Detection of the Type of Left and Right Returns-to-Scales Using Envelopment DEA Models in Crisp and Fuzzy Environments: An Application for Predicting Changes in the Stock Market

This paper aims to analyze left and right returns-to-scales in crisp and fuzzy data envelopment analysis (DEA). Since all previous envelopment DEA models for assessing left and right returns-to-scales are parametric, they are prone to encountering infeasibility problems, producing incorrect or different solutions for determining the type of returns-to-scale because of the different choices of parameter values. This misdiagnosis will lead to poor management decisions. Due to the mentioned problems, the issue of one-sided returns-to-scale has also not been studied in inaccurate environments. The present paper first proposes an alternative method of left and right returns-to-scales determination with crisp data to address this problem. This approach develops two non-parametric envelopment DEA models for analyzing left and right returns-to-scales. Then, the proposed method is extended to the fuzzy environment where data are considered more realistic. Due to its major advantages, credibility measure is used for solving fuzzy DEA models built to determine left and right returns-to-scales. As an application of the proposed method, data of companies in the Iran stock market are collected for 2014–2019 as fuzzy data and frontier units are analyzed by one-sided returns-to-scale.


Introduction
Data Envelopment Analysis (DEA) is a method for estimating the relative efficiency of a set of decision-making units (DMU) with multiple inputs and outputs [3,5]. This method has extensive use in the performance evaluation of factories, organizations, and generally homogeneous systems.
Returns-To-Scale (RTS) is one of the economical concepts that can be analyzed by DEA. RTS in DEA is determined using sensitivity analysis of the outputs of DMUs to changes in inputs. The three possible types of returns-to-scale, namely increasing returns-to-scale, decreasing returns-toscale, and constant returns-to-scale, were first introduced by Banker [2]. Banker and Thrall [4] and Färe and Grosskopf [7] proposed some methods for determining returns-to-scale based on the optimal solution of multiplier or envelopment DEA models. A study by Khodabakhshi et al. [16] introduced a method for determining returns-to-scale in nondeterministic DEA based on additive models. Other researchers, including Jahanshahloo and Soleimani-damaneh [15], Zarepisheh et al. [28], Hosseinzadeh Lotfi et al. [14], and Fukuyama and Matousek [9], have also proposed a variety of methods for estimating returns-to-scale. Given that the type of RTS can be determined by scale elasticity [17], Färe et al. [8] measured scale elasticity by combining the results of multiplier DEA models. Podinovski et al. [25] derived scale elasticity in DEA by parametric linear programming models. A report on the latest research in this area can be found in Krivonozhko et al. [17].
Return-to-scale is a local characteristic. Therefore, RTS may have one type of behavior in one direction while behaving differently in the other. In this way, it is not possible to express a general view of returns-to-scale. Given that in data envelopment analysis, inputs can increase (move to the right) or decrease (move to the left), concepts of left returns-toscale and right returns-to-scale are extended forms of ordinary returns-to-scale that present the effect of increase and decrease inputs on outputs, respectively. One-sided RTS may be important in economics which can serve as useful tools to survey the effect of expansionary or contractionary policies on DMUs. Golany and Yu [11] were the first to provide a model for determining the type of left and right returns-toscales of a DMU. The models proposed by these researchers were parametric programming models developed with the help of envelopment DEA models. The major problem of these models is their strong dependence on parameter setting.
Parametric models create two main problems: (1) sometimes models become infeasible, and (2) sometimes by changing the value of the model parameter used, the type of the one-sided RTS may also change. In other words, the type of right RTS may be increasing by using a value for the parameter of the method, while using another value for the same parameter may be decreasing. This problem has existed in all the methods developed to determine left and right returns-toscales. Taking an alternative approach, Eslami and Khoveyni [6] modified multiplier DEA models to develop nonlinear models to estimate left and right returns-to-scales. Allahyar and Rostamy-Malkhalifeh [1] modified the models of Golany and Yu [11] into alternative parametric programming models for examining the left and right returns-to-scales of strongly efficient DMUs to solve infeasibility. However, these models have the second problem. Omidi et al. [22] proposed a series of parametric linear programming models for examining left and right returns-to-scales by conducting a parametric sensitivity analysis on the ∑ n j=1 j = 1 constraint. As mentioned, a common drawback of all parametric models is their dependence on parameter setting, as using different parameters is likely to lead to different results.
On the other hand, different parametric methods may lead to different results for determining the type of one-sided returns-to-scale of a unit. In other words, the type of right RTS may increase using a parametric method, while using another method may decrease. Therefore, there are two main problems in using parametric methods to determine onesided RTS as follows: Although there are non-parametric methods for determining one-sided RTS, they are not without flaws. To measure the left and right scales elasticity, another way to determine left and right RTS type, Podinovski et al. [25] and Podinovski and Forsund [24] proposed envelopment parametric DEA models may face infeasibility. They used dual models of the proposed models when the parameter is equal to one. Therefore, they resorted to multiplier DEA models to determine one-sided scale elasticity and RTS, and as a result, the problem of non-determination of one-sided scale elasticity and RTS by envelopment DEA models remains. Mirbolouki and Allahyar [21] attempted to provide non-parametric envelopment models to determine one-sided RTS. They gave an example to state some of the problems of using parametric models to determine one-sided RTS. Mirbolouki and