An adaptive ejection pool with toggle-rule diversification approach for the capacitated team orienteering problem

In the capacitated team orienteering problem (CTOP), we are given a set of homogeneous vehicles and a set of customers each with a service demand value and a profit value. A vehicle can get the profit of a customer by satisfying its demand, but the total demand of all customers in its route cannot exceed the vehicle capacity and the length of the route must be within a specified maximum. The problem is to design a set of routes that maximizes the total profit collected by the vehicles. In this article, we propose a new heuristic algorithm for the CTOP using the ejection pool framework with an adaptive strategy and a diversification mechanism based on toggling between two priority rules. Experimental results show that our algorithm can match or improve all the best known results on the standard CTOP benchmark instances proposed by Archetti et al. (2008).     

A new bilevel formulation for the vehicle routing problem and a solution method using a genetic algorithm

The Vehicle Routing Problem (VRP) is one of the most well studied problems in operations research, both in real life problems and for scientific research purposes. During the last 50 years a number of different formulations have been proposed, together with an even greater number of algorithms for the solution of the problem. In this paper, the VRP is formulated as a problem of two decision levels. In the first level, the decision maker assigns customers to the vehicles checking the feasibility of the constructed routes (vehicle capacity constraints) and without taking into account the sequence by which the vehicles will visit the customers. In the second level, the decision maker finds the optimal routes of these assignments. The decision maker of the first level, once the cost of each routing has been calculated in the second level, estimates which assignment is the better one to choose. Based on this formulation, a bilevel genetic algorithm is proposed. In the first level of the proposed algorithm, a genetic algorithm is used for calculating the population of the most promising assignments of customers to vehicles. In the second level of the proposed algorithm, a Traveling Salesman Problem (TSP) is solved, independently for each member of the population and for each assignment to vehicles. The algorithm was tested on two sets of benchmark instances and gave very satisfactory results. In both sets of instances the average quality is less than 1%. More specifically in the set with the 14 classic instances proposed by Christofides, the quality is 0.479% and in the second set with the 20 large scale vehicle routing problems, the quality is 0.826%. The algorithm is ranked in the tenth place among the 36 most known and effective algorithms in the literature for the first set of instances and in the sixth place among the 16 algorithms for the second set of instances. The computational time of the algorithm is decreased significantly compared to other heuristic and metaheuristic algorithms due to the fact that the Expanding Neighborhood Search Strategy is used.     

Branch-and-price-and-cut for the multiple traveling repairman problem with distance constraints

In this paper, we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel; the resulting problem is called the multiple traveling repairmen problem with distance constraints (MTRPD). In the MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers. Each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once. The objective is to minimize the total waiting time of all customers after the vehicles leave the depot. To optimally solve the MTRPD, we propose a new exact branch-and-price-and-cut algorithm, where the column generation pricing subproblem is a resource-constrained elementary shortest-path problem with cumulative costs. An ad hoc label-setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem. Computational results show the effectiveness of the proposed method. The optimal solutions to 179 out of 180 test instances are reported in this paper. Our computational results serve as benchmarks for future researchers on the problem.     

Swarm intelligence-based hyper-heuristic for the vehicle routing problem with prioritized customers

The vehicle routing problem (VRP) is a combinatorial optimization management problem that seeks the optimal set of routes traversed by a vehicle to deliver products to customers. A recognized problem in this domain is to serve ‘prioritized’ customers in the shortest possible time where customers with known demands are supplied by one or several depots. This problem is known as the Vehicle Routing with Prioritized Customers (VRPC). The purpose of this work is to present and compare two artificial intelligence-based novel methods that minimize the traveling distance of vehicles when moving cargo to prioritized customers. Various studies have been conducted regarding this topic; nevertheless, up to now, few studies used the Cuckoo Search-based hyper-heuristic. This paper modifies a classical mathematical model that represents the VRPC, implements and tests an evolutionary Cuckoo Search-based hyper-heuristic, and then compares the results with those of our proposed modified version of the Clarke Wright (CW) algorithm. In this modified version, the CW algorithm serves all customers per their preassigned priorities while covering the needed working hours. The results indicate that the solution selected by the Cuckoo Search-based hyper-heuristic outperformed the modified Clarke Wright algorithm while taking into consideration the customers’ priority and demands and the vehicle capacity.

