An Integrated Slacks-Based Measure of Super-Efficiency with Input Saving and Output Surplus Scaling Factors and its Application in Paper Chemical Mills

Data envelopment analysis (DEA) as a nonparametric programming approach has been widely extended and applied in many areas. Conventional DEA models can well measure the efficiency of inefficient decision-making units (DMUs) but cannot further discriminate the efficient DMUs. A lot of methods are proposed to address this problem. One of themost importantmethods is the slacks-based measure of super-efficiency model (S-SBM model) developed by Tone in 2002. However, the projection for a DMU on the efficient frontier identified by S-SBM model may not be strongly Pareto-efficient that makes the super-efficiency score misestimated. /is paper revises the usual slacks-based measure of super-efficiency by incorporating input saving and output surplus scaling factors into the objection function for measuring DMUs. We integrate SBM model and S-SBM model effectively and yield input saving and output surplus scaling factors as well as input and output slacks under only one integrated model. According to the study, the projection reference point identified by our method is strongly Pareto-efficient. Meanwhile, how each decision variable influences the efficiency score for a specific DMU is revealed and illustrated through two numerical examples and an empirical study in paper chemical mills.


Introduction
Over the past several decades, the data envelopment analysis (DEA) initially proposed by Charnes et al. has proved to be an effective data-oriented programming method for measuring the efficiencies of a group of homogenous decision making units (DMUs) [1]. e technique of DEA depicts a best-practice production efficient frontier formed by observed DMUs and provides a benchmark or reference point on this frontier for each DMU to compute its efficiency score. Using DEA does not assume any production function or presuppose any specific weight restriction to inputs and outputs. So in the recent years, DEA has been widely extended and applied in many fields [2]. e efficiency value obtained by the classic CCR model indicates how efficiently a DMU has performed when compared with other DMUs so as to determine its efficient level within the group of all DMUs. e CCR model works as a radial model and has both input-and output-orientation styles which permits the DMU under assessment to proportionally reduce all its inputs for producing its given outputs, or to proportionally expand all its outputs by using its given inputs.
However, when the evaluation target group includes a considerable number of DMUs, more than one unit will always get the same efficient score of unity. In this case, the CCR model cannot further differentiate the efficiency performances of these DMUs and cannot provide more recognizing information on them. For example, it does not detect whether the evaluated DMU is weakly efficient. As Chen pointed out, the "radial" efficiency model may make some DMU measured against a weakly efficient point on the efficient frontier in the production possibility set [3].
Specifically, the weakly efficient reference point for the current evaluated DMU may still have a positive amount of input excesses or output shortfalls, for it is not the strongly Pareto-efficient reference point. From the view of DEA, the efficiency score obtained by the weakly efficient reference point makes the evaluated DMU misestimated with respect to its strongly Pareto-efficient reference point on the efficient frontier.
So far, many methods have been developed and studied in order to enhance the cognition levels and discrimination abilities to distinguish DMUs, such as the cross-efficiency technique [4,5], the benchmark ranking method, and others [6]. ese newly developed methods are mainly built to solve problems resulting from the original CCR model in a certain aspect. erein, Andersen and Petersen creatively developed the first radial super-efficiency model under the assumption of constant returns to scale (CRS) to reassess the efficient DMUs under CCR model [7]. ey exclude the DMUs being evaluated from the reference set by the envelopment linear program so as to retrieve the called super-efficiency scores for those efficient DMUs, while, for the variable returns to scale (VRS) super-efficiency model, Seiford and Zhu found that the problem of infeasibility may occur [8]. Chen further ascribed the sources of super-efficiency for an efficient DMU to its achieved positive amounts of input saving and output surplus regarding its efficient reference point on the superefficiency frontier [9]. Cook et al. derived a revised VRS super-efficiency model which could generate optimal solutions for some efficient DMUs that is infeasible in the original VRS super-efficiency model [10]. e resulting super-efficiency scores gained by their model could be described from both inputs and outputs aspects to some degree. Later, Lee et al. introduced a two-stage process to address the infeasibility issue under VRS [11]. Next, this two-stage process was merged into a single linear program by the work of Chen and Liang [12]. More recently, Lee and Zhu settled the infeasibility caused by zero input data and decompose the acquired super-efficiency score into three indices: radial efficiency index, input saving index, and output surplus index [13]. e above super-efficiency models are all of a radial type and they all have both input-and output-oriented forms. Tone built a popular nonradial model named slacksbased measure (SBM), which uses a fractional objection function depending on input and output slacks instead of a simple radial efficiency variable [14]. e SBM model only has one style, for there is no distinction between inputorientation and output-orientation under SBM. e efficiency score computed by the SBM model is also between 0 and 1. Compared to the traditional radial super-efficiency DEA models under CRS, the SBM model avoids the problem of infeasibility. Moreover, input slacks and output slacks in the SBM model can be utilized to detect the input excesses and output shortfalls of a given DMU, respectively. However, for SBM-efficient DMUs, Tone designed a super-efficiency model (S-SBM model) to examine their super-efficiency scores in order to allow SBM-efficient DMUs to be also ranked and compared [15]. Liu and Chen developed a HypoSBM model to distinguish the worstperformance DMUs from the bad ones [16]. Du et al. extended the SBM super-efficiency model to the additive slacks-based DEA model [17]. Fang et al. established a twostage process, which was an alternative disposal treatment for the SBM method proposed by Tone [18]. ey demonstrated that their two-stage approach generated the same results as Tone's models. Chen indicated that there exists a discontinuous gap between the SBM score and S-SBM score for a DMU who has a weakly reference point under S-SBM model [3]. In his study, an ambidextrous joint computation model (J-SBM model) for slacks-based measure was provided.
However, J-SBM model has two noted shortcomings. First, it fails to represent all input saving and output surplus scaling factors explicitly for each DMU. Second, it has a more complicated operational procedure and needs a three-stage computational process. e current paper further investigates this topic and presents a modified slacks-based measure of super-efficiency with input saving and output surplus scaling factors to reevaluate these DMUs. Our approach is devoted to detecting the specific scaling factors for each input and output as well as input and output slacks explicitly by one integrated model. Meanwhile, the phenomenon of discontinuousness between SBM score and S-SBM score for the same DMU can be eliminated.
e results indicate that the projection obtained through our proposed model is strongly Paretoefficient. And in this approach, we can see clearly how each decision variable influences the final efficiency score for a specific DMU. e structure of this paper is organized as follows. Section 2 briefly reviews several kinds of slacks-based models. Section 3 presents a modified slacks-based measure of super-efficiency with input saving and output surplus scaling factors to determine DMUs' efficiency scores. In Section 4, two numerical examples are applied to compare our approach with the previous models. Section 5 applies our approach to an empirical example where the performance of 32 paper chemical mills in China is evaluated. e main conclusions and remarks are given in Section 6.

