摘要
在经济管理领域中,“黑箱”问题是普遍存在的。例如,政府在制定相关政策时,并不能准确预知该政策实施后会给社会带来什么样的影响,而只能在政策真正实施后对政策所带来的影响进行观察,此时,政府制定的政策对社会的影响过程是一个“黑箱”,其输入变量是“政策”,输出变量是“社会影响”。同样,企业在进行定价决策时,往往不能准确预知新价格可能会带来的市场反应,而只能在价格公开后去观察客户与竞争对手的反应。此时,企业价格策略的作用过程也是一个“黑箱”,其输入变量是“价格”,其输出变量是“客户与竞争对手的反应”。这些管理中的“黑箱”,我们可以知道其作用过程的一些规律和机制,却没有量化的函数表达式。
由于“黑箱”现象的普遍性,在管理优化过程中,人们经常需要面对由“黑箱”现象所带来的困扰。当管理优化模型中存在黑箱函数时,对优化决策系统输入的控制变量x1 , x2 ,…, xn,我们只能观测到黑箱函数f ( x1 , x2 ,…, xn)的值(信息)并利用观测到的信息进行求解。然而,这种观测(信息的获得)往往代价不菲,或者观测次数受到限制。因此,如何在黑箱函数条件下建立管理优化模型并给出行之有效的求解方法是一个有实际应用背景的问题。
尽管黑箱函数条件给管理优化问题的求解带来了一定的障碍,但并非无法求解。华罗庚教授推广普及的黄金分割法就是一种一维单峰黑箱函数条件下的最优化方法,在管理实践中已被广泛接受与应用,成为黑箱条件下一维优化的有力工具。本文对多维单峰黑箱函数条件下的一些极值问题与多维单调黑箱函数条件下的一些平衡问题进行了讨论,是对华罗庚推广普及黄金分割法的继承与发扬。
论文首先给出了选题的背景,认为黑箱是管理中的普遍现象,人们在管理优化中必然需要面对黑箱函数障碍。当管理优化模型中存在黑箱函数时,寻求一些只用函数值的管理优化方法是很有必要的。本研究的思路是将一些管理优化问题写为变分不等式模型,从而可以用一些求解变分不等式的只用函数值的方法对问题进行求解。在介绍了一些管理优化问题的变分不等式形式以及变分不等式在管理优化中的一些应用之后,提出本文的意义在于其可以为目标函数表达式未知时或对? f ( x) =0的求解非常困难时的管理优化决策提供参考,并给出了本文研究的主要方法、目标与内容。
其次,论文总结了相关的变分不等式算法,介绍了不同条件、不同结构变分不等式问题的各类求解方法,包括基本投影法、预测-校正方法、结构变分不等式的求解方法与结构可分变分不等式的求解方法。对于结构可分的变分不等式,本文在交替方向法的基础上提出了一种下降方法,并给出了相应的算法收敛性证明与数值试验结果。
第三,论文给出了管理优化目标为a≤f ( u )≤b时的管理优化变分不等式模型。以物流运输服务价格调整模型与能源价格调整模型为例,当管理优化的目标为a≤f ( u )≤b且f ( u )为黑箱函数时,利用决策变量与相关表达式之间的互补关系建立了隐式互补模型。在一定的假设之下,论文证明了函数f ( u )的单调性与连续性,从而可以利用一些直接迭代方法(仅需函数f ( u )的观测结果)进行求解。对给出的算例,论文给出了不同算法的计算结果并进行了比较。
第四,论文给出了管理优化目标为max( or min) f ( x )时的Box约束管理优化模型。以一类基于空间价格均衡的物流配送量优化模型与一类价格调整模型为例,在一定的假设条件下,分析了目标函数最大或最小时决策变量与黑箱函数之间的互补关系,从而将其改写为互补模型,并对模型的假设条件与求解方法等进行了讨论。对给定的算例,论文给出了计算结果。
第五,论文给出了在线性约束条件下管理优化目标为max( or min) f ( x )时的管理优化模型。以物流配送量优化问题为例,分别讨论了存在配送中心能力限制、边容量限制、配送中心与边容量双重限制条件下的配送量优化模型。对模型的假设条件与求解方法等进行了讨论。对给定的算例,论文给出了不同限制条件下的优化结果。
最后对全文内容及研究结论和创新之处进行了总结,并对文中有待进一步深入研究的地方提出日后继续研究的方向和展望。
In the economical and managerial field,‘black box’is a common phenomenon. For example, when a government is making some policy, it can’t predict exactly what effect the policy will make on the society and can only observe the effect after the policy is implemented. So the affecting process of the policy made by the government on the society is a‘black box’. The input variable is‘policy’and the output variable is‘social effect’. Similarly, when an enterprise is making a pricing policy, it can’t know exactly what social reaction the policy will lead to and can only observe the reactions of customers and competitors after the price is announced. Then the affecting process of the pricing policy is also a‘black box’. The input variable is‘price’and the output variable is‘the reactions of customers and competitors’. We can know some law of those‘black boxes’. In spite of that, the exact operation mechanisms are unknown.
