An ICE aggregation method based on optimal rally points in fuzzy environment

Abstract

In order to avoid the hesitation of choosing between aggressive and benevolent strategies, we propose two cross-efficiency models to get interval cross-efficiency (ICE) from the relatively neutral angle in fuzzy environment, and then propose a novel aggregation method for ICE to solve the full ranking of Decision-Making Units (DMUs). Firstly, regard the expected value of fuzzy data as the input and output of Data Envelopment Analysis (DEA) method based on fuzzy set theory. Secondly, construct the cross-efficiency models based on the fuzzy expected values from the relatively neutral angle, and generate the lower and upper bounds of ICE for all DMUs, which determines the interval cross-efficiency matrix (ICEM). Thirdly, project all ICE onto the plane as points, then seek the optimal rally point for each DMU based on ICEM as the comprehensive ICE. Fourthly, rank the comprehensive ICE to obtain the complete ranking of DMUs by using the optimal number sorting method. Finally, the proposed model is applied to the evaluation of manufacturing enterprises, and the results are compared with different models to prove its effectiveness.

Keywords

Interval cross-efficiency DEA fuzzy sets fuzzy numbers the optimal rally point aggregation method

1 Introduction

Data Envelopment Analysis (DEA) is a non-parametric method, initially developed by Charnes et al. [1], generally used to evaluate the performance of a set of homogeneous Decision-Making Units (DMUs). The relative efficiency of each DMU can be computed based on multiple inputs and multiple outputs. DMU is viewed as inefficient if it can produce more outputs by keeping the current inputs unchanged, or can use fewer inputs while keeping the current outputs unchanged. The CCR model is popular with the assumption of constant return on scale (CRS), and this technique has been developed for many years in a wide range of applications, such as assessment of the banking crisis [2], the evaluation of agricultural energy and environmental efficiency [3], application to hospital performance [4], research performance evaluation of university [5], assessment on the effectiveness of maritime safety control based on the main safety factors [6, 7]. However, the CCR model blindly seeks the optimal weights for self-evaluation efficiency, which will lead to over-optimism about the DMU evaluated. Generally speaking, multiple DMUs are often evaluated as DEA efficiency based on the CCR model, so the optimal DMUs cannot be further distinguished. To improve the discrimination of this technology, Sexton et al. [8] suggested the cross-efficiency evaluation. This technology emphasizes that the performance of each DMU is not only determined by its own weight, but also combined with the weight of all other DMUs. The advantage of cross-efficiency is that it both considers self-evaluation and peer-evaluation, eliminates unrealistic weight schemes, and finally obtains a complete ranking.

Although the cross-efficiency improves the ability to distinguish DMUs, the optimal weight set obtained from the traditional DEA model may not be unique, resulting in the inconsistency of ranking. In order to exist and have only one set of optimal solutions, scholars have introduced the secondary goal function, which can find a unique satisfying solution while keeping the optimal self-evaluation efficiency unchanged. Generally known, Doyle and Green [9] proposed the benevolent and aggressive models which addressed the problem of non-unique optimal weight in traditional DEA. At the same time, many studies are constantly design the secondary goal functions from different angles to improve the stability of optimal weight, so that cross efficiency has become a powerful sorting tool and has been widely developed and applied. Liang et al. [10] applied game mechanics to cross-efficiency, treating each DMU as a participant to maximize its own cross-efficiency while minimizing the impact on other DMUs. Wang & Chin [11] proposed a neutral cross-efficiency model, considering that each DMU seeks the optimal set of weight just from its own view, not caring too much about other DMUs from aggressive or benevolent perspectives. Subsequently, Wang & Chin [12] viewed the CCR efficiency as the ideal point, and proposed some DEA cross-efficiency models by varying the objective function based on various forms of deviation. The theory of cross efficiency is developing continuously, and it is also applied in real life. Essid et al. [13] developed a novel portfolio selection framework based on a combination of the Maverick index and the game cross-efficiency approach for stock portfolio selection. Wang et al. [14] were the first to use game cross-efficiency to analyze provincial energy efficiency in the construction industry. Wang et al. [15] extended the game cross-efficiency to two-stage DEA to evaluate the performance of industrial water systems. Green port is one of the goals and tasks of sustainable development [16], Wang et al. [17] used the cross-efficiency model to evaluate the efficiency of green ports, and introduced a new development model of port groups. Although cross-efficiency evaluation has become a useful extension tool for DEA, the cross-efficiency scores obtained may not be Pareto optimum, so Wu et al. [18] suggested a cross-efficiency evaluation approach based on Pareto improvement to resolve this problem. Furthermore, owing to the complexities and flexibility of real-world scenarios, researchers may not use cross-efficiency assessment as the only measure, but rather a combination of diverse behavioral decision theories to solve decision-making issues. Liu et al. [19] took into account decision makers’ risk attitudes and developed a cross-efficiency evaluation based on prospect theory research to reflect the illogical psychology of decision-makers under risk. An assessment approach for regret cross-efficiency was proposed by Gong et al. [20], a multi-objective combination selection model based on regret theory is constructed by integrating the cross-efficiency into the mean-variance skewness framework.

The models discussed above all deal with crisp data, but in the real world, there are often full of fuzziness and uncertainty, which need to structure the effective models to solve the complex reality system. Kao & Liu [21] developed the technology of dealing with imprecise input and introduced the concept of fuzziness, the main idea of this technology is to transform the fuzzy DEA model into the traditional clear DEA model by α - cut method. Conversely, Guo & Tanaka [22] proposed a fuzzy DEA model to deal with the efficiency evaluation of given fuzzy data, and extended the crisp efficiency based on the CCR model to a fuzzy number to reflect the uncertainty in the actual evaluation. There are lots of original information are eliminated by using α - cut method to handle the fuzzy data, which will lead to the distortion of evaluation, so Zerafat Angiz [23] proposed the novel concept of “local α - cut”, and evaluated the uncertain efficiency by a multi-objective linear programming. Wang & Chin [24] weighted the fuzzy input and fuzzy output in order to simplify the processing of the fuzziness, used the expected value to measure the optimistic and pessimistic efficiency of DMUs, and geometrically averaged the two efficiencies to determine the optimal DMU. It can be seen from the above that CCR model has been extended to fuzzy environment and has been widely studied. Similarly, the cross-efficiency DEA models in uncertain environment have been further developed to overcome the inherent defects of the CCR model. For example, Chen & Wang [25] proposed a new fuzzy cross-efficiency ranking method based on the traditional DEA model, which obtained the ICE on the basis of retaining the original fuzzy information. Dotoli et al. [26] combined the cross-efficiency DEA model and fuzzy logic framework to solve the uncertainty, estimated the fuzzy inputs and outputs by triangular fuzzy distribution, and calculated the fuzzy efficiency of each DMU by the weight obtained from the compromise between the set goals. Due to the complexity of fuzzy environment, it is difficult to evaluate the performance of DMUs with crisp efficiency value, and the efficiency of each DMU is expressed as interval cross-efficiency (ICE), so it is necessary to propose a new method for aggregating ICE to obtain the final rankings of DMUs. Wang et al. [29] got the ICE based on benevolent and aggressive strategies, and aggregate ICE by introducing generalized Fermat Torricelli point, considered the consensus of group decision-making and avoided the subjective preference of decision-makers. Wang & Yang [30] determined the ICE based on the best and worst relative efficiency, and using the Hurwicz criterion approach to aggregate ICE for the final ranking of DMUs. Wang et al. [31] constructed a distance entropy model for obtaining the optimal weight of ICM, and get the full ranking of all alternatives based on the relative Euclidean distance from the positive solution. Fang & Yang [32] introduced the cumulative prospect theory into the aggregation process of ICE, the parameters about DM’s attitude towards risk are taken as the reference point, the comprehensive weight of each scenario is obtained by using the similarity measurement method to obtain the complete ranking.