Preliminaries
Suppose there are n DMUs, {DMU k (k � 1, 2, . . . , n)}. Let x k � (x 1k , . . . , x mk ) and y k � (y 1k , . . . , y sk ) denote the input and output vectors of the kth DMU. e ith input of the kth DMU is denoted as x ik and the rth output of the kth DMU is denoted as y rk , respectively. λ j is the intensity coefficient for the kth DMU which means its contribution to forming the efficient frontier. Assume that all input and output data are positive.

Tone's SBM Measure.
Tone developed the following SBM model to evaluate the relative efficiency of DMU k [14], where input and output slacks are denoted, respectively, as z − i (i � 1, . . . , m) and z + r (r � 1, . . . , s). Unlike the CCR model, the objective function in SBM model has a fractional form which directly includes input and output slacks.
λ j y rj � y rk + z + r , r � 1, . . . , s, where DMU k is called SBM-efficient if and only if z − i � z + r � 0 for all i and r, that is, ρ * k � 1; otherwise, it is called SBM-inefficient. In order to further discriminate SBMefficient DMUs with the same SBM efficiency score of 1, Tone introduced a SBM super-efficiency model which was referred to as S-SBM model in the following formula [15].
It should be noticed that the S-SBM model can only be used for the DMU whose SBM efficiency score ρ * � 1.
rough model (2), these SBM-efficient DMUs get its superefficiency scores. However, model (2) cannot discriminate SBM-inefficient DMUs for they will get the same efficiency score of 1.

