# Investigating the Sustainability of Return to Scale Classification in a Two-Stage Network Based on DEA Models

Authors: Maryam Sarparast; Farhad Hosseinzadeh Lotfi; Alireza Amirteimoori
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

Purpose. The purpose of this study is to sensitivity analysis analyze the returns to scale in two-stage network based on DEA models. The main focus of the firms has always been to obtain the maximum output with the least available resources, which points to the improvement of the firm’s performance and the importance of returns to scale and technical improvement. Design/Methodology/Approach. This study examines the sensitivity of returns to scale classifications in a two-stage DEA network. A new input-oriented model was progressed to identify the efficient decision-making units in the two-stage network, after which a new method of determining the returns to scale classifications in the efficient DMUs in two-stage network (constant, increasing, or decreasing returns to scale) was established. Findings. The stability of the returns to scale classifications in the two-stage network was analyzed. A stability region for changes in primary inputs and final outputs is only determined especially for DMUs that are efficient so that it maintains the classification of the returns to scale units. The results are shown by numerical examples. Practical Implications. The sensitivity analysis of returns to scale classifications is one of the most significant issues in data envelopment analysis (DEA), which plays an essential role in management decisions. Originality/Value. Using this model can help improve the performance of companies by using new tools and also improve the quality of work and increase acceptance competition.