Because the universality of the‘black box’, managers have to face the troubles brought by the phenomenon when optimizing the managerial problems. When an optimizations model has a‘black box’function, to a control variable of the optimal policy, we can only solve the model by observing the value of the‘black box’. However, the observation is an expensive process or the number of such observation is limited. Therefore, how to build management optimization models which are based on‘black box’and how to present an effective algorithm are issues with practical applications.
‘Black box’brings some obstacles to the solving processes. In spite of that, the problems are not insurmountable. The golden section method presented by Luogen Hua is an optimization method which is used under the conditions involving one dimension, one hump, and‘black box’. The golden section method, a powerful tool for one-dimensional optimization, has been widely accepted. In this dissertation, we carried down and developed the idea of golden section and focused on two kinds of problems, one is management optimization models and algorithms under the conditions involving multi dimensions, one hump, and‘black box’, the other is equilibrium models and algorithms under the conditions involving multi dimensions, monotonous, and‘black box’.
Firstly, the thesis introduces the backgrounds, the significance, the innovative achievements and the motivations of choosing this topic.‘Black box’is a common phenomenon in the economical and managerial field and managers have to face the troubles brought by‘black box’. It is highly necessary to find some some solutions without derivatives when there are‘black boxes’. The thesis introduces some applications of variational inequality in optimization and the equivalent variational inequality forms of some optimizations. The significance or contribution of this thesis is to provide a reference for the decision when some functions are black boxes or it is difficult to solve the system of equations ? f ( x) = 0.
Secondly, the thesis reviews some algorithms which are used to solve variational inequalities and introduces various kinds of algorithms to different condition and structure, including the basic projection methods, the Prediction-Correction methods, the methods for structured variational inequalities, and the methods for variational inequalities with separate structures. To variational inequalities with separate structures, the thesis presents a descend method based on the alternating directions method, and some numerical results demonstrate that the new method is effective in practice.
Thirdly, the thesis presents some variational inequality models whose objectives of optimization are a≤f ( u )≤b. Take the process of the logistics provider readjusting the prices and the government readjusting the resource prices for instance, when the objective is a≤f ( u )≤b and f ( u ) is a‘black box’, we can build an implicit complementarity model according to the complement relationship between decision variables and some expressions. Under the given assumptions, the thesis proves that f ( u ) is continuous and monotone, thereby the direct iterative algorithm, which only needs the value of f ( u ), can be used in the process of solving. To the given numerical experiments, the thesis provides the computation results of various algorithms and makes a comparison.
Fourthly, the thesis presents some variational inequality models when the objective of management optimization is max( or min) f ( x )and there is no restriction. Take the models of a kind of distribution optimizationof the models of a kind of distribution optimizations based on spatial price equilibrium and the models of a kind of price readjusting for instance, the thesis analyzes the complement relationship between the decision variables and the black-box functions when the objective function reaches the maximum or minimum point. And then an implicit complementarity model can be build.
Finally the thesis provides the computation results of the given numerical experiments. Fifthly, the thesis presents some variational inequality models with linear constraints whose objectives of management optimization are max( or min) f ( x ). Take the models of distribution optimization for instance, the thesis presents the distribution models separately with capacity constraints from the centers of distribution, links, and both of them. The assumptions and algorithms are discussed. To the given numerical experiments, the optimal distribution plan with various capacity constraints is given.
Finally, the thesis gives conclusions to the contents and innovative achievements of the research, and presents the future scope, purpose and prospect of this topic in further studies.
