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研究生： 呂庭維
Lu, Ting-Wei
論文名稱： Braess運輸問題的均衡解
On Equilibrium Solutions of the Braess Transportation Problem
指導教授： 許瑞麟
Sheu, Ruey-Lin
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 78
中文關鍵詞： 使用者均衡 、分配式網路 、路徑規劃 、Braess悖論
外文關鍵詞： user equilibrium, distribution network, route planning, the Braess paradox
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    • 在這篇論文裡，我們聚焦在Dietrich Braess所提出的四節點五路徑的路網模型。在依賴車流量的線性旅程函數以及總車流給定的前提之下，不論平衡達到與否，我們對所有可能的車流分配模式，做一個完整分類，以直接獲得正確的平衡模式。我們的研究建立在Marguerit Frank原先的工作之上。Frank給出Braess悖論存在性在代數上的充分和必要條件，但我們的分類更廣泛的覆蓋所有的情況，包括悖論不發生的時候，使得添加新路徑的效用能夠被判斷出來。

    In this thesis, we focus on the four-node and five-link network model proposed by Dietrich Braess. Given a flow-dependent linear traveling function on each link and the total flow, we can completely classify the flow distribution pattern regardless whether the equilibrium of the network does happen or not. Our study is based on the original work of Marguerit Frank in which algebraic necessary and sufficient conditions for the existence of the Braess Paradox were derived. Our classification extends to the cases when the paradox does not happen so that the effectiveness of the newly added link is able to be judged.

    CONTENTS 1. Introduction 7 2. The Braess Problem and the Equilibrium Solutions 14 2.1 Patterns of Equilibrium Condition in the Braess Problem 14 2.2 The Equilibrium Solution and its Constraints in Problem I 17 2.3 The Equilibrium Solution and its Constraints in Problem II 20 3. The Sequence of Equilibrium Patterns Determined by the Demand 33 3.1 Classify the Equilibrium Patterns by Critical Points 33 3.2 The Sequence of Equilibrium Patterns in Problem I 40 3.3 The Sequence of Equilibrium Patterns in Problem II for  > 0 42 3.4 The Sequence of Equilibrium Patterns in Problem II for  < 0 55 3.5 The Sequence of Equilibrium Patterns in Problem II for  = 0 67 4. Necessary and Sufficient Condition to The Braess Paradox 71 5. Discussion and Future Directions 77 Bibliography 78

    [1] D. Braess, A. Nagurney, and T. Wakolbinger, On a paradox of traffic planning, Transportation Science 39 (2005), no. 4, 446-450.
    [2] P. D. D. D. Braess,  Uber ein paradoxon aus der verkehrsplanung, Unternehmensforschung 12 (1968), no. 1, 258-268.
    [3] S. C. Dafermos, An extended traffic assignment model with applications to two-way traffic, Transportation Science 5 (1971), no. 4, 366-389.
    [4] C. F. Daganzo and Y. She , On stochastic models of traffic assignment, Transportation Science 11 (1977), no. 3, 253-274.
    [5] X. Di, X. He, X. Guo, and H. X. Liu, Braess paradox under the boundedly rational user equilibria, Transportation Research Part B: Methodological 67 (2014), 86-108.
    [6] M. Frank, The Braess paradox, Mathematical Programming 20 (1981), no. 1, 283-302.
    [7] H. Mahmassani and R. Herman, Dynamic user equilibrium departure time and route choice on idealized traffic arterials, Transportation Science 18 (1984), no. 4, 362-384.
    [8] A. Nagurney, The negation of the Braess paradox as demand increases: The wisdom of crowds in transportation networks, EPL (Europhysics Letters) 91 (2010), no. 4, 48002.
    [9] R. Steinberg and W. I. Zangwill, The prevalence of Braess' paradox, Transportation Science 17 (1983), no. 3, 301-318.

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