A modified slacks‐based ranking method handling negative data in data envelopment analysis

Performance ranking for a set of comparable decision‐making units (DMUs) with multiple inputs and outputs is an important and often‐discussed topic in data envelopment analysis (DEA). Conventional DEA models distinguish efficient units from inefficient ones but cannot further discriminate the efficient units, which all have a 100% efficiency score. Another weakness of these models is that they cannot handle negative inputs and/or outputs. In this paper, a new modified slacks‐based measure is proposed that works in the presence of negative data and provides quantitative data that helps decision makers obtain a full ranking of DMUs in situations where other methods fail. In addition, the new method has the properties of unit invariance and translation invariance, and it can give targets for inefficient DMUs to guide them to achieve full efficiency. Two numerical examples are analysed to demonstrate the usefulness of the new method.     

Data envelopment analysis with classification and regression tree – a case of banking efficiency

Abstract: Data envelopment analysis (DEA) is a non‐parametric method for measuring the efficiency and productivity of decision‐making units (DMUs). On the other hand data mining techniques allow DMUs to explore and discover meaningful, previously hidden information from large databases. Classification and regression (C&R) is the commonly used decision tree in data mining. DEA determines the efficiency scores but cannot give details of factors related to inefficiency, especially if these factors are in the form of non‐numeric variables such as operational style in the banking sector. This paper proposes a framework to combine DEA with C&R for assessing the efficiency and productivity of DMUs. The result of the combined model is a set of rules that can be used by policy makers to discover reasons behind efficient and inefficient DMUs. As a case study, we use the proposed methodology to investigate factors associated with the efficiency of the banking sector in the Gulf Cooperation Council countries.     

A new data envelopment analysis method for ranking decision making units: an application in industrial parks

Relative efficiency of decision‐making units (DMUs) is assessed by classical data envelopment analysis (DEA) models. DEA is a popular technique for efficiency evaluation. There might be a couple of efficient DMUs. Classical DEA models cannot fully rank efficient DMUs. In this paper, a novel technique for fully ranking all DMUs based on changing reference set using a single virtual inefficient DMU is proposed. To this end, the first concept of virtual DMU is defined as average of all inefficient DMUs. Virtual DMU is a proxy of all inefficient DMUs. This new method proposes a new ranking method that takes into account impact of efficient DMUs on virtual DMU and impact of efficient DMUs on influences of other efficient DMUs. A case study is given to show applicability of the proposed approach.     

The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis

Making optimal use of available resources has always been of interest to humankind, and different approaches have been used in an attempt to make maximum use of existing resources. Limitations of capital, manpower, energy, etc., have led managers to seek ways for optimally using such resources. In fact, being informed of the performance of the units under the supervision of a manager is the most important task with regard to making sensible decisions for managing them. Data envelopment analysis (DEA) suggests an appropriate method for evaluating the efficiency of homogeneous units with multiple inputs and multiple outputs. DEA models classify decision making units (DMUs) into efficient and inefficient ones. However, in most cases, managers and researchers are interested in ranking the units and selecting the best DMU. Various scientific models have been proposed by researchers for ranking DMUs. Each of these models has some weakness(es), which makes it difficult to select the appropriate ranking model. This paper presents a method for ranking efficient DMUs by the voting analytic hierarchy process (VAHP). The paper reviews some ranking models in DEA and discusses their strengths and weaknesses. Then, we provide the method for ranking efficient DMUs by VAHP. Finally we give an example to illustrate our approach and then the new method is employed to rank efficient units in a real world problem.     

A new DEA approach to fully rank DMUs with an application to MBA programs

Data envelopment analysis (DEA) can be used to evaluate the efficiencies of decision‐making units (DMUs) in various areas like education, healthcare, and energy. Several DEA methods are proposed for this purpose; however, some of these methods cannot provide a full ranking and others often overlook some considerations that arise with special characteristics of DMUs. We propose a new DEA‐based approach to achieve a full ranking of DMUs. Our approach takes various issues into account such as the initial efficiency score of the DMU, the DMUs that should be removed from the set for it to become efficient (if any) and its effects on the efficiency scores of other DMUs. We demonstrate the shortcomings of several other DEA methods and discuss how our approach overcomes these. We apply our approach to evaluate 50 MBA programs from Financial Times 2018 rankings and compare the results with the evaluations of other methods. As opposed to some methods, our approach has the advantage of differentiating between all efficient DMUs as well as inefficient ones. In addition, the results demonstrate that we can achieve a consistent ranking that considers different aspects of the problem setting. The generated scores are also used to sort DMUs in classes of preference.     