     

Metaheuristics for the team orienteering problem

The Team Orienteering Problem (TOP) is the generalization to the case of multiple tours of the Orienteering Problem, known also as Selective Traveling Salesman Problem. A set of potential customers is available and a profit is collected from the visit to each customer. A fleet of vehicles is available to visit the customers, within a given time limit. The profit of a customer can be collected by one vehicle at most. The objective is to identify the customers which maximize the total collected profit while satisfying the given time limit for each vehicle. We propose two variants of a generalized tabu search algorithm and a variable neighborhood search algorithm for the solution of the TOP and show that each of these algorithms beats the already known heuristics. Computational experiments are made on standard instances.     

An iterated local search algorithm for the vehicle routing problem with convex time penalty functions

We propose an iterated local search algorithm for the vehicle routing problem with time window constraints. We treat the time window constraint for each customer as a penalty function, and assume that it is convex and piecewise linear. Given an order of customers each vehicle to visit, dynamic programming (DP) is used to determine the optimal start time to serve the customers so that the total time penalty is minimized. This DP algorithm is then incorporated in the iterated local search algorithm to efficiently evaluate solutions in various neighborhoods. The amortized time complexity of evaluating a solution in the neighborhoods is a logarithmic order of the input size (i.e., the total number of linear pieces that define the penalty functions). Computational comparisons on benchmark instances with up to 1000 customers show that the proposed method is quite effective, especially for large instances.     

An exact algorithm for a single-vehicle routing problem with time windows and multiple routes

This paper describes an exact algorithm for solving a problem where the same vehicle performs several routes to serve a set of customers with time windows. The motivation comes from the home delivery of perishable goods, where vehicle routes are short and must be combined to form a working day. A method based on an elementary shortest path algorithm with resource constraints is proposed to solve this problem. The method is divided into two phases: in the first phase, all non-dominated feasible routes are generated; in the second phase, some routes are selected and sequenced to form the vehicle workday. Computational results are reported on Euclidean problems derived from benchmark instances of the classical vehicle routing problem with time windows.     

线型/圈型网络上单台车辆分群调度问题

本文研究线型/圈型网络上单台车辆分群调度问题。给定一个线型/圈型网络,若干客户分布其中。所有客户被划分成若干个子集,每个子集称为一个群。每个客户有一个释放时间和一个服务时间。给定一台车辆,其需要服务所有客户,且每个群内的客户连续服务。问题的要求是计算一个时间表,使得车辆能够按要求服务完所有客户并返回初始出发位置所花费的时间最少。针对该问题,就线型网络和圈型网络,分别给出一个7/4和一个13/7近似算法。     

Heuristics and memetic algorithm for the two-dimensional loading capacitated vehicle routing problem with time windows

This work deals with a new combinatorial optimization problem, the two-dimensional loading capacitated vehicle routing problem with time windows which is a realistic extension of the well known vehicle routing problem. The studied problem consists in determining vehicle trips to deliver rectangular objects to a set of customers with known time windows, using a homogeneous fleet of vehicles, while ensuring a feasible loading of each vehicle used. Since it includes NP-hard routing and packing sub-problems, six heuristics are firstly designed to quickly compute good solutions for realistic instances. They are obtained by combining algorithms for the vehicle routing problem with time windows with heuristics for packing rectangles. Then, a Memetic algorithm is developed to improve the heuristic solutions. The quality and the efficiency of the proposed heuristics and metaheuristic are evaluated by adding time windows to a set of 144 instances with 15–255 customers and 15–786 items, designed by Iori et al. (Transport Sci 41:253–264, 2007) for the case without time windows.     

Optimised crossover genetic algorithm for capacitated vehicle routing problem

This paper presents a genetic algorithm for solving capacitated vehicle routing problem, which is mainly characterised by using vehicles of the same capacity based at a central depot that will be optimally routed to supply customers with known demands. The proposed algorithm uses an optimised crossover operator designed by a complete undirected bipartite graph to find an optimal set of delivery routes satisfying the requirements and giving minimal total cost. We tested our algorithm with benchmark instances and compared it with some other heuristics in the literature. Computational results showed that the proposed algorithm is competitive in terms of the quality of the solutions found.     

