CONGESTION AND NON-CONGESTION AREAS: IDENTIFY AND MEASURE CONGESTION IN DEA

. Detecting the weak and strong congestion statuses of decision-making units (DMUs) and measuring them via data envelopment analysis (DEA) is an important issue that has been discussed in several studies and with different views. The efficiency frontier is a concept derived from the underlying production possibility set (PPS), and the congestion concept is related to them. Still, researchers have defined congestion for each DMU in many previous studies and ignored that congestion is linked with the underlying production technology. In the congestion measurement matter, this paper presents two new insights into a congestion area and non-congestion area for production technology and two new mathematical definitions of congestion based on the PPS properties and detecting the weak and strong congestion status of DMUs (CSOD). We prove our definitions are equal to the original definition of congestion. First, we describe the congestion and non-congestion areas based on the PPS; then provide full details of how to measure congestion built on these new insights. Our approaches are very accurate and fast to calculate; they are theoretically elementary and efficient in performance. Our proposed methods can deal with both non-negative and negative data. Finally, an empirical example is provided to illustrate our approaches.


Introduction
If the resources of a decision-making unit (DMU) are overused, this excess consumption may adversely affect its performance. When units use resources in this way, without considering optimality, they only pay attention to the benefits/costs and ignore other adverse effects they have on the company, such as congestion. So, to improve the performance of a DMU, it is essential to identify and detect the presence of congestion as a first step and to plan and formulate appropriate scientific and practical solutions to eliminate such congestion as the second step in production analysis. For this purpose, Färe and Svensson [12] introduced congestion insights in the economy.
Inefficiency is a necessary condition for congestion. Congestion is an extreme of inefficiency. It is necessary to note that congestion is a production frontiers' property and not a DMU's. So, congestion is a characteristic of PPS, not a DMU. To improve the DMU's performance, it is essential to recognize the existence of inefficiency, and definition of strong congestion was different from Tone and Sahoo [23]. To identify the CSOD, first time, they defined the non-negative convex set of input-output slack vectors that correspond to DMUs in the congestionbased DEA technology, that all dominate the DMU under evaluation. Next time, they proved that the problem under consideration reduces to finding a maximal element an element with the maximum number of positive components of this set. So, they developed a single-stage LP model to determine congestion. But, despite the great accuracy and power in identifying CSOD, yet, their method is unable to measure the amount of congestion.
While Mehdiloozad et al. [19] identify weak and strong CSOD with indicator max ( = 1, . . . , ) without measuring the values of the input's congestion. This is the best reason to provide our proposed method. So, in improving the performance of Mehdiloozad et al. [19], we develop an approach; exactly equivalent to Mehdiloozad et al. [19] (occur and efficient in CSOD) and identify weak and strong CSOD by measuring the amount of input congestion. For this aim, we work on input space. Therefore, we formulate the two non-negative convex sets of inputs space as congestion area ( C )/non-congestion area ( NCA ) by defining specific DMUs named CSP, which, this CSP concept has some special properties. So, our aim for developing this paper is to fill in this null.
This paper provides two new mathematical definitions for congestion and offers two alternative approaches to the discussion. Briefly, this paper focuses on identifying and measuring congestion in the DEA measurement framework. We define two new concepts in the inputs space as congestion area (CA) and non-congestion area based on PPS property and made our definitions based on these two concepts. Also, we present a new CSP concept and formulate them, then, in inputs space formulate C based on CSP's input. So, to find CSOD and measure congested DMU's congestion, we formulate our suggested model. Finally, we present two numerical examples to explain new insights and compare the results of our method with Cooper et al. [9], Noura et al. [20], Tone and Sahoo [23], and Mehdiloozad et al. [19].
