Multistage decision making based on prioritization of hesitant multiplicative preference relations

Abstract

Hesitant multiplicative preference relation (HMPR) is a straightforward and efficient tool for representing hesitant fuzzy information in decision making. The aim of this paper is to develop a method to obtain priority vectors from HMPRs in the context of multistage decision-making (MSDM). To start the investigation, a simple linear programming model motivated by the idea of orness is purposed to calculate the relative weights of different stages. Based on these obtained weights, we develop a least square deviation method as well as a convergent iterative algorithm for prioritizing and ranking the MSDM- HMPR problems. The prominent property of this method is then studied. Finally, a practical example concerning the selection of logistics service providers is given to illustrate the feasibility and applicability of the proposed approach.

Keywords

Hesitant fuzzy set hesitant multiplicative preference relation multistage decision making priority ranking least square deviation method

1 Introduction

Multiplicative preference relation (MPR), also known as pairwise comparison matrix (PCM) [1], is a practical yet widespread methodology to express preference of a decision maker (DM) when s/he needs to rank a set of alternatives. By human intrinsic ability to express perceptions in a hierarchical way [2], the MPRs allow a DM to discriminate between two alternatives and provide judgments in terms of the important intensity of one alternative over the other.

MPRs have been studied extensively and have found huge applications in decision-making problems since the original work of Saaty et al. [1]. However, due to inherent complexity and uncertainty of real-world decisions, it is often impractical for DMs to provide precise data when making pairwise comparisons. Under such cases, it is more natural or easier for DMs to express judgments in a fuzzy way. To typify this fuzziness, different fuzzy preference relations have been studied [3–7 , 39–43], such as interval MPR [4], intuitionistic MPR [5], intuitionistic fuzzy preference relation (IFPR) [3 , 24], triangular IFPR [7], linguistic preference relation [8, 9] etc. Recently, Torra and Narukawa [10, 11] introduced the hesitant fuzzy sets (HFSs) as an extension of fuzzy set, which are featured by a membership function that preserves a few different values from interval [0, 1]. The motivation for introducing HFSs is that when determining the membership degree of an element to a reference set, sometimes the difficulty lies in that there is a doubt among a set of possible values. Wang et al. [12] introduced the dominance relations and the opposition relations for HFSs and studied several desirable properties. Liao et al. [13] defined the hesitant fuzzy preference relation (HFPR) where the pairwise judgments over alternatives are expressed by hesitant fuzzy elements. The multiplicative consistency of HFPRs and its application in group decision making (GDM) were also discussed. Furthermore, Xu et al. [14] developed an intuitionistic fuzzy analytic hierarchy process (AHP) method based on the multiplicative consistency.

Inspired by the HFSs, Xia and Xu [29] introduced the concepts of hesitant multiplicative preference relation (HMPR), which can efficiently perform the advantages of HFSs of treating uncertainty. After a sort of preference relations is established, how to derive reliable priority vectors from given preference relations is an important issue. From existing literatures, however, seldom research has been done for the HMPRs.

Prioritization methods vary greatly with various kinds of preference relations. Saaty [1, 17] purposed the well-known eigenvector method (EM) and the least-square method (LSM) to obtain priority vectors from a multiplicative preference relation. Ramanathan [18] put forward a DEAHP method for weight derivation from pairwise comparison matrices, which regards each alternative (criterion) as a decision-making unit. Wang et al. [19] presented a goal programming method (GPM) to obtain interval weights from interval multiplicative preference relations. Xu et al. [21] defined an intuitionistic fuzzy preference relation (IFPR) as a preference structure, whose elements are intuitionistic fuzzy values, and developed a series of prioritization methods, such as the error-analysis-based method [20], linear goal programming method, and fractional programming methods [21, 22] with IFPRs based on two kinds of consistencies, the additive consistency and the multiplicative consistency. Zhang and Wu [23] designed a hesitant goal programming model to derive priorities based on α-normalization principle, which degenerates a HMPR into a MPR via removing some elements of HFEs from the HMPR. Although numerous researches have been done on how to get priorities from different preference relations, it is noted that no method is superior to others in all cases [24] and the choice of priority methods depends upon the objective of problem analysis.

The aforementioned researches show that quite a lot of attention has been paid to the prioritization of preference relations. Nevertheless, all these above studies are absorbed in single or static stage. In reality, most decision-making problems are dynamic, such as series financing, venture investments, mechanical dynamic evaluation etc. The original information of these problems is usually collected from multiple stages and may vary with each stage [25]. Thus, in order to make a reliable judgment, DMs need to study the distinction and connection rooted in different stages since the information in different stages may be ever-changing. It is essential to aggregate the information from different stages to estimate the comprehensive performance of alternatives. Therefore, How to determine the relative weights of each stage for information fusion is a key issue of multistage decision-making (MSDM). Xu [25] defined the dynamic weighted averaging (DWA) operators, which take time into consideration, and proposed three weight determination methods to acquire relative weights associated with DWA operators. Moreover, Xu and Yager [26] extended the DWA operators to intuitionistic fuzzy environment and introduced the dynamic intuitionistic fuzzy weighted averaging (DIFWA) operators and some methods such as normal distribution-based method and average age method to obtain the relative weights of DIFWA operators. Hao et al. [27] developed the minimum adjacent deviation method to determine the aggregation of multistage linguistic information in an uncertain surrounding. Liao and Xu [28] presented an improved maximum entropy method and the minimum average deviation method associated with dynamic hesitant fuzzy weighted averaging (DHFWA) operators. Based on the two methods, an approach for solving the hesitant fuzzy multi-criterion MSDM problems was developed.