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## Body

## 1. Introduction

Data envelopment analysis (DEA) is a nonparametric approach to evaluating the relative efficiency of decision-making units (DMUs), having similar inputs and outputs. This method was developed by Charnes et al. [1] to enable the application of the same inputs to produce the same outputs and was extended by Banker et al. [2] through their proposed BCC (Banker, Charnes, and Cooper) model. Meanwhile, in classical DEA methods, systems are regarded as a black box, calculations are limited by the final output, and initial inputs and internal processes are disregarded. These procedures, however, have been tentatively illustrated as occasionally incorrect. Wang et al. [3], for example, showed that banking operations are underlain by two procedures, namely, capital collection and investment. Along the same vein, Charnes et al. [4] indicated that army recruitment proceeds through two processes, that is, stimulating awareness through advertisements and creating contracts. The DEA technique has been extensively employed in research. Chen and Zhu [5] used DEA to analyze the consequences of information technology (IT) on the performance of a firm and then developed a new methodology that measures the marginal advantages of IT to illustrate its effects. Accordingly, Seiford and Zhu [6] examined the difference between the profitability and marketability of 55 US commercial banks on the basis of a two-stage production process. In other words, the comparison of small and large banks revealed that the latter are more profitable, whereas the former are more marketable. That is, the lower profitability of small banks was attributed to their failure to achieve substantially effective performance in both the profitability and marketability dimensions. In this respect, the present authors proposed a two-stage approach to identifying strategies for improving bank performance.Ebrahimnejad and Hosseinzadeh Lotfi [7] have used Zionts-Wallenius’s method to form a new model according to the general MOLP problems and combined-oriented CCR model, followed by Ebrahimnejad et al. [8] along with his colleagues recommended a new method based on integrated DEA and simulation to reach the group consensus ranking. Maddahi et al. [9] provided a new technique of secondary goals to evaluate cross-efficiency using a selection of wight symmetrical to its corresponding input or outputs to overcome the lacks of previous method. Tavana et al. [10] schemed a new method regarding an equivalence relationship between desirable and undesirable inputs and outputs simultaneously with uncontrollable variables.Fare and Grosskopf [11] were the first to present the concept of network DEA, wherein each network unit in a network system produces resources for subunits, which consume the generated resources. Thus, each subunit in a DMU comprises several initial, middle, and final inputs (outputs) originating from within or outside of a unit. Likewise, Zhu [12] developed DEA models to identify diverse amounts in order to calculate the revenue performance of 500 fortune companies. In many models, the performance of best-practice frontier companies is determined on the basis of the construction of reference share measures. Lewis and Sexton [13], also, in order to measure the efficiency of units produced in a two-stage network, developed a two-stage network DEA model. Later in 2004, using the model proposed, they investigated the efficiency of organizations with complex inner construction [14]. In the given model, units consist of an entire network that comprises all produced subunits.Luo [15] used DEA to evaluate the profitability and marketability of large banks. Chen and Zhu [5] extended a two-stage DEA model to ascertain the efficient frontier of a two-stage production process and used the extended version to evaluate IT’s indirect effects on firm performance. In the same line, Chen [16] presented an integrated structural frame to decompose the performance of a dynamic production network. Lu and Hung [17] also determined the operating performance of 40 fabless integrated circuit firms in Taiwan via a two-stage production process and designed a new approach to studying the difference between profitability and marketability.Kao and Hwang [18] put forward a model based on the standard DEA model to measure the efficiency of an entire process that can be decomposed into the product of the efficiencies of two subprocesses. The authors also compared their model with the two-stage approach of Chen and Zhu [5]. Intrigued by the interrelationship of processes within a system, Kao and Hwang [18] examined the structural network DEA model and introduced an original changeable network to a series system, in which each stage can function as a parallel system. Similarly, it is shown by Kao [19] that parallel process systems are efficient only if all their component processes are equally efficient. The author applied DEA to analyze the efficiency of the process stage that causes inefficiency in a system and proposed a method for enhancing system performance.Rostamy-Malkhalifeh et al. [20] peruse a new model based on SBN model with three central, decentral, and mix mechanism evaluating supply chain performance. Rostamy-Malkhalifeh and Esmaeili [21] also suggested a more accurate model to investigate efficiency interval of data. However, some indicators can also be negative.Du et al. [22] provided a new measurement of the performance of a two-stage system on the basis of the concepts of bargaining games. In this regard, Liang et al. [23] developed and investigated a two-stage model using game theory phenomena. They presented a linear model in which overall efficiency is a product of the separate efficiencies of substages. Chen et al. [24] also put forward a new approach to measuring the stability of product design performance via a two-stage network DEA. They used the DEA approach to discover the most eco-efficient way to secure improved environmental performance. They also provided a monotonous conceptual model that works in conjunction with particle swarm optimization in examining crowd movement in computer graphics.Despotis et al. [25] paid attention to the failure of the multiplicative method in which the efficiency estimates obtained was not unique. As a result, they presented a new approach to evaluate unique and unbiased efficiency scores for the separate stages. Then, they expanded and enveloped models to determine their previous multiplier efficiency assessment model was usefully justified. Despotis et al. [26] worked on the supply chains and used network DEA to have a new definition of overall system efficiency based on the “weak-link.” To estimate the individual stage efficiencies and overall system efficiency in two-stage processes. They have used a multiobjective programming framework. The priority of their paper is the estimation of unique and unbiased efficiency scores, and if it is required to operate the efficiency of the actual assessments in line with specific optimization given to the stages.Bernstein and Parmeter [27] studied returns to scale with the findings of two influential studies on returns to scale in the United States electricity generation market. They also compared the main results using both local linear nonparametric regression, a technique robust to parametric functional form assumptions, as well as an updated data set. They showed quantitative findings across all of the estimators that were deployed differ somewhat regarding the magnitude of returns to scale.Peykani and Mohammadi [28] introduced novel robust data envelopment analysis models capacity or potentiality of investigation in the presence of discrete and continuous uncertainties. A year later, in 2019, they presented a new approach, FDEA, for scale efficiency and stock ranking. Put differently, the model was offered to measure the efficiency of stocks when negative data and uncertainties within input/output parameters exist, Peykani et al. [29–31].In a nutshell, most DEA experts have considered the sensitivity of data chaos and returns to scale as main issues in DEA. Accordingly, Ali and Seiford [32], for instance, focused on the “sensitivity of DEA to models and variable sets in a hypothesis test setting.” An ongoing endeavor is Charles et al. [4] inquiry into the sensitivity of a DEA model when changes are introduced to individual outputs. Seiford and Zhu [6] have also extensively analyzed the sensitivity of the efficiency classification in DEA and presented a new strategy for estimating and classifying returns to scale. They also formulated linear programming problems for the sensitivity analysis of returns to scale classification in DEA. Khaleghi et al. [33], in the same way, studied the structure of a two-stage system, proposed returns to scale in DEA network structures, and examined returns to scale and scale elasticity in a two-stage DEA. Research on returns to scale has also delved into the types of returns to scale changes that are based on suitable and equal modifications to all produced elements.Färe et al. [34] determined returns to scale types due to the levels of efficiency and provided a model that classifies the returns to scale of a DMU into increasing, constant, and decreasing returns to scale. Because the returns to scale in DEA are local in nature, research on the stability of such returns is also of a local scope.Forghani and Najafi [35] introduced a new article that this paper examines the combined model for two-stage DEA and considers the sensibility analysis of DMUs on the overall limit. Actually, vital and sufficient conditions to preserve a DMU’s efficiency categories are progressed when different data variations are applied to all DMUs. Allahyar and Rostamy-Malkhalifeh [36] suggested a new method based on solving two LP models. This method is able to evaluate returns to scale to the right and left of the given unit in all conditions. In the same vein, Khodakarami et al. [37] in their article “concurrent estimation of efficiency, effectiveness, and returns to scale” studied the efficiency, effectiveness, and return to scale of DMUs simultaneously. Machado et al. [38] research the hypothesis that economies of scale are a typical feature of the generation market in Brazil used by the cost structure of the electricity generation companies by using a translog cost function in Brazil during the period 2000–2010. Hatami-Marbini et al. [39] have proposed A lexicographic multiobjective linear programming (MOLP) approach to solve the fuzzy models proposed [39]. In this study proposed a novel fully fuzzified DEA (FFDEA) approach where, in addition to input and output data, all the variables are considered fuzzy, including the resulting efficiency scores. Nasseri et al. [49] introduced two virtual fuzzy DMUs, namely fuzzy ideal DMU and fuzzy anti-ideal DMU in the fuzzy DEA framework that they combined the best and worst fuzzy efficiencies in order to find a fuzzy relative closeness of each DMU to FADMU to provide a full ranking of performances for the DMUs.Neralic and Wendell [50] also provided an algorithm approach to sensitivity in DEA for the CCR and additive models that provide sufficient conditions that preserve the efficiency of the input and/or outputs of DMUs. Nastion and et al. [47] prepared an article entitled “sensitivity analysis in data envelopment analysis for interval data” make the efficiency of DEA modeling better and provide a model to measure the upper and lower limits for each DMUs. Due to Chapparo et al., advocates of DEA are often parenthetically proposing the superiority of certain criteria in particular robustness.”Table1 compares the researches that were conducted about sensitivity analysis, scale efficiency, and networking. As can be seen, “sensitivity analysis of returns to scale in two-stage network” is one of the issues that is less discussed in data envelopment analysis. Much research has been conducted on network, returns to scale, or sensitivity analysis based on DEA, but the sensitivity analysis of returns to scale in network and two-stage network has not been performed independently and completely. The current study examined the sensitivity of returns to scale classifications in a two-stage DEA network. The present writers developed an input-oriented model that determines the efficient DMUs in the network and designed a method that identifies the kind of returns to scale in the efficient DMUs. Finally, linear programming problems were formulated for the sensitivity analysis of returns to scale classifications in two-stage network systems.Table 1
Comparative table of studies related to sensitivity analysis, returns to scale, and network.
AuthorsObjectNetwork DEARTSSensitivityanalysisChen [16]A network-DEA model with new efficiency measures to incorporate the dynamic effect in production networks✓Kao [19]A new methodology for evaluating sustainable product design performance with two-stage network data envelopment analysis✓Chen et al. [24]A new methodology for evaluating sustainable product design performance with two-stage network data envelopment analysis✓Chen [16]A network-DEA model with new efficiency measures to incorporate the dynamic effect in production networks✓Jahanshahloo et al. [40]Sensitivity and stability analysis in DEA✓Lewis and Sexton [13]Two-stage DEA: An application to major league baseball✓Lewis and Sexton [14]Network DEA: Efficiency analysis of organizations with complex internal structure✓✓Rostamy-Malkhalifeh et al. [20]A new non-radial network DEA model for evaluating performance supply chain”✓Allahyar and Rostamy-Malkhalifeh [36]An improved approach for estimating returns to scale in DEA✓Khaleghi et al. [33]Returns to scale and scale elasticity in two-stage DEA✓✓Khodabakhshi et al. [41]An additive model approach for estimating returns to scale in imprecise data envelopment analysis✓Khodakarami et al. [37]Concurrent estimation of efficiency, effectiveness and returns to scale✓✓Zarepisheh and Soleimani-damaneh [42]A dual simplex-based method for determination of the right and left returns to scale in DEA✓✓Nasution et al. [43]Sensitivity analysis in data envelopment analysis for interval data✓Hibiki and Sueyoshi [44]DEA sensitivity analysis by changing a reference set: regional contribution to Japanese industrial development”✓Banker and Thrall [45]Estimation of returns to scale using data envelopment analysis✓Avkiran and McCrystal [46]Sensitivity analysis of network DEA: NSBM versus NRAM✓✓Despotis et al. [25]Composition versus decomposition in two-stage network DEA: a Reverse approach, journal of productivity analysis✓Forghani and Najafi [35]Sensitivity analysis in two-stage DEA✓✓Despotis et al. [25]The “weak-link” approach to network DEA for two-stage processesNasution et al. [47]Polak-ribiere updates analysis with binary and linear function in determining coffee exports in Indonesia✓Peykani and Mohammadi [28]Interval network data envelopment analysis model for classification of investment companies in the presence of uncertain data✓Tavassoli et al. [48]Assessing sustainability of suppliers: A novel stochastic-fuzzy DEA model✓Peykani and Mohammadi [30]Window network data envelopment analysis: An application to investment companies”✓Peykani et al. [31]An adjustable fuzzy chance-constrained network DEA approach with application to ranking investment firms✓This paperAlternative method of sensitivity analysis of returns to scale in two-stage network: based on DEA models✓✓✓The rest of the paper is organized as follows: Section2 presents the basic DEA model (i.e., the BCC model) and an introduction to the fundamental concepts of the efficient DMUs in a two-stage network. Section 3 develops DEA model in input oriented for determining the efficient DMUs in a two-stage network, have investigated sensitivity analysis method for the RTS estimation in two-stage networks. Section 4 provides a simple numerical example that explains the sensitivity analysis of returns to scale, and Section 5 concludes the paper.


