Ranking DMUs by using interval efficiencies in data envelopment analysis

Abstract

Efficiency is a relative measure that allows assessment across different ranges. Evaluating the performance of decision-making units (DMUs) from an optimistic perspective yields the best relative efficiency (optimistic efficiency), which establishes an efficiency frontier. Conversely, evaluating from a pessimistic perspective produces the worst relative efficiency (pessimistic efficiency) and creates an inefficiency frontier. This study examines the efficiency of DMUs in two scenarios and proposes models for adjustment coefficient. The pessimistic and optimistic efficiencies are adjusted to the lower and upper bounds of the DMUs based on the adjustment coefficient, enabling determination of efficiency intervals for all DMUs, as well as evaluation and ranking. A Hurwicz criterion-based approach is introduced and applied to compare and rank the interval efficiencies of DMUs. Two numerical examples are examined using the proposed DEA adjustment coefficient models to demonstrate its potential application and validity.

Keywords

Data envelopment analysis interval efficiency adjustment coefficient model ranking

1 Introduction

Data envelopment analysis (DEA) is an effective tool to evaluate the performance of decision-making units (DMUs) based on multiple inputs and outputs. In 1978, Charnels, Cooper and Rhodes measured the efficiency (CCR efficiency) by the ratio of total weighted outputs to total weighted inputs on condition that the similar ratios of each DMU do not exceed the value of 1. Therefore, the CCR ratio model is identified as the best relative efficiency or optimistic efficiency.

Meanwhile, self-evaluation-based CCR efficiency leads to multiple DMUs with an efficiency score of one, making it impossible to completely rank all DMUs. To this end, researchers have proposed numerous DEA methods to comprehensively rank the performance of all DMUs. Among these methods, DEA cross-efficiency has gained widespread usage in evaluating efficiency, mainly due to its robust discriminatory ability [1]. However, cross-efficiency may be less effective as the DEA optimal weights are not unique. Therefore, in order to address the issue of non-uniqueness in cross-efficiency, secondary goal models were built to make them more unique, some classic secondary goal models such as the aggressive and benevolent cross-efficiency models [2, 3], the neutral cross-efficiency model [4], and the DEA game cross-efficiency model [5]. In addition, some researchers have used the aggregation method and aggregation weights of decision-makers (DM) in the cross-efficiency evaluation. Fu and Li [6] delved into the endogenous preference structure within the cross-efficiency matrix. Shi et al. [7] proposed a neutral cross-efficiency evaluation method based on interval reference points to account for bounded rationality. Wu et al. [8] took a unique approach by synthesizing information from multiple perspectives, introducing an innovative synthesis method that ranks DMUs using Shannon entropy of cross-efficiency scores derived from satisfaction and consensus perspectives. Pendharkar [9] applied the principle of maximum decisional efficiency to compute cross-efficiency scores for input-oriented and output-oriented frontier efficiency models, thereby making more precise analysis. For more information of cross-efficiency aggregation, readers can refer to the literature [1 , 10–12].

Traditional DEA models are typically constructed to optimize the performance of DMUs from an optimistic perspective. The best relative efficiency, or optimistic efficiency, is defined as the maximum ratio of weighted outputs to weighted inputs under certain constraints. However, it is still important to consider the worst relative efficiency, or pessimistic efficiency, which represents the minimum ratio of weighted outputs to inputs under those same constraints [13]. The consideration of only the best or worst relative efficiency while neglecting the other can lead to biased results. Hence, applying both perspectives can comprehensively evaluate the two extreme performances of every DMU [14].

In recent years, researchers have been trying to combine the two extreme efficiencies to evaluate DMU performance. To be specific, Doyle et al. [15] and Entani et al. [16] measured DMU efficiency from both optimistic and pessimistic perspectives. They shared a similar model structure. To define the efficiency interval for each DMU, their models should be the pessimistic efficiency of each DMU under the constraint that the maximum optimistic efficiency remains 1. As a result, only one input and one output were used in their models, to calculate the pessimistic efficiency of each DMU. Furthermore, both models failed to identify DEA inefficient units. Therefore, to make up for such deficiencies, Wang et al. [14] proposed the worst relative efficiency model by minimizing the DMU efficiency on condition that efficiency is no less than 1. In doing so, they tried to get the pessimistic efficiency as well as a geometric average efficiency. However, Wang et al. also failed to identify DEA efficient and inefficient units. After that, Wang and Yang [17] introduced a virtual anti-ideal DMU to find the lower bound of the efficiency interval, with upper bound set as 1. Unfortunately, their model did not include the property of unit-invariance [18]. Recently, Liu and Wang [18] considered the best and worst DMUs to build DEA models from both optimistic and pessimistic perspectives, to obtain normalized efficiency intervals. However, evaluating normalized interval efficiency would change the ranking order of DMUs, which has no proper explanation. In addition, as for DMUs with imprecise input/output data, Jahed et al. [19] proposed an imprecise DEA model from both optimistic and pessimistic perspectives to measure DMU performance. Additionally, they developed a new fuzzy DEA model to deal with fuzzy data. Azizi and Jahed [20], Ahmady et al. [21] and Azizi et al. [22] conducted more studies on measuring DMU performance with imprecise data. In addition to theoretical research, readers will find numerous applied studies on the double frontier cross-efficiency models that may be of interest [13 , 23–30].

At present, there are not enough studies on how to evaluate DEA and optimize performance measurement. Therefore, this study proposes a simpler and more effective way from both optimistic and pessimistic perspectives to measure DMU performance and efficiency within intervals. On such basis, new DEA adjustment coefficient models are built to identify the range of interval efficiency and pessimistic interval cross-efficiency via the most unfavorable weights of each DMU. In doing so, the pessimistic efficiencies of DMUs are adjusted to the lower bounds of efficiencies, so the best and worst relative efficiencies form up an interval to comprehensively measure DMU performance. To testify the validity of such approach, this study puts forth a theorem on measuring the best and worst relative efficiencies within a unified DEA model framework.

In this paper, we consider the efficiency of DMUs under two scenarios, and propose DEA adjustment coefficient models. The main contributions of this study can be summarized as follows. First, it can identify both DEA efficient and inefficient DMUs; the former constitutes the efficiency frontier, and the latter the inefficiency frontier, covering all the DEA unspecified DMUs. Second, this study puts forth a theorem proving that a unified DEA model produces consistent efficiency intervals which further prove the effectiveness of our method. Third, the adjustment coefficient only needs to be solved once to adjust the pessimistic efficiency of each DMU, so as to obtain the lower bound of efficiency interval— simpler than other interval DEA models. Last, the efficiency intervals are in line with the optimistic and pessimistic efficiencies with consistent ranking orders.

