A novel cross-efficiency evaluation method under hesitant fuzzy environment

Abstract

In the cross-efficiency model, when the decision making unit (DMU) self-evaluation efficiency value is optimal, the sets of input and output weights exhibit non-uniqueness, and the choice among alternative optimal solutions results in different peer- evaluation values, consequently leading to different cross-efficiency scores and rankings. In addition, the interval value constitutes the cross-efficiency value, which increases the uncertainty of the decision- making due to the information fuzziness. To solve this problem, a group decision-making method for cross-efficiency is proposed on the basis of hesitant fuzzy sets (HFSs). This method selects five optimal solutions. Moreover, given that the attitudes of decision-makers range from pessimistic to optimistic, this method introduces HFSs to the cross-efficiency matrix, and obtains the multi-objective optimization method to rank the alternatives based on the relative closeness degree. Finally, a classical numerical example is provided to illustrate the potential applications of the proposed method and its effectiveness in ranking DMUs.

Keywords

cross-efficiency group decision-making hesitant fuzzy set attribute weight relative closeness degree

1 Introduction

In real-world practice, we may face tasks and activities in which it is necessary to employ decision- making processes. Owing to the inherent complexity and uncertainty of the decision situation or the existence of multiple attributes, decision-making problems are complex and difficult. Data envelopment analysis (DEA) is a non-parametric methodology for evaluating the relative efficiency of group decision making units (DMUs) that utilize multiple inputs to produce multiple outputs. The CCR model, which was the first classical DEA model, was developed by Charnes, Cooper, and Rhodes [1] in 1978. This model, which was used to solve a set of optimal weights in order to maximize the efficiency of the evaluated DMU, measured the efficiency of the DMU from the perspective of self-evaluation, without considering the attitude of other DMUs. Therefore, Sexton et al. [27] proposed a cross- efficiency evaluation method based on a self- evaluation and peer-evaluation system, which has attracted significant research attention. Cross-efficiency evaluation has been extensively applied in various research fields, including resource allocation [35], economic benefit evaluation [31], portfolio research [23, 36], energy efficiency research [20], and research and development project selection [16].

Since the sets of input and output weights are not unique when the self-evaluation efficiency value of the DMU is at its maximum, the multi-optimal solution directly leads to the diversity of peer- evaluation values in the cross-efficiency matrix, resulting in instability of the cross-efficiency matrix, which ultimately affects the overall efficiency value of the DMU. Doyle and Green [8] introduced different secondary goals in the cross-efficiency evaluation method, and proposed the aggressive and benevolent cross-efficiency models. The aggressive model selects the optimal input and output weights that maximize the efficiency value of the evaluated DMU while minimizing the efficiency value of the integrated unit composed of other DMUs. Such a model is often used in situations of mutual competition or hostility. The benevolent model does the opposite. Traditional aggressive and benevolent strategies calculate different integrated efficiency values, which, in turn, affect the ranking of DMUs, thus making the choice of strategies more difficult. In reality, it is not reasonable to directly divide the DMU into opponents or allies. Wang and Chin [34] proposed a neutral cross-efficiency model. On the premise of satisfying the maximum efficiency value of the evaluated DMU, the optimal input and output weights that are most conducive to its minimum single output efficiency are selected irrespective of their impact on other evaluated DMUs. These three models established a secondary goal of cross-efficacy from three different perspectives and introduced a new perspective to researchers. Wu and Liang [9] further presented modified benevolent model and aggressive models, by either maximizing or minimizing the evaluation efficiency value of each DMU to solve the range of peer-evaluation under the multi-optimal solution. They also investigated the cross-efficiency value. All DMUs are regarded as either allies or enemies, which does not fully meet the individual desires of the DMU. Therefore, Yang et al. [4] proposed a DEA cross-efficiency method that takes into consideration of both competition and cooperation among DMUs. All DMUs are divided into several groups. For a given DMU, its optimal preferential weights can maximize the aggregated efficiency of its allies and minimize the aggregated efficiency of its adversaries. And Hou [21] indicated that in reality, in addition to competition and cooperation, DMUs may also involve asymmetrical relationships. We may consider both competition and cooperation at the same time. Therefore, the cross-efficiency theory under competition and cooperation has been proposed. Lim [24] presented the modified benevolent strategy and aggressive strategy, and introduced min-max and max-min functions into the secondary goal of cross-efficiency, such that when selecting weights, the efficiency value of the worst-performing DMU is increased as much as possible, or the efficiency of the best- performing DMU is reduced. In addition, Liang [14] introduced a non-cooperative game theory and proposed a DEA game cross-efficiency model. Bi et al. [5] constrained the weights in the cross-efficiency model by constructing the sets of input and output weights that favor the over-efficiency of the DMU. Li et al. [2] proposed a new DEA cross-efficiency evaluation model based on TOPSIS from the perspective of minimizing ideal DMUs. Wu [10] considered the cross-efficiency model in terms of the Pareto optimal solution.

The efficiency value is usually crisp in a cross-efficiency matrix. However, the crisp values under different models are not the same, which may cause some important information to be lost when evaluating and ranking DMUs. In 2002, Entani [25, 26] proposed that the efficiency value should be taken as the interval number, introducing the concept of the interval efficiency of the DEA model, which can achieve a more reasonable evaluation and ranking of DMUs. Wu et al. [9], Wang et al. [18], and Ramón [19], also constructed different interval values to rank DMUs in terms of interval efficiency evaluation. Wang et al. [13] analyzed the 3-parameter interval cross-efficiency values of DMUs, which are described by the optimal efficiency value, the worst efficiency value, and the most probable efficiency value.

