Richard F. Hartl

A survey of the maximum principles for optimal control problems with state constraints

This paper gives a survey of the various forms of Pontryagins maximum principle for optimal control problems with state variable inequality constraints. The relations between the different sets of optimality conditions arising in these forms are shown. Furthermore, the application of these maximum principle conditions is demonstrated by solving some illustrative examples.

An improved Ant System algorithm for theVehicle Routing Problem

The Ant System is a distributed metaheuristic that combines an adaptive memory with alocal heuristic function to repeatedly construct solutions of hard combinatorial optimizationproblems. In this paper, we present an improved ant system algorithm for the Vehicle RoutingProblem with one central depot and identical vehicles. Computational results on fourteenbenchmark problems from the literature are reported and a comparison with five othermetaheuristic approaches for solving Vehicle Routing Problems is given.

Applying the ANT System to the Vehicle Routing Problem

In this paper we use a recently proposed metaheuristic, the Ant System, to solve the Vehicle Routing Problem in its basic form, i.e., with capacity and distance restrictions, one central depot and identical vehicles. A “hybrid” Ant System algorithm is first presented and then improved using problem-specific information (savings, capacity utilization). Experiments on various aspects of the algorithm and computational results for fourteen benchmark problems are reported and compared to those of other metaheuristic approaches such as Tabu Search, Simulated Annealing and Neural Networks.

D-Ants: savings based ants divide and conquer the vehicle routing problem

This paper presents an algorithm that builds on the Savings based Ant System presented in [Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2002), Morgan Kaufmann, San Francisco, 2002] and enhances its performance in terms of computational effort. This is achieved by decomposing the problem and solving only the much smaller subproblems resulting from the decomposition.The computational study and statistical analysis conducted both on standard benchmark problem instances as well as on new large scale Vehicle Routing Problem instances will show that the approach does not only improve the efficiency, but also improves the effectiveness of the algorithm leading to a fast and powerful problem solving tool for real world sized Vehicle Routing Problems.

A variable neighborhood search heuristic for periodic routing problems

The aim of this paper is to propose a new heuristic for the Periodic Vehicle Routing Problem (PVRP) without time windows. The PVRP extends the classical Vehicle Routing Problem (VRP) to a planning horizon of several days. Each customer requires a certain number of visits within this time horizon while there is some flexibility on the exact days of the visits. Hence, one has to choose the visit days for each customer and to solve a VRP for each day. Our method is based on Variable Neighborhood Search (VNS). Computational results are presented, that show that our approach is competitive and even outperforms existing solution procedures proposed in the literature. Also considered is the special case of a single vehicle, i.e. the Periodic Traveling Salesman Problem (PTSP). It is shown that slight changes of the proposed VNS procedure is also competitive for the PTSP.

A Variable Neighborhood Search for the Multi Depot Vehicle Routing Problem with Time Windows

The aim of this paper is to propose an algorithm based on the philosophy of the Variable Neighborhood Search (VNS) to solve Multi Depot Vehicle Routing Problems with Time Windows. The paper has two main contributions. First, from a technical point of view, it presents the first application of a VNS for this problem and several design issues of VNS algorithms are discussed. Second, from a problem oriented point of view the computational results show that the approach is competitive with an existing Tabu Search algorithm with respect to both solution quality and computation times.

Ant colony optimization for the two-dimensional loading vehicle routing problem

In this paper a combination of the two most important problems in distribution logistics is considered, known as the two-dimensional loading vehicle routing problem. This problem combines the loading of the freight into the vehicles, and the successive routing of the vehicles along the road network, with the aim of satisfying the demands of the customers. The problem is solved by different heuristics for the loading part, and by an ant colony optimization (ACO) algorithm for the overall optimization. The excellent behavior of the algorithm is proven through extensive computational results. The contribution of the paper is threefold: first, on small-size instances the proposed algorithm reaches a high number of proven optimal solutions, while on large-size instances it clearly outperforms previous heuristics from the literature. Second, due to its flexibility in handling different loading constraints, including items rotation and rear loading, it allows us to draw qualitative conclusions of practical interest in transportation, such as evaluating the potential savings by permitting more flexible loading configurations. Third, in ACO a combination of different heuristic information usually did not turn out to be successful in the past. Our approach provides an example where an ACO algorithm successfully combines two completely different heuristic measures (with respect to loading and routing) within one pheromone matrix.

Pareto ant colony optimization with ILP preprocessing in multiobjective project portfolio selection

Abstract One of the most important, common and critical management issues lies in determining the “best” project portfolio out of a given set of investment proposals. As this decision process usually involves the pursuit of multiple objectives amid a lack of a priori preference information, its quality can be improved by implementing a two-phase procedure that first identifies the solution space of all efficient (i.e., Pareto-optimal) portfolios and then allows an interactive exploration of that space. However, determining the solution space is not trivial because brute-force complete enumeration only solves small instances and the underlying NP-hard problem becomes increasingly demanding as the number of projects grows. While meta-heuristics in general provide an attractive compromise between the computational effort necessary and the quality of an approximated solution space, Pareto ant colony optimization (P-ACO) has been shown to perform particularly well for this class of problems. In this paper, the beneficial effect of P-ACO’s core function (i.e., the learning feature) is substantiated by means of a numerical example based on real world data. Furthermore, the original P-ACO approach is supplemented by an integer linear programming (ILP) preprocessing procedure that identifies several efficient portfolio solutions within a few seconds and correspondingly initializes the pheromone trails before running P-ACO. This extension favors a larger exploration of the search space at the beginning of the search and does so at a low cost.

Metaheuristics for vehicle routing problems with three-dimensional loading constraints

This paper addresses an important combination of three-dimensional loading and vehicle routing, known as the Three-Dimensional Loading Capacitated Vehicle Routing Problem. The problem calls for the combined optimization of the loading of freight into vehicles and the routing of vehicles along a road network, with the aim of serving customers with minimum traveling cost. Despite its clear practical relevance in freight distribution, the literature on this problem is very limited. This is because of its high combinatorial complexity. We solve the problem by means of an Ant Colony Optimization algorithm, which makes use of fast packing heuristics for the loading. The algorithm combines two different heuristic information measures, one for routing and one for packing. In numerical tests all publicly available test instances are solved, and for almost all instances new best solutions are found.

Heuristics for the multi-period orienteering problem with multiple time windows

We present the multi-period orienteering problem with multiple time windows (MuPOPTW), a new routing problem combining objective and constraints of the orienteering problem (OP) and team orienteering problem (TOP), constraints from standard vehicle routing problems, and original constraints from a real-world application. The problem itself comes from a real industrial case. Specific route duration constraints result in a route feasibility subproblem. We propose an exact algorithm for this subproblem, and we embed it in a variable neighborhood search method to solve the whole routing problem. We then provide experimental results for this method. We compare them to a commercial solver. We also adapt our method to standard benchmark OP and TOP instances, and provide comparative tables with state-of-the-art algorithms.