A new method for complete ranking of DMUs

So far numerous models have been proposed for ranking the efficient decision-making units (DMUs) in data envelopment analysis (DEA). But, the most shortcoming of these models is their two-stage orientation. That is, firstly we have to find efficient DMUs and then rank them. Another flaw of some of these models, like AP-model (A procedure for ranking efficient units in data envelopment analysis, Management Science, 39 (10) (1993) 1261–1264), is existence of a non-Archimedean number in their objective function. Besides, when there is more than one weak efficient unit (or non-extreme efficient unit) these models could not rank DMUs. In this paper, we employ hyperplanes of the production possibility set (PPS) and propose a new method for complete ranking of DMUs in DEA. The proposed approach is a one stage method which ranks all DMUs (efficient and inefficient). In addition to ranking, the proposed method determines the type of efficiency for each DMU, simultaneously. Numerical examples are given to show applicability of the proposed method.     

Ranking efficient DMUs using the Tchebycheff norm

One problem that has been discussed frequently in data envelopment analysis (DEA) literature has been lack of discrimination in DEA applications, in particular when there are insufficient DMUs or the number of inputs and outputs is too high relative to the number of units. This is an additional reason for the growing interest in complete ranking techniques. In this paper a method for ranking extreme efficient decision making units (DMUs) is proposed. The method uses L_∞(or Tchebycheff) Norm, and it seems to have some superiority over other existing methods, because this method is able to remove the existing difficulties in some methods, such as Andersen and Petersen [2] (AP) that it is sometimes infeasible. The suggested model is always feasible.     

Ranking efficient decision making units in data envelopment analysis based on reference frontier share

Data envelopment analysis (DEA) is a powerful technique for performance evaluation of decision making units (DMUs). Ranking efficient DMUs based on a rational analysis is an issue that yet needs further research. The impact of each efficient DMU in evaluation of inefficient DMUs can be considered as additional information to discriminating among efficient DMUs. The concept of reference frontier share is introduced in which the share of each efficient DMU in construction of the reference frontier for evaluating inefficient DMUs is considered. For this purpose a model for measuring the reference frontier share of each efficient DMU associated with each inefficient one is proposed and then a total measure is provided based on which the ranking is made. The new approach has the capability for ranking extreme and non-extreme efficient DMUs. Further, it has no problem in dealing with negative data. These facts are verified by theorems, discussions and numerical examples.     

A ranking method based on a full-inefficient frontier

Since in evaluating by traditional data envelopment analysis (DEA) models many decision making units (DMUs) are classified as efficient, a large number of methods for fully ranking both efficient and inefficient DMUs have been proposed. In this paper a ranking method is suggested which basically differs from previous methods but its models are similar to traditional DEA models such as BCC, additive model, etc. In this ranking method, DMUs are compared against an full-inefficient frontier, which will be defined in this paper. Based on this point of view many models can be designed, and we mention a radial and a slacks-based one out of them. This method can be used to rank all DMUs to get analytic information about the system, and also to rank only efficient DMUs to discriminate between them.     

Super-efficiency in DEA by effectiveness of each unit in society

One of the most important topics in management science is determining the efficiency of Decision Making Units (DMUs). The Data Envelopment Analysis (DEA) technique is employed for this purpose. In many DEA models, the best performance of a DMU is indicated by an efficiency score of one. There is often more than one DMU with this efficiency score. To rank and compare efficient units, many methods have been introduced under the name of super-efficiency methods. Among these methods, one can mention Andersen and Petersen’s (1993) [1] super-efficiency model, and the slack-based measure introduced by Tone (2002) [4]. Each of the methods proposed for ranking efficient DMUs has its own advantages and shortcomings. In this paper, we present a super-efficiency method by which units that are more effective and useful in society have better ranks. In fact, in order to determine super-efficiency by this method, the effectiveness of each unit in society is considered rather than the cross-comparison of the units. To do so, we divide the inputs and outputs into two groups, desirable and undesirable, at the discretion of the manager, and assign weights to each input and output. Then we determine the rank of each DMU according to the weights and the desirability of inputs and outputs.     

Ranking all units in data envelopment analysis

The motivation of this study is to propose an equitable method for ranking decision making units (DMUs) based on the data envelopment analysis (DEA) concept. For this purpose, first, the minimum and maximum efficiency values of each DMU are computed under the assumption that the sum of efficiency values of all DMUs is equal to unity. Then, the rank of each DMU is determined in proportion to a combination of its minimum and maximum efficiency values.     

An epsilon-free approach for finding the most efficient unit in DEA

Data envelopment analysis (DEA), considering the best condition for each decision making unit (DMU), assesses the relative efficiency and partitions DMUs into two sets: efficient and inefficient. Practically, in traditional DEA models more than one efficient DMU are recognized and these models cannot rank efficient DMUs. Some studies have been carried out aiming at ranking efficient DMUs, although in some cases only discrimination of the most efficient unit is desirable. Furthermore, several investigations have been done for finding the most CCR-efficient DMU. The basic idea of the majority of them is to introduce an integrated model which achieves an optimal common set of weights (CSW). These weights help us identify the most efficient unit in an identical condition.     

