A number of methodological extensions to the original DEA model have resulted from the application of the DEA technique to various real life problems. An important development in this regard is known as the “weight restrictions approach” that allows
for human preferences to be taken into accounting while running a DEA model. In simple words, weight restrictions refer to incorporation of additional constraints or restrictions, on weights assigned to different inputs and/or outputs, by the DEA models. Due to the relevance of weights restrictions approach for the current study, the next section provides an overview of some basic aspects related to this approach.
3.10.1.
Motivations for Incorporation of Weight restrictions
The development of the weight restrictions approach is a result of two major motivating factors. The first factor is based on the unbounded nature of the basic DEA models. The original CCR and BCC models allow complete flexibility in the assignment of weights to different input and output variables. Such flexibility sometimes leads to certain undesirable consequences, such as insufficient discrimination. Consequently, a large number of DMUs in an analysis could appear efficient; through assignment of zero, very large, or very small weights to different variables. Such assignment of weights implies either exclusion of, or unnecessary importance being assigned to, some of the variables; which is difficult to justify many times in reality (Dyson and Thanassoulis, 1988).
Insufficient discrimination is a common phenomenon, especially for situations where the total number of DMUs is relatively small, as compared to the number of inputs and outputs included in the analysis. This issue is also known the curse of dimensionality;
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whereby, few firms have many dimensions, represented by a number of inputs and outputs (Coelli et al., 2005). In addition to a relatively smaller number of DMUs, other possible reasons responsible for the insufficient discrimination include; unusual mix of input/output variables for some subsets of DMUs, uniformity of performance of different DMUs, and comparatively different scale sizes being exhibited by some of the DMUs under the VRS assumption (Podinovski and Thanassoulis, 2007).
A number of approaches have been proposed to deal with the problem of insufficient discrimination34. One of the earlier developments in this regard includes the super efficiency model proposed by Andersen and Petersen (1993) for improving the discrimination for the Pareto efficient DMUs. Other common approaches for improving discrimination include cross efficiency (Doyle and Green, 1993, Green et al., 1996), and multiple criteria DEA (Li and Reeves, 1999). Despotis (2002) introduced an approach for discriminating among the globally efficient DMUs, through a second stage of DEA; while more recently, Bal et al. (2010) recommended a weighted goal programming DEA model for improving discrimination. Podinovski and Thanassoulis (2007), and Angulo-Meza and Lins (2002) also provide informative readings by discussing various methods for improving discrimination.
The second commonly cited reason for introduction of the weight restrictions approach is the need for incorporating value judgments in DEA based assessments. Allen et al. (1997) define value judgements as “logical constructs, incorporated within an efficiency assessment study, reflecting the Decision Maker’s (DM) preferences in the process of assessing efficiency”. Major incentives for incorporation of such value judgments
include; incorporation of prior perceptions about efficiency of different DMUs, the need
34 An informative discussion can be found in ANGULO-MEZA, L. & LINS, M. P. E. 2002. Review of methods for increasing discrimination in data envelopment analysis. Annals of Operations Research, 116,
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to relate values of inputs and outputs in certain real life problems35, and taking into consideration the existing information about values of different input and output variables.
Taking into account the existing information about values of different variables has particular relevance for situations, where estimation of overall efficiency is the ultimate goal (Thompson et al., 1990). For such estimation, access to information about prices or worth of inputs and outputs is a pre-requisite. In the absence of relevant information on prices, value judgments can be used as a proxy for missing information about input prices or outputs’ worth. This is especially helpful in situations involving assessment of
not-for profit or public sector organizational units, producing outputs that cannot be evaluated easily in terms of traditional profit and return measures (Sherman, 1984). Finally, the use of value judgments is also recommended for aligning the imputed input/output values, with the economic concept of input/output substitution (Allen et al., 1997, Thanassoulis, 2001, Cooper et al., 2011).
3.10.2.
Major Approaches for Restricting Weights
A number of approaches have been proposed over the years for dealing with, what Cook and Seiford (2009) called, unacceptable or undesirable weighting schemes, resulting from unbounded DEA models. There are several sub-categories of weight restrictions, such as; absolute weight restrictions, cone ratios and assurance regions. One of the earliest developments in this domain was introduced by Dyson and Thanassoulis (1988), who proposed lower and upper bounds to be imposed on individual multipliers. Other examples of this type of weight restrictions were provided by Roll et al. (1991), Cook et al. (1991), and Roll and Golany (1993). The second category of weight restrictions is known as the cone ratio approach, under which multipliers are required to
35 An interesting example is provided in THANASSOULIS, E. 1995. Assessing police forces in England and Wales using data envelopment analysis. European Journal of Operational Research, 87, 641-657.
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belong to closed cones (Charnes et al., 1989b). A third category of weight restrictions was proposed by Thompson et al. (1986), who introduced the concept of “Assurance Regions (AR)” and homogeneous linear weight restrictions. This concept was further
improved by Thompson et al. (1990). Alternative generalizations of the AR approach can also be found in Allen et al. (1997), Thanassoulis and Allen (1998), Cook and Zhu (2007), and Cook and Zhu (2008).
Instead of restricting the actual weights, another proposed practice involves placing restrictions on virtual inputs and outputs. Virtual inputs and outputs are, in essence, normalized weights that reflect the extent to which the efficiency score of a DMU is affected by certain variables. Wong and Beasley (1990), introduce different ways in which such restrictions could be placed on virtual inputs and outputs. The first method is to place restriction only for the DMU under assessment, while the relative virtual values for remaining DMUs are left free. The second method is to place restrictions in respect of average DMUs. The third approach is to place restrictions on all the DMUs included in an assessment. This last method is considered to be computationally expensive, because of the larger number of constraints involved.
3.10.3.
Problems with the Weight Restrictions Approach
The weight restrictions approach is one of the most common methodologies to deal with insufficient discrimination problem and to incorporate value judgments in basic DEA models (Thanassoulis, 2001). Despite the frequent application of weights restrictions, certain problems associated with this approach have been identified.
A basic limitation associated with this approach is related to the manner in which weight restrictions are incorporated in a DEA model. As discussed earlier, there are two mutually dual linear programming (LP) models, that can be solved to calculate the relative efficiency scores of DMUs, i.e. the envelopment and the multiplier models
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(Cooper et al., 2007). A basic distinction between these models is the availability of different frameworks, for interpreting radial efficiency scores of the DMUs under assessment. The multiplier model provides the managerial meaning of efficiency, as the relative position of the DMU with reference to other DMUs, given the most favourable weights or prices of different inputs and outputs. The envelopment model, on the other hand, provides the technological meaning of efficiency scores, in the form of a possible radial improvement factor for different inputs and outputs (Podinovski, 2007b).
Of these two forms of linear DEA models, the envelopment form offers a clear economic meaning; but the incorporation of weights restrictions is actually done in the multiplier form, and not in the envelopment form. Consequently, in some situations, the efficient radial target identified by the DEA model may not be feasible in reality; or in other words, may not be technologically producible (Allen et al., 1997, Thanassoulis and Allen, 1998). A similar concern in this regard is raised by Podinovski and Thanassoulis (2007), who point out that the use of weight restrictions in many cases may be motivated by a desire to achieve higher discirmination; even at the cost of using unrealistic profile of optimal weights for different inputs and outputs. In such cases, it is quite possible that there would be no link between the proposed weights restrictions and the technological realities associated with specific production process being analysed; thus rendering such analysis meaningless.