As to the MSDM with hesitant multiplicative preference relations (MSDM-HMPRs), as far as we know, no research has been done on this issue. The purpose of this paper is to investigate the MSDM- HMPRs problems where all the decision information is in the form of HMPRs and generates at different stages. We use a strong ordering constraint in determining the relative weights of multiple stages, which can make the weights more logical. On the other hand, a novel method and a convergent iterative algorithm are developed to derive priority vectors from HFPRs. This method is efficient and flexible. It allows the DMs to assign suitable weights to different stages to reflect their preferences in MSDM-HMPRs problems.

The remainder of the paper is organized as follows. Section 2 briefly describes the HFS, MPR and HMPR. In Section 3, a method called linear ordering minimax deviation is presented to determine the weight vector for different stages. Based on the weight vector, Section 4 develops a method for prioritization under the framework of MSDM-HMPRs. Section 5 reports a case study on Fujian Motor Industry Group analysis to illustrate the proposed approach. The paper is concluded in Section 6.

2 Preliminaries

Definition 1. [10, 29] Let X be a reference set. A hesitant fuzzy set E on X is defined in terms of a function h_E (x) that returns a subset of [0, 1] when applied to X : E ={ 〈 x, 1pt1pth_E (x) |x ∈ X 〉 }, where h_E (x) is a set of values in [0, 1] that denote the possible membership degrees of the element x ∈ X to the set E. h_E is called hesitant fuzzy element (HFE) denoted by h_E ={ γ^λ|λ = 1, 2, …, |h_E| }. We use the symbol |h_E| to symbolize the number of elements in h_E.

Definition 2. [29, 30] Given three HFEs h, h₁ and h₂, respectively, the basic operations on HFEs are:

h₁ ⊕ h₂ = ⋃ _{γ₁∈h₁,γ₂∈h₂} {γ₁ + γ₂ - γ₁γ₂};

h₁ ⊗ h₂ = ⋃ _{γ₁∈h₁,γ₂∈h₂} {γ₁ γ₂};

h₁ ⊖ h₂ = ⋃ _{γ₁∈h₁,γ₂∈h₂} {ξ}, where $ξ = {\begin{matrix} \frac{γ_{1} - γ_{2}}{1 - γ_{2}} & if γ_{1} \geq γ_{2} and γ_{2} \neq 1 \\ 0 & otherwise \end{matrix}$

h₁ ø h₂ = ⋃ _{γ₁∈h₁,γ₂∈h₂} {ξ}, where $ξ = {\begin{matrix} \frac{γ_{1}}{γ_{2}} & if γ_{1} \geq γ_{2} and γ_{2} \neq 0 \\ 0 & otherwise \end{matrix}$

h^α = ⋃ _γ∈h {γ^α} (α > 0);

αh = ⋃ _γ∈h {1 - (1 - γ) ^α} (α > 0).

Remark 1. The number of elements in different HFEs may be different and these elements are often out of order. To avoid this situation, Xia and Xu [24] arrange the elements of HFEs in an increasing order where γ^λ means the λth smallest element in h_E.

Definition 3. [32] A multiplicative preference relation R on the set X = {X₁, X₂, … , X_m} is characterized by a matrix R = (r_ik) _m×m with

$\begin{matrix} r_{ik} & = & 1 / r_{ki}, r_{ii} = 1, r_{ii} \in [\frac{1}{9}, 9], \\ i, k = 1, 2, \dots, m \end{matrix}$ (2.1) where r_ik is the preference degree of alternative X_i over X_k measured by the 1–9 scale.

A multiplicative preference relation R = (r_ik) _m×m is called perfectly consistent if it always satisfies the following transitivity condition [1, 26]: $r_{ik} = r_{ij} r_{jk}, for i, j, k = 1, 2, \dots, m .$ (2.2)

Similarly, it has been pointed out that if there exists a weight vector W = (w₁, w₂, …, w_m) ^T satisfying $\sum_{i = 1}^{m} w_{i} = 1$ and 0 ≤ w_i ≤ 1, for i = 1, 2, ... ,m, such that $r_{ik} = w_{i} / w_{k}, for i, k = 1, 2, \dots, m .$ (2.3) then R is perfectly consistent. In case that a MPR is inconsistent, there exists no priority vector that can make Equation (2.3) hold accurately.

Definition 4. [16] Let X = {X₁, X₂, …, X_m} be a reference set. A hesitant multiplicative preference relation (HMPR) on X is defined as Z = (z_ik) _m×m, where $z_{ik} = {γ_{ik}^{λ} | λ = 1, 2, \dots, | z_{ik} |}$ is a hesitant multiplicative element (HME) implying the intensity of preferences of X_i over X_k with the following conditions (i, k = 1, 2, …, m): ${\begin{matrix} γ_{ik}^{λ} \cdot γ_{ki}^{λ} = 1, γ_{ii}^{λ} = 1, \\ | z_{ik} | = | z_{ki} | \\ γ_{ik}^{λ} \leq γ_{ik}^{λ + 1}, (i < k) \end{matrix}$ (2.4) where $γ_{ik}^{λ}$ is the λth smallest element in z_ik.