One of the congestion area advantages is that it does not need to calculate congestion for all DMUs when management wants to determine; which DMUs have congestion or not? Which DMUs work well or not? One advantage of the mathematical definition of congestion is that; can be used in modeling and calculations because the traditional definition of congestion is theoretical and cannot be used in calculations. This definition is also accurate only for DMUs on the production function frontier and inaccurate for inefficient DMUs that are not on the production frontier. That is the mean for inefficient DMUs, and they can also increase some or all inputs and reduce some or all outputs simultaneously. So, what is the difference between congestion and inefficiency? Therefore, the mathematical definitions we provide are practical and accurate for all efficient and inefficient DMUs. Note that the traditional definition of congestion in economics is accurate and acceptable because, in economics, they work only with the production frontier. Still, in DEA, researchers work with all DMUs (on the frontier or below the frontier) (Tab. 1).

Preliminary assumption and definitions
Data Envelopment Analysis (DEA) is an efficiency evaluation technique with extensive use in efficiency analysis over the past three decades. With theoretical and practical progress in this branch of management science, DEA has become a perfect tool for assessing the performance and efficiency of decision-making units (DMUs) of a different variety. Since basic DEA does not involve calculating the efficiency frontier, congestion in DMU inputs cannot be determined in that way. Throughout non-parametric DEA, using the production possibility set (PPS) instead of the production frontier, one could investigate the existence of congestion. Suppose that it is homogenous DMU ( = 1, . . . , ) in a constant time interval, and ( 1 , . . . , ) and = ( 1 , . . . , ) are the input and output vectors of DMU , respectively. As mentioned in the introduction, congestion is a production-related situation that can be viewed as a severe technical inefficiency case. Before proceeding to the main discussion, we need to define two critical concepts based on Cooper et al. [4] and Cooper et al. [6].
Definition 2.1 (Technical efficiency). A DMU ( ∈ ) is a technically efficient DMU if the optimal solution of the following BBC model (2.1a) [1] is * = 1 to evaluate it. Table 1. Comparison of the advantages of our method over previous models.
Definition 2.2. Congestion occurs when with convex technology, increasing one or more inputs reduces one or more outputs without improvement in any other inputs or outputs. Conversely, decreasing one or more inputs increases one or more outputs, worsening no other input or output (this is an economic definition of congestion).
In the last part of the introduction as explained, in DEA, this is not an accurate definition and needs improvement (Because it does not distinguish between inefficient and congested DMUs, also this definition is not rigorous enough since it ignores that congestion should be a concept focusing on the frontier). Thus, we introduce our mathematical definition of congestion to fill this gap.

Technology set
The technology is defined as the following, Notice that the technologies convex and were defined in the upper section. Assume that the outputs and inputs are non-negative. So, convex is the decussating of all Technologies ⊂ that satisfy the syllogism of (i) involving observations, (ii) strong disposability of outputs, and (iii) convexity of PPS, and is the decussating of all technologies ⊂ that satisfy the syllogism of (i) involving of observations, (ii) strong disposability of outputs, (ii) strong disposability of inputs, and (iii) convexity of PPS.
The congestion measurement is discussed in output-oriented DEA models. The congestion concept does not fit the traditional DEA literature, because, the congestion concept was started by economic researchers. Furthermore, congestion is modeled and measured by replacing the axiom of strong input disposability with the axiom of weak input disposability. In other words, congestion occurs when the strong disposability of inputs is not available. Therefore, to model congestion, this principle must be omitted from the fundamental principles of DEA. A PPS called convex should be constructed to act as the reference for the congestion measurement in DEA. The explicit DEA-base representations of and convex in the VRS are then defined: By definition, technology convex satisfies all the syllogisms that define technology . Because is the smallest technology that satisfies this syllogism, so convex ⊆ .

Background of congestion
In compaction research within the DEA framework, there are two basic ideas that we will describe below.
