Pareto approach for DEA cross efficiency evaluation based on interval programming

Abstract

Efficiency assessment by using data envelopment analysis (DEA) in interval environment is studied. Two parameters with regard to the input and output are introduced to characterize the variability of the production possibility sets. The extended production facets with different production possibility sets are determined. The inclusion relation between different extended production facets is discussed, and self-evaluation models are constructed to calculate the interval efficiency of the decision-making units (DMUs) with the optimal production facet. By setting self-evaluation as a target, the aggressive and benevolent cross-efficiency models are established based on the likelihood between the values of self-evaluation and peer evaluation. The analysis of the models yields the interval cross-efficiency matrices and the weight allocation method that is more advantageous to the DMU for aggregating the interval cross-efficiency matrices. An example is used to illustrate the applications of the models.

Keywords

Interval programming Pareto optimal DEA cross-efficiency

1 Introduction

Management efficiency is widely used by organizations for making amendments. Charnes et al. [1] developed data envelopment analysis (DEA) to measure the relative efficiency of decision-making units (DMUs) with inputs and outputs. Owing to the rapid development of theory, DEA has recently found extensive applications [2 –25].

In conventional DEA models, such as the Charnes-Cooper-Rhodes (CCR) and Banker-Chanes-Cooper (BCC) models, each DMU was evaluated by choosing the most favorable weights to itself. With this self-evaluation, each DMU was allowed to consider its own circumstances only. The results classified the DMUs into two groups: efficient and inefficient based on the Pareto facets. In general, several efficient DMUs lead to no inferior-to-superior relationship among them. Considering this limitation, Sexton et al. [2] originally proposed cross-efficiency evaluation. Unlike the conventional models, the DMU was evaluated not only by self-evaluation but also by peer evaluation. In most studies, the overall efficiency of each DMU was calculated with the optimal weights of all DMUs by the CCR model. Compared with the conventional models, cross-efficiency evaluation can effectively distinguish the priority and rank the DMUs in a unique order. Because the optimal solutions of the CCR model are not unique, it is necessary to set a secondary goal to select an appropriate solution. The benevolent and aggressive models were proposed by Doyle and Green [3]. In the former, each DMU considered the other DMUs partners and selected the weights to simultaneously maximize their efficiency and maintain its own efficiency. In the latter, each DMU maintained its own efficiency but minimized the efficiency of the other DMUs, that is, the other DMUs were considered competitors. Liang et al. [4] introduced game theory to explain the competition and cooperation among the DMUs and obtained the Nash equilibrium solution of the model. Liang et al. extended Doyle and Green’s models by considering deviations from the ideal efficiency [5]. Moreover, since not every DMU was on the Pareto facet, Wang and Chin [6] improved the models by considering deviations from the CCR efficiency instead of ideal efficiency. Wu et al. [7] extended secondary goal models for weight selection. In their approach, each DMU had desirable and undesirable cross-efficiency targets. Different models were proposed to ensure that the results cloud be simultaneously close to the desirable target and far from the undesirable target. Moreover, some secondary goal function was developed to decrease the number of zero weights [8 –10]. Wu et al. [11] made a Pareto improvement model for cross-efficiency evaluation.

Owing to the development of soft computing technologies, an increasingly large number of decision-making tools have been developed to cope with the problem of uncertain data, such as interval and fuzzy data. In addition to being an important method of multi-attribute decision-making, DEA has greatly contributed in the uncertain decision-making area since Cooper et al. [12] proposed the imprecise DEA (IDEA). With the development of uncertainty theory, IDEA has gradually developed as well. In accordance with the principles of interval linear programming, Despotis and Simirlis [13] computed the interval efficiency by introducing parameters to transform the nonlinear model to a linear model, and developed a model for handling interval and fuzzy data. Wang et al. [14] used the unified production facet to measure the interval efficiency of DMU with interval data. Lee et al. [15] extended IDEA to the two-stage additive model. Based on the CCR model, Kao [16] constructed a pair of bi-level mathematical programming problems, which was transformed into an ordinary linear programming problem by productivity concepts and variable substitution tools. The objective function value of the model was the interval efficiency of DMU. Chen et al. [17] computed the interval cross efficiency with interval and fuzzy data. In particular, the fuzzy data was converted into interval data by the tolerance and α-level approaches. Mariagrazia Dotoli et al. [18] considered triangular fuzzy numbers in DEA, and the model used fuzzy triangular efficiencies to assess DMUs. This retained as much fuzzy information as possible until the final ranking stage. Hossein Azizi et al. [19] formulated the ISBM model using the same production frontier from different viewpoints. It assessed the overall efficiency of DMUs in the presence of interval data with slack values. Hatami-Marbini, Adel et al. [20] proposed a new four-step fuzzy DEA method to re-shape weakly efficient facets along with revisiting the efficiency score of DMUs in terms of perturbing the weakly efficient facets. Wei and Wang [21] extended the robust efficiency analysis procedure to the situation where precise information on some input and output data is unavailable.

In existing works about IDEA, there are two problems worth considering. Namely, most IDEA models were formulated by different constraint sets to measure the interval efficiency of each DMU with uncertain data [13, 17]. This implies that each DMU is evaluated in a different production possibility set. Therefore, different production facets are used in the process of efficiency evaluation. This is unfavorable for comparing efficiencies. In uncertain environments, production possibility sets would vary across the inputs and outputs of imprecise data. The selection of the production facets would affect efficiency assessment. Moreover, to the best of our knowledge, cross-efficiency DEA has seldom been introduced in uncertain environments. In order to obtain an overall evaluation of DMUs, the assessments should combine self-evaluation and peer evaluation. However, the original models of cross-efficiency evaluation with crisp data cannot be directly used to handle interval data. Uncertainty theory is required in order to establish a reasonable model and ranking scheme. For example, Muren et al. [20] considered imprecise and vague information by typical fuzzy logic to retain the information about uncertainty. Chen et al. [17] ranked the DMUs by minimizing the maximum regret value.

In this paper, inputs and outputs of DMUs with interval data will be considered. Two parameters are introduced to represent the different production sets, and the DMUs achieving Pareto optimal according to these production sets contribute to the production facets. It provides variable reference facets for decision makers. Moreover, the inclusion relation among them will be analyzed. The inclusion relation between different extended production facets provides a theoretical basis for self-evaluation with the optimal production facet, which was proposed by Wang et al. [14] and Azizi et al. [19]. The cross-efficiency evaluation models with interval data will be formulated based on possibility degree theory. The relation between different evaluations, which is described by the upper and lower bounds of the interval efficiencies with the definition of likelihood, makes the cross-efficiency models more reasonable, simple and easy to be used than the models in [17]. The operation (max) of two intervals is introduced to ensure that the results are consistent with conventional results by crisp data. Then, the likelihood method is utilized to aggregate the cross-efficiency matrix. The results of the aggregation will be ranked by that method.

This paper is organized as follows: In Section 2, definitions and operations of interval numbers are introduced. In Section 3, the production possibility sets and production facets are defined. In Section 4, the cross-evaluation models with interval data under the maximum production possibility set are formulated. In Section 5, the cross-evaluation matrixes are aggregated and the DMUs are ranked. Section 6 contains numerical illustrations. Section 7 concludes the paper.

2 Preliminaries on interval data

Definition 1. [23] Let $\bar{a} = [a^{L}, a^{U}]$ be an interval number, where a^L, a^U ∈ R. I (R) represents a set of interval numbers. If a^L = a^U, then $\bar{a}$ degenerates into a real number. The length of $\bar{a}$ is $m (\bar{a}) = a^{U} - a^{L}$ , the midpoint of $\bar{a}$ is $w (\bar{a}) = (a^{U} + a^{L}) / 2$ .

Let $\bar{a}, \bar{b} \in I (R)$ be two interval numbers. Arithmetic operations can be defined between them as follows [24]:

$\bar{a} + \bar{b} = [a^{L} + b^{L}, a^{U} + b^{U}];$

$\bar{a} - \bar{b} = [a^{L} - b^{U}, a^{U} - b^{L}];$

$\bar{a} \times \bar{b} = [a^{L} b^{L}, a^{U} b^{U}];$

$\bar{a} / \bar{b} \approx [a^{L} / b^{U}, a^{U} / b^{L}] .$

Moreover, $\bar{a} = \bar{b}$ if and only if a^L = b^L, a^U = b^U.

To compare two interval numbers, we introduce the likelihood to express the priority between them. $\bar{a} \geq \bar{b}$ denotes that $\bar{b}$ is no greater than $\bar{a}$ . The likelihood of $\bar{a} \geq \bar{b}$ is defined as [22, 26]: $p (\bar{a} \geq \bar{b}) = \frac{max {0, a^{U} - b^{L}} - max {0, a^{L} - b^{U}}}{a^{U} - a^{L} + b^{U} - b^{L}}$ (1)

It is obvious that $0 \leq p (\bar{a} \geq \bar{b}) \leq 1$ and $p (\bar{a} \geq \bar{b}) + p (\bar{b} \geq \bar{a}) = 1$ .

Some useful properties of the likelihood are as follows [24, 26]:

$p (\bar{a} \geq \bar{b}) = 1$ if and only if a^L ≥ b^U;

$p (\bar{a} \geq \bar{b}) = 0$ if and only if a^U ≤ b^L;

$p (\bar{b} \geq \bar{a}) \geq 0.5$ if and only if $w (\bar{a}) \geq w (\bar{b})$ , $p (\bar{a} \geq \bar{b}) = 0.5$ if and only if $w (\bar{a}) = w (\bar{b})$ .

