GraphTheoretic Techniques for the Study of Structures or Networks in Engineering
View this Special IssueResearch Article  Open Access
Dong Guo, ZhengQun Cai, "SuperEfficiency Infeasibility in the Presence of Nonradial Measurement", Mathematical Problems in Engineering, vol. 2020, Article ID 6264852, 7 pages, 2020. https://doi.org/10.1155/2020/6264852
SuperEfficiency Infeasibility in the Presence of Nonradial Measurement
Abstract
The original radial VRS superefficiency model in DEA excludes the DMU under evaluation from the reference set. However, it must lead to the problem of infeasibility specifically when the DMU under consideration is at the extremity of the frontier. In this paper, a modified nonradial VRS superefficiency model is established. The superefficiency model in the presence of nonradial measurement still maintains some good properties, and the original radial VRS superefficiency model infeasibility can also be detected through it. Our model with nonradial measurement can help decision makers allocate input resources and arrange production activities because it finds an efficient benchmark DMU, which is different from the reference DMU under radial measurement.
1. Introduction
Data envelopment analysis (DEA), first introduced by Charnes et al. [1] and extended by Banker et al. [2], is an effective nonparametric technique for measuring the relative efficiency of peer decisionmaking units (DMUs) with multiple inputs and outputs. Their initial models based upon the constant returns to scale (CRS) and variable returns to scale (VRS) are commonly referred to as the CCR model and the BCC model, respectively. They compute scalar efficiency scores with a range of zero to unity which indicate how efficient each DMU has performed as compared to other DMUs in converting inputs to outputs and determine efficient level or position for each DMU under evaluation among all DMUs. DMUs that obtain a score of unity are deemed as efficient and on the DEA (bestpractice) frontier, while other DMUs are treated as inefficient. Nowadays, DEA has become a popular method without any prior complicated weight assumptions since its advent and has been rapidly applied in improving the performances of different kinds of entities engaged in different activities and contexts [3, 4], such as human resources planning [5, 6], fixed cost allocation [7, 8], and resource sharing [9, 10].
However, when some DMUs all get scores of unity, these efficient DMUs cannot be distinguished further through the CCR model alone. To break the tie of efficient DMUs and further enhance the discrimination power of DEA, Andersen and Petersen [11] proposed a new model according to the CCR model, which is called superefficiency model where the DMU under evaluation is excluded from the reference set. It allows efficient DMUs to have efficiency scores larger than or equal to unity (under inputoriented superefficiency model), and for inefficient DMUs, the superefficiency model yields scores that are identical to those received from the CCR model. Analogously, based upon the variable returns to scale (VRS) model of Banker et al. [12], the VRS superefficiency model can be obtained. But under the condition of VRS, the superefficiency model may be infeasible when some efficient DMUs are under evaluation, while the superefficiency model under CRS does not suffer the problem of infeasibility. In face of this trouble, much effort has been focused on solving the problem of VRS superefficiency model’s infeasibility.
Seiford and Zhu [13] indicate that infeasibility must occur in the case of the variable returns to scale (VRS) superefficiency model and further provide the necessary and sufficient conditions for infeasibility of superefficiency models. Lovell and Rouse [14] assign a userdefined scaling factor to find a feasible solution for those efficient DMUs for which feasible solutions are unavailable in the VRS superefficiency model. Chen [15] shows that in order to fully characterize the superefficiency, both inputoriented and outputoriented superefficiency DEA models are needed when infeasibility occurs. Chen [15] further points out that superefficiency can be regarded as input saving/output surplus achieved by an efficient DMU. Cook et al. [16] develop a modified VRS superefficiency model that yields optimal solutions and superefficiency scores that characterize the extent of superefficiency in both inputs and outputs. Lee et al. [17] develop a twostage process to address the VRS infeasibility issue. In the first stage, they test whether a VRS superefficiency model is infeasible by investigating the existence of output surplus (input saving) when infeasibility occurs in the inputoriented (outputoriented) VRS superefficiency model. In the second stage, they proposed a modified VRS superefficiency model to yield a superefficiency score that characterizes both the radial efficiency and input saving/output surplus. Chen and Liang [18] further prove that the twostage process can be solved in a single linear program. However, when a DMU has zero data, these models may still be infeasible. Thrall [19] and Zhu [20] point out that the CRS superefficiency model can also be infeasible when an efficient DMU has zero input values. The same conclusion can be applied to nonCRS superefficiency models. Lee et al. [21] first point out that zero output data will not lead to infeasibility of the outputoriented superefficiency models developed in the studies of Cook et al. [16], Lee et al. [17], and Chen and Liang [18]. This is because the output side of the constraints can always be satisfied. Therefore, they only assume that some inputs are zero for some efficient DMUs. Then, they revise the model of Lee et al. [17], and the revised model will be feasible when zero data exist in inputs.
