近十年來，許多學者將模糊數學應用在排程理論，稱之為模糊排程。使用符合人類思考方式的模糊數來表示排程的輸入變數，再結合模糊數學與排程理論，設計適當的排程解決方案，以使排程更簡易可行且富有彈性。零工式排程是最一般化的排程問題，本文探討當加工時間為不確定時之零工式排程問題，應用模糊數表現其加工時間之不確定性，稱之為模糊零工式排程問題。移動瓶頸法為解零工式排程問題的最佳啟發式演算法之一，其乃以單機排程問題為基礎，所以本文先提出模糊單機排程法，解決模糊單機排程問題。再以模糊單機排程法為基礎，設計模糊移動瓶頸法，以解決模糊零工式排程問題。並証明傳統移動瓶頸法為本文所提方法之特例。
在設計模糊移動瓶頸法時，我們發現如何應用模糊數的比較是相當重要的，模糊數之比較有二個目的，以二個模糊數之比較為例，第一個目的是找出這二個模糊數的較大 (小)值，稱之為模糊最大 (小)值運算。第二個目的是決定這二個模糊數的大小順序，稱之為模糊數排序。在模糊排程中，這兩種模糊運算法的選用關係著整個模糊排程的結果。本文欲探討並比較用在設計模糊排程解法上的各種模糊最大 (小)值運算與模糊數排序運算，並各提出一模糊最大 (小)值與模糊數排序運算的新方法，使在設計模糊排程解法時，有另一種選擇方案。並証明確定數之最大 (小)值與排序運算為本文所提方法之特例。

Fuzzy algebraic calculus is a generalization of the usual tolerance analysis, so it can be applied in any scientific domain where quantities which are vaguely known have to be combined, provided that this uncertainty may be quantified. Scheduling theory can be fuzzified in a computable manner using this approach. Job shop scheduling is a kind of scheduling problems. One of the most successful heuristic procedures developed for job shop problems is the shifting bottleneck heuristic. It considers each machine as a single machine problem with the processing times, the release times and the due dates and with the maximum lateness to minimized.
Fuzzy numbers were used with uncertain time requirements for a job shop problem. We called it “fuzzy job shop scheduling”. In this paper, the membership functions of the scheduled tasks are assumed to be triangular, and a new fuzzy single machine procedure and a new fuzzy shifting bottleneck heuristic procedure are proposed to solve fuzzy job shop scheduling problems. Also, the conventional single machine procedure and the conventional shifting bottleneck heuristic procedure are shown as a special case of the fuzzy single machine procedure and the fuzzy shifting bottleneck heuristic procedure with special membership functions being assigned.
In the fuzzy shifting bottleneck heuristic procedure, it is very important about the use of the fuzzy maximum (minimum) method and fuzzy ranking method. The purpose of the fuzzy maximum (minimum) method is to find the fuzzy maximum (minimum) between the fuzzy numbers. The purpose of the fuzzy ranking method is to rank the orders of the fuzzy numbers. In this paper, these two methods are surveyed and compared. And the new fuzzy maximum (minimum) method and fuzzy ranking method are proposed. Also, the conventional maximum (minimum) method and the conventional ranking method are shown as a special case of the fuzzy maximum (minimum) method and fuzzy ranking method with special membership functions being assigned. The aim of this paper was to present an alternative way to deal with this kind of problems, taking into consideration the fact that, in many real production situations, the tasks' variables are fuzzy rather than precise entities.