Through the study of the existing literature, we find that there are still problems that can be solved. Firstly, the cross-efficiency DEA models in fuzzy environment need to be further developed, because many researchers pay more attention to the cross-efficiency evaluation of DMUs in the clear environment, ignoring the fuzzy DEA models are more realistic; Secondly, in the existing of fuzzy cross-efficiency DEA models, many studies make the defuzzification, determine the cross-efficiency as a crisp value instead of the ICE, which leads to the loss of fuzzy information and the reliability of the evaluation results of DMUs; Thirdly, most of the existing ICE determination is based on benevolent or aggressive strategies, so the obtained cross-efficiency interval is too extreme; Fourthly, the simple average operator is often used when aggregating the ICE, which often leads to the loss of interval information and ignores the consensus between the cross-efficiency. To fill the gaps in the existing literature, the paper’s main contributions can be summarized as follows. Firstly, based on fuzzy set theory, the expected value of fuzzy data is introduced into the cross-efficiency model as the input and output of DEA method, which reflects the uncertainty in reality. Secondly, determine the ICE from the relatively neutral perspective in the fuzzy environment, that is, for the benevolent cross-efficiency evaluation, the aggressive strategy is combined to minimize the maximum deviation from the ideal point, while for the aggressive cross-efficiency evaluation, the benevolent strategy is combined to maximize the minimum deviation from the ideal point. Thirdly, we consider the consensus between the cross efficiency when aggregating the generated ICE, project ICE of DMU into the two-dimensional coordinate plane, and take the point with the smallest Euclidean distance from each point as the consensus point of all cross-efficiency intervals, referred to as the optimal rally point of DMU. Finally, the comprehensive ICE of all DMUs is fully sorted by the optimal number sorting method instead of averaging them, which makes the ranking results satisfy the order-preserving property.

The remainders of this paper are organized as follows. Section 2 presents the basic of fuzzy set and fuzzy DEA cross-efficiency. Section 3 proposes the new generation method of ICE model and the aggregation method of the generalized interval CEM. Section 4 contains a case study of the application to the evaluation of manufacturing enterprises. The results are compared with different models from different angles in Section 5. Section 6 gives some conclusion of this paper and puts forward some prospects.

2 Preliminaries

In order to understand the proposed method, this section briefly reviews some basic concepts of fuzzy set and fuzzy DEA cross-efficiency.

2.1 Fuzzy set and related operations

Fuzzy set theory was first proposed by Zadeh [33], has been continuously developed and improved, and was widely used in different fields. According to the research of Luo [34], the related definitions and theorems of fuzzy numbers are given.

Definition 1 A fuzzy subset $\tilde{A}$ in set X is defined as $\tilde{A} = {(x, μ_{\tilde{A}} (x)); x \in X}$ by membership function $\tilde{A} = μ_{\tilde{A}} (x) : X \to [0, 1]$ , where $μ_{\tilde{A}} (x) \in [0, 1]$ demonstrates membership degree of x in set $\tilde{A}$ .

Definition 2 A fuzzy set is a fuzzy number if and only if its membership function as follows: $f_{\tilde{A}} (x) = {\begin{matrix} L_{\tilde{A}} (x), & a < x < b, \\ 1, & b < x < c, \\ T_{\tilde{A}} (x), & c < x < d, \\ 0, & otherwise . \end{matrix}$ (1)

Where, $L_{\tilde{A}} (x)$ is a strictly increasing function of right continuity, $T_{\tilde{A}} (x)$ is a strictly decreasing function of left continuous. $\tilde{A}$ is a nonnegative fuzzy number if a ⩾ 0. For ease of description, fuzzy number is abbreviated as $\tilde{A} = [a, b, c, d; L_{A} (x), T_{A} (x)]$ . $μ_{{\tilde{A}}_{1}} (x) = {\begin{matrix} (x - a) / (b - a), & a < x < b, \\ 1, & b < x < c, \\ (d - x) / (d - c), & c < x < d, \\ 0, & otherwise . \end{matrix}$ (2)

Definition 3 $\tilde{A}$ is trapezoidal fuzzy number when $L_{\tilde{A}} (x) = (x - a) / (b - a), T_{\tilde{A}} (x) = (d - x) / (d - c)$ , denoted by $\tilde{A} = (a, b, c, d)$ . In particular, $\tilde{B}$ is triangular fuzzy number if b = c, denoted by $\tilde{B} = (a, b, d)$ . The trapezoidal and triangular fuzzy numbers defined by the following membership functions, respectively: $μ_{{\tilde{A}}_{2}} (x) = {\begin{matrix} (x - a) / (b - a), & a ⩽ x ⩽ b, \\ (d - x) / (d - b), & b ⩽ x ⩽ d, \\ 0, & otherwise . \end{matrix}$ (3)

Definition 4 [35] For any nonnegative trapezoidal fuzzy number $\tilde{A} = (a_{L}, a_{M}, a_{N}, a_{U})$ and $\tilde{B} = (b_{L}, b_{M}, b_{N}, b_{U})$ , the arithmetic operations as follows:

$\tilde{A} + \tilde{B} = (a_{L} + b_{L}, a_{M} + b_{M}, a_{N} + b_{N}, a_{U} + b_{U});$

$\tilde{A} - \tilde{B} = (a_{L} - b_{U}, a_{M} - b_{N}, a_{N} - b_{M}, a_{U} - b_{L});$

let λ ⩾ 0, then $λ \tilde{A} = (λ a_{L}, λ a_{M}, λ a_{N}, λ a_{U});$

$\tilde{A} \times \tilde{B} \approx (a_{L} b_{L}, a_{M} b_{M}, a_{N} b_{N}, a_{U} b_{U});$

$\tilde{A} / \tilde{B} \approx (a_{L} / b_{U}, a_{M} / b_{N}, a_{N} / b_{M}, a_{U} / b_{L}) .$

Definition 5 [36] Let ξ be a fuzzy variable with membership function $μ : R \to [0, 1]$ and r be a real number. The possibility and the necessity of {ξ⩾ r } are respectively defined by $Pos {ξ ⩾ r} = sup_{x ⩾ r} μ (x),$ (4) $Nec {ξ ⩾ r} = 1 - Pos {ξ ⩾ r} = 1 - sup_{x < r} μ (x),$ (5)

Based on (4) and (5), the credibility measure is defined as $Cr (ξ ⩾ r) = \frac{1}{2} (Pos {ξ ⩾ r} + Nec {ξ ⩾ r}),$ (6)

The fuzzy expected value of ξ can be defined as $E [ξ] = \int_{0}^{\infty} Cr (ξ ⩾ r) dr - \int_{- \infty}^{0} Cr (ξ ⩽ r) dr,$ (7)

In particular, if ξ is a trapezoidal fuzzy variable (r₁, r₂, r₃, r₄), then the expected value ξ is (1/4)(r₁ + r₂ + r₃ + r₄), if ξ is a triangular fuzzy variable (r₁, r₂, r₃), then the expected value ξ is (1/4)(r₁ + 2r₂ + r₃).

