多用户多准则随机交通均衡理论与应用研究
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摘要
进入本世纪以来，我国交通需求得到了前所未有的迅猛增长，原有的道路设施和传统的交通管理手段已不能适应新的需要，交通堵塞成为许多城市的顽疾和发展瓶颈。要有效解决和根本治理这种情况，必须科学管理和有效利用交通基础设施，合理调节交通流量的时空分布，实现交通需求与供给的协调发展。这要求管理实践者和理论工作者首先掌握出行者的出行规律和交通流量分配理论。
另一方面，随着全球化竞争的加剧和科技发展的日新月异，供应链管理和网络经济已引起各国政府、企业界和学术界的极大关注。供应链管理是企业降低成本、提高效率、提升竞争力的有效途径。当前，供应链管理者迫切希望有一套理论来刻画和分析供应链网络的内部结构和竞争行为。网络外部性是网络经济的一个重要特征，也是影响网络产品差异化竞争的重要因素。网络产品厂商必须充分了解和有效利用网络外部性的特征及影响，来构造有利的经营战略和赢利模式。交通均衡配流模型和变分不等式方法是研究供应链网络均衡和网络产品市场均衡的有效工具。
本文基于上述背景展开研究。本文首先在非对称路段出行成本及准则权重与出行者类别相关的假设下，运用随机效用理论和变分不等式方法，研究了多用户多准则随机均衡原理和均衡配流问题。分别证明了具有固定需求和弹性需求的多用户多准则随机均衡配流模型等价于一个变分不等式问题，首次阐述了该变分不等式解的存在性与唯一性条件，及满足上述条件的一类出行成本和准则权重，给出了求解算法和计算实例。
本文其次针对固定需求和出行者的时间价值为离散分布的双准则随机交通均衡，分别研究了依费用度量和依时间度量的多用户双准则随机系统最优和最优收费问题。分别建立了基于费用和基于时间的随机系统最优的最优化模型，阐述了该模型解的唯一性条件及等价的变分不等式问题，研究了一阶最优收费的可行性，即能否依边际定价原则，通过收取与出行者类别无关的道路收费使多用户多准则随机均衡流与随机系统最优流一致。由于一阶最优收费不适用于依时间度量的随机系统最优情况，本文进而给出了一个最优化模型来得到此时的非歧视性道路收费。最后分别给出了算例。
本文接着研究了由多个生产商、多个零售商和若干需求市场组成的多商品流供应链网络均衡问题。针对不同生产商生产的同种产品为除产地、品牌差异外的完全同质产品，运用Nash均衡方法分析生产商之间、零售商之间的竞争行为，将产地、品牌差异的影响视为随机变量，运用基于随机效用理论和多项式logit模型的弹性需求随机均衡配流的变分不等式模型来刻画需求市场均衡，分别得到了供应链网络各层均衡及整体均衡的条件、经济解释和变分不等式模型。针对不同生产商生产的同种产品为非同质产品，且消费者对质量、品牌等产品属性的偏好为离散分布，将消费者的产品选择视为出行者的随机路径选择，运用多用户多准则弹性需求随机均衡配流的变分不等式模型来刻画需求市场均衡，建立了多用户多准则产品随机选择下多商品流供应链网络均衡的变分不等式模型。最后针对上述两种情况，分别给出了对应的求解方法和算例。
本文随后假定消费者对网络外部性的偏好为连续分布，并综合购买成本和网络外部性大小选择网络产品，研究了网络产品市场均衡和两厂商竞争均衡问题。由于网络外部性的存在，消费者购买网络产品所获得的效用是相互依赖的，网络产品市场均衡是消费者选择的不动点。本文将消费者的网络产品选择视为出行者的确定性路径选择，结合交通流量分配技术和无穷维变分方法，首次建立了刻画网络产品市场均衡的无穷维变分不等式模型，并研究了均衡解的存在性和唯一性条件，给出了求解算法和算例。针对两家网络产品厂商首先同时沿长度为1的线性城市选址，然后进行价格竞争的两阶段完全信息动态博弈问题，本文将网络外部性引入线性运输成本下的Hotelling模型，运用无穷维双准则交通均衡配流的变分不等式模型来刻画两家厂商的市场划分，给出了市场划分的存在性和唯一性条件，得到了厂商竞争均衡的存在性条件及两阶段均衡决策，推广了一般Hotelling模型的结论。在厂商定位在两端情况下，分析了网络外部性特征及单位运输成本对两厂商竞争均衡的影响。
上述前三部分的主要分析工具为有限维变分不等式方法，第四部分以无穷维变分不等式方法为主要分析工具，通过研究相应映射的单调性证明了解的存在性与唯一性。变分不等式应用领域非常广泛。近年来广义变分不等式研究引起了专家学者的重视，并取得了一定的研究成果，但现有研究还不够深入。因此本文最后研究了一类广义变分不等式问题，即η-变分不等式问题。建立了广义invex单调映射和广义invex cocoercive映射的定义，分析了这些单调映射之间的关系和微分特性，然后将其用于刻画η-变分不等式的解集特征。
本文综合了交通运筹学、管理科学、经济学等交叉学科的理论与方法，将随机效用理论、消费者选择理论、厂商竞争理论与变分不等式方法相结合，以交通均衡流量分配技术和变分不等式方法为主线，研究了多用户多准则随机均衡、最优道路收费、供应链网络均衡模型、网络产品市场均衡及η-变分不等式问题。研究结果具有一定理论价值和实践意义。
The beginning of 21th century sees the unprecedented fast growth of traffic demand in China. The original road facilities and the traditional transportation management measures can't meet the need. Traffic jam has been a stubborn problem and development bottleneck of many cities. To solve this problem completely, we must manage and utilize the transportation infrastructure scientifically and effectively, regulate the time and space distribution of traffic flow reasonably, and realize the coordination development of transportation demand and supply. It requires the managers and the scholars to understand the principle of passenger behaviors and the theory of traffic assignment very well.
On the other hand, the increasingly fierce global competition and rapid technological progress have motivated the governments, the enterprises and the academies to pay great attention to the supply chain management and the network economy. Supply chain management is an effective measure for the enterprises to reduce cost, improve efficiency and promote competitive power. At present, supply chain managers urgently need a set of theories to depict and analyze the internal structure and the competitive behaviors of the supply chain network. Network externalities are both the essential characteristic of the network economy and the key factors affecting differentiation of network products. Network product manufacturers must fully understand and effectively utilize the character and influence of network externalities to construct beneficial business strategy and profitable mode. Traffic assignment model and variational inequality (VI) method are powerful tools to study the supply chain network equilibrium and the market equilibrium of network products.
