Restricting virtual weights in data envelopment analysis

Abstract

The consequences of the use of absolute weights restrictions (i.e. restricting the multipliers) on the efficiency score and targets of a DEA model have been explored elsewhere, the same is not true for the use of restrictions on the virtuals (i.e. the product of the input/output factor by its multiplier). In this paper, a reflection on the uses of virtual weights restrictions is presented. The reasons for using virtual weights restrictions instead of absolute weights restrictions, in particular cases, are explained. Following a critique of Wong and Beasley's [J. Oper. Res. Soc. 41 (1990) 829] first proposed method for constraining the virtuals in DEA, a new classification scheme for virtual weights restrictions is presented, which brings the concept of assurance regions into virtual weights restrictions. It is shown that the use of simple virtual restrictions and virtual assurance regions are preferable to the use of the more generally advocated WB's proportional virtual weights restrictions. In recognition of levels of decision making at the unit, and external to the unit, the use of the terms unit of assessment (UOA) and controller is proposed. It is concluded that the use of virtual assurance regions applying to the target UOA can be a natural representation of preference structures and translate established patterns between the input–output divide. Also, the meaning of the efficiency score and targets in this approach most approximate traditional DEA. Alternatives to using virtual weights restrictions are considered, namely using absolute weights restrictions with a virtual meaning. Finally, an empirical example is offered.

Introduction

The application of DEA to concrete situations has motivated the use of weights restrictions to curb the complete freedom of variation of weights allowed by the original DEA model. The problem of allowing total flexibility of the weights is that the values of the weights obtained by solving the unrestricted DEA program are often in contradiction to prior views or additional available information.

Thompson et al. (1986) were the first authors to propose the use of weights restrictions to increase discrimination of the results of a DEA problem to support the siting of a laboratory, where only six alternatives were under consideration. Their technique included the imposition of acceptable bounds on ratios of multipliers (weights), to solve a choice problem. Dyson and Thanassoulis (1988) were concerned about the omission of particular inputs or outputs from the efficiency score, through the assignment of zero weights. They suggested imposing meaningful bounds directly on individual multipliers based on average input levels per unit of output. Charnes et al. (1990), in another approach to the problem, suggested transforming input–output data to simulate weights restrictions, where DMUs are assessed on the basis of the input–output levels of pre-selected DMUs which were a priori recognised by experts as being efficient. Halme et al. (1999) offer, still, another alternative to incorporating the decision maker's preferences into the assessment of DMUs. They develop a procedure that begins by aiding the decision maker in searching for the most preferred combination of inputs and outputs of DMUs which are efficient in DEA––the most preferred solution. Then, efficiency scores are calculated, which can be interpreted as the relative difference in value between the most preferred solution and the unit under investigation.

One of the problems with directly restricting the multipliers, i.e. absolute weights restrictions (assurance regions, Thompson et al., 1986, or simple absolute weights restrictions on the multipliers, Dyson and Thanassoulis, 1988) is that they are dependent on the units of measurement of the inputs and outputs. Virtual input/output, the product of the input/output level and optimal weight for that input/output, however, is dimensionless. The virtual inputs and outputs of a DMU reveal the relative contribution of each input and output to its efficiency rating. The higher the level of virtual input/output, the more important that input/output is in the efficiency rating of the DMU concerned. Therefore use of virtual inputs and outputs can help to identify strong and weak areas of performance. Additionally, frequently, it is hard to give a meaning to absolute weights restrictions. It is often difficult to ascertain meaningful bounds directly on individual multipliers, in the case of simple absolute weights restrictions, or to calculate meaningful marginal rates of substitution, in the case of assurance regions (see Allen et al., 1997a, for a comprehensive discussion). Virtual weights restrictions are, in most occasions, more intuitive for the decision maker. In order to avoid the problems of absolute weights restrictions, Wong and Beasley (1990) proposed the use of virtual weights restrictions, and in particular, the use of proportional virtual weights restrictions, which were intended to make it easier for the decision maker to quantify value judgements in terms of percentage values. Roll and Golany (1993), to avoid the dependence on the units of measurement of the input and output factors in absolute weights restrictions, proposed instead the normalisation of the input–output data. One of the disadvantages of this method is that once results are obtained they must be transformed back to the original form in order to interpret the results. Also, absolute weights restriction can be a problem for the analyst when dealing with managers who do not necessarily understand DEA. In which case, it is easier to elicit from management virtual weights restrictions in terms of the proportional importance of the factors. The Roll and Golany approach overcomes some of the problems with absolute weights restrictions, but does not allow direct comparisons of the relative contributions of inputs and outputs to the efficiency rating.

A comprehensive review of the evolution, development and research directions on the use of weights restrictions can be found in Allen et al. (1997a). In this review the consequences for the interpretation of the results from DEA models with weights restrictions has been analysed for absolute weights restrictions. The analysis of the pros and cons of the use of virtual weights restrictions and how it compares with the use of absolute weights restrictions are proposed as a further direction of research. This paper proposes to contribute to that analysis.

The intention of incorporating value judgements might be, as seen above, to incorporate prior views or information regarding the assessment of efficient DMUs. On the other hand, there might be two levels of decision making, the DMU (for instance, a department or university), and the corporate top management or external evaluator (for instance the State, or the applicant in relation to a university or department in Sarrico et al. (1997)). The DMU might use its value judgements if it wants to use the assessment for benchmarking itself against other DMUs (see, for instance, Sarrico and Dyson, 2000).

However, if an external agent does the evaluation, the expressions DMU and decision maker might be misleading, as the decision maker is, in fact, at a different level. In this case, the DEA assessment becomes a game between what can be called the unit of assessment (UOA) trying to show itself in the best possible light, and a higher level decision maker, i.e. a controller imposing its preference structure. Sarrico et al. (1997) and Sarrico and Dyson (2000) have used DEA in the assessment of UK universities' performance, where the university or department is the UOA with the applicant or the State, respectively, being the controller. A similar situation occurs in the regulated industries where the regulator is the controller. Virtual weights restrictions are particularly appealing in these circumstances when `outside' judgements need to be translated into weights restrictions in a DEA model. The higher the level of a virtual input or output, the more important that input or output for the efficiency rating of the DMU concerned. Absolute weights, however, do not normally have an obvious meaning to the controller.

Allen et al. (1997a) point out that the substantial changes to the UOA's current mix of input and output levels indicated by the imposition of weights restrictions might be beneficial. It might lead to the conclusion that the current mix is inadequate given the controller's preferences. The same goes for the deterioration of current levels of some inputs and outputs.

Although there are often references in the literature to WB's first proposed methods of restricting the virtuals in DEA, neither WB's original paper nor subsequent literature explore the consequences of their use in the interpretation of the efficiency score and targets thus obtained. Section 2 proposes to do that. In Section 3 it is shown how WB's restrictions could be translated into equivalent absolute weights restrictions, and how easily some of their methods lead to infeasible problems. In Section 4, the authors support the use of virtual weights restrictions, albeit of a different kind of WB's, for particular cases. They then propose a new classification of virtual weights restrictions, which introduce the concept of virtual assurance regions. In Section 5 the consequences of using the authors proposed virtual weights restrictions are explored, and their advantage over WB's ascertained. Finally, in Section 6 an empirical example, which illustrates the use of a range of types of weights restrictions including virtual assurance regions, is presented.