A TWO-STAGE STRUCTURE WITH UNDESIRABLE OUTPUTS: SLACKS-BASED AND ADDITIVE SLACKS-BASED MEASURES DEA MODELS

. The slacks-based measure (SBM) and additive SBM (ASBM) models are two widely used DEA models acting based on inputs and outputs slacks and giving efficiency scores between zero and unity. In this paper, we use both models with the application of the weak disposability axiom for outputs to evaluate efficiency in a two-stage structure in the presence of undesirable outputs. In the external evaluation, the SBM model is reformulated as a linear program and the ASBM model is reformulated as a second-order cone program (SOCP) that is a convex programming problem. In the internal evaluation, the SBM model for a specific choice of weights is linearized while the ASBM model is presented as an SOCP for arbitrary choice of weights. Finally, the proposed models are applied on a real dataset for which efficiency comparison and Pearson correlation coefficients analysis show advantages of the ASBM model to the SBM model.


Introduction
Data envelopment analysis (DEA) is a widely used tool for efficiency analysis, performance evaluation, and benchmarking.It identifies a set of best units from a given set of decision-making units (DMUs) with multiple inputs and outputs [6].There are several DEA models measuring the efficiency of DMUs based on inputs and outputs slacks.Charnes et al. developed the first additive DEA model in which the objective function is defined as the summation of all inputs and outputs slacks [4].While the additive model can identify efficient DMUs, it fails to produce a comparable DEA score or composite index.To overcome this drawback, Green et al. improved this model with an additive slacks-based measure (ASBM) that assigns an efficiency score between zero and unity to each DMU and unit invariance property is also assured [14].Chen and Zhu have shown that ASBM is equivalent to Russell graph measure and thus reformulated it as a second order cone program (SOCP) [5].Further, they have extended it for network DEA structures.Gerami et al. developed a novel geometric interpretation of value efficiency to ASBM modeling under the hypothesis of variable returns to scale [12].The authors solved the ASBM model without changing its objective function and constraints.They used convenient transformations to linearize the ASBM model and then they extended the ASBM model to compute value efficiency.
In classical production theory, the aim is to minimize inputs and maximize produced outputs, also, in most DEA models, the resource assignment is in such a way that all inputs are applied entirely to produce good (desirable) outputs and bad (undesirable) outputs.In specific cases, when there are undesirable outputs in the production system, an increase in the production of this type of outputs is not desirable.In such situations, one of the best solutions is to apply weak disposability axiom for outputs in the productivity possibility set (PPS).Weak disposability means that in order to decrease the amount of undesirable outputs, the desirable outputs have to decrease commensurately, too.While Färe et al. utilized one abatement factor for all DMUs, Kuosmanen used a distinct abatement factor for each DMU [10,19].Later, Kuosmanen et al. demonstrated that using a uniform abatement factor may result in non-convex PPS.They also asserted that their introduced PPS is the correct minimum extrapolation technology that satisfies the strong disposability of desirable outputs and inputs; weak disposability of all outputs and convexity axioms [20].In recent decades, scientists have focused on the performance evaluation of the production units by taking into account internal structure and relations of sub-processes.In the sequel, we review some of the studies related to the network SBM models in DEA that takes into account undesirable outputs as well.
Fukuyama et al. proposed a slacks-based inefficiency measure for a two-stage system with bad outputs and applied it for the evaluation of banks [11].Bian et al. presented the efficiency evaluation of Chinese regional industrial systems with undesirable factors using a two-stage SBM approach [2].They provided analytical suggestions for improving the performance of a regional industrial system by identifying its inefficient internal stages; that helped to find out the main sources of the inefficiency arises from the internal stages within the system.This analysis cannot be done using conventional environmental DEA models.Song et al. carried out systematic research on efficiency evaluation using the SBM model considering undesirable outputs, and expanded it to analysis of network structures [27].Their model also calculated desirable and undesirable outputs separately.Zhu et al. utilized an SBM approach on the network DEA model to evaluate the eco-efficiency of products [35].Moreover, Lozano proposed an efficiency model based on the SBM model for a general network with undesirable outputs based on variable returns to scale (VRS), non-decreasing returns to scale (NDRS), and non-increasing returns to scale (NIRS) technologies [21].Cui et al. designed a network epsilon-based model with managerial disposability to evaluate efficiency of the airlines under CNG2020 approach [7].In another study, Cui et al. presented a network range adjusted model with weak-G disposability model to compute the environmental efficiency of 29 global airlines [8].Zhou et al. extended a mixed network structure two-stage SBM model to measure the detailed efficiencies and find out the weak link of the system and applied it successfully to the provincial dataset of China during 2006-2015 [34].Hu et al. proposed an extended two-stage SBM model with undesirable output and a feedback variable and then applied it to a real example from oil industry [16].Cui et al. presented a novel model based on the modified SBM model in a network structure to evaluate the environmental efficiencies of airlines when the inputs or outputs of airlines are negative [9].The authors dealt with undesirable outputs by turning them into negative data for efficiency scales.Shi et al. proposed a network SBM model with undesirable outputs to evaluate the performance of production processes having complex structures containing both series and parallel processes [26].Yang et al. enlarged a dynamic network SBM model to recognize the proficiency of regional industrial water systems that wastewater is considered as an undesirable intermediate output and total volume of industrial wastewater discharged is supposed as an undesirable output, which prepared effective information for managers to handle the inefficiencies of the subsystems and periods [32].
In this paper, we study the performance evaluation of DMUs in a two-stage structure with undesirable outputs by the SBM model of Tone and the ASBM model of Green et al. from both external and internal perspectives [14,29].The SBM model is defined in a ratio form of aggregated output slacks to input slacks.To deal with undesirable outputs, we apply the weak disposability of Kuosmanen to reduce pollutant outputs that should be decreased [19].For internal evaluation, we use the weighted average of stages' efficiencies for the overall efficiency of DMUs as in Kao [18].The unknown weights are functions of slack variables and may vary from one DMU to another.The value of the variables is determined after solving the proposed SBM model.On the other hand, the proposed ASBM models for both internal and external evaluations are presented as SOCP problems for arbitrary weights.Some advantages of the proposed SBM and ASBM models are as follows: -Both models apply weak disposability of Kuosmanen [19], to reduce undesirable outputs.
-The SBM model in the external evaluation is linear while the ASBM model is an SOCP that is also a convex program.-The SBM model in the internal evaluation is linear for a specific choice of weights while the ASBM model is again an SOCP.-In the internal evaluation of the ASBM model, weights can be determined by the decision-maker preference.
-The proposed models are consistent, unit invariant, and translation invariant.
-Using different abatement factors in disposability set improves the discrimination power of models.
-Since there is no need to determine the weights for combining stage efficiencies, one can locate different stage efficiencies of Pareto optimal equivalence by varying combining weights from zero to unity.
The remainder of the paper is organized as follows.In Section 2, we define the weak disposability principle of outputs.In Section 3, the external and internal evaluations of DMUs via the SBM and the ASBM models in the presence of undesirable outputs are introduced.Section 4 compares the proposed models on a real numerical example and our conclusions follow in Section 5.

