Lam [1] developed the following mixed integer linear Data Envelopment Analysis (DEA) model to find the most efficient Decision-Making Units (DMU).

Where M is a positive large number. It is easy to see that in the optimal solution for only one we have which implies the efficiency score of DMUk is more than or equal to one and the rest less than or equal to one. It should be here noted that a positive implies that the efficiency score of DMUk is strictly more than one, which shows this DMU is the most efficient DMU. However, we will show that this DMU may not be the only most efficient DMU and we may have several most efficient DMUs. Lam [1] proposed the following model of Amin and Toloo [3] to find a proper value for the epsilon to use in the Model (1).

Salahi and Toloo [2] discussed that Model (2) may give an unsuitable value for the In other words, Model (1) may be unable to find the most efficient DMU by utilizing the maximum value for the epsilon obtained by solving the Model (2). Therefore, they proposed the following Model (3) to find a suitable value for the epsilon to use in Model (1).

They claimed that by using the optimal objective value of Model (3), Model (1) gives exactly one unit as the most efficient (best DMU). In the following example, we show that Model (1) may give different DMUs as the most efficient DMU.

Example 1. Consider the following three DMUs, each uses a single input to produce two outputs.

Solving the Model (3) with gives Now, solving the Model (1) with and gives the following optimal solution.

This optimal solution implies that the efficiency score of the DMU1 is strictly greater than one and the rest less than or equal to one. In other words, this DMU is the most efficient DMU. We add the constraint to the Model (1) and resolve it to give the following optimal solution.

The above optimal solution implies that the Model (1) has multiple optimal solutions and so the DMU2 is also the most efficient DMU. As a result, the proposed approach by Lam [1] and Salahi and Toloo [2] cannot discriminate between DMU1 and DMU2. Therefore, in some cases we may have several most efficient DMUs, and so the Model (1) randomly reports one of them as the best DMU based on the solver that is used to solve it. One method to discriminate between such best DMUs is using weight restrictions.

In the following, we show that the second step (Model (1)) is redundant. In other words, the most efficient DMU can be obtained just by solving the Model (3). Indeed, we show that in Model (3), , which implies this model gives a DMU with an efficiency score of strictly greater than one and the rest less than or equal to one.

Theorem 1. In the Model (3), . In other words, solving the Model (3) gives the most efficient DMU. Proof: according to Cooper, Seiford, & Tone [4], suppose DMUk is an extremely efficient DMU and consider the following super-efficiency model of Andersen & Petersen [5] for this DMU.

Since DMUk is an extremely efficient DMU, the optimal solution of Model (4) implies with positive weights of Now, let and suppose that M is a large enough

positive number such that , in this case, we show that is

a feasible solution to Model (3) that completes the proof.

Let in this case, for the first type of constraints of Model (3) we have:

For the second type of constraint s of Model (3), we have:

For the third type of constraint s of Model (3), we have:

Therefore, the weights is a feasible solution for Model (3) that implies

Note 1: the optimal value of objective function of Model (4) may go to infinity. In such a situation, we could select feasible positive weights with a bonded to prove the Theorem 1.

Based on Theorem 1, applying Model (3) to the data presented in Table 1 gives the following optimal solution.

This optimal solution implies that the DMU1 is the most efficient DMU. Solving the Model (3) with the additional restriction of gives which implies DMU2 is another most efficient DMU. These results are consistence with the results of the models of Salahi and Toloo [2].

By considering the discussed issues, in the next section we propose an improved approach to find the best DMU by solving only one model.

The main aim of Model (1) is to find a DMU with an efficiency score of greater than one and the rest less than or equal to one. For this purpose, we propose the following Model (5) to find the most efficient DMU.

It is obvious that the Model (5) is always feasible. In the following we validate it by some theorem.

Theorem 2. In the Model (5), which implies

Proof: since Model (5) has fewer constraints compared to Model (3), its optimal solution must be greater than or equal to the optimal objective function of the Model (3). Therefore, and hence Now, considering the first and second type constraints of the Model (5) implies:

That completes the proof.

Theorem 3. The optimal objective function of Model (5) is strictly positive.

Proof: it is obvious by considering the proof of the Theorem 1.

Theorem 4. Solving the Model (5) gives a DMU with an efficiency score of strictly greater than one and the rest less than or equal to one.

Proof: let in the optimal solution we have so we conclude:

Note 2: our proposed approach solves only one model with fewer constraints to find the best DMU; however, the proposed approach by Lam [1] and Salahi & Toloo [2] solves two models with some more constraints to find the best DMUs. Therefore, our method is more efficient compared with the two mentioned approaches. In the next section, we will verify Model (5) by a real numerical example.

In this real numerical example, we show how our approach can find the most efficient DMU just by solving Model (5). Table 2 depicts the data set of nineteen Facility Layout Designs (FLDs) taken from Ertay et al. [6]. Solving the Model (5) with gives the following optimal solution.

This optimal solution implies that the FLD10 is the most efficient DMU. Solving the Model (5) with the additional constraint of gives Reducing the optimal objective value shows that there is no alternative solution and hence the FLD10 is the best DMU. Now, we apply the proposed models by Lam [1] and Salahi & Toloo [2] to find the best FLD. Solving the Model (3) gives As we proved in Theorem 1, this optimal solution implies that the FLD7 is the most efficient DMU. Anyway, solving Model (1) verifies Theorem 1 and determines this FLD as the most efficient DMU. The optimal weights, efficiency scores and full ranking of FLDs are given in Table 3. It should be here noted that, as reported in Lam [1], using in the Model (1) gives the FLD10 as the best DMU.

The results in Table 3 reveal the following points:

I. The Model (5) implies that the efficiency score of the FLD10 is 1.275, which shows the distance between the efficiency score of the first two top-ranking FLDs is equal to 0.275. This value for the proposed approach by Lam [1] is equal to 1.054-1=0.054, which shows the discrimination power of the Model (5) is more than the Model (1).

II. As reported by Toloo & Salahi [7], all of the proposed approaches by Andersen & Petersen [5], Foroughi [8], Wang and Jiang [9], Toloo [10] and Toloo and Salahi [7] imply that FLD10 is the best DMU that verify the result of the Model (5).

As mentioned in Note 2, our approach solves only one model to find the best FLD; however, the proposed approach by Lam [1] solves two models with more constraints to obtain the best FLD.

This paper studied the two-step method proposed by Lam [1] and Salahi & Toloo [2] to find the most efficient units. It is mathematically proved that the model of the first step in Salahi & Toloo [2] is enough to find the best DMU, and so the second step is redundant. We improved the model of the first step and, proposed a modified model and showed that it could find the best DMU with considerably less calculation volume. Also, the results of the real numerical example showed that the discrimination power of the modified model is higher than the models of Lam [1]. Also, we compared our result with five existing approaches in the literature that verified the Model (5).

Bohlool Ebrahimi was responsible for the conceptualization and design of the study, as well as drafting the manuscript. Szabolcs Fischer contributed to the development of the methodology and provided critical revisions. Milos Milovancevic performed the numerical analysis and assisted in the interpretation of results.

This research received no specific funding from any public, commercial, or not-for-profit organization.

The data and materials supporting this study are available from the corresponding author upon reasonable request.

The authors declare no conflicts of interest regarding the publication of this paper.