This dissertation is to launch the research under the above-mentioned background. Firstly, based on random utility theory and VI method, the dissertation studies the mul-ticlass, multicriteria stochastic user equilibrium (MMSUE) condition and MMSUE assignment problem, under the assumption that the cost functions are unsymmetrical and the weights of criteria are class-dependent. It has been proved that MMSUE assignment problem with fixed demand or elastic demand is equivalent to a VI problem. The qualitative properties of the VI problems, such as existence and uniqueness of solution, are presented, together with one type of cost functions and weights of criteria meeting the
above-mentioned conditions. Two corresponding solution algorithms and numerical examples are presented, respectively.
Secondly, considering bicriteria stochastic network equilibrium with fixed demand and discrete value of time, the dissertation studies the multiclass, bicriteria stochastic system optimum and optimal tolling problem, measured in time and cost respectively. Two respective optimization models of stochastic system optimum, together with their uniqueness conditions of solution and equivalent VI problems, are put forward. The feasibility of first-best tolling, i.e. whether it can drive a MMSUE flow pattern to a system optimum flow by adopting identical tolls to all user classes based on the principle of marginal-cost pricing, is studied. Since first-best tolling isn't suitable for time-based stochastic system optimum, the dissertation presents an optimization model to find the feasible toll pattern. Two examples are provided, respectively.
Thirdly, two three-level competitive supply chain network equilibrium models with multi-commodity are studied. Competitive behaviors of manufactures and retailers are analyzed by using Nash equilibrium theory. The stochastic user equilibrium VI model with elastic demand based on stochastic utility theory and multinomial logit model is utilized to study the demand market equilibrium by treating the influence of region and brand as stochastic variable. The equilibrium models of each level and whole network are developed by variational inequality method, along with their equilibrium conditions and economic interpretations, in consideration of homogeneous products except for product differentiation of region and brand. Under the assumption that multiclass consumers with discrete prefence for product quality, brand and so on, the MMSUE VI model with elastic demand is utilized to characterize the demand market equilibrium by treating product choice of consumers as stochastic path choice of passengers. A multiclass, multicriteria, multi-commodity flow supply chain network equilibrium model with stochastic choice is set up in consideration of heterogeneous products. Two corresponding solution algorithms and numerical examples are presented, respectively.