Fang's Models.
Fang et al. provided a two-stage process which brings in the same efficiency scores as those obtained by Tone's two models [18]. In the first stage, they replace x ik , (3) for detecting both input savings and output surpluses for all DMUs first.
rough Model (3), the optimal w − * i and w + * r for each input and output for all DMUs can be obtained. For SBM-efficient DMUs in model (1), there will exist at least one i or r, so that w − * i > 0 or w + * r > 0 in model (3), while, for SBM-inefficient DMUs in model (1), they have no positive input saving and output surplus for each corresponding input and output at all. So these DMUs will get w − i � w + r � 0 for all i and r. en, plug w − * i and w + * r into the following model, and a slacks-based measure is reconstructed as model (4) exhibits.
Obviously, model (3) is equivalent to model (2). So model (3) also determines the super-efficiency scores for SBM-efficient DMUs. e optimal SBM efficiency scores and the optimal slacks (s − * i , s + * r ) for SBM-inefficient DMUs can be computed based on the optimal (w − * i , w + * i ) by model (4).
At last, the final efficiency score for DMU k based on SBM is defined as Journal of Chemistry Chen investigated a data set in Tone's which can be seen in Table 1 and found the reference points generated by model (2) or (3) might not be Pareto-efficient [3]. is resulted in a discontinuous gap between the SBM score and S-SBM score. e author indicated that the issue of discontinuity makes troubles to rationalize the efficiency score. So he established an ambidextrous joint computation model (6) (J-SBM model) to find the Pareto-efficient point for each DMU.
M is a large enough positive number.

(6)
In model (6), b 1 and b 2 are two binary variables that are used to control which one of the three kinds of constraint conditions (6.1), (6.2), and (6.3) is chosen. For the SBMinefficient DMUs, b 1 � b 2 � 1, constraint condition (6.1) is active; now model (6) works as the SBM model. For SBMefficient DMUs, model (6) first acts as the S-SBM model under active constraint condition (6.2) when b 1 � 0, b 2 � 1. In the meantime, if the super-efficiency reference point is not Pareto-efficient for the evaluated DMU, model (6) will activate constraint condition (6.3), and the corresponding super-efficiency score will be corrected for the DMU. Additionally, Chen further proved that the reference points for all DMUs under model (6) are Pareto-efficient [3].
As can be seen from the above procedures in operating model (6), the constraint condition (6.3) is set intentionally to correct the misestimated efficiency score due to the less Pareto-efficient reference point caused by the S-SBM model (2) or (3). For these DMUs who have a weakly efficient reference point under model (2) or (3), however, the achievement of its super-efficiency needs a three-stage process. For instance, when DMU E in Table 1 gets its superefficiency of 1.25 under the constraint condition (6.3), the constraint conditions (6.1), (6.2) have been inspected before.

A Modified Slacks-Based Measure of Super-efficiency
In this section, we first establish an equivalent form of S-SBM model, which explicitly contains input saving and output surplus scaling factors. Note that x ik ≥ x ik for all i and y rk ≤ y rk for all r in model (2). Let x ik � x ik + t i x ik , t i ≥ 0 and y rk � y rk − β r y rk ≥ 0, 0 ≤ β r ≤ 1, or w − i � x ik + t i x ik , t i ≥ 0 and w + r � y rk − β r y rk ≥ 0, 0 ≤ β r ≤ 1; then model (2) or model (3) will become the following model (7).
e difference between models (7) and (3) is that model (7) can not only measure the input saving t i x ik and the output surplus β r y rk , but also present the specific scaling factors t i for x ik and β r for y rk of DMU k .
Let (t * i , β * r ) be the optimal solution of model (7). Based on model (7), the standard SBM model (1) can be revised as follows: Similarly, we apply model (7) to all DMUs before the utilization of model (8) on inefficient DMUs. We define the final efficiency score for DMU k based on SBM with the following piecewise function: where (s − * i , s + * i ) is the optimal solution in model (8). e above two-stage process is identical to that of Fang et al. Model (7) may also suffer from the problem that the projection reference point for a specific DMU may not be Pareto-efficient. As Chen mentioned, the SBM and S-SBM efficiency scores are calculated in two separate models with two different objection functions that have two different projecting styles for the evaluated DMU [3]. e SBM scores are achieved by their input and output slacks, while the S-SBM scores depend only on the reference point (may not be Pareto-efficient) on the super-efficiency frontier. erefore, the whole evaluation system is not an integrated one Journal of Chemistry and is divided into two styles, which results in the problem of a discontinuous gap. In order to integrate the whole evaluation system, we maintain the basic sense of SBM model and S-SBM model into one model and ensure that the desired model has the uniform projecting way for the evaluated DMU. So, we develop the following modified SBM measure based on model (7). , which not only includes the input and output slacks in SBM measure, but also includes input saving and output surplus scaling factors in S-SBM model. It is able to comprehensively explain the reason for the resulting efficiency scores; that is to say, when input saving scaling factors or output surplus scaling factors appear together with input or output slacks, they all contribute to the efficiency score.
Hence, the efficiency score can be defined as is the optimal solution of model (10). Second, we integrate SBM model and S-SBM model under the rule that the evaluated DMU must have the minimum super-efficiency score if it has at least one t * i > 0 or β * r > 0. So we add the controllable term of M × (1 + (1/m) m i�1 t i /1 − (1/s) s r�1 β r ) to the objective function so as to make t i and β r as small as possible for all i and r.
Here, M (we set it 10 7 so as to make it large enough in the empirical study) is a predetermined large number used mainly for magnifying the effect of input saving and output surplus scaling factors on the objective function under the condition of minimum S-SBM score. Simultaneously, it can control the optimal values of decision variables. Note the constraints in model (10): A severe problem which may occur is that when t i > 0 and s − i > 0 for a certain i (or β r > 0 and s + r > 0 for a certain r) simultaneously exist, there is an offsetting contradiction between t i x ik and s − i (or β r y rk and s + r ). To avoid this kind of situation, the controllable term of M × ((1 + (1/m) m i�1 t i )/(1 − (1/s) s r�1 β r )) added to the objective function is able to ensure that either t * i > 0 or s − * i > 0 for a certain i (either β * r > 0 or s + * r > 0 for a certain r) and both of them are not in existence.
Model (10) has close connections with the SBM model and S-SBM model.

