A Range Adjusted Measure of Super-Efficiency in Integer-Valued Data Envelopment Analysis with Undesirable Outputs

2.1 Classic Additive DEA Model and Range Adjusted Measure

Charnes, et al. first proposed model (1) to evaluate the performance of DMU_k (the DMU under evaluation) in 1985^[10]. The model was named “additive DEA model” because its objective is to maximize the sum of all slacks. This model simultaneously deals with the slacks for inputs and desirable outputs, so it can identify all inefficiencies in inputs and desirable outputs^[10].

where s⁻_j indicates the slack related to the jth input; sp+denotes the slack related to the pth desirable output; and λ_i is the weight forDMU_i. Note that the original additive DEA model proposed by Charnes, et al. is under the CRS (constant returns to scale) condition while model (1) assumes VRS (variable returns to scale)^[10]. The original CRS additive DEA model can be obtained by deleting ∑i=1qλi=1from model (1). According to model (1), DMU_k is efficient if and only if all slacks equal to 0 (sj−∗=sp+∗=0).We let the superscript * indicate the optima obtained from DEA models. Obviously, model (1) cannot generate efficiency scores for DMUs. In addition, it treats integer-valued variables as real-valued variables and cannot deal with undesirable outputs. Therefore, this model is not good enough for our problem.

Cooper, et al. developed the RAM-DEA model (model (2)) based on the additive DEA model, which maximized the weighted sum of all slacks^[12]. Different from the additive DEA model, the RAM-DEA model can generate efficiency scores. θk∗is the efficiency score of DMU_k. Moreover, the RAM-DEA model has five properties that should be satisfied. 1)0≤θk∗≤1,which means that the efficiency scores obtained from model (2) are between zero and one. 2)θk∗=1(sj−∗=sp+∗=0)if and only if DMU_k is fully Pareto-Koopmans efficient. 3)θk∗is invariant to the units of inputs and outputs and invariant to alternative optima. 4)θk∗is invariant to transformation (θk∗is not affected by a change in the origin of coordinate values). 5) θ^∗_k has strong monotonicity^{[20, 21, 22]}. However, model (2) cannot handle integer-valued variables or undesirable outputs. Therefore, we update this model and extend the updated RAM-DEA model to a super-efficiency RAM-DEA model.

where Rj−=maxixji−minixjiis the range of the jth input and R⁺_p = max_i{y_pi} − min_i {y_pi} is the range of the pth desirable output. The other mathematical notations are the same as those in model (1).

2.2 Integer-Valued RAM-DEA Model with Undesirable Outputs

Suppose x_ji ∈ X^I (j = 1, 2, · · · , g), y_pi ∈ Y ^I (p = 1, 2, · · · , o), z_hi ∈ Z^I (h = 1, 2, · · · , e), x_ji ∈ X^NI (j = g + 1, g + 2, · · · , r), y_pi ∈ Y ^NI (p = o + 1, o + 2, · · · ,m), and z_hi ∈ Z^NI (h = e+1, e+2, · · · , t), where XI∈Z+g,YI∈Z+o,ZI∈Z+eindicate the subsets of corresponding integer-valued variables while X ^NI, Y ^NI, Z ^NI denote the subsets of corresponding real-valued variables. Then, the DEA production possibility set (PPS) assuming VRS for our problem can be represented by

The efficiency score of DMU_k relative to the T_IUDEA reference technology can be computed by solving the proposed integer-valued RAM-DEA model with undesirable outputs (RAM-IUDEA, model 4). In the model, λ_i is the weight of DMU_i; Rj−=maxixji−minixjiis the range of the jth input; Rp+=maxiypi−miniypiis the range of the pth desirable output; Rh−=maxizhi−minizhiis the range of the hth undesirable output; sjI−,sj−,spI+,sp+,ςhI−,and ςh−indicate the slacks related to integer-valued inputs, real-valued inputs, integer-valued desirable outputs, real-valued desirable outputs, integer-valued undesirable outputs, and real-valued undesirable outputs, respectively; ^Z is the set of integer values and therefore sjI−∗,spI+∗,and ςhI−∗must be integers.

