A study of optimal weights of Data Envelopment Analysis – Development of a
context-dependent DEA-R model
Ching-Kuo Wei
a, Liang-Chih Chen
b, Rong-Kwei Li
b, Chih-Hung Tsai
c,⇑, Hsiao-Ling Huang
da
Department of Health Care Administration, Oriental Institute of Technology, 58, Sec. 2, Sihchuan Rd., Pan-Chiao City, Taipei County 22061, Taiwan
b
Department of Industrial Engineering and Management, National Chiao-Tung University, Hsinchu, Taiwan
c
Department of Information Management, Yuanpei University, No. 306, Yuanpei Street, Hsin-Chu, Taiwan
d
Department of Healthcare Management, Yuanpei University, No. 306, Yuanpei Street, Hsin-Chu, Taiwan
a r t i c l e
i n f o
Keywords:
Data Envelopment Analysis Redundant restraints on weight Context-dependent
Cluster analysis Medical center
a b s t r a c t
The weight is one of the main issues of Data Envelopment Analysis (DEA), and relevant theoretical research indicates that many DEA mathematical models include redundant restraints on weight, result-ing in underestimated efficiency, pseudo inefficiency, and difficulty in representresult-ing specific Input/Output relationships. This study proposes a context-dependent DEA-R model to address shortcomings resulting from redundant restraints on the weights of an efficient decision making unit (DMU), and converts the optimal weight to analyze the influences of redundant restraints on weights. The evaluation results of Taiwan medical centers show that the efficiency of the DMU is underestimated and pseudo inefficiency may occur due to redundant restraints on weight. Moreover, optimal weights are used as variables to conduct cluster analysis in order to determine the information of the weights. The results of cluster anal-ysis indicate that it can assist DMUs in understanding the relationships between DMUs, and contribute to the development of a unique survival strategy for hospitals.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Discussions of how to improve the efficiency of organizations are very important, and one of the most representative methods of efficiency evaluation is Data Envelopment Analysis (DEA) (Wei, Chen, Li, & Tsai, 2011), which is constructed on the basis on two concepts, namely, non-dominance solutions, as proposed by Italian economistPareto (1927), and relative efficiency through production frontier evaluation, as proposed by Farrell (1957). Based on these two concepts, Charnes, Cooper, and Rhodes
(1978)developed a mathematical programming approach to
calcu-late the efficiency of decision making units (DMUs), where DMUs with efficiency value equal to 1 are considered efficient, and those less than 1 are considered inefficient. This method uses a mathe-matical programming approach to determine the efficiency of a DMU, which are collectively referred to as DEA, and the first DEA mathematical model is called a CCR. DEA is characterized by each DMU’s ability to select its most favorable weight, and evaluate its relative efficiency among a set of DMUs. In addition, DEA is both objective and subjective in its method of efficiency evaluation, as
the subjective opinions of experts and decision makers can be incorporated.
In the field of DEA, many researches have discussed the issue of weight. Some research discusses how to incorporate preferences or expert opinions into weight restrictions, such as Dyson and Thanassoulis (1988), Thompson, Langemeier, Lee, Lee, and Thrall
(1990), and Wong and Beasley (1990). Other researches have
fo-cused on how to modify the models and limit weight within a rea-sonable range, such as the assurance region (AR) concept proposed
byThompson, Singleton, Thrall, and Smith (1986); the cone ratio
concept and its applications byCharnes, Cooper, Wei, and Huang
(1989) and Charnes, Cooper, Huang, and Sun (1990); the common
weight concept proposed byRoll, Cook, and Golany (1991), and fur-ther developed by Roll and Golany (1993). Despic´, Despic´, and
Paradi (2007)pointed out that the CCR model included an
imper-ceptible redundant restraint on weights, making it difficult to rep-resent the weight relationship and the influences of single Input/ Output, thus, the novel DEA-R model was proposed by Despic´
et al. (2007) to avoid such problems. Moreover, other research
pointed out that such restraints could lead to underestimations and pseudo inefficiency, namely, an efficient DMU being judged as inefficient. However, when DMUs are evaluated by an ordinary DEA-R model, the efficiency level of all efficient DM is 1 and the optimal weight has multiple solutions. Identical efficiency scores of DMUs make evaluating the influence difficult to understand,
0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.10.001
⇑ Corresponding author. Address: College of Management and Language, Yuanpei University, Department of Information Management, No. 306, Yuanpei Street, Hsin-Chu 30015, Taiwan. Tel.: +886 3 6102338; fax: +886 3 6102343.
E-mail addresses:[email protected],[email protected](C.-H. Tsai).
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namely, how redundant restraints of weights affect an efficient DMU, and the degree of advantage of the efficient DMU. Moreover, multiple solutions may lead to analysis errors. Thus, a context-dependent concept is introduced to develop the original DEA-R model into a context-dependent DEA-R model, which is applied to discuss the influence of redundant restraints on weights, as ap-plied in the case study of medical centers. Cluster analysis is also employed to provide an in-depth analysis of the weights.
In sum, the purposes of this research include: (1) develop a con-text-dependent DEA-R model and the conversion of CCR weights to DEA-R weights, and then, compare the evaluation results to gain an understanding of the influence of redundant restraints on weight for efficient DMUs; (2) conduct cluster analysis for further under-standing of the differences of the various models in weight selec-tion; (3) perform a case study to make suggestions for medical organizations, enabling them to offer additional high-quality med-ical services.
This study is organized as follows. Section 1 describes the motives and purpose of this research; Section 2 introduces the context-dependent CCR and context-dependent DEA-R models, and explains the reasons for selecting each model; Section3 con-tains the case description and efficiency evaluation, as well as explaining the reasons for and suitability of the case selection. In addition, by analyzing the evaluation results, the influence of redundant weight restraints is clarified. Section4conducts cluster analysis of the optimal weight of the two models in order to pro-vide further insight into the information hidden within the weights; and Section5offers conclusions.
2. Description of an efficient evaluation model
In this study, context-dependent CCR and context-dependent DEA-R models are selected for evaluation. This study selects the model based on CCR, as it is the first equation of DEA and one of the most popular models. However, the CCR model includes redun-dant weight restraints, making it difficult to represent specific In-put/Output weight relationships, and possibly resulting in underestimated efficiency or pseudo inefficiency. Hence, the DEA-R model, which is the model without redundant weight re-straints, is selected to develop an extended model. To understand the degree of advantage of an efficient DMU, and the influences of redundant weight restraints on an efficient DMU, it must be evaluated through the extended model. Prior to the concept of con-text-dependency,Andersen and Petersen (1993)proposed a con-cept for model extension, which boasted super efficiency, to evaluate an efficient DMU. However, because efficient DMUs are not evaluated against the same reference set as the super efficient model,Seiford and Zhu (2003)proposed the concept of context-dependency, which identifies all DMUs through an efficiency frontier, then removes these DMUs and determines a second-level efficiency frontier from the remaining DMUs, and continues until no DMUs remain. Efficient DMUs of a first-level efficiency frontier can be applied to the second-level, or other levels of efficiency frontiers, in order to evaluate their efficiencies. To overcome the shortcomings of the context-dependent CCR model, Morita,
Hirokawa, and Zhu (2005) developed a context-dependent SBM
model, and evaluated 14 Japanese power companies. This study develops a context-dependent model based on DEA-R to address the possible problems of the context-dependent CCR model.
