二层决策问题的研究及区间决策方法的应用
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摘要
多层规划是研究递阶系统优化问题的基本模型,众多研究者对此进行了深入的研究,并且已广泛地应用在许多领域中,如社会经济、工程技术、管理部门及军事等领域。二层规划是多层规划的重要组成部分,是近年来受到众多领域广泛关注的前沿课题,现实的决策系统大都可以看成二层决策系统。论文的研究对象为二层线性规划问题中的价格控制问题和食用豆腐生产经营过程中的区间决策问题,其主要工作如下:
首先,在对国内外研究现状进行了深入分析的基础上,对二层决策系统中的上层无约束的二层线性规划做了详细介绍,给出了相应的基本模型、定义及在不同的假设条件下它们的相关性质。
其次,引入模糊结构元理论,对有关模糊结构元的基本概念做了简明扼要的介绍,给出了基于模糊结构元理论的模糊数的排序方法及运算法则,利用结构元方法将有模糊变量的价格控制问题转化为传统的价格控制问题,并给出了求解算法;而后给出了优面的定义和性质以及最优解和优面的关系,对二层价格控制问题的优面算法步骤进行了设计,并通过算例说明了该算法的有效性。
最后,由于在豆腐生产过程中存在着许多不确定性,自然环境的改变、生产条件的变化以及市场资源的变化,给豆腐作坊的净收益带来了不确定的因素,于是豆腐作坊的最优净收益就呈现出区间的特性。第五章针对豆腐作坊生产经营豆腐的问题,建立了改变生产规模的区间决策模型和不改变生产规模的区间决策模型,给出了区间决策问题的基本定理及求解方法,并对实际问题进行了求解,对豆腐作坊给出了结果分析和建议。
Multilevel programming is fundamental model used to study hierarchical system problem, many researchers proceed thorough research to the hierarchical system, and it has been extensively applied in society economy, engineering technique, manage department and military etc. Bilevel programming is the main part of the multilevel programming, attracting the extensive attention in multitudinous in recent years. In fact, most of the realistic decision making systems can be treated as bilevel decision making systems. The research object of this paper is the price control problem of bilevel linear programming and interval decision making problem for making and management of tofu, main works are as follows:
Firstly, in this thesis, according to the thorough research to domestic and international present condition, the bilevel linear programming without constraint in the upper level is elaborated in the beginning, the basic model, definition and theorem are given and the interrelated properties are discussed respectively under different hypothesis.
Secondly, the theory of fuzzy structured element is induced. The basic concept and the basic theorem are briefly introduced. Sequencing and operations of fuzzy numbers based on the theory of fuzzy structured element are given in the paper. By using the method of fuzzy structured element, we transform price control problem with fuzzy variables into traditional price control problem, and give the algorithm. In addition, we give the definitions and some properties of optimal plane, and give the relationship between optimal solution and optimal plane. Then, the optimal plane algorithm steps of the price control problem is designed, and an example is adopted to verify the effectiveness of the proposed algorithm.
Finally, in making of tofu, with the changing of the nature envoronment, the producting condition and market source, the wet profit of tofu in the tofu factory is not distinct, so the wet profit of tofu in the tofu factory takes on interval character. In the fifth chapter, for the problem about making and management of tofu in the tofu factory, we establish interval decision making model of changed producting scale and unchanged one, and give some theorems and a method for solving the interval decision making problem. Then, we solve the above real problem and give the result analysis and advice to the tofu factory.
首先,在对国内外研究现状进行了深入分析的基础上,对二层决策系统中的上层无约束的二层线性规划做了详细介绍,给出了相应的基本模型、定义及在不同的假设条件下它们的相关性质。
其次,引入模糊结构元理论,对有关模糊结构元的基本概念做了简明扼要的介绍,给出了基于模糊结构元理论的模糊数的排序方法及运算法则,利用结构元方法将有模糊变量的价格控制问题转化为传统的价格控制问题,并给出了求解算法;而后给出了优面的定义和性质以及最优解和优面的关系,对二层价格控制问题的优面算法步骤进行了设计,并通过算例说明了该算法的有效性。
最后,由于在豆腐生产过程中存在着许多不确定性,自然环境的改变、生产条件的变化以及市场资源的变化,给豆腐作坊的净收益带来了不确定的因素,于是豆腐作坊的最优净收益就呈现出区间的特性。第五章针对豆腐作坊生产经营豆腐的问题,建立了改变生产规模的区间决策模型和不改变生产规模的区间决策模型,给出了区间决策问题的基本定理及求解方法,并对实际问题进行了求解,对豆腐作坊给出了结果分析和建议。
Multilevel programming is fundamental model used to study hierarchical system problem, many researchers proceed thorough research to the hierarchical system, and it has been extensively applied in society economy, engineering technique, manage department and military etc. Bilevel programming is the main part of the multilevel programming, attracting the extensive attention in multitudinous in recent years. In fact, most of the realistic decision making systems can be treated as bilevel decision making systems. The research object of this paper is the price control problem of bilevel linear programming and interval decision making problem for making and management of tofu, main works are as follows:
Firstly, in this thesis, according to the thorough research to domestic and international present condition, the bilevel linear programming without constraint in the upper level is elaborated in the beginning, the basic model, definition and theorem are given and the interrelated properties are discussed respectively under different hypothesis.