引文
[1] Auslender A., Teboulle M. and Ben-Tiba S., A logarithmic-quadratic proximal method for variational inequalities, Computational Optimization and Applications 12 (1999) 31-40.
[2] Bauschke H.H. and Combettes P.L., A weak-to-strong convergence principle for Fejer-monotone methods in Hilbert spaces, Math. Oper. Res. 26(2), 248-264 (2001)
[3] Bertsekas, D.P. and Gafni E.M., Projection methods for variational inequalities with applications to the traffic assignment problem, Mathematical Programming Study 17 (1982) 139-159.
[4] Bonnans J.F., Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Applied Mathematics and Optimization 29 (1994) 161-186.
[5] Burachik R.S. and Iusem A.N., A generalized proximal point algorithm for the variational inequality problem in a Hilbert space, SIAM Journal on Optimization 8 (1998) 197-216.
[6] Carbone A. and Isac G., The generalized order complementarity problem: Applications to economics and an existence result, Nonlinear studies, 5(1998) ,139-151
[7] Censor Y., A.N. Iusem and S.A. Zenios, An interior point method with Bregman functions for variational inequality problem with paramonotone operators, Mathematical Programming 81 (1998) 373-400.
[8] Chen G. and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming 64 (1994) 81-101.
[9] Cui Y. and He B.S., A class of projection and contraction methods for asymmetric linear variational inequalities and their relations to Fukushima’s descent method, Computers and Mathematics with Applications 38 (1999) 151-164.
[10] Dafermos S., Traffic equilibrium and variational inequalities, TransportationScience 14 (1980) 42-54.
[11] Daniele P., Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model, Journal of global optimization 28(2004) 283-295
[12] Dial R.B, Network-optimized road pricing: Part I: A parable and a model Operations Research. Linthicum: 47(1999) 54 -64
[13] Dial R.B, Network-optimized road pricing: Part II: Algorithms and examplesOperations Research. Linthicum: 47(1999) 327-336
[14] Dong J., Zhang D. and Nagurney A. , A supply chain network equilibrium model with random demands, European Journal of Operational Research, 156(2004) 194-212
[15] Downs A., Point of view: implementing peak-hour road pricing at fullscale: Finding solution to practical problems. TR News 167 (1993) 7-9.
[16] Eaves B.C. On the basic theorem of complementarity, Mathematical Programming, 1(1971), 68-75
[17] Ebiefung A. and Kostreva M.M., Production equilibrium point in multi-unit manufacturing systems and the vertical linear complementarity problem, Annals of operations research,124(2003) 183-192
[18] Eckstein J. and Bertsekas D.P., On the Douglas-Rachford splitting method and the proximal points algorithm for maximal monotone operators, Mathematical Programming 55 (1992) 293-318.
[19] Eckstein J. and Fukushima M., Some reformulation and applications of the alternating direction method of multipliers, Large Scale Optimization: State of the Art, W.W.Hager et al eds., Kluwer Academic Publishers, (1994)115-134,
[20] Eckstein J., Approximate iterations in Bregman-function-based proximal algorithms, Mathematical Programming 83 (1998) 113-123.
[21] Eckstein J., Some saddle-function splitting methods for convex programming, Optimization Methods and Software 4 (1994) 75-83.
[22] Eckstein J., Some saddle-function splitting methods for convex programming, Optimization Methods and Software 4 (1994) 75-83.
[23] Facchinel F. and Pang J.S. Finite-dimensional variational inequalities and complementarity problem, Springer-Verlag, 2003.
[24] Ferris M.C., Pang J.S., Engineering and Econcomic Applications of Complementarity Problems, SIAM Review39(1997) 669-713,
[25] Florian M. and Spiess H., The convergence of diagonalization algorithms for asymmetric network equilibrium problems, Transportation Research 16B (1982) 447-483.
[26] Fortin M. and Glowinski R., Eds., Augmented Lagrangian Methods: Applications to the solution of Boundary-Valued Problems, North-Holland, Amsterdam, (1983).
[27] Fukushima M., Application of the alternating directions method of multipliers to separable convex programming problems, Computational Optimization and Applications 2 (1992) 93-111.
[28] Gabay D. and Mercier B., A dual algorithm for the solution of nonlinear variational problems via finite-element approximations, Computers and Mathematics with Applications 2(1) (1976) 17-40.
