Using Enhanced Russell Model to Solve Inverse Data Envelopment Analysis Problems

Abstract

This paper studies the inverse data envelopment analysis using the nonradial enhanced Russell model. Necessary and sufficient conditions for inputs/outputs determination are introduced based on Pareto solutions of multiple-objective linear programming. In addition, an approach is investigated to identify extra input/lack output in each of input/output components (maximum/minimum reduction/increase amounts in each a of input/output components). In addition, the following question is addressed: if among a group of DMUs, it is required to increase inputs and outputs to a particular unit and assume that the DMU maintains its current efficiency level with respect to other DMUs, how much should the inputs and outputs of the DMU increase? This question is discussed as inverse data envelopment analysis problems, and a technique is suggested to answer this question. Necessary and sufficient conditions are established by employing Pareto solutions of multiple-objective linear programming as well.

1. Introduction

Data envelopment analysis (DEA) is a nonparametric method based on linear programming for computing relative efficiencies of a decision making unit (DMU) by comparing it with other DMUs such that they produce the same multiple outputs by consuming the same multiple inputs. This method was first introduced by Charnes et al. (CCR model) [1] and extended by Banker et al. (BCC model) [2]. In the two recent decades, a wide range of research in operations research field has been allocated to this technique; see, for example, [3–5].

Relationships between DEA and MOLP can be applied as instruments in strategic planning and management control. These two types of models have much in common. However, DEA is to assess past performances as part of the management control function while MOLP is to plan future performances. These relationships have been used and developed by many scholars; see, for example, [6–12].

Inverse DEA is one of the most noticeable subjects both practically and theoretically. This concept first was discussed by Zhang and Cui [13]. Some of the questions are introduced by Wei et al. [14] in inverse DEA filed. They considered inverse DEA to answer this question: “if among a group of DMUs, we increase certain inputs to a particular unit and assume that the DMU maintains its current efficiency level with respect to other DMUs, how much should the outputs of the DMU increase?” In order to answer this question, Wei et al. [14] and Yan et al. [15] offered a linear programming problem to estimate the outputs when the unit under assessment is weakly efficient and a MOLP model when the unit under assessment is inefficient.

Moreover, in the following, Hadi-Vencheh et al. [16, 17] attempted to answer this question: “if among a group of DMUs, we increase certain outputs to a particular unit and assume that the DMU maintains its current efficiency level with respect to other DMUs, how much should the inputs of the DMU increase?” In their studies, the proposed models by Wei et al. [14] have been developed. After introducing inverse DEA, some of researchers studied it theoretically and practically [15–21].

In all investigations done on inverse DEA, researchers considered radial models such as CCR [1], BCC [2], ST [22], and FG [23] models. As it is known, the nonradial models have some different properties compared with the radial models. Therefore, in some cases, answering the questions presented in the literature inverse DEA with the nonradial models can provide more appropriate information. Consequently, for more suitable analysis, one of the nonradial models, for instance, additive models or slack-based models, can be considered. In this research, to solve some of the above problems, the nonradial Enhanced Russell Model (ERM) [24] is considered. In addition, a new problem in inverse DEA field is introduced: “if among a group of DMUs for a particular unit, the decision maker is required to increase inputs and outputs, in which, the DMU maintains its current efficiency level with respect to other DMUs, how much should the inputs and outputs of the DMU increase?” Pareto solutions of MOLP are used to solve this problem.

The paper is organized as follows. In Section 2, DEA models are reviewed and extended and the problem is stated in DEA terms. It is shown that how the inverse DEA problem (increment of the outputs) can be converted to and solved by a multiobjective programming problem, when the DMU is ERM efficient. If there exists a lack in each of the output components, its amount is specified. In Section 3, the question proposed by Wei et al. [14] is answered, when the DMU is ERM efficient. Likewise, if there exists an extra in each of the input components, its amount is specified. The new problem is solved in Section 4. For a special decision making unit providing that the ERM-efficiency score remains unchanged, necessary and sufficient conditions are introduced based on Pareto solutions of multiple-objective linear programming problems to find the minimum and maximum increase of input and output vectors, respectively. In Section 5, two examples are used to illustrate our calculation method. Finally, Section 6 demonstrates some conclusions and suggestions for future research.

2. Estimate Inputs

Let us consider a set of DMU_𝑠 {DMU_𝑗∶𝑗=1,…,𝑛}, in which DMU_𝑗 produces multiple outputs 𝑦_𝑟𝑗(𝑟=1,…,𝑠), by utilizing multiple inputs 𝑥_𝑖𝑗(𝑖=1,…,𝑚). Let input and output for DMU_𝑗 be denoted by 𝑋_𝑗=(𝑥_1𝑗,𝑥_2𝑗,…,𝑥_𝑚𝑗)^𝑡 and 𝑌_𝑗=(𝑦_1𝑗,𝑦_2𝑗,…,𝑦_𝑠𝑗)^𝑡, respectively. Also, suppose 𝑋_𝑗≩0 and 𝑌_𝑗≩0. The nonradial enhanced Russell model (ERM) [24] is considered for measuring the relative efficiency of the unit under assessment DMU_𝑜, 𝑜∈{1,2,…,𝑛}, as follows: 𝜌^∗_𝑜()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃_𝑖𝑥_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑_𝑟𝑦_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠,𝜆∈Ω, (1) where Ω=𝜆∣𝜆=𝜆₁,…,𝜆_𝑛,𝛿₁^𝑛_𝑗=1𝜆_𝑗+𝛿₂(−1)^𝛿3𝜈=𝛿₁,𝜈≥0,𝜆_𝑗≥0,𝑗=1,…,𝑛. (2)

Here 𝛿₁, 𝛿₂, and 𝛿₃ are parameters with 0-1 values. It is obvious that if 𝛿₁=0, then model (1) is under constant returns to scale (CRC), if 𝛿₁=1 and 𝛿₂=0, then model (1) is under variable returns to scale (VRS), if 𝛿₁=𝛿₂=1 and 𝛿₃=0, then model (1) is under a nonincreasing returns to scale (NIRS), and if 𝛿₁=𝛿₂=𝛿₃=1, then model (1) is under a nondecreasing returns to scale (NDRS) assumption of the production technology.

Remark 1. Although in this study under discussion results satisfy CRC, VRS, NIRS, and NDRS assumption of the production technology, but the CRC model is considered only for simplicity.