Remark 2. The definition of HMPR has a little difference from what defined in [23], which assumes $γ_{ik}^{λ} \cdot γ_{ki}^{(| z_{ki} | - λ + 1)} = 1$ , i, k = 1, 2, …, m. In other words, all $γ_{ik}^{λ}$ are ranked in an increasing order in [23], while in this paper, $γ_{ik}^{λ} \leq γ_{ik}^{λ + 1}$ , if i < k; and $γ_{ik}^{λ} \geq γ_{ik}^{λ + 1}$ , if i > k.

3 Determination of the multistage relative weights

3.1 Approach to fuzzy data

In what follows, a brief description of MSDM- HMPR problems is presented. There are n stages s_j, j = 1, 2, …, n, whose weight vector is denoted as D = (d₁, d₂, …, d_n) ^T satisfying $\sum_{j = 1}^{n} d_{j} = 1$ , d_j ≥ 0. Let s_j denote a stage that has j periods gap away from the current time. That is to say, s₁ is the most recent stage, while s_n is the oldest one. Let Z = {Z¹, Z², …, Zⁿ} be a set of HMPRs, where $Z^{j} = (z_{ik}^{j})_{m \times m} = {γ_{ik, j}^{λ} | λ = 1, 2, \dots, | z_{ik} |}$ is generated at the stage s_j by DMs to express pairwise judgments on alternatives set X = {X₁, X₂, …, X_m}, j = 1, 2, …, n.

After allocating the weights to each stage, a process of aggregation of sequential data is conducted to rank the alternatives. Numerous methods for weight determination have been investigated, such as the minimum variability model [34], the improved maximum entropy [28], and the minimum adjacent deviation method [27] etc. After careful studies, we find that these above methods are quite complicated and require a quadratic or logarithmic programming model to be designed. To circumvent this limitation, we are dedicated to developing a sound yet simple method for obtaining sequential weights based on the measurement of orness [35] defined below,

Definition 5. [35] Let D = (d₁, d₂, …, d_n) ^T be a weight vector, satisfying $\sum_{j = 1}^{n} d_{j} = 1$ , d_j ≥ 0, the orness of D is defined as $orness (D) = α = \frac{1}{n - 1} \sum_{j = 1}^{n} (n - j) d_{j}$ (3.1) where 0 ≤ α ≤ 1, indicating the degree to which the aggregation seems to an or operation. Xu [36] pointed out that the orness associated with the weight vector D can be viewed as DM’s preference for multistage or sequential time.

In order to minimize the disparities between two adjacent weights under a given level of orness, Wang et al. [37, 38] developed the minimax disparity approach to generate weights. In this approach, when α ≤ 0.5, i.e., a pessimistic or neutral DM will give weight allocation as d₁ ≤ d₂ ≤ , …, ≤ d_n, j = 1, 2, …, n. Such a weight allocation has no problems when dealing with single stage decision analysis. However, under multistage decision-making scenarios, new (fresh) data are constantly generated and are utilized to update the old data. DMs may have some preferences for fresh data and put more emphasis on more recent data [39]. Basically, the newer the information, the larger the weight. For example, when predicting market competition percentage using Markov process theory, we postulate that the probability distribution of the current stage depends only on the previous stage while has nothing to do with the earlier stages. In other words, when assigning weights to multiple stages, it is reasonable to request that d_j ≥ d_i when j < i.

From the above analysis, we adopt a strong ordering constraint d₁ ≥ 2d₂ ≥ … ≥ (n - 1) d_n-1 ≥ nd_n, which was suggested by Noguchi et al. [40]. This constraint stands to reason in MSDM scenario because it has two features that d₁ ≥ d₂ ≥ … ≥ d_n as well as d₁ - d₂ > d₂ - d₃ … > d_n-1 - d_n, and both of them are highly in accordance with the multistage weighting requirement.

Inspired by the strong ordering constraint, the following linear ordering minimax deviation (LOMD) model is formulated:

$\begin{matrix} Min & δ \\ s . t . & \frac{1}{n - 1} \sum_{j = 1}^{n} (n - j) d_{j} = orness (D) \\ \sum_{j = 1}^{n} d_{j} = 1 \\ orness (D) - α - δ \leq 0, \\ orness (D) - α + δ \geq 0, \\ d_{1} \geq 2 d_{2} \geq \dots \geq {nd}_{n} \geq 0 . \end{matrix}$ (3.2) where 0 ≤ α ≤ 1 representing the sequential preference given by the DM. Evidently, the weights d_j, j = 1, 2, …, n, are dependent on the value of α and their ranking orders. The main advantage of the LOMD model is the use of simple linear programming to obtain the sequential weights that do not follow a regular distribution.

Theorem 1. Model (3.2) has feasible solutions if and only if α - δ ≤ 2n - 1/ - 3n - 3.