A DEA‐based method of allocating the fixed cost as a complement to the original input

This paper focuses on the problem of how to divide a fixed cost as a complement to an original input among decision‐making units (DMUs) equitably. Using the data envelopment analysis (DEA) technique, this paper concerns the problem from the perspective of efficiency analysis. It is found that not all DMUs can become efficient under common weights if a low enough fixed cost is assigned. Therefore, the global modified additive DEA (MAD) model is introduced. By optimizing the global MAD‐efficiency, a new allocation method and the corresponding algorithm to ensure the uniqueness of the allocation result is designed. The proposed method can be used under both constant returns to scale and variable returns to scale for nonnegative data; it is suitable for the situation where the costs play a great role in the production of DMUs. Numerical results show the validity and advantages of our method.     

Additive centralized and Stackelberg DEA models for two‐stage system with shared resources

Data envelopment analysis (DEA) is a nonparametric programming method for evaluating the efficiency performance of decision making units (DMUs) with multiple inputs and outputs. The classic DEA model cannot provide accurate efficiency measurement and inefficiency sources of DMUs with complex internal structure. The network DEA approach opens the “black box” of DMU by taking its internal operations into consideration. The complexities of DMU's internal structure involve not only the organization of substages, but also the inputs allocation and the operational relations among the individual stages. This paper proposes a set of additive DEA models to evaluate and decompose the efficiency of a two‐stage system with shared inputs and operating in cooperative and Stackelberg game situations. Under the assumptions of cooperative and noncooperative gaming, the proposed models are able to highlight the effects of strategic elements on the efficiency formation of DMUs by calculating the optimal proportion of the shared inputs allocated to each stage. The case of information technology in the banking industry at the firm level, as discussed by Wang, is revisited using the developed DEA approach.     

Pareto–Koopmans efficiency in two‐stage network data envelopment analysis in the presence of undesirable intermediate products and nondiscretionary factors

Many studies have recently been conducted on the evaluation of system performance with a two‐stage network structure in data envelopment analysis (DEA) literature. One of the topics of interest to researchers has been the mitigation of undesirable products or nondiscretionary factors into their corresponding possible production set (PPS) and their impact on overall efficiency calculations. Determination of decision‐making units (DMUs) with Pareto–Koopmans efficiency status is decisive in identifying benchmark units. The calculated overall efficiency status is compromised when both undesirable products and nondiscretionary factors are present. This work utilizes an axiomatic approach. A novel PPS for a two‐stage network in presence of undesirable intermediate products and nondiscretionary exogenous inputs is introduced. Based on this PPS and by focusing on the principle of mathematical dominance, new models for evaluating overall and divisional efficiencies are presented. In addition, by proposing a two‐step network DEA approach, a necessary and sufficient condition for detection of DMUs with Pareto–Koopmans efficiency status is provided. And by introducing a two‐step algorithm, a novel technique for determining overall efficiency conditions is produced. Finally, the proposed technology is applied to a practical example, and outcomes are discussed.     

A procedure for large-scale DEA computations

Data envelopment analysis (DEA), a performance evaluation method, measures the relative efficiency of a particular decision making unit (DMU) against a peer group. Most popular DEA models can be solved using standard linear programming (LP) techniques and therefore, in theory, are considered as computationally easy. However, in practice, the computational load cannot be neglected for large-scale—in terms of number of DMUs—problems. This study proposes an accelerating procedure that properly identifies a few “similar” critical DMUs to compute DMU efficiency scores in a given set. Simulation results demonstrate that the proposed procedure is suitable for solving large-scale BCC problems when the percentage of efficient DMUs is high. The computational benefits of this procedure are significant especially when the number of inputs and outputs is small, which are most widely reported in the literature and practices.     

Right and left returns to scales in data envelopment analysis: Determining type and measuring value

This study identifies types and values of right and left returns to scales (RTSs) of efficient decision making units (DMUs) in data envelopment analysis (DEA). In this research, we first introduce a new approach to estimate types of right and left returns to scales of efficient DMUs and then, values of right and left returns to scales of these DMUs are measured by presenting two new DEA models.     

