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第9期 黄晓霞：模糊环境下资金预算的机会约束NPV目标规划模型 .959 经过6000次迭代后，基于模糊模拟的遗传算 文所设计的算法能很好地解决所提出的资金预算的 法给出的最优资金分配计划为x*=(1,0,1,0),该 机会约束目标规划模型. 计划能很好地满足第一目标，但离第二目标值还有 5.81的负偏差.这意味着，如果投资者选择项目1 参考文献 和项目2，在95%的可信性下，投资资金不会超过预 [1]Weingartner H M.Mathematical Programming and the Analysis 算，虽然在90%的可信性下投资总收益的净现值不 of Capital Budgeting Problems.Prentice Hall:Englewood Press. 1963 能达到预期设定的目标值3000万元，但是在90% [2]Megginson W L.Corporate Finance Theory.London:Pearson 的可信性下，投资总收益的净现值可以达到2419 Education Limited.1997 万元 [3]MyersS C.A note on linear programming and capital budgeting. Jin,1972,27:89 表1投资支出 [4]Padberg M.Optimal project selection when borrowing and lending Table 1 Investment outlay rates differ.Math Comput Modell.1999.29:63 项目 投资支出/10元 项目 投资支出/10元 [5]Zadeh L.Fuzzy sets.Inf Control.1965.8:338 1 51=(2.5,3.4) 3 5=(2,3,4) [6]Liu B.Liu Y K.Expected value of fwzy variable and fwzy ex 2 2=(3,4,5) 9=(3,4.5,5) pected value models.IEEE Trans Fwzzy Syst.2002.10(4):445 [7]刘宝碇，赵瑞清，王纲.不确定规划及应用北京：清华大学 表2回收期每年末的投资回收期净现金流 出版社，2003 Table 2 Annual net cash flow [8]Lorie J H.Savage L J.Three problems in capital rationing J 项目 年净现金流/105元 项目寿命/a Business,1955,28:229 [9]Kumar N V,Ganesh L S.An empirical analysis of the use of the 1 h=(1.5,2,3) 10 analytic hierarchy process for estimating membership values in a 2 2=(2,3,3.5) F fwzzy set.Fuzzy Sets Syst.1996.96:1 3 1%=(1,2.5,3) 10 [10]Avineri E.Prashker J.Ceder A.Transportation projeets selee- 7=(2,2.5,4) tion process using fuzzy sets theory.Fuzy Sets Syst.2000. 116,35 [11]Charnes A,Cooper WW.Chanceconstrained programming 5结论 Manage Sci.1959.6:73 [12]Holland J H.Adaptation in Natural and Artificial Systems.Ann 考虑了资金的时间价值，给出了当期初投资金 Arbor:University of Michigan Press.1975 额和投资回收期的净现金流为模糊变量时，资金预 [13]Huang X.Chance-constrained programming models for capital 算的机会约束目标规划模型，并且采用基于模糊模 budgeting with NPV as fuzzy parameters.J Comput Appl 拟的遗传算法解决提出的优化模型.算例显示，本 Math,2007,198:149 Chance-constrained goal programming model for capital budgeting problems in fuzzy environment HUANG Xiaoxia Economics and Management School.University of Science and Technology Beijing.Beijing 100083.China ABSTRACI The net present value method was extended to fuzzy environment.A chance constrained goal pro gramming model for capital budgeting in fuzzy environment was proposed.A fuzzy simulation based on genetic algorithm was employed to provide a general solution for the new model problem.A numerical example was pre- sented to illustrate the modeling idea and the effectiveness of the proposed algorithm KEY WORDS capital budgeting:fuzzy environment;fuzzy goal programming;chance constrained program- ming经过6000次迭代后基于模糊模拟的遗传算 法给出的最优资金分配计划为 x ∗＝（1010）．该 计划能很好地满足第一目标但离第二目标值还有 5∙81的负偏差．这意味着如果投资者选择项目1 和项目2在95％的可信性下投资资金不会超过预 算虽然在90％的可信性下投资总收益的净现值不 能达到预期设定的目标值3000万元但是在90％ 的可信性下投资总收益的净现值可以达到2419 万元． 表1 投资支出 Table1 Investment outlay 项目 投资支出／106 元 1 ξ1＝（2∙53∙4） 2 ξ2＝（345） 项目 投资支出／106 元 3 ξ3＝（234） 4 ξ4＝（34∙55） 表2 回收期每年末的投资回收期净现金流 Table2 Annual net cash flow 项目 年净现金流／106 元 项目寿命／a 1 η1＝（1∙523） 10 2 η2＝（233∙5） 8 3 η3＝（12∙53） 10 4 η4＝（22∙54） 8 5 结论 考虑了资金的时间价值给出了当期初投资金 额和投资回收期的净现金流为模糊变量时资金预 算的机会约束目标规划模型并且采用基于模糊模 拟的遗传算法解决提出的优化模型．算例显示本 文所设计的算法能很好地解决所提出的资金预算的 机会约束目标规划模型． 参 考 文 献 ［1］ Weingartner H M．Mathematical Programming and the Analysis of Capital Budgeting Problems．Prentice—Hall：Englewood Press 1963 ［2］ Megginson W L．Corporate Finance Theory．London：Pearson Education Limited1997 ［3］ Myers S C．A note on linear programming and capital budgeting． J Fin197227：89 ［4］ Padberg M．Optimal project selection when borrowing and lending rates differ．Math Comput Modell199929：63 ［5］ Zadeh L．Fuzzy sets．Inf Control19658：338 ［6］ Liu BLiu Y K．Expected value of fuzzy variable and fuzzy expected value models．IEEE Trans Fuzzy Syst200210（4）：445 ［7］ 刘宝碇赵瑞清王纲．不确定规划及应用．北京：清华大学 出版社2003 ［8］ Lorie J HSavage L J．Three problems in capital rationing．J Business195528：229 ［9］ Kumar N VGanesh L S．An empirical analysis of the use of the analytic hierarchy process for estimating membership values in a fuzzy set．Fuzzy Sets Syst199696：1 ［10］ Avineri EPrashker JCeder ATransportation projects selection process using fuzzy sets theory．Fuzzy Sets Syst2000 116：35 ［11］ Charnes ACooper W W．Chance-constrained programming． Manage Sci19596：73 ［12］ Holland J H．Adaptation in Natural and Artificial Systems．Ann Arbor：University of Michigan Press1975 ［13］ Huang X．Chance-constrained programming models for capital budgeting with NPV as fuzzy parameters． J Comput Appl Math2007198：149 Chance-constrained goal programming model for capital budgeting problems in fuzzy environment HUA NG Xiaoxia Economics and Management SchoolUniversity of Science and Technology BeijingBeijing100083China ABSTRACT The net present value method was extended to fuzzy environment．A chance-constrained goal programming model for capital budgeting in fuzzy environment was proposed．A fuzzy simulation based on genetic algorithm was employed to provide a general solution for the new model problem．A numerical example was presented to illustrate the modeling idea and the effectiveness of the proposed algorithm． KEY WORDS capital budgeting；fuzzy environment；fuzzy goal programming；chance-constrained programming 第9期 黄晓霞： 模糊环境下资金预算的机会约束 NPV 目标规划模型 ·959·