The structure of this study is as follows: Section 2 introduces how basic DEA models measure the best and worst relative efficiencies of DMUs. Section 3 briefly explains cross-efficiency, pessimistic cross-efficiency and interval cross-efficiency models, then puts forward the pessimistic interval cross-efficiency models. Section 4 proposes DEA adjustment coefficient models to measure interval efficiencies of DMUs. Section 5 introduces the interval evaluation approach to compare and rank interval efficiencies. Section 6 presents two examples of the DEA adjustment coefficient models. Finally, Section 7 is the conclusion.

2 DEA models for measuring two extreme DMU performances

2.1 The best relative efficiency— optimistic efficiency

In terms of evaluating n DMUs, DMU_j (j = 1, …, n) consumes positive value x_ij (i = 1, …, m) of m inputs and produces positive value y_rj (r = 1, …, s) of r outputs. The optimistic efficiency is measured by the CCR model [31] which calculates the best relative efficiencies of DMUs within the range of ≤1. For a given DMU₀, the optimistic efficiency θ₀ is obtained through a linear programming (LP) as follows: $\begin{matrix} Max θ_{0} = \sum_{r = 1}^{s} u_{r} y_{r 0} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{i 0} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} ⩽ 0, j = 1, \dots, n, \\ u_{r}, v_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m, \end{matrix}$ (1)

where u_r (r = 1, …, s) and v_i (i = 1, …, m) are decision variables. Suppose there are positive weights to make $θ_{0}^{*} = 1$ , DMU₀ is deemed DEA efficient or optimistic efficient; otherwise, it is deemed DEA non-efficient or optimistic non-efficient. For n DMUs, LP model (1) is solved n times, one DMU each time. Consequently, at least one DMU is identified as DEA efficient, and all DEA efficient DMUs form an efficiency frontier.

2.2 The worst relative efficiency— pessimistic efficiency

Pessimistic efficiency, or the worst relative efficiency, is measured by minimizing within the range of ≥1. The pessimistic efficiency of DMU₀ is measured by the following LP model [14]: $\begin{matrix} Min φ_{0} = \sum_{r = 1}^{s} u_{r} y_{r 0} \\ s . t . \sum_{i = 1}^{m} v_{i} x_{i 0} = 1, \\ \sum_{r = 1}^{s} u_{r} y_{rj} - \sum_{i = 1}^{m} v_{i} x_{ij} ⩾ 0, j = 1, \dots, n, \\ u_{r}, v_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (2)

Suppose there are positive weights to make $φ_{0}^{*} = 1$ , DMU₀ is deemed DEA inefficient or pessimistic inefficient; otherwise, it is deemed DEA non-inefficient or pessimistic non-inefficient. For n DMUs, LP model (2) is solved n times, one DMU each time. Consequently, at least one DMU is identified as DEA inefficient, and all DEA inefficient DMUs form an inefficiency frontier.

3 DEA cross-efficiency models

3.1 Concept of cross-efficiency

Efficiency is assessed with the optimal input and output weights of each DMU in CCR model, hence it is hard to distinguish DEA efficient units. To fill this gap, Sexton et al. [32] first proposed an approach to evaluate cross-efficiency, elevating self-evaluation of DEA models to peer-evaluation.

For n DMUs, DEA always finds n sets of optimal input and output weights denoted by $u_{rk}^{*}$ and $v_{ik}^{*}$ , k = 1, …, n. Suppose $E_{jk} = \frac{\sum_{r = 1}^{s} u_{rk}^{*} y_{rj}}{\sum_{i = 1}^{m} v_{ik}^{*} x_{ij}}, j, k = 1, \dots, n .$ (3)

Then, E_jk (k = 1, …, n) are deemed cross-efficiencies of DMU_j, and calculated via a set of the optimal input and output weights of DMU_k— $u_{rk}^{*}$ and $v_{ik}^{*}$ . In particular, E_jj is the optimistic efficiency of DMU_j— $E_{jj} = θ_{j}^{*}$ .

It is, in other words, peer-evaluation, and E = (E_jk) _n×n is the cross-efficiency matrix. Each DMU such as DMU_j has n cross-efficiencies: E_j1, …, E_jn, so the average cross-efficiency score of DMU_j is: ${\bar{E}}_{j} = \frac{1}{n} \sum_{k = 1}^{n} E_{jk}, j = 1, \dots, n .$ (4)

3.2 Concept of pessimistic cross-efficiency

Current studies define cross-efficiency via the optimal weights of DMU inputs and outputs. However, based on model (2), cross-efficiency can also be defined by the most unfavorable weights of inputs and outputs. In this study, it is named “pessimistic cross-efficiency” to distinguish it from cross-efficiency. Suppose $H_{jk} = \frac{\sum_{r = 1}^{s} {\tilde{u}}_{rk}^{*} y_{rj}}{\sum_{i = 1}^{m} {\tilde{v}}_{ik}^{*} x_{ij}}, j, k = 1, \dots, n .$ (5)

Then, H_jk (k = 1, …, n) are referred to as the pessimistic cross-efficiencies of DMU_j, calculated via the most unfavorable weights of inputs and outputs of DMU_k— ${\tilde{u}}_{rk}^{*}$ and ${\tilde{v}}_{ik}^{*}$ . In particular, H_jj is the pessimistic efficiency of DMU_j, and $H_{jj} = φ_{j}^{*}$ . So, the average pessimistic cross-efficiency score of DMU_j is: ${\bar{H}}_{j} = \frac{1}{n} \sum_{k = 1}^{n} H_{jk}, j = 1, \dots, n .$ (6)

3.3 Interval cross-efficiency models

Unfortunately, cross-efficiency evaluation suffers one major deficiency. To be specific, the optimal or most unfavorable weights of input and output generated by model (1) or (2) may be non-unique and dependent on the software used, leading to inconsistent cross-efficiency scores [33]. For that, four categories of secondary goal models were proposed: benevolent, aggressive, neutral, and other [34]. Yang et al. [35] proposed an interval cross-efficiency model after Liang et al. [5] based on the two most used ones— benevolent model and aggressive model.