Ranking efficient DMUs using cooperative game theory

The problem of ranking Decision Making Units (DMUs) in Data Envelopment Analysis (DEA) has been widely studied in the literature. Some of the proposed approaches use cooperative game theory as a tool to perform the ranking. In this paper, we use the Shapley value of two different cooperative games in which the players are the efficient DMUs and the characteristic function represents the increase in the discriminant power of DEA contributed by each efficient DMU. The idea is that if the efficient DMUs are not included in the modified reference sample then the efficiency score of some inefficient DMUs would be higher. The characteristic function represents, therefore, the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient units is dropped from the sample. Alternatively, the characteristic function of the cooperative game can be defined as the change in the efficiency scores of the inefficient DMUs that occurs when a given coalition of efficient DMUs are the only efficient DMUs that are included in the sample. Since the two cooperative games proposed are dual games, their corresponding Shapley value coincide and thus lead to the same ranking. The more an efficient DMU impacts the shape of the efficient frontier, the higher the increase in the efficiency scores of the inefficient DMUs its removal brings about and, hence, the higher its contribution to the overall discriminant power of the method. The proposed approach is illustrated on a number of datasets from the literature and compared with existing methods.     

New analytical hierarchical process/data envelopment analysis methodology for ranking decision‐making units

The analytical hierarchical process/data envelopment analysis (AHP/DEA) methodology for ranking decision‐making units (DMUs) has some problems: it illogically compares two DMUs in a DEA model; it is not compatible with DEA ranking in the case of multiple inputs/multiple outputs; and it leads to weak discrimination in cases where the number of inputs and outputs is large. In this paper, we propose a new two‐stage AHP/DEA methodology for ranking DMUs that removes these problems. In the first stage, we create a pairwise comparison matrix different from AHP/DEA methodology; the second stage is the same as AHP/DEA methodology. Numerical examples are presented in the paper to illustrate the advantages of the new AHP/DEA methodology.     

Common set of weights in data envelopment analysis under prospect theory

Data envelopment analysis (DEA) is a data‐driven tool for performance evaluation, measuring decision‐making units (DMUs) and designating them with specific weightings. The standard DEA model typically sets up that decision‐makers (DMs) are wholly rational to select the most favourable weights to obtain the maximum performance score, but does not take into account their attitude toward risk during the assessment. The prospect theory generally matches humans' psychological behaviours. Thus, our study captures the non‐rational behaviours of DMs, performing under risk scenarios, in order to construct a novel common‐weights DEA model that maximizes the total prospect value, which can vary more steeply for losses than for gains, hence obtaining a more realistic common weight scheme. Our proposed model not only generates DMUs, with higher total prospect values, but also greater degrees of satisfaction. The current study shows that the prospect theory can be aptly extended to the DEA research area, supplying a proper guideline for future DEA research.     

Ranking units in DEA based on efficiency intervals and decision‐maker's preferences

Data Envelopment Analysis (DEA) uses the best favorable weight set for the inputs and outputs of each decision‐making unit (DMU) to obtain its best possible score. Hence, this score can be considered as an upper bound of the real efficiency score. If we also use the least favorable weight set of each DMU, then a lower bound of the efficiency score can also be obtained. So, instead of one score, we can find an interval that gives all possible values of the efficiency score for each DMU. The aim of this paper is to propose an approach for determining efficiency intervals and setting up a full ranking of DMUs based on these intervals. We incorporate explicitly the decision‐maker's preferences in two phases. The first phase is for obtaining efficiency intervals, by introducing some restrictions on the input and output weights. The second one is for ranking the intervals based on the combination of the lower and the upper bounds of the efficiency intervals. The developed formulations will be illustrated through some numerical examples.     

DEA efficiency analysis: A DEA approach with double frontiers

Data envelopment analysis (DEA) is a method for measuring efficiency of peer decision-making units (DMUs). Conventional DEA evaluates the performance of each DMU using a set of most favourable weights. As a result, traditional DEA models can be considered methods for the analysis of the best relative efficiency or analysis of the optimistic efficiency. DEA efficient DMUs obtained from conventional DEA models create an efficient production frontier. Traditional DEA can be used to identify units with good performance in the most desirable scenarios. There is a similar approach that evaluates the performance indicators of each DMU using a set of most unfavourable weights. Accordingly, such models can be considered models for analysing the worst relative efficiency or pessimistic efficiency. This approach uses the inefficient production frontier for determining the worst relative efficiency that can be assigned to each DMU. DMUs lying on the inefficient production frontier are referred to as DEA inefficient while those neither on the efficient frontier nor on the inefficient frontier are declared DEA inefficient. It can be argued that both relative efficiencies should be considered simultaneously and any approach with only one of them would be biased. This paper proposed the integration of both efficiencies as an interval so that the overall performance score would belong to this interval. It was shown that efficiency interval provided more information than either of the two efficiencies, which was illustrated using two numerical examples.     

