Marius M. Solomon

Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints

This paper considers the design and analysis of algorithms for vehicle routing and scheduling problems with time window constraints. Given the intrinsic difficulty of this problem class, approximation methods seem to offer the most promise for practical size problems. After describing a variety of heuristics, we conduct an extensive computational study of their performance. The problem set includes routing and scheduling environments that differ in terms of the type of data used to generate the problems, the percentage of customers with time windows, their tightness and positioning, and the scheduling horizon. We found that several heuristics performed well in different problem environments; in particular an insertion-type heuristic consistently gave very good results.

A new optimization algorithm for the vehicle routing problem with time windows

The vehicle routing problem with time windows VRPTW is a generalization of the vehicle routing problem where the service of a customer can begin within the time window defined by the earliest and the latest times when the customer will permit the start of service. In this paper, we present the development of a new optimization algorithm for its solution. The LP relaxation of the set partitioning formulation of the VRPTW is solved by column generation. Feasible columns are added as needed by solving a shortest path problem with time windows and capacity constraints using dynamic programming. The LP solution obtained generally provides an excellent lower bound that is used in a branch-and-bound algorithm to solve the integer set partitioning formulation. Our results indicate that this algorithm proved to be successful on a variety of practical sized benchmark VRPTW test problems. The algorithm was capable of optimally solving 100-customer problems. This problem size is six times larger than any reported to date by other published research.

Survey Paper—Time Window Constrained Routing and Scheduling Problems

We have witnessed recently the development of a fast growing body of research focused on vehicle routing and scheduling problem structures with time window constraints. It is the aim of this paper to survey the significant advances made for the following classes of routing problems with time windows: the single and multiple traveling salesman problem, the shortest path problem, the minimum spanning tree problem, the generic vehicle routing problem, the pickup and delivery problem including the dial-a-ride problem, the multiperiod vehicle routing problem and the shoreline problem. Having surveyed the state-of-the-art in this area, we then offer some perspectives on future research.

Vehicle Routing Problem with Time Windows

In this chapter we discuss the Vehicle Routing Problem with Time Windows in terms of its mathematical modeling, its structure and decomposition alternatives. We then present the master problem and the subproblem for the column generation approach, respectively. Next, we illustrate a branch-and-bound framework and address acceleration strategies used to increase the efficiency of branch-and-price methods. Then, we describe generalizations of the problem and report computational results for the classic Solomon test sets. Finally, we present our conclusions and discuss some open problems.

2-Path Cuts for the Vehicle Routing Problem with Time Windows

This paper introduces a strong valid inequality, the 2-path cut, to produce better lower bounds for the vehicle routing problem with time windows. It also develops an effective separation algorithm to find such inequalities. We next incorporate them as needed in the master problem of a Dantzig-Wolfe decomposition approach. In this enhanced optimization algorithm, the coupling constraints require that each customer be serviced. The subproblem is a shortest path problem with time window and capacity constraints. We apply branch and bound to obtain integer solutions. We first branch on the number of vehicles if this is fractional, and then on the flow variables. The algorithm has been implemented and tested on problems of up to 100 customers from the Solomon datasets. It has succeeded in solving to optimality several previously unsolved problems and a new 150-customer problem. In addition, the algorithm proved faster than algorithms previously considered in the literature. These computational results indicate the effectiveness of the valid inequalities we have developed.

A Unified Framework for Deterministic Time Constrained Vehicle Routing and Crew Scheduling Problems

Time constrained routing and scheduling is of significant importance across land, air and water transportation. These problems are also encountered in a variety of manufacturing, warehousing and service sector environments. Their mathematical complexity and the magnitude of the potential cost savings to be achieved by utilizing O.R. methodologies have attracted researchers since the early days of the field. Witness to this are the pioneering efforts of Dantzig and Fulkerson (1954), Ford and Fulkerson (1962), Appelgren (1969, 1971), Levin (1971), Madsen (1976) and Orloff (1976). Much of the methodology developed has made extensive use of network models and algorithms.