Proof. According to the strong ordering constraint, when all the weights are nonzero, they can be rewritten as the following equation: d_j = d_n + (n - j) δ, j = 1, 2, … n. Since $\sum_{j = 1}^{n} d_{j} = 1$ , namely,

$\sum_{j = 1}^{n} [d_{n} + (n - j) δ] = 1$ , which is equivalent to δ = 2 (1 - nd_n)/ - n (n - 1). Due to the fact that, $\begin{matrix} \frac{1}{n - 1} \sum_{j = 1}^{n} (n - j) d_{j} \\ = \frac{1}{n - 1} [\sum_{j = 1}^{n} (n - j) d_{n} \end{matrix} + \sum_{j = 1}^{n} δ (n - j)^{2}]$ and $δ = \frac{2 (1 - {nd}_{n})}{n (n - 1)}$ , this gives $\frac{1}{n - 1} \sum_{j = 1}^{n} (n - j) d_{j} = \frac{2 n - 1}{3 (n - 1)} - \frac{n (n + 1) d_{n}}{6 (n - 1)}$

Thus, $\begin{matrix} orness (D) & = & \frac{1}{n - 1} \sum_{j = 1}^{n} (n - j) d_{j} \\ = & \frac{2 n - 1}{3 (n - 1)} - \frac{n (n + 1) d_{n}}{6 (n - 1)} . \end{matrix}$

After rearranging, we have $d_{n} = \frac{6 (n - 1)}{n (n + 1)} [\frac{2 n - 1}{3 (n - 1)} - orness (D)]$

From model (3.2), we know that feasible solutions exist only when d_j ≥ 0, j = 1, 2, …, n, which require $(2 n - 1) / (3 n - 3) \geq orness (D)$

Since orness (D) - α + δ ≥ 0, it yields α - δ ≤ (2n - 1)/ - (3n - 3). This completes the proof.

4 A priority method for MSDM-HMPRs

Let W = (w₁, w₂, … w_m) ^T be the priority vector for HMPRs $Z^{j} = (z_{ik}^{j})_{m \times m}$ , where $\sum_{i = 1}^{m} w_{i} = 1$ , w_i ≥ 0. As each element in $z_{ik}^{j}$ indicates a possible degree of preference relations in the jth stage for the pairwise comparison between alternatives X_i and X_k, the consistent preferences can be obtained by Equation (2.3):

$\begin{matrix} \frac{w_{i}}{w_{k}} & = & γ_{ik, 1}^{j} or γ_{ik, 2}^{j}, \dots, γ_{ik, | z_{ik}^{j} |}^{j}, \\ i, k = 1, 2, \dots, m . \end{matrix}$ (3.3) where $γ_{ik, λ}^{j}$ is the λth smallest element in $z_{ik}^{j}$ .

Let $T (z_{ik}^{j}) = γ_{ik, 1}^{j}$ or $γ_{ik, 2}^{j}, \dots,$ or $γ_{ik, | z_{ik}^{j} |}^{j}$ , then we have $w_{i} / w_{k} = T (z_{ik}^{j}), i, k = 1, 2, \dots, m .$ (3.4)

Since consistent HMPRs are not always available there nearly exists no priority vector that can make Equation (3.4) hold for inconsistent HMPRs. Nevertheless, the second best thing that a DM can do is to find a priority vector closest to its corresponding perfectly consistent HMPR. Thus, it is desirable that the deviation between the given HMPRs and its consistent HMPRs is as small as possible. Hence, we establish the following least square deviation model to find an optimal priority vector W = (w₁, w₂, … w_m) ^T,

$\begin{matrix} Min F (W) = \sum_{j = 1}^{n} d_{j} \sum_{i = 1}^{m} \sum_{k = 1}^{m} \frac{[T (z_{ik}^{j}) - (w_{i} / w_{k})]^{2}}{w_{i} / w_{k}} \\ s . t . {\begin{matrix} \sum_{i = 1}^{m} w_{i} = 1 \\ w_{i} > 0, i = 1, 2, \dots m \end{matrix} \\ Ω_{w} = {W = (w_{1}, w_{2}, \dots w_{m})^{T} | \sum_{i = 1}^{m} w_{i} = 1, w_{i} > 0} \end{matrix}$ (3.5)

The idea of the above model is to minimize the overall square deviation from Equation (3.4). Thus, we call the model as the least square deviation method. With respect to the solution of the above model, we have the following theorem.

Theorem 2. F (W) has a unique minimum point $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots w_{m}^{*})^{T} \in Ω_{w}$ , which is the solo solution of the following system of equations in Ω_w

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{i} (1 + T (z_{ki}^{j})^{2})}{w_{k}} \\ = \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}} \end{matrix}$ (3.6)

Proof. Since the vector space Ω_w is bounded and F (W) ≥0 is a continuous function in Ω_w, the solution space of F (W) is finite and has an infimum. So, there must exist a minimum point $W^{*} = (w_{1}^{*}, w_{2}^{*}, \dots w_{m}^{*})^{T} \in Ω_{w}$ enabling F (W) to achieve its minimum value.

To locate the minimum point, we define the following Lagrangian function: $L (W, θ) = F (W) + θ (\sum_{i = 1}^{n} w_{i} - 1)$ where θ is the Lagrangian multiplier.

Let $\frac{\partial L (W, θ)}{\partial w_{i}} = 0$ , for i = 1, 2, …, m. This gives,

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} [\frac{(1 + T (z_{ki}^{j})^{2})}{w_{k}} - \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}^{2}}] \\ + θ = 0, i = 1, 2, \dots, m \end{matrix}$ (3.7)

Multiplying Equation (3.7) by w_i yields

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} [\frac{w_{i} (1 + T (z_{ki}^{j})^{2})}{w_{k}} - \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}}] \\ + θ w_{i} = 0, i = 1, 2, \dots, m \end{matrix}$ (3.8)

Side by side summation of the Equation (3.8) gives

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} \\ d_{j} [\frac{w_{i} (1 + T (z_{ki}^{j})^{2})}{w_{k}} - \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}}] \\ + θ \sum_{i = 1}^{m} w_{i} = 0 \end{matrix}$ (3.9)

Since $\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sum_{i = 1}^{m} \\ d_{j} [\frac{w_{i} (1 + T (z_{ki}^{j})^{2})}{w_{k}} - \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}}] = 0 \end{matrix}$ and $\sum_{i = 1}^{m} w_{i} = 1 \neq 0$ , then we have θ = 0. Therefore, Equation (3.7) becomes

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} \\ d_{j} [\frac{(1 + T (z_{ki}^{j})^{2})}{w_{k}} - \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}^{2}}] = 0 \end{matrix}$ (3.10)

That is

Self-evidently, the minimum point W^* is a solution to Equation (3.6). If the solution is unique in Ω_w, then W^* can be uniquely determined.