Let $\bar{a}, \bar{b}, \bar{c}$ be three interval numbers. If $p (\bar{a} \geq \bar{b}) \geq 0.5$ and $p (\bar{b} \geq \bar{c}) \geq 0.5$ , then $p (\bar{a} \geq \bar{c}) \geq 0.5$ .

Properties (a) and (b) indicate that the two interval numbers do not overlap. Property (c) shows the relationship between the likelihood and the midpoint of the interval number. Property (d) is the transitivity of likelihood.

Definition 2. [27] Let $\bar{α} = (\bar{a_{1}}, \bar{a_{2}}, \dots, \bar{a_{n}})$ , where $\bar{a_{i}} \in I (R), i = 1, \dots, n$ . $\bar{α}$ denotes an n-dimensional interval number vector. Iⁿ (R) is the set of all n-dimensional interval number vectors.

M_n×m (I (R)) denotes the set of all n × m matrices over I (R). In particular, let M_n (I (R)) = M_n×n (I (R)). For A ∈ M_n (I (R)), $\bar{a_{ij}}$ denotes the element of I (R) in the (i, j)-entry of A.

3 Production facet for DEA with interval data

It is assumed that there is a set of n DMUs with m inputs X_ij (i = 1, ⋯ , m) and s outputs Y_rj (r = 1, ⋯ , s). Due to the uncertainty, it is assumed that the data of the inputs and outputs are interval numbers. Hence, X_ij and Y_ij can be replaced with $\bar{X_{ij}} = [x_{ij}^{L}, x_{ij}^{U}]$ and $\bar{Y_{ij}} = [y_{ij}^{L}, y_{ij}^{U}]$ , respectively. $x_{ij} (γ) = x_{ij}^{L} + γ (x_{ij}^{U} - x_{ij}^{L})$ (0 ≤ γ ≤ 1) denotes the point of $\bar{X_{ij}}$ , and $y_{rj} (δ) = y_{rj}^{L} + δ (y_{rj}^{U} - y_{rj}^{L})$ (0 ≤ δ ≤ 1) the point of $\bar{Y_{ij}}$ . Let X_j (γ) = (x_1j (γ) , x_2j (γ) , ⋯ , x_mj (γ)), Y_j (δ) = (y_1j (δ) , y_2j (δ) , ⋯ , y_sj (δ)) be two crisp vectors. Thus, (X_j (γ) , Y_j (δ)) can be considered a representative point of DMU_j. In particular, (X_j (0) , Y_j (1)) is the optimal point of DMU_j, whereas (X_j (1) , Y_j (0)) can be regarded as the worst point of DMU_j.

The representative points of DMUs constitute a reference set

$\begin{matrix} S (γ, δ) & = & {(X_{1} (γ), Y_{1} (δ)), (X_{2} (γ), Y_{2} (δ)), \\ \dots, (X_{n} (γ), Y_{n} (δ))} \end{matrix}$ (2)

S (γ, δ) generates the convex cone C (S (γ, δ)), which can be expressed as follows:

$\begin{matrix} C (S (γ, δ)) \\ = {\sum_{j = 1}^{n} λ_{j} (X_{j} (γ), Y_{j} (δ)) | λ_{j} \geq 0, j = 1, \dots, n} \end{matrix}$ (3)

Definition 3. [30] If the vectors ω⁰ > 0, μ⁰ > 0, ω⁰ ∈ R^m, μ⁰ ∈ R^s satisfy that (ω^0T, μ^0T) is the normal vector of a facet of C (S (γ, δ)), and C (S (γ, δ)) is on the same side of the normal vector of the facet, the facet is called the effective production facet of the S (γ, δ).

Definition 4. When ω^0TX - μ^0TY > 0, (X, Y) is said to be on the negative side of the production facet; when ω^0TX - μ^0TY < 0, (X, Y) is said to be on the positive side of the production facet.

Obviously, if (X, Y) ∈ C (S (γ, δ)), then ω^0TX - μ^0TY ≥ 0.

As is known, in DEA models, the production facet is used to reflect the optimal production status of a decision-making unit. In traditional DEA with crisp data, it is obvious that the boundary points of a production possibility set are defined as the efficient production facet, since the DMUs on the boundary of the production possibility set achieve the Pareto Optimal. However, the effective production facets of S (γ, δ) are not the optimal production state of all DMUs. In general, it is deemed necessary to determine the efficient DMU by considering the efficiency of its optimal point. Furthermore, the production reference sets are uncertain when the input and output values of DMUs are interval numbers. Consequently, it is necessary to consider whether the optimal point of DMU₀ achieves the Pareto Optimal with respect to the different production reference sets.

The linear programming problems based on the S (γ, δ) are now constructed as follows: $Max μ^{T} Y_{0} (1)$ (4) $\begin{matrix} s . t . & ω^{T} X_{j} (γ) - μ^{T} Y_{j} (δ) \geq 0; j = 1, \dots, n \\ ω^{T} X_{0} (0) = 1; \\ ω, μ \geq 0, \end{matrix}$ where the subscript 0 is the DMU under evaluation, denoted by DMU₀. The optimal value of model (4) is denoted by $θ_{0}^{U} (γ, δ)$ . In particular, when the production reference set consists of the optimal points, model (4) takes the following form: $Max μ^{T} Y_{0} (1)$ (5) $\begin{matrix} s . t . & ω^{T} X_{j} (0) - μ^{T} Y_{j} (1) \geq 0; j = 1, \dots, n \\ ω^{T} X_{0} (0) = 1; \\ ω, μ \geq 0 . \end{matrix}$

In model (4), the point (X₀ (0) , Y₀ (1)) is not always included in S (γ, δ), and the constraint conditions ω^TX_j (γ) - μ^TY_j (δ) ≥0 are only for S (γ, δ). Thus, the optimal value of model (4) will be larger than one. If $θ_{0}^{U} (γ, δ) < 1$ , then the efficiency of DMU₀ should be improved. If $θ_{0}^{U} (γ, δ) = 1$ , then the optimum status of S (γ, δ) is attained at the optimal point of DMU₀. If $θ_{0}^{U} (γ, δ) > 1$ , then the technique and experience of the optimal point of DMU₀ are higher than those of the existing reference set S (γ, δ). Compared with model (4), model (5), which is constructed with the production reference set by the optimal points of DMUs, can be considered the conventional CCR model, and its optimal value is denoted by $θ_{0}^{U_{CCR}} (0, 1)$ .

As C (S (γ, δ)) is finitely multidimensional, it can form the data envelopment of S (γ, δ). Then, the production possibility set according to S (γ, δ) is defined as follows:

$\begin{matrix} T (γ, δ) & = & {(X, Y) | \sum_{j = 1}^{n} Y_{j} (δ) λ_{j} \geq Y, \\ \sum_{j = 1}^{n} X_{j} (γ) λ_{j} \leq X, | λ_{j} \geq 0, \\ j = 1, \dots, n} \end{matrix}$ (6)

Theorem 1.Ifγ₁ ≤ γ₂andδ₁ ≥ δ₂, thenT (γ₁, δ₁) ⊇ T (γ₂, δ₂).

Proof. ∀ (X, Y) ∈ T (γ₂, δ₂), by the definition of T (γ₂, δ₂) and the condition of the theorem, it holds that $X \geq \sum_{j = 1}^{n} X_{j} (γ_{2}) λ_{j} \geq \sum_{j = 1}^{n} X_{j} (γ_{1}) λ_{j}$ and $Y \leq \sum_{j = 1}^{n} Y_{j} (δ_{2}) λ_{j} \leq \sum_{j = 1}^{n} Y_{j} (δ_{1}) λ_{j} .$

Therefore, (X, Y) ∈ T (γ₁, δ₁), that is, T (γ₁, δ₁) ⊇ T (γ₂, δ₂). This completes the proof.

Corollary 1.Ifγ = 0, δ = 1, then the production possibility setT (0, 1) can be regarded as the maximum production possibility set, as it contains all the representative points of every DMU.

Since (X₀ (0) , Y₀ (1)) is not in T (γ, δ), T^/ (γ, δ) can be represented as follows:

$\begin{matrix} T^{/} (γ, δ) \\ = {(X, Y) | \sum_{j = 1}^{n} Y_{j} (δ) λ_{j} + Y_{0} (1) λ_{0} \geq Y, \\ \sum_{j = 1}^{n} X_{j} (γ) λ_{j} + X_{0} (0) λ_{0} \leq X, | λ_{j} \geq 0, \\ j = 0, 1, \dots, n} \end{matrix}$ (7)

In particular, if γ = 0, δ = 1, then T (0, 1) = T^/ (0, 1).

The multi-objective programming problem ${VP}_{(γ, δ)} {\begin{matrix} min (x_{1}, \dots, x_{m}, - y_{1}, \dots, - y_{s}) \\ X, Y \in T^{/} (γ, δ) \end{matrix}$ (8) is now considered and its Pareto effective solution is discussed as follows:

Theorem 2.Whenγ ≠ 0, δ ≠ 1, if (X₀ (0) , Y₀ (1)) is on the positive side of the effective production facet, with (ω^0T, μ^0T) as its normal vector, then $θ_{0}^{U} (γ, δ) > 1$ .

Proof. (ω^0T, μ^0T) is the normal vector; hence, the equation of the effective production facet is ω^0TX - μ^0TY = 0. Since (X_j (γ) , Y_j (δ)) ∈ T (γ, δ), it follows that ω^0TX_j (γ) - μ^0TY_j (δ) ≥0, j = 1, ⋯ , n. (X₀ (0) , Y₀ (1)) is on the positive side of the effective production facet, that is ω^0TX₀ (0) - μ^0TY₀ (1) <0. Let $ω^{*} = \frac{1}{ω^{0 T} X_{0} (0)} ω^{0}, μ^{*} = \frac{1}{ω^{0 T} X_{0} (0)} μ^{0}$ . Then, $μ^{* T} Y_{0} (1) = \frac{1}{ω^{0 T} X_{0} (0)} μ^{0 T} Y_{0} (1) = \frac{μ^{0 T} Y_{0} (1)}{ω^{0 T} X_{0} (0)} > 1$ . This implies that $θ_{0}^{U} (γ, δ) > 1$ , completing the proof.