Definition 2 (ERM-efficiency, see [24]). The optimal value 𝜌^∗_𝑜 of the model (1) is called the ERM-efficiency score of DMU_𝑜. DMU_𝑜 is ERM efficient, if and only if 𝜌^∗_𝑜=1 (this condition is equivalent to 𝜃^∗_𝑖=1 and 𝜑^∗_𝑟=1 for each 𝑖=1,…,𝑚, 𝑟=1,…,𝑠 in any optimal solution).

In [13] Zhang and Cui introduced inverse DEA. Since then this problem has allocated to itself some of researches in DEA field; see, for example [5, 14, 16–19, 21]. Based upon investigated results by Hadi-Vencheh et al. [16], this question is considered: suppose that the DMU_𝑜 is ERM efficient, if the ERM-efficiency score of DMU_𝑜 remains unchanged, but the outputs increase, how much should the inputs of the DMU_𝑜 increase? 

To answer the above question, until the end of this section, presume that the outputs of DMU_𝑜 are increased from 𝑌_𝑜 to 𝛽_𝑜=𝑌_𝑜+Δ𝑌_𝑜, where Δ𝑌_𝑜≩0. The objective of the problem is to estimate the input vector on the condition that DMU_𝑜 is still ERM efficient. In fact, 𝛼^∗_𝑜=𝛼^∗_1𝑜,𝛼^∗_2𝑜,…,𝛼^∗_𝑚𝑜^𝑡=𝑋_𝑜+Δ𝑋_𝑜,Δ𝑋_𝑜≩0. (3)

Assume DMU_𝑛+1 represents DMU_𝑜 after modification of the inputs and outputs. The following model is considered to calculate the ERM efficiency of DMU_𝑛+1: 𝜌_𝑜^+∗()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗+𝛼^∗_𝑖𝑜𝜆_𝑛+1≤𝜃_𝑖𝛼^∗_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗+𝛽_𝑟𝑜𝜆_𝑛+1≥𝜑_𝑟𝛽_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛+1. (4)

Along with [16], the following definition is considered.

Definition 3. If the optimal values of models (1) and (4) are equal, it is said that the ERM efficiency remains unchanged; that is, eff(𝛼^∗_𝑜,𝛽_𝑜)=eff(𝑋_𝑜,𝑌_𝑜).

To answer the above question, based on the results of [16], the following MOLP model is considered: min𝛼_1𝑜,…,𝛼_𝑚𝑜s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜𝛼,𝑟=1,…,𝑠_𝑖𝑜≥𝑥_𝑖𝑜𝜆,𝑖=1,…,𝑚_𝑗≥0,𝑗=1,…,𝑛, (5) where (𝜃^∗=(𝜃^∗₁,…,𝜃^∗_𝑚), 𝜑^∗=(𝜑^∗₁,…,𝜑^∗_𝑠)) is an optimal solution to problem (1).

Definition 4 (see [16]). Let (𝜆^∗,𝛼^∗_𝑜) be a feasible solution to problem (5). If there is no feasible solution (𝜆,𝛼_𝑜) of (5) such that 𝛼_𝑖𝑜≤𝛼^∗_𝑖𝑜 for all 𝑖=1,…,𝑚 and 𝛼_𝑖𝑜<𝛼^∗_𝑖𝑜 for at least one 𝑖, then it is said the (𝜆^∗,𝛼^∗_𝑜) is a Pareto (strongly efficient) solution to problem (5).

Definition 5 (see [14]). Let (𝜆^∗,𝛼^∗_𝑜) be a feasible solution to problem (5). If there is no feasible solution (𝜆,𝛼_𝑜) of (5) such that 𝛼_𝑖𝑜<𝛼^∗_𝑖𝑜 for all 𝑖=1,…,𝑚, then it is said the (𝜆^∗,𝛼^∗_𝑜) is a weakly Pareto (efficient) solution to problem (5).

Theorem 6. Suppose that the 𝐷𝑀𝑈_𝑜 is 𝐸𝑅𝑀 efficient and (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1) (𝜃^∗_𝑖=1,𝜑^∗_𝑟=1 for all 𝑖,𝑟 and 𝜌^∗_𝑜=1). Let (𝜆^∗,𝛼^∗_𝑜) be a Pareto solution to problem (5) such that 𝛼^∗_𝑜>𝑋_𝑜, and (𝜆^+∗,𝜃^+∗,𝜑^+∗) is an optimal solution to problem (4) with the optimal value of 𝜌_𝑜^+∗. In addition, suppose that the optimal value and an optimal solution of the problem 𝐾^∗_𝑜1=max𝑠^𝑠_𝑟=1𝑘_𝑟𝑜s.t.^𝑛_𝑗=1𝜆_𝑗^+∗+𝜆^+∗_𝑛+1𝜆^∗_𝑗𝑦_𝑟𝑗≥𝑘_𝑟𝑜𝜑^∗_𝑟𝛽_𝑟𝑜=𝑘_𝑟𝑜𝛽_𝑟𝑜,𝑘𝑟=1,…,𝑠_𝑟𝑜≥1,𝑟=1,…,𝑠 (6) are 𝐾^∗_𝑜 and 𝑘^∗_𝑜=(𝑘^∗_1𝑜,…,𝑘^∗_𝑠𝑜), respectively. If the inputs of the 𝐷𝑀𝑈_𝑜 are increased to 𝛼^∗_𝑜, then(i)if 𝐾^∗_𝑜=1 (it is clear 𝑘^∗_𝑟𝑜=1, for each 𝑟), then 𝑒𝑓𝑓(𝛼^∗_𝑜,𝛽_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜),(ii)if 𝐾^∗_𝑜>1, then 𝑒𝑓𝑓(𝛼^∗_𝑜,𝑘^∗_𝑜⋆𝛽_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜).

