A three-cycle decision-making selection mechanism with intuitionistic trapezoidal fuzzy preference relations

Abstract

Preference relations as information representation form are widely used to manage complexity and uncertainty in practical decision-making cases. Intuitionistic trapezoidal fuzzy numbers (ITrFNs) can convey more fuzzy messages than other fuzzy numbers. This paper defines intuitionistic trapezoidal fuzzy preference relations (ITrFPRs) to adequately describe the preferences of decision makers (DMs). To optimize the decision-making selection process, this paper constructs a three-cycle decision-making selection mechanism with ITrFPRs. To build this mechanism, firstly, a possibility degree formula is presented to rank ITrFNs. Secondly, ITrFPRs, consistent ITrFPRs, acceptable consistent ITrFPRs and a new consistency index are defined. Thirdly, a novel consistency transformation method is built to elevate the consistency of ITrFPRs. Fourthly, considering the relationship between consistency conditions and priority weight, a priority weight derivation method is exploited to derive intuitionistic trapezoidal fuzzy weights. Ultimately, a three cycle selection mechanism integrated with consistency checking, consistency improvement and priority weight derivation procedures is constructed and applied into the case of trip itinerary selection. Sensitive and comparative analyses are carried out to illustrate the effectiveness and applicability of the proposed mechanism.

Keywords

Intuitionistic trapezoidal fuzzy preference relation consistency priority weight decision-making

1 Introduction

As a convenient tool in modeling decision-making issues, preference relations as the judgment matrix or pairwise comparison matrix can depict decision makers’ (DMs’) preference information only requiring comparisons between each pair of alternatives at a time [1]. There are many types of preference relations, including fuzzy preference relation [2], interval fuzzy preference relation [3], intuitionistic fuzzy preference relation (IFPR) [3, 4], triangular fuzzy preference relation (TFPR) [5], trapezoidal fuzzy preference relation [6], multiplicative preference relation [7], interval multiplicative preference relation [8], intuitionistic multiplicative preference relation [9 –11], triangular fuzzy multiplicative preference relation [12]. Based on this tendency, more applications in the field of decision making can be developed [13 –17]. As for preference relation, its research tendency mainly includes consistency improvement and the priority weight derivation methods.

Consistency is one of the critical topics in preference relations [18]. The lack of consistency in preference relations is caused by the judgment limitation of DMs and complexity of context, which would lead to negative effect on the decision making results [19]. Lots of consistency improvement methods have emerged to resolve this problem [20]. Consistency improvement methods are developed based on two types of transitivity: additive transitivity [21] and multiplicative transitivity [22]. Transitivity based consistency improvement methods with different preference relations have been extensively studied [5]. To handle additive inconsistency, Liu et al. [5] investigated a group decision making model with triangular fuzzy additive reciprocal matrices based on additive approximation consistency. Meng et al. [4] presented an additive consistency concept of linguistic intuitionistic preference relations. With respect to multiplicative transitivity, Wu et al. [3] explored isomorphic multiplicative transitivity for IFPRs. Li and Wang [19] designed a concept of expected multiplicative consistent probabilistic hesitant fuzzy preference relation based on multiplicative transitivity. Xu et al. [23] defined ordinal and multiplicative consistency for reciprocal preference relations. Using the concept of both the ordinal and multiplicative consistencies in building consistency for preference relations not only can overcome inconsistency but also they are appropriate for both strict and non-strict circumstances. Based on discussion above, additive transitivity and multiplicative transitivity are basics of consistency improvement method for preference relations [19]. In order to develop a scientific consistency improvement method, using additive transitivity and multiplicative transitivity together to reflect and improve the consistency level should be specifically discussed.

Another worthy research topic of preference relations is priority weight derivation method. Recently, numerous research has investigated priority weight derivation method based on consistent conditions [24]. According to the principle of minimizing deviation, Wu et al. [3] developed a multiplicative consistency-based multi-objective programming model to generate priority vector from the IFPRs. Based on additive consistency, Meng and Chen [25] proposed a simple method to compute triangular fuzzy priority weight vector from triangular fuzzy reciprocal preference relations. Zhang and Pedrycz [26] proposed some fractional programming models of intuitionistic multiplicative preference relations to derive the intuitionistic fuzzy weights vector. Based on the existing isomorphism, Wu [27] defined multiplicative consistency and developed a multiobjective programming model to derive the priority vector of IFPRs. The advantages of this kind of model was not losing information compared with existing methods [28]. From the literature discussed above, priority weight derivation methods are commonlyinvestigated based on mathematic models, which is the trend to improve consistency because of their advantages in preserving original and objective information. In this paper, the priority weight derivation method was specifically analyzed to build consistency improvement method for preference relations.

Owing to the inevitably fuzziness and uncertainty in reality, it is hard for DMs to furnish evaluation on several alternatives in the form of crisp numbers [29]. Fuzzy judgments were widely used in preference relations to better describe practical situations [30]. Intuitionistic fuzzy numbers (IFNs) [31], triangular fuzzy numbers (TFNs) [5] and intuitionistic TFNs (ITFNs) [32] are the typical representatives. TFNs are a special form of intuitionistic trapezoidal fuzzy numbers (ITrFNs), while ITrFNs can convey more uncertain and fuzzy information than TFNs. The membership and non-membership degrees of ITrFNs explicitly reflect the fuzzy concept relative to trapezoidal fuzzy numbers rather than simple fuzzy concept of “excellent” or “good”, when comparing to the IFNs [33]. Many basics of ITrFNs have been investigated. Different types of aggregation operators of ITrFNs were explored to fuse information for resolving MCDM issues [34]. MCDM problems cannot be solved by utilizing aggregation operators alone. Possibility degree is an effective means to rank fuzzy numbers and derive the optimal alternative. Some shortcomings of existing possibility degree were analyzed, then an improved possibility degree was presented to rank IFNs [35]. Garg and Kumar [36] proposed a new ranking method called possibility degree measure to rank different linguistic IFNs. A new possibility degree method was also extended into solving problems under linguistic interval-valued intuitionistic fuzzy set environment [37]. New possibility degree measure should be investigated to ITrFNs for generating reliable ranking results.

Given the superiorities of preference relations and ITrFNs, this paper introduces ITrFNs to preference relations, and defines intuitionistic trapezoidal fuzzy preference relations (ITrFPRs). A three-cycle decision-making selection mechanism with ITrFPRs is constructed to ensure a reasonable result. Main contributions can be specifically highlighted as follows: (1) this paper newly defines ITrFPRs characterized by ITrFNs, whcih can convey more comprehensive and realistic information. (2) This paper constructs an applicable three-cycle decision-making selection mechanism. The consistency checking and consistency improvement procedures ensure a scientific selection by diminishing contradictive and asymmetry information. Meanwhile, this paper uses both additive transitivity and multiplicative transitivity to reflect and improve the consistency level making it appropriate for both strict and non-strict circumstances. (3) This paper also constructs a priority weight derivation model. It is an interactive process. Most original and objective information in practice can be preserved by generating the priority weight directly. The ranking result can better match thereality.

