Applying hierarchical grey relation clustering analysis to geographical
information systems – A case study of the hospitals in Taipei City
, Chin-Tsai Linb
, Kua-Hsin Pengc
, Chiu-Chin Huangd
Department of Healthcare Management, Yuanpei University, No. 306, Yuanpei St., Hsin Chu 30015, Taiwan
Graduate School of Management, Ming Chuan University, No. 250, Zhong Shan N. Rd., Sec. 5, Taipei 111, Taiwan
Graduate Institute of Management Science, National Chiao Tung University, No. 1001, University Rd., Hsin Chu 30010, Taiwan
Department of Senior Service Management, Ming-Hsin University of Science and Technology, No. 1, Xinxing Rd., Xinfeng Hsinchu 30401, Taiwan
a r t i c l e
i n f o
Grey relational analysis Grey clustering analysis Hierarchical clustering analysis Geographical information system Medical resource
a b s t r a c t
Deng proposed grey clustering analysis (GCA) in 1987. Later, Jin presented a new method in 1993, called grey relational clustering (GRC) method that combined grey relational analysis with clustering. However, the GRC method cannot use a tree diagram to make appropriate classiﬁcation decisions without re-computation. This study thus attempts to combine GRC and hierarchical clustering analysis. Given the existence of an excess of medical resources in the Taipei area, this study attempts to understand the degree of concentration of medical resources in this area. Speciﬁcally, this study applies a geograph-ical information system (GIS) to present the geographgeograph-ical distribution of hospitals in Taipei. Additionally, a new-type of cluster analysis, known as hierarchical grey relation clustering analysis, is used to analyze the distribution of hospitals and understand how they compete with one another. The analytical results demonstrate that hierarchical grey relation clustering analysis is a suitable method of analyzing geo-graphical position. Tree diagrams can help policymakers make appropriate classiﬁcation decisions with-out re-computation. The study results can inform hospitals of their competitors and help them to develop appropriate responses. Additionally, the analytical results can also provide a reference to government or hospital policymakers to help them position hospitals in areas, thus achieving a better distribution of medical resources in Taipei.
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According to the grey system theory proposed by Deng in 1982, the population system, which comprises population and environ-ment, is a grey system, because the structure, function and interac-tion mechanism between the related factors that inﬂuence the system are extremely complex and relevant information is lacking. The grey systems theory is mainly utilized to study systems that model uncertainty, analyze system relations, establish models, and make forecasts and decisions (Tsai, Hsiao, & Liang, 2005). The grey clustering analysis (GCA) proposed by Deng in 1987. A cluster refers to a group of objects that are clustered according to some rule. Clusters thus by nature have a certain degree of homo-geneity. However,Jin (1993)described various procedures, includ-ing factor relational analysis (Deng, 1989), fuzzy clustering, systematic clustering, grey clustering (Feng, 1992), etc., that are applied to multi-target objects. Because of the complex factors
and confused nature of the information involved, a new method, known as the grey relational clustering (GRC) method, which com-bines grey relational analysis and clustering, is devised. Grey relation clustering distinguishes itself through simplicity, effec-tiveness and ﬂexibility. However, the GRC method cannot use a tree diagram to make appropriate decisions to classify without re-computation. This study thus tries to combine the GRC and a hierarchical clustering analysis.
Computer science and technology recently have developed very rapidly, for example in the ﬁelds of geographical information sys-tem (GIS), remote sensing (RS), global positioning syssys-tem (GPS), and so on (Shen et al., 2004). The speed of development of GIS has been especially fast. Different deﬁnitions of GIS exist in the foreign literature. Smith, Menon, Star, and Estes (1987)deﬁned the GIS as a database system in which most data are spatially in-dexed, and on which a set of procedures are operated to answer queries regarding spatial entities in the database. Blakemore (1986)deﬁned GIS as a computer packages which integrates the storage, manipulation, analysis, modeling and mapping of digital spatial information. Malpica, Alonso, and Sanz (2007) presented that GIS can be deﬁned as a system of hardware and software used for the input, storage, retrieval, mapping, display and analysis of
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⇑ Corresponding author. Tel.: +886 3 610 2321; fax: +886 3 610 2323. E-mail addresses:firstname.lastname@example.org(W.-H. Wu),email@example.com. edu.tw (C.-T. Lin), firstname.lastname@example.org (K.-H. Peng), cchuang@must. edu.tw(C.-C. Huang).
