資料包絡分析法是多投入、產出系統的效率分析工具。一般的生產資料除了明確的定量資料外，也常遇到定性資料。因為定性資料不適合做代數運算，因此傳統的資料包絡分析模式僅能處理定量資料。目前處理定性資料的方式都是先把其量化，再以傳統的資料包絡分析模式計算效率；但其量化的方式、計算過程和結果往往與定性資料的特性不符。
那一個等級的量化值應該多少才恰當呢？顯然不同的人有不同的答案，然而應該有一個區間是大家都可以接受，但慢慢遠離這個區間，接受的人就愈來愈少，直到最後根本沒有人可以接受。由這個觀點來看，等級就很適合用模糊數來表示。因此本文提出把定性資料量化為模糊數，以呈現定性觀測值的不明確性；並發展二階數學規劃，實踐模糊集的擴展原則，以求解模糊效率特定的α-截集。利用數學規劃求解一組足夠數量的α-截集，即可決定模糊效率，所得的模糊效率能反應定性觀測值的不明確性。
為了提高模糊數代表定性資料概念的可行性，本文也提出以資料包絡分析法為工具，整合主客觀的生產資料和量尺等級的順序資訊，找出定性資料的可能量化值，並據以決定代表等級的模糊數。雖然模糊數可以表達定性資料的不明確性，但較適當的方式是定性觀測值導出的效率值也是定性。因此本文用多中心物件集群法，配合郝氏多夫距離把模糊效率分群，再轉為定性效率。定性效率除了與定性觀測值在資料型態具有一致性外，也有利於使用統計方法分析影響效率的重要因子。
上述處理定性觀測值的概念不僅可以用在資料包絡分析法，本文也把它推廣到多準則決策分析的「多屬性價值分析」模式。最後以二個案例驗證發展的模式與方法。結果顯示，模糊效率可以反映定性資料的不明確性，而且其辨識有效率單位的能力高於傳統的點效率(或明確效率)。

Data envelopment analysis (DEA) is a useful tool to measure the relative efficiency of decision-making units in multiple-input and multiple-output production systems. The original DEA models only deal with quantitative data, because the algebraic operations on qualitative data are meaningless. Hence, the qualitative data is quantified before doing efficiency analysis.
Currently, methods dealing with qualitative data often have shortcomings. For example, the quantified observations are crisp or interval, and the resulting efficiencies are crisp or with too wide ranges. None of them truly reflects the imprecise properties of qualitative observations. Therefore, this paper proposes using fuzzy numbers to represent qualitative data, then develops a two-level mathematical program to implement fuzzy extension principle and find the α-cut of fuzzy efficiency from fuzzy efficiency models. Then adequate α-cuts can determine the fuzzy efficiency. Furthermore, to provide persuadable fuzzy numbers representing qualitative data, this paper proposed a method that DEA models act as a tool to integrate production data and precedence information among qualitative terms to generate possible values of these terms. Based on the collection of possible values, the shape parameters of fuzzy numbers for qualitative terms are determined.
Although fuzzy efficiencies show the imprecise properties of qualitative data, it is logical that the efficiencies derived by qualitative observations are qualitative. Therefore, this paper also proposes a method based on k-medoid method along with Hausdorff distance to partition fuzzy efficiencies into groups and then transform them into qualitative data. Qualitative efficiencies keep the format consistent with qualitative observations and allow users to apply the statistical techniques to find key factors relevant to efficiencies. In addition, the paper boils down the techniques dealing with qualitative data in DEA to multiple criteria decision making. Finally, this paper applies the methodology to two cases to demonstrate their validity.

1. Arbel, A., Approximate articulation of preference and priority derivation, European Journal of Operational Research, 43, pp. 317-326 (1989).
2. Banker, R. D., Charnes, A., and Cooper, W. W., Some models for estimating technical and scale inefficiencies in data envelopment analysis, Management Science, 30, pp. 1078-1092 (1984).
3. Beasley, J. E., Comparing university departments, OMEGA, 18, pp. 171-183 (1990).
4. Bilgiç, T., and Türkşen, I. B., Measurement of membership functions: Theoretical and empirical work, in International Handbook of Fuzzy Sets and Possibility Theory, eds. H. Prade, D. Dubois and H. J. Zimmermann, Norwell, MA: Kluwer Academic (1998).
5. Chakraborty, C., and Chakraborty, D., A theoretical development on a fuzzy distance measure for fuzzy numbers, Mathematical and Computer Modelling, 43, pp. 254-261 (2006).
6. Chang, H., Determinants of hospital efficiency: The case of central government-owned hospitals in Taiwan, OMEGA, 26, pp. 307-317 (1998).