The structure of this paper is as follows. Section 2 briefly reviews some basics. In Section 3, a novel possibility degree formula is proposed to rank ITrFNs. We propose a consistency checking and a consistency improvement method. Section 4 investigates another ITrFPRs consistency conversion formula in the form of vector. A priority weight deviation method is developed. Then, a three-cycle decision-making selection mechanism is summarized. In Section 5, an illustrative example, sensitive and comparative analyses in practical context are provided. In Section 6, conclusions are finally provided.

2 Preliminaries

In this section, some related basic concepts of ITrFPRs are reviewed.

2.1 IFNs and ITrFNs

Definition 1. [1] Let X be the universe of discourse. Then, an intuitionistic fuzzy sets (IFSs) B in X is defined by $B = (〈 x, μ_{B} (x), υ_{B} (x) 〉 | x \in X),$ (1) where μ_B (x) , υ_B (x) ∈ [0, 1], 0 ≤ μ_B (x) + υ_B (x) ≤ 1. ∀x ∈ X, μ_B (x), υ_B (x) and π (x) = 1 - μ_B (x) - υ_B (x) are the membership, nonmembership degrees and degree of indeterminancy of x belonging to B, respectively, where 0 ≤ π (x) ≤ 1.

For a given element x in the universe of discourse X, (μ_B (x) , υ_B (x)) is an IFN. For convenience, an IFN $\tilde{b}$ can be expressed as $\tilde{b} = (μ_{\tilde{b}}, υ_{\tilde{b}})$ , where $μ_{\tilde{b}} \in [0, 1]$ , $υ_{\tilde{b}} \in [0, 1]$ and $0 \leq μ_{\tilde{b}} + υ_{\tilde{b}} \leq 1$ .

Definition 2. [38] An interval-valued IFS (IVIFS) defined on the universe of discourse X is $B = (〈 x, μ_{B} (x), υ_{B} (x) 〉 | x \in X),$ where μ_B (x) ⊆ [0, 1] and υ_B (x) ⊆ [0, 1] are the membership and nonmembership degrees interval of x belonging to B, satisfying: $sup μ_{B} (x) + sup υ_{B} (x) \leq 1, \forall x \in X .$

For convenience, an interval-valued IFN (IVIFN) can be simply expressed as T = ([a, b] , [c, d]), where [a, b] ⊆ [0, 1], [c, d] ⊆ [0, 1] and b + d ≤ 1.

Definition 3. [39] Let T₁ = ([a₁, b₁] , [c₁, d₁]) and T₂ = ([a₂, b₂] , [c₂, d₂]) be two IVIFNs. The possibility degree of T₁ ≥ T₂ is defined as $\begin{matrix} P (T_{1} \geq T_{2}) = \frac{1}{2} {max [1 - max (\frac{b_{2} - a_{1}}{b_{1} - a_{1} + b_{2} - a_{2}}, 0), 0] \\ + max [1 - max (\frac{d_{2} - c_{1}}{d_{1} - c_{1} + d_{2} - c_{2}}, 0), 0]} \end{matrix}$

2.2 ITrFNs

Definition 4. [33] Let r = ([a, b, c, d] ; μ_r, υ_r) be an ITrFN. The membership function of A is defined as $μ_{r (x)} = {\begin{matrix} \frac{x - a}{b - a} μ_{r}, & a \leq x \leq b \\ μ_{r}, & b \leq x \leq c \\ \frac{d - x}{d - c} μ_{r}, & c \leq x \leq d \\ 0, & others \end{matrix}$ (2) ) its non-membership function is $υ_{r (x)} = {\begin{matrix} \frac{b - x + υ_{r} (x - a_{1})}{b - a_{1}}, & a_{1} \leq x \leq b \\ υ_{r}, & b \leq x \leq c \\ \frac{x - c + υ_{r} (d_{1} - x)}{d_{1} - c}, & c \leq x \leq d_{1} \\ 0, & others \end{matrix}$ (3) where 0 ≤ μ_r ≤ 1 and 0 ≤ υ_r ≤ 1. Especially, if b = c, then r = ([a, b, c, d] ; μ_r, υ_r) is degenerated to an ITFN; if μ_r = 1, υ_r = 0, then A is reduced to be a trapezoidal fuzzy number.

Definition 5. [40] Let r₁ = ([a₁, b₁, c₁, d₁] ; μ₁, υ₁) and r₂ = ([a₂, b₂, c₂, d₂] ; μ₂, υ₂) be two ITrFNs, $∥ r_{1} ∥ = \frac{| a_{1} | + | b_{1} | + | c_{1} | + | d_{1} |}{4}$ , $r_{2} = \frac{| a_{2} | + | b_{2} | + | c_{2} | + | d_{2} |}{4}$ (θ ≥ 0). Thus, the operations of ITrFNs are defined as $\begin{matrix} (1) r_{1} + r_{2} = {[a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}, d_{1} + d_{2}]; \\ \frac{∥ r_{1} ∥ μ_{1} + ∥ r_{2} ∥ μ_{2}}{∥ r_{1} ∥ + ∥ r_{2} ∥}, \frac{∥ r_{1} ∥ υ_{1} + ∥ r_{2} ∥ υ_{2}}{∥ r_{1} ∥ + ∥ r_{2} ∥}}, \end{matrix}$ when $∥ r_{1} ∥ = ∥ r_{2} ∥ = 0, μ_{r_{1} + r_{2}} = \frac{μ_{1} + μ_{2}}{2}, μ_{r_{1} + r_{2}} = \frac{υ_{1} + υ_{2}}{2}$ , $r_{1} \cdot r_{2} = ([min (a_{1} a_{2}, a_{1} d_{2}, d_{1} a_{2}, d_{1} d_{2}),$ $\begin{matrix} (2) min (b_{1} b_{2}, b_{1} c_{2}, c_{1} b_{2}, c_{1} c_{2}), max (b_{1} b_{2}, b_{1} c_{2}, \\ c_{1} b_{2}, c_{1} c_{2}), \\ max (a_{1} a_{2}, a_{1} d_{2}, d_{1} a_{2}, d_{1} d_{2})]; μ_{1} μ_{2}, υ_{1} + υ_{2} \\ - υ_{1} υ_{2}), \end{matrix}$ $(3) θ r_{1} = ([θ a_{1}, θ b_{1}, θ c_{1}, θ d_{1}]; 1 - (1 - μ_{1})^{θ}, υ_{1}^{θ}),$ $(4) r_{1}^{θ} = ([a_{1}^{θ}, b_{1}^{θ}, c_{1}^{θ}, d_{1}^{θ}]; μ_{1}^{θ}, 1 - (1 - υ_{1})^{θ}) .$

Definition 6. [34] Let r₁ = ([a₁, b₁, c₁, d₁] ; μ₁, υ₁) and r₂ = ([a₂, b₂, c₂, d₂] ; μ₂, υ₂) be two ITrFNs. The normalized Hamming distance between two ITrFNs is defined as follows: $\begin{matrix} d (r_{1}, r_{2}) = \\ \frac{1}{8} (| (1 + μ_{1} - υ_{1}) a_{1} - (1 + μ_{2} - υ_{2}) a_{2} | \\ + | (1 + μ_{1} - υ_{1}) b_{1} - (1 + μ_{2} - υ_{2}) b_{2} | \\ + | (1 + μ_{1} - υ_{1}) c_{1} - (1 + μ_{2} - υ_{2}) c_{2} | \\ + | (1 + μ_{1} - υ_{1}) d_{1} - (1 + μ_{2} - υ_{2}) d_{2} |) . \end{matrix}$ (4)

2.3 Fuzzy preference relations and consistency properties

Definition 7. [1] Let R = (r_ij) _n×n be a preference relation on a finite alternatives set X ={ x₁, …, x_n }. If

$\begin{matrix} 0 \leq r_{ij} \leq 1, r_{ij} + r_{ji} = 1, r_{ii} = 0.5 \\ for all i, j = 1, 2, \dots, n, \end{matrix}$ (5) then R is called a fuzzy preference relation.

