Rank order data in DEA: A general framework

Abstract

In data envelopment analysis (DEA), performance evaluation is generally assumed to be based upon a set of quantitative data. In many real world settings, however, it is essential to take into account the presence of qualitative factors when evaluating the performance of decision making units (DMUs). Very often rankings are provided from best to worst relative to particular attributes. Such rank positions might better be presented in an ordinal, rather than numerical sense. The paper develops a general frame work for modeling and treating qualitative data in DEA and provides a unified structure for embedding rank order data into the DEA framework. The existing techniques are discussed and their equivalence is demonstrated. Both continuous and discrete projection models are provided. It is shown that qualitative data can be treated in conventional DEA methodology.

Introduction

In the data envelopment analysis (DEA) model of Charnes et al. (1978), each member of a set of decision making units (DMUs) is to be evaluated relative to its peers. This evaluation is generally assumed to be based on a set of quantitative output and input factors. In many real world settings, however, it is essential to take into account the presence of qualitative factors when rendering a decision on the performance of a DMU. Very often it is the case that for a factor such as management competence, one can, at most, provide a ranking of the DMUs from best to worst relative to this attribute. The capability of providing a more precise, quantitative measure reflecting such a factor is generally beyond the realm of reality. In some situations such factors can be legitimately ‘quantified’, but very often such quantification may be superficially forced as a modeling convenience.

In situations such as that described, the ‘data’ for certain influence factors (inputs and outputs) might better be represented as rank positions in an ordinal, rather than numerical sense. Refer again to the management competence example. In certain circumstances, the information available may permit one only to put each DMU into one of L categories or groups (e.g. ‘high’, ‘medium’ and ‘low’ competence). In other cases, one may be able to provide a complete rank ordering of the DMUs on such a factor.

Cook et al., 1993, Cook et al., 1996 first presented a modified DEA structure, incorporating rank order data. The 1996 article applied this structure to the problem of prioritizing a set of research and development projects, where both inputs and outputs were defined on a Likert scale. An alternative to the Cook et al approach was provided in Cooper et al. (1999) in the form of the imprecise DEA (IDEA) model. While various forms of imprecise data were examined, one major component of that research focused on ordinal (rank order) data.

In the current paper, we present a unified structure for embedding rank order or Likert scale data into the DEA framework. To provide a practical setting for the methodology to be developed herein, Section 2 briefly discusses the R&D project selection problem as presented in more detail in Cook et al. (1996), and the Korean Telephone offices problem of Kim et al. (1999). Section 3 presents a continuous projection model, based on the conventional radial model of Charnes et al. (1978). In Section 4 this approach is compared to the IDEA methodology of Cooper et al. (1999). We demonstrate that IDEA for Likert scale data is in fact equivalent to the earlier approach of Cook et al. (1996). Section 5 develops a discrete projection methodology that guarantees projection to points on the Likert scale. Conclusions and further directions are addressed in Section 6.