Network-based method for ranking of efficient units in two-stage DEA models

This study presents a methodology that is able to further discriminate the efficient decision-making units (DMUs) in a two-stage data envelopment analysis (DEA) context. The methodology is an extension of the single-stage network-based ranking method, which utilizes the eigenvector centrality concept in social network analysis to determine the rank of efficient DMUs. The mathematical formulation for the method to work under the two-stage DEA context is laid out and then applied to a real-world problem. In addition to its basic ranking function, the exercise highlights two particular features of the method that are not available in standard DEA: suggesting a benchmark unit for each input/intermediate/output factor, and identifying the strengths of each efficient unit. With the methodology, the value of DEA greatly increases.     

Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis

It has been widely recognized that data envelopment analysis (DEA) lacks discrimination power to distinguish between DEA efficient units. This paper proposes a new methodology for ranking decision making units (DMUs). The new methodology ranks DMUs by imposing an appropriate minimum weight restriction on all inputs and outputs, which is decided by a decision maker (DM) or an assessor in terms of the solutions to a series of linear programming (LP) models that are specially constructed to determine a maximin weight for each DEA efficient unit. The DM can decide how many DMUs to be retained as DEA efficient in final efficiency ranking according to the requirement of real applications, which provides flexibility for DEA ranking. Three numerical examples are investigated using the proposed ranking methodology to illustrate its power in discriminating between DMUs, particularly DEA efficient units.     

Fuzzy cross-efficiency evaluation: a possibility approach

Cross-efficiency evaluation is an extension of data envelopment analysis (DEA) aimed at ranking decision making units (DMUs) involved in a production process regarding their efficiency. As has been done with other enhancements and extensions of DEA, in this paper we propose a fuzzy approach to the cross-efficiency evaluation. Specifically, we develop a fuzzy cross-efficiency evaluation based on the possibility approach by Lertworasirikul et al. (Fuzzy Sets Syst 139:379–394, 2003a) to fuzzy DEA. Thus, a methodology for ranking DMUs is presented that may be used when data are imprecise, in particular for fuzzy inputs and outputs being normal and convex. We prove some results that allow us to define “consistent” cross-efficiencies. The ranking of DMUs for a given possibility level results from an ordering of cross-efficiency scores, which are real numbers. As in the crisp case, we also develop benevolent and aggressive fuzzy formulations in order to deal with the alternate optima for the weights.     

Output–input ratio analysis and DEA frontier

Data envelopment analysis (DEA) is a mathematical programming technique for identifying efficient frontiers for peer decision making units (DMUs). The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, we present mathematical properties which characterize the inherent relationships between DEA frontier DMUs and output–input ratios. It is shown that top-ranked performance by ratio analysis is a DEA frontier point. This in turn allows identification of membership of frontier DMUs without solving a DEA program. Such finding is useful in streamlining the solution of DEA.     

A distance-based measure of super efficiency in data envelopment analysis: an application to gas companies

Conventional data envelopment analysis (DEA) assists decision makers in distinguishing between efficient and inefficient decision making units (DMUs) in a homogeneous group. Standard DEA models can not provide more information about efficient units. Super-efficiency DEA models can be used in ranking the performance of efficient DMUs and overcome this obstacle. Because of the possible infeasibility, the use of super efficiency models has been restricted. This research proposes a methodology to determine a distance-based measure of super-efficiency. The proposed methodology overcomes the infeasibility problem of the existing ranking methodologies. The applicability of the proposed model is illustrated in the context of the analysis of gas companies?? performance.     

Optimal system design series-network DEA models

There is an urgent need in a wide range of fields such as logistics and supply chain management to develop effective approaches to measure and/or optimally design a network system comprised of a set of units. Data envelopment analysis (DEA) researchers have been developing network DEA models to measure decision making units’ (DMUs’) network systems. However, to our knowledge, there are no previous contributions on the DEA-type models that help DMUs optimally design their network systems. The need to design optimal systems is quite common and is sometimes necessary in practice. This research thus introduces a new type of DEA model termed the optimal system design (OSD) network DEA model to optimally design a DMUs (exogenous and endogenous) input and (endogenous and final) output portfolios in terms of profit maximization given the DMUs total available budget. The resulting optimal network design through the proposed OSD network DEA models is efficient, that is, it lies on the frontier of the corresponding production possibility set.     

