Modeling undesirable factors in data envelopment analysis

Data envelopment analysis is a mathematical programming technique for identifying efficient frontiers for peer decision making units with multiple inputs and multiple outputs. These performance factors (inputs and outputs) are classified into two groups: desirable and undesirable. Obviously, undesirable factors in production process should be reduced to improve the performance. In the current paper, we present a data envelopment analysis (DEA) model in which can be used to improve the relative performance via increasing undesirable inputs and decreasing undesirable outputs.     

Data envelopment analysis with nonlinear virtual inputs and outputs

An underlying assumption in DEA is that the weights coupled with the ratio scales of the inputs and outputs imply linear value functions. In this paper, we present a general modeling approach to deal with outputs and/or inputs that are characterized by nonlinear value functions. To this end, we represent the nonlinear virtual outputs and/or inputs in a piece-wise linear fashion. We give the CCR model that can assess the efficiency of the units in the presence of nonlinear virtual inputs and outputs. Further, we extend the models with the assurance region approach to deal with concave output and convex input value functions. Actually, our formulations indicate a transformation of the original data set to an augmented data set where standard DEA models can then be applied, remaining thus in the grounds of the standard DEA methodology. To underline the usefulness of such a new development, we revisit a previous work of one of the authors dealing with the assessment of the human development index on the light of DEA.     

Equivalence in two-stage DEA approaches

Data envelopment analysis (DEA) is a linear programming problem approach for evaluating the relative efficiency of peer decision making units (DMUs) that have multiple inputs and outputs. DMUs can have a two-stage structure where all the outputs from the first stage are the only inputs to the second stage, in addition to the inputs to the first stage and the outputs from the second stage. The outputs from the first stage to the second stage are called intermediate measures. This paper examines relations and equivalence between two existing DEA approaches that address measuring the performance of two-stage processes.     

Gradual technical and scale efficiency improvement in DEA

In data envelopment analysis (DEA) an inefficient unit can be projected onto an efficient target that is far away, i.e. reaching the target may demand large reductions in inputs and increases in outputs. When the inputs and outputs modifications planned are large, it may be troublesome to carry them out all at once. In order to help an inefficient unit reach a distant target, a strategy of gradual improvements with successive, intermediate targets has been proposed. This paper extends such approach to the variable returns to scale (VRS) case. In the VRS scenario we distinguish between units that are technical efficient and those that are not. On the one hand, for those units that are not technical efficient the proposed approach determines successive intermediate targets leading to the technical efficiency frontier, i.e. the priority for those units is to attain technical efficiency. On the other hand, for those units that are technical efficient but not scale efficient the proposed approach computes a sequence of targets ending in the global efficiency frontier, i.e. when technical efficiency is guaranteed the goal is then to attain global efficiency. In both cases, the successive targets are obtained by iteratively solving specific DEA models that take into account given bounds on the rates of change in inputs and outputs that the unit can implement in each step.     

对DEA中有效单元排序的一种新方法

文章对数据包络分析(DEA)的强有效性问题提出了一种新的研究方法.利用有效值和负有效值来构造复合输入和输出这种方法可以实现有效决策单元的完全排序.文章还给出了新方法中模型的一些性质.最后,用两个例子来检验此方法并和其他模型的计算结果进行了比较.     

Theory of integer-valued data envelopment analysis

Conventional data envelopment analysis (DEA) models assume real-valued inputs and outputs. In many occasions, some inputs and/or outputs can only take integer values. In some cases, rounding the DEA solution to the nearest whole number can lead to misleading efficiency assessments and performance targets. This paper develops the axiomatic foundation for DEA in the case of integer-valued data, introducing new axioms of “natural disposability” and “natural divisibility”. We derive a DEA production possibility set that satisfies the minimum extrapolation principle under our refined set of axioms. We also present a mixed integer linear programming formula for computing efficiency scores. An empirical application to Iranian university departments illustrates the approach.     

Sensitivity and stability analysis on the first and second levels of efficiency score relative to data error

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 a category DMUs and finds the stability radius for all efficient DMUs. By means of combining some classic DEA models and with the condition that the efficiency scores of efficient DMUs remain unchanged, we are able to determine what perturbations of the data can be tolerated before efficient DMUs become inefficient. Our approach generalizes the conventional sensitivity analysis approach in which the inputs of efficient DMUs increase and their outputs decrease, while the inputs of inefficient DMUs decrease and their outputs increase. We find the maximum quantity of perturbations of data so that all first level efficient DMUs remain at the same level.     

Data envelopment analysis with imprecise data

In original data envelopment analysis (DEA) models, inputs and outputs are measured by exact values on a ratio scale. Cooper et al. [Management Science, 45 (1999) 597–607] recently addressed the problem of imprecise data in DEA, in its general form. We develop in this paper an alternative approach for dealing with imprecise data in DEA. Our approach is to transform a non-linear DEA model to a linear programming equivalent, on the basis of the original data set, by applying transformations only on the variables. Upper and lower bounds for the efficiency scores of the units are then defined as natural outcomes of our formulations. It is our specific formulation that enables us to proceed further in discriminating among the efficient units by means of a post-DEA model and the endurance indices. We then proceed still further in formulating another post-DEA model for determining input thresholds that turn an inefficient unit to an efficient one.     

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.     