Allahyar [21] formulated a non-parametric integer envelopment DEA model for analyzing left and right returns-to-scales. In addition to being nonlinear, this model requires identifying a series of reference units, which is one of the complex processes of DEA. Unfortunately, one can also show by some examples that models of Mirbolouki and Allahyar [21] may determine the left and right returns-to-scales incorrectly. The problems expressed in the existing methods for determining one-sided RTS motivated the authors of this paper to research to find a method without parameters to determine the type of one-sided RTS. To solve the mentioned problems, the present paper proposes two non-parametric linear programming models, one for examining the left returns-toscale and the other for examining the right returns-to-scale of a DMU. The proposed models are linear programming problems, guarantee a globally optimal solution, and nonparametric models, which eliminate the impact of parameter setting on the analysis of returns-to-scale.
In the world around us, data are often shared and collected using the spoken language with vague verbal expressions such as "good," "moderate," "bad," "preferably," "almost," "young," and "beautiful." These inherently vague data can be called fuzzy data. Also, when a large amount of data is available, they can be aggregated into fuzzy data, and the behavior of a system can be analyzed based on the aggregated information. The concept of the fuzzy set was first introduced by Zadeh [26], who developed the classic concept of the membership function. Fuzzy data envelopment analysis (FDEA) was firstly introduced by Girod and Triantis [10].
Furthermore, its use in theory and practice became more and more over time. For a comprehensive overview of FDEA, refer to Zhou and Xu [29]. This paper investigates the left and right returns-to-scales in the fuzzy environment when input and output data are fuzzy triangular numbers. The main reasons for this research are: (A) Given that the determination of the left and right RTS in the classics DEA has some ambiguities, as mentioned in the previous paragraphs, the left and right RTS issue has not been studied in imprecise DEA. (B) We often encounter large amounts of data in economic matters; aggregating them into inaccurate data is one way to examine their behavior. Therefore, this is necessary that RTS as an economic concept is studied in inaccurate environments.
This paragraph is presented to summarize what has been explained so far and express the main motivation and innovation of the article. None of the existing methods for determining the left and right returns-to-scales have been successful. The authors of this paper have recognized that this problem is the parametric nature of the proposed mathematical models. This defect is illustrated by examples in this paper. Therefore, based on the concept of one-sided RTS, new definitions, methods, and models are presented for determining the left and right returnsto-scales that are not dependent on the parameter. On the other hand, due to the Objections of determining the onesided returns-to-scales in the presence of accurate data, this issue has not been studied in the fuzzy environment. To fill this gap, this paper introduces the concepts of left and right returns-to-scales in the presence of fuzzy data and determines the one-sided RTS in an application problem with fuzzy data.
In this paper, credibility measure is used to defuzzificate the proposed FDEA models. There are four main reasons for this choice. First, this theory simultaneously utilizes two basic concepts called "possibility measure" and "necessity measure." Thus, the credibility measure introduced by Liu [18] has all features of the possibility and necessity measures. Second, credibility theory can determine the crisp solution for each -credibility value and determine uncertain solutions for each overall evaluation. Third, the credibility measure has self-duality property. Fourth, burdensome computations of equivalent crisp models in credibility theory are less than that in -cut method. Researchers can refer to Payan [23] for more information on the benefits of using credibility measure in fuzzy DEA.
Using a fuzzy approach, the concepts of classic management can be extended in a variety of management tasks, including decision-making, policy-making, and planning, leading to enhanced precision and applicability. The fuzzy set theory also facilitates the formulation of models to process qualitative data and thus allows us to make better use of primarily fuzzy data, such as knowledge, experience, and human judgment, in the development of DEA models. Zhou and Xu [29] presented comprehensive research on using fuzzy DEA in various real-world applications. One of the successful applications of FDEA is in the stock basket and market. Gupta et al. [12] used credibility FDEA for portfolio efficiency and benchmarking. Returns-to-scale is an interesting concept for researchers in economics, finance, and management as it can provide valuable insights into how DMUs adapt to changes in their economic conditions. Information about returns-to-scale can help one decide whether to expand or restrict an activity with a low-risk economy. Also, in real-world problems, a large amount of data needs to be aggregated and then analyzed. Thus, as an economical application of left and right returns-to-scales in a fuzzy environment, data of companies in the Tehran Stock Exchange is collected for 2014-2019 as fuzzy data and behavior of units are analyzed by the fuzzy left and right returns-to-scales.
The rest of this paper is structured as follows. The second section discusses the economic meaning of left and right returns-to-scales and offers a new approach for these concepts. This method is then used as a basis for the formulation of non-parametric DEA models to analyze left and right returns-to-scales. The third section extends the proposed method to fuzzy DEA and uses credibility theory for solving corresponding models. In the next section, an example shows the advantage of the proposed method to the existing methods in the literature. An illustrative example is then provided to explore the fuzzy proposed method, and the method is also used to analyze the onesided returns-to-scale of Iranian Stock Exchange Companies while their data in the period 2014-2019 collected as triangular fuzzy data. The fifth section summarizes the results of the paper and provides some suggestions for future research.

