變分不等式均衡路網流量敏感性分析路網資訊獨立性之研究

Abstract

均衡路網流量的問題不滿足Tobin所提出的變分不等式敏感性分析定理的唯一性條件，因此無法直接應用在均衡路網流量的敏感性分析上。以線性規劃方法，利用路段流量求出一組路徑流量後，對該組路徑流量作敏感性分析，解決了唯一性的問題。但是，該研究所假設的非退化特性，使得正流量的路徑數必須小於或等於正流量路段數加上起迄點數的和。在非退化性的假設下，Tobin和Friesz證明了路網敏感性分析和選擇哪一組路徑組合或哪一組流量組合都無關。而在較大型的路網當中，路徑的數目遠大於路段的數目，因此在一般的路網狀態下，Tobin和Friesz所證明的路徑流量獨立性並不存在。 卓訓榮為解決非退化性的問題，以廣義反矩陣方法將可行解空間由路徑轉到路段，再以路段的可行解空間分析敏感性問題，也避免了唯一性的困擾。卓訓榮亦提出最短距離方法，在已知一組路徑正流量解及微擾程度並不大的假設下進行可行解集合的轉換。 本研究證明以廣義反矩陣方法和最短距離方法所求得之敏感性分析具有路徑流量及路段╱路徑投引矩陣選取的獨立性，所謂路徑流量的獨立性是指選取相同的路段╱路徑投引矩陣，路徑流量不相同；而路段╱路徑投引矩陣的獨立性是指選取不同的路段╱路徑投引矩陣，路徑流量亦不相同。最後，本研究亦提出應用例題說明線性規劃法、廣義反矩陣法以及最短距離法的獨立性。

In the solving process of the linear programming problem method provided by Tobin and Friesz, there is an assumption of the nondegeneracy. Under the assumption of the nondegeneracy, they have proved the independency of the path flow and the independency of the path pattern. The nondegeneracy in the linear programming problem method means that the number of the path with positive flow is less than or equal to the number of the arcs plus the number of origin-destination pairs in the network. Under this assumption, Yagar and Yang, and Yang and Lam have applied in the traffic control. But, in the real network, the independency of the path flow does not exist. The methods of generalized inverse matrix and the minimum distance have no nondegenerate assumption, and have been used in solving the network design problem by Cho and Lee, and Cho and Lo. In this paper, the independency of the path flow and the independency of the arc/path incidence matrix using the generalized inverse method and the minimum distance method are proved. The independency of the path flow and the arc/path incidence matrix means that the network sensitivity analysis can get the same result in choosing different path flow or different arc/path incidence matrix. The numerical example is presented in this paper.