$\begin{matrix} Max \sum_{r = 1}^{s} u_{rk} y_{rj} = E_{jk}^{U} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ij} = 1; \\ θ_{k}^{*} \times \sum_{i = 1}^{m} v_{ik} x_{ik} - \sum_{r = 1}^{s} u_{rk} y_{rk} = 0; \\ \sum_{i = 1}^{m} v_{ik} x_{ij} - \sum_{r = 1}^{s} u_{rk} y_{rj} ⩾ 0, j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (7) $\begin{matrix} Min \sum_{r = 1}^{s} u_{rk} y_{rj} = E_{jk}^{L} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ij} = 1; \\ θ_{k}^{*} \times \sum_{i = 1}^{m} v_{ik} x_{ik} - \sum_{r = 1}^{s} u_{rk} y_{rk} = 0; \\ \sum_{i = 1}^{m} v_{ik} x_{ij} - \sum_{r = 1}^{s} u_{rk} y_{rj} ⩾ 0, j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (8)

Based on the self-evaluation efficiency of DMU_k, DMU_j seeks both maximum and minimum cross-efficiency via model (7) and (8). Results show that no matter what strategy DM adopts, the cross-efficiency score of DMU_j stays within the interval $[E_{jk}^{L}, E_{jk}^{U}]$ . When $E_{jk}^{L} = E_{jk}^{U}$ , the interval cross-efficiency matrix is simplified as a traditional cross-efficiency matrix. The elements in the diagonal represent the self-evaluation efficiency of each DMU, in line with $E_{jj}^{L} = E_{jj}^{U} = θ_{j}^{*} (j = 1, \dots, n)$ .

3.4 Pessimistic interval cross-efficiency models

Current secondary goal models are basically from the optimistic perspective. Based on DMU self-evaluation efficiency, the weights for calculating cross-efficiency scores are determined by various secondary goals. However, few studies tried to obtain the non-unique cross-efficiency scores via cross-efficiency evaluation from the pessimistic perspective. Hence, this study draws from the research of Liang et al. [5] and proposes an interval cross-efficiency model from the pessimistic perspective. $\begin{matrix} Min \sum_{r = 1}^{s} u_{rk} y_{rj} = H_{jk}^{L} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ij} = 1; \\ θ_{k}^{*} \times \sum_{i = 1}^{m} v_{ik} x_{ik} - \sum_{r = 1}^{s} u_{rk} y_{rk} = 0; \\ \sum_{i = 1}^{m} v_{ik} x_{ij} - \sum_{r = 1}^{s} u_{rk} y_{rj} ⩽ 0, j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (9) $\begin{matrix} Min \sum_{r = 1}^{s} u_{rk} y_{rj} = E_{jk}^{L} \\ s . t . \sum_{i = 1}^{m} v_{ik} x_{ij} = 1; \\ θ_{k}^{*} \times \sum_{i = 1}^{m} v_{ik} x_{ik} - \sum_{r = 1}^{s} u_{rk} y_{rk} = 0; \\ \sum_{i = 1}^{m} v_{ik} x_{ij} - \sum_{r = 1}^{s} u_{rk} y_{rj} ⩾ 0, j = 1, \dots, n; j \neq k; \\ u_{rk}, v_{ik} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (10)

Based on the pessimistic efficiency of DMU_k, DMU_j seeks both maximum and minimum cross-efficiency scores via model (9) and (10). Results show that no matter what strategy DM adopts, the cross-efficiency score of DMU_j stays within the interval $[L_{jk}^{L}, E_{jk}^{U}]$ . When $E_{jk}^{L} = E_{jk}^{U}$ , the interval cross-efficiency matrix is simplified as a pessimistic cross-efficiency matrix. The elements in the diagonal represent the pessimistic efficiency of each DMU, in line with $E_{jj}^{L} = E_{jj}^{U} = θ_{j}^{*} (j = 1, \dots, n)$ .

4 DEA adjustment coefficient models for measuring interval efficiencies of DMU

4.1 Constructing the efficiency interval

Optimistic and pessimistic efficiencies cannot be directly compared as they are measured in different intervals. In theory, optimistic and pessimistic efficiencies should form an efficiency interval, integrating both optimistic and pessimistic perspectives. To do so, the pessimistic efficiency should be further adjusted. Next, in order to measure the efficiency interval, this study introduces the adjustment coefficient β (0 < β ⩽ 1). Thus, the adjusted pessimistic efficiency can be expressed as ${\tilde{φ}}_{j}^{*} = β φ_{j}^{*}$ (j = 1, …, n), in line with ${\tilde{φ}}_{j}^{*} = β φ_{j}^{*} ⩽ θ_{j}^{*} (j = 1, \dots, n)$ or $β ⩽ min_{j \in {1, \dots, n}} {θ_{j}^{*} / φ_{j}^{*}}$ . So, the efficiency interval for DMU_j can be expressed as $[β φ_{j}^{*}, θ_{j}^{*}] (j = 1, \dots, n)$ . Obviously, the adjustment coefficient β determines the lower bound of the efficiency interval of each DMU, so how to determine the adjustment coefficient in a reasonable and accurate manner is crucial.

4.2 DEA adjustment coefficient models for crisp data

The range of DMU efficiency interval is affected by the value of adjustment coefficient β. To this end, this study constructs the fractional programming (FP) model below to determine the value of β: $\begin{matrix} Max β \\ s . t . β φ_{j}^{*} ⩽ \frac{\sum_{r = 1}^{s} u_{r} y_{rj}}{\sum_{i = 1}^{m} v_{i} x_{ij}} ⩽ θ_{j}^{*}, j = 1, \dots, n, \\ β H_{j}^{max} ⩽ θ_{j}^{*}, \\ β φ_{j}^{*} ⩽ E_{j}^{min}, \\ u_{r}, v_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m, \end{matrix}$ (11) where $θ_{j}^{*}$ is the optimistic efficiency of DMU_j determined by model (1), $φ_{j}^{*}$ the pessimistic efficiency determined by model (2), and $E_{j}^{min} = \min_{k \in {1, \dots, n}} {E_{jk}}$ the minimum cross-efficiency score. E_jk in {} is identified by model (3), $H_{j}^{max} = \max_{k \in {1, \dots, n}} {H_{jk}}$ the maximum pessimistic cross-efficiency score, and H_jk in {} identified by model (5). Since the cross-efficiency and pessimistic cross-efficiency scores are non-unique, building various secondary goal models would result in different cross-efficiency and pessimistic cross-efficiency scores, further influencing the uniqueness of $E_{j}^{min}$ and $H_{j}^{max}$ . For that, this study employs the interval cross-efficiency model (7) and (8) to identify the lower and upper bound of $E_{j}^{min}$ , that is, $E_{j}^{L min} ⩽ E_{j}^{min} ⩽ E_{j}^{U min}$ , where $E_{j}^{L min} = \min_{k \in {1, \dots, n}} {E_{jk}^{L}}$ and $E_{j}^{U min} = \min_{k \in {1, \dots, n}} {E_{jk}^{U}}$ . The lower and upper bound of $H_{j}^{max}$ are identified by the pessimistic interval cross-efficiency model (9) and (19)— $H_{j}^{L max} ⩽ H_{j}^{max} ⩽ H_{j}^{U max}$ , where $H_{j}^{L max} = \max_{k \in {1, \dots, n}} {H_{jk}^{L}}$ and $H_{j}^{U max} = \max_{k \in {1, \dots, n}} {H_{jk}^{U}}$ . Such method stands out as it integrates all possible weight sets in the weight space while calculating cross-efficiency and pessimistic cross-efficiency scores, and the DM does not need to decide on certain secondary goal models, thus avoiding the non-uniqueness of weight sets. Therefore, the maximum value of adjustment coefficient β can be obtained by the FP model below: $\begin{matrix} Max β \\ s . t . β φ_{j}^{*} ⩽ \frac{\sum_{r = 1}^{s} u_{r} y_{rj}}{\sum_{i = 1}^{m} v_{i} x_{ij}} ⩽ θ_{j}^{*}, j = 1, \dots, n, \\ β H_{j}^{L max} ⩽ θ_{j}^{*}, \\ β φ_{j}^{*} ⩽ E_{j}^{U min}, \\ u_{r}, v_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m, \end{matrix}$ (12)