[29] Gabay D., Applications of the method of multipliers to variational inequalities, in: Fortin M. and Glowinski R., eds., Augmented Lagrangian Methods: Applications to the Solution of Boundary-valued Problems, M.Fortin and R. Glowinski, eds. (Amsterdam:North Holland) 299-331.
[30] Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer-Verlag, NewYork, 1984.
[31] Glowinski R. and P. Le Tallec,“Augmented Lagrangian and operator-splitting method in nonlinear mechanics”, SIAM Studies in Applied Mathematics, Philadelphia, PA, 1989.
[32] Han D.R. and He B.S., A new accuracy criterion for approximate proximal point algorithms, Journal of Mathematical Analysis and Applications 263(2) (2001) 243-254.
[33] Han D.R. and He B.S., An implementable step size rule for Goldstein, Levitin-Polyak projection method, Numerical Mathematics, A Journal of Chinese University 23 (2001) 56-62.
[34] Han D.R. and Lo H.K., A new modified extra-gradient method for variational inequality problems, Neural, Parallel, and Scientific Computations 8 (2000) 401-412.
[35] Han D.R. and Lo H.K., A new stepsize rule in He and Zhou’s alternating direction method, Applied Mathematics Letters 15(2) (2002) 181-185.
[36] Han D.R., A modified alternating direction method for variational inequality problems, Applied Mathematics and Optimization 45(1) (2002) 63-74.
[37] Harker P.T. and Pang J.S., A damped-Newton method for the linear complementarity problem, Lectures in Applied Mathematics 26 (1990) 265-284.
[38] Harker P.T. and Pang J.S., Finite dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical Programming 48 (1990) 161-220.
[39] He B.S., A modified projection and contraction method for a class of linear complementarity problems. Journal of Computational Mathematics 14 (1996), 54-63.
[40] He B.S. and Liao L. Z., Improvements of some projection methods for monotone nonlinear variational inequalities. Journal of Optimization Theory and Applications112(2002) 111-128.
[41] He B.S. and Stoer J., Solution of projection problems over polytopes, Numerische Mathematik 61 (1992) 73-90.
[42] He B.S. and Yang H. Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities, Operations Research Letters 23 (1998) 151-161.
[43] He B.S. and Yang H., A neural-network model for monotone linear asymmetric variational inequalities, IEEE Transactions on Neural Networks 11 (2000) 3-16.
[44] He B.S. and Zhou J., A modified alternation direction method for convex minimization problems, Applied Mathematics Letters 13 (2000) 122-130.
[45] He B.S., A class of projection and contraction methods for monotone variational inequalities, Applied Mathematics and Optimization 35 (1997) 69-76.
[46] He B.S., A modified projection and contraction method for a class of linearcomplementarity problems, Journal of Computational Mathematics 14 (1996) 54-63.
[47] He B.S., A new method for a class of linear variational inequalities, Mathematical Programming 66 (1994) 137-144.
[48] He B.S., A projection and contraction method for a class of linear complementarity problem and its application in convex quadratic programming, Applied Mathematics and Optimization 25 (1992) 247-262.
[49] He B.S., Inexact implicit methods for monotone general variational inequalities, Mathematical Programming 86 (1999) 199-216.
[50] He B.S., New contraction methods for linear inequalities, Linear Algebra and Its Applications 207 (1994) 115-133.
[51] He B.S., On a class of iterative projection and contraction methods for linear programming, Journal of Optimization Theory and Applications 78 (1993) 247-266.
[52] He B.S., On some projection and contraction method for solving monotone variational inequalities, Mathematica Numerica Sinica 18 (1996) 54-60.
[53] He B.S., Solution and applications of a class of general linear variational inequalities, Science in China, Series A 39(1996) 395-404.
[54] He B.S., Solving a class of linear projection equations, Numerische Mathematik, 68 (1994) 71-80.
[55] He B.S., Liao L. Z., Han D. R. and Yang H., A new inexact alternating directions method for monotone variational inequalities , Math. Program.,92(2002) 103–118.
[56] He B.S., Liao L.Z. and Yang Z.H., A new approximate proximal point algorithm for maximal monotone operator, Science in China, 46(2003) 200-206
[57] He B.S., Solving a class of linear projection equations, Numerische Mathematik 68(1994) 71-80 .