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geographic data. Additionally,Antenucci, Brown, Croswell, Kevany, and Archer (1991)deﬁned GIS as a computer system that stores and links non-graphic attributes or geographically-referenced data that possess graphic map features to permit various information processing and display operations, and map production, analysis, and modeling. The above demonstrates that different deﬁnitions of GIS existence ﬁnds that GIS encompasses a fundamental and universally applicable set of value-added tools for capturing, trans-forming, managing, analyzing, and presenting geographically refer-enced information (Tim, 1995).Cheng, Li, and Yu (2007)suggested that among the variety of GIS technology applications includes re-source management, land surveying and business planning.Ford, Grifﬁths, and Watson (2005)indicated that GIS provides an effec-tive means of analyzing the disparate data related to the cob
build-ings. Bhana (1999) also demonstrated that GIS enables the
determination of the precise longitude and latitude at which a facility is positioned, and thus can determine the proper distribu-tion of facilities and services. Savas Durduran (2010) presented that GIS has the ability to hold a vast amount of data that can be easily stored, shared analyzed and managed and also provides a platform for spatial data analyses. Consequently, spatial clustering has already been applied to GIS recently. The purpose of clustering is to divide objects into subclasses. Moreover, clustering applica-tions are commonly applied to GIS; for example, Wu, Bruggen, Subbarao, and Pennings (2001) presented clustering analysis in geographic information systems on the interpolated disease inci-dence for different periods.
An average of the hospital bed number in 2009 in Taiwan coun-ties, the area with the greatest excess of medical resources is Taipei City, with 15,788 beds (Department of Health, 2010). The above data demonstrate that Taiwan suffers from a problem of the un-even geographical distribution of medical resources. This study also ﬁnds that Taipei City has an excess of medical resources. Ar-nold (1991)stated that the increased number of hospitals exagger-ates the competitiveness in a health care business. Goldstein, Ward, Leong, and Butler (2002) described that hospital location is strongly related to performance. According to the above state-ment, the hospital should not only pay careful attention to choos-ing operatchoos-ing location, but must also assess the local geographical distribution of medical resources level of competition. To under-stand the degree of concentration of medical resources in the Tai-pei City, this study applies GIS to present the geographical locations of hospitals in Taipei City. Additionally, the new type of cluster analytic call hierarchical grey relation clustering analysis is applied to analyze the distribution of hospitals and to under-stand how hospitals compete with one another.
The remainder of this paper is organized as follows. Section2 then describes the calculation procedure of the GIS and the hierar-chical relation grey clustering analysis. Next, to provide a clearer explanation of calculation procedure, Section 2 of this study selects 10 hospitals in Taipei for the analysis of hospital distribution to demonstrate the effectiveness of the proposed assessment proce-dure. Section 3 then describes the analysis results for the distribu-tion of hospitals in Taipei City. Finally Secdistribu-tion 4 presents conclusions.
This study thus attempts to combine GRC and hierarchical clus-tering analysis. The hierarchical grey relation clusclus-tering analysis calculation procedure is presented below:
Let xjkdenote the kth coordinate axis for the jth hospital, and
let xj represent the indices series of the jth hospital, which is
written as: xj¼ ðxj1;xj2Þ
Step 1. Calculate the difference series
DijðkÞ ¼ jxiðkÞ xjðkÞj ð1Þ
where i = 1, 2, . . . , m, j
ei, k = 1, 2.