In addition, input oriented models, which take the reduction of inputs as an improved strategy for the inefficient DMU, are adopted in this study, as increased output does not correspond with increased revenues under Taiwan’s medical payment system, which executes a global budget system. In the input oriented mod-el, an efficient DMU has a strong advantage if it can maintain its efficiency with large increases of inputs; otherwise, an efficient
DMU would have less advantage. In other words, there are many resources available for the DMU with a strong advantage position. In practice, such resources have not yet been applied to unlimited outputs, such as medical research.
2.1. Context-dependent CCR model
In the context-dependent CCR model, l denotes lth level of the evaluation; o denotes the object DMU; hl
odenotes the efficiency of
the DMU of the lth level evaluation; xijor xiodenotes the i-th input
of the DMU; yrjor yrodenotes the rth output of the DMU;
v
liodenotesthe ith input weight of the DMU in lth level evaluation; ul
rodenotes
the rth output weight of DMU in lth level evaluation; Jl denotes DMU as the reference set in lth level evaluation; and
e
is an Archime-dean number. In the context-dependent model, the basic model is repetitively used to evaluate the efficiency of each DMU in every le-vel until the completion of efficiency calculations at every lele-vel. The input oriented CCR, which is the basic model of the input oriented context-dependent CCR model, is expressed below:max hlo¼ Xs r¼1 ul ro yro s:t: X m i¼1
v
l io xijP Xs r¼1 ul ro yrjj 2 J l Xm i¼1v
l io xio¼ 1v
l io;u l roPe
>0 ð1ÞThe context-dependent CCR model can calculate the efficiency hl oof
the various levels, as per the following steps:
Step 1: set l as 1, let all DMUs be contained in J1, namely, all DMUs
are taken as the baseline of the first level. After calculating the efficiency of all DMUs, through the CCR model, incor-porate those DMUs with an efficiency value equal to 1 into E1.
Step 2: let Jl+1= Jl
El. Stop this step when no DMU exists in Jl+1
otherwise enter into Step 3.
Step 3: take Jl+1as the baseline of l + 1 level. After calculating the efficiency of all DMUs through the CCR model (1), incorpo-rate those DMUs, with an efficiency value equal to 1, into El+1, and then enter into Step 4.
Step 4: let l = l +1, and return to Step 2. 2.2. Context-dependent DEA-R model
In the context-dependent DEA-R model, l denotes lth evaluation level; o denotes the object DMU; hl
odenotes the efficiency of the
DMU in the l-th evaluation level; xijor xio denotes the ith input
of the DMU; yrjor yrodenotes the rth output of the DMU; Wliro
de-notes the weight of the ith input versus the rth output ratio of the object DMU in the lth evaluation level; and Jldenotes the DMU as the baseline in the lth level evaluation. In the context-dependent DEA-R model (2), DEA-R is repetitively used to evaluate the ciency of each DMU in every level until the completion of the effi-ciency calculations of every level. The input oriented DEA-R model, which the input oriented context-dependent DEA-R is based on, is expressed below: max hlo s:t: X m i¼1 Xs r¼1 Wliro ðXij=YrjÞ ðXio=YroÞ P hlo; j 2 J l Xm i¼1 Xs r¼1 Wl iro¼ 1 Wl iroP0; h l oP0 ð2Þ
The context-dependent DEA-R model can calculate the efficiency hl o
of the various levels, as per the following steps:
Step 1: set l as 1, let all DMUs be contained in J1, namely, all DMUs
are taken as the baseline of the first level. After calculating the efficiency of all DMUs, through the DEA-R model, incorporate those DMUs with an efficiency value equal to 1 into E1.
Step 2: let Jl+1= Jl
El. Stop this step when no DMU exists in Jl+1
otherwise enter into Step 3.
Step 3: take Jl+1as the baseline of l + 1 level evaluation. After cal-culating the efficiency of all DMUs through the DEA-R model (1), incorporate those DMUs, with an efficiency value equal to 1, into El+1, and then enter into Step 4.
Step 4: let l = l +1, and return to Step 2.
Next, the two context-dependent models are applied to evalu-ate the same case, and then, compare their results.
2.3. Corresponding weights
Because the numbers of CCR and DEA-R weights are different, it is not suitable to directly compare CCR weights with DEA-R weights, for instance, in a case where there are 2 input variables and 3 output variables, there are five CCR weights, and six DEA-R weights, thus, the two models’ weights are not suited for direct comparison. In order to compare the two models’ weights and determine the influence of the redundant weight restraints, this study converts the CCR weights to corresponding weights accord-ing to the variable relationships of the weights. Because the input variable i is related to DEA-R weight Wir, and the CCR weight
v
i, andthe output variable r are related to DEA-R weight Wir and CCR
weight ur, the corresponding CCR weight to the DEA weight could
be set as W0
ir=
v
ixi uryr t; and,Pmi¼1v
i xio¼ 1; andPmi¼1Ps r¼1
Wir¼ 1. Therefore, t ¼ 1 Psr¼1uryr
and the transformation of the CCR weight to its corresponding weight, w0
ir ¼ ð
v
ixi uryrÞPs r¼1uryr
;are obtained.
After the corresponding weights are obtained, and each CCR weight with a corresponding DEA-R weight is converted, there are DEA-R weights remaining, as the CCR weights cannot cover all. By analyzing the corresponding weights, it is found that, the corresponding weights include the restraints, of which w11:w21...:wm1= w12:w22...:wm2= = w1s:w2s...:wms=
v
1:v
2...:v
mare input variables, and w11:w12...:w1s= w21:w22. . .:w2s] = =
wm1:wm2...:wms= u1:u2...:us are output variables. Moreover, these
restraints are the mathematical representation of the redundant weight restraints and could explain why the corresponding CCR weights could not cover all DEA-R weights.