[58] He B.S., Wang S.L. and Yang H., A modified variable-penalty alternating directions method for monotone variational inequalities, J. Computational Mathematics, 21(2003) 495-504.
[59] He B.S., Yang H. and Wang S.L., Alternating directions method with self-adaptiveparameter for monotone variational inequalities, J. Optim. Theory Appli., 106, 349-368, (2000).
[60] He B.S., Yuan X. M. and Zhang J. Z., Comparison of two kinds of prediction- correction methods for monotone variational inequalities, Computational Optimization a nd Applications 27 (2004) 247-267.
[61] Kanzow C. and Qi H.D., A QP-free constrained Newton-type method for variational inequality problems, Mathematical Programming 85 (1999) 81-106.
[62] Khobotov E. N.. Modification of the extragradient method for solving variational inequalities and certain optimization problems. U.S.S.R. Comput. Math. Phys., 1987, (27): 120～127.
[63] Kincaid D. and Cheney W., Numerical Ananysis: Mathematics of Scientic Computing, 3rd Edition, Thomson Learning, USA.
[64] Kolstad C.D. and Mathiesen L.,Computing cournot-nash equilibria, Operations research, 39(1991) 739-748
[65] Kontogiorgis S. and Meyer R.R., A variable-penalty alternating directions method for convex optimization, Mathematical Programming 83 (1998) 29-53.
[66] Korpelevich G. M., The extra-gradient method for finding saddle points and other problems, Matecon 12 (1976) 747-756.
[67] Lam W. H. K. and Huang H. J. A combined activity/travel choice model for congested road networks with queues Transportation. Amsterdam, 29 (2002) Vol. 29, 5
[68] Larsson T. and Patriksson M.: An augmented Lagrangean dual algorithm for link capacity constrained traffic assignment problems. Transportation Research 29B(1995) 433-455.
[69] Metzler C., Hobbs B. and Pang J.S., Nash-cournot equilibria in power markets on a linearized DC network with arbitrage: Formulations and properties, Networks and spatial economics, 3(2003) 123-150
[70] Nagurney A. and Dhanda K.K., Marketable pollution permits in oligopolistic markets with transaction cost, Operations research, 48(2000) 424-435
[71] Nagurney A. and Dhanda K.K., Noncompliant oligopolistic firms and marketablepollution permits: Statics and dynamics, Annals of operations research, 95(2000) 285-312
[72] Nagurney A. and Ramanujam P., Transportation network policy modeling with goal targets and generalized penalty functions, Transportation Science 30 (1996) 3-13.
[73] Nagurney A. and Toyasaki F., Supply chain supernetworks and environmental criteria [J ] . Transportation Research : Part D , 8(2003) 185 - 213.
[74] Nagurney A. and Zhang D., Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts, 1996.
[75] Nagurney A. and Zhao L., Variational inequalities and networks in the formulation and computation of market equilibria and disequilibria : the case of direct demand functions[J ] . Transportation Science , 27(1993) 4 - 15.
[76] Nagurney A., Cruz J., Matsypura D., Dynamics of global supply chain supernetworks. Mathematical and Computer Modelling 37(2003) 963–983.
[77] Nagurney A., Dong J. and Mokhtarian P.L., A space-time network for telecommuting versus commuting decision-making, Regional science, 82(2003) 451-473.
[78] Nagurney A., Dong J. and Zhang D., A supply chain network equilibrium model, transportation Research E38(2002) 281-303.
[79] Nagurney A., Ke K., Cruz J., Hancock K. and Southworth F., Dynamics of supplychains: A multilevel (logistical/informational/ˉnancial) network perspective, Environment @ Planning B 29 (2000) 795-818.
[80] Nagurney A., Loo J., Dong J. and Zhang D., Supply chain networks and electronic commerce: a theoretical perspective, Netnomics, 4(2002) 187-220.
[81] Nagurney A., Zhang D. and Dong J., Spatial economic with multicriteria producers and consumers: Statics and dynamic, The annals of regional science, 36 (2002),79-105
[82] Noor M.A., Some recent advances in variational inequalities, Part I : Basic concepts, New Zealand Journal of Mathematics 26 (1997) 53-80.
[83] Pang J.S., Complementarity problems, Handbook of Global Optimization, R. Horst and P. Pardalos (eds.), Kluwer Academic Publishers, Nowell, MA, 271-338, 1994.
