A new DEA‐based method for fully ranking all decision‐making units

Abstract: Decision makers always lay great emphasis on performance evaluation upon a group of peer business units to pick out the best performer. Standard data envelopment analysis models can evaluate the relative efficiency of decision‐making units (DMUs) and distinguish efficient ones from inefficient ones. However, when there are more than one efficient DMU, it is impossible to rank all of them solely according to standard efficiency scores. In this paper, a new method for fully ranking all DMUs is proposed, which is based on the combination of each efficient DMU's influence on all the other DMUs and the standard efficiency scores. This method is effective in helping decision makers differentiate all units' performance thoroughly and select the best performer.     

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.     

Single‐phase slack‐based centralized DEA for resource reallocation

Data envelopment analysis (DEA) is an effective method for measuring the relative efficiency of a set of homogeneous decision‐making units (DMUs). Yu et al.’s study proposed an extended centralized DEA (CDEA) model that utilizes a two‐phase process for reallocating resources to project not only in each DMU (e.g., branch company) but also in the central DMU (e.g., headquarters, central authority) on the production frontier. However, evaluating the two‐phase model using the approach of Yu et al. may present some challenges because of inconsistent benchmarks. To solve this issue, we modified a single‐phase slack‐based CDEA that considers transfer‐in and transfer‐out slacks to facilitate the reallocation and adjustment of resources. Our modified single‐phase slack‐based CDEA is demonstrated with a numerical example illustrating input resource reallocation. Results show that the modified single‐phase CDEA model is effective to deal with a more realistic inconsistency reference set and provide much more reallocation ability than the two‐phase approach.     

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.     

基于SBM区间模型的决策单元相似度

在数据包络分析方法的研究中,当决策单元的输入输出值为区间数时,区间效率的测算取决于生产参照集的选择.对此,针对非径向非角度的松弛测度模型(SBM),通过比较计算认为,以最佳生产状态为统一参照来测算决策单元的区间效率最有利于排序.同时,利用区间数相似度理论建立模糊相似矩阵,进而提出基于参照单元区间效率的相似度排序方法,对决策单元聚类和排序.最后通过算例表明了所提出方法的可行性和有效性.     

DEA nonradial efficiency based on vector properties

Classic data envelopment analysis (DEA) models determine the efficiency of productive units, called decision making units (DMUs). DEA uses as its methodology the equiproportional reduction of inputs or increase of outputs and the finding of a single target for each DMU. This target does not incorporate the preference of the decision maker. Later works propose obtaining alternative targets based on nonradial projections on the efficiency frontier that are obtained through nonproportional variations of inputs or outputs. However, the efficiencies are not calculated for these alternative targets. This impedes a comparison among the DMUs. Thus, diverse nonradial efficiency indexes have been proposed based on mathematical averages or weighted averages that do not consider the vectorial characteristics of the efficiency. In this work, we present a nonradial efficiency index based on the initial concept of efficiency associated with each alternative (nonradial) target obtained through a multiobjective model of an inefficient DMU.     

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.     

Improving the discrimination power and weights dispersion in the data envelopment analysis

Data envelopment analysis (DEA) has been a very popular method for measuring and benchmarking relative efficiency of peer decision making units (DMUs) with multiple input and outputs. Beside of its popularity, DEA has some drawbacks such as unrealistic input–output weights and lack of discrimination among efficient DMUs. In this study, two new models based on a multi-criteria data envelopment analysis (MCDEA) are developed to moderate the homogeneity of weights distribution by using goal programming (GP). These goal programming data envelopment analysis models, GPDEA-CCR and GPDEA-BCC, also improve the discrimination power of DEA.     

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.     

A robust hybrid artificial neural network double frontier data envelopment analysis approach for assessing sustainability of power plants under uncertainty

To assess sustainability of power plants, this paper presents a novel hybrid method. To this end, self‐organizing map method of artificial neural networks is employed. Then, a double frontier data envelopment analysis is developed to rank power plants in each cluster of decision‐making units. Because outputs of power plants might be uncertain, a robust optimization approach is incorporated into proposed double frontier data envelopment analysis model to present ranks that are robust against different uncertainties. A case study is given to validate the proposed model. The case study shows that the proposed model can present improvement solutions that guide power plants towards efficient frontier and far from inefficient frontier. Given the results, decision makers can decide on which power plants should be closed and which power plants should be expanded.     

Minimizing deviations of input and output weights from their means in data envelopment analysis

In this paper, we propose a model that minimizes deviations of input and output weights from their means for efficient decision-making units in data envelopment analysis. The mean of an input or output weight is defined as the average of the maximum and the minimum attainable values of the weight when the efficient decision making unit under evaluation remains efficient. Alternate optimal weights usually exist in the linear programming solutions of efficient decision-making units and the optimal weights obtained from most of the linear programming software are somewhat arbitrary. Our proposed model can yield more rational weights without a priori information about the weights. Input and output weights can be used to compute cross-efficiencies of decision-making units in peer evaluations or group decision-making units, which have similar production processes via cluster analysis. If decision makers want to avoid using weights with extreme or zero values to access performance of decision-making units, then choosing weights that are close to their means, may be a rational choice.     

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.     