Theorem 3. (1)Whenγ ≠ 0, δ ≠ 1, if the optimal solution of model (4) $θ_{0}^{U} (γ, δ) \geq 1$ , then (X₀ (0) , Y₀ (1)) is the Pareto effective solution ofVP_(γ, δ), whereγ ∈ (0, 1] , δ ∈ [0, 1);

(2) When γ = 0, δ = 1, $θ_{0}^{U^{CCR}} = 1$ if and only if (X₀ (0) , Y₀ (1)) is the Pareto effective solution of VP_(0,1).

Proof. (1) Let (ω^*, μ^*) be a feasible solution of model (4) and $θ_{0}^{U} (γ, δ)$ be its optimal solution. When $θ_{0}^{U} (γ, δ) > 1$ , it holds that ω^*X₀ (0) - μ^*Y₀ (1) ≤0. If (X₀ (0) , Y₀ (1)) is not the Pareto effective solution of VP_(γ, δ), then there exists (X, Y) ∈ T^/ (γ, δ) for (X, Y) < (X₀ (0) , Y₀ (1)); consequently, ω^*TX - μ^*TY < ω^*TX₀ (0) - μ^*TY₀ (1) ≤0. This implies that ω^*T (X - X₀ (0)) - μ^*T (Y - Y₀ (1)) <0.

Furthermore, $\begin{matrix} ω^{* T} (\sum_{j = 1}^{n} X_{j} (γ) λ_{j} + (λ_{0} - 1) X_{0} (0)) \\ - μ^{* T} (\sum_{j = 1}^{n} Y_{j} (δ) λ_{j} + (λ_{0} - 1) Y_{0} (1)) \\ = \sum_{i = 1}^{n} λ_{j} (ω^{* T} X_{j} (γ) - μ^{* T} Y_{j} (δ)) + (λ_{0} - 1) \\ (ω^{* T} X_{0} (0) - μ^{* T} Y_{0} (1)) < 0 . \end{matrix}$

It is now proved that if (X, Y) < (X₀ (0) , Y₀ (1)), then 0 ≤ λ₀ < 1. It is assumed that λ₀ ≥ 1. Since $\sum_{j = 1}^{n} X_{j} (γ) λ_{j} \geq 0$ , it follows that X ≥ X₀ (0), contradicting (X, Y) < (X₀ (0) , Y₀ (1)). Consequently, 0 ≤ λ₀ < 1. Since (ω^*TX₀ (0) - μ^*TY₀ (1)) ≤0, it follows that (λ₀ - 1) (ω^*TX₀ (0) - μ^*TY₀ (1)) ≥0. This shows that there at least exists j ∈ {1, ⋯ , n} such that ω^*TX_j (γ) - μ^*TY_j (δ) <0, which contradicts the assumption.

(2) As mentioned earlier, when γ = 0, δ = 1, the problem is consistent with the traditional CCR model, and the proof may be found in [28 –30]. This completes the proof.

The case γ ≠ 0, δ ≠ 1 is first considered. By the proof of Theorem 4, it follows that if the optimal points of DMUs are on the positive side of the production facet, then the optimal solution of model (4) is greater than 1 and achieves the Pareto Optimal of VP_(γ, δ). E (γ, δ) denotes the extended production facet composed of the DMUs that achieve the Pareto Optimality. Thus, there are different extended productions facets corresponding to different production possibilities sets. In particular, if γ = 0, δ = 1, then the definition of production facet E (0, 1) is the same as in the CCR model.

Theorem 4.Whenγ ≠ 0, δ ≠ 1, ifγ₁ ≤ γ₂, δ₁ ≥ δ₂, thenE (γ₁, δ₁) ⊆ E (γ₂, δ₂), whereγ₁, γ₂ ∈ (0, 1] , δ₁, δ₂ ∈ [0, 1).

Proof. Let (X₀ (0) , Y₀ (1)) be in E (γ₁, δ₁), so that $θ_{0}^{U} (γ_{1}, δ_{1}) > 1$ . If (ω^0T, μ^0T) is the optimal solution of (4) when γ = γ₁, δ = δ₁, then (ω^0T, μ^0T) is the feasible solution of (2) with γ = γ₂, δ = δ₂, since ω^0TX_j (γ₂) - μ^0TY_j (δ₂) ≥ ω^0TX_j (γ₁) - μ^0TY_j (δ₁) ≥0.

It is noted that $\begin{matrix} θ_{0}^{U} (γ_{2}, δ_{2}) \geq θ_{0}^{U} (γ_{1}, δ_{1}) \\ = ω^{0 T} X_{0} (0) - μ^{0 T} Y_{0} (1) > 1 . \end{matrix}$

By the proof of Theorem 3, this shows that (X₀ (0) , Y₀ (1)) is in E (γ₂, δ₂), that is, E (γ₁, δ₁) ⊆ E (γ₂, δ₂). This completes the proof.

Corollary 2.If $θ_{0}^{U^{CCR}} = 1,$ then $θ_{0}^{U} (γ, δ) \geq 1$ , whereγ, δ ∈ [0, 1].

By Corollary 2 and Theorem 3, it holds that the DMUs in E (0, 1) are also in E (γ, δ), for all 0 ≤ γ ≤ 1, 0 ≤ δ ≤ 1. E (0, 1) represents the optimal configuration of DMUs in the current optimal production state. Thus, E (0, 1) is defined to be the best production facet.

4 Cross-efficiency of DMUs with interval data

According to Wang et al. [14], the efficiencies of DMUs are not comparable in different production possibility sets, as different production facets are adopted in the assessment. According to the conclusion of Section 3, T (0, 1) is the maximum production possibility sets, and E (0, 1) is the best production facet with T (0, 1). That is, all points of the DMUs will be included in T (0, 1), and all DMUs in E (0, 1) achieve Pareto Optimality under any condition. Therefore, for the unified and comprehensive evaluation, interval DEA models with the same production possibility set will be developed.

4.1 Self-evaluation models

Let $\bar{θ_{d}} = \frac{μ_{d}^{T} \bar{Y_{d}}}{ω_{d}^{T} \bar{X_{d}}}$ be the efficiency of DMU_d by self-evaluation. Using the operation rules on interval data, it holds that $\bar{θ_{d}} = [θ_{d}^{L}, θ_{d}^{U}] = [\frac{μ_{d}^{T} Y_{d} (0)}{ω_{d}^{T} X_{d} (1)}, \frac{μ_{d}^{T} Y_{d} (1)}{ω_{d}^{T} X_{d} (0)}] .$ (9)

We formulate a pair of linear programming (LP) models to measure the interval efficiency of DMU_d under T (0, 1) as follows: $Max μ^{T} Y_{d} (1)$ (10) $\begin{matrix} s . t . & ω^{T} X_{j} (0) - μ^{T} Y_{j} (1) \geq 0, j = 1, \dots, n; \\ ω^{T} X_{d} (0) = 1; \\ ω, μ \geq 0 . \end{matrix}$ and $Max μ^{T} Y_{d} (0)$ (11) $\begin{matrix} s . t . & ω^{T} X_{j} (0) - μ^{T} Y_{j} (1) \geq 0, j = 1, \dots, n; \\ ω^{T} X_{d} (1) = 1; \\ ω, μ \geq 0 . \end{matrix}$

The optimal values of models (10) and (11) are denoted by $θ_{d}^{U}$ and $θ_{d}^{L}$ , respectively. $θ_{d}^{L}$ and $θ_{d}^{U}$ form the best relative interval efficiency. In fact, the above models (10) and (11) are the same as models (5) and (6) in [14], respectively. Notably, $θ_{d}^{U}$ equals $θ_{d}^{U^{CCR}}$ of model (5) in Section 3, and $θ_{d}^{L}$ can be considered to be $θ_{d}^{L^{CCR}}$ , as explained in [14]. Hence, $\bar{θ_{d}}$ can be regarded as the CCR interval efficiency of DMU_d.