In the first step, they proved that the relative interior points of the minimum face (a dimension that contains all the projections) have congestion if one of the vertices spanning this face has congestion, and each of the points displays the equivalent congestion (weak or strong). In the next step, they have defined DMU's congestion at its MAX-projection (By congestion-identification model and maximal element of the non-negative convex set of all possible optimal intensity vectors MAX-projection can be achieved) the GRS can be identified. Finally, with the help of results from Mehdiloozad et al. ( [18,19]), they developed an approach (a unique single-stage LP model) to identify the MAX-projection by using the GRS. The calculations of Mehdiloozad et al. [19] are summarized as follows: From solving model (4.5) for input and output values, the columns representing the maximal elements (alpha-maximal (in the number of inputs), beta-maximal (in the number of outputs)) are obtained (refer to paragraph 9 below model (4.5) and Cor. 1.3 from [19]) that, only the positive component is important not their value, that is, suppose we have two inputs and one output, if all three components are positive (two alpha-maxima for two inputs and one Beta-maximum for output) is strong congestion. Also, if the third component (Beta-maximal) is positive but the first or second component (two alpha-maximal) is zero, the congestion is weak.
As we have already explained, the basis of our new definition of congestion is the decomposition of the PPS (and, more precisely, the decomposition of the DMUs' inputs area). Therefore, we divide the inputs' area into two special subsets of the congestion area and the non-congestion area. Then define each area and describe the properties and each area. These properties provide two mathematical definitions of congestion (unlike the previous definition of congestion, which is an economic definition).
This research focused on detecting and measuring congestion in the DEA measurement framework. The authors proposed a mathematical definition of congestion and congestion zone linked to the production possibility set instead of each DMU. The advantage of our paper over others is the identification and measurement of the amount of congestion in the construction of the congestion area. That means, without talking about the congestion of each DMU, we specify an area where the inputs of each DMU in that area have congestion. In this case, identifying congested DMUs and also measuring their congestion can be easily identified and measured.

Congestion area
Congestion is a frontier concept (like returns to scale), and any argument on its evaluation is bounded to the DMUs, which are efficient only. We define the congestion of one inefficient DMU at its efficient projection that it is obtained by congestion, based on the DEA model. Before discussing our proposed method, we must present the definitions of some new congestion-related concepts and propose a new mathematical definition for the concept of congestion itself.
Definition 4.1 (Congestion area). A convex area created by DMU's congested inputs is called the congestion area, So, the congestion area is an area in the inputs space that the congested inputs located in this area and denoted by congestion . So, DMU : ( , ) has congestion, thus, ∈ congestion .
Concerning Definition 4.1 which includes congested inputs of congested DMUs, it refers to equation (4.14) and does not build on the preceding Definition 2.2 of congestion, but is directly related to it. This area does not include all DMUs. As we will show in the following, this area is infinite and convex in a way that, from the bottom, it is bounded to no congestion and technically efficient DMUs, such as DMU ( ∈ ). Such points are called congestion starting points (CSPs). CSPs are a type of technically efficient DMUs -that are weakly but not strongly efficient DMUs -with the highest input consumption among the efficient (with maximum output). The most important feature of the CSP is that the production frontier, after CSP, by increasing some inputs leads to reducing some outputs, provided that other inputs and outputs remain unchanged, or by increasing all inputs leads to reducing all outputs. Hence, there are supporting hyperplanes on convex at the CSP, that all components have negative partial derivatives. We can use the following model (4.4) to distinguish (1) CSPs and (2) supporting hyperplanes from the technically efficient DMUs (the CSPs). Assume that DMU is a technically efficient DMU in evaluating with the model (2.1a) and ( * 1 , . . . , * , * 1 , . . . , * , * 0 ) is an optimal solution for the model (4.4). Note that the condition of congestion for a DMU is that; in evaluation by BCC model (2.1a), this DMU should not be technically efficient (otherwise, model (4.4) will be infinite, and the supporting hyperplane will be Fig. 1b). So, not only the model (4.4) can obtain the supporting hyperplanes of convex (Figs. 1a and 1b) but also can find CSPs. So, for any technically efficient point (on the model (2.1a)), we run the following model to obtain CSPs and convex supporting hyperplanes. Suppose the point ( , ) is located on the upper frontier of convex . In this sense, for each distinct point in convex such as ( , ) we have . By eliminating the input principle's disposability in the PPS, increasing some inputs will not necessarily result in increasing or remaining unchanged outputs; somehow, this increase in inputs will also reduce some outputs to be fully described in the next sections. This issue is also debatable, to reduce some inputs. Reducing some inputs of the technically efficient DMUs can lead to a decrease, remain unchanged, or even increase some outputs (congestion).
