A column generation algorithm for the vehicle routing problem with soft time windows

The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms are added to the solution cost whenever a vehicle serves a customer outside of his time window. We present a branch-and-price algorithm which is the first exact optimization algorithm for this problem.     

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).     

Simple Heuristics for the Vehicle Routeing Problem with Soft Time Windows

We describe three simple heuristics for the vehicle routeing problem with customer time windows that can be violated by paying appropriate penalties. The customer demands are known, and a homogeneous fleet of vehicles stationed at a single depot is considered. The penalty payable to a customer is assumed to be a linear function of the amount of time window violation. Upper limits are imposed on both the penalty payable and the waiting time allowed at any customer. At each customer in a route, the PC-based heuristics simultaneously determine the actual time to begin service, and the next customer to serve. To achieve this, each heuristic uses different measures to compare the cost of waiting and penalty payable, with the benefit obtained by leaving immediately for the next customer. Computational results on a set of benchmark problems show that the procedure is efficient and enables significant reduction in the number of vehicles required and/or the total route distances while controlling both customer penalties and waiting times.     

A synthesis of assignment and heuristic solutions for vehicle routing with time windows

This paper proposes a three-stage method for the vehicle-routing problem with time window constraints (VRPTW). Using the Hungarian method the optimal customer matching for an assignment approximation of the VRPTW, which is a travel time-based relaxation that partially respects the time windows, is obtained. The assignment matching is transformed into feasible routes of the VRPTW via a simple decoupling heuristic. The best of these routes, in terms of travelling and vehicle waiting times, form part of the final solution, which is completed by the routes provided by heuristic methods applied to the remainder of the customers. The proposed approach is tested on a set of standard literature problems, and improves the results of the heuristic methods with respect to total travel time. Furthermore, it provides useful insights into the effect of employing optimal travel time solutions resulting from the assignment relaxation to derive partial route sets of the VRPTW.     

A computational comparison of algorithms for the inventory routing problem

The inventory routing problem is a distribution problem in which each customer maintains a local inventory of a product such as heating oil and consumes a certain amount of that product each day. Each day a fleet of trucks is dispatched over a set of routes to resupply a subset of the customers. In this paper, we describe and compare algorithms for this problem defined over a short planning period, e.g. one week. These algorithms define the set of customers to be serviced each day and produce routes for a fleet of vehicles to service those customers. Two algorithms are compared in detail, one which first allocates deliveries to days and then solves a vehicle routing problem and a second which treats the multi-day problem as a modified vehicle routing problem. The comparison is based on a set of real data obtained from a propane distribution firm in Pennsylvania. The solutions obtained by both procedures compare quite favorably with those in use by the firm.Part of this work was performed while this author was visiting the University of Waterloo.     

A branch-price-and-cut algorithm for the vehicle routing problem with time windows and multiple deliverymen

We address a variant of the vehicle routing problem with time windows that includes the decision of how many deliverymen should be assigned to each vehicle. In this variant, the service time at each customer depends on the size of the respective demand and on the number of deliverymen assigned to visit this customer. In addition, the objective function consists of minimizing a weighted sum of the total number of routes, number of deliverymen and traveled distance. These characteristics make this variant very challenging for exact methods. To date, only heuristic approaches have been proposed for this problem, and even the most efficient optimization solvers cannot find optimal solutions in a reasonable amount of time for instances of moderate size when using the available mathematical formulations. We propose a branch-price-and-cut method based on a new set partitioning formulation of the problem. To accelerate the convergence of the method, we rely on an interior-point column and cut generation process, a strong branching strategy and a mixed-integer programming-based primal heuristic. Additionally, a hierarchical branching strategy is used to take into account the different components of the objective function. The computational results indicate the benefits of using the proposed exact solution approach. We closed several instances of the problem and obtained upper bounds that were previously unknown in the literature.     

Routing relatively few customers per route

This paper presents a modification of the well-known Solomon (1987) sequential insertion heuristic I1 for the Vehicle Routing Problem with Time Windows (VRPTW). VRPTW involves servicing customers within a pre-specified service time window by homogeneously capacitated vehicles from a single depot. By using two new measures for the additional time needed to insert a customer in the route, significantly better solutions are obtained for relatively short routes compared to the original Solomon (1987) sequential insertion heuristic I1. Because high-quality initial heuristics often allow local searches and metaheuristics to achieve better solutions more quickly, the new approach is likely to help generating better solutions to practical routing problems.     

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.     

A multi-depot period vehicle routing problem arising in the utilities sector

This paper considers the resource planning problem of a utility company that provides preventive maintenance services to a set of customers using a fleet of depot-based mobile gangs. The problem is to determine the boundaries of the geographic areas served by each depot, the list of customers visited each day and the routes followed by the gangs. The objective is to provide improved customer service at minimum operating cost subject to constraints on frequency of visits, service time requirements, customer preferences for visiting on particular days and other routing constraints. The problem is solved as a Multi-Depot Period Vehicle Routing Problem (MDPVRP). The computational implementation of the complete planning model is described with reference to a pilot study and results are presented. The solution algorithm is used to construct cost-service trade-off curves for all depots so that management can evaluate the impact of different customer service levels on total routing costs.     

Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

Efficient techniques for the multi-period vehicle routing problem with time windows within a branch and price framework

In many appointment-based logistics systems customer orders may be served within a set of consecutive periods/days (i.e. a period window). In this case, the Multi-Period Vehicle Routing Problem (MPVRP) is relevant, and its efficient solution may lead to significant operational improvements. In this paper we investigate the MPVRP with Time Windows (MPVRPTW). The latter are time intervals within each period of the period window, during which service may be provided to the customer. A general model and an exact method to solve the MPVRPTW are presented. The solution method is based on column generation. Furthermore, two novel, efficient techniques to accelerate the solution procedure of the MPVRPTW are proposed. These techniques exploit the structure of the multi-period setting in order to identify similarities within the different subproblems and, thus, avoid solving every subproblem at each iteration. The experimental study analyzed the performance of the proposed methods systematically for various problem parameters, such as geographical distribution of customers and period window patterns. In most cases, the new methods improve significantly the efficiency of convergence to the optimal solution as compared to the classical method.     

