Uncertain Data Envelopment Analysis for Cross Efficiency Evaluation with Imprecise Data

Abstract

Self evaluation and peer evaluation in data envelopment analysis (DEA) are effective means to comprehensively reflect the efficiencies of decision-making units (DMUs). However, when some of the inputs and outputs of DMUs cannot be accurately observed, the traditional evaluation methods will lose their applicability. This paper attempts to treat the imprecise inputs and outputs as uncertain variables based on uncertainty theory and hence to propose a new uncertain DEA model for cross-efficiency evaluation via the evaluation of both self efficiency and peer efficiency. Moreover, the equivalent form and the proof of the new model are also presented for accurate calculation. Finally, a numerical example is given to illustrate the evaluation results.

1. Introduction

Achieving higher efficiency is one of the most significant goals of a decision-making unit (DMU) in economy and management. Farrell [1] proposed the first model in an effort to measure productive efficiency of DMUs in 1957. Afterwards, Charnes et al. [2] put forward a data envelopment analysis (DEA) approach, known as CCR (Charnes, Cooper and Rhode) model to evaluate the relative efficiencies of DMUs with multiple inputs and multiple outputs. Furthermore, Banker et al. [3] then modified the CCR model by Shephard distance function and introduced the BCC (Banker, Charnes, and Cooper) model. In later research, a large variety of extended DEA models were created and promoted, such as network DEA models [4], which considered the internal structure inside DMUs; slacks-based measure (SBM) models [5], which dealt with the problem of over input but under output in DMUs; and super-efficiency DEA models [6], which sorted efficient units. All of the above efforts on DEA made it a dominant method in the field of efficiency evaluation of DMUs.

However, the above-mentioned models mainly focused on self evaluation, which means that DMUs evaluate their efficiencies by themselves, and did not consider the competition among peer units. In this case, these models may have a flaw in that they drive DMUs to overstate advantages or avoid disadvantages of their inputs and outputs to achieve better performances. As a result, self evaluation may lead to higher efficiencies of DMUs than they should actually be or the appearance of mavericks. In order to overcome the shortcoming, Sexton et al. [7] extended the classic DEA model by evaluating both self efficiency and peer efficiency, which is known as cross-efficiency evaluation. Subsequently, some other extensions were developed based on different assumptions, such as aggressive and benevolent strategies for cross-efficiency [8], neutral DEA model for cross-efficiency [9], and other secondary goals for cross-efficiency [10]. Afterwards, the evaluation method of cross efficiency has been significantly applied in various fields, such as environmental analysis [11,12], economic and strategic choices [13,14], and transportation [15].

Nevertheless, in many practical issues, such as oil reserves, bridge emphasis, and securities’ returns [16,17], the input and output variables of DMUs are often not accurately obtained for technical or economic reasons. Generally, they can only be estimated based on the knowledge and preference of domain experts. Due to the uncertainty of human behavior, the distribution function obtained from the human-estimated data could not be in line with the real frequency. Consequently, these quantities ought to be ascribed to uncertain variables in accordance with the uncertainty theory introduced by Liu [18] in 2007. Subsequently, the first uncertain DEA model [19,20] with imprecise inputs and outputs was proposed based on uncertainty theory. In order to further judge whether the DMU is in a state of increasing returns to scale, decreasing returns to scale, or constant returns to scale, the uncertain DEA models for scale-efficiency evaluation were proposed and improved [21,22]. Afterward, with the application of uncertain DEA in practical problems [16], some other uncertain DEA models were proposed such as the network DEA models [23], which can identify the internal structure of DMUs, and the uncertain random DEA models [24], which contain both random variables and uncertain variables. However, the above uncertain DEA models mainly focused on self evaluation, and can only distinguish the DMUs into efficient and inefficient units, rather than ranking them. Motivated by these models, we attempt to propose a new uncertain DEA model to evaluate cross-efficiency in this paper.