## 2. Background

In DEA,n observed DMUs with homological m inputs and s outputs are evaluated. Let xij (i = 1, … , m) be the ith input and yrj be the rth output provided to DMUj (j = 1, …, n). In this study, the basic DEA model (i.e., BCC) is used in the following multiplier form:(1)ep=Max∑r=1suryrp+u0∑i=1mvixip,s.t:∑r=1suryrj+u0∑i=1mvixij≤1,j=1,…,n,ur≥ε,r=1,…,s,vi≥ε,i=1,…,m,where ϵ>0 is the non-Archimedean constant, vi and ur are the non-negative variables that represent weights assigned to the ith input and rth output for the unit under assessment, and u0 is a variable with no restrictions on the identification of returns to scale. Here, u0<0 indicates decreasing returns to scal, and u0>0 denotes increasing returns to scale [51].Let us consider the two-stage network system in Figure1, where in Zdp represents the first-stage dth output that is produced with inputs (xip, i = 1, …, m) and the second-stage dth input that is consumed for output production (yrp, r = 1, …, s) and yrj be the jth output provided to DMUj (j = 1, …, n). In this study, the basic DEA model (i.e., BCC) is used in multiplier form.Figure 1
Two-stage network unit.Fare and Grosskopf [11] defined a new production possibility set (PPS) for a network system.The PPS of the two-stage network was used as follows: the PPS in the following equation was employed to prove the theories in this research.(2)PPSn−vx,z,yx≤∑j=1nλjxj&z≥∑j=1nλjzj&z≤∑j=1nμjzj&y≥∑j=1nμyj,λj&μj≥0;j=1,…,n..Kao and Hwang [18] proposed an input-oriented multiplier model to evaluate the efficiency of a two-stage network program. They derived the efficiency score of each stage by using the CCR (Charnes, Cooper, and Rhodes) multiplier model.(3)θp1=Max∑d=1Dwdzdp∑i=1mvixip,s.t:∑d=1Dwdzdj∑i=1mvixij,≤1,j=1,…,n,wd≥ε,d=1,…,D,vi≥ε,i=1,….m,θp2=Max∑r=1suryrp∑d=1Dwdzdp,s.t:∑r=1suryrj∑d=1Dwdzdj≤1,j=1,…,n,ur≥ε,r=1,…,s,wd≥ε,d=1,…,D.As can be seen,θp1≤1andθp2≤1, on this basis and with the help of the efficiency score of each stage, the overall efficiency of the pth unit is measured as follows:(4)θpoveral=Max∑r=1suryrp∑i=1mvixip,s.t:∑d=1Dwdzdj∑i=1mvixij≤1,j=1,…,n,∑r=1suryrj∑d=1Dwdzdj≤1,j=1,…,n,∑r=1suryrj∑i=1mvixij≤1,j=1,…,n,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….m.The efficiency score is between 0 and 1. The dual version of equation (4) is as follows:(5)θp∗=Minθ−1si+1sw+1so,s.t:∑j=1nλj+δjxij+sii=θxipi=1,…,m,∑j=1nδjzdj−swd=0,d=1,…,D,λj≥0,j=1,…,n,where θp∗ represents the overall efficiency score of DMUp in envelopment form. In what follows, we introduce our input-oriented DEA model, which identifies the efficient DMUs in the two-stage network in a variable returns to scale environment, and our method for detecting the type of returns to scale in the network.


## 3. Methodology

### 3.1. Evaluating the Performance of the Two-Stage Network

Let DMUp be the unit under evaluation in the two-stage system. The level of efficiency of each stage in a variable returns to scale environment is obtained through the following models in an input-oriented manner.(6)θp1=Max∑d=1Dwdzdp∑i=1mvixip−v0,s.t:∑d=1Dwdzdj∑i=1mvixij−v0≤1,j=1,…,n,wd≥ε,d=1,…,D,vi≥εi=1,….m,and(7)θp2=Max∑r=1suryrp+u0∑d=1Dwdzdp,s.t:∑r=1suryrj+u0∑d=1Dwdzdj≤1,j=1,…,n,wd≥ε,d=1,…,D,ur≥ε,i=1,….s.As can be seen,θp1≤1andθp2≤1, let(8)θp=θp1×θp2=∑d=1Dwdzdp∑i=1mvixip−v0×∑r=1suryrp+u0∑d=1Dwdzdp,where ∑d=1Dwdzdj>0;j=1…n; so,(9)θp=∑r=1suryrp+u0∑i=1mvixip−v0,where θ is the efficiency score of the two-stage network under variable returns to scale. The following equation is presented to evaluate the efficiency of the network under variable returns to scale. This equation is based on the abovementioned concept.(10)Maxθp=∑r=1suryrp+u0∑i=1mvixip−v0,s.t.∑r=1suryrj+u0∑d=1Dwdzdj≤1,j=1,…,n,∑d=1Dwdzdj∑i=1mvixij−v0≤1,j=1,…,n,∑r=1suryrj+u0∑i=1mvixij−v0≤1,j=1,…,n,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.Model (9) is a linear fractional programming problem that can be easily converted into a linear format through the method proposed by Charnes and Cooper [52]. The specific conversion proceeds as follows: ∑i=1mvixip−v0−1=t, and let tvi=vi, tur=ur, twd=wd, tu0=u0, and tv0=v0. Hence, model (9) can be expressed in the following form:(11)Maxθp=∑r=1suryrp+u0,s.t.∑i=1mvixip−v0=1,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s.where ∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.) are redundant constraints. Therefore, model (10) can be written in a more convenient form as follows:(12)Maxθp=∑r=1suryrp+u0,s.t.∑i=1mvixip−v0=1,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n,∑d=1Dwdzdj−∑i=1mvixij+v0≤0j=1,…,n,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.The dual version of model (10) is as follows(13)Minθ−1si+1sw+1so,s.t.∑j=1nλj+δjxij+si=θxip,i=1,…,m,∑j=1nδjzdj−sw=0,d=1,…,D,∑j=1nλjyrj−so=yrp,r=1,…,s,∑j=1nλj=1,∑j=1nλj+δj=θ,λj,μj≥0,j=1,…,n.Theorem 1.
In stage 1 and 2, DMUp is efficient, if DMUp is entirely efficient under model (11).Proof.
The proof is obvious.


### 3.2. Classifications of Returns to Scale

Let us suppose that there arem two-stage networks, and we want to estimate the nature of returns to scale.Definition 1.
Suppose that thepth network is located on the frontier of Tn−v.(i)
Thepth network has increasing returns to scale if and only if δ∗>0 so that(14)∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈intTn−v.(ii)
Thepth network has decreasing returns to scale if and only if δ∗>0 so that(15)∀δ;0<δ<δ∗⟶∀δ;0<δ<δ∗⟶1−δxp,1−δzp,1−δyp∈intTn−v.(iii)
Thepth network has constant returns to scale if and only if δ∗>0 so that(16)a∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1−δz,1−δyp∈∂Tn−v,b∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δz,1−δyp∉Tn−v,c∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1+δz,1−δyp∉Tn−v,ord∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v.Theorem 2.
Suppose that DMUp is efficient under model (10), and v∗,w∗,u∗,v0∗,u0∗ is an arbitrary optimal solution to model (10). Then,(i)
If in any optimal solution,u0∗+v0∗>0, then DMUp has increasing returns to scale(ii)
If in any optimal solution,u0∗+v0∗<0, then DMUp has decreasing returns to scale(iii)
If in some optimal solutions,u0∗+v0∗=0, then DMUp has constant returns to scale.Proof.
Case (i): Let us assume thatv∗,w∗,u∗,v0∗,u0∗ is an optimal solution from model (10) for assessing DMUp. Because DMUp is efficient,(17)u∗yp−v∗xp+u0∗+v0∗=0,u∗yp−w∗zp+u0∗=0,w∗zp−v∗xp+v0∗=0.We define the following model:(18)P1=u∗1+δyp−v∗1+δxp+u0∗+v0∗,P2=u∗1+δyp−w∗1+δzp+u0∗,P3=w∗1+δzp−v∗1+δxp+v0∗.Thus,(19)P1=1+δu∗yp−v∗xp+u0∗+v0∗−u0∗+v0∗δ=0−u0∗+v0∗δ<0.Ifu0∗+v0∗>0, then 1+δyp,1+δzp,1+δxp∈intTn−v. According to the definition, therefore, DMUp has increasing returns to scale.Case (ii): This case can be proved in a similar manner as that for case (i).Case (iii): Let us suppose thatv∗,w∗,u∗,v0∗,u0∗ is an optimal solution from model for assessing DMUp. Therefore, u∗yp−v∗xp+u0∗+v0∗=0 so that u0∗+v0∗=0. In this case, we have the following model:(20)u∗yp−v∗xp+u0∗+v0∗+δu∗yp−v∗xp+u0∗+v0∗−δu0∗+v0∗=0,u∗yp−v∗xp+u0∗+v0∗−δu∗yp−v∗xp+u0∗+v0∗+δu0∗+v0∗=0,
So that(21)u∗1+δyp−v∗1+δxp+u0∗+v0∗=0,u∗1−δyp−v∗1−δxp+u0∗+v0∗=0.
Thus,(22)u∗1+δyp+01+δzp−v∗1+δxp+u0∗+v0∗=0,u∗1−δyp+01−δzp−v∗1−δxp+u0∗+v0∗=0.
Ergo, pointsxp,zp,yp, 1+δxp,1+δzp,1+δyp, and 1−δxp,1−δzp,1−δyp are on hyperplane u∗yp−v∗xp+u0∗+v0∗=0, and(23)xp,zp,yp=121+δxp,1+δzp,1+δyp+121−δxp,1−δzp,1−δyp.
That is,xp,zp,yp is on the segment connecting points 1+δxp,1+δzp,1+δyp and 1−δxp,1−δzp,1−δyp. Thereupon, these points are characterized by the following conditions for Tn−v:(a)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v,(b)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δz,1−δyp∉Tn−v,(c)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1+δzp,1−δyp∉Tn−v, or(d)
δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v.
This completes the proof. A similar proof can be put forward.