Basic Definition
Suppose there are n DMUs. DMU j (j = 1, ..., n) consumes the m-element input vector x j = x 1j , ..., x mj to produce the s-element output vector y j = y 1j , ..., y sj . According to the principles of DEA, the production possibility set (PPS) for variable returns-to-scale is: Based on this PPS, BCC model has been proposed by Banker et al. [3] for measuring the efficiency of units as: (1)  Returns-to-scale is a microeconomic concept reflecting the extent to which the outputs of a system increase following a certain increase in its inputs. Depending on the system, returns-to-scale could be increasing, decreasing, or constant. Returns-to-scale is said to be constant when an increase in inputs causes the same proportion of increase in outputs. Returns-to-scale is called increasing when outputs increase by a larger proportion than the increase in inputs and is called decreasing when the opposite is true. Organizations can use returns-to-scale as a measure to determine the extent to which they can benefit from expanding or scaling up their operations.
Since the returns-to-scale discussion also encompasses the impact of declining inputs on outputs, which from an economic perspective, can be interpreted as downsizing or contraction, the concepts of left and right returns-to-scales have been introduced to cover both aspects of this discussion. The assessment of the right returns-to-scale means examining the effect of an increase in independent indices (inputs) on dependent indices (outputs). On the contrary, the assessment of left returns-to-scale means examining the effect of a decrease in independent indices (inputs) on dependent indices (outputs). In the context of economics, if an increase (decrease) in independent indices leads to a greater increase in dependent indices, it is said that the right returns-to-scale (left returns-to-scale) is increasing; if an increase (decrease) in independent indices results in a smaller increase in dependent indices, it is said that the right returns-to-scale (left returns-to-scale) is decreasing, and if an increase (decrease) in independent indices causes the same amount of increase in dependent indices, it is said that the right returns-to-scale (left returns-to-scale) is constant.
Suppose DMU o is an efficient unit and is a non-negative scalar, and: [13]) If DMU o is efficient, then the following is true for its left and right returns-to-scales:

Non-parametric Models
Based on Definition 2, to determine the type of left and right returns-to-scales of an efficient DMU, one should examine the changes in its size in a small neighborhood around the unit in such a way that the unit would stay on the efficient frontier. The geometric description of left and right returnsto-scales is provided in Fig. 1: Consider the efficient DMU C. As shown in Fig. 1, the neighborhood of C partially overlaps with the production possibility set. To determine the right returns-to-scale, one has to examine the extent of change in output following an increase in input, which in the Fig. 1, refers to the zone named CHF. As the Fig. 1 shows, after an increase in input, the output must also increase so that we arrive at the efficient frontier. This increase in outputs could be smaller, larger or equal to the amount of increase in input, in which case we have decreasing, increasing, or constant right returns-toscale, respectively. To determine the left returns-to-scale, we must examine the extent of change in the output following a decrease in the input. However, since such change would take us outside the production possibility set, our proposal is to examine the changes in the zone CIJ. In this zone, one must examine the changes in the input after a decrease in Page 5 of 24 26 the output. It can be seen that to reach the efficient frontier after a decrease in the output, the input must decrease as well. The ratio of this decrease determines the type of left returns-to-scale. This paper claims that the magnitude of decrease in input, which could be smaller, larger, or equal to the magnitude of decrease in output, can show whether the left returns-to-scale is increasing, decreasing, or constant. This approach to determine returns-to-scale guarantees the models to be feasible, as it never allows going outside the production possibility set. Therefore, the mathematical models developed based on this approach will not have the problem of infeasibility. For a mathematical description of the above claim, consider the following sets: Suppose DMU o is efficient. The set of all proportional input/output changes inside the production possibility set, which in this paper is denoted by P(x o , y o ) , is expressed as follows: Therefore, the mathematical description of the zone CHF in Fig. 1, which represents the new position of the unit following the increase in output and input, is as follows: Now, to make sure that the new position of the unit after changes in inputs and outputs is still on the efficient frontier, we set: Similarly, the mathematical description of the zone CIJ region in Fig. 1, which represents the new position of the unit following the decrease in output and input, is expressed as follows: To make sure that the new position of the unit after the decrease in inputs and outputs is still on the efficient frontier, we set: Based on the above sets, the left and right returns-toscales of the efficient unit DMU o are defined as follows.

