (i,j=1,2,\ldots ,n;i<j)\\ \end{array}\right. \end{aligned} \end{aligned}$$

(20)

Solving Eq. (20), the interval priority weights $w_i=[w_i^-,w_i^+]$ $(i=1,2,\ldots ,n)$ are derived from IMPR $R=(r_{ij})_{n\times n}$.

According to the above analysis, the TFPWs of TFMPR ${\widetilde{R}}$ can be determined from the extracted IMPRs $\dot{R}$ and $\ddot{R}$ which are derived from the TFMPR ${\widetilde{R}}$ in the majority and minority cases, respectively. Set $R=\dot{R}$, solving Eq. (20) yields the interval priority weights $w_i=[w_i^-,w_i^+]$ $(i=1,2,\ldots ,n)$, denoted as $\dot{w}_i=[\dot{w}_i^-,\dot{w}_i^+]$. Set $R=\ddot{R}$, solving Eq. (20) yields the interval priority weights $w_i=[w_i^-,w_i^+]$ $(i=1,2,\ldots ,n)$, denoted as $\ddot{w}_i=[\ddot{w}_i^-,\ddot{w}_i^+]$.

To unify the interval priority weights $\dot{w}_i$ and $\ddot{w}_i$ into the TFPWs $\tilde{w}_i=(w_i^l,w_i^m,w_i^u)$ $(i=1,2,\ldots ,n)$, the four tuples ${\hat{w}}_i=(w_i^1,w_i^2,w_i^3,w_i^4)$ is formed firstly, which is a permutation of $(\dot{w}_i^-,\dot{w}_i^+,\ddot{w}_i^-,\ddot{w}_i^+)$ satisfying that $w_i^1\le w_i^2\le w_i^3\le w_i^4$. Then, a goal programming model is established as follows:

$$\begin{aligned} \begin{aligned}&\min \ \sum _{i=1}^n{(|w_i^l-w_i^1|+|w_i^m-w_i^2|+|w_i^m-w_i^3|+|w_i^u-w_i^4|)}\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}\sum _{i=1}^nw_i^m=1, 0\le w_i^l,w_i^m,w_i^u\le 1\ (i=1,2,\ldots ,n)\\ &{}w_i^l+\sum _{j=1,j\ne i}^nw_j^u\ge 1\, w_i^u+\sum _{j=1,j\ne i}^nw_j^l\le 1\ (i=1,2,\ldots ,n)\\ \end{array}\right. \end{aligned} \end{aligned}$$

(21)

where the constraints are the conditions of normalized TFPWs.

Set some parameters $p_i^{1+}=(|w_i^l-w_i^1|+w_i^l-w_i^1)/2$, $p_i^{1-}=(|w_i^l-w_i^1|-w_i^l+w_i^1)/2$, $p_i^{2+}=(|w_i^m-w_i^2|+w_i^m-w_i^2)/2$, $p_i^{2-}=(|w_i^m-w_i^2|-w_i^m+w_i^2)/2$, $p_i^{3+}=(|w_i^m-w_i^3|+w_i^m-w_i^3)/2$, $p_i^{3-}=(|w_i^m-w_i^3|-w_i^m+w_i^3)/2$, $p_i^{4+}=(|w_i^u-w_i^4|+w_i^u-w_i^4)/2$, $p_i^{4-}=(|w_i^u-w_i^4|-w_i^u+w_i^4)/2$. Equation (21) is converted into a linear programming model as follows:

$$\begin{aligned} \begin{aligned}&\min \ \sum _{i=1}^n{(p_i^{1+}+p_i^{1-}+p_i^{2+}+p_i^{2-}+p_i^{3+}+p_i^{3-}+p_i^{4+}+p_i^{4-}})\\&\hbox {s.t.}\quad \left\{ \begin{array}{ll} &{}p_i^{1+}-p_i^{1-}=w_i^l-w_i^1\, p_i^{2+}+p_i^{2-}=w_i^m-w_i^2\ (i=1,2,\ldots ,n)\\ &{}p_i^{3+}+p_i^{3-}=w_i^m-w_i^3\, p_i^{4+}+p_i^{4-}=w_i^u-w_i^4\ (i=1,2,\ldots ,n)\\ &{}\sum _{i=1}^nw_i^m=1, 0\le w_i^l\le w_i^m\le w_i^u\le 1\ (i=1,2,\ldots ,n)\\ &{}w_i^l+\sum _{j=1,j\ne i}^nw_j^u\ge 1\, w_i^u+\sum _{j=1,j\ne i}^nw_j^l\le 1\ (i=1,2,\ldots ,n)\\ &{}p_i^{1+},p_i^{1-},p_i^{2+},p_i^{2-},p_i^{3+},p_i^{3-},p_i^{4+},p_i^{4-}\ge 0\ (i=1,2,\ldots ,n) \end{array}\right. \end{aligned} \end{aligned}$$