DEA model with shared resources and efficiency decomposition

Data envelopment analysis (DEA) has proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, DMUs have a two-stage network process with shared input resources used in both stages of operations. For example, in hospital operations, some of the input resources such as equipment, personnel, and information technology are used in the first stage to generate medical record to track treatments, tests, drug dosages, and costs. The same set of resources used by first stage activities are used to generate the second-stage patient services. Patient services also use the services generated by the first stage operations of housekeeping, medical records, and laundry. These DMUs have not only inputs and outputs, but also intermediate measures that exist in-between the two-stage operations. The distinguishing characteristic is that some of the inputs to the first stage are shared by both the first and second stage, but some of the shared inputs cannot be conveniently split up and allocated to the operations of the two stages. Recognizing this distinction is critical for these types of DEA applications because measuring the efficiency of the production for first-stage outputs can be misleading and can understate the efficiency if DEA fails to consider that some of the inputs generate other second-stage outputs. The current paper develops a set of DEA models for measuring the performance of two-stage network processes with non splittable shared inputs. An additive efficiency decomposition for the two-stage network process is presented. The models are developed under the assumption of variable returns to scale (VRS), but can be readily applied under the assumption of constant returns to scale (CRS). An application is provided.     

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.     

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.     

Deriving the DEA frontier for two-stage processes

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently DEA has been extended to examine the efficiency of two-stage processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model provides not only an overall efficiency score for the entire process, but as well yields an efficiency score for each of the individual stages. Due to the existence of intermediate measures, the usual procedure of adjusting the inputs or outputs by the efficiency scores, as in the standard DEA approach, does not necessarily yield a frontier projection. The current paper develops an approach for determining the frontier points for inefficient DMUs within the framework of two-stage DEA.     

On some generalization of the DEA models

Inadequate results may arise in some instances of DEA model applications. For example, a data envelopment analysis (DEA) model may show ‘a notoriously inefficient unit’ as an efficient one. In addition, too many efficient units may appear in some DEA models. An elegant and subtle approach was proposed to deal with these problems, which is based on incorporating domination cones in DEA models. Yu, Wei and Brockett suggested the generalized DEA (GDEA) model that unifies and extends most of the well-known DEA models based on using domination cones. In this paper, we propose a model that is more general than the GDEA model, on the one hand, as it covers situations that the GDEA model cannot describe. On the other hand, our model enables one to construct step-by-step any model from the family of the GDEA models by incorporating artificial units and rays in the space of inputs and outputs in the Banker, Charnes, Cooper (BCC) model, which makes the process of model construction visible and more understandable. Moreover, we show that any GDEA model can be approximated by some BCC model.     

A bargaining game model for measuring performance of two-stage network structures

Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs), where the internal structures of DMUs are treated as a black-box. Recently DEA has been extended to examine the efficiency of DMUs that have two-stage network structures or processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model not only provides an overall efficiency score for the entire process, but also yields an efficiency score for each of the individual stages. The current paper develops a Nash bargaining game model to measure the performance of DMUs that have a two-stage structure. Under Nash bargaining theory, the two stages are viewed as players and the DEA efficiency model is a cooperative game model. It is shown that when only one intermediate measure exists between the two stages, our newly developed Nash bargaining game approach yields the same results as applying the standard DEA approach to each stage separately. Two real world data sets are used to demonstrate our bargaining game model.     

Scaling units via the canonical correlation analysis in the DEA context

This paper deals with the evaluation of decision making units which have multiple inputs and outputs. A new method (CCA/DEA) is developed where the Canonical Correlation Analysis (CCA) is utilized to provide a full rank scaling for all the units rather than a categorical classification (for efficient and inefficient units) as done by the Data Envelopment Analysis (DEA). The CCA/DEA approach is an attempt to bridge the gap between the frontier approach of DEA and the average tendencies of statistics (econometrics). Nonparametric statistical tests are employed to validate the consistency between the classification from the DEA and the postclassification that was generated by the CCA/DEA.     

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.     

Measuring the performance of manufacturing firms with super slacks based model of data envelopment analysis: An application of 500 major industrial enterprises in Turkey

Data envelopment analysis (DEA) is a mathematical programming technique which has a wide application area. There are many applications of DEA to measure firms’ performance. Balance sheet data is frequently used in order to measure performance of firms through DEA. So it is the characteristic of balance sheets that assets and liabilities amount to the same value. When the data for inputs and outputs are selected from both assets and liabilities sections of the balance sheet, it is important that more attention be paid to the analysis since values assigned to inputs and outputs could be included in assets and liabilities at the same time. Such a situation could create problems concerning the conclusions drawn as a result of analysis.     

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.     

DEA and the discriminant analysis of ratios for ranking units

The purpose of this study is to develop a new method which provides for given inputs and outputs the best common weights for all the units that discriminate optimally between the efficient and inefficient units as pregiven by the Data Envelopment Analysis (DEA), in order to rank all the units on the same scale. This new method, Discriminant Data Envelopment Analysis of Ratios (DR/DEA), presents a further post-optimality analysis of DEA for organizational units when their multiple inputs and outputs are given. We construct the ratio between the composite output and the composite input, where their common weights are computed by a new non-linear optimization of goodness of separation between the two pregiven groups. A practical use of DR/DEA is that the common weights may be utilized for ranking the units on a unified scale. DR/DEA is a new use of a two-group discriminant criterion that has been presented here for ratios, rather than the traditional discriminant analysis which applies to a linear function. Moreover, non-parametric statistical tests are employed to verify the consistency between the classification from DEA (efficient and inefficient units) and the post-classification as generated by DR/DEA.