Consistency analysis and priority weights of multiplicative trapezoidal fuzzy preference relations based on multiplicative consistency and logarithmic least square model

Abstract

Consistency and the priority vector are two important issues in preference relations. As one of preference relations, multiplicative trapezoidal fuzzy preference relation (MTFPR) is an effective form of vague and imprecise information when decision maker (DM) express his/her opinions by comparing alternatives or criteria with each other in group decision making (GDM). Therefore, it is meaningful to discuss the consistency and the method for deriving priority vector of MTFPRs. In this paper, we define multiplicative consistency of MTFPRs and investigate the necessary and sufficient conditions of multiplicative consistent MTFPRs. Some properties of multiplicative consistent MTFPRs are studied in detail. Based on the necessary and sufficient conditions of multiplicative consistent MTFPRs, two consistent measurement matrices (CMMs) are developed to define an acceptably multiplicative consistent MTFPR. A logarithmic least square model is further constructed for deriving a normalized trapezoidal fuzzy priority vector from a MTFPR. Three numerical examples including a GDM problem are analyzed to demonstrate the validity of the proposed models.

Keywords

Group decision making multiplicative trapezoidal fuzzy preference relations multiplicative consistency logarithmic least square priority model

1 Introduction

Group decision making (GDM) is one of the most common activities in the world. During the process of GDM, decision maker (DM) is usually asked to give his/her preference information by comparing alternatives or criteria with each other. Preference relation is known as pairwise comparisons, which is a very efficient and popular tool to express DMs’ preference information. The Analytic Hierarchy Process (AHP) [1] is one of the extensively used GDM methods. However, with the high-speedy development of social economy, the GDM problems are more complex and uncertain. Hence, traditional AHP cannot deal with these GDM problems. Fuzzy set theory [2] is regarded as an extension of numeric values and it can be used to express the DMs’ preference information, which are imprecise and vague. Based on the fuzzy sets, the Fuzzy AHP was proposed and it was regarded as an extension of the traditional AHP. Fuzzy AHP is used to deal with the GDM problems, where DMs provide their preference by utilizing interval number, triangular fuzzy number and trapezoidal fuzzy number. Van Laarhoven and Pedrycz [3] proposed triangular fuzzy preference relations (TFPRs) whose elements are fuzzy numbers with triangular membership functions, and they also put forward the fuzzy priority theory. Buckley [4] introduced multiplicative trapezoidal fuzzy preference relations (MTFPRs) in the AHP method to model subjective uncertainty. In the past few decades, the Fuzzy AHP has been widely studied in theories [3– 5 , 7] and numerous successful applications [8 –11].

There are two crucial issues worth investigating in GDM, which are consistency and priority vectors of preference relations. Consistency is performed to ensure that preference relations are neither random nor illogical in pairwise comparisons. Lack of consistency in decision making with preference relations can lead to inconsistent conclusions. Consistency is a cardinal transitivity in the strength of preferences. Research on consistency for various preference relations are summarized in Table 1. As can be seen from Table 1, the concepts of additive consistency and multiplicative consistency of preference relations with fuzzy numbers are the direct extension or improvement of multiplicative preference relation [1] and fuzzy preference relation [20]. For example, the additive consistency in [12, 13] is the direct extension of fuzzy preference relations, and the multiplicative consistency in [4 , 14] is the direct extension of consistent multiplicative preference relations. However, Dubois [15] pointed out that one cannot obtain the definition of consistent multiplicative preference relation by using a direct extension of consistent reciprocal preference relation, and the consistent relations of ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \approx {\tilde{a}}_{ik}$ for all i < k < j or i > k > j should be satisfied.

Table 1
Discussions of related works addressing the consistency for different preference relations (PRs)

PRs References Consistency Mathematical expressiones

MPRs A = (a_ij) _n×n Saaty [1] Multiplicative consistency a_ija_jk = a_ik

Xia and Chen [16] Multiplicative consistency a_ija_kl = a_lja_ki (∀ i, j, k)

FPRs R = (r_ij) _n×n Tanino [20] Additive consistency r_ij + r_jk + r_ki = 1.5 (∀ i, j, k)

IMFPRs $\bar{A} = ({\bar{a}}_{ij})_{n \times n}$ Wei et al. [7] Multiplicative consistency ${\bar{a}}_{ij} \otimes {\bar{a}}_{jk} = {\bar{a}}_{ik} \otimes {\bar{a}}_{jj} (\forall i, j, k)$

IFPRs $\bar{R} = ({\bar{r}}_{ij})_{n \times n}$ Gong et al. [12] Additive consistency ${\bar{r}}_{ij} \oplus {\bar{r}}_{jk} = {\bar{r}}_{ik} \oplus \bar{0.5}$

Gong et al. [12] Multiplicative consistency $(\frac{1}{{\bar{r}}_{ij}} - 1) (\frac{1}{{\bar{r}}_{jk}} - 1) (\frac{1}{{\bar{r}}_{ki}} - 1)$

$= (\frac{1}{{\bar{r}}_{ji}} - 1) (\frac{1}{{\bar{r}}_{kj}} - 1) (\frac{1}{{\bar{r}}_{ik}} - 1)$

TFRPRs $\hat{A} = ({\hat{a}}_{ij})_{n \times n}$ Gogus and Boucher [14] Multiplicative consistency ${\hat{a}}_{ij} \otimes {\hat{a}}_{jk} = {\hat{a}}_{ik}$

Xu and Wang [17] Multiplicative consistency ${\hat{a}}_{ij} \otimes {\hat{a}}_{jk} \approx {\hat{a}}_{ik} (i \leq j \leq k)$

Wang [18] Multiplicative consistency ${\hat{a}}_{ik} {\hat{a}}_{kj} {\hat{a}}_{ji} = {\hat{a}}_{ij} {\hat{a}}_{jk} {\hat{r}}_{ki}$

TFCPRs $\hat{R} = ({\hat{r}}_{ij})_{n \times n}$ Gong [19] Multiplicative consistency ${\hat{r}}_{ik} {\hat{r}}_{kj} {\hat{r}}_{ji} = {\hat{r}}_{ij} {\hat{r}}_{jk} {\hat{r}}_{ki} (i \neq j \neq k)$

Wang and Chen [13] Additive consistency ${\hat{r}}_{ij} \oplus {\hat{r}}_{jk} \oplus {\hat{r}}_{ki} = \hat{1.5} (\forall i, j, k)$

ATFPRs $\tilde{R} = ({\tilde{r}}_{ij})_{n \times n}$ Gong et al. [12] Additive consistency ${\tilde{a}}_{ij} \oplus {\tilde{a}}_{jm} = {\tilde{a}}_{im} \oplus {\tilde{a}}_{jj} (\forall i, j, m)$

MTFPRs $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ Buckley [4] Multiplicative consistency ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \approx {\tilde{a}}_{ik} (\forall i, j, k)$

PRs	References	Consistency	Mathematical expressiones
MPRs A = (a_ij) _n×n	Saaty [1]	Multiplicative consistency	a_ija_jk = a_ik
	Xia and Chen [16]	Multiplicative consistency	a_ija_kl = a_lja_ki (∀ i, j, k)
FPRs R = (r_ij) _n×n	Tanino [20]	Additive consistency	r_ij + r_jk + r_ki = 1.5 (∀ i, j, k)
IMFPRs $\bar{A} = ({\bar{a}}_{ij})_{n \times n}$	Wei et al. [7]	Multiplicative consistency	${\bar{a}}_{ij} \otimes {\bar{a}}_{jk} = {\bar{a}}_{ik} \otimes {\bar{a}}_{jj} (\forall i, j, k)$
IFPRs $\bar{R} = ({\bar{r}}_{ij})_{n \times n}$	Gong et al. [12]	Additive consistency	${\bar{r}}_{ij} \oplus {\bar{r}}_{jk} = {\bar{r}}_{ik} \oplus \bar{0.5}$
	Gong et al. [12]	Multiplicative consistency	$(\frac{1}{{\bar{r}}_{ij}} - 1) (\frac{1}{{\bar{r}}_{jk}} - 1) (\frac{1}{{\bar{r}}_{ki}} - 1)$
			$= (\frac{1}{{\bar{r}}_{ji}} - 1) (\frac{1}{{\bar{r}}_{kj}} - 1) (\frac{1}{{\bar{r}}_{ik}} - 1)$
TFRPRs $\hat{A} = ({\hat{a}}_{ij})_{n \times n}$	Gogus and Boucher [14]	Multiplicative consistency	${\hat{a}}_{ij} \otimes {\hat{a}}_{jk} = {\hat{a}}_{ik}$
	Xu and Wang [17]	Multiplicative consistency	${\hat{a}}_{ij} \otimes {\hat{a}}_{jk} \approx {\hat{a}}_{ik} (i \leq j \leq k)$
	Wang [18]	Multiplicative consistency	${\hat{a}}_{ik} {\hat{a}}_{kj} {\hat{a}}_{ji} = {\hat{a}}_{ij} {\hat{a}}_{jk} {\hat{r}}_{ki}$
TFCPRs $\hat{R} = ({\hat{r}}_{ij})_{n \times n}$	Gong [19]	Multiplicative consistency	${\hat{r}}_{ik} {\hat{r}}_{kj} {\hat{r}}_{ji} = {\hat{r}}_{ij} {\hat{r}}_{jk} {\hat{r}}_{ki} (i \neq j \neq k)$
	Wang and Chen [13]	Additive consistency	${\hat{r}}_{ij} \oplus {\hat{r}}_{jk} \oplus {\hat{r}}_{ki} = \hat{1.5} (\forall i, j, k)$
ATFPRs $\tilde{R} = ({\tilde{r}}_{ij})_{n \times n}$	Gong et al. [12]	Additive consistency	${\tilde{a}}_{ij} \oplus {\tilde{a}}_{jm} = {\tilde{a}}_{im} \oplus {\tilde{a}}_{jj} (\forall i, j, m)$
MTFPRs $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$	Buckley [4]	Multiplicative consistency	${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \approx {\tilde{a}}_{ik} (\forall i, j, k)$

MPRs: Multiplicative preference relations; FPRs: Fuzzy preference relations; IMFPRs: Interval multiplicative fuzzy preference relations; IFPRs: Interval fuzzy preference relations; TFRPRs: Triangular fuzzy reciprocal preference relations; TFCPRs: Triangular fuzzy complementary preference relations; ATFPRs: Additive trapezoidal fuzzy preference relations; MTFPRs: Multiplicative trapezoidal fuzzy preference relations.