7. Charnes, A., Cooper, W. W., and Rhodes, E., Measuring the efficiency of decision making units, European Journal of Operational Research, 2, pp. 429-444 (1978).
8. Chen, C. B., and Klein, C. M., A simple approach to ranking a group of aggregated fuzzy utilities, IEEE Transactions on Systems Man and Cybernetics, 27, pp. 26-35 (1997).
9. Chen, S. J., and Chen, S. M., Fuzzy risk analysis based on similarity measures of generalized fuzzy numbers, IEEE Transactions on Fuzzy Systems, 11, pp. 45-56 (2003).
10. Chen, S. M., Yeh, M. S., and Hsiao, P. Y., A comparison of similarity measures of fuzzy values, Fuzzy Sets and Systems, Vol 72, pp. 79-89 (1995).
11. Cook, W. D., Kress, M., and Seiford, L. M., On the use of ordinal data in data envelopment analysis, Operational Research Society, 44, pp. 133-140 (1993).
12. Cook, W. D., Kress, M., and Seiford, L. M., Data envelopment analysis in the presence of both quantitative and qualitative factors, Journal of the Operational Research Society, 47, pp. 945-953 (1996).
13. Cook, W. D., and Seiford, L. M., Data envelopment analysis (DEA) - Thirty years on, European Journal of Operational Research, 192, pp. 1-17 (2009).
14. Cook, W. D., and Zhu, J., Rank order data in DEA: A general framework, European Journal of Operational Research, 174, pp. 1021-1038 (2006).
15. Cooper, W. W., Park, K. S., and Yu, G., IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management Science, 45, pp. 597-607 (1999).
16. Cooper, W. W., Park, K. S., and Yu, G., An illustrative application of IDEA (Imprecise Data Envelopment Analysis) to a Korean mobile telecommunication company, Operations Research, 49, pp. 807-820 (2001).
17. Despotis, D. K., and Smirlis, Y. G., Data envelopment analysis with imprecise data, European Journal of Operational Research, 140, pp. 24-36 (2002).
18. Dia, M., A model of fuzzy data envelopment analysis, INFOR, 42, pp. 267-279 (2004).
19. Dyson, R. G., and Shale, E. A., Data envelopment analysis, operational research and uncertainty, Journal of the Operational Research Society, 61, pp. 25-34 (2010).
20. Emrouznejad, A., Parker, B. R., and Tavares, G., Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA, Socio-Economic Planning Sciences, 42, pp. 151-157 (2008).
21. Farrell, M. J., The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, 120, pp. 253-281 (1957).
22. Fawcett, S. E., Ellram, L. M., and Ogden, J. A., Supply Chain Management: From Vision to Implementation, Upper Saddle River, NJ: Prentice Hall (2007).
23. Friedrich, C., Fohrer, N., and Frede, H. G., Quantification of soil properties based on external information by means of fuzzy-set theory, Journal of Plant Nutrition and Soil Science, 165, pp. 511-516 (2002).
24. Grzegorzewski, P., Distances between intuitionistic fuzzy sets and/or interval-valued fuzzy sets based on the Hausdorff metric, Fuzzy Sets and Systems, 148, pp. 319-328 (2004).
25. Guha, D., and Chakraborty, D., A new approach to fuzzy distance measure and similarity measure between two generalized fuzzy numbers, Applied Soft Computing, 10, pp. 90-99 (2010).
26. Gullo, F., Ponti, G., and Tagarelli, A., Clustering Uncertain Data via k-medoids, Lecture Notes in Artificial Intelligence, 5291, 229-242, Berlin: Springer (2008).
27. Guo, P., and Tanaka, H., Fuzzy DEA: A perceptual evaluation method, Fuzzy Sets and Systems, 119, pp. 149-160 (2001).
28. Hollander, M., and Wolfe, D. A., Nonparametric Statistical Methods, New York: John Wiley and Sons (1973).
29. Huang, D. S., and Pan, W., Incorporating biological knowledge into distance-based clustering analysis of microarray gene expression data, Bioinformatics, 22, pp. 1259-1268 (2006).
30. Hung, W., and Yang, M., Similarity measures of intuitionistic fuzzy sets based on Hausdorff distance, Pattern Recognition Letters, 25, pp. 1603-1611 (2004).
31. Huttenlocher, D., Klanderman, G., and Rucklidge, W., Comparing images using the Hausdorff distance, IEEE Transactions on Pattern Analysis and Machine Intelligence, pp. 850-863 (1993).
32. Hwang, S., and Chang, T., Using data envelopment analysis to measure hotel managerial efficiency change in Taiwan, Tourism Management, 24, pp. 357-369 (2003).
33. Kao, C., Evaluation of junior colleges of technology: The Taiwan case, European Journal of Operational Research, 72, pp. 43-51 (1994).