Suppose W = (w₁, w₂, …, w_m) ^T ∈ Ω_w and V = (v₁, v₂, … v_m) ^T ∈ Ω_w are two solutions to Equation (3.6). Let μ_i = w_i/ - v_i, i = 1, 2, … m and $μ_{l} = max_{i \in {1, 2, \dots m}} μ_{i}$ . If there exists some k ∈ {1, 2, … m} such that μ_k < μ_l, this gives

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{v_{l} (1 + T (z_{kl}^{j})^{2})}{v_{k}} \\ < \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{l} (1 + T (z_{kl}^{j})^{2})}{w_{k}} \end{matrix}$ (3.12) and

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{v_{k} (1 + T (z_{lk}^{j})^{2})}{v_{l}} \\ > \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{k} (1 + T (z_{lk}^{j})^{2})}{w_{l}} \end{matrix}$ (3.13)

In terms of Equations (3.6), (3.12) and (3.13), we have

$\begin{matrix} \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{k} (1 + T (z_{lk}^{j})^{2})}{w_{l}} \\ < \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{l} (1 + T (z_{kl}^{j})^{2})}{w_{k}} \end{matrix}$ (3.14) which contradicts Equation (3.6). Therefore, it is false that μ_k < μ_l. i.e. μ_k = μ_l for k = 1, 2, … m.

Since $\sum_{i = 1}^{m} w_{i} = 1$ , and $\sum_{i = 1}^{m} v_{i} = 1$ , it can be concluded that w_i = v_i, i = 1, 2 … m. This completes the proof.

Note that Equation (3.6) is a group of nonlinear equations and solving this model directly is time consuming. In order to solve Equation (3.6), we have developed a convergent iterative technique as below:

Algorithm 1. Let $Z^{j} = (z_{ik}^{j})_{m \times m}$ , j = 1, 2, … n, be the initial HMPR given by the DM.

Step 1. Setting an initial priority vector W (0) = (w₁ (0) , w₂ (0) , … w_m (0)) ^T ∈ Ω_w and appointing a tiny error parameter φ (0 < φ < 1), such as φ = 0.001. Let L = 0;

Step 2. Computing $\begin{matrix} η_{i} (W (L)) = \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{i} (1 + T (z_{ki}^{j})^{2})}{w_{k}} \\ - \sum_{j = 1}^{n} \sum_{k = 1}^{m} d_{j} \frac{w_{k} (1 + T (z_{ik}^{j})^{2})}{w_{i}}, \\ i = 1, 2, \dots m \end{matrix}$

If |η_i (W (L)) | ≤ φ holds for all i = 1, 2, …, m, then let optimal weight vector W^* = W (L) and stop; otherwise, go to Step 3.

Step 3. Determining the number p that $| η_{p} (W (L)) | = max_{i \in {1, 2, \dots m}} | η_{i} (W (L)) |$ , and calculating ${\begin{matrix} S (L) = \frac{\sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T (z_{pk}^{j})^{2}) \frac{w_{k} (L)}{w_{p} (L)}}{\sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T (z_{kp}^{j})^{2}) \frac{w_{p} (L)}{w_{k} (L)}} \\ y_{i} (L) = {\begin{matrix} S (L) w_{p} (L), i = p \\ w_{i} (L), i \neq p \end{matrix} \\ y_{i} (L + 1) = y_{i} (L) / \sum_{i = 1}^{m} y_{i} (L), i = 1, 2, \dots m \end{matrix}$

Step 4. Let L = L + 1 and switch to Step 2.

Theorem 3. For any 0 < φ < 1, the algorithm 1 turns out to be convergent.

Proof. Consider the variation of W (L), that is, shift W (L) to W (L + 1). Assume that t > 0, and ${\begin{matrix} G (t) = F [y (L)] \\ y (L) = (w_{1} (L), \dots w_{p - 1} (L), \\ t w_{p} (L), w_{p + 1} (L) \dots w_{n} (L)) \end{matrix}$ (3.15)

We have $\begin{array}{l} G (t) \\ = \sum_{j = 1}^{n} {\sum_{k = 1, k \neq p}^{m} d_{j} \frac{[T (z_{p k}^{j}) - \frac{t w_{p} (L)}{w_{k} (L)}]}{w_{p} (L) / t w_{k} (L)}}^{2} \\ + \sum_{j = 1}^{n} {\sum_{i = 1, i \neq p}^{m} d_{j} \frac{[T (z_{i p}^{j}) - \frac{w_{i} (L)}{t w_{p} (L)}]}{w_{i} (L) / t w_{p} (L)}}^{2} \\ + \sum_{j = 1}^{n} {\sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} \sum_{\begin{matrix} i = 1, \\ i \neq p \end{matrix}}^{m} d_{j} \frac{[T (z_{i k}^{j}) - \frac{w_{i} (L)}{w_{k} (L)}]}{w_{i} (L) / w_{k} (L)}}^{2} \end{array}$ (3.16)

