CENTRALIZED RESOURCE ALLOCATION TO CREATE NEW MOST PRODUCTIVE SCALE SIZE (MPSS) DMUS

. Data envelopment analysis (DEA) is a mathematical programming - based technique to evaluate the performance of a homogeneous group of decision-making units (DMUs) with multiple inputs and outputs. One of the DEA applications involves aggregating input resources and reallocating them to create efficient DMUs.The present study employs the centralized resource allocation (CRA) approach to develop a model for creating new DMUs. These new DMUs are the most productive scale size (MPSS), and all new DMUs lie on a strong supporting hyperplane. In this case, a dual model is used to obtain the strong supporting hyperplane which all new DMUs lie on. This hyperplane is derived by solving the dual model and generating a common set of weights. Then, it is shown that all new DMUs lie on a strong supporting hyperplane, and an MPSS facet is the intersection of this hyperplane with the production possibility set (PPS).


Introduction
Data envelopment analysis (DEA) is a non-parametric approach to evaluate the relative performance of a set of DMUs.It consumes multiple inputs to produce multiple outputs.Farrell [17] first introduced a method to assess the performance of DMUs.One of the substantial shortcomings of the Farrell method was related to consideronly one output.Charnes et al. [10] developed Farrells work for the case of multiple inputs and outputs.Their work, called the CCR model, introduced the data envelopment analysis concept.Then, Banker et al. [8] extended the CCR model by incorporating the concept of variable returns to scale and thus introduced the BCC model.Afterward, DEA was widely used in different performance evaluation problems, and various models were developed to measure the efficiency and assess the performance of DMUs.Several researchers have also proposed ways to improve the efficiency of inefficient DMUs.In this regard, Golany et al. [18][19][20] developed the centralized resource allocation (CRA) approach for achieving this purpose.Then, Lozano and Villa [26][27][28] suggested a radial and non-radial model for CRA.In this approach, the total input consumed by all DMUs is reallocated to them, so that all DMUs become efficient.Also, it is assumed that DMUs are controlled by a centralized common decision-maker to manage inputs of all DMUs and reallocate them to DMUs.A point related to the CRA is an equitable allocation of the resources in which the decision-maker tends to allocate resources or fixed costs to the controlled DMUs fairly.
In addition, Cook and Kress [11] proposed a DEA method for the equitable allocation of fixed costs based on the two principles of invariance and Pareto-optimality.Athanassopoulos [3] suggested a technique based on the DEA and goal programming method.In this study, some bounds of the total consumption have been estimated for each input using a series of LP models.Although these bounds may not be reachable, their deviation can be minimized to find a resource reallocation.Then, Athanassopoulos [4] developed a goal programming approach using multiplier DEA models.In this case, the weights were pre-calculated by an additive model to prevent the models nonlinearity.Beasley [9] proposed a non-linear model for allocating resources.Based on this study, the input and output levels of DMUs have been determined to maximize the average efficiency.Thus, a non-linear fractional model was developed considering some bounds for inputs and outputs.However, the deficiency of the proposed method was related to the issue that the projection point of DMUs was not necessarily at the efficient frontier.Asmild et al. [2] recommended a model for CRA based on the BCC model.This model applies the process only for inefficient DMUs, and thus efficient and inefficient DMUs are first supposed to be determined.
Cook and Zhu [12] extended the approach proposed by Cook and Kress [11] and found a practical method for CRA problems.Jahanshahloo et al. [24] evolved a method for the CRA problem and obtained a solution without solving a linear programming problem and only with a simple formula.This approach was based on the invariance principle presented by Cook and Kress [11].Korhonen et al. [25] suggested a multi-objective linear programming method for the CRA problem.In this case, it was assumed that a central unit controlled all DMUs simultaneously.Toloo [32] demonstrated the role of non-Archimedean epsilon in determining the mostefficient DMU.Also, Hatami-Marbini and Toloo [22] proposed several models for the ratio data.Toloo et al. [33] published an article entitled Robust optimization and its duality in data envelopment analysis.Also, Salahi et al. [30] published an article entitled A new robust optimization approach to common weights formulation in DEA.Salahi et al. [29] published an article entitled Robust Russell and enhanced Russell measures in DEA.
Hadi-Vencheh et al. [21] proposed a CRA approach using inverse DEA.In addition, the authors can recommend other studies on CRA for further review, such as Fang [15] and Tone [34].DU et al. [14] used the cross-efficiency concept in DEA to address resource allocation problems.The literature survey showed that the most significant shortcoming of the previous studies ignored the growing potential of the DMU, while it plays a crucial role in allocating extra resources.Indeed, the extra-resource allocation to a DMU must affect its output.Amirteimori and Kordrostami [1] developed a method for implementing demand and supply changes in a centralized decision-making process.Finally, Wu et al. [35] suggested a CRA model based on MOLP considering desirable and undesirable inputs and outputs.Another effort to cope with the CRA problem was performed by Zhang et al. [37].In this study, a linear model with bounded variables was proposed for extra resource allocation.Tao et al. [31] employed the theories of network flows to examine CRA for all individual units in a DEA framework.Zhang et al. [36] suggested a two-stage scheme for CRA and distributed power control regarding NR V2X integrated with NOMA technology.Fang et al. [5] introduced the Nash bargaining game theory to develop a new CRA model.The authors considered the overall goals of the organization and competition among DMUs.
Also, many researchers have investigated the development and management of centralized resource reallocation as one of the classic research topics in the management and economy sciences.In particular, Managers strive to maximize productivity by consuming rare and expensive input resources and preventing waste of these valuable resources.In this regard, the present study proposes a new approach to obtaining new units with maximum productivity through the reallocation of centralized resources.Thus, a fractional model is first developed to derive the most productive scale size (MPSS) units.Then, these are converted into a linear model by transforming the Charnes-Cooper variable.Afterward, a dual model is solved, and it was shown that all new units are located on the BCC-CCR joint frontiers.The proposed model has several advantages over other centralized resource reallocation models.First, the new units can be obtained by solving only one linear and non-radial model.Second, the new units are all MPSS ones.By consuming valuable and expensive equipment, it is especially important to achieve maximum productivity.This paper is organized into five sections.Section 1 includes an introduction to DEA and a literature review.Then, Section 2 presents the basic definitions and proposed models to identify MPSS units and CRA schemes.In Section 3, the proposed CRA model aims to constitute new MPSS units.Also, this section discusses some properties of the dual model.Section 4 gives some numerical examples.Finally, Section 5 represents conclusions and suggestions for future research.