Next, corresponding efficiency is calculated by placing the cor-responding weights into the DEA-R model, and then, the DEA-R efficiency, with redundant weight restraints, is calculated by plac-ing weight restraints on the DEA-R model weights. Then, the influ-ences of the differinflu-ences of summing methods, differinflu-ences of weight selections, and redundant weight restraints are distin-guished by calculating four diverse efficiencies: CCR efficiency, cor-responding efficiency, DEA-R efficiency with restraints, and DEA-R efficiency without restraints. The difference between the CCR effi-ciency and the corresponding effieffi-ciency could represent the influ-ences of the differinflu-ences between the CCR summing method, Ps
r¼1urYro
Pm i¼1viXio
, and the DEA-R summing method,Pmi¼1Psr¼1WirðXij =YrjÞ
ðXio=YroÞ. In
addition, the difference between corresponding efficiency and the DEA-R efficiency, with redundant weight restraints could represent the influence of differences between CCR and DEA-R weight selec-tions. Finally, the difference between DEA-R efficiency, with and without redundant restrain, could represent the influence of the
redundant restraint on the weight, however, because of the added restraint, it is inferable that the efficiency with the redundant re-straint on the weight is no greater than the efficiency without the restraint. To confirm this inference of redundant restraint on weight, medical originations will be evaluated.
3. Comparison of efficiency and optimal weight 3.1. Case description
This study evaluates data of Taiwan’s medical centers from 2007, taken from the ‘‘The Statistical Annual Report of Medical Care Institutions Status & Hospitals Utilization’’, as collected by the Department of Health. The medical institutions in Taiwan are worthy of discussion as the coverage of medical insurance has reached 99%, and 95% of medical institutions can deliver high qual-ity medical services to the insurant at reasonable cost. Moreover, the growing demands of the public for medical treatments means, that competent authorities must adopt a global budget system to cover both public health and financial integrity. Therefore, how to offer high-quality medical services with limited public resources has become a hot topic, and DEA has been widely used to evaluate the efficiency of medical institutions at various levels. According to the investigation of the Department of Health, the utilization rate of large medical institutions is higher than its peers.Hu and Huang
(2004) pointed out that with the implementation of healthcare
insurance, large medical institutions in Taiwan have attracted increasing numbers of patients, thus, only such large high-level medical centers are selected for this study, which consists of 21 medical centers, including 7 state-owned hospitals (33%), and 14 privately owned hospitals (67%).
Sickbeds and doctors are selected as input, while outpatients, inpatients, and surgery are selected as output. The data are listed
in Table 1, and the correlation coefficients of the variables are
listed inTable 2. The number of Input and Output variables, which is smaller than half the number of DMUs, and the correlation coef-ficients of Input/Output variables, which is larger than 0.7, and thus, does not violate empirical rules, and variable selection is unquestioned.
3.2. Comparison of the efficiency of context-dependent CCR and context-dependent DEA-R
This study compares the efficiency levels of a context-depen-dent CCR with a context-depencontext-depen-dent DEA-R model to explain why the context-dependent DEA-R is developed. In addition, in Section
3.3, the weight of DMUs with different efficiency levels are ana-lyzed to learn the influence of redundant weight restraints for DMU efficiency. The efficient frontier of the DMU, the first-level efficiency of CCR and DEA-R, the second-level efficiency of CCR and DEA-R, as well as the differences of CCR and DEA-R efficiency are listed inTable 3.
Observations of efficient DMUs show that by definition, DMUs 02, 03, 13, 14, and 21 are first level CCR and DEA-R, with efficiency levels equal to 1 are on the first level of the efficiency frontier, namely, efficient DMUs, as marked in bold inTable 3. All efficient DMUs are removed in order to re-determine the efficient frontier through the remaining DMUs, in order to calculate efficiency. To distinguish various efficient frontiers, the efficiency frontier deter-mined the first time is called the first level efficient frontier, and from the second time is called the second level efficient frontier, and so on. Like an efficient frontier, the relative efficiency obtained from the first level efficient frontier is called the first level effi-ciency, and that obtained from the second level efficient frontier is called the second level efficiency.
Then ensure that all second-level efficiency levels of efficient DMUs are larger than 1. The efficient DMU with a higher second le-vel efficiency denotes a DMU with obvious advantages; otherwise, the efficient DMU with a lower second level efficiency is a DMU with an insignificant advantage. If the increased input exceeds the second level efficiency, the DMU will drop from a first level to third level efficient frontier, take DMU 13 as an example; the CCR second level efficiency of DMU 13 is 1.1901, which denotes that DMU 13 remains at the second level efficient frontier when sickbeds of DMU 13 increase from 1130 to 1345 (=1130 1.1901), and doctors increase from 421 to 501. However, if the increase exceeds this value, then DMU 13 drops to a third level effi-cient frontier. DMU 21 is an effieffi-cient DMU with a maximum second level efficiency of 2.0536. DMU 2 is an efficient DMU with
mini-mum second level efficiency of 1.0540. This result means that although both DMU 21 and DMU 02 are efficient DMUs, their advantages have a large difference. DMUs with fewer advantages may drop from the first level to the third level efficient frontier more easily than DMUs with stronger advantages. Thus, when evaluating according to a context-dependent model, the evaluator must identify the efficient DMUs; moreover, there must be an understanding of the degree of the advantages of the efficient DMUs.
According to the second-level efficiency of inefficient DMUs 11, 16, 17, and 20 are CCR second level, with efficiencies equal to 1; and DMUs 11, 16, 17, 19, and 20 are DEA-R second level, with effi-ciencies equal to 1. According to the definition, a DMU whose sec-ond level efficiency is equal to 1 is on the secsec-ond level efficient
Table 1
Input and output data of Taiwan medical centers in 2007. DMU Input1 Sickbed Input2 Doctor Output1 Outpatient Output2 Inpatient Output3 Surgery DMU Input1 Sickbed Input2 Doctor Output1 Outpatient Output2 Inpatient Output3 Surgery 01 3721 1158 2,319,835 1,009,763 81,855 11 1311 415 1,387,916 364,970 36,209 02 2909 976 2,455,352 854,531 80,085 12 1250 542 1,053,882 318,096 20,846 03 2661 708 1,877,506 691,048 41,424 13 1130 421 1,856,101 329,073 31,196 04 2632 1156 2,104,800 666,980 41,371 14 1053 300 1,367,840 287,960 30,426 05 2062 552 1,646,344 434,422 32,737 15 985 307 598,405 248,012 17,029 06 1771 549 1,677,396 390,950 36,787 16 925 309 341,951 260,572 16,087 07 1676 515 1,388,045 412,189 33,124 17 921 391 1,090,327 213,138 24,911 08 1658 602 1,987,233 409,152 21,573 18 981 310 877,364 242,451 15,016 09 1515 573 1,486,432 364,272 26,273 19 776 326 963,372 174,565 20,203 10 1406 473 1,163,799 346,212 24,143 20 756 329 1,309,539 196,162 14,194 21 340 167 1,209,475 49,624 6891 Table 2
Correlation coefficients of Input and Output Variables.