### 3.3. Sensitivity Analysis for Classifications of Returns to Scale

We first pointed out that increasing or decreasing outputs in a specific DMU can alter returns to scale classifications. If DMUp exhibits either increasing or decreasing returns to scale, an increase in output can change its returns to scale.For sensitivity analysis, we partitioned the set of efficient DMUs into the following classes.(24)E=jDMUjis.efficiont,E1=jj∈Eandin.all.optimal.solutionu0+v0>0,E2=jj∈Ein.all.optimal.solutionu0+v0<0,E3=jj∈E,there.exist.an.optimalsolution.in.whichu0+v0=0,


#### 3.3.1.p∈E1

Let us suppose that DMUp (xp, zp, yp) has increasing returns to scale and zp is the first-stage output produced by using xp as input. In the second stage, it is used as an input to produce yp. We are interested in the sensitivity analysis of the classification of returns to scale in DMUp. For DMUp, therefore, we have the following perturbed data:(25)xp=xp+α;0≤α≤minjminixij−xip,yp=yp+β;0≤β≤minjminryrj−yrp.If DMUp has increasing returns to scale, according to the concept above, a model for sensitivity analysis is presented as follows:(26)Maxα+β,s.t:∑i=1mvi+∑r=1sur=1,∑r=1suryrp+β−∑i=1mvixip+α+u0+v0=θp∗−1,∑r=1suryrp+β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdp+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjminryrj−yrpvi≥εi=1,….m,wd≥εd=1,…,D,ur≥εr=1,….s,where ϵ>0 is the non-Archimedean constant and θp∗ is an optimal solution obtained from model (13) Model (26) is a nonlinear programming equation. By using restriction ∑i=1mvi+∑r=1sur=1, model (26) is reduced to the following form:(27)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip+1−∑i=1mviβ−1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp+1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(28)Maxα+β,s.t.,(28a)∑r=1suryrp−∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα+u0+v0≤θp∗−1,(28b)−∑r=1suryrp+∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα−u0−v0≤1−θp∗,(29)∑r=1suryrp−∑d=1Dwdzdp+β−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrp,0≤β≤minjmaxryrj−yrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minjmaxryrj−yrp so that 0≤viβ≤vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With regard to these conditions, the following model is offered:(30)Maxα+β,s.t.(30a)∑r=1suryrp−∑i=1mvixip+β+∑r=1surkp′−α+u0+v0≤θp∗−1,(30b)−∑r=1suryrp+∑i=1mvixip−β+α+∑i=1mvikp−u0−v0≤1−θp∗,(31)∑r=1suryrp−∑d=1Dwdzdp+β+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surkp′+v0≤0∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε.0≤α≤kp′,0≤β≤kpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.Theorem 4.
The optimal solution obtained from model (28) is equal to the optimal solution derived from model (30).Proof.
Let us assume thatu∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is an obtained optimal solution from model (30). As can be seen, u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for model (28). According to constraint (28b) of model (28), then, we have the following model:(32)α∗≤1−θp∗+r=1∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗I.
Suppose thatu^,w^,v^,u^0,v^0,α^,β^ is an optimal solution from model (28). If α^≤α∗, then there is nothing to prove, and α∗ is best solution for model (28). Otherwise, α^>α∗. We adopt α^=α∗+δ so that δ>0. According to (I), we have the following model:(33)α^−δ≤1−θp∗+∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗II.
As shown,u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for models (30) and (30a). These yields the following model:(34)−α∗≤−∑r=1sur∗yrp+∑i=1mvi∗xip−β∗−∑r=1sur∗kp′−u0∗−v0∗+1−θp∗.
We indicate the sum of (II) and (III) as follows:(35)α^−α∗≤−∑i=1mvr∗β+∑r=1sur∗α−∑r=1sur∗kp′+δ<δ.
Thus,α^<α∗+δ=α^. This is a contradiction. The same can be proved for β^<β∗+δ=β^. Hereupon, the α∗ and β∗ obtained from model (30) are the best solutions for model (28).


#### 3.3.2.p∈E2

Let us assume thatp∈E2; therefore, DMUp has decreasing returns to scale. In this case, decreasing outputs cannot change the unit’s returns to scale. Assessing the sensitivity of returns to scale classifications therefore necessitates the following disturbed data:(36)xp=xp−α,yp=yp−β.Given the confusion in the inputs and outputs of DMUp, we put forward the following model for identifying the best interval at which DMUp continues to earn decreasing returns to scale.(37)Maxα+βs.t.∑i=1mvi+∑r=1sur=1,∑r=1suryrp−β−∑i=1mvixip−α+u0+v0=θp∗−1,∑r=1suryrp−β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥εi=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.In the equation above,θp∗ is the optimal solution obtained from model (10). Model (37) is a nonlinear programming equation. Using restriction ∑i=1mvi+∑r=1sur=1 reduces model (37) to the following form:(38)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−1−∑i=1mviβ+1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp−1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(39)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα+u0+v0≤θp∗−1,−∑r=1suryrp+∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0≤−ε.0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minryrp so that 0<viβ<vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With respect to these conditions, we offer the following model:(40)Maxα+β,s.t:∑r=1suryrp−∑i=1mvixip−β+∑i=1mvikp+α+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β−α+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mvikp+u0≤0∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ϵ.0≤α≤kp′0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵd=1,…,D,ur≥ϵ,r=1,….s.Theorem 5.
The optimal solution obtained from model (40) is equal to the number of the model (39).Proof.
The proof is analogous to that for Theorem4.


#### 3.3.3.p∈E3

If DMUp has constant returns to scale, u0∗,v0∗ is an optimal solution under model (10) so that u0∗+v0∗=0. Increasing or decreasing outputs can change classifications of constant returns to scale. Therefore, we used the two models below to determine that for α and β, α is the amount of disturbance input when outputs change into β.(41)Maxα1+β1,s.t:∑r=1suryrp−∑i=1mvixip+β1+∑r=1surkp′−α1+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip−β1+α1+∑i=1mvikp−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp+β1+u0≤0∑d=1Dwdzdp−∑i=1mvixip−α1+∑r=1surkp′+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0=0,0≤α1≤kp′,0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s,and(42)Maxα2+β2,s.t:∑r=1suryrp−∑i=1mvixip−β2+∑i=1mvikp+α2+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β2−α2+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β2+∑i=1mvikp+u0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0=0,0≤α≤fp′,0≤β≤fp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s.thus,(43)kp=minjminryrj−yrp,kp′=minjmaxixij−xip.fp=minryrpfp′=minjminixij−xip.By choosingα∗=minα1.α2 and β∗=minβ1.β2, increasing and decreasing inputs per α; 0≤α≤α∗ and increasing and decreasing outputs per β; 0≤β≤β∗ cannot change the type of constant returns to scale in the evaluated unit.