Remark 7. 𝑘^∗_𝑜⋆𝛽_𝑜=(𝑘^∗_1𝑜𝛽_1𝑜,…,𝑘^∗_𝑠𝑜𝛽_𝑠𝑜) and (𝑘^∗_𝑜⋆𝛽_𝑜)−𝛽_𝑜=(𝑘^∗_1𝑜𝛽_1𝑜−𝛽_1𝑜,…,𝑘^∗_𝑠𝑜𝛽_𝑠𝑜−𝛽_𝑠𝑜). Note that the 𝑘^∗_𝑟𝑜𝛽_𝑟𝑜−𝛽_𝑟𝑜 indicates the lack-output amount in 𝑟th output component of the DMU_𝑜. In other words, for the decision maker to preserve the ERM-efficiency score of the DMU_𝑜 while the inputs increase from 𝑋_𝑜 to 𝛼^∗_𝑜, they are required to increase the outputs from 𝑌_𝑜 to 𝑘^∗_𝑜⋆𝛽_𝑜.

Proof. Because (𝜆^∗,𝛼^∗_𝑜) is a feasible solution of (5), the following inequalities are held: ^𝑛_𝑗=1𝜆^∗_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼^∗_𝑖𝑜=𝛼^∗_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆^∗_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜=𝛽_𝑟𝑜,𝑟=1,…,𝑠𝛼^∗_𝑖𝑜≥𝑥_𝑖𝑜,𝑖=1,…,𝑚𝜆^∗_𝑗≥0,𝑗=1,…,𝑛. (7)(8)(9)(10)
With regard to inequalities (7), (8), and (10) it is obvious that (𝜆,𝜃^∗,𝜑^∗) is a feasible solution to problem (4), where 𝜆=(𝜆^∗,0)≥0, 𝜃^∗=(𝜃^∗₁,…,𝜃^∗_𝑚)∈𝑅^𝑚₊, and 𝜑^∗=(𝜑^∗₁,…,𝜑^∗_𝑠)∈𝑅^𝑠₊, therefore, 𝜌_𝑜^+∗≤𝜌^∗_𝑜.
Using inequalities (7) and (8) in problem (4), the following results are obtained: 𝜃_𝑖^+∗𝛼^∗_𝑖𝑜≥^𝑛_𝑗=1𝜆_𝑗^+∗𝑥_𝑖𝑗+𝜆^+∗_𝑛+1𝛼^∗_𝑖𝑜=^𝑛_𝑗=1𝜆_𝑗^+∗𝑥_𝑖𝑗+𝜆^+∗_𝑛+1^𝑛_𝑗=1𝜆^∗_𝑗𝑥_𝑖𝑗,𝜃_𝑖^+∗𝛼^∗_𝑖𝑜≥^𝑛_𝑗=1𝜆_𝑗^+∗+𝜆^+∗_𝑛+1𝜆^∗_𝑗𝑥_𝑖𝑗,𝑖=1,…,𝑚𝜑_𝑟^+∗𝛽_𝑟𝑜≤^𝑛_𝑗=1𝜆_𝑗^+∗𝑦_𝑟𝑗+𝜆^+∗_𝑛+1𝛽_𝑟𝑜≤^𝑛_𝑗=1𝜆_𝑗^+∗𝑦_𝑟𝑗+𝜆^+∗_𝑛+1^𝑛_𝑗=1𝜆^∗_𝑗𝑦_𝑟𝑗,𝜑_𝑟^+∗𝛽_𝑟𝑜≤^𝑛_𝑗=1𝜆_𝑗^+∗+𝜆^+∗_𝑛+1𝜆^∗_𝑗𝑦_𝑟𝑗,𝑟=1,…,𝑠. (11)(12)
Set 𝜆_𝑗=𝜆_𝑗^+∗+𝜆^+∗_𝑛+1𝜆^∗_𝑗 for each 𝑗=1,…,𝑛, and it is obviously seen that 𝜆=(𝜆₁,…,𝜆_𝑛)≥0.
Now by contradiction assume that 𝜌_𝑜^+∗<𝜌^∗_𝑜.
Since 𝜌_𝑜^+∗<𝜌^∗_𝑜, then (1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖^+∗<1 or (1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟^+∗>1, so that there are two cases.
(a) Let (1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖^+∗<1. There exists at least one 𝑡, 1≤𝑡≤𝑚, such that 𝜃_𝑡^+∗<1. With regard to inequalities (11), the following inequality is obtained: ^𝑛_𝑗=1𝜆_𝑗𝑥_𝑡𝑗≤𝜃_𝑡^+∗𝛼^∗_𝑡𝑜<𝛼^∗_𝑡𝑜. (13)
On the other hand, since, for each 𝑖=1,…,𝑚, 𝛼^∗_𝑖𝑜>𝑥_𝑖𝑜, therefore if 𝜂_𝑡=min𝛼^∗_𝑡𝑜−^𝑛_𝑗=1𝜆_𝑗𝑥_𝑡𝑗,𝛼^∗_𝑡𝑜−𝑥_𝑡𝑜, (14) then 𝜂_𝑡>0. Now define 𝛼_𝑖𝑜=𝛼^∗_𝑖𝑜𝛼,if𝑖≠𝑡,_∗𝑖𝑜−𝜂_𝑡,if𝑖=𝑡. (15)
Taking (14) into consideration, the following inequalities are obtained: 𝜂_𝑡≤𝛼^∗_𝑡𝑜−^𝑛_𝑗=1𝜆_𝑗𝑥_𝑡𝑗⟹^𝑛_𝑗=1𝜆_𝑗𝑥_𝑡𝑗≤𝛼^∗_𝑡𝑜−𝜂_𝑡=𝜃^∗_𝑡𝛼_𝑡𝑜,𝜂_𝑡≤𝛼^∗_𝑡𝑜−𝑥_𝑡𝑜⟹𝑥_𝑡𝑜≤𝛼^∗_𝑡𝑜−𝜂_𝑡,⟹𝑥_𝑡𝑜≤𝛼_𝑡𝑜, (16)(17) which implies that 𝛼_𝑜≥𝑋_𝑜.
Based on 𝜆≥0 and (11), (12), and (16) and 𝛼_𝑜≥𝑋_𝑜, (𝜆,𝛼_𝑜) is a feasible solution to problem (5), where 𝛼_𝑖𝑜≤𝛼^∗_𝑖𝑜 for all 𝑖=1,…,𝑚, and 𝛼_𝑖𝑜<𝛼^∗_𝑖𝑜 for some 𝑖=1,…,𝑚. This contradicts the assumption that (𝜆^∗,𝛼^∗_𝑜) is a Pareto solution to problem (5).
(b) Let (1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟^+∗>1. There exists at least one 𝑡, 1≤𝑡≤𝑠, such that 𝜑_𝑡^+∗>1. By (12), the following strict inequality is obtained: ^𝑛_𝑗=1𝜆_𝑗𝑦_𝑡𝑗≥𝜑_𝑡^+∗𝛽_𝑡𝑜>𝛽_𝑡𝑜 (18) and, therefore, there exists a 𝜇_𝑡>1 that satisfies ^𝑛_𝑗=1𝜆_𝑗𝑦_𝑡𝑗≥𝜇_𝑡𝛽_𝑡𝑜=𝜇_𝑡𝑘^∗_𝑡𝑜𝛽_𝑡𝑜. (19)
Now define 𝑘_𝑟𝑜=𝑘^∗_𝑟𝑜𝑘=1,if𝑟≠𝑡,_∗𝑟𝑜𝜇_𝑡=𝜇_𝑡,if𝑟=𝑡. (20)
It is obvious that 𝑘_𝑜=(𝑘_1𝑜,…,𝑘_𝑠𝑜) is a feasible solution to problem (6), such that 𝐾^∗_𝑜=1𝑠^𝑠_𝑟=1𝑘_𝑟𝑜<1𝑠^𝑠_𝑟=1𝑘_𝑟𝑜=𝐾_𝑜. (21)
This contradicts the assumption expressing that the optimal value of problem (6) is 𝐾^∗_𝑜. Consequently, in each case 𝜌_𝑜^+∗<𝜌̸^∗_𝑜, and since 𝜌_𝑜^+∗⩽𝜌^∗_𝑜, therefore, 𝜌_𝑜^+∗=𝜌^∗_𝑜.
(ii) with replacing the notation 𝛽_𝑟𝑜 by 𝑘^∗_𝑟𝑜𝛽_𝑟𝑜 for each 𝑟=1,…,𝑠, in problem (6), clearly, optimal value of problem (6) is 1. Therefore, according to part (i), then eff(𝛼^∗_𝑜,𝑘^∗_𝑜⋆𝛽_𝑜)=eff(𝑋_𝑜,𝑌_𝑜).