An integrated model for logistics network design

In this paper we introduce a new formulation of the logistics network design problem encountered in deterministic, single-country, single-period contexts. Our formulation is flexible and integrates location and capacity choices for plants and warehouses with supplier and transportation mode selection, product range assignment and product flows. We next describe two approaches for solving the problem---a simplex-based branch-and-bound and a Benders decomposition approach. We then propose valid inequalities to strengthen the LP relaxation of the model and improve both algorithms. The computational experiments we conducted on realistic randomly generated data sets show that Benders decomposition is somewhat more advantageous on the more difficult problems. They also highlight the considerable performance improvement that the valid inequalities produce in both solution methods. Furthermore, when these constraints are incorporated in the Benders decomposition algorithm, this offers outstanding reoptimization capabilities.

Chapter 2 Time constrained routing and scheduling

Publisher Summary Time constrained routing and scheduling problems are encountered in a variety of industrial and service sector applications, ranging from logistics and transportation systems to material handling systems in manufacturing. The traveling salesman problem with time windows has applications in single and multiple vehicle problems. The vehicle routing problem with time windows has many industrial applications including those where dock availability is a bottleneck such as for distribution centers. This chapter describes the significant advances made in time constrained routing and scheduling. In terms of solution methodology capable of solving realistic size problems, this field has seen a natural progression from ad-hoc methods to simple heuristics, to optimization-based heuristics and recently optimal algorithms. The chapter discusses fixed schedule problems and develops in detail the Dantzig-Wolfe decomposition/column generation approach which will then be applied to many of the other problem types. The vehicle routing problem with time windows and several important problem variants including the multiple traveling salesman problems is explored. The chapter examines a unified framework for fleet and crew scheduling problems.

Crew pairing at Air France

Abstract In the airline industry, crew schedules consist of a number of pairings. These are round trips originating and terminating at the same crew home base composed of legal work days, called duties, separated by rest periods. The purpose of the airline crew pairing problem is to generate a set of minimal cost crew pairings covering all flight legs. The set of pairings must satisfy all the rules in the work convention and all the appropriate air traffic regulations. The resulting constraints can affect duty construction, may restrict each pairing, or be imposed on the overall crew schedule. The pairing problem is formulated as an integer, nonlinear multi-commodity network flow problem with additional resource variables. Nonlinearities occur in the objective function as well as in a large subset of constraints. A branch-and-bound algorithm based on an extension of the Dantzig-Wolfe decomposition principle is used to solve this model. The master problem becomes a Set Partitioning type model, as in the classical formulation, while pairings are generated using resource constrained shortest path subproblems. This primal approach implicitly considers all feasible pairings and also provides the optimality gap value on a feasible solution. A nice feature of this decomposition process is that it isolates all nonlinear aspects of the proposed multi-commodity model in the subproblems which are solved by means of a specialized dynamic programming algorithm. We present the application and implementation of this approach at Air France. It is one of the first implementations of an optimal approach for a large airline carrier. We have chosen a subproblem network representation where the duties rather than the legs are on the arcs. This ensures feasibility relative to duty restrictions by definition. As opposed to Lavoie, Minoux and Odier (1988), the nonlinear cost function is modeled without approximations. The computational experiments were conducted using actual Air France medium haul data. Even if the branch-and-bound trees were not fully explored in all cases, the gaps certify that the computed solutions are within a fraction of one percentage point of the optimality. Our results illustrate that our approach produced substantial improvements over solutions derived by the expert system in use at Air France. Their magnitude led to the eventual implementation of the approach.

AN OPTIMIZATION-BASED HEURISTIC FOR VEHICLE ROUTING AND SCHEDULING WITH SOFT TIME WINDOW CONSTRAINTS

The Vehicle Routing and Scheduling Problem with Time Window constraints is formulated as a mixed integer program, and optimization-based heuristics which extend the cluster-first, route-second algorithm of Fisher and Jaikumar are developed for its solution. We present a new formulation based on the treatment of the time window constraints as soft constraints that can be violated at a cost and we heuristically decompose the problem into an assignment/clustering component and a series of routing and scheduling components. Numerical results based on randomly generated and benchmark problem sets indicate that the algorithm compares favorably to state-of-the-art local insertion and improvement heuristics.