34. Kao, C., Interval efficiency measures in data envelopment analysis with imprecise data, European Journal of Operational Research, 174, pp. 1087-1099 (2006).
35. Kao, C., and Lin, P. H., Qualitative factors in data envelopment analysis: A fuzzy number approach, European Journal of Operational Research, 211, pp. 586-593 (2011).
36. Kao, C., and Liu, S. T., Fuzzy efficiency measures in data envelopment analysis, Fuzzy Sets and Systems, 113, pp. 427-437 (2000).
37. Kaufman, L., and Rousseeuw, P., Finding Groups in Data: An Introduction to Cluster Analysis. Wiley's Series in Probability and Statistics, New York: John Wiley and Sons (2005).
38. Keeney, R. L., and Raiffa, H., Decisions with Multiple Objectives, New York: John Wiley and Sons (1976).
39. Keller, M., Procurement and Inventory: Unilever's Procurement Function & Inventory Requirements at a Glance, Munich: GRIN Verlag (2010).
40. Klir, G. J., and Yuan, B., Fuzzy Sets and Fuzzy Logic: Theory and Applications, Upper Saddle River, NJ: Prentice Hall (1995).
41. Lee, H. S., An optimal aggregation method for fuzzy opinions of group decision, in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, pp. 314-319 (1999).
42. Medaglia, A. L., Fang, S. C., Nuttle, H. L. W., and Wilson, J. R., An efficient and flexible mechanism for constructing membership functions, European Journal of Operational Research, 139, pp. 84-95 (2002).
43. Medasani, S., Kim, J., and Krishnapuram, R., An overview of membership function generation techniques for pattern recognition, International Journal of Approximate Reasoning, 19, pp. 391-417 (1998).
44. Myatt, G. J., Making Sense of Data: A Practical Guide to Exploratory Data Analysis and Data Mining, Hoboken: John Wiley & Sons (2007).
45. Pöyhönen, M. A., Hämäläinen, R. P., and Salo, A. A., An experiment on the numerical modelling of verbal ratio statements, Journal of Multi Criteria Decision Analysis, 6, pp. 1-10 (1997).
46. Pappis, C. P., and Karacapilidis, N. I., A comparative assessment of measures of similarity of fuzzy values, Fuzzy Sets and Systems, 56, pp. 171-174 (1993).
47. Park, K. S., Efficiency bounds and efficiency classifications in DEA with imprecise data, Journal of the Operational Research Society, 58, pp. 533-540 (2007).
48. Park, K. S., Duality, efficiency computations and interpretations in imprecise DEA, European Journal of Operational Research, 200, pp. 289-296 (2010).
49. Punj, G., and Stewart, D. W., Cluster analysis in marketing research: Review and suggestions for application, Journal of Maketing Research, 20, pp. 134-148 (1983).
50. Sadi-Nezhad, S., and Damghani, K. K., Application of a fuzzy TOPSIS method base on modified preference ratio and fuzzy distance measurement in assessment of traffic police centers performance, Applied Soft Computing, 10, pp. 1028-1039 (2010).
51. Santini, S., and Jain, R., Similarity measures, Pattern Analysis and Machine Intelligence, IEEE Transactions on Intelligence, 21, pp. 871-883 (1999).
52. Seydel, J., Data envelopment analysis for decision support, Industrial Management and Data Systems, 106, p. 81 (2006).
53. Sharma, S., Applied Multivariate Techniques, New York: John Wiley and Sons (1996).
54. Tavares, G., A bibliography of data envelopment analysis (1978-2001), Technical Report Rutcor Research Report RRR 01-02, Rutgers Center for Operational Reseach, Rutgers Univ. (2002).
55. Tversky, A., Features of similarity, Psychological Review, 84, p. 327 (1977).
56. Xu, Z., and Xia, M., Distance and similarity measures for hesitant fuzzy sets, Information Sciences (2011).
57. Yoon, K. P., and Hwang, C., Multiple Attribute Decision Making: An Introduction, Thousand Oaks: Sage (1995).
58. Zadeh, L. A., Fuzzy sets, Information and control, 8, pp. 338-353 (1965).
59. Zadeh, L. A., The concept of linguistic variable and its application to approximation to approximate reasoning, Information Sciences, 8, pp. 199-249 (1975).
60. Zhu, J., Imprecise DEA via standard linear DEA models with a revisit to a Korean mobile telecommunication company, Operations Research, 52, pp. 323-329 (2004).
61. Zimmermann, H. J., Fuzzy Set Theory and Its Applications, Boston: Kluwer (1991).
62. Zwick, R., Carlstein, E., and Budescu, D. V., Measures of similarity among fuzzy concepts: A comparative analysis, International Journal of Approximate Reasoning, 1, pp. 221-242 (1987).