A Range Adjusted Measure of Super-efficiency in Integer-valued Data Envelopment Analysis with Undesirable Outputs

DEA (Data Envelopment Analysis) models can be divided into two groups: radial DEA and non-radial DEA, and the latter has higher discriminatory power than the former. The Range Adjusted Measure (RAM) is an effective and widely used non-radial DEA approach. However, to the best of our knowledge, there is no literature on the integer-valued super-efficiency RAM-DEA model, especially when undesirable outputs are included. We first propose an integer-valued RAM-DEA model with undesirable outputs and then extend this model to an integer-valued super-efficiency RAM-DEA model with undesirable outputs. Compared with other DEA models, the two novel models have many advantages: (1) they are non-oriented and non-radial DEA models, which enable decision makers to simultaneously and non-proportionally improve inputs and outputs; (2) they can handle integer-valued variables and undesirable outputs, so the results obtained are more reliable; (3) the results can be easily obtained as it is based on linear programming; (4) the integer-valued super-efficiency RAM-DEA model with undesirable outputs can be used to accurately rank efficient DMUs. The proposed models are applied to evaluate the efficiency of China’s regional transportation systems (RTSs) considering the number of transport accidents (an undesirable output). The results help decision makers improve the performance of inefficient RTSs and analyze the strengths of efficient RTSs.


Introduction
1 Data envelopment analysis (DEA) is generally regarded as an effective nonparametric technique to evaluate the relative 2 efficiency (performance) of decision making units (DMUs) [1,2] . DEA models can be divided into two groups: radial DEA 3 and non-radial DEA [3,4]. Radial DEA models, i.e., the CCR model by Charnes et al. [5] and the BCC model by Banke et al. [6] , 4 only allow proportional reductions of inputs or increases of outputs to improve the performance of DMUs, while non-radial studied the joint decomposition of RAM-UDEA and Luenberger productivity indicators and applied this approach to 48 evaluate the atmospheric environmental performance in China [42] . Eftekharian et al. proposed a RAM-UDEA model to 49 measure the environmental efficiency of industrial industries in Iran [43] . Wang and Yuan proposed a RAM-UDEA model to 50 measure the energy and CO2 (carbon dioxide) emission efficiencies of cigarette companies in China [44] . Chen et al. used the 51 RAM-UDEA approach to compare congestion effects of 46 countries along the Belt and Road [45] . Yuan et al. presented a 52 RAM-UDEA model to assess the inclusive and sustainable industrial development in China [46] . 53 However, conventional radial and non-radial DEA models cannot be used to distinguish better performers from efficient 54 DMUs because the efficiency scores of all efficient DMUs must be equal to 1. To differentiate the efficient DMUs, it is 55 necessary to measure super-efficiency. Super-efficiency DEA models can be developed by removing the DMU under 56 evaluation from the reference set [47,48] allowing their efficiency (super-efficiency) scores greater than one. In this way, 57 decision makers can rank the efficient DMUs based on their super-efficiency scores. Andersen and Petersen proposed the 58 first radial super-efficiency DEA model [49] . Radial super-efficiency DEA models can only proportionally change inputs and 59 outputs and therefore their discriminatory ability is weaker than non-radial super-efficiency DEA models [50,51] . In the field of 60 non-radial super-efficiency DEA, Du et al. [52] , Guo et al. [53] , Yu and Hsu [54] , among others studied super-efficiency additive 61 DEA models, and Tone [55] , Tran et al. [56] , Chen et al. [57] , and many other scholars researched super-efficiency SBM-DEA 62 models. Many studies show that super-efficiency additive DEA models are always feasible while conventional radial super-63 efficiency DEA models would be infeasible under the assumption of VRS (Variable Returns to Scale) [52,54] .

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To the best of our knowledge, however, there is no literature on the super-efficiency RAM-DEA model, especially 3 the results obtained from our model are more reliable; (3) the results can be easily obtained because the model is based on 73 linear programming; (4) the Super-RAM-IUDEA model can be used to accurately rank efficient DMUs.