In addition, the aggregation of self-evaluation values and peer-evaluation values of cross-efficiency is also a complex issue. The initial cross-efficiency is calculated by the mean value. Wang [32, 33] introduced the ordered weighted averaging operator (OWA) into the aggregation of cross-efficiency values and further studied the method of attribute aggregation. Huang et al. [37] determined the weight of each DMU by maximizing the deviation. Yang [6] gathered cross-efficiencies through the evidential- reasoning approach (ER). Fan et al. [11] used error transfer and entropy to process efficiency values. Zhang et al. [22] proposed an adaptive group evaluation method of DMUs to evaluate cross- efficiency, which can synchronously and iteratively adjust the weights of the DMU of “experts” and the weights given by the model when calculating efficiency values.

In this study, we select the modified aggressive model [9], the aggressive model [8], the neutral model [34], the benevolent model [8], and the modified benevolent model [9] as group decision models. These five classic cross-efficiency models represent five group decision-making experts whose attitudes range from pessimistic to optimistic, and introduce hesitant fuzzy sets (HFSs) into the cross- efficiency matrix. The efficiency values obtained by different optimal solutions can be considered the results of group decisions. The remainder of this manuscript is organized as follows. In Section 2, we provide a brief introduction to the different cross- efficiency models, and the multi-objective attribute weight optimization method is introduced in order to calculate the attribute weight of the efficiency value. We also present the method of ranking DMUs under hesitant fuzzy information. The calculation steps are listed in Section 3, numerical examples examined in Section 4, and conclusions presented in Section 5.

2 Models and methods

2.1 Cross-efficiency models

2.1.1 CCR model and traditional cross-efficiency model

In the case of multiple inputs and multiple outputs, Charnes et al. [1] proposed a CCR model in order to calculate the effectiveness of DMUs. Consider n DMUs that are evaluated in terms of m inputs and s outputs. Let x_i,j and y_r,j be their input and output values for i = 1,...,m; r = 1,...,s; and j = 1,...,n. The efficiencies of the n DMUs are measured by the following CCR model:

$\begin{matrix} θ_{kk} & = & max \sum_{r = 1}^{s} μ_{rk} y_{rk} \\ s . t . & \sum_{i = 1}^{m} ν_{ik} x_{ik} = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, j = 1, 2, \dots, n, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m, \end{matrix}$ (1) where υ_k = (υ_1k, υ_2k, ⋯ , υ_mk) ^T and μ_k = (μ_1k, μ_2k, ⋯ , μ_sk) ^T are input and output weights. Model (1) is solved for each DMU separately. As a consequence, there are n sets of input and output weights available for the n DMUs; we can obtain the self- evaluation efficiency values of the n DMUs.

The CCR model divides the DMU into 2 categories: effective DMUs (efficiency value of 1) and ineffective DMUs (efficiency value < 1). Such a classification results in all DMUs being not completely ranked or further distinguished. Sexton et al. [27] proposed a cross-efficiency model that determines a unique set of input and output weights for each DMU and then calculates its efficiencies by using all the sets of weights. Accordingly, each DMU has multiple yet different efficiency scores, whose average reflects the overall performance of the DMU. Based on average cross-efficiencies, DMUs are evaluated and ranked. The specific model is expressed as follows: $θ_{kd} = \frac{\sum_{r = 1}^{s} μ_{rk} y_{rd}}{\sum_{i = 1}^{s} ν_{ik} x_{id}}, θ_{d} = \frac{1}{n} \sum_{k = 1}^{n} θ_{kd},$ (2) where θ_kd is referred to as the cross-efficiency of DMU_d, which is calculated by setting the optimal set of input and output weights to model (2) and the optimal set of input and output weights is obtained by setting DMU_k to model (1). DMU_k is the evaluation of DMU and DMUd is the evaluated DMU. The obtained efficiency, which is the peer-evaluation and self-evaluation efficiency value together, constitutes the cross-efficiency matrix. The overall efficiency value θ_d is obtained by solving the mean value of the n–1 evaluation efficiency values of the d column and the self-evaluation efficiency values.

2.1.2 Benevolent and aggressive cross-efficiency models

There are multiple optimal solutions for the input- output weights of the CCR model (1), and we can obtain different peer-evaluation efficiency values while selecting different optimal solutions to calculate the cross-efficiency model (2). Doyle and Green [8] introduced a secondary goal in cross- efficiency assessment methods and proposed an aggressive model and a benevolent model. The idea of an aggressive model is to maximize the weight of the evaluated DMU while selecting the optimal set of weights to minimize the efficiency of the integrated units of other DMU, and is often used in situations of mutual competition or hostility. The idea of the benevolent model is just the opposite.