## 3.1. Evaluating the Performance of the Two-Stage Network

Let DMUp be the unit under evaluation in the two-stage system. The level of efficiency of each stage in a variable returns to scale environment is obtained through the following models in an input-oriented manner.(6)θp1=Max∑d=1Dwdzdp∑i=1mvixip−v0,s.t:∑d=1Dwdzdj∑i=1mvixij−v0≤1,j=1,…,n,wd≥ε,d=1,…,D,vi≥εi=1,….m,and(7)θp2=Max∑r=1suryrp+u0∑d=1Dwdzdp,s.t:∑r=1suryrj+u0∑d=1Dwdzdj≤1,j=1,…,n,wd≥ε,d=1,…,D,ur≥ε,i=1,….s.As can be seen,θp1≤1andθp2≤1, let(8)θp=θp1×θp2=∑d=1Dwdzdp∑i=1mvixip−v0×∑r=1suryrp+u0∑d=1Dwdzdp,where ∑d=1Dwdzdj>0;j=1…n; so,(9)θp=∑r=1suryrp+u0∑i=1mvixip−v0,where θ is the efficiency score of the two-stage network under variable returns to scale. The following equation is presented to evaluate the efficiency of the network under variable returns to scale. This equation is based on the abovementioned concept.(10)Maxθp=∑r=1suryrp+u0∑i=1mvixip−v0,s.t.∑r=1suryrj+u0∑d=1Dwdzdj≤1,j=1,…,n,∑d=1Dwdzdj∑i=1mvixij−v0≤1,j=1,…,n,∑r=1suryrj+u0∑i=1mvixij−v0≤1,j=1,…,n,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.Model (9) is a linear fractional programming problem that can be easily converted into a linear format through the method proposed by Charnes and Cooper [52]. The specific conversion proceeds as follows: ∑i=1mvixip−v0−1=t, and let tvi=vi, tur=ur, twd=wd, tu0=u0, and tv0=v0. Hence, model (9) can be expressed in the following form:(11)Maxθp=∑r=1suryrp+u0,s.t.∑i=1mvixip−v0=1,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s.where ∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.) are redundant constraints. Therefore, model (10) can be written in a more convenient form as follows:(12)Maxθp=∑r=1suryrp+u0,s.t.∑i=1mvixip−v0=1,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n,∑d=1Dwdzdj−∑i=1mvixij+v0≤0j=1,…,n,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.The dual version of model (10) is as follows(13)Minθ−1si+1sw+1so,s.t.∑j=1nλj+δjxij+si=θxip,i=1,…,m,∑j=1nδjzdj−sw=0,d=1,…,D,∑j=1nλjyrj−so=yrp,r=1,…,s,∑j=1nλj=1,∑j=1nλj+δj=θ,λj,μj≥0,j=1,…,n.Theorem 1.
In stage 1 and 2, DMUp is efficient, if DMUp is entirely efficient under model (11).Proof.
The proof is obvious.


## 3.2. Classifications of Returns to Scale

Let us suppose that there arem two-stage networks, and we want to estimate the nature of returns to scale.Definition 1.
Suppose that thepth network is located on the frontier of Tn−v.(i)
Thepth network has increasing returns to scale if and only if δ∗>0 so that(14)∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈intTn−v.(ii)
Thepth network has decreasing returns to scale if and only if δ∗>0 so that(15)∀δ;0<δ<δ∗⟶∀δ;0<δ<δ∗⟶1−δxp,1−δzp,1−δyp∈intTn−v.(iii)
Thepth network has constant returns to scale if and only if δ∗>0 so that(16)a∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1−δz,1−δyp∈∂Tn−v,b∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δz,1−δyp∉Tn−v,c∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1+δz,1−δyp∉Tn−v,ord∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v.Theorem 2.
Suppose that DMUp is efficient under model (10), and v∗,w∗,u∗,v0∗,u0∗ is an arbitrary optimal solution to model (10). Then,(i)
If in any optimal solution,u0∗+v0∗>0, then DMUp has increasing returns to scale(ii)
If in any optimal solution,u0∗+v0∗<0, then DMUp has decreasing returns to scale(iii)
If in some optimal solutions,u0∗+v0∗=0, then DMUp has constant returns to scale.Proof.
Case (i): Let us assume thatv∗,w∗,u∗,v0∗,u0∗ is an optimal solution from model (10) for assessing DMUp. Because DMUp is efficient,(17)u∗yp−v∗xp+u0∗+v0∗=0,u∗yp−w∗zp+u0∗=0,w∗zp−v∗xp+v0∗=0.We define the following model:(18)P1=u∗1+δyp−v∗1+δxp+u0∗+v0∗,P2=u∗1+δyp−w∗1+δzp+u0∗,P3=w∗1+δzp−v∗1+δxp+v0∗.Thus,(19)P1=1+δu∗yp−v∗xp+u0∗+v0∗−u0∗+v0∗δ=0−u0∗+v0∗δ<0.Ifu0∗+v0∗>0, then 1+δyp,1+δzp,1+δxp∈intTn−v. According to the definition, therefore, DMUp has increasing returns to scale.Case (ii): This case can be proved in a similar manner as that for case (i).Case (iii): Let us suppose thatv∗,w∗,u∗,v0∗,u0∗ is an optimal solution from model for assessing DMUp. Therefore, u∗yp−v∗xp+u0∗+v0∗=0 so that u0∗+v0∗=0. In this case, we have the following model:(20)u∗yp−v∗xp+u0∗+v0∗+δu∗yp−v∗xp+u0∗+v0∗−δu0∗+v0∗=0,u∗yp−v∗xp+u0∗+v0∗−δu∗yp−v∗xp+u0∗+v0∗+δu0∗+v0∗=0,
So that(21)u∗1+δyp−v∗1+δxp+u0∗+v0∗=0,u∗1−δyp−v∗1−δxp+u0∗+v0∗=0.
Thus,(22)u∗1+δyp+01+δzp−v∗1+δxp+u0∗+v0∗=0,u∗1−δyp+01−δzp−v∗1−δxp+u0∗+v0∗=0.
Ergo, pointsxp,zp,yp, 1+δxp,1+δzp,1+δyp, and 1−δxp,1−δzp,1−δyp are on hyperplane u∗yp−v∗xp+u0∗+v0∗=0, and(23)xp,zp,yp=121+δxp,1+δzp,1+δyp+121−δxp,1−δzp,1−δyp.
That is,xp,zp,yp is on the segment connecting points 1+δxp,1+δzp,1+δyp and 1−δxp,1−δzp,1−δyp. Thereupon, these points are characterized by the following conditions for Tn−v:(a)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v,(b)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δz,1−δyp∉Tn−v,(c)
∀δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∈∂Tn−v&1−δxp,1+δzp,1−δyp∉Tn−v, or(d)
δ;0<δ<δ∗⟶1+δxp,1+δzp,1+δyp∉Tn−v&1−δxp,1−δzp,1−δyp∈∂Tn−v.
This completes the proof. A similar proof can be put forward.


