Data Envelopment Analysis with Uncertain Inputs and Outputs

Abstract

Data envelopment analysis (DEA), as a useful management and decision tool, has been widely used since it was first invented by Charnes et al. in 1978. On the one hand, the DEA models need accurate inputs and outputs data. On the other hand, in many situations, inputs and outputs are volatile and complex so that they are difficult to measure in an accurate way. The conflict leads to the researches of uncertain DEA models. This paper will consider DEA in uncertain environment, thus producing a new model based on uncertain measure. Due to the complexity of the new uncertain DEA model, an equivalent deterministic model is presented. Finally, a numerical example is presented to illustrate the effectiveness of the uncertain DEA model.

1. Introduction

Data envelopment analysis is a mathematical programming technique that measures the relative efficiency of decision making units with multiple inputs and outputs, which was initialized by Charnes et al. [1]. This was followed by variety of theory research work, including Banker et al. [2], Charnes et al. [3], Petersen [4], and Tone [5]. More DEA papers can refer to Seiford [6] in which 500 references are documented.

The original DEA models assume that inputs and outputs are measured by exact values. However, in many situations, such as in a manufacturing system, a production process, or a service system, inputs and outputs are volatile and complex so that they are difficult to measure in an accurate way. Thus many researchers tried to model DEA with various uncertain theories. Probability theory is the earliest theory which was used to establish the stochastic DEA models. Sengupta [7] generalized the stochastic DEA model using the expected value. Banker [8] incorporated statistical elements into DEA, thus developing a statistical method. Many papers [9–13] have employed the chance-constrained programming to DEA in order to accommodate stochastic variations in data. Fuzzy theory is another theory which was used to deal with the uncertainty in DEA. As one of the DEA initiators, Cooper et al. [14–16] introduced how to deal with imprecise data such as bounded data, ordinal data, and ratio bounded data in DEA. Kao and Liu [17] developed a method to find the membership functions of the fuzzy efficiency scores when some inputs or inputs are fuzzy numbers. Entani et al. [18] proposed a DEA model with an interval efficiency by the pessimistic and the optimistic values. Many researchers have introduced possibility measure [19] into DEA [20, 21].

A lot of surveys showed that human uncertainty does not behave like fuzziness. For example, we say “the input is about 10.” Generally, we employ fuzzy variable to describe the concept of “about 10;" then there exists a membership function, such as a triangular one (9, 10, 11). Based on this membership function, we can obtain that “the input is exactly 10" with possibility measure 1. On the other hand, the opposite event of “not exactly 10" has the same possibility measure. The conclusion that “not 10" and “exactly 10" have the same possibility measure is not appropriate. This inspired Liu [22] to found an uncertainty theory which has become a branch of axiomatic mathematics for modeling human uncertainty. This paper will apply the uncertainty theory to DEA to deal with human uncertainty, thus producing some uncertain DEA models.

In this paper, we will assume the inputs and outputs are uncertain variables and propose some uncertain DEA models. The rest of this paper is organized as follows. Section 2 will introduce some basic concepts and properties about uncertain variables. Then an uncertain DEA model as well as its equivalent crisp model will be presented in Section 3. Finally, a numerical example will be given to illustrate the uncertain DEA model in Section 4.

2. Preliminaries

Uncertainty theory was founded by Liu [22] in 2007 and refined by Liu [23] in 2010. As extensions of uncertainty theory, uncertain process, and uncertain differential equations [24], uncertain calculus [25] were proposed. Besides, uncertain programming was first proposed by Liu [26] in 2009, which wants to deal with the optimal problems involving uncertain variable. This work was followed by an uncertain multiobjective programming, an uncertain goal programming [27], and an uncertain multilevel programming [28]. Since that, uncertainty theory was used to solve variety of real optimal problems, including finance [29–31], reliability analysis [32, 33], graph [34, 35], and train scheduling [36, 37]. In this section, we will state some basic concepts and results on uncertain variables. These results are crucial for the remainder of this paper.

Let Γ be a nonempty set, and let Ł be a 𝜎-algebra over Γ. Each element Λ∈ Ł is assigned a number 𝑀{Λ}∈[0,1]. In order to ensure that the number 𝑀{Λ} has certain mathematical properties, Liu [22] presented the four axioms.

Axiom 1. 𝑀{Γ}=1 for the universal set Γ.

Axiom 2. 𝑀{Λ}+𝑀{Λ^𝑐}=1 for any event Λ.