(22)

Solving Eq. (22), the normalized TFPWs $\tilde{w}_i=(w_i^l,w_i^m,w_i^u)$ $(i=1,2,\ldots ,n)$ are obtained. To rank TFPWs $\tilde{w}_i$ $(i=1,2,\ldots ,n)$, the arithmetic mean values $P(\tilde{w}_i)$ are obtained as [20]:

$$\begin{aligned} P(\tilde{w}_i)=(w_i^l+2w_i^m+w_i^u)/4\ (i=1,2,\ldots ,n). \end{aligned}$$

(23)

Thus, the ranking order of alternatives is generated in descending order of the arithmetic mean values $P(\tilde{w}_i)$ $(i=1,2,\ldots ,n)$.

Example 2

Determine the priority weights from TFMPR provided in Example 1.

The extracted IMPR $\dot{R}$ and $\ddot{R}$ are generated in Example 1. Set $R=\dot{R}$, solving Eq. (20) yields the interval priority weights as $\dot{w}_1=[0.1324,0.4653]$, $\dot{w}_2=[0.1489,0.3812]$, $\dot{w}_3=[0.1588,0.2482]$, $\dot{w}_4=[0.2269,0.2383]$. Set $R=\ddot{R}$, solving Eq. (20) yields the interval priority weights as $\ddot{w}_1=[0.25,0.25]$, $\ddot{w}_2=[0.25,0.25]$, $\ddot{w}_3=[0.25,0.25]$, $\ddot{w}_4=[0.25,0.25]$.

Combining $\dot{w}_i$ and $\ddot{w}_i$, the four tuples ${\hat{w}}_i$ are formed. Then, solving Eq. (22) yields the TFPWs as $\tilde{w}_1=(0.1324,0.2500,0.4653)$, $\tilde{w}_2=(0.1489,0.2500,0.3812)$, $\tilde{w}_3=(0.1588,0.2500,0.2500)$, $\tilde{w}_4=(0.2269,0.2500,0.2500)$.

Using Eq. (23), the arithmetic mean values $P(\tilde{w}_i)$ $(i=1,2,3,4)$ are obtained as $P(\tilde{w}_1)=0.2744$, $P(\tilde{w}_2)=0.2575$, $P(\tilde{w}_3)=0.2272$, $P(\tilde{w}_4)=0.2442$.

Thus, the ranking order of alternatives is generated as $x_1\succ x_2\succ x_4\succ x_3$.

Based on the method in [9], the optimal triangular fuzzy multiplicative weights vector is obtained from the TFMPR ${\widetilde{R}}$ as follows:

$$\begin{aligned} \tilde{W}^*=(\tilde{w}_1^*,\tilde{w}_2^*,\tilde{w}_3^*,\tilde{w}_4^*)= \left( \begin{array}{c} (0.6667,1,1.5000),(0.7500,1,1.3333),\\ (0.8000,1,1.2500),(0.8333,1,1.2000).\\ \end{array}\right) \end{aligned}$$

Using Eq. (5.15) in [9], the geometric mean-based defuzzification values $d_i$ $(i=1,2,3,4)$ are obtained as $d_1=d_2=d_3=d_4=1$. Thus, the ranking order of alternatives are generated as $x_1\sim x_2\sim x_3\sim x_4$. These alternatives cannot be distinguished by [9]. The middle values $m_{ij}$ of the elements $\tilde{r}_{ij}$ in TFMPR ${\widetilde{R}}$ are equal to 1 and the left value $l_{ij}$ and the right value $u_{ij}$ are reciprocal. However, the left value $l_{ij}$ and the right value $u_{ij}$ are different for different elements $\tilde{r}_{ij}$. Thus compared with the result in [9], the result $x_1\succ x_2\succ x_4\succ x_3$ obtained by this paper is logical and reasonable.