Simplifying and rearranging Equation (3.16) gives $\begin{array}{l} G (t) \\ = \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T {(z_{p k}^{j})}^{2}) \frac{w_{k} (L)}{t w_{p} (L)} \\ + \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T {(z_{k p}^{j})}^{2}) \frac{t w_{p} (L)}{w_{k} (L)} \\ + \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} \sum_{\begin{matrix} i = 1, \\ i \neq p \end{matrix}}^{m} \\ d_{j} {[T (z_{i k}^{j}) - \frac{w_{i} (L)}{w_{k} (L)}]}^{2} \frac{w_{k} (L)}{w_{i} (L)} \\ - \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (T {(z_{p k}^{j})}^{2} + T {(z_{k p}^{j})}^{2}) \end{array}$ (3.17)

For simplicity, let $\begin{matrix} b_{1} & = & \sum_{j = 1}^{n} \sum_{k = 1, k \neq p}^{m} d_{j} (1 + T (z_{pk}^{j})^{2}) \frac{w_{k} (L)}{w_{p} (L)} \\ b_{2} & = & \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T (z_{kp}^{j})^{2}) \frac{w_{p} (L)}{w_{k} (L)} \\ b_{3} & = & \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} \sum_{\begin{matrix} i = 1, \\ i \neq p \end{matrix}}^{m} \\ d_{j} {[T (z_{ik}^{j}) - \frac{w_{i} (L)}{w_{k} (L)}]}^{2} \frac{w_{k} (L)}{w_{i} (L)} \\ - \sum_{j = 1}^{n} \sum_{k = 1, k \neq p}^{m} d_{j} (T (z_{pk}^{j})^{2} + T (z_{kp}^{j})^{2}) \end{matrix}$

Then, Equation (3.17) can be equivalently denoted as $G (t) = b_{1} / t + t b_{2} + b_{3}$ (3.18)

Letting $\frac{\partial G (t)}{\partial t} = 0$ , we have $\begin{matrix} t^{*} & = & \sqrt{\frac{\sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T (z_{pk}^{j})^{2}) \frac{w_{k} (L)}{w_{p} (L)}}{\sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} (1 + T (z_{kp}^{j})^{2}) \frac{w_{p} (L)}{w_{k} (L)}}} \\ G (t^{*}) & = & 2 \sqrt{b_{1} b_{2}} + b_{3} \end{matrix}$ where t^* represents for the minimum point, and G (t^*) is the minimum (optimal) value of G (t).

Obviously, when t^* = 1, then $\begin{array}{l} η_{p} (W (L)) \\ = \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} \frac{w_{k} (L) (1 + T {(z_{p k}^{j})}^{2})}{w_{p} (L)} \\ - \sum_{j = 1}^{n} \sum_{\begin{matrix} k = 1, \\ k \neq p \end{matrix}}^{m} d_{j} \frac{w_{p} (L) (1 + T {(z_{k p}^{j})}^{2})}{w_{k} (L)} \end{array}$ (3.19) which also holds for k = p. As p is the subscript that maximizes the |η_p (W (L)) |, so we obtain |η_i (W (L)) | = 0, i = 1, 2, … m. Hence, W^* = W (L). When t^* ≠ 1, then $F (W (L)) - F (y (L)) = G (1) - G (t^{*}) > 0$ (3.20)

Because F (W) is a homogeneous function, F (y (L)) = F (W (L + 1)). Formula (3.20) implies that F (W (L)) > F (W (L + 1)), for any L ≥ 0. That is to say, F (W (L)) is convergent and monotonically decreasing sequence with an infimum in Ω_w.

It is observed that HMPRs are a generalization of MPRs, and can be reduced to MPRs when all the numbers of HFEs are equal to one. Thus, the proposed method in this section can be seen as an extension of the approach in Wang et al. [41], which can deal with more complicated preference relations.

5 A case study of logistics service provider selection

This case study concerns the Fujian Motor Industry Group (FMIG), which was established in Fuzhou in 1985 and is now a flagship automotive manufacturer in southeast China. The FMIG is a holding company with four main subsidiaries: the Fujian Tractor, the Fujian Benz, the Southeast Motors and the Kinglong Bus manufacturer. With rapid business growth and expansion, FMIG has abundant capital facilities; however, it faces a deteriorating logistics bottleneck problem in recent years. So, it desires to select a competent third-party logistics service provider (3P-LSP) for saving operation cost and upgrading core competence. After releasing an announcement, four candidate logistics providers: Shenghui (x₁), Deppon (x₂), Cosco (x₃), Hoau (x₄) are invited for further assessment.