Hence, the supporting hyperplanes on the convex on the point ( , ) will be as follows, That specifies the rate and change type relative to the change . Obviously, an increased decrease when = < 0, that is an indicator for entering the CA. Suppose the point ( , ) is located on the upper frontier of convex . The component has congestion if there are supporting hyperplanes on the convex on the ( , ) as Then, we say that the point ( , ) related to the th component of input is a CSP. DMU has congestion if and only if it has congestion in at least one of its input components. So, to determine the technical efficient points (CSPs) as ( , ) and upper supporting hyperplanes on the convex (upper frontier of convex ) we used the following model: Suppose that ( * , * , 0 ) = ( * 1 , . . . , * , * 1 , . . . , * , * 0 ) is an optimal solution to model (4.4) and ∑︀ * − ∑︀ * + * 0 = 0 is one of the supporting hyperplanes on the convex in the technically efficient ( , ) . If the optimal solution ( * , * , * 0 ) has * = 0 (∀ ), so ( , ) not only the DMU does not have congestion but also not CSP. We have * 0 = ∞. Otherwise, if the optimal solution ( * , * , 0 ) has negative components of * , then not only the technical efficient DMU ( , ) is the landing on convex frontier, but also ( , ) is a CSP. We have * 0 > 0. Highlight 1. The technically efficient DMU ( , ) (that is weakly technically efficient DMU ) is a CSP if and only if in evaluating with the model (4.4) we have * 0 ≥ 0, * 0 ̸ = 0. In other words, ( , ) is a CSP, if on multiple model (4.4) there exists an optimal solution such as ( * , * , * 0 ) that * 0. By solving model (4.4), the supporting hyperplanes of convex can be obtained. Because of the convex constituent supporting hyperplanes that envelopment all observation and including CSPs have maximum 0 compared to all computable supporting hyperplanes (consider Figs. 1a and 1b).
Highlight 2. There is another point; the authors have assumed DMU ( , ) as a technically efficient DMU, and introduced model (4.4). However, it is unknown that of the technically efficient DMUs should be considered DMU in the model (4.4)? Several technically efficient DMUs can be used as DMU in the model (4.4). It is clear; first, we run models (2.1a), (2.1b) and calculated the set of all technically efficient DMUs. Also, sometimes, the observation sets of technically efficient DMUs may not include weakly efficient and DMUs are strongly efficient, so, to find CSP in this case, the all technically efficient DMUs which have the maximum inputs are CSP.
So, to find CSPs between the set of technically efficient DMUs, and the supporting hyperplanes corresponding optimal solution ( * , * , 0 ) with maximum 0 as convex frontier, we run model (4.4) ( * 0 ≥ 0, * 0 ̸ = 0). So, the coordinate of the CSP input vector is located on the lower bound of congestion frontier. Besides, the necessary condition that ( , ) as a landing location of the DEA production function frontier is * 0 > 0. It should be noted that, if in some optimal solutions of the above model * < 0, then the production frontier after the point ( , ) with increasing one or more inputs more than decreases some outputs without improving (worsening) other inputs or outputs.  If production technology has congestion property, then there is at least one CSP. Alternatively, vice versa, there is at least one CSP so, production technology has congestion properties. . Furthermore, if the model has an optimal solution, it is either zero, positive, or unbounded. In other words, it will never be negative. As described in the introduction, the economic definition of congestion (Def. 2.2) has some wrong in DEA. Still, using the set of CSPs, this definition can be expressed: Remark 4.5. A DMU has congestion if an increase in one or more inputs over one or more CSPs input. It is associated with decreases in one or more outputs without improving (worsening) other inputs or outputs.
