A Solution Algorithm for Long Haul Freight Network Design Using Shipper-Carrier Freight Flow Prediction with Explicit Capacity Constraints

Freight transportation has long been recognized as an important foundation of economic strength. Previous studies use traditional methods to examine a set of scenarios. However, due to the complexity of transportation projects which can have substitution effects in a network the number of resulting scenarios may be more than can be examined on a case by case basis. In this paper, a sequential shipper-carrier freight flow prediction model is examined. Additionally, an explicit capacity constraint is used to divert the traffic volume from congested links. A branch and bound method is applied to obtain a solution to our model. We discuss the benefits and limitations of our method, examine its computational efficiency and provide a numerical example. The results show that project selection by the traditional case by case analysis method cannot capture the complexity of freight transportation network improvements and yields the sub-optimal solution. INTRODUCTION Freight transportation has long been recognized as an important foundation of economic strength. Demands for long haul freight movements continue to grow due to increasing international trade. This increase drives the need for major infrastructure improvements at the local, state and federal level. Many states have undertaken recent freight planning studies ((NJDOT and PBQD, 2004), and (USDOT and FHWA, 2005)). Previous studies use traditional methods to examine a set of scenarios. However, due to the complexity of transportation projects which can have substitution effects in a network, the number of resulting scenarios may be more than can be examined on a case by case basis. A model which can deal with the combinatorial problem and consider traffic flow behaviors which change corresponding to the projects selected should be developed. Such a model is referred to as a network design model. In such an optimization problem, the existing network is provided along with a set of proposed improvement projects and the relevant budget limitations. An objective function is used to evaluate the efficiency of alternative networks. The output of the model is the set of projects that perform best under the budget constraints. In recent year, the network design models for real applications have been developed. For example, (Ben-Ayed, et al., 1992) applies network design to the Tunisian highway network and (Kuby, Xu and Xie, 2001) applies their model to the Chinese railway network. However, none of earlier research considers explicit freight behavior and a multimodal perspective. Our work seeks to develop a model and corresponding solution algorithm which carefully considers these two aspects of the long haul multi-modal freight network design problem. The primary focus of our paper is freight route choice behavior. A sequential shipper-carrier freight flow prediction model is used and a branch and bound method is applied to obtain a solution. Additionally, an explicit capacity constraint is used to divert the traffic volume from congested links which are unreliable. We discuss our technique, its benefits and limitations and examine its computational efficiency. LITERATURE REVIEW The network design problem can be modeled as an integer programming optimization problem considering dealing the network improvement decisions. (Magnanti and Wong, 1984) shows that many combinatorial transportation planning problems are special forms of network design problems. Network design problems can have different settings. In this paper, we model a capacitated discrete budget allocation problem with non-linear routing costs. Given an existing network and budget constraints, the problem is to choose the set of improvement alternatives to optimize the network efficiency criteria. In most previous studies, the highway network design problem has been the focus and route choice behavior is limited to the passenger movements. In order to formulate the long haul freight network design problem, freight route choice behaviors have to be considered and the existing solution algorithms need to be adjusted accordingly. We begin our literature review with papers related to freight route choice behaviors. Then the solution algorithms for discrete budget design problems will be reviewed. The freight flow prediction models are used to reflect the freight route choices on the transportation network. One of differences between freight and passenger movements is that the freight route choices are cooperative decisions made by multiple agents. The two primary agents in the decisions are the shipper and the carrier. The shipper is a transportation customer who wants to move commodities from one point to another. The carrier provides transportation services for these demands. There are three general approaches used to predict freight flows (Harker, 1987). The first one is the econometric model which uses time series and/or cross sectional data to estimate structural relationship between supply and demand for transportation services. That model is very useful for studying the impact of various policies on the transportation market but it cannot detail the flows on transportation links. The spatial price equilibrium model focuses instead on producers, consumers, and shippers. That model predicts freight flow demand by equilibrating demand and supply in different regions through a simplified network representation. Each node represents a region and direct links represent costs to travel among all regions. The last model is the freight network equilibrium model. That type of model is the focus of our work since it uses the real network to predict freight flows and thus gives specific information for each link. This information is used to select the links that should receive improvements. The freight network equilibrium is similar to passenger traffic assignment except that many agents are considered. Two distinctive earliest models referred by (Harker, 1987) are (Roberts, 1966) focuses on the shipper with constant unit costs and (Peterson and Fullerton, 1975) focuses on the carrier and nonlinear unit costs with a user equilibrium assumption. The first model that considers multiple agents is (Friesz, 1981). The paper is explained later in (Friesz, Gottfried and Morlok, 1986). The model considers both shippers and carriers explicitly by sequentially loading travel demand onto the transportation service network and then loading this service demand onto the physical transportation network. In order to improve the interaction between shippers and carriers, (Friesz, Viton and Tobin, 1985) improves the model and loads both networks simultaneously. (Harker and Friesz, 1986a) and (Harker and Friesz, 1986b) introduce the consumer and the producer onto the freight network equilibrium by combining it with a spatial price equilibrium model. In his model, the freight flow is impacted by both travel costs and the commodity prices in each region. In return, the commodity prices are varied by demand and supply for the commodities. Recently, (Fernandez, Cea and Soto, 2003) develops a new modeling approach to a simultaneous shipper-carrier model with more advance trip distribution and mode choice formulations. At least two freight flow prediction models have been developed and tested using real data. (Friesz, Gottfried and Morlok, 1986) uses the shipper-carrier freight flow prediction model for the US rail network. (Guelat, Florian and Crainic, 1990) assumes that shippers’ behaviors is already included in OD estimation. Therefore, their model is similar to the carrier model of (Friesz, Gottfried and Morlok, 1986). Their transportation network is modified with virtual links and nodes to provide intermodal transfers. The Brazil transportation network is presented in that paper. Network design problems can have many settings such as introduced in (Magnanti and Wong, 1984), (Friesz, Viton and Tobin, 1985) ,and (Yang and Wang, 1998). The problems can have fixed or varied costs, budget constraints, or capacity constraints. If the users’ route choice behaviors cannot be controlled, bi-level optimization is usually used to simulate the different objectives of the users and the transportation agencies. In the bi-level setting, the upper level model represents the decision makers who choose the projects based on the social benefit while the lower level model represents network users who travel under route choice behavior assumptions. General network design problems are known to be NPcomplete ((Johnson, Lenstra and Rinnooy Kan, 1978)) which means there is no known algorithm to solve problems efficiently to optimality. In this paper we examine the network design problem with discrete decision variables. The models are usually bi-level since the route choices are decided by network users. The solution algorithms for such discrete optimization problems typically use implicit enumeration and incremental solution improvements. The optimization techniques for this problem type can be Bender’s decomposition ((Hoang, 1982)), and branch and bound algorithms ((Boyce, Farhi and Weischedel, 1973) and (Leblanc and Boyce, 1986)). Recently meta-heuristics have also been used ((Friesz, et al., 1992), and (Friesz, et al., 1993)). In our research, we use a branch and bound algorithm. Branch and bound algorithms construct a binary search tree to enumerate all possible solutions but accelerate the search process by pruning the search tree and eliminating nodes that cannot contain the optimal solution. At each node, some decisions are fixed and others are explored. A lower bound will be obtained for each node. If it is worse than the current best solution, the node will be excluded from future consideration. The algorithm stops when all nodes are excluded (also known as fathomed) or explored. The current best solution at the end is the optimal solution. The branch and bound algorithm can be viewed as the upper level model while the lower level model is used to calculate the lower bound. For network design problems with fixed link costs (Boyce, Farhi and Weischedel, 1973) proposes that a lower bound for a node is the upper level objective value when all undecided projects are set to be implemented. Network users are assumed to use the shortest paths. Tighter lower bounds are proposed by (Hoang, 1973). (Dionne and Florian, 1979) proposes several improvements such as a specialized algorithm to calculate shortest paths when a single arc has been deleted from the network. The network design problem is more difficult for the congested network with nonlinear link cost functions. Several earlier researchers study highway network design for passenger movements in which the cost functions are usually assumed to be strictly increasing convex functions. For highway network design, the typical network design objective is to minimize the total delay while the road users optimize their own benefits causing the traffic conditions to converging to user equilibrium condition. (Leblanc, 1975) uses a branch-and-bound algorithm to solve small problems optimally. The lower bound is calculated similar to (Boyce, Farhi and Weischedel, 1973). The pitfall of the algorithm is the existence of Braess’ Paradox which implies that the selected network improvements do not always result in an improvement in the objective function. In order to avoid the pitfall, the traffic volumes are assigned using system optimal routing instead of user equilibrium routing. This makes the lower bound looser. In order to deal with larger networks, several heuristics are developed. (Poorzahedy and Turnquist, 1982) modifies the network design problem by replacing the objective function with Beckman’s Formulation. The replacement gets rid of the Braess’ Paradox problem. The paper provides arguments as to why the modified problem can be used as an approximation for the original problem. (Haghani and Daskin, 1983) proposes an extraction algorithm for network design. In their algorithm, the links which have less traffic volume than a specific point will be excluded from consideration and the travel demand table is updated accordingly. Fewer links result in a faster traffic assignment algorithm. However, they report that the time needed to update the travel demand table may offset this benefit. An alternative for larger networks is to use a decomposition method which moves from larger networks to smaller sub-networks. (Solanki, Gorti and Southworth, 1998) clusters the sub-networks in a hierarchical order and performs network design for each cluster separately. Although, that paper uses a fixed cost network, it can be adjusted to be applied to networks with nonlinear link costs. METHODOLOGY Our network design model is a bi-level model. An upper level model decides how to improve a given network by selecting a set of improvements. The lower model represents the transportation network users’ behavior and determines traffic flows on the network. For the upper level model, implicit enumeration is performed by the branch and bound algorithm. In the branch and bound algorithm, the lower level model works as a sub-model which predicts traffic flows at each node of the branch and bound search tree. A shipper-carrier freight flow prediction model is used for the lower problem. The following variables are generally used in this paper: Subscripts l for the shipper networka link in a transportation service network for the carrier network a link in a physical transportation network