Step 2. Calculate the maximum and minimum of the difference series
Dmax¼ max 8j2i max8k jxiðkÞ xjðkÞj ð2Þ
Dmin¼ min 8j2i min8k jxiðkÞ xjðkÞj ð3Þ where i = 1, 2, . . . , m, j
ei, k = 1, 2. Maximum Minimum Grey Relation Coefficient Grey Relation Grade Develop Matrix G
Combine the Most Near Two Points Difference
Hierarchical Grey Relation Clustering Analysis calculation
Repeat all steps until all
data are in one cluster Database Transfer Analysis Finish Inquiries Coordinate Axle
Fig. 1. The procedure of analysis.
The coordinate axle data of Hospital.
ID Hospital name X axle (xj1) Y axle (xj2)
1 Shin-Kong 121,520,417 25,096,045 2 Cathay 121,553,800 25,036,809 3 Mackay 121,522,597 25,058,372 4 Wan-Fang 121,557,646 24,999,680 5 Veterans 121,520,569 25,120,203 6 Yang-Ming 121,531,859 25,105,276 7 Chunghsing 121,509,434 25,050,916 8 Jen-Ai 121,545,103 25,037,545 9 He-Ping 121,507,082 25,035,793 10 Women and Children 121,519,331 25,028,799
The result of the difference series (where the Shin-Kong hospital is regarded as the standard series).
Hospital Difference series
Dij(1) Dij(2) Shin-Kong 0 0 Cathay 33,383 59,236 Mackay 2180 37,673 Wan-Fang 37,229 96,365 Veterans 152 24,158 Yang-Ming 11,442 9231 Chunghsing 10,983 45,129 Jen-Ai 24,686 58,500 He-Ping 13,335 60,252
Step 3. Calculate the grey relation coefﬁcient
cðxiðkÞ; xjðkÞÞ ¼
1¼ 0:1 i ¼ 1; 2; . . . ; m; j 2 i; k ¼ 1; 2 ð4Þ
where f value can be adjusted in accordance with need. Step 4. Calculate the grey relation grade to develop matrix R
Cij¼ 1 k Xk k¼1
cðxiðkÞ; xjðkÞÞ i ¼ 1; 2; . . . ; m; j 2 i; k ¼ 1; 2 ð5Þ R ¼ ð
CijÞ; i; j ¼ 1; 2; . . . ; m
Step 5. Develop matrix G
Matrix G which is presented below and is known as the grey similar matrix, lays the foundation for grey relational clustering,
G ¼ ½gij; i; j ¼ 1; 2; . . . ; m ð6Þ
where gij¼ ðCijþCijÞ=2.
CijandCjiare grey grades with the formCij=
In where, parenthesis, the former is a reference series, the latter is one of compared series (Deng, 1989).
Step 6. Identify the two points (the hospital) of the most near Identify the two hospitals of the most near, and the center value (based on the X and Y coordinate axes) is then calculated by com-bining the two nearest hospitals.
ij gij ð7Þ
Step 7. Repeat steps 1–6 until all data are in one cluster
Summary of the grey relation coefﬁcient (taking the Shin-Kong hospital as the standard series).
Hospital Grey relation coefﬁcient
c(xi(1), xj(1)) c(xi(2), xj(2)) Shin-Kong 1.0000 1.0000 Cathay 0.2240 0.1399 Mackay 0.8155 0.2037 Wan-Fang 0.2056 0.0909 Veterans 0.9845 0.2851 Yang-Ming 0.4572 0.5107 Chunghsing 0.4673 0.1760 Jen-Ai 0.2808 0.1414 He-Ping 0.4195 0.1379
Women and Children 0.8987 0.1253
Summary of the grey relation grade (matrix R).