第壹章 緒論 1
第一節 研究動機與目的 1
第二節 研究步驟 3
第三節 研究大綱 4
第貳章 模糊理論與模糊排程之文獻 5
第一節 模糊理論 5
第二節 模糊排程之相關文獻 9
第參章 模糊最大 (小)值 15
第一節 不同的輸入變數型態 16
第二節 模糊最大 (小)值之探討 19
第三節 求模糊最大值之新方法 24
第四節 求模糊最小值之新方法 41
第五節 模糊最大 (小)值方法之比較 59
第肆章 模糊數排序 63
第一節 模糊數排序之探討 63
第二節 模糊數排序的新方法 80
第三節 模糊數排序方法之比較 87
第伍章 模糊零工式排程 90
第一節 模糊單機排程法 90
第二節 模糊移動瓶頸法 96
第陸章 結論與建議 114
參考文獻 116

[1]陸海文 (2001)，"模糊數排序法之探討"，國立成功大學工業管理研究所未出版博士論文。
[2]莊宗南、蔡燿全 (2002)，三角模糊數在單機排程的應用，台大管理論叢，12(2)，頁1-32。
[3]蔡燿全、莊宗南 (1999)，A new max operator on triangular fuzzy sets，模糊系統學刊，第五卷，第二期，頁71-78。
[4]Adams, J., E. Balas and D. Zawack (1988), "The shifting bottleneck procedure for job shop scheduling," Management Science, 34, pp. 391-401.
[5]Adamopoulos, G. I. and C.P. Pappis (1996), "A fuzzy-linguistic approach to a multi-criteria sequencing problem," European Journal of Operational Research, 92, pp. 628-636.
[6]Adamo J.M. (1980), " Fuzzy decision trees, " Fuzzy Sets and Systems, 4, pp. 207-219.
[7]Baldwin, J. and N. Guild (1977) " Comparison of fuzzy sets on the same decision space, " Fuzzy Sets and Systems, 2, pp. 213-233.
[8]Bardossy, A., L. Duckstein (1995), Fuzzy Rule-based Modeling with Applications to Geophysical, Biological and Engineering Systems, CRC Press, Boca Raton, FL.
[9]Bass, S. and H. Kwakernaak (1977), "Rating and ranking of multiple-aspect alternatives using fuzzy sets, " Automatica, 13, pp. 47-58.
[10]Bortolan G. and R. Degani (1985), "Areview of some methods for ranking fuzzy subsets," Fuzzy Sets and Systems, 15, pp. 1-19.
[11]Buckley, J. and S. Chanas (1989), "A fast method of ranking alternatives using fuzzy numbers," Fuzzy Sets and Systems, 30, pp. 337-339.
[12]Campbell, H.G., R.A. Dudek, M.L. Smith, "A heuristic algorithm for the n-job m-machine sequencing problem, " Management Science, 16, pp. 630-637.
[13]Chang, W. (1981), "Ranking of fuzzy utilities with triangular membership function," Proceeding of the International Conference on Policy Analysis and Information System,263-272.
[14]Chen, S.H. (1985), "Ranking fuzzy numbers with maximizing set and minimizing set," Fuzzy Sets and Systems, 17, pp. 113-129.
[15]Chen, S.J. and C.L. Hwang (1992), Fuzzy Multiple Attribute Decision Making: Methods and Applications, Springer, New York.
[16]Cheng, C.H (1998), "A new approach for ranking fuzzy numbers by distance method," Fuzzy Sets and Systems, 95, pp. 307-317.
[17]Chen, C.B. and C.M. Klein (1997), "A simple approach to ranking a group of aggregated fuzzy utilities, " IEEE Transactions on Systems, Man, and Cybernetics, 27, pp. 26-35.
[18]Choobineh, F. and H. Li (1993), "An index for ordering fuzzy numbers," Fuzzy Sets and Systems, 54, pp. 287-294.
[19]Coffman, E.G. and P.J. Denning (1973), Operating Systems Theory, New Jersey, Prentice-Hall.
[20]Delgado, M., J.L. Verdegay, and M.A. Villa (1988), "A procedure for ranking fuzzy numbers using fuzzy relations," Fuzzy Sets and Systems, 26, pp. 49-62.
[21]Dubois, D. and Henri Prade (1978), "Operations on fuzzy numbers," Int. J. Systems Sci., 9, pp. 613-626.