To quantify the level of consistency of preference relations with fuzzy numbers, some scholars have already done this work. According to the normal distribution, Dong et al. [6] introduced average-case consistency measure for interval-valued reciprocal preference relations. Based on midpoint and two endpoints of triangular fuzzy numbers, Gogus and Boucher [14] constructed two matrices A^m and A^g, which are derived from triangular fuzzy reciprocal preference relation $\hat{A}$ . The elements of A^m are the median values of triangular fuzzy numbers in $\hat{A}$ and the elements of A^g are the geometric mean of upper bound and lower bound of triangular fuzzy numbers in $\hat{A}$ . In line with the eigenvalue method (EVM), if A^m and A^g are consistent, then $\hat{A}$ is consistent. Consider upper bound and lower bound of interval number in interval multiplicative fuzzy preference relation $\bar{A}$ , Liu [21] constructed two multiplicative preference relations B and C. Based on the EVM, if B and C are consistent, then $\bar{A}$ is consistent. Liu et al. [22] constructed three multiplicative preference relations A^L, A^M and A^R by triangular multiplicative preference relation $\hat{A}$ . In a similar way, if A^L, A^M and A^R are consistent, then $\hat{A}$ is consistent. Wang [18] pointed out the drawbacks of model developed by [22] and defined two multiplicative preference relations A^glu and A^m, which are derived from the geometric mean of two endpoints and midpoint of triangular fuzzy number. According to A^glu and A^m, if A^glu and A^m are consistent, then $\hat{A}$ is consistent. Wu et al. [51] proposed a fuzzy group decision making model based on a logarithmic compatibility measure with MTFPRs based on continuous ordered weighted geometric averaging operator.

Another crucial issue is the approach for generating priority vectors of preference relations, which is the basis of the ranking of alternatives. A number of priority methods have been developed to derive priority vectors from different preference relations. They are summarized in Table 2. As shown in Table 2, the optimization model is widely used to derive the priority vectors from various preference relations. These optimization models are developed by minimizing the deviation between preferences and its priority weights. Some straightforward construction methods [4 , 29] are also used to obtian priority vectors of different preference relations. Based on Table 2, we can see that few investigations have been devoted to the issue on the method of deriving priority vectors of MTFPRs. Based on the deviations between ${\tilde{a}}_{ij}$ and ${\tilde{w}}_{i} ø {\tilde{w}}_{j} (\forall i, j = 1, 2, \dots, n)$ , which are trapezoidal fuzzy numbers, Wu et al. [34] developed a logarithmic least square model to derive the priority vectors. It should be noted that if i = j, ${\tilde{w}}_{i} ø {\tilde{w}}_{j}$ should be equal to (1, 1, 1, 1) in the actual decision making problems. However, based on the division operation of trapezoidal fuzzy numbers, ${\tilde{w}}_{i} ø {\tilde{w}}_{j}$ is not equal to (1, 1, 1, 1) given in [34]. Therefore, the objective function of logarithmic least square model proposed in [34] should not consider the case of i = j. It means that the logarithmic least square model proposed in [34] is not perfect.

Table 2

Methods for deriving priority weights of different PRs

PRs	References	Methods
Multiplicative PRs	Saaty [1]	Eigenvalue method
	Wang et al. [24]	Chi-square method
Fuzzy PRs	Gong [19]	Least-square method
	Wang et al. [24]	Chi-square method
Interval multiplicative PRs	Conde and Pérez [25]	Multi-objective optimization model
	Lan et al. [26]	Straightforward construction method
	Wang and Elhag [27]	Goal programming method
	Wang et al. [23]	Two-stage logarithmic goal programming
Interval fuzzy PRs	Genc et al. [28]	Straightforward approach
	Liu et al. [29]	Straightforward construction method
	Wang and Chen [30]	Logarithmic least square method
	Wang and Li [31]	Goal programming method
	Xu and Chen [32]	Linear programming method
Triangular fuzzy PRs	Wang [18]	Logarithmic least square method
	Van Laarhoven and Pedrycz [3]	Optimization model
Trapezoidal fuzzy PRs	Buckley [4]	Geometric mean method
	Zhou et al. [33]	A least deviation model
	Wu et al. [34]	Logarithmic least square model
Incomplete hesitant fuzzy PR	Zhang et al. [39, 40]	Logarithmic least square model

Based on the analyses mentioned above, it is obvious that consistency and priority vectors have mainly been defined and applied in preference relations with crisp data, interval numbers, triangular fuzzy numbers. The two issues of MTFPRs are still rare and imperfect. Compared with crisp data, interval numbers, triangular fuzzy numbers, the trapezoidal fuzzy numbers are practical in more complicated decision-making environment. Meanwhile, Gong et al. [15] said “it is vital to research trapezoidal fuzzy numbers theoretically and practically”. Therefore, it is necessary to investigate the decision making problems with trapezoidal fuzzy numbers in GDM. Motivated by these, the goals of this paper consist in discussing the consistency and priority vectors of MTFPRs, which are as follows:

A multiplicative consistency is firstly put forward for MTFPRs. Some properties are provided for completely consistent MTFPRs.

Based on the necessary and sufficient conditions of multiplicative consistent MTFPRs, an approach is developed to check for the multiplicative consistency degree of MTFPRs.

A logarithmic least square priority model, developed from multi-objective logarithmic least square model, is established for generating a normalized trapezoidal fuzzy priority vector from MTFPR.

The rest of this paper is organized as follows. Section 2 reviews some of the basic concepts. Section 3 defines the multiplicative consistency for MTFPRs and some properties of multiplicative consistent MTFPRs are discussed. Then, two consistent measurement matrices (CMMs) are introduced to check for the multiplicative consistency degree of MTFPRs. Section 4 develops a logarithmic least square model to derive normalized trapezoidal fuzzy priority vector from MTFPR. The proposed methods are illustrated by three illustrative examples including a GDM problem with MTFPRs in Section 5. Finally, Section 6 summarizes the main conclusions of the paper.

2 Preliminaries

In this section, we mainly review the basic knowledge regarding trapezoidal fuzzy numbers (TFNs), their arithmetic operations, multiplicative preference relation, and multiplicative trapezoidal fuzzy preference relations (MTFPRs).

2.1 Trapezoidal fuzzy numbers and their arithmetic operations

Let $\tilde{a}$ be a quaternary array denoted as $\tilde{a} = (a_{1}, a_{2}, a_{3}, a_{4})$ and a₁ ≤ a₂ ≤ a₃ ≤ a₄. Then $\tilde{a}$ is called a TFN, if its membership function $μ_{\tilde{a}} (x) : R \to [0, 1]$ satisfies: $μ_{\tilde{a}} (x) = {\begin{matrix} f_{\tilde{a}}^{L} & a_{1}, \leq x \leq a_{2}, \\ 1, & a_{2} \leq x \leq a_{3}, \\ f_{\tilde{a}}^{R}, & a_{3} \leq x \leq a_{4}, \\ 0, & otherwise . \end{matrix}$ (1) where $f_{\tilde{a}}^{L} = \frac{x - a_{1}}{a_{2} - a_{1}}$ is called the left membership function and $f_{\tilde{a}}^{R} = \frac{x - a_{4}}{a_{3} - a_{4}}$ is called the right membership function.

In particular, if a₂ = a₃, then $\tilde{a}$ is called triangular fuzzy number, if a₁ = a₂ and a₃ = a₄, then $\tilde{a}$ is called interval fuzzy number, if a₁ = a₂ = a₃ = a₄, then $\tilde{a}$ degenerates into an ordinary real number. Especially, if TFN $\tilde{a}$ satisfies a₁ > 0 and a₂ - a₁ = a₄ - a₃, then $\tilde{a}$ is a positive symmetric TFN. For convenience, throughout this paper, we assume that all TFNs are positive TFNs. Let $\tilde{a} = (a_{1}, a_{2}, a_{3}, a_{4})$ and $\tilde{b} = (b_{1}, b_{2}, b_{3}, b_{4})$ be two TFNs. Some arithmetic operations between $\tilde{a}$ and $\tilde{b}$ are listed as follows [35]:

$\tilde{a} \oplus \tilde{b} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}, a_{4} + b_{4})$ ;

$\tilde{a} ⊖ \tilde{b} = (a_{1} - b_{4}, a_{2} - b_{3}, a_{3} - b_{2}, a_{4} - b_{1})$ ;

$\tilde{a} \otimes \tilde{b} \approx (a_{1} \times b_{1}, a_{2} \times b_{2}, a_{3} \times b_{3}, a_{4} \times b_{4})$ ;

$\tilde{a} ø \tilde{b} \approx (\frac{a_{1}}{b_{4}}, \frac{a_{2}}{b_{3}}, \frac{a_{3}}{b_{2}}, \frac{a_{4}}{b_{1}})$ ;

$(\tilde{a})^{λ} \approx ((a_{1})^{λ}, (a_{2})^{λ}, (a_{3})^{λ}, (a_{4})^{λ})$ , λ ∈ R⁺;

$log (\tilde{a}) \approx (log (a_{1}), log (a_{2}), log (a_{3}), log (a_{4}))$ .

2.2 Multiplicative preference relation

A widely used preference relation is called multiplicative preference relation, which is first proposed by Saaty [1]. Let X ={ x₁, x₂, ⋯ , x_n } be a finite set of alternatives. Then, multiplicative preference relation can be formally defined as Definition 1.

Definition 1. ([1]) A multiplicative preference relation on the set X is defined as a matrix A = (a_ij) _n×n ⊂ X × X, if it satisfies the following conditions: $a_{ij} \times a_{ji} = 1, a_{ii} = 1, \forall i, j = 1, 2, \dots, n,$ (2) where a_ij indicates the preference degree of the ith alternative over the jth alternative. Specially, a_ij = 1 is interpreted as indifference between x_i and x_j, a_ij > 1 indicates x_i ≻ x_j and a_ij < 1 indicates x_j ≻ x_i. Note that “≻” means preferred to. Multiplicative transitivity is a vital basis for multiplicative preference relation in GDM. Let A = (a_ij) _n×n be as before. If $a_{ij} = a_{ik} \times a_{kj}, i, j = 1, 2, \dots, n,$ (3) then A is called consistent multiplicative preference relation [1].

2.3 Multiplicative trapezoidal fuzzy preference relation

In order to deal with the complex and uncertain decision-making problems, Buckley [4] extended the multiplicative preference relations to fuzzy environment and introduced the MTFPRs. Let X be as before. Then, MTFPR can be defined as follows.