Secondly, the theory of fuzzy structured element is induced. The basic concept and the basic theorem are briefly introduced. Sequencing and operations of fuzzy numbers based on the theory of fuzzy structured element are given in the paper. By using the method of fuzzy structured element, we transform price control problem with fuzzy variables into traditional price control problem, and give the algorithm. In addition, we give the definitions and some properties of optimal plane, and give the relationship between optimal solution and optimal plane. Then, the optimal plane algorithm steps of the price control problem is designed, and an example is adopted to verify the effectiveness of the proposed algorithm.
Finally, in making of tofu, with the changing of the nature envoronment, the producting condition and market source, the wet profit of tofu in the tofu factory is not distinct, so the wet profit of tofu in the tofu factory takes on interval character. In the fifth chapter, for the problem about making and management of tofu in the tofu factory, we establish interval decision making model of changed producting scale and unchanged one, and give some theorems and a method for solving the interval decision making problem. Then, we solve the above real problem and give the result analysis and advice to the tofu factory.
引文
1 J. Bracken, J. M. McGill. Mathematical Programs with Optimization Problems in the Constraitns. Operations Research, 1973,(21):37-44
2 T. Basar, J. Olsder. Dynamic Noncooperative Game Theory. London, Academic Press, 1982:75-86
3 J. Kornal, T. Liptak. Two-level Planning. Econometric, 1965,(33):141-169
4 J. Freeland, N. Baker. Good Partitioning in A Hierarchical Organization. Management Science, 1975,(3):673-678
5 R. G. Cassidy, M. J. L. Kirby, W. M. Raike. Efficient Distribution of Resource Through Three Levels of Government. Manegement Science, 1971,(17):462-473
6 W. J. Baumol, T. Fabian. Decomposition, Pricing for Decentralization and External Economics. Management Science, 1964,(11):1-31
7 W. Candler, R. Norton. Multilevel Programming. Technical Report, World Bank Development Research Center, Washington D.C. 1977:20-25
8 H. Stackelberg. The Theory of the Market Economy. New York, Oxford: Oxford University Press, 1952:30-37
9 Migdalas. Bilevel Programming in Traffic Planning: Models, Methods and Challenge. Journal of Global Optimization, 1995,(7):381-405
10 Sonia, Ankit Khandelwal, M.C. Puri. Bilevel Time Minimizing Transportation Problem. Discrete Optimization, 2008,5(5):714-723
11 J. Zh. Zhang, D. T. Zhu. A Bilevel Programming Method for Pipe Network Optimization. Journal of Global Optimization, 1996,6(3):838-857
12 Marcotte. Network Design Problem with Congestion Effects: A Case of Bilevel Programming. Mathematical Programming, 1986,34(6):142-162
13 F. Amat, B. Mccarl. A Representation and Economic Interpretation of A Two-level Programming Problem. Journal of Operations Research Society, 1981,32(4):783-792
14 J. H. Ryu, V. Dua. A Bilevel Programming Framework for Enterprise-Wide Process Networks Uncertainty. Computers and Chemical Engineering, 2004,28(6-7):1121-1129
15 P. Marcotte, G. Savard and F. Semet. A Bilevel Programming Approach to the Traveling Salesman Problem. Operations Research Letters, 2004,32(3):240-248
16 L. Mangasarian. Misclassification Minimization. Journal of Globle Optimization, 1994, 5(7):309-323