A branch and cut algorithm for the location-routing problem with simultaneous pickup and delivery

This paper addresses a location-routing problem with simultaneous pickup and delivery (LRPSPD) which is a general case of the location-routing problem. The LRPSPD is defined as finding locations of the depots and designing vehicle routes in such a way that pickup and delivery demands of each customer must be performed with same vehicle and the overall cost is minimized. We propose an effective branch-and-cut algorithm for solving the LRPSPD. The proposed algorithm implements several valid inequalities adapted from the literature for the problem and a local search based on simulated annealing algorithm to obtain upper bounds. Computational results, for a large number of instances derived from the literature, show that some instances with up to 88 customers and 8 potential depots can be solved in a reasonable computation time.     

An Ant Colony Algorithm for the Capacitated Vehicle Routing

The Vehicle Routing Problem (VRP) requires the determination of an optimal set of routes for a set of vehicles to serve a set of customers. We deal here with the Capacitated Vehicle Routing Problem (CVRP) where there is a maximum weight or volume that each vehicle can load. We developed an Ant Colony algorithm (ACO) for the CVRP based on the metaheuristic technique introduced by Colorni, Dorigo and Maniezzo. We present preliminary results that show that ant algorithms are competitive with other metaheuristics for solving CVRP.     

The multi-trip vehicle routing problem

The basic vehicle routing problem is concerned with the design of a set of routes to serve a given number of customers, minimising the total distance travelled. In that problem, each vehicle is assumed to be used only once during a planning period, which is typically a day, and therefore is unrepresentative of many practical situations, where a vehicle makes several journeys during a day. The present authors have previously published an algorithm which outperformed an experienced load planner working on the complex, real-life problems of Burton's Biscuits, where vehicles make more than one trip each day. This present paper uses a simplified version of that general algorithm, in order to compare it with a recently published heuristic specially designed for the theoretical multi-trip vehicle routing problem.     

The Clustered Orienteering Problem

In this paper we study a generalization of the Orienteering Problem (OP) which we call the Clustered Orienteering Problem (COP). The OP, also known as the Selective Traveling Salesman Problem, is a problem where a set of potential customers is given and a profit is associated with the service of each customer. A single vehicle is available to serve the customers. The objective is to find the vehicle route that maximizes the total collected profit in such a way that the duration of the route does not exceed a given threshold. In the COP, customers are grouped in clusters. A profit is associated with each cluster and is gained only if all customers belonging to the cluster are served. We propose two solution approaches for the COP: an exact and a heuristic one. The exact approach is a branch-and-cut while the heuristic approach is a tabu search. Computational results on a set of randomly generated instances are provided to show the efficiency and effectiveness of both approaches.     

Single vehicle routing problems with a predefined customer sequence, compartmentalized load and stochastic demands

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers according to a particular customer order. The demands of the customers for each product are assumed to be random variables with known distributions. Each product type is stored in its dedicated compartment in the vehicle. Using a suitable dynamic programming algorithm we find the policy that satisfies the demands of the customers with the minimum total expected cost. We also prove that this policy has a specific threshold-type structure. Furthermore, we investigate a corresponding infinite-time horizon problem in which the service of the customers does not stop when the last customer has been serviced but it continues indefinitely with the same customer order. It is assumed that the demands of the customers at different tours have the same distributions. It is shown that the discounted-cost optimal policy and the average-cost optimal policy have the same threshold-type structure as the optimal policy in the original problem. The theoretical results are illustrated by numerical examples.     

An interactive GRAMPS algorithm for the heterogeneous fixed fleet vehicle routing problem with and without backhauls

In this article, a visual interactive approach based on a new greedy randomised adaptive memory programming search (GRAMPS) algorithm is proposed to solve the heterogeneous fixed fleet vehicle routing problem (HFFVRP) and a new extension of the HFFVRP, which is called heterogeneous fixed fleet vehicle routing problem with backhauls (HFFVRPB). This problem involves two different sets of customers. Backhaul customers are pickup points and linehaul customers are delivery points that are to be serviced from a single depot by a heterogeneous fixed fleet of vehicles, each of which is restricted in the capacity it can carry, with different variable travelling costs.     

