Roberto Roberti

Recent exact algorithms for solving the vehicle routing problem under capacity and time window constraints

This paper provides a review of the recent developments that had a major impact on the current state-of-the-art exact algorithms for the vehicle routing problem (VRP). The paper reviews mathematical formulations, relaxations and recent exact methods for two of the most important variants of the VRP: the capacitated VRP (CVRP) and the VRP with time windows (VRPTW). The paper also reports a comparison of the computational performances of the different exact algorithms for the CVRP and VRPTW.

New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem

In this paper, we describe an effective exact method for solving both the capacitated vehicle routing problem (cvrp) and the vehicle routing problem with time windows (vrptw) that improves the method proposed by Baldacci et al. [Baldacci, R., N. Christofides, A. Mingozzi. 2008. An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Math. Programming115(2) 351--385] for the cvrp. The proposed algorithm is based on the set partitioning (SP) formulation of the problem. We introduce a new route relaxation called ng-route, used by different dual ascent heuristics to find near-optimal dual solutions of the LP-relaxation of the SP model. We describe a column-and-cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem. The new ng-route relaxation and the different dual solutions achieved allow us to generate a reduced SP problem containing all routes of any optimal solution that is finally solved by an integer programming solver. The proposed method solves four of the five open Solomons vrptw instances and significantly improves the running times of state-of-the-art algorithms for both vrptw and cvrp.

An Exact Algorithm for the Two-Echelon Capacitated Vehicle Routing Problem

In the two-echelon capacitated vehicle routing problem (2E-CVRP), the delivery to customers from a depot uses intermediate depots, called satellites. The 2E-CVRP involves two levels of routing problems. The first level requires a design of the routes for a vehicle fleet located at the depot to transport the customer demands to a subset of the satellites. The second level concerns the routing of a vehicle fleet located at the satellites to serve all customers from the satellites supplied from the depot. The objective is to minimize the sum of routing and handling costs. This paper describes a new mathematical formulation of the 2E-CVRP used to derive valid lower bounds and an exact method that decomposes the 2E-CVRP into a limited set of multidepot capacitated vehicle routing problems with side constraints. Computational results on benchmark instances show that the new exact algorithm outperforms the state-of-the-art exact methods.

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.

Models and algorithms for the Asymmetric Traveling Salesman Problem: an experimental comparison

This paper surveys the most effective mathematical models and exact algorithms proposed for finding the optimal solution of the well-known Asymmetric Traveling Salesman Problem (ATSP). The fundamental Integer Linear Programming (ILP) model proposed by Dantzig, Fulkerson and Johnson is first presented, its classical (assignment, shortest spanning r-arborescence, linear programming) relaxations are derived, and the most effective branch-and-bound and branch-and-cut algorithms are described. The polynomial ILP formulations proposed for the ATSP are then presented and analyzed. The considered algorithms and formulations are finally experimentally compared on a set of benchmark instances.

New State-Space Relaxations for Solving the Traveling Salesman Problem with Time Windows

The traveling salesman problem with time windows (TSPTW) is the problem of finding in a weighted digraph a least-cost tour starting from a selected vertex, visiting each vertex of the graph exactly once according to a given time window, and returning to the starting vertex. This nP-hard problem arises in routing and scheduling applications. This paper introduces a new tour relaxation, called ngL-tour, to compute a valid lower bound on the TSPTW obtained as the cost of a near-optimal dual solution of a problem that seeks a minimum-weight convex combination of nonnecessarily elementary tours. This problem is solved by column generation. The optimal integer TSPTW solution is computed with a dynamic programming algorithm that uses bounding functions based on different tour relaxations and the dual solution obtained. An extensive computational analysis on basically all TSPTW instances (involving up to 233 vertices) from the literature is reported. The results show that the proposed algorithm solves all but one instance and outperforms all exact methods published in the literature so far.