[22]Dubois, D. and H. Prade (1983), "Ranking of fuzzy numbers in the setting of possibility theory," Information Sciences, 30, pp. 183-224.
[23]Dubois, D. and Henri Prade (1997), "A synthetic view of belief revision with uncertain inputs in the framework of possibility theory," International Journal of Approximate Reasoning, 17, pp. 295-324.
[24]Dubois, D. and Henri Prade (1999), "Fuzzy sets in approximate reasoning, part1: inference with possibility distributions," Fuzzy Sets and Systems 100 Supplement, pp. 73-132.
[25]Dumitru, V., F. Luban (1982), "Membership functions, some mathematical programming models and production scheduling, " Fuzzy Sets and Systems, 8, pp. 19-33.
[26]Efstathiou, J. and R. Tong (1980), "Ranking fuzzy sets using linguistic preference relations," Proc. 10th International Symposium on Multiple-Valued Logic,Northwestern University, Evanston, 137-142.
[27]Feng, C. (1995), "Fuzzy multicriteria decision-making in distribution of factories, an application of approximate reasoning," Fuzzy Sets and Systems, 71, pp. 197-205.
[28]Fortemps, P. and Roubens, M. (1996), "Ranking and defuzzification methods based on area compensation," Fuzzy Sets and Systems 82, 319-330.
[29]Freeling, A.N.S. (1980), "Fuzzy sets and decision analysis, " IEEE Transaction on Systems, Man, and Cybernetics, 10, pp. 341-354.
[30]Gazdik, I. (1983), "Fuzzy-network planning," IEEE Transactions on Reliability, 32:3, pp. 304-313.
[31]Grabot, B., L. Geneste (1994), "Dispatching rules in scheduling: A fuzzy approach," International Journal of Production Research, 32:4, pp. 903-915.
[32]Graham, I. and P.L. Jones (1988), Expert Systems - Knowledge, Uncertainty and Decision, Boston, Chapman and Computing, pp. 117-158.
[33]Gupta, S.K. and J. Kyparisis (1987), "Single machine Scheduling Research," Omega, 15, pp.207-227.
[34]Han, S., H. Ishii and S. Fujii (1994), "One machine scheduling problem with fuzzy duedates," European Journal of Operational Research, 79, pp. 1-12.
[35]Hapke, M. and R. Slowinski (1996), "Fuzzy priority heuristics for project scheduling," Fuzzy Sets and Systems, 83, pp. 291-299.
[36]Hong, T. P. and Chuang, T. N. (1998), "Fuzzy scheduling on two-machine flow shop, " Journal of Intelligent & Fuzzy Systems, vol. 6, pp. 471-481
[37]Hong, T.P., C.H. Huang and K.M. Yu (1998), "LPT scheduling for fuzzy tasks," Fuzzy Sets and Systems, 97, pp. 277-286.
[38]Huang, C.C. (1989), "A study on the fuzzy ranking and its application on the decision support systems, " Ph.D. dissertation, Tamkang Univ., Taiwan, R.O.C.
[39]Ignall, E.J. L.E. Schrage (1965), "Applications of the branch and bound technique to some flow shop scheduling problems," Operations Research, 13, pp. 400-412.
[40]Ishibuchi, H., N. Yamamoto, S. Misaki, H. Tanaka (1994), "Local search algorithms for flow shop scheduling with fuzzy due dates, " International Journal of Production Economics 33, pp. 53-66.
[41]Ishii, H. And M. Tada (1995), "Single machine scheduling problem with fuzzy precedence relation," European Journal of Operational Research, 87, pp. 284-288.
[42]Ishii, H., M. Tada and T. Masuda (1992), "Two scheduling problems with fuzzy duedates," Fuzzy Sets and Systems, 46, pp. 339-347.
[43]Johnson, S.M. (1954), "Optimal two-three-stage production schedules with setup times included, " Naval Research Logistics Quarterly, 1, pp. 61-68.
