Improvement of efficiency intervals based on DEA by adjusting inputs and outputs

We improve the efficiency interval of a DMU by adjusting its given inputs and outputs. The Interval DEA model has been formulated to obtain an efficiency interval consisting of evaluations from both the optimistic and pessimistic viewpoints. DMUs which are not rated as efficient in the conventional sense are improved so that their lower bounds become as large as possible under the condition that their upper bounds attain the maximum value one. The adjusted inputs and outputs keep each other balanced by improving the lower bound of efficiency interval, since the lower bound becomes small if all the inputs and outputs are not proportioned. In order to improve the lower bound of efficiency interval, different target points are defined for different DMUs. The target point can be regarded as a kind of benchmark for the DMU. First, a new approach to improvement by adjusting only outputs or inputs is proposed. Then, the combined approach to improvement by adjusting both inputs and outputs simultaneously is proposed. Lastly, numerical examples are shown to illustrate our proposed approaches.     

Common set of weights in data envelopment analysis: a linear programming problem

Data Envelopment Analysis is used to determine the relative efficiency of Decision Making Units as the ratio of weighted sum of outputs by weighted sum of inputs. To accomplish the purpose, a DEA model calculates the weights of inputs and outputs of each DMU individually so that the highest efficiency can be estimated. Thus, the present study suggests an innovative method using a common set of weights leading to solving a linear programming problem. The method determines the efficiency score of all DMUs and rank them too.     

Super-efficiency DEA in the presence of infeasibility

It is well known that super-efficiency data envelopment analysis (DEA) approach can be infeasible under the condition of variable returns to scale (VRS). By extending of the work of Chen (2005), the current study develops a two-stage process for calculating super-efficiency scores regardless whether the standard VRS super-efficiency mode is feasible or not. The proposed approach examines whether the standard VRS super-efficiency DEA model is infeasible. When the model is feasible, our approach yields super-efficiency scores that are identical to those arising from the original model. For efficient DMUs that are infeasible under the super-efficiency model, our approach yields super-efficiency scores that characterize input savings and/or output surpluses. The current study also shows that infeasibility may imply that an efficient DMU does not exhibit super-efficiency in inputs or outputs. When infeasibility occurs, it can be necessary that (i) both inputs and outputs be decreased to reach the frontier formed by the remaining DMUs under the input-orientation and (ii) both inputs and outputs be increased to reach the frontier formed by the remaining DMUs under the output-orientation. The newly developed approach is illustrated with numerical examples.     

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.     

An inverse optimization model for imprecise data envelopment analysis

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies.     

Determining sources of relative inefficiency in heterogeneous samples: Methodology using Cluster Analysis,DEA and Neural Networks

Data Envelopment Analysis (DEA) is a powerful data analytic tool that is widely used by researchers and practitioners alike to assess relative performance of Decision Making Units (DMU). Commonly, the difference in the scores of relative performance of DMUs in the sample is considered to reflect their differences in the efficiency of conversion of inputs into outputs. In the presence of scale heterogeneity, however, the source of the difference in scores becomes less clear, for it is also possible that the difference in scores is caused by heterogeneity of the levels of inputs and outputs of DMUs in the sample. By augmenting DEA with Cluster Analysis (CA) and Neural Networks (NN), we propose a five-step methodology allowing an investigator to determine whether the difference in the scores of scale heterogeneous DMUs is due to the heterogeneity of the levels of inputs and outputs, or whether it is caused by their efficiency of conversion of inputs into outputs. An illustrative example demonstrates the application of the proposed methodology in action.     

A modified super-efficiency DEA model for infeasibility

The super-efficiency data envelopment analysis (DEA) model is obtained when a decision making unit (DMU) under evaluation is excluded from the reference set. This model provides for a measure of stability of the “efficient” status for frontier DMUs. Under the assumption of variable returns to scale (VRS), the super efficiency model can be infeasible for some efficient DMUs, specifically those at the extremities of the frontier. The current study develops an approach to overcome infeasibility issues. It is shown that when the model is feasible, our approach yields super-efficiency scores that are equivalent to those arising from the original model. For efficient DMUs that are infeasible under the super-efficiency model, our approach yields optimal solutions and scores that characterize the extent of super-efficiency in both inputs and outputs. The newly developed approach is illustrated with two real world data sets.     

