more inputs to produce the same amount of output compared with Hospitals A and B. Hospital C is thus relatively inefficient. The efficiency measure of each hospital is the relative distance of the hospital to the frontier. In this example, the efficiency of Hospital C is OCV/OC. The ratio OCV/OC represents the potential proportionate reduction in the inputs of Hospital C to produce the given output to make the hospital efficient. Therefore, Hospital C is considered efficient if the ratio OCV/OC equals to 1 and inefficient if the ratio is greater than 0 but less than 1. This efficiency measurement technique can be reformulated into a linear programming problem. The following linear programming equation (Eq. (1)) is used for evaluating efficiency ZC of Hospital C in a set of n hospitals (Charnes, Cooper, & Rhodes, 1978). minimize: ZC subject to:
X n j¼1
xijkjVxiCZC i ¼ 1;...;m; X n j¼1 kjyrjzyrC; r ¼ 1;...;s; kjz0; j ¼ 1;...;n; ð1Þ
where xij and yrj are observed values of the ith input and the rth output for hospital j. Hospital CVis virtually created as a ‘‘projected hospital’’ for Hospital C. The inputs (outputs) of Hospital CVare the linear combinations of corresponding inputs (outputs) of all hospitals in the sample. The linear combinations are formed by weight kj. The outputs of Hospital CVmust equal to or greater than the outputs produced by Hospital C. The efficiency of Hospital C is the potential proportionate reduction in
Fig. 2. Efficiency measurement of the DEA model.
B. Watcharasriroj, J.C.S. Tang / Journal of High Technology Management Research 15 (2004) 1–16 7
its inputs by factor ZC relative to Hospital CV. For evaluating the efficiency of all hospitals in the sample, Eq. (1) is solved n times giving n sets of kj, one set for each hospital to determine its relative efficiency. Hospitals that have efficiency scores of 1 are relatively technically efficient. Those with the scores of less than 1 are relatively inefficient. The second step of the analytical framework is to examine size effects on hospital efficiency by determining the efficiency distribution of large and small hospitals. From DEA perspective, all hospitals in the sample are assumed to have the same production technology and thus face the same best-practice frontier. In fact, there is the possibility that large and small hospitals may practice different production technologies due to, for example, different operating environments. In this case, large and small hospitals may have their own separate frontiers that are different. To allow for this possibility, an investigation of frontier difference is introduced similar to the work of