Input1 Sickbed Input2 Doctor Output1 Outpatient Output2 Inpatient Output3 Surgery
Input1 Sickbed 1.000** Input2 Doctor 0.940** 1.000** Output1 Outpatient 0.790** 0.793** 1.000** Output2 Inpatient 0.986** 0.933** 0.786** 1.000** Output3 Surgery 0.897** 0.832** 0.775** 0.932** 1.000** ** P-value < 0.01 (two-tails). Table 3
Efficient Frontier, Efficiency and Efficiency Difference of DMUs.
DMU CCR DEA-R Difference between CCR and DEA
L First level efficiency Second level efficiency L First level efficiency Second level efficiency First level efficiency Second level efficiency
01 3 0.9555 0.9915 3 0.9555 0.9915 0 0 02 1 1 1.0534 1 1 1.0540 0 0.0006 03 1 1 1.1099 1 1 1.1099 0 0 04 5 0.8634 0.9075 5 0.8640 0.9091 0.0006 0.0016 05 3 0.8135 0.8949 3 0.8152 0.8949 0.0018 0 06 3 0.7843 0.8750 3 0.7869 0.8908 0.0026 0.0157 07 3 0.8708 0.9101 3 0.8708 0.9101 0 0 08 3 0.8453 0.9101 3 0.8483 0.9362 0.0030 0.0261 09 4 0.8216 0.8698 3 0.8228 0.8731 0.0012 0.0033 10 4 0.8436 0.8827 4 0.8470 0.8838 0.0034 0.0011 11 2 0.9801 1 2 0.9817 1 0.0016 0 12 4 0.8675 0.9120 4 0.8683 0.9133 0.0008 0.0013 13 1 1 1.1901 1 1 1.2636 0 0.0736 14 1 1 1.3084 1 1 1.3322 0 0.0238 15 4 0.8860 0.9186 4 0.8860 0.9186 0 0 16 2 0.9609 1 2 0.9609 1 0 0 17 2 0.9361 1 2 0.9361 1 0 0 18 3 0.8650 0.8893 3 0.8687 0.8893 0.0037 0 19 3 0.9089 0.9983 2 0.9105 1 0.0016 0.0017 20 2 0.9400 1 2 0.9907 1 0.0507 0 21 1 1 2.0536 1 1 2.0536 0 0
frontier. Dive DMUs, which are on the second level efficient fron-tier, are found using DEA-R, whereas, only 4 DMUs are found using CCR. The results show that DMU 19, on the second-level efficient frontier, that cannot be found by CCR is similar to pseudo ineffi-ciency that some DMUs of first-level efficient frontier cannot be found. Thus, this study infers that a context-dependent CCR, like CCR, may lead to pseudo inefficiency.
Finally, analysis of the differences between efficiencies of the context-depend CCR and context-depend R shows that DEA-R could be used to calculate efficiencies equal to or higher than that of CCR. There are 11 DMUs (DMU 04–06, 08–12, 18–20) whose first level DEA-R efficiency is larger than CCR efficiency, and 10 DMUs (DMU 02, 04, 06, 08–10, 12–14, 19) whose second level DEA-R effi-ciency is larger than CCR effieffi-ciency. The differences of effieffi-ciency are listed in last two columns ofTable 3, and marked with a border. The result of DEA-R efficiency being higher than CCR efficiency confirms the previous theoretical research. The practical meaning of this result is that DEA-R could locate more available resources without affecting the judgment of an efficient frontier. Such results infer that the context-dependent CCR may lead to pseudo ineffi-cient results, and indicates that the context-dependent DEA-R model is a better choice than context-dependent CCR, and develop-ing the context-dependent DEA-R is worthy of research.
3.3. Comparison of optimal weights for context-dependent CCR and context-dependent DEA-R
To explain why the efficiency of a context-dependent DEA-R model is higher than or equal to that of the context-dependent CCR model, the optimal weights of the two models are compared. Due to the difference of the numbers of context-dependent DEA-R weights and context-dependent CCR weights, such as, 6 weights for DEA-R and 5 weights for CCR in this case, the CCR weight must be converted into a corresponding weight. As per the description in Section 2.3, the corresponding weight is w0
ir¼ ð
v
ixi uryrÞ=Ps
r¼1uryr:Next, the corresponding weight is placed into the
DEA-R model to calculate the corresponding efficiency. Since the second level efficiency has a greater difference and is capable of evaluating the advantage degree of an efficient DMU, the second level optimal weight is taken for comparison. The optimal CCR and DEA-R weight sets are represented inTable 4. Corresponding weight and corre-sponding efficiency are listed inTable 5.
3.3.1. DMU with the same efficiencies
Take DMU 21, with the highest efficiency, as an example, where the DEA-R efficiency of DMU 21 is no different than the CCR effi-ciency, where the corresponding weight set of the CCR weight is w11¼ ð
v
1x1 u1y1Þ=Ps
r¼1uryr¼ 1 2:0536=2:0536 ¼ 1; w21= 0,
w12= 0, w22= 0, w13= 0, w23= 0. In this way, the CCR
correspond-ing weight of DMU 21 is the same as the DEA-R weight, meancorrespond-ing that CCR and DEA-R show consistent viewpoints on the weight selection for DMU 21.When the efficiencies of DMU 01, 03, 05, 15, 18, and 2 are evaluated, neither the efficiency, nor the corre-sponding weight calculated for these two models show any differ-ences from the DEA-R optimal weight. However, when DMU 11, 16, 17, and 20 are evaluated, the second level efficiencies are equal to 1 and CCR corresponding weights are not consistent with DEA-R optimal weights. Thus, the following conclusions can be drawn: (1) the consistency of efficiency between the two models is not correlated with either the high or low efficiency; (2) when the effi-ciency is 1, the influence of weight on effieffi-ciency is unclear due to the multiple solutions of the optimal weight; and (3) when the context-dependent efficiency is not 1, and the efficiency levels of CCR and DEA-R are consistent, the weights of two models must be consistent.