Although Theorem 6 satisfies for all units that be ERM-efficient, but the following theorem is the converse version of Theorem 6 that satisfies for all DMU_𝑠.

Theorem 8. Suppose that the (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1) with the optimal value of 𝜌^∗_𝑜. Let (𝜆,𝛼_𝑜) be a feasible solution to problem (5). If 𝑒𝑓𝑓(𝛼_𝑜,𝛽_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜), then (𝜆,𝛼_𝑜) must be a Pareto solution to problem (5).

Proof. If (𝜆,𝛼_𝑜) is not a Pareto solution to problem (5), then there would exist another feasible solution of (5), (𝜆,𝛼_𝑜) in which, 𝛼_𝑖𝑜≤𝛼_𝑖𝑜 for 𝑖=1,…,𝑚 and 𝛼_𝑖𝑜<𝛼_𝑖𝑜 for at least one 𝑖∈{1,…,𝑚}. Let 𝐼={𝑖∣𝛼_𝑖𝑜<𝛼_𝑖𝑜}. Then ^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜<𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖∈𝐼,_𝑛𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜≤𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖∉𝐼,_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜,𝑟=1,…,𝑠,𝜆_𝑗≥0,𝑗=1,…,𝑛. (22)(23)(24)(25)
By inequality (22) there exists 𝜇_𝑖<1 for all 𝑖∈𝐼, such that ^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜇_𝑖𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖∈𝐼. (26)
For each 𝑖=1,…,𝑚 and 𝑟=1,…,𝑠, define 𝜃_𝑖=𝜇_𝑖𝜃^∗_𝑖𝜃if𝑖∈𝐼,_∗𝑖,if𝑖∉𝐼,𝜑_𝑟=𝜑^∗_𝑟,𝑟=1,…,𝑠. (27)
Based on inequalities (23)–(26), (𝜆,𝜃,𝜑) is a feasible solution to problem (1) (only restrictions on the right have replaced the 𝑋_𝑜 and 𝑌_𝑜 by 𝛼_𝑜 and 𝛽_𝑜 in (1), resp.), where 𝜃=(𝜃₁,…,𝜃_𝑚) and 𝜑=(𝜑₁,…,𝜑_𝑠). Therefore 𝜌_𝑜=(1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟<(1/𝑚)∑^𝑚_𝑖=1𝜃^∗_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑^∗_𝑟=𝜌^∗_𝑜, (28) which is against the assumption expressing that eff(𝛼_𝑜,𝛽)=𝜌^∗_𝑜.

Along with Theorem 2 in Section 5 in [14] the following theorem is considered.

Theorem 9. Suppose that the 𝐷𝑀𝑈_𝑜 is 𝐸𝑅𝑀 efficient and the inputs and the outputs for 𝐷𝑀𝑈_𝑜 are going to increase from 𝑋_𝑜 and 𝑌_𝑜 to 𝛼_𝑜=𝑋_𝑜+Δ𝑋_𝑜 and 𝛽_𝑜=𝑌_𝑜+Δ𝑌_𝑜, respectively, where there are Δ𝑋_𝑜≩0, Δ𝑌_𝑜≩0. It is worth mentioning that it is required that the (𝛼_𝑜,𝛽_𝑜) belongs to the production possibility set. Consider the following problem: 𝜌_𝑜()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃_𝑖𝛼_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑_𝑟𝛽_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛. (29)
If the inputs and outputs of 𝐷𝑀𝑈_𝑜 are increased from 𝑋_𝑜 to 𝑋_𝑜=(𝜃₁𝑥_1𝑜,…,𝜃_𝑚𝑥_𝑚𝑜) and 𝑌_𝑜=(𝜑₁𝑦_1𝑜,…,𝜑_𝑠𝑦_𝑠𝑜), respectively, and (𝜆,𝜃,𝜑) is an optimal solution to problem (29) with the optimal value of 𝜌, then 𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜).