Definition 8. [27, 41] Let R = (r_ij) _n×n be a fuzzy preference relation. If it satisfies the additive transitivity $r_{ij} = r_{ik} - r_{jk} + 0.5, for all i, j = 1, 2, \dots, n,$ (6) then R is said to be an additive consistent fuzzy preference relation.

The additive transitivity can also be denoted in the form of vector [42]: $r_{ij} = 0.5 (w_{i} - w_{j} + 1), for all i, j = 1, 2, \dots, n .$ (7)

Definition 9. [27] Let R = (r_ij) _n×n be a fuzzy preference relation. If it satisfies the multiplicative transitivity $r_{ij} r_{jk} r_{ki} = r_{ji} r_{ik} r_{kj}, for all i, j = 1, 2, \dots, n,$ (8) then the fuzzy preference relation R is said to be a multiplicative consistent fuzzy preference relation.

The multiplicative transitivity in the vector form is expressed as [27]: $r_{ij} = \frac{w_{i}}{w_{i} + w_{j}},$ (9)

Definition 10. [43] Let R = (r_ij) _n×n be a fuzzy preference relation, and $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ be its corresponding consistent fuzzy preference relation. Then, the consistency index is defined as $CI (r) = [1 - \frac{2}{n (n - 1)} \sum_{i, j = 1, i < j}^{n} d (r_{ij}, {\bar{r}}_{ij})] .$ (10)

2.4 IFPRs and TFPRs

Definition 11. [1] Let X ={ x₁, …, x_n } be a finite alternatives set, and R = (r_ij) _n×n ⊂ X × X be a fuzzy preference relation. If for all i, j = 1, 2, …, n, it satisfies $0 \leq μ_{ij} + υ_{ij} \leq 1, μ_{ji} = υ_{ij}, υ_{ji} = μ_{ij}, μ_{ii} = υ_{ii},$ (11) then R is called an IFPR, where r_ij is an IFN composed by the certainty degree μ_ij and υ_ij; μ_ij is the preferred degree of alternative x_i to x_j and υ_ij is the non-preferred degree of alternative x_i to x_j.

Definition 12. [25] Let X ={ x₁, …, x_n } be a finite alternatives set, and R = (r_ij) _n×n ⊂ X × X be a fuzzy preference relation. If for all i, j = 1, 2, …, n, it satisfies $\begin{matrix} r_{ij} = [a_{ij}, b_{ij}, c_{ij}], c_{ij} \geq b_{ij} \geq a_{ij} \geq 0 \\ a_{ij} + c_{ji} = b_{ij} + b_{ji} = c_{ij} + a_{ji} = 1, \\ a_{ii} = b_{ii} = c_{ii} = 0.5 . \end{matrix}$ (12) then R is called a TFPR, where r_ij represents the triangular fuzzy preference degree x_i over x_j.

3 Consistency checking and improvement procedures

This section establishes consistency checking and improvement procedures of the three-cycle decision-making selection mechanism with ITrFPRs. We provide a new ranking method and definition of ITrFNs. Consistency conditions, consistency index and a consistency transformation method of ITrFPRs are presented.

3.1 A new ranking method of ITrFNs

Motivated by the core idea of the possibility degree of IVIFNs defined in Definition 3, this paper develops a new ranking method of ITrFNs as follows:

Definition 13. Let r₁ = ([a₁, b₁, c₁, d₁] ; μ₁, υ₁) and r₂ = ([a₂, b₂, c₂, d₂] ; μ₂, υ₂) be two ITrFNs. Then a possibility degree is defined as $\begin{matrix} P (r_{1} \geq r_{2}) = \frac{1}{4} {max [1 - max (\frac{b_{2} - a_{1}}{b_{1} - a_{1} + b_{2} - a_{2}}, 0), 0] + max [1 - max (\frac{c_{2} - b_{1}}{c_{1} - b_{1} + c_{2} - b_{2}}, 0), 0] \\ + max [1 - max (\frac{d_{2} - c_{1}}{d_{1} - c_{1} + d_{2} - c_{2}}, 0), 0] + max [1 - max (\frac{υ_{2} - μ_{1}}{υ_{1} - μ_{1} + υ_{2} - μ_{2}}, 0), 0]} \end{matrix}$ (13)

Especially, if b₁ = c₁ and b₂ = c₂, then the ITrFNs are reduced to ITFNs.

Property 1. Let r₁ = ([a₁, b₁, c₁, d₁] ; μ₁, υ₁) and r₂ = ([a₂, b₂, c₂, d₂] ; μ₂, υ₂) be two ITrFNs, then

0 ≤ P (r₁ ≥ r₂) ≤1, 0 ≤ P (r₂ ≥ r₁) ≤1,

r₁ ≥ r₂, P (r₁ ≥ r₂) =1 ; r₂ ≥ r₁, P (r₂ ≥ r₁) =1,

r₁ ≥ r₂, P (r₁ ≥ r₂) =0; r₂ ≥ r₁, P (r₂ ≥ r₁) =0,

P (r₁ ≥ r₂) + P (r₂ ≥ r₁) =1; $P (r_{1} \geq r_{1}) = \frac{1}{2}$ .

3.2 Consistency checking procedure with ITrFPRs

Definition 14. Let R = (r_ij) _n×n be a fuzzy preference relation on a finite set of alternatives X ={ x₁, x₂, …, x_n }. If for any i, j = 1, 2, …, n, it satisfies $\begin{matrix} a_{ij} + d_{ji} = d_{ij} + a_{ji} = b_{ij} + c_{ji} = c_{ij} + b_{ji} = 1, \\ a_{ij} \leq b_{ij} \leq c_{ij} \leq d_{ij}, a_{ii} = b_{ii} = c_{ii} = d_{ii} = 0.5, \\ μ_{ij}, υ_{ij} \in [0, 1], μ_{ij} + υ_{ij} \leq 1, μ_{ij} = μ_{ji}, \\ υ_{ij} = υ_{ji}, μ_{ii} = 1, υ_{ii} = 0 \end{matrix}$ (14) then R is called an ITrFPR.

The ITrFPR is characterized by r_ij = ([a_ij, b_ij, c_ij, d_ij] ; μ_ij, υ_ij), where μ_ij represents x_i is superior to x_j with the certainty degree [a_ij, b_ij, c_ij, d_ij]; υ_ij represents x_i is inferior to x_j with the certainty degree [a_ij, b_ij, c_ij, d_ij] π_ij = 1 - μ_ij - υ_ij represents the uncertainty degree of [a_ij, b_ij, c_ij, d_ij] to determine the superior one over each pair of alternatives. Given the consistency conditions, based on [43, 44], we generalize new consistency conditions to make them applicable to ITrFPRs.