Theorem 1
If DMU o is efficient, then the following is true for its left and right returns-to-scales: is increasing. 5. If x ( y ) − < y for some y < 1 , then left returns-to-scale is decreasing. 6. If x ( y ) − = y for some y < 1 , then left returns-to-scale is constant.
Proof See Appendix A.
Based on Theorem 1 and Eqs. (6), (7), we propose the model (10) for determining the type of right returns-to-scale for an efficient unit such as DMU o : It is clear that o = 1, j = 0, j ≠ o, x = y = 1 is a feasible solution of model (10). Also, constraint ∑ n j=1 j y j ≥ y y o along with ∑ n j=1 j = 1 ensure that the model (10) is not unbounded. Therefore, model (10) as a linear programming problem has optimal solution.
Using the optimal solution of model (10), the type of right returns-to-scale DMU o (efficient unit) can be determined as follows: Theorem 2 If DMU o is efficient, then the following is true for its right returns-to-scale: 1. If there exists an optimal solution of model (10) in which * y > * x > 1 , then right returns-to-scale is increasing. 2. If for every optimal solution of model (10), * y = * x = 1 , then right returns-to-scale is decreasing. 3. If there exists an optimal solution of model (10) By a similar manner and using Theorem 1 and Eqs. (8) and (9), we propose the following model (11) for determining the type of left returns-to-scale of an efficient unit such as DMU o : By a similar discussion, model (11) is feasible and bounded. Using the optimal solution of model (11), the type of left returns-to-scale of DMU o can be determined as follows: Theorem 3 If DMU o is efficient, then the following is true for its left returns-to-scale: 1. If for every optimal solution of model (11), * x = * y = 1 , then left returns-to-scale is increasing. 2. If there exists an optimal solution of model (11) in which * x < * y < 1 , then left returns-to-scale is decreasing. 3. If there exists an optimal solution of model (11) in which * x = * y < 1 , then left returns-to-scale is constant.
Based on the concepts, models and theorems mentioned in this section, following algorithm is suggested for determining the left and right returns-to-scales of efficient units as: Step 0. Put o = 0. Step Step 2. Solve model (2) to evaluate DMU o .
Step 2a. If DMU o is efficient based on definition 1, go to step 3.
Step 2b. If DMU o is not efficient based on definition 1, go to step 1.

One-Sided RTS with Fuzzy Data
The fuzzy set theory was first introduced by Zadeh [26] and has since been used by many researchers in various fields of science. Zadeh [27] later proposed the concept of possibility measure to determine the probability of a fuzzy event.
Although being widely used in many applications, possibility measure does not have a self-duality property. Thus, a self-dual measure was needed in both theory and practice.
To resolve this issue, Liu and Liu [20] proposed the concept of credibility measure. Introduced by Liu [18] and revised by Liu [19] credibility theory can be described as a branch of mathematics dedicated to the study of fuzzy phenomena. The next subsection is mainly focused on the concepts, definitions, and theorems of credibility theory.

Credibility Measure
Let Θ be a nonempty set and P(Θ) be the power set of Θ , and Pos be a set function on P(Θ) . The Pos set function is called a possibility measure if it satisfies the following conditions: where I is an arbitrary set of indices. Based on the possibility measure, Liu and Liu [20] proposed a self-dual set function called the credibility measure (Cr) . Assuming the triplet (Θ, P(Θ), Pos) as a possibility space, for any A ∈ P(Θ) , the credibility measure of the event A is defined as: where A c is the complement of A. The credibility measure satisfies the following axioms: 5. Consider the nonempty sets Θ = Θ 1 × ... × Θ n and Cr satisfies the above axioms; then for every ( 1 , ..., n ) ∈ Θ , Cr{( 1 , ..., n )} = Cr{ 1 } ∧ ⋯ ∧ Cr{ n }.
is a set function on (Θ, P(Θ), Pos) , which is defined as: Then, Pos is a possibility measure on the possibility space (Θ, P(Θ), Pos) , as it satisfies both of the possibility measure conditions. The possibility measure and the credibility measure of the event B = { 2 , 4 , 5 } are given by:

Definition 4
If is a fuzzy variable on the credibility space (Θ, P(Θ), Cr) , then its membership function is given by: The fuzzy variable with equally distributed possibility on [a, b] , where a and b are crisp numbers and a < b , has the following membership function: where a L , a C , and a U are crisp numbers and a L < a C < a U has the following membership function: If is a fuzzy variable with the membership function , then for any subset B of real numbers, we have:

Theorem 6
If is the triangular fuzzy variable as = (a L , a C , a U ) , then for every credibility level α, the crisp (deterministic) equivalent of constraint Cr{ ≥ r} ≥ will be: is a triangular fuzzy variable, then for each credibility level, the crisp (deterministic) equivalent of the constraint Cr{ ≤ r} ≥ is:

Fuzzy Left and Right Returns-to-Scales
Suppose there are n fuzzy DMUs. Fuzzy DMU j (j = 1, ..., n) consumes the m fuzzy inputs as a fuzzy vector x j = x 1j , ...,x mj to produce the s fuzzy outputs as a vector ỹ j = ỹ 1j , ...,ỹ sj . Fuzzy PPS in variable returns-to-scale assumption is as: Based on this PPS, fuzzy BCC model will be as: Using credibility theory, model (13) is converted to the following crisp model (14) as: (14), we have: As an extension of crisp models (10) and (11), two fuzzy DEA models (15) and (16) are presented to determine fuzzy right and left returns-to-scales of fuzzy DMU o as follows: To determine left and right returns-to-scales for each -credibility value of -efficient DMU o , which are named by -left and -right returns-to-scales, models (15) and (16) are transformed to following crisp linear programming problem by applying credibility measure as follows: * = 1, S − * = 0, S + * = 0.

Theorem 9
If DMU o is -efficient ( < 0.5 ) and , x , y ≥ 0 is the optimal solution of model (19), then for determining the type of right returns-to-scale of -efficient DMU o : 1. If there exists an optimal solution of model (19) in which y > x > 1 , then -right returns-to-scale is increasing. 2. If for every optimal solution of model (19), y = x = 1 , then -right returns-to-scale is decreasing. 3. If there exists an optimal solution of model (19) in which y = x > 1 , then -right returns-to-scale is constant.
Proof Proof is similar to Theorem 2.

Theorem 10
If DMU o is -efficient ( < 0.5 ) and , x , y ≥ 0 is the optimal solutions of model (20), then for determining the type of left returns-to-scale of -efficient DMU o : 4. If for every optimal solution of model (20), then -left returns-to-scale is increasing. 5. If there exists an optimal solution of model (20) in which x < y < 1 , then -left returns-to-scale is decreasing.
6. If there exists an optimal solution of model (20) in which x = y < 1 , then -left returns-to-scale is constant.
Proof Proof is similar to Theorem 3.
To determine the type of right and left returns-to-scales, -efficient DMU o , in which ≥ 0.5 , models (21) and (22) must be solved, respectively. Similar theorems as Theorems 9 and 10 can be stated and proved to estimate the type of right and left returns-to-scales of -efficient DMU o . Table 1 shows the specifications of 14 DMUs, each with two inputs and two outputs, in the example of Mirbolouki and Allahyar [21]. They used the example to compare their method with the methods of Golany and Yu [11], Allahyar and Rostamy-Malkhalifeh [1] and Eslami and Khoveyni [6]. Since the methods of Golany and Yu [11], Allahyar and Rostamy-Malkhalifeh [1] and Eslami and Khoveyni [6] are parametric, this comparison illustrated the major drawback of these methods, which is the dependence of their assessments of left and right returns-to-scales and their results on parameter settings. Table 2 shows that not only the results yield to one-sided RTS in the different methods are different, but also, different results have been obtained for one-sided RTS by different parameters in each method.