4 Group decision making with TFMPRs

In this section, the GDM problem with TFMPRs is described. Then, experts’ weights are determinated by a linear program. Finally, a method is proposed to solve GDM with TFMPRs.

4.1 Description of group decision-making problem with TFMPRs

For a GDM problem, let $X=\{x_1,x_2,\ldots ,x_n\}$ be a set of alternative and $E=\{e_1,e_2,\ldots ,e_k\}$ be a set of experts. Each expert $e_t$ is able to provide his/her preference for each pair of alternatives and gives the individual TFMPR ${\widetilde{R}}^t=(\tilde{r}_{ij}^t)_{n\times n}$ with $\tilde{r}_{ij}^t=(l_{ij}^t,m_{ij}^t,u_{ij}^t)$ $(t=1,2,\ldots ,k)$.

Set ${\varvec{{V}}}=\{{{\varvec{{v}}}=(v_1,v_2,\ldots ,v_k)^\mathrm {T}|v_t\ge 0(t=1,2,\ldots ,k),\sum _{t=1}^kv_t=1}\}$. Suppose that experts’ weight vector is ${\varvec{{v}}}\in {\varvec{{V}}}$. Using the operations of TFNs, the collective TFMPR ${\widetilde{R}}=(\tilde{r}_{ij})_{n\times n}$ with $\tilde{r}_{ij}=(l_{ij},m_{ij},u_{ij})$ is obtained where

$$\begin{aligned} \tilde{r}_{ij}=\prod _{t=1}^k\tilde{r}{_{ij}^t}^{v_t}= \left( \prod _{t=1}^kl{_{ij}^t}^{v_t},\prod _{t=1}^km{_{ij}^t}^{v_t},\prod _{t=1}^ku{_{ij}^t}^{v_t}\right) . \end{aligned}$$

(24)

Denote the normalized TFPW vector of alternatives by $\tilde{{\varvec{{w}}}}=(\tilde{w}_1,\tilde{w}_2,\ldots ,\tilde{w}_n)^\mathrm {T}$ with $\tilde{w}_i=(w_i^l,w_i^m,w_i^u)$ $(i=1,2,\ldots ,n)$, which needs to be obtained from the collective TFMPR ${\widetilde{R}}$ for ranking alternatives.

4.2 Determination of experts’ weights

For an individual TFMPR ${\widetilde{R}}^t=(\tilde{r}_{ij}^t)_{n\times n}$ with $\tilde{r}_{ij}^t=(l_{ij}^t,m_{ij}^t,u_{ij}^t)$ $(t=1,2,\ldots ,k)$, the evaluation vector of alternative $x_i$ provided by expert $e_t$ is expressed as $\tilde{{\varvec{{r}}}}_i^t=(\tilde{r}_{i1}^t,\tilde{r}_{i2}^t,\ldots ,\tilde{r}_{in}^t)^\mathrm {T}$. Corresponding, the positive ideal vector $\tilde{{\varvec{{p}}}}_i=(\tilde{p}_{i1},\tilde{p}_{i2},\ldots ,\tilde{p}_{in})^\mathrm {T}$ and the negative ideal vector $\tilde{{\varvec{{g}}}}_i=(\tilde{g}_{i1},\tilde{g}_{i2},\ldots ,\tilde{g}_{in})^\mathrm {T}$ of alternative $x_i$ are constructed where

$$\begin{aligned}&\tilde{p}_{ij}=\left( l_{ij}^p,m_{ij}^p,u_{ij}^p\right) =\left( \max \limits _{t=1,2,\ldots ,k}l_{ij}^t,\max \limits _{t=1,2,\ldots ,k}m_{ij}^t,\max \limits _{t=1,2,\ldots ,k}u_{ij}^t\right) . \end{aligned}$$

(25)

$$\begin{aligned}&\tilde{g}_{ij}=\left( l_{ij}^g,m_{ij}^g,u_{ij}^g\right) =\left( \min \limits _{t=1,2,\ldots ,k}l_{ij}^t,\min \limits _{t=1,2,\ldots ,k}m_{ij}^t,\min \limits _{t=1,2,\ldots ,k}u_{ij}^t\right) . \end{aligned}$$

(26)