[44]Jackson, J.R. (1955), "Scheduling a production line to minimize maximum tardiness, " Research Report 43, Management Science Research Project, University of California, Los Angeles.
[45]Jain, R. (1976), "Decision making in the presence of fuzzy variables," IEEE Transactions on Systems, Man, and Cybernetics, 6, pp. 698-703.
[46]Kandel, A. (1992), Fuzzy Expert Systems, Boca Raton, CRC Press, pp. 8-19.
[47]Kaufmann, A. and M.M. Gupta (1984), Introduction to Fuzzy Arithmetic Theory and Applications, New York, Van Nostrand Reinhold.
[48]Kerre, E. (1982), " The use of fuzzy set theory in electrocardiological diagnostics, in: M. Gupta and E. Sanchez, Ed., " Approximate Reasoning in Decision Analysis, North-Holland, Amsterdam. pp. 277-282.
[49]Kim, K. and Park, K. (1990), "Ranking fuzzy numbers with index of optimism," Fuzzy Sets and Systems 35, 143-150.
[50]Klein, C.M. (1989), "Fuzzy shortest paths," Fuzzy Sets and Systems, pp. 27-41.
[51]Kolodziejczyk, W. (1986), "Orlovsky’s concept of decision-making with fuzzy preference relation further results," Fuzzy Sets and Systems, 19, pp. 11-20.
[52]Koulamas C. (1994), "The total tardiness problem: review and extensions," Operations Research, 42, pp.1025-1041.
[53]Kuroda, M. and Z. Wang (1996), "Fuzzy job shop scheduling," International Journal of Economics, 44, pp. 45-51.
[54]Lee, E. S. and R. J. Li (1988), "Comparison of fuzzy number based on the probability measure of fuzzy events," Computers Math. Applic., 15:10, pp. 887-896.
[55]Lee, J., A. Tiao, J. Yen (1994), "A fuzzy rule-based approach to real–time scheduling," Proceedings of IEEE International Conference on Fuzzy Systems, pp. 1394-1399.
[56]Li, Y., P.B. Luh, X. Guan (1994), "Fuzzy optimization-based Scheduling of identical machines with possible breakdown, " Proceedings of IEEE International Conference on Robotics and Automation, 4, pp. 3447-3452.
[57]Liou T.S and M.J. Wang (1992), "Ranking fuzzy numbers with integral value," Fuzzy Sets and Systems, 50, pp. 247-255.
[58]Mabuchi G. (1988), "An approach to the comparison of fuzzy subsets with an α-cut dependent index," IEEE Transactions on Systems, Man, and Cybernetics, 18, pp. 264-272.
[59]McCahone, C.S. (1987), Fuzzy set theory applied to production and inventory control, ph.D. Thesis, Department of Industrial Engineering, Kansas State University.
[60]McCahone, C.S. and E.S. Lee (1992), "Fuzzy job sequencing for a flow shop," European Journal of Operational Research, 62, pp. 294-301.
[61]McCahone, C. and E.S. Lee (1990), "Job sequencing with fuzzy processing times," Computers Math. Applic., 19:7, pp. 31-41.
[62]Morton, T.E. and D.W. Pentico (1993), Heuristic Scheduling Systems with Applications to Production Systems and Project Management, New York, John Wiley & Sons Inc.
[63]Murakami, S. and S. Maeda (1983), "Fuzzy decision analysis on the development of centralized regional energy control systems," Reprints IFAC Conf. On Fuzzy Information, Knowledge Representation and Decision Analysis, pp. 353-358
[64]Nakamura, K. (1986), "Preference relation on a set of fuzzy utilities as a basis for decision making, " Fuzzy Sets and Systems, 20, pp. 147-162.
[65]Nasution, S.H. (1994), "Fuzzy critical path method," IEEE Transactions on Systems, Man and Cybernetics, 24:1, pp. 48-57.