Counter Example
In the following, we solve the same example with the method proposed in this paper and show that the results of the method of Mirbolouki and Allahyar [21] are not correct. Also, the method of Mirbolouki and Allahyar [21] requires reference units, which are difficult to obtain, and their models are also nonlinear. Through a brief comparison, we will show that since the linear programming models presented in this paper are based on a simple definition of left and right returns-to-scales, it indeed works more effectively than the method of Mirbolouki and Allahyar [21].   Tables 3 and 4 present the results of the method of Mirbolouki and Allahyar [21] and the method presented in this paper regarding the type of left and right returns-to-scales of efficient units of data in Table 1. In this example, efficient DMUs are units 2, 3, 4, 6, 7, 9,10,11,14. The results reported in Tables 3 and 4 show a difference in the type of left returns-to-scale of the fourth DMU. While the method of Mirbolouki and Allahyar [21] has identified its returns-toscale as increasing, our method has identified it as decreasing. Since the evaluated DMUs have four input/output entries, the type of returns-to-scale cannot be determined by a geometrical representation. By solving the model (11) for unit 4, it is clear that (0.15x 4 , 0.16y 4 ) is a point on the efficient frontier and therefore belongs to the production possibility set. Due to the fact that DMU 4 is an efficient unit, the line segment connecting these two points belongs to the PPS. Suppose is a value less than one but close to it, so ( x 4 , y 4 ) is a point below this line segment. As a result, ( x 4 , y 4 ) is an interior point of PPS. Thus, Therefore, left returns-to-scale of DMU 4 is decreasing while Mirbolouki and Allahyar's method [21] tells this point has increasing returns-to-scale. This fact shows that results of Mirbolouki and Allahyar's method [21] cannot be reliable to determine left and right RTS.

Illustrative Example
To illustrate the performance of the proposed models, it is used to solve a problem including 9 DMUs and each unit with one fuzzy triangular input and one fuzzy triangular output provided in Table 5.
Efficiency values and the type of returns-to-scale of efficient units are reported in Table 6, for different values of -credibility. As shown in Table 6, unit A is efficient for all -credibility levels, unit E for ≥ 0.2 , unit I for ≥ 0.4 , unit C for ≥ 0.5 and unit B for ≥ 0.7 . Results of determine the type of left and right returns-to-scales of efficient units at different -credibility levels are reported in Table 6, based on the optimal solutions of models (19), (20), (21) and (22).
To analyze what happened in the models (19), (20), (21) and (22) and reported in Table 6, the following Figs. 2, 3 and 4 are pictured. There are three groups of DMUs in Fig. 2  and their outputs are the upper bounds of fuzzy outputs of data in Table 5. These units are shown, respectively, by AL, BL, …, IL. For example, AL = (x l A , y u A ) = (1, 4) . The second group consists the units that their inputs and outputs are, respectively, the middle points of fuzzy inputs and outputs. These units are shown by purple points and named by AC, …, IC. The third group is consisted of units that their inputs are the upper bounds of fuzzy inputs and their outputs are the lower bounds of fuzzy outputs which are displayed in red color by AU,…,IU. Based on this classification, strong efficient frontier for each group of data is drawn in Fig. 2. Now, we survey treatment of proposed models by analyzing one-sided returns-to-scale of unit A, as an example. Model (19) is used to determine right returns-to-scale of DMU A, for 0 ≤ ≤ 0.5 . In this model, by increasing , all units except DMUA move from BL, CL, … IL towards BC,…, IC, respectively. Direction of these movements is shown in Fig. 3, for efficient units B, C, I and L. Unlike other  Fig. 3, strong efficient frontier from piecewise linear function AU.IL.BL.EL, for = 0 , has been changed to piecewise linear function AC.CC.IC.EC, for = 0.5 . We see that unit A is efficient in this condition. Now, we consider a fix value of credibility level as = 0 . Based on the optimal solution of model (19) for evaluation unit A in = 0 , we have A x = 1.27, A y = 4.75 and so this is possible that increasing in output of A be more than its increasing in input. As a result, right returns-to-scales of DMU A is increasing, for = 0 . By comparing strong efficient frontier in = 0 , which is shown in blue segments, with CCR frontier O.IL, this is trivial that right returns-toscales of DMU A is increasing which confirm the analysis of optimal solution of model (19).

. The first group which is displayed in blue color is considered units that their inputs are the lower bounds of fuzzy inputs
In general, this is trivial from Fig. 3 that by changing position of unit A from AU to AC and other units from blue positions toward purple positions, unit A is efficient and its right returns-to-scale is increasing. Similar analysis can be done for other units, for 0 ≤ ≤ 0.5. Figure 4 is drawn to analyze right returns-to-scale of DMU A in while > 0.5 . Based on the constraints of model (21), all units except DMU A put in their center in while = 0.5 , which are shown with purple points in Fig. 4, and by increasing value move toward red points. Direction of these movements is shown in Fig. 4, for efficient units C, I and E. But unit A, which is in evaluating DMU, moves form AC to AL. Changing strong efficient frontier will be from piecewise linear function AC.CC.IC.EC to piecewise linear function AL.IU.EE. Unit A is efficient for > 0.5 and this is trivial from Fig. 4 that its right returns-to-scale in comparing with CCR efficient frontier is increasing, for each > 0.5 . These observations are coincided with the results in Table 6. Similar analyzes can be performed for the right returnsto-scale of other units. The left returns-to-scale of all units may also be discussed, intuitively.

