Single vehicle routing with a predefined customer sequence and multiple depot returns

The optimal routing of a single vehicle with limited capacity that delivers one product to n clients according to a predefined client sequence can be determined using dynamic programming. In the present paper we propose and investigate three practical variations of this problem: (i) the case of multiple-product deliveries when each product is stored in its own compartment in the vehicle, (ii) the case of multiple-product deliveries when all products are stored together in the vehicle’s single compartment, and (iii) the case in which the vehicle picks up from and delivers a single product to each customer. Suitable dynamic programming algorithms that find the optimal routing of the vehicle are developed for each case. The efficiency of the algorithms is studied by solving large problem sets.     

Stochastic single vehicle routing problem with delivery and pick up and a predefined customer sequence

In this paper we study the routing of a single vehicle that delivers products and picks up items with stochastic demand. The vehicle follows a predefined customer sequence and is allowed to return to the depot for loading/unloading as needed. A suitable dynamic programming algorithm is proposed to determine the minimum expected routing cost. Furthermore, the optimal routing policy to be followed by the vehicle’s driver is derived by proposing an appropriate theorem. The efficiency of the algorithm is studied by solving large problem sets.     

Stochastic single vehicle routing with a predefined customer sequence and multiple depot returns

We study the routing of a single vehicle that delivers multiple products under stochastic demand. Specifically, we investigate two practical variations of this problem: (i) The case in which each product type is stored in its dedicated compartment in the vehicle, and (ii) the case in which all products are stored together in the vehicle’s single compartment. Suitable dynamic programming algorithms are proposed to determine the minimum expected (routing) cost for each case. Furthermore, the optimal routing policy is derived by developing appropriate theorems. The efficiency of the algorithms is studied by solving large problem sets.     

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.     

A branch-and-price algorithm for a vehicle routing with demand allocation problem

We investigate the vehicle routing with demand allocation problem where the decision-maker jointly optimizes the location of delivery sites, the assignment of customers to (preferably convenient) delivery sites, and the routing of vehicles operated from a central depot to serve customers at their designated sites. We propose an effective branch-and-price (B&P) algorithm that is demonstrated to greatly outperform the use of commercial branch-and-bound/cut solvers such as CPLEX. Central to the efficacy of the proposed B&P algorithm is the development of a specialized dynamic programming procedure that extends works on elementary shortest path problems with resource constraints in order to solve the more complex column generation pricing subproblem. Our computational study demonstrates the efficacy of the proposed approach using a set of 60 problem instances. Moreover, the proposed methodology has the merit of providing optimal solutions in run times that are significantly shorter than those reported for decomposition-based heuristics in the literature.     

A branch-and-price algorithm for the capacitated vehicle routing problem with stochastic demands

This article introduces a new exact algorithm for the capacitated vehicle routing problem with stochastic demands (CVRPSD). The CVRPSD can be formulated as a set partitioning problem and it is shown that the associated column generation subproblem can be solved using a dynamic programming scheme. Computational experiments show promising results.     

A column generation approach for location-routing problems with pickup and delivery

In this paper we formulate an integer programming model for the Location and Routing Problem with Pickup and Delivery. We propose a column generation scheme and implement, for the subproblem, a label-setting algorithm for the shortest path with pickup and delivery and time windows problem. We also propose a set of heuristics to speed up this process. To validate the model, we implement the column generation scheme and test it on different instances developed in this paper. We also provide an analysis of how the costs of opening depots and the fixed cost of routes affect the optimal solution.     

Dynamic pickup and delivery problems

In the last decade, there has been an increasing body of research in dynamic vehicle routing problems. This article surveys the subclass of those problems called dynamic pickup and delivery problems, in which objects or people have to be collected and delivered in real-time. It discusses some general issues as well as solution strategies.     

Branch-and-cut with lazy separation for the vehicle routing problem with simultaneous pickup and delivery

We propose a branch-and-cut algorithm for the VRPSPD where the constraints that ensure that the capacities are not exceeded in the middle of a route are applied in a lazy fashion. The algorithm was tested in 87 instances with 50–200 customers, finding improved lower bounds and several new optimal solutions.     