2.2 Fuzzy DEA CCR model

Suppose there are n DMUs to be evaluated in terms of m inputs to produce s outputs. Let x_ij(i = 1, . . . , m) and y_rj(r = 1, . . . , s) be the input and output data of DMU_j(j = 1, . . . , n). Without losing generality, all input and output are assumed to be uncertain and characterized by trapezoidal fuzzy numbers ${\tilde{x}}_{ij} = (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{N}, x_{ij}^{U})$ and ${\tilde{y}}_{rj} = (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{N}, y_{rj}^{U})$ with $x_{ij}^{L} ⩾ 0$ and $y_{rj}^{L} ⩾ 0$ for i = 1, . . . m, r = 1, . . . , s, j = 1, . . . , n. Suppose the weights of input and output are crisp, the total fuzzy weighted output (FWO) and the total fuzzy weighted input (FWI) of DMU_j are given by [24] $\begin{matrix} {FEO}_{j} = \sum_{r = 1}^{s} u_{r} {\tilde{y}}_{rj} = \sum_{r = 1}^{s} u_{r} \times (y_{rj}^{L}, y_{rj}^{M}, y_{rj}^{N}, y_{rj}^{U}) \\ \approx (\sum_{r = 1}^{s} u_{r} y_{rj}^{L}, \sum_{r = 1}^{s} u_{r} y_{rj}^{M}, \sum_{r = 1}^{s} u_{r} y_{rj}^{N}, \sum_{r = 1}^{s} u_{r} y_{rj}^{U}), \end{matrix}$ (8) $\begin{matrix} {FWI}_{j} = \sum_{i = 1}^{m} v_{i} {\tilde{x}}_{ij} = \sum_{i = 1}^{m} v_{i} (x_{ij}^{L}, x_{ij}^{M}, x_{ij}^{M}, x_{ij}^{U}) \\ \approx (\sum_{i = 1}^{m} v_{i} x_{ij}^{L}, \sum_{i = 1}^{m} v_{i} x_{ij}^{M}, \sum_{i = 1}^{m} v_{i} x_{ij}^{N}, \sum_{i = 1}^{m} v_{i} x_{ij}^{U}), \end{matrix}$ (9) which can be considered as two trapezoidal fuzzy variables, whose fuzzy weighted expected values can therefore be determined as $\begin{matrix} E ({FWO}_{j}) = \frac{1}{4} \sum_{r = 1}^{s} u_{r} (y_{rj}^{L} + y_{rj}^{M} + y_{rj}^{N} + y_{rj}^{U}) \\ = \sum_{r = 1}^{s} u_{r} \times \frac{1}{4} (y_{rj}^{L} + y_{rj}^{M} + y_{rj}^{N} + y_{rj}^{U}) = \sum_{r = 1}^{s} u_{r} E ({\tilde{y}}_{rj}) \end{matrix}$ (10) $\begin{matrix} E ({FWI}_{j}) = \frac{1}{4} \sum_{i = 1}^{m} v_{i} (x_{ij}^{L} + x_{ij}^{M} + x_{ij}^{N} + x_{ij}^{U}) \\ = \sum_{i = 1}^{m} v_{i} \times \frac{1}{4} (x_{ij}^{L} + x_{ij}^{M} + x_{ij}^{N} + x_{ij}^{U}) = \sum_{i = 1}^{m} v_{i} E ({\tilde{x}}_{ij}) . \end{matrix}$ (11)

In particular, if ${\tilde{x}}_{ij}, {\tilde{y}}_{rj}$ are triangular fuzzy numbers and the FWO_j, FWI_j of them are triangular fuzzy variables, whose fuzzy weighted expected values can therefore be determined as $\begin{matrix} E ({FWO}_{j}) = \frac{1}{4} \sum_{r = 1}^{s} u_{r} (y_{rj}^{L} + 2 y_{rj}^{M} + y_{rj}^{U}) \\ = \sum_{r = 1}^{s} u_{r} \times \frac{1}{4} (y_{rj}^{L} + 2 y_{rj}^{M} + y_{rj}^{U}) = \sum_{r = 1}^{s} u_{r} E ({\tilde{y}}_{rj}) \end{matrix}$ (12) $\begin{matrix} E ({FWI}_{j}) = \frac{1}{4} \sum_{i = 1}^{m} v_{i} (x_{ij}^{L} + 2 x_{ij}^{M} + x_{ij}^{U}) \\ = \sum_{i = 1}^{m} v_{i} \times \frac{1}{4} (x_{ij}^{L} + 2 x_{ij}^{M} + x_{ij}^{U}) = \sum_{i = 1}^{m} v_{i} E ({\tilde{x}}_{ij}) . \end{matrix}$ (13)

Consider a target DMU_k(k = 1, . . . , n), the self-efficiency of DMU_k based on fuzzy expected value of fuzzy output and input in fuzzy environment can be computed by CCR model which was formulated by Charnes et al. [1] as follows $\begin{matrix} max θ_{kk} = \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) / \sum_{i = 1}^{m} v_{i} E ({\tilde{x}}_{ik}) \\ s . t . \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) / \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) ⩽ 1, j = 1, . . ., n, \\ u_{rk}, v_{ik} ⩾ 0, r = 1, . . ., s, i = 1, . . ., m . \end{matrix}$ (14)

Model (14) can be converted into an equivalent linear programming formulation by the research of Charnes & Cooper [37], the following is the converted solution $\begin{matrix} max θ_{kk} = \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) \\ s . t . \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 1, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) ⩽ 0, j = 1, . ., n, \\ u_{rk}, v_{ik} ⩾ 0, r = 1, . . ., s, i = 1, . . ., m . \end{matrix}$ (15)

Solving Model (15) n times, the self-efficiency value of each DMU is obtained, the value of self-efficiency between zeros and one, the DMU can be view as DEA efficient with the value of one, otherwise, the DMU is DEA inefficient. but there are two obvious disadvantages of CCR model, (i) the self-efficiency of each DMU is obtained which select the best weight to calculate it from the perspective most conducive to oneself, it is also called relative efficiency or optimistic efficiency; (ii) the performance of DEA efficient DMUs calculated by CCR model cannot be further distinguished, that is, DMUs cannot be completely sorted.

2.3 Fuzzy DEA Cross-efficiency models

Suppose there are n DMUs to be evaluated with m inputs and soutputs in the fuzzy environment. Let ${\tilde{x}}_{ij} (i = 1, . . ., m)$ represent the fuzzy input, and ${\tilde{y}}_{rj} (r = 1, . . ., s)$ represent the fuzzy output for DMU_j(j = 1, . . . , n), for any DMU_k(k = 1, . . . , n) under evaluation, the best relative efficiency of DMU_k under self-efficiency is $θ_{kk}^{*} = \sum_{r = 1}^{s} u_{rk}^{*} E ({\tilde{y}}_{rk})$ , where $u_{rk}^{*} (r = 1, . . ., s)$ and $v_{ik}^{*} (i = 1, . . ., m)$ are the optimal solutions of Model (15), for any DMU_j, the k-th cross-efficiency is $θ_{kj} = \sum_{r = 1}^{s} u_{rk}^{*} E ({\tilde{y}}_{rj}) / \sum_{i = 1}^{m} v_{ik}^{*} E ({\tilde{x}}_{ij})$ which represents the peer-efficiency of DMU_k to DMU_j, the average cross-efficiency of DMU_j can be donated by ${\bar{θ}}_{j} = \frac{1}{n} \sum_{j = 1}^{n} θ_{kj}$ .