Definition 2. ([4]) Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be a matrix, where ${\tilde{a}}_{ij} = (a_{ij 1,} a_{ij 2,} a_{ij 3,} a_{ij 4})$ is a TFN. For ∀t = 1, 2, 3, 4, if $a_{ijt} \times a_{ji (5 - t)} = 1, {\tilde{a}}_{ii} = (1, 1, 1, 1),$ (4) then $\tilde{A}$ is called MTFPR on set X, where ${\tilde{a}}_{ij}$ indicates the trapezoidal fuzzy preference degree of ith alternative over jth alternative.

Note that for convenience, throughout this paper, we let M_n be the set of all n × n MTFPRs.

In line with the multiplicative transitivity of multiplicative preference relation, the definition of multiplicative transitivity corresponding to MTFPR is formally defined as follows.

Definition 3. ([4]) For ∀i, j, k, MTFPR $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is consistent if and only if ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \approx {\tilde{a}}_{ik}$ .

In the following, we will illustrate that Definition 3 is not prefect.

Proof. Let ${\tilde{a}}_{ij} = (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4})$ , ${\tilde{a}}_{jk} = (a_{jk 1}$ , a_jk2, a_jk3, a_jk4), ${\tilde{a}}_{ik} = (a_{ik 1}, a_{ik 2}, a_{ik 3}, a_{ik 4})$ , if ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \approx {\tilde{a}}_{ik}$ , based on the multiplication of TFNs, we have $a_{ijt} \times a_{jkt} = a_{ikt}, t = 1, 2, 3, 4 .$ Especially, if i = k, the above equations also should hold, it means that $a_{ijt} \times a_{jit} = a_{iit} = 1, t = 1, 2, 3, 4 .$ By the definition of TFN, we have $a_{ij 1} \leq a_{ij 2} \leq a_{ij 3} \leq a_{ij 4}, a_{ji 1} \leq a_{ji 2} \leq a_{ji 3} \leq a_{ji 4} .$ Thus, $\begin{matrix} a_{ii 1} = a_{ij 1} \times a_{ji 1} \leq a_{ij 2} \times a_{ji 2} \\ \leq a_{ij 3} \times a_{ji 3} \leq a_{ij 4} \times a_{ji 4} \\ = a_{ii 4} = 1 . \end{matrix}$ (5) Equation (5) holds if and only if a_ij1 = a_ij2 = a_ij3 = a_ij4, a_ji1 = a_ji2 = a_ji3 = a_ji4, which means that $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is a multiplicative preference relation, that is, Definition 3 is not proper to measure whether a MTFPR is consistent. In this paper, DMs provide their preference relations by using the linguistic terms, which are given in Table 3 [10]. Based on the relations between linguistic terms and TFNs, which are positive symmetric TFNs, the MTFPRs can be obtained by transforming from linguistic matrices.

Table 3

The linguistic variables and TFNs for the evaluation [10]

Linguistic variables	Abbreviation	Trapezoidal fuzzy number
Equally important	EI	(1,1,1,1)
Intermediate WI and EI	IWIEI	(1,3/2,5/2,3)
Weakly important	WI	(2,5/2,7/2,4)
Intermediate WI and ES	IWIES	(3, 7/2,9/2,5)
Essentially important	ES	(4,9/2,11/2,6)
Intermediate ES and VS	IESVS	(5,11/2,13/2,7)
Very strongly important	VS	(6,13/2,15/2,8)
Intermediate VS and AI	IVSAI	(7,15/2,17/2,9)
Absolutely important	AI	(8,17/2,9,9)
Reciprocal Absolutely important	RA	(1/9,1/9,2/17,1/8)
Intermediate RA and RV	IRARV	(1/9,2/17,2/15,1/7)
Reciprocal Very strongly important	RV	(1/8,2/15,2/13,1/6)
Intermediate RV and RE	IRVRE	(1/7,2/13,2/11,1/5)
Reciprocal Essentially important	RE	(1/6,2/11,2/9,1/4)
Intermediate RE and RW	IRERW	(1/5,2/9,2/7,1/3)
Reciprocal Weakly important	RW	(1/4,2/7,2/5,1/2)
Intermediate RW and EI	IRWEI	(1/3,2/5,2/3,1)

3 Multiplicative consistency and properties of multiplicative trapezoidal fuzzy preference relation

This section introduces a multiplicative consistency of MTFPRs based on multiplicative transitivity equation. With the conditions of multiplicative consistency for MTFPRs, a method to check for MTFPRs’ multiplicative consistency is proposed.

3.1 Multiplicative consistency of multiplicative trapezoidal fuzzy preference relation

Due to the drawback of definition of MTFPRs’ consistency, we introduce the definition of multiplicative consistent MTFPRs in this subsection.

Definition 4. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be a MTFPR, where ${\tilde{a}}_{ij} = (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4})$ . For i, j, k = 1, 2, ⋯ , n, if it satisfies the following multiplicative transitivity: ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \otimes {\tilde{a}}_{ki} = {\tilde{a}}_{ik} \otimes {\tilde{a}}_{kj} \otimes {\tilde{a}}_{ji},$ (6) then, $\tilde{A}$ is called a multiplicative consistent MTFPR.

For all TFNs ${\tilde{a}}_{ij} \in \tilde{A}$ degenerate crisp values, i.e., a_ij1 = a_ij2 = a_ij3 = a_ij4, then the MTFPR degenerates into a multiplicative preference relation, and Equation (6) degenerates into Equation (3), which is equivalent to the original multiplicative consistency proposed by Saaty [1].

Based on Equation (6), the necessary condition of multiplicative consistency for MTFPRs is given as follows:

Theorem 1. If MTFPR $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ satisfies the multiplicative consistency, then the following equation holds: $\sqrt[4]{T_{ij}} \times \sqrt[4]{T_{jk}} = \sqrt[4]{T_{ik}},$ (7) where T_ij = a_ij1 × a_ij2 × a_ij3 × a_ij4, T_jk = a_jk1 × a_jk2 × a_jk3 × a_jk4 and T_ik = a_ik1 × a_ik2 × a_ik3 × a_ik4.

Proof. Suppose that $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ satisfies the multiplicative consistency. According to Definition 4, we have ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \otimes {\tilde{a}}_{ki} = {\tilde{a}}_{ik} \otimes {\tilde{a}}_{kj} \otimes {\tilde{a}}_{ji}$ , which implies that the following equations hold: $a_{ij 1} \times a_{jk 1} \times a_{ki 1} = a_{ik 1} \times a_{kj 1} \times a_{ji 1};$ (8) $a_{ij 2} \times a_{jk 2} \times a_{ki 2} = a_{ik 2} \times a_{kj 2} \times a_{ji 2};$ (9) $a_{ij 3} \times a_{jk 3} \times a_{ki 3} = a_{ik 3} \times a_{kj 3} \times a_{ji 3};$ (10) $a_{ij 4} \times a_{jk 4} \times a_{ki 4} = a_{ik 4} \times a_{kj 4} \times a_{ji 4} .$ (11) By applying the reciprocity of a_ijta_ji(5-t) = 1 (t = 1, 2, 3, 4), then for ∀i, j, k = 1, 2, ⋯ , n, we obtain $T_{ij} \times T_{jk} = T_{ik} .$ (12) Taking the $\frac{1}{4}$ th root of both sides and it follows that $\sqrt[4]{T_{ij}} \times \sqrt[4]{T_{jk}} = \sqrt[4]{T_{ik}} .$ In line with the previous analysis, Equation (7) is necessary condition of multiplicative consistency of MTFPR.

In order to directly analyze the properties of MTFPR, based on Equation (6), we state the following theorem on the necessary and sufficient condition of multiplicative consistency corresponding to MTFPRs.

Theorem 2. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be as before. For i, j, k = 1, 2, ⋯ , n, $\tilde{A}$ is multiplicative consistent if and only if the following two equations hold: $a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4},$ (13) $a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} .$ (14)

Proof. Suppose that $\tilde{A}$ is a multiplicative consistent MTFPR. According to Definition 4, we have ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \otimes {\tilde{a}}_{ki} = {\tilde{a}}_{ik} \otimes {\tilde{a}}_{kj} \otimes {\tilde{a}}_{ji} .$ In line with the multiplication of TFNs, we obtain $a_{ij 1} \times a_{jk 1} \times a_{ki 1} = a_{ik 1} \times a_{kj 1} \times a_{ji 1};$ (15) $a_{ij 2} \times a_{jk 2} \times a_{ki 2} = a_{ik 2} \times a_{kj 2} \times a_{ji 2};$ (16) $a_{ij 3} \times a_{jk 3} \times a_{ki 3} = a_{ik 3} \times a_{kj 3} \times a_{ji 3};$ (17) $a_{ij 4} \times a_{jk 4} \times a_{ki 4} = a_{ik 4} \times a_{kj 4} \times a_{ji 4} .$ (18) For t = 1, 4, by applying the reciprocity of a_jkt × a_kj(5-t) = 1, a_kit × a_ik(5-t) = 1, a_jit × a_ji(5-t) = 1 to Equations (15) and (18), then we obtain $a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4} .$ Similarly, for t = 2, 3, by applying the reciprocity of a_jkt × a_kj(5-t) = 1, a_kit × a_ik(5-t) = 1, a_jit × a_ji(5-t) = 1 to Equations (16) and (17), we obtain $a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} .$ On the other hand, by reversing the process of the necessary proof, we have $\begin{matrix} a_{ij 1} \times a_{jk 1} \times a_{ki 1} = a_{ik 1} \times a_{kj 1} \times a_{ji 1}; \\ a_{ij 2} \times a_{jk 2} \times a_{ki 2} = a_{ik 2} \times a_{kj 2} \times a_{ji 2}; \\ a_{ij 3} \times a_{jk 3} \times a_{ki 3} = a_{ik 3} \times a_{kj 3} \times a_{ji 3}; \\ a_{ij 4} \times a_{jk 4} \times a_{ki 4} = a_{ik 4} \times a_{kj 4} \times a_{ji 4} . \end{matrix}$ That is, we have ${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \otimes {\tilde{a}}_{ki} = {\tilde{a}}_{ik} \otimes {\tilde{a}}_{kj} \otimes {\tilde{a}}_{ji}$ . Hence, based on Definition 5, $\tilde{A}$ is multiplicative consistent MTFPR. Theorem 2 indicates that the condition of multiplicative consistency defined by Equation (6) can be equivalently expressed by the two conditions given by Equations (13) and (14).