Definition 15. Let R = (r_ij) _n×n be an ITrFPR characterized by r_ij = ([a_ij, b_ij, c_ij, d_ij] ; μ_ij, υ_ij). If for any i, j, k = 1, 2, …, n, there exits $a_{ij} + d_{ij} + a_{ki} + d_{ki} + a_{jk} + d_{jk} = 3,$ (15)

$b_{ij} + c_{ij} + b_{ki} + c_{ki} + b_{jk} + c_{jk} = 3$ (16)

$μ_{ij} μ_{jk} μ_{ki} υ_{ji} υ_{kj} υ_{ik} = μ_{ji} μ_{kj} μ_{ik} υ_{ij} υ_{jk} υ_{ki}$ (17) then R = (r_ij) _n×n is called a consistent ITrFPR.

Definition 16. Let R = (r_ij) _n×n be an ITrFPR, and $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ be a consistent ITrFPR characterized by ${\bar{r}}_{ij} = ([{\bar{a}}_{ij}, {\bar{b}}_{ij}, {\bar{c}}_{ij}, {\bar{d}}_{ij}]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij})$ . The consistency index (SAI) of ITrFPRs is denoted as follows: $SAI (r) = 1 - \frac{\sum_{i = 1}^{n} \sum_{j = 1}^{n} d (r_{ij}, {\bar{r}}_{ij})}{\sum_{i = 1}^{n} \sum_{j = 1}^{n} d (r_{ij}, {\bar{r}}_{ij}) + \sum_{i = 1}^{n} \sum_{j = 1}^{n} d (r_{ij}, {\bar{r}}_{ij}^{c})}$ (18) where ${\bar{r}}_{ij}^{c} = ([{\bar{a}}_{ij}, {\bar{b}}_{ij}, {\bar{c}}_{ij}, {\bar{d}}_{ij}]; {\bar{υ}}_{ij}, {\bar{μ}}_{ij})$ is the complementary set of ${\bar{r}}_{ij}$ ; $d (r_{ij}, {\bar{r}}_{ij})$ is the distance between elements in ITrFPRs and consistent ITrFPRs; $d (r_{ij}, {\bar{r}}_{ij}^{c})$ is the distance between elements in ITrFPRs and the consistent ITrFPRs.

Definition 17. Let R = (r_ij) _n×n be an ITrFPR, and its elements be expressed by r_ij = ([a_ij, b_ij, c_ij, d_ij] ; μ_ij, υ_ij), i, j = 1, 2, …, n. If the consistency index satisfies SAI (r) ≤ ς, then R = (r_ij) _n×n is an acceptably consistent ITrFPR, where ς is the acceptable consistency threshold.

3.3 Consistency improvement procedure with ITrFPRs

Based on additive transitivity, operational laws and a positive number β discussed by Xu et al. [45], we present a transformation method to modify and improve the consistency of ITrFPRs.

Theorem 1. Suppose there exist alternatives X = {x₁, x₂, …, x_n} on a finite set, and an arbitrary ITrFPR R = (r_ij) _n×n ⊂ X × X characterized by r = (r₁, r₂, …, r_n), where r_i = ([a_i, b_i, c_i, d_i] ; μ_i, υ_i) (i = 1, 2, …, n). Then,the ITrFPR R = (r_ij) _n×n can be converted into a completely consistent ITrFPR $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ by

$\begin{matrix} {\bar{r}}_{ij} = ([{\bar{a}}_{ij}, {\bar{b}}_{ij}, {\bar{c}}_{ij}, {\bar{d}}_{ij}]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}) \\ = ([β (a_{i} - d_{j}) + 0.5, β (b_{i} - c_{j}) + 0.5, \\ β (c_{i} - b_{j}) + 0.5, β (d_{i} - a_{j}) + 0.5]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}) \end{matrix}$ (19)

where ${\bar{μ}}_{ij} = \frac{∥ r_{i} ∥ μ_{i} + ∥ r_{j} ∥ μ_{j}}{∥ r_{i} ∥ + ∥ r_{j} ∥}$ ${\bar{υ}}_{ij} = \frac{∥ r_{i} ∥ υ_{i} + ∥ r_{j} ∥ υ_{j}}{∥ r_{i} ∥ + ∥ r_{j} ∥}$ .

Especially, if i = j, then r_ii = ([0.5, 0.5, 0.5, 0.5] ;1, 0). Particularly, an element r_i = ([a_i, b_i, c_i, d_i] ; μ_r, υ_r) of ITrFPRs would degenerate to a trapezoidal fuzzy number when μ_r = 1, ν_r = 0.

For the space limitation, the proof is omitted here.

In particular, if the coefficient β = 1 of the trapezoid segment in the ITrFN, then the conversion formula is equivalent to the additive transitivity condition in Equation (6). Consequently, Equation (19) is efficient to modify and improve the consistency of ITrFPR.

4 A consistency based priority weight derivation procedure

This section presents a priority weight derivation procedure based on a consistency conversion formula in the vector form. Integrated with consistency checking, consistency improvement and priority weight derivation procedures, a three-cycle decision-making selection mechanism is constructed.

4.1 A consistency transformation formula with ITrFPRs

To generate the priority weight of ITrFPRs, we define a consistency conversion formula in the vector form of ITrFPRs. Let $w_{i} = ([ω_{i}^{a}, ω_{i}^{b}, ω_{i}^{c}, ω_{i}^{d}]; ω_{i}^{μ}, ω_{i}^{υ})$ (i = 1, 2, …, n) be a normalized intuitionistic trapezoidal fuzzy weight of R = (r_ij) _n×n. Where $[ω_{i}^{a}, ω_{i}^{b}, ω_{i}^{c}, ω_{i}^{d}]$ (i = 1, 2, …, n) is a trapezoidal fuzzy vector satisfying $0 \leq ω_{i}^{a} \leq ω_{i}^{b} \leq ω_{i}^{c} \leq ω_{i}^{d} \leq 1$ and $0 \leq \sum_{i = 1}^{n} ω_{i}^{a} \leq 1 \leq \sum_{i = 1}^{n} ω_{i}^{d}$ . $[ω_{i}^{μ}, ω_{i}^{υ}]$ (i = 1, 2, …, n) is an intuitionistic fuzzy vector satisfying $ω_{i}^{μ}, ω_{i}^{υ} \in [0, 1]$ , $ω_{i}^{μ} + ω_{i}^{υ} \leq 1$ , ${\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{μ} \leq ω_{i}^{υ}$ and $ω_{i}^{μ} + n - 2 \geq \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{n} ω_{j}^{u} \cdot ω_{i}^{u}$ and $ω_{i}^{υ}$ denote the membership and non-membership degrees to the alternative x_i with respect to the fuzzy concept $[ω_{i}^{a}, ω_{i}^{b}, ω_{i}^{c}, ω_{i}^{d}]$ , respectively. Based on consistency conditions in Equations (7 and 9), the consistent conversion formula of ITrFPRs is displayed as follows:

Theorem 2. Assume there exists a normalized intuitionistic trapezoidal fuzzy weight vector w = (w₁, w₂, …, w_n). Then a consistent ITrFPR $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ can be calculated by $\begin{matrix} {\bar{r}}_{ij} = ([{\bar{a}}_{ij}, {\bar{b}}_{ij}, {\bar{c}}_{ij}, {\bar{d}}_{ij}]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}) \\ = {\begin{matrix} ([0.5, 0.5, 0.5, 0.5]; 1, 0) & i = j \\ ([\frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5, \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5, \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5, \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}) & i \neq j \end{matrix} \end{matrix}$ (20) $\begin{matrix} {\bar{μ}}_{ij} = \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1}, \\ {\bar{υ}}_{ij} = \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1}, \end{matrix}$ where $0 \leq ω_{i}^{a} \leq ω_{i}^{b} \leq ω_{i}^{c} \leq ω_{i}^{d} \leq 1$ , $0 \leq \sum_{i = 1}^{n} ω_{i}^{a} \leq 1 \leq \sum_{i = 1}^{n} ω_{i}^{d}$ , $ω_{i}^{μ}, ω_{i}^{υ} \in [0, 1]$ , $ω_{i}^{μ} + ω_{i}^{υ} \leq 1$ , $\sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{n} ω_{j}^{u} \leq 1 ω_{i}^{μ}$ and $ω_{i}^{μ} + n - 2 \geq \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{n} ω_{j}^{u}$ for all i = 1, 2, …, n; β is a positive number discussed by Xu et al. [45]; λ is a tolerance parameter reflecting the consistency and deviation tolerance degree of DMs.

The proof is similar to Theorem 1 and omitted here.

4.2 A priority weight derivation procedure with ITrFPRs

In real decision-making process, a DM cannot always give an ITrFPR which entirely satisfies consistency. In this case, the deviation between the given judgment matrix R = (r_ij) _n×n and the consistent ITrFPR $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ is provided to represent this difference. Elements in R = (r_ij) _n×n and $\bar{R} = {({\bar{r}}_{ij})}_{n \times n}$ is expressed by r_ij = ([a_ij, b_ij, c_ij, d_ij] ; μ_ij, υ_ij) and $\begin{matrix} {\bar{r}}_{ij} = ([{\bar{a}}_{ij}, {\bar{b}}_{ij}, {\bar{c}}_{ij}, {\bar{d}}_{ij}]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}) \\ = ([\frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5, \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5, \\ \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5, \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5]; {\bar{μ}}_{ij}, {\bar{υ}}_{ij}), \end{matrix}$

where ${\bar{μ}}_{ij} = \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1}$ , and ${\bar{υ}}_{ij} = \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1}$ .

Therefore, the deviation is shown as $d_{ij}^{a} = \frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5 - a_{ij} i, j = 1, 2, \dots, n; i \neq j$ $d_{ij}^{b} = \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5 - b_{ij} i, j = 1, 2, \dots, n; i \neq j$ $d_{ij}^{c} = \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5 - c_{ij} i, j = 1, 2, \dots, n; i \neq j$ $d_{ij}^{d} = \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5 - d_{ij} i, j = 1, 2, \dots, n; i \neq j$ $\begin{matrix} ɛ_{ij} = \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - μ_{ij} \\ i, j = 1, 2, \dots, n; i \neq j \\ ζ_{ij} = \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - υ_{ij} \\ i, j = 1, 2, \dots, n; i \neq j \end{matrix}$

The smaller the absolute deviations is, the higher the consistency of ITrFPR will be. Hence, a fractional programming model is built as follows: $\begin{matrix} Model 1 : Min J = \sum_{i = 1}^{n} \sum_{j = 1}^{n} | d_{ij}^{a} | + | d_{ij}^{b} | + | d_{ij}^{c} | + | d_{ij}^{d} | + | ɛ_{ij} | + | ζ_{ij} | \\ s . t . {\begin{matrix} \frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5 - a_{ij} - d_{ij}^{a} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5 - b_{ij} - d_{ij}^{b} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5 - c_{ij} - d_{ij}^{c} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5 - d_{ij} - d_{ij}^{d} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - μ_{ij} - ɛ_{ij} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - υ_{ij} - ζ_{ij} = 0 i, j = 1, 2, \dots, n; i \neq j \\ 0 \leq ω_{i}^{a} \leq ω_{i}^{b} \leq ω_{i}^{c} \leq ω_{i}^{d} \leq 1, 0 \leq \sum_{i = 1}^{n} ω_{i}^{a} \leq 1 \leq \sum_{i = 1}^{n} ω_{i}^{d} i, j = 1, 2, \dots, n; i \neq j \\ ω_{i}^{μ}, ω_{i}^{υ} \in [0, 1], ω_{i}^{μ} + ω_{i}^{υ} \leq 1 i, j = 1, 2, \dots, n; i \neq j \\ {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{μ} \leq ω_{i}^{υ}, ω_{i}^{μ} + n - 2 \geq {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{υ} i, j = 1, 2, \dots, n; i \neq j \end{matrix} \end{matrix}$

To simplify the fractional programming model, let us first review the simplifying calculation formula given by Wang and Li [46] as follows:

Let $d_{ij}^{γ +} = \frac{| d_{ij}^{γ} | + d_{ij}^{γ}}{2}$ and $d_{ij}^{γ -} = \frac{| d_{ij}^{γ} | - d_{ij}^{γ}}{2}$ then $d_{ij}^{γ} = d_{ij}^{γ +} + d_{ij}^{γ -}$ and $∥ d_{ij}^{γ} | = d_{ij}^{γ +} + d_{ij}^{γ -}$ , respectively, where $d_{ij}^{γ +} \geq 0$ , $d_{ij}^{γ -} \geq 0$ , and $d_{ij}^{γ +} d_{ij}^{γ -} = 0$ (γ = a, b, c, d, μ, υ). After simplifying calculation, the fractional programming model can be rewritten as. $\begin{matrix} Model 2 : Min J = \sum_{i = 1}^{n} \sum_{j = 1}^{n} d_{ij}^{a +} + d_{ij}^{a -} + d_{ij}^{b +} + d_{ij}^{b -} + d_{ij}^{c +} + d_{ij}^{c -} + d_{ij}^{d +} + d_{ij}^{d -} + ɛ_{ij}^{+} + ɛ_{ij}^{-} + ζ_{ij}^{+} + ζ_{ij}^{-} \\ s . t . {\begin{matrix} \frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5 - a_{ij} - d_{ij}^{a +} + d_{ij}^{a -} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5 - b_{ij} - d_{ij}^{b +} + d_{ij}^{b -} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5 - c_{ij} - d_{ij}^{c +} + d_{ij}^{c -} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5 - d_{ij} - d_{ij}^{d +} + d_{ij}^{d -} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - μ_{ij} - ɛ_{ij}^{+} + ɛ_{ij}^{-} = 0 i, j = 1, 2, \dots, n; i \neq j \\ \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - υ_{ij} - ζ_{ij}^{+} + ζ_{ij}^{-} = 0 i, j = 1, 2, \dots, n; i \neq j \\ 0 \leq ω_{i}^{a} \leq ω_{i}^{b} \leq ω_{i}^{c} \leq ω_{i}^{d} \leq 1, 0 \leq \sum_{i = 1}^{n} ω_{i}^{a} \leq 1 \leq \sum_{i = 1}^{n} ω_{i}^{d} i, j = 1, 2, \dots, n; i \neq j \\ ω_{i}^{μ}, ω_{i}^{υ} \in [0, 1], ω_{i}^{μ} + ω_{i}^{υ} \leq 1 i, j = 1, 2, \dots, n; i \neq j \\ {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{μ} \leq ω_{i}^{υ}, ω_{i}^{μ} + n - 2 \geq {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{υ} i, j = 1, 2, \dots, n; i \neq j \\ d_{ij}^{a +} \geq 0, d_{ij}^{a -} \geq 0, d_{ij}^{b +} \geq 0, d_{ij}^{b -} \geq 0, d_{ij}^{c +} \geq 0, d_{ij}^{c -} \geq 0, d_{ij}^{d +} \geq 0, d_{ij}^{d -} \geq 0 \\ d_{ij}^{a +} d_{ij}^{a -} = 0, d_{ij}^{b +} d_{ij}^{b -} = 0, d_{ij}^{c +} d_{ij}^{c -} = 0, d_{ij}^{d +} d_{ij}^{d -} = 0 i, j = 1, 2, \dots, n; i \neq j \\ ɛ_{ij}^{+} \geq 0, ɛ_{ij}^{-} \geq 0, ζ_{ij}^{+} \geq 0, ζ_{ij}^{-} \geq 0, ɛ_{ij}^{+} ɛ_{ij}^{-} = 0, ζ_{ij}^{+} ζ_{ij}^{-} = 0 i, j = 1, 2, \dots, n; i \neq j \end{matrix} \end{matrix}$