model (12) attempts to find a set of non-negative weights $u_{r}^{*} (r = 1, \dots, s)$ and $v_{i}^{*} (i = 1, \dots, m)$ . The efficiency of DMU_j (j = 1, …, n) is between the adjusted pessimistic efficiency and optimistic efficiency, where the objective function value β is maximized. This study then performs the Charnes-Cooper transform [36] on DMU— $t = 1 / \sum_{i = 1}^{m} v_{i} x_{io}$ , ∀o∈ { 1, …, n }, μ_r = tu_r (r = 1, …, s), and ω_i = tv_i (i = 1, …, m), and model (12) is transformed into the LP model below: $\begin{matrix} Max β \\ s . t . \sum_{r = 1}^{s} μ_{r} y_{rj} - θ_{j}^{*} \sum_{i = 1}^{m} ω_{i} x_{ij} ⩽ 0, \\ \sum_{r = 1}^{s} μ_{r} y_{rj} - β φ_{j}^{*} \sum_{i = 1}^{m} ω_{i} x_{ij} ⩾ 0, j = 1, \dots, n, \\ \sum_{i = 1}^{m} ω_{i} x_{io} = 1, \\ β H_{j}^{L max} ⩽ θ_{j}^{*}, \\ β φ_{j}^{*} ⩽ E_{j}^{U min}, \\ μ_{r}, ω_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (13)

Theorem 1. Model (12) is equal to model (13) in the following scenario:

1) If $u_{r}^{*} (r = 1, \dots, s)$ and $v_{i}^{*} (i = 1, \dots, m)$ are the optimal solutions of model (12), $μ_{r}^{*} = t^{*} u_{r}^{*} (r = 1, \dots, s)$ , $ω_{i}^{*} = t^{*} v_{i}^{*} (i = 1, \dots, m)$ are the optimal solutions of model (13), and the optimal values of model (12) and (13) are equal, where $t^{*} = 1 / \sum_{i = 1}^{m} ω_{i} x_{io}$ .

2) If $μ_{r}^{*} (r = 1, \dots, s)$ and $ω_{i}^{*} (i = 1, \dots, m)$ are the optimal solutions of model (13), $μ_{r}^{*} (r = 1, \dots, s)$ , $ω_{i}^{*} (i = 1, \dots, m)$ are the optimal solutions of model (12), and the optimal values of model (12) and (13) are equal.

Proof. The same constraints shared by model (12) and (13) prove the validity of Theorem 1. □

In accordance with model (13), the objective function value β is only correlated with all DMUs as a whole. In other words, if the DMU_o to be evaluated under the constraints of model (13) changes, the optimal objective function remains the same, but the weights μ_r (r = 1, …, s) and ω_i (i = 1, …, m) may vary.

Model (12) and (13) are DEA adjustment coefficient models. The adjustment coefficient β is attained via model (13), to adjust the pessimistic efficiencies of all DMUs, then the adjusted pessimistic and optimistic efficiencies constitute an efficiency interval $[β φ_{j}^{*}, θ_{j}^{*}] (j = 1, \dots, n)$ which is defined as follows:

Definition 1. DMU_j refers to DEA efficient only if $θ_{j}^{*} = 1$ ; otherwise DEA inefficient.

Definition 2. DMU_j refers to DEA inefficient only if $β φ_{j}^{*} = β$ ; otherwise non-DEA inefficient.

Definition 3. DMU_j refers to DEA unspecified only if DMU_j is neither DEA efficient nor DEA inefficient.

Definition 4. DMU_j refers to DEA special only if DMU_j is both DEA efficient and DEA inefficient.

All DEA efficient DMUs constitute an efficient frontier, and all DEA inefficient DMUs constitute an inefficient frontier. Unspecified DMUs, on the other hand, are covered by both efficient and inefficient frontiers. And some DMUs fall in the category of both DEA efficient and inefficient, so they have the largest efficiency interval [β, 1], resulting in the biggest uncertainty in evaluation.

Theorem 2. Let $θ_{j}^{*}$ be the optimistic efficiency of DMU_j determined by model (1), $φ_{j}^{*}$ the pessimistic efficiency determined by model (2), β the adjustment coefficient identified by model (12) or (13), and E_jk and H_jk the cross-efficiencies identified by model (3) and (5), so $[β φ_{j}^{*}, θ_{j}^{*}]$ is the optimal efficiency interval of DMU_j determined by the following upper and lower bounds with FP models: $\begin{matrix} Max / Min \frac{\sum_{r = 1}^{s} u_{r} y_{r 0}}{\sum_{i = 1}^{m} v_{i} x_{i 0}} \\ s . t . β ⩽ \frac{\sum_{r = 1}^{s} u_{r} y_{rj}}{\sum_{i = 1}^{m} v_{i} x_{ij}} ⩽ θ_{j}^{*}, j = 1, \dots, n, \\ u_{r}, v_{i} ⩾ 0, r = 1, \dots, s; i = 1, \dots, m . \end{matrix}$ (14)

Proof of Theorem 2 is provided in the Appendix A.

5 Interval efficiency evaluation

The adjusted pessimistic efficiency and optimistic efficiency are the lower and upper bounds of the interval efficiency of a DMU. In other words, $[β φ_{j}^{*}, θ_{j}^{*}]$ is the interval efficiency of DMU_j. So, to give full play to it and comprehensively assess the performance of all DMUs, this study uses the Hurwicz criterion approach (HCA) introduced by Wang and Yang [17] to compare and rank the interval efficiencies of DMUs.

