Guy Desaulniers

Shortest Path Problems with Resource Constraints

In most vehicle routing and crew scheduling applications solved by column generation, the subproblem corresponds to a shortest path problem with resource constraints (SPPRC) or one of its variants.

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.

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.

Tabu Search, Partial Elementarity, and Generalized k-Path Inequalities for the Vehicle Routing Problem with Time Windows

The vehicle routing problem with time windows consists of delivering goods at minimum cost to a set of customers using an unlimited number of capacitated vehicles assigned to a single depot. Each customer must be visited within a prescribed time window. The most recent successful solution methods for this problem are branch-and-price-and-cut algorithms where the column generation subproblem is an elementary shortest-path problem with resource constraints (ESPPRC). In this paper, we propose new ideas having the potential to improve such a methodology. First, we develop a tabu search heuristic for the ESPPRC that allows, in most iterations, the generation of negative reduced cost columns in a short computation time. Second, to further accelerate the subproblem solution process, we propose to relax the elementarity requirements for a subset of the nodes. This relaxation, however, yields weaker lower bounds. Third, we introduce a generalization of the k-path inequalities and highlight that these generalized inequalities can, in theory, be stronger than the traditional ones. Finally, combining these ideas with the most recent advances published in the literature, we present a wide variety of computational results on the Solomons 100-customer benchmark instances. In particular, we report solving five previously unsolved instances.

Multi-depot vehicle scheduling problems with time windows and waiting costs

The multi-depot vehicle scheduling problem with time windows (MDVSPTW) consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost. Each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots. The problem is formulated as an integer nonlinear multi-commodity network flow model with time variables and is solved using a column generation approach embedded in a branch-and-bound framework. This paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times. This new and more realistic cost structure gives rise to a nonlinear objective function in the model. Optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem (UBSP) and freight transport scheduling problem (FTSP). The results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used, and can efficiently treat the case with exact waiting costs.

Branch-and-Price-and-Cut for the Split-Delivery Vehicle Routing Problem with Time Windows

This paper addresses the split-delivery vehicle routing problem with time windows (SDVRPTW) that consists of determining least-cost vehicle routes to service a set of customer demands while respecting vehicle capacity and customer time windows. The demand of each customer can be fulfilled by several vehicles. For solving this problem, we propose a new exact branch-and-price-and-cut method, where the column generation subproblem is a resource-constrained elementary shortest-path problem combined with the linear relaxation of a bounded knapsack problem. Each generated column is associated with a feasible route and a compatible delivery pattern. As opposed to existing branch-and-price methods for the SDVRPTW or its variant without time windows, integrality requirements in the integer master problem are not imposed on the variables generated dynamically, but rather on additional variables. An ad hoc label-setting algorithm is developed for solving the subproblem. Computational results show the effectiveness of the proposed method.

A Branch-and-Price Method for a Liquefied Natural Gas Inventory Routing Problem

We consider a maritime inventory routing problem in the liquefied natural gas (LNG) business, called the LNG inventory routing problem (LNG-IRP). Here, an actor is responsible for the routing of the fleet of special purpose ships, and the inventories both at the liquefaction plants and the regasification terminals. Compared to many other maritime inventory routing problems, the LNG-IRP includes some complicating aspects such as (1) a constant rate of the cargo evaporates each day and is used as fuel during transportation; (2) variable production and consumption of LNG, and (3) a variable number of tanks unloaded at the regasification terminals. The problem is solved by a branch-and-price method. In the column generation approach, the master problem handles the inventory management and the port capacity constraints, while the subproblems generate the ship route columns. Different accelerating strategies are implemented. The proposed method is tested on instances inspired from real-world problems faced by a major energy company.

Accelerating Strategies in Column Generation Methods for Vehicle Routing and Crew Scheduling Problems

This paper focuses on accelerating strategies used in conjunction with column generation to solve vehicle routing and crew scheduling problems. We describe techniques directed at speeding up each of the five phases of the solution process: pre-processor, subproblem, master problem, branch-and-bound, and post-optimizer. In practical applications, these methods often were key elements for the viability of this optimization approach. The research cited here shows their use led to computational gains, and notably to solutions that could not have been obtained otherwise due to practical problem complexity and size. In particular, we present recent methods directed at the integer programming aspect of the approach that were instrumental in substantially reducing the integrality gap found in certain applications, thereby helping to efficiently produce excellent quality solutions.

An efficient algorithm to find a shortest path for a car-like robot

We study the problem of finding a shortest path for a car-like mobile robot with a minimal turning radius constraint. This vehicle can move either forward or backward in an unconstrained environment. By applying necessary conditions on the segment lengths of shortest paths, we partition the configuration space into elements such that a single path type is associated with 150 elements and two path types are associated with the other 11 elements. A shortest path from a fixed initial configuration to any final configuration in an element can always be found among the paths of types associated with that element. We then present an algorithm based on this partition and we give results showing that our algorithm is 15 times faster on average than the Reeds-Shepp algorithm (1990).

OPERATIONAL CAR ASSIGNMENT AT VIA RAIL CANADA

Assigning locomotives and cars to a set of scheduled trains is a complex but important problem for passenger railways. This task is normally carried out in several phases in which increasing levels of detail are considered. This paper describes a modeling and solution methodology for a car assignment problem that arises when individual car routings that satisfy all operational constraints must be determined. This methodology considers typical constraints such as maintenance requirements but also more complex constraints such as minimum connection times that depend on the positions of the individual cars in a train consist. The problem is solved heuristically by a branch-and-bound method in which the linear relaxations are solved by column generation. Simulation experiments performed on realistic data show that the solution approach yields good quality solutions in very short computing times. A software system based on this approach is now being evaluated by VIA Rail Canada.