Nevertheless, these models mentioned above so far only consider radial efficiency. The current paper extends the work of Lee and Zhu [21] to nonradial measurement. And we find that the superefficiency model in the presence of nonradial measurement still maintains some good properties. In fact, superefficiency with nonradial measurement can also help decision makers more allocate input resources according to underlying preferences and resource availability.
The remainder of this paper is organized as follows: Section 2 looks back upon several radial superefficiency models that had been established and the problem of superefficiency infeasibility. In Section 3, we develop our superefficiency model with nonradial measurement and demonstrate that the model has some good properties. Meanwhile, we point out the difference between radial and nonradial measurement. In Section 4, the newly developed approach is applied to a data set on the 15 Illinois strip mines. In the end, main conclusions are given in Section 5.
2. Radial SuperEfficiency Models
Suppose there are n DMUs {}. Each consumes a set of m inputs, , in the production of a set of s outputs, . Based upon the VRS model of Banker et al. [12], the inputoriented VRS superefficiency model for efficient can be expressed aswhere the under consideration is excluded from the reference set. Obviously, one of the circumstances where model (1) is infeasible if the DMU under consideration has the largest outputs, regardless of the input values. In fact, as pointed out by Lee et al. [17], when the outputs of the evaluated DMU is outside the production possibility set spanned by the outputs of the remaining DMUs, the infeasibility of inputoriented superefficiency will occur.
Consider the same simple numerical example given in the study of Lee et al. [17] in Table 1. Owing that DMU_{D} has the largest outputs and DMU_{E} has zero input data, they are both infeasible under model (1). Lee et al. [17] develop a new superefficiency model and twostage process to investigate the issue of the VRS superefficiency infeasibility. Chen and Liang [18] establish a single model and integrate the twostage process into an equivalent “one model” approach:where is a userdefined large positive number, and in the study of Cook et al. [16], is set equal to 10^{5}. When all inputs are positive, model (1) is feasible if and only if all in model (2). The superefficiency score is defined as , where . However, owing to zero data X_{2} = 0 in DMU_{E}, as depicted in Figure 1, DMU_{E} remains infeasible under model (2).
Subsequently, Lee et al. [21] consider the situation when zero input data exist. They propose model (3) to address this issue about zero input data and define the superefficiency score as when neither one of the sets and is empty, in which and . They demonstrate that models (2) and (3) yield the same results when data are positive:where . The radial scaling factor , superefficiency score, and more details of results by using model (3) that are computed by Lee et al. [21] are listed in Table 2. Here, DMU_{E} can be feasible under model (3).

3. A Modified Nonradial SuperEfficiency Model
In the real production processes, decision makers tend to arrange production plan reasonably according to possession of the resources and production capability. Investment of input resources must be consistent with resource availability and underlying preferences (for example, decision makers may be more willing to use cheaper to help lower costs). Radial scaling with inputs, i.e., decreasing inputs, proportionally is not always met in the actual production processes. Therefore, it is necessary and meaningful to incorporate nonradial measurement into superefficiency model at some point. So, we modify model (3) to the situation of nonradial measurement. Consider the following model for :
Under nonradial measurement, we permit that each input element could be scaled down by an exclusive scaling factor, , instead of the common proportional scaling factor for all inputs in model (3). Obviously, model (4) still works when zero input data exist. Based on model (4), some good properties can be derived as follows.
Theorem 1. In model (4), and .
Proof. Note that . Therefore, . Also, there exists ; thus, . So, when is large enough, must exist; then, . Furthermore, .