Application
In this section, we examine the one-sided returns-to-scale of 75 profitable Companies in the Iranian stock market. By determining the left and right returns-to-scales, one can make predictions of companies due to input and output changes, which are usually influenced by political, economic, and social developments. For a fairer assessment of returns-to-scale, data from  Table 7. To summarize and aggregate the data, we converted them into fuzzy triangular numbers so that the smallest value of each index for each unit was considered the lower bound, the largest of them was given as the upper bound. The average of these five numbers for each index per unit was calculated to the center of the fuzzy triangular number. The information obtained is listed in Tables 9 and 10, and reported in Appendix B. Table 8 shows the results of models (19), (20), (21) and (22) in determining the left and right returns-to-scales of each efficient unit for some -credibility values.
The data of this problem are related to the real information of Iranian stock exchange companies in the years 2014-2019. Given that the aggregation of this data is done in the form of fuzzy numbers and there are two optimistic and pessimistic views to examine the units in the fuzzy environment, these views may not be fair and do not correspond to the reality of the data. To give a fair view of the performance of the units in the fuzzy environment, in this paper, the credibility values are considered close to 0.5. Therefore, the range proposed in this article is [0.4, 0.6] . Of course, a wider range can be considered, such as [0.3, 0.7] general evaluation, which includes [0, 1] . In this paper, returns-toscale on the stock exchange for credibility value of 0.4 as a pessimistic view, 0.5 as a fair view, and 0.6 as an optimistic view are reported in Table 8.
It is clear from the results of Table 8 that when we move from a pessimistic view with a credibility value of 0.4 to an optimistic view with a credibility value of 0.6, the number of efficient units increases, which is a reasonable result. Five efficient units in = 0.4 , 19 efficient units in = 0.5 (approximately four times the pessimistic state), and 42 efficient units in = 0.6 (more than half of the units) are shown in Table 8.
The results in Table 8 show that the most efficient units for any credibility measure have a constant returns-to-scale or a descending returns-to-scale. For = 0.4 , the number of such units is 80% of the total efficient units, and for credibility values 0.5 and 0.6, it is more than 98% of the total efficient units. As stated in economic analysis, successful Fig. 3 Analysis right returns-to-scales when -credibility level is increased from 0 to 0.5 companies and economic organizations usually move after a period towards performance stability (constant returns-toscale) or recession (descending returns-to-scale). This means that while they are efficient, further development will not lead to greater profitability. Therefore, the number of units with increasing right returns-to-scale is very small compared to the total number of efficient units. Nevertheless, these units have the advantage that they can still be further developed and profitable while maintaining efficiency. Table 8 states that unit 12 has an increasing right returns-to-scale for all credibility values examined in this paper. The development of such a unit will be more profitable for the company. As a result, if the company pursues a development policy and investment is aware of this policy, it will receive high profits if it invests.
There is a completely different economic analysis for companies with decreasing left returns-to-scale and abundant in this example. Expansion policies in such companies' waste costs, but profitability and revenue will be reduced to a lesser extent when they decrease costs. Therefore, in such companies, the investment risk will be high.
In the case of a company like company 3, which in each of the pessimistic, fair, and optimistic perspectives has to decrease right returns-to-scale and increase left returns-toscale, the economic recommendation is not to change the input indicators and maintain managerial stability. Therefore, if the investment is deprived of the managerial stability of these companies, it will experience safe investment in such companies. Similar analysis can be expressed for other companies in terms of different credibility values and based on the type of their one-sided returns-to-scales.