Remark 4.6. A DMU has weak congestion if an increase in some or (all) inputs over some (all) inputs of CSPs, is associated with decreases in some or (all) outputs without improving (worsening) other inputs or outputs.
Remark 4.7. A DMU has strong congestion if an increase in all inputs is more than all CSP's inputs, It is associated with decreases in all outputs without improving (worsening) other inputs or outputs. As described above, CSPs themselves have no congestion.
Remark 4.8. CSPs are a type of technically efficient DMU -that are weakly but not strongly efficient DMU -with maximum outputs; thus, they do not have congestion.
In the DEA literature, the concept of congestion is expressed for the unit under evaluation, while any obvious point does not necessarily refer to the production frontier. To address the problem, instead of doing calculations on observed points, the calculations are done on the production frontier's projection points. To calculate the congestion of the observed point ( , ) its projection point is an optimal solution for a model (2.1b). To specify whether one DMU has congestion or not (without congestion measurement on inputs components), the model (4.5) based on the congestion definition of Cooper et al. [4,6] and the property of the convex PPS is presented: + is an output slack of the model (4.5). Also, * and + are an optimal solution and the optimum output slack of the model (2.1b). This model can identify the congested DMUs (from inefficient DMUs evaluated by model (2.1b)) and does not talk about their congestion size. So,  Figures 2a and 2b below, that the convex production technology ( convex ) is composed of five DMUs. As can be seen in the Figure 2, DMU is CSP because an increase in input more than the input of DMU causes a decreased output from the maximum level. Also, DMU is technically efficient in . As per Definition 4.9, DMU is not congested since DMU = ( = 8, = 6) and = 8 / ∈ . But DMU is congested since DMU = ( = 9, = 4) ∈ convex and also (︀ is an efficient point and there is ≥ 0 [9] and ( ) ∈ {(1), (2), . . . , ( )} such that:  Proof. Assume that 1 , 2 ∈ C , and ∈ (0, 1). Under these assumptions, it must be proved that ( ) ∈ C , where ( ) = 1 + (1 − ) 2 . Since ∈ ; ( = 1, 2) then, there is > 0 such that: (4.9) Then we have ( 1 , *1 1 + 1+* ) / ∈ convex . So, for every ( ) that It is clear that  So, ( ( ), ( ) ( ) + + ( )) / ∈ convex . Hence, it is enough to prove that ( ( ), ( )) ∈ convex . If the relation is true, then the theorem proved, assume, the relation is not correct then, according to the output disposability, there are 0 < < 1 such that ( ( ), ( )) ∈ convex Hence, it is enough: (4.14) So, ( ( ), ( ) ( ) This is obvious that congestion is a polytope, but a polytope may be convex (or maybe non-convex), so this above theorem must be proved that congestion ( ) is convex.  Proof. This is obvious according to the definition of .
A necessary condition for the DMU to be congested is that the input's vector should be inside . Note that Theorem 4.13 does not constitute sufficient conditions for the DMUs to be congested. Indeed, it could be positioned by the frontier of the set that represents congestion relative to some but not all inputs; that is the definition of weak congestion. The remark below provides the necessary and sufficient conditions for a DMU to be recognized as having congestion. Figure 3a shows the frontier and congestion area in the input space of DMUs and PPS projection on the input planes. Note that this projection of PPS is not from Farrell's frontier. Indeed, some points within the projection area could be technically efficient. As shown in Figure 3a, the congestion area is a subset of the input space of the PPS. According to the above theorems, any point such as DMU that is positioned within the congestion area has congestion.