Aggressive model:

$\begin{matrix} min & \sum_{r = 1}^{s} μ_{rk} (\sum_{j = 1, j \neq k}^{n} y_{rj}) \\ s . t . & \sum_{i = 1}^{m} ν_{ik} (\sum_{j = 1, j \neq k}^{n} x_{ij}) = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rk} - θ_{kk}^{*} \sum_{i = 1}^{m} ν_{ik} x_{ik} = 0, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, \\ j = 1, 2, \dots, n, j \neq k, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m . \end{matrix}$ (3) Benevolent model:

$\begin{matrix} max & \sum_{r = 1}^{s} μ_{rk} (\sum_{j = 1, j \neq k}^{n} y_{rj}) \\ s . t . & \sum_{i = 1}^{m} ν_{ik} (\sum_{j = 1, j \neq k}^{n} x_{ij}) = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rk} - θ_{kk}^{*} \sum_{i = 1}^{m} ν_{ik} x_{ik} = 0, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, \\ j = 1, 2, \dots, n, j \neq k, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m . \end{matrix}$ (4)

2.1.3 Neutral cross-efficiency model

In reality, it is unreasonable to regard other DMUs as either enemies or allies. Wang and Chin [34] proposed a neutral cross-efficiency evaluation method: Each DMU determines the weights only from its own perspective, regardless of their impacts on the other DMUs. The model is expressed as follows:

$\begin{matrix} max & δ \\ s . t . & \sum_{i = 1}^{m} ν_{ik} x_{ik} = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rk} = θ_{kk}^{*}, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, \\ j = 1, 2, \dots, n; j \neq k, \\ μ_{rk} y_{rk} - δ \geq 0, δ \geq 0, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m . \end{matrix}$ (5)

The goal function in the above model indicates that DMU_k searches for a set of input and output weights to maximize its efficiency as a whole while at the same time making each of its outputs as efficient as possible to produce sufficient efficiency as an individual. It is not affected by other DMUs and can objectively describe the information of the DMU. Finally, the optimal weights obtained can be used to evaluate other DMUs in order to obtain peer- evaluation efficiency values.

2.1.4 Modified benevolent and aggressive models

The benevolent model, aggressive model, and neutral model described in the preceding sections use the secondary goal to obtain the optimal weights under self-evaluation, which can be the public weight used to calculate the peer-evaluation efficiency values of other DMU, and to finally calculate the comprehensive efficiency value. Their common characteristic is that each row uses a set of public weights to calculate the efficiency value, and a total of n sets of weight coefficients are selected. The disadvantage is that the selection of public weights may not be unique, which will result in instability of the obtained cross-efficiency matrix. Instead of considering the public weights under the secondary goal, Wu and Liang [9] calculated the boundary value of the cross-efficiency value of the DMU by maximizing or minimizing the peer-evaluation efficiency of the evaluated DMU to obtain the boundary value of the cross-efficiency value of the DMU. The model is expressed as follows:

Modified benevolent model:

$\begin{matrix} θ_{kd} & = max \sum_{r = 1}^{s} μ_{rk} y_{rd} \\ s . t . & \sum_{i = 1}^{m} ν_{ik} x_{id} = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rk} - θ_{kk}^{*} \sum_{i = 1}^{m} ν_{ik} x_{ik} = 0, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, \\ j = 1, 2, \dots, n, j \neq k, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m . \end{matrix}$ (6)

The efficiency value calculated by the above model is the maximum efficiency value that can be achieved by the evaluated DMU_d under the premise of satisfying the maximum efficiency value of the evaluation DMU_k. The efficiency value is unique and is not affected by the multiple optimal solutions of the DMU_k. The aggressive model takes the minimum value of the above objective function, and the peer-efficiency value obtained at this time is also the minimum efficiency value that can be obtained by the DMU_d under the premise of the maximum efficiency value of the evaluation DMU_k. This value is also unique. The model is expressed as follows:

Modified aggressive model:

$\begin{matrix} θ_{kd} & = & min \sum_{r = 1}^{s} μ_{rk} y_{rd} \\ s . t . & \sum_{i = 1}^{m} ν_{ik} x_{id} = 1, \\ \sum_{r = 1}^{s} μ_{rk} y_{rk} - θ_{kk}^{*} \sum_{i = 1}^{m} ν_{ik} x_{ik} = 0, \\ \sum_{r = 1}^{s} μ_{rk} y_{rj} - \sum_{i = 1}^{m} ν_{ik} x_{ij} \leq 0, \\ j = 1, 2, \dots, n, j \neq k, \\ μ_{rk} \geq 0, r = 1, 2, \dots, s, \\ υ_{ik} \geq 0, i = 1, 2, \dots, m . \end{matrix}$ (7)

From the viewpoint of the overall DMU, the benevolent model and the aggressive model calculate the peer-evaluation efficiency value based on optimism and pessimism, respectively. The neutral model calculates the evaluation efficiency value from the perspective of peer evaluation. However, the modified benevolent model and the modified aggressive model are, respectively, based on an optimistic view and a pessimistic view from the individual consideration of the DMU to calculate peer-evaluation efficiency values. These five models are ranked from the perspective of pessimism to optimism: the modified aggressive model (7) to the aggressive model (4) to the neutral model (5) to the benevolent model (3) to the modified benevolent model (6).

2.2 HFSs

In the actual decision-making process, due to the complexity of the objective world and the limitation of human cognition, the information that people obtain is constantly changing. Zadeh [15] introduced the fuzzy set in 1965, using the membership degree to describe the degree of uncertainty. Since then, many experts have studied other extended forms of fuzzy sets. Atanassov [12] proposed intuitionistic fuzzy sets (IFSs), which consist of membership degree and non-membership degree, and effectively combine the information of support, opposition, and hesitation. Torra [28, 29] considered that the decision- maker may hesitate when judging the degree of membership of the object, and proposed the HFS, which fully reflects the multiple membership degrees that cannot reach the same consensus. Actually, the HFS, which aims to solve the difficulties in describing the hesitancy in practical evaluation, has been widely used in multi-criteria decision-making (MCDM) problems, and scientists have conducted further research [7,17, 7,17]. In addition, Song et al. [3] introduced the probabilistic hesitant fuzzy set (P- HFS) in multi-criteria decision-making problems.