Shin-Kong Cathay Mackay Wan-Fang Veterans Yang-Ming Chunghsing Jen-Ai He-Ping Women and Children Shin-Kong 1.0000 0.1616 0.4402 0.1779 0.6602 0.5068 0.2600 0.1873 0.2553 0.5067 Cathay 0.1820 1.0000 0.1941 0.5016 0.1962 0.2293 0.2322 0.7028 0.5228 0.3713 Mackay 0.5096 0.2449 1.0000 0.2131 0.5095 0.3582 0.4133 0.2764 0.3122 0.4864 Wan-Fang 0.1483 0.4339 0.1226 1.0000 0.1681 0.1907 0.1224 0.2882 0.1663 0.2158 Veterans 0.6348 0.1458 0.4220 0.1681 1.0000 0.4488 0.2372 0.1715 0.2379 0.4858 Yang-Ming 0.4840 0.1920 0.2584 0.2105 0.4815 1.0000 0.1745 0.2465 0.1812 0.2643 Chunghsing 0.3217 0.2649 0.3865 0.1952 0.3340 0.2414 1.0000 0.2851 0.5701 0.3863 Jen-Ai 0.2111 0.7042 0.2222 0.3657 0.2283 0.2892 0.2520 1.0000 0.5049 0.3864 He-Ping 0.2787 0.5214 0.2500 0.2214 0.2984 0.2154 0.5304 0.5018 1.0000 0.4969 Women and Children 0.5120 0.3524 0.4136 0.2660 0.5117 0.2893 0.3252 0.3644 0.4774 1.0000
Grey similar matrix G.
Shin-Kong Cathay Mackay Wan-Fang Veterans Yang-Ming Chunghsing Jen-Ai He-Ping Women and Children Shin-Kong 1.0000 Cathay 0.1718 1.0000 Mackay 0.4749 0.2195 1.0000 Wan-Fang 0.1631 0.4677 0.1679 1.0000 Veterans 0.6475 0.1710 0.4658 0.1681 1.0000 Yang-Ming 0.4954 0.2106 0.3083 0.2006 0.4652 1.0000 Chunghsing 0.2908 0.2486 0.3999 0.1588 0.2856 0.2080 1.0000 Jen-Ai 0.1992 0.7035 0.2493 0.3270 0.1999 0.2679 0.2685 1.0000 He-Ping 0.2670 0.5221 0.2811 0.1938 0.2682 0.1983 0.5503 0.5034 1.0000
Women and Children 0.5094 0.3619 0.4500 0.2409 0.4987 0.2768 0.3557 0.3754 0.4872 1.0000
Summary of the combined processes.
Cluster 1 2 3 4 5
Grey similar value gij 0.7035 0.6475 0.6374 0.5123 0.4823
Combination Cathay General hospital, Jen-Ai hospital Shin-Kong, Veterans General hospital Cluster 2, Yang-Ming hospital Chunghsing hospital, He-Ping Cluster 3, Mackay Memorial hospital Cluster 6 7 8 9
Grey similar value gij 0.3721 0.2846 0.1855 0.1060
Combination Cluster 5, Women and Children hospital
To explain calculation procedure more clearly, this study selects 10 hospitals in Taipei for the analysis of hospital distribution to demonstrate the effectiveness of the proposed assessment
proce-dure. This study applies GIS to present the geographical locations of 10 hospitals in Taipei City. Additionally, hierarchical grey rela-tion clustering analysis is applied to analyze the degree of concen-tration of geographical position. The calculation procedure is presented below (as show inFig. 1):
Step 1. Transfer the data into the PAPAGO! SDK software (GIS)
Transfer the hospital address into the PAPAGO! SDK software (GIS).
Step 2. Calculate the X, Y coordinate axle data
The PAPAGO! SDK software can automatically make inquiries regarding the X and Y coordinate axle data according to hospital name or address. In order to explain more clearly, this study chose just 10 hospitals in Taipei City and analyzed their distribution (as listed inTable 1).
Fig. 3. The geographical position of the 10 sample hospitals in Taipei.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 2 8 4 1 5 6 3 10 7 9 Hospital ij g
Step 3. Calculate the difference series
Shin-Kong hospital is taken as the standard series, the kth difference series of the ﬁrst and second hospitals is D12(k) =
|x1(k) x2(k)|, k = 1, 2 using formula(1). When k = 1, the ﬁrst
dif-ference series of the ﬁrst and second hospitals is
D12ð1Þ ¼ jx1ð1Þ x2ð1Þj ¼ j121; 520; 417 121; 553; 800j ¼ 33; 383 k ¼ 2; D12ð2Þ ¼ jx1ð2Þ x2ð2Þj ¼ j25; 096; 045 25; 036; 809j ¼ 59; 236
Treat the Shin-Kong hospital as the standard series, and the cal-culation results for the difference series as listed inTable 2.