Two-stage structure in the presence of undesirable outputs
In this section, first we discuss the weak disposability of outputs due Kuosmanen [19], for the classical DEA models, then we present its extension for the two-stage structure in this paper.

Weak disposability of outputs
The classical DEA approaches rely on minimizing inputs and maximizing outputs to improve the efficiency score.However, in some applications, there are undesirable measures (for instance emission of harmful substances in air, energy wasted in power plant) that should be minimized.Hailu and Veeman in [15] have developed non-parametric productivity analysis approach to comprise undesirable outputs.They presented a monotonicity condition on their technology and asserted it is premier to "weak disposability" concept in DEA.Fare and Grosskopf showed that using monotonicity conditions in the steal of weak disposability is inconvenient with natural law [10].Assume that there are  DMUs to be evaluated, indexed by  = 1, . . .,  and the vectors of inputs, desirable outputs and undesirable outputs are   ( = 1, . . ., ),   ( = 1, . . .,  ) and   ( = 1, . . ., ), respectively.The production technology can be illustrated by: Definition 2.1.Desirable and undesirable outputs are weakly disposable if and only if (, ) ∈  () and 0 ≤  ≤ 1 imply (, ) ∈  (),  ∈ R  (for more details see [25]).
Fare and Grosskopf [10] presented a technology under VRS satisfying weak disposability supposition that the contraction parameter  in their approach corresponds to Definition 2.1.This parameter permits simultaneous contraction of good and bad outputs.Kuosmanen mentioned that Fare's approach applies a uniform abatement factor to all firms [19].Applying different abatement factor for each DMU can have two main advantages.First, the corresponding models are linear.Second, the presented performance measurement approaches have more discrimination power.To allow non-uniform abatement factors of the individual firms, Kuosmanen suggested the following production technology: Free disposability of the inputs and good outputs is formulated through the use of inequality restrictions regarding  and .The nonlinear technology   is further transformed into an equivalent linear form by Kuosmanen letting   =   +   and   =     [19].We use this technology to model undesirable measures in a two-stage production process in the next subsection.

