本文選題：多目標(biāo)二層規(guī)劃 + 罰函數(shù)　； 參考：《長江大學(xué)》2016年碩士論文

【摘要】：二層規(guī)劃問題為一類具有兩層階梯結(jié)構(gòu)的系統(tǒng)決策問題,在該問題的數(shù)學(xué)模型中,包含了具有不同目標(biāo)函數(shù)與約束條件的上,下兩層優(yōu)化問題.他們既彼此獨(dú)立,又相互影響.具體表現(xiàn)為：上層問題的約束條件與下層問題的最優(yōu)解密切相關(guān),下層問題的最優(yōu)解又受上層給定的決策變量所影響.因?yàn)槎䦟右?guī)劃問題為一類NP-hard問題,所以其在基礎(chǔ)理論和求解算法上的發(fā)展都較為緩慢,但這并沒有影響它在各種實(shí)際問題中的應(yīng)用.目前為止,二層規(guī)劃問題已經(jīng)被廣泛應(yīng)用于各種生活領(lǐng)域.如市場競爭,環(huán)境保護(hù),交通網(wǎng)設(shè)計(jì),資源分配,物流管理,價(jià)格控制等.本文在已有的研究基礎(chǔ)上,首先對二層規(guī)劃問題的理論及算法的發(fā)展作了簡短的綜述,然后介紹了與本文研究內(nèi)容相關(guān)的基礎(chǔ)知識,最后針對兩類多目標(biāo)二層規(guī)劃問題,設(shè)計(jì)了相應(yīng)的數(shù)值求解算法,并通過相關(guān)數(shù)值實(shí)驗(yàn)檢驗(yàn)了算法的可行性.論文主要安排如下.第一章簡要的介紹了二層規(guī)劃問題的數(shù)學(xué)模型,并從理論算法和實(shí)際應(yīng)用兩個(gè)方面介紹了二層規(guī)劃問題的研究背景及發(fā)展現(xiàn)狀.在理論算法上介紹了求解二層規(guī)劃問題的幾種常用方法.包括罰函數(shù)法,極點(diǎn)搜索法,智能求解算法,分支定界法等.并對上述方法的求解思路及優(yōu)缺點(diǎn)作了簡單的概括與總結(jié).在實(shí)際應(yīng)用方面介紹了二層規(guī)劃問題在交通和管理中的應(yīng)用.最后介紹了本文各章節(jié)的具體安排.第二章給出了與本文相關(guān)的一系列預(yù)備知識,具體內(nèi)容包括：相關(guān)的數(shù)學(xué)概念,如閉集,凸集,連續(xù)函數(shù),可微函數(shù),局部極小(大)值點(diǎn)等；線性及非線性二層規(guī)劃的數(shù)學(xué)模型及其解的性質(zhì)；多目標(biāo)優(yōu)化問題的數(shù)學(xué)模型,最優(yōu)性條件及主要目標(biāo)求解方法；模糊集概念及確定隸屬函數(shù)的方法.為第三,四章求解多目標(biāo)二層規(guī)劃問題提供理論依據(jù).第三章針對上層是多目標(biāo)下層是單目標(biāo)的一類非線性多目標(biāo)二層規(guī)劃問題,設(shè)計(jì)了主要目標(biāo)求解法.第一節(jié)給出了此類二層規(guī)劃問題的數(shù)學(xué)模型及pareto-最優(yōu)解的概念,并對該模型中的相關(guān)變量作了簡要說明.第二節(jié)在假設(shè)下層問題為凸規(guī)劃問題的基礎(chǔ)上,利用下層問題的K-T最優(yōu)性條件,將原多目標(biāo)二層規(guī)劃問題轉(zhuǎn)化為帶互補(bǔ)約束的多目標(biāo)優(yōu)化問題.將多目標(biāo)優(yōu)化問題的互補(bǔ)約束條件作為罰項(xiàng),構(gòu)造該多目標(biāo)規(guī)劃問題的罰問題.通過證明該罰問題的收斂性可知該罰問題的pareto-最優(yōu)解一定是原問題的pareto-最優(yōu)解.隨后設(shè)計(jì)了求解該罰問題的主要目標(biāo)法,并給出了詳細(xì)的求解步驟.第三節(jié)通過求解相關(guān)算例,可證明本文所設(shè)計(jì)的主要目標(biāo)求解法是有效且可行的.第四節(jié)總結(jié)了該算法的優(yōu)點(diǎn)與不足.第四章針對上層是單目標(biāo),下層是多目標(biāo)的一類線性多目標(biāo)二層規(guī)劃問題,即半向量二層規(guī)劃問題.設(shè)計(jì)了模糊決策求解法.求解思路為：首先,利用線性加權(quán)法將下層多目標(biāo)規(guī)劃轉(zhuǎn)化為單目標(biāo)優(yōu)化問題,可則將半向量二層規(guī)劃問題轉(zhuǎn)化為單目標(biāo)二層規(guī)劃問題.其次,利用模糊集理論,構(gòu)造對應(yīng)的隸屬函數(shù)用于描述上,下兩層目標(biāo)函數(shù)的滿意度.然后,構(gòu)造新的模糊目標(biāo)評價(jià)函數(shù),并在此基礎(chǔ)上給出了該模糊決策求解法的具體求解步驟.相關(guān)數(shù)值實(shí)驗(yàn)可證明：本章所設(shè)計(jì)的模糊決策法是可行的.第五章分析總結(jié)了本文所設(shè)計(jì)的兩種算法的優(yōu)缺點(diǎn).
[Abstract]:The two layer programming problem is a class of system decision-making problem with two layers of ladder structure. In the mathematical model of the problem, it contains the upper and lower two layers of optimization problems with different objective functions and constraints. They are independent and mutual influence. The concrete performance is that the constraints of the upper questions and the optimal solution of the lower level are closely related. The optimal solution of the lower level problem is influenced by the decision variables of the upper level. Because the two layer programming problem is a class of NP-hard problems, the development of the basic theory and the solution algorithm is slow, but it does not affect its application in various practical problems. So far, the two layer planning problem has been widely used. In various fields of life, such as market competition, environmental protection, traffic network design, resource allocation, logistics management, price control, etc. This paper, based on the existing research, makes a brief overview of the theory and algorithm development of the two layer planning problem, and then introduces the basic knowledge related to the content of this study, and finally to the two categories. The corresponding numerical solution algorithm is designed for the target two layer programming problem, and the feasibility of the algorithm is tested by the relevant numerical experiments. The thesis is mainly arranged as follows. The first chapter briefly introduces the mathematical model of