For expert $e_t$, the closer the distance between $\tilde{{\varvec{{r}}}}_i^t$ and the positive ideal vector $\tilde{{\varvec{{p}}}}_i$, the farther away from the negative ideal vector $\tilde{{\varvec{{g}}}}_i$, the better the alternative $x_i$. Motivated by TOPSIS, the closeness degree $C_i^t$ of alternative $x_i$ by expert $e_t$ is defined as

$$\begin{aligned} C_i^t=\frac{d\left( \tilde{{\varvec{{r}}}}_i^t,\tilde{{\varvec{{p}}}}_i\right) }{d\left( \tilde{{\varvec{{r}}}}_i^t,\tilde{{\varvec{{p}}}}_i\right) +d\left( \tilde{{\varvec{{r}}}}_i^t,\tilde{{\varvec{{g}}}}_i\right) }\ (i=1,2,\ldots ,n;t=1,2,\ldots ,k). \end{aligned}$$

(27)

where $d(\tilde{{\varvec{{r}}}}_i^t,\tilde{{\varvec{{p}}}}_i)$ and $d(\tilde{{\varvec{{r}}}}_i^t,\tilde{{\varvec{{g}}}}_i)$ are the distance between $\tilde{{\varvec{{r}}}}_i^t$ and $\tilde{{\varvec{{p}}}}_i$, as well as $\tilde{{\varvec{{g}}}}_i$, respectively, calculated by Eq. (3). It is clear that the smaller the value of $C_i^t$, the better the alternative $x_i$ by expert $e_t$.

With respect to alternative $x_i$, the group utility index $S(x_i)$ and the individual regret index $R(x_i)$ are defined as follows:

$$\begin{aligned} S(x_i)= & {} \sum _{t=1}^kv_tC_i^t\quad (i=1,2,\ldots ,n), \end{aligned}$$

(28)

$$\begin{aligned} R(x_i)= & {} \max \left\{ v_tC_i^t|t=1,2,\ldots ,k\right\} \quad (i=1,2,\ldots ,n). \end{aligned}$$

(29)

The group utility index $S(x_i)$ reflects the average group evaluation of alternative $x_i$. The individual regret index $R(x_i)$ expresses the worst evaluation in a particular expert. It is obvious that the smaller the values of $S(x_i)$ and $R(x_i)$,the better the alternative $x_i$. Combining $S(x_i)$ and $R(x_i)$, the compromise index $Q(x_i)$ of alternative $x_i$ is defined as

$$\begin{aligned} Q(x_i)=\lambda S(x_i)+(1-\lambda ) R(x_i), \end{aligned}$$

(30)

where compromise attitude $\lambda \in [0,1]$ is the weight for the strategy of the maximum group utility. If $\lambda >0.5$, the compromise solution is selected with ’voting by majority rule’. If $\lambda =0.5$, the compromise solution is selected with ’consensus’. If $\lambda <0.5$, the compromise solution is selected with ’veto’.

Based on the above analysis, the bigger the value of the compromise index $Q(x_i)$, the better the alternative $x_i$. Thus to determine experts’ weights, a multi-objective programming model is constructed by minimizing the compromise indices of alternatives as follows:

$$\begin{aligned} \begin{aligned}&\min \ Q(x_i)\ (i=1,2,\ldots ,n)\\&\hbox {s.t.}\quad {\varvec{{v}}}\in {\varvec{{V}}} \end{aligned} \end{aligned}$$

(31)

Set $\eta _i=\max \{v_tC_i^t|t=1,2,\ldots ,k\}\,(i=1,2,\ldots ,n)$. By linear sum of equal weights and Eqs. (28)–(30), Eq. (31) is converted into a linear program as

$$\begin{aligned} \begin{aligned}&\min \ \lambda \sum _{i=1}^n\sum _{t=1}^kv_tC_i^t+(1-\lambda )\sum _{i=1}^n\eta _i\\&\hbox {s.t.}\quad {\varvec{{v}}}\in {\varvec{{V}}},v_tC_i^t\le \eta _i\ (i=1,2,\ldots ,n; t=1,2,\ldots ,k) \end{aligned} \end{aligned}$$

(32)

Experts’ weights $v_t$ $(t=1,2,\ldots ,k)$ are obtained by solving Eq. (32). Then, the collective TFMPR ${\widetilde{R}}=(\tilde{r}_{ij})_{n\times n}$ is derived by Eq. (24).

