The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs

The most popular weight restrictions are assurance regions (ARs), which impose ratios between weights to be within certain ranges. ARs can be categorized into two types: ARs type I (ARI) and ARs type II (ARII). ARI specify bounds on ratios between input weights or between output weights, whilst ARII specify bounds on ratios that link input to output weights. DEA models with ARI successfully maximize relative efficiency, but in the presence of ARII the DEA models may under-estimate relative efficiency or may become infeasible. In this paper we discuss the problems that can occur in the presence of ARII and propose a new nonlinear model that overcomes the limitations discussed. Also, the dual model is described, which enables the assessment of relative efficiency when trade-offs between inputs and outputs are specified. The application of the model developed is illustrated in the efficiency assessment of Portuguese secondary schools.     

Assessing the relative efficiency of decision making units using DEA models with weight restrictions

In models of data envelopment analysis (DEA), an optimal set of input and output weights is generally assumed to represent the assessed decision making unit (DMU) in the best light in comparison to all the other DMUs. The paper shows that this may not be correct if absolute weight bounds or some other weight restrictions are added to the model. A consequence may be that the model will underestimate the relative efficiency of DMUs. The incorporation of weight restrictions in a maximin DEA model is suggested. This model can be further converted to more operational forms, which are similar to the classical DEA models.     

Congestion analysis in DEA inputs under weight restrictions

Data envelopment analysis (DEA), which is used to determine the efficiency of a decision-making unit (DMU), is able to recognize the amount of input congestion. Moreover, the relative importance of inputs and outputs can be incorporated into DEA models by weight restrictions. These restrictions or a priori weights are introduced by the decision maker and lead to changes in models and efficiency interpretation. In this paper, we present an approach to determine the value of congestion in inputs under the weight restrictions. Some discussions show how weight restrictions can affect the congestion amount.     

DEA models for the explicit maximisation of relative efficiency

It has recently been demonstrated that incorporating weight bounds and other non-homogeneous restrictions in DEA models may lead to underestimation of the maximum relative efficiency of decision making units. This paper suggests a way of avoiding this by replacing the objective function in DEA models by the relative efficiency of the assessed unit and converting the resulting models to linear forms. An alternative approach based on incorporating weight restrictions in the recently introduced maximin DEA model is also considered. It is shown that imposing weight bounds in the maximin model is equivalent to imposing bounds on ratios of individual weights.     

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.     

Productivity change and innovation in Norwegian electricity distribution companies

Regulators of electricity distribution networks have typically applied Data Envelopment Analysis (DEA) to cross-section data for benchmarking purposes. However, the use of panel data to analyse the impact of regulatory policies on productivity change over time is less frequent. The main purpose of this paper is to construct a Malmquist productivity index to examine the recent productivity change experienced by Norwegian distribution companies between 2004 and 2007. The Malmquist index is decomposed in order to explore the sources of productivity change, and to identify the innovator companies that pushed the frontier forward each year. The input and output variables considered are those used by the Norwegian regulator. In order to reflect appropriately the exogenous conditions where the companies operate, the efficiency model used in this paper incorporates geography variables as outputs of the DEA model. Unlike the model used by the regulator, we included virtual weight restrictions in the DEA formulation to correct the biases in the DEA results that may be associated to a judicious choice of weights by some of the companies.     

Value-based DEA models: application-driven developments

Data envelopment analysis (DEA) is an approach based on linear programming to assess the relative efficiency of peer decision-making units (DMUs). Typically, each DMU is free to choose the weights of the factors used in its evaluation. However, the evaluator's preferences may not warrant so much freedom. Several approaches have been proposed to allow the incorporation of managerial preferences in DEA, but few address the additive DEA model specifically. This paper presents additive DEA models that use multi-criteria decision analysis concepts to incorporate managerial preferences, and presents the corresponding preference elicitation protocols. The models developed allow the incorporation of preferences at different levels: on valuing performance improvements, on introducing weight restrictions, and on finding adequate targets. These were application-driven developments, resulting from discussing modelling options and preliminary results with the top-level management of a retail chain in the context of an assessment of stores’ performance, also described in this paper.     

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.     

A multiplier bound approach to assess relative efficiency in DEA without slacks

In this paper, we propose a new approach to deal with the non-zero slacks in data envelopment analysis (DEA) assessments that is based on restricting the multipliers in the dual multiplier formulation of the used DEA model. It guarantees strictly positive weights, which ensures reference points on the Pareto-efficient frontier, and consequently, zero slacks. We follow a two-step procedure which, after specifying some weight bounds, results in an “Assurance Region”-type model that will be used in the assessment of the efficiency. The specification of these bounds is based on a selection criterion among the optimal solutions for the multipliers of the unbounded DEA models that tries to avoid the extreme dissimilarity between the weights that is often found in DEA applications. The models developed do not have infeasibility problems and we do not have problems with the alternate optima in the choice of weights that is made. To use our multiplier bound approach we do not need a priori information about substitutions between inputs and outputs, and it is not required the existence of full dimensional efficient facets on the frontier either, as is the case of other existing approaches that address this problem.     