This study provides a fresh DEA model for cross-efficiency evaluation from the perspective of imprecise data. To this end, some of the essential and fundamental knowledge of uncertainty theory for modeling is presented in the second section. The third section intends to present a new uncertain DEA model for cross-efficiency evaluation by introducing self efficiency and peer efficiency in advance, and the equivalent forms of them is also given. A numerical example to test the new model is illustrated in the fourth section. Finally, concluding remarks will be made in the final section.

2. Preliminaries

In this section, the essential and fundamental knowledge of uncertainty theory is brought forth in order to deal with uncertain variables and establish models in the next section.

There are a number of measurable sets in $\mathcal{L}$, each of which also is called element $\Lambda$, composing a nonempty set $\Gamma$. Accordingly, ($\Gamma$, $\mathcal{L}$, M) is set as a measure space where the uncertain measure M is defined as a set function on a $\sigma$-algebra $\mathcal{L}$ and satisfies the following axioms proposed by Liu [18].

Normality Axiom: $M\{\Gamma \}=1$ satisfied the universal set Γ.

Duality Axiom: $M\{\Lambda \}+M\left\{{\Lambda }^c\right\}=1$ satisfied any event Λ.

Subadditivity Axiom: For every countable sequence of events ${\Lambda }_1,{\Lambda }_2,\cdots$, we obtain

$${\displaystyle M\left\{\bigcup _{i=1}^{\infty }{\Lambda }_i\right\}\le \sum _{i=1}^{\infty }M\left\{{\Lambda }_i\right\}.}$$

Then, Liu [25] proposed the fourth axiom to define product uncertain measure in 2009.

Product Axiom: The product uncertain measure M in uncertainty spaces $({\Gamma }_k,{\mathcal{L}}_k,M_k)$ is an uncertain measure meeting

$${\displaystyle M\left\{\prod _{k=1}^{\infty }{\Lambda }_k\right\}=\underset{k=1}{\overset{\infty }{\bigwedge }}{M}_k\left\{{\Lambda }_k\right\}}$$

where ${\Lambda }_k$ are arbitrarily chosen events from ${\mathcal{L}}_k$ for $k=1,2,\cdots$, respectively.

Subsequently, some notions, symbols, and theorems will be elucidated, which are germane to the modeling and calculation in the next section.

Assume the uncertain variable $\xi$ exists with an uncertainty distribution expressed as $\Phi$. According to uncertainty theory [18],

$$\Phi \left(x\right)=M\{\xi \le x\}$$

for any real number x.

With respect to a regular uncertainty distribution $\Phi \left(x\right)$, which is greater than 0 and less than 1, the inverse function ${\Phi }^{-1}\left(\alpha \right)$ exists on the range of open interval $(0,1)$. The expected value shows the measurement of uncertain variables and can be regarded as the average from the point of an uncertain measure. The expected value of an uncertain variable $\xi$ can be calculated by following formula:

$${\displaystyle E[\xi ]={\int }_0^1{\Phi }^{-1}\left(\alpha \right)\mathrm{d}\alpha .}$$

Since a linear uncertain variable labeled with $\xi \sim L(a,b)$ is the most common variable used to express imprecise data in uncertain models, the uncertainty distribution is as follows

$$\Phi \left(x\right)=\left\{\begin{array}{cc}{\displaystyle 0,}& \mathrm{if} x\le a\hfill \\ {\displaystyle \frac{x-a}{b-a},}& \mathrm{if} a<x\le b\hfill \\ {\displaystyle 1,}& \mathrm{if} x>b,\hfill \end{array}\right.$$

which has an expected value

$${\displaystyle E[\xi ]=\frac{a+b}{2}.}$$

As we know, efficiency is the ratio of inputs and outputs in data-envelopment analysis, but when contains uncertain variables, it is necessary to depend on the expected value to express the measurement of the efficiency.