Due to the equations a_ij + d_ji = d_ij + a_ji = b_ij + c_ji = c_ij + b_ji = 1 and μ_ij = μ_ji, υ_ij = υ_ji, the deviation of the elements in the upper triangular matrix are equal to the deviation in the lower side. It is obvious that we can simplify Model 2 into

$\begin{matrix} Model 3 : Min J = \sum_{i = 1}^{n - 1} \sum_{j = i + 1}^{n} d_{ij}^{a +} + d_{ij}^{a -} + d_{ij}^{b +} + d_{ij}^{b -} + d_{ij}^{c +} + d_{ij}^{c -} + d_{ij}^{d +} + d_{ij}^{d -} + ɛ_{ij}^{+} + ɛ_{ij}^{-} + ζ_{ij}^{+} + ζ_{ij}^{-} \\ s . t . {\begin{matrix} \frac{β}{2} (ω_{i}^{a} - ω_{j}^{d}) + 0.5 - a_{ij} - d_{ij}^{a +} + d_{ij}^{a -} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ \frac{β}{2} (ω_{i}^{b} - ω_{j}^{c}) + 0.5 - b_{ij} - d_{ij}^{b +} + d_{ij}^{b -} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ \frac{β}{2} (ω_{i}^{c} - ω_{j}^{b}) + 0.5 - c_{ij} - d_{ij}^{c +} + d_{ij}^{c -} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ \frac{β}{2} (ω_{i}^{d} - ω_{j}^{a}) + 0.5 - d_{ij} - d_{ij}^{d +} + d_{ij}^{d -} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ \frac{2 ω_{i}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - μ_{ij} - ɛ_{ij}^{+} + ɛ_{ij}^{-} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ \frac{2 λ ω_{j}^{μ}}{(ω_{i}^{μ} - ω_{i}^{υ}) + λ (ω_{j}^{μ} - ω_{j}^{υ}) + λ + 1} - υ_{ij} - ζ_{ij}^{+} + ζ_{ij}^{-} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ 0 \leq ω_{i}^{a} \leq ω_{i}^{b} \leq ω_{i}^{c} \leq ω_{i}^{d} \leq 1, 0 \leq \sum_{i = 1}^{n} ω_{i}^{a} \leq 1 \leq \sum_{i = 1}^{n} ω_{i}^{d} i = 1, 2, \dots, n - 1 \\ ω_{i}^{μ}, ω_{i}^{υ} \in [0, 1], ω_{i}^{μ} + ω_{i}^{υ} \leq 1 i = 1, 2, \dots, n - 1 \\ {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{μ} \leq ω_{i}^{υ}, ω_{i}^{μ} + n - 2 \geq {\sum_{j = 1}}_{j \neq i}^{n} ω_{j}^{υ} i = 1, 2, \dots, n - 1 \\ d_{ij}^{a +} \geq 0, d_{ij}^{a -} \geq 0, d_{ij}^{b +} \geq 0, d_{ij}^{b -} \geq 0, d_{ij}^{c +} \geq 0, d_{ij}^{c -} \geq 0, d_{ij}^{d +} \geq 0, d_{ij}^{d -} \geq 0 \\ d_{ij}^{a +} d_{ij}^{a -} = 0, d_{ij}^{b +} d_{ij}^{b -} = 0, d_{ij}^{c +} d_{ij}^{c -} = 0, d_{ij}^{d +} d_{ij}^{d -} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \\ ɛ_{ij}^{+} \geq 0, ɛ_{ij}^{-} \geq 0, ζ_{ij}^{+} \geq 0, ζ_{ij}^{-} \geq 0, ɛ_{ij}^{+} ɛ_{ij}^{-} = 0, ζ_{ij}^{+} ζ_{ij}^{-} = 0 i = 1, 2, \dots, n - 1; j = i + 1, \dots, n \end{matrix} \end{matrix}$

By solving Model 3, we can obtain an optimal intuitionistic trapezoidal fuzzy weight denotedby w = (w₁, w₂, ⋯ w_n) where $w_{i} = ([ω_{i}^{a}$ , $ω_{i}^{b}, ω_{i}^{c}, ω_{i}^{d}]; ω_{i}^{μ}, ω_{i}^{υ})$ . If the optimal objective value J is equal to 0, the original R = (r_ij) _n×n is a consistent.

4.3 A three-cycle decision-making selection mechanism

As aforementioned, this section constructs a three-cycle decision-making selection mechanism integrated with the consistency checking, consistency improvement and priority weight derivation procedures, which is shown as follows:

Step 1. Given an ITrFPR R = (r_ij) _n×n.

Let an expert e give an R = (r_ij) _n×n, and w = (w₁, w₂, …, w_n) be an intuitionistic trapezoidal fuzzy weight on the order set N ={ 1, 2, …, n } derived from R = (r_ij) _n×n.

Step 2. Generate the priority weights of the ITrFPR.

Let β = 1, λ = 1, and construct the vector matrix based on the consistency conversion formula in Equation (20). Based on the Model 3, generate the intuitionistic trapezoidal fuzzy weights of the alternatives.

Step 3. Construct a consistent ITrFPR $\bar{R}$ .

By utilizing Equation (20), the corresponding consistent $\bar{R}$ can be constructed.

Step 4. Calculate the consistency index of $\bar{R}$ .