Note that the boundaries of the congestion area do not necessarily indicate the presence of congestion. In Figure 3a, for example, DMU does not have congestion because it is located on the lower border that includes technically efficient points as CSPs (it means DMU is a technically efficient DMU and CSP). Also, DMU does not have congestion, and it is only technically efficient DMU and CSP. So, CSPs are technically efficient points and have no congestion because each point ( , ) ∈ on the border is a convex combination of at least two CSPs, so any point on the border is technically efficient and has no congestion also, the points ( , ) ∈ located on the border between congestion areas ( ) and non-congestion area have no congestion and any points on this border is CSP. Therefore, the points on the PPS can be divided into three classes: (1) without congestion, (2) with weak congestion, (3) with strong congestion.
Remark 4.14. If a DMU does has weak congestion, some but not all, of the inputs, are within .  Remark 4.16. Each congested DMU is within the congestion area. Since the congestion area is convex and was made by inputs of congested DMUs. So, the convex composition of both points within the area and each DMUs is congested (Fig. 4).
That is important to note that the blue area in Figure 3b is a part of congestion area and some previous congestion measurement methods based on DMUs (not production frontier) same as Noura et al. [20] lack congestion in DMUs located this area. It means these methods cannot identify congestion from DMUs whose inputs are located in this blue area. If the DMU does has congestion, the distance of a DMU from the congestion area bottom frontier can represent its congestion. Model (19) computes the so-called non-radial distance of a DMU with congestion. This model derived from the additive DEA model can determine the congestion of DMU's components.
(2) DMU does has weak congestion if * > 0 If we had used the radial model to compute congestion, it would be impossible to obtain component congestion. Similarly, the SBM model (for efficiency evaluation) can construct a model for measuring relative and non-radial congestion of DMUs. The relative congestion model is presented below: It is necessary to mention that; Congestion occurs in the input vectors of the DMUs. Therefore, when we say DMU has congestion, which means, the input of this DMU has congestion. So, when we say a DMU is CSP, it means the input of this DMU is CSP (consider inputs space). This is easy to understand when there is only one input-output, but when dealing with multiple inputs-outputs, it can be described as follows: (︀ 4 , 4 )︀ third and fourth inputs congestion measure of DMU 4 and (︀ 5 )︀ congestion measure of five inputs of DMU 5 , respectively. Other input components have no congestion. DMU 1 has strong congestion, which means all 5 inputs components have congestion, DMU 2 has weak congestion only in 2 1 , DMU 3 has weak congestion in 3 1 , 3 2 , DMU 4 has weak congestion in 4 3 , 4 4 and DMU 5 has weak congestion in 5 5 .
) obtained by subtracting congestion values from their corresponding input components is the CSP unit. More precisely, the input components of this virtual unit are the congestion boundary and with the turbulence of each of these components plus epsilon, the said component will be congested. Then, in multiple inputs, components of input (some or all) will play the role of CSP. To describe precisely, in the above example, for the first component If in a collection of congested DMUs, one or more DMUs have strong congestion, then all their input components are CSP, otherwise, only the congested input component of the DMUs are CSP. The is also created by the same CSP components. Of course, there may not be strong congestion, in which case the congestion zone will be made up of a combination of compact components that make up CSP and not all inputs components. Congested DMUs are not CSPs because CSPs are technically efficient points. Also, CSPs are a border between congestion and non-congestion. In single input (dimension one) all CSPs are one point and this border is one point, in two input all points located on the convex composition of two CSPs are CSP and this border is a line, and for 3 input this border is a hyperplane and for more than 3 inputs this is a convex multi-dimensional.

Second insight of congestion measurement; non-congestion area
This section describes new insights into congestion based on non-congestion DMU's inputs space and proves some fundamental theorems and a second new mathematical definition of congestion. Before discussing our proposed method, we must present the definitions of several new congestion-related concepts and propose a new mathematical definition for the concept of congestion itself. Building the non-congestion area is enough to find DMUs efficiently in the BCC-O and get the convex composition of these efficient DMUs. The DMUs in this compound are convex within the non-congestion area and lack congestion. Other DMUs are congested.