Definition 1. (see Refs. 28 and 29). HFS. Let X be a non-empty set, x ∈ X, then call A ={ 〈 x, h_A (x) 〉 |x ∈ X } a hesitant fuzzy set in which h_A (x) is called the hesitant fuzzy element (HFE), which is a set of several values in the interval [0,1], indicating that the membership degree of x belonging to A consists of a set of possible values.

Definition 2. Ranking HFE. Let h (x) be an HFE that is defined on x ∈ X. The ith small element of the plurality of membership degrees constitutes the HFE that is represented by h^{τ
_i}, and is ranked and expressed as h (x) = (h^{τ
₁} (x) , h^{τ
₂} (x) , ⋯ , h^{τ
_l} (x)), where l represents the number of the elements in h (x), which is also called the length of the HFE.

Definition 3. (see Refs.7 and 17). Score function and deviation function of HFE. Let h (x) be an HFE; then, $s (h) = \frac{1}{l} \sum_{i = 1}^{l} h^{τ_{i}} (x)$ is called the score function of the HFE h; $v (h) = \frac{1}{l} \sqrt{\sum_{1 \leq j < i \leq l} (h^{τ_{i}} (x) - h^{τ_{j}} (x))^{2}}$ is the deviation function of the HFE h.

2.3 Determination of weights under multi-attribute decision-making

The cross-efficiency matrix is formed by the group decision; the information in the matrix consists of HFSs. Each evaluation DMU is regarded as an attribute and the evaluated DMU is regarded as an evaluation target. For the hesitant fuzzy multi- attribute decision-making problem with completely unknown weights, the characteristics of the cross- efficiency matrix are composed of HFSs that were comprehensively considered, and the multi-objective optimization method of attribute weights of Liu [30] was introduced to calculate the weight of attributes.

(1) It is an HFS from the standpoint of each constituent unit that forms a cross-efficiency matrix. The more concentrated the evaluation value of the set, the smaller the deviation function, which means that the more consistent the evaluation value provided by the decision-maker, the more it can reflect the true level of the DMU. The larger the evaluation value of the set, the larger the score function, which means that the larger the efficiency value of the DMU, the better the DMU. Based on this determined objective function, the modeling is performed as follows:

$\begin{matrix} M 1 & = & max f_{1} (W) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} w_{i} \\ s . t . & \sum_{i = 1}^{m} w_{i}^{2} = 1, 0 \leq w_{i} \leq 1 . \end{matrix}$ (8)

(2) From the horizontal direction of the cross- efficiency matrix, the difference is represented by distance, and each DMU is under the same attribute w_i. The greater the contribution of the attribute in distinguishing the DMU, the greater the weight. Based on this determined objective function, the model is expressed as follows:

$\begin{matrix} M 2 & = & max f_{2} (W) \\ = & \frac{1}{n} \sum_{i = 1}^{m} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} w_{i} \\ = & \frac{1}{n} \sum_{i = 1}^{m} \sqrt{\sum_{1 \leq p, q \leq n} \frac{1}{l} \sum_{k = 1}^{l} | h_{ip}^{τ_{k}} - h_{iq}^{τ_{k}} |} w_{i} \\ s . t . & \sum_{i = 1}^{m} w_{i}^{2} = 1, 0 \leq w_{i} \leq 1 . \end{matrix}$ (9)

(3) From the vertical direction of the cross- efficiency matrix, the more similar the evaluation result of an attribute and other attribute values, the smaller the influence on the ranking of the DMU, and the smaller the attribute weight. In addition, the similarity is represented by the relevance degree. Based on this determined objective function, the modeling is performed as follows:

\begin{array}{l} M 3 = \max f_{3} (W) = \sum_{i = 1}^{m} \sum_{t = 1}^{m} (1 - r_{i t}) w_{i} \\ = \sum_{i = 1}^{m} \sum_{t = 1}^{m} w_{i} (1 - \frac{\sum_{a = 1}^{n} (\frac{1}{l} \sum_{k = 1}^{l} h_{i a}^{τ_{k}} \cdot h_{t a}^{τ_{k}})}{\max {\sum_{a = 1}^{n} (\frac{1}{l} \sum_{k = 1}^{l} {(h_{i a}^{τ_{k}})}^{2}, \frac{1}{l} \sum_{k = 1}^{l} {(h_{t a}^{τ_{k}})}^{2})}}) \\ s . t . \sum_{i = 1}^{m} w_{i}^{2} = 1, 0 \leq w_{i} \leq 1. \end{array}

(10)

Considering the above three objectives comprehensively, the sets of weights of the three objectives are (α, β, γ), respectively, representing the balance coefficients of the three objectives, and the combined model obtained is expressed as follows:

$\begin{matrix} M & = & max f (W) = α \sum_{i = 1}^{m} \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} w_{i} \\ + β \frac{1}{n} \sum_{i = 1}^{m} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} w_{i} \\ + γ \sum_{i = 1}^{m} \sum_{t = 1}^{m} (1 - r_{it}) w_{i} \\ s . t . & \sum_{i = 1}^{m} w_{i}^{2} = 1, 0 \leq w_{i} \leq 1 . \end{matrix}$ (11) This study solves the final weight of the model by constructing the Lagrangian auxiliary function: $L (w, λ) = f (W) + \frac{1}{2} λ (\sum_{i = 1}^{m} w_{i}^{2} - 1) .$ (12)