17 C. Shi, G. Zhang, J. Lu and H. Zhou. An Extended Branch and Bound Algorithm for Linear Bilevel Programming. Applied Mathematics and Computation, 2006,180(9): 529-537
18 J. Bracken, J. E. Falk and J. T. McGill. The Equivalent of Two Mathematical Programs with Optimization Problems in the Constraints. Operations Research, 1974,(22): 1102-1104
19 Y. Lv, T. Hu, G. Wang, Z. Wan. A Neural Network Approach for Solving Nonlinear Bilevel Programming Problem. Computers and Mathematics with Applications, 2008,55(6):2823-2829
20刘兵兵.一类非线性二层混合整数规划问题全局最优解的遗传算法.燕山大学学报, 2007,31(6):554-557
21 Migdalas, P. M. Pardalas. Editorial: Hierarchical and Bilevel Programming. Journal of Global Optimization, 1996,8(5):209-215
22 G. Zhang, C. Shi, J. Lu. An Extended Kth-best Approach for Referential-uncooperative Bilevel Multi-follower Decision Making. International Journal of Computational Intelligence Systems, 2008,1(3):205-214
23 C. Shi, H. Zhou, J. Lu and G. Zhang. The Kth-best Approach for Linear Bilevel Multifollower Programming with Partial Shared Variables Among Followers. Applied Mathematics and Computation, 2007,188(2):1686-1698
24 P. Hansen, B. Jaumard and G. Savard. New Branch-and-Bound Rules for Linear Bilevel Programming. SIAM Journal on Scientific and Statistic Computing, 1992,13(5): 1194-1217
25 D. J. White, G. Anandalingam. A Penalty Function Approach for Solving Bilevel Linear Programs. Journal of Global Optimization, 1993,3(5):393-419
26 Y. Lv, Z. Chen, Z. Wan. A Penalty Function Method Based on Bilevel Programming for Solving Inverse Optimal Value Problems. Applied Mathematics Letters, 2010,(23): 170-175
27雷万明,邓先礼.用罚函数求解二层线性规划的方法.汕头大学学报, 2000,15(2): 74-78
28 J. F. Bard. An Investigation of the Linear Three-Level Programming Problem. IEEE Transactions on Systems, Man. and Cybemetics, 1984,14(5):711-716
29吕一兵,姚天祥,陈忠.求解线性二层规划的一种全局优化方法.长江大学学报(自然科学版), 2008,5(4):7-10
30董银红,王广民,刘洪海.二层线性规划问题的全局收敛算法.武汉大学学报, 2006,6(12):105-107
31 G. Savarad, J. Gauvin. The Steepest Descent Direction for the Nonlinear Bilevel Programming Problem. Operations Research Letters, 1994,15(7):275-282
32 C. D. Kolstad, S. L. Lasdon. Derivative Evaluation and Computations Experience with Large Bilevel Mathematical Program. Journal of Optimization Theory and Applications, 1990,65(10):485-499
33 J. F.Bard. Convex Two-level Optimization. Mathematical Programming, 1988,40:15-27
34 K. Shimizu, E. Aiyoshi. A New Computational Method for Stackelberg and Min-Max Problem by Use of A Pealty Method. IEEE Transactions on Automatic Control, 1981,26(2):460-466
35 K. Shimizu, M. Lu. A Global Optimization Method for the Stackelberg Problem with Convex Functions Via Problem Transformation and Concave Programming. IEEE Transactions on Systems C, 1995,25(12):1635-1640
36徐凌,张圣贵.基于过滤信赖域算法的非线性二层规划求解方法.福建师范大学学报(自然科学版), 2010,26(2):5-9
37郑寒凝,张圣贵.一类非线性二层规划的平衡点算法.福建师范大学学报(自然科学版), 2010,26(2):10-15
38 K. M. Lan, U. P. Wen, H. S. Shih, E. Stanley Lee. A Hybrid Neural Network Approach to Bilevel Programming Problems. Applied Mathematics Letters, 2007,20(7):880-884
39吴睿.非线性二层规划的一种混合粒子群算法.衡水学院学报, 2010,12(1):14-16
40金俊培,盖钧益.大豆地方品种在豆腐产量品质及有关加工性状上的遗传变异[J].南京农业大学学报, 1995,18(1):5-9
41刘志胜,李里特,辰巳英三.大豆原料对豆腐得率和质构的影响.中国农业大学学报, 1999,(6):101-105
42滕春贤,李智慧.二层规划的理论与应用.第二版.北京:科学出版社, 2002:15-33
43王先甲,冯尚友.二层系统最优化理论.北京:科学出版社, 1995:25-60
44魏权龄,闫洪.广义最优化理论和模型.北京:科学出版社, 2003:153-196
45刘志勇,滕春贤,陈东彦.关于二层价格控制决策问题的探讨.统计与决策, 2007,(20):29-31
46周水生,刘三阳.价格控制问题的基本性质.应用数学与计算数学学报, 1998,12 (2):53-58
