Oli B.G. Madsen

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 heuristic algorithm for a dial-a-ride problem with time windows, multiple capacities, and multiple objectives

The paper describes a system for the solution of a static dial-a-ride routing and scheduling problem with time windows (DARPTW). The problem statement and initialization of the development project was made by the Copenhagen Fire-Fighting Service (CFFS). The CFFS needed a new system for scheduling elderly and disabled persons, involving about 50.000 requests per year. The problem is characterized by, among other things, multiple capacities and multiple objectives. The capacities refer to the fact that a vehicle may be equipped with e.g. normal seats, children seats or wheel chair places. The objectives relate to a number of concerns such as e.g. short driving time, high vehicle utilization or low costs. A solution algorithm REBUS based on an insertion heuristics was developed. The algorithm permits in a flexible way weighting of the various goals such that the solution reflects the users preferences. The algorithm is implemented in a dynamic environment intended for on-line scheduling. Thus, a new request for service is treated in less than 1 second, permitting an interactive user interface.

An Optimization Algorithm for the Vehicle Routing Problem with Time Windows Based on Lagrangian Relaxation

Our paper presents a new optimization method for the Vehicle Routing Problem with Time Windows (VRPTW). The VRPTW is a generalization of the Vehicle Routing Problem, where the service of a customer must start within a given time interval—a so-called time window. Our method is based on a Lagrangian relaxation of the constraint set requiring that each customer must be serviced. The master problem consists of finding the optimal Lagrangian multipliers and the subproblem is a Shortest Path Problem with Time Windows and Capacity Constraints. The optimal multipliers are found using a method exploiting the benefits of subgradient methods as well as a bundle method. The method has been implemented and tested on a series of well-known benchmark problems of size up to 100 customers. Our algorithm turns out to be very competitive compared to algorithms considered in the literature, and we have succeeded in solving several previously unsolved problems.

A comparative study of heuristics for a two-level routing-location problem

Abstract In the process of distributing newspapers, transfer points play the role of depots. When designing a newspaper distribution system, three types of decisions are to be made: 1. (1) number and locations of transfer points, 2. (2) Design of tours originating at the printing office to serve the transfer points, 3. (3) design of tours emanating from transfer points to serve retailers Thus, the problem is a combined location/routing problem. The question addressed is the design of a suitable heuristic for solving the problem. This paper gives a comparison of three different procedures: 1. (A) a tour construction method with implicit transfer point location, 2. (B) an Alternate Location-Allocation procedure for transfer point location (1) followed by Savings procedures four routing (2) and (3), 3

Vehicle Routing with Time Windows: Two Optimization Algorithms

We describe two optimization methods for vehicle routing problems with time windows. These are a K-Tree relaxation with time windows added as side constraints and a Lagrangian decomposition in which variable splitting is used to divide the problem into two subproblems—a semi-assignment problem and a series of shortest path problems with time windows and capacity constraints. We present optimal solutions to problems with up to 100 customers.

Partially dynamic vehicle routing—models and algorithms

In this paper we propose a framework for dynamic routing systems based on their degree of dynamism. Next, we consider its impact on solution methodology and quality. Specifically, we introduce the Partially Dynamic Travelling Repairman Problem and describe several dynamic policies to minimize routing costs. The results of our computational study indicate that increasing the dynamic level results in a linear increase in route length for all policies studied. Furthermore, a Nearest Neighbour policy performed, on the average, uniformly better than the other dispatching rules studied. Among these, a Partitioning policy produced only slightly higher average route lengths.

Booking policy for flights with two types of passengers

Abstract An overbooking model for a fixed nonstop flight with two types of passengers is presented. The model takes cancellations, and reservations prior to departure into consideration as well as ‘no-shows’ (passengers who fail to arrive for flights without notice), denied boardings and downgrading of passengers. The model treats the airline booking process as a Markovian nonhomogeneous sequential decision process. The model is solved by two dimensional stochastic dynamic programming. Computational experiences and numerical results from a real-life case, including different types of sensivity analysis, are presented.

Lagrangian duality applied to the vehicle routing problem with time windows

This paper considers the vehicle routing problem with time windows, where the service of each customer must start within a specified time interval. We consider the Lagrangian relaxation of the constraint set requiring that each customer must be served by exactly one vehicle yielding a constrained shortest path subproblem. We present a stabilized cutting-plane algorithm within the framework of linear programming for solving the associated Lagrangian dual problem. This algorithm creates easier constrained shortest path subproblems because less negative cycles are introduced and it leads to faster multiplier convergence due to a stabilization of the dual variables. We have embedded the stabilized cutting-plane algorithm in a branch-and-bound search and introduce strong valid inequalities at the master problem level by Lagrangian relaxation. The result is a Lagrangian branch-and-cut-and-price (LBCP) algorithm for the VRPTW. Making use of this acceleration strategy at the master problem level gives a significant speed-up compared to algorithms in the literature based on traditional column generation. We have solved two test problems introduced in 2001 by Gehring and Homberger with 400 and 1000 customers respectively, which to date are the largest problems ever solved to optimality. We have implemented the LBCP algorithm using the ABACUS open-source framework for solving mixed-integer linear-programs by branch, cut, and price.

Methods for solving combined two level location-routing problems of realistic dimensions

Abstract Very few distribution studies have dealt with combined location-routing problems, the problems of locating depots from which customers are served by tours rather than individual trips. This paper gives a survey of methods solving combined location-routing problems. Some methods are analyzed and three new heuristic methods are developed, implemented and compared. A newspaper delivery system consisting of 4500 customers is solved. The results seem to indicate that an alternate location-allocation-savings procedure and a saving-drop procedure are promising.