Later, to analogize, take the Cathay, Mackay, Wan-Fang, Veter-ans, Yang-Ming, Chunghsing, Jen-Ai, He-Ping, Women and Children hospital as the standard series, and the difference series is calcu-lated individually.
Step 4. Calculate the maximum and minimum of the difference series
Calculate the maximum and minimum using formula(2) and
8j2i max8k jxiðkÞ xjðkÞj ¼ 96; 365
8j2i min8k jxiðkÞ xjðkÞj ¼ 0
Later to analogize, take the Cathay, Mackay, Wan-Fang, Veter-ans, Yang-Ming, Chunghsing, Jen-Ai, He-Ping, and Women and Children hospital as the standard series, and calculate the maxi-mum and minimaxi-mum of the difference series individually.
Step 5. Calculate grey relation coefﬁcient
Take the Shin-Kong hospital as the standard series, then the kth difference series of the ﬁrst and second hospitals is
D12ðkÞþ1Dmax; k ¼ 1; 2 using formula(4). When k = 1, the ﬁrst grey relation coefﬁcient difference series of the ﬁrst and second hospi-tals is,
cðx1ð1Þ; x2ð1ÞÞ ¼
Dmax ¼ 0:1 96; 365 33; 383 þ 0:1 96; 365¼ 0:2240 By analogy, when k = 2
cðx1ð2Þ; x2ð2ÞÞ ¼
Dmax¼ 0:1 96; 365 59; 236 þ 0:1 96; 365¼ 0:1399 Taking Shin-Kong hospital as the standard series, the calcula-tion result of the grey relacalcula-tion coefﬁcient as listed inTable 3.
Subsequently by way of analogy, repeat steps 1–3. Take the Cat-hay, Mackay, Wan-Fang, Veterans, Yang-Ming, Chunghsing, Jen-Ai, He-Ping, and Women and Children hospital as the standard series, and calculate the grey relation coefﬁcient individually.
Step 6. Calculate grey relation grade to obtain matrix R The grey relation grade is C12¼1k
cðx1ðkÞ; x2ðkÞÞ; k ¼ 1; 2 using formula(5).
C12¼ 1 k Xk k¼1
cðx1ðkÞ; x2ðkÞÞ ¼ 1 2ð0:2440 þ 0:1399Þ ¼ 0:1820 Later by way of analogy, calculate the grey relation grade indi-vidually (as listed inTable 4).
Step 7. Development matrix G
Table 5lists the grey similar matrix G, where according to for-mula(6), for example i = 1, j = 2, then
C21Þ=2 ¼ ð0:1820 þ 0:1616Þ=2 ¼ 0:1718
Step 8. Identify the two points (the hospital) of the most near As shown the bold values inTable 5, Cathay and Jen-Ai hospital are identiﬁed as the two nearest points, max
ij gij¼ g28¼ 0:7035.
The center value (based on the X and Y coordinate axes) is then cal-culated by combining Cathay and Jen-Ai hospital, and the results are presented X coordinate axle value = 121,549,451, Y coordinate axle value = 25,037,177.
Step 9. Repeat steps 1–6 until all data are in one cluster
Table 6summarizes the combined processes. Based onTable 6, a tree diagram can be produced, as shown inFig. 2.Fig. 2shows that decision makers wish to create ﬁve clusters, and thus deter-mine the threshold gij= 0.5 in the form of a real line. That is, the
2 (Cathay) and 8 (Jen-Ai) hospitals are in the ﬁrst cluster, the 4 (Wan-Fang) is in the second cluster, the 1 (Shin-Kong), 5 (Veter-ans), 6 (Yang-Ming) are in the third cluster, the 3 (Mackay) is in the fourth cluster, the 10 (Women and Children) is in the ﬁfth clus-ter, and the 7 (Chunghsing), and 9 (He-Ping) are in the sixth cluster. The results of analysis also show the distribution situation of the 10 sample hospitals in Taipei (as shown inFig. 3).