Definition 5. Let α be a parameter in [0, 1] and $A_{k} = [β φ_{k}^{*}, θ_{k}^{*}]$ be the interval efficiency of DMU_k, where $β φ_{k}^{*}$ and $θ_{k}^{*}$ are the adjusted pessimistic efficiency and optimistic efficiency of DMU_k, then the Hurwicz index value of A_k is defined as follows: $H (A_{k}) = α θ_{k}^{*} + (1 - α) β φ_{k}^{*}, k = 1, \dots, n,$ (15) where the parameter α can serve as either the DM or assessor’s optimism level. If α = 1 or 0, the Hurwicz index value becomes the optimistic efficiency or adjusted pessimistic efficiency. When α > 0.5, the DM or assessor is optimistic; meanwhile, when α < 0.5, the DM or assessor becomes pessimistic. If α = 0.5, the DM or assessor picks up a neutral attitude, and the Hurwicz index value is the average of optimistic efficiency and adjusted pessimistic efficiency. It is recommended to set the parameter α at 0.5 in cases where the assessor’s level of optimism cannot be determined when applying our approach.

In this study, there are two properties of HCA:

Property 1. If $θ_{i}^{*} > θ_{j}^{*}$ and $β φ_{i}^{*} > β φ_{j}^{*}$ , H (A_i) > H (A_j) with α ∈ [0, 1].

Property 2. Suppose A_j is included in A_i, then $θ_{i}^{*} ⩾ θ_{j}^{*}$ and $β φ_{i}^{*} ⩽ β φ_{j}^{*}$ with at least one inequality. Let $α_{0} = \frac{β φ_{j}^{*} - β φ_{i}^{*}}{(θ_{i}^{*} - β φ_{i}^{*}) - (θ_{j}^{*} - β φ_{j}^{*})}$ , then

H (A_i) > H (A_j) only if the parameter α is in (α₀, 1] ;

H (A_i) < H (A_j) only if the parameter α is in [0, α₀) ;

H (A_i) = H (A_j) only if the parameter α = α₀.

According to Property 1, if two interval efficiencies DMU_i and DMU_j are not embedded, the one with the higher lower and upper bounds is better. Property 2, on the other hand, shows that the ranking order of DMU_i and DMU_j would be affected by parameter α. Furthermore, Wang and Yang [17] conducted a sensitivity analysis on α via the following theorem.

Theorem 3. Let $A_{i} = [β φ_{i}^{*}, θ_{i}^{*}] (i = 1, \dots, n)$ be an interval efficiency set. With a given optimism level α₀, if the ranking is A_{i
₁} ≻ A_{i
₂} ≻ … ≻ A_{i
_n}, there is an interval for optimism level α determined by (α_L, α_R) ∩ [0, 1], where $α_{L} = max_{j} {\frac{β φ_{i_{j + 1}}^{*} - β φ_{i_{j}}^{*}}{| (θ_{i_{j}}^{*} - β φ_{i_{j}}^{*}) - (θ_{i_{j + 1}}^{*} - β φ_{i_{j + 1}}^{*}) |} | (θ_{i_{j}}^{*} - β φ_{i}^{*}) - (θ_{i_{j + 1}}^{*} - β φ_{i_{j + 1}}^{*}) > 0},$ $α_{R} = min_{j} {\frac{β φ_{i_{j + 1}}^{*} - β φ_{i_{j}}^{*}}{| (θ_{i_{j}}^{*} - β φ_{i_{j}}^{*}) - (θ_{i_{j + 1}}^{*} - β φ_{i_{j + 1}}^{*}) |} | (θ_{i_{j}}^{*} - β φ_{i}^{*}) - (θ_{i_{j + 1}}^{*} - β φ_{i_{j + 1}}^{*}) < 0} .$

When α varies within the above interval, the ranking of interval efficiencies remains unchanged.

6 Numerical examples

In this section, two examples are included in this study to demonstrate the better discriminating power of DEA adjustment coefficient models over other interval DEA models to measure DMU performance. One is a numerical example comparing the calculation process with other interval DEA models. The other is an empirical example demonstrating practices of performance measurement.

Example 1. A numerical example with 5 DMUs, each having one input and two outputs. The data set is from Liu and Wang [18], as shown in Table 1, and all inputs are normalized to 1.

Table 1
Data for 5 DMUs with one input and two outputs

DMU Input (X₁) Output 1 (Y₁) Output 2 (Y₂)

1 1 2 7

2 1 4 6

3 1 5 15

4 1 9 16

5 1 15 7

DMU	Input (X₁)	Output 1 (Y₁)	Output 2 (Y₂)
1	1	2	7
2	1	4	6
3	1	5	15
4	1	9	16
5	1	15	7

Table 2 shows the optimistic and pessimistic efficiencies of each DMU along with the interval efficiency attained by the models proposed by Entani et al. [16] (see Appendix B), Wang and Yang [17] (see Appendix C), Liu and Wang [18] (see Appendix D), and this study.

Table 2

Relative efficiencies for the 5 DMUs with one input and two outputs

DMU	Optimistic efficiency	Pessimistic efficiency	Model of Entani et al.	Model of Wang and Yang	Model of Liu and Wang	Model of this study
1	0.4375	1.0000	[0.1333, 0.4375]	[0.3750, 0.4375]	[0.0000, 0.1000]	[0.3389, 0.4375]
2	0.4068	1.0000	[0.2667, 0.4068]	[0.3750, 0.3952]	[0.0000, 0.1538]	[0.3389, 0.4068]
3	0.9375	2.1875	[0.3333, 0.9375]	[0.8203, 0.9375]	[0.2308, 0.9000]	[0.7413, 0.9375]
4	1.0000	2.5625	[0.6000, 1.0000]	[0.9609, 1.0000]	[0.5385, 1.0000]	[0.8684, 1.0000]
5	1.0000	1.1667	[0.4375, 1.0000]	[0.4375, 0.7944]	[0.1000, 1.0000]	[0.3954, 1.0000]

Table 2 shows that, from the optimistic perspective, DMU₄ and DMU₅ are evaluated as DEA efficient by model (1), and DMU₁, DMU₂ and DMU₃ as optimistic non-efficient. It is believed that the performance of efficient units is better than that of optimistic non-efficient units, as the former units are on the efficiency frontier composed of both efficient units. In accordance with optimistic efficiency, the 5 DMUs are ranked DMU₄ ∼ DMU₅ ≻ DMU₃ ≻ DMU₁ ≻ DMU₂, where the symbol ‘∼’ represents indifference and ‘≻’ means ‘superior to’.

However, from the pessimistic perspective, DMU₁ and DMU₂ are evaluated as DEA inefficient by model (2), and DMU₃, DMU₄ and DMU₅ are evaluated as pessimistic non-inefficient, as shown in Table 2. It is believed that the performance of inefficient units is poorer than that of pessimistic non-inefficient units, as the former units are on the inefficiency frontier composed of both inefficient units. In accordance with pessimistic efficiency, the 5 DMUs are ranked DMU₄ ≻ DMU₃ ≻ DMU₅ ≻ DMU₁ ∼ DMU₂.