Weak disposability in a two-stage structure
Consider a two-stage structure in which each DMU is composed of two sub-DMUs (stages) sequentially as follows: Again assume that there are  DMUs to be evaluated, indexed by  = 1, . . .,  and each DMU has a two-stage structure as given in Figure 1.In the first stage, by consuming  inputs   ( = 1, . . ., ),  desirable outputs   ( = 1, . . .,  ) and  1 undesirable outputs  1 1 ( 1 = 1, . . .,  1 1 ) are produced.In addition to these,  desirable outputs   ( = 1, . . ., ) also are produced, called intermediate products that are inputs to the second stage.In stage two, in addition to the intermediate products,  other inputs  2 ℎ (ℎ = 1, . . ., ) also are consumed to produce  desirable outputs  2  ( = 1, . . ., ) and  2 undesirable outputs  2 2 ( 2 = 1, . . .,  2 ).Based on (   1 ), the PPS for the two-stage structure given in Figure 1 for modeling undesirable outputs under VRS is as follows: where, non-uniform abatement factor  1  ,  2  allows simultaneous restriction of desirable and undesirable outputs.Desirable inputs and desirable outputs are strongly disposable, but outputs (bad and good) are weakly disposable.Now, based on the   and by adding slacks, we have the following constraints: ) ) ) ) Note that (2.2) is the union of restrictions of slack variables for each stage without considering the connection between the stages.For this reason, (2.2) can be let as the restrictions for solving VRS-SBM and ASBM models for each stage.In addition,   's are intermediate products of the two-stage network and can be hidden from the external viewers.Now, considering the continuity of activities between the stages at the two-stage production system, without loss of generality, we suppose that the intermediate products (i.e.,   ) consumed by a DMU are not completely generated within the system.Also, as the first stage produces and second consumes, the amount of the first stage is either larger or equal to the second stage amount i.e., the following constraint: (2.3)

Efficiency evaluation
In this section, we evaluate the efficiency of the two-stage structure of Figure 1 from both external and internal perspectives following Kao's framework [17,18], for defining the overall efficiency of the structure.

External evaluation
In external evaluation approach, the overall efficiency is considered without slack variables of intermediates which are only visible by the insiders.In this case, first we calculate the overall efficiency and then the efficiencies of the first and second stages are obtained as in [18,31].Thus the SBM model based on (2.2) is as follows: Model (3.1) has nonlinear constraints and nonlinear objective function.One of the advantages of using distinct abatement factors is that model (3.1) can be linearized by the following change of variables due to [19]: Further let and where is an infinitesimal positive real number.Now, let After solving model (3.3), to obtain the efficiencies of the first and second stages, the slack variables of the intermediate products are calculated as follows [5]: Then, the efficiencies of the first and second stages are obtained as follows: ) The overall efficiency of the ASBM model in external evaluation is as follows: Constraint sets (2.2a) to (2.2j) In the next theorem, we reformulate model (3.6) as an SOCP that is a convex program.
Theorem 3.1.Problem (3.6) is equivalent to the following SOCP: (3.10) (︀ Similarly, inequality (3.9) can be written as follows: )︁ is an optimal solution of model (3.7).To obtain the efficiencies of the first and second stages, first we calculate the slack variables of the intermediate products as follows: Then, the efficiencies of the first and second stages are obtained as follows: ) In general, external evaluation can be seen as a non-cooperative process, where overall efficiency and stages efficiencies are computed in a serial way.In the sequel, we discuss internal evaluation that is able to gain overall efficiency and stages efficiencies together.