EFFICIENT ABSTRACTION SELECTION IN REINFORCEMENT LEARNING

This article addresses reinforcement learning problems based on factored Markov decision processes (MDPs) in which the agent must choose among a set of candidate abstractions, each build up from a different combination of state components. We present and evaluate a new approach that can perform effective abstraction selection that is more resource‐efficient and/or more general than existing approaches. The core of the approach is to make selection of an abstraction part of the learning agent's decision‐making process by augmenting the agent's action space with internal actions that select the abstraction it uses. We prove that under certain conditions this approach results in a derived MDP whose solution yields both the optimal abstraction for the original MDP and the optimal policy under that abstraction. We examine our approach in three domains of increasing complexity: contextual bandit problems, episodic MDPs, and general MDPs with context‐specific structure. © 2013 Wiley Periodicals, Inc.     

Improving the discrimination of data envelopment analysis models in multiple time periods

The trade‐offs approach is an advanced tool for the improvement of the discrimination of data envelopment analysis (DEA) models; this can improve the traditional meaning of efficiency as a radial improvement factor for inputs or outputs. Therefore, the Malmquist index – the prominent index for measuring the productivity change of decision making units (DMUs) in multiple time periods that use DEA models with variable returns to scale and constant returns to scale technologies – can be improved by using the trade‐offs technology. Hence, an expanded Malmquist index can be defined as an improved method of a traditional Malmquist index that uses the production possibility set, which could present more discrimination of DMUs, in the presence of the trade‐offs technology. In addition, similar to a traditional Malmquist index, it breaks down into different components. An illustrative example is presented to show the ability of the suggested method of presenting the Malmquist index from a computational point of view.     

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.     

Optimal direct mailing modelling based on data envelopment analysis

Data envelopment analysis (DEA) is a mathematical programming technique that is frequently used for measuring and benchmarking efficiency of the homogenous decision‐making units (DMUs). This paper proposes a new use of DEA for customers scoring and particularly their direct mailing modelling. Moreover, because DEA models suffer from some weaknesses, that is, unrealistic weighting scheme of the inputs and outputs and incomplete ranking among efficient DMUs, the present paper compares different ways of solving these problems and concludes that common set of weights method, as a result of some advantages, outperforms other procedures.     

Data Envelopment Analysis Based on Choquet Integral

Data envelopment analysis (DEA) is a widely used technique in decision making. The existing DEA models always assume that the inputs (or outputs) of decision‐making units (DMUs) are independent with each other. However, there exist positive or negative interactions between inputs (or outputs) of DMUs. To reflect such interactions, Choquet integral is applied to DEA. Self‐efficiency models based on Choquet integral are first established, which can obtain more efficiency values than the existing ones. Then, the idea is extended to the cross‐efficiency models, including the game cross‐efficiency models. The optimal analysis of DEA is further investigated based on regret theory. To estimate the ranking intervals of DMUs, several models are also established. It is founded that the models considering the interactions between inputs (or outputs) can obtain wider ranking intervals.     

决策单元特殊关系的挖掘与建立

基于偏序集理论的数据包络分析方法,通过引进适当的偏序关系,挖掘出决策单元之间的特殊关系。然而,随着决策单元所选取的投入产出指标个数的增加,决策单元之间的偏序关系变得越来越少。对此,通过引进决策单元之间的距离和适当的样本决策单元,建立决策单元之间的特殊关系,最终生成决策单元之间的格论关系,并引进相关定理及其算法。最后通过仿真结果表明了所提出算法的有效性和实用性。     

A dimensional decomposition approach to identifying efficient units in large-scale DEA models

In this paper, we propose the use of a dimensional decomposition procedure together with lexicographic parametric programming to reduce computational burden when identifying the efficient decision making units in data envelopment analysis (DEA). The use of lexicographic parametric programming makes it possible to develop an efficient algorithm for the problems with few inputs and outputs. Based on this we propose the procedure which first partitions the original problem dimensionally into sub-problems and then identifies the efficient units of the sub-problems. Since those units are a subset of the weakly efficient solutions of the original problem, they are used as an initial approximation for the efficient units of the original problem. The efficiency of the approach is illustrated by numerical results.     

Allocating the fixed cost based on data envelopment analysis in view of the Shapley value

This paper considers fixed cost allocation in view of cooperative game theory and proposes an approach based on data envelopment analysis while incorporating the perspectives of coalition efficiency and the Shapley value. To do this, we first build two models to evaluate coalition efficiencies before and after cost allocation, and we prove that all coalitions can be efficient after fixed cost allocation. Then, following the premise that each coalition makes itself efficient without reducing the efficiencies of other decision-making units' preallocation efficiency, we propose a model that determines the acceptable range of each coalition's allocated fixed cost. Furthermore, a model is constructed to determine the final cost allocation based on three principles: efficiency, monotonicity, and similarity. Moreover, the Shapley value is employed to obtain the cost allocated to each decision-making unit (DMU). The proposed approach considers the relationships among DMUs across their forming coalitions to determine their interaction types and then generates a fixed cost allocation result that possesses the features of the Shapley value. This process makes the fixed cost allocation more acceptable. Finally, a simple numerical example and an empirical case are provided to illustrate the calculation process of the proposed approach and compare our approach with the traditional methods.