Proof. When the inputs and outputs of the DMU_𝑜 convert to 𝑋_𝑜 and 𝑌_𝑜, respectively, the ERM-efficiency score of the DMU equals the optimal value of the problem below 𝜌_𝑜()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗+𝜆_𝑛+1𝑥_𝑖𝑜≤𝜃_𝑖𝑥_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗+𝜆_𝑛+1𝑦_𝑟𝑜≥𝜑_𝑟𝑦_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛+1. (30)
Suppose that (𝜆,𝜃,𝜑) is an optimal solution to problem (30) with the optimal value of 𝜌_𝑜. To prove the theorem, 𝜌_𝑜=1 should be shown. By contradiction assume that 𝜌_𝑜<1.
Problem (30) is nonlinear, but it can be converted to the following linear programming problem by employing the 𝑡=1/(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟>0: 𝜌_𝑜1=min𝑚^𝑚_𝑖=1𝜃_𝑖s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗+𝜆_𝑛+1𝑥_𝑖𝑜≤𝜃_𝑖𝑥_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗+𝜆_𝑛+1𝑦_𝑟𝑜≥𝜑_𝑟𝑦_𝑟𝑜,𝑟=1,…,𝑠_𝑠𝑟=1𝜑_𝑟=𝑠,𝑡≥𝜖,𝜃_𝑖≤𝑡,𝜑_𝑟≥𝑡,𝑖=1,…,𝑚,𝑟=1,…,𝑠𝜆_𝑗≥0,𝑗=1,…,𝑛+1, (31) where (𝜃,𝜑)=(𝑡𝜃,𝑡𝜑), 𝜆_𝑗=𝑡𝜆_𝑗 for 𝑗=1,…,𝑛, and 𝜖 is non-Archimedean infinitesimal. Note that 𝑡≥𝜖 is a redundant constraint in problem (31) and hence can be omitted. The dual problem (31) is as follows: max𝑠𝑧s.t.𝑈^𝑡𝑌_𝑗−𝑉^𝑡𝑋_𝑗𝑈≤0,𝑗=1,…,𝑛_𝑡𝑌_𝑜−𝑉^𝑡𝑋_𝑜≤0,𝑣_𝑖𝑥_𝑖𝑜−𝜏_𝑖=1𝑚,𝑖=1,…,𝑚−𝑢_𝑟𝑦_𝑟𝑜+𝑤_𝑟+𝑧=0,𝑟=1,…,𝑠𝑒_𝑚𝜏−𝑒_𝑠𝑊=0,𝑉≥0,𝑈≥0,𝜏≥0,𝑧∈𝑅, (32)(33)(34) where 𝑒_𝑚=(1,…,1)∈𝑅^𝑚 and 𝑒_𝑠=(1,…,1)∈𝑅^𝑠. Assume (𝑉,𝑈,𝜏,𝑊,𝑧) is an optimal solution to the above problem. Based on the duality Theorem, the following relation is held: 𝑠𝑧=𝜌_𝑜<1. (35)
Also, constraint sets (33) are added together and result in −𝑈^𝑡𝑌_𝑜+𝑒_𝑠𝑊+𝑠𝑧+𝑉^𝑡𝑋_𝑜−𝑒_𝑚𝜏=1; (36) taking (35) into consideration, the following inequality is obtained: 𝑠𝑧=𝜌_𝑜<1=−𝑈^𝑡𝑌_𝑜+𝑒_𝑠𝑊+𝑠𝑧+𝑉^𝑡𝑋_𝑜−𝑒_𝑚𝜏, (37) and, therefore 𝑈^𝑡𝑌_𝑜−𝑉^𝑡𝑋_𝑜<0. (38)
Since the variable 𝜆_𝑛+1 is corresponding to the constraint (32) in the above problem that is unbinding at each optimal solution, therefore, according to the complementary slackness theorem, 𝜆_𝑛+1 must be 0, and so 𝜆_𝑛+1=0.
Consequently, 𝜌_𝑜 is also the optimal value of the following problem: 𝜌_𝑜()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃_𝑖𝑥_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑_𝑟𝑦_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛. (39)
Now, consider the following problem: 𝜌_𝑜()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃_𝑖𝜃_𝑖𝑥_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑_𝑟𝜑_𝑟𝑦_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖𝜃_𝑖≤1,𝜑_𝑟𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛. (40)
Clearly, (𝜆,𝜃,𝜑) is a feasible solution to problem (40), and so 𝜌_𝑜≤(1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟𝜑_𝑟. (41)
On the other hand, since 𝜌_𝑜<1, there exists at least one 𝑡, 1≤𝑡≤𝑚 or 1≤𝑡≤𝑠, such that 𝜃_𝑡<1, or 𝜑_𝑡>1. Therefore, in each case 𝜌_𝑜≤(1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟𝜑_𝑟<(1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟=𝜌_𝑜. (42)
Obviously, problem (40) is just problem (29) (replacing the notation 𝜃_𝑖𝜃_𝑖 by 𝜃_𝑖 and 𝜑_𝑟𝜑_𝑟 by 𝜑_𝑟). Therefore, in each case, the optimal value of problem (29) would be 𝜌_𝑜<𝜌_𝑜, which contradicts the fact that the maximum value of (29) is 𝜌_𝑜.

3. Estimate Outputs

In this section, the problem provided by Wei et al. [14] is considered. Based on the results of their study, this question is addressed: suppose that the DMU_𝑜 is ERM efficient, if the ERM-efficiency score of DMU_𝑜 remains unchanged while the inputs increase, and how much should the outputs of the DMU_𝑜 increase?

To solve the above problem, to the end of this section, presume that the inputs of DMU_𝑜 are increased from 𝑋_𝑜 to 𝛼_𝑜=𝑋_𝑜+Δ𝑋_𝑜, where Δ𝑋_𝑜≩0. The aim of the problem is to estimate the output vector provided that the DMU_𝑜 is still ERM efficient. In fact, 𝛽^∗_𝑜=𝛽^∗_1𝑜,𝛽^∗_2𝑜,…,𝛽^∗_𝑠𝑜^𝑡=𝑌_𝑜+Δ𝑌_𝑜,Δ𝑌_𝑜≩0. (43)

Assume DMU_𝑛+1 represents DMU_𝑜 after modification of the inputs and outputs. Along the line of [14], the following model is considered to calculate the ERM efficiency of DMU_𝑛+1: 𝜌_𝑜^+∗()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗+𝛼_𝑖𝑜𝜆_𝑛+1≤𝜃_𝑖𝛼_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗+𝛽^∗_𝑟𝑜𝜆_𝑛+1≥𝜑_𝑟𝛽^∗_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛+1. (44)

Based on Definition 3 it is said that the ERM efficiency remains unchanged, if and only if eff(𝛼_𝑜,𝛽^∗_𝑜)=eff(𝑋_𝑜,𝑌_𝑜). To answer the above question, along the line of [14], the following MOLP model is considered: max𝛽_1𝑜,…,𝛽_𝑠𝑜s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜𝛼,𝑟=1,…,𝑠_𝑖𝑜≥𝑥_𝑖𝑜𝜆,𝑖=1,…,𝑚_𝑗≥0,𝑗=1,…,𝑛, (45) where (𝜃^∗,𝜑^∗) is an optimal solution to problem (1).