By determining the partial derivative, we can obtain the attribute weight: $w_{i} = \frac{α \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} + β \frac{1}{n} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} + γ \sum_{t = 1}^{m} (1 - r_{it})}{\sqrt{{(α \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} + β \frac{1}{n} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} + γ \sum_{t = 1}^{m} (1 - r_{it}))}^{2}}} .$ (13)

Normalization of attribute weights yields $w_{i}^{*} = \frac{α \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} + β \frac{1}{n} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} + γ \sum_{t = 1}^{m} (1 - r_{it})}{\sum_{i = 1}^{m} (α \sum_{j = 1}^{n} \frac{s (h_{ij})}{s (h_{ij}) + v (h_{ij})} + β \frac{1}{n} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{ip}, h_{iq})} + γ \sum_{t = 1}^{m} (1 - r_{it}))} .$ (14)

2.4 Ranking DMUs under hesitant fuzzy information

The decision system comprises n DMUs and m attributes and the decision information consists of HFSs. The closeness degree between the DMU and positive (negative) ideal points is calculated based on the hesitant weighted Euclidean distance.

We can use closeness degree to calculate the pros and cons of the schemes.

$D (Y_{j}) = \frac{d^{-} (Y_{j} Y_{-})}{d^{-} (Y_{j} Y_{-}) + d^{+} (Y_{j} Y_{+})},$ (15) in which $d^{-} (Y_{j} Y_{-}) = \sum_{i = 1}^{n} w_{i} \sqrt{\frac{1}{l} \sum_{k = 1}^{l} {| h_{ij}^{τ_{k}} - h_{i -}^{τ_{k}} |}^{2}}$ , d⁺ (Y_jY₊) $= \sum_{i = 1}^{n} w_{i} \sqrt{\frac{1}{l} \sum_{k = 1}^{l} {| h_{ij}^{τ_{k}} - h_{i +}^{τ_{k}} |}^{2}}$ , $h_{i +}^{τ_{k}}$ , and $h_{i -}^{τ_{k}}$ , respectively, indicate the positive and negative ideal DMUs that constitute the HFS under the ith attribute.

The DMUs are sorted according to the final calculation result. The larger the D (Y_j), the better the DMU becomes.

3 Calculation steps

Regarding the problem of ranking and selecting of DMUs with multiple inputs and outputs, the cross- efficiency model of the self-evaluation and peer- evaluation system is used for the evaluation. Since there are multiple optimal solutions for the input- output weights obtained from the calculation of self- evaluation values, which cross-efficiency model should be chosen to calculate the efficiency values is controversial because the global cross-efficacy values calculated with different models may be different, which can eventually affect the sorting result such that an alternative might be a better option.

In fact, taking the multiple optimal solutions into account, the different cross-efficiency models can fully reflect the relevant information of the DMU. The five cross-efficiency models in Table 1 represent the five decision-makers whose attitudes range from pessimistic to optimistic. The questions are then transformed into group decision problems. The efficiency values obtained independently by these five models may not have the same ranking results, making it difficult to reach an agreement. By collating the efficiency matrices calculated by the five models, the cross-efficiency matrix composed of HFSs can be obtained. In the cross-efficiency matrix, the evaluation DMU is regarded as an attribute and the data characteristics between group decision- makers are taken into account. The attribute weight multi-objective method is introduced to determine the attribute weights. Finally, the closeness degree between the DMU and the positive and negative ideal DMUs are calculated in order to rank the DMUs.

Table 1
Selection of cross-efficiency models

Cross-efficiency model Modified aggressive model Aggressive model Neutral model Benevolent model Modified benevolent model

Self-evaluation Peer-evaluation MAX Peer-evaluation value of every DMU is minimum MAX Peer-evaluation value of all DMUs are minimum MAX Not considered MAX Peer-evaluation value of all DMUs are maximum MAX Peer-evaluation value of every DMU is maximum

Decision-maker’s attitude Transition from pessimism to optimism

Cross-efficiency model	Modified aggressive model	Aggressive model	Neutral model	Benevolent model	Modified benevolent model
Self-evaluation Peer-evaluation	MAX Peer-evaluation value of every DMU is minimum	MAX Peer-evaluation value of all DMUs are minimum	MAX Not considered	MAX Peer-evaluation value of all DMUs are maximum	MAX Peer-evaluation value of every DMU is maximum
Decision-maker’s attitude		Transition from pessimism to optimism

We assume that there are N DMUs, m inputs, and s outputs. The specific steps used to order the DMUs:

Step 1: Construct a positive ideal DMU and a negative ideal DMU in the existing DMUs. The positive ideal DMU consists of a minimum amount of inputs and a maximum amount of outputs. Conversely, it constitutes a negative ideal DMU and forms an evaluation system consisting of N+2 DMUs.

Step 2: Calculate the cross-efficiency matrix under the above five models to form a comprehensive cross-efficiency matrix under a group decision. The efficiency value θ_kd in the matrix consists of an HFS, indicating that the efficiency value of the group decision is made by DMU_k to evaluate DMU_d. The DMU_k is called an evaluation DMU and the DMU_d is called an evaluated DMU. We sort the elements of the HFS in ascending order, and obtain the cross-efficiency matrix composed of the hesitant fuzzy information which has already been collated.

Step 3: Consider the evaluation DMU in the cross-efficiency matrix as an attribute and calculates the weight of each attribute under the cross- efficiency matrix with formula (14).

Step 4: According to formula (15), calculate the closeness degree formed by the distance between each evaluated DMU and the positive ideal DMU and then the negative ideal DMU, and then sort the DMUs by closeness degree.