Theorem 1

(Liu [26]). For uncertain variables ξ and η possessing regular uncertainty distributions Φ and Ψ, they are set to be independent and positive. The expected value of $\frac{\xi }{\eta }$ is shown as follows,

$${\displaystyle E[\frac{\xi }{\eta }]={\int }_0^1\frac{{\Phi }^{-1}\left(\alpha \right)}{{\Psi }^{-1}(1-\alpha )}\mathrm{d}\alpha .}$$

The above is all the essential and fundamental knowledge of uncertainty theory required for modeling in the next section.

3. An Uncertain DEA Model for Cross-Efficiency Evaluation

Sometimes, the inputs and outputs cannot be accurately obtained in practice, such as oil reserves and bridge strengthening, so data will be estimated based on the knowledge and preferences of domain experts. Therefore, the distribution functions of the estimated data may be inconsistent with the real frequencies, and we should regard these quantities as uncertain variables [19].

There are a limited number of DMUs; assume that the number is n. Any DMU${}_j$ ($1\le j\le n$) constituting the whole can exploit uncertain input vector ${\stackrel{\sim }{\mathit{x}}}_{\mathit{j}}$ to produce uncertain output vector ${\stackrel{\sim }{\mathit{y}}}_{\mathit{j}}$. For every DMU, we define self-efficiency as the expectation of uncertain outputs and the inputs ratio of assigned weights and artificially restrict it within the upper limit of 1, i.e.,

$$E\left[\frac{{\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{j}}}{{\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{j}}}\right]\le 1,j=1,2,\dots ,n,$$

(1)

where $\mathit{v}$ and $\mathit{u}$ are non-negative weight vectors of ${\stackrel{\sim }{\mathit{x}}}_{\mathit{j}}$ and ${\stackrel{\sim }{\mathit{y}}}_{\mathit{j}}$, respectively. A target DMU${}_k$ is regarded as self-efficient if it is capable of acquiring a batch of the most salutary weights (${\stackrel{\sim }{\mathit{v}}}^{*}$, ${\stackrel{\sim }{\mathit{u}}}^{*}$) to ensure that the expected ratio of DMU${}_k$ reaches 1, i.e.,

$$E\left[\frac{{{\mathit{v}}^{*}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}}{{{\mathit{u}}^{*}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}}\right]=1$$

with constraint (1). For the purpose of testifying if DMU${}_k$ is self-efficient, the following model may be utilized,

$$\left\{\begin{array}{c}{\displaystyle \underset{\mathit{u},\mathit{v}}{max}{\eta }_{{}_{kk}}=E\left[\frac{{\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}}{{\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}}\right]}\hfill \\ subject to:\hfill \\ {\displaystyle E\left[\frac{{\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{j}}}{{\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{j}}}\right]\le 1,j=1,2,\dots ,n,}\hfill \\ {\displaystyle \mathit{u},\mathit{v}\ge 0,}\hfill \end{array}\right.$$

(2)

where $\mathit{u}$ and $\mathit{v}$ are non-negative weight vectors of ${\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}$ and ${\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}$, respectively, and ${\stackrel{\sim }{\mathit{x}}}_{\mathit{j}}$ and ${\stackrel{\sim }{\mathit{y}}}_{\mathit{j}}$ are uncertain input and output vectors DMU${}_j$, $j=1,2,\dots ,n$, respectively.

Definition 1

(Self-efficiency). DMU${}_k$ is considered to be self-efficient if the supreme value ${\eta }_{{}_{kk}}^{*}$ of (2) is 1.

