An application of DEA for comparative analysis and ranking of regions in Serbia with regards to social-economic development

In this paper, we use data envelopment analysis (DEA) to estimate how well regions in Serbia utilize their resources. Based on data for four inputs and four outputs we applied an output-oriented CCR DEA model and it appears that 17 out of 30 regions are efficient. For each inefficient unit, DEA identifies the sources and level of inefficiency for each input and output. An output-oriented set of targets is determined for 13 inefficient regions. In addition, the possibilities of combining DEA and linear discriminant analysis (LDA) in evaluating performance are explored. The efficient regions are ranked using a cross efficiency matrix and an output-oriented version of Andersen–Petersen’s DEA model and the results are analyzed and compared.     

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.     

Radial DEA models without inputs or without outputs

In this paper we consider radial DEA models without inputs (or without outputs), and radial DEA models with a single constant input (or with a single constant output). We demonstrate that (i) a CCR model without inputs (or without outputs) is meaningless; (ii) a CCR model with a single constant input (or with a single constant output) coincides with the corresponding BCC model; (iii) a BCC model with a single constant input (or a single constant output) collapses to a BCC model without inputs (or without outputs); and (iv) all BCC models, including those without inputs (or without outputs), can be condensed to models having one less variable (the radial efficiency score) and one less constraint (the convexity constraint).     

Target setting in data envelopment analysis using MOLP

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) can be used as tools in management control and planning. The existing models have been established during the investigation of the relations between the output-oriented dual DEA model and the minimax reference point formulations, namely the super-ideal point model, the ideal point model and the shortest distance model. Through these models, the decision makers’ preferences are considered by interactive trade-off analysis procedures in multiple objective linear programming. These models only consider the output-oriented dual DEA model, which is a radial model that focuses more on output increase. In this paper, we improve those models to obtain models that address both inputs and outputs. Our main aim is to decrease total input consumption and increase total output production which results in solving one mathematical programming model instead of n models. Numerical illustration is provided to show some advantages of our method over the previous methods.     

Multiobjective data envelopment analysis

Data envelopment analysis (DEA) is popularly used to evaluate relative efficiency among public or private firms. Most DEA models are established by individually maximizing each firm's efficiency according to its advantageous expectation by a ratio. Some scholars have pointed out the interesting relationship between the multiobjective linear programming (MOLP) problem and the DEA problem. They also introduced the common weight approach to DEA based on MOLP. This paper proposes a new linear programming problem for computing the efficiency of a decision-making unit (DMU). The proposed model differs from traditional and existing multiobjective DEA models in that its objective function is the difference between inputs and outputs instead of the outputs/inputs ratio. Then an MOLP problem, based on the introduced linear programming problem, is formulated for the computation of common weights for all DMUs. To be precise, the modified Chebychev distance and the ideal point of MOLP are used to generate common weights. The dual problem of this model is also investigated. Finally, this study presents an actual case study analysing R&D efficiency of 10 TFT-LCD companies in Taiwan to illustrate this new approach. Our model demonstrates better performance than the traditional DEA model as well as some of the most important existing multiobjective DEA models.     

A new proposed model of restricted data envelopment analysis by correlation coefficients

The concept of efficiency in data envelopment analysis (DEA) is defined as weighted sum of outputs/weighted sum of inputs. In order to calculate the maximum efficiency score, each decision making unit (DMU)’s inputs and outputs are assigned to different weights. Hence, the classical DEA allows the weight flexibility. Therefore, even if they are important, the inputs or outputs of some DMUs can be assigned zero (0) weights. Thus, these inputs or outputs are neglected in the evaluation. Also, some DMUs may be defined as efficient even if they are inefficient. This situation leads to unrealistic results. Also to eliminate the problem of weight flexibility, weight restrictions are made in DEA. In our study, we proposed a new model which has not been published in the literature. We describe it as the restricted data envelopment analysis ((ARIII(COR))) model with correlation coefficients. The aim for developing this new model, is to take into account the relations between variables using correlation coefficients. Also, these relations were added as constraints to the CCR and BCC models. For this purpose, the correlation coefficients were used in the restrictions of input–output each one alone and their combination together. Inputs and outputs are related to the degree of correlation between each other in the production. Previous studies did not take into account the relationship between inputs/outputs variables. So, only with expert opinions or an objective method, weight restrictions have been made. In our study, the weights for input and output variables were determined, according to the correlations between input and output variables. The proposed new method is different from other methods in the literature, because the efficiency scores were calculated at the level of correlations between the input and/or output variables.     