Fourthly, under the assumption that multiclass comsumers with continuous probability distribution of preference for network externalities choose network products according to both total purchase cost and network externalities, the dissertation studies the market equilibrium of network products and the competitive equilibrium between two manufactures. Since consumers' utilities are interdependent because of network ex-
ternalities, the market equilibrium of network products is one fixed point of customer choice. By treating network product choice of consumers as determinant path choice of passengers, the dissertation combines traffic assignment technicque and infinite dimensional variational inequality method to formulate the market equilibrium of network products, i.e. infinite dimensional VI formulation, whose qualitative properties such as existence and uniqueness of solution is also studied. A solution algorithm is put forward together with its application to solving a numerical example. Considering two-stage dynamic simultaneous game of two network product manufacturers, i.e. locating along a linear city with unit length firstly and pricing later, the dissertation utilizes infinite dimensional bicriteria traffic network equilibrium VI model to characterize the market share by Hotelling model with network externalities and linear transportation cost functions. The existence and uniqueness of market share are put forwarded. The two-stage equilibrium decisions of manufactures are presented together with the existence condition, which generalizes the results of general Hotelling model. Under the assumption that two manufactures locate the distinct ends of city, the dissertation analyzes how the character of network externalities and unit transportation cost influence the competitive equilibrium between two manufactures.
Finite dimensional VI method is a key analysis tool for the above-mentioned parts first to third, as well as infinite dimensional VI method for the fourth part. The dissertation proves the existence and uniqueness of solution by studying the monotonicity of corresponding mappings. VI method is widely used in many fields. In recent years, many specialists and scholars have investigated generalized VI problems and obtained some results. But further research is needed. The dissertation finally studies a type of generalized VI problem, i.e. η-variational inequality problem. New classes of generalized invex monotone mappings and invex cocoercive mappings are introduced. Their relationships and differential properties are studied. The solution set of η-variational inequality problem is charactered by using of the new generalized monotonicity.
The dissertation combines the theories and methods in some interrelated fields such as Traffic Operational Research, Management Science, Economics and so on. It integrates random utility theory, consumer choice theory, firm competition theory with variational inequality method. Based on the main thread of traffic assignment technology and variational inequality method, the dissertation studies MMSUE, optimal
tolling, supply chain network models, market equilibrium of network products and η-variational inequality problem. The results have both theoretical value and practical significance to a certian extent.
另一方面，随着全球化竞争的加剧和科技发展的日新月异，供应链管理和网络经济已引起各国政府、企业界和学术界的极大关注。供应链管理是企业降低成本、提高效率、提升竞争力的有效途径。当前，供应链管理者迫切希望有一套理论来刻画和分析供应链网络的内部结构和竞争行为。网络外部性是网络经济的一个重要特征，也是影响网络产品差异化竞争的重要因素。网络产品厂商必须充分了解和有效利用网络外部性的特征及影响，来构造有利的经营战略和赢利模式。交通均衡配流模型和变分不等式方法是研究供应链网络均衡和网络产品市场均衡的有效工具。