Interval cross efficiency for fully ranking decision making units using DEA/AHP approach

Data envelopment analysis (DEA) is a popular technique for measuring the relative efficiency of a set of decision making units (DMUs). Fully ranking DMUs is a traditional and important topic in DEA. In various types of ranking methods, cross efficiency method receives much attention from researchers because it evaluates DMUs by using self and peer evaluation. However, cross efficiency score is usual nonuniqueness. This paper combines the DEA and analytic hierarchy process (AHP) to fully rank the DMUs that considers all possible cross efficiencies of a DMU with respect to all the other DMUs. We firstly measure the interval cross efficiency of each DMU. Based on the interval cross efficiency, relative efficiency pairwise comparison between each pair of DMUs is used to construct interval multiplicative preference relations (IMPRs). To obtain the consistency ranking order, a method to derive consistent IMPRs is developed. After that, the full ranking order of DMUs from completely consistent IMPRs is derived. It is worth noting that our DEA/AHP approach not only avoids overestimation of DMUs’ efficiency by only self-evaluation, but also eliminates the subjectivity of pairwise comparison between DMUs in AHP. Finally, a real example is offered to illustrate the feasibility and practicality of the proposed procedure.     

Finding the efficiency score and RTS characteristic of DMUs by means of identifying the efficient frontier in DEA

Data envelopment analysis (DEA) is basically a linear programming based technique used for measuring the relative performance of organizational units, referred to as decision-making units (DMUs), where the presence of multiple inputs and outputs makes comparisons difficult. The ability of identifying frontier DMUs prior to the DEA calculation is of extreme importance to an effective and efficient DEA computation. In this paper, a method for identifying the efficient frontier is introduced. Then, the efficiency score and returns to scale (RTS) characteristic of DMUs will be produced by means of the equation of efficient frontier.     

Sensitivity analysis of DEA models for simultaneous changes in all the data

In data envelopment analysis (DEA) efficient decision making units (DMUs) are of primary importance as they define the efficient frontier. The current paper develops a new sensitivity analysis approach for the basic DEA models, such as, those proposed by Charnes, Cooper and Rhodes (CCR), Banker, Charnes and Cooper (BCC) and additive models, when variations in the data are simultaneously considered for all DMUs. By means of modified DEA models, in which the specific DMU under examination is excluded from the reference set, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalises the usual sensitivity analysis approach developed in which perturbations of the data are only applied to the test DMU while all the remaining DMUs remain fixed. In our framework data are allowed to vary simultaneously for all DMUs across different subsets of inputs and outputs. We study the relations of the infeasibility of modified DEA models employed and the robustness of DEA models. It is revealed that the infeasibility means stability. The empirical applications demonstrate that DEA efficiency classifications are robust with respect to possible data errors, particularly in the convex DEA case.     

A note on DEA vs principal component analysis: An improvement to Joe Zhu's approach

This paper considers a previous article published by Zhu in the European Journal of Operational Research which describes a joint use of data envelopment analysis (DEA) and principal component analysis (PCA) in ranking of decision making units (DMUs). In Zhu's empirical study, DEA and PCA yield a consistent ranking. However, this paper finds that in certain instances, DEA and PCA may yield inconsistent rankings. The PCA procedure adopted by Zhu is slightly modified in this article by incorporating other important features of ranking that Zhu has not considered. Numerical results reveal that both approaches show a consistency in ranking with DEA when the data set has a small number of efficient units. But, when a majority of the DMUs in the sample are efficient, only the modified approach produces consistent ranking with DEA.     

Programmatic efficiency comparisons between unequally sized groups of DMUs in DEA

Data envelopment analysis (DEA) is a popular non-parametric technique for determining the efficiency of a homogeneous set of decision-making units (DMUs). In many practical cases, there is some doubt if the all the DMUs form a single group with a common efficiency distribution. The Mann–Whitney rank statistic has been used in DEA both to test if two groups of DMUs come from a common efficiency distribution and also to test if the two groups have a common frontier, each of which are likely to have important but different policy implications for the management of the groups. In this paper it is demonstrated that where the Mann–Whitney rank statistic is used for the second of these it is likely to overestimate programmatic inefficiency, particularly of the smaller group. A new non-parametric statistic is proposed for the case of comparing the efficient frontiers of two groups, which overcomes the problems we identify in the use of the Mann–Whitney rank statistic for this purpose.     

Slack-based ranking method: an interpretation to the cross-efficiency method in DEA

The cross-efficiency method is generally utilized to rank decision-making units (DMUs) in data envelopment analysis (DEA) based on peer-evaluation logic. This brief note provides a method of using the available information from the linear program outputs to calculate the ranking of all DMUs with fewer computations and offers an alternative interpretation to the cross-efficiency method based on slack analysis in DEA.     

A new DEA ranking system based on changing the reference set

This research proposes a new ranking system for extreme efficient DMUs (Decision Making Units) based upon the omission of these efficient DMUs from reference set of the inefficient DMUs. We state and prove some facts related to our model. A numerical example where the proposed method is compared with traditional ranking approaches is shown.