Evaluating the performances of decision-making units based on interval efficiencies

Efficiency could be not only the ratio of weighted sum of outputs to that of inputs but also that of weighted sum of inputs to that of outputs. When the previous efficiency measures the best relative efficiency within the range of no more than one, the decision-making units (DMUs) who get the optimum value of one perform best among all the DMUs. If the previous efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform worst among all the DMUs. When the later efficiency is measured within the range of no more than one, the DMUs who get the optimum value of one perform worst among all the DMUs. If the later efficiency is measured within the range of no less than one, the DMUs who get the optimum value of one perform best among all the DMUs. This paper mainly studies an interval DEA model with later efficiency, in which efficiency is measured within the range of an interval, whose upper bound is set to one and the lower bound is determined by introducing a virtual ideal DMU, whose performance is definitely superior to any DMUs. The efficiencies, obtained from interval DEA model, turn out to be all intervals and are referred to as interval efficiencies, which combine the best and the worst relative efficiency in a reasonable manner to give an overall assessment of performances for all DMUs. Assessor's preference information on input and output weights is also incorporated into interval DEA model reasonably and conveniently. Through an example, some differences are found from the ranking results obtained from interval DEA model and bounded DEA model using the Hurwicz criterion approach to rank the interval efficiencies.     

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.     

A generalized DEA model for inputs/outputs estimation

In this paper we discuss the question: among a group of decision making units (DMUs), if a DMU changes some of its input (output) levels, to what extent should the unit change outputs (inputs) such that its efficiency index remains unchanged? In order to solve this question we propose a solving method based on Data Envelopment Analysis (DEA) and Multiple Objective Linear Programming (MOLP). In our suggested method, the increase of some inputs (outputs) and the decrease due to some of the other inputs (outputs) are taken into account at the same time, while the other offered methods do not consider the increase and the decrease of the various inputs (outputs) simultaneously. Furthermore, existing models employ a MOLP for the inefficient DMUs and a linear programming for weakly efficient DMUs, while we propose a MOLP which estimates input/output levels, regardless of the efficiency or inefficiency of the DMU. On the other hand, we show that the current models may fail in a special case, whereas our model overcomes this flaw. Our method is immediately applicable to solve practical problems.     

Classifying and characterizing efficiencies and inefficiencies in data development analysis

DEA (Data Envelopment Analysis) attempts to identify sources and estimate amounts of inefficiencies contained in the outputs and inputs generated by managed entities called DMUs (=Decision Making Units). Explicit formulation of underlying functional relations with specified parametric forms relating inputs to outputs is not required. An overall (scalar) measure of efficiency is also obtained for each DMU from the observed values of its multiple inputs and outputs without requiring uses of a priori weights. There are many different ways of specifying DEA reference sets. A partition into 6 classes is provided for such observations in which 3 are scale inefficient and 3 are scale efficient with the latter containing substs of DMUs that are also technically (=zero waste) efficient.     

Data Envelopment Analysis with Preference Structure

It is important to consider the decision making unit (DMU)'s or decision maker's preference over the potential adjustments of various inputs and outputs when data envelopment analysis (DEA) is employed. On the basis of the so-called Russell measure, this paper develops some weighted non-radial CCR models by specifying a proper set of ‘preference weights’ that reflect the relative degree of desirability of the potential adjustments of current input or output levels. These input or output adjustments can be either less or greater than one; that is, the approach enables certain inputs actually to be increased, or certain outputs actually to be decreased. It is shown that the preference structure prescribes fixed weights (virtual multiplier bounds) or regions that invalidate some virtual multipliers and hence it generates preferred (efficient) input and output targets for each DMU. In addition to providing the preferred target, the approach gives a scalar efficiency score for each DMU to secure comparability. It is also shown how specific cases of our approach handle non-controllable factors in DEA and measure allocative and technical efficiency. Finally, the methodology is applied with the industrial performance of 14 open coastal cities and four special economic zones in 1991 in China. As applied here, the DEA/preference structure model refines the original DEA model's result and eliminates apparently efficient DMUs.     

Finding strong defining hyperplanes of Production Possibility Set

Production Possibility Set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. Data Envelopment Analysis models implicitly use PPS to evaluate relative efficiency of Decision Making Units (DMUs). Although DEA models can determine the efficiency of a DMU, they cannot present efficient frontiers of PPS. In this paper, we propose a method for finding all Strong Defining Hyperplanes of PPS (SDHP). They are equations that form efficient surfaces. These equations are useful in Sensitivity and Stability Analysis, the status of Returns to Scale of a DMU, incorporating performance information into the efficient frontier analysis and so on.     