## 3.3. Sensitivity Analysis for Classifications of Returns to Scale

We first pointed out that increasing or decreasing outputs in a specific DMU can alter returns to scale classifications. If DMUp exhibits either increasing or decreasing returns to scale, an increase in output can change its returns to scale.For sensitivity analysis, we partitioned the set of efficient DMUs into the following classes.(24)E=jDMUjis.efficiont,E1=jj∈Eandin.all.optimal.solutionu0+v0>0,E2=jj∈Ein.all.optimal.solutionu0+v0<0,E3=jj∈E,there.exist.an.optimalsolution.in.whichu0+v0=0,


### 3.3.1.p∈E1

Let us suppose that DMUp (xp, zp, yp) has increasing returns to scale and zp is the first-stage output produced by using xp as input. In the second stage, it is used as an input to produce yp. We are interested in the sensitivity analysis of the classification of returns to scale in DMUp. For DMUp, therefore, we have the following perturbed data:(25)xp=xp+α;0≤α≤minjminixij−xip,yp=yp+β;0≤β≤minjminryrj−yrp.If DMUp has increasing returns to scale, according to the concept above, a model for sensitivity analysis is presented as follows:(26)Maxα+β,s.t:∑i=1mvi+∑r=1sur=1,∑r=1suryrp+β−∑i=1mvixip+α+u0+v0=θp∗−1,∑r=1suryrp+β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdp+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjminryrj−yrpvi≥εi=1,….m,wd≥εd=1,…,D,ur≥εr=1,….s,where ϵ>0 is the non-Archimedean constant and θp∗ is an optimal solution obtained from model (13) Model (26) is a nonlinear programming equation. By using restriction ∑i=1mvi+∑r=1sur=1, model (26) is reduced to the following form:(27)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip+1−∑i=1mviβ−1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp+1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(28)Maxα+β,s.t.,(28a)∑r=1suryrp−∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα+u0+v0≤θp∗−1,(28b)−∑r=1suryrp+∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα−u0−v0≤1−θp∗,(29)∑r=1suryrp−∑d=1Dwdzdp+β−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrp,0≤β≤minjmaxryrj−yrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minjmaxryrj−yrp so that 0≤viβ≤vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With regard to these conditions, the following model is offered:(30)Maxα+β,s.t.(30a)∑r=1suryrp−∑i=1mvixip+β+∑r=1surkp′−α+u0+v0≤θp∗−1,(30b)−∑r=1suryrp+∑i=1mvixip−β+α+∑i=1mvikp−u0−v0≤1−θp∗,(31)∑r=1suryrp−∑d=1Dwdzdp+β+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surkp′+v0≤0∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε.0≤α≤kp′,0≤β≤kpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.Theorem 4.
The optimal solution obtained from model (28) is equal to the optimal solution derived from model (30).Proof.
Let us assume thatu∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is an obtained optimal solution from model (30). As can be seen, u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for model (28). According to constraint (28b) of model (28), then, we have the following model:(32)α∗≤1−θp∗+r=1∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗I.
Suppose thatu^,w^,v^,u^0,v^0,α^,β^ is an optimal solution from model (28). If α^≤α∗, then there is nothing to prove, and α∗ is best solution for model (28). Otherwise, α^>α∗. We adopt α^=α∗+δ so that δ>0. According to (I), we have the following model:(33)α^−δ≤1−θp∗+∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗II.
As shown,u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for models (30) and (30a). These yields the following model:(34)−α∗≤−∑r=1sur∗yrp+∑i=1mvi∗xip−β∗−∑r=1sur∗kp′−u0∗−v0∗+1−θp∗.
We indicate the sum of (II) and (III) as follows:(35)α^−α∗≤−∑i=1mvr∗β+∑r=1sur∗α−∑r=1sur∗kp′+δ<δ.
Thus,α^<α∗+δ=α^. This is a contradiction. The same can be proved for β^<β∗+δ=β^. Hereupon, the α∗ and β∗ obtained from model (30) are the best solutions for model (28).


### 3.3.2.p∈E2

Let us assume thatp∈E2; therefore, DMUp has decreasing returns to scale. In this case, decreasing outputs cannot change the unit’s returns to scale. Assessing the sensitivity of returns to scale classifications therefore necessitates the following disturbed data:(36)xp=xp−α,yp=yp−β.Given the confusion in the inputs and outputs of DMUp, we put forward the following model for identifying the best interval at which DMUp continues to earn decreasing returns to scale.(37)Maxα+βs.t.∑i=1mvi+∑r=1sur=1,∑r=1suryrp−β−∑i=1mvixip−α+u0+v0=θp∗−1,∑r=1suryrp−β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥εi=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.In the equation above,θp∗ is the optimal solution obtained from model (10). Model (37) is a nonlinear programming equation. Using restriction ∑i=1mvi+∑r=1sur=1 reduces model (37) to the following form:(38)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−1−∑i=1mviβ+1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp−1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(39)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα+u0+v0≤θp∗−1,−∑r=1suryrp+∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0≤−ε.0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minryrp so that 0<viβ<vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With respect to these conditions, we offer the following model:(40)Maxα+β,s.t:∑r=1suryrp−∑i=1mvixip−β+∑i=1mvikp+α+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β−α+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mvikp+u0≤0∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ϵ.0≤α≤kp′0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵd=1,…,D,ur≥ϵ,r=1,….s.Theorem 5.
The optimal solution obtained from model (40) is equal to the number of the model (39).Proof.
The proof is analogous to that for Theorem4.


### 3.3.3.p∈E3

If DMUp has constant returns to scale, u0∗,v0∗ is an optimal solution under model (10) so that u0∗+v0∗=0. Increasing or decreasing outputs can change classifications of constant returns to scale. Therefore, we used the two models below to determine that for α and β, α is the amount of disturbance input when outputs change into β.(41)Maxα1+β1,s.t:∑r=1suryrp−∑i=1mvixip+β1+∑r=1surkp′−α1+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip−β1+α1+∑i=1mvikp−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp+β1+u0≤0∑d=1Dwdzdp−∑i=1mvixip−α1+∑r=1surkp′+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0=0,0≤α1≤kp′,0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s,and(42)Maxα2+β2,s.t:∑r=1suryrp−∑i=1mvixip−β2+∑i=1mvikp+α2+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β2−α2+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β2+∑i=1mvikp+u0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0=0,0≤α≤fp′,0≤β≤fp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s.thus,(43)kp=minjminryrj−yrp,kp′=minjmaxixij−xip.fp=minryrpfp′=minjminixij−xip.By choosingα∗=minα1.α2 and β∗=minβ1.β2, increasing and decreasing inputs per α; 0≤α≤α∗ and increasing and decreasing outputs per β; 0≤β≤β∗ cannot change the type of constant returns to scale in the evaluated unit.