4 Example analysis

Based on the data in Table 2, Wu [9] considered all multiple optimal solutions, calculated the critical value of the efficiency scores by using the modified benevolent model and the modified aggressive model, assuming that the efficiency scores are evenly distributed across the interval, and calculated the final solution based on the possibility. The efficiency values are finally sorted, as shown in Table 3. The efficiency values calculated from the three cross- efficiency models in columns 2–4 are significantly different, indicating that choosing different cross-efficiency models can affect the efficiency of the DMU. The volatility of the ranking results is not significant, which is related to the characteristics of the input-output information of the DMU and the number and input-output indicators of the DMU. In this study, the efficiency values calculated under the five cross-efficiency models constitute an HFS, and the HFEs are composed of five efficiency scores. Finally, we use the closeness degree to sort the DMUs.

Table 2
Inputs and outputs of DMUs

DMU X1 X2 X3 Y1 Y2

1 7 7 7 4 4

2 5 9 7 7 7

3 4 6 5 5 7

4 5 9 8 6 2

5 6 8 5 3 6

DMU	X1	X2	X3	Y1	Y2
1	7	7	7	4	4
2	5	9	7	7	7
3	4	6	5	5	7
4	5	9	8	6	2
5	6	8	5	3	6

Table 3

Cross-efficiencies and rankings under different models

DMU	Primary cross-efficiency model	Modified aggressive model	Modified benevolent model	Interval efficiency value based on probability
1	0.4743(5)	0.4473(5)	0.5845(5)	0.7615(5)
2	0.8793(2)	0.8629(2)	0.9295(2)	3.5000(2)
3	0.9856(1)	0.9571(1)	1.0000(1)	4.5000(1)
4	0.5554(4)	0.5400(3)	0.7100(3)	2.2751(3)
5	0.5587(3)	0.4900(4)	0.6386(4)	1.4734(4)

Note: Figures in parentheses indicate ranking results of efficiency values for each DMU under different models.

(1) This study constructs positive and negative ideal DMUs. 6⁺ represents a positive ideal DMU. The input indicator selects the minimum value of the indicator value of all DMUs and the output indicator selects the maximum value. 7^– represents a negative ideal DMU. The selection of indicator values is contrary to the positive ideal DMU. The new DMU information system created is shown in Table 4.

Table 4

Input and output information of adding positive and negative ideal DMUs

DMU	X1	X2	X3	Y1	Y2
1	7	7	7	4	4
2	5	9	7	7	7
3	4	6	5	5	7
4	5	9	8	6	2
5	6	8	5	3	6
6⁺	4	6	5	7	7
7^-	7	9	8	3	2

For convenience of description, the five models in Table 1 are represented as A, B, C, D, and E, respectively. Such models are respectively calculated to obtain the efficiency values of the DMU and ranked, as shown in Table 5. The positive and negative ideal DMUs are not considered when ranking. It is clear that the positive ideal DMU has the largest efficiency value and the negative ideal DMU has the smallest efficiency value. We can see from Table 5 that the rankings of DMUs by the different models vary.

Table 5

Calculation results and rankings of different cross-efficiency models

DMU	A		B		C		D		E
1	0.3848	(4)	0.3848	(5)	0.4079	(4)	0.4173	(5)	0.4315	(5)
2	0.7116	(2)	0.7497	(2)	0.7255	(2)	0.7190	(2)	0.7497	(2)
3	0.7959	(1)	0.8218	(1)	0.8571	(1)	0.8165	(1)	0.9184	(1)
4	0.3160	(5)	0.4771	(3)	0.4064	(5)	0.4605	(4)	0.5153	(4)
5	0.4184	(3)	0.4457	(4)	0.5126	(3)	0.4786	(3)	0.6276	(3)
6⁺	1.0000		1.0000		1.0000		1.0000		1.0000
7^-	0.1985		0.2275		0.2219		0.2361		0.2451

Note: Figures in parentheses indicate ranking results of efficiency values for each DMU under different models.

(2) Under different cross-efficiency models, it is actually a different choice of multiple optimal solutions that will affect the ordering of DMUs. Taking the impact of multiple optimal solutions on the efficiency value of DMUs into account, we introduce hesitant fuzzy information. The five cross- efficiency models in Table 1 are selected to represent decision-makers with attitudes from pessimistic to optimistic, and the HFS consists of different efficiency scores, and finally forms a group decision comprehensive cross-efficiency matrix, as shown in Table 6.