47李荣钧著.模糊多准则决策理论与应用.北京:科学出版社, 2002:8-23
48郭嗣琮.模糊值函数分析学的结构元方法简介(I).数学的实践与认识, 2008,38 (2):88-93
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53郭嗣琮.由模糊结构元线性生成的模糊数运算.辽宁工程技术大学学报, 2006,(1): 154-157
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55 Y. Lv, T. Hu, Z. Wan. A Penalty Function Method for Solving Weak Price Control Problem. Applied Mathematics and Computation, 2007,(8):1520-1525
56涂为员.线性规划的优面算法.南昌大学学报:理科版, 2002,26(4):320-322
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59阮国桢,杨丰梅,汪寿阳.线性二级价格控制问题的单纯形算法.系统工程理论与实践, 1996,16(12):38-43
60苏继颖.大豆制品的营养及发展趋势.中国油脂, 2006,31(8):40-41
61孔凡真.豆制品的改进方向:卫生、营养、风味改良和快餐化.粮食科技与经济, 2003,(4):49
62达庆利,刘新旺.区间数线性规划及其满意解.系统工程理论与实践, 1999,(4):3-7
63 Tanaka H. On Fuzzy Mathematical Programming. Joutnal of Cybernetics, 1984,46(4): 37-46
64 Rommel Fanger H. Linear Programming with Fuzzy Objective. Fuzzy Sets and Systems, 1989,(29):31-48
65 Stefan C, Dorota K. Multiobjective Programming in Optimization of Interval Objective Functions—A Generalized Approach. European Journal of Operational Research, 1996, (94):594-598
66 Stefan C, Doreta K. A Concept of the Optimal Solution of the Transporation Problem with Fuzzy Cost Coefficients. Fuzzy Sets and Systems, 1996,82(7):299-305
67 Tong S. Interval Number and Fuzzy Number Linear Programming. Fuzzy Sets and Systems, 1994,66(5):301-306
68徐泽水,达庆利.区间数排序的可能度法及其应用.系统工程学报, 2003, (1):67-70
69史加荣,刘三阳,熊文涛.区间线性规划的一种新解法.系统工程理论与实践, 2005,(2):101-106
2 T. Basar, J. Olsder. Dynamic Noncooperative Game Theory. London, Academic Press, 1982:75-86
3 J. Kornal, T. Liptak. Two-level Planning. Econometric, 1965,(33):141-169
4 J. Freeland, N. Baker. Good Partitioning in A Hierarchical Organization. Management Science, 1975,(3):673-678
5 R. G. Cassidy, M. J. L. Kirby, W. M. Raike. Efficient Distribution of Resource Through Three Levels of Government. Manegement Science, 1971,(17):462-473
6 W. J. Baumol, T. Fabian. Decomposition, Pricing for Decentralization and External Economics. Management Science, 1964,(11):1-31
7 W. Candler, R. Norton. Multilevel Programming. Technical Report, World Bank Development Research Center, Washington D.C. 1977:20-25
8 H. Stackelberg. The Theory of the Market Economy. New York, Oxford: Oxford University Press, 1952:30-37
9 Migdalas. Bilevel Programming in Traffic Planning: Models, Methods and Challenge. Journal of Global Optimization, 1995,(7):381-405
10 Sonia, Ankit Khandelwal, M.C. Puri. Bilevel Time Minimizing Transportation Problem. Discrete Optimization, 2008,5(5):714-723
11 J. Zh. Zhang, D. T. Zhu. A Bilevel Programming Method for Pipe Network Optimization. Journal of Global Optimization, 1996,6(3):838-857
12 Marcotte. Network Design Problem with Congestion Effects: A Case of Bilevel Programming. Mathematical Programming, 1986,34(6):142-162
13 F. Amat, B. Mccarl. A Representation and Economic Interpretation of A Two-level Programming Problem. Journal of Operations Research Society, 1981,32(4):783-792
14 J. H. Ryu, V. Dua. A Bilevel Programming Framework for Enterprise-Wide Process Networks Uncertainty. Computers and Chemical Engineering, 2004,28(6-7):1121-1129
15 P. Marcotte, G. Savard and F. Semet. A Bilevel Programming Approach to the Traveling Salesman Problem. Operations Research Letters, 2004,32(3):240-248
16 L. Mangasarian. Misclassification Minimization. Journal of Globle Optimization, 1994, 5(7):309-323