The analytical result that the geographical position distribution of hospitals in Taipei City,Fig. 4shows that decision makers wish to create eight clusters, and thus determine the threshold gij= 0.71
in line form.Table 7lists the ﬁndings of this study regarding estab-lish year, bed number, hospital property (private or public) and distribution district. AsTable 7andFig. 3show, the 5 (Veterans), 12 (Cheng Hsin), 1 (Shin-Kong) and 7 (Yang-Ming) hospitals con-centrated in Shihlin districts comprise the ﬁrst cluster, and the 9 (Jen-Ai), 17 (Country), 20 (Clinic), 2 (Cathay), 18 (Chung Shan), 6 (Chang Gung), 13 (Taiwan Adventist), 30 (Po Jen), 41 (Song Shan), 19 (Show Chwan), 29 (Pei Ling), 25 (Women and Children) and 15 (Zhongxiao) hospitals concentrated in the Daan, Songshan, Sinyi,
Jhongshan and Nangang districts comprise the second cluster. Both these clusters are located in the southern area of Taipei, where agriculture developed relatively early and brought economic pros-perity and a sharp increase in population, leading to medical re-sources being relatively plentiful compared to other areas. Furthermore, the 39 (Jen Kang), 40 (Taipei Medical University) and 14 (Songde) hospitals concentrated in Sinyi district comprise the third cluster, while the 21 (Fu Cyun), 22 (Tai An), 23 (Disease Control and Prevention), 24 (Cing Sheng) and 3 (Mackay) hospitals concentrated in Jhongshan district are in the fourth cluster. Addi-tionally, the 4 (Wan-Fang) and 28 (Jin Mei) hospitals in Wunshan district comprise the ﬁfth cluster, while the 11 (Women and Chil-dren) 26 (Postal), 38 (National Taiwan University), 33 (Yan Chai), 36 (Bei Hu), 10 (He-Ping), 32 (West Garden), 37 (Wan Hua), 8 (Chunghsing), and 16 (Taipei) hospitals concentrated in the Da-tong, Jhongjheng, Daan and Wanhua district form the sixth cluster. Because these areas are close to Taipei County, where medical re-sources are relatively scarce, policymakers have tended to set up hospitals in these areas, causing considerable centralization of medical resources. Furthermore, the 42 (Cathay Neihu), 43 (Tri-Service), 27 (Kang Ning) and 31 (Zih Sheng Tang) hospitals that are concentrated in the Neihu and Nangang districts comprise the seventh cluster, while the 34 (Gan Dau) and 35 (Yat Sen)
Relevant information of 43 hospitals in Taipei.