4.2 Peer-evaluation models and cross-efficiency matrix

In conventional peer-evaluation, each DMU always evaluates other DMUs by ensuring its best relative efficiency. The cross-efficiency of a given DMU_j, j = 1, ⋯ , n, obtained with the weights of DMU_d, is as follows: $θ_{dj} = \frac{μ_{d}^{T} Y_{j}}{ω_{d}^{T} X_{j}} .$ (12)

However, an adjustment is required for DMUs with interval data. In that case, the best relative efficiency is the interval efficiency, unlike in the case of DMUs with crisp data. Therefore, the cross efficiency of the given DMU should be interval data. Let $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ be the optimal solutions of model (10), and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ the optimal solutions of model (11). As is known, $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ are the best weights for the optimal point of DMU_d, and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ are the weights that are most beneficial to the worst point of DMU_d. They represent the weight configuration of DMU_d in two extreme states under the maximum production set. It is one-sided to use $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ or $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ alone to measure the cross-efficiency $\bar{θ_{dj}}$ . Thus, the cross-efficiency of DMU_i must be computed by both the weights $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ . The peer evaluation is denoted by $[\frac{μ_{d}^{U^{*}} Y_{j} (0)}{ω_{d}^{U^{*}} X_{j} (1)}, \frac{μ_{d}^{U^{*}} Y_{j} (1)}{ω_{d}^{U^{*}} X_{j} (0)}]$ , which reflects that DMU_d assesses DMU_j by the weights corresponding to the best state. By contrast, $[\frac{μ_{d}^{L^{*}} Y_{j} (0)}{ω_{d}^{L^{*}} X_{j} (1)}, \frac{μ_{d}^{L^{*}} Y_{j} (1)}{ω_{d}^{L^{*}} X_{j} (0)}]$ is obtained by the weights of the best relative efficiency of the worst point of DMU_d, which reflects that DMU_d assesses DMU_j by the weights corresponding to the worst state. The operation ∨ (max) of two intervals is introduced to define the cross-efficiency $\bar{θ_{dj}}$ as follows:

$\begin{matrix} \bar{θ_{dj}} & = & [θ_{dj}^{L}, θ_{dj}^{U}] = [\frac{μ_{d}^{L^{*}} Y_{j} (0)}{ω_{d}^{L^{*}} X_{j} (1)}, \frac{μ_{d}^{L^{*}} Y_{j} (1)}{ω_{d}^{L^{*}} X_{j} (0)}] \lor [\frac{μ_{d}^{U^{*}} Y_{j} (0)}{ω_{d}^{U^{*}} X_{j} (1)}, \frac{μ_{d}^{U^{*}} Y_{j} (1)}{ω_{d}^{U^{*}} X_{j} (0)}] \\ = & [max (\frac{μ_{d}^{L^{*}} Y_{j} (0)}{ω_{d}^{L^{*}} X_{j} (1)}, \frac{μ_{d}^{U^{*}} Y_{j} (0)}{ω_{d}^{U^{*}} X_{j} (1)}), max (\frac{μ_{d}^{L^{*}} Y_{j} (1)}{ω_{d}^{L^{*}} X_{j} (0)}, \frac{μ_{d}^{U^{*}} Y_{j} (1)}{ω_{d}^{U^{*}} X_{j} (0)})] \end{matrix}$ (13)

In the existing cross-efficiency model with crisp data, the peer-evaluation of a DMU is equal to its self-evaluation. Theorem 5 will show the consistency between the interval data and the crisp data.

Theorem 5.Ifd = j, then $\bar{θ_{dd}} = \bar{θ_{d}}$ .

Proof. It is assumed that $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ are the optimal solutions of model (10) for DMU_d, and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ are the optimal solutions of model (11) for DMU_d. Let $ω_{d}^{U^{/}} = \frac{ω_{d}^{U^{*}}}{ω_{d}^{U^{*}} X_{d} (1)}, μ_{d}^{U^{/}} = \frac{μ_{d}^{U^{*}}}{ω_{d}^{U^{*}} X_{d} (1)},$ so that they satisfy $\begin{matrix} ω_{d}^{U^{/}} X_{j} (0) - μ_{d}^{U^{/}} Y_{j} (1) \\ = \frac{1}{ω_{d}^{U^{*}} X_{d} (1)} (ω_{d}^{U^{*}} X_{j} (0) - μ_{d}^{U^{*}} Y_{j} (1) \geq 0), \\ j = 1, \dots, n; \\ ω_{d}^{U^{/}} X_{d} (1) = \frac{ω_{d}^{U^{*}}}{ω_{d}^{U^{*}} X_{d} (1)} X_{d} (1) = 1 . \end{matrix}$

It is obvious that $ω_{d}^{U^{/}}$ and $μ_{d}^{U^{/}}$ are the feasible solutions to model (11) for DMU_d. Therefore, $\frac{μ_{d}^{U^{*}} Y_{j} (0)}{ω_{d}^{U^{*}} X_{j} (1)} = \frac{μ_{d}^{U^{/}} Y_{d} (0)}{ω_{d}^{U^{/}} X_{d} (1)} \leq \frac{μ_{d}^{L^{*}} Y_{d} (0)}{ω_{d}^{L^{*}} X_{d} (1)}$

Similarly, $\frac{μ_{d}^{L^{*}} Y_{j} (1)}{ω_{d}^{L^{*}} X_{j} (0)} \leq \frac{μ_{d}^{U^{*}} Y_{j} (1)}{ω_{d}^{U^{*}} X_{j} (0)}$ . Then, by the definition of $\bar{θ_{dd}}$ , the conclusion is established.

Theorem 6.Assume that $\bar{θ_{j}}$ is the CCR interval efficiency of theDMU_j, and the $\bar{θ_{dj}}$ is the target cross-efficiency ofDMU_jrelative toDMU_d. Then for ∀d ∈ (1, 2, ⋯ , n), we have $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) \leq 0.5$ .

Proof. We assume that $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ are the optimal solutions of models (10) and (11) for DMU_d, respectively; $μ_{j}^{U^{*}}, ω_{j}^{U^{*}}$ are the optimal solution of model (10) for DMU_j. As in the proof of Theorem 5, let $ω_{d}^{U^{/}} = \frac{ω_{d}^{U^{*}}}{ω_{d}^{U^{*}} X_{j} (0)}$ , $μ_{d}^{U^{/}} = \frac{μ_{d}^{U^{*}}}{ω_{d}^{U^{*}} X_{j} (0)}$ , $ω_{d}^{L^{/}} = \frac{ω_{d}^{L^{*}}}{ω_{d}^{L^{*}} X_{j} (0)}$ , $μ_{d}^{L^{/}} = \frac{μ_{d}^{L^{*}}}{ω_{d}^{L^{*}} X_{j} (0)}$ be the feasible solutions of models (10) for DMU_j, so that

$\begin{matrix} \frac{μ_{d}^{U^{*}} Y_{j} (1)}{ω_{d}^{U^{*}} X_{j} (0)} & = & \frac{μ_{d}^{U^{/}} Y_{j} (1)}{ω_{d}^{U^{/}} X_{j} (0)} \leq \frac{μ_{j}^{U^{*}} Y_{j} (1)}{ω_{j}^{U^{*}} X_{j} (0)}, \\ \frac{μ_{d}^{L^{*}} Y_{j} (1)}{ω_{d}^{L^{*}} X_{j} (0)} & = & \frac{μ_{d}^{L^{/}} Y_{j} (1)}{ω_{d}^{L^{/}} X_{j} (0)} \leq \frac{μ_{j}^{U^{*}} Y_{j} (1)}{ω_{j}^{U^{*}} X_{j} (0)} . \end{matrix}$

That is, $θ_{dj}^{U} \leq θ_{j}^{U}$ . Similarly, we have $θ_{dj}^{L} \leq θ_{j}^{L}$ . This implies that $w (\bar{θ_{j}}) \geq w (\bar{θ_{dj}})$ . By property (c) of the likelihood, we obtain $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) \leq 0.5$ . This completes the proof.

Corollary 3.Ifd = j, then $P (\bar{θ_{dd}} \geq \bar{θ_{d}}) = 0.5$ .

By Theorem 6, the position relationship of $\bar{θ_{dj}}$ and $\bar{θ_{j}}$ is shown in Fig. 1

Fig.1

Position relationship of $\bar{θ_{dj}}$ and $\bar{θ_{j}}$ ; (a) left, (b) right.

Theorem 7.Assume that $\bar{θ_{j}} = [θ_{j}^{L}, θ_{j}^{U}]$ is the CCR-efficiency of the DMU_dand $\bar{θ_{dj}} = [θ_{dj}^{L}, θ_{dj}^{U}]$ is the cross-efficiency of DMU_jrelative to DMU_d. When $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) \geq 0$ , if $θ_{dj}^{L}$ is close to $θ_{j}^{L}$ and $θ_{dj}^{U}$ is close to $θ_{j}^{U}$ , then $P (\bar{θ_{dj}} \geq \bar{θ_{j}})$ will be close to 0.5.

Proof. It is assumed that $\bar{θ_{j}}$ is the CCR-efficiency of DMU_d and $\bar{θ_{dj}}$ is the target cross-efficiency of DMU_j relative to DMU_d. By Theorem 6, $θ_{dj}^{L} \leq θ_{j}^{L}$ and $θ_{dj}^{U} \leq θ_{j}^{U}$ . If $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) > 0$ , as shown in Fig. 1(a), the following holds: $\begin{matrix} P (\bar{θ_{dj}} \geq \bar{θ_{j}}) \\ = \frac{max {0, θ_{dj}^{U} - θ_{j}^{L}} - max {0, θ_{dj}^{L} - θ_{j}^{U}}}{θ_{dj}^{U} - θ_{dj}^{L} + θ_{j}^{U} - θ_{j}^{L}} \\ = \frac{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U})}{m (\bar{θ_{j}}) + (θ_{dj}^{U} - θ_{dj}^{L})} . \end{matrix}$

Then $\begin{matrix} \frac{1}{P (\bar{θ_{dj}} \geq \bar{θ_{j}})} & = & \frac{m (\bar{θ_{j}}) + (θ_{dj}^{U} - θ_{dj}^{L})}{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U})} \\ = & \frac{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U}) + θ_{j}^{U} - θ_{dj}^{L}}{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U})} \\ = & 1 + \frac{θ_{j}^{U} - θ_{dj}^{L}}{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U})} \\ = & 1 + \frac{m (\bar{θ_{j}}) + θ_{j}^{L} - θ_{dj}^{L}}{m (\bar{θ_{j}}) - (θ_{j}^{U} - θ_{dj}^{U})} . \end{matrix}$

Obviously, the above equations show that if $θ_{dj}^{L}$ is close to $θ_{j}^{L}$ and $θ_{dj}^{U}$ is close to $θ_{j}^{U}$ , then $P (\bar{θ_{dj}} \geq \bar{θ_{j}})$ will be close to 0.5. This completes the proof.