Theorem 1. For any SBM-inefficient DMU
Proof. For SBM-inefficient DMU k , there must be t * i � 0 and β * r � 0 for all i and r in model (10). In this case, model (10) degenerates into SBM model (1).

□ Theorem 2. For SBM-efficient DMU k , if the reference point under S-SBM model for DMU
Proof. For SBM-inefficient DMU k , suppose the reference point of DMU k in S-SBM model is Pareto-efficient, there must be s − * i and s + * r � 0 for all i and r in model (10). At this moment, model (10) degenerates into model (7) which is equivalent to S-SBM model (2).
However, for other SBM-efficient DMUs, if the reference point under S-SBM model is not Pareto-efficient, model (10) will revise the super-efficiency score obtained by S-SBM model (7) but just make sure that model (10) can identify the Pareto-efficient reference for these DMUs. Proof. For a SBM-inefficient DMU, model (10) serves as the standard SBM model (1); the reference point identified the standard SBM model as Pareto-efficient, and so does model (10).
For a SBM-efficient DMU, there at least exists one input i ∈ 1, . . . , m { } or one output r ∈ 1, . . . , s { } such that at least one scaling factor t * i > 0 or β * r > 0. Meanwhile, the existence of input and output slacks allows the evaluated DMU to decrease its inputs and increase its outputs. us, the reference point identified by model (10) is as follows:

Illustration Examples
Two numerical examples from Tone are used to verify our approach through comparing with the SBM model, S-SBM model, and J-SBM model. In Table 1, the efficiency scores derived by the SBM model, S-SBM model, and J-SBM model are presented and those identified through our approach are listed in the last column. Table 2 shows detailed optimal solutions under J-SBM model (6) and we rewrite them in the form of input and output surplus scaling factors so as to facilitate comparison. Table 3 presents detailed optimal solutions under our model (10). As shown in Tables 2 and 3, these DMUs get the same efficiency scores but may be a different optimal solution for each decision variable. For example, DMU D gets the super-efficiency score of 1.25 under J-SBM model (6), and the achievement of superefficiency score seems reflected in its input saving scaling factors t * 1 � 0.234 and t * 2 � 0.266, while under our model (10) it not only owns input saving scaling factors t * 1 � 0.137 and t * 2 � 0.163, but also has the output surplus scaling factor β * 1 � 0.08. e reason is that due to the different projecting style of these two models, DMU D gets different reference points, which are both Pareto-efficient projected points. And for DMU E , J-SBM model (6) and our model (10) identified the same Pareto-efficient reference point. Our model also overcomes the problem of the discontinuous gap and finds its input saving scaling factor t * 1 � 1 and input slack s − * 2 � 2, so the rational super-efficiency score is In Table 4, we mainly compare the approach from Tone's or Fang's and ours using data set 2 from Tone. For the first four DMUs (1-4), two approaches gain the same efficiency scores. But for DMU 5 , our approach detects that it has both input saving scaling factor t * 1 � 0.4 and output surplus scaling factor β * 2 � 0.475. Besides, it has input slack s − * 2 � 1.9 and output slack s + * 1 � 0.4. is situation means that S-SBM model does not project DMU 5 at a Pareto-efficient targeted point and its efficiency score should be revised. According to our approach, the efficiency score should be modified as Journal of Chemistry 7