Based on the results of model (2), all DMUs can be divided into two categories of self-efficient units and self-inefficient units. However, the model (2) may allow the DMUs to overstate advantages or avoid disadvantages of their inputs and outputs to achieve better efficiencies. Therefore, in order to eliminate this shortcoming, we attempted to evaluate the efficiencies of target DMU${}_k$ against to other peer DMUs (DMU${}_j$, $j=1,2,\dots ,n$ but $j\ne k$) by assigning the weight vectors of the inputs and outputs of the peer units to those of DMU${}_k$. We thereby define peer efficiency as the expected ratio of uncertain outputs and inputs of DMU${}_k$ weighted by DMU${}_j$. The formula is shown as follows:

$${\eta }_{{}_{jk}}=E\left[\frac{{\mathit{v}}_{\mathit{j}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}}{{\mathit{u}}_{\mathit{j}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}}\right],k=1,2,\dots ,n,j=1,2,\dots ,n,j\ne k,$$

(3)

where ${\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}$ and ${\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}$ are uncertain input vectors and uncertain output vectors of DMU${}_k$, and ${\mathit{v}}_{\mathit{j}}$ and ${\mathit{u}}_{\mathit{j}}$ are non-negative weight vectors of DMU${}_j$. The DMU${}_k$ evaluated by DMU${}_j$ is regarded as peer-efficient if the expected ratio ${\eta }_{{}_{jk}}$ can reach up to 1. Obviously, each DMU will have n-1 peer efficiencies.

Definition 2

(Peer-efficiency). DMU${}_k$ is considered to be peer-efficient to DMU${}_j$, if the optimal value ${\eta }_{{}_{jk}}^{*}$ of (3) achieves to 1.

The self-efficiencies and peer-efficiencies obtained from model (2) and (3) of all DMUs can be presented in an n-by-n matrix (shown in

Table 1. Cross-efficiency matrix for n DMUs.

Target DMU${}_{\mathit{k}}$	DMU${}_{\mathit{j}}$
	1	2	⋯	k	⋯	n
1	${\eta }_{{}_{11}}$	${\eta }_{{}_{12}}$	⋯	${\eta }_{{}_{1k}}$	⋯	${\eta }_{{}_{1n}}$
2	${\eta }_{{}_{21}}$	${\eta }_{{}_{22}}$	⋯	${\eta }_{{}_{2k}}$	⋯	${\eta }_{{}_{2n}}$
⋯	⋯	⋯	⋯	⋯	⋯	⋯
k	${\eta }_{{}_{k1}}$	${\eta }_{{}_{k2}}$	⋯	${\eta }_{{}_{kk}}$	⋯	${\eta }_{{}_{kn}}$
⋯	⋯	⋯	⋯	⋯	⋯	⋯
n	${\eta }_{{}_{n1}}$	${\eta }_{{}_{n2}}$	⋯	${\eta }_{{}_{nk}}$	⋯	${\eta }_{{}_{nn}}$

), which is named cross-efficiency matrix. In the the cross-efficiency matrix, the self-efficiencies are marked on the diagonal, and the peer-efficiencies lie on the other positions. The values in $k_{th}$ column represent efficiencies of DMU${}_k$ weighted by DMU${}_j$ ($j=1,2,...,n$), and the values in $k_{th}$ line represent the efficiencies of DMU${}_j$ ($j=1,2,...,n$) weighted by DMU${}_k$.

According to the efficiencies shown in the matrix, we can calculate the cross-efficiencies of DMU${}_k$ in two ways by averaging its self-efficiency and peer-efficiencies horizontally and vertically. We hence define horizontal cross-efficiency ${\overline{\eta }}_{H_k}$ as the Formula (4) and vertical cross-efficiency ${\overline{\eta }}_{V_k}$ as the Formula (5):

$${\overline{\eta }}_{H_k}=\frac{1}{n}(\sum _{j=1}^n{\eta }_{{}_{kj}}),k=1,2,\dots ,n.$$

(4)

$${\overline{\eta }}_{V_k}=\frac{1}{n}(\sum _{j=1}^n{\eta }_{{}_{jk}}),k=1,2,\dots ,n.$$

(5)

Definition 3

(Horizontal cross-efficiency). The horizontal cross-efficiency of DMU${}_k$ is considered as the average of its self-efficiency and n-1 peer-efficiencies horizontally.