Theorem 3. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be as before. $\tilde{A}$ is multiplicative consistency if and only if the following two equations are equivalent:

a_ij1 × a_ij4 = a_ik1 × a_ik4 × a_kj1 × a_kj4, a_ij2 × a_ij3 = a_ik2 × a_ik3 × a_kj2 × a_kj3, i, j, k = 1, 2, ⋯ , n.

a_ij1 × a_ij4 = a_ik1 × a_ik4 × a_kj1 × a_kj4, a_ij2 × a_ij3 = a_ik2 × a_ik3 × a_kj2 × a_kj3, ∀i < j < k.

Proof. Obviously, (a) ⇒ (b). (b) ⇒ (a). Based on the definition of MTFPR, we have ${\tilde{a}}_{ii} = (1, 1, 1, 1)$ and a_ijt × a_ji(5-t) = 1 (t = 1, 2, 3, 4) for i, j, k = 1, 2, ⋯ , n, we know (a) holds true when any two or three of indexes i, j, k are equal. We discuss six possible index orders for i ≠ j ≠ k. (1) i < j < k. Obviously, we have (b) ⇒ (a). (2) i < k < j. Based on (b), we have $\begin{matrix} a_{ik 1} \times a_{ik 4} = a_{ij 1} \times a_{ij 4} \times a_{jk 1} \times a_{jk 4}, \\ a_{ik 2} \times a_{ik 3} = a_{ij 2} \times a_{ij 3} \times a_{jk 2} \times a_{jk 3} . \end{matrix}$ Since a_ijt × a_ji(5-t) = 1 (t = 1, 2, 3, 4), we obtain $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$ (3) j < i < k. According to (b), we have $\begin{matrix} a_{ji 1} \times a_{ji 4} = a_{jk 1} \times a_{jk 4} \times a_{ki 1} \times a_{ki 4}, \\ a_{ji 2} \times a_{ji 3} = a_{jk 2} \times a_{jk 3} \times a_{ki 2} \times a_{ki 3} . \end{matrix}$ Since a_jit × a_ij(5-t) = 1, a_jkt × a_kj(5-t) = 1, a_kit × a_ik(5-t) = 1 for t = 1, 2, 3, 4, we obtain $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$ (4) j < k < i. In line with (b), we have $\begin{matrix} a_{jk 1} \times a_{jk 4} = a_{ji 1} \times a_{ji 4} \times a_{ik 1} \times a_{ik 4}, \\ a_{jk 2} \times a_{jk 3} = a_{ji 2} \times a_{ji 3} \times a_{ik 2} \times a_{ik 3} . \end{matrix}$ For t = 1, 2, 3, 4, we have a_jit × a_ij(5-t) = 1, a_jkt × a_kj(5-t) = 1. Hence, we obtain $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$ (5) k < i < j. It follows from (b) that $\begin{matrix} a_{ki 1} \times a_{ki 4} = a_{kj 1} \times a_{kj 4} \times a_{ji 1} \times a_{ji 4}, \\ a_{ki 2} \times a_{ki 3} = a_{kj 2} \times a_{kj 3} \times a_{ji 2} \times a_{ji 3} . \end{matrix}$ For t = 1, 2, 3, 4, we have a_kit × a_ik(5-t) = 1, a_jit × a_ij(5-t) = 1. Hence, we obtain $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$ (6) k < j < i. By (b), we have $\begin{matrix} a_{kj 1} \times a_{kj 4} = a_{ki 1} \times a_{ki 4} \times a_{ij 1} \times a_{ij 4}, \\ a_{kj 2} \times a_{kj 3} = a_{ki 2} \times a_{ki 3} \times a_{ij 2} \times a_{ij 3} . \end{matrix}$ For t = 1, 2, 3, 4, we have a_kit × a_ik(5-t) = 1. Hence, we obtain $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$

Corollary 1. For ∀i < j < k, $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is a multiplicative consistent MTFPR if and only if $\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$

The two conditions can be used to judge the degree of multiplicative consistency for MTFPR $\tilde{A}$ , and the discriminant method will be explained in more detail in the following subsection.

Let D = {d₁, d₂, ⋯ , d_m} be a set of DMs. Each DM provides his/her MTFPR ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n} (k = 1, 2, \dots, m)$ . Assume that L = (l₁, l₂, . . . , l_m) ^T is the weighting vector of DMs, satisfying l_k ≥ 0, $\sum_{k = 1}^{m} l_{k} = 1$ . Based on the geometric mean [4], the average matrix ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n}$ is determined as follows: ${\tilde{a}}_{ij} = ⨂_{k = 1}^{m} ({\tilde{a}}_{ij}^{(k)})^{l_{k}}, \forall i, j = 1, 2, . . ., n .$ (19)Theorem 4. If the matrix ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n}$ is the average matrix of ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ , then

${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ ;

If ${\tilde{A}}^{(k)} (k = 1, 2, \dots, m)$ is multiplicative consistency for all k, then ${\tilde{A}}^{G}$ is also multiplicative consistency.

Proof. (1) For i ≠ j (i, j = 1, 2, ⋯ , n) and ∀t = 1, 2, 3, 4, we have

$\begin{matrix} a_{ijt} \times a_{ji (5 - t)} & = & \prod_{k = 1}^{m} {(a_{ijt}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{ji (5 - t)}^{(k)})}^{l_{k}} \\ = & \prod_{k = 1}^{m} {(a_{ijt}^{(k)} \times a_{ji (5 - t)}^{(k)})}^{l_{k}} \\ = & 1 . \end{matrix}$ On the other hand, for i = j, we have ${\tilde{a}}_{ii} = (1, 1$ , 1, 1). Thus, based on Definition 2, ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n}$ is a MTFPR, which means that ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n} \in M_{n}$ .

(2) Since all ${\tilde{A}}^{(k)}$ have multiplicative consistency, as Theorem 2, we can obtain

$\begin{matrix} a_{ij 1}^{(k)} \times a_{ij 4}^{(k)} = a_{ik 1}^{(k)} \times a_{ik 4}^{(k)} \times a_{kj 1}^{(k)} \times a_{kj 4}^{(k)}, \\ a_{ij 2}^{(k)} \times a_{ij 3}^{(k)} = a_{ik 2}^{(k)} \times a_{ik 3}^{(k)} \times a_{kj 2}^{(k)} \times a_{kj 3}^{(k)} . \end{matrix}$ It follows that

$\begin{matrix} \prod_{k = 1}^{m} {(a_{ij 1}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{ij 4}^{(k)})}^{l_{k}} = \prod_{k = 1}^{m} {(a_{ik 1}^{(k)})}^{l_{k}} \\ \times \prod_{k = 1}^{m} {(a_{ik 4}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{kj 1}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{kj 4}^{(k)})}^{l_{k}}, \\ \prod_{k = 1}^{m} {(a_{ij 2}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{ij 3}^{(k)})}^{l_{k}} = \prod_{k = 1}^{m} {(a_{ik 2}^{(k)})}^{l_{k}} \\ \times \prod_{k = 1}^{m} {(a_{ik 3}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{kj 2}^{(k)})}^{l_{k}} \times \prod_{k = 1}^{m} {(a_{kj 3}^{(k)})}^{l_{k}} . \end{matrix}$ Based on Equation (19), we have

$\begin{matrix} a_{ij 1} \times a_{ij 4} = a_{ik 1} \times a_{ik 4} \times a_{kj 1} \times a_{kj 4}, \\ a_{ij 2} \times a_{ij 3} = a_{ik 2} \times a_{ik 3} \times a_{kj 2} \times a_{kj 3} . \end{matrix}$ By Theorem 2, ${\tilde{A}}^{G}$ is multiplicative consistent MTFPR.

Theorem 4 indicates that the average matrix ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n}$ , which is derived from ${\tilde{A}}^{(k)} = ({\tilde{a}}_{ij}^{(k)})_{n \times n}$ by using the geometric mean, is MTFPR. Hence, the average matrix ${\tilde{A}}^{G} = ({\tilde{a}}_{ij})_{n \times n}$ is also called group MTFPR. It guarantees continuity of MTFPRs in GDM. On the other hand, Theorem 4 also indicates that if all MTFPRs ${\tilde{A}}^{(k)}$ provided by DMs have multiplicative consistency, then the average matrix ${\tilde{A}}^{G}$ is also multiplicative consistent MTFPR. It guarantees the consistency of MTFPR in GDM.

3.2 Degree of multiplicative consistency for multiplicative trapezoidal fuzzy preference relations

Because each element in MTFPR incorporates more than one value, Saaty’s approach is not applicable to check for the multiplicative consistency of MTFPR. Therefore, it is vital to find a reasonable approach to check for MTFPRs’ multiplicative consistency. In the following, we will develop a discriminant method to realize it.

Inspired by the necessary and sufficient condition of multiplicative consistency corresponding to MTFPR, two matrices A₁ and A₂, derived from MTFPR $\tilde{A}$ , are constructed to check for the multiplicative consistency of $\tilde{A}$ . Then, two consistent measurement matrices (CMMs) A₁ and A₂ can be formally defined as Definition 5.

Definition 5. Assume that $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ is a MTFPR, where ${\tilde{a}}_{ij} = (a_{ij 1}, a_{ij 2}, a_{ij 3}, a_{ij 4})$ is a TFN. Let A₁ = (a_1,ij) _n×n and A₂ = (a_2,ij) _n×n are two matrices. If $a_{1, ij} = \sqrt{a_{ij 1} a_{ij 4}}, a_{2, ij} = \sqrt{a_{ij 2} a_{ij 3}},$ (20) then we call A₁ and A₂ are CMMs of MTFPR $\tilde{A}$ .

The former is determined by taking the geometric means of the interval [a_ij1, a_ij4] of ${\tilde{a}}_{ij}$ in MTFPR $\tilde{A}$ , and the latter is derived by taking the geometric means of the interval [a_ij2, a_ij3] composed by a_ij2 and a_ij3 of ${\tilde{a}}_{ij}$ in MTFPR $\tilde{A}$ .

According to the reciprocity of MTFPRs, the following theorem is directly obtained.

Theorem 5. The two CMMs A₁ and A₂ defined above are two multiplicative preference relations.

Based on the guidelines [1] of checking for degree of consistency, the consistency ratio (CR) of each matrix is computed and checked if it is less than 0.1.

