Productivity Growth and Efficiency Measurements in Fuzzy Environments with an Application to Health Care

Health care organizations must continuously improve their productivity to sustain long-term growth and profitability. Sustainable productivity performance is mostly assumed to be a natural outcome of successful health care management. Data envelopment analysis (DEA) is a popular mathematical programming method for comparing the inputs and outputs of a set of homogenous decision making units (DMUs) by evaluating their relative efficiency. The Malmquist productivity index (MPI) is widely used for productivity analysis by relying on constructing a best practice frontier and calculating the relative performance of a DMU for different time periods. The conventional DEA requires accurate and crisp data to calculate the MPI. However, the real-world data are often imprecise and vague. In this study, the authors propose a novel productivity measurement approach in fuzzy environments with MPI. An application of the proposed approach in health care is presented to demonstrate the simplicity and efficacy of the procedures and algorithms in a hospital efficiency study conducted for a State Office of Inspector General in the United States. DOI: 10.4018/ijfsa.2012040101 2 International Journal of Fuzzy System Applications, 2(2), 1-35, April-June 2012 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. In addition to comparing the relative performance of a set of DMUs at a specific period, the conventional DEA can also be used to calculate the productivity change of a DMU over time using the Malmquist productivity index, hereafter referred to as MPI. The MPI was first introduced by Malmquist (1953). Cave et al. (1982a, 1982b) proposed a MPI, which calculated the relative performance of a DMU for different time periods using a parametric method. Färe et al. (1989) and Färe, Ggrosskopf, and Lovell (1994) proposed a non-parametric Malmquist index for productivity analysis that relied on constructing a best practice frontier and computing the distance of individual observations from the frontier. Productivity is measured by the MPI and defined as the ratio between efficiency, as calculated by the DEA, for the same DMU in two different time periods. Several modifications for calculating MPI have been proposed in the literature. Jacobs et al. (2006, Ch. 6) provided a comprehensive review of the MPI in health care. MPI is a very useful method for calculating the productivity change in the DMUs and many applications have been reported in the literature (Chang et al., 2009; Chen, 2003; Chen & Ali, 2004; Emrouznejad & Thanassoulis, 2010; Fiordelisi & Molyneux, 2010; Hashimoto et al., 2009; Kao, 2010; Liu & Wang, 2008; Odeck, 2000, 2006, 2009; Oliveira et al., 2009; Swanson Kazley & Ozcan, 2009; Tsekouras et al., 2004; Zhou et al., 2010). In health care, the growing trends of rising costs have forced the government agencies and health care providers to be more concerned with their profitability and productivity. MPI has been widely used in health care to evaluate productivity change in hospitals, nursing homes, dialysis providers, and pharmacies, among others (Chang et al., 2011; Färe et al., 1995; Ouellette & Vierstraete, 2004; Ozgen, 2006; Retzlaff-Roberts et al., 2004; Kirigia et al., 2004; O’Neill et al., 2008; Kirigia et al., 2008). Hollingsworth (2008) provides a comprehensive review of the DEA literature in health care. The conventional MPI requires precise measurement of the inputs and outputs. One of the main challenges associated with the application of the MPI is the difficulty in quantifying some of the input and output data in real-world problems where the observed values are often imprecise or vague. Imprecise or vague data may be the result of unquantifiable, incomplete and non obtainable information. One way to manipulate uncertain data for the MPI is to represent the imprecise or vague values by membership functions of the fuzzy sets theory. In this study, we propose a novel productivity measurement approach with MPI in Fuzzy Environments. We show the application of the proposed approach in a hospital efficiency study conducted for a State Office of Inspector General in the United States. The remainder of this paper is organized as follows. The next section presents the relative preliminaries and definitions including the basic DEA model, the MPI with precise data, and the Malmquist index under VRS and scale efficiency change. Fuzzy set definitions are given and the MPI under VRS and scale efficiency change with fuzzy data is presented. The applicability of the proposed framework in health care and exhibits the efficacy of the procedures and algorithms is then discussed. The final section consists of the conclusion and future research directions. PRELIMINARIES AND DEFINITIONS This section presents a review of the basic DEA models and MPI with precise data followed by some basic definitions of the fuzzy sets theory. The Basic DEA Models The DEA model was initially introduced by Charnes et al. (1978) as the CCR model to measure technical efficiency with the assumption of constant returns to scale (CRS). Banker et al. (1984) subsequently extended the CCR model to accommodate a more flexible variable returns to scale (VRS) by relaxing the constant returns to scale assumption in their model known as the BCC model. Since then, DEA has been International Journal of Fuzzy System Applications, 2(2), 1-35, April-June 2012 3 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. widely applied to measure the relative efficiency of a group of homogeneous DMUs. Let us consider n DMUs where each DMUj ( j n = 1 2 , ,..., ) uses a vector of inputs xij ( i m = 1 2 , ,..., ) to produce a vector of outputs yrj (r s = 1 2 , ,..., ). The CCR and BCC models are introduced by the following linear programming models for problems with precise input and output data: The precise CCR model (1a):