An Exact Algorithm for the Multitrip Vehicle Routing Problem

The multitrip vehicle routing problem MTVRP is a variant of the capacitated vehicle routing problem where each vehicle can perform a subset of routes, called a vehicle schedule, subject to maximum driving time constraints. Despite its practical importance, the MTVRP has received little attention in the literature. Few heuristics have been proposed, and only an exact algorithm has been presented for a variant of the MTVRP with customer time window constraints and unlimited driving time for each vehicle. We describe two set-partitioning-like formulations of the MTVRP. The first formulation requires the generation of all feasible routes, whereas the second formulation is based on the generation of all feasible schedules. We study valid lower bounds, based on the linear relaxations of both formulations enforced with valid inequalities, that are embedded into an exact solution method. The computational results show that the proposed exact algorithm can solve MTVRP instances taken from the literature, with up to 120 customers.

A new lower bound for curriculum-based course timetabling

Abstract In this paper, we propose a new method to compute lower bounds for curriculum-based course timetabling (CTT), which calls for the best weekly assignment of university course lectures to rooms and time slots. The lower bound is obtained by splitting the objective function into two parts, considering one separate problem for each part of the objective function, and summing up the corresponding optimal values (or, in some cases, lower bounds on these values), found by formulating the two parts as Integer Linear Programs (ILPs). The solution of one ILP is obtained by using a column generation procedure. Experimental results show that the proposed lower bound is often better than the ones found by the previous methods in the literature, and also much better than those found by other new ILP formulations illustrated in this paper. The proposed approach is able to obtain improved lower bounds on real-world benchmark instances from the literature, used in the international timetabling competitions ITC2002 and ITC2007, proving for the first time that some of the best-known heuristic solutions are indeed optimal (or close to the optimal ones).

The Fixed Charge Transportation Problem: An Exact Algorithm Based on a New Integer Programming Formulation

The fixed charge transportation problem generalizes the well-known transportation problem where the cost of sending goods from a source to a sink is composed of a fixed cost and a continuous cost proportional to the amount of goods sent. In this paper, we describe a new integer programming formulation with exponentially many variables corresponding to all possible flow patterns to sinks. We show that the linear relaxation of the new formulation is tighter than that of the standard mixed integer programming formulation. We describe different classes of valid inequalities for the new formulation and a column generation method to compute a valid lower bound embedded into an exact branch-and-price algorithm. Computational results on test problems from the literature show that the new algorithm outperforms the state-of-the-art exact methods from the literature and can solve instances with up to 70 sources and 70 sinks. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2014.1947 . This paper was accepted by Dimitris Bertsimas, optimization.

Exact Algorithms for Different Classes of Vehicle Routing Problems

We deal with five problems arising in the field of logistics: the Asymmetric TSP (ATSP), the TSP with Time Windows (TSPTW), the VRP with Time Windows (VRPTW), the Multi-Trip VRP (MTVRP), and the Two-Echelon Capacitated VRP (2E-CVRP). The ATSP requires finding a lest-cost Hamiltonian tour in a digraph. We survey models and classical relaxations, and describe the most effective exact algorithms from the literature. A survey and analysis of the polynomial formulations is provided. The considered algorithms and formulations are experimentally compared on benchmark instances. The TSPTW requires finding, in a weighted digraph, a least-cost Hamiltonian tour visiting each vertex within a given time window. We propose a new exact method, based on new tour relaxations and dynamic programming. Computational results on benchmark instances show that the proposed algorithm outperforms the state-of-the-art exact methods. In the VRPTW, a fleet of identical capacitated vehicles located at a depot must be optimally routed to supply customers with known demands and time window constraints. Different column generation bounding procedures and an exact algorithm are developed. The new exact method closed four of the five open Solomon instances. The MTVRP is the problem of optimally routing capacitated vehicles located at a depot to supply customers without exceeding maximum driving time constraints. Two set-partitioning-like formulations of the problem are introduced. Lower bounds are derived and embedded into an exact solution method, that can solve benchmark instances with up to 120 customers. The 2E-CVRP requires designing the optimal routing plan to deliver goods from a depot to customers by using intermediate depots. The objective is to minimize the sum of routing and handling costs. A new mathematical formulation is introduced. Valid lower bounds and an exact method are derived. Computational results on benchmark instances show that the new exact algorithm outperforms the state-of-the-art exact methods.