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.     

A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis

Data envelopment analysis (DEA) is a linear programming methodology to evaluate the relative technical efficiency for each member of a set of peer decision making units (DMUs) with multiple inputs and multiple outputs. It has been widely used to measure performance in many areas. A weakness of the traditional DEA model is that it cannot deal with negative input or output values. There have been many studies exploring this issue, and various approaches have been proposed.     

Integrated data envelopment analysis: Global vs. local optimum

Chiou et al. (2010) (A joint measurement of efficiency and effectiveness for non-storable commodities: integrated data envelopment analysis approaches. European Journal of Operational Research 201, 477–489) propose an integrated data envelopment analysis model in measuring decision making units (DMUs) that have a two-stage internal network structure with multiple inputs, outputs, and consumptions. They claim that any optimal solutions determined by their DEA model are a global optimum, not a local optimum. We show that such a conclusion is a false statement due to their misuse of Hessian matrix in examining the concavity of the objective function, and their DEA model is actually a non-convex optimization problem. As a result, their DEA model is unusable in practice due to a lack of efficient algorithm for this particular non-convex DEA model. We further show that Chiou et al.’s (2010) model is a special case of a well-known two-stage network DEA model, and it can be transformed into a parametric linear program for which an approximate global optimal solution can be obtained by solving a sequence of linear programs in combination with a simple search algorithm.     

只有输出(入)的DEA改进模型及其应用

研究了只有输出(入)的DEA分析方法,针对只有输出(入)DEA模型的不足,重新定义了只有输出(入)的DEA评价方法的有效性,并改进了模型。相对已有的只有输出(入)的DEA模型,该模型充分利用了决策单元的诸输出(入),提高了DEA评价的效果。作为应用,运用新模型对武警防暴队形优选问题进行了有效性分析。     

In the determination of weight sets to compute cross-efficiency ratios in DEA

Data envelopment analysis (DEA) measures the production performance of decision-making units (DMUs) which consume multiple inputs and produce multiple outputs. Although DEA has become a very popular method of performance measure, it still suffers from some shortcomings. For instance, one of its drawbacks is that multiple solutions exist in the linear programming solutions of efficient DMUs. The obtained weight set is just one of the many optimal weight sets that are available. Then why use this weight set instead of the others especially when this weight set is used for cross-evaluation? Another weakness of DEA is that extremely diverse or unusual values of some input or output weights might be obtained for DMUs under assessment. Zero input and output weights are not uncommon in DEA. The main objective of this paper is to develop a new methodology which applies discriminant analysis, super-efficiency DEA model and mixed-integer linear programming to choose suitable weight sets to be used in computing cross-evaluation. An advantage of this new method is that each obtained weight set can reflect the relative strengths of the efficient DMU under consideration. Moreover, the method also attempts to preserve the original classificatory result of DEA, and in addition this method produces much less zero weights than DEA in our computational results.     

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.     

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.     

Applying inverse DEA and cone constraint to sensitivity analysis of DMUs with undesirable inputs and outputs

In this paper, the inverse data envelopment analysis (DEA) with the preference of cone constraints will be discussed in a way that in the decision-making units, the undesirable inputs and outputs exist simultaneously. Supposing that the efficiency level does not change, if the unit under assessment increases the level of the desirable outputs and decreases the level of the undesirable outputs, how will it affect the amount of the desirable input level and the undesirable input level? To answer this question, the application of the inverse DEA with preference of cone constraints is suggested. The suggested approach, while maintaining the efficiency level, increases the level of its undesirable input and decreases the level of its desirable input by selection of strongly efficient solutions or some weakly efficient solutions of the multiple objective linear programming (MOLP) model. While maintaining the efficiency level, the suggested approach by selection of strongly efficient solution or some of the weakly efficient solutions of the MOLP model can increase the undesirable input level and decrease the desirable input level. Similarly, the suggested approach can be applied if the decision-making unit increases its undesirable input level and decreases the desirable input level so that the undesirable output level decreases and the desirable output level increases while maintaining the efficiency level. As an illustration, two numerical examples are rendered.     

