DEA models with undesirable inputs and outputs

Data Envelopment Analysis (DEA) models with undesirable inputs and outputs have been frequently discussed in DEA literature, e.g., via data transformation. These studies were scatted in the literature, and often confined to some particular applications. In this paper we present a systematic investigation on model building of DEA without transferring undesirable data. We first describe the disposability assumptions and a number of different performance measures in the presence of undesirable inputs and outputs, and then discuss different combinations of the disposability assumptions and the metrics. This approach leads to a unified presentation of several classes of DEA models with undesirable inputs and/or outputs.     

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).     

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.     

On classifying inputs and outputs in DEA: A revised model

Cook and Zhu [Cook, W.D., Zhu, J., 2007. Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research 180, 692–699] introduced a new method to determine whether a measure is an input or an output. In practice, however, their method may produce incorrect efficiency scores due to a computational problem as result of introducing a large positive number to the model. This note introduces a revised model that does not need such a large positive number.     

Measuring production with random inputs and outputs using DEA and certainty equivalent

This study evaluates production operations with inputs/outputs under random influences. We introduce a measurement of efficiency using utility function families. Applying Data Envelopment Analysis (DEA) and the certainty equivalent, the proposed measurement is capable of accommodating various risk attitudes of evaluators.     

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.     

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.     

Measurement of simultaneous scale and mix changes in inputs and outputs using DEA facets and RTS

We show a new use of the efficient facets in DEA. Specifically, once we have identified all facets of the DEA technology, we are able to estimate the potential changes in some inputs and outputs, while fixing other inputs and outputs, ranges of simultaneous scale and mix changes in inputs and outputs, while proportionally increasing or decreasing other inputs and outputs, and, finally, the RTS. The proposed algorithms are applied to corporate planning processes of chemical companies.     

A note on a variant of radial measure capable of dealing with negative inputs and outputs in DEA

The recent contribution by Cheng et al. (2013) presents a variant of the traditional radial input- and output-oriented efficiency measures whereby original values are replaced with absolute values. This comment spells out that this article contains some imprecisions and therefore presents some further results.     

Classifying inputs and outputs in data envelopment analysis

In conventional data envelopment analysis it is assumed that the input versus output status of each of the chosen performance measures is known. In some situations, however, certain performance measures can play either input or output roles. We refer to these performance measures as flexible measures. This paper presents a modification of the standard constant returns to scale DEA model to accommodate such flexible measures. Both an individual DMU model and an aggregate model are suggested as methodologies for deriving the most appropriate designations for flexible measures. We illustrate the application of these models in two practical problem settings.     

Enhancing the inclusion of non-discretionary inputs in DEA

The aim of this paper is to provide an alternative approach for estimating efficiency when a set of decision-making units uses non-discretionary inputs in the productive process. To test the influence of these variables, our proposal uses a multi-stage approach based on Tobit regressions. In order to avoid potential bias, a bootstrap procedure is used to estimate these regressions. This methodology allows enhancing other models previously proposed to introduce non-controllable inputs in data envelopment analysis (DEA) overcoming, thus, some of their main shortcomings. We illustrate our framework with an empirical application on Spanish high schools where non-controllable factors play a major role to explain educational achievements.     

Network DEA approach to airports performance assessment considering undesirable outputs

In this paper, a directional distance approach is proposed to deal with network DEA problems in which the processes may generate not only desirable final outputs but also undesirable outputs. The proposed approach is applied to the problem of modelling and benchmarking airport operations. The corresponding network DEA model considers two process (Aircraft Movement and Aircraft Loading) with two final outputs (Annual Passenger Movement and Annual Cargo handled), one intermediate product (Aircraft Traffic Movements) and two undesirable outputs (Number of Delayed Flights and Accumulated Flight Delays). The proposed approach has been applied to Spanish airports data for year 2008 comparing the computed directional distance efficiency scores with those obtained using a conventional, single-process directional distance function approach. From this comparison, it can be concluded that the proposed network DEA approach has more discriminatory power than its single-process counterpart, uncovering more inefficiencies and providing more valid results.     

Nonlinear inputs and diminishing marginal value in DEA

The conventional model structures presented in the data envelopment analysis (DEA) literature view all variables as behaving in a linear fashion, meaning that regardless of the amounts, large or small, of a variable held by the set of decision-making units, we apply the same multiplier to those various amounts. In certain situations this linearity assumption is not appropriate, and the conventional models need to be altered to accommodate nonlinear representations. In the current paper, we propose a modified DEA structure that captures certain forms of nonlinear behaviour within the additive DEA model, namely those that exhibit diminishing marginal value.     

An extended assignment problem considering multiple inputs and outputs

The existing assignment problems for assigning n jobs to n individuals are limited to the considerations of cost or profit incurred by each possible assignment. However, in real applications, various inputs and outputs are usually concerned in an assignment problem, such as a general decision-making problem. This paper develops a procedure for resolving assignment problems with multiple incommensurate inputs and outputs for each possible assignment. The concept of the relative efficiency in using various resources, instead of cost or profit, is adopted for each possible assignment of the problem. Data envelopment analysis (DEA) is employed in this paper to measure the efficiency of one assignment relative to that of the others according to a set of decision-making units. A composite efficiency index, consisting of two kinds of relative efficiencies under different comparison bases, is defined to serve as the performance measurement of each possible assignment in the problem formulation. A mathematical programming model for the extended assignment problem is proposed, which is then expressed as a classical integer linear programming model to determine the assignments with the maximum efficiency. A numerical example is used to demonstrate the approach.     

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.     

Using a group support system to aid input–output identification in DEA

In data envelopment analysis (DEA), operating units are compared on their outputs relative to their inputs. The identification of an appropriate input–output set is of decisive significance if assessment of the relative performance of the units is not to be biased. This paper reports on a novel approach used for identifying a suitable input–output set for assessing central administrative services at universities. A computer-supported group support system was used with an advisory board to enable the analysts to extract information pertaining to the boundaries of the unit of assessment and the corresponding input–output variables. The approach provides for a more comprehensive and less inhibited discussion of input–output variables to inform the DEA model.     

Using SSM to rethink the analysis of energy efficiency initiatives

This paper reflects an attempt to rethink the process of analysis of energy efficiency initiatives using soft systems methodology (SSM) as a problem structuring tool. The aim of the work is to provide public and private initiative promoters or evaluators with a structured support for a more informed decision regarding the implementation of energy efficiency measures. The SSM approach contributed with the identification of all market players and their relations, as well as the insight into the deficiencies of current methodologies. Some future work directions are also proposed.     

Uncertain data envelopment analysis with imprecisely observed inputs and outputs

Data envelopment analysis (DEA) is a powerful analytical tool in operations research and management for measuring and estimating the efficiency of decision-making units. Both the inputs and the outputs are assumed to be known constants in the classical DEA models. However, in many cases, those data (e.g., carbon emissions and social benefit) cannot be measured in a precise way. Therefore, in this article, the inputs and outputs are considered as uncertain variables and a new uncertain DEA model is introduced. The sensitivity and stability of the new model are also analyzed. Finally, a numerical example of the new model is documented.     

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.