Theorem 7 may be used to explain the value of the likelihood $P (\bar{θ_{dj}} \geq \bar{θ_{j}})$ when $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) \geq 0$ . It indicates that the proximity of peer-evaluation to self-evaluation can be estimated by the values of $θ_{dj}^{L}$ and $θ_{dj}^{U}$ . However, when $P (\bar{θ_{dj}} \geq \bar{θ_{j}}) = 0$ , if $θ_{dj}^{L}$ is close to $θ_{j}^{L}$ and $θ_{dj}^{U}$ is close to $θ_{j}^{U}$ , then $w (\bar{θ_{dj}})$ is close to $w (\bar{θ_{j}})$ . Thus, $\bar{θ_{dj}}$ is close to the $\bar{θ_{j}}$ .

By the definition of cross-efficiency, secondary goal models must be developed to solve the non-unique optimal weights problem in DEA self-efficiency assessment. Since Doyle and Green [3] proposed secondary goal models (benevolent and aggressive), these models have been extended by various methods [4 –11]. Liang et al. [5], Wang [6], and Wu et al. [7] proposed target efficiency and compared it with cross-efficiency, and the maximization or minimization of the gap between them was used to set up the secondary goal programming. Even though the quadratic programming models mentioned above were used for crisp data, the ideas of these models can be used to handle interval data as well. In this paper, the CCR interval efficiencies of the DMUs are set as their target efficiencies. According to Theorem 7, the likelihood $P (\bar{θ_{dj}} \geq \bar{θ_{j}})$ represents the priority of the value of peer-evaluation over the value of self-evaluation. In this paper, more realistic alternative models for DEA cross-efficiency evaluation from both benevolent and aggressive points of view will be developed by considering the priority between self-evaluation and peer-evaluation of the assessed DMU. As discussed above, it is necessary to formulate two pairs of models to compute the interval cross efficiencies of DMUs. To this end, let $(\bar{θ_{1}^{*}}, \bar{θ_{2}^{*}}, \dots, \bar{θ_{n}^{*}})$ be the CCR interval efficiencies of n DMUs. The models will be constructed as follows: $Min / Max \sum_{j = 1}^{n} s_{dj}^{U_{1}} + s_{dj}^{U_{2}}$ (14) $\begin{matrix} s . t . & ω_{d}^{U^{T}} X_{j} (0) - μ_{d}^{U^{T}} Y_{j} (1) \geq 0, \\ j = 1, \dots, n; \\ ω_{d}^{U^{T}} X_{d} (0) = 1; \\ Y_{j} (1) = θ_{d}^{U *}; \\ ω_{d}^{U^{T}} X_{j} (0) - θ_{j}^{U *} μ_{d}^{U^{T}} Y_{j} (1) + s_{dj}^{U_{1}} = 0, \\ j = 1, \dots, n; \\ ω_{d}^{U^{T}} X_{j} (1) - θ_{j}^{U *} μ_{d}^{U^{T}} Y_{j} (0) + s_{dj}^{U_{2}} = 0, \\ j = 1, \dots, n; \\ ω_{d}^{U^{T}}, μ_{d}^{U^{T}} \geq 0, s_{dj}^{U_{1}}, s_{dj}^{U_{2}} \geq 0, \\ j = 1, \dots, n . \end{matrix}$ and $Min / Max \sum_{j = 1}^{n} s_{dj}^{L_{1}} + s_{dj}^{L_{2}}$ (15) $\begin{matrix} s . t . & ω_{d}^{L^{T}} X_{j} (0) - μ_{d}^{L^{T}} Y_{j} (1) \geq 0, \\ j = 1, \dots, n; \\ ω_{d}^{L^{T}} X_{d} (1) = 1; \\ μ_{d}^{L^{T}} Y_{j} (0) = θ_{d}^{L *}; \\ ω_{d}^{L^{T}} X_{j} (0) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (1) + s_{dj}^{L_{1}} = 0, \\ j = 1, \dots, n; \\ ω_{d}^{L^{T}} X_{j} (1) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (0) + s_{dj}^{L_{2}} = 0, \\ j = 1, \dots, n; \\ ω_{d}^{L^{T}}, μ_{d}^{L^{T}} \geq 0, s_{dj}^{L_{1}}, s_{dj}^{L_{2}} \geq 0, \\ j = 1, \dots, n . \end{matrix}$

The constraint conditions $ω_{d}^{L^{T}} X_{d} (1) = 1$ and $μ_{d}^{L^{T}} Y_{j} (0) = θ_{d}^{L *}$ in model (14) indicate that the weights for peer-evaluation are selected from the optimal solutions of model (11) for the DMU_d. $s_{dj}^{L_{1}}$ and $s_{dj}^{L_{2}}$ are the deviations between the interval cross-efficiency of DMU_j and the lower CCR interval efficiency of DMU_d. Based on the proof of Theorem 6, it holds that $s_{dj}^{L_{1}}, s_{dj}^{L_{2}} \geq 0$ ; therefore, the constraint conditions $ω_{d}^{L^{T}} X_{j} (0) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (1) + s_{dj}^{L_{1}} = 0$ and $ω_{d}^{L^{T}} X_{j} (1) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (0) + s_{dj}^{L_{2}} = 0$ are reasonable for all j = 1, ⋯ , n. Moreover, the constraint conditions $\begin{matrix} ω_{d}^{L^{T}} X_{j} (0) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (1) + s_{dj}^{L_{1}} = 0, \\ ω_{d}^{L^{T}} X_{j} (1) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (0) + s_{dj}^{L_{2}} = 0, \\ j = 1, \dots, n \end{matrix}$ show that the interval cross-efficiencies of all DMUs are evaluated by $ω_{d}^{L^{T}}, μ_{d}^{L^{T}}$ . This implies that DMU_d evaluates the other DMUs by the weights of its worst points. The objective of model (15) is to minimize or maximize the total deviations between the interval cross-efficiency of all DMUs and the lower CCR interval efficiency of DMU_d. By contrast, in model (14), DMU_d evaluates the other DMUs by the weights of its best points, where $ω_{d}^{U^{T}} X_{d} (0) = 1$ and $μ_{d}^{U^{T}} Y_{j} (1) = θ_{d}^{U *}$ in model (14) show that DMU_d assesses other DMUs by retaining its own best relative efficiency. $s_{dj}^{U_{1}}$ and $s_{dj}^{U_{2}}$ are the deviations between the interval cross-efficiency of DMU_j and the upper CCR interval efficiency of DMU_d. Based on the proof of Theorem 6, $s_{dj}^{U_{1}}, s_{dj}^{U_{2}} \geq 0$ ; therefore, the constraint conditions $ω_{d}^{U^{T}} X_{j} (0) - θ_{j}^{U *} μ_{d}^{U^{T}} Y_{j} (1) + s_{dj}^{U_{1}} = 0$ an $ω_{d}^{U^{T}} X_{j} (1) - θ_{j}^{U *} μ_{d}^{U^{T}} Y_{j} (0) + s_{dj}^{U_{2}} = 0$ are reasonable for all j = 1, ⋯ , n. Moreover, the constraint conditions $\begin{array}{l} ω_{d}^{L^{T}} X_{j} (0) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (1) + s_{d j}^{L_{1}} = 0, \\ _{d}^{L^{T}} X_{j} (1) - θ_{j}^{L *} μ_{d}^{L^{T}} Y_{j} (0) + s_{d j}^{L_{2}} = 0, \\ j = 1, \dots, n \end{array}$ show that the interval cross-efficiencies of all DMUs are evaluated by $ω_{d}^{U^{T}}, μ_{d}^{U^{T}}$ . Thus, the objective of model (14) is to minimize or maximize the total deviations between the interval cross-efficiency of all DMUs and the upper CCR interval efficiency of DMU_d. Let $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ be the optimal solutions of models (14) and (15), respectively. $\bar{θ_{dj}}$ is computed by formula (13) with $μ_{d}^{U^{*}}, ω_{d}^{U^{*}}$ and $μ_{d}^{L^{*}}, ω_{d}^{L^{*}}$ . If the objections of models (14) and (15) are the minimizations (resp., maximization), then we consider the pair of modes as benevolent (resp., aggressive) with the peer-evaluations of all DMUs being as close as possible to their self-evaluations.

To a certain extent, the results of models (14) and (15) can be used to indicate the prior relationship between cross efficiency and CCR interval efficiency. Based on the prior relationship, a model will be formulated for aggregating the cross-efficiency matrix.

5 Methods for aggregating the cross-evaluation matrix and ranking the DMUs

As a result of the above models, there are n cross-efficiency scores for each DMU, which form cross-efficiency matrixes. In traditional models, the average cross-efficiency score for each DMU as

$\begin{matrix} \bar{E_{j}} & = & \frac{1}{n} \sum_{d = 1}^{n} \bar{θ_{dj}} \\ = & [\frac{1}{n} \sum_{d = 1}^{n} θ_{dj}^{L}, \frac{1}{n} \sum_{d = 1}^{n} θ_{dj}^{U}] \end{matrix}$ (16) is adopted. However, the models merely aggregate them without considering their relative importance. In this section, a new aggregation method based on the likelihood between the interval cross-efficiency and the CCR interval efficiency will be proposed.