The traditional cross-efficiency model integrates self-evaluation and peer-evaluation efficiency to comprehensively evaluate DMU, which reduces the blind optimism to a certain extent, but it is still unable to overcome the problem that the optimal weight of CCR model is not unique, which makes the cross-efficiency inconsistent. In order to avoid the above defects, many studies have introduced the secondary goal functions, which can find the only satisfactory solution while keeping the optimal self-evaluation efficiency unchanged, the well-known are benevolent, aggressive [38] and neutral [11] cross-efficiency models. The secondary goals cross-efficiency models can be shown below based on fuzzy expected values of input and output in the fuzzy environment. $\begin{matrix} max \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1, j \neq k}^{n} E ({\tilde{y}}_{rj})) \\ s . t . \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1, j \neq k}^{n} E ({\tilde{x}}_{ij})) = 1, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 0, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) ⩽ 0, j = 1, . . ., n; j \neq k, \\ u_{rk} ⩾ 0, v_{ik} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (16) and, $\begin{matrix} max δ \\ s . t . \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1, j \neq k}^{n} E ({\tilde{x}}_{ij})) = 1, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) = θ_{kk}^{*}, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) ⩽ 0, j = 1, . . ., n; j \neq k, \\ u_{rk} E ({\tilde{y}}_{rk}) - δ ⩾ 0, r = 1, . . ., s, \\ u_{rk} ⩾ 0, v_{ik} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m, \\ δ ⩾ 0 . \end{matrix}$ (17)

Model (16) is the fuzzy benevolent cross-efficiency model which strives to maximize the cross-efficiency values of other DMUs under the condition of unchanging the optimal self-efficiency of the under evaluation DMU_k. In the contrary, the aggressive cross-efficiency model can be obtained by changing the objective function of Model (16) from Maximize to Minimize, the view of this model is that there is a competitive relationship between each DMU. Model (17) is the neutral cross-efficiency model proposed by Wang & Chin [11] who hold the view that the DMU should not care too much about how to aggressive or benevolent to other DMUs, it just focuses on whether the weight is optimal for itself.

2.4 Fuzzy ICE model

Though secondary goals formulations can further distinguish effective DMUs to a certain extent, but sometimes impossible to completely sort DMUs. To eliminate inconsistent rankings, motivated by Liang et al. [10], Yang et al. [27] created a novel ICE model on the basic of game cross-efficiency DEA model. Based on above, we combined with the fuzzy theory, expand the ICE model, introduce the expected value of fuzzy output and fuzzy input, and the model can be determined as, $\begin{matrix} θ_{kj}^{L} = min u_{rk} E ({\tilde{y}}_{rj}) or \\ θ_{kj}^{U} = max u_{rk} E ({\tilde{y}}_{rj}) \\ s . t . \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) = 1, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) ⩽ 0, j = 1, . . ., n; j \neq k, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) ⩾ 0, \\ u_{rk} ⩾ 0, v_{ik} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (18)

In Model (18), DMU_j seeks the maximize or minimize cross-efficiency under unchanging the optimal self-efficiency of DMU_k evaluated. The optimal solutions $θ_{kj}^{L *}$ or $θ_{kj}^{U *}$ constitute the lower and upper bounds of ${DMU}_{j}^{'} s$ cross-efficiency which determined by aggressive and benevolent strategies. That is, the cross-efficiency evaluation value of DMU_k to DMU_j is kept within the range of $[θ_{kj}^{L *}, θ_{kj}^{U *}]$ . Particularly, the interval become a real number if there is $θ_{kj}^{L *} = θ_{kj}^{U *}$ . All DMUs′ cross-efficiency scores can be obtained by solving the model n times which generates the interval cross-efficiency matrix (ICEM), as shown in Table 1. Specially, diagonal elements are real numbers, there is $θ_{jj}^{L *} = θ_{jj}^{U *} = θ_{jj}^{*}$ for j = 1, . . . n .

Table 1

The exhibition of ICEM

DMU_j\ DMU_k	1	2	. . .	n
1	$[θ_{11}^{L }, θ_{11}^{U }]$	$[θ_{12}^{L }, θ_{12}^{U }]$	. . .	$[θ_{1 n}^{L }, θ_{1 n}^{U }]$
2	$[θ_{21}^{L }, θ_{21}^{U }]$	$[θ_{22}^{L }, θ_{22}^{U }]$	. . .	$[θ_{2 n}^{L }, θ_{2 n}^{U }]$
. . .	. . .	. . .	. . .	. . .
n	$[θ_{n 1}^{L }, θ_{n 1}^{U }]$	$[θ_{n 2}^{L }, θ_{n 2}^{U }]$	. . .	$[θ_{nn}^{L }, θ_{nn}^{U }]$
The optimal	$[θ_{1}^{L }, θ_{1}^{U }]$	$[θ_{2}^{L }, θ_{2}^{U }]$	. . .	$[θ_{n}^{L }, θ_{n}^{U }]$
rally points

3 Proposed the generation and aggregation methods of interval cross-efficiency

In this section, we first analyze the shortcomings of the existing model, and then explain our motivation to propose the model. As can be seen from the model reviewed above, in the fuzzy environment, most of the upper and lower bounds of ICE are determined by benevolent and aggressive strategies, that is, the efficiency value determined under the aggressive strategy is taken as the upper bound of the interval, and the value calculated under the benevolent strategy is taken as the upper bound of the interval, this approach will make the range too extreme. What’s more, the aggregation of ICE mostly adopts a simple average operator, that is, the subjective weight of decision-makers is forced to be used, which leads to the disappearance of the correlation between cross-efficiency and optimal weight. To solve the above problems, we propose a new method to generate ICE from the neutral strategy in fuzzy environment, which integrate aggressive and benevolent strategies to eliminate extremes, then aggregate the ICE like the shown in Table 1 based on the rally points and obtain the full ranking of DMUs finally.

3.1 Proposed the model for generating ICE with fuzzy data

The ICE model shown in Model (18) adopts extreme aggressive and benevolent strategies, next we will develop the more realistic models from the relatively neutral perspective to generated the cross-efficiency of DMUs under fuzzy environment.

Suppose we have n DMUs to be evaluated according to m inputs to produce s outputs, let ${\tilde{x}}_{ij} (i = 1, . ., m)$ and ${\tilde{y}}_{rj} (r = 1, . . ., s)$ be the input and output data of DMU_j(j = 1, . . . , n) which assumed as fuzzy numbers. We use Model (15) to calculate the optimal self-efficiency $θ_{kk}^{*}$ of fuzzy input and output data for any DMU_k(k = 1, . . . , n), let $(θ_{11,}^{*} θ_{22,}^{*} . . ., θ_{nn}^{*})$ be the ideal point for n DMUs, we combine the aggressive and benevolent perspectives, we firstly maximize the deviation from the ideal point from aggressive strategy, then minimize the maximum deviation from benevolent strategy, the comprehensive strategy makes the evaluation model avoid extremes and more reasonable. We can also maximize the minimum deviation from ideal points. The models are expressed as follows: $\begin{matrix} \min max d_{jk} \\ s . t . \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1}^{n} E ({\tilde{y}}_{rj})) + \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1}^{n} E ({\tilde{x}}_{ij})) = n, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - θ_{jj}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) + d_{jk} = 0, j = 1, . . ., n; j \neq k, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 0, \\ u_{rk}, v_{ik} d_{jk} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (19) and, $\begin{matrix} max min d_{jk} \\ s . t . \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1}^{n} E ({\tilde{y}}_{rj})) + \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1}^{n} E ({\tilde{x}}_{ij})) = n, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - θ_{jj}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) + d_{jk} = 0, j = 1, . . ., n; j \neq k, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 0, \\ u_{rk}, v_{ik} d_{jk} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (20)

