Data envelopment analysis with missing data: an application to University libraries in Taiwan
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Data envelopment analysis with missing data: an application to University libraries in Taiwan C Kao1* and S-Tai Liu2 1
National Cheng Kung University, Taiwan; 2Van Nung Institute of Technology, Taiwan
In measuring the relative ef®ciencies of a set of decision making units (DMUs) via data envelopment analysis (DEA), detailed inputs and outputs are usually involved. However, there are cases where some DMUs are unable to provide all the necessary data. This paper adopts the concept of a membership function used in fuzzy set theory for representing imprecise data. The smallest possible, most possible, and largest possible values of the missing data are derived from the observed data to construct a triangular membership function. With the membership function, a fuzzy DEA model can be utilized to calculate the ef®ciency scores. Since the ef®ciency scores are fuzzy numbers, they are more informative than crisp ef®ciency scores calculated by assuming crisp values for the missing data. As an illustration, the ef®ciency scores of the 24 University libraries in Taiwan, with three missing values, are calculated to show the extent that the actual amount of resources and services provided by each University is away from the technically ef®cient amount of resources and services. This methodology can also be applied to calculate the relative ef®ciencies of the DMUs with imprecise linguistic data. Keywords: data envelopment analysis; fuzzy sets; libraries
Introduction Data envelopment analysis (DEA) developed by Charnes et al1 is a methodology for measuring relative ef®ciencies within a group of decision making units (DMUs) which utilise several inputs to produce a set of outputs. In the last twenty years, several models and a great variety of applications have been reported.2±4 As its name indicates, DEA is data based. Different from other methodologies such as regression analysis, this methodology uses only a single set of observations per DMU. As a result, any data which is missing will make this approach inapplicable. This occasionally happens when a DMU is unable to provide certain data or the data is simply lost. In statistical analysis there are formulas for estimating missing values in randomized blocks, latin squares, etc.5 In DEA, however, none of the existing models is able to handle missing data. One approach widely used in the absence of data is to ask the experts for their subjective estimates of the data. Since considerable uncertainty is involved, three estimates, viz, the most pessimistic, the most optimistic, and the most likely estimates, are usually requested. To deal quantitatively with imprecision, the concepts and techniques of probability theory could be employed. However, due to the lack of suf®cient historical data, a probability estimate is not quite meaningful. Moreover, within this context, the *Correspondence: Dr C Kao, Department of Industrial Management, National Cheng Kung University, Ta
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