Let $\bar{ϑ} = (\bar{θ_{11}}, \bar{θ_{22}}, \dots, \bar{θ_{nn}})$ be the CCR interval efficiency vector of the n DMUs, and $\bar{σ_{d}} = (\bar{θ_{d 1}}, \bar{θ_{d 2}}, \dots, \bar{θ_{dn}})$ the interval cross-efficiency vector of the n DMUs, which are evaluations for all DMUs by DMU_d. As the aim of the evaluated DMU is a more favorable final evaluation vector, the two interval efficiency vectors should be as close to each other as possible. Wang et al. [31] proposed the weighted least-square deviation model to make the deviations as small as possible. The model for handling interval data will now be constructed. In practical applications, there are great differences between peer-evaluation efficiency and self-evaluation efficiency. In this case, both the DMU and the decision makers have reason to reject such unfair peer evaluation results. This amounts to assigning a zero weight to these poor peer-evaluated efficiencies. Therefore, the corresponding model is established to solve that problem. Let $τ_{d} = \sum_{j = 1}^{n} P (\bar{θ_{dj}} \geq \bar{θ_{jj}})$ be the general likelihood between $\bar{ϑ}$ and $\bar{φ_{d}}$ . The above models can be set up as follows: $Min J = \sum_{d = 1}^{n} (w_{d} τ_{d})^{2}$ (17) $\begin{matrix} s . t . & \sum_{d = 1}^{n} w_{d} = 1 \\ 0 \leq w_{d} \leq 1, d = 1, 2, \dots, n \end{matrix}$

The optimal solution of model (17) is $w_{d}^{*} = \frac{1 / τ_{d}^{2}}{\sum_{j = 1}^{n} (1 / τ_{j}^{2})}$ (18)

By property (b) of the likelihood, τ_d = 0 indicates that $\bar{θ_{dj}}$ and $\bar{θ_{jj}}$ are disjoint for all j = 1, ⋯ , n. This implies that the peer evaluation provided by DMU_d has no reference value. Thus, zero weight is assigned by the optimal solution of model (17).

Let w₁, w₂ ⋯ , w_n be the aggregation weights computed by model (17). The cross-efficiency value for each DMU by aggregating can be expressed as follows:

$\begin{matrix} \bar{E_{j}} & = & \sum_{d = 1}^{n} \bar{θ_{dj}} w_{d} \\ = & [\sum_{d = 1}^{n} θ_{dj}^{L} w_{d}, \sum_{d = 1}^{n} θ_{dj}^{U} w_{d}] \end{matrix}$ (19)

By aggregating the cross-efficiency matrix, we obtain the efficiency vector $\bar{E} = (\bar{E_{1}}, \bar{E_{2}}, \dots, \bar{E_{n}})$ . Based on the efficiency vector, the n DMUs can be ranked by the likelihood ranking method [19].

The steps of the cross-efficiency method based on likelihood are as follows.

Step 1: Computing the interval self-efficiency of each DMU by models (10) and (11).

Step 2: Computing the cross-efficiency matrix from the benevolent or aggressive perspective by models (14) and (15).

Step 3: Aggregating the cross-efficiency matrix using the weigh vector obtained by model (17).

Step 4: Ranking the efficiency vector $\bar{E}$ by the likelihood ranking method [19].

6 Numerical example

In this section, the feasibility of the above methods is proved through the analysis of a numerical example from Kao and Liu [32]. The example is about 24 banks in Taiwan. The commercial banks could be regarded as financial intermediaries whose main business is to borrow funds from depositors and lend to others. There are three input indicators and three output indicators, as follows:

x₁= Total deposit;

x₂= Interest expense;

x₃= Non-interest expense;

y₁= Total loans;

y₂= Interest income;

y₃= Non-interest income.

The data set for the 24 banks is shown in Table 1.

Table 1
Interval forecasts of financial data for the 24 commercial banks in Taiwan

Banks x ₁ x ₂ x ₃ y ₁ y ₂ y ₃

1 L 788670.6 40241.93 11811.94 724380.14 60822.39 7094.72

U 840859.3 43683.96 12022.59 773314.72 66231.62 7623.20

2 L 926235.9 42863.30 15496.88 786268.25 66067.14 12826.69

U 1014339.3 49421.62 17952.67 850782.56 72605.03 14022.96

3 L 895985.4 40469.85 13030.99 770236.24 57395.59 11691.72

U 989805.9 43463.23 13986.15 861676.05 64209.40 13079.72

4 L 458981.8 29869.43 6267.73 418079.49 44354.53 5663.31

U 516654.9 32997.12 6924.03 467712.46 49620.15 6335.638

5 L 235351.1 7881.37 2820.19 169336.03 11427.47 1618.14

U 259995.1 8706.64 3115.50 189439.03 12784.10 1810.24

6 L 256277.5 8499.21 1163.29 200432.66 11234.13 2845.69

U 283112.9 9389.18 1285.10 224227.34 12567.80 3183.52

7 L 108792.8 5421.99 1405.51 80058.74 7848.88 302.15

U 120184.7 5791.03 1596.83 89563.04 8612.78 338.02

8 L 78795.8 4052.71 2488.02 47904.99 4975.08 249.43

U 85261.1 4430.68 2720.07 53079.75 5459.29 279.05

9 L 383560.8 27531.87 5352.50 325799.31 35344.23 5393.14

U 411675.2 29694.05 5744.83 353146.19 3831.94 5963.52

10 L 507635.2 22708.68 3727.91 402910.43 37609.65 3298.46

U 560790.8 25086.55 4118.27 441709.21 42074.53 3543.68

11 L 166251.0 8518.76 3621.04 147175.58 11443.13 1671.40

U 182253.8 8960.14 3886.46 167355.33 12593.50 1869.82

12 L 176709.8 8324.76 1554.94 158536.00 11591.02 710.44

U 194858.3 8804.19 1644.49 177494.95 12967.06 794.78

13 L 432487.9 21002.18 2693.84 349537.63 29012.39 4799.48

U 477774.5 23201.36 2975.92 391033.55 32456.64 5369.26

14 L 717622.8 32432.93 5207.24 591874.45 45500.26 3017.95

U 770223.5 35154.16 5577.30 662139.76 50901.89 3376.23

15 L 101281.3 5491.09 4927.33 78813.65 7421.86 578.59

U 111886.6 6066.08 5443.28 87607.24 8302.96 647.27

16 L 126969.3 7023.18 3063.38 122170.19 9147.28 1698.28

U 141594.1 7758.59 3384.15 136673.82 10233.21 1899.90

17 L 145850.9 7933.35 5981.42 127122.12 12139.73 757.21

U 164181.9 8573.00 6607.75 144379.11 13437.21 8487.11

18 L 143347.3 8101.26 2799.39 126680.92 11828.34 2366.53

U 165099.7 8949.56 2904.32 144429.80 13232.56 2647.48

19 L 190173.5 9307.44 661.98 145200.48 13106.07 2027.19

U 210086.9 10282.04 731.29 162438.18 14661.98 2267.85

20 L 216899.7 9514.11 1910.87 149165.08 12403.45 4760.57

U 239611.8 10510.35 2110.96 166873.45 13875.95 5016.70

21 L 131203.4 6496.85 3852.75 101543.04 8989.75 1593.30

U 141507.4 7041.09 4119.01 108383.65 9644.39 1718.14

22 L 214511.8 10666.97 4651.90 171767.41 19395.85 3022.84

U 239220.0 11895.62 5187.71 194236.44 21933.04 3418.26

23 L 155200.1 9242.28 14248.46 91728.20 10657.76 1038.97

U 167934.4 10000.6 15577.34 105261.87 12003.14 1190.32

24 L 153476.5 7816.278 1619.78 144453.15 11601.73 918.05

U 166608.1 8307.35 1721.55 157371.73 12320.21 969.98

Banks		x ₁	x ₂	x ₃	y ₁	y ₂	y ₃
1	L	788670.6	40241.93	11811.94	724380.14	60822.39	7094.72
	U	840859.3	43683.96	12022.59	773314.72	66231.62	7623.20
2	L	926235.9	42863.30	15496.88	786268.25	66067.14	12826.69
	U	1014339.3	49421.62	17952.67	850782.56	72605.03	14022.96
3	L	895985.4	40469.85	13030.99	770236.24	57395.59	11691.72
	U	989805.9	43463.23	13986.15	861676.05	64209.40	13079.72
4	L	458981.8	29869.43	6267.73	418079.49	44354.53	5663.31
	U	516654.9	32997.12	6924.03	467712.46	49620.15	6335.638
5	L	235351.1	7881.37	2820.19	169336.03	11427.47	1618.14
	U	259995.1	8706.64	3115.50	189439.03	12784.10	1810.24
6	L	256277.5	8499.21	1163.29	200432.66	11234.13	2845.69
	U	283112.9	9389.18	1285.10	224227.34	12567.80	3183.52
7	L	108792.8	5421.99	1405.51	80058.74	7848.88	302.15
	U	120184.7	5791.03	1596.83	89563.04	8612.78	338.02
8	L	78795.8	4052.71	2488.02	47904.99	4975.08	249.43
	U	85261.1	4430.68	2720.07	53079.75	5459.29	279.05
9	L	383560.8	27531.87	5352.50	325799.31	35344.23	5393.14
	U	411675.2	29694.05	5744.83	353146.19	3831.94	5963.52
10	L	507635.2	22708.68	3727.91	402910.43	37609.65	3298.46
	U	560790.8	25086.55	4118.27	441709.21	42074.53	3543.68
11	L	166251.0	8518.76	3621.04	147175.58	11443.13	1671.40
	U	182253.8	8960.14	3886.46	167355.33	12593.50	1869.82
12	L	176709.8	8324.76	1554.94	158536.00	11591.02	710.44
	U	194858.3	8804.19	1644.49	177494.95	12967.06	794.78
13	L	432487.9	21002.18	2693.84	349537.63	29012.39	4799.48
	U	477774.5	23201.36	2975.92	391033.55	32456.64	5369.26
14	L	717622.8	32432.93	5207.24	591874.45	45500.26	3017.95
	U	770223.5	35154.16	5577.30	662139.76	50901.89	3376.23
15	L	101281.3	5491.09	4927.33	78813.65	7421.86	578.59
	U	111886.6	6066.08	5443.28	87607.24	8302.96	647.27
16	L	126969.3	7023.18	3063.38	122170.19	9147.28	1698.28
	U	141594.1	7758.59	3384.15	136673.82	10233.21	1899.90
17	L	145850.9	7933.35	5981.42	127122.12	12139.73	757.21
	U	164181.9	8573.00	6607.75	144379.11	13437.21	8487.11
18	L	143347.3	8101.26	2799.39	126680.92	11828.34	2366.53
	U	165099.7	8949.56	2904.32	144429.80	13232.56	2647.48
19	L	190173.5	9307.44	661.98	145200.48	13106.07	2027.19
	U	210086.9	10282.04	731.29	162438.18	14661.98	2267.85
20	L	216899.7	9514.11	1910.87	149165.08	12403.45	4760.57
	U	239611.8	10510.35	2110.96	166873.45	13875.95	5016.70
21	L	131203.4	6496.85	3852.75	101543.04	8989.75	1593.30
	U	141507.4	7041.09	4119.01	108383.65	9644.39	1718.14
22	L	214511.8	10666.97	4651.90	171767.41	19395.85	3022.84
	U	239220.0	11895.62	5187.71	194236.44	21933.04	3418.26
23	L	155200.1	9242.28	14248.46	91728.20	10657.76	1038.97
	U	167934.4	10000.6	15577.34	105261.87	12003.14	1190.32
24	L	153476.5	7816.278	1619.78	144453.15	11601.73	918.05
	U	166608.1	8307.35	1721.55	157371.73	12320.21	969.98