An approximation algorithm for the pickup and delivery vehicle routing problem on trees

This paper presents an approximation algorithm for a vehicle routing problem on a tree-shaped network with a single depot where there are two types of demands, pickup demand and delivery demand. Customers are located on nodes of the tree, and each customer has a positive demand of pickup and/or delivery.Demands of customers are served by a fleet of identical vehicles with unit capacity. Each vehicle can serve pickup and delivery demands. It is assumed that the demand of a customer is splittable, i.e., it can be served by more than one vehicle. The problem we are concerned with in this paper asks to find a set of tours of the vehicles with minimum total lengths. In each tour, a vehicle begins at the depot with certain amount of goods for delivery, visits a subset of the customers in order to deliver and pick up goods and returns to the depot. At any time during the tour, a vehicle must always satisfy the capacity constraint, i.e., at any time the sum of goods to be delivered and that of goods that have been picked up is not allowed to exceed the vehicle capacity. We propose a 2-approximation algorithm for the problem.     

Lasso solution strategies for the vehicle routing problem with pickups and deliveries

This paper considers the vehicle routing problem with pickups and deliveries (VRPPD) where the same customer may require both a delivery and a pickup. This is the case, for instance, of breweries that deliver beer or mineral water bottles to a set of customers and collect empty bottles from the same customers. It is possible to relax the customary practice of performing a pickup when delivering at a customer, and postpone the pickup until the vehicle has sufficient free capacity. In the case of breweries, these solutions will often consist of routes in which bottles are first delivered until the vehicle is partly unloaded, then both pickup and delivery are performed at the remaining customers, and finally empty bottles are picked up from the first visited customers. These customers are revisited in reverse order, thus giving rise to lasso shaped solutions. Another possibility is to relax the traditional problem even more and allow customers to be visited twice either in two different routes or at different times on the same route, giving rise to a general solution. This article develops a tabu search algorithm capable of producing lasso solutions. A general solution can be reached by first duplicating each customer and generating a Hamiltonian solution on the extended set of customers. Test results show that while general solutions outperform other solution shapes in term of cost, their computation can be time consuming. The best lasso solution generated within a given time limit is generally better than the best general solution produced with the same computing effort.     

High-Level Relay Hybrid Metaheuristic Method for Multi-Depot Vehicle Routing Problem with Time Windows

This paper presents an efficient hybrid metaheuristic solution for multi-depot vehicle routing with time windows (MD-VRPTW). MD-VRPTW involves the routing of a set of vehicles with limited capacity from a set of depots to a set of geographically dispersed customers with known demands and predefined time windows. The present work aims at using a hybrid metaheuristic algorithm in the class of High-Level Relay Hybrid (HRH) which works in three levels and uses an efficient genetic algorithm as the main optimization algorithm and tabu search as an improvement method. In the genetic algorithm various heuristics incorporate local exploitation in the evolutionary search. An operator deletion- retrieval strategy is executed which shows the efficiency of the inner working of the proposed method. The proposed algorithm is applied to solve the problems of the standard Cordeau??s Instances. Results show that proposed approach is quite effective, as it provides solutions that are competitive with the best known in the literature.     

A tabu search algorithm for the single vehicle routing allocation problem

In the single vehicle routing allocation problem (SVRAP) we have a single vehicle, together with a set of customers, and the problem is one of deciding a route for the vehicle (starting and ending at given locations) such that it visits some of the customers. Customers not visited by the vehicle can either be allocated to a customer on the vehicle route, or they can be isolated. The objective is to minimize a weighted sum of routing, allocation and isolation costs. One special case of the general SVRAP is the median cycle problem, also known as the ring star problem, where no isolated vertices are allowed. Other special cases include the covering tour problem, the covering salesman problem and the shortest covering path problem. In this paper, we present a tabu search algorithm for the SVRAP. Our tabu search algorithm includes aspiration, path relinking and frequency-based diversification. Computational results are presented for test problems used previously in the literature and our algorithm is compared with the results obtained by other researchers. We also report results for much larger problems than have been considered by others.