Approximation Algorithms for Capacitated k-Travelling Repairmen Problems

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem

The multi-depot k-traveling repairman problem

In this paper, we study the multi-depot k-traveling repairman problem. This problem extends the traditional traveling repairman problem to the multi-depot case. Its objective, similar to the single depot variant, is the minimization of the sum of the arrival times to customers. We propose two distinct formulations to model the problem, obtained on layered graphs. In order to find feasible solutions for the largest instances, we propose a hybrid genetic algorithm where initial solutions are built using a splitting heuristic and a local search is embedded into the genetic algorithm. The efficiency of the mathematical formulations and of the solution approach are investigated through computational experiments. The proposed models are scalable enough to solve instances up to 240 customers

An Effective Metaheuristic for Multiple Traveling Repairman Problem with Distance Constraints

Multiple Traveling Repairman Problem with Distance Constraints (MTRPD) is an extension of the NP-hard Multiple Traveling Repairman Problem. In MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers with the following constraints. First, each vehicle's travel distance is limited by a threshold. Second, each customer must be visited exactly once. Our goal is to find the visiting order that minimizes the sum of waiting times. To solve MTRPD we propose to combine the Insertion Heuristic (IH), Variable Neighborhood Search (VNS), and Tabu Search (TS) algorithms into an effective two-phase metaheuristic that includes a construction phase and an improvement phase. In the former phase, IH is used to create an initial solution. In the latter phase, we use VNS to generate various neighborhoods, while TS is employed to mainly prohibit from getting trapped into cycles. By doing so, our algorithm can support the search to escape local optima. In addition, we introduce a novel neighborhoods’ structure and a constant time operation which are efficient for calculating the cost of each neighboring solution. To show the efficiency of our proposed metaheuristic algorithm, we extensively experiment on benchmark instances. The results show that our algorithm can find the optimal solutions for all instances with up to 50 vertices in a fraction of seconds. Moreover, for instances from 60 to 80 vertices, almost all found solutions fall into the range of 0.9 %-1.1 % of the optimal solutions' lower bounds in a reasonable duration. For instances with a larger number of vertices, the algorithm reaches good-quality solutions fast. Moreover, in a comparison to the state-of-the-art metaheuristics, our proposed algorithm can find better solutions

Agriculture fleet vehicle routing: A decentralised and dynamic problem

To date, the research on agriculture vehicles in general and Agriculture Mobile Robots (AMRs) in particular has focused on a single vehicle (robot) and its agriculture-specific capabilities. Very little work has explored the coordination of fleets of such vehicles in the daily execution of farming tasks. This is especially the case when considering overall fleet performance, its efficiency and scalability in the context of highly automated agriculture vehicles that perform tasks throughout multiple fields potentially owned by different farmers and/or enterprises. The potential impact of automating AMR fleet coordination on commercial agriculture is immense. Major conglomerates with large and heterogeneous fleets of agriculture vehicles could operate on huge land areas without human operators to effect precision farming. In this paper, we propose the Agriculture Fleet Vehicle Routing Problem (AF-VRP) which, to the best of our knowledge, differs from any other version of the Vehicle Routing Problem studied so far. We focus on the dynamic and decentralised version of this problem applicable in environments involving multiple agriculture machinery and farm owners where concepts of fairness and equity must be considered. Such a problem combines three related problems: the dynamic assignment problem, the dynamic 3-index assignment problem and the capacitated arc routing problem. We review the state-of-the-art and categorise solution approaches as centralised, distributed and decentralised, based on the underlining decision-making context. Finally, we discuss open challenges in applying distributed and decentralised coordination approaches to this problem

The cumulative capacitated vehicle routing problem with min-sum and min-max objectives: An effective hybridisation of adaptive variable neighbourhood search and large neighbourhood search

The cumulative capacitated vehicle routing problem (CCVRP) is a relatively new variant of the classical capacitated vehicle routing problem in which the objective is to minimise the sum of arrival times at customers (min-sum) instead of the total route distance. While the literature for the CCVRP is scarce, this problem has useful applications especially in the area of supplying humanitarian aid after a natural disaster. In this paper, a two-stage adaptive variable neighbourhood search (AVNS) algorithm that incorporates large neighbourhood search (LNS) as a diversification strategy is proposed. When tested on the benchmark data sets, the results show that the proposed AVNS is highly competitive in producing new best known solutions to more than half of the instances. An alternative but related objective that minimises the maximum arrival time (min-max) is also explored in this study demonstrating the flexibility and the effectiveness of the proposed metaheuristic. To the best of our knowledge, this is the first study that exploits the min-max objective of the CCVRP in addition to providing extensive computational results for a large number of instances for the min-sum. As a by-product of this study, managerial insights for decision making are also presented

Solving a Location, Allocation, and Capacity Planning Problem with Dynamic Demand and Response Time Service Level

Logistic systems with uncertain demand, travel time, and on-site processing time are studied here where sequential trip travel is allowed. The relationship between three levels of decisions: facility location, demand allocation, and resource capacity (number of service units), satisfying the response time requirement, is analysed. The problem is formulated as a stochastic mixed integer program. A simulation-based hybrid heuristic is developed to solve the dynamic problem under different response time service level. An initial solution is obtained from solving static location-allocation models, followed by iterative improvement of the three levels of decisions by ejection, reinsertion procedure with memory of feasible and infeasible service regions. Results indicate that a higher response time service level could be achieved by allocating a given resource under an appropriate decentralized policy. Given a response time requirement, the general trend is that the minimum total capacity initially decreases with more facilities. During this stage, variability in travel time has more impact on capacity than variability in demand arrivals. Thereafter, the total capacity remains stable and then gradually increases. When service level requirement is high, the dynamic dispatch based on first-come-first-serve rule requires smaller capacity than the one by nearest-neighbour rule

The cumulative capacitated vehicle routing problem with min-sum and min-max objectives: An effective hybridisation of adaptive variable neighbourhood search and large neighbourhood search

The cumulative capacitated vehicle routing problem (CCVRP) is a relatively new variant of the classical capacitated vehicle routing problem in which the objective is to minimise the sum of arrival times at customers (min-sum) instead of the total route distance. While the literature for the CCVRP is scarce, this problem has useful applications especially in the area of supplying humanitarian aid after a natural disaster. In this paper, a two-stage adaptive variable neighbourhood search (AVNS) algorithm that incorporates large neighbourhood search (LNS) as a diversification strategy is proposed. When tested on the benchmark data sets, the results show that the proposed AVNS is highly competitive in producing new best known solutions to more than half of the instances. An alternative but related objective that minimises the maximum arrival time (min-max) is also explored in this study demonstrating the flexibility and the effectiveness of the proposed metaheuristic. To the best of our knowledge, this is the first study that exploits the min-max objective of the CCVRP in addition to providing extensive computational results for a large number of instances for the min-sum. As a by-product of this study, managerial insights for decision making are also presented