Theorem 10. Suppose that the DMU_𝑜 is ERM efficient and (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1). Let (𝜆^∗,𝛽^∗_𝑜) be a Pareto solution to problem (45) such that 𝛽^∗_𝑜>𝑌_𝑜, and (𝜆^+∗,𝜃^+∗,𝜑^+∗) is an optimal solution to problem (4) with the optimal value of 𝜌_𝑜^+∗. Also, suppose that the optimal value and an optimal solution of the following problem: 𝐾^∗_𝑜1=max𝑚^𝑚_𝑖=1𝑘_𝑖𝑜s.t.^𝑛_𝑗=1𝜆_𝑗^+∗+𝜆^+∗_𝑛+1𝜆^∗_𝑗𝑥_𝑖𝑗≤𝑘_𝑖𝑜𝜃^∗_𝑖𝛼_𝑖𝑜=𝑘_𝑖𝑜𝛼_𝑖𝑜,𝑘𝑖=1,…,𝑚_𝑖𝑜≤1,𝑖=1,…,𝑚 (46) are 𝐾^∗_𝑜 and 𝑘^∗_𝑜=(𝑘^∗_1𝑜,…,𝑘^∗_𝑚𝑜). If the outputs of DMU_𝑜 are increased to 𝛽^∗_𝑜, then(i)if 𝐾^∗_𝑜=1, then 𝑒𝑓𝑓(𝛼_𝑜,𝛽^∗_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜),(ii)if 𝐾^∗_𝑜<1, then 𝑒𝑓𝑓(𝑘^∗_𝑜⋆𝛼_𝑜,𝛽^∗_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜).

Remark 11. 𝑘^∗_𝑜⋆𝛼_𝑜=(𝑘^∗_1𝑜𝛼_1𝑜,…,𝑘^∗_𝑚𝑜𝛼_𝑚𝑜) and 𝛼_𝑜−(𝑘^∗_𝑜⋆𝛼_𝑜)=(𝛼_1𝑜−𝑘^∗_1𝑜𝛼_1𝑜,…,𝛼_𝑚𝑜−𝑘^∗_𝑚𝑜𝛼_𝑚𝑜). Note that the 𝛼_𝑖𝑜−𝑘^∗_𝑖𝑜𝛼_𝑖𝑜 indicates extra-input amount in 𝑖th input component of the DMU. In other words, for the decision maker to preserve the ERM-efficiency score of the DMU_𝑜 while the outputs increase from 𝑌_𝑜 to 𝛽^∗_𝑜, they are required to increase the inputs from 𝑋_𝑜 to 𝑘^∗_𝑜⋆𝛼_𝑜.

Proof. The proof is similar to the proof of Theorem 6.

Theorem 12. Assume (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1) with the optimal value of 𝜌^∗_𝑜. Let (𝜆,𝛽_𝑜) be a feasible solution to problem (45). If 𝑒𝑓𝑓(𝛼_𝑜,𝛽_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜), then (𝜆,𝛽_𝑜) must be a Pareto solution to problem (45).

Proof. The proof is similar to the proof of Theorem 8.

4. Estimate the Minimum Increase of Inputs and the Maximum Increase of Outputs

In order to present suitable patterns to the decision maker to increase inputs and outputs for an ERM-efficient DMU, under preserving the ERM-efficiency index, this new question in field of inverse DEA is addressed: if DMU_𝑜 is ERM efficient, while the inputs and outputs are required to be increased, how much should the inputs and outputs of the DMU_𝑜 be increased? In other words, under preserving the ERM-efficiency score, how much should the minimum and maximum of input and output vectors of the DMU_𝑜 increase, respectively?

By answering the question, the decision maker may be able to take better decisions in order to extend decision making units. That is to say that the decision maker can take necessary actions by choosing a suitable strategy for spreading an ERM-efficient DMU.

The aim of addressing this question is estimating the minimum increase of inputs and the maximum increase of outputs provided that the DMU_𝑜 is still ERM efficient. In fact, 𝛼^∗_𝑜=𝛼^∗_1𝑜,𝛼^∗_2𝑜,…,𝛼^∗_𝑚𝑜^𝑡=𝑋_𝑜+Δ𝑋_𝑜,Δ𝑋_𝑜𝛽≩0,_∗𝑜=𝛽^∗_1𝑜,𝛽^∗_2𝑜,…,𝛽^∗_𝑠𝑜^𝑡=𝑌_𝑜+Δ𝑌_𝑜,Δ𝑌_𝑜≩0. (47)

Remark 13. Note that it is required to make the rate of increase inputs and outputs of the unit under assessment bounded; otherwise, it is possible that at least one of the components of 𝛼^∗_𝑜 or 𝛽^∗_𝑜 is unbounded.

Suppose that DMU_𝑛+1 represents DMU_𝑜 after modification of the inputs and outputs. Based on results of [14, 16], the following model is considered to estimate the ERM efficiency of DMU_𝑛+1: 𝜌_𝑜^+∗()∑=min1/𝑚^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗+𝛼^∗_𝑖𝑜𝜆_𝑛+1≤𝜃_𝑖𝛼^∗_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗+𝛽^∗_𝑟𝑜𝜆_𝑛+1≥𝜑_𝑟𝛽^∗_𝑟𝑜𝜃,𝑟=1,…,𝑠_𝑖≤1,𝜑_𝑟𝜆≥1,𝑖=1,…,𝑚,𝑟=1,…,𝑠_𝑗≥0,𝑗=1,…,𝑛+1. (48)

Definition 14. If the optimal values of problems (1) and (48) are equal, it is said that the ERM efficiency remains unchanged; that is, eff(𝛼^∗_𝑜,𝛽^∗_𝑜)=eff(𝑋_𝑜,𝑌_𝑜).