To obtian the CR, the weighting vector of each matrix has to be calculated. Based on multiplicative preference relations, Saaty’s approach [36] for deriving the weighting vectors can be used. According to Saaty’s approach [36], the weighting vectors of CMMs A₁ and A₂, denoted as w¹ and w², can be computed by $\begin{matrix} w^{1} = (w_{1}^{1}, \dots, w_{i}^{1}, \dots, w_{n}^{1})^{T}, \\ w^{2} = (w_{1}^{2}, \dots, w_{i}^{2}, \dots, w_{n}^{2})^{T}, \end{matrix}$ where $w_{i}^{1} = \frac{1}{n} \sum_{j = 1}^{n} (\frac{a_{1, ij}}{\sum_{i = 1}^{n} a_{1, ij}})$ and $w_{i}^{2} = \frac{1}{n}$ $\sum_{j = 1}^{n} (\frac{a_{2, ij}}{\sum_{i = 1}^{n} a_{2, ij}})$ . Then, the larger eigenvalue of each CMM is computed by the following equations $\begin{matrix} λ_{1, max} = \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{1, ij} (w_{j}^{1} / w_{i}^{1}), \\ λ_{2, max} = \frac{1}{n} \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{2, ij} (w_{j}^{2} / w_{i}^{2}) . \end{matrix}$ According to Saaty’s procedure, consistency indices (CIs), which represent the deviation from prefect consistent multiplicative preference relations, are given by $C I_{1} = \frac{λ_{1, max} - n}{n - 1}, C I_{2} = \frac{λ_{2, max} - n}{n - 1} .$ Before calculating the CRs of two CMMs, one point should be considered. Under TFNs environment, DMs cannot be expected to give only integer number to express their preference opinions It is different from Saaty’s consistency checking method. As we all know, the two CMMs may contain non-integer number. Based on Definition 5, we know that a_1,ij and a_2,ij of A₁ and A₂ not belong to the set {1/9, 1/8, ⋯ , 8, 9}. The set is used in random index (RI) estimation by Saaty. Hence, Saaty’s RIs cannot be used to determine the degree of multiplicative consistency of MTFPRs given by DMs.

The following steps give the process of generating the RIs.

Step 1. Random linguistic matrices generation with different sizes.

Step 2. Transform linguistic matrices into MTFPRs.

Step 3. Computing the CMMs.

Step 4. Computing the corresponding CIs.

Step 5. Calculating the mean of these CIs to obtain RIs of each size.

We generate RIs of two CMMs derived from MTFPRs of order 1–14 using a sample size of 10000, and RIs are given in Table 4.

Table 4

Random indices of CMMs with different size

Size of the matrix	1	2	3	4	5	6	7
RI ₁	0	0	0.4879	0.8255	1.0232	1.1436	1.2313
RI ₂	0	0	0.4946	0.8317	1.0311	1.1642	1.2401
Size of the matrix	8	9	10	11	12	13	14
RI ₁	1.2867	1.323	1.3525	1.376	1.3953	1.4098	1.4211
RI ₂	1.2932	1.334	1.364	1.3858	1.4078	1.4208	1.4324

The CR, defined as $CR = \frac{CI}{RI}$ , indicates the ratio of CI to the average RI for the same size matrix. In the case of TFNs, CRs of both the CMMs must be calculated.

Based on Theorem 2, the following corollary is obtained.

Corollary 2. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ be as before. $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ has multiplicative consistency if and only if the two CMMs A₁ and A₂ are consistent defined by Equation (3).

In real-world decision-making problems, the DMs’ preference opinions are subjective, and the preference opinions are provided by pairwise comparisons. Hence, it is difficult for the DMs to provide a multiplicative consistent MTFPR. Following Saaty’s rule [1], the CR is less than or equal to 0.1. Then, it is considered acceptable consistency (satisfactory consistency). In what follows, we focus on defining acceptable multiplicative consistency of MTFPRs.

Definition 6. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ , A₁ and A₂ be as before. If the two CMMs A₁ and A₂ satisfy Saaty’s acceptable consistency, then MTFPR $\tilde{A}$ is said to be an acceptably multiplicative consistent MTFPR, otherwise, MTFPR $\tilde{A}$ is unacceptably multiplicative consistent.

As can be seen from Definition 6, an acceptably multiplicative consistent MTFPR reduces to an acceptably consistent multiplicative preference relation if all TFNs degenerate into non-fuzzy numbers. In order to judge whether $\tilde{A}$ is an acceptably multiplicative consistent MTFPR or not, we can check for whether two CMMs A₁ and A₂ are Saaty’s acceptable consistency. Namely, if CR (A₁) <0.1 and CR (A₂) <0.1, then $\tilde{A}$ satisfies acceptable multiplicative consistency; otherwise, $\tilde{A}$ has unacceptable multiplicative consistency.

4 Logarithmic least square model for generating normalized trapezoidal fuzzy priority vector

In this section, a logarithmic least square model is introduced to derive a normalized trapezoidal fuzzy priority vector from a MTFPR.

Let $\tilde{W} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})^{T}$ be a weighting vector, where ${\tilde{w}}_{i} = (w_{i 1}, w_{i 2}, w_{i 3}, w_{i 4})$ is positive TFN, then $\tilde{W}$ is called a normalized trapezoidal fuzzy priority vector if the following conditions hold: $0 \leq w_{i 1} \leq w_{i 2} \leq w_{i 3} \leq w_{i 4} \leq 1,$ (21) $\sum_{i = 1}^{n} w_{i 1} \leq 1, \sum_{i = 1}^{n} w_{i 4} \geq 1 .$ (22) Based on the division of TFNs, we have ${\tilde{w}}_{i} ø {\tilde{w}}_{j} \approx (\frac{w_{i 1}}{w_{j 4}}, \frac{w_{i 2}}{w_{j 3}}, \frac{w_{i 3}}{w_{j 2}}, \frac{w_{i 4}}{w_{j 1}})$ . For any normalized trapezoidal fuzzy priority vector $\tilde{W}$ , let ${\tilde{c}}_{ij} = {\begin{matrix} (1, 1, 1, 1), i = j, \\ (\frac{w_{i 1}}{w_{j 4}}, \frac{w_{i 2}}{w_{j 3}}, \frac{w_{i 3}}{w_{j 2}}, \frac{w_{i 4}}{w_{j 1}}), i \neq j . \end{matrix}$ (23) then, we have the following theorem.

Theorem 6. Let $\tilde{C} = ({\tilde{c}}_{ij})_{n \times n}$ , where ${\tilde{c}}_{ij}$ is defined by Equation (23), then

$\tilde{C} = ({\tilde{c}}_{ij})_{n \times n} \in M_{n}$ ,

$\tilde{C} = ({\tilde{c}}_{ij})_{n \times n}$ has multiplicative consistency.

Proof. (1) Obviously, for i = j, we have ${\tilde{c}}_{ii} = (1, 1, 1, 1)$ . For i ≠ j and t = 1, 2, 3, 4, we obtain ${\tilde{c}}_{ijt} \times {\tilde{c}}_{ji (5 - t)} = \frac{w_{it}}{w_{j (5 - t)}} \times \frac{w_{j (5 - t)}}{w_{it}} = 1$ . According to Definition 2, $\tilde{C}$ is a MTFPR, i.e., $\tilde{C} = ({\tilde{c}}_{ij})_{n \times n} \in M_{n}$ .

(2) By Equation (23), we have $c_{ij 1} \times c_{ij 4} = \frac{w_{i 1}}{w_{j 4}} \times \frac{w_{i 4}}{w_{j 1}} = \frac{w_{i 1}}{w_{k 4}} \times \frac{w_{i 4}}{w_{k 1}} \times \frac{w_{k 1}}{w_{j 4}} \times \frac{w_{k 4}}{w_{j 1}} = c_{ik 1} \times c_{ik 4} \times c_{kj 1} \times c_{kj 4}$ and $c_{ij 2} \times c_{ij 3} = \frac{w_{i 2}}{w_{j 3}} \times \frac{w_{i 3}}{w_{j 2}} = \frac{w_{i 2}}{w_{k 3}} \times \frac{w_{i 3}}{w_{k 2}} \times \frac{w_{k 2}}{w_{j 3}} \times \frac{w_{k 3}}{w_{j 2}} = c_{ik 2} \times c_{ik 3} \times c_{kj 2} \times c_{kj 3}$ . In line with Theorem 2, $\tilde{C} = ({\tilde{c}}_{ij})_{n \times n}$ is a multiplicative consistent MTFPR.

Corollary 3. Let $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ and $\tilde{W}$ be as before. For i ≠ j and t = 1, 2, 3, 4, if $a_{ijt} = \frac{w_{it}}{w_{j (5 - t)}}, i, j = 1, 2, \dots, n,$ (24) then, MTFPR $\tilde{A}$ has multiplicative consistency.

For i, j = 1, 2, ⋯ , n, i ≠ j and t = 1, 2, 3, 4, Equation (24) can be rewritten into the following equations: $\sqrt{a_{ij 1} \times a_{ij 4}} = \sqrt{\frac{w_{i 1}}{w_{j 4}} \times \frac{w_{i 4}}{w_{j 1}}},$ (25) $\sqrt{a_{ij 2} \times a_{ij 3}} = \sqrt{\frac{w_{i 2}}{w_{j 3}} \times \frac{w_{i 3}}{w_{j 2}}},$ (26) $\prod_{t = 1}^{4} a_{ijt} = \prod_{t = 1}^{4} w_{it} / w_{j (5 - t)} .$ (27) For i, j = 1, 2, ⋯ , n, i ≠ j, it is obvious that Equations (25)-(27) can be rewritten as the following logarithmic expression:

$\begin{matrix} ln (\sqrt{a_{ij 1} \times a_{ij 4}}) = ln (\sqrt{w_{i 1} \times w_{i 4}}) \\ - ln (\sqrt{w_{j 1} \times w_{j 4}}), \end{matrix}$ (28) $\begin{matrix} ln (\sqrt{a_{ij 2} \times a_{ij 3}}) = ln (\sqrt{w_{i 2} \times w_{i 3}}) \\ - ln (\sqrt{w_{j 2} \times w_{j 3}}), \end{matrix}$ (29)

$\begin{matrix} \sum_{t = 1}^{4} ln a_{ijt} = \sum_{t = 1}^{4} ln w_{it} - ln w_{j (5 - t)} . \end{matrix}$ (30) Equations (28)-(30) can remain the multiplicative consistency of MTFPR $\tilde{A}$ . However, due to the inherent limitations of thinking by DMs, the MTFPRs provided by DMs are not always multiplicative consistency. It means that

