Aristide Mingozzi

State‐space relaxation procedures for the computation of bounds to routing problems

It is well-known that few combinatorial optimization problems can be solved effectively by dynamic programming alone, since the number of vertices of the state space graph is enormous. What we are proposing here is a general relaxation procedure whereby the state-space associated with a given dynamic programming recursion is relaxed in such a way that the solution to the relaxed recursion provides a bound which could be embedded in general branch and bound schemes for the solution of the problem. This state space relaxation method is analogous to Langrangian relaxation in integer programming. This paper gives a survey of this new methodology, and gives, as examples, applications to the traveling salesman problem (TSP), the timeconstrained TSP and the vehicle routing problem (VRP). Valid state space relaxations are discussed for these problems and several bounds are derived in each case. Subgradient optimization and “state space ascent” are discussed as methods of maximizing the resulting lower bounds. More details of the procedures surveyed in this paper can be found in [2 ,3 ,41.

An Exact Method for the Vehicle Routing Problem with Backhauls

We consider the problem in which a fleet of vehicles located at a central depot is to be optimallyused to serve a set of customers partitioned into two subsets of linehaul and backhaul customers. Each route starts and ends at the depot and the backhaul customers must be visited afterthe linehaul customers. A new (0-1) integer programming formulation of this problem is presented. We describe a procedure that computes a valid lower bound to the optimal solution cost by combining different heuristic methods for solving the dual of the LP-relaxation of the exact formulation. An algorithm for the exact solution of the problem is presented. Computational tests on problems proposed in the literature show the effectiveness of the proposed algorithms in solving problems up to 100 customers.

A Bionomic Approach to the Capacitated p-Median Problem

This paper advocates the use of the bionomic algorithm, a recently proposed metaheuristic technique, as an effective method to solve capacitated p-median problems (CPMP). Bionomic algorithms already proved to be an effective framework for finding good solutions to combinatorial optimization problems, when good local optimization algorithms are available. The paper also presents an effective local search technique for the CPMP. Computational results show the effectiveness of the proposed approach, when compared to the best performing heuristics so far presented in the literature.

The Rollon--Rolloff Vehicle Routing Problem

In this paper, a sanitation routing problem called the Rollon--Rolloff Vehicle Routing Problem (RRVRP) is defined. In the RRVRP, tractors move large trailers between locations and a disposal facility. The trailers are so large that the tractor can only transport one trailer at a time. In this paper, the RRVRP is defined, a mathematical programming formulation is presented, and two lower bounds and four heuristic algorithms are developed and tested on 20 different problems. Conclusions are derived and recommendations for further research are presented.

A heuristic procedure for the crew rostering problem

Abstract This paper deals with the problem of planning work schedules in a given time horizon so as to evenly distribute the workload among the drivers in a mass transit system. An integer programming formulation of this problem is given. An iterative heuristic algorithm is described which makes use of a lower bound derived from the mathematical formulation. Furthermore, the algorithm at each iteration solves a multilevel bottleneck assignment problem for which a new procedure that gives asymptotically optimal solutions is proposed. Computational results for both the rostering problem and the multilevel bottleneck assignment problem are given.

A new method for solving capacitated location problems based on a set partitioning approach

Abstract We consider the capacitated p-median problem (CPMP) in which a set of n customers must be partitioned into p disjoint clusters so that the total dissimilarity within each cluster is minimized and constraints on maximum cluster capacities are met. The dissimilarity of a cluster is computed as the sum of the dissimilarities existing between each entity of the cluster and the median associated to such cluster. In this paper we present an exact algorithm for solving the CPMP based on a set partitioning formulation of the problem. A valid lower bound to the optimal solution cost is obtained by combining two different heuristic methods for solving the dual of the LP-relaxation of the exact formulation. Computational tests on problems proposed in the literature and on new sets of test problems show the effectiveness of the proposed algorithm. Scope and purpose A basic location problem consists of locating a number of facilities or depots to supply a set of customers. The objective is to minimize the cost of locating the facilities and assigning the customers to them. This problem has been extensively studied in the literature and is commonly referred to as the plant location problem, or facility location problem. When each potential facility has a constraint on the maximum demand that it can supply and the number of facilities to locate is specified, the problem is known as the Capacitated p-median problem (CPMP). The purpose of this paper is to present a new exact algorithm for the CPMP and evaluate its computational performance on a set of test problems taken from the literature and on a new set of test problems.

A Set Partitioning Approach to the Crew Scheduling Problem

The crew scheduling problem (CSP) appears in many mass transport systems (e.g., airline, bus, and railway industry) and consists of scheduling a number of crews to operate a set of transport tasks satisfying a variety of constraints. This problem is formulated as a set partitioning problem with side constraints (SP), where each column of the SP matrix corresponds to a feasible duty, which is a subset of tasks performed by a crew. We describe a procedure that, without using the SP matrix, computes a lower bound to the CSP by finding a heuristic solution to the dual of the linear relaxation of SP. Such dual solution is obtained by combining a number of different bounding procedures. The dual solution is used to reduce the number of variables in the SP in such a way that the resulting SP problem can be solved by a branch-and-bound algorithm. Computational results are given for problems derived from the literature and involving from 50 to 500 tasks.

An exact algorithm for the Traveling Salesman Problem with Deliveries and Collections

In this paper, we describe a new integer programming formulation for the Traveling Salesman Problem with mixed Deliveries and Collections (TSPDC) based on a two-commodity network flow approach. We present new lower bounds that are derived from the linear relaxation of the new formulation by adding valid inequalities, in a cutting-plane fashion. The resulting lower bounds are embedded in a branch-and-cut algorithm for the optimal solution of the TSPDC. Computational results on different classes of test problems taken from the literature indicate the effectiveness of the proposed method.

A set partitioning approach to the multiple depot vehicle scheduling problem

We address the problem of scheduling a fleet of vehicles, stationed in different depots, in such a way to perform a set of time-tabled trips and to minimize a given objective function. We consider the consti that requires each vehicle to return to the starting depot. This problem, known as Multiple Depot V& Scheduling (MD-VSP), is NP-hard. In this paper we formulate the MD-VSP as a Set Partitioning Prot with side constraints (SP). We describe a procedure that, without using the SP matrix, computes a lower bound to the MD-VSP by finding a heuristic solution to the dual of the continuous relaxation of SP. dual solution obtained is used to reduce the number of variables in the SP in such a way that the resu SP problem can be solved by usual branch and bound techniques. The computational results show effectiveness of the proposed method.

The project scheduling problem with irregular starting time costs

We consider the problem of scheduling a set of activities satisfying precedence constraints in order to minimize the sum of the costs associated with the starting times of the activities. We consider the case in which the activity starting time cost functions are irregular functions. This problem can be solved in polynomial time either when the precedence graph has a special structure or when the starting time cost function of each activity is monotonic. A (0-1) integer programming formulation of this problem is presented and used to derive valid lower bounds to the optimal solution cost. An exact branch-and-bound algorithm is described. Computational results show the effectiveness of the proposed exact algorithm in solving problems up to 100 activities.