Theory of Integer-Valued Data Envelopment Analysis

Data envelopment analysis (DEA) is a mathematical programming approach for evaluating performance of decision making units (DMUs) that convert multiple inputs into multiple outputs. Conventional DEA models assume real-valued inputs and outputs. However, there are many occasions in which some inputs and/or outputs must only take integer values. For example, in efficiency evaluation of university departments, such inputs as the number of professors and such outputs as the number of published articles are restricted to the whole numbers. While the rounding of performance targets to the nearest whole number does not necessarily make a big difference for large departments, for small departments it can be a major issue. For example, suppose a department has 3 full professors, and the DEA analysis suggests the efficient level of professors is 2.4. Such result raises a dilemma: there is no evidence that 2 professors would suffice to meet the educational and scientific objectives of the department, but rounding 2.4 up to 3 does not save any resources even though the efficiency score of the department is only 0.8. The need to deal with integer-valued data in DEA naturally occurs when one uses categorical or ordinal data (Banker and Morey 1986; Kamakura 1988; Rousseau and Semple 1993; among others), but restricting to the whole numbers can be important even when the input-output variables are defined on the interval or ratio scales. Lozano and Villa (2006) were the first to address the issue at a more general level, proposing a mixed integer linear programming (MILP) DEA model to guarantee the required integrality of the computed targets. However, this pioneering article has two major shortcomings. First, the theoretical foundation of Lozano and Villa’s model is ambiguous. Clearly, assuming integer valued inputs and outputs immediately violates the standard convexity, free disposability and returns to scale properties of DEA. Thus, Lozano and Villa’s model is not consistent with the minimum extrapolation principle (Banker et al. 1984), which is the foundation of all DEA models. Second, Lozano and Villa’s MILP formulation for computing efficiency scores can lead to overestimated efficiency results, as shown below by numerical examples. This paper tackles both these problems. We develop a new axiomatic foundation for integer-valued DEA models, and show that the production possibility set proposed by Lozano and Villa (2006) is consistent with the proposed set of axioms. We also proposed a modification of the classic Farrell input efficiency measure, and derive a MILP formulation for computing it. The rest of the paper is organized as follows. We start by introducing the new notions of natural disposability” and natural divisibility” in the next section. In Section 3 we derive the associated DEA production sets that satisfy the fundamental minimum extrapolation principle (Banker et al. 1984). Section 4 generalizes the method to the hybrid case where both real and integer valued inputs and outputs are present. In Section 5 we adapt the Farrell input efficiency measure to the integer DEA setting, and show how the efficiency score can be computed by solving a MILP problem, which differs from that of Lozano and Villa (2006) in two important respects. An application to the efficiency evaluation of 42 departments of the Islamic Azad University, Karaj Branch (IAUK) illustrates the method in Section 6. Section 7 presents our concluding remarks and suggests avenues for future research

Keywords:

Data Envelopment Analysis , Efficiency , Mixed Integer Linear Programming , Production Theory