本文基于上述背景展开研究。本文首先在非对称路段出行成本及准则权重与出行者类别相关的假设下，运用随机效用理论和变分不等式方法，研究了多用户多准则随机均衡原理和均衡配流问题。分别证明了具有固定需求和弹性需求的多用户多准则随机均衡配流模型等价于一个变分不等式问题，首次阐述了该变分不等式解的存在性与唯一性条件，及满足上述条件的一类出行成本和准则权重，给出了求解算法和计算实例。
本文其次针对固定需求和出行者的时间价值为离散分布的双准则随机交通均衡，分别研究了依费用度量和依时间度量的多用户双准则随机系统最优和最优收费问题。分别建立了基于费用和基于时间的随机系统最优的最优化模型，阐述了该模型解的唯一性条件及等价的变分不等式问题，研究了一阶最优收费的可行性，即能否依边际定价原则，通过收取与出行者类别无关的道路收费使多用户多准则随机均衡流与随机系统最优流一致。由于一阶最优收费不适用于依时间度量的随机系统最优情况，本文进而给出了一个最优化模型来得到此时的非歧视性道路收费。最后分别给出了算例。
本文接着研究了由多个生产商、多个零售商和若干需求市场组成的多商品流供应链网络均衡问题。针对不同生产商生产的同种产品为除产地、品牌差异外的完全同质产品，运用Nash均衡方法分析生产商之间、零售商之间的竞争行为，将产地、品牌差异的影响视为随机变量，运用基于随机效用理论和多项式logit模型的弹性需求随机均衡配流的变分不等式模型来刻画需求市场均衡，分别得到了供应链网络各层均衡及整体均衡的条件、经济解释和变分不等式模型。针对不同生产商生产的同种产品为非同质产品，且消费者对质量、品牌等产品属性的偏好为离散分布，将消费者的产品选择视为出行者的随机路径选择，运用多用户多准则弹性需求随机均衡配流的变分不等式模型来刻画需求市场均衡，建立了多用户多准则产品随机选择下多商品流供应链网络均衡的变分不等式模型。最后针对上述两种情况，分别给出了对应的求解方法和算例。
本文随后假定消费者对网络外部性的偏好为连续分布，并综合购买成本和网络外部性大小选择网络产品，研究了网络产品市场均衡和两厂商竞争均衡问题。由于网络外部性的存在，消费者购买网络产品所获得的效用是相互依赖的，网络产品市场均衡是消费者选择的不动点。本文将消费者的网络产品选择视为出行者的确定性路径选择，结合交通流量分配技术和无穷维变分方法，首次建立了刻画网络产品市场均衡的无穷维变分不等式模型，并研究了均衡解的存在性和唯一性条件，给出了求解算法和算例。针对两家网络产品厂商首先同时沿长度为1的线性城市选址，然后进行价格竞争的两阶段完全信息动态博弈问题，本文将网络外部性引入线性运输成本下的Hotelling模型，运用无穷维双准则交通均衡配流的变分不等式模型来刻画两家厂商的市场划分，给出了市场划分的存在性和唯一性条件，得到了厂商竞争均衡的存在性条件及两阶段均衡决策，推广了一般Hotelling模型的结论。在厂商定位在两端情况下，分析了网络外部性特征及单位运输成本对两厂商竞争均衡的影响。
上述前三部分的主要分析工具为有限维变分不等式方法，第四部分以无穷维变分不等式方法为主要分析工具，通过研究相应映射的单调性证明了解的存在性与唯一性。变分不等式应用领域非常广泛。近年来广义变分不等式研究引起了专家学者的重视，并取得了一定的研究成果，但现有研究还不够深入。因此本文最后研究了一类广义变分不等式问题，即η-变分不等式问题。建立了广义invex单调映射和广义invex cocoercive映射的定义，分析了这些单调映射之间的关系和微分特性，然后将其用于刻画η-变分不等式的解集特征。
本文综合了交通运筹学、管理科学、经济学等交叉学科的理论与方法，将随机效用理论、消费者选择理论、厂商竞争理论与变分不等式方法相结合，以交通均衡流量分配技术和变分不等式方法为主线，研究了多用户多准则随机均衡、最优道路收费、供应链网络均衡模型、网络产品市场均衡及η-变分不等式问题。研究结果具有一定理论价值和实践意义。
The beginning of 21th century sees the unprecedented fast growth of traffic demand in China. The original road facilities and the traditional transportation management measures can't meet the need. Traffic jam has been a stubborn problem and development bottleneck of many cities. To solve this problem completely, we must manage and utilize the transportation infrastructure scientifically and effectively, regulate the time and space distribution of traffic flow reasonably, and realize the coordination development of transportation demand and supply. It requires the managers and the scholars to understand the principle of passenger behaviors and the theory of traffic assignment very well.
On the other hand, the increasingly fierce global competition and rapid technological progress have motivated the governments, the enterprises and the academies to pay great attention to the supply chain management and the network economy. Supply chain management is an effective measure for the enterprises to reduce cost, improve efficiency and promote competitive power. At present, supply chain managers urgently need a set of theories to depict and analyze the internal structure and the competitive behaviors of the supply chain network. Network externalities are both the essential characteristic of the network economy and the key factors affecting differentiation of network products. Network product manufacturers must fully understand and effectively utilize the character and influence of network externalities to construct beneficial business strategy and profitable mode. Traffic assignment model and variational inequality (VI) method are powerful tools to study the supply chain network equilibrium and the market equilibrium of network products.