Centralized DEA models with the possibility of downsizing

While traditional data envelopment analysis (DEA) models assess the relative efficiency of similar, independent decision making units (DMUs) centralized DEA models aim at reallocating inputs and outputs among the units setting new input and output targets for each one. This system point of view is appropriate when the DMUs belong to a common organization that allocates their inputs and appropriates their outputs. This intraorganizational perspective opens up the possibility that greater technical efficiency for the organization as a whole might be achieved by closing down some of the existing DMUs. In this paper, we present three centralized DEA models that take advantage of this possibility. Although these models involve some binary variables, we present efficient solution approaches based on Linear Programming. We also present some numerical results of the proposed models for a small problem from the literature.     

A modified slacks-based measure model for data envelopment analysis with ‘natural’ negative outputs and inputs

This paper is primarily concerned with data envelopment analysis (DEA) of systems where negative outputs and negative inputs arise naturally. Examples of situations in which both negative inputs and negative outputs occur are given. More attention has been paid, in the literature, to the former type of problem. Most available DEA software does not solve this type of problem or copes with negative outputs and possibly negative inputs by assigning zero weights to them. A modified slacks-based measure (MSBM) model is presented, in which both negative outputs and negative inputs occur. The MSBM model overcomes the lack of translation invariance in the slacks-based measure model by drawing on the ideas from the range directional model (RDM). The MSBM model takes into account individual input and output slacks, which provides more precise evaluation of inefficient decision-making units (DMUs). It therefore, generally leads to lower efficiencies for inefficient DMUs than the RDM.     

An extended slacks-based measure model for data envelopment analysis with negative data

Data envelopment analysis (DEA) is a non-parametric approach based on linear programming that has been widely applied for evaluating the relative efficiency of a set of homogeneous decision-making units (DMUs) with multiple inputs and outputs. The original DEA models use positive input and output variables that are measured on a ratio scale, but these models do not apply to the variables in which negative data can appear. However, with the widespread use of interval scale data and undesirable data, the emphasis has been directed towards the simultaneous consideration of the positive and negative data in DEA models. In this paper, using the slacks-based measure, we propose an extended model to evaluate the efficiency of DMUs, even if some variables are measured on an interval scale and some on a ratio scale. Moreover, the extended model allows for the presence of all interval-scale variables, which are capable of taking both negative and positive values.     

Output-specific input-assurance regions in DEA

The use of assurance region (AR) constraints to restrict multipliers in data envelopment analysis (DEA) is well-established, and has been discussed at length in the literature. The conventional assumption in imposing such restrictions is that they apply universally. Specifically, AR constraints on input multipliers are intended to control the relative importance of the individual inputs in terms of how they impact the entire bundle of outputs. In many settings the relative importance of inputs is different for some of the outputs than for others. A typical example of this in the financial services sector is where the importance of sales staff versus service staff is different in regard to sales outputs than is true for service outputs. In this paper we develop a general DEA framework that incorporates multiple input-AR structures that cater to multiple output classes. We examine the cases of both divisible and indivisible inputs, and as well as mutually exclusive and overlapping output sets. The concepts are applied to a financial services situation.     

Restricting weights in supplier selection decisions in the presence of dual-role factors

One of the uses of data envelopment analysis (DEA) is supplier selection. Weight restrictions allow for the integration of managerial preferences in terms of relative importance levels of various inputs and outputs. As well, in some situations there is a strong argument for permitting certain factors to simultaneously play the role of both inputs and outputs. The objective of this paper is to propose a method for selecting the best suppliers in the presence of weight restrictions and dual-role factors. This paper depicts the supplier selection process through a DEA model, while allowing for the incorporation of decision maker’s preferences and considers multiple factors which simultaneously play both input and output roles. The proposed model does not demand exact weights from the decision maker. This paper presents a robust model to solve the multiple-criteria problem. A numerical example demonstrates the application of the proposed method.     

Avoiding infeasibility in DEA models with weight restrictions

The flexibility of weights assigned to inputs and outputs is a key aspect of DEA modeling. However, excessive weight variability and implausible weight values have led to the development of DEA models that incorporate weight restrictions, reflecting expert judgment. This in turn has created problems of infeasibility of the corresponding linear programs. We provide an existence theorem that establishes feasibility conditions for DEA multiplier programs with weight restrictions. We then propose a linear model that tests for feasibility and a nonlinear model that provides minimally acceptable adjustments to the original restrictions that render the program feasible. The analysis can be applied to restrictions on weight ratios, or to restrictions on virtual inputs or outputs.