A directional distance based super-efficiency DEA model handling negative data

This paper develops a new radial super-efficiency data envelopment analysis (DEA) model, which allows input–output variables to take both negative and positive values. Compared with existing DEA models capable of dealing with negative data, the proposed model can rank the efficient DMUs and is feasible no matter whether the input–output data are non-negative or not. It successfully addresses the infeasibility issue of both the conventional radial super-efficiency DEA model and the Nerlove–Luenberger super-efficiency DEA model under the assumption of variable returns to scale. Moreover, it can project each DMU onto the super-efficiency frontier along a suitable direction and never leads to worse target inputs or outputs than the original ones for inefficient DMUs. Additional advantages of the proposed model include monotonicity, units invariance and output translation invariance. Two numerical examples demonstrate the practicality and superiority of the new model.     

Ranking DMUs by l1-norm with fuzzy data in DEA

The relative efficiency of a DMU is the result of comparing the inputs and outputs of the DMU and those of other DMUs in the PPS (production possibility set). If the inputs and outputs are fuzzy, the DMUs cannot be easily evaluated and ranked using the obtained efficiency scores. In this paper, presenting a new idea for ranking of DMUs with fuzzy data. And finally, we introduce a numerical example.     

Data envelopment analysis vs. principal component analysis: An illustrative study of economic performance of Chinese cities

This article compares two approaches in aggregating multiple inputs and multiple outputs in the evaluation of decision making units (DMUs), data envelopment analysis (DEA) and principal component analysis (PCA). DEA, a non-statistical efficiency technique, employs linear programming to weight the inputs/outputs and rank the performance of DMUs. PCA, a multivariate statistical method, combines new multiple measures defined by the inputs/outputs. Both methods are applied to three real world data sets that characterize the economic performance of Chinese cities and yield consistent and mutually complementary results. Nonparametric statistical tests are employed to validate the consistency between the rankings obtained from DEA and PCA.     

A new framework for the solution of DEA models

We provide an alternative framework for solving data envelopment analysis (DEA) models which, in comparison with the standard linear programming (LP) based approach that solves one LP for each decision making unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier–Motzkin (F–M) elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F–M method improves on existing methods of (i) establishing the returns to scale status of each DMU, (ii) calculating cross-efficiencies and (iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems.     

Network DEA: Additive efficiency decomposition

In conventional DEA analysis, DMUs are generally treated as a black-box in the sense that internal structures are ignored, and the performance of a DMU is assumed to be a function of a set of chosen inputs and outputs. A significant body of work has been directed at problem settings where the DMU is characterized by a multistage process; supply chains and many manufacturing processes take this form. Recent DEA literature on serial processes has tended to concentrate on closed systems, that is, where the outputs from one stage become the inputs to the next stage, and where no other inputs enter the process at any intermediate stage. The current paper examines the more general problem of an open multistage process. Here, some outputs from a given stage may leave the system while others become inputs to the next stage. As well, new inputs can enter at any stage. We then extend the methodology to examine general network structures. We represent the overall efficiency of such a structure as an additive weighted average of the efficiencies of the individual components or stages that make up that structure. The model therefore allows one to evaluate not only the overall performance of the network, but as well represent how that performance decomposes into measures for the individual components of the network. We illustrate the model using two data sets.     

Unoriented two-stage DEA: The case of the oscillating intermediate products

Data envelopment analysis (DEA) allows us to evaluate the relative efficiency of each of a set of decision-making units (DMUs). However, the methodology does not permit us to identify specific sources of inefficiency because DEA views the DMU as a “black box” that consumes a mix of inputs and produces a mix of outputs. Thus, DEA does not provide a DMU manager with insight regarding the internal source of the organization’s inefficiency.     

有关判断决策单元的DEA有效性的新方法的探讨

为了判断决策单元是否(弱)DEA有效并克服现有的模型及[1]中模型在解决上述问题时的不足之处,本文将讨论的新模型是由CCR模型与CCGSS模型变来的,且定理的证明不同于[1]．还讨论了文中新模型的最优解的存在性,此外,研究了所有决策单元的输入输出的变化对某决策单元有效性的影响．