Axiom 3. For every countable sequence of events Λ₁,Λ₂,…, we have 𝑀^∞_𝑖=1Λ_𝑖≤^∞_𝑖=1𝑀Λ_𝑖. (1)

Axiom 4. Let (Γ_𝑘,Ł_𝑘,𝑀_𝑘) be uncertainty spaces for 𝑘=1,2,…. Then the product uncertain measure 𝑀 is an uncertain measure satisfying 𝑀^∞_𝑘=1Λ_𝑘=^∞_𝑘=1𝑀_𝑘Λ_𝑘, (2) where Λ_𝑘 are arbitrarily chosen events from Ł_𝑘 for 𝑘=1,2,…, respectively.

If the set function 𝑀 satisfies the first three axioms, it is called an uncertain measure.

Definition 1 (see Liu [22]). Let Γ be a nonempty set, let Ł be a 𝜎-algebra over Γ, and let 𝑀 be an uncertain measure. Then the triplet (Γ,Ł,𝑀) is called an uncertainty space.

Definition 2 (see Liu [22]). An uncertain variable 𝜉 is a measurable function from an uncertainty space (Γ,Ł,𝑀) to the set of real numbers; that is, for any Borel set 𝐵 of real numbers, the set {𝜉∈𝐵}=𝛾∈Γ∣𝜉𝛾∈𝐵 (3) is an event.

Definition 3 (see Liu [22]). The uncertainty distribution Φ of an uncertain variable 𝜉 is defined by Φ(𝑥)=𝑀{𝜉≤𝑥} (4) for any real number 𝑥.

Example 4. The linear uncertain variable 𝜉∼ℒ(𝑎,𝑏) has an uncertainty distribution Φ(𝑥)=()0,if𝑥≤𝑎,𝑥−𝑎(𝑏−𝑎),if𝑎≤𝑥≤𝑏,1,if𝑥≥𝑏. (5)

Example 5. An uncertain variable 𝜉 is called zigzag if it has a zigzag uncertainty distribution Φ(𝑥)=()0,if𝑥≤𝑎,𝑥−𝑎2()()𝑏−𝑎,if𝑎≤𝑥≤𝑏,𝑥+𝑐−2𝑏2(𝑐−𝑏),if𝑏≤𝑥≤𝑐,1,if𝑥≥𝑐 (6) denoted by 𝒵(𝑎,𝑏,𝑐), where 𝑎,𝑏,𝑐 are real numbers with 𝑎<𝑏<𝑐.

Definition 6 (see Liu [25]). The uncertain variables 𝜉₁,𝜉₂,…,𝜉_𝑛 are said to be independent if 𝑀^𝑛_𝑖=1𝜉_𝑖∈𝐵_𝑖=^𝑛_𝑖=1𝑀𝜉_𝑖∈𝐵_𝑖 (7) for any Borel sets 𝐵₁,𝐵₂,…,𝐵_𝑛.

Definition 7 (see Liu [23]). An uncertainty distribution Φ of an uncertain variable 𝜉 is said to be regular if its inverse function Φ⁻¹(𝛼) exists and is unique for each 𝛼∈(0,1). In this case, the inverse function Φ⁻¹(𝛼) is called the inverse uncertainty distribution of 𝜉.

Example 8. The inverse uncertainty distribution of a zigzag uncertain variable 𝒵(𝑎,𝑏,𝑐) is Φ⁻¹(𝛼)=()()()1−2𝛼𝑎+2𝛼𝑏,if𝛼≤0.5,2−2𝛼𝑏+2𝛼−1𝑐,if𝛼>0.5. (8)

Theorem 9 (see Liu [23]). Let 𝜉₁,𝜉₂,…,𝜉_𝑛 be independent uncertain variables with regular uncertainty distributions Φ₁,Φ₂,…,Φ_𝑛, respectively. If 𝑓 is a strictly increasing function, then 𝜉=𝑓𝜉₁,𝜉₂,…,𝜉_𝑛 (9) is an uncertain variable with inverse uncertainty distribution Ψ⁻¹(𝛼)=𝑓Φ₁⁻¹(𝛼),Φ₂⁻¹(𝛼),…,Φ_𝑛⁻¹(𝛼). (10)