Finding common weights based on the DM's preference information

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). The flexibility in selecting the weights in standard DEA models deters the comparison among DMUs on a common base. Moreover, these weights are not suitable to measure the preferences of a decision maker (DM). For dealing with the first difficulty, the concept of common weights was proposed in the DEA literature. But, none of the common weights approaches address the second difficulty. This paper proposes an alternative approach that we term as ‘preference common weights’, which is both practical and intellectually consistent with the DEA philosophy. To do this, we introduce a multiple objective linear programming model in which objective functions are input/output variables subject to the constraints similar to the equations that define production possibility set of standard DEA models. Then by using the Zionts–Wallenius method, we can generate common weights as the DM's underlying value structure about objective functions.     

Zero weights and non-zero slacks: Different solutions to the same problem

This paper re-assesses three independently developed approaches that are aimed at solving the problem of zero-weights or non-zero slacks in Data Envelopment Analysis (DEA). The methods are weights restricted, non-radial and extended facet DEA models. Weights restricted DEA models are dual to envelopment DEA models with restrictions on the dual variables (DEA weights) aimed at avoiding zero values for those weights; non-radial DEA models are envelopment models which avoid non-zero slacks in the input-output constraints. Finally, extended facet DEA models recognize that only projections on facets of full dimension correspond to well defined rates of substitution/transformation between all inputs/outputs which in turn correspond to non-zero weights in the multiplier version of the DEA model. We demonstrate how these methods are equivalent, not only in their aim but also in the solutions they yield. In addition, we show that the aforementioned methods modify the production frontier by extending existing facets or creating unobserved facets. Further we propose a new approach that uses weight restrictions to extend existing facets. This approach has some advantages in computational terms, because extended facet models normally make use of mixed integer programming models, which are computationally demanding.     

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.     

Weight-restricted DEA in action: from expert opinions to mathematical models

A common problem in real-world DEA applications is that all inputs and outputs may not be equally relevant to the organizations analysed and their stakeholders. In many cases, one is also faced with a data set where the decision-making units do not clearly outnumber the quantity of inputs and outputs. This study reports an application where DEA embellished with weight restrictions is used to analyse the efficiency of public organizations to overcome the above-mentioned problems. Whereas there are numerous documented applications of weight-restricted DEA in the literature, the process of defining the actual weight restrictions is seldom described. However, that part — defining the actual weights restrictions based on price, preference or value information — is the most difficult step involved in using the weight-restricted DEA. Comparing various weight restriction schemes with real data suggests that the ability to consider and include preference information in DEA adds important insights into the analysis.     

The law of one price in data envelopment analysis: Restricting weight flexibility across firms

The Law of One Price (LoOP) states that all firms face the same prices for their inputs and outputs under market equilibrium. Taken here as a normative condition for ‘efficiency prices’, this law has powerful implications for productive efficiency analysis, which have remained unexploited thus far. This paper shows how LoOP-based weight restrictions can be incorporated in Data Envelopment Analysis (DEA). Utilizing the relation between industry-level and firm-level cost efficiency measures, we propose to apply a set of input prices that is common for all firms and that maximizes the cost efficiency of the industry. Our framework allows for firm-specific output weights and for variable returns-to-scale, and preserves the linear programming structure of the standard DEA. We apply the proposed methodology to the evaluation of the research efficiency of economics departments of Dutch Universities. This application shows that the methodology is computationally tractable for practical efficiency analysis, and that it helps in deepening the DEA analysis.     

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.     

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.     

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.     

The extension and integration of the inverse DEA method

The inverse DEA (Data Envelopment Analysis) method is primarily used to analyse the changing relationship between the inputs and outputs of a DMU (Decision-Making Unit) when its efficiency is kept constant or set to a target value. However, the existing inverse DEA method cannot be applied directly to estimate all the changing relationships. For example, the existing DEA models fail to estimate the input variations when the supervisor wants to maintain the DMU’s output-oriented efficiency during the downscaling of production. This paper analyses all the possible changing relationships that need to be solved by the inverse DEA method and develops different models for both the output and input orientations, accomplishing the extension and integration of the inverse DEA model. For illustration of our results, a numerical example is given.     

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.     

General and multiplicative non-parametric corporate performance models with interval ratio data

The increasing intensity of global competition has led organizations to utilize various types of performance measurement tools for improving the quality of their products and services. Data envelopment analysis (DEA) is a methodology for evaluating and measuring the relative efficiencies of a set of decision making units (DMUs) that use multiple inputs to produce multiple outputs. All the data in the conventional DEA with input and/or output ratios assumes the form of crisp numbers. However, the observed values of data in real-world problems are sometimes expressed as interval ratios. In this paper, we propose two new models: general and multiplicative non-parametric ratio models for DEA problems with interval data. The contributions of this paper are fourfold: (1) we consider input and output data expressed as interval ratios in DEA; (2) we address the gap in DEA literature for problems not suitable or difficult to model with crisp values; (3) we propose two new DEA models for evaluating the relative efficiencies of DMUs with interval ratios, and (4) we present a case study involving 20 banks with three interval ratios to demonstrate the applicability and efficacy of the proposed models where the traditional indicators are mostly financial ratios.