Calculate the consistency index of $\bar{R}$ referring to Equation (18). Determine $\bar{R}$ whether can satisfy the acceptable consistency or not. If the consistency index SAI (r) < ς, then go to Step 5. If the consistency index SAI (r) ≥ ς, then turn to Step 8 directly.

Step 5. Aggregate the average preference degree of each alternative in $\bar{R}$ .

Utilize the arithmetic average operator to aggregate the ith row. The result of the aggregation indicates the average preference degree r_i = ([a_i, b_i, c_i, d_i] ; μ_i, υ_i) of alternative x_i over other alternatives, i = 1, 2, …, n.

Step 6. Construct an improved consistent ITrFPR ${\bar{R}}^{'}$ .

Assume β = 1, then the improved consistent ITrFPR can be built by the consistency transformation method defined in Equation (19).

Step 7. Calculate the consistency index of ${\bar{R}}^{'}$ .

By calculating the consistency index of the ITrFPR ${\bar{R}}^{'}$ defined in Equation (18), judge the transformation method whether can improve the consistency or not. If the consistency index SAI (r) < ς, then repeat Step 5 and Step 6. If the consistency index SAI (r) ≥ ς, then conduct next step.

Step 8. Generate the priority weights of ${\bar{R}}^{'}$ .

The detailed calculation process is identical to Step 2.

Step 9. Rank all the alternatives and select the optimal one(s).

Based on Equation (14), we will gain a possibility degree matrix which can be used to determine the ranking order of all alternatives. According to the descending order of w_i (i = 1, 2, …, m), we can select the optimal one(s).

5 Illustration of decision-making selection mechanism

In this section, the selection mechanism is employed into a practical example of a trip itinerary selection problem to highlight its applicability and effectiveness. And a sensitivity analysis and comparative analysis are provided to further verify it.

5.1 An illustrative example

Qunar.com is one of the biggest travel search engine websites in China. The website would recommend different trip itineraries to users according to their preference. Though the users fill in the travel mode, travel days, and travel theme information on the website, there are several trip itineraries would be recommended to users. Assume a user e want to be a free walker travelling in Thailand from June 1, 2018 to June 6, 2018. After filling in these basic information, four types of trip itineraries are recommended for him by the website. For the user, due to his judgment limitation, it is hard for him to directly select an optimal itinerary considering all influential factors. In order to pick out a satisfactory alternative, the user is invited to give his preference in the form of ITrFNs comparing each pair of itineraries. For notation simplification, we denote and call these potential itineraries by x₁, x₂, x₃, and x₄. Then, the following ITrFPR R is formed. $\begin{matrix} R = (\begin{matrix} ([0.5, 0.5, 0.5, 0.5]; 1, 0) & ([0.2, 0.3, 0.4, 0.6]; 0.3, 0.6) \\ ([0.4, 0.6, 0.7, 0.8]; 0.3, 0.6) & ([0.5, 0.5, 0.5, 0.5]; 1, 0) \\ ([0.3, 0.4, 0.5, 0.6]; 0.6, 0.4) & ([0.1, 0.2, 0.4, 0.5]; 0.6, 0.3) \\ ([0.2, 0.4, 0.6, 0.7]; 0.4, 0.6) & ([0.3, 0.5, 0.7, 0.8]; 0.5, 0.5) \end{matrix} \\ \begin{matrix} ([0.4, 0.5, 0.6, 0.7]; 0.6, 0.4) & ([0.3, 0.4, 0.6, 0.8]; 0.4, 0.6) \\ ([0.5, 0.6, 0.8, 0.9]; 0.6, 0.3) & ([0.2, 0.3, 0.5, 0.7]; 0.5, 0.5) \\ ([0.5, 0.5, 0.5, 0.5]; 1, 0) & ([0.1, 0.3, 0.4, 0.6]; 0.5, 0.5) \\ ([0.4, 0.6, 0.7, 0.9]; 0.5, 0.5) & ([0.5, 0.5, 0.5, 0.5]; 1, 0) \end{matrix}) \end{matrix}$

Step 1. Given an ITrFP R = (r_ij) _n×n.

The R = (r_ij) _n×n is given above.

Step 2. Generate the priority weight of the ITrFPR.

Let β = 1 and λ = 1, a consistent ITrFPR is constructed by Equation (20). Then, construct a model by Model 3. By solving this model with optimization computer package Matlab, we can derive the objective function value J^* = 1.9694. The result of the intuitionistic trapezoidal fuzzy weights can be derived as $w_{1} = ([0.0686, 0.2256, 0.2256, 0.4686]; 0.1521, 0.7731)$ $w_{2} = ([0.2686, 0.4256, 0.6256, 0.6686]; 0.3032, 0.6716)$ $w_{3} = ([0.0001, 0.0256, 0.2256, 0.2686]; 0.1516, 0.7726)$ $w_{4} = ([0.0686, 0.4252, 0.4483, 0.4686]; 0.2880, 0.7069)$

Step 3. Construct a consistent ITrFPR $\bar{R}$ .

By utilizing Equation (20), the corresponding consistent ITrFPR $\bar{R}$ can be constructed.

Step 4. Calculate the consistency index of $\bar{R}$ .

Let ς = 0.98 and utilize Equation (18), the consistency index of the ITrFPR is obtained as SAI (r) =0.9698. Owing to the consistency index SAI (r) =0.9698 < 0.98, the given ITrFPR is inconsistent, then go to Step 5.

Step 5. Aggregate the average preference degree of each alternative in $\bar{R}$ .

Utilize the arithmetic average operator to aggregate the ith row.

Step 6. Construct an improved consistent ITrFPR ${\bar{R}}^{'}$ .

Let β = 1, and use the transformation method in Equation (19) to repair the consistency of the ITrFPR. Thus, the improved consistent ITrFPR is obtained.

Step 7. Calculate the consistency index of the ${\bar{R}}^{'}$ .

Calculate the consistency index of ${\bar{R}}^{'}$ SAI (r) =0.9926 by Equation (18). Owing to SAI (r) =0.9926 ≥ 0.98, it indicates that ${\bar{R}}^{'}$ is acceptably consistent and the transformation method has improved the consistency of the $\bar{R}$ .

Step 8. Generate the priority weights of ${\bar{R}}^{'}$ .

Similar to Step 2, priority weights of ${\bar{R}}^{'}$ are computed.

Step 9. Rank all the alternatives and select the optimal one(s).

According to Equation (14), we can obtain the possibility degree matrix and ranking value are computed as V (w₁) =0.2174, V (w₂) =0.1905, V (w₃) =0.2948 and V (w₄) =0.2973. The ranking is x₄ ≻ x₃ ≻ x₁ ≻ x₂.

5.2 Sensitivity analysis

By applying the proposed procedure in the illustrative example with different λ, we can obtain different models in Step 2 and different consistent ITrFPRs in Step 3. The influence results of tolerance parameter λ to the consistency index, objective function value and ranking result are shown in Figs. 1–4.

Fig.1

Influence results of tolerance parameter under 1 to SAI (r) and J^*.

Fig.2

Influence results of tolerance parameter over 1 to SAI (r) and J^*.

Fig.3

Influence results of tolerance parameter under 1 to ranking values.