Example 10. Let 𝜉 be an uncertain variable with regular uncertainty distribution Φ. Since 𝑓(𝑥)=𝑎𝑥+𝑏 is a strictly increasing function for any constants 𝑎>0 and 𝑏, the inverse uncertainty distribution of 𝑎𝜉+𝑏 is Ψ⁻¹(𝛼)=𝑎Φ₁⁻¹(𝛼)+𝑏. (11)

Example 11. Let 𝜉₁,𝜉₂,…,𝜉_𝑛 be independent uncertain variables with regular uncertainty distributions Φ₁,Φ₂,…,Φ_𝑛, respectively. Since 𝑓𝑥₁,𝑥₂,…,𝑥_𝑛=𝑥₁+𝑥₂+⋯+𝑥_𝑛 (12) is a strictly increasing function, the sum 𝜉=𝜉₁+𝜉₂+⋯+𝜉_𝑛 (13) is an uncertain variable with inverse uncertainty distribution Ψ⁻¹(𝛼)=Φ₁⁻¹(𝛼)+Φ₂⁻¹(𝛼)+⋯+Φ_𝑛⁻¹(𝛼). (14)

Theorem 12 (see Liu [23]). Assume the constraint function 𝑔(𝑥,𝜉₁,𝜉₂,…,𝜉_𝑛) is strictly increasing with respect to 𝜉₁,𝜉₂,…,𝜉_𝑘 and strictly decreasing with respect to 𝜉_𝑘+1,𝜉_𝑘+2,…,𝜉_𝑛. If 𝜉₁,𝜉₂,…,𝜉_𝑛 are independent uncertain variables with uncertainty distributions Φ₁,Φ₂,…,Φ_𝑛, respectively, then the chance constraint 𝑀𝑔𝑥,𝜉₁,𝜉₂,…,𝜉_𝑛≤0≥𝛼 (15) holds if and only if 𝑔𝑥,Φ₁⁻¹(𝛼),…,Φ_𝑘⁻¹(𝛼),Φ⁻¹_𝑘+1(1−𝛼),…,Φ_𝑛⁻¹(1−𝛼)≤0. (16)

3. DEA Model

In many situations, inputs and outputs are volatile and complex so that they are difficult to measure in an accurate way. This inspired many researchers to apply probability to DEA. As we know, probability or statistics needs a large amount of historical data. In the vast majority of real cases, the sample size is too small (even no sample) to estimate a probability distribution. Then we have to invite some domain experts to evaluate their degree of belief that each event will occur. This section will give some researches to empirical uncertain DEA using the theory introduced in Section 2. The new symbols and notations are given as follows: DMU_𝑖: the 𝑖th DMU,  𝑖=1,2,…,𝑛; DMU₀: the target DMU;  𝑥_𝑘=(𝑥_𝑘1,𝑥_𝑘2,…,𝑥_𝑘𝑝): the uncertain inputs vector of DMU_𝑘,  𝑘=1,2,…,𝑛; Φ_𝑘𝑖(𝑥): the uncertainty distribution of 𝑥_𝑘𝑖,  𝑘=1,2,…,𝑛,  𝑖=1,2,…,𝑝; 𝑥₀=(𝑥₀₁,𝑥₀₂,…,𝑥_0𝑝): the inputs vector of the target DMU₀; Φ_0𝑖(𝑥): the uncertainty distribution of 𝑥_0𝑖,  𝑖=1,2,…,𝑝; 𝑦_𝑘=(𝑦_𝑘1,𝑦_𝑘2,…,𝑦_𝑘𝑞): the uncertain outputs vector of DMU_𝑘,  𝑘=1,2,…,𝑛; Ψ_𝑘𝑗(𝑥): the uncertainty distribution of 𝑦_𝑘𝑗,  𝑘=1,2,…,𝑛,  𝑗=1,2,…,𝑞; 𝑦₀=(𝑦₀₁,𝑦₀₂,…,𝑦_0𝑞): the outputs vector of the target DMU₀; Ψ_0𝑗(𝑥): the uncertainty distribution of 𝑦_0𝑗,  𝑗=1,2,…,𝑞.

3.1. Uncertainty Distributions of Inputs and Outputs

Liu and Ha [38] proposed a questionnaire survey for collecting expert’s experimental data. It is based on expert’s experimental data rather than historical data. The starting point is to invite one expert who is asked to complete a questionnaire about the meaning of an uncertain input (output) 𝜉 like “How many is the input (output)."

We first ask the domain expert to choose a possible value 𝑥 that the uncertain input 𝜉 may take and then quiz him, “How likely is 𝜉 less than or equal to 𝑥?"