FMIG invites three experts (i.e., DMs) e₁, e₂ and e₃ to form the decision group. They are asked to conduct pairwise comparisons for evaluating the four candidates. For a comprehensive review, the decision group will examine the performances of the following criteria of candidates: quality, delivery, logistics cost, rejection rate and technique in 2012–2014 (s_j, j = 3, 2, 1) of each 3P-LSP, and gives a consensus preferential measure for sequential periods as α = 0.6. Since each expert has a different expertise, the comparison results may be varied from each other. In order to retain the original decision information, the group intends to use the hesitant multiplicative preference relations to express their judgments. That is to say, each paired comparison value provided by the expert group is a HFE containing three numbers in [1/9, 9]. Let Z (s_i), (i = 1, 2, 3) be the HMPRs over the three years 2012–2014, respectively, which are provided as follow: $\begin{matrix} Z (s_{1}) & = & (\begin{matrix} {1} & {2, \frac{1}{3}, \frac{2}{3}} & {\frac{5}{7}, 3, 1} & {\frac{6}{7}, \frac{3}{5}, \frac{1}{4}} \\ {\frac{1}{2}, 3, \frac{3}{2}} & {1} & {1, \frac{2}{3}, 3} & {2, \frac{7}{2}, \frac{3}{2}} \\ {\frac{7}{5}, \frac{1}{3}, 1} & {1, \frac{3}{2}, \frac{1}{3}} & {1} & {\frac{8}{5}, \frac{4}{7}, \frac{5}{6}} \\ {\frac{7}{6}, \frac{3}{5}, 4} & {\frac{1}{2}, \frac{2}{7}, \frac{2}{3}} & {\frac{5}{8}, \frac{7}{4}, \frac{6}{5}} & {1} \end{matrix}) \\ Z (s_{2}) & = & (\begin{matrix} {1} & {\frac{2}{3}, 2, \frac{5}{2}} & {\frac{3}{4}, \frac{7}{2}, \frac{5}{2}} & {3, \frac{1}{3}, \frac{3}{2}} \\ {\frac{3}{2}, \frac{1}{2}, \frac{2}{5}} & {1} & {3, 2, \frac{1}{2}} & {\frac{5}{2}, 1, 2} \\ {\frac{4}{3}, \frac{2}{7}, \frac{2}{5}} & {\frac{1}{3}, \frac{1}{2}, 2} & {1} & {\frac{1}{2}, \frac{1}{3}, 1} \\ {\frac{1}{3}, 3, \frac{2}{3}} & {\frac{2}{5}, 1, \frac{1}{2}} & {2, 3, 1} & {1} \end{matrix}) \\ Z (s_{3}) & = & (\begin{matrix} {1} & {\frac{4}{5}, \frac{5}{3}, \frac{5}{7}} & {2, 1, \frac{5}{3}} & {\frac{3}{4}, 3, \frac{8}{9}} \\ {\frac{5}{4}, \frac{3}{5}, \frac{7}{5}} & {1} & {2, \frac{7}{3}, 3} & {1, \frac{1}{2}, \frac{5}{2}} \\ {\frac{1}{2}, 1, \frac{3}{5}} & {\frac{1}{2}, \frac{3}{7}, \frac{1}{3}} & {1} & {3, \frac{7}{3}, \frac{7}{2}} \\ {\frac{4}{3}, \frac{1}{3}, \frac{9}{8}} & {1, 2, \frac{2}{5}} & {\frac{1}{3}, \frac{3}{7}, \frac{2}{7}} & {1} \end{matrix}) \end{matrix}$

Step 1. Rearrange the elements of $z_{ik}^{j}$ in each HMPR Z (s_i) in increasing order when i ≤ k (i, k = 1, 2, 3, 4, j = 1, 2, 3), which are listed below: $\begin{matrix} Z (s_{1}) & = & (\begin{matrix} {1} & {\frac{1}{3}, \frac{2}{3}, 2} & {\frac{5}{7}, 1, 3} & {\frac{1}{4}, \frac{3}{5}, \frac{6}{7}} \\ {3, \frac{3}{2}, \frac{1}{2}} & {1} & {1, \frac{2}{3}, 3} & {\frac{3}{2}, 2, \frac{7}{2}} \\ {\frac{7}{5}, 1, \frac{1}{3}} & {1, \frac{3}{2}, \frac{1}{3}} & {1} & {\frac{4}{7}, \frac{5}{6}, \frac{8}{5}} \\ {4, \frac{3}{5}, \frac{7}{6}} & {\frac{2}{3}, \frac{1}{2}, \frac{2}{7}} & {\frac{7}{4}, \frac{6}{5}, \frac{5}{8}} & {1} \end{matrix}) \\ Z (s_{2}) & = & (\begin{matrix} {1} & {\frac{2}{3}, 2, \frac{5}{2}} & {\frac{3}{4}, \frac{5}{2}, \frac{7}{2}} & {\frac{1}{3}, \frac{3}{2}, 3} \\ {\frac{3}{2}, \frac{1}{2}, \frac{2}{5}} & {1} & {\frac{1}{2}, 2, 3} & {1, 2, \frac{5}{2}} \\ {\frac{4}{3}, \frac{2}{5}, \frac{2}{7}} & {2, \frac{1}{2}, \frac{1}{3}} & {1} & {\frac{1}{3}, \frac{1}{2}, 1} \\ {3, \frac{2}{3}, \frac{1}{3}} & {1, \frac{1}{2}, \frac{2}{5}} & {3, 2, 1} & {1} \end{matrix}) \\ Z (s_{3}) & = & (\begin{matrix} {1} & {\frac{5}{7}, \frac{4}{5}, \frac{5}{3}} & {1, \frac{5}{3}, 2} & {\frac{3}{4}, \frac{8}{9}, 3} \\ {\frac{7}{5}, \frac{5}{4}, \frac{3}{5}} & {1} & {2, \frac{7}{3}, 3} & {\frac{1}{2}, 1, \frac{5}{2}} \\ {1, \frac{3}{5}, \frac{1}{2}} & {\frac{1}{2}, \frac{3}{7}, \frac{1}{3}} & {1} & {\frac{7}{3}, \frac{7}{2}, 3} \\ {\frac{4}{3}, \frac{9}{8}, \frac{1}{3}} & {2, 1, \frac{2}{5}} & {\frac{3}{7}, \frac{2}{7}, \frac{1}{3}} & {1} \end{matrix}) \end{matrix}$

