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

Workload Equity in Vehicle Routing Problems: A Survey and Analysis

Over the past two decades, equity aspects have been considered in a growing number of models and methods for vehicle routing problems (VRPs). Equity concerns most often relate to fairly allocating workloads and to balancing the utilization of resources, and many practical applications have been reported in the literature. However, there has been only limited discussion about how workload equity should be modeled in VRPs, and various measures for optimizing such objectives have been proposed and implemented without a critical evaluation of their respective merits and consequences. This article addresses this gap with an analysis of classical and alternative equity functions for biobjective VRP models. In our survey, we review and categorize the existing literature on equitable VRPs. In the analysis, we identify a set of axiomatic properties that an ideal equity measure should satisfy, collect six common measures, and point out important connections between their properties and those of the resulting Pareto-optimal solutions. To gauge the extent of these implications, we also conduct a numerical study on small biobjective VRP instances solvable to optimality. Our study reveals two undesirable consequences when optimizing equity with nonmonotonic functions: Pareto-optimal solutions can consist of non-TSP-optimal tours, and even if all tours are TSP optimal, Pareto-optimal solutions can be workload inconsistent, i.e. composed of tours whose workloads are all equal to or longer than those of other Pareto-optimal solutions. We show that the extent of these phenomena should not be underestimated. The results of our biobjective analysis are valid also for weighted sum, constraint-based, or single-objective models. Based on this analysis, we conclude that monotonic equity functions are more appropriate for certain types of VRP models, and suggest promising avenues for further research.Comment: Accepted Manuscrip

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

The Multi-Depot Minimum Latency Problem with Inter-Depot Routes

The Minimum Latency Problem (MLP) is a class of routing problems that seeks to minimize the wait times (latencies) of a set of customers in a system. Similar to its counterparts in the Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP), the MLP is NP-hard. Unlike these other problem classes, however, the MLP is customer-oriented and thus has impactful potential for better serving customers in settings where they are the highest priority. While the VRP is very widely researched and applied to many industry settings to reduce travel times and costs for service-providers, the MLP is a more recent problem and does not have nearly the body of literature supporting it as found in the VRP. However, it is gaining significant attention recently because of its application to such areas as disaster relief logistics, which are a growing problem area in a global context and have potential for meaningful improvements that translate into reduced suffering and saved lives. An effective combination of MLP\u27s and route minimizing objectives can help relief agencies provide aid efficiently and within a manageable cost. To further the body of literature on the MLP and its applications to such settings, a new variant is introduced here called the Multi-Depot Minimum Latency Problem with Inter-Depot Routes (MDMLPI). This problem seeks to minimize the cumulative arrival times at all customers in a system being serviced by multiple vehicles and depots. Vehicles depart from one central depot and have the option of refilling their supply at a number of intermediate depots. While the equivalent problem has been studied using a VRP objective function, this is a new variant of the MLP. As such, a mathematical model is introduced along with several heuristics to provide the first solution approaches to solving it. Two objectives are considered in this work: minimizing latency, or arrival times at each customer, and minimizing weighted latency, which is the product of customer need and arrival time at that customer. The case of weighted latency carries additional significance as it may correspond to a larger number of customers at one location, thus adding emphasis to the speed with which they are serviced. Additionally, a discussion on fairness and application to disaster relief settings is maintained throughout. To reflect this, standard deviation among latencies is also evaluated as a measure of fairness in each of the solution approaches. Two heuristic approaches, as well as a second-phase adjustment to be applied to each, are introduced. The first is based on an auction policy in which customers bid to be the next stop on a vehicle\u27s tour. The second uses a procedure, referred to as an insertion technique, in which customers are inserted one-by-one into a partial routing solution such that each addition minimizes the (weighted) latency impact of that single customer. The second-phase modification takes the initial solutions achieved in the first two heuristics and considers the (weighted) latency impact of repositioning nodes one at a time. This is implemented to remove potential inefficient routing placements from the original solutions that can have compounding effects for all ensuing stops on the tour. Each of these is implemented on ten test instances. A nearest neighbor (greedy) policy and previous solutions to these instances with a VRP objective function are used as benchmarks. Both heuristics perform well in comparison to these benchmarks. Neither heuristic appears to perform clearly better than the other, although the auction policy achieves slightly better averages for the performance measures. When applying the second-phase adjustment, improvements are achieved and lead to even greater reductions in latency and standard deviation for both objectives. The value of these latency reductions is thoroughly demonstrated and a call for further research regarding customer-oriented objectives and evaluation of fairness in routing solutions is discussed. Finally, upon conclusion of the results presented in this work, several promising areas for future work and existing gaps in the literature are highlighted. As the body of literature surrounding the MLP is small yet growing, these areas constitute strong directions with important relevance to Operations Research, Humanitarian Logistics, Production Systems, and more