the two layer planning problem, and introduces the research background and development of the two layer planning problem from two aspects of the theoretical algorithm and the practical application. In the theoretical algorithm, some common methods for solving two layer programming problems are introduced, including the penalty function method, the pole search method, the intelligent solution algorithm, the branch and bound method and so on. The solution ideas and advantages and disadvantages of the above methods are briefly summarized and summarized. The two layer planning problems are introduced in traffic and management in practical application. The second chapter gives a series of preparatory knowledge related to this article. The specific contents include: related mathematical concepts, such as closed sets, convex sets, continuous functions, differentiable functions, local minimum (large) value points, linear and nonlinear two layer programming mathematical models and their solutions. The mathematical model of the objective optimization problem, the optimality condition and the main objective solution method, the fuzzy set concept and the method of determining the membership function, provide the theoretical basis for the third, fourth chapter to solve the multi-objective two layer programming problem. The third chapter is designed for a class of nonlinear multi-objective two layer programming problems with a multi objective lower layer is a single objective. The first section gives the mathematical model of the two layer programming problems and the concept of pareto- optimal solution, and gives a brief description of the related variables in the model. The second section, on the basis of the assumption that the lower level is a convex programming problem, uses the K-T optimality condition of the lower level problem to turn the original multiobjective two layer programming problem into a problem. The multi objective optimization problem with complementary constraints is transformed into a penalty term of the complementary constraint condition of the multi-objective optimization problem, and the penalty problem of the multi-objective programming problem is constructed. By proving the convergence of the penalty problem, it is known that the pareto- optimal solution of the penalty problem must be the best solution of the original problem. Then the main solution to the penalty problem is designed. The third section can prove that the main objective solution method of this paper is effective and feasible. The fourth section summarizes the advantages and disadvantages of the algorithm. The fourth chapter is a class of linear multi-objective two layer programming problem with a single objective and the lower layer is multiobjective, that is, half The problem of fuzzy decision solving is designed. The solution method is designed as follows: first, the linear weighted method is used to transform the lower multi-objective programming into a single objective optimization problem. The semi vector two layer programming problem can be transformed into a single target two layer programming problem. Secondly, the corresponding membership function is constructed by using the fuzzy set theory. The satisfaction of the next two layers of objective function is given. Then, a new fuzzy objective evaluation function is constructed, and on this basis, the concrete solution steps of the fuzzy decision solving method are given. The relevant numerical experiments prove that the fuzzy decision method designed in this chapter is feasible. The fifth chapter analyses the advantages and disadvantages of the two algorithms designed in this paper.

【學(xué)位授予單位】：長江大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O221

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