Unlike Model (18), the first constraint condition above models is fixed and unified, so that the weights of different target DMUs meet the same constraint and have strong comparability. For ease of calculation, let α_k = max d_jk and β_k = min d_jk, above models are transformed into linear programming models below: $\begin{matrix} \min α_{k} \\ s . t . \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1}^{n} E ({\tilde{y}}_{rj})) + \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1}^{n} E ({\tilde{x}}_{ij})) = n, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - θ_{jj}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) + d_{jk} = 0, j = 1, . . ., n; j \neq k, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 0, \\ α_{k} - d_{jk} ⩾ 0, j = 1, . . ., n; j \neq k, \\ u_{rk}, v_{ik} d_{jk} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (21) and, $\begin{matrix} max β_{k} \\ s . t . \sum_{r = 1}^{s} u_{rk} (\sum_{j = 1}^{n} E ({\tilde{y}}_{rj})) + \sum_{i = 1}^{m} v_{ik} (\sum_{j = 1}^{n} E ({\tilde{x}}_{ij})) = n, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rj}) - θ_{jj}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ij}) + d_{jk} = 0, j = 1, . . ., n; j \neq k, \\ \sum_{r = 1}^{s} u_{rk} E ({\tilde{y}}_{rk}) - θ_{kk}^{*} \sum_{i = 1}^{m} v_{ik} E ({\tilde{x}}_{ik}) = 0, \\ d_{jk} - β_{k} ⩾ 0, j = 1, . . ., n; j \neq k, \\ u_{rk}, v_{ik} d_{jk} ⩾ 0, r = 1, . . ., s; i = 1, . . ., m . \end{matrix}$ (22)

The optimal solutions of Model (21) are $u_{rk (min max)}^{*}, v_{ik (min max)}^{*}$ , the cross-efficiency of DMU_j be determined by $θ_{kj (min max)}^{*} = \sum_{r = 1}^{s} u_{rk (min max)}^{*} E ({\tilde{y}}_{rj}) / \sum_{i = 1}^{m} v_{ik (min max)}^{*} E ({\tilde{x}}_{ij})$ . Similarly, the cross-efficiency of DMU_j calculated by Model (22) is $θ_{kj (max min)}^{*} = \sum_{r = 1}^{s} u_{rk (max min)}^{*} E ({\tilde{y}}_{rj}) / \sum_{i = 1}^{m} v_{ik (max min)}^{*} E ({\tilde{x}}_{ij})$ . Let $θ_{kj}^{L} = min (θ_{kj (max min)}^{*}, θ_{kj (max min)}^{*})$ , $θ_{kj}^{U} = max (θ_{kj (max min)}^{*}, θ_{kj (max min)}^{*})$ , then $[θ_{kj}^{L *}, θ_{kj}^{U *}]$ are the lower and upper bounds of ICE for DMU_j, all DMUs′ ICE calculated by the proposed model and build the ICEM shown in Table 1.

3.2 Proposed the ICE aggregation method for all DMUs

The cross-efficiency of each DMU is interval in ICEM, we cannot use the simple average operator to aggregate the ICEM, Qiu & Li [39] proposed the definition of the optimal rally point as below,

Definition 6 Suppose there are n(n > 3) points on the plane, if there is a point P^*, the sum of Euclidean distances from other points on the same plane is the smallest, then P^* is the optimal rally point, the smallest total Euclidean distance can be determined as $\begin{matrix} D = min \sum_{i = 1}^{n} | P^{*} P_{i} | = \\ min (\sqrt{(x^{*} - a_{1})^{2} + (y^{*} - b_{1})^{2}} + . . . + \\ \sqrt{(x^{*} - a_{n})^{2} + (y^{*} - b_{n})^{2}}) . \end{matrix}$ (23)

Motivated by this idea, we project the ICE of each DMU as points to the two-dimensional plane, that is, the lower bounds $θ_{kj}^{L *}$ are regarded as abscissa and the upper bounds $θ_{kj}^{U *}$ as ordinate. There are n ICEs are projected as n points, the optimal rally point achieves Pareto optimality seen in Fig. 1, indicating that it can synthesize the overall opinions of each evaluated DMU, aggregate the self-evaluation and peer-evaluation efficiency, and obtain a comprehensive ICE for each DMU.

Fig. 1

The optimal rally point of DMU_j.

Finding the optimal rally points is a complex nonlinear problem when there are multiple random points on the plane, and as the number of random points increases the amount of calculation will increase exponentially, so it is difficult to obtain the result with general solution methods. We employ the Genetic Algorithm Toolbox which be good at solving nonlinear optimization to address this problem is effective.

3.3 The ranking of comprehensive ICE

To acquire the ultimate ICE ranking solutions, we must rank the comprehensive ICE shown in the end of the row in Table 1. We use the optimal number sorting method proposed by Gao [40] for the comprehensive sorting of interval numbers, which has the advantage of order preservation over other methods, as shown below,

Definition 7 Let a = [a^-, a⁺] = {x|0 ⩽ a^- ⩽ a⁺}. Here, a is name as a nonnegative interval. In particular, if a^- = a⁺,then a is a non-negative real number.

Definition 8 Let a = [a^-, a⁺] and b = [b^-, b⁺], and let l_a = a⁺ - a^- and l_b = b⁺ - b^-,then the degree of the possibility of a ⩾ b can be represented as, t $P (a ⩾ b) = {\begin{matrix} \begin{matrix} 0, \\ \frac{a^{+} - b^{-}}{(a^{+} - a^{-}) + (b^{+} - b^{-})}, \\ 1, \end{matrix} & \begin{matrix} b^{-} > a^{+} \\ b^{-} ⩽ a^{+}, a^{-} ⩽ b^{+} \\ b^{+} < a^{-} \end{matrix} \end{matrix}$ (24)

The possibility Formulas (24) is uniform possibility distribution functions, and it has the following properties:

0 ⩽ p(a ⩾ b) ⩽ 1, 0 ⩽ p(b ⩾ a) ⩽ 1 .

p(a ⩾ b) + p(b ⩾ a) = 1 . In particular, p(a ⩾ a) = p(b ⩾ b) = 0.5 .

Definition 9 In the interval array $a_{i} = [a_{i}^{-}, a_{i}^{+}] (i = 1, . . ., n)$ , let their possibility matrix for comparing with each other be P =(p_ij) _n×n, call the number of roots greater than 0.5 in the elements p_ij(j = 1, . . . , n) of the i^th row of the matrix P the number of the superiority of the interval a_i, denoted as τ(a_i).

Theorem 1 In the interval array $a_{i} = [a_{i}^{-}, a_{i}^{+}] (i = 1, . . ., n)$ , let’s the number of the superiority of a_i, then a sufficient necessary condition for τ(a_i) > τ(a_k) is that the interval number $a_{i} = [a_{i}^{-}, a_{i}^{+}]$ is to be greater than $a_{k} = [a_{k}^{-}, a_{k}^{+}]$ , i.e. a_i ≻ a_k.

Based on the above definitions and theorems, the procedure of the optimal number sorting method is as follows:

Step 1 Calculate the possibility matrix P =(p_ij) _n×n, where P_ij = P { a_i ⩾ a_j } , i, j = 1, . . . , n;

Step 2 Calculate the numbers of superiority. $τ (a_{i}) = \sum_{j = 1}^{n} sgn {P_{ij} > 0.5}$ where sgn { P_ij > 0.5 } = 1, sgn { P_ij ⩽ 0.5 } = 0;

Step 3 Sort according to the size of the superiority numbers. The sorting rule is, if τ(a_i) > τ(a_j),then a_i ≻ a_j, i = 1, . . . , n, i ≠ j.

3.4 Procedures for the proposed integrated ranking approach

Based on the presentation of the above knowledge, the specific implementation process of the ICE aggregation method with fuzzy data based on optimal rally points under uncertain environment show in Fig. 2, the proposed method uses standard algorithm format with pseudo-codes shown in Algorithm I.

