Yvan Dumas

The pickup and delivery problem with time windows

Abstract The vehicle routing problem ( vrp ) involves the design of a set of minimum cost routes for a fleet of vehicles which services exactly once a set of customers with known demands. The pickup and delivery problem with time windows ( pdptw ) is a generalization of the vrp which is concerned with the construction of optimal routes to satisfy transportation requests, each requiring both pickup and delivery under capacity, time window and precedence constraints. This paper presents an exact algorithm which solves the pickup and delivery problem when transporting goods. This algorithm uses a column generation scheme with a constrained shortest path as a subproblem. The algorithm can handle multiple depots and different types of vehicles.

A Dynamic Programming Solution of the Large-Scale Single-Vehicle Dial-A-Ride Problem with Time Windows

SYNOPTIC ABSTRACTThe single-vehicle dial-a-ride problem with time window constraints for both pick-up and delivery locations, and precedence and capacity constraints, is solved using a forward dynamic programming algorithm. The total distance is minimized. The development of criteria for the elimination of infeasible states results in solution times which increase linearly with problem size.

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.

A Request Clustering Algorithm for Door-to-Door Handicapped Transportation

This paper examines whether there is a substantial additional payoff to be derived from using mathematical optimization techniques to globally define a set of mini-clusters. Specifically, we present a new approximate method to mini-clustering that involves solving a multi-vehicle pick-up and delivery problem with time windows by column generation. To solve this problem we have enhanced an existing optimal algorithm in several ways. First, we present an original network design based on lists of neighboring transportation requests. Second, we have developed a specialized initialization procedure which reduces the processing time by nearly 40%. Third, the algorithm was easily generalized to multi-dimensional capacity. Finally, we have also developed a heuristic to reduce the size of the network, while incurring only small losses in solution quality. This allows the application of our approach to much larger problems. To be able to compare the results of optimization-based and local heuristic mini-clustering, we also present a parallel insertion approach to mini-clustering. Our computational results highlight the success of our approach. On test problems with up to 250 requests, our optimization-based method reduced the travel time within the mini-clusters by an average of 10% over the parallel insertion approach. Furthermore, for an actual problem with 2545 requests, a substantial 5.9% improvement in total traveling time was achieved over the heuristic.

Solving the discrete lotsizing and scheduling problem with sequence dependent set-up costs and set-up times using the Travelling Salesman Problem with time windows

Abstract In this paper we consider the Discrete Lotsizing and Scheduling Problem with sequence dependent set-up costs and set-up times (DLSPSD). DLSPSD contains elements from lotsizing and from job scheduling, and is known to be NP-Hard. An exact solution procedure for DLSPSD is developed, based on a transformation of DLSPSD into a Travelling Salesman Problem with Time Windows (TSPTW). TSPTW is solved by a novel dynamic programming approach due to Dumas et al. (1993). The results of a computational study show that the algorithm is the first one capable of solving DLSPSD problems of moderate size to optimality with a reasonable computational effort.

Technical Note—Optimizing the Schedule for a Fixed Vehicle Path with Convex Inconvenience Costs

We present an algorithm that solves the problem of finding the vehicle schedule which minimizes total inconveniences for travel along a fixed path, where service times at nodes are constrained by time windows and where inconvenience is modeled using convex functions of the service times. This problem occurs as the last step or as a sub-problem in many common approaches to solving routing and scheduling problems. We show that the complexity of the algorithm, expressed as a number of unidimensional minimizations, is on the order of the number of nodes for convex inconvenience functions. For linear and quadratic functions, this complexity is linear in the number of nodes. We present extensions to the case where linear costs are applied to waiting time, and also to the case where the service time variables are discrete.

The Multiple Vehicle DIAL-A-RIDE Problem

This article examines the use of mathematical optimization algorithms to construct vehicle routes in a large scale multi-vehicle many-to-many system in the context of the transportation of the handicapped. The main contributions of this paper are the innovative concept of the mini-clusters, the generalisation of the column generation algorithm for the m-TSP with time windows (Desrosiers, Soumis and Desrochers [11]) to solve problems with several depots and availability constraints, and in addition, a method of decomposition into time slices to handle very large problems.

Bombardier Flexjet Significantly Improves Its Fractional Aircraft Ownership Operations

The fractional aircraft market is the fastest growing segment of the business aircraft industry. A fractional aircraft operation is complex--essentially an unscheduled airline in a constantly changing environment. Bombardier Flexjet implemented a comprehensive three-module optimization system to simultaneously maximize its use of aircraft, crews, and facilities. AD OPT Technologies designed the modules, using the GENCOL optimizer developed at GERAD, which employs a column-generation approach to decompose large-scale mixed-integer nonlinear programming problems. Since inception, the project has generated savings in excess of

Time Constrained Routing and Scheduling

54 million with projected additional savings of

OPTIMIZING THE SCHEDULE FOR A FIXED VEHICLE PATH WITH CONVEX INCONVENIENCE COSTS

27 million annually, primarily by lowering crew levels (20 percent), aircraft inventory (40 percent), and supplemental charter aircraft usage (five percent) while increasing aircraft utilization (10 percent). The quality of customer service has remained consistently high, with significant reduction in supply.