$\begin{matrix} ln (\sqrt{a_{ij 1} \times a_{ij 4}}) \approx ln (\sqrt{w_{i 1} \times w_{i 4}}) \\ - ln (\sqrt{w_{j 1} \times w_{j 4}}), \end{matrix}$ (31) $\begin{matrix} ln (\sqrt{a_{ij 2} \times a_{ij 3}}) \approx ln (\sqrt{w_{i 2} \times w_{i 3}}) \\ - ln (\sqrt{w_{j 2} \times w_{j 3}}), \end{matrix}$ (32)

$\begin{matrix} \sum_{t = 1}^{4} ln a_{ijt} \approx \sum_{t = 1}^{4} ln w_{it} - ln w_{j (5 - t)} . \end{matrix}$ (33) To obtain an appropriate decision-making result based on an acceptable multiplicative consistent MTFPR $\tilde{A}$ , we hope to find a normalized trapezoidal fuzzy priority vector $\tilde{W} = ({\tilde{w}}_{1}, {\tilde{w}}_{2}, \dots, {\tilde{w}}_{n})^{T}$ , which is close to $\tilde{A}$ as much as possible. Obviously, there is deviation between the left and right sides of the Equations (31)-(33). Based on the above analyses, the multi-objective logarithmic least square model is constructed to derive the normalized trapezoidal fuzzy priority vector from the $\tilde{A}$ . $\min ob j_{1} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} {(δ_{ij})}^{2}$ (34) $\min ob j_{2} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} {(φ_{ij})}^{2}$ (35) $\min ob j_{3} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} \sum_{t = 1}^{4} {(φ_{ij, t})}^{2}$ (36) $s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i 1} \leq 1, \\ \sum_{i = 1}^{n} w_{i 4} \geq 1, \\ 0 \leq w_{i 1} \leq w_{i 2} \leq w_{i 3} \leq w_{i 4} \leq 1, \\ i = 1, 2, \dots, n, \end{matrix}$ (37) where $δ_{ij} = ln (\sqrt{a_{ij 1} \times a_{ij 4}}) - ln (\sqrt{w_{i 1} \times w_{i 4}}) + ln (\sqrt{w_{j 1} \times w_{j 4}})$ , φ_ij,t = ln a_ijt - ln w_it + ln w_j(5-t), $φ_{ij} = ln (\sqrt{a_{ij 2} \times a_{ij 3}}) - ln (\sqrt{w_{i 2} \times w_{i 3}}) + ln (\sqrt{w_{j 2} \times}$ $\bar{w_{j 3}})$ and the constraints are to guarantee that the $\tilde{W}$ is a normalized trapezoidal fuzzy priority vector defined by Equations (21) and (22).

Notice that since $\sum_{i = 1}^{n} w_{i 1} \leq 1$ and $\sum_{i = 1}^{n} w_{i 4} \geq 1$ , it is feasible to find vectors α = (α₁, ⋯ , α_n) and β = (β₁, ⋯ , β_n) (α_i ≠ β_i) such that $\sum_{i = 1}^{n} α_{i} = 1$ and $\sum_{i = 1}^{n} β_{i} = 1$ , where w_i1 ≤ α_i ≤ w_i4 and w_i1 ≤ β_i ≤ w_i4. Let $\sum_{i = 1}^{n} \sqrt{w_{i 1} \times w_{i 4}} = \sum_{i = 1}^{n} α_{i}$ and $\sum_{i = 1}^{n} \sqrt{w_{i 2} \times w_{i 3}} = \sum_{i = 1}^{n} β_{i}$ , we have $\sum_{i = 1}^{n} \sqrt{w_{i 1} \times w_{i 4}} = 1,$ (38) $\sum_{i = 1}^{n} \sqrt{w_{i 2} \times w_{i 3}} = 1 .$ (39) It is feasible to incorporate the constraints defined by Equations (38) and (39) into Equation (37) for deriving the normalized trapezoidal fuzzy priority vector of a MTFPR $\tilde{A}$ . Meanwhile, we can easily solve the minimization problems (34) and (35). Let $w_{i}^{14} = \sqrt{w_{i 1} \times w_{i 4}}, i = 1, 2, \dots, n .$ (40) $w_{i}^{23} = \sqrt{w_{i 2} \times w_{i 3}}, i = 1, 2, \dots, n .$ (41) Then, a solution of the minimization problems Equations (34)-(36) with the constrains defined by Equations (37)-(39) is found by solving the following three logarithmic least square models, which are shown as follows: $\min ob j_{1} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} {({\bar{δ}}_{ij})}^{2}$ (42) $s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i}^{14} = 1, \\ w_{i}^{14} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ $\min ob j_{2} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} {({\bar{φ}}_{ij})}^{2}$ (43) $s . t . {\begin{matrix} \sum_{i = 1}^{n} w_{i}^{23} = 1, \\ w_{i}^{23} \geq 0, i = 1, 2, \dots, n . \end{matrix}$ $\min ob j_{3} = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} \sum_{t = 1}^{4} {(φ_{ij, t})}^{2}$ (44) $s . t . {\begin{matrix} 0 \leq w_{i 1} \leq w_{i 2} \leq w_{i 3} \leq w_{i 4} \leq 1, \\ \sum_{i = 1}^{n} w_{i 1} \leq 1, \\ \sum_{i = 1}^{n} w_{i 4} \geq 1, \\ w_{i}^{* 14} = \sqrt{w_{i 1} \times w_{i 4}}, \\ w_{i}^{* 23} = \sqrt{w_{i 2} \times w_{i 3}}, \\ i = 1, 2, \dots, n, \end{matrix}$ where ${\bar{δ}}_{ij} = ln (\sqrt{a_{ij 1} \times a_{ij 4}}) - ln (w_{i}^{14}) + ln (w_{j}^{14})$ , ${\bar{φ}}_{ij} = ln (\sqrt{a_{ij 2} \times a_{ij 3}}) - ln (w_{i}^{23}) + ln (w_{j}^{23})$ , $(w_{1}^{* 14},$ $w_{2}^{* 14}, \dots, w_{n}^{* 14})$ and $(w_{1}^{* 23}, w_{2}^{* 23}, \dots, w_{n}^{* 23})$ are the optimal solutions of (42) and (43), respectively.

It is worth noticing that models defined by Equations (42) and (43) are logarithmic least square models for deriving priority vectors from the two multiplicative preference relations A₁ and A₂, respectively. Based on the necessary conditions for the existence of an extremum, we have $w_{i}^{* 14} = \frac{{(\prod_{j = 1}^{n} a_{1, ij})}^{\frac{1}{n}}}{\sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{1, ij})}^{\frac{1}{n}}}, i = 1, 2, \dots, n .$ (45) $w_{i}^{* 23} = \frac{{(\prod_{j = 1}^{n} a_{2, ij})}^{\frac{1}{n}}}{\sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{2, ij})}^{\frac{1}{n}}}, i = 1, 2, \dots, n .$ (46) where a_1,ij and a_2,ij are defined in Definition 5.

As previously analyzed, model Equation (44) can be equivalently formulated as follows: $\begin{matrix} \min obj = \sum_{i = 1}^{n} \sum_{j = 1, j \neq i}^{n} \sum_{t = 1}^{4} {(φ_{ij, t})}^{2} \\ s . t . {\begin{matrix} 0 \leq w_{i 1} \leq w_{i 2} \leq w_{i 3} \leq w_{i 4} \leq 1, \\ \sum_{i = 1}^{n} w_{i 1} \leq 1, \sum_{i = 1}^{n} w_{i 4} \geq 1, \\ \sqrt{w_{i 2} \times w_{i 3}} = \frac{{(\prod_{j = 1}^{n} a_{2, ij})}^{\frac{1}{n}}}{\sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{2, ij})}^{\frac{1}{n}}}, \\ \sqrt{w_{i 1} \times w_{i 4}} = \frac{{(\prod_{j = 1}^{n} a_{1, ij})}^{\frac{1}{n}}}{\sum_{i = 1}^{n} {(\prod_{j = 1}^{n} a_{1, ij})}^{\frac{1}{n}}}, \\ i = 1, 2, \dots, n . \end{matrix} \end{matrix}$ By solving model Equation (47), we can obtain the optimal solution denoted by $w_{i 1}^{*}, w_{i 2}^{*}, w_{i 3}^{*}, w_{i 4}^{*}$ . Thus, a normalized trapezoidal fuzzy priority vector ${\tilde{W}}^{*} = ({\tilde{w}}_{1}^{*}, \dots, {\tilde{w}}_{i}^{*}, \dots, {\tilde{w}}_{n}^{*})^{T}$ is derived from the MTFPR $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ , where ${\tilde{w}}_{i}^{*} = (w_{i 1}^{*}, w_{i 2}^{*}, w_{i 3}^{*}, w_{i 4}^{*})$ , i = 1, 2, ⋯ , n.

5 Numerical examples

This section provides three numerical examples including a GDM problem to illustrate the validity of the proposed methods.

Example 1. Consider a 3 × 3 MTFPR: $\begin{matrix} \tilde{A} = (\begin{matrix} (1, 1, 1, 1) \\ (0.8034, 0.9166, 1.1786, 1.3389) \\ (0.4979, 0.6060, 0.8484, 0.9959) \end{matrix} \\ \begin{matrix} \begin{matrix} (0.7469, 0.8485, 1.0909, 1.2448) \\ (1, 1, 1, 1) \\ (0.5739, 0.6346, 0.7500, 0.8033) \end{matrix} \\ \begin{matrix} (1.0041, 1.1786, 1.6501, 2.0083) \\ (1.2448, 1.3333, 1.5759, 1.7425) \\ (1, 1, 1, 1) \end{matrix}) . \end{matrix} \end{matrix}$ Based on Definition 4 and Theorem 2, we will check for whether the conditions of multiplicative consistency are satisfied. The results are given in Table 5. Note that Equation (6) holds true when any two or three of indexes i, j, k are equal. Thus, we only verify the case when i ≠ j ≠ k. As can be seen from Table 5, the results state that the MTFPR $\tilde{A}$ satisfies all requirements of multiplicative consistency. Meanwhile, based on Definition 5, we have two CMMs A₁ and A₂ of MTFPR $\tilde{A}$ as follows: $\begin{matrix} A_{1} = (\begin{matrix} 1.0000 & 0.9642 & 1.4201 \\ 1.0371 & 1.0000 & 1.4728 \\ 0.7042 & 0.6790 & 1.0000 \end{matrix}), \\ A_{2} = (\begin{matrix} 1.0000 & 0.9621 & 1.3946 \\ 1.0394 & 1.0000 & 1.4495 \\ 0.7171 & 0.6899 & 1.0000 \end{matrix}) . \end{matrix}$ The two CMMs A₁ and A₂ all satisfy the Saaty’s completely consistency, which is defined by Equation (3). It confirms the conclusion of Corollary 2.

