یک مطالعه در مورد توسعه مدل بهره وری مقایسه ای مبتنی بر نسبت ورودی محور

Data envelopment analysis (DEA) is a representative method to estimate efficient frontiers and derive efficiency. However, in a situation with weight restrictions on individual input–output pairs, its suitability has been questioned. Therefore, the main purpose of this paper is to develop a mathematical method, which we call the input-oriented ratio-based comparative efficiency model, DEA-R-I, to derive the input-target improvement strategy in situations with weight restrictions. Also, we prove that the efficiency score of DEA-R-I is greater than that of CCR-I, which is the first and most popular model of DEA, in input-oriented situations without weight restrictions to claim the DEA-R-I can replace the CCR model in these situations. We also show an example to illustrate the necessity of developing the new model. In a nutshell, we developed DEA-R-I to replace CCR-I in all input-oriented situations because it sets a more accurate weight restriction and yields a more achievable strategy.

Data envelopment analysis (DEA) is one popular method for identifying efficient frontiers and evaluating efficiency. An efficient frontier is based on the concept of a non-dominated condition, which was first expressed by the Italian economist Pareto in 1927. This concept was adapted to production by Koopmans in 1951 and to evaluate efficiency by Farrell in 1957 (Cooper et al., 2002). Charnes, Cooper, and Rhodes (1978) applied linear programming (LP) to identify efficient frontiers and measure productivity. This method, which measures productivity by LP, is called “data envelopment analysis”. They derived both an output-oriented (CCR-O) model and an input-oriented (CCR-I) model, which are not only the first but also most popular models of DEA. Many scholars have used DEA as the representative method to estimate an efficient frontier and measure productivity (Amirteimoori, 2007 and Jahanshahloo et al., 2005a). Over the past two decades, DEA has been established as a robust and valuable methodology (Chen and Ali, 2002 and Liu and Chuang, 2009). One advantage of DEA is objective weight selection, and there are many studies that focus on weight (Bernroider and Stix, 2007, Jahanshahloo et al., 2004, Jahanshahloo et al., 2005b, Lotfi et al., 2007 and Wang et al., 2008). However, when applying the typical DEA model, which is based on (∑vx)/(∑uy)∑vx/∑uy or (∑uy)/(∑vx)∑uy/∑vx, to a situation with weight restrictions on individual input–output pairs, its suitability is questionable. We take a case in hospitals as the example of the necessity of weight restrictions. Sickbeds, physicians, outpatients, inpatients, and surgery are important variables for hospital performance evaluations, where the sickbed variable contributes only to the inpatient and surgery variables but not the outpatient variable. In this situation, it is hard to assign a suitable weight restriction to an outpatient-sickbed pair. Golany and Roll (1989) argue that input–output pairs must correspond to an isotonicity assumption to avoid this problem. However, an isotonicity assumption represents a statistical rather than a causal relationship. For example, the statistical relationship between outpatient services and sickbeds is high, but the causal relationship between them is low. Therefore, conformance to the isotonicity assumption does not always avoid this problem. Dyson et al. (2001) argue that handling weight restrictions is still a pitfall in DEA applications from a theoretical perspective. Despic, Despic and Paradi (2007) claim that this kind of problem is difficult to solve with a typical DEA model and therefore developed DEA-R, a model to solve the problem of weight restriction. DEA-R is expressed as follows: javascript:void(0); However, the DEA-R model developed by Despic et al. (2007) is an output-oriented model (we call it DEA-R-O). In some situations, we need an input-oriented model to provide an input-target improvement strategy with weight restrictions. Using Taiwan’s private hospitals as an example again, the output was bounded by National Health Insurance; they have to adopt an input-targeted improvement strategy (reduce inputs), rather than an output-targeted strategy, to improve their efficiency. Therefore, a new mathematical method of deriving the input strategy (we call it input-oriented DEA-R, or DEA-R-I) has been developed. In addition, the DEA-R-I seems to substitute for CCR-I in input-oriented situations without weight restrictions because the efficiency score of DEA-R-I is greater than or equal to than CCR-I when the relationship between DEA-R-O and CCR-O is unclear. According to Despic et al. (2007), the efficiency score of DEA-R-O with no weight restrictions is sometimes higher and sometimes lower than the efficiency score of CCR-O. This drawback prevents DEA-R-O from replacing CCR-O in a situation without weight restrictions. But, based on our study, we found two factors that cause this efficiency score discrepancy. The first is a more flexible selection of optimum weight, which affects the efficiency score of DEA-R-O higher than the efficiency score of CCR-O, while the second is the sum of the output-oriented ratio View the MathML source∑w×yx, which affects the efficiency score of DEA-R-O less than the efficiency score of CCR-O. Since we will use the sum of the input-oriented ratio View the MathML source∑w×xy to replace the sum of the output-oriented ratio View the MathML source∑w×yx in computing the efficiency score in the DEA-R-I mathematical method, we suggest that the efficiency score of DEA-R-I will always be greater than or equal to the efficiency score of CCR-I (CCR input-oriented). This also means that the strategies of DEA-R-I are easier to achieve than the strategies of CCR-I because the strategy derived from the higher efficiency score needs fewer changes. If we can prove this hypothesis, the CCR-I model can be replaced by DEA-R-I because DEA-R-I provides a more accurate efficiency score in situations with weight restrictions and a better strategy in situations without weight restrictions. Because of the above reasons, the first goal of this paper is to develop a mathematical method (we call it DEA-R-I) to derive the input-target improvement strategy in a situation with weight restrictions. The second goal is to prove that the input-target improvement strategy developed by DEA-R-I is always better than the CCR-I model in a situation without weight restrictions. Therefore, we can claim that the DEA-R-I model can replace the CCR-I model in all input-oriented situations.

In this paper, we developed an input-oriented ratio-based model (DEA-R-I) for calculating efficiency scores and identifying input-target improvement strategies in situations with weight restrictions. We also show further proof of our model in order to claim that this model can replace the CCR-I model in situations without weight restrictions. A numerical example shows the difference between DEA-R-O and DEA-R-I to support our claim that the development of the DEA-R-I model is necessary for input-oriented situations with weight restrictions. This example further supports the claim that DEA-R-I can also provide easier improvement strategies than CCR-I in situations without weight restrictions. Because of its accuracy in situations with weight restrictions and its better strategy, we claim that DEA-R-I can replace CCR-I in all input-oriented situations.