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The purpose of this paper is to present a new deterministic metaheuristic based on a modification of Variable Neighborhood Search of Mladenovic and Hansen (1997) for solving the vehicle routing problem with time windows. Results are reported for the standard 100, 200 and 400 customer data sets by Solomon (1987) and Gehring and Homberger (1999) and two real-life problems by Russell (1995). The findings indicate that the proposed procedure outperforms other recent local searches and metaheuristics. In addition four new best-known solutions were obtained. The proposed procedure is based on a new four-phase approach. In this approach an initial solution is first created using new route construction heuristics followed by route elimination procedure to improve the solutions regarding the number of vehicles. In the third phase the solutions are improved in terms of total traveled distance using four new local search procedures proposed in this paper. Finally in phase four the best solution obtained is improved by modifying the objective function to escape from a local minimum. (Metaheuristics; Vehicle Routing; Time Windows) 1.

The Vehicle Routing Problem with Time windows (VRPTW) is an extension of the capacity constrained Vehicle Routing Problem (VRP). The VRPTW is NP-Complete and instances with 100 customers or more are very hard to solve optimally. We represent the VRPTW as a multi-objective problem and present a genetic algorithm solution using the Pareto ranking technique. We use a direct interpretation of the VRPTW as a multi-objective problem, in which the two objective dimensions are number of vehicles and total cost (distance). An advantage of this approach is that it is unnecessary to derive weights for a weighted sum scoring formula. This prevents the introduction of solution bias towards either of the problem dimensions. We argue that the VRPTW is most naturally viewed as a multi-modal problem, in which both vehicles and cost are of equal value, depending on the needs of the user. A result of our research is that the multi-objective optimization genetic algorithm returns a set of solutions that fairly consider both of these dimensions. Our approach is quite effective, as it provides solutions competitive with the best known in the literature, as well as new solutions that are not biased toward the number of vehicles. A set of well-known benchmark data are used to compare the effectiveness of the proposed method for solving the VRPTW.

Abstract. This paper studies vehicle routing problems on asymmetric metrics. Our starting point is the directed k-TSP problem: given an asymmetric metric (V, d), a root r ∈ V and a target k ≤ |V |, compute the minimum length tour that contains r and at least k other vertices. We present a polynomial time O(log 2 n · log k)-approximation algorithm for this problem. We use this algorithm for directed k-TSP to obtain an O(log 2 n)-approximation algorithm for the directed orienteering problem. This answers positively, the question of poly-logarithmic approximability of directed orienteering, an open problem from Blum et al. [2]. The previously best known results were quasi-polynomial time algorithms with approximation guarantees of O(log 2 k) for directed k-TSP, and O(log n) for directed orienteering (Chekuri & Pal [4]). Using the algorithm for directed orienteering within the framework of Blum et al. [2] and Bansal et al. [1], we also obtain poly-logarithmic approximation algorithms for the directed versions of discounted-reward TSP and the vehicle routing problem with time-windows. 1

Abstract. In this paper we analyse a new evolutionary approach to the vehicle routing problem. We present Genetic Vehicle Representation (GVR), a two-level representational scheme designed to deal in an effective way with all the information that candidate solutions must encode. Experimental results show that this method is both effective and robust, allowing the discovery of new best solutions for some well-known benchmarks. 1