4.3 Group decision-making method with TFMPRs

Based on the above analysis, the steps of GDM method with TFMPRs are concluded in the following.

Step 1:

The positive ideal vector $\tilde{{\varvec{{p}}}}_i$ and negative ideal vector $\tilde{{\varvec{{g}}}}_i$ of alternative $x_i$ are constructed by Eqs. (25) and (26), respectively.

Step 2:

Calculate the closeness degrees $C_i^t$ $(i=1,2,\ldots ,n; t=1,2,\ldots ,k)$ by Eq. (27).

Step 3:

Solving Eq. (32), experts’ weights $v_t$ $(t=1,2,\ldots ,k)$ are obtained.

Step 4:

Using Eq. (24), the collective TFMPR ${\widetilde{R}}$ is derived.

Step 5:

If the feasible regions of Eqs. (12) and (14) are not empty, solving Eqs. (12) and (14) obtains the optimal solutions $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;$ $i<j)$, respectively. Otherwise, solving Eqs. (15) and (16) obtains the optimal solutions $\alpha _{ij}$ and $\beta _{ij}$ $(i,j=1,2,\ldots ,n;$ $i<j)$, respectively. Then using Eq. (8), the corresponding IMPRs $\dot{R}$ and $\ddot{R}$ are generated.

Step 6:

Set $R=\dot{R}$, solving Eq. (20) yields the interval priority weights $\dot{w}_i$. Set $R=\ddot{R}$, solving Eq. (20) yields the interval priority weights $\ddot{w}_i$.

Step 7:

Combining $\dot{w}_i$ and $\ddot{w}_i$, the four tuples ${\hat{w}}_i$ are formed, which are a permutation of $(\dot{w}_i^-,\dot{w}_i^+,\ddot{w}_i^-,\ddot{w}_i^+)$ satisfying that $w_i^1\le w_i^2\le w_i^3\le w_i^4$.

Step 8:

Solving Eq. (22) yields the normalized TFPWs $\tilde{w}_i$ $(i=1,2,\ldots ,n)$.

Step 9:

Calculate the arithmetic mean values $P(\tilde{w}_i)$ $(i=1,2,\ldots ,n)$ by Eq. (23).

Step 10:

In descending order of the arithmetic mean values $P(\tilde{w}_i)(i=1,2,\ldots ,n)$, the ranking order of alternatives is generated.

Remark 1

The complexity of computation of the proposed method depends on the linear programming models, i.e., Eqs. (12), (14), (15), (16), (20), (22) and (32). Using Karmarkar’s algorithm to solve the linear programming models, the time complexity of Eqs. (12), (14) and (20) is $O(n^9)$, where n is the number of alternatives. Similarly, the time complexity of Eqs. (15) and (16) is $O(n^{12.5})$. The time complexity of Eq. (22) is $O(n^{4.5})$. Time complexity of Eq. (32) is $O(k^{4.5}n^{4.5})$, where k is the number of experts. Meanwhile, the space complexity of Eqs. (12), (14), (15), (16), (20), (22) and (32) is O(1). Although the worst-case complexity is a little big, the processing times for solving such linear programming models are very quick since many mature softwares can be applied to solve these models effectively.

5 A performance evaluation example of precise poverty alleviation and comparison analysis

In this section, a performance evaluation example of precise poverty alleviation is provided to illustrate the applicability and implementation process of the GDM method proposed in this paper. The comparison analysis of computational results are also conducted to show the effectiveness and superiority of the proposed method. The Spearman’s rank-correlation test is performed to further compare the ranking results obtained by different methods.

5.1 Performance evaluation of precise poverty alleviation

To evaluate the precise poverty alleviation performance of four poor villages $\{x_1,x_2,x_3,x_4\}$, a poverty alleviation office in the poverty areas hires three experts with senior technical titles to set up a decision group $E=\{e_1,e_2,e_3\}$. Experts $e_t$ $(t=1,2,3)$ provide their preference information on alternatives by TFNs and elicit the corresponding TFMPRs ${\widetilde{R}}^t$ as follows:

$$\begin{aligned}&{\widetilde{R}}^1= \left( \begin{array}{cccc} (1,1,1)&{}\quad (3,4,5)&{}\quad (1/3,1,2)&{}\quad (1/2,2,3)\\ (1/5,1/4,1/3)&{}\quad (1,1,1)&{}\quad (1/6,8/15,3)&{}\quad (1/2,3/4,4)\\ (1/2,1,3)&{}\quad (1/3,15/8,6)&{}\quad (1,1,1)&{}\quad (7/9,1,4)\\ (1/3,1/2,2)&{}\quad (1/4,4/3,2)&{}\quad (1/4,1,9/7)&{}\quad (1,1,1)\\ \end{array}\right) \\&{\widetilde{R}}^2=\left( \begin{array}{cccc} (1,1,1)&{}\quad (1/2,2,3)&{}\quad (1/5,1/3,3)&{}\quad (1/7,3,5)\\ (1/3,1/2,2)&{}\quad (1,1,1)&{}\quad (1,9/7,4)&{}\quad (1/9,1/5,3)\\ (1/3,3,5)&{}\quad (1/4,7/9,1)&{}\quad (1,1,1)&{}\quad (1/11,1/3,5)\\ (1/5,1/3,7)&{}\quad (1/3,5,9)&{}\quad (1/5,3,11)&{}\quad (1,1,1)\\ \end{array}\right) \\&{\widetilde{R}}^3= \left( \begin{array}{cccc} (1,1,1)&{}\quad (1/5,2/3,5)&{}\quad (1,3/2,3)&{}\quad (3/4,1,3)\\ (1/5,3/2,5)&{}\quad (1,1,1)&{}\quad (4/3,7/2,5)&{}\quad (1,7/5,4)\\ (1/3,2/3,1)&{}\quad (1/5,2/7,3/4)&{}\quad (1,1,1)&{}\quad (1/2,4/3,3)\\ (1/3,1,4/3)&{}\quad (1/4,5/7,1)&{}\quad (1/3,3/4,2)&{}\quad (1,1,1)\\ \end{array}\right) \end{aligned}$$

Using the method proposed in this paper, the corresponding experts’ weights and ranking results are calculated for different values of $\lambda$ in Table 1.

Table 1 Experts’ weights and ranking results for different $\lambda$

It is found that the ranking results are not the same when the values of parameter $\lambda$ are different. Thus, it is necessary to introduce the parameter $\lambda$ in the decision making.

5.2 Ranking reversal test of the proposed GDM with TFMPRs

The ranking reversal proposed in [21] is a common criterion to measure the performance of the decision-making methods. A reasonable method should avoid the rank reversal by adding or deleting of an alternative. In other words, if an alternative is added or deleted from the GDM problem, then any other two alternatives should keep the same ranking order. To test the rank reversal of the proposed GDM method, the original performance evaluation example of precise poverty alleviation is decomposed into Subproblem 1 by deleting alternative $x_4$ and Subproblem 2 by deleting alternative $x_2$ from the original problem, respectively. Thus, there exists an alternative set $\{x_1,x_2,x_4\}$ in Subproblem 1 and there exists an alternative set $\{x_2,x_3,x_4\}$ in Subproblem 2. Set $\lambda =0.5$. Using the proposed GDM method, the corresponding computation results are, respectively, generated in Table 2.

It is evident from Table 2 that the ranking order is $x_1\succ x_2\succ x_4$ for Subproblem 1 and $x_2\succ x_3\succ x_4$ for Subproblem 2, which are consistent with the ranking $x_1\succ x_2\succ x_3\succ x_4$ for the original problem. The ranking for alternatives $x_1$, $x_2$ and $x_4$ is $x_1\succ x_2\succ x_4$ before alternative $x_3$ is introduced, keeps the same as the order after $x_3$ is added. The same results can be observed for alternatives $x_2$, $x_3$ and $x_4$. There is no any rank reversal phenomenon for any two alternatives by addition or deletion of an alternative. Thus, the proposed method can well avoid the rank reversal. The examination of the rank reversal shows the validity and practicability of the method proposed in this paper.

Table 2 Ranking results of two subproblems

5.3 Comparison analysis with existing methods

There exists little research on GDM with TFMPRs. By transforming TFMPR into the corresponding triangular fuzzy additive reciprocal preference relation (TFARPR), methods in Wang and Tong [22] and Liu et al. [23] are applied to illustrate the advantage of the method proposed in this paper.