Table 6

Comprehensive cross-efficiency matrix under group decisions

	Evaluated DMU(DMU)
DMU	1	2	3	4	5	6⁺	7^-
Evaluation DMU (attributet)	1	0.4898	0.6667	0.7143	0.1905	0.3214	1.0000	0.1905
		0.4898	0.6667	0.7779	0.3810	0.3930	1.0000	0.2381
		0.4898	0.6667	0.7981	0.4597	0.4157	1.0000	0.2578
		0.4898	0.6667	0.8571	0.4867	0.4821	1.0000	0.2645
		0.4898	0.6667	1.0000	0.5714	0.6429	1.0000	0.2857
	2	0.3265	0.8000	0.7143	0.2286	0.2857	1.0000	0.1633
		0.3265	0.8000	0.7551	0.4571	0.3265	1.0000	0.2041
		0.3265	0.8000	0.7569	0.6175	0.3283	1.0000	0.2327
		0.3265	0.8000	0.8571	0.6204	0.4286	1.0000	0.2332
		0.3265	0.8000	1.0000	0.6857	0.5714	1.0000	0.2449
	3	0.3265	0.6667	1.0000	0.1786	0.5714	1.0000	0.1633
		0.3265	0.6882	1.0000	0.1847	0.5714	1.0000	0.1633
		0.3915	0.7275	1.0000	0.2005	0.6517	1.0000	0.1758
		0.4478	0.8000	1.0000	0.2286	0.7282	1.0000	0.1847
		0.4898	0.8000	1.0000	0.2286	0.8571	1.0000	0.1905
	4	0.3265	0.8000	0.7143	0.6857	0.2857	1.0000	0.2449
		0.3265	0.8000	0.7143	0.6857	0.2857	1.0000	0.2449
		0.3265	0.8000	0.7143	0.6857	0.2857	1.0000	0.2449
		0.3265	0.8000	0.7143	0.6857	0.2857	1.0000	0.2449
		0.3265	0.8000	0.7143	0.6857	0.2857	1.0000	0.2449
	5	0.4082	0.7143	1.0000	0.1786	0.8571	1.0000	0.1786
		0.4082	0.7143	1.0000	0.1786	0.8571	1.0000	0.1786
		0.4082	0.7143	1.0000	0.1786	0.8571	1.0000	0.1786
		0.4082	0.7143	1.0000	0.1786	0.8571	1.0000	0.1786
		0.4082	0.7143	1.0000	0.1786	0.8571	1.0000	0.1786
	6⁺	0.3265	0.6667	0.7143	0.1786	0.2857	1.0000	0.1633
		0.3265	0.6975	0.7523	0.3707	0.3423	1.0000	0.2287
		0.4230	0.7037	0.7709	0.4987	0.4366	1.0000	0.2262
		0.4324	0.8000	0.8571	0.5952	0.5612	1.0000	0.2615
		0.4898	0.8000	1.0000	0.6857	0.8571	1.0000	0.2857
	7^-	0.4898	0.6667	0.7143	0.5714	0.3214	1.0000	0.2857
		0.4898	0.6667	0.7143	0.5714	0.3214	1.0000	0.2857
		0.4898	0.6667	0.7143	0.5714	0.3214	1.0000	0.2857
		0.4898	0.6667	0.7143	0.5714	0.3214	1.0000	0.2857
		0.4898	0.6667	0.7143	0.5714	0.3214	1.0000	0.2857

In Table 6, the horizontal value represents the evaluated DMU or the DMU, the vertical value represents the evaluation DMU or attribute, and θ_kd is composed of an HFS, in which HFEs are the efficiency values in the matrix obtained from the five different cross-efficiency models. When the evaluation DMU is DMU1 and the evaluated DMU is DMU4, θ₁₄ is calculated as (0.1905, 0.3810, 0.4597, 0.4867, 0.5714), constituting HFSs, indicating that the opinion of group experts obtained by the ascending order of the efficiency value of DMU1 in calculating DMU4 that is 0.1905, 0.3810, 0.4597, 0.4867, and 0.5714. The diagonal line indicates the self- evaluation efficiency value, such that the group decision results are consistent, the values are the same and are the maximum values of each column. The evaluated DMU⁶⁺ is a positive ideal DMU, which is located on the effective frontier. Therefore, regardless of whether the self-evaluation value or the peer-evaluation value is calculated, the result is consistent and the value is 1. The evaluated DMU7^– is a negative ideal DMU and the efficiency value is the smallest value among all evaluated DMUs.

(3) This study considers the evaluation DMU in the cross-efficiency matrix as an attribute and calculates the weight of the attribute. The weights of attributes are considered from three perspectives:

a. The larger the score function of each HFS, the better the DMU; and the smaller the deviation function, the more consistent the group decision, and the better the DMU. Thus, when all DMUs are considered under the same attribute, the larger the score function of the HFS of the DMU, and the smaller the deviation function, the greater the weight of the attribute.

b. The more inconsistent the information of different DMUs under the same attribute, and the stronger the distinguishing ability of the attribute to the DMU, the greater the attribute weight.

c. The smaller the association between the attribute and other attributes, and the more influence the attribute has on the result of the evaluation, the greater the attribute weight.

We can set the target balance factor as $(α, β, γ) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$ . Therefore, the weight of each attribute under the comprehensive cross-efficiency matrix is calculated by formula (14):

w_{i}^{*} = \frac{\frac{1}{3} \sum_{j = 1}^{7} \frac{s (h_{i j})}{s (h_{i j}) + v (h_{i j})} + \frac{1}{3} \times \frac{1}{7} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{i p}, h_{i q})} + \frac{1}{3} \sum_{t = 1}^{7} (1 - r_{i t})}{\sum_{i = 1}^{7} (\frac{1}{3} \sum_{j = 1}^{7} \frac{s (h_{i j})}{s (h_{i j}) + v (h_{i j})} + \frac{1}{3} \times \frac{1}{7} \sqrt{\sum_{1 \leq p, q \leq n} d^{2} (h_{i p}, h_{i q})} + \frac{1}{3} \sum_{t = 1}^{7} (1 - r_{i t}))},

w^{*} = (0.1349, 0.1349, 0.1434, 0.1504, 0.1580, 0.1277, 0.1508) .

(4) This study calculates the closeness degree of the DMU according to the distance between the DMU and the positive and negative ideal points. The closeness degree of each DMU is calculated as follows:

$\begin{matrix} D (Y_{j}) = \frac{d^{-} (Y_{j} Y_{7})}{d^{-} (Y_{j} Y_{7}) + d^{+} (Y_{j} Y_{6})}, \\ D (Y) = (0.2343, 0.6514, 0.7873, 0.2762, 0.3604, 1, 0) . \end{matrix}$

It can be seen from the above calculation results that the closeness degree of the positive ideal DMU is 1 and the closeness degree of the negative ideal DMU is 0, which is consistent with reality.