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The remaining of this paper is organized as follows. The RAM-IUDEA and Super-RAM-IUDEA models are presented 75 in Section 2. The proposed models are applied to evaluate the efficiency (super-efficiency) of China's regional transportation 76 systems in Section 3. A concluding comment and the limitation of this research are summarized in Section 4. We also 77 highlight the contributions of this paper in the last section.   [62,63] , and treating undesirable outputs as inputs [13] . Compared with these 133 approaches, the SBM approach is simple and easy to understand [58] . The constraints related to integer-valued variables are 134 inequalities because Model (4) integers. This method makes the efficient frontier of DMUs with integer-153 valued variables more reliable [17][18][19] .  and all slacks equal to 0. Proposition 1 has been proved.

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(2) *1 k   because the second part of the right side of the objective function in model (11)

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In this section, we apply our novel RAM-DEA models to evaluate the performance of China's regional transportation 218 systems (RTSs) considering several integer-valued variables including the number of transport accidents (an undesirable 219 output). As far as we know, limited attention has been paid to this topic [67][68][69][70] .

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Let 1 x , 2 x , 1 y , and 1 z denote the "labor", "fixed capital investment", "value-added", and "transport accidents", 230 respectively. As shown in Table 1, the data about the RTSs of 31 provinces in mainland China in 2017 are collected from the 231 National Bureau of Statistics of China [65] . As the data related to all the variables in 2019 and the data about "fixed capital 232 investment" in 2018 are not available, we use the dataset in 2017. Note that Nei Mongol is also called Inner Mongolia.

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The differences between China's RTSs are huge. Tibet, the least developed and sparsely populated area in China, has 234 the fewest "labor", the least "value-added", and the fewest "transport accidents" in its RTS. Ningxia, one of the least 235 developed areas in China, has the least "fixed capital investment". Guangdong, the most developed and populous area in 236 China, has the most "labor", the most "value-added", and also the most "transport accidents" in its RTS. Sichuan, whose 237 GDP ranks the 6th and population ranks the 4th in mainland China, has the most "fixed capital investment".

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In order to differentiate efficient China's RTSs the Super-RAM-IUDEA model (model 11) is applied. As shown in Table   268 4, the performance of Hebei's RTS is the best, followed by Shanghai, Guangdong, Tibet, Ningxia, Shandong, Shanxi, Jiangsu,

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Liaoning, and Tianjin.       Table 7). The labor inputs savings in the RTS of Tibet should be 307 32325 (as shown in Table 5), but the result obtained from the Super-RAM-UDEA model is 32324.190 (as shown in Table 8).

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The reason for the differences is that the PPS (as shown in formula 3) for DMUs could be changed if decision makers do not 309 consider their integer-valued variables when measuring efficiency. Therefore, our integer-valued models are better than 310 traditional real-valued models.

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We also propose an integer-valued RAM-DEA model with undesirable outputs besides our integer-valued super-330 efficiency RAM-DEA model. However, the objective function of this model is to maximize the slacks of the selected 331 variables so that the resulting targets for inefficient DMUs are the farthest. In the future, we will research the RAM-DEA 332 approach to find the closest targets for DMUs.

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The integer-valued super-efficiency RAM-DEA model is proposed under the assumption of strong disposability of 334 undesirable outputs. This assumption is based on our belief that the undesirable output (the number of transport accidents) in our 335 case study can be disposed of. Although the disposal of the undesirable output involves some costs, it is possible to be done with 336 an acceptable increase in the costs of production. However, there are some undesirable outputs which may be weakly 337 disposable, e.g., the amount of carbon emissions. In the future, it is necessary to develop an integer-valued super-efficiency 338 RAM-DEA model considering both strongly and weakly disposable undesirable outputs at the same time.