Definition 4

(Vertical cross-efficiency). The vertical cross-efficiency of DMU${}_k$ is considered as the average of its self-efficiency and n-1 peer-efficiencies vertically.

For calculation, the models (2) and (3) should be transformed into accurate models, and the equivalent forms of model (2) and (3) are proved as follows:

Theorem 2.

For each DMU${}_j$ ($j=1,2,\dots ,n$) with uncertain inputs ${\stackrel{\sim }{x}}_{j1}$, ${\stackrel{\sim }{x}}_{j2}$,⋯, ${\stackrel{\sim }{x}}_{jm}$ and uncertain outputs ${\stackrel{\sim }{y}}_{j1}$, ${\stackrel{\sim }{y}}_{j2}$,⋯,${\stackrel{\sim }{y}}_{js}$, let the uncertain variables are independent and possess regular uncertainty distributions ${\Phi }_{j1},{\Phi }_{j2},\cdots ,{\Phi }_{jm}$ and ${\Psi }_{j1},{\Psi }_{j2},\cdots ,{\Psi }_{js}$, respectively. In this context, the model (1) is transformed into the following computable equivalent form:

$$\left\{\begin{array}{c}{\displaystyle \underset{\mathit{v},\mathit{u}}{max}{\eta }_{{}_{kk}}={\int }_0^1\frac{{\displaystyle \sum _{r=1}^s}{v}_{kr}{\Psi }_{kr}^{-1}\left(\alpha \right)}{{\displaystyle \sum _{i=1}^m}{u}_{ki}{\Phi }_{ki}^{-1}(1-\alpha )}\mathrm{d}\alpha }\hfill \\ subject to:\hfill \\ {\displaystyle {\int }_0^1\frac{{\displaystyle \sum _{r=1}^s}{v}_{kr}{\Psi }_{jr}^{-1}\left(\alpha \right)}{{\displaystyle \sum _{i=1}^m}{u}_{ki}{\Phi }_{ji}^{-1}(1-\alpha )}\mathrm{d}\alpha \le 1,j=1,2,\dots ,n,}\hfill \\ {\displaystyle \mathit{u}=(u_1,u_2,\cdots ,u_s)\ge 0,}\hfill \\ {\displaystyle \mathit{v}=(v_1,v_2,\cdots ,v_m)\ge 0,}\hfill \end{array}\right.$$

(6)

where ${\Phi }_{k1},{\Phi }_{k2},\cdots ,{\Phi }_{km}$ and ${\Psi }_{k1},{\Psi }_{k2},\cdots ,{\Psi }_{ks}$ are the regular uncertainty distributions of ${\stackrel{\sim }{x}}_{k1}$, ${\stackrel{\sim }{x}}_{k2}$,⋯, ${\stackrel{\sim }{x}}_{km}$ and ${\stackrel{\sim }{y}}_{k1}$, ${\stackrel{\sim }{y}}_{k2}$,⋯, ${\stackrel{\sim }{y}}_{ks}$, respectively.

Proof. 

On the basis of Theorem 1, since the function $\left({\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}\right)/\left({\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}\right)$ is strictly increasing to ${\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}$ and strictly decreasing to ${\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}$ for each k, it can be obtained that the inverse uncertainty distribution of $\left({\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}\right)/\left({\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}\right)$ is

$${\displaystyle F_k^{-1}\left(\alpha \right)={\int }_0^1\frac{{\displaystyle \sum _{r=1}^s}{v}_{kr}{\Psi }_{kr}^{-1}\left(\alpha \right)}{{\displaystyle \sum _{i=1}^m}{u}_{ki}{\Phi }_{ki}^{-1}(1-\alpha )}\mathrm{d}\alpha .}$$