An integrated data envelopment analysis and free disposal hull framework for cost-efficiency measurement using rough sets

Traditional cost-efficiency analysis methods require exact and precise values for inputs, outputs and input prices. However, this is not the case in many real-life applications. This study proposes a rough cost-efficiency approach to the problem of ranking efficient decision making units (DMUs). Based on rough set theory, a nonparametric methodology for cost-efficiency analysis is developed. The merits of this methodology include computational ease and the capacity to incorporate data uncertainty. Furthermore, it applies to both convex data envelopment analysis (DEA) and non-convex free disposal hull (FDH) technologies under different returns-to-scale assumptions. A numerical example and a real-life case study in the Japanese banking industry demonstrate the applicability of the proposed framework. In particular, the rankings of the DMUs resulting from the proposed models are compared with those obtained using the maximum technical efficiency loss index.     

Expansion of the cross-efficiency evaluation by using the equations forming the efficient frontier in DEA

In this paper, we propose an algorithm to calculate cross-efficiency scores which used the equations forming the efficient frontier in data envelopment analysis (DEA). In many standard DEA models, each decision-making unit (DMU) is evaluated by using the advantageous weight for itself. Then, many DMUs are evaluated as efficient, and those efficient DMUs are not ranked by the models. The cross-efficiency evaluation is a method to rank DMUs by using the advantageous weights for all DMUs. Previously, the cross-efficiency scores based on different ideas are calculated by solving multiple linear or nonlinear programming problems. However, it is often hard to solve such a nonlinear programming problem. Therefore, by analysing the efficient frontier, we construct an algorithm to calculate alternative cross-efficiency scores.     

A learning performance evaluation with benchmarking concept for English writing courses

This paper adopts data envelopment analysis (DEA), a robust and reliable evaluation method widely applied in various fields to explore the key indicators contributing to the learning performance of English freshmen writing courses in a university of Taiwan from the academic year 2004 to 2006. The results of DEA model applied in learning performance change our original viewpoint and reveal that some decision-making units (DMUs) with higher actual values of inputs and outputs have lower efficiency because the relative efficiency of each DMU is measured by their distance to the efficiency frontier. DMUs may refer to different facet reference sets according to their actual values located in lower or higher ranges. In the managerial strategy of educational field, the paper can encourage inefficient DMUs to always compare themselves with efficient DMUs in their range and make improvement little by little. The results of DEA model can also give clear indicators and the percentage of which input and output items to improve. The paper also demonstrates that the benchmarking characteristics of the DEA model can automatically segment all the DMUs into different levels based on the indicators fed into the performance evaluation mechanism. The efficient DMUs on the frontier curve can be considered as the boundaries of the classification which are systematically defined by the DEA model according to the statistic distribution.     

Estimating most productive scale size with imprecise-chance constrained input–output orientation model in data envelopment analysis

Varieties of data envelopment analysis (DEA) models have been formulated to assess performance of decision making units (DMUs) in various fields with different data such as: deterministic, interval, fuzzy, etc. Classic DEA requires that values of all inputs and outputs are known exactly. However, this assumption may not be true, since in practice, data can not be precisely measured. Furthermore, a realistic situation is no longer realistic when imprecise and uncertain information are neglected to analyze efficiency of DMUs and measurement errors and data entry errors, etc.     

Measuring efficiency and productive performance of colleges at the university of Santo Tomas: a nonparametric approach

This paper estimates relative efficiency and productive performance of 13 colleges at the University of Santo Tomas (UST), using data envelopment analysis (DEA) – Malmquist indices and a multi‐stage model. DEA is a management evaluation tool that assists with identifying the most efficient and inefficient decision‐making units (DMUs) in the best practice frontier. Total factor productivity (TFP) is measured for a sample of 13 colleges at UST over the period 1998–2003. Empirical results show that the main contributing factor to TFP growth is efficiency change. That is, UST colleges are technically operating efficiently in the frontier technology; though there is a downward shift in the technological advancement. Our results further imply that with the use of output–input mix, UST colleges as a whole have recorded a higher level of technical efficiency than innovation. These new findings contribute significantly to the existing literature on efficiency and productive performance in the education sector.