Fig. 2

The specific implementation process of proposed method.

Algorithm I

For each DMU_k in {DMU₁, DMU₂, . . . , DMU_n}

For fuzzy outputs ${\tilde{y}}_{rj} (r = 1, . . ., s)$ and fuzzy inputs ${\tilde{x}}_{ij} (i = 1, . . ., m)$

If ${\tilde{x}}_{ij}, {\tilde{y}}_{rj}$ are trapezoidal fuzzy numbers

Calculate the fuzzy weighted expected values shown in Model (10) and (11)

Elif ${\tilde{x}}_{ij}, {\tilde{y}}_{rj}$ are triangular fuzzy numbers

Calculate the fuzzy weighted expected values shown in Model (12) and (13)

Endif

Calculate the lower bounds and upper bounds of ICE shown by Model (21) and (22).

Aggregate of ICEM shown in Formula (23)

For each optimal rally point obtained from Formula (23)

Calculate the possibility matrix shown in Formula (24)

Calculate the numbers of superiority

Endfor

Obtain the full ranking of DMUs

Endfor

4 A case study: the application to the evaluation of manufacturing enterprises

We apply the proposed model to the evaluation of ten manufacturing enterprises, the original data comes from the research of Zhou et al. [41]. There are ten manufacturing enterprises with three inputs and two output and they have the same products. Inputs include manufacturing cost (MC), number of employees (NOE) and floor space (FS). Outputs include gross output value (GOV) and product quality (PQ). The data of GOV and MC are regarded as trapezoidal fuzzy numbers due to the difficulty of collecting crisp data. PQ is reflected by customers’ fuzzy linguistic terms, such as “Very Good”, “Good”, “Average”, “Poor” and “Very Poor”, which can be regarded as triangular fuzzy numbers. All the input and output data for the ten manufacturing enterprises are listed in Table 2. Based on the input and output data in Table 2, the lower bounds and upper bounds of the ICE can be determined by solving Model (21) and (22), and all DMUs′ ICE construct the generalized interval CEM like Table 3. Project all ICE into two-dimensional plane coordinates, and the use Genetic Algorithm Toolbox to find the optimal rally points of all DMUs based on Formula (23), the solutions are shown in the end of Table 3, these points are considered as comprehensive ICE of DMUs.

Table 2
Input and output data for ten manufacturing enterprises

DMU Input Output

MC(¥ 1000,000) NOE FS(1000ft²) GOV(¥ 1000,000) PQ

A (21,21.3,21.7,22.1) 1780 17.3 (147.5,147.9,148,148.7) (3,4,5)

B (14.1,14.5,14.6,15) 1430 16.4 (125.8,126.2,127.2,128.1) (1,2,3)

C (25,25.5,25.7,26.1) 2630 11.2 (179,180,182.6,184.5) (3,4,5)

D (22,22.5,23.5,24) 2000 10.5 (149.7,152.7,154,155) (3,4,5)

E (14.8,15,15.2,15.6) 1570 9.5 (138.9,142.6,143.3,145.4) (1,2,3)

F (19.6,20,20.3,21) 1670 4.8 (140.5,143.1,144.6,145.7) (2,3,4)

G (22,22.4,22.6,23.2) 1890 6.2 (164.5,168.7,170.8,175.4) (2,3,4)

H (24,24.6,25.2,25.5) 2350 11.1 (176.7,179.6,181.2,185.3) (3,4,5)

I (15.8,16.3,16.8,17.6) 1750 9.8 (139.8,146.2,148.3,150) (1,2,3)

J (14.9,15.3,15.8,16) 1690 8.5 (140,142.8,143.5,144.5) (2,3,4)

DMU	Input	Output
A	(21,21.3,21.7,22.1)	1780	17.3	(147.5,147.9,148,148.7)	(3,4,5)
B	(14.1,14.5,14.6,15)	1430	16.4	(125.8,126.2,127.2,128.1)	(1,2,3)
C	(25,25.5,25.7,26.1)	2630	11.2	(179,180,182.6,184.5)	(3,4,5)
D	(22,22.5,23.5,24)	2000	10.5	(149.7,152.7,154,155)	(3,4,5)
E	(14.8,15,15.2,15.6)	1570	9.5	(138.9,142.6,143.3,145.4)	(1,2,3)
F	(19.6,20,20.3,21)	1670	4.8	(140.5,143.1,144.6,145.7)	(2,3,4)
G	(22,22.4,22.6,23.2)	1890	6.2	(164.5,168.7,170.8,175.4)	(2,3,4)
H	(24,24.6,25.2,25.5)	2350	11.1	(176.7,179.6,181.2,185.3)	(3,4,5)
I	(15.8,16.3,16.8,17.6)	1750	9.8	(139.8,146.2,148.3,150)	(1,2,3)
J	(14.9,15.3,15.8,16)	1690	8.5	(140,142.8,143.5,144.5)	(2,3,4)

Table 3

The generalized ICEM and the optimal rally points

DMU_k\ DMU_j		A	B	C	D	E	F	G	H	I	J
A	L	1.000	0.622	0.677	0.890	0.567	0.799	0.706	0.757	0.509	0.790
	U	1.000	0.904	0.830	0.949	0.946	1.000	0.992	0.914	0.870	1.000
B	L	1.000	0.987	0.815	0.918	1.000	0.982	1.000	0.901	0.918	0.994
	U	1.000	0.987	0.815	0.918	1.000	0.982	1.000	0.901	0.918	0.994
C	L	0.806	0.495	0.881	0.965	0.644	1.000	0.865	0.901	0.603	1.000
	U	0.806	0.495	0.881	0.965	0.644	1.000	0.865	0.901	0.603	1.000
D	L	0.967	0.600	0.828	1.000	0.653	0.956	0.837	0.889	0.598	0.956
	U	1.000	0.739	0.850	1.000	0.802	1.000	0.929	0.922	0.736	1.000
E	L	0.889	0.958	0.763	0.830	1.000	0.935	0.978	0.843	0.918	0.936
	U	0.999	0.982	0.820	0.921	1.000	0.983	1.000	0.905	0.922	1.000
F	L	0.370	0.195	0.571	0.610	0.337	1.000	0.774	0.577	0.327	0.565
	U	0.995	0.900	0.830	0.947	0.946	1.000	0.993	0.913	0.870	1.000
G	L	0.830	0.861	0.755	0.822	0.963	0.963	1.000	0.834	0.885	0.911
	U	0.972	0.902	0.829	0.934	0.961	1.000	1.000	0.910	0.892	1.000
H	L	1.000	0.739	0.850	1.000	0.802	1.000	0.929	0.922	0.736	1.000
	U	1.000	0.740	0.850	1.000	0.802	1.000	0.930	0.922	0.737	1.000
I	L	0.711	0.769	0.824	0.796	1.000	0.965	1.000	0.859	0.946	1.000
	U	0.712	0.769	0.824	0.797	1.000	0.965	1.000	0.859	0.946	1.000
J	L	0.857	0.824	0.790	0.813	0.846	0.768	0.751	0.812	0.782	1.000
	U	0.995	0.900	0.830	0.947	0.946	1.000	0.993	0.913	0.870	1.000
The optimal rally points	L	0.890	0.750	0.790	0.879	0.847	0.963	0.906	0.843	0.782	0.956
	U	0.984	0.821	0.830	0.941	0.938	0.995	0.962	0.905	0.866	1.000