To prove that the NC is convex, we prove that the convex combination of both points of this area is within This is inconsistent with the assumption.
According to this theorem, we proved that the convex combination of both non-congestion area points is in the non-congestion area: the convex combination of two points that are not congested, not congested.
We know that if the DMU is ( * BCC-o ) efficient, it does not have congestion. So, the convex combination of the ( * BCC-o ) efficient points do not have congestion. Then we consider the convex combination of ( * BCC-o ) effective points and define that any point outside this area has congestion.
Note that to identify congested DMUs, we divide the PPS into two areas of congestion and non-congestion areas. We also classified the DMUs into two groups congested and non-congested (NC). Then, by computing the non-congestion area and a non-radial model, we identify the congested DMUs and calculate congestion of the congested DMUs simply and accurately. Given that compaction is a feature of the production frontier, and we also use post-PPS properties to identify congested DMUs, we can say that our method is also very accurate and fast. We test our model with the best methods available to identify congestion DMUs and show that our method is the most accurate to prove our claim.
We can express the congestion area defined in the previous section: So, the non-congestion area creates: The minimum distance of the DMU from the non-congestion area's frontier represents the amount of its congestion measure. The model (5.6) computes the non-radial distance of congested DMU from the noncongestion area. This model, derived from the additive DEA model, can determine the amount of congestion in the presence of both negative and non-negative data.
Remark 5.5. If DMU does not has congestion, then ∈ NC . In other words, if DMU does has congestion, thus ( ) / ∈ NC .
Note that C and NC are areas in inputs space. But to better represent, also showing the DMUs which are not on the border, we symbolically display it as PPS, to maintain its integrity. The yellow color in Figure 7a  Remark 5.9. According to the definition of the and NC , our two approaches are exactly equivalent to each other. Table 2 shows the data about China's textile and automobile industries from 1981 through 1997, which  compiled. In this example, variables are defined: is the output measured in units of one million renminbi at 1991 prices, is the capital price measured in units of one million renminbi at 1991 prices, and is the work done in units of 1000 people.

Numerical example (The textile industries and auto industries in China)
The results are summarized in Tables 3 and 4, which are obtained by solving the model of Cooper et al. [9], Noura et al. [20], Mehdiloozad et al. [19], and our proposed method using the data in Table 2  In the next step, by running model (4.4) for the set E, we try to find technically efficient DMUs (CSPs) with maximum * 0 . So, the set ( ) will be identified. We analyzed the sample data with three approaches Mehdiloozad et al. [19], Noura et al. [20], and Cooper et al. [9]. Also, to check in accuracy and performance of our method and for comparison, the results of our method with the results of solving other methods are presented in Tables 4 and 5. It should be noted that the values under the columns of Noura et al. [20] and Cooper et al. [9] show the congestion value of each component,  Table 4. Also, Table 5 is the result of our methods and comparison with four methods Cooper et al. [9] and Noura et al. [20], Mehdiloozad et al. [19], and Tone and Sahoo [23].
Using the data of Tables 4 and 5 and comparing the congestion values calculated by our proposed algorithm with the information obtained from the Noura et al. [20], Cooper et al. [9], and Mehdiloozad et al. [19] models. According to Table 3 Table 5, the result of our two proposed methods is equal to Mehdiloozad et al. [19] exactly (in the weak and strong CSOD identification). According to Table 4, our method's results (value of congested DMUs) are illustrated. To find the CSP, congestion area, congested DMUs, and congestion in each input component, we run our purposed algorithm. In addition, according to the results specified in Table 4, the results of our two proposed methods are completely consistent with each other, so it can be said that the methods are equivalent.