In Table 1, “L” is the lower limit of the interval number, and “U” is the upper limit of the interval number.

The three inputs and three outputs are expressed in intervals rather than single values due to their uncertain nature. In order to illustrate the change of extended production facets, four different production possibility sets are chosen. The best relative efficiency of the optimal points of the 24 DMUs are shown in Table 2. The third, fifth, seventh, and ninth columns indicate whether the DMU attains the Pareto Optimal under the different production possibility sets. It follows that E (1, 0) consists of all DMUs, which implies that each DMU has reached the Pareto optimal by taking the worst points of all DMUs as a reference. It is clear that the results of this evaluation are meaningless. Similarly, it is noted that E (0.5, 0.5) is composed of DMU₁, DMU₂, DMU₃, DMU₄, DMU₅, DMU₆, DMU₉, DMU₁₀, DMU₁₁, DMU₁₂, DMU₁₃, DMU₁₄, DMU₁₆, DMU₁₇, DMU₁₈, DMU₁₉, DMU₂₀, DMU₂₂, and DMU₂₄; E (0.1, 0.8) is composed of DMU₁, DMU₃, DMU₄, DMU₆, DMU₉, DMU₁₀, DMU₁₂, DMU₁₃, DMU₁₄, DMU₁₆, DMU₁₇, DMU₁₈, DMU₁₉, DMU₂₀, DMU₂₂, and DMU₂₄; E (0, 1) is composed of DMU₃, DMU₄, DMU₆, DMU₉, DMU₁₀, DMU₁₂, DMU₁₃, DMU₁₄, DMU₁₆, DMU₁₇, DMU₁₉, DMU₂₀, and DMU₂₂. Therefore, E (1, 0) ⊇ E (0.5, 0.5) ⊇ E (0.1, 0.8) ⊇ E (0, 1), and E (0, 1) is the best production facet.

Table 2

Relative efficiency under different production reference sets

Banks	$θ_{0}^{U}$	Y/N	$θ_{0}^{U}$	Y/N	$θ_{0}^{U}$	Y/N	$θ_{0}^{U}$	Y/N
	(1, 0)		(0.5, 0.5)		(0.1, 0.8)		(0, 1)
1	1.753	Y	1.069	Y	1.006	Y	0.980	N
2	1.805	Y	1.084	Y	0.982	N	0.977	N
3	1.911	Y	1.104	Y	1.067	Y	1.000	Y
4	1.770	Y	1.122	Y	1.034	Y	1.000	Y
5	1.728	Y	1.106	Y	0.915	N	1.000	N
6	1.605	Y	1.111	Y	1.032	Y	1.000	Y
7	1.692	Y	0.969	N	0.904	N	0.878	N
8	1.435	Y	0.791	N	0.735	N	0.710	N
9	3.933	Y	2.115	Y	0.968	Y	0.929	Y
10	1.974	Y	1.111	Y	1.032	Y	1.000	Y
11	1.575	Y	1.069	Y	0.992	N	0.971	N
12	1.660	Y	1.105	Y	1.031	Y	1.000	Y
13	1.646	Y	1.111	Y	1.032	Y	1.000	Y
14	1.672	Y	1.077	Y	1.007	Y	1.000	Y
15	1.611	Y	0.963	N	0.900	N	0.871	N
16	1.552	Y	1.116	Y	1.033	Y	1.000	Y
17	1.805	Y	1.951	Y	1.238	Y	1.000	Y
18	1.740	Y	1.125	Y	1.023	Y	0.992	N
19	1.727	Y	1.111	Y	1.032	Y	1.000	Y
20	1.555	Y	1.103	Y	1.029	Y	1.000	Y
21	1.582	Y	0.938	N	0.868	N	0.849	N
22	2.191	Y	1.122	Y	1.035	Y	1.000	Y
23	1.384	Y	0.817	N	0.754	N	0.729	N
24	1.679	Y	1.087	Y	1.025	Y	0.864	N

In this paper, self-evaluation of DMU is based on the best production facet. Based on self-evaluation, we introduce benevolent and aggressive models to select the appropriate weights for evaluating other DMUs. The evaluation results of the 24 banks are presented in matrix form: the benevolent cross-evaluation matrix and aggressive cross-evaluation matrix. More weight will be distributed to the DMU whose peer evaluation is closer to the self-evaluation. Thus, the cross evaluation matrices are aggregated by model (17), and the results are shown in the second and the fourth columns of Table 3. Moreover, we compute the average score of the cross-evaluation matrix. The data in the third column is the average vector of the benevolent cross-evaluation matrix. The average vector of the aggressive cross evaluation matrix is shown in the fifth column of Table 3. As shown in Table 3, the results of aggregation and the average are not significantly different in the benevolent cross-evaluation matrix. However, the difference between the fourth and the fifth columns is greater. This is due to the fact that the DMU regards peers as competitors and may provide poor peer evaluation. For example, τ₂ is 10.836 and τ₁₉ is 1.172, which implies that the peer evaluation given by DMU₂ is far lower than self-evaluation, whereas DMU₁₉ provides the peer evaluation that is closest to self-evaluation. Hence, DMU₁₉ should have the largest aggregation weight, in contrast with DMU₂, whose cross-efficiencies should be given the lowest aggregation weight.

Table 3

Results of different aggregates of cross-efficiency matrices

Banks	Benevolent	Benevolent	Aggressive	Aggressive
	vector	Average	vector	Average
		vector		vector
1	(0.790,0.911)	(0.827,0.952)	(0.690,0.796)	(0.794,0.915)
2	(0.731,0.896)	(0.766,0.938)	(0.619,0.772)	(0.735,0.904)
3	(0.740,0.900)	(0.787,0.957)	(0.632,0.765)	(0.756,0.919)
4	(0.776,0.968)	(0.780,0.972)	(0.713,0.885)	(0.757,0.944)
5	(0.666,0.824)	(0.722,0.892)	(0.578,0.715)	(0.692,0.855)
6	(0.745,0.911)	(0.815,0.988)	(0.689,0.851)	(0.788,0.974)
7	(0.673,0.814)	(0.694,0.840)	(0.627,0.755)	(0.672,0.813)
8	(0.523,0.627)	(0.544,0.653)	(0.422,0.506)	(0.514,0.618)
9	(0.477,0.734)	(0.568,0.721)	(0.243,0.670)	(0.530,0.730)
10	(0.770,0.941)	(0.795,0.969)	(0.773,0.950)	(0.782,0.955)
11	(0.708,0.852)	(0.756,0.913)	(0.570,0.677)	(0.717,0.864)
12	(0.757,0.914)	(0.807,0.973)	(0.696,0.832)	(0.781,0.941)
13	(0.735,0.909)	(0.765,0.945)	(0.711,0.878)	(0.748,0.924)
14	(0.737,0.888)	(0.779,0.940)	(0.698,0.843)	(0.757,0.914)
15	(0.581.0.716)	(0.619,0.762)	(0.424,0.524)	(0.577,0.711)
16	(0.708,0.879)	(0.757,0.940)	(0.551,0.683)	(0.714,0.886)
17	(0.669,0.937)	(0.708,0.967)	(0.507,0.700)	(0.663,0.919)
18	(0.734,0.928)	(0.755,0.958)	(0.634,0.784)	(0.727,0.917)
19	(0.738,0.910)	(0.759,0.936)	(0.764,0.944)	(0.747,0.925)
20	(0.694,0.846)	(0.718,0.879)	(0.637,0.781)	(0.697,0.854)
21	(0.650,0.751)	(0.686,0.792)	(0.512,0.592)	(0.649,0.750)
22	(0.766,0.966)	(0.775,0.978)	(0.675,0.852)	(0.750,0.945)
23	(0.445,0.547)	(0.465,0.573)	(0.306,0.376)	(0.436,0.538)
24	(0.803,0.927)	(0.843,0.977)	(0.729,0.832)	(0.814,0.942)