A multi-commodity, capacitated pickup and delivery problem: The single and two-vehicle cases

We explore dynamic programming solutions for a multi-commodity, capacitated pickup and delivery problem. Cargo flows are given by an origin/destination matrix which is not necessarily symmetric. This problem is a generalization of several known pickup and delivery problems, as regards both problem structure and objective function. Solution approaches are developed for the single-vehicle and two-vehicle cases. The fact that for each cargo that goes from a node i to another node j there may be a cargo going in the opposite direction provides the motivation for the two-vehicle case, because one may conceivably consider solutions where no cargoes that travel in opposite directions between node pairs are carried by the same vehicle. Yet, it is shown that such scenarios are generally sub-optimal. As expected, the computational effort of the single vehicle algorithm is exponential in the number of cargoes. For the two-vehicle case, said effort is of an order of magnitude that is not higher than that of the single-vehicle case. Some rudimentary examples are presented or both the single-vehicle and two-vehicle cases so as to better illustrate the method.     

Replenishment routing problems between a single supplier and multiple retailers with direct delivery

We consider the replenishment routing problems of one supplier who can replenish only one of multiple retailers per period, while different retailers need different periodical replenishment. For simple cases satisfying certain conditions, we obtain the simple routing by which the supplier can replenish each retailer periodically so that shortage will not occur. For complicated cases, using number theory, especially the Chinese remainder theorem, we present an algorithm to calculate a feasible routing so that the supplier can replenish the selected retailers on the selected periods without shortages.     

City-courier routing and scheduling problems

This research seeks to propose innovative routing and scheduling strategies to help city couriers reduce operating costs and enhance service level. The strategies are realized by constructing a new type of routing and scheduling problem. The problem directly takes into account the inherent physical and operating constraints associated with riding in city distribution networks, which makes the problem involve multiple objectives and visiting specified nodes and arcs. Through network transformations, this study first formulates the city-courier routing and scheduling problem as a multi-objective multiple traveling salesman problem with strict time windows (MOMTSPSTW) that is NP-hard and new to the literature, and then proposes a multi-objective Scatter Search framework that seeks to find the set of Pareto-optimal solutions to the problem. Various new and improved sub-procedures are embedded in the solution framework. This is followed by an empirical study that shows and analyzes the results of applying the proposed method to a real-life city-courier routing and scheduling problem.     

Vehicle scheduling with combinable delivery and pickup operations

Motivated by the logistics operations in an express delivery company, we develop and study a new scheduling model. The problem seems similar to scheduling with sequence-dependent setup times and scheduling with release times, however, the ability to combine or separate the job operations makes our problem unique.     

Heuristic algorithms for a vehicle routing problem with simultaneous delivery and pickup and time windows in home health care

This paper addresses a vehicle scheduling problem encountered in home health care logistics. It concerns the delivery of drugs and medical devices from the home care company’s pharmacy to patients’ homes, delivery of special drugs from a hospital to patients, pickup of bio samples and unused drugs and medical devices from patients. The problem can be considered as a special vehicle routing problem with simultaneous delivery and pickup and time windows, with four types of demands: delivery from depot to patient, delivery from a hospital to patient, pickup from a patient to depot and pickup from a patient to a medical lab. Each patient is visited by one vehicle and each vehicle visits each node at most once. Patients are associated with time windows and vehicles with capacity. Two mixed-integer programming models are proposed. We then propose a Genetic Algorithm (GA) and a Tabu Search (TS) method. The GA is based on a permutation chromosome, a split procedure and local search. The TS is based on route assignment attributes of patients, an augmented cost function, route re-optimization, and attribute-based aspiration levels. These approaches are tested on test instances derived from existing VRPTW benchmarks.     