According to Theorem 1, we prove

$${\displaystyle E\left[\frac{{\mathit{v}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}}{{\mathit{u}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}}\right]={\int }_0^1\frac{{\displaystyle \sum _{r=1}^s}{v}_{kr}{\Psi }_{kr}^{-1}\left(\alpha \right)}{{\displaystyle \sum _{i=1}^m}{u}_{ki}{\Phi }_{ki}^{-1}(1-\alpha )}\mathrm{d}\alpha ,}$$

$k=1,2,\dots ,n.$

The theorem is then verified. Similarly, the Formula (3) for peer efficiency is equivalent to the following form:

$$E\left[\frac{{\mathit{v}}_{\mathit{j}}^T{\stackrel{\sim }{\mathit{y}}}_{\mathit{k}}}{{\mathit{u}}_{\mathit{j}}^T{\stackrel{\sim }{\mathit{x}}}_{\mathit{k}}}\right]={\int }_0^1\frac{{\displaystyle \sum _{r=1}^s}{v}_{jr}{\Psi }_{kr}^{-1}\left(\alpha \right)}{{\displaystyle \sum _{i=1}^m}{u}_{ji}{\Phi }_{ki}^{-1}(1-\alpha )}\mathrm{d}\alpha ,$$

(7)

$k=1,2,\dots ,n,j=1,2,\dots ,n.$

Then, the horizontal cross-efficiency and the vertical cross-efficiency can be calculated in accordance with the Formulas (4) and (5). □

4. A Numerical Example

This section illustates a numerical example with fifteen DMUs to corroborate the applicability of the model proposed in the previous section. In this example, each DMU has two imprecise inputs and two imprecise outputs, classified as uncertain variables. In conformity with uncertainty theory, we set every variable to have linear uncertainty distributions represented as $\mathcal{L}$$(a,b)$, shown in

Table 2. 15 DMUs with two inputs and two outputs.

DMU${}_{\mathit{j}}$	Uncertain Output 1	Uncertain Output 2	Uncertain Input 1	Uncertain Input 2
1	$\mathcal{L}$$(51,65)$	$\mathcal{L}(45,55)$	$\mathcal{L}(8,13)$	$\mathcal{L}(10,15)$
2	$\mathcal{L}(64,71)$	$\mathcal{L}(60,69)$	$\mathcal{L}(17,20)$	$\mathcal{L}(12,19)$
3	$\mathcal{L}(55,60)$	$\mathcal{L}(80,96)$	$\mathcal{L}(10,19)$	$\mathcal{L}(10,16)$
4	$\mathcal{L}(47,55)$	$\mathcal{L}(79,95)$	$\mathcal{L}(15,24)$	$\mathcal{L}(12,21)$
5	$\mathcal{L}(49,62)$	$\mathcal{L}(54,67)$	$\mathcal{L}(10,16)$	$\mathcal{L}(13,19)$
6	$\mathcal{L}(46,52)$	$\mathcal{L}(55,68)$	$\mathcal{L}(11,19)$	$\mathcal{L}(16,22)$
7	$\mathcal{L}(52,60)$	$\mathcal{L}(67,80)$	$\mathcal{L}(13,21)$	$\mathcal{L}(10,17)$
8	$\mathcal{L}(54,67)$	$\mathcal{L}(65,75)$	$\mathcal{L}(11,16)$	$\mathcal{L}(14,20)$
9	$\mathcal{L}(45,55)$	$\mathcal{L}(62,76)$	$\mathcal{L}(10,16)$	$\mathcal{L}(8,12)$
10	$\mathcal{L}(48,57)$	$\mathcal{L}(54,63)$	$\mathcal{L}(11,17)$	$\mathcal{L}(9,18)$
11	$\mathcal{L}(56,69)$	$\mathcal{L}(64,76)$	$\mathcal{L}(8,12)$	$\mathcal{L}(10,20)$
12	$\mathcal{L}(60,78)$	$\mathcal{L}(63,70)$	$\mathcal{L}(6,13)$	$\mathcal{L}(7,15)$
13	$\mathcal{L}(58,66)$	$\mathcal{L}(80,97)$	$\mathcal{L}(5,11)$	$\mathcal{L}(12,25)$
14	$\mathcal{L}(47,58)$	$\mathcal{L}(68,78)$	$\mathcal{L}(12,19)$	$\mathcal{L}(14,22)$
15	$\mathcal{L}(44,53)$	$\mathcal{L}(53,69)$	$\mathcal{L}(14,20)$	$\mathcal{L}(9,15)$

.

