Multi-Mode and Single Goods Transportation Network Equilibrium Model Based on Super Network

Abstract

We expatiate and summarize the partition method about transportation mode, and then we analyze the function concept of generalized transportation cost in details in this paper. After the equilibrium model of regional transportation network and the algorithm of the multi-mode and single goods have been given, the problems of path search in the network are carefully studied. Based on path oriented model, we list two kinds of effective path search methods to specific problems in this paper, namely, the directional tree algorithm of efficient path and the directional hierarchical spatial reasoning algorithm of efficient path. Finally, we get the model and algorithm of the multi-mode transportation network equilibrium based on super network. Introduction With the development of the super network theory, people gradually realize that there are a large number of significant interactions between the basic networks which form the super network. For example, regional transportation network, information network and financial network are closely related with each other. The history of the goods information and cash flow situation stored in information network provide decision-making information for deciding the subsequent transport and cash flow. The transportation of goods inevitably occurs with the flow of funds and the generation of new information. And the flow of funds tends to affect the goods transportation and the generation of information. Due to the complexity of network, the effect of the network policy has extensive influences, both in society and in economics. Just as the social justice problem and the fair use of the money problem brought about by the congestion charging policy. To know the above features of the super network is useful for us to define the super network, and avoid possible bias in research. We focus on the issue of network equilibrium while there are different transportation modes co-existed in regional transportation network, and make use of the super network and analyze the in-equation, finally get the corresponding network equilibrium model and algorithm in this paper. The core work of this paper is to convert multi-mode network model into the network model with employing single charge standard by setting reasonable generalized cost function. With transformation, the original complicated problems are simplified. The problems can be solved by using the commonly used algorithm of traffic flow distribution. Another focus of this paper is to add spatial price equilibrium principle among regional transportation network flows, and make network equilibrium reflect the intrinsic equilibrium driving force. Model and Algorithm of Traffic Flow Distribution under Multi-Mode Transportation The Setting of Generalized Sections’ Cost Function Under the Multi-Mode Condition. Before analyzing generalized sections’ cost function, while it exists multi-mode there. Firstly, the basic network parameters and constraints should be presented. Suppose that the network that under inspection is ) , ( L N G , in this network, N is a collection of nodes in network; L is the collection of sections in network. In the network, nodes are divided into three categories: the starting point of transport R r  , the end of the transport S s , and ordinary nodes ) ( S R N i    . The common 3rd International Conference on Machinery, Materials and Information Technology Applications (ICMMITA 2015) © 2015. The authors Published by Atlantis Press 123 sections in the network are represented by L b a  , , and the general paths in the network are represented by P q p  , . The above capital letters represent relevant elements collection. Suppose the flow on a section is a f , the flow on the path p is p x . Suppose That the number of sections and the number of paths in the network are respectively n and np , and the total flow on the sections and paths is represented by the following vectors } , , { 1 n a a f f f   and } , , { 1 np p p x x x   respectively. Suppose that any O-D’s flow on the rs section (r is the starting point, s is the end) is rs d , and all paths on this section are represented by rs P . Among them, the equation (1) is the flow rs d , which represents any O-D’s flow on the rs section, and it equals the sum of all the flows on the alternative paths connecting O and D. In formula (2), ap  is the correlation coefficient between section a and path p. If section a belongs to path p, we suppose ap  ’s value is 1; Otherwise, 0. Formula (2) means that the flow on any section is equal to the sum of flow that passed through this section. Formula (3) represents logic constraints of path flow, to ensure that the flow rate value is not negative. Based on the relationship between formula (2) and (3), it is easy to infer that the flow on any section of the path is nonnegative. Next we will analyze the generalized transportation cost function related to specific section. Suppose that there are H kinds of cost indicators on any path to be used as a path choice of goods, and one of the possible path is h. Suppose ha c is the cost measurement of path h, and is related with section a, then we can get the following equation, ) ( f c c ha ha  L a  (4) Here, we suppose the function ) ( f cha is a continuous function of flow. On the basis of formula (4), we can define the generalized transport cost for the road as   h ha a f c c ) ( L a  (5) And generalized transport cost of section ) ( f ca is also a continuous function. At the same time, the generalized transport cost for the path is ap a h ha p c c    P p  (6) Based on the formula (4) and the relationship between the flow on the sections and the sum flow of paths on this section, the generalized transportation cost can be expressed as the function of section flow or paths flow of this section, as shown in the following equation, ) ( ) ( x c f c c p p p   P p  (7) There are many choices for the cost items of specific sections of the generalized transportation. We can choose according to the goods transported. Besides, the transportation time, transportation cost, the reliability of transportation and the opportunity cost of transportation, etc., can be the alternative indicators of cost. It is important to note that in formula (4), the assumed function has already considered the weights of different indicators. Therefore, there is no need to add the weight of each index in formula (5). Before the model of multi-mode regional transportation network equilibrium is given, the path selection principle in the freight should be analyzed first. In practical application, both of the two principles are studied and put into practical use. For example, the famous software STAN freight distribution adapts the principle of the system optimal. System optimal principles play an important role in transportation policy analysis, while the user optimal principle has better explanation for decentralized decision in actual transportation market area. In STAN software, using the optimal system, for the author, is to achieve better convergence of model and the easier implementation of algorithm. Here, we will adapt a more realistic network transportation principle, namely, user optimal allocation principle. User optimal allocation principle is equivalent to the utility maximization principle in economics, at the same time, it can be explained by Game Theory. This principle can be concretely embodied in