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

Approximation Algorithms for Capacitated Minimum Forest Problems in Wireless sensor Networks with a Mobile Sink

To deploy a wireless sensor network for the purpose of large-scale monitoring, in this paper, we propose a heterogeneous and hierarchical wireless sensor network architecture. The architecture consists of sensor nodes, gateway nodes, and mobile sinks. Th

Approximation algorithms for regret minimization in vehicle routing problems

In this thesis, we present new approximation algorithms as well as hardness of approximation results for NP-hard vehicle routing problems related to public transportation. We consider two different problem classes that also occur frequently in areas such as logistics, robotics, or distribution systems. For the first problem class, the goal is to visit as many locations in a network as possible subject to timing or cost constraints. For the second problem class, a given set of locations is to be visited using a minimum-cost set of routes under some constraints. Due to the relevance of both problem classes for public transportation, a secondary objective must be taken into account beyond a low operation cost: namely, it is crucial to design routes that optimize customer satisfaction in order to encourage customers to use the service. Our measure of choice is the regret of a customer, that is the time comparison of the chosen route with the shortest path to a destination. From the first problem class, we investigate variants and extensions of the Orienteering problem that asks to find a short walk maximizing the profit obtained from visiting distinct locations. We give approximation algorithms for variants in which the walk has to respect constraints on the regret of the visited vertices. Additionally, we describe a framework to extend approximation algorithms for Orienteering problems to consider also a second budget constraint, namely node demands, that have to be satisfied in order to collect the profit. We obtain polynomial time approximation schemes for the Capacitated Orienteering problem on trees and Euclidean metrics. Furthermore, we study variants of the School Bus problem (SBP). In SBP, a given set of locations is to be connected to a destination node with both low operation cost and a low maximum regret. We note that the Orienteering problem can be seen as the pricing problem for SBP and it often appears as subroutine in algorithms for SBP. For tree-shaped networks, we describe algorithms with a small constant approximation factor and complement them by showing hardness of approximation results. We give an overview of the known results in arbitrary networks and we prove that a general variant cannot be approximated unless P = NP. Finally, we describe an integer programming approach to solve School Bus problems in practice and present an improved bus schedule for a private school in the lake Geneva region

A Vehicle Routing Problem with Payload-Range Dependency by Fuel Consumption

In this research, a new variant of vehicle routing problem is introduced. Fuel consumption constitutes a significant component of transportation costs especially when large volumes of goods are transported using means of transportation such as aircrafts. Hence, the objective of this research is to perform efficient routing of a heterogeneous fleet of vehicles such that fuel consumption costs are minimized. Reduced fuel consumption also reduces greenhouse gases emission and creates a positive impact on the environment. Another unique characteristic studied is the dependence between load carried by a vehicle and the maximum distance it can travel without stopping. Weight of fuel is considered along with the load carried for vehicle capacity constraints. Split delivery and time window constraints are also considered. A mathematical model for the new problem has been developed. It has been implemented to solve a real-world case study for express delivery of goods. An initial solution greedy algorithm and a tabu search heuristic algorithm have also been developed in order to solve large scale instances of the problem. Comparison with optimal solution suggests that a good solution can be obtained using the heuristic algorithm in relatively short time