Under the premise of adding ideal DMUs, we reorder the DMUs using the method of [9], and the same ranking results are obtained. What is shown in Table 7 proves the effectiveness of the proposed method. However, there are irrationalities in [9]: (i) When constructing the interval cross-efficiency matrix, only the efficiency boundary value of the DMU is considered, which is equivalent to only considering the 2 extreme cases of multiple optimal solutions, resulting in evaluation results that can be too fuzzy; (ii) the cross-efficiency value obtained by simple summation and then an average of the interval cross-efficiency matrix does not consider the correlation of the information; (iii) it is assumed that the efficiency score of each DMU is a uniform distribution across the interval values. According to the possibility of ranking, this assumption is not feasible in practice. In comparison, the method proposed in this study uses the information of multiple optimal solutions to the maximum extent, introduces a multi-objective weight optimization method, comprehensively considers the correlation of information in the cross-efficiency matrix under group decision, calculates the weights of attributes, and, based on the closeness degree between the DMU and the positive and negative ideal points, ranks the DMUs. As a result, the evaluation results are more acceptable.

Table 7

Comparison of ranking results

DMU		1	2	3	4	5	6⁺	7^-
Literature [9]	Comprehensive value	1.799	4.5	5.5	2.323	4.378	6.5	0.5
	Rank	5	2	1	4	3	–	–
This study	Comprehensive value	0.2343	0.6514	0.7873	0.2762	0.3604	1	0
	Rank	5	2	1	4	3	–	–

(5) This study applies the sensitivity analysis of model correlation coefficients. The different balance coefficients are selected to calculate the attribute weight combination and closeness degree, as shown in Table 8.

Table 8

Sensitivity of different balance coefficients

Balance coefficient	Weight of different attributes	Closeness degree
(1/3,1/3,1/3)	(0.1349,0.1349,0.1434,0.1504,0.1580,0.1277,0.1508)	(0.2343,0.6514,0.7873,0.2762,0.3604)
(1,0,0)	(0.1377,0.1370,0.1409,0.1517,0.1517,0.1293,0.1517)	(0.2338,0.6512,0.7847,0.2794,0.3556)
(0,1,0)	(0.1273,0.1439,0.1664,0.1398,0.1726,0.1327,0.1172)	(0.2341,0.6563,0.8026,0.2612,0.3850)
(0,0,1)	(0.1019,0.1029,0.1635,0.1378,0.2336,0.1035,0.1568)	(0.2408,0.6503,0.8133,0.2423,0.4101)

According to the closeness degree calculated by different balance coefficients, the DMUs are ranked and the ranking results are consistent, DMU3>DMU2>DMU5>DMU4>DMU1, indicating that the ranking of the DMUs in this study is stable under the consideration of multiple targets. However, the selection of the balance coefficient still affects the decision-making process, which is reflected in the calculation results of the attribute weights: the coefficients of variation of each attribute weight are, respectively, 12.3%, 14%, 7.3%, 4.3%, 18.7%, 10.7%, and 12.4%.

5 Conclusions

Cross-efficiency evaluation is a significant method for comparing and ranking DMUs. In this study, we utilized and investigated group decision models, including the modified aggressive, aggressive, neutral, benevolent, and modified benevolent models. The five classic cross-efficiency models represent five group decision-making experts whose attitudes range from pessimistic to optimistic. We introduced HFSs into a cross-efficiency matrix, in which the HFSs were composed of efficiency scores obtained from the aforementioned five models. The efficiency values obtained from the different optimal solutions can be considered the result of a group decision; the data characteristics of the cross-efficiency matrix composed of HFSs were then comprehensively considered, and the multi-objective attribute weight optimization method introduced in order to calculate the attribute weight of the efficiency value. We presented a modified closeness formula to calculate the relative closeness degree of DMUs between the positive ideal DMU and the negative ideal DMU, and then ranked the DMUs. Numerical examples have been tested to demonstrate the potential applications of the new cross-efficiency group decision- making method and its effectiveness in discriminating among DMUs.

In the future, we plan to more fully utilize the in- formation of multiple optimal solutions. However, there are still deficiencies in the proposed method. Although the closeness degree can be used to select the best-ordering DMUs, its final value is no longer an efficiency value. Further consideration should be given as to how to use the obtained attribute weights and HFSs to calculate the case in which the final value is still efficient.

Acknowledgments

This work was supported by the Fund for Shanxi “1331 Project” Key Innovative Research Team.

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A novel cross-efficiency evaluation method under hesitant fuzzy environment

Abstract

Keywords

1 Introduction

2 Models and methods

2.1 Cross-efficiency models

2.1.1 CCR model and traditional cross-efficiency model

2.3 Determination of weights under multi-attribute decision-making

Table 2 Inputs and outputs of DMUs DMU X1 X2 X3 Y1 Y2 1 7 7 7 4 4 2 5 9 7 7 7 3 4 6 5 5 7 4 5 9 8 6 2 5 6 8 5 3 6

Acknowledgments

References

Table 2
Inputs and outputs of DMUs

DMU X1 X2 X3 Y1 Y2

1 7 7 7 4 4

2 5 9 7 7 7

3 4 6 5 5 7

4 5 9 8 6 2

5 6 8 5 3 6