Theorem 2. Model (4) is identical with model (3) when .
In other words, when , the objective function values of model (3) and model (4) are identical.
Theorem 3. Model (4) is always feasible.
Proof. According to the proof for feasibility of model (3) of Lee et al. [21], model (3) is always feasible. Suppose is a feasible solution of model (3), let , then, is just a feasible solution to model (4). In other words, the feasible region of model (4) is not empty. Thus, model (4) is always feasible.
Theorem 4. Model (1) is infeasible if and only if(i)Some when , where is the optimal solution in model (4)(ii)Some or some when , where is the optimal solution in model (4)
Proof. (i)When , the first constraint condition in model (4) becomes , . Let , then model (4) could turn out to be Based upon the studies of Cook et al. [16], Lee et al. [17] and Chen and Liang [18], the proposition is obviously true, i.e., model (1) is infeasible if and only if some in model (4).(ii)When , and in model (4) indicate that model (1) is feasible. So, model (1) is infeasible, and there must be some or some in model (4). On the contrary, if model (1) is feasible, this means and is a feasible solution to model (4), which directly contradict the fact that some or is the optimal solution in model (4).
Theorem 5. Results of model (4) is not larger than that of model (3).
Proof. Adding the constraint condition of to model (4), model (4) has become model (3). So, the feasible region of model (3) is smaller than model (4). A relaxation in one of the constraints may yield a better optimal solution inside the new polyhedron in space. Therefore, the result of model (4) is not larger than that of model (3).
Theorem 6. Models (4) and (3) yield the same amount of input saving and output surplus when zero input data exist.
Proof. For any , suppose has zero input data , we have . Due to and the minimization of objective function, there must exist. So, the first constraint condition in model (4) will be, which is identical with that in model (3) when .
Theorem 6 also indicates the way where the DMU under evaluation in model (4) is projected to the frontier formed by other DMUs is the same as that in model (3) when the DMU has zero input values.
The superefficiency model (4) with nonradial measurement of ours also determines an efficient referent (benchmark) DMU that is on the frontier formed by the remaining DMUs. In order to fully characterize the superefficiency by input saving/output surplus achieved by an efficient DMU if it exists when the VRS superefficiency feasibility is present, and based upon Cook et al. [16] and Lee et al. [17, 21], we also denote and . Then input savings index and output savings index can be defined in the following manner:where and are the cardinality of the sets R and I, respectively.
Unlike superefficiency score of Lee et al. [21], we put as the nonradial efficiency. Then, the superefficiency score which could also characterize the superefficiency in both inputs and outputs can be defined asThe efficiency measure consists of three parts: the nonradial efficiency , the input saving index , and the output surplus index . As described by Lee et al. [21], when the set is not empty, the input savings index reflects how far the DMU_{k} is below the dashed horizontal efficient boundary (see the dashed line through A in Figure 1). And the output surplus index reflects how far the DMU_{k} is above the dashed efficient boundary.
We use Figure 1 to illustrate the difference from the method of Lee et al. [21], mainly including the different projection mode to the (bestpractice) frontier formed by other DMUs and the different nonradial superefficiency result obtained. For instance, for DMU_{B}, the efficient frontier is made up of broken lines CA and AE. DMU_{B} is on the left of the frontier which implies that its superefficiency should be greater than 1. DMU_{B} achieves radial efficiency of 1.4 and gets its projective point (efficient referent DMU) at on the frontier through scaling down proportionally under radial measurement. While under nonradial measurement, DMU_{B} gets its projective point (efficient referent DMU) at ; that is to say, DMU_{B} takes DMU_{A} as its benchmark under nonradial measurement. Meanwhile, DMU_{B} gains its nonradial efficiency of . For DMU_{C}, on account that it has the largest input X_{2} = 4 and in model (4), it is infeasible under model (1), but it is on the frontier and gets radial efficiency of 1 under model (3). It gets its projective point on the frontier at under nonradial measurement, i.e., DMU_{C} puts DMU_{B} as its bestpractice benchmark. Now, DMU_{C} earns its nonradial efficiency of . DMU_{C} has the same amount of output surplus of under both measurements which equals the distance from DMU_{B} to DMU_{C}. Finally, for DMU_{E}, it has zero input data and in model (4), it is also infeasible under model (1). However, it reaches the efficient referent (benchmark) DMU_{A} in the same type of way and derives the same input saving of under both measurements. Concretely, DMU_{E} goes through point (2, 0) on Xaxis by radial or nonradial scaling and then moves upwards by 1 (). But it has different radial efficiency of 0.666667 and nonradial efficiency of 0.33349. Table 3 shows more specific details and final superefficiency result of model (4) for each DMU under nonradial measurement.