Production possibility set and MPSS in DEA
Consider  DMUs with coordinates (x  , y  ) in which the jth DMU consumes  inputs as the components of the vector x  = ( 1 , ..,   ) to produce  outputs as the components of the vector y  = ( 1 , ..,   ).Hence, the PPS can be defined as  = { (x, y) |x   y}.
The PPS with constant returns to scale is as follows: Also, the PPS for the case of variable returns to scale is as follows: Fare, Groskopf, and Lovell [16] and Cooper et al. [13] introduced non-increasing returns to scale (NIRS) and non-decreasing returns to scale (NDRS) models and called FG and ST models, respectively.DMUs can be assessed by different DEA models.This procedure is accomplished using various production technologies discussed above.Recently, the CCR and BCC models have broadly been used by researchers.The CCR and BCC models for evaluating DMU  are as below, respectively: where  − and  + are slack variables for input and output CCR models.Also s − and s + are slack variables for input and output BCC models, respectively.
In this research, the symbol  > 0 utilized in models (2.1) and (2.2) signifies that the variable  is minimized in the first stage, and then the sum of slack variables is maximized in the second stage while assuming  =  * , where  * is the optimal value in the first stage.
Assume that ( * ,  * ,s −* ,s +* ) is an optimal solution to model (2.2).Then, DMU  is efficient if and only if the following conditions are satisfied in each optimal solution: Nowadays, analysis of economic scales is of particular importance.The MPSS concept in DEA was first produced by Banker [6].The MPSS region shows that points in PPS whose ratio of their output to input are the highest values.Also, DMUs in the MPSS region have the best size.Due to variations in rare, expensive, or valuable inputs (e.g., large equipment), these DMUs are important to optimize outputs to inputs ratios.
Definition 2.1.The unit (x  ,y  ) ∈   is an MPSS if and only if: Based on the above definition, Banker [7] proposed the following model to identify whether or not the point (x  ,y  ) is an MPSS: ) is an optimal solution to model (2.3).Then, the unit (x  ,y  ) is an MPSS if and only if the following conditions are satisfied in each optimal solution: Consider the following output-oriented CCR model to evaluate DMU  : Assume that ( * ,  * , s −* , s +* ) is an optimal solution to model (2.4) (2.5) Assume that ( * ,  * ,  * , s −* , s +* ) is an optimal solution to model (2.5).Then, the unit (x  ,y  ) is an MPSS if and only if the following conditions are satisfied in each optimal solution:

Resource allocation problem
Consider a situation in which an organization consists of homogeneous units, and each unit consumes several input resources to produce output resources.If the organization faces extra resources in the subsequent period of activity, it requires reallocating input resources to constitute new efficient units.In this regard, Lozano and Villa [26]  ( = 1, 2, . . ., ), respectively.Then, the authors proposed the following radial model: Here, s −  and s +  are slack variables for aggregated inputs and outputs, respectively.Also,  is the non-Archimedean infinitesimal.In addition,   is the proportion of new DMU  from the old DMU  .Now, if it is assumed that ( * ,  * , s −* , s +* ) is an optimal solution to model (2.5), the projection points represent  new efficient DMUs, which are formulated as follows:

Proposed model
Consider  DMUs with coordinates (x  , y  ), where:  According to Table 2, the approach proposed by Lozano and Villa [26] projects all DMUs onto    .Figure 2 depicts the projection of all DMUs with Lozano and Villas model.Also,   is BCC efficient, but it is not CCR efficient.Therefore,    / ∈   ∩   .This situation demonstrated that Lozano and Villas approach could not provide MPSS projections for DMUs.
Also, the proposed model by Zhang et al.The optimal solution of model (3.3) with the DMUs of Table 1 is expressed as follows: Therefore, the strong supporting hyperplane including new DMUs with the proposed approach is given by the following equation: Example 4.2.Consider two DMUs with two inputs and a single output (Lozano and Villa [26]) as presented in Table 3.
The results of the CCR and BCC models for DMUs in Table 3 are presented in Table 4.
According to the solutions of CCR and BCC models, it is implied that DMUs D, E, G, H, and J are CCR and BCC efficient.This shows that {, , , , } ∈   ∩   .Thus, by Definition 2.2, DMUs D, E, G, H, and J are MPSS.Now, solving Model (2.6) proposed by Lozano and Villa [26] and Model (3.2) for DMUs in Table 3, we obtain new DMUs according to Table 5.As seen, Lozano and Villa's model [26] projects DMU B to the point (x, ȳ = (6.71,9.12, 4.82, 3.18), which is BCC efficient but CCR inefficient.Therefore, (x, ȳ) / ∈   ∩   , and, by Definition 2.2, the point (x, ȳ) is not an MPSS, while the new DMUs built by the proposed approach are not MPSS.As a result, Lozano and Villa's model [26] cannot construct MPSS units, whereas the proposed approach provides new MPSS units.

Conclusions
In the present study, a new method has been proposed for the centralized resource reallocation to construct new MPSS units.Since the input indices include rare and high-value equipment and resources, it is necessary to achieve maximum productivity by reallocating and re-aggregating these resources.This situation prevents wasting these resources and makes the best use of the potential of these resources.Thus, the re-aggregation and reallocation of the centralized resources have been performed to construct new MPSS units.In this regard, we proposed a method that yielded new MPSS units that would be homogeneous.Afterward, a dual model was used to show that all the newly constructed units were located at the CCR-BCC joint frontiers.Also, the joint boundary between CCR and BCC included the points with maximum productivity.The proposed model has several advantages over conventional models, including (I) the new efficient units can be obtained by solving only one linear and non-radial model and (II) unlike other prevalent used centralized resource reallocation models, all the new units are homogeneous MPSS ones.Therefore, these units can provide the ground for full utilization of the available potential of the inputs.
In this research, two numerical examples were presented.The first example, including a single input and a single output and taken from Zhang et al. [37], aimed to compare the proposed approach with other centralized resource reallocation approaches.Because the sum of inputs for the new MPSS units determined by Model (3.2) is greater than that of DMUs, future studies may consider centralized resource reallocation focusing on the lowest or highest point on the MPSS facet.This situation contradicts the optimality of ( * ,  * ,  * ) in the model (7).

Figure 1 .
Figure 1.Projection (x  ,y  )to the intersection of BCC and CCR frontiers.

Figure 2 .
Figure 2. Distribution of the new DMUs generated from Lozano and Villa's model.
[37] projects the efficient DMUs of A, B, F, H, and I and the weak efficient    onto the MPSS facet.But it does not project inefficient DMUs of C, D, E, and G onto the frontier of the MPSS region.Figure3shows the projection of all DMUs using Zhang et al.'s model.As shown in Figure 4, the proposed model in this research (model 8) projects all DMUs to point H. Also, this figure indicates that    ∈   ∩   , and thus    is an MPSS point.

Figure 3 .
Figure 3. Distribution of the new DMUs generated from Zhang et al.'s model.

Figure 4 .
Figure 4. Distribution of the new DMUs generated by the proposed approach.

Table 2 .
[37] of new DMUs generated from the models proposed by Lozano and Villa[26], Zhang et al.[37], and the proposed approach.

Table 4 .
Optimal solutions of CCR and BCC models.

Table 5 .
[37] of new DMUs generated from the models proposed by Lozano and Villa[26], Zhang et al.[37], and the proposed approach.