A goal programming approach to vehicle routing problems with soft time windows

The classical vehicle routing problem involves designing a set of routes for a fleet of vehicles based at one central depot that is required to serve a number of geographically dispersed customers, while minimizing the total travel distance or the total distribution cost. Each route originates and terminates at the central depot and customers demands are known. In many practical distribution problems, besides a hard time window associated with each customer, defining a time interval in which the customer should be served, managers establish multiple objectives to be considered, like avoiding underutilization of labor and vehicle capacity, while meeting the preferences of customers regarding the time of the day in which they would like to be served (soft time windows). This work investigates the use of goal programming to model these problems. To solve the model, an enumeration-followed-by-optimization approach is proposed which first computes feasible routes and then selects the set of best ones. Computational results show that this approach is adequate for medium-sized delivery problems.     

Real-time split-delivery pickup and delivery time window problems with transfers

In this research we present the design and implementation of heuristics for solving split-delivery pickup and delivery time window problems with transfer (SDPDTWP) of shipments between vehicles for both static and real-time data sets. In the SDPDTWP each shipment is constrained with the earliest possible pickup time from the origin and the latest acceptable delivery time to a destination. Split-deliveries occur when two or more vehicles service the same origin or destination. The proposed heuristics were applied to both static and real-time data sets. The heuristics computed a solution, in a few seconds, for a static problem from the literature, achieving an improvement of 60% in distance in comparison to the published solution. In the real-time SDPDTWP problems, requests for pickup and delivery of a package, breakdown of a truck or insertion of a truck can occur after the vehicle has left the origin and is enroute to service the customers. Thirty data sets, each consisting of one to seven real-time customer or truck events, were used to test the efficiency of the heuristics. The heuristics obtained solutions to real-time data sets in under five seconds of CPU time.      

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.     

Maritime crude oil transportation - A split pickup and split delivery problem

The maritime oil tanker routing and scheduling problem is known to the literature since before 1950. In the presented problem, oil tankers transport crude oil from supply points to demand locations around the globe. The objective is to find ship routes, load sizes, as well as port arrival and departure times, in a way that minimizes transportation costs. We introduce a path flow model where paths are ship routes. Continuous variables distribute the cargo between the different routes. Multiple products are transported by a heterogeneous fleet of tankers. Pickup and delivery requirements are not paired to cargos beforehand and arbitrary split of amounts is allowed. Small realistic test instances can be solved with route pre-generation for this model. The results indicate possible simplifications and stimulate further research.     

Solving the multi-criteria time-dependent routing and scheduling problem in a multimodal fixed scheduled network

Multi-criteria routing and scheduling in a multimodal fixed scheduled network with time-dependent travel times involves the determination of the non-dominated itineraries (i.e., paths enhanced with scheduled departures) under the following constraints: (i) visiting a given set of intermediate stops in a specified sequence, and (ii) strict time windows on the origin, the destination and the intermediate stops. The objective of this paper is to present the formulation and algorithmic solution for the multi-criteria itinerary planning problem that takes into account the aforementioned features. The algorithmic approach proposed is based on the decomposition of the problem to a sequence of elementary itinerary sub-problems, solved by a dynamic programming algorithm. The computational performance of the algorithms on a set of large scale test problems indicates non-prohibitive time requirements and encourages its integration into travel planning decision support systems.     

The pickup and delivery problem with transfers: Formulation and a branch-and-cut solution method

In this paper, a strict formulation of a generalization of the classical pickup and delivery problem is presented. Here, we add the flexibility of providing the option for passengers to transfer from one vehicle to another at specific locations. As part of the mathematical formulation, we include transfer nodes where vehicles may interact interchanging passengers. Additional variables to keep track of customers along their route are considered. The formulation has been proven to work correctly, and by means of a simple example instance, we conclude that there exist some configurations in which a scheme allowing transfers results in better quality optimal solutions. Finally, a solution method based on Benders decomposition is addressed. We compare the computational effort of this application with a straight branch and bound strategy; we also provide insights to develop more efficient set partitioning formulations and associated algorithms for solving real-size problems.