François Soumis

Routing with time windows by column generation

Consider a set of trips where each trip is specified a priori by a place of origin, a destination, a duration, a cost, and a time interval within which the trip must begin. The trips may include visits to one or more specific points. Our problem is to determine the number of vehicles required, together with their routes and schedules, so that each trip begins within its given time interval, while the fixed costs related to the number of vehicles, and the travel costs between trips, are minimized. The problem is a generalization of the m-traveling salesman problem. We use column generation on a set partitioning problem solved by simplex and branch-and-bound; columns are generated by a shortest path algorithm with time windows on the nodes. Numerical results for several school bus transportation problems with up to 151 trips are discussed.

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.

A Method for Optimally Solving the Rostering Problem

This paper presents a prototype method for optimally solving the rostering problem, i.e., constructing work schedules for airline crew members. The main goal is to show the possibilities of solving the rostering problem using optimal methods. The prototype uses a column generation approach embedded in a branch and bound algorithm to solve the rostering problem. To the knowledge of the authors, this prototype is the first optimal method for solving the rostering problem.

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).

A column generation method for optimal load management via control of electric water heaters

Electric water heaters have been the focus of several previous studies where they are viewed as energy storing devices to be considered for peak load shaving within device interruption based load management programs. It is well known that such controls, when applied indiscriminately, can cause payback phenomena which could actually worsen existing peak loads. Thus, optimal switching strategies must be devised. A column generation based optimization scheme which combines the advantages of linear programming and dynamic programming approaches, both previously proposed in load management literature, is presented. The method is successfully tested on a representative set of residential electric water heating load mixes. The results are analyzed in the paper, both from the computational and load management point of view. >

Methods for routing with time windows

Abstract Consider a set of trips where each trip is specified a priori by a place of origin, a destination, a duration, a cost and a time interval within which the trip must begin. The trips may include visits to one or more specific points. Our problem is to determine the number of vehicles required together with their routes and schedules, so that each trip begins within his given time interval, while the fixed costs related to the number of vehicles, and the travel costs between trips are minimized. The problem is a generalization of the m-travelling salesman problem. We compare numerical results for 3 algorithms developed by our research team: 1. (1) Column generation on a set partitioning problem solved by simplex and branch-and-bound; columns are generated by a shortest path algorithm with time constraints on the nodes. 2. (2) Adaptation of the Carpaneto-Toth algorithm for the asymmetric travelling salesman problem: solution of network problems by relaxing scheduling constraints, and branch-and-bound on flow variables. 3. (3) Solution of network problems by relaxing scheduling constraints and branch-and-bound based on dividing the time windows.

Optimal Strip Sequencing Strategies for Flexible Manufacturing Operations in Two and Three Dimensions

In this paper, optimal strip strategies are developed for a variety of two-dimensional and three-dimensional sequencing problems arising in flexible manufacturing. These strategies are appropriate for CNC drilling operations, NC punching operations, and circuit board population, for example. Seven different metrics are considered.

Optimal sequencing rules for some large-scale flexible manufacturing problems under the Manhattan and Chebychev metrics

The purpose of this paper is to develop optimal tool partitioning policies and strip sequencing strategies for a class of flexible manufacturing problems. The problems under consideration involve a large number of operations to be performed by a series of tools on a two-dimensional object. For example, these operations could consist of drilling holes in a metallic sheet. Tools are arranged in a carousel or along a toolbar according to a predetermined sequence. Operations are performed by repeatedly moving the sheet to bring the hole locations under the tool. During each pass, as all operations involving a series of consecutive tools are executed, two main problems are to be solved: (1) how to move the sheet during each pass, (2) how to partition the tools into blocks of consecutive tools. A strip strategy is used to move the sheet. Given this policy, optimal strip widths and tool partitioning policies are determined jointly. Analytical solutions are derived under two metrics corresponding to different operating modes. A numerical example is provided.

Optimal policies for some discrete sequencing problems arising in flexible manufacturing

Abstract This paper describes strip sequencing models for large scale traveling salesman problems arising in flexible manufacturing operations in two or three discrete dimensions, under a Manhattan metric. Asymptotically optimal values are derived for the strip width that minimizes unit expected manufacturing time.