Demetrio Laganà

A decomposition-based heuristic for the multiple-product inventory-routing problem

Abstract The inventory-routing problem is an integrated logistics planning problem arising in situations where customers transfer the responsibility for inventory replenishment to the vendor. The vendor must then decide when to visit each customer, how much to deliver and how to sequence customers in vehicle routes. In this paper, we focus on the case where several different products have to be delivered by a fleet of vehicles over a finite and discrete planning horizon. We present a three-phase heuristic based on a decomposition of the decision process of the vendor. In the first phase, replenishment plans are determined by using a Lagrangian-based method. These plans do not specify delivery sequences for the vehicles. The sequencing of the planned deliveries is performed in the second phase in which a simple procedure is employed to construct vehicle routes. The third phase incorporates planning and routing decisions into a mixed-integer linear programming model aimed at finding a good solution to the integrated problem. Computational experiments show that our heuristic is effective on instances with up to 50 customers and 5 products.

An optimization-based heuristic for the Multi-objective Undirected Capacitated Arc Routing Problem

The Multi-objective Undirected Capacitated Arc Routing Problem (MUCARP) is the optimization problem aimed at finding the best strategy for servicing a subset of clients localized along the links of a logistic network, by using a fleet of vehicles and optimizing more than one objective. In general, the first goal consists in minimizing the total transportation cost, and in this case the problem brings back to the well-known Undirected Capacitated Arc Routing Problem (UCARP). The motivation behind the study of the MUCARP lies in the study of real situations where companies working in the logistic distribution field deal with complex operational strategies, in which different actors (trucks, drivers, customers) have to be allocated within an unified framework by taking into account opposite needs and different employment contracts. All the previous considerations lead to the MUCARP as a benchmark optimization problem for modeling practical situations. In this paper, the MUCARP is heuristically tackled. In particular, three competitive objectives are minimized at the same time: the total transportation cost, the longest route cost (makespan) and the number of vehicles (i.e., the total number of routes). An approximation of the optimal Pareto front is determined through an optimization-based heuristic procedure, whose performances are tested and analyzed on classical benchmark instances.

Modeling and solving the mixed capacitated general routing problem

We tackle the mixed capacitated general routing problem (MCGRP) which generalizes many other routing problems. We propose an integer programming model for the MCGRP and extend some inequalities originally introduced for the capacitated arc routing problem (CARP). Identification procedures for these inequalities and for some relaxed constraints are also discussed. Finally, we describe a branch and cut algorithm including the identification procedures and present computational experiments over instances derived from the CARP.

A constructive heuristic for the undirected rural postman problem

This paper describes a constructive heuristic for the well-known Undirected Rural Postman Problem. At each iteration, the procedure inserts a connected component of the required edges and performs a local postoptimization. Computational results on a set of benchmark instances with up to 350 vertices show that the proposed procedure is competitive with the classical Frederickson procedure. Its use is recommended when a high-quality solution is needed in a short amount of time (e.g., in laser plotter applications).

Ant colony optimization for the arc routing problem with intermediate facilities under capacity and length restrictions

The aim of this paper is to introduce a new ant colony optimization procedure for the Arc Routing Problem with Intermediate Facilities under Capacity and Length Restrictions (CLARPIF), a variant of the Capacitated Arc Routing Problem (CARP) and of the Capacitated Arc Routing Problem with Intermediate Facilities (CARPIF). Computational results show that this algorithm is capable of providing substantial improvements over other known heuristics.

The mixed capacitated general routing problem under uncertainty

We study the General Routing Problem defined on a mixed graph and with stochastic demands. The problem under investigation is aimed at finding the minimum cost set of routes to satisfy a set of clients whose demand is not deterministically known. Since each vehicle has a limited capacity, the demand uncertainty occurring at some clients affects the satisfaction of the capacity constraints, that, hence, become stochastic. The contribution of this paper is twofold: firstly we present a chance-constrained integer programming formulation of the problem for which a deterministic equivalent is derived. The introduction of uncertainty into the problem poses severe computational challenges addressed by the design of a branch-and-cut algorithm, for the exact solution of limited size instances, and of a heuristic solution approach exploring promising parts of the search space. The effectiveness of the solution approaches is shown on a probabilistically constrained version of the benchmark instances proposed in the literature for the mixed capacitated general routing problem.