Abstract—This paper considers a heterogeneous team of cooperating unmanned aerial vehicles (UAVs) drawn from several distinct classes and engaged in a search and action mission over a spatially extended battlefield with targets of several types. During the mission, the UAVs seek to confirm and verifiably destroy suspected targets and discover, confirm, and verifiably destroy unknown targets. The locations of some (or all) targets are unknown apriori, requiring them to be located using cooperative search. In addition, the tasks to be performed at each target location by the team of cooperative UAVs need to be coordinated. The tasks must, therefore, be allocated to UAVs in real time as they arise, while ensuring that appropriate vehicles are assigned to each task. Each class of UAVs has its own sensing and attack capabilities, so the need for appropriate assignment is paramount. In this paper, an extensive dynamic model that captures the stochastic nature of the cooperative search and task assignment problems is developed, and algorithms for achieving a high level of performance are designed. The paper focuses on investigating the value of predictive task assignment as a function of the number of unknown targets and number of UAVs. In particular, it is shown that there is a tradeoff between search and task response in the context of prediction. Based on the results, a hybrid algorithm for switching the use of prediction is proposed, which balances the search and task response. The performance of the proposed algorithms is evaluated through Monte Carlo simulations. Index Terms—Cooperative search, path planning, task allocation, unmanned aerial vehicle. I.

Genetic Vehicle Representation (GVR) is a new two-level representational scheme designed to encode all the information required by potential solutions for the vehicle routing problem. In a previous paper we described a set of experiments performed with several instances from the Capacitated Vehicle Routing Problem (CVRP). In this preliminary investigation, GVR proved to be both effective and robust. In this work we extend the application of this new genetic representation to the vehicle routing problem with time windows, a variant that adds additional time constraints to the original definition. We present the results of a comprehensive set of tests that show that GVR is also efficient with this alternative, allowing the evolutionary computation algorithm to reach optimal solutions for some well know benchmarks. 1.

Abstract. In this paper, we study the following vehicle routing problem: given n vertices in a metric space, a specified root vertex r (the depot), and a length bound D, find a minimum cardinality set of r-paths that covers all vertices, such that each path has length at most D. This problem is NP-complete, even when the underlying metric is induced by a weighted star. We present a 4-approximation for this problem on tree metrics. On general metrics, we obtain an O(log D) approximation algorithm,andalsoan(O(log 1),1+ɛ) bicriteria approximation. All these

Uninhabited autonomous vehicles(UAVs) are an increasingly important part of battlefield environments, and may soon be common in civilian applications such as disaster relief, environmental monitoring and planetary exploration. Such vehicles may be airborne, land-based or aquatic, though the focus so far has been on airborne vehicles for military applications, and this is the focus of the research presented here. We consider a heterogeneous group of UAVs drawn from several distinct classes and engaged in a search and destroy mission over an extended battlefield. During the mission, the UAVs perform Search, Confirm, Attack and Battle Damage Assessment (BDA) tasks at various locations. The tasks are determined in real-time by the actions of all UAVs and their consequences (e.g., sensor readings), so that the task dynamics are stochastic. The tasks must, therefore, be allocated to UAVs in real-time as they arise, while ensuring that appropriate vehicles are assigned to each task. Each class of UAVs has its own sensing and attack capabilities, so the need for appropriate assignment is paramount. We present a simple...

The Vehicle Routing Problem with Time Windows (VRPTW) is a well-know and complex combinatorial problem, which has received considerable attention in recent years. This problem has been addressed using many different techniques including both exact and heuristic methods. The VRPTW benchmark problems of Solomon (1987) have been most commonly chosen to evaluate and to compare all solutions proposed in the literature. Results from exact methods have been improved considerably because of parallel implementations and modern branch-and-cut techniques. However, 25 out of the 56 high order instances from Solomon's test set still remain unsolved. Additionally, in many cases a prohibitive time is needed to find the exact solution. Many efficient heuristic methods have been developed to make possible a good solution in a reasonable amount of time. Unfortunately, while all publications on exact methods have considered the total traveled distance as their main objective, almost all of the heuristic attempts have considered the total number of vehicles as their main objective. Consequently, it is more difficult to compare and to take advantage of the strong points from each approach. With travelled distance as the main objective, this paper proposes a robust heuristic approach for the VRPTW problem using an efficient Genetic Algorithm and a MIP formulation. With this innovative approach it is possible to compare the results with the exact methods published. Additionally, computational results show that the heuristic approach proposed here outperforms all previously known heuristic methods published, in terms of minimization of traveled distance.