Denote the expert’s belief degree by 𝛼. An expert’s experimental data (𝑥,𝛼) is thus acquired from the domain expert.

Repeating the above process, we can obtain the following expert’s experimental data: 𝑥₁,𝛼₁,𝑥₂,𝛼₂,…,𝑥_𝑛,𝛼_𝑛 (17) that meet the following consistence condition (perhaps after a rearrangement): 𝑥₁<𝑥₂<⋯<𝑥_𝑛,0≤𝛼₁≤𝛼₂≤⋯𝛼_𝑛≤1. (18)

Based on those expert’s experimental data, Liu and Ha [38] suggested an empirical uncertainty distribution, Φ(𝑥)=0,if𝑥≤𝑥₁,𝛼_𝑖+𝛼_𝑖+1−𝛼_𝑖𝑥−𝑥_𝑖𝑥_𝑖+1−𝑥_𝑖,if𝑥_𝑖≤𝑥≤𝑥_𝑖+1,1≤𝑖<𝑛,1,if𝑥>𝑥_𝑛. (19)

Assume there are 𝑚 domain experts and each produces an uncertainty distribution. Then we may get 𝑚 uncertainty distributions Φ₁(𝑥),Φ₂(𝑥),…,Φ_𝑚(𝑥). The Delphi method was originally developed in the 1950s by the RAND Corporation based on the assumption that group experience is more valid than individual experience. Wang et al. [39] recast the Delphi method as a process to determine the uncertainty distribution. The main steps are listed as follows.

Step 1. The 𝑚 domain experts provide their expert’s experimental data, 𝑥_𝑖𝑗,𝛼_𝑖𝑗,𝑗=1,2,…,𝑛_𝑖,𝑖=1,2,…,𝑚. (20)

Step 2. Use the 𝑖th expert’s experimental data (𝑥_𝑖1,𝛼_𝑖1),(𝑥_𝑖2,𝛼_𝑖2),…,(𝑥_𝑖𝑛𝑖,𝛼_𝑖𝑛𝑖) to generate the 𝑖th expert’s uncertainty distribution Φ_𝑖.

Step 3. Compute Φ(𝑥)=𝑤₁Φ₁(𝑥)+𝑤₂Φ₂(𝑥)+⋯+𝑤_𝑚Φ_𝑚(𝑥), where 𝑤₁,𝑤₂,…,𝑤_𝑚 are convex combination coefficients.

Step 4. If |𝛼_𝑖𝑗−Φ(𝑥_𝑖𝑗)| are less than a given level 𝜀>0, then go to Step 5. Otherwise, the 𝑖th expert receives the summary (Φ and reasons) and then provides a set of revised expert’s experimental data. Go to Step 2.

Step 5. The last Φ is the uncertainty distribution of the input (output).

3.2. Uncertain DEA Model

Similar to traditional DEA model [3], the objective of the uncertain DEA model is to maximize the total slacks in inputs and outputs subject to the constraints. Then the uncertain DEA model can be given as follows: max^𝑝_𝑖=1𝑠⁻_𝑖+^𝑞_𝑗=1𝑠⁺_𝑗subjectto:𝑀^𝑛_𝑘=1𝑥_𝑘𝑖𝜆_𝑘≤𝑥_0𝑖−𝑠⁻_𝑖𝑀≥𝛼,𝑖=1,2,…,𝑝,_𝑛𝑘=1𝑦_𝑘𝑗𝜆_𝑘≥𝑦_0𝑗+𝑠⁺_𝑗≥𝛼,𝑗=1,2…,𝑞,_𝑛𝑘=1𝜆_𝑘𝜆=1,_𝑘𝑠≥0,𝑘=1,2,…,𝑛,_−𝑖𝑠≥0,𝑖=1,2…,𝑝,_+𝑗≥0,𝑗=1,2,…,𝑞. (21)

Definition 13 (𝛼-efficiency). DMU₀ is 𝛼-efficient if 𝑠_𝑖^−∗ and 𝑠_𝑗^+∗ are zero for 𝑖=1,2,…,𝑝 and 𝑗=1,…,𝑞, where 𝑠_𝑖^−∗ and 𝑠_𝑗^+∗ are optimal solutions of (21).

Since the uncertain measure is involved, this definition is different from traditional efficiency definition. For instance, as determined by the choice of 𝛼, there is a risk that DMU₀ will not be efficient even when the condition of Definition 13 is satisfied.