Consequently, assessment results vary with different perspectives, as shown in Table 2. Especially for DMU₃ and DMU₅, while evaluated based on optimistic efficiency, the latter performs better than the former; while evaluated based on pessimistic efficiency, the former performs better than the latter. As a result, assessment results vary in a biased and unreliable manner.

Therefore, to come up with a comprehensive evaluation method, Entani et al. [16] put forth interval efficiency evaluation models to measure the interval efficiency of each DMU from both optimistic and pessimistic perspectives (results are shown in Table 2). From Table 2, the upper bound of interval efficiency of each DMU equals CCR efficiency, while the lower bound varies in different examples. Nonetheless, such models accurately identify DEA efficient production frontier but not the inefficient production frontier. For example, DMU₁ is identified as DEA inefficient with the smallest lower bound efficiency of 0.1333, but DMU₂ is not identified as DEA inefficient.

The interval efficiencies included in Table 2 are measured by the bounded DEA models [17], in which the lower bound of the constraint is attained by introducing a virtual DMU— anti-ideal DMU (ADMU). ADMU consumes the most inputs to produce the least outputs. Moreover, according to Table 1, ADMU’s input is the maximum value of the inputs of the five DMUs— 1, while each ADMU output is the minimum of the outputs of the five DMUs— 2 and 6 respectively. Next, the CCR efficiency of ADMU can be calculated— 0.375, in line with the lower bound of the constraint of the bounded DEA models. However, the bounded DEA models have similar deficiencies as Entani et al.’s models— they can only identify DEA inefficient production frontier but not efficient production frontier. For example, DMU₄ is identified as DEA efficient with the biggest upper bound efficiency of 1, but DMU₅ is not identified as DEA efficient. In the meantime, the upper bound of interval efficiency of each DMU obtained by the bounded DEA models is not exactly equal to the CCR efficiency. For instance, DMU₂ and DMU₅ are 0.4068 and 1 in terms of CCR efficiencies, while their upper bounds of interval efficiency are 0.3952 and 0.7944. Furthermore, as Liu and Wang [18] pointed out, the bounded DEA models do not meet unit-invariance. For example, when the input and outputs of DMU₁ are divided by 2, the ADMU’s input is still 1, while the outputs are modified to 1 and 3.5. In addition, the CCR efficiency of ADMU will change from 0.3750 to 0.2188.

Table 2 shows Liu and Wang’s interval efficiency evaluation results. Liu and Wang build two DEA models from both optimistic and pessimistic perspectives which consider both the best and worst DMUs to obtain the best and worst normalized efficiencies respectively, and the two normalized efficiencies are employed as the lower and upper bounds of the efficiency interval of each DMU. In this case, though Liu and Wang’s models can accurately identify DEA inefficient units like DMU₁ and DMU₂, as well as efficient units like DMU₄ and DMU₅, the models still produce unreasonable results. For instance, the optimistic efficiency of DMU₁ is 0.4375, larger than that of DMU₂–0.4068, ranking higher than DMU₂. However, the best normalized efficiency of DMU₁ is 0.1, smaller than that of DMU₂ of 0.1538, ranking lower than DMU₂. Therefore, such models are inconsistent when it comes to DMU ranking.

In contrast, this study proposes new models for reevaluation. In detail, this study first determines the value of the adjustment coefficient β (β = 0.3389) via model (12) or (13), then multiplies the pessimistic efficiency to get the lower bound of interval efficiency of each DMU to constitute efficiency interval with optimistic efficiency. Such results of assessing the interval efficiency, as presented in Column 7 of Table 2, can also be obtained by model (14). Table 2 also shows that the adjustment coefficient models identify not only two DEA efficient units— DMU₄ and DMU₅, but also two inefficient units— DMU₁ and DMU₂. Such results are consistent with the results of optimistic efficiency model (1) and optimistic efficiency model (2). In addition, compared with Liu and Wang’s models [18], the models proposed in this study produce more accurate and reasonable efficiency intervals of the 5 DMUs, with consistency in the ranking of DMU₁ and DMU₂ from a simpler calculation process, because model (12) or (13) only needs to be used once.

To rank the 5 DMUs, the Hurwicz index values can be calculated to generate the ranking order under three extreme optimism levels with α = 0, 0 . 5, 1, as shown in Table 3. Also from Table 3, the 5 DMUs are ranked DMU₄ ≻ DMU₃ ≻ DMU₅ ≻ DMU₁ ∼ DMU₂ when α = 0, meaning that the DM or assessor only considers the worst DMU performance with a pessimistic attitude. When α = 0.5, the DM or assessor is neutral and ranks the 5 DMUs as DMU₄ ≻ DMU₃ ≻ DMU₅ ≻ DMU₁ ≻ DMU₂. When α = 1, the 5 DMUs is ranked DMU₄ ∼ DMU₅ ≻ DMU₃ ≻ DMU₁ ≻ DMU₂, meaning that the DM or assessor only considers the best DMU performance with an optimistic attitude.

Table 3

Hurwicz index values and ranking orders for the 5 DMUs under three optimism levels

DMU	Hurwicz index values
	α = 0	α = 0.5	α = 1
1	0.3389 (4)	0.3882 (4)	0.4375 (4)
2	0.3389 (4)	0.3729 (5)	0.4068 (5)
3	0.7413 (2)	0.8394 (2)	0.9375 (3)
4	0.8684 (1)	0.9342 (1)	1.0000 (1)
5	0.3954 (3)	0.6977 (3)	1.0000 (1)

Example 2. Evaluating the innovation performance of 12 R&D institutes in a province of Eastern China. To clearly present results, only the input and output data in 2021 is included in this study, as listed in Table 4. Specifically, two inputs and four outputs are selected to describe the innovation performance:

Table 4

Data of 12 R&D institutes in a province of Eastern China

DMU	x₁	x₃	y₁	y₂	y₃	y₄
1	2189	2150830	23	3536	28	15530
2	2413	2102809	28	2677	12	16385
3	2794	2179348	2	5751	13	14925
4	2818	1254213	33	1878	8	12500
5	2303	1524399	76	2062	13	14000
6	3679	2190971	29	1892	65	12480
7	2442	1966802	9	3665	2	35860
8	2836	1199259	9	2780	4	15069
9	2257	1149589	22	2489	12	61340
10	2249	1473336	11	3110	14	10670
11	1506	1636283	20	1976	28	10052
12	3988	1472129	14	1542	10	10052

Inputs:

x₁: the number of R&D personnel;

x₂: the internal expenditure of R&D funds (unit: 1,000 Renminbi (RMB));

Outputs:

y₁: the number of patents granted;

y₂: papers published in journals indexed by Science Citation Index (SCI), Engineering Index (EI), and Conference Proceedings Citation Index-Science (CPCI-S);

y₃: published scientific and technological works;

y₄: technology market turnover (unit: 1,000 RMB).

