## A DEA approach to derive individual lower and upper bounds for technical and allocative efficiency

This article introduces concepts of Data Envelopment Analysis as applied to technical efficiency. It serves as an introduction to the JORS paper **A DEA approach to derive individual lower and upper bounds for the technical and allocative components of the overall profit efficiency**, authored by J L Ruiz and I Sirvent. The paper is available to OR Society members in the *Journal of the Operational Research Society* (2011) **62,** 1907–1916, published online 24 November 2010, The paper is freely quoted in this article, and its authors have approved this note.

### Introduction

In this paper, Ruiz and Sirvent propose a slack-based data envelopment analysis approach to be used in economic efficiency analyses when the objective is profit maximization. The focus is on the measurement of the technical component of the overall efficiency with the purpose of guaranteeing the achievement of the Pareto efficiency. As a result, we will be able to estimate correctly the allocative component in the sense that this latter only reflects the improvements that can be accomplished by reallocations along the Pareto-efficient frontier. Some new measures of technical and allocative efficiency in terms of both profit ratios and differences of profits are defined.

Technical efficiency refers to all of the sources of waste which can be eliminated without worsening any other input and/or output, and can be distinguished from that generally referred to as economic efficiency, which involves recourse to information on prices and/or costs. In the analyses of the mentioned economic efficiency the objectives are usually cost minimization, revenue maximization or profit maximization. In all three cases, these analyses are mainly based on the measures of overall efficiency and its technical and allocative components. Overall efficiency reflects the comparison of the unit under assessment with the best plan (with the given prices). The measurement of allocative efficiency presumes technical efficiency. Thus, allocative efficiency is that remaining in the overall efficiency (the residual) once the unit being evaluated has achieved technical efficiency, and it can be seen as the improvements that can be accomplished by reallocations along the efficient frontier.

Cost minimization and revenue maximization have been the cases more deeply investigated. In this paper we address the profit maximization case. One of the main difficulties with many of the existing approaches, not only in profit settings but also in those only concerned with either costs or revenues, is that they are based on a Farrell-type measurement of technical efficiency, which only considers equiproportionate input contractions and/or equiproportionate output expansions, and so, they do not account for the inefficiency in the slacks. Therefore, these measures do not guarantee the achievement of the technical efficiency (in the sense of Pareto), which is a previous requirement to the measurement of the allocative efficiency. As a consequence, the corresponding allocative component will be incorrectly measured, since it will reflect not only the gains in profit that can be accomplished by substitutions along the efficient frontier but also it will be accounting for some technical inefficiency.

### Main ideas

In this paper the authors propose a slack-based DEA approach to the measurement of the technical and allocative components of the overall profit efficiency. This guarantees the achievement of the technical efficiency so that the allocative efficiency component, which is defined as a residual, can be correctly estimated and it only reflects the profit improvements that can be accomplished by reallocations along the efficient frontier. By using this approach, they are able to provide a decomposition of the overall efficiency both when this is defined as a ratio of profits and when we use differences of profits. The key issue in this approach is the measurement of the profit that the unit under assessment may achieve operating efficiently in a technical sense. This allows us to define efficiency measures that reflect the comparisons between the actual profit and that profit, on one hand, and between that profit at the technical efficiency and the maximum profit, on the other.

### An introduction to the maths

Consider a set of *n* organisation (DMUs, or Decision Making Units), each of which which use *m* inputs (with costs *c*) to produce *s* outputs (with prices* p*). The production possibility set (taken as data from these comparable organisations) is *T*={(*x*,*y*)?R+*m*+*s*/*x* can produce *y*}; these represent all known possibilities of production.

Overall profit efficiency measures are usually the result of comparing the actual profit of DMU0, ?0, with the maximum profit that this unit could achieve operating in a fully efficient manner, ?0*. This maximum profit can be actually expressed as ?0*=*p*0*ty**-*c*0*tx**, and can be obtained as the optimal value of the following problem:

i.e. the maximum profit to be made at the same prices and costs, by varying the resources and output used.

In order to decompose the overall efficiency, the key is in defining the technical efficiency component, since the allocative one is the resulting residual. To do it, we can proceed in a similar manner as with the overall measures and compare the actual profit of DMU0, ?0, with the maximum profit, ?0*T*, that this unit could achieve operating in a technically efficient manner. We take here as a definition of technical efficiency, a measure where all sources of waste that can be eliminated without worsening any other input and/or output (a Pareto definition). This profit ?0*T* can be obtained as the optimal value of the following LP problem, which determines the maximum profit that DMU0 could achieve by reducing their inputs and/or expanding their outputs:

Note that models (1) and (2) only differ in the set of comparable points considered. While (1) seeks the highest profit within the whole set of possibilities of production, (2) restricts the search to the points dominating DMU0, since no variable can worsen when technical inefficiency is assessed. It is also to be noted that ?0T is obtained without imposing any previously specified direction to be followed to reach the efficient frontier (radial, hyperbolic or the one determined by a pre-specified directional vector), since there is no need to assume any requirement other than dominance when the technical efficiency component is to be measured. For this reason, in solving (2) we simply maximize the profit among all the efficient points dominating DMU0.

The allocative efficiency is then defined as the difference between the full efficiency and the technical efficiency.

With the approach above described it may be possible to perform efficiently in a technical sense with a lower level of profit, and this would obviously affect the subsequent measurement of the allocative efficiency. In order to deal with this situation, the authors derive individual lower and upper bounds for the technical component of the overall efficiency, which will be associated with the minimum and the maximum profits that the unit under assessment may achieve performing in a manner technically efficient. To do it, they exploit the fact that no assumption is made on the path the unit under assessment should follow to reach the efficient frontier, and extend the ideas in the minimum distance to the frontier approach in Aparicio et al. (2007) to their use with the allocation models. Obviously, the two bounds derived for the technical efficiency determine another couple of bounds for the corresponding values of the allocative efficiency component since this latter is defined as a residual. Thus, we will have a range (instead of a single value) for potential profit improvements that can be accomplished by achieving the technical and allocative efficiencies. This information broadens the range of possibilities for the explanation of the overall efficiency in terms of its technical and allocative efficiency components.