In terms of the aforementioned original data, the self-efficiency of each DMU can be calculated through model (2). The values of self-efficiencies are presented on the diagonal of the cross-efficiency matrix shown in

Table 3. Self-efficiencies of 15 DMUs in cross-efficiency matrix.

DMU${}_{\mathit{k}}$	DMU${}_{\mathit{j}}$
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0.7323
2		0.6642
3			1.0000
4				0.7737
5					0.5837
6						0.5362
7							0.8068
8								0.6775
9									1.0000
10										0.6790
11											0.7661
12												1.0000
13													1.0000
14														0.6359
15															0.7620

. From Table 3, it can be seen that the values of DMU${}_3$, DMU${}_9$, DMU${}_{12}$, and DMU${}_{13}$ are 1, and then they are considered self-efficient, while the values of the other DMUs are not, and then they are considered to be self-inefficient.

In addition, we can obtain the peer efficiencies of all DMUs by calculating the model (3). The values of peer efficiencies are presented on the other positions of the cross-efficiency matrix shown in

Table 4. Cross-efficiency matrix for 15 DMUs.

DMU${}_{\mathit{k}}$	DMU${}_{\mathit{j}}$
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	0.7323	0.4807	0.5384	0.3516	0.5638	0.4314	0.4441	0.5859	0.5367	0.5001	0.8105	1.0000	1.0000	0.4482	0.3832
2	0.6936	0.6642	0.7664	0.5605	0.5455	0.4179	0.6908	0.5666	0.8369	0.6324	0.6810	0.9989	0.5877	0.4920	0.6648
3	0.6316	0.6071	1.0000	0.7697	0.5943	0.5073	0.7862	0.6453	0.9877	0.6643	0.7740	0.9771	0.8396	0.6280	0.7126
4	0.6218	0.6168	0.9985	0.7737	0.5811	0.4935	0.7975	0.6295	1.0000	0.6647	0.7481	0.9625	0.7876	0.6148	0.7332
5	0.7033	0.5081	0.6778	0.4708	0.5837	0.4688	0.5331	0.6182	0.6566	0.5477	0.8199	1.0000	0.9984	0.5176	0.4596
6	0.6488	0.5696	0.9769	0.7372	0.6201	0.5362	0.7370	0.6770	0.9300	0.6482	0.8367	1.0000	0.9979	0.6496	0.6447
7	0.6616	0.6672	0.9626	0.7401	0.5813	0.4768	0.8068	0.6207	1.0000	0.6818	0.7341	1.0000	0.7118	0.5882	0.7622
8	0.6487	0.5697	0.9782	0.7383	0.6205	0.5368	0.7378	0.6775	0.9311	0.6486	0.8374	1.0000	1.0000	0.6504	0.6453
9	0.6266	0.6329	0.9815	0.7611	0.5726	0.4801	0.8010	0.6170	1.0000	0.6664	0.7294	0.9628	0.7399	0.5979	0.7472
10	0.6565	0.6487	0.9791	0.7525	0.5898	0.4902	0.8018	0.6332	0.9986	0.6790	0.7530	1.0000	0.7590	0.6050	0.7455
11	0.6283	0.4779	0.7076	0.5063	0.5493	0.4550	0.5418	0.5891	0.6745	0.5291	0.7661	0.9174	0.9474	0.5182	0.4662
12	0.7153	0.5287	0.6235	0.4252	0.5622	0.4342	0.5183	0.5864	0.6300	0.5431	0.7683	1.0000	0.8237	0.4748	0.4573
13	0.5063	0.3580	0.5727	0.4068	0.4603	0.3924	0.4181	0.4992	0.5246	0.4155	0.6741	0.7427	1.0000	0.4391	0.3496
14	0.6514	0.5717	0.9562	0.7198	0.6135	0.5263	0.7279	0.6676	0.9164	0.6435	0.8246	0.9975	0.9658	0.6359	0.6393
15	0.6614	0.6670	0.9624	0.7400	0.5812	0.4767	0.8067	0.6205	1.0000	0.6816	0.7340	1.0000	0.7116	0.5880	0.7620