This dissertation is to launch the research under the above-mentioned background. Firstly, based on random utility theory and VI method, the dissertation studies the mul-ticlass, multicriteria stochastic user equilibrium (MMSUE) condition and MMSUE assignment problem, under the assumption that the cost functions are unsymmetrical and the weights of criteria are class-dependent. It has been proved that MMSUE assignment problem with fixed demand or elastic demand is equivalent to a VI problem. The qualitative properties of the VI problems, such as existence and uniqueness of solution, are presented, together with one type of cost functions and weights of criteria meeting the
above-mentioned conditions. Two corresponding solution algorithms and numerical examples are presented, respectively.
Secondly, considering bicriteria stochastic network equilibrium with fixed demand and discrete value of time, the dissertation studies the multiclass, bicriteria stochastic system optimum and optimal tolling problem, measured in time and cost respectively. Two respective optimization models of stochastic system optimum, together with their uniqueness conditions of solution and equivalent VI problems, are put forward. The feasibility of first-best tolling, i.e. whether it can drive a MMSUE flow pattern to a system optimum flow by adopting identical tolls to all user classes based on the principle of marginal-cost pricing, is studied. Since first-best tolling isn't suitable for time-based stochastic system optimum, the dissertation presents an optimization model to find the feasible toll pattern. Two examples are provided, respectively.
Thirdly, two three-level competitive supply chain network equilibrium models with multi-commodity are studied. Competitive behaviors of manufactures and retailers are analyzed by using Nash equilibrium theory. The stochastic user equilibrium VI model with elastic demand based on stochastic utility theory and multinomial logit model is utilized to study the demand market equilibrium by treating the influence of region and brand as stochastic variable. The equilibrium models of each level and whole network are developed by variational inequality method, along with their equilibrium conditions and economic interpretations, in consideration of homogeneous products except for product differentiation of region and brand. Under the assumption that multiclass consumers with discrete prefence for product quality, brand and so on, the MMSUE VI model with elastic demand is utilized to characterize the demand market equilibrium by treating product choice of consumers as stochastic path choice of passengers. A multiclass, multicriteria, multi-commodity flow supply chain network equilibrium model with stochastic choice is set up in consideration of heterogeneous products. Two corresponding solution algorithms and numerical examples are presented, respectively.
Fourthly, under the assumption that multiclass comsumers with continuous probability distribution of preference for network externalities choose network products according to both total purchase cost and network externalities, the dissertation studies the market equilibrium of network products and the competitive equilibrium between two manufactures. Since consumers' utilities are interdependent because of network ex-
ternalities, the market equilibrium of network products is one fixed point of customer choice. By treating network product choice of consumers as determinant path choice of passengers, the dissertation combines traffic assignment technicque and infinite dimensional variational inequality method to formulate the market equilibrium of network products, i.e. infinite dimensional VI formulation, whose qualitative properties such as existence and uniqueness of solution is also studied. A solution algorithm is put forward together with its application to solving a numerical example. Considering two-stage dynamic simultaneous game of two network product manufacturers, i.e. locating along a linear city with unit length firstly and pricing later, the dissertation utilizes infinite dimensional bicriteria traffic network equilibrium VI model to characterize the market share by Hotelling model with network externalities and linear transportation cost functions. The existence and uniqueness of market share are put forwarded. The two-stage equilibrium decisions of manufactures are presented together with the existence condition, which generalizes the results of general Hotelling model. Under the assumption that two manufactures locate the distinct ends of city, the dissertation analyzes how the character of network externalities and unit transportation cost influence the competitive equilibrium between two manufactures.
Finite dimensional VI method is a key analysis tool for the above-mentioned parts first to third, as well as infinite dimensional VI method for the fourth part. The dissertation proves the existence and uniqueness of solution by studying the monotonicity of corresponding mappings. VI method is widely used in many fields. In recent years, many specialists and scholars have investigated generalized VI problems and obtained some results. But further research is needed. The dissertation finally studies a type of generalized VI problem, i.e. η-variational inequality problem. New classes of generalized invex monotone mappings and invex cocoercive mappings are introduced. Their relationships and differential properties are studied. The solution set of η-variational inequality problem is charactered by using of the new generalized monotonicity.
The dissertation combines the theories and methods in some interrelated fields such as Traffic Operational Research, Management Science, Economics and so on. It integrates random utility theory, consumer choice theory, firm competition theory with variational inequality method. Based on the main thread of traffic assignment technology and variational inequality method, the dissertation studies MMSUE, optimal
tolling, supply chain network models, market equilibrium of network products and η-variational inequality problem. The results have both theoretical value and practical significance to a certian extent.
引文
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