An Empirical Study
After high speed development on industries and economics during the past several decades, China has accumulated a lot of air environment problems and the Chinese government is paying more and more attention to ecological and environmental assessment. Recent years have seen a lot of studies based on DEA to measure China's environmental pollution problems. For example, Zhou et al. [19,20] construct a set of DEA models with the integral and zero-sum gain constraints for calculating air quality and a new nonradial directional distance function to scale the performance of water use and wastewater emission, respectively. In this section, the method that we have developed is applied to examine the efficiencies and rankings of 32 paper chemical mills along the Huai River in China. In this empirical study, four input and output variables are considered to evaluate each mill's performance. e inputs of each paper mill include labor and capital, and good output as paper products as well as bad output as biochemical oxygen demand (BOD). e detailed data set of these 32 paper chemical mills is shown in Table 5.
In papermaking industry, the good products are always produced with bad products and it is impossible to increase good outputs and decrease bad outputs meanwhile. First, the attribute of "the smaller the better" for bad outputs is consistent with inputs. Second, bad outputs cannot create any new profit and dealing with them (such as sewage treatment and air purification) always comes at a price which the mill should afford [21]. erefore, here we simply treat the bad outputs as inputs from the cost point of view.
We use model (10) to assess the performance of 32 paper chemical mills and the computed efficiency scores and ranking results are also displayed in Table 5. e last column gives their ranks according to efficiency scores obtained by our method. Of the 32 paper mills, only three mills, named mills 9, 12, and 25, are identified to obtain super-efficiency scores and they are ranked as the top three accordingly. For example, mill 9 and 12 get super-efficiency of 1.0734 and 1.4546, respectively, due to that the scaling factor of output surplus in one good output (paper) is 0.0734 and 0.4546 separately and there exist no slacks in all inputs and outputs. In contrast, 29 mills fail to get super-efficiency scores. For instance, mill 6 ranks dead last because it does not have any scaling factor of each input or output. Meanwhile, positive slacks are found to exist in two inputs and one bad output. Although mill 31 achieves the input saving scaling factor of 0.0229 in the first input (labor) and the output surplus scaling factor of 0.1077 in the good output (paper), its efficiency score is identified as 0.8538 and is ranked eighth finally. e reason for this is that there exists a slack amount of 13.7051 in the bad output (BOD), which brings more negative impact to efficiency scores than the positive effect brought by input and output scaling factors.  Table 3: Results from our model (10).   As can be seen from Table 5, the efficiency score or the super-efficiency score for each mill can be obtained by the integrated model (10). is course by our method prevents transformations among three submodels of the method by Chen. e final efficiency scores ψ * k (k � 1, . . . , 32) are affected synthetically by all decision variables that include input saving and output surplus scaling factors, input and output slacks. So we can fully differentiate the performances of all 32 paper mills and can provide a complete ranking criterion for all DMUs to be compared with regard to these efficiency scores.

Conclusions and Remarks
e classic SBM model proposed by Tone projects the DMU under evaluation at a Pareto-efficient reference point on the production frontier, while the S-SBM model proposed by Tone may not apply and the reference point on the superefficiency frontier by S-SBM model may not be Pareto-efficient. Chen found an approach through transformations among three submodels to handle this issue.
In the present paper, we build a universal model to realize the integration of SBM model and S-SBM model. Our approach not only shows the specific scaling factors for each input and output of a specific DMU explicitly, but also provides input saving, output surplus, and slacks information simultaneously only in one model. And we can see clearly how the efficiency scores are obtained by the optimal value for each decision making variable. us, more recognizing information on DMUs is revealed via two numerical examples and an empirical study in paper chemical mills. e composite efficiency scores for 32 mills are represented and ranked by integrating the effect of all decision variables, which include input saving, output surplus scaling factors, and input and output slacks. e current paper also overcomes the problem of a discontinuous gap between SBM score and S-SBM score found by Chen. Model (10) is proved to make sure that for each DMU under evaluation it supplies a Pareto-efficient reference point on the efficient frontier. is is achieved by incorporating input and output slacks into S-SBM model and controlling optimal values for each variable through the use of big M. For one specific input (output), input saving scaling factor (output surplus scaling factor) and input slack (output slack) are not permitted to appear at the same time. Especially, for a certain DMU, we can judge whether the reference point in S-SBM model is Pareto-efficient through our model. at is, the reference point ((1 + t * i )x ik , (1 − β * r )y rk ) for DMU k in S-SBM model (7) is not Pareto-efficient if and only if there exists s − * i ′ > 0 (i ′ ≠ i ∈ (1, . . . , m)) or s + * r′ > 0 (r ′ ≠ r ∈ (1, . . . , m)) under model (10).

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare no conflicts of interest.