Table 5
Satisfaction of multiplicative consistency

i j k Equation (6) Equation (13) Equation (14)

${\tilde{a}}_{ij} \otimes {\tilde{a}}_{jk} \otimes {\tilde{a}}_{ki} = {\tilde{a}}_{ik} \otimes {\tilde{a}}_{kj} \otimes {\tilde{a}}_{ji}$

1 2 3 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 0.9297=0.9297 0.9256=0.9256

1 3 2 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 2.0170=2.0170 1.9448=1.9448

2 1 3 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 1.0760=1.0760 1.0803=1.0803

2 3 1 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 2.1691=2.1691 2.1011=2.1011

3 2 1 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 0.4610=0.4610 0.4760=0.4760

3 1 2 (0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600) 0.4959=0.4959 0.5141=0.5141

i j k	Equation (6)	Equation (13)	Equation (14)
1 2 3	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.9297=0.9297	0.9256=0.9256
1 3 2	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	2.0170=2.0170	1.9448=1.9448
2 1 3	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	1.0760=1.0760	1.0803=1.0803
2 3 1	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	2.1691=2.1691	2.1011=2.1011
3 2 1	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.4610=0.4610	0.4760=0.4760
3 1 2	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.4959=0.4959	0.5141=0.5141

Obviously, Equations (6) and (7) hold true when any two or three of indexes i, j, k are equal. Thus, we only give the results of Equations (6) and (7) when i ≠ j ≠ k. These results are shown in Table 6. As shown in Table 6, we know the $\tilde{A}$ is multiplicative consistent MTFPR and Equation (7) holds, which confirms the conclusion of Theorem 1.

Table 6

Satisfaction of necessary condition corresponding to multiplicative consistency

i j k	Equation (6)	Equation (7)
1 2 3	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	1.4072=1.4072
1 3 2	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.9632=0.9632
2 1 3	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	1.4611=1.4611
2 3 1	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	1.0383=1.0383
3 2 1	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.7106=0.7106
3 1 2	(0.4630,0.6856,1.4586,2.1600)=(0.4630,0.6856,1.4586,2.1600)	0.6844=0.6844

Example 2. Consider a 4 × 4 MTFPR $\tilde{B} =$ : $(\begin{matrix} (1, 1, 1, 1) & (1, \frac{3}{2}, \frac{5}{2}, 3) & (7, \frac{15}{2}, \frac{17}{2}, 9) & (3, \frac{7}{2}, \frac{9}{2}, 5) \\ (\frac{1}{3}, \frac{2}{5}, \frac{2}{3}, 1) & (1, 1, 1, 1) & (6, \frac{13}{2}, \frac{15}{2}, 8) & (4, \frac{9}{2}, \frac{11}{2}, 6) \\ (\frac{1}{9}, \frac{2}{17}, \frac{2}{15}, \frac{1}{7}) & (\frac{1}{8}, \frac{2}{15}, \frac{2}{13}, \frac{1}{6}) & (1, 1, 1, 1) & (\frac{1}{3}, \frac{2}{5}, \frac{2}{3}, 1) \\ (\frac{1}{5}, \frac{2}{9}, \frac{2}{7}, \frac{1}{3}) & (\frac{1}{6}, \frac{2}{11}, \frac{2}{9}, \frac{1}{4}) & (1, \frac{3}{2}, \frac{5}{2}, 3) & (1, 1, 1, 1) \end{matrix}) .$ Based on Definition 5, we have the two CMMs B₁ and B₂ of MTFPR $\tilde{B}$ as follows: $\begin{matrix} B_{1} = (\begin{matrix} 1.0000 & 1.7321 & 7.9373 & 3.8730 \\ 0.5774 & 1.0000 & 6.9282 & 4.8990 \\ 0.1260 & 0.1443 & 1.0000 & 0.5774 \\ 0.2582 & 0.2041 & 1.7321 & 1.0000 \end{matrix}), \\ B_{2} = (\begin{matrix} 1.0000 & 1.9365 & 7.9844 & 3.9686 \\ 0.5164 & 1.0000 & 6.9821 & 4.9749 \\ 0.1252 & 0.1432 & 1.0000 & 0.5144 \\ 0.2520 & 0.2010 & 1.9365 & 1.0000 \end{matrix}) . \end{matrix}$ It is easy to verify that B₁ and B₂ are both inconsistent. Based on Corollary 2, we know $\tilde{B}$ is an inconsistent MTFPR. On the other hand, we have CR (B₁) =0.0218 and CR (B₂) =0.0293, which are less than the acceptable threshold 0.1. It means that B₁ and B₂ are acceptably consistent. In line with Definition 6, $\tilde{B}$ is an acceptably consistent MTFPR.

Example 3. To demonstrate our proposed approach, let us consider a GDM problem concerning the selection of best investment for an investment company which was discussed in [34]. In the example, three DMs, denoted as D = {d₁, d₂, d₃}, give his or her preference opinion over a set of alternatives S = {S₁, S₂, S₃, S₄} with different linguistic terms given in Table 3. These linguistic terms are transformed into TFNs that are used to construct MTFPRs, which are listed as follows:

${\tilde{A}}^{(1)} =$ $(\begin{matrix} (1, 1, 1, 1) & (2, \frac{5}{2}, \frac{7}{2}, 4) & (5, \frac{11}{2}, \frac{13}{2}, 7) & (6, \frac{13}{2}, \frac{15}{2}, 8) \\ (\frac{1}{4}, \frac{2}{7}, \frac{2}{5}, \frac{1}{2}) & (1, 1, 1, 1) & (6, \frac{13}{2}, \frac{15}{2}, 8) & (\frac{1}{9}, \frac{2}{17}, \frac{2}{15}, \frac{1}{7}) \\ (\frac{1}{7}, \frac{2}{13}, \frac{2}{11}, \frac{1}{5}) & (\frac{1}{8}, \frac{2}{15}, \frac{2}{13}, \frac{1}{6}) & (1, 1, 1, 1) & (\frac{1}{5}, \frac{2}{9}, \frac{2}{7}, \frac{1}{3}) \\ (\frac{1}{8}, \frac{2}{15}, \frac{2}{13}, \frac{1}{6}) & (7, \frac{15}{2}, \frac{17}{2}, 9) & (3, \frac{7}{2}, \frac{9}{2}, 5) & (1, 1, 1, 1) \end{matrix}),$ ${\tilde{A}}^{(2)} =$ $(\begin{matrix} (1, 1, 1, 1) & (1, \frac{3}{2}, \frac{5}{2}, 3) & (\frac{1}{6}, \frac{2}{11}, \frac{2}{9}, \frac{1}{4}) & (\frac{1}{6}, \frac{2}{11}, \frac{2}{9}, \frac{1}{4}) \\ (\frac{1}{3}, \frac{2}{5}, \frac{2}{3}, 1) & (1, 1, 1, 1) & (\frac{1}{5}, \frac{2}{9}, \frac{2}{7}, \frac{1}{3}) & (5, \frac{11}{2}, \frac{13}{2}, 7) \\ (4, \frac{9}{2}, \frac{11}{2}, 6) & (3, \frac{7}{2}, \frac{9}{2}, 5) & (1, 1, 1, 1) & (6, \frac{13}{2}, \frac{15}{2}, 8) \\ (4, \frac{9}{2}, \frac{11}{2}, 6) & (\frac{1}{7}, \frac{2}{13}, \frac{2}{11}, \frac{1}{5}) & (\frac{1}{8}, \frac{2}{15}, \frac{2}{13}, \frac{1}{6}) & (1, 1, 1, 1) \end{matrix}),$ ${\tilde{A}}^{(3)} =$ $(\begin{matrix} (1, 1, 1, 1) & (\frac{1}{4}, \frac{2}{7}, \frac{2}{5}, \frac{1}{2}) & (\frac{1}{5}, \frac{2}{9}, \frac{2}{7}, \frac{1}{3}) & (\frac{1}{7}, \frac{2}{13}, \frac{2}{11}, \frac{1}{5}) \\ (2, \frac{5}{2}, \frac{7}{2}, 4) & (1, 1, 1, 1) & (1, \frac{3}{2}, \frac{5}{2}, 3) & (8, \frac{17}{2}, 9, 9) \\ (3, \frac{7}{2}, \frac{9}{2}, 5) & (\frac{1}{3}, \frac{2}{5}, \frac{2}{3}, 1) & (1, 1, 1, 1) & (\frac{1}{9}, \frac{2}{17}, \frac{2}{15}, \frac{1}{7}) \\ (5, \frac{11}{2}, \frac{13}{2}, 7) & (\frac{1}{9}, \frac{2}{9}, \frac{2}{17}, \frac{1}{8}) & (7, \frac{15}{2}, \frac{17}{2}, 9) & (1, 1, 1, 1) \end{matrix}) .$ In line with Definition 5, we have the two CMMs $A_{1}^{(k)}$ , $A_{2}^{(k)}$ of MTFPRs ${\tilde{A}}^{(k)} (k = 1, 2, 3)$ , and we have $\begin{matrix} CR (A_{1}^{(1)}) = 0.6280, CR (A_{2}^{(1)}) = 0.6562, \\ CR (A_{1}^{(2)}) = 0.6866, CR (A_{2}^{(2)}) = 0.6576, \\ CR (A_{1}^{(3)}) = 0.8090, CR (A_{2}^{(3)}) = 0.7932 . \end{matrix}$ Obviously, they are all greater than the acceptable threshold of Satty’s approach. In line with Definition 6, ${\tilde{A}}^{(k)} (k = 1, 2, 3)$ are unacceptably multiplicative consistent MTFPRs. These results are in accord with the results given in [34].

In [34], Wu et al. developed a method to improve the consistency of MTFPRs. Based on the method, the adjusted MTFPRs ${\bar{A}}^{(k)}$ , which have property of acceptable multiplicative consistency - see Appendix 1. In line with Definition 5, we have: $\begin{matrix} CR ({\bar{A}}_{1}^{(1)}) = 0.0248, CR ({\bar{A}}_{2}^{(1)}) = 0.0247, \\ CR ({\bar{A}}_{1}^{(2)}) = 0.0647, CR ({\bar{A}}_{2}^{(2)}) = 0.0708, \\ CR ({\bar{A}}_{1}^{(3)}) = 0.0222, CR ({\bar{A}}_{2}^{(3)}) = 0.0197, \end{matrix}$ where ${\bar{A}}_{1}^{(k)}$ and ${\bar{A}}_{2}^{(k)} (k = 1, 2, 3)$ are the CMMs of MTFPRs ${\bar{A}}^{(k)}$ , respectively, and defined by Equation (20). It is clear that they are all less than the acceptable threshold 0.1. Based on Definition 6, ${\bar{A}}^{(k)} (k = 1, 2, 3)$ are acceptably multiplicative consistent MTFPRs. These results are in accord with the results given in [34].