## 3.3.1.p∈E1

Let us suppose that DMUp (xp, zp, yp) has increasing returns to scale and zp is the first-stage output produced by using xp as input. In the second stage, it is used as an input to produce yp. We are interested in the sensitivity analysis of the classification of returns to scale in DMUp. For DMUp, therefore, we have the following perturbed data:(25)xp=xp+α;0≤α≤minjminixij−xip,yp=yp+β;0≤β≤minjminryrj−yrp.If DMUp has increasing returns to scale, according to the concept above, a model for sensitivity analysis is presented as follows:(26)Maxα+β,s.t:∑i=1mvi+∑r=1sur=1,∑r=1suryrp+β−∑i=1mvixip+α+u0+v0=θp∗−1,∑r=1suryrp+β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdp+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjminryrj−yrpvi≥εi=1,….m,wd≥εd=1,…,D,ur≥εr=1,….s,where ϵ>0 is the non-Archimedean constant and θp∗ is an optimal solution obtained from model (13) Model (26) is a nonlinear programming equation. By using restriction ∑i=1mvi+∑r=1sur=1, model (26) is reduced to the following form:(27)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip+1−∑i=1mviβ−1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp+1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(28)Maxα+β,s.t.,(28a)∑r=1suryrp−∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα+u0+v0≤θp∗−1,(28b)−∑r=1suryrp+∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα−u0−v0≤1−θp∗,(29)∑r=1suryrp−∑d=1Dwdzdp+β−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε,0≤α≤minjmaxixij−xip,0≤β≤minjmaxryrj−yrp,0≤β≤minjmaxryrj−yrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minjmaxryrj−yrp so that 0≤viβ≤vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With regard to these conditions, the following model is offered:(30)Maxα+β,s.t.(30a)∑r=1suryrp−∑i=1mvixip+β+∑r=1surkp′−α+u0+v0≤θp∗−1,(30b)−∑r=1suryrp+∑i=1mvixip−β+α+∑i=1mvikp−u0−v0≤1−θp∗,(31)∑r=1suryrp−∑d=1Dwdzdp+β+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+∑r=1surkp′+v0≤0∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠pu0+v0≥ε.0≤α≤kp′,0≤β≤kpvi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.Theorem 4.
The optimal solution obtained from model (28) is equal to the optimal solution derived from model (30).Proof.
Let us assume thatu∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is an obtained optimal solution from model (30). As can be seen, u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for model (28). According to constraint (28b) of model (28), then, we have the following model:(32)α∗≤1−θp∗+r=1∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗I.
Suppose thatu^,w^,v^,u^0,v^0,α^,β^ is an optimal solution from model (28). If α^≤α∗, then there is nothing to prove, and α∗ is best solution for model (28). Otherwise, α^>α∗. We adopt α^=α∗+δ so that δ>0. According to (I), we have the following model:(33)α^−δ≤1−θp∗+∑r=1sur∗yrp−∑i=1mvi∗xip+β∗−∑i=1mvi∗β+∑r=1sur∗α+u0∗+v0∗II.
As shown,u∗,w∗,v∗,u0∗,v0∗,α∗,β∗ is a feasible solution for models (30) and (30a). These yields the following model:(34)−α∗≤−∑r=1sur∗yrp+∑i=1mvi∗xip−β∗−∑r=1sur∗kp′−u0∗−v0∗+1−θp∗.
We indicate the sum of (II) and (III) as follows:(35)α^−α∗≤−∑i=1mvr∗β+∑r=1sur∗α−∑r=1sur∗kp′+δ<δ.
Thus,α^<α∗+δ=α^. This is a contradiction. The same can be proved for β^<β∗+δ=β^. Hereupon, the α∗ and β∗ obtained from model (30) are the best solutions for model (28).


## 3.3.2.p∈E2

Let us assume thatp∈E2; therefore, DMUp has decreasing returns to scale. In this case, decreasing outputs cannot change the unit’s returns to scale. Assessing the sensitivity of returns to scale classifications therefore necessitates the following disturbed data:(36)xp=xp−α,yp=yp−β.Given the confusion in the inputs and outputs of DMUp, we put forward the following model for identifying the best interval at which DMUp continues to earn decreasing returns to scale.(37)Maxα+βs.t.∑i=1mvi+∑r=1sur=1,∑r=1suryrp−β−∑i=1mvixip−α+u0+v0=θp∗−1,∑r=1suryrp−β−∑d=1Dwdzdp+u0≤0,∑d=1Dwdzdp−∑i=1mvixip−α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥εi=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.In the equation above,θp∗ is the optimal solution obtained from model (10). Model (37) is a nonlinear programming equation. Using restriction ∑i=1mvi+∑r=1sur=1 reduces model (37) to the following form:(38)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−1−∑i=1mviβ+1−∑r=1surα+u0+v0=θp∗−1,∑r=1suryrp−∑d=1Dwdzdp−1−∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+1−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ε,0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥ε,r=1,….s.thus,(39)Maxα+β,s.t.∑r=1suryrp−∑i=1mvixip−β+∑i=1mviβ+α−∑r=1surα+u0+v0≤θp∗−1,−∑r=1suryrp+∑i=1mvixip+β−∑i=1mviβ−α+∑r=1surα−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mviβ+u0≤0,∑d=1Dwdzdp−∑i=1mvixip+α−∑r=1surα+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0≤−ε.0≤α≤minjmaxixij−xip,0≤β≤minryrp,vi≥ε,i=1,….m,wd≥ε,d=1,…,D,ur≥εr=1,….s.Letkp′=minjmaxixij−xip and kp=minryrp so that 0<viβ<vikp (for all i). Hence, 0≤∑i=1mviβ≤∑i=1mvikp and 0≤∑r=1surα≤∑r=1surkp′. With respect to these conditions, we offer the following model:(40)Maxα+β,s.t:∑r=1suryrp−∑i=1mvixip−β+∑i=1mvikp+α+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β−α+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β+∑i=1mvikp+u0≤0∑d=1Dwdzdp−∑i=1mvixip+α+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0≤−ϵ.0≤α≤kp′0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵd=1,…,D,ur≥ϵ,r=1,….s.Theorem 5.
The optimal solution obtained from model (40) is equal to the number of the model (39).Proof.
The proof is analogous to that for Theorem4.


## 3.3.3.p∈E3

If DMUp has constant returns to scale, u0∗,v0∗ is an optimal solution under model (10) so that u0∗+v0∗=0. Increasing or decreasing outputs can change classifications of constant returns to scale. Therefore, we used the two models below to determine that for α and β, α is the amount of disturbance input when outputs change into β.(41)Maxα1+β1,s.t:∑r=1suryrp−∑i=1mvixip+β1+∑r=1surkp′−α1+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip−β1+α1+∑i=1mvikp−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp+β1+u0≤0∑d=1Dwdzdp−∑i=1mvixip−α1+∑r=1surkp′+v0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0,j=1,…,n.&j≠p,u0+v0=0,0≤α1≤kp′,0≤β≤kp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s,and(42)Maxα2+β2,s.t:∑r=1suryrp−∑i=1mvixip−β2+∑i=1mvikp+α2+u0+v0≤θp∗−1−∑r=1suryrp+∑i=1mvixip+β2−α2+∑r=1surkp′−u0−v0≤1−θp∗,∑r=1suryrp−∑d=1Dwdzdp−β2+∑i=1mvikp+u0≤0,∑r=1suryrj−∑d=1Dwdzdj+u0≤0,j=1,…,n.&j≠p,∑d=1Dwdzdj−∑i=1mvixij+v0≤0,j=1,…,n.&j≠p,∑r=1suryrj−∑i=1mvixij+u0+v0≤0j=1,…,n.&j≠p,u0+v0=0,0≤α≤fp′,0≤β≤fp,vi≥ϵ,i=1,….m,wd≥ϵ,d=1,…,D,ur≥ϵ,r=1,….s.thus,(43)kp=minjminryrj−yrp,kp′=minjmaxixij−xip.fp=minryrpfp′=minjminixij−xip.By choosingα∗=minα1.α2 and β∗=minβ1.β2, increasing and decreasing inputs per α; 0≤α≤α∗ and increasing and decreasing outputs per β; 0≤β≤β∗ cannot change the type of constant returns to scale in the evaluated unit.