Model (4) has all the advantages of the classic RAM-DEA model and is able to identify all inefficiencies in integer-valued and real-valued inputs, integer-valued and real-valued desirable outputs, and integer-valued and real-valued undesirable outputs. Note that we use the slacks-based measure (SBM) approach (∑i=1qzhiλi≤zhk−ςhI−,h=1,2,⋯,e;∑i=1qzhiλi=zhk−

ςh−,h=e+1,e+2,⋯,t,whereςhI−andςh−are the slacks related to integer-valued undesirable outputs and real-valued undesirable outputs, respectively) and formulate RAM-DEA models to handle undesirable outputs^[58]. There are several approaches to deal with undesirable outputs, such as the assumption of weak disposability proposed by Färe, et al.^[59], direction distance function (DDF) suggested by Chung, et al.^[60,61], linear or non-linear monotonic decreasing transformation^[62,63], and treating undesirable outputs as inputs^[13]. Compared with these approaches, the SBM approach is simple and easy to understand^[58]. The constraints related to integer-valued variables are inequalities because ∑i=1qxjiλi∗,∑i=1qypiλi∗,and∑i=1qzhiλi∗may not be integers (where0≤λi∗≤1)whilexjk−sjI−∗,ypk+spI+∗,andzhk−ςhI−∗must be integers. This method can effectively deal with integer-valued variables when measuring efficiency, so it has been widely used^[52,64].

Model (4) is the first RAM-DEA model that can deal with undesirable outputs and integer-valued variables. The main advantages of the novel model are: 1) It is a non-oriented and non-radial DEA model, which enables decision makers to simultaneously and non-proportionally improve inputs and outputs; 2) It can handle integer-valued variables and undesirable outputs, and as such the results obtained from our model are more reliable; 3) The results can be easily obtained because the model is based on linear programming.

2.3 Integer-Valued Super-Efficiency RAM-DEA Model with Undesirable Outputs

However, as the efficiency scores of all efficient DMUs must equal to one, it is impossible to differentiate efficient DMUs using model (4). To develop an integer-valued super-efficiency RAM-DEA model with undesirable outputs (Super-RAM-IUDEA), we cannot just delete the evaluated DMU from the reference set because the model may be infeasible for efficient DMUs if we do so. In this context, we develop model (11) to calculate the super-efficiency score φk∗of efficient DMU_k. In the model, ξjI−,ξj−,ξpI+,ξp+,τhI−,andτh−denote DMU_k’s integer-valued inputs savings, real-valued inputs savings, integer-valued desirable outputs surpluses, real-valued desirable outputs surpluses, integer-valued undesirable outputs savings, and real-valued undesirable outputs savings, respectively. The other mathematical notations are the same as those in model (4). Model (11) has all the advantages of model (4) and can be used to accurately rank efficient DMUs.

3.1 Data

As shown in Table 1, the data about the RTSs of 31 Province / Autonomous region / Municipality in Chinese mainland in 2017 are collected from the National Bureau of Statistics of China^[65].

x₁, x₂, y₁ and z₁ denote the “labor”, “fixed capital investment”, “value-added” and “transport accidents”, respectively. As the data related to all the variables in 2019 and the data about “fixed capital investment” in 2018 are not available, we use the dataset in 2017. Note that Nei Mongol is also called Inner Mongolia. The differences between China’s RTSs are huge.

Tibet, the least developed and sparsely populated area in China, has the fewest “labor”, the least “value-added”, and the fewest “transport accidents” in its RTS. Ningxia, one of the least developed areas in China, has the least “fixed capital investment”. Guangdong, the most developed and populous area in China, has the most “labor”, the most “value-added”, and also the most “transport accidents” in its RTS. Sichuan, whose GDP ranks the 6th and population ranks the 4th in Chinese mainland, has the most “fixed capital investment”.

3.2 Efficiency

The RAM-IUDEA model (model 4) is used to measure the efficiency of China’s RTSs, and the results are shown in Table 2. Ten RTSs are efficient DMUs, i.e., Tianjin, Hebei, Shanxi, Liaoning, Shanghai, Jiangsu, Shandong, Guangdong, Tibet, and Ningxia. Note that the RTSs of Tibet, Ningxia, and Guangdong are regarded as efficient DMUs while the performance of Sichuan’s RTS is the worst. The four RTSs have the maximum or minimum values of the selected variables as we mentioned in Subsection 3.1.

The inputs waste, desirable outputs surpluses, and undesirable outputs savings for inefficient China’s RTSs are shown in Table 3. s1I−,s2−,s1+andς1I−respectively indicate the labor inputs waste, the fixed capital investment inputs waste, the value-added outputs shortage, and the transport accidents outputs excess. Decision makers can further calculate the targets (to eliminate slacks) for DMUs based on the results and equations (5)∼(10). For example, Beijing’s RTS has the largest number of labor to cut, Yunnan’s RTS has the largest amount of fixed capital investment to reduce, Qinghai’s RTS has the largest amount of value-added to increase, and Guizhou’s RTS has the largest number of transport accidents to reduce.