To solve this new question, that is, to estimate the minimum increase of inputs and the maximum increase of outputs, along the line of [14, 16], the following MOLP problem is considered: min𝛼_1𝑜,…,𝛼_𝑚𝑜max𝛽_1𝑜,…,𝛽_𝑠𝑜s.t.^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖=1,…,𝑚_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜𝛼,𝑟=1,…,𝑠_𝑖𝑜≥𝑥_𝑖𝑜𝛽,𝑖=1,…,𝑚_𝑟𝑜≥𝑦_𝑟𝑜𝛽,𝑟=1,…,𝑠_𝑜∈Γ,𝛼_𝑜𝜆∈Λ,_𝑗≥0,𝑗=1,…,𝑛, (49) where 𝜃^∗ and 𝜑^∗ are an optimal solution to problem (1), Γ and Λ are bounded sets, and show the maximum rate of increase in inputs and outputs of the DMU_𝑜, respectively, such that by the decision maker are considered.

Theorem 15. Suppose that the 𝐷𝑀𝑈_𝑜 is ERM efficient and (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1). Let (𝜆^∗,𝛼^∗_𝑜,𝛽^∗_𝑜) be a Pareto solution to problem (49) such that 𝛼^∗_𝑜>𝑋_𝑜,𝛽^∗_𝑜≩𝑌_𝑜. If the inputs and outputs of DMU_𝑜 are increased to 𝛼^∗_𝑜 and 𝛽^∗_𝑜, respectively, then 𝑒𝑓𝑓(𝛼^∗_𝑜,𝛽^∗_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜).

Proof. Assume (𝜆^+∗,𝜃^+∗,𝜑^+∗) is an optimal solution to problem (48) with the optimal value of 𝜌^+∗. The proof is similar to the proof of Theorem 6, when replacing the notation 𝛽_𝑜 by 𝛽^∗_𝑜. The only difference is in case (b), that is, when (1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟^+∗>1. In this case, there exists at least one 𝑝, 1≤𝑝≤𝑠, such that 𝜑_𝑝^+∗>1. Taking (12) into consideration, the following inequality is obtained: ^𝑛_𝑗=1𝜆_𝑗𝑦_𝑝𝑗≥𝜑_𝑝^+∗𝛽^∗_𝑝𝑜>𝛽^∗_𝑝𝑜, (50) and, thus, there exists a 𝜂_𝑝>0 that satisfies ^𝑛_𝑗=1𝜆_𝑗𝑦_𝑝𝑗≥𝛽^∗_𝑝𝑜+𝜂_𝑝. (51)
Now, define 𝛽_𝑟𝑜=𝛽^∗_𝑟𝑜𝜂,if𝑟≠𝑝,_𝑟+𝛽^∗_𝑟𝑜,if𝑟=𝑝. (52)
Based on 𝜆≥0, inequality (9), (11), and (51), and 𝛽_𝑜≩𝛽^∗_𝑜≩𝑌_𝑜, (𝜆,𝛼^∗_𝑜,𝛽_𝑜) is a feasible solution to problem (49), where 𝛽_𝑟𝑜≥𝛽^∗_𝑟𝑜 for all 𝑟=1,…,𝑠, and 𝛽_𝑟𝑜>𝛽^∗_𝑟𝑜 for some 𝑟=1,…,𝑠. This contradicts the assumption expressing that the (𝜆^∗,𝛼^∗_𝑜,𝛽^∗_𝑜) is a Pareto solution to problem (49). Therefore, in any two cases, because 𝜌_𝑜^+∗<𝜌̸^∗_𝑜 and 𝜌_𝑜^+∗⩽𝜌^∗_𝑜, therefore 𝜌_𝑜^+∗=𝜌^∗_𝑜.

Theorem 16. Suppose that the (𝜆^∗,𝜃^∗,𝜑^∗) is an optimal solution to problem (1) with the optimal value of 𝜌^∗_𝑜. Let (𝜆,𝛼_𝑜,𝛽_𝑜) be a feasible solution to the problem (49). If 𝑒𝑓𝑓(𝛼_𝑜,𝛽_𝑜)=𝑒𝑓𝑓(𝑋_𝑜,𝑌_𝑜), then (𝜆,𝛼_𝑜,𝛽_𝑜) must be a Pareto solution to problem (49).

Proof. If (𝜆,𝛼_𝑜,𝛽_𝑜) is not a Pareto solution to problem (49), then there would exist another feasible solution of problem (5), (𝜆,𝛼_𝑜,𝛽), such that 𝛼_𝑖𝑜≤𝛼_𝑖𝑜 for all 𝑖 and 𝛽_𝑟𝑜≥𝛽_𝑟𝑜 for all 𝑟; furthermore, there exists at least one 𝑖 in which 𝛼_𝑖𝑜<𝛼_𝑖𝑜 or at least one 𝑟 such that 𝛽_𝑟𝑜>𝛽_𝑟𝑜. Let 𝐼={𝑖∣𝛼_𝑖𝑜<𝛼_𝑖𝑜} and 𝑂={𝑟∣𝛽_𝑟𝑜>𝛽_𝑟𝑜}. Therefore ^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜<𝜃^∗_𝑖𝛼_𝑖𝑜𝑖∈𝐼_𝑛𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜃^∗_𝑖𝛼_𝑖𝑜≤𝜃^∗_𝑖𝛼_𝑖𝑜𝑖∉𝐼_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜>𝜑^∗_𝑟𝛽_𝑟𝑜𝑟∈𝑂_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜑^∗_𝑟𝛽_𝑟𝑜≥𝜑^∗_𝑟𝛽_𝑟𝑜𝑟∉𝑂𝜆_𝑗≥0,𝑗=1,…,𝑛. (53)(54)(55)(56)(57)
Based on inequalities (53) and (56) there exists 𝜇_𝑖<1 for 𝑖∈𝐼 and 𝜂_𝑟>1 for 𝑟∈𝑂, such that ^𝑛_𝑗=1𝜆_𝑗𝑥_𝑖𝑗≤𝜇_𝑖𝜃^∗_𝑖𝛼_𝑖𝑜,𝑖∈𝐼_𝑛𝑗=1𝜆_𝑗𝑦_𝑟𝑗≥𝜂_𝑟𝜑^∗_𝑟𝛽_𝑟𝑜,𝑟∈𝑂. (58)(59)
For each 𝑖=1,…,𝑛 and 𝑟=1,…,𝑠, define 𝜃_𝑖=𝜇_𝑖𝜃^∗_𝑖𝜃,if𝑖∈𝐼,_∗𝑖,if𝑖∉𝐼,𝜑_𝑟=𝜂_𝑟𝜑^∗_𝑟𝜑,if𝑟∈𝑂,_∗𝑟,if𝑟∉𝑂. (60)
By (54) and (56)–(59), (𝜆,𝜃,𝜑) is a feasible solution of problem (1) (only restrictions on the right have replaced the 𝑋_𝑜 and 𝑌_𝑜 by 𝛼_𝑜 and 𝛽_𝑜 in problem (1)) where 𝜃=(𝜃₁,…,𝜃_𝑚) and 𝜑=(𝜑₁,…,𝜑_𝑠). Therefore 𝜌_𝑜=(1/𝑚)∑^𝑚_𝑖=1𝜃_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑_𝑟<(1/𝑚)∑^𝑚_𝑖=1𝜃^∗_𝑖(1/𝑠)∑^𝑠_𝑟=1𝜑^∗_𝑟=𝜌^∗_𝑜 (61) which is against the assumption that is eff(𝛼_𝑜,𝛽_𝑜)=eff(𝑋_𝑜,𝑌_𝑜)=𝜌^∗_𝑜.