[66]Ozelkan, E.C. and L. Duckstein (1999), "Optimal fuzzy counterparts of scheduling rules," European Journal of Operational Research, 113, pp. 593-609.
[67]Philippe F. and M. Roubens (1996), "Ranking and defuzzification methods based on area compensation," Fuzzy Sets and Systems, 82, pp. 319-330.
[68]Pinedo, M. (1995), Scheduling Theory, Algorithms, and Systems, New Jersey, Prentice-Hall.
[69]Sakawa, M. and T. Mori (1999), "An efficient genetic algorithm for job-shop scheduling problems with fuzzy processing time and fuzzy duedate," Computers & Industrial Engineering, 36, pp. 325-341.
[70]Sakawa, M. and R. Kubota (2000), "Fuzzy programming for multiobjective job shop scheduling with fuzzy processing time and fuzzy duedate through genetic algorithms," European Journal of Operational Research, 120, pp. 393-407.
[71]Stanfield, P.M., R.E. King and J.A. Joines (1996), "Scheduling arrivals to a production in a fuzzy environment," European Journal of Operational Research, 93, pp. 75-87.
[72]Tong R.M. and P.P. Bonissone (1984,) "Linguistic solutions to fuzzy decision problems, " In:TIMS/Studies in the Management Science, 20,323-334.
[73]Tseng, T.Y. and C.M. Klein (1989), "New algorithm for the ranking procedure in fuzzy decision making," IEEE Transactions on systems, Man, and Cybernetics, 19, pp. 1289-1296.
[74]Tsujimura, Y., S.H. Park, I.S. Chang, M. Gen (1993), "An effective method for solving flow shop scheduling problems with fuzzy processing times," Computers & Industrial Engineering, 25, pp. 239-242.
[75]Tsujimura, Y., M. Gen and E. Kubota (1995), "Solving fuzzy assembly-line balancing problem with genetic algorithms," Computers & Industrial Engineering, 29, pp. 543-547.
[76]Tsukamoto, Y., P.N Nikiforuk, and M. M. Gupta (1983), "On the comparison of fuzzy sets using fuzzy chopping, In: H. Akashi (Ed.)," Control Science and Technology for Progress of Society, New York, 46-51.
[77]Watson, S.R., J.J. Weiss, and M.L. McDonnell (1979), " Fuzzy decision analysis, " IEEE Transactions on Systems, Man, and Cybernetics, 9, pp. 1-9.
[78]Yager, R.R. (1978), "Ranking fuzzy subsets over the unit interval, " Proceedings of the 1978 CDC,1435-1437.
[79]Yager, R.R. (1980), "On choosing between fuzzy subsets, " Kybernetes, 9, pp. 151-154.
[80]Yager, R.R (1980), "On a general class of fuzzy connectives," Fuzzy Sets and Systems, 4, pp. 235-242.
[81]Yager, R.R (1981), "A procedure for ordering fuzzy subsets of the unit interval", Information Sciences, 24, pp. 143-161.
[82]Zadeh, L.A. (1965), "Fuzzy sets," Information and Control, 8, pp. 338-353.
[83]Zadeh, L.A. (1978), "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems, 1, pp. 3-28.
[84]Zadeh, L.A. (1988), "Fuzzy logic," IEEE Computer, 21:4, pp. 83-93.
[85]Zhu, Q. and E.S. Lee (1992), Comparison and ranking of fuzzy numbers, in J. Kacprzyk and M. Fedrizi (eds.), Fuzzy Regression Analysis, Omnitech Press, Warsaw, and Physica - Verlag, Heidelberg.
[86]Zimmermann, H.J. (1987), Fuzzy Sets, Decision Making, and Expert Systems, Boston, Kluwer Academic Publishers.
[87]Zimmermann, H.J. (1991), Fuzzy Set Theory and Its Applications, Boston, Kluwer Academic Publisher.