The results show that our novel RAM-IUDEA model is able to generate the integer-valued slacks (and targets) for integer-valued variables. This advantage of our model is important for managers because it makes it simple for them to make and implement an efficiency improvement plan. Moreover, integer-valued DEA models are more reliable than real-valued DEA models as we explained in Section 1 and Section 2.

3.3 Super-Efficiency

In order to differentiate efficient China’s RTSs the Super-RAM-IUDEA model (model (11)) is applied. As shown in Table 4, the performance of Hebei’s RTS is the best, followed by Shanghai, Guangdong, Tibet, Ningxia, Shandong, Shanxi, Jiangsu, Liaoning, and Tianjin.

Table 5 shows the inputs savings, desirable outputs surpluses, and undesirable outputs savings for efficient RTSs. ξ1I−,ξ1−,ξ1+andτ1I−represent the labor inputs savings, fixed capital investment inputs savings, value-added outputs surpluses, and transport accidents outputs savings, respectively. The results not only show the rankings of efficient RTSs but also enable decision makers to identify the strengths of efficient RTSs. Moreover, the results are reliable because the models can generate integer values for integer-valued variables and real values for real-valued variables.

Finally, as shown in Table 6, we get the performance rankings of China’s RTSs.

3.4 Discussion

Decision makers should take undesirable outputs into account when measuring the performance of DMUs with undesirable outputs. Otherwise, the results may be problematic. If we do not consider the “transport accidents” when measuring the efficiency of China’s RTSs, the results would be inaccurate as shown in Table 7. For example, the RTS of Shanghai is regarded as an inefficient DMU if we apply the integer-valued super-efficiency RAM-DEA model without undesirable outputs (Super-RAM-IDEA). Shanghai is the most densely populated area and the amount of value-added by Shanghai’s RTS ranks the 11th in Chinese mainland. But the number of transport accidents in Shanghai is only 709, which is less than the other provinces except Tibet. Therefore, the ranking of Shanghai’s RTS should not be only the 18th (the results obtained from the Super-RAM-IDEA model).

In addition, decision makers cannot simply treat integer-valued variables as reals and round the real-valued solutions to integers. If we apply the real-valued super-efficiency RAM-DEA model with undesirable outputs (Super-RAM-UDEA) to measure the super-efficiency of DMUs with integer-valued variables, the resulting inputs savings, desirable outputs surpluses, and undesirable outputs savings for efficient DMUs may be wrong. Similarly, if we apply the real-valued RAM-DEA model with undesirable output (RAM-UDEA) to measure the efficiency of DMUs with integer-valued variables, the resulting inputs waste, desirable outputs shortage, and undesirable outputs excess for inefficient DMUs may be also erroneous. The results obtained from RAM-UDEA and Super-RAM-UDEA models are shown in Table 8 and Table 9, where``N''_indicates that the real-valued solutions for the RTS obtained from the RAM-UDEA or Super-RAM-UDEA models cannot be rounded to our integer-valued solutions.

The results show that there are huge differences between the results for China’s RTSs efficiency measurement obtained from our integer-valued models and those resulting from real-valued models. As to the inefficient RTSs, about 76% of RTSs’ real-valued solutions obtained from the RAM-UDEA cannot be rounded to the integer-valued solutions. For instance, the labor inputs slack in the RTS of Heilongjiang should be 140954 (as shown in Table 3), but the result obtained from the Super-RAM-UDEA model is 140956.800 (as shown in Table 8). The labor inputs savings in the RTS of Tibet should be 32325 (as shown in Table 5), but the result obtained from the Super-RAM-UDEA model is 32324.190 (as shown in Table 9). The reason for the differences is that the PPS (as shown in formula (3)) for DMUs could be changed if decision makers do not consider their integer-valued variables when measuring efficiency. Therefore, our integer-valued models are better than traditional real-valued models.

As stated above, the results may be problematic if decision makers did not take undesirable outputs into account when measuring the performance of DMUs with undesirable outputs or restrict variables to integers when measuring the performance of DMUs with integer-valued variables. The proposed RAM-IUDEA and Super-RAM-IUDEA models can be used to deal with integer-valued variables and undesirable outputs, which can help decision makers accurately measure the efficiency and super-efficiency of DMUs.