Firstly, according to Meng and Chen [11], a TFMPR ${\widetilde{R}}=(r_{ij})_{n\times n}$ with $\tilde{r}_{ij}=(l_{ij},m_{ij},u_{ij})$ is transformed into the corresponding TFARPR ${\widehat{R}}=({\hat{r}}_{ij})_{n\times n}$ with ${\hat{r}}_{ij}=(rl_{ij},rm_{ij},ru_{ij})$ where

$$\begin{aligned} {\hat{r}}_{ij}=\frac{\log _9(\tilde{r}_{ij}+1)}{2}=\left( \frac{\log _9(l_{ij}+1)}{2},\frac{\log _9(m_{ij}+1)}{2},\frac{\log _9(u_{ij}+1)}{2}\right) \end{aligned}$$

(33)

Based on Eq. (33), the above individual TFMPRs ${\widetilde{R}}^{t}(t=1,2,3)$ are transformed into the corresponding TFARPRs ${\widehat{R}}^t$.

Set experts’ weight vector is $(\frac{1}{3},\frac{1}{3},\frac{1}{3})^{\mathrm {T}}$. Using method in Wang and Tong [22], the ranking order of alternatives is derived as $x_1\succ x_4\succ x_3\succ x_2$, which is different from the results obtained by the proposed methods in this paper.

Considering that each expert gives the ($n-1$) restricted preference values on alternative set X by using triangular fuzzy numbers in Liu et al. [23], the above TFARPRs ${\widetilde{B}}^t=(\tilde{b}_{ij}^t)_{4\times 4}\,(t=1,2,3)$ are further transformed into the following incomplete TFARPRs.

$$\begin{aligned} {\widetilde{B}}^1= & {} \left( \begin{array}{cc} (0.5000, 0.5000, 0.5000)&{}(0.7500, 0.8155, 0.8662)\\ \times &{}(0.5000, 0.5000, 0.5000)\\ \times &{}\times \\ \times &{}\times \\ \end{array} \right. \\&\left. \begin{array}{cc} \times &{}\times \\ (0.0923,0.3570,0.7500)&{}\times \\ (0.5000,0.5000,0.5000)&{}(0.4428,0.5000,0.8155)\\ \times &{}(0.5000,0.5000,0.5000)\\ \end{array} \right) \\ {\widetilde{B}}^2= & {} \left( \begin{array}{cc} (0.5000,0.5000,0.5000)&{}(0.3423,0.6577,0.7500)\\ \times &{}(0.5000,0.5000,0.5000)\\ \times &{}\times \\ \times &{}\times \\ \end{array}\right. \\&\left. \begin{array}{cc} \times &{}\times \\ (0.5000,0.5572,0.8155)&{}\times \\ (0.5000,0.5000,0.5000)&{}(0,0.2500,0.8662)\\ \times &{}(0.5000,0.5000,0.5000)\\ \end{array}\right) \\ {\widetilde{B}}^3= & {} \left( \begin{array}{cc} (0.5000,0.5000,0.5000)&{}(0.1338,0.4077,0.8662)\\ \times &{}(0.5000,0.5000,0.5000)\\ \times &{}\times \\ \times &{}\times \\ \end{array}\right. \\&\left. \begin{array}{cc} \times &{}\times \\ (0.5655,0.7851,0.8662)&{}\times \\ (0.5000,0.5000,0.5000)&{}(0.3423,0.5655,0.7500)\\ \times &{}(0.5000,0.5000,0.5000)\\ \end{array}\right) \end{aligned}$$

Using Liu et al. [23] to solve the incomplete TFAMRPRs ${\widetilde{B}}^t$, the ranking of four alternatives is obtained as $x_1\succ x_4\succ x_2\succ x_3$. The result is the same as the result obtained by the proposed method when $\lambda =0$ and different from the results obtained by the proposed method when $\lambda =0.5$ and $\lambda =1$.

The results obtained by the methods in this paper and [22, 23] are listed in Table 3. From Table 3, it can be seen that alternative $x_1$ is the best in all results. It illustrates the effectiveness of the proposed method in this paper. However, the differences among the ranking results are the orders of alternatives $x_2$, $x_3$ and $x_4$.

Table 3 Ranking results in different methods

5.4 Spearman’s rank-correlation test of the obtained results

To further compare the ranking results obtained by methods in this paper and [22, 23], Spearman’s rank-correlation test is performed in the following.

