组合Ω2的计算结果 表6
任务 运行时间 最早开始时间（ES） 最早结束时间（EF） 最迟结束时间（LF） 最迟开始时间（LS） 浮动时间 关键线路Ⅰ I 0 0 0 0 0 0 1 4 0 4 6 2 2 2 5 0 5 5 0 0 3 4 5 9 10 6 1 4 5 6 3 4 5 5 5 9 8 9 14 9 9 15 6 5 10 1 0 1 I-2-5-8-9-E 7 5 8 13 15 10 2 8 6 9 15 15 9 0 9 5 15 20 20 15 0 E 0 20 20 20 20 0
组合Ω3的计算结果 表7
任务 运行时间 最早开始时间（ES） 最早结束时间（EF） 最迟结束时间（LF） 最迟开始时间（LS） 浮动时间 关键线路Ⅰ I 0 0 0 0 0 0 1 7 0 7 7 0 0 2 2 0 2 7 5 5 3 6 7 13 13 7 0 4 5 6 5 3 8 2 2 13 7 5 21 13 17 21 8 14 13 6 12 0 I-1-3-6-9-E 7 7 7 21 21 14 7 8 4 7 21 21 17 10 9 3 21 24 24 21 0 E 0 24 24 24 24 0
α α=0 不同α和λ值时的关键路线 表8 关键线路 λ λ=0 I-2-5-8-9-E λ=0.5 I-2-5-8-9-E λ=1 I-2-5-8-9-E λ=0 I-2-3-6-9-E λ=0.5 I-2-5-8-9-E λ=1 I-2-5-8-9-E I-2-5-8-9-E α=0.5 α=1 5、总结
PERT（计划评审技术）是规划和协调大型项目使用最广泛的技术，PERT的主要假设
是，一个项目活动工期可以精确地估计并且在统计上相对独立，因此，在这个假设不成立时，PERT技术会导致错误的估计和管理对策的不当。在这个研究中，FPERT方法是用来解决这个问题的方法。三角模糊数表示一个项目网络的活动工期，在每个活动中模糊界限可以用来计算开始和结束时间。根据模糊的自由时间得出的关键性指数可以用来计算每个活动和路线的关键度，而第二个指标则是用来分析在不可预知的环境中符合一个项目网络所需时间的可能性。利用该模型，项目网络的状态信息可以直接获得，敏感性分析也可以通过改变关键路径上的活动持续时间来进行。
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参考资料：
[1] S. Avraham, Project segmentation – a tool for project management, International Journal of Project Management 15 (1997) 15–19.
[2] A. Azaron, C. Perkgoz, M. Sakawa, A genetic algorithm approach for the time-cost trade-off in PERT networks, Applied Mathematics and Computation 168 (2005) 1317–1339.
[3] A. Azaron, H. Katagiri, M. Sakawa, K. Kato, A. Memariani, A multi-objective resource allocation problem in PERT networks,European Journal of Operational Research 172 (2006) 838–854.
[4] S. Chanas, J. Kamburowski, The use of fuzzy variables in PERT, Fuzzy Sets and Systems 5 (1981) 11–19.
[5] S. Chanas, P. Zielinski, Critical path analysis in the network with fuzzy task times, Fuzzy Sets and Systems 122 (2001) 195–204.
[6] S.M. Chen, T.H. Chang, Finding multiple possible critical paths using fuzzy PERT, IEEE Transactions on Systems, man and Cybernetics – Part B: Cybernetics 31 (2001) 930–937. [7] S.J. Chen, C.L. Hwang, Fuzzy Multiple Attribute Decision Making-Methods and Applications, Springer-Verlag, Berlin Heidelberg,1992.
[8] C.W. Chiu, H.L. Ping, C.T. Yingn, Resource-constrained project management using enhanced theory of constraint, International Journal of Project Management 20 (2002) 561–567.
[9] C.W. Dawson, R.J. Dawson, Generalised task-on-the-node networks for managing uncertainty in projects, International Journal of Project Management 13 (1995) 353–362. [10] R.J. Dawson, R.J. Dawson, Practical proposals for managing uncertainty and risk in project planning, International Journal of Project Management 16 (1998) 299–310. [11] D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academy Press, 1980. [12] D. Dubois, H. Fargier, V. Galvagonon, On latest starting times and floats in task networks with ill-known durations, European Journal of Operational Research 147 (2003) 266–280. [13] H. Fargier, V. Galvagnon, Fuzzy PERT in series-parallel graphs, in: Prooceedings of Nineth International Conference on Fuzzy Systems, 2000, pp. 717–722.
[14] S.M.T. Fatemi Ghomi, E. Teimouri, Path critical index and task critical index in PERT networks, European Journal of Operational Research 141 (2002) 147–152. [15] S.M.T. Fatemi Ghomi, M. Rabbani, A new structural mechanism for reducibility of stochastic PERT networks, European Journal of Operational Research 145 (2003) 394–402.
[16] K.R. Graham, Critical chain: the theory of constraints applied to project management, International Journal of Project Management 18 (2000) 173–177.
[17] M. Giovanni, Measuring uncertainty and criticality in network planning by PERT-path technique, International Journal of Project Management 15 (1997) 377–387.
[18] M. Hapke, R. Slowinski, Fuzzy project scheduling system for software development, Fuzzy Sets and Systems 67 (1994) 101–107.
[19] M. Hapke, R. Slowinski, Fuzzy priority heuristics for project scheduling, Fuzzy Sets and Systems 83 (1996) 291–299.
[20] J. Heizer, B. Render, Principles of Operations Management, Prentice-Hall, 1999. [21] K.N. Jha, K.C. Iyer, Critical determinants of project coordination, International Journal of Project Management 24 (2006) 314–322.
[22] A. Kaufmann, M.M. Gupta, Introduction to Fuzzy Arithmetic: Theory and Applications, International Thomson Computer Press, London, 1991.
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[23] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, International Inc., 1995.
[24] D. Kuchta, Use of fuzzy numbers in project risk (criticality) assessment, International Journal of Project Management 19 (2001) 305–310.
[25] E.S. Lee, R.J. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Computers and Mathematics with Applications 15 (1988) 887–896.
[26] D.L. Mon, C.H. Cheng, H.C. Lu, Application of fuzzy distributions on project management, Fuzzy Sets Systems 73 (1995) 227–234.
[27] H. Neil, The prediction and control of project duration: a recursive model, International Journal of Project Management 19 (2001)401–409. [28] I.M. Premachandrak, An approximation of the task duration distribution in PERT, Computers and Operations Research 28 (2001)443–452.
[29] G. Roy, P. Nava, S. Israel, Integrating system analysis and project management tools, International Journal of Project Management 20 (2002) 461–468.
[30] W.J. Stevenson, Operation Management, seventh ed., McGraw-Hill, 2002.
[31] W.C. Wang, Impact of soft logic on the probabilistic duration of construction projects, International Journal of Project Management 23 (2005) 600–610.
[32] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.
[33] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU) – an outline, Information Sciences 17 (2005) 1–40.
[34] H.J. Zimmerman, Fuzzy Set Theory and Its Applications, second ed., Kluwer Academic Publishers, Boston, 1991.
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