Since 𝑗=0 is one of the DMU_𝑗, we can always get a solution with 𝜆₀=1, 𝜆_𝑗=0  (𝑗≠0), and all slacks zero. Thus this uncertain DEA model has feasible solution and the optimal value 𝑠_𝑖^−∗=𝑠_𝑗^+∗=0 for all 𝑖,𝑗.

3.3. Deterministic Model

Model (21) is an uncertain programming model, which is too complex to compute directly. This section will give its equivalent crisp model to simplify the computation process.

Theorem 14. Assume that 𝑥_1𝑖,𝑥_2𝑖,…,𝑥_𝑛𝑖 are independent uncertain inputs with uncertainty distribution Φ_1𝑖,Φ_2𝑖,…,Φ_𝑛𝑖 for each 𝑖, 𝑖=1,2,…,𝑝, and 𝑦_1𝑖,𝑦_2𝑖,…,𝑦_𝑛𝑖 are independent uncertain outputs with uncertainty distribution Ψ_1𝑗,Ψ_2𝑗,…,Ψ_𝑛𝑗 for each 𝑗, 𝑗=1,2,…,𝑞. Then 𝑀^𝑛_𝑘=1𝑥_𝑘𝑖𝜆_𝑘≤𝑥_0𝑖−𝑠⁻_𝑖𝑀≥𝛼,𝑖=1,2,…,𝑝,_𝑛𝑘=1𝑦_𝑘𝑗𝜆_𝑘≥𝑦_0𝑗+𝑠⁺_𝑗≥𝛼,𝑗=1,2…,𝑞 (22) holds if and only if ^𝑛_{𝑘=1,𝑘≠0}𝜆_𝑘Φ⁻¹_𝑘𝑖(𝛼)+𝜆₀Φ⁻¹_0𝑖(1−𝛼)≤Φ⁻¹_0𝑖(1−𝛼)−𝑠⁻_𝑖,𝑖=1,2,…,𝑝,_{𝑛𝑘=1,𝑘≠0}𝜆_𝑘Ψ⁻¹_𝑘𝑗(1−𝛼)+𝜆₀Ψ⁻¹_0𝑗(𝛼)≥Ψ⁻¹_0𝑗(𝛼)+𝑠⁻_𝑗,𝑗=1,2,…,𝑞. (23)

Proof. Without loss of generality, let 𝑖=1 and 𝑥₀=𝑥₁; then we will consider the equation 𝑀^𝑛_𝑘=1𝑥_𝑘1𝜆_𝑘≤𝑥₁₁−𝑠⁻_𝑖≥𝛼. (24)
Rewrite (24) as 𝑀^𝑛_𝑘=2𝑥_𝑘1𝜆_𝑘−1−𝜆₁𝑥₁₁≤−𝑠⁻_𝑖≥𝛼. (25)
Since −(1−𝜆₁)𝑥₁₁ is an uncertain variable which is decreasing with respect to 𝑥₁₁, its inverse uncertainty distribution is Υ⁻¹₁₁(𝛼)=−1−𝜆₁Φ⁻¹₁₁(1−𝛼),0<𝛼<1. (26)
For each 2≤𝑘≤𝑛,  𝑥_𝑘1𝜆_𝑘 is an uncertain variable whose inverse uncertainty distribution is Υ⁻¹_𝑘1(𝛼)=𝜆_𝑘Φ⁻¹_𝑘1(𝛼),0<𝛼<1. (27)
It follows from the operational law that the inverse uncertainty distribution of the sum ∑^𝑛_𝑘=2𝑥_𝑘1𝜆_𝑘−(1−𝜆₁)𝑥₁₁ is Υ⁻¹(𝛼)=^𝑛_𝑘=1Υ⁻¹₂₁(𝛼)=^𝑛_𝑘=2𝜆_𝑘Φ⁻¹_𝑘1(𝛼)−1−𝜆₁Φ⁻¹₁₁(1−𝛼),0<𝛼<1. (28) From which we may derive the result immediately for 𝑖=1 and 𝑥₀=𝑥₁. Similarly, we can get other results.
Following Theorem 14, the uncertain DEA model can be converted to the crisp model as follows: max^𝑝_𝑖=1𝑠⁻_𝑖+^𝑞_𝑗=1𝑠⁺_𝑗subjectto:^𝑛_{𝑘=1,𝑘≠0}𝜆_𝑘Φ⁻¹_𝑘𝑖(𝛼)+𝜆₀Φ⁻¹_0𝑖(1−𝛼)≤Φ⁻¹_0𝑖(1−𝛼)−𝑠⁻_𝑖,𝑖=1,2,…,𝑝,_{𝑛𝑘=1,𝑘≠0}𝜆_𝑘Ψ⁻¹_𝑘𝑗(1−𝛼)+𝜆₀Ψ⁻¹_0𝑗(𝛼)≥Ψ⁻¹_0𝑗(𝛼)+𝑠⁺_𝑗,𝑗=1,2,…,𝑞,_𝑛𝑘=1𝜆_𝑘𝜆=1,_𝑘𝑠≥0,𝑘=1,2,…,𝑛,_−𝑖𝑠≥0,𝑖=1,2…,𝑝,_+𝑗≥0,𝑗=1,2,…,𝑞 (29) which is a linear programming model. Thus it can be easily solved by many traditional methods.