4. An Empirical Example
We apply our approach to the data set which contains a set of 15 Illinois strip mines in Lee et al. [21]. The data set is shown in Table 4, which has only one output of tons of coal produced and eight inputs. Of the inputs, one is labor in thousand miner days, and the other variables include three capital variables (K1, K2, and K3) and four geological variables (T1, D1, T2, and D2). In Table 4, K1 = bucket capacity of draglines, K2 = dipper capacity of power shovels, K3 = earth moving capacity of wheel excavators, T1 = thickness of first (upper) seam mined, D1 = depth to first seam mined, T2 = thickness of second (lower) seam mined, and D2 = depth to second seam mined. In this particular data set, there are many zero inputs.

Table 5 shows superefficiency score and solutions details of each DMU under four models, respectively. But the very first point which needs to be made is that we set , here, to seek its optimal solutions of all DMUs under model (4) with nonradial measurement. Because under nonradial measurement, it should adjust the value of userdefined large positive number to suit for the specific data set to find its optimal solution. From Table 5, we discover that there are 5 DMUs that are infeasible under the original superefficiency model (1). From Theorem 4 and the solution details of model (4), we know that DMU_{1}, DMU_{2}, and DMU_{4} have nonzero output surplus because of the existence of and DMU_{10} and DMU_{13} have nonzero input saving because of the existence of and , respectively. For example, DMU_{10} is infeasible under model (1) because the second input gets input savings of 25.49864 () compared with the factual second input amount of zero. That is the reason why DMU_{10} is infeasible under model (1) and model (2). For DMU_{13}, it gets output surplus of 5 () under model (2), so it is infeasible under model (1). However, under model (3) and model (4), output surplus is eliminated and input saving of the third input with the value of 0.40368 () comes into being. The result under model (4) with nonradial measurement is smaller than that of model (3) for each DMU. Using model (3) and model (4), the same amount of input saving or output surplus derived under both radial and nonradial measurement can also be seen in Table 5.

5. Conclusions
Based upon the previous work, the current paper extends radial superefficiency model to the circumstances of nonradial measurement and establishes a modified nonradial superefficiency model. We find that the superefficiency model with nonradial measurement still maintains some good properties and the original radial VRS superefficiency model can also be detected whether it is infeasible through our nonradial superefficiency model. In the actual production processes, due to underlying preferences and resource availability, superefficiency model with nonradial measurement can help decision makers seek an efficient benchmark different from the reference under radial measurement. It is beneficial for decision makers to allocate input resources, make production plan, and arrange production activities. This paper only discusses the inputoriented nonradial model, for outputoriented situation, and it can be discussed in the same way.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was partially supported by the Humanities and Social Science Youth Foundation of the Ministry of Education of China (Grant No. 20YJC630029); the Natural Science Foundation of Anhui Province of China (Grant No. 2008085MG228); and the Scientific Research Starting Foundation of Anhui Jianzhu University (Grant No. 2019QDZ09).
References
 A. Charnes, W. W. Cooper, and E. Rhodes, “Measuring the efficiency of decision making units,” European Journal of Operational Research, vol. 2, no. 6, pp. 429–444, 1978. View at: Publisher Site  Google Scholar
 R. D. Banker, A. Charnes, and W. W. Cooper, “Some models for estimating technical and scale inefficiencies in DEA,” Management Science, vol. 30, no. 9, pp. 1078–1092, 1984. View at: Publisher Site  Google Scholar
 W. W. Cooper, L. M. Seiford, and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEASolver Software, Kluwer Academic Publishers, Boston, MA, USA, 2000.
