An integer programming model and directed Steiner-forest based heuristic for routing less-than-truckload freight

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Abstract

Less-than-truckload (LTL) freight transportation is a vital part of Canada’s economy, with revenues running into billions of dollars and a cascading impact on many other industries. LTL operators have to deal with large volumes of shipments and uncertainty in demand patterns. In an industry that already has low profit margins, it is therefore vitally important to make good quality routing decisions without expending a lot of time. The optimization of such LTL freight networks results in large scale mathematical programming problems. In this paper, we present a novel Integer Linear Programming (ILP) formulation and heuristics for routing LTL freight. Experiments in collaboration with our industry partner indicate that our proposals can significantly lower routing costs for Canadian LTL operators. In addition, our approach can be useful for generating load plans on larger American LTL networks.

Introduction

Less Than Truckload (LTL) freight transportation is a multi-billion dollar industry, with its operations having a significant impact on other industries (Hernández et al., 2011, Özkaya et al., 2010). The principal task of LTL operators is to collect freight from customers for delivery to the intended recipients within a specified timeline (i.e., within a service contract, or at a service level). Normally, freight is picked up from the customers using small trucks and delivered to a regional End-of-line (EOL) terminal. Freight from EOL terminals is carried by line-haul trucks to the nearest break-bulk terminal (Katayama and Yurimoto, 2016, Powell and Sheffi, 1989). At each break-bulk, freight is consolidated (i.e., sorted and reloaded) into trailers for transportation to an adjacent break-bulk (i.e., to a break-bulk to which direct service is available). Freight may travel through several intermediate break-bulks before eventually arriving at its destination EOL terminal, from where it is dispatched in small trucks to the respective consignees. This last distribution operation is often referred to as last-mile delivery. In countries like Canada and Spain, break-bulks also serve as EOL terminals (Barcos et al., 2010).

Typically, the volume of freight traveling from a break-bulk to a destination EOL terminal is less than the volume capacity of a standard trailer (hence the name, “Less-than-truckload”). Consolidation of freight at the break-bulks is required in order to properly utilize trailer capacity, since dedicating an entire trailer for each individual shipment is not cost effective. Note that trailers fully packed with freight (all of it bound for the same destination) are not considered part of the LTL problem because such trailers do not participate in freight consolidation. LTL networks are carefully designed to allow for effective consolidation of freight at each break-bulk. Our goal in this paper is to minimize the cost of routing LTL freight given an operational network and a service contract.

Each indivisible unit of shipment at a break-bulk is called a skid. A skid is a wooden container marked with its weight and volume, a unique “pro-bill” number, a destination, and a delivery deadline. Skids can differ in weight and volume but are much smaller in size than the standard trailer capacity, and need to be packed into trailers rented from third-party carriers. LTL operators often have long-term contracts with these third-party carriers to guarantee the availability of trailers on popular routes at reasonable prices. In addition to trailer rental costs, there are some handling costs associated with packing skids into trailers at each break-bulk.

To summarize, the essence of the LTL routing problem is to pack skids at each break-bulk into trailers to minimize transportation and handling costs, while ensuring that the route chosen for each skid meets its delivery deadline. In addition, it is desirable that the average trailer capacity utilization exceeds a certain threshold. More detailed descriptions of the design and operation of LTL networks can be found in Crainic et al. (1998), Erera et al. (2013b), and Powell and Sheffi (1983).

In this paper, we present a novel ILP formulation for the LTL freight routing problem. The key difference between our approach and most existing approaches is that we do not restrict skids having the same origin and destination to travel along the same route. Such restrictions are commonly used for routing freight on large LTL networks, and are implemented using load plans as described later in Section 2. Our ILP formulation can easily be adapted to handle larger LTL routing problems, and we use it as a foundation for developing a hybrid heuristic which is based on the notion of a minimum cost directed Steiner-forest (Feldman et al., 2012). To the best of our knowledge, we are the first to extend the notion of a directed Steiner-forest to a time–space network (Kennington and Nicholson, 2010, Erera et al., 2013a) and use it for routing LTL freight. The rest of the paper is organized as follows. We start with a brief survey of existing literature in Section 2. Our ILP formulation and hybrid heuristic appear in Section 3. Experimental results are presented in Section 4. We conclude and propose future research directions in Section 5.

Section snippets

Literature review

Crainic et al. (1998) present an excellent survey of all aspects of the LTL freight scheduling problem, starting from network design and strategic issues to operational decisions such as routing (which is our focus). The survey describes some typical problems facing LTL routing such as the classic “back-haul problem”, the unpredictability of future demand, and the need for making routing decisions in real time. Of these, the back-haul problem can be understood as the accumulation of trailers at

Model and heuristic algorithms

Before presenting our ILP model, we note that every node in our industry partner’s LTL network plays the dual role of an EOL terminal and a break-bulk. We only consider freight movement between these nodes, i.e., every skid travels between a pair of these nodes. In addition, our implementation completes the following preliminary tasks that are required for building the ILP model:

  • All possible routes between any two break-bulks are enumerated in advance. For a pair of origin–destination

Experimental evaluation

In order to evaluate the performance of our algorithms, we ran them on our industry partner’s Canadian LTL network with real data from their operations in March 2016. Their network has 17 nodes and 67 arcs. We routed the skids on several days using all three algorithms (namely the Greedy heuristic, the directed Steiner-forest heuristic, and CPLEX). In addition, we also changed some parameters (see Appendix A.1) and studied the effect of those changes on solution quality.

All our tests were

Conclusions and future work

In this paper, we proposed an Integer Linear Programming (ILP) model for routing LTL freight. We tested our model on our industry partner’s Canadian LTL network and found that it can significantly lower their routing costs. Our ILP model can also be solved with a few carefully selected skids to generate a load plan and a lower bound on the routing cost for each day. The lower bounds we found were always more than 50% of CPLEX’s solution cost on our industry partner’s LTL network.

Our model can

Acknowledgments

We would like to thank the Natural Sciences and Engineering Research Council (NSERC), the Discovery and Engage grant programs, and the Canada Foundation for Innovation (CFI) .

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