4. A Numerical Example

This example wants to illustrate the uncertain DEA model. For simplicity, we will only consider five DMUs with two inputs and two outputs which are all zigzag uncertain variables denoted by 𝒵(𝑎,𝑏,𝑐). Table 1 gives the information of the DMUs.

Table 1 

DMUs with two uncertain inputs and two uncertain outputs.

For illustration, let DMU₁ be the target DMU; then the uncertain DEA model (29) can be written as max𝑠⁻₁+𝑠⁻₂+𝑠⁺₁+𝑠⁺₂subjectto:⁵_𝑘=2𝜆_𝑘Φ⁻¹_𝑘1(𝛼)+𝜆₁Φ⁻¹₁₁(1−𝛼)≤Φ⁻¹₁₁(1−𝛼)−𝑠⁻₁,_5𝑘=2𝜆_𝑘Φ⁻¹_𝑘2(𝛼)+𝜆₁Φ⁻¹₁₂(1−𝛼)≤Φ⁻¹₁₂(1−𝛼)−𝑠⁻₂,_5𝑘=2𝜆_𝑘Ψ⁻¹_𝑘1(1−𝛼)+𝜆₁Ψ⁻¹₁₁(𝛼)≥Ψ⁻¹₁₁(𝛼)+𝑠⁺₁,_5𝑘=2𝜆_𝑘Ψ⁻¹_𝑘2(1−𝛼)+𝜆₁Ψ⁻¹₁₂(𝛼)≥Ψ⁻¹₁₂(𝛼)+𝑠⁺₂,_5𝑘=1𝜆_𝑘𝜆=1,_𝑘𝑠≥0,𝑘=1,2,…,5,₋₁𝑠≥0,₋₂𝑠≥0,₊₁𝑠≥0,₊₂≥0. (30)

Table 2 shows the results of evaluating DMUs with confidence level 𝛼=0.6. The results can be interpreted in the following way: DMU₁ and DMU₃ are inefficient, whereas DMU₂, DMU₄, and DMU₅ are efficient. Moreover, DMU₃ is more efficient than DMU₁ from the total slacks ∑^𝑝_𝑖=1𝑠_𝑖^−∗+∑^𝑞_𝑗=1𝑠_𝑗^+∗, since they are both inefficient.

Table 2 

Results of evaluating the DMUs with 𝛼=0.6.

Uncertain efficiencies obtained from model (30) for different confidence levels 𝛼 are shown in Table 3. DMU₁ is inefficient at all confidence levels, whereas DMU₂, DMU₄, and DMU₅ are always efficient at all levels. It can be seen that the number of the efficient DMUs is affected by the confidence level 𝛼. The higher the confidence level 𝛼 is, the bigger the number of efficient DMUs is. This phenomena indicate that uncertain DEA is more complex than the traditional DEA because of the inherent uncertainty contained in inputs and outputs.

Table 3 

Results of evaluating the DMUs with different confidence level 𝛼.

5. Conclusion

Due to its widely practical used background, data envelopment analysis (DEA) has become a pop area of research. Since the data cannot be precisely measured in some practical cases, many papers have been published when the inputs and outputs are uncertain. This paper has given some researches to uncertain DEA model. A new DEA model as well as its equivalent deterministic model was presented. For illustration, a numerical example was designed.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by National Natural Science Foundation of China (nos. 71201005 and 61104132).