Fig.4

Influence results of tolerance parameter over 1 to ranking values.

As shown in Figs. 1, 2, consistency index is slightly different with the variation of λ value, which means λ has an impact on consistency level of different ITrFPRs. Generally, consistency index values remain stable around SAI (r) =0.97 along with the growth of λ. Even so, most of them cannot reach ς = 0.98. When λ = 0.1, it has the highest consistency level. It means that other than λ = 0.1 no matter what value of λ, ITrFPRs are inconsistent and we need to conduct the consistency improvement process. The objective function values are in the trend of decrease with different λ under 1, reaching the bottom point when λ = 0.6. According to Figs. 3, 4, the order between x₁ and x₂ is ranked as the third and fourth positions along with λ. Although the position of alternative x₄ and x₃ are different, their ranking values are almost the same. That is to say, x₄ ≻ x₃ ≻ x₁ ≻ x₂ yielded by this paper is reasonable and reliable.

What discussed above indicates the tolerance parameter λ affects the consistency improvement and priority weight derivation procedures. The reason is that ITrFPRs are obtained by means of different consistency transformation formula defined in Equation (20) with different λ. In practical application, experts have different requirements to consistency level under different consistency tolerance degree and deviation tolerance degree. Thus, λ can be set according to different DMs’ consistency requirements. In the ideal situation, we assume that the deviation tolerance degree can reach as smaller as possible to eliminate the deviation influence and the acceptable consistency tolerance degree is high to ensure a valid ranking result. To match this situation, the objective function value could be set as the minimum value 0.7860 and the acceptable consistency threshold as ς = 0.98. If and only if λ = 0.6 can satisfy the first condition.

5.3 Comparative analysis

ITrFNs is an extensional form of TFNs. When μ_r = 1, υ_r = 0, b = c, it is apparent that ITrFNs can degenerate to TFNs. When the membership degree μ_r = 1, it indicates that the DM’s preference degree over two alternatives expressed by the TFNs without uncertainty. Inspired by expectation value theory, we utilize the score function of IFNs and transform ITrFNs into TFNs to avoid information loss. ITrFPRs characterized by ITrFNs can compare with TFPRs by performing this normalization steps. By this way, the original ITrFPRs R shown in the illustrative example can be rewritten as TFPRs R′ and consistency improved ITrFPRs ${\bar{R}}^{'}$ To illustrate the rationality and effectiveness of the proposed method, we conduct a comparative analysis with Approach I in the research of [47] by conducting R′ and ${\bar{R}}^{″}$ , respectively.

Comparative analysis with the rewritten original ITrFPR R′

Given the practical characteristics of the problem in illustrative example, we only adopt general objective layer and alternative layer to the hierarchy structure but omit the multiple DMs aggregation and incompleteness recovery steps in Approach I

Comparative analysis using with rewritten consistency improved ITrFPR ${\bar{R}}^{″}$

Using ${\bar{R}}^{″}$ , the normalized priority weights can be calculated as w₁ = 0.9049, w₂ = 0.9346, w₃ = 0.8933, w₄ = 1. By operating identical steps in Approach I with the rewritten original ITrFPR R′. Results can be visually observed in Table 1.

Table 1
Results with different methods

Methods Ranking of alternative The best alternatives

The procedure in this paper x₄ ≻ x₃ ≻ x₁ ≻ x₂ x ₄

The Approach I given by [47]

Using R′ x₄ ≻ x₂ ≻ x₃ ≻ x₁ x ₄

Using ${\bar{R}}^{″}$ x₄ ≻ x₂ ≻ x₁ ≻ x₃ x ₄

Methods	Ranking of alternative	The best alternatives
The procedure in this paper	x₄ ≻ x₃ ≻ x₁ ≻ x₂	x ₄
The Approach I given by [47]
Using R′	x₄ ≻ x₂ ≻ x₃ ≻ x₁	x ₄
Using ${\bar{R}}^{″}$	x₄ ≻ x₂ ≻ x₁ ≻ x₃	x ₄

The obtained result shows that there is a slight difference between the order of alternatives x₁, x₂ and x₃. The optimal alternative of all the methods are x₄, which is identical to the result in our paper. As we conduct the same procedures with [47] using the original and the consistency improved, generating identical optimal alternative reveals that the proposed consistency improvement procedure positively works. It can be further concluded that the selection mechanism can get over the influence caused by contradiction and asymmetry information to the final result.

The advantages of the proposed selection mechanism can be mainly generalized as follows: (1) the three-cycle decision-making selection mechanism ensures a reliable final result because this mechanism is integrated with consistency checking and consistency improvement procedures, thereby well diminishing contradictive and asymmetry information. (2) The selection mechanism is of agility and applicability for DMs by setting different tolerance parameter in Model 3. The tolerance parameter in the consistency conversion formula can be set based on real situations according to different deviation and consistency tolerance degree of DMs. It is suitable to derive priority weights in practical decision-making problems by considering tolerance parameter. (3) The interactive process in the Model 3 and mechanism cycling procedures make the ranking result better match the reality. The priority weight derivation model is an interactive process. It can preserve more original and objective information by generating the priority weight directly.

6 Conclusion

The prominent target of this paper is to furnish a selection mechanism that can optimize the decision-making selection with ITrFPRs. Motivated by the characteristics of the ITrFPRs, the consistency transformation procedure, consistency checking procedure and priority weight derivation procedure are provided and arranged to the three-cycle decision-making selection mechanism.

The superiorities of the three-cycle decision-making mechanism are categorically drawn. Firstly, it ensures the effectiveness and reasonability of the final result in achieving higher consistency level. Secondly, it is of flexibility and applicability since DMs can choose different tolerance parameters. Distinctive DMs’ consistency and deviation tolerance degree can be taken into consideration. Thirdly, it can be found that the selection procedure is closer to practical decision-making context because of the interactive process contained in the selection mechanism and the priority weight derivation model.

It must be acknowledged that the proposed selection mechanism still has some limitations. Group decision making and consensus should also be specifically discussed in the future [48]. The proposed three-cycle decision-making selection mechanism can also be extended into other environment, such as IFSs [49, 50], Pythagorean fuzzy sets [51 –54], interval-valued fuzzy soft sets [55] and neutrosophic soft sets [56], dual hesitant fuzzy soft sets [57].

Footnotes

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions to help improve the overall quality of this paper. This work was supported by the National Natural Science Foundation of China (No.71571193).

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A three-cycle decision-making selection mechanism with intuitionistic trapezoidal fuzzy preference relations

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 IFNs and ITrFNs

3.1 A new ranking method of ITrFNs

4.1 A consistency transformation formula with ITrFPRs

4.3 A three-cycle decision-making selection mechanism

5 Illustration of decision-making selection mechanism

5.1 An illustrative example

5.2 Sensitivity analysis

Table 1 Results with different methods Methods Ranking of alternative The best alternatives The procedure in this paper x4 ≻ x3 ≻ x1 ≻ x2 x 4 The Approach I given by [47] Using R′ x4 ≻ x2 ≻ x3 ≻ x1 x 4 Using R ¯ ″ x4 ≻ x2 ≻ x1 ≻ x3 x 4

Footnotes

Acknowledgments

References