5. Illustrative Examples

Example 1. Consider a problem of three DMU with two inputs 𝑥₁, 𝑥₂ and two outputs 𝑦₁, 𝑦₂. The data of inputs, outputs, and ERM-efficiency score are shown in Table 1.
As can be seen, DMU_𝐶 is an ERM-efficient DMU. If the decision maker is interested in increasing the output vector from 𝑌_𝐶=(5,5) to 𝛽_𝐶=(6,6), then, to solve the model (5) by employing the weight-sum method [25], a Pareto solution for this MOLP is generated as 𝛼^∗_𝐶=(1.20,2.40). Based on the model (6), 𝐾^∗_𝐶=1, eff(𝑋_𝐶,𝑌_𝐶)=eff(𝛼^∗_𝐶,𝛽_𝐶)=1, which means that the ERM-efficiency remains unchanged. Hence, due to Theorem 6, when the outputs of DMU_𝐴 increase to 𝛽_𝐶=(6,6) if the decision maker would like to preserve the efficiency score of this DMU, the inputs should increase to 𝛼^∗_𝐶=(1.20,2.40).
Also, if the decision maker is interested in increasing the output vector of 𝑌_𝐶=(5,5) to 𝛽_𝐶=(5.5,6), then to solve the model (5) by employing the weight-sum method [25], must the input vector of the 𝑋_𝐶=(1,2) increase to 𝛼^∗_𝐶=(1.20,2.40). Based on model (6), since 𝐾^∗_𝐶=11.5/5>1 and 𝑘^∗_𝐶=(6/5,1), we have eff(𝑋_𝐶,𝑌_𝐶)≠eff(𝛼^∗_𝐶,𝛽_𝐶)=0.96; therefore, according to part (ii) of Theorem 6, we have eff(𝑋_𝐶,𝑌_𝐶)=eff(1.20,2.40,6/5∗5.5,1∗6)=1. This indicates that new DMU in output component the first to amount “0.5” of the lack produce. In other words, using model (4), we have eff(𝑋_𝐶,𝑌_𝐶)=eff(1.20,2.40,6/5∗5.5,1∗6)=1.

Table 1 

The data and ERM efficiency in Example 1.

Example 2. Consider a problem of four DMU with one input 𝑥 and one output 𝑦. The data of input, output, and ERM-efficiency scores are shown in Table 2.
As can be seen, DMU_𝐵 is an ERM-efficient DMU. Assume the decision maker identified rate of increase input and output for this DMU, respectively, as 2≤𝑥≤7, 4≤𝑦≤10. In order to propose patterns to the decision maker to increase input and output for this DMU, under preserving the ERM-efficiency index, based on model (49) the following MOLP model is considered: min𝛼_𝐵max𝛽_𝐵s.t.𝜆₁+2𝜆₂+5𝜆₃+6𝜆₄≤𝜃^∗_𝐵𝛼_𝐵𝜆₁+4𝜆₂+5𝜆₃+7𝜆₄≥𝜑^∗_𝐵𝛽_𝐵𝛼_𝐵≥2,𝛽_𝐵𝛼≥4_𝐵≤7,𝛽_𝐵𝜆≤10_𝑗≥0,𝑗=1,…,4. (62)
Using the weight-sum method [25], a Pareto solution is generated for this MOLP as (𝛼^∗_𝐵,𝛽^∗_𝐵)=(5,10). Therefore, according to Theorem 15, when the inputs and outputs of DMU_𝐵 increase to 5 and 10, respectively, then the ERM-efficiency score of this DMU is 1.

Table 2 

The data in Example 2.

6. Conclusion

In the present paper, a typical inverse optimization problem on the nonradial enhanced Russell model has been studied: two main questions on inverse DEA have been discussed and the models have been proposed to estimate the input (output) levels of a given DMU when some or all its output (input) levels were increased under the constant ERM-efficiency score. To determine sufficient and necessary conditions of estimated inputs (outputs), Pareto solutions of the MOLP were used. Moreover, in finding inputs (outputs), if there exists lack output (extra input) in each of a output (input) components, the amount of the lack (extra) is specified by auxiliary models. For an ERM-efficient DMU, necessary and sufficient conditions were introduced to find the minimum and maximum increase of input and output levels, respectively, provided that the ERM-efficiency score remains unchanged. Therefore, the patterns can be presented to the decision maker in order to increase inputs and outputs (extending decision making units) for an ERM-efficient DMU such that the ERM-efficiency remains unchanged. In other words, inverse DEA can be used in theoretical and practical purposes such as strategic planning, management control, resource allocation, and ranking. The sufficient conditions were established only for the ERM-efficient DMU, but the given necessary conditions were for each ERM-efficient or ERM-inefficient DMU. Finding sufficient conditions for ERM-inefficient DMU_𝑠 can be a suitable research field. In addition, using nonradial models under intertemporal dependence, solving the problems introduced by Wei et al. [14], Hadi-Vencheh et al. [16], and the new problem discussed in this paper can be a useful research topic.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.