3.3.2. DMU with different efficiencies
Observe the DMUs which CCR efficiency level differ from DEA-R’s efficiency level. Taking DMU 13, with the largest difference of efficiency levels as an example, the corresponding weight of the context-dependent CCR, which is (w11, w21, w12, w22, w13, w23) =
(0, 0.805, 0, 0, 0, 0.195), shows an obvious difference with the opti-mal weight of context-dependent DEA-R, which is (w11, w21, w12,
w22, w13, w23) = (0.590, 0, 0, 0, 0, 0.410). As previous argument in
Sections2 and 3stated, the differences of CCR’s and DEA-R’s effi-ciency levels are attributed to three factors: the difference of the summing method, the difference of weight selection, and redun-dant weight restraints. To distinguish the influences of the three factors, the corresponding weight is first placed into the constraint of context-dependent DEA-R to calculate the corresponding effi-ciency. Then, the difference of corresponding efficiency, in relation to the CCR efficiency, is caused by the different of summing meth-ods, and the difference of corresponding efficiency, in relation to the DEA-R efficiency, is caused by the difference of weight selec-tion and redundant weight restraints. Take DMU 08 as an example, the difference of the corresponding weight’s efficiency of 1.2266, for the CCR efficiency of 1.1901, is caused from the different sum-ming methods, and the difference of the corresponding efficiency with DEA-R’s efficiency of 1.2636, is caused from weight. The total difference between the two models is 0.0736, of which 0.0365 (49.7%) is caused by the summing method, and 0.0370 (50.3%) is caused by weight. The influences of the summing method and weight are listed inTable 5. Although the influence of summing method and weight are distinguished, the influences of weight selection and redundant weight restraints are unclear. Therefore, after understanding the mean of the redundant weight restraints, defining the influence of redundant weight restraints will follow. 3.3.3. The influence of redundant restraints on weight
It is found that the corresponding weight of CCR to DEA-R has the following characteristic: if one output variable has no advan-tage, then, no corresponding weights related to this output will be selected. In other words, CCR includes the assumption that re-strains on input weights conform to w11:w21...:wm1= w12:w22...:
wm2= = w1s:w2s...:wms=
v
1:v
2...:v
m, and output weights conformto w11:w12...:w1s= = wm1:wm2...:wms= u1:u2...:us. Take the
corre-sponding weight of DMU 13 as an example, if input 1 has no advan-tage, no advantage exists no matter the input, 1 produces output 1 or output 2, the corresponding DEA-R weight, will not confirm this restraint. Take optimal DEA-R weight of DMU 13 as an example. A doctor attracting an inpatient has an advantage, the weight relat-ing to the doctor and inpatient could be selected even if the outpa-tient has no advantage. These restraints are the mathematic representation of redundant restraints on weight.
Because corresponding weights, which are the weights selected by the CCR model, are not optimal weights in DEA-R, the DEA-R efficiency and optimal DEA-R weight, with redundant weight re-straints, are computed. Moreover, the different between corre-sponding efficiency and DEA efficiency with redundant restraints on weight is caused by the influence of the different models’ weight selections. The difference between DEA efficiency, both with and without redundant restraint, is caused by the influence of the redundant weight restraints. In this case, redundant weight restraints affect the efficiencies of DMU 08 and 13. The weight with redundant restraint on weight, CCR efficiency, corresponding effi-ciency, DEA-R efficiency with redundant restraint, DEA-R efficiency without redundant restraint, and the influences of three factors are as listed inTable 6.
In sum: (1) the difference between CCR and DEA-R efficiencies are caused by the summing methods, weight selections, and redundant weight restraints. In addition, the influences of the three factors could be distinguished by CCR efficiency, corresponding
efficiency, DEA-R efficiency with redundant restraint, DEA-R effi-ciency without redundant restraint; and (2) The difference of the summing method and weight selection comes from the differences between the CCR and DEA-R models, redundant weight restraints are a shortcoming of the CCR model, as the redundant restraint causes an underestimation of CCR efficiency.
3.4. Restrictions of single Input/Output relationship
After discussing the underestimation of efficiency and pseudo inefficiency caused by redundant weight restraints in the CCR model, which makes it difficult for CCR to represent the relation-ship of single Input/Output. In the other words, because there is only one CCR weighted input variable, it is difficult to simulta-neously represent relationships with several output variables. However, there are equivalent DEA-R weights of input variables for each output variable; therefore, it is easy to simultaneously
represent relationships with several output variables by setting the DEA-R weight of the input variables for specific output variables.
Take the following case as an example to explain a single Input/ Output relationship, represented by the DEA-R model. For a medi-cal center, a doctor is an important input variable providing inpa-tients, surgery, and outpatient services; however, sickbeds are the input only contributing to inpatient and surgery, but not outpa-tient services. Because there is only one CCR weight for the input sickbed variable, it is difficult to simultaneously represent different relationships between a doctor with three output variables and sickbeds with three output variables. Unlike CCR weight, three DEA-R weights of each input variable could represent the relation-ships of each input variable with inpatient, surgery, and outpatient service, respectively. Therefore, the DEA-R weight of input sickbed for output outpatient service is set as 0 to represent the different relationship between the input variable sickbed and the output
Table 4
Second Level CCR and DEA-R Optimal Weight.
DMU Second level CCR optimal weight Second level DEA-R optimal weight
Input1 Input2 Output1 Output2 Output3 I 1/O 1 I 2/O 1 I 1/O 2 I 2/O 2 I 1/O 3 I 2/O 3
1 0.000 1.000 0.000 0.992 0.000 0.000 0.000 0.000 1.000 0.000 0.000 2 1.000 0.000 0.000 1.022 0.031 0.000 0.000 0.979 0.000 0.021 0.000 3 0.000 1.000 0.000 1.110 0.000 0.000 0.000 0.000 1.000 0.000 0.000 4 1.000 0.000 0.014 0.894 0.000 0.008 0.000 0.992 0.000 0.000 0.000 5 0.000 1.000 0.000 0.895 0.000 0.000 0.000 0.000 1.000 0.000 0.000 6 0.000 1.000 0.574 0.301 0.000 0.000 0.843 0.000 0.000 0.000 0.157 7 0.000 1.000 0.000 0.910 0.000 0.000 0.000 0.000 1.000 0.000 0.000 8 1.000 0.000 0.109 0.801 0.000 0.455 0.000 0.000 0.545 0.000 0.000 9 1.000 0.000 0.089 0.780 0.000 0.149 0.000 0.851 0.000 0.000 0.000 10 1.000 0.000 0.014 0.869 0.000 0.007 0.000 0.993 0.000 0.000 0.000 11 1.000 0.000 0.020 0.406 0.574 0.106 0.000 0.000 0.000 0.098 0.797 12 1.000 0.000 0.014 0.898 0.000 0.007 0.000 0.993 0.000 0.000 0.000 13 0.000 1.000 0.958 0.000 0.232 0.590 0.000 0.000 0.000 0.000 0.410 14 0.000 1.000 0.991 0.000 0.318 0.000 0.845 0.000 0.000 0.000 0.155 15 0.000 1.000 0.000 0.919 0.000 0.000 0.000 0.000 1.000 0.000 0.000 16 1.000 0.000 0.000 1.000 0.000 0.000 0.000 1.000 0.000 0.000 0.000 17 1.000 0.000 0.275 0.000 0.725 0.266 0.000 0.000 0.000 0.734 0.000 18 0.000 1.000 0.000 0.889 0.000 0.000 0.000 0.000 1.000 0.000 0.000 19 0.884 0.116 0.422 0.000 0.577 0.575 0.000 0.000 0.000 0.167 0.259 20 1.000 0.000 1.000 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 21 1.000 0.000 2.054 0.000 0.000 1.000 0.000 0.000 0.000 0.000 0.000 Table 5
Corresponding weight, corresponding efficiency, and influences of summing method and weight.