Next, the optimal number sorting method proposed by Gao [39] is used to obtain the ultimate ranking of DMUs. Firstly, the possibility matrix P =(P_ij) _n×n can be established based on Formulas (24) expressed in Table 4, and the number of superiority of DMU_j show in the rightmost column. From Table 4, we have τ(A) =7, τ(B) =0, τ(C) =1, τ(D) =5, τ(E) =4, τ(F) =9, τ(G) =6, τ(H) =3, τ(I) =2, τ(J) =8. The ultimate ranking of ten manufacturing enterprises are determined as F ≻ J ≻ A ≻ G ≻ D ≻ E ≻ H ≻ I ≻ C ≻ B

Table 4

The possibility matrix and number of superiority of DMU_j

DMU _j	A	B	C	D	E	F	G	E	F	G	τ(DMU_j)
A	0.500	1.000	1.000	0.672	0.739	0.166	0.519	0.902	1.000	0.203	7
B	0.000	0.500	0.276	0.000	0.000	0.000	0.000	0.000	0.248	0.000	0
C	0.000	0.724	0.500	0.000	0.000	0.000	0.000	0.000	0.386	0.000	1
D	0.328	1.000	1.000	0.500	0.615	0.000	0.294	0.792	1.000	0.000	5
E	0.261	1.000	1.000	0.385	0.500	0.000	0.216	0.620	0.888	0.000	4
F	0.834	1.000	1.000	1.000	1.000	0.500	1.000	1.000	1.000	0.515	9
G	0.481	1.000	1.000	0.706	0.784	0.000	0.500	1.000	1.000	0.057	6
H	0.098	1.000	1.000	0.208	0.380	0.000	0.000	0.500	0.841	0.000	3
I	0.000	0.752	0.614	0.000	0.112	0.000	0.000	0.159	0.500	0.000	2
J	0.797	1.000	1.000	1.000	1.000	0.485	0.943	1.000	1.000	0.500	8

5 Comparison analysis

In order to fully illustrate the effectiveness of the proposed model, we analyze it from three aspects.

5.1 Comparison of different cross-efficiency strategies

In order to study the difference of DMUs’ ranking between different cross-efficiency strategies, we compare the proposed with aggressive and benevolent cross-efficiency [10]. The ranking solution of aggressive cross-efficiency is F ≻ J ≻ G ≻ D ≻ H ≻ C ≻ A ≻ E ≻ I ≻ B. The ranking solution of benevolent cross-efficiency is A = E = G = J≻ B ≻ F ≻ D = H ≻ I ≻ C. The rankings of ten manufacturing enterprises with different cross-efficiency strategies are shown in Fig. 3. Obviously, the benevolent cross-efficiency model has poor ability to distinguish DMUs, it holds that A, E, G and J have the same performance and better than others, the situation does not seem to be convincing, nor does it contribute to the complete ranking of ten manufacturing enterprises, indicating that the benevolent cross-efficiency model the stability of differentiated DMUs needs to be improved in fuzzy environment. The proposed method distinguishes ten manufacturing enterprises with the order of F ≻ J ≻ A ≻ G ≻ D ≻ E ≻ H ≻ I ≻ C ≻ B, comparing with the ranking results of the aggressive cross-efficiency model, we can see that there are two manufacturing enterprises F and J have the same ranking under two different strategies, both in top two, and the last manufacturing enterprise also remains the same. However, the ranking of other manufacturing enterprises is slightly different, because the proposed method is to evaluate the performance of the DMU from a relatively neutral perspective, partially weakens the aggressive and benevolent nature and makes the results of ranking more convincing.

Fig. 3

The rankings of DMUs with different cross-efficiency strategies.

5.2 Comparison of interval cross-efficiency aggregation methods

For aggregating the ICE determined by Model (21) and (22), we project all ICEs onto the plane as points, then seek the optimal rally point for each DMU based on ICEM as the comprehensive ICE. In order to highlight the advantages of this aggregation method, we compare this method with other ICE aggregation method, including the average operator, the ICE aggregation method based on Shannon Entropy [31] and the Hurwicz criterion aggregation method [30].

Firstly, we demonstrate that the optimal rally points aggregation method is superior to the average operator aggregation ICE. The average operator uses a simple aggregation rule, which averages the upper and lower bounds of ICE respectively, so as to obtain the comprehensive ICE of each DMU. In order to intuitively show that the optimal rally points can better reflect the comprehensive ICE, we compare the shortest Euclidean distance based on the rally points and the average distance based on the average operator, as can be seen from Fig. 4 the shortest Euclidean distance is always less than the average distance, illustrates that using the optimal rally points as the comprehensive ICE can better synthesize the interval efficiency of self-evaluation and peer-evaluation, which is more scientific.

Fig. 4

The comparison between shortest distance and average distance.

Secondly, we analyze the difference and stability of different ICE aggregation methods. We use the aggregation method based on Shannon Entropy proposed by Wang et al. [31] to aggregate the generalized ICEs in Table 3, the ranking of ten manufacturing enterprises is shown in the third column of Table 5. Similarly, the Hurwicz criterion aggregation method proposed by Wang and Yang [30] existed the difference in the ranking of DMUs when holding different risk attitudes. The parameter α represent the Decision Maker’s attitude towards risk, when α = 0, 0.5, 1, we rank ten manufacturing enterprises respectively, the ranking solutions are shown in the last three columns of Table 5. We show the ranking results in Fig. 5 so as to more intuitively show the differences between different aggregation methods, we can see that the aggregation method proposed by Wang er al. [31] has a big difference with other aggregation methods. For example, we think F and B are abnormal, because the other aggregation method all think F have the best evaluation performance, and B has the worst evaluation performance, but Wang et al.’s aggregation method is not quite reasonable to has the opposite result. Through the in-depth study of this method, we believe that the reason why the aggregation effect is not ideal is that too much original interval cross-efficiency information is lost in the aggregation process, resulting in the distortion of information. What’s more, the aggregation method proposed by Wang and Yang [30] is not absolutely stable, the ranking of DMUs will slightly change when Decision Maker hold the different risk attitudes, extremely, the method cannot distinguish the performance of all DMUs when α = 1 and it becomes invalid.

Table 5

The rankings of DMUs with different aggregation methods

DMU	The optimal rally points aggregation method	Wang et al. [31]	α = 0	α = 0.5 Wang and Yang [30]	α = 1
A	3	9	7	7	1
B	10	2	10	10	7
C	9	8	5	6	10
D	5	7	3	3	1
E	6	5	8	8	1
F	1	10	1	1	1
G	4	1	2	2	1
H	7	6	4	5	9
I	8	4	9	9	8
J	2	3	6	4	1

Fig. 5

The rankings of DMUs between different ICE aggregation methods.

So, the optimal rally points aggregation method has three advantages by comparing with different aggregation methods:

We use the optimal rally point to aggregate the generated ICM, finding that there is the smallest total Euclidean distance from other cross-efficiencies for each DMU, indicating that the optimal rally point can gather all opinions of ICEs and represent the consensus results among the intervals.

In the process of using the optimal rally point aggregation method to aggregate ICEs, just projected the interval information onto the plane synchronously and the original interval is not modified. The original interval information is retained to the greatest extent, making the sorting results more scientific and credible.

The optimal rally point aggregation method does not involve risk attitude of Decision Makers, at the same time, the result of aggregation remains in a stable state, it shows that this method has strong robustness and stability in the process of aggregation of ICE.