. From Table 4, we can see the peer efficiencies of DMU${}_{12}$ evaluated by DMU${}_1$, DMU${}_5$, DMU${}_6$, DMU${}_7$, DMU${}_8$, DMU${}_{10}$, and DMU${}_{15}$ are 1; i.e., DMU${}_{12}$ is regarded peer efficient to these DMUs. However, according to Table 3, the self-efficiencies of these DMUs are less than 1, which means the sets of favorable weights found by themselves cannot make themselves self-efficient but can make DMU${}_{12}$ peer-efficient. Therefore, we can conclude that DMU${}_{12}$ has the most excellent perfermance among the peer units in terms of efficiency values. The results for DMU${}_9$ and DMU${}_{13}$ can be interpreted in a similar way.

Subsequently, we can calculate the horizontal cross-efficiency and vertical cross-efficiency of DMU${}_k$ by averaging its self-efficiency and peer-efficiencies horizontally and vertically according to Formulas (4) and (5). Furthermore, the results are indicated in

Table 5. Horizontal and vertical cross-efficiencies of 15 DMUs.

DMU${}_{\mathit{j}}$	Horizontal Cross-Efficiencies	Vertical Cross-Efficiencies
1	0.5871	0.6525
2	0.6533	0.5712
3	0.7417	0.8455
4	0.7349	0.6302
5	0.6376	0.5746
6	0.7473	0.4749
7	0.7330	0.6766
8	0.7480	0.6165
9	0.7277	0.8415
10	0.7394	0.6097
11	0.6183	0.7661
12	0.6061	0.9705
13	0.5173	0.8580
14	0.7372	0.5632
15	0.7329	0.6115

, revealing that the horizontal cross-efficiency of DMU${}_8$ is 0.7480, which is the largest among the other DMUs. The result means that, by having been weighted with the weights of DMU${}_8$, all DMUs can achieve better overall performance. In other words, the weight vectors of DMU${}_8$ is the most favorable to all DMUs in terms of efficiency evaluation.

Moreover, the vertical cross-efficiency values represent the efficiencies of the target DMU relative to the evaluation of all peer units. From Table 5, it is obvious that the vertical cross-efficiency of DMU${}_{12}$ is 0.9705, which is the largest. Then, we can conclude that DMU${}_{12}$ is the best performer in terms of efficiency evaluation among all DMUs.

5. Conclusions

This paper introduced a new uncertain DEA model to evaluate DMUs with imprecise inputs and outputs under the action of uncertainty theory and cross-efficiency evaluation. The new model took the evaluation of self-efficiency and peer-efficiency into account simultaneously and obtained the horizontal cross-efficiency and vertical cross-efficiency by averaging efficiencies horizontally and vertically in order to analyze the performances of all DMUs and a single DMU. Subsequently, we used a numerical example to evaluate DMUs with uncertain variables and calculated the self-efficiency and peer-efficiency of 15 DMUs. At the same time, the DMUs were divided into efficient and inefficient units and sorted by horizontal cross efficiency and vertical cross efficiency. The new uncertain DEA model was verified by the numerical example. Afterward, the applications of the cross efficiency evaluation was launched in the further research of economics and environmental science by virtue of the fresh uncertain DEA model.