Utilize Equation (19) to aggregate all individual MTFPRs and obtain the average matrix ${\tilde{A}}^{G}$ based on the importance of the three DMs (i.e. l₁ = 0.43, l₂ = 0.40 and l₃ = 0.17), where

${\tilde{A}}^{G} =$ $\begin{matrix} (\begin{matrix} (1, 1, 1, 1) & (0 . 94901 . 13141 . 46951 . 6453) \\ (0 . 60780 . 68050 . 88391 . 0538) & (1, 1, 1, 1) \\ (0 . 94500 . 98651 . 07921 . 1331) & (0 . 77300 . 83730 . 99091 . 0999) \\ (1 . 01271 . 05461 . 14661 . 1972) & (0 . 90160 . 93340 . 99871 . 0330) \end{matrix} \\ \begin{matrix} (0 . 88250 . 92661 . 01361 . 0582) & (0 . 83530 . 87220 . 94820 . 9875) \\ (0 . 90921 . 00921 . 19431 . 2937) & (0 . 96801 . 00131 . 07131 . 1091) \\ (1, 1, 1, 1) & (0 . 78220 . 81050 . 87700 . 9152) \\ (1 . 09271 . 14021 . 23381 . 2784) & (1, 1, 1, 1) \end{matrix}) . \end{matrix}$ By solving model Equation (47), we obtain normalized trapezoidal fuzzy priority weights as follows: $\begin{matrix} {\tilde{w}}_{1}^{*} = (0.2345, 0.2489, 0.2724, 0.2825), \\ {\tilde{w}}_{2}^{*} = (0.2119, 0.2251, 0.2543, 0.2739), \\ {\tilde{w}}_{3}^{*} = (0.2230, 0.2261, 0.2357, 0.2422), \\ {\tilde{w}}_{4}^{*} = (0.2691, 0.2694, 0.2694, 0.2694) . \end{matrix}$ According to the ranking approach proposed in [37], the $N ({\tilde{w}}_{i}^{*})$ is calculated for each normalized trapezoidal fuzzy priority weights ${\tilde{w}}_{i}^{*}$ as $\begin{matrix} N ({\tilde{w}}_{1}^{*}) = 0.2599, N ({\tilde{w}}_{2}^{*}) = 0.2408, \\ N ({\tilde{w}}_{3}^{*}) = 0.2315, N ({\tilde{w}}_{4}^{*}) = 0.2693 . \end{matrix}$ The $N ({\tilde{w}}_{i}^{*})$ are ranked in descending order as follows: $N ({\tilde{w}}_{4}^{*}) > N ({\tilde{w}}_{1}^{*}) > N ({\tilde{w}}_{2}^{*}) > N ({\tilde{w}}_{3}^{*}) .$ Rank all the alternatives x_i, (i = 1, 2, 3, 4) in accordance with the $N ({\tilde{w}}_{i}^{*})$ , and we have $x_{4} ≻ x_{1} ≻ x_{2} ≻ x_{3} .$ It can be seen that the ranking order obtained by method [34] is different from that obtained by the proposed method. Compared with the method developed in [34], we observe the primary reasons may come from three aspects:

Wu et al. [34] measured the degree of MTFPRs’ consistency by computing the deviation between the MTFPR and its expected fuzzy preference relation. However, the consistency level of a MTFPR should not be changed with its corresponding expected fuzzy preference relation.

In this paper, the discriminant method of MTFPR’s consistent level defined in Definition 6 is stable and reliable since it only depends on the information of the original MTFPR.

On the other hand, two approaches in [34] and this paper for deriving trapezoidal fuzzy priority vector are different. Wu et al. [34] constructed a programming model that only minimizes the deviation between ${\tilde{a}}_{ij}$ in MTFPR $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ and ${\tilde{w}}_{i} ø {\tilde{w}}_{j}$ . However, the programming model should not consider the case when i = j. In this paper, a logarithmic least square model is developed to improve this method. The proposed method is more flexible than the approach proposed in [34].

For different preference relations, consistency measure [43 , 50], consensus [45, 46] and ranking approach [47] have been investigated. However, this paper mainly discusses the multiplicative consistency and priority vector of preference relations with trapezoidal fuzzy numbers. The proposed methods have the following characteristics:

The proposed multiplicative consistency discriminant method for MTFPR does not compute its corresponding expected fuzzy preference relation. It is developed by using the original information of MTFPR.

To measure the level of multiplicative consistency more accurately, we developed two CMMs, generated two sets of random indices, and defined the concept of MTFPR with acceptable multiplicative consistency.

The normalized trapezoidal fuzzy priority weights are obtained by utilizing the developed logarithmic least square model, which is widely used to derive priority vectors from different preference relations. Meanwhile, the normalized trapezoidal fuzzy priority weights lie in the interval [0,1], which can be used to solve multi-criteria decision making problems.

6 Conclusion

Application of fuzzy set theory to GDM with preference relations have been successful and some important issues in GDM with preference relations have been studied in some literatures. But some issues of GDM under TFNs environment have not been discussed. In this paper, we have addressed on multiplicative consistency and priority vector of MTFPRs.

We have started by defining the multiplicative consistency of MTFPRs, and some priorities of multiplicative consistent MTFPRs have been discussed. Based on the necessary and sufficient condition of multiplicative consistent MTFPRs, two CMMs of MTFPR have been introduced. For measuring the consistent degree of CMMs, there is a need for new RIs by using simulation to further discuss. In line with the two CMMs and the new RIs, an approach to check for the consistency of MTFPRs has been developed.

Another issues is the priority vector of MTFPRs. In order to derive reasonable priority vector of MTFPRs, a logarithmic least square model has been presented to obtain a normalized trapezoidal fuzzy priority vector from MTFPR.

Further research would be focused on the following studies:

The concept of preference relation with interval type-2 fuzzy information [38] would be defined. Then, the consistency measure and the priorities may be investigated by means of the multiplicative consistency and the approach for deriving priority vector proposed by this paper.

Inspired by Li et al. [48], some open problems will be discussed in the future, including incomplete MTFPR with self-confidence levels, a unified framework connected the various preference relations [41, 42] and the decision results analysis for different consistency concepts of MTFPR.

With the development of information and network technology, social network is often involved in large-scale decision making problem. Xiao et al. [49] developed a new framework to deal with personalized individual semantics and consensus in large-scale GDM using linguistic distribution preference relations. Hence, the consistency and consensus analysis of MTFPR under social network will be addressed in the future.

Footnotes

Appendix 1.

$\begin{matrix} {\bar{A}}^{(1)} = (\begin{matrix} (1, 1, 1, 1) & (1.6978, 1.8849, 2.2049, 2.3436) \\ (0.4267, 0.4535, 0.5305, 0.5890) & (1, 1, 1, 1) \\ (0.3640, 0.3734, 0.3992, 0.4169) & (0.4761, 0.4924, 0.5276, 0.5474) \\ (0.4961, 0.5117, 0.5482, 0.5703) & (1.7482, 1.8148, 1.9316, 1.9831) \end{matrix} \\ \begin{matrix} (2.3986, 2.5049, 2.6784, 2.7470) & (1.7534, 1.8241, 1.9544, 2.0158) \\ (1.8269, 1.8955, 2.0310, 2.1003) & (0.5043, 0.5177, 0.5510, 0.5720) \\ (1, 1, 1, 1) & (0.3845, 0.4033, 0.4534, 0.4875) \\ (2.0514, 2.2054, 2.4792, 2.6008) & (1, 1, 1, 1) \end{matrix}), \end{matrix}$ $\begin{matrix} {\bar{A}}^{(2)} = (\begin{matrix} (1, 1, 1, 1) & (0.8849, 1.1599, 1.6350, 1.8489) \\ (0.5408, 0.6116, 0.8622, 1.1301) & (1, 1, 1, 1) \\ (2.0916, 2.1990, 2.4150, 2.5332) & (1.8571, 2.0261, 2.3131, 2.4344) \\ (1.6294, 1.7257, 1.9241, 2.0334) & (0.5497, 0.5770, 0.6264, 0.6500) \end{matrix} \\ \begin{matrix} (0.3948, 0.4141, 0.4547, 0.4781) & (0.4918, 0.5197, 0.5795, 0.6137) \\ (0.4108, 0.4323, 0.4936, 0.5385) & (1.5384, 1.5965, 1.7330, 1.8191) \\ (1, 1, 1, 1) & (2.0429, 2.0976, 2.1949, 2.2328) \\ (0.4479, 0.4556, 0.4767, 0.4895) & (1, 1, 1, 1) \end{matrix}), \end{matrix}$ $\begin{matrix} {\bar{A}}^{(3)} = (\begin{matrix} (1, 1, 1, 1) & (0.2500, 0.2857, 0.4000, 0.5000) \\ (2, 2.5, 3.5, 4) & (1, 1, 1, 1) \\ (1.6637, 1.7873, 2.0547, 2.1922) & (0.3333, 0.4, 0.6667, 1) \\ (2.0525, 2.1052, 2.2375, 2.2907) & (0.5319, 0.5290, 0.5544, 0.5801) \end{matrix} \\ \begin{matrix} (0.4562, 0.4867, 0.5595, 0.6011) & (0.4365, 0.4469, 0.4750, 0.4872) \\ (1, 1.5, 2.5, 3) & (1.7299, 1.8036, 1.8903, 1.8801) \\ (1, 1, 1, 1) & (0.4947, 0.5073, 0.5389, 0.5535) \\ (1.8067, 1.8556, 1.9714, 2.0215) & (1, 1, 1, 1) \end{matrix}) . \end{matrix}$

Acknowledgements

The work was supported by National Natural Science Foundation of China (Nos. 71771001, 71701001, 71501002, 71871001, 71901001, 71901088), Natural Science Foundation for Distinguished Young Scholars of Anhui Province (No. 1908085J03), Research Funding Project of Academic and technical leaders and reserve candidates in Anhui Province (No.2018H179), College Excellent Youth Talent Support Program (gxyq2019236), Key Research Project of Humanities and Social Sciences in Colleges and Universities of Anhui Province (SK2019A0013), Social Science Innovation and Development Research Project in Anhui Province (2019CX094).

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