Table 5 shows that, based on model (1), DMU₁, DMU₃, DMU₅, DMU₆, DMU₉ and DMU₁₁ are evaluated as DEA efficient, constituting an efficiency frontier. The other six units, on the other hand, are evaluated as optimistic non-efficient. However, when evaluated by model (2), DMU₂, DMU₃, DMU₆, DMU₇, DMU₈, DMU₁₁ and DMU₁₂ are identified as DEA inefficient, constituting an inefficiency frontier, while the other five as pessimistic non-inefficient, as shown in Table 5. The two results are, however, based on optimistic perspective and pessimistic perspective respectively. As proved above, evaluation from either optimistic or pessimistic perspective is unreliable and even opposite. In particular, DMU₃, DMU₆ and DMU₁₁ are, from the optimistic perspective, identified as DEA efficient, implying the best performance. Meanwhile, from the pessimistic perspective, they are identified as DEA inefficient, implying the worst performance.

Table 5

Relative efficiencies of 12 R&D institutes in a province of Eastern China

DMU	Optimistic efficiency	Pessimistic efficiency	Model of Entani et al.	Model of Wang and Yang	Model of Liu and Wang	Model of this study
1	1.0000	1.1118	[0.1353, 1.0000]	[0.3399, 1.0000]	[0.0174, 1.0000]	[0.1501, 1.0000]
2	0.7696	1.0000	[0.1460, 0.7696]	[0.2771, 0.7590]	[0.0000, 0.6870]	[0.1350, 0.7696]
3	1.0000	1.0000	[0.0184, 1.0000]	[0.2771, 1.0000]	[0.0000, 1.0000]	[0.1350, 1.0000]
4	0.8210	1.1339	[0.1527, 0.8210]	[0.3142, 0.8210]	[0.0477, 0.6720]	[0.1531, 0.8210]
5	1.0000	1.2742	[0.1721, 1.0000]	[0.3531, 1.0000]	[0.0529, 1.0000]	[0.1720, 1.0000]
6	1.0000	1.0000	[0.1068, 1.0000]	[0.2771, 1.0000]	[0.0000, 1.0000]	[0.1350, 1.0000]
7	0.9351	1.0000	[0.0343, 0.9351]	[0.2771, 0.8942]	[0.0000, 0.9172]	[0.1350, 0.9351]
8	0.9439	1.0000	[0.0759, 0.9439]	[0.2771, 0.9439]	[0.0000, 0.8972]	[0.1350, 0.9439]
9	1.0000	1.8135	[0.2860, 1.0000]	[0.5025, 1.0000]	[0.2304, 1.0000]	[0.2448, 1.0000]
10	0.9294	1.0843	[0.1357, 0.9294]	[0.3307, 0.9294]	[0.0130, 0.8648]	[0.1464, 0.9294]
11	1.0000	1.0000	[0.1151, 1.0000]	[0.3116, 1.0000]	[0.0000, 1.0000]	[0.1350, 1.0000]
12	0.5245	1.0000	[0.0927, 0.5245]	[0.2771, 0.5245]	[0.0000, 0.2563]	[0.1350, 0.5245]

In evaluating each interval efficiency, the lower and upper bounds of each DMU are respectively assessed from both optimistic and pessimistic perspectives. In detail, Table 5 shows the models proposed by Entani et al. only accurately identify DEA efficient production frontier but not inefficient production frontier. For instance, DMU₃ is identified as DEA inefficient with the smallest lower bound of interval efficiency of 0.0184, but DMU₂, DMU₆, DMU₇, DMU₈, DMU₁₁ and DMU₁₂ are not identified as DEA inefficient. Similar problem occurs in the evaluation via the bounded DEA model [17], as shown in Table 5. To be specific, DMU₂, DMU₃, DMU₆, DMU₇, DMU₈ and DMU₁₂ are identified as DEA inefficient with the smallest lower bound of interval efficiency of 0.2771, but DMU₁₁ is not identified as DEA inefficient.

In this case, though the models proposed by Liu and Wang [18] can accurately identify DEA efficient units with the biggest upper bound of interval efficiency as 1, as well as DEA inefficient units with the smallest lower bound of interval efficiency as 0, as shown in Table 5, the models still produce unreasonable results. For example, the optimistic efficiency of DMU₄ is 0.8210, larger than that of DMU₂— 0.7696, ranking higher than DMU₂. However, the best normalized efficiency of DMU₄ is 0.6720, smaller than that of DMU₂— 0.6870, ranking lower than DMU₂. A similar result occurs between DMU₇ and DMU₈. Therefore, Liu and Wang’ models are inconsistent in DMU ranking.

According to Table 5, the adjustment coefficient models proposed in this study identify not only the six DEA efficient units with the biggest upper bound of interval efficiency as 1, but also the seven DEA inefficient units with the smallest lower bound of interval efficiency as 0.1350. Such results are consistent with those attained by the optimistic efficiency model (1) and the optimistic efficiency model (2). In addition, the efficiency intervals of the 12 DMUs obtained by our models are more reasonable, as the ranking orders are consistent between DMU₂ and DMU₄ as well as DMU₇ and DMU₈.

As mentioned above, the four types of interval efficiency evaluations are from both optimistic and pessimistic perspectives. To comprehensively assess and rank the performances of the 12 DMUs, combining both perspectives is crucial. For that, this study calculates the Hurwicz index values of the four types of interval efficiencies with α = 0.5, to obtain the ranking orders of the 12 DMUs, as shown in Table 6.

In Table 6, DMU₉, DMU₅, DMU₁ and DMU₁₁ are ranked 1st, 2nd, 3rd and 4th by all four evaluation methods. However, results vary in some DMUs. To further look into how similar the ranking orders are to one another, this study calculates the Spearman’s rank correlation coefficients, as shown in Table 7. Obviously, the Spearman’s coefficients of the ranking of the 12 DMUs via the four methods are all greater than 0.8— results are similar to one another. In addition, Wang and Yang’s models produce the closest results of our models, and their rank correlation coefficient reaches 0.9860 in the correlation matrix. In fact, according to the two results above, the top 4 and bottom 4 DMUs are the same, as shown in Table 6.