Two-phase branch-and-cut for the mixed capacitated general routing problem

The Mixed Capacitated General Routing Problem (MCGRP) is defined over a mixed graph, for which some vertices must be visited and some links must be traversed at least once. The problem consists of determining a set of least-cost vehicle routes that satisfy this requirement and respect the vehicle capacity. Few papers have been devoted to the MCGRP, in spite of interesting real-world applications, prevalent in school bus routing, mail delivery, and waste collection. This paper presents a new mathematical model for the MCGRP based on two-index variables. The approach proposed for the solution is a two-phase branch-and-cut algorithm, which uses an aggregate formulation to develop an effective lower bounding procedure. This procedure also provides strong valid inequalities for the two-index model. Extensive computational experiments over benchmark instances are presented.

Solving the shortest path tour problem

In this paper, we study the shortest path tour problem in which a shortest path from a given origin node to a given destination node must be found in a directed graph with non-negative arc lengths. Such path needs to cross a sequence of node subsets that are given in a fixed order. The subsets are disjoint and may be different-sized. A polynomial-time reduction of the problem to a classical shortest path problem over a modified digraph is described and two solution methods based on the above reduction and dynamic programming, respectively, are proposed and compared with the state-of-the-art solving procedure. The proposed methods are tested on existing datasets for this problem and on a large class of new benchmark instances. The computational experience shows that both the proposed methods exhibit a consistent improved performance in terms of computational time with respect to the existing solution method.

The Express heuristic for probabilistically constrained integer problems

Integer problems under joint probabilistic constraints with random coefficients in both sides of the constraints are extremely hard from a computational standpoint since two different sources of complexity are merged. The first one is related to the challenging presence of probabilistic constraints which assure the satisfaction of the stochastic constraints with a given probability, whereas the second one is due to the integer nature of the decision variables. In this paper we present a tailored heuristic approach based on alternating phases of exploration and feasibility repairing which we call Express (Explore and Repair Stochastic Solution) heuristic. The exploration is carried out by the iterative solution of simplified reduced integer problems in which probabilistic constraints are discarded and deterministic additional constraints are adjoined. Feasibility is restored through a penalty approach. Computational results, collected on a probabilistically constrained version of the classical 0–1 multiknapsack problem, show that the proposed heuristic is able to determine good quality solutions in a limited amount of time.

The Undirected Capacitated General Routing Problem with Profits

In this paper we introduce and study the Undirected Capacitated General Routing Problem with Profits (UCGRPP). This problem is defined on an undirected graph where a subset of vertices and edges correspond to customers, which are associated with a given profit and demand. The profit of each customer can be collected at most once. A fleet of homogeneous capacitated vehicles is given to serve the customers. The objective is to find the vehicle routes that maximize the difference between the total collected profit and the traveling cost in such a way that the demand collected by each vehicle does not exceed the capacity and the total duration of each route is not greater than a maximum given time limit. We propose a mathematical formulation of the problem and introduce valid inequalities to strengthen the corresponding continuous relaxation. Moreover, we provide an aggregate formulation that allows us to introduce further inequalities. Then, we propose a two–phase exact algorithm for the solution of the UCGRPP. In the first phase, a branch-and-cut algorithm is used to solve the aggregate formulation and to identify a cut pool of aggregate valid inequalities to be used in the second phase, where a branch-and-cut algorithm is implemented to optimally solve the UCGRPP. Computational results on a large set of problem instances show that the use of the aggregate formulation is effective, making the two-phase exact algorithm able to optimally solve a large number of instances.