5.3 Comparison of different ranking methods with fuzzy inputs and outputs

For the sake of proving the validity of the proposed model, we will compare it with models of Zhou et al. [41], Wang et al. [42] and Saati et al. [43]. Zhou et al. [41] proposed a generalized fuzzy DEA with assurance regions (GFDEA) which reduced to many subclasses of fuzzy DEA by setting three binary parameters δ₁, δ₂ and δ₃ different combinations, especially, when δ₁ = 0, it reduces to the fuzzy CCR model, the ranking result of DMUs is listed in columns 3 of Table 6. For convenience, let $x_{ij}^{M}' = (x_{ij}^{M} + x_{ij}^{N}) / 2$ and $y_{rj}^{M}' = (y_{rj}^{M} + y_{rj}^{N}) / 2$ , we transform the input indicator MC and the output indicator GOV into triangular fuzzy numbers, then obtain the fuzzy efficiencies of the ten manufacturing enterprises based on the model of Wang et al. [42], the ranking solution is shown in in columns 4 of Table 6. What’s more, the classic sorting method for solving fuzzy input data is α - cut proposed by Saati et al. [43], we consider α = 0, 0.5, 1, the rankings of ten manufacturing enterprises are listed in the last three columns of Table 6. In order to further explore the correlation between different models, we calculated the Pearson correlation coefficients between them shown in Table 7, we can see that the correlation coefficients between the proposed model and different models are greater than 0.5 expect the model of Zhou et al. [41], with the largest average correlation coefficient. The model of Zhou et al. [41] considers that F has the best evaluation performance among ten manufacturing enterprises, this is inconsistent with other models, because when δ₁ = 0, it reduces to the fuzzy CCR model, and it is viewed as self-evaluation which is often too optimistic and overestimates its own relative efficiency, unlike the cross-efficiency model, which combines self-evaluation and peer-evaluation, making the evaluation more objective.

Table 6
The ranking of DMUs with different ranking methods

DMU The proposed model Zhou et al. [41] Wang et al. [42] α = 0 [43] α = 0.5 [43] α = 1 [43]

A 3 7 5 5 4 1

B 10 3 6 8 9 7

C 9 10 10 7 7 10

D 5 9 7 3 3 1

E 6 1 4 9 8 1

F 1 4 3 1 2 1

G 4 2 2 4 5 1

H 7 8 9 6 6 9

I 8 6 8 10 10 8

J 2 5 1 2 1 1

DMU	The proposed model	Zhou et al. [41]	Wang et al. [42]	α = 0 [43]	α = 0.5 [43]	α = 1 [43]
A	3	7	5	5	4	1
B	10	3	6	8	9	7
C	9	10	10	7	7	10
D	5	9	7	3	3	1
E	6	1	4	9	8	1
F	1	4	3	1	2	1
G	4	2	2	4	5	1
H	7	8	9	6	6	9
I	8	6	8	10	10	8
J	2	5	1	2	1	1

Table 7

Pearson correlation coefficients between the different models

	Proposed	Zhou et al. [41]	Wang [42]	α = 0 [43]	α = 0.5 [43]	α = 1 [43]	AVERAGE
Proposed	1.000	0.188	0.745	0.818	0.867	0.819	0.740
Zhou et al. [41]	0.188	1.000	0.709	-0.091	-0.139	0.456	0.354
Wang [42]	0.745	0.709	1.000	0.539	0.552	0.828	0.729
α = 0 [43]	0.818	-0.091	0.539	1.000	0.964	0.596	0.638
α = 0.5 [43]	0.867	-0.139	0.552	0.964	1.000	0.651	0.649
α = 1 [43]	0.819	0.456	0.828	0.596	0.651	1.000	0.725

6 Conclusion

This paper research the aggregation method of ICE in fuzzy environment, and gives a novel ranking method of ICE DEA model with fuzzy output and input data.

Firstly, for the original fuzzy data, we calculate the fuzzy expected values based on fuzzy set theory, and introduce into the cross-efficiency model as the input and output of DEA method. Secondly, we construct the cross-efficiency DEA model based on the fuzzy expected values from the relatively neutral perspective which determines the upper and lower bounds of ICE, so as to form the generalized interval CEM. Thirdly, we project each cross-efficiency interval of DMU into the two-dimensional coordinate plane, and take the point with the smallest Euclidean distance from each point as the consensus point of all ICEs, referred to as the optimal rally point of DMU. Finally, the comprehensive ICE of all DMUs is fully sorted by the optimal number sorting method instead of averaging them, which makes the ranking results satisfy the order-preserving property. By comparing the final ranking results with other models, we can see that the proposed model in this paper has the following advantages:

We extend cross-efficiency to fuzzy sets, and construct the fuzzy cross-efficiency model from the relatively neutral angle, comparing the interval generation determined by aggressive and benevolent cross-efficiency DEA model, the proposed method can avoid the extremes of intervals and make the determined intervals more objective and fairer.

We project the ICE into points in two-dimensional plane and find the point with the smallest distance from the sum of all points as the optimal rally point, which is regarded as the aggregation interval of ICE. Comparing with different aggregation methods, we can find that the optimal rally point can better reflect the consensus between all ICEs and has higher acceptance. In addition, the original interval information is retained to the greatest extent, making the sorting results more scientific and credible.

We calculate Pearson correlation coefficients between the proposed model and different models, finding that there is a strong correlation with other classical models, but at the same time, there are subtle differences due to their unique advantages, indicating the proposed model is reasonable and effective.

At present, there are still some limitations in this paper. Firstly, we only discuss one kind of fuzzy information, that is, fuzzy number. In future research, we should continue to explore the fusion methods under a variety of fuzzy information. Secondly, we take the fuzzy expected value of input-output into DEA model which will eliminate the fuzzy information and lose the original fuzzy information, future research will consider how to retain the original information and improve the DEA model.

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 61773123).

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DMU	Input			Output
	MC(¥ 1000,000)	NOE	FS(1000ft²)	GOV(¥ 1000,000)	PQ
A	(21,21.3,21.7,22.1)	1780	17.3	(147.5,147.9,148,148.7)	(3,4,5)
B	(14.1,14.5,14.6,15)	1430	16.4	(125.8,126.2,127.2,128.1)	(1,2,3)
C	(25,25.5,25.7,26.1)	2630	11.2	(179,180,182.6,184.5)	(3,4,5)
D	(22,22.5,23.5,24)	2000	10.5	(149.7,152.7,154,155)	(3,4,5)
E	(14.8,15,15.2,15.6)	1570	9.5	(138.9,142.6,143.3,145.4)	(1,2,3)
F	(19.6,20,20.3,21)	1670	4.8	(140.5,143.1,144.6,145.7)	(2,3,4)
G	(22,22.4,22.6,23.2)	1890	6.2	(164.5,168.7,170.8,175.4)	(2,3,4)
H	(24,24.6,25.2,25.5)	2350	11.1	(176.7,179.6,181.2,185.3)	(3,4,5)
I	(15.8,16.3,16.8,17.6)	1750	9.8	(139.8,146.2,148.3,150)	(1,2,3)
J	(14.9,15.3,15.8,16)	1690	8.5	(140,142.8,143.5,144.5)	(2,3,4)

An ICE aggregation method based on optimal rally points in fuzzy environment

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Fuzzy set and related operations

3.1 Proposed the model for generating ICE with fuzzy data

5.1 Comparison of different cross-efficiency strategies

Table 6 The ranking of DMUs with different ranking methods DMU The proposed model Zhou et al. [41] Wang et al. [42] α = 0 [43] α = 0.5 [43] α = 1 [43] A 3 7 5 5 4 1 B 10 3 6 8 9 7 C 9 10 10 7 7 10 D 5 9 7 3 3 1 E 6 1 4 9 8 1 F 1 4 3 1 2 1 G 4 2 2 4 5 1 H 7 8 9 6 6 9 I 8 6 8 10 10 8 J 2 5 1 2 1 1

Footnotes

Acknowledgments

References