After effective aggregation, the cross-efficiency matrix is transformed into vector form. Notably, the self-evaluation efficiency vector is measured by the maximum production possibility set. Table 4 shows the various efficiency vectors and sorting of the 24 banks by likelihood ranking methods. As is known, when the input and output of DMUs is real data, the results of the cross-efficiency models will satisfy the following properties: (I) Self-efficiency is higher than cross-efficiency for every DMU; (II) the diagonal elements of the cross-efficiency matrix are the n self-efficiencies. These two properties reflect that self-evaluation efficiency is higher than peer evaluation, and the results of self-evaluation remain unchanged in peer evaluation. These are the basic principles of peer evaluation. Therefore, the cross-efficiency values are best to satisfy these two properties by interval numbers. In this paper, Theorem 6 shows that the upper and lower bounds of interval efficiency of each DMU are smaller than the values obtained by the self-evaluation model. This conclusion is consistent with property (I). Moreover, Theorem 5 shows that the diagonal elements of both the benevolent and aggressive cross-efficiency matrices are the self-efficiency vector. This conclusion is in accordance with property (II). Therefore, the cross-efficiency matrix of both the aggressive and benevolent models satisfy the two properties. Furthermore, it follows from Table 4 that the upper and lower limits of interval cross-efficiencies of DMUs computed by benevolent models are larger than those obtained from aggressive models. This shows that the benevolent (resp., aggressive) model is appropriate for maximizing (resp., minimizing) the interval cross efficiency of the other DMUs. There is large difference in ranking orders between the benevolent and aggressive models. For example, even though little difference is found in the interval efficiencies of DMU₁₃ between the benevolent and aggressive models, DMU₁₃ is ranked 14th by the aggressive model, whereas it is ranked 21st by the benevolent model. This shows that different evaluation strategies will result in different ranking orders. Therefore, the decision-makers have more flexibility in choosing their models according to their decision preferences.

Table 4

Comparison of efficiency in different models

Banks	Self-evaluation	rank	Aggressive	rank	Benevolent	rank	Cross-evaluation	rank
	by model (8) and (9)		Cross-evaluation		Cross-evaluation		by chen’s model [17]
1	(0.85,0.98)	21	(0.69,0.79)	15	(0.79,0.91)	22	(0.88,0.97)	20
2	(0.79,0.97)	12	(0.61,0.77)	11	(0.73,0.89)	13	(0.87,0.95)	18
3	(0.83,1.00)	16	(0.63,0.76)	12	(0.74,0.90)	16	(0.87,0.96)	19
4	(0.80,1.00)	15	(0.71,0.88)	22	(0.77,0.96)	24	(0.90,0.99)	23
5	(0.80,1.00)	9	(0.57,0.71)	9	(0.66,0.82)	7	(0.72,0.87)	7
6	(1.00,1.00)	24	(0.68,0.85)	18	(0.74,0.91)	18	(0.83,0.96)	15
7	(0.72,0.87)	6	(0.62,0.75)	10	(0.67,0.81)	6	(0.71,0.84)	4
8	(0.59,0.71)	2	(0.42,0.50)	2	(0.52,0.62)	2	(0.53,0.67)	1
9	(0.82,0.92)	3	(0.24,0.67)	4	(0.43,0.73)	3	(0.51,0.91)	5
10	(0.82,1.00)	23	(0.77,0.95)	24	(0.77,0.94)	20	(0.84,0.96)	17
11	(0.80,0.97)	7	(0.57,0.67)	7	(0.70,0.85)	9	(0.80,0.93)	9
12	(0.84,1.00)	17	(0.69,0.83)	17	(0.75,0.91)	19	(0.81,0.97)	13
13	(0.80,1.00)	18	(0.71,0.87)	21	(0.73,0.90)	14	(0.86,0.94)	16
14	(0.82,1.00)	13	(0.69,0.84)	19	(0.73,0.88)	12	(0.79,0.93)	8
15	(0.70,0.87)	4	(0.42,0.52)	3	(0.58.0.71)	4	(0.58,0.81)	3
16	(0.80,1.00)	8	(0.55,0.68)	6	(0.70,0.87)	9	(0.82,0.97)	11
17	(0.79,1.00)	10	(0.50,0.70)	8	(0.66,0.93)	11	(0.76,0.96)	10
18	(0.77,0.99)	11	(0.63,0.78)	13	(0.73,0.92)	17	(0.89,0.97)	24
19	(0.80,1.00)	19	(0.76,0.94)	23	(0.73,0.91)	15	(0.84,0.95)	14
20	(0.85,1.00)	14	(0.63,0.78)	14	(0.69,0.84)	8	(0.81,0.96)	12
21	(0.73,0.84)	5	(0.51,0.59)	5	(0.65,0.75)	5	(0.72,0.85)	6
22	(0.85,0.98)	21	(0.69,0.79)	15	(0.79,0.91)	22	(0.88,0.97)	20
23	(0.79,0.97)	12	(0.61,0.77)	11	(0.73,0.89)	13	(0.87,0.95)	18
24	(0.83,1.00)	16	(0.63,0.76)	12	(0.74,0.90)	16	(0.87,0.96)	19

There is a connection between the α-level based approach of fuzzy DEA category and interval DEA category since α-level based approach generally converts the fuzzy DEA model into a pair of parametric models for calculating the lower and upper bounds of the efficiency at a given α-lever, that is to say, interval DEA models can be a special case of the α-level based approach of fuzzy DEA models for a given α-level. So the results obtained from the method in this paper are compared with the results obtained by Chen’s method [17]. The conclusions are as follows:

In this paper, the models are proposed from both aggressive and benevolent perspectives. Compared with Chen’s model, our evaluation methods are more comprehensive. For example, as shown in Table 4, DMU₁₃, DMU₁₄, DMU₁₈, and DMU₂₀, are differently ranked by the aggressive and benevolent models. On the contrary, the results in Chen’s model do not reflect this difference.

From Table 4, it follows that the interval cross efficiencies calculated by Chen’s model do not fully satisfy property (I). For instance, the lower bounds of interval efficiencies of DMU₁, DMU₂, DMU₃, DMU₄, DMU₁₀, DMU₁₃, DMU₁₆, DMU₁₈, DMU₁₉, DMU₂₂, and DMU₂₄ are higher than the corresponding results of the self-evaluation. Moreover, the diagonal elements of the cross-efficiency matrix calculated by Chen’s model do not correspond to the self-efficiency vector. This implies that the results of Chen’s model do not conform with property (II). On the contrary, in this paper, the interval cross efficiency has been redefined and the models have been constructed by considering properties (I) and (II). This ensures that the models do not violate the basic principles of peer evaluation.

From the above it follows that the maximum production possibility set contains all points of DMUs, and the corresponding production facet contains the least effective DMUs. Therefore, it is more fair and reasonable to evaluate the self-efficiencies of DMUs under the maximum production possibility set. The aggressive and benevolent cross-efficiency models, which are set up by making the self-evaluation efficiencies as targets, can be consistent with the properties of the traditional cross efficiency model. Moreover, they can effectively discriminate the DMUs and evaluate the DMUs from different perspectives.

7 Conclusion

Many DEA models have been developed to evaluate the performance of DMUs with interval inputs and outputs. In general, when the input and output of DMUs are interval data, the production conditions are variable, that is, the production possibility sets are variable. In the conventional setting with the production possibility set in a fixed manner, the models are unable to reflect the efficiency changes and characterize the facets of DMUs. When the production possibility set is no longer well-defined, the results are paradoxical and self-evaluation and peer evaluation of DMUs are not well conducted. Moreover, the conventional models for interval data have only been proposed to evaluate DMUs by themselves, and seldom consider peer evaluations.

In this paper, the theoretical framework for interval DEA models is enriched as follows: (1) The input and output parameters are introduced to characterize the variability of production possibility set, and extended production facets are determined according to the different sets. Furthermore, the relationship of the different facets is discussed. It follows that as the number of points in the production possibility set increases, fewer effective units are evaluated. Therefore, it is more reasonable to evaluate the decision units in such a facet according to the production possibility set with maximum number of points. Thus, the interval self-efficiencies of DMUs are calculated under the maximum production possible set. (2) Aggressive and benevolent cross-efficiency models are established by making the self-evaluation efficiencies as targets. According to the interval efficiency value, the likelihood is used to reflect the difference between the self-evaluation and the peer evaluation. The operation (max) of two intervals is introduced to ensure that the results are consistent with conventional results by crisp data. The pairs of cross-efficiency models constructed from both maximization and minimization of the difference values between the self-evaluation and the peer evaluation can provide more comprehensive and reasonable evaluation for decision makers. Moreover, the weight allocation method, which is more advantageous to DMU, is used to aggregate the interval cross-efficiency matrices.

An example is used to illustrate the models. The results not only explain the effect of the parameter selection on the extended production facets, but also demonstrate the applicability of our cross-efficiency models, which provides more choices for the decision-makers. Therefore, the benevolent and aggressive models with interval data in this paper can be seen as improvements and extensions of the traditional cross-efficiency models with real data. This makes them meaningful contributions to uncertain DEA cross-efficiency evaluation.

Finally, interval efficiency provides a more comprehensive assessment of DMU than the traditional DEA efficiency. Therefore, it is expected to be more widely applied in even more diverse contexts. Notably, the variability of production possibility sets can be considered with the returns to scale. The model can also be extended to a stochastic model by considering that the distribution of the points in the interval is random. This is omitted due to space limitations.

Footnotes

Acknowledgments

This research is supported by National Natural Science Foundation of China (Nos. 71371053 and 71471125), Humanities and Social Science Foundation of the Ministry of Education (No. 14YJC630056) and Fujian Provincial Education Department (No. Jb14001).

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