 W. W. Cooper, L. M. Seiford, and J. Zhu, Handbook on Data Envelopment Analysis, Springer, Berlin, Germany, 2011.
 M.M. Yu, C.C. Chern, and B. Hsiao, “Human resource rightsizing using centralized data envelopment analysis: evidence from Taiwan’s Airports,” Omega, vol. 41, no. 1, pp. 119–130, 2013. View at: Publisher Site  Google Scholar
 A. Varmaz, A. Varwig, and T. Poddig, “Centralized resource planning and yardstick competition,” Omega, vol. 41, no. 1, pp. 112–118, 2013. View at: Publisher Site  Google Scholar
 F. Li, Q. Zhu, and Z. Chen, “Allocating a fixed cost across the decision making units with twostage network structures,” Omega, vol. 83, pp. 139–154, 2019. View at: Publisher Site  Google Scholar
 Q. An, P. Wang, A. Emrouznejad, and J. Hu, “Fixed cost allocation based on the principle of efficiency invariance in twostage systems,” European Journal of Operational Research, vol. 283, no. 2, pp. 662–675, 2020. View at: Publisher Site  Google Scholar
 D. Gong, S. Liu, and X. Lu, “Modelling the impacts of resource sharing on supply chain efficiency,” International Journal of Simulation Modelling, vol. 14, no. 4, pp. 744–755, 2015. View at: Publisher Site  Google Scholar
 Q. An, Y. Wen, T. Ding, and Y. Li, “Resource sharing and payoff allocation in a threestage system: integrating network DEA with the Shapley value method,” Omega, vol. 85, pp. 16–25, 2019. View at: Publisher Site  Google Scholar
 P. Andersen and N. C. Petersen, “A procedure for ranking efficient units in data envelopment analysis,” Management Science, vol. 39, no. 10, pp. 1261–1264, 1993. View at: Publisher Site  Google Scholar
 R. D. Banker, A. Charnes, and W. W. Cooper, “Some models for estimating technical and scale inefficiencies in data envelopment analysis,” Management Science, vol. 30, no. 9, pp. 1078–1092, 1984. View at: Publisher Site  Google Scholar
 L. M. Seiford and J. Zhu, “Infeasibility of superefficiency data envelopment analysis models,” INFOR, vol. 37, pp. 174–187, 1999. View at: Publisher Site  Google Scholar
 C. A. K. Lovell and A. P. B. Rouse, “Equivalent standard DEA models to provide superefficiency scores,” Journal of the Operational Research Society, vol. 54, no. 1, pp. 101–108, 2003. View at: Publisher Site  Google Scholar
 Y. Chen, “Measuring superefficiency in DEA in the presence of infeasibility,” European Journal of Operational Research, vol. 161, no. 2, pp. 545–551, 2005. View at: Publisher Site  Google Scholar
 W. D. Cook, L. Liang, Y. Zha, and J. Zhu, “A modified superefficiency DEA model for infeasibility,” Journal of the Operational Research Society, vol. 60, no. 2, pp. 276–281, 2009. View at: Publisher Site  Google Scholar
 H.S. Lee, C.W. Chu, and J. Zhu, “Superefficiency DEA in the presence of infeasibility,” European Journal of Operational Research, vol. 212, no. 1, pp. 141–147, 2011. View at: Publisher Site  Google Scholar
 Y. Chen and L. Liang, “Superefficiency DEA in the presence of infeasibility: one model approach,” European Journal of Operational Research, vol. 213, no. 1, pp. 359360, 2011. View at: Publisher Site  Google Scholar
 R. M. Thrall, “Chapter 5 duality, classification and slacks in DEA,” Annals of Operations Research, vol. 66, no. 2, pp. 109–138, 1996. View at: Publisher Site  Google Scholar
 J. Zhu, “Robustness of the efficient DMUs in data envelopment analysis,” European Journal of Operational Research, vol. 90, no. 3, pp. 451–460, 1996. View at: Publisher Site  Google Scholar
 H.S. Lee and J. Zhu, “Superefficiency infeasibility and zero data in DEA,” European Journal of Operational Research, vol. 216, no. 2, pp. 429–433, 2012. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Dong Guo and ZhengQun Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.