Additionally, this study further calculates the Hurwicz index values of the two types of interval efficiencies and ranking orders of each DMU when α equals 0.25 and 0.75 respectively. In accordance with Table 8, when α shifts from 0.5 to 0.25 or 0.75, the Hurwicz index values of the two types vary accordingly, and the ranking orders also change. For example, the ranking order of DMU₃, DMU₄, DMU₆, DMU₇ and DMU₁₀ changes in Wang and Yang’s models, while three DMUs— DMU₇, DMU₈ and DMU₁₀ change in our models. The three levels of optimism presented herein correspond to distinct scenarios, each yielding a unique perspective on performance evaluation. In practice, the parameter α assumes a critical role and is subject to determination by the DM or assessor. The HCA offers a flexible means of comparing and ranking interval efficiencies, considering the DM’s or assessor’s level of optimism, while also allowing for sensitivity analysis. Such analysis enables a deeper understanding of how rankings respond to fluctuations in the level of optimism, illustrating the stability of rankings across various scenarios.

Table 6

Hurwicz index values of four types of interval efficiencies with α = 0.5

DMU	Model of Entani et al.	Model of Wang and Yang	Model of Liu and Wang	Model of this study
1	0.5677 (3)	0.6700 (3)	0.5087 (3)	0.5750 (3)
2	0.4578 (11)	0.5181 (11)	0.3435 (11)	0.4523 (11)
3	0.5092 (8)	0.6386 (5)	0.5000 (4)	0.5675 (4)
4	0.4869 (9)	0.5676 (10)	0.3599 (10)	0.4870 (10)
5	0.5861 (2)	0.6765 (2)	0.5265 (2)	0.5860 (2)
6	0.5534 (5)	0.6386 (5)	0.5000 (4)	0.5675 (4)
7	0.4847 (10)	0.5857 (9)	0.4586 (7)	0.5351 (9)
8	0.5099 (7)	0.6105 (8)	0.4486 (8)	0.5395 (7)
9	0.6430 (1)	0.7513 (1)	0.6152 (1)	0.6224 (1)
10	0.5326 (6)	0.6301 (7)	0.4389 (9)	0.5379 (8)
11	0.5576 (4)	0.6558 (4)	0.5000 (4)	0.5675 (4)
12	0.3086 (12)	0.4008 (12)	0.1282 (12)	0.3298 (12)

Table 7

Spearman’s rank correlation coefficients of the four ranking results

Model of Entani et al.	Model of Wang and Yang	Model of Liu and Wang	Model of this study
Model of Entani et al.	1	0.9545	0.8706	0.9196
Model of Wang and Yang	0.9545	1	0.9650	0.9860
Model of Liu and Wang	0.8706	0.9650	1	0.9790
Model of this study	0.9196	0.9860	0.9790	1

Table 8

Hurwicz index values of two types of interval efficiencies for 12 DMUs

DMU	Model of Wang and Yang			Model of this study
α = 0.25	α = 0.50	α = 0.75	α = 0.25	α = 0.50	α = 0.75
1	0.5049(3)	0.6700 (3)	0.8350(3)	0.3626(3)	0.5750 (3)	0.7875(3)
2	0.3976(11)	0.5181 (11)	0.6385(11)	0.2937(11)	0.4523 (11)	0.6110(11)
3	0.4578(6)	0.6386 (5)	0.8193(5)	0.3513(4)	0.5675 (4)	0.7838(4)
4	0.4409(9)	0.5676 (10)	0.6943(10)	0.3201(10)	0.4870 (10)	0.6540(10)
5	0.5148(2)	0.6765 (2)	0.8383(2)	0.3790(2)	0.5860 (2)	0.7930(2)
6	0.4578(6)	0.6386 (5)	0.8193(5)	0.3513(4)	0.5675 (4)	0.7838(4)
7	0.4314(10)	0.5857 (9)	0.7399(9)	0.3350(9)	0.5351 (9)	0.7351(8)
8	0.4438(8)	0.6105 (8)	0.7772(8)	0.3372(8)	0.5395 (7)	0.7417(7)
9	0.6269(1)	0.7513 (1)	0.8756(1)	0.4336(1)	0.6224 (1)	0.8112(1)
10	0.4804(5)	0.6301 (7)	0.7797(7)	0.3421(7)	0.5379 (8)	0.7336(9)
11	0.4837(4)	0.6558 (4)	0.8279(4)	0.3513(4)	0.5675 (4)	0.7838(4)
12	0.3390(12)	0.4008 (12)	0.4627(12)	0.2324(12)	0.3298 (12)	0.4271(12)

7 Conclusion

A comprehensive evaluation of DMU performance needs to be conducted from multiple perspectives. This study aims to optimize and extend performance evaluation, spot the deficiencies of current evaluation approaches, and propose a simpler yet more effective method, to assess both the best and worst relative efficiencies in a unified DEA model. Specifically, we first introduce the concepts of cross-efficiency and interval cross-efficiency models, then explore the notion of pessimistic cross-efficiency and construction of pessimistic interval cross-efficiency models. To obtain upper and lower bounds for efficiency intervals of DMUs, we propose DEA adjustment coefficient models. We also present a theorem that facilitates the measurement of interval efficiencies within a unified DEA framework. Finally, to validate the proposed DEA models, two illustrative examples are demonstrated.

Given interval efficiencies offer a more comprehensive assessment of DMUs’ performance compared to traditional DEA efficiency, they hold significant potential for applications. It is worth noting that the input-oriented DEA adjustment coefficient models developed in this study can be easily adapted for other scenarios, such as output-oriented, BCC (Banker- Charnes-Cooper) and additive DEA models. Furthermore, it can also be applied to model interval input and output data, which would be a major focus of future study.

Footnotes

Acknowledgments

This research was supported by the Key Project of Philosophy and Social Science Planning of Zhejiang Province (23NDJC055Z), the Philosophy and Social Science Planning Project of Anhui Province (AHSKQ2019D024), the Higher School Outstanding Young Talent Support Project of Anhui Province (gxyqZD2020105), the Doctoral Initiation Fund Project of Chongqing Normal University (No. 23XWB008) and the Talent Research Start-up Fund project of Tongling University (2021tlxyrc20).

The Appendix is available in the electronic version of this article: .

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Ranking DMUs by using interval efficiencies in data envelopment analysis

Abstract

Keywords

1 Introduction

2 DEA models for measuring two extreme DMU performances

2.1 The best relative efficiency— optimistic efficiency

3.1 Concept of cross-efficiency

4.1 Constructing the efficiency interval

4.2 DEA adjustment coefficient models for crisp data

Table 1 Data for 5 DMUs with one input and two outputs DMU Input (X1) Output 1 (Y1) Output 2 (Y2) 1 1 2 7 2 1 4 6 3 1 5 15 4 1 9 16 5 1 15 7

Footnotes

Acknowledgments

References

Table 1
Data for 5 DMUs with one input and two outputs

DMU Input (X₁) Output 1 (Y₁) Output 2 (Y₂)

1 1 2 7

2 1 4 6

3 1 5 15

4 1 9 16

5 1 15 7