Internal evaluation
Internal evaluation is a process of efficiency aggregation that can be considered as a contrary method of external evaluation, where the overall efficiency and stage efficiencies are calculated in a sequential procedure [18].However, in internal evaluation, overall efficiency, and stages' efficiencies are obtained simultaneously.In the internal evaluation, the overall efficiency also includes the intermediate product, thus we have the following two constraints: Based on (2.2), (2.3) and (3.15), the overall efficiency of the SBM model in internal evaluation is as follows [18,30]: ) where  1 and  2 are weights of the first and second stages, respectively with  1 +  2 = 1.If we let and use Charnes-Cooper transformation [3], we get the following equivalent LP as in the external case: ) where  is an infinitesimal positive real number.Let (︁  − )︁ be the optimal solutions of model (3.16) and (3.19), respectively.Then we have After solving model (3.19), the efficiency of the subunits can be also observed.
Similarly, based on (2.2), (2.3) and (3.15), the ASBM model for the internal evaluation is as follows: where weights are selected by the decision maker such that 0 <  1 ≤ 1.In the next theorem, similar to the external evaluation, equivalent SOCP formulation of model (3.20) is presented.
Theorem 3.2.Model (3.20) is equivalent to the following SOCP: Proof.The proof is similar to the proof of Theorem 3.1.
After solving model (3.21), the overall efficiency and efficiency of subunits can be viewed simultaneously.
In this section, we discussed SBM and ASBM models from external and internal perspectives.The SBM models are in the form LP for both cases, while using specific weights for the internal evaluation.However, the nonlinear ASBM models are reformulated as SOCPs for both external and internal evaluations independent of weights.
In the next section, we use a real dataset from the literature to evaluate both models.

Case study and discussion
In this section, the presented models are applied on a real dataset to show advantages of the ASBM model.All models are solved using CVX software package Grant et al. [13].The data for this example, given in Table 1, is taken from Lozano et al. [22], to evaluate 39 Spanish airports in the presence of adverse outputs.Airport processes are divided into two stages: the aircraft movement process and the aircraft loading process.In the first stage, it has three inputs, Total runway area ( 1 ), Apron capacity ( 2 ), Number of boarding gates ( 3 ) and two undesirable outputs, Number of delayed flights ( 1 1 ), Accumulated flight delays ( 1 2 ), and an intermediate product, Aircraft traffic movements ( 1 ).In the second stage, in addition to the intermediate product, it also has two other inputs, Number of baggage belts ( 2 1 ), Number of check-in counters ( 2 2 ), and two final desirable outputs, Annual passenger movements ( 2 1 ) and Cargo handled ( 2 2 ), (Fig. 2).

External evaluation
The results of the external evaluation of the SBM and ASBM models are reported in Tables 2 and 3, respectively.The results show that efficient DMUs are the same in both models and overall efficiencies of the ASBM model are always greater than or equal of those of the SBM model.However, it can be seen that the obtained rankings from the SBM and the ASBM are different for some DMUs.
Moreover, from the results shown in Table 2 it can be seen that 51.2% and 48.7% of DMUs in overall system and stage 2, respectively, have efficiency equivalent to zero.Also, in stage 2, the efficiency scores for   16 and   27 are greater than one.This means that the efficiencies for the most DMUs in stage 2 and overall system are not reasonable.In Table 3 2 and 3 illustrate that external ASBM model is more preferable than the external SBM model.Furthermore, Pearson correlation coefficients test results for both models at significant level 0.01 are given in Tables 4-5.As can be seen from Table 4, for the external SBM model the overall efficiency is significantly correlated with only stage 2, which also confirmed the overall system inefficiency sources.However, Table 5 shows that the overall efficiency of the ASBM model is correlated with both stages.In other words, comparing the efficiency scores of Tables 2 and 3 illustrate that ASBM model is a powerful alternative to SBM in the case of external estimation due to the consistency and similarities in ranking found as discussed.
In the sequel, we apply internal evaluations models to the dataset in order to detect the overall system's inefficiency sources.In summary, the decomposition of the overall system into two stages can help managers to recognize the real causes of inefficiencies in the network structures.