DMU Corresponding weight Corresponding efficiency Influences
I 1/O 1 I 2/O 1 I 1/O 2 I 2/O 2 I 1/O 3 I 2/O 3 Weight (%) Summing method (%)
1 0.000 0.000 0.000 1.000 0.000 0.000 0.9915 – – – – 2 0.000 0.000 0.971 0.000 0.029 0.000 1.0535 0.0005 84% 0.0001 16% 3 0.000 0.000 0.000 1.000 0.000 0.000 1.1099 – – – – 4 0.015 0.000 0.985 0.000 0.000 0.000 0.9080 0.0011 71% 0.0005 29% 5 0.000 0.000 0.000 1.000 0.000 0.000 0.8949 – – – – 6 0.000 0.656 0.000 0.344 0.000 0.000 0.8779 0.0129 82% 0.0029 18% 7 0.000 0.000 0.000 1.000 0.000 0.000 0.9101 – – – – 8 0.120 0.000 0.880 0.000 0.000 0.000 0.9159 0.0203 78% 0.0058 22% 9 0.103 0.000 0.897 0.000 0.000 0.000 0.8702 0.0029 88% 0.0004 12% 10 0.016 0.000 0.984 0.000 0.000 0.000 0.8829 0.0009 81% 0.0002 19% 11 0.020 0.000 0.406 0.000 0.574 0.000 1.0000 – – – – 12 0.016 0.000 0.984 0.000 0.000 0.000 0.9123 0.0010 79% 0.0003 21% 13 0.000 0.805 0.000 0.000 0.000 0.195 1.2266 0.0370 50% 0.0365 50% 14 0.000 0.757 0.000 0.000 0.000 0.243 1.3145 0.0177 74% 0.0061 26% 15 0.000 0.000 0.000 1.000 0.000 0.000 0.9186 – – – – 16 0.000 0.000 1.000 0.000 0.000 0.000 1.0000 – – – – 17 0.275 0.000 0.000 0.000 0.725 0.000 1.0000 – – – – 18 0.000 0.000 0.000 1.000 0.000 0.000 0.8893 – – – – 19 0.373 0.049 0.000 0.000 0.511 0.067 1.0000 0.0000 0% 0.0017 100% 20 1.000 0.000 0.000 0.000 0.000 0.000 1.0000 – – – – 21 1.000 0.000 0.000 0.000 0.000 0.000 2.0536 – – – –
variable outpatient service. The second level efficiency and optimal weight, with restrictions on the weights of sickbeds/outpatients, are as listed inTable 7.
The optimal weights of DMUs 01–03, 05–07, 14–16, and 18, as well as the second level efficiency, remain unchanged. The optimal weights of DMUs 04, 08–13, 17, and 19–21, which weights do not conform to restricted conditions, have changed accordingly. Among these 11 DMUs, 9 second level efficiencies are lower than previous efficiencies. The second level efficiencies of DMU 11 and 20 remain unchanged; however, the optimal weights lack of changes is attributed to the multiple solutions of optimal weights. Moreover, since multiple solutions of optimal weights would prob-ably lead to analytical errors, future studies should carefully ana-lyze those DMU with an efficiency of 1. Redundant weight restraints not only cause the underestimation of efficiency, but also the problem of representing the ability of a single Input/Output
relationship, thus, the DEA-R based model is worthy of develop-ment to avoid underestimates and accurately represent the rela-tionship of single Input/Output.
4. Cluster analysis
Wu, Liang, and Yang (2009)applied cluster analysis by taking
individual cross efficiencies of DMUs as variables. When cluster analysis takes cross efficiency as its variables, the DMUs with sim-ilar efficiency are classified into the same cluster. That research in-spired this study to conduct further analysis on optimal weights by cluster analysis. When cluster analysis takes optimal weight as a variable, the DMUs with similar optimal weight are classified into the same cluster. In addition, the averages of the optimal weights are compared to learn the characteristics of the cluster. The infor-mation differences between the two models may be obtained by
Table 6
Optimal weight and Efficiency with redundant restraint on weight and influences of factors.
DMU Optimal DEA-R weight with redundant restraint on weight DEA-R efficiency CCR efficiency Influences of factors I 1/O 1 I 2/O 1 I 1/O 2 I 2/O 2 I 1/O 3 I 2/O 3 Without restraint With restraint Corresponding CCR Restraint on Model different
Weight Sum 1 0.000 0.000 0.000 1.000 0.000 0.000 0.9915 0.9915 0.9915 0.9915 – – – 2 0.000 0.000 0.979 0.000 0.021 0.000 1.0540 1.0540 1.0535 1.0534 16% 84% 0% 3 0.000 0.000 0.000 1.000 0.000 0.000 1.1099 1.1099 1.1099 1.1099 – – – 4 0.008 0.000 0.992 0.000 0.000 0.000 0.9091 0.9091 0.9080 0.9075 29% 71% 0% 5 0.000 0.008 0.000 0.993 0.000 0.000 0.8949 0.8949 0.8949 0.8949 – – – 6 0.000 0.843 0.000 0.000 0.000 0.157 0.8908 0.8908 0.8779 0.8750 18% 82% 0% 7 0.000 0.000 0.000 1.000 0.000 0.000 0.9101 0.9101 0.9101 0.9101 – – – 8 0.191 0.285 0.210 0.314 0.000 0.000 0.9362 0.9263 0.9159 0.9101 22% 40% 38% 9 0.149 0.000 0.851 0.000 0.000 0.000 0.8731 0.8731 0.8702 0.8698 12% 88% 0% 10 0.007 0.000 0.993 0.000 0.000 0.000 0.8838 0.8838 0.8829 0.8827 19% 81% 0% 11 0.006 0.000 0.993 0.000 0.000 0.000 1.0000 1.0000 1.0000 1.0000 – – – 12 0.007 0.000 0.993 0.000 0.000 0.000 0.9133 0.913 0.9123 0.9120 21% 79% 0% 13 0.392 0.147 0.000 0.000 0.336 0.126 1.2636 1.2437 1.2266 1.1901 50% 23% 27% 14 0.000 0.845 0.000 0.000 0.000 0.155 1.3322 1.3322 1.3145 1.3084 26% 74% 0% 15 0.000 0.000 0.000 1.000 0.000 0.000 0.9186 0.9186 0.9186 0.9186 – – – 16 0.000 0.000 0.848 0.152 0.000 0.000 1.0000 1.0000 1.0000 1.0000 – – – 17 0.405 0.034 0.059 0.005 0.459 0.038 1.0000 1.0000 1.0000 1.0000 – – – 18 0.000 0.000 0.000 1.000 0.000 0.000 0.8893 0.8893 0.8893 0.8893 – – – 19 0.401 0.008 0.214 0.004 0.366 0.007 1.0000 1.0000 1.0000 0.9983 100% 0% 0% 20 0.241 0.172 0.172 0.123 0.170 0.121 1.0000 1.0000 1.0000 1.0000 – – – 21 1.000 0.000 0.000 0.000 0.000 0.000 2.0536 2.0536 2.0536 2.0536 – – – Table 7
Efficiency and optimal weight with the restriction on the weight of sickbeds/outpatient.