The ranking value $d_i$ of alternative $x_i$ $(i=1,2,3,4)$ is the order in ranking of alternatives. For example, in ranking result $x_1\succ x_4\succ x_2\succ x_3$, the ranking values of alternative $x_1$, $x_2$, $x_3$ and $x_4$ are 1, 3, 4 and 2, respectively. Motivated by the idea of [20], the test statistics $r_s$ and Z are defined to evaluate the difference between two ranking orders as

$$\begin{aligned} r_s=1-6\sum _{i=1}^n\frac{(d_i)^2}{n(n^2-1)}, \quad Z=r_s\sqrt{(n-1)}. \end{aligned}$$

(34)

where n is the number of alternatives and $d_i$ is the ranking difference of alternative $x_i$ between two different ranking order.

According to [24], the closer the value of $r_s$ is to $+1$ or $-1$, the stronger the relation between two rankings order. Moreover, a level of significance $\sigma$ is used to compare with Z. In general, $Z=1.645$ is the predefined value which corresponds to the level of significance $\sigma =0.5$. If $Z\ge 1.645$, then it can be concluded that there is evidence of a positive relation between two rankings; otherwise, the two rankings can be accepted as dissimilar. Using the test statistics $r_s$ and Z, the comparisons of rankings of alternative are given in Table 4. The three ranking results $x_1\succ x_4\succ x_2\succ x_3$, $x_1\succ x_2\succ x_3\succ x_4$ and $x_1\succ x_4\succ x_3\succ x_2$ are denoted as ranking A, B and C, respectively.

Table 4 Comparisons of ranking order in different methods

It can be seen from Table 4 that in the all rankings, the values of $r_s$ are not close to $+1$ or $-1$ and the values of Z are lower than 1.645. Such rankings obtained by methods [22, 23] and the proposed method are significantly different from statistics. Moreover, the differences between the methods in [22, 23] and the method proposed in this paper are theoretically analyzed as follows:

• (1) Methods [22, 23] focus on the GDM with TFARPRs. The additive approximation consistency and multiplicative consistency of TFARPR are defined, respectively, in [22, 23], which are the extend of consistency of FPR and ignore the characterization of TFARPR itself. This paper investigates the geometric consistency of TFMPR by considering the geometric consistency of IMPR, which is comprehensive and reasonable.

• (2) To rank alternatives, the overall values of alternatives are obtained by the aggregation operators in methods [22, 23], which could result in the information loss. In this paper, the two interval priority weight vectors are derived from the extracted two IMPRs. Then, the TFPWs are determined by a linear programming model and applied to generate the ranking order of alternatives. The determination of TFPWs is credible and logical.

• (3) Wang and Tong [22] assumed that experts’ weights are provided in advance, which is subjective. Liu et al. [23] obtained experts’ weights using the compatibility degree. Motivated by TOPSIS and VIKOR, this paper defines the compromise indices by the closeness degrees of alternatives. Then using a linear programming model, experts’ weights are determinated by considering experts’ compromise attitude, which is objective and feasible.

6 Conclusions

In this paper, the geometric consistency of TFMPR is defined, and a method is proposed to solve GDM with TFMPRs. The primary advantages of the method proposed in this paper are outlined as follows:

• (1) The defined geometric consistency of TFMPR is rational and has solid theoretical basis because it is defined by analyzing the relationship between TFN and interval. Considering the majority case and minority case, two geometrically consistent IMPRs are extracted from the TFMPR by the linear programming models or the adjusted programming models.

• (2) Using a linear program, the two interval priority weight vectors are derived from the two extracted IMPRs, respectively. Aggregating the two interval priority weight vectors, the TFPWs are determinated, which is comprehensive.

• (3) The compromise indices of alternatives are derived by the group utility indices and individual regret indices of alternatives. By minimizing the compromise indices of alternatives to construct a multi-objective programming model, experts’ weights are obtained, which considers experts’ compromise attitude flexibly and is in accordance with the real decision situations.

• (4) A method is proposed to solve GDM with TFMPRs. Although the proposed method focuses on the TFMPRs, it can also be extended to the GDM with FPRs. Therefore, the proposed method has a wide range of applications.

In the future, the consistency of incomplete TFMPR is deserved to investigate. At the same time, it is important and necessary to propose a method for solving GDM problems with incomplete TFMPRs.