## 4. Application

### 4.1. Numerical Example

We prepared a simple numerical example that explains the sensitivity analysis for the returns to scale classifications derived in Section3. Table 2 shows a set of synthetic data for 10 two-stage DMUs with a single input, a single intermediate value, and a single final output. Note that the number of internal z is an output of the first stage and an input for the second stage.Table 2
Synthetic data for 10 two-stage DMUs.
DMU12345678910Input for stage 1 (x)2.545.58116.59452Intermediate measure (z)3.55.56.5864.585.542Final output (y)7105.513978961Among the DMUs evaluated in the first stage, DMU1, DMU2, DMU3, DMU4, and DMU10 are efficient under model (6); in the second stage, DMU1, DMU2, DMU4, DMU8, and DMU10 are efficient under model (7) (See Table 3, columns 2 and 3). The fourth column of Table 3 presents the scores of efficiencies of the two-stage network under model (10). The units that are efficient in both stages are also efficient under the overall two-stage network model (i.e., model (10)), that is, DMU1, DMU2, DMU4, and DMU10 are efficient under model (10) and E=1,2,4,10 is a set of index-efficient DMUs. The fifth and sixth columns of Table 3 show the min and max u0∗+v0∗, which indicate that (u0∗, v0∗) is the optimal solution obtained from model (10) for the evaluation of the DMUs’ efficiency scores. According to Theorem 2, DMU1 has constant returns to scale, DMU10 has increasing returns to scale, and DMU2 and DMU4 have decreasing returns to scale. Thus, the set of efficient DMUs may be partitioned into classes E1 = {10}, E2 = {2,4}, and E3 = {1} on the basis of the classification in Section 3.Table 3
Table of result.
θp1∗ (6)θp2∗ (7)θpoveral∗ (10)Maxu0∗+v0∗ (10)Minu0∗+v0∗ (10)DMU11.00001.00001.00001.6429−0.3810DMU21.00001.00001.0000−0.2420−0.0001DMU31.00000.48080.4808−0.1670−0.2230DMU41.00001.00001.0000−0.75402.0000DMU50.75000.80560.6042−0.0001−0.0001DMU60.63380.77780.4930−0.2040−0.6150DMU70.96000.50000.5208−0.0001−0.0001DMU80.36361.00000.8788−0.2420−0.7270DMU90.37500.75000.52700.17600.1760DMU101.00001.00001.00002.87502.8750DMU10 has constant returns to scale; thus, increasing output can change the type of returns in this unit. To identify the perfect enhancement of outputs so that DMU10 maintains increasing returns to scale, we used model (30), wherein α∗=0.17,β∗=6 is an optimal solution for assessing DMU10.To analyze the sensitivity of the returns to scale classifications of DMU1, models (40) and (41) should be used. Because DMU1 has constant returns to scale and because increasing and decreasing outputs can change the types of returns in the unit, α1∗=1 and β1∗=1 were used as the optimal solution from model (40), and α2∗=0.29 and β2∗=3.5 served as the best solution from model (41). Thus, increasing or decreasing inputs in each α; 0≤α≤min0.26,1 and increasing or decreasing outputs in each β; 0≤β≤min1,3.5 cannot change the type of constant returns to scale in DMU10.DMU2 and DMU4 have decreasing returns to scale. Accordingly, model (40) was used for the sensitivity analysis of returns to scale classifications. The analysis yielded α∗=1.8 and β∗=2.25 as the best input and output reduction values, respectively, for DMU2 to maintain decreasing returns to scale. Additionally, α∗=2.67 and β∗=1.67 are optimal solutions from model (40) in the sensitivity analysis of the returns to scale classifications in DMU4.


## 4.1. Numerical Example

We prepared a simple numerical example that explains the sensitivity analysis for the returns to scale classifications derived in Section3. Table 2 shows a set of synthetic data for 10 two-stage DMUs with a single input, a single intermediate value, and a single final output. Note that the number of internal z is an output of the first stage and an input for the second stage.Table 2
Synthetic data for 10 two-stage DMUs.
DMU12345678910Input for stage 1 (x)2.545.58116.59452Intermediate measure (z)3.55.56.5864.585.542Final output (y)7105.513978961Among the DMUs evaluated in the first stage, DMU1, DMU2, DMU3, DMU4, and DMU10 are efficient under model (6); in the second stage, DMU1, DMU2, DMU4, DMU8, and DMU10 are efficient under model (7) (See Table 3, columns 2 and 3). The fourth column of Table 3 presents the scores of efficiencies of the two-stage network under model (10). The units that are efficient in both stages are also efficient under the overall two-stage network model (i.e., model (10)), that is, DMU1, DMU2, DMU4, and DMU10 are efficient under model (10) and E=1,2,4,10 is a set of index-efficient DMUs. The fifth and sixth columns of Table 3 show the min and max u0∗+v0∗, which indicate that (u0∗, v0∗) is the optimal solution obtained from model (10) for the evaluation of the DMUs’ efficiency scores. According to Theorem 2, DMU1 has constant returns to scale, DMU10 has increasing returns to scale, and DMU2 and DMU4 have decreasing returns to scale. Thus, the set of efficient DMUs may be partitioned into classes E1 = {10}, E2 = {2,4}, and E3 = {1} on the basis of the classification in Section 3.Table 3
Table of result.
θp1∗ (6)θp2∗ (7)θpoveral∗ (10)Maxu0∗+v0∗ (10)Minu0∗+v0∗ (10)DMU11.00001.00001.00001.6429−0.3810DMU21.00001.00001.0000−0.2420−0.0001DMU31.00000.48080.4808−0.1670−0.2230DMU41.00001.00001.0000−0.75402.0000DMU50.75000.80560.6042−0.0001−0.0001DMU60.63380.77780.4930−0.2040−0.6150DMU70.96000.50000.5208−0.0001−0.0001DMU80.36361.00000.8788−0.2420−0.7270DMU90.37500.75000.52700.17600.1760DMU101.00001.00001.00002.87502.8750DMU10 has constant returns to scale; thus, increasing output can change the type of returns in this unit. To identify the perfect enhancement of outputs so that DMU10 maintains increasing returns to scale, we used model (30), wherein α∗=0.17,β∗=6 is an optimal solution for assessing DMU10.To analyze the sensitivity of the returns to scale classifications of DMU1, models (40) and (41) should be used. Because DMU1 has constant returns to scale and because increasing and decreasing outputs can change the types of returns in the unit, α1∗=1 and β1∗=1 were used as the optimal solution from model (40), and α2∗=0.29 and β2∗=3.5 served as the best solution from model (41). Thus, increasing or decreasing inputs in each α; 0≤α≤min0.26,1 and increasing or decreasing outputs in each β; 0≤β≤min1,3.5 cannot change the type of constant returns to scale in DMU10.DMU2 and DMU4 have decreasing returns to scale. Accordingly, model (40) was used for the sensitivity analysis of returns to scale classifications. The analysis yielded α∗=1.8 and β∗=2.25 as the best input and output reduction values, respectively, for DMU2 to maintain decreasing returns to scale. Additionally, α∗=2.67 and β∗=1.67 are optimal solutions from model (40) in the sensitivity analysis of the returns to scale classifications in DMU4.


## 5. Conclusion

In DEA models in the past, DMUs were known as a black box, and calculations were narrowed to the final output and initial inputs. But today, the researchers have a variety of attitudes toward the two-stage networks. This paper, has developed a DEA model to evaluate the efficiency of a two-stage network in a variable returns to scale area. A new definition of the kind of returns to scale in two-stage network is planned and a method to determine the kinds of returns to scale in efficient DMUs is expended. DEA model have been used in sensitivity analysis for returns to scale categorization in the tow-stage DMU network. At last, an easy and plain numerical example is provided to describe the analysis.An interesting challenge to the future studies will be the development of the models to the analysis of two-stage system performance, the identification of subprocesses, and the classification of returns to scale.There is expanded literature in DEA analysis and DEA networks. This manuscript considers the analysis of two-stage network returns to scale, in which the exact data is used, while the real handicaps in a real life convey ambiguity such as multiple meanings of expressions, Fuzzy sets theory can be effectively used to control the data ambiguity and vagueness in network DEA issues.It is possible to propose in future researches to analyze and investigate the sensitivity of returns to scale in two-stage network if there is any Fuzzy data. The reason is that returns to scale in an important management issue. It considers the decrease of expense as an advantage because of the increase in production volume, and another weakness of classic models of DEA and network DEA is that there would not be permission of random changes in inputs and outputs.To enter the random deviations of input and output data like the measurement errors and invalid data modeling is not allowed and finally the sensitivity of result of these deviations can be an appropriate proposal to analyze the returns to scale in network DEA with random data in future.



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