DMU Optimal DEA-R weight with the restriction on the weight of sickbeds/outpatient DEA-R efficiency
I 1/O 1 I 2/O 1 I 1/O 2 I 2/O 2 I 1/O 3 I 2/O 3 Without restriction With restriction
1 0.000 0.000 0.000 1.000 0.000 0.000 0.9915 0.9915 2 0.000 0.000 0.979 0.000 0.021 0.000 1.0540 1.0540 3 0.000 0.000 0.000 1.000 0.000 0.000 1.1099 1.1099 4 0.000 0.010 0.990 0.000 0.000 0.000 0.9091 0.9068 5 0.000 0.000 0.000 1.000 0.000 0.000 0.8949 0.8949 6 0.000 0.843 0.000 0.000 0.000 0.157 0.8908 0.8908 7 0.000 0.000 0.000 1.000 0.000 0.000 0.9101 0.9101 8 0.000 0.699 0.000 0.301 0.000 0.000 0.9362 0.9227 9 0.000 0.006 0.994 0.000 0.000 0.000 0.8731 0.8631 10 0.000 0.007 0.993 0.000 0.000 0.000 0.8838 0.8835 11 0.000 0.009 0.991 0.000 0.000 0.000 1.0000 1.0000 12 0.000 0.009 0.991 0.000 0.000 0.000 0.9133 0.9111 13 0.000 0.805 0.000 0.000 0.000 0.195 1.2636 1.2267 14 0.000 0.845 0.000 0.000 0.000 0.155 1.3322 1.3322 15 0.000 0.000 0.000 1.000 0.000 0.000 0.9186 0.9186 16 0.000 0.000 1.000 0.000 0.000 0.000 1.0000 1.0000 17 0.000 0.000 0.000 0.000 1.000 0.000 1.0000 0.9793 18 0.000 0.000 0.000 1.000 0.000 0.000 0.8893 0.8893 19 0.000 0.000 0.000 0.000 1.000 0.000 1.0000 0.9426 20 0.000 1.000 0.000 0.000 0.000 0.000 1.0000 1.0000 21 0.000 1.000 0.000 0.000 0.000 0.000 2.0536 1.8195
comparing the results of cluster analysis, taking CCR and DEA-R optimal weights as variable. Hence, SPSS is used as a tool to con-duct cluster analysis.
4.1. Cluster analysis with optimal weight of context-dependent CCR as a variable
Cluster analysis, with an optimal weighted context-dependent CCR as a variable, is as shown inFig. 1. When DMUs are divided into six clusters: where the first cluster contains DMU 21; the sec-ond cluster contains DMUs 06, 13, and 14; the third cluster con-tains DMUs 01, 03, 05, 07, 15, and 18; the fourth cluster concon-tains DMUs 11, 17, and 19; the fifth cluster contains DMUs 02, 04, 08– 10, 12, and 16; and the sixth cluster contains DMU 20. The clusters to which the DMU belong are listed inTable 8. The average optimal CCR weights of Clusters are listed inTable 9, which result reveals the characteristics of each cluster. Among the input variables, the
DMU in first cluster only selects the weight of the sickbeds; among the output variable, the DMU in first cluster only selects the weight of outpatient, whose value is significantly higher than other clus-ters; hence, the medical institutions in first cluster are character-ized by attracting patients according to the scale of sickbeds.
Among the input variable, the DMUs in second cluster only se-lects the weight of doctors; among the output variables of DMUs in the second cluster, the weight of outpatient is the most important; hence, the medical institutions in second cluster are characterized by attracting patients via doctors. Among the input variables, the DMUs in the third cluster only selects the weight of doctors; among the output variables, the DMUs in third cluster selects only the weight of inpatients whose value is higher than the other clus-ters; hence, the medical institutions in the third cluster are charac-terized by attracting inpatients via doctors. Among the input variable of the DMUs in the fourth cluster, the weight of the sick-beds is most important; among the output variables of DMUs in the fourth cluster, the weight of surgery, which value is signifi-cantly higher than other clusters, is most important; hence, the medical institutions in the fourth cluster are characterized by transforming sickbeds into surgery. Among the input variable, the DMUs in the fifth cluster only selects the weight of the sick-beds; among the output variable of the DMUs in the fifth cluster, the weight of inpatients is most important; hence the medical institutions in the fifth cluster are characterized by transforming sickbeds into inpatients. Among the weights of the input variables, the DMUs in the sixth cluster selects only the weight of the sick-beds; among the output variables, the DMUs in the sixth cluster se-lects only the weight of the outpatients, which is lower than that of the first cluster. Hence, the medical institutions in the sixth clus-ters are characterized by attracting patients according to sickbeds. It is found that most DMUs select one input and one output as their advantage.
4.2. Cluster analysis with optimal weight of context-dependent DEA-R as a variable
Cluster analysis diagram, with the optimal weight of a context-dependent DEA-R as a variable, is as shown inFig. 2. When DMUs are divided into six clusters, DMUs 13 and 19–21 are in the first cluster; DMUs 06 and 14 are in the second cluster; DMUs 01, 03, 05, 07, 08, 15, and 18 are in the third cluster; DMU 11 is in the fourth cluster; DMUs 02, 04, 09, 10, 12, and 16 are in the fifth clus-ter; and DMU 17 is in the sixth cluster. The clusters in which the DMUs belong are as listed inTable 8.
The averages of the optimal DEA-R weights of the Clusters are as listed inTable 9. This result reveals the characteristics of each cluster. The average w11of the DMUs in the first cluster is higher
than that of the other clusters, thus, the advantages of medical institutions in the first cluster are focused on attracting outpatients according to sickbeds. The average w21of the DMUs in the second
cluster is higher than that of other clusters, thus, the medical
insti-Fig. 1. Cluster analysis diagram with CCR weight as a variable.
Table 8 Clusters of DMUs.