Internal evaluation
The results of the internal evaluation for both models are given in Tables 6 and 7, respectively.Note that in the ASBM model, the weight of each stage is considered 0.5.As can be seen, in internal evaluation, as in external evaluation, efficient units are the same in both models, and also the overall efficiency of the ASBM model is always greater than or equal to the SBM model.Moreover, the rankings of the two models differ for several DMUs.Here unlike the external evaluation, the efficiency is calculated in terms of the weighted average subunits.Thus, it is possible to get different rankings for the ASBM model for different weights, while the weights of the SBM model are fixed.This can be considered as an advantage of the ASBM model to the SBM model.As well, Table 6 displays that 38.4% and 46.1% of DMUs in the overall system and stage 2, respectively, have an efficiency equivalent to zero which is not the case in Table 7 for the ASBM model.Figure 3 shows that the overall efficiency of the ASBM model is between efficiencies of stage 1 and stage 2 which is not the case in the external evaluation (for example see DMUs 4 and 5 in Tab. 3).Although this observation holds for the SBM model as can be seen in Table 6 for fixed weights, but for the ASBM it holds for arbitrary choice of weights.Also, Figure 4 shows the radar chart of the ranking changes of the ASBM model for three different weights compared to the SBM ranking.It can be seen that internal efficiency scores of the most DMUs in ASBM model increases with the weight.
Furthermore, Pearson correlation coefficients between the two stages and the overall system in the internal evaluation are reported on Tables 8-9.For the SBM model, the overall efficiency is significantly correlated only with stage 2, while for ASBM it is correlated with both stages.This might be due to specific choice of weights for the SBM model, where the weights of the second stage in most cases are higher than those of the first stage.Indeed, in the SBM model these weights present virtual restrictions that changes the feasible region and thus the resulting SBM models stages efficiencies may not be one of the Pareto solutions.However, the ASBM model allows managers to specify constant weights, and can locate different stage efficiencies of Pareto optimal equivalence by varying combining weight from zero to unity.Therefore, ASBM is more than a reliable alternative to SBM in the case of internal evaluation.

Conclusions
In this paper, we studied the SBM and ASBM models to evaluate the efficiency of decision-making units in a two-stage structure in the presence of undesirable outputs under the weak disposability of Kuosmanen [19].The weak disposability axiom for all outputs is beneficial for the performance measurement of the production units.In order to perform a comprehensive analysis, we evaluated the efficiency from two managerial perspectives, namely the external and internal ones.We proposed two equivalent LPs for the SBM in both evaluations, where in the internal case this is achieved for specific weights.However, the ASBM models, in both evaluations, are reformulated as SOCPs that are convex programs independent of weights choice, which is an important computational achievement.Moreover, the flexibility of weighs enable decision-makers to incorporate managerial preferences in the evaluation process.The application of the proposed models for a real dataset showed that in the external and internal cases the SBM model gives unreasonable efficiency scores for about 50% and 40% of DMUs in overall and stage 2.Moreover, Pearson correlation coefficients showed that in both external and internal evaluations, the SBM model relies more on the second stage which is not the case in the ASBM model where it relies on both stages almost equally.In addition, the proposed ASBM model in the internal evaluation can identify the desirable outputs from the undesirable outputs, thereby avoiding the need for weighting with managers priorities.In recent studies a secondary goal has been added to the DEA models in order to consider DMUs fairness mentality that plays an important role in behavioral economics [36].Thus one may combine it with our proposed ASBM models in the presence of undesirable outputs.Moreover, the idea of uncontrollable inputs [33], nondiscretionary factors [24] and uncertainty in dataset [1,23,28], have also pervasive applications, so including them in the proposed models would be an absorbing future research directions.

Figure 1 .
Figure 1.A two-stage structure in the presence of undesirable outputs.

Figure 2 .
Figure 2. The structure of Spanish airports.

Figure 3 .
Figure 3. Overall and stages efficiencies of the ASBM model for internal evaluation.

Figure 4 .
Figure 4.The internal Efficiency scores for different weights of ASBM and SBM model.

Table 1 .
Inputs and outputs data of 39 Spanish airports.

Table 2 .
Results of external evaluation in the SBM model.

Table 3 .
Results of external evaluation in the ASBM model.

Table 4 .
Pearson correlation coefficients between overall and two stages efficiency in the SBM model.

Table 5 .
Pearson correlation coefficients between overall and two stages efficiency in the ASBM model.

Table 6 .
Results of internal evaluation of the SBM model.

Table 7 .
Results of internal evaluation of the ASBM model.

Table 8 .
Pearson correlation coefficients between overall and two stages efficiency in the SBM model.

Table 9 .
Pearson correlation coefficients between overall and two stages efficiency in the ASBM model.