Cluster Hospital name Establish year Hospital property Bed number Distribution district
1 Veterans 1958 Public 2908 Shihlin
Cheng Hsin 1967 Private 757 Shihlin
Shin Kong 1986 Private 921 Shihlin
Yang Ming 1950 Public 600 Shihlin
2 Cathay 1977 Private 772 Daan
Jen Ai 1954 Public 757 Daan
Country 1965 Private 136 Daan
Chung Shan 1976 Private 217 Daan
Show Chwan 1996 Public 74 Daan
Clinic 1973 Private 227 Daan
Chang Gung 1976 Private 3900 Songshan
Taiwan Adventist 1955 Private 450 Songshan
Pei Ling 1946 Private 85 Songshan
Po Jen 1976 Private 348 Songshan
Song Shan 1949 Public 500 Songshan
Zhongxiao 1988 Public 350 Nangang
Womens 1976 Private 49 Jhongshan
3 Songde 1969 Public 526 Sinyi
Jen Kang 1977 Private 96 Sinyi
Taipei Medical University 1976 Public 900 Sinyi
4 Mackay 1980 Private 1168 Jhongshan
Fu Cyun 1977 Private 89 Jhongshan
Tai An 2003 Private 101 Jhongshan
Disease Control and Prevention 2000 Public 20 Jhongshan
Cing Sheng 1975 Private 10 Jhongshan
5 Wan Fang 1997 Public 800 Wunshan
Jin Mei 1979 Private 95 Wunshan
6 Chunghsing 1905 Public 563 Datong
Taipei 1968 Public 709 Datong
He Ping 1967 Public 567 Jhongjheng
Women and Children 1974 Public 250 Daan
Postal 1946 Public 46 Daan
National Taiwan University 1946 Public 2564 Jhongjheng
West Garden 1971 Private 247 Wanhua
Yan Chai 1967 Private 87 Wanhua
Bei Hu 1949 Public 47 Jhongjheng
Wan Hua 1996 Private 98 Jhongjheng
7 Kang Ning 1970 Private 476 Neihu
Cathay Neihu 1997 Private 170 Neihu
Tri-Service 1990 Public 1721 Neihu
Zih Sheng Tang 1971 Public 30 Nangang
8 Gan Dau 2000 Public 243 Beitou
hospitals concentrated in Beitou district comprise the eighth clus-ter. Because transportation is difﬁcult, these areas contain sparse populations, and relatively few hospitals are located there. Addi-tionally, the analytical results also demonstrate the distribution of the hospitals in Taipei City (as shown inFig. 5).
To summarize, the above analysis indicates an uneven geo-graphical distribution of medical resources in Taipei City, with a concentration in the central city and a lack of resources in the sub-urbs. The distribution of medical resources is excessively central-ized in Daan, Songshan, Sinyi, Jhongshan and Wanhua districts, creating excessive competition among hospitals. Meanwhile, the Wunshan and Beitou districts suffer a lack of medical resources. The lack of medical resources in certain districts causes consider-able public inconvenience.
Table 7demonstrates that many new hospitals have tended to locate in areas that already have abundant medical resources. For example, the district of Jhongshan, which already had ample med-ical services, saw the establishment of the Tai An hospital in 2003, the similarly well served district of Wanhua was selected as the location of the new Wanhua hospital in1996, and so on. This situ-ation exists because medium and small hospitals have limited funding and reputation and face a survival challenge if they locate
in remote areas with limited numbers of patients. Government hopes to achieve a more even distribution of medical resources. Therefore, governments are encouraging large and public hospitals such as Tri-Service, Wan-Fang, Gan Dau, etc. to set up in areas that lack medical resources.
The research results demonstrate that hierarchical grey relation clustering analysis can effectively analyze geographical position. This study presents the hierarchical grey relation clustering analy-sis, which does not need to determine the threshold, number of clusters or choice of initial cluster centers. Tree diagrams (as shown inFig. 4) can help policymakers make appropriate classiﬁ-cation decisions without re-computation.
The distribution of medical resources in Taipei City is too cen-tralized. This phenomenon results in the border districts such as Wunshan and Beitou suffering from a lack of medical resources. In other words, the geographical distribution of medical resources in Taipei is uneven. Private medium and small hospitals tend to establish themselves in areas that already possess abundant
medical resources, because remote areas that have fewer medical resources typically also have a lower population and thus less demand for medical resources. Unfortunately, excessive centraliza-tion of medical resources creates intense competicentraliza-tion among hos-pitals and erodes proﬁts. Therefore, policymakers must carefully consider how to reduce competition among hospitals and help hos-pitals to coexist. The results of this study can inform hoshos-pitals of their competition and help them develop appropriate responses to competitor. The government tends to support locating hospitals in remote areas to provide improved access to medical resources for the entire population. Governments thus encourage large hos-pitals and public hoshos-pitals to establish in areas lacking medical re-sources, for example between Beitou and Shihlin districts or between Daan and Songshan districts for a more even distribution of medical resources.
The authors would like to thank the National Science Council of the Republic of China, Taiwan, for ﬁnancially supporting this research under Contract No. NSC 98-2410-H-264-004.
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