A memetic algorithm for the multiperiod vehicle routing problem with profit

In this paper, we extend upon current research in the vehicle routing problem whereby labour regulations affect planning horizons, and therefore, profitability. We call this extension the multiperiod vehicle routing problem with profit (mVRPP). The goal is to determine routes for a set of vehicles that maximizes profitability from visited locations, based on the conditions that vehicles can only travel during stipulated working hours within each period in a given planning horizon and that the vehicles are only required to return to the depot at the end of the last period. We propose an effective memetic algorithm with a giant-tour representation to solve the mVRPP. To efficiently evaluate a chromosome, we develop a greedy procedure to partition a given giant-tour into individual routes, and prove that the resultant partition is optimal. We evaluate the effectiveness of our memetic algorithm with extensive experiments based on a set of modified benchmark instances. The results indicate that our approach generates high-quality solutions that are reasonably close to the best known solutions or proven optima, and significantly better than the solutions obtained using heuristics employed by professional schedulers.     

Product line selection and pricing under a share-of-surplus choice model

Product line selection and pricing decisions are critical to the profitability of many firms, particularly in today’s competitive business environment in which providers of goods and services are offering a broad array of products to satisfy customer needs.We address the problem of selecting a set of products to offer and their prices when customers select among the offered products according to a share-of-surplus choice model. A customer’s surplus is defined as the difference between his utility (willingness to pay) and the price of the product. Under the share-of-surplus model, the fraction of a customer segment that selects a product is defined as the ratio of the segment’s surplus from this particular product to the segment’s total surplus across all offered products with positive surplus for that segment.We develop a heuristic procedure for this non-concave, mixed-integer optimization problem. The procedure utilizes simulated annealing to handle the binary product selection variables, and a steepest-ascent-style procedure that relies on certain structural properties of the objective function to handle the non-concave, continuous portion of the problem involving the prices. We also develop a variant of our procedure to handle uncertainty in customer utilities. In computational studies, our basic procedures perform extremely well, producing solutions whose objective values are within about 5% of those obtained via enumerative methods. Our procedure to handle uncertain utilities also performs well, producing solutions with expected profit values that are roughly 10% higher than the corresponding expected profits from solutions obtained under the assumption of deterministic utilities.     

A heuristic method for dispatching repair men

A company has to provide service to its customers. A service consists of a visit to the customer plus the spending of some given time at the scene. The future customer demand is not known but the probability distribution for the demand may be known. When a customer call comes in, the company must immediately specify a time window within which the start of service will be provided. The problem is for a fixed service level to determine an optimal strategy of route design and time window setting so that the total distance travelled is minimized over the time horizon given. A heuristic method BARTOC (Booking Algorithm for Routing and Timing Of Customers) to solve the problem mentioned above is suggested. BARTOC is based on a cluster-first route-second approach. Some computational results are presented. The results indicate that BARTOC produces high quality solutions.Peter Matthiesen, Inc.Dano Chemo, Inc.     

A memetic algorithm for the team orienteering problem

The team orienteering problem (TOP) is a generalization of the orienteering problem. A limited number of vehicles is available to visit customers from a potential set. Each vehicle has a predefined running-time limit, and each customer has a fixed associated profit. The aim of the TOP is to maximize the total collected profit. In this paper we propose a simple hybrid genetic algorithm using new algorithms dedicated to the specific scope of the TOP: an Optimal Split procedure for chromosome evaluation and local search techniques for mutation. We have called this hybrid method a memetic algorithm for the TOP. Computational experiments conducted on standard benchmark instances clearly show our method to be highly competitive with existing ones, yielding new improved solutions in at least 5 instances.     

A Mathematical Programming Approach to Optimal Markovian Switching of Poisson Arrival Streams to Queueing Systems

Motivated by certain situations in manufacturing systems and communication networks, we look into the problem of maximizing the profit in a queueing system with linear reward and cost structure and having a choice of selecting the streams of Poisson arrivals according to an independent Markov chain. We view the system as a MMPP/GI/1 queue and seek to maximize the profits by optimally choosing the stationary probabilities of the modulating Markov chain. We consider two formulations of the optimization problem. The first one (which we call the PUT problem) seeks to maximize the profit per unit time whereas the second one considers the maximization of the profit per accepted customer (the PAC problem). In each of these formulations, we explore three separate problems. In the first one, the constraints come from bounding the utilization of an infinite capacity server; in the second one the constraints arise from bounding the mean queue length of the same queue; and in the third one the finite capacity of the buffer reflect as a set of constraints. In the problems bounding the utilization factor of the queue, the solutions are given by essentially linear programs, while the problems with mean queue length constraints are linear programs if the service is exponentially distributed. The problems modeling the finite capacity queue are non-convex programs for which global maxima can be found. There is a rich relationship between the solutions of the PUT and PAC problems. In particular, the PUT solutions always make the server work at a utilization factor that is no less than that of the PAC solutions.     

The Orienteering Problem with Time Windows

The orienteering problem with time windows, denoted by OPTW, belongs to a class of routeing and scheduling problems that arise in physical distribution. It may be modelled as a problem on a graph. It considers a set of nodes (customers), each with an associated profit and service duration (time window), and a set of arcs, each with an associated travel time. The objective of the problem is to construct an acyclic path beginning