普羅比式交通量指派問題求解之分析研究

Abstract

隨機性普羅比模式是一種不連續性的選擇模式，首先由數學心理學家Thurstone 在1927年提出，在當時由於計算上的困難而無法運用在實際的問題上，直到近二、三十年才漸漸被接受。本研究利用普羅比模式求解交通量指派問題，主要有兩個目的，一為利用普羅比模式的共變異矩陣描述路網上路經間相互重疊之關係，二為找到適當的方法求解這類的問題。 首先，描述普羅比模式與羅吉特模式兩者之差異，比較兩種模式在求解重疊性路網上的優缺點，說明選擇普羅比模式求解交通量指派問題之動機。然而介紹求解普羅比模式主要的三種方法：數值積分法、蒙地卡羅法、數值逼近法，比較個別方法之優缺點，最後選擇蒙地卡羅法與數值逼近法。 求解普羅比模式前，必須先建構問題本身的共變異矩陣，我們在第三章提出一個新方法來構建路網的共變異矩陣，適用於各種重疊性的路網，且保證構建之矩陣為正定矩陣，方便兩種方法求解各路徑被選擇之機率。在第四章描述蒙地卡羅法與數值逼近法詳細的求解步驟，一些比較細節的技術則於附錄詳加說明，第五章，我們提出了幾個實際的路網問題來做交通量指派，可以從這些實例中發現，並非兩種方法皆適用於求解大型的路網，及蒙地卡羅法在求解大型路網的一些特性。最後於第六章提出此研究的結論與建議。

The objectives of this thesis are twofold. The first is to describe a general form constructing covariance matrix for Multinomial Pobit (MNP) model on traffic assignment problems. The second is to find out one approach to perform the calculation well. This paper begins with describing two models, which are multinomial logit model and multinomial probit model. We analysis the advantages and disadvantages of the two models on solving networks with overlapping routes. Then select the multinomial probit model to solve traffic assignment problems. We also describe some approaches for solving MNP model of traffic assignment problems, including the numerical integration, the Monte Carlo simulation and the approximation method. Because the first approach, the numerical integration method, can not solve networks with more than three alternatives. We only use the other two approaches. Using the MNP model to solve traffic assignment problems, we need to construct the covariance matrix of the network firstly. This paper use a new idea to construct the covariance matrix illustrated in Chapter 3. The formulas are suitable for any kind of network with overlapping routes and guarantee the final covariance matrix be positive definite. By the character, one can use both Monte Carlo simulation and approximation method to calculate the probabilities of routes. In Chapter 4, the calculation procedure of the two methods are described obviously and some techniques like random number generation are presented in Appendix. We provide several examples in Chapter 5. In that chapter, one can find that not both methods are suitable for solving this kind of traffic assignment problems. Finally, we draw conclusions in Chapter 6.