[84] Parise Q., New economic interpretations of complementarity problem. Southern economic journal, 46(1979) 568-578
[85] Raghavan T.E.S. and Syed Z., Computing stationary nash c of undiscounted single-controller stochastic games, Mathematics of operations research, 27(2002) 383-400
[86] Solodov M.V. and Svaiter B.F., A hybrid approximate extragradient-proximal point algorithm using the enlargement of a maximal monotone operator, Set-Valued Analysis 7 (1999) 323-345.
[87] Solodov M.V. and Svaiter B.F., A hybrid projection-proximal point algorithm, Journal of Convex Analysis 6(1) (1999) 59-70.
[88] Solodov M.V. and Svaiter B.F., A new projection method for variational inequality problems, SIAM, Journal on Control and Optimization 37 (1999) 765-776.
[89] Solodov M.V. and Tseng P., Modified projection-type methods for monotone variational inequalities, SIAM Journal of Control and Optimization 34 (1996) 1814-1830.
[90] Sun D.F., A class of iterative methods for solving nonlinear projection equations, Journal of Optimization Theory and Applications 91 (1996) 123-140.
[91] Sun D.F., A new step-size skill for solving a class of nonlinear projection equations, Journal of Computational Mathematics 13 (1995) 357-368.
[92] Tseng P., Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM Journal on Control and Optimization 29 (1991) 119-138.
[93] Tseng P., A modified forward-backward splitting method for maximal monotone mappings, SIAM. J. Control. Optim. 38 (2000), 431-446
[94] Tseng P.,“Alternating Projection-Proximal Methods for Convex Programming and Variational Inequalities”, SIAM Journal on Optimization 7 (1997) 951-967. [86] Vickrey W., Point of view: principles and applications of congestion pricing. TRNews 167(1993) 4-5.
[95] Wakolbinger T. and Nagurney A., Dynamic supernetworks for the integration of social networks and supply chains with electronic commerce: modeling and analysis of buyer-seller relationships with computations, Netnomics 6(2004) 153-185.
[96] Wang S.L., Yang H. and He B.S., Solving a class of asymmetric variational inequalities via a new alternating direction method, Computers and Mathematics with Applications 40 (2000) 927-937.
[97] Yang H. and Bell M. G. H., Traffic restraint, road pricing and network equilibrium. Transportation Research 31B(1997) 303-314.
[98] Yang H. and Huang H. J., Principle of marginal-cost pricing: How does it work in a general network? Transprotation Research 32A (1998) 45-54.
[99] Yang H., Meng Q. and Lee D. H.,Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions. Transportation Research 38B (2004) 477-493.
[100] Ye C.H. and Yuan X.M., A descent method for structured monotone variational inequalities, to appear on Optimization Methods and Software, 2006
[101]邓宁,张守凤,江涛.供应链网络均衡模型,价值工程.2004(02),39-43
[102]黄亦恺,缪立新.供应链网络平衡模型设计.中国物资与采购.2005(08),60-63
[103]何炳生,求解单调变分不等式的投影收缩算法-基于信息的求解方法.南京大学数学系讲义, 2006.
[104]陆化普,殷亚峰.交通网络均衡分析的变分不等式方法.公路交通科技. 1997(06). 24-29
[105]华罗庚.华罗庚科普著作选集.上海.上海教育出版社. 1984
[106]钱颂迪.运筹学,北京,清华大学出版社, 1993.
[107]王晛,李渝曾,张少华.求解电力市场均衡模型的非线性互补方法.电力系统自动化.2004(01).7-11
[108]徐明华,叶彩鸿,何炳生.一类资源价格调控的数学模型和它的求解方法.系统工程,2005(10),92-96.
[109]杨超.运筹学,北京,科学出版社, 2004.
[110]叶彩鸿,徐明华,何炳生.一类价格调整问题的数学模型及其求解方法.数量经济技术经济研究. 2006(03), 121-128.
[111]张铁柱,刘志勇,滕春贤,胡运权,多商品流供应链网络均衡模型的研究,系统工程理论与实践, 2005 (07),61-66
[112]“出口尿素就是出口能源”的警示.中国出口贸易网. http://www.cnexpnet.com/html/1/200522390204/