17 C. Shi, G. Zhang, J. Lu and H. Zhou. An Extended Branch and Bound Algorithm for Linear Bilevel Programming. Applied Mathematics and Computation, 2006,180(9): 529-537
18 J. Bracken, J. E. Falk and J. T. McGill. The Equivalent of Two Mathematical Programs with Optimization Problems in the Constraints. Operations Research, 1974,(22): 1102-1104
19 Y. Lv, T. Hu, G. Wang, Z. Wan. A Neural Network Approach for Solving Nonlinear Bilevel Programming Problem. Computers and Mathematics with Applications, 2008,55(6):2823-2829
20刘兵兵.一类非线性二层混合整数规划问题全局最优解的遗传算法.燕山大学学报, 2007,31(6):554-557
21 Migdalas, P. M. Pardalas. Editorial: Hierarchical and Bilevel Programming. Journal of Global Optimization, 1996,8(5):209-215
22 G. Zhang, C. Shi, J. Lu. An Extended Kth-best Approach for Referential-uncooperative Bilevel Multi-follower Decision Making. International Journal of Computational Intelligence Systems, 2008,1(3):205-214
23 C. Shi, H. Zhou, J. Lu and G. Zhang. The Kth-best Approach for Linear Bilevel Multifollower Programming with Partial Shared Variables Among Followers. Applied Mathematics and Computation, 2007,188(2):1686-1698
24 P. Hansen, B. Jaumard and G. Savard. New Branch-and-Bound Rules for Linear Bilevel Programming. SIAM Journal on Scientific and Statistic Computing, 1992,13(5): 1194-1217
25 D. J. White, G. Anandalingam. A Penalty Function Approach for Solving Bilevel Linear Programs. Journal of Global Optimization, 1993,3(5):393-419
26 Y. Lv, Z. Chen, Z. Wan. A Penalty Function Method Based on Bilevel Programming for Solving Inverse Optimal Value Problems. Applied Mathematics Letters, 2010,(23): 170-175
27雷万明,邓先礼.用罚函数求解二层线性规划的方法.汕头大学学报, 2000,15(2): 74-78
28 J. F. Bard. An Investigation of the Linear Three-Level Programming Problem. IEEE Transactions on Systems, Man. and Cybemetics, 1984,14(5):711-716
29吕一兵,姚天祥,陈忠.求解线性二层规划的一种全局优化方法.长江大学学报(自然科学版), 2008,5(4):7-10
30董银红,王广民,刘洪海.二层线性规划问题的全局收敛算法.武汉大学学报, 2006,6(12):105-107
31 G. Savarad, J. Gauvin. The Steepest Descent Direction for the Nonlinear Bilevel Programming Problem. Operations Research Letters, 1994,15(7):275-282
32 C. D. Kolstad, S. L. Lasdon. Derivative Evaluation and Computations Experience with Large Bilevel Mathematical Program. Journal of Optimization Theory and Applications, 1990,65(10):485-499
33 J. F.Bard. Convex Two-level Optimization. Mathematical Programming, 1988,40:15-27
34 K. Shimizu, E. Aiyoshi. A New Computational Method for Stackelberg and Min-Max Problem by Use of A Pealty Method. IEEE Transactions on Automatic Control, 1981,26(2):460-466
35 K. Shimizu, M. Lu. A Global Optimization Method for the Stackelberg Problem with Convex Functions Via Problem Transformation and Concave Programming. IEEE Transactions on Systems C, 1995,25(12):1635-1640
36徐凌,张圣贵.基于过滤信赖域算法的非线性二层规划求解方法.福建师范大学学报(自然科学版), 2010,26(2):5-9
37郑寒凝,张圣贵.一类非线性二层规划的平衡点算法.福建师范大学学报(自然科学版), 2010,26(2):10-15
38 K. M. Lan, U. P. Wen, H. S. Shih, E. Stanley Lee. A Hybrid Neural Network Approach to Bilevel Programming Problems. Applied Mathematics Letters, 2007,20(7):880-884
39吴睿.非线性二层规划的一种混合粒子群算法.衡水学院学报, 2010,12(1):14-16
40金俊培,盖钧益.大豆地方品种在豆腐产量品质及有关加工性状上的遗传变异[J].南京农业大学学报, 1995,18(1):5-9
41刘志胜,李里特,辰巳英三.大豆原料对豆腐得率和质构的影响.中国农业大学学报, 1999,(6):101-105
42滕春贤,李智慧.二层规划的理论与应用.第二版.北京:科学出版社, 2002:15-33
43王先甲,冯尚友.二层系统最优化理论.北京:科学出版社, 1995:25-60
44魏权龄,闫洪.广义最优化理论和模型.北京:科学出版社, 2003:153-196
45刘志勇,滕春贤,陈东彦.关于二层价格控制决策问题的探讨.统计与决策, 2007,(20):29-31
46周水生,刘三阳.价格控制问题的基本性质.应用数学与计算数学学报, 1998,12 (2):53-58
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