Conclusions and Suggestions
This paper presented two envelopment DEA models for assessing left and right returns-to-scales of decision-making units in classic DEA and fuzzy DEA. The geometric interpretation of the proposed approach was explained, based on  which the supporting theory of the proposed method and models were expressed and proved by theorems. The main advantages of the proposed method are the linearity and non-parametric nature of both models. These features lead to correct detection of the type of one-sided RTS and prevent infeasibility and different results. The non-parametric models of this paper provided the development of one-sided returns-to-scale to inaccurate data. In this paper, the data related to 75 stock exchange companies in Iran were aggregated in the form of triangular fuzzy data, and the type of their left and right returns-to-scales was determined. In this way, it is possible to determine the one-sided returns-toscale in the presence of other inaccurate data such as interval, stochastic data, etc. However, this article, like any other scientific research, has limitations, the main of which are: (a) In this paper, a simple input-output structure for decision-making units was considered to determine the left and right returns to scales, while in practice, we are faced with more complex (network) structures and processes.
(b) Among the effective indicators in the performance of stock exchange companies, there are also negative indicators that were not considered in this article. (c) Also, due to the unavailability of the values of some indicators in some years in stock exchange companies, these indicators were not considered in the evaluation.
Based on the theoretical and practical links expressed in this article, new scientific research can be done. Given the relationship between scale elasticity and returns-toscale, measuring crisp and fuzzy scale elasticity is a good research topic for the future. Another useful research in the stock market is to extend the proposed fuzzy DEA to fuzzy negative DEA. In this way, companies with negative profit (losses) can also be examined in measuring performance and one-sided returns-to-scale.
(Part 5). We prove that if x ( y ) − < y , then the left returns-to-scale of DMU o according to Definition 2 is decreasing.
According to the definition of x ( y ) − , we have: Since x ( y ) − < y , therefore: Thus, according to Part 5 of Definition 2, left returns-toscale is decreasing. (Part 6). We prove that if x ( y ) − = y , then the left returns-to-scale of DMU o according to Definition 2 is constant.
According to the definition of x ( y ) − , we have: Since x ( y ) − = y , therefore: Thus, according to Part 6 of Definition 2, left returns-toscale is constant.
Proof of Theorem 2 Proof: 1) Suppose * , * x , * y is the optimal solution of model (10) in the evaluation of DMU o . Since in this case * y > * x > 1 , according to Part 1 of Theorem 1, right returns-to-scale is increasing.
Proof of Theorem 3 Proof: 1) Suppose * , * x , * y is the optimal solution of model (11) in the evaluation of DMU o . In this case * y = * x = 1 and therefore the optimal solution of model (11) will be zero. Now, suppose y < 1 . We claim that if the optimal solution of model (9) is x ( y ) − , then x ( y ) − > y , and otherwise x ( y ) − ≤ y . Therefore * , x ( y ) − , y is a feasible solution of model (11). On the o t h e r h a n d , i f x ( y ) − < y , t h e n : 2) Suppose * , * x , * y is the optimal solution of model (10) in the evaluation of DMU o . Since * y = * x = 1 , the optimal solution of model (10) will be zero. Now suppose x > 1 . We claim that if the optimal solution of model (7) is y ( x ) + , then y ( x ) + < x , and otherwise y ( x ) + ≥ x . Therefore * , x , y ( x ) + is a feasible solution of model (10). On the other hand, if y ( x ) + > x , then: which contradicts * , * x , * y being optimal. But if y ( x ) + = x , then the optimal solution of model (10) will be * , x , y ( x ) + . Since x > 1 , this contradicts the assumptions of Part 2. Therefore, for x > 1 , we will have y ( x ) + < x . Thus, according to Part 2 of Theorem 1, right returns-to-scale is decreasing.
3) Suppose the optimal solution of model (10) in the evaluation of DMU o is * , * x , * y , where * y = * x > 1 . Now assume that x = * x > 1 . We claim that if the optimal solution of model (7) is y ( x ) + , then y ( x ) + = x . It is not possible to have y ( x ) + because * , * x , * y is a feasible solution of model (7). Now we assume, for the sake of contradiction, that y ( x ) + < x . Based on this assumption, * , x , y ( x ) + is a feasible solution of model (10) and also: x , * y being optimal. Therefore y ( x ) + = x and thus according to Part 3 of Theorem 1, right returns-to-scale is constant.
which contradicts * , * x , * y being optimal. But if x ( y ) − = y , then the optimal solution of model (11) will be * , x ( y ) − , y . Since y < 1 , this contradicts the assumption of Part 1. Therefore, for y < 1 , we will have x ( y ) − > y . Thus, according to Part 4 of Theorem 1, left returns-to-scale is increasing.
2) Suppose * , * x , * y is the optimal solution of model (11) in the evaluation of DMU o . Since in this case * x < * y < 1 , according to Part 5 of Theorem 1, left returnsto-scale is decreasing.
3) Suppose the optimal solution of model (11) in the evaluation of DMU o is * , * x , * y , where * y = * x < 1 . Now assume that y = * y < 1 . We claim that if the optimal solution of model (9) is x ( y ) − , then x ( y ) − = y . Since * , * x , * y is a feasible solution of model (9), it is impossible to have x ( y ) − > y Now we assume, for the sake of contradiction, that x ( y ) − < y . Based on this assumption, * , x ( y ) − , y is a feasible solution of model (11) and also: which contradicts * , * x , * y being optimal. As a result, y ( x ) + = x and therefore according to Part 6 of Theorem 1, left returns-to-scale is constant.

Author Contributions
The study is written by all authors. All authors reviewed the results and approved the final version of the manuscript.
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Availability of Data and Materials
The data of this article are reported in Tables 9 and 10 in Appendix B.

Conflict of interest
The authors declare there are no relevant financial and non-financial competing interests to the manuscript.
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