DMU Cluster with CCR weight
Cluster with DEA-R weight
DMU Cluster with CCR weight Cluster with DEA-R weight 1 3 3 11 4 4 2 5 5 12 5 5 3 3 3 13 2 1 4 5 5 14 2 2 5 3 3 15 3 3 6 2 2 16 5 5 7 3 3 17 4 6 8 5 3 18 3 3 9 5 5 19 4 1 10 5 5 20 6 1 21 1 1 Table 9
Average Efficiency and Average Optimal Weight of Clusters.
CCR DEA-R
C N Efficiency Average optimal weight C N Efficiency Average optimal weight
I 1 I 2 O 1 O 2 O 3 I 1/O 1 I 2/O 1 I 1/O 2 I 2/O 2 I 1/O 3 I 2/O 3
1 1 2.0536 1.000 0.000 2.054 0.000 0.000 1 4 1.3293 0.791 0.000 0.000 0.000 0.042 0.167 2 3 1.1245 0.000 1.000 0.841 0.100 0.183 2 2 1.1115 0.000 0.844 0.000 0.000 0.000 0.156 3 6 0.9524 0.000 1.000 0.000 0.952 0.000 3 7 0.9501 0.065 0.000 0.000 0.935 0.000 0.000 4 3 0.9994 0.961 0.039 0.239 0.135 0.625 4 1 1.0000 0.106 0.000 0.000 0.000 0.098 0.797 5 7 0.9336 1.000 0.000 0.034 0.895 0.004 5 6 0.9389 0.028 0.000 0.968 0.000 0.003 0.000 6 1 1.0000 1.000 0.000 1.000 0.000 0.000 6 1 1.0000 0.266 0.000 0.000 0.000 0.734 0.000
tutions in the second cluster are characterized by attracting outpa-tients through doctors. According to the average optimal weight of the DMUs in the third cluster that, the average w22is higher than
that of other clusters, thus, the medical institutions in the third cluster are composed of DMUs that have advantages in the man-agement of inpatients and doctors. In the fourth cluster, w23 is
higher than that of the other clusters, thus, the medical institutions in the fourth cluster are composed of DMUs that have advantages in surgery performed by doctors. In the fifth cluster, w12is higher
than that of the other clusters, thus, the medical institutions in the fifth cluster dominate in attracting inpatients according to sick-beds. In the sixth cluster, w13is higher than that of the other
clus-ters, indicating that these medical institutions dominate in surgery according to sickbeds. It is found that most DMUs select one input and one output as their advantage.
When analyzing the correlation of clustering and efficiency of a context-dependent model; it is found from the clustering that highly efficient DMUs could generate a squeezing effect, and other DMUs will try to increase the weight of other outputs to evade the advantages of the efficient DMU. In the case study, such a squeez-ing effect is particularly obvious in the cluster analysis of context-dependent CCR weights; as almost all hospitals strive to increase the weight of inpatients, and eliminate any significant advantages of DMU 21 on outpatients. Unlike the context-dependent CCR, the context-dependent DEA-R could more flexibly select weights, en-abling DMUs without significant advantages to become efficient models, via single Input/Output.
5. Conclusions
As the main feature of DEA, weight selection has been exten-sively discussed, both in practice and theory. Since 2007, a series of theoretical researches discussed redundant the weight restraints of CCR, and have overcome the shortcomings through DEA-R. This study researched redundant weight restraints by developing an ex-tended R model, and then, converted the CCR weights to DEA-R weights, based on research. This research obtained the following results: (1) context-dependent DEA-R model was developed by combining the basic DEA-R model with the context-dependent concept; (2) the CCR weight was converted to corresponding DEA-R weights, and then, the DEA-R model was added with weight restraints that discussed the influences of redundant weight re-straint restrictions upon the underestimation of efficiency; (3) the weight restriction on single Input/Output was used to repre-sent a single Input/Output relationship; and (4) the weight was further discussed according to cluster analysis in order to learn the relationships between DMUs.
First, a context-dependent DEA-R model was developed to eval-uate the efficiency of an efficient DMU, while the original DEA-R cannot analyze the degrees of the advantages contained in an effi-cient DMU. In this case, the advantage of the highest efficiency DMU of 2.0536, is double that of the advantage of an efficient DMUs of the lowest efficiency, 1.0540. This indicated the practical value in developing a context-dependent DEA-R model, as based on the original DEA-R, which was unable to evaluate the degrees of the advantages of an efficient DMU. Secondly, the difference in efficiency is caused by three factors, which are the different sum-ming methods, the different weight selection methods, and redun-dant weight restraints. Through the conversion of CCR weights to the corresponding weight, and added weight restraints of DEA-R, the influences of the three factor variables are distinguished. In addition, the redundant restraints on weight tend to cause an underestimation in efficiency. Thirdly, constraints on single In-put/Output are added to the DEA-R model in order to represent the single Input/Output relationship. In sum, the second and third results indicate the necessity of developing a context-dependent DEA-R, as based on the context-dependent CCR, including the redundant weight restraints, which tend to cause underestima-tions in efficiency and fail to represent the ability of an Input/Out-put relationship.
Finally, the optimal weight is taken as a variable of the cluster analysis in this study. The case study shows the correlation of clus-tering and efficiency of a context-dependent CCR, where it was found from the clustering that an efficient DMU, with an obviously higher level of efficiency, could generate a squeezing affect; and other DMUs would try to increase the weights of other outputs in order to evade the advantages of an efficient DMU. In the case study, such a squeezing effect was particularly obvious in the clus-ter analysis of context-dependent CCR weights, as almost all hospi-tals strive to increase the weights of hospitalization in order to negate the significant advantages of DMU 21 on clinics. Unlike the context-dependent CCR, the context-dependent DEA-R has greater flexibly in selecting weights, enabling DMUs without sig-nificant advantages to become efficient, via single Input/Output. Thus, by conducting cluster analysis, weight provides information on the advantages of the DMUs; moreover, it provides information regarding relationships with other DMUs, such as the characteris-tics of a DMU or the squeezing effect.
In practice, many Taiwanese hospitals have been accredited as medical centers, and as such could acquire a greater global budget, which in turn, would support more research and development. However, governments find it difficult to develop unique hospitals through pooling resources. By using a context-dependent effi-ciency evaluation model, the competent medical authorities could distinguish the efficient hospitals, and support their unique devel-opment for the benefit of the public, while maintaining fiscal integ-rity. Moreover, exceptional medical centers could not stand out among its peers under general blanketing policies; therefore, in addition to acquiring the most dominate situation through obser-vations of optimal weights, cluster analysis facilitates the correla-tion of a global context, allowing competent authorities and medical centers to achieve long-lasting success, growth, and devel-opment, based on performance evaluations, where exceptional ef-fort is recognized through analysis.
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