【正文】 Wang, 2008). Notably, these objectives are normally fuzzy owing to inplete and unavailable information. Accordingly, three fuzzy objective functions are simultaneously considered during the formulation of the fuzzy MOLP model, as follows: Minimize total project costs (1)where the terms are used to calculate total direct costs. Total direct costs include total normal cost and total crashing cost, obtained using additional direct resources such as overtime, personnel and equipment. Generally, the major direct costs such as overtime, personnel and equipment, depend either on activity times or on project pletion time, although materials costs are fixed during the planning horizon. Total direct costs increase with decreasing project duration. The terms denote total indirect costs, including administration, contractual penalties, depreciation, financial and other variable overhead costs that can be avoided by reducing total pletion time.Minimize total pletion timeThe symbol ‘’ is the fuzzified version of ‘=’ and refers to the fuzzification of the aspiration levels. For each objective function in the original fuzzy MOLP model, this work assumes that the DM has such fuzzy objective as, “the objective function should be essentially equal to some value”. In realworld PM decisions, Eps. (1)– (3) are generally fuzzy and incorporate the variations in the DM’s judgments relating to the solutions of the fuzzy optimization problem. ConstraintsConstraints on the time between events i and j Constraints on the crashing time for activity (i, j)Constraint on the total budgetNonnegativity constraints on decision variables2. Solution methodology Twophase fuzzy goal programming approach (TPFGP) Phase IIn phase I, the original fuzzy MOLP problem designed above can be solved using the fuzzy decisionmaking concept of Bellman and Zadeh (1970), together with the FGP technique of Hannan (1981). The piecewise linear membership functions are specified for representing all the fuzzy goals involved, and the minimum operator is adopted to aggregate fuzzy sets. By introducing the auxiliary variable, the original fuzzy MOLP problem can be converted into an equivalent ordinary LP model that can be solved efficiently using the simplex method. Appendix A detailed the derivation of the equivalent。 Liang, 2006。 Liang, 2004a。 Gen, 2007。 Ellis, 1990。 Gungor, 2001。 Haouari, 2005。 Wang amp。 Zimmermann, 1978). Assumption 6 represents that the indirect costs can be divided into fixed costs and variable costs. Fixed costs represent the indirect costs under normal conditions and remain constant regardless of project duration. Meanwhile, variable costs, which are used to measure savings or increases in variable indirect costs, vary directly with the difference between actual pletion and normal duration of the project (Liang, 2006。 Wang amp。 Lai amp。附錄Fuzzy multiobjective project management decisions using twophase fuzzy goal programming approach1. Problem formulation Problem description, assumptions and notationThe fuzzy multiobjective PM decision problem examined in this work can be described as follows. Assume a project involves n interrelated activities that must be executed in a certain order before the entire task can be pleted. In realworld PM decisions, the values of the objective functions cannot be accurately measured because some information regarding the environmental coefficients and related parameters is inplete and/or unobtainable over the project planning horizon. Hence, this work focuses on developing a fuzzy mathematical programming technique to solve multiobjective PM decision problems in a fuzzy environment. The fuzzy MOLP model formulated here attempts to simultaneously minimize total project costs, total pletion time and total crashing costs associated with direct costs, indirect and contractual penalty costs, duration of activities and the constraint of available budget. These objective functions are required to be optimized simultaneously by the project managers in the framework of fuzzy aspiration levels. The fuzzy mathematical programming model is based on the following assumptions. (1) All of the objective functions are fuzzy with imprecise aspiration levels.(2) All of the objective functions and constraints are linear equations.(3) The normal time and shortest possible time for each activity and the cost of pleting the activity in the normal time and crash time are certain.(4) The available total budget is known over the duration of the project.(5) The piecewise linear membership functions are specified for fuzzy goals, and the minimum operator and the average operator are sequentially used to aggregate fuzzy sets in twophase solution procedure.(6) The total indirect costs can be divided into two categories, fixed costs and variable costs, and the variable costs per unit time are the same regardless of project pletion time.Assumption 1 relates to the fuzziness of the objective functions in practical PM decision problems, and incorporates the variations in the DM judgments regarding the solutions of fuzzy optimization problems in a framework of imprecise aspiration levels. Assumptions 2–4 indicate that the linearity, proportionality and certainty properties must be technically satisfied as a standard LP form (Carlsson amp。我們互相關(guān)心，互相幫助，四年時(shí)光說(shuō)長(zhǎng)不長(zhǎng)，但我們結(jié)下了畢生的友誼。能在各位老師的指引下成長(zhǎng)，我感到十分的幸運(yùn)，在此謹(jǐn)向各位老師表示深深的敬意和感謝。其次感謝所有的專業(yè)課老師，四年的大學(xué)生活，有幸得到各位老師的指導(dǎo)。論文所取得的成果無(wú)不凝聚著馬老師的心血和汗水。首先感謝我的導(dǎo)師馬濤馬老師。 Control Letters 40 (2000) ：43～57致 謝時(shí)光飛逝，轉(zhuǎn)眼間四年學(xué)習(xí)生涯即將結(jié)束。ez , Marcela Riquelme . Hybrid predictive control for realtime optimization of public transport systems’ operations based on evolutionary multiobjective optimization [ J ]. Transportation Research Part C 18 (2010)： 757–769[ 15 ] TienFu Liang. Fuzzy multiobjective project management decisions using twophase fuzzy goal programming approach [J]. Computers amp。ez , Freddy Milla , Alfredo Nn E. Cort233。通過(guò)