Step 2. Determine the relative weights of multistage. Let D = (d₁, d₂, d₃) ^T be the weight vector of stages s_j (j = 1, 2, 3). According to Model (3.2) and given orness degree α = 0.6, we establish the following linear optimization model: $\begin{matrix} Min & δ \\ s . t & d_{1} + 0.5 d_{2} - 0.6 + δ \geq 0 \\ d_{1} + 0.5 d_{2} - 0.6 - δ \leq 0 \\ d_{1} \geq 2 d_{2} \geq 3 d_{3} > 0 \\ \sum_{j = 1}^{3} d_{j} = 1 \end{matrix}$

The solution of this model is D = (0.5455, 0 . 2727, 0 . 1818) ^T.

Step 3. Construct the following nonlinear optimazation model by Equation (3.5).

Step 4. Solving the above model with Algorithm 1 by setting W (0) = (0 . 25, 0 . 25, 0 . 25, 0 . 25) ^T and φ = 0.01.

Step 5. Output the optimal weight vector W^* = (0 . 2350, 0 . 3525, 0 . 1773, 0 . 2352).

According to the optimal weight vector W^*, the ranking order of the four 3P-LSPs is x₂ ≻ x₄ ≻ x₁ ≻ x₃. Therefore, it is advisable for FMIG to choose Deppon logistics company (x₂) as its logistics provider.

As can be seen from this case, the HMPRs are powerful in handling multistage and GDM problems because they can intuitively indicate uncertain or multi-valued preference information. If the experts alter their sequential measurement from α = 0.6 to 0.85, this alteration yields D = (0 . 70, 0 . 30, 0 . 00) ^T. Thanks to d₃ = 0, this is in accordance with Theorem 1 as $α - δ = 0.85 - 0.01 > \frac{5}{6}$ . Accordingly, the new optimal weight vector is obtained as W^* = (0 . 2208, 0 . 2940, 0 . 2648, 0 . 2204), which gives the ranking order x₂ ≻ x₃ ≻ x₁ ≻ x₄, slightly different from the ranking obtained with α = 0.6. Despite of the slight difference, the Deppon (x₂) is still the best candidate.

In order to verify the proposed approach, we compare the ranking results with the one achieved by the approach of Zhang and Wu [23]. It should be pointed out that the α-normalization-based goal programming model suggested by Zhang and Wu [23] is only applicable to the single stage (static) decision-making with HMPRs. To facilitate our analysis, we consider the same relative weight for multistage in comparison analysis.

From the performance comparison in Table 1, it is observed that the relative weights have only minor differences between the two approaches and have the same ranking order for the four logistics service providers.

In this case, the nearest stage s₁ should be the most vital as discussed in Section 2. Let us have a look at the preference information Z (s₁). We may have more interest on x₂ as it is ranked in the first place. The hesitant fuzzy values in the second row of Z (s₁) respectively are ${\frac{1}{2}, 3, \frac{3}{2}}$ , {1}, ${1, \frac{2}{3}, 3}$ and ${2, \frac{7}{2}, \frac{3}{2}}$ . We find that most of these values are greater than or equal to 1. Besides, in the second and the third stages, most of the evaluation values of x₂ are also greater than or equal to the others. Intuitively, this manifests that x₂ outweighs the other three logistics companies. Namely, the outcome obtained from the proposed method is in accordance with our perception. So, this case analysis verifies the validity and the robustness of our proposed approach. Besides, another merit of the proposed approach is that it does not require the weights of the DMs in multi-person (group) decision making, while most of the existing methods require them to be specified and may yield an unpersuasive outcome.

6 Conclusions

Hesitant multiplicative preference relations where the elements are hesitant fuzzy values are powerful and effective in representing DMs’ preferences over alternatives by pairwise comparison. In this paper, an approach and a convergent iterative algorithm are developed to obtain priority vector from multistage HMPRs information. The notable feature of this approach is that it fully considers the DMs’ hesitancy about pairwise comparison of alternatives in dynamic environment. Specifically, the LOMD model is first proposed using orness, a measurement of DM’s preference for sequences, to produce the relative weights associated with different stages. To highlight the importance of new (fresh) data, a strong ordering constraint is considered in the LOMD model. Based on the obtained weights and the concept of consistency, the least square deviation method is then developed to prioritize and rank alternatives for MSDS-HMPR problems. In addition, the proposed approach is illustrated with a practical logistics service provider selection problem and the analysis shows that the proposed approach is also applicable to multi-person decision making. Finally, how the sequential parameter in LOMD model would affect the final results is discussed. For future research, there are still some issues that need to be explored. For example, the priority vectors derived from MSDS-HMPRs are numerical values rather than HFEs or dual hesitant fuzzy elements (DHFEs) [44 –46]. Besides, if the DMs provide incomplete HMPRs, it would be worthwhile to investigate how to measure and repair the incomplete HMPRs.

Footnotes

Acknowledgments

This research was partly supported by the National Natural Science Foundation of China (NSFC) under Grant 70925004, and the Social Science Foundation of Fujian Province under Grant FJ2015C111. The authors are very grateful to the Associate Editor and anonymous referees for their constructive comments that have helped to improve the quality of this paper to its current standard.

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