Also, to identify the congestion state of DMUs by NC , consider to following steps related to relation (5.4)-(5.6). The results are presented in Table 5

Conclusion
This paper focuses on detecting and measuring congestion in the DEA measurement framework. First, this paper aims at investigating the weak and strong congestion statuses of decision-making units (DMUs) by identifying a congestion zone (CZ) and non-congestion zone (NC) for production technology. Second, unlike the methods of Tone and Sahoo (2004) which considered multiple projections of DMU , Mehdiloozad et al. [19] thought of projections of DMU , maximal elements and used definition and properties of the face, minimum face, convex hull, and cone, we want to provide the new mathematical well-defined and efficient perspective of congestion relying on the concept of PPS only. The alternative approaches, based on the congestion and non-congestion areas, can identify weak and strong CSOD and efficiently calculate the amount of congestion of each input component.
Recently, Khoveyni et al. [17] have created a new method to develop the Tone and Sahoo [23] approach in identifying the weak and strong CSOD with both negative-positive data. However, as mentioned, their methods do not identify the strong congestion status of DMUs. Correctly. Meanwhile, they did not attempt been made to deal with the incidence of multiple optimal projections. However, as reasoned by Sueyoshi and Sekitani [22], specifying the weak-strong CSOD (inefficient DMUs) is probably problematic under such an incidence. So, this paper is interested in the precise recognition of the weak and strong CSOD under the decomposition of PPS with both non-negative and negative data. Since Tone and Sahoo's [23] description of strong congestion includes commensurate changes in inputs-outputs, based on this definition, we create a new mathematical definition and new insight of congestion to create it stable with both negative and non-negative data.
In this paper, we proved that the convex combination of both points of the non-congestion area is in the non-congestion area; the convex combination of two not congested points lacks congestion. Also, we explain that if the DMU is (( * BCC-o )) efficient; it does not have congestion and the convex combination of the (( * BCC-o )) efficient points do not have congestion. Then we consider the convex combination of (( * BCC-o )) effective points and define that any point outside this area has congestion. In this paper, first, define a new definition of congestion as the point ( , ) ∈ convex does has congestion if and only if ( , ) ∈ NC and second, we prove that the necessary and sufficient condition for a DMU to does has congestion is that its input vector should not be within NC . DMU has congestion if / ∈ NC . In this paper, to identify congested DMUs, we divide the PPS into two areas of congestion and non-congestion area. We also classified the DMUs into two groups congested and non-congested DMUs. Finally, we identify the congested DMUs and calculate congested DMUs simply and accurately by computing the non-congestion area and a non-radial model. Given that congestion is a feature of the production frontier, and we also use post-PPS properties to identify congested DMUs, it can be said that our method is also very accurate and fast. We test our model with the best methods available to identify congestion DMUs and show that our method is the most accurate to prove our claim.

Algorithm
(A) First, we run model (2.1a) and (2.1b) BCC-o to create a set of technically efficient DMUs. So, assume that we have an optimal solution as (︀ * , * , −* , +* )︀ . (a) If * = 1, −* = 0, +* = 0, then ( , ) is strongly efficient DMU and not congested and but, maybe CSP (if weak efficiency or technically efficient frontier does not exist). (b) If * = 1, −* ̸ = 0 or +* ̸ = 0, then ( , ) is technically efficient DMU and not congested and also, this is a candidate for CSP. So we go to Section C. (B) If * ̸ = 1, then ( , ) is not efficient DMU, and we run model (4.5). We run the model (C) We run model (4.4) for the set of DMUs obtained from step A-b. The set of solutions obtained from this model's running gives rise to the index ( ) (the set of CSP DMUs). If in the optimal solution, be equal to zero; there is no CSP, and, finally, there is no congestion area. Otherwise, if some be equal to positive, thus, and these DMUs are CSP and upper supporting hyperplanes (upper frontier) of convex so, we can create the CF and go to step 3. Also, we make the set of ( ) with that are positive (CSP DMUs). It means ( ) is the index of DMUs (CSPs) that have positive . finally, If in the optimal solution, be equal unbounded; there is no CSP and congestion area. then we go to D, (D) We creat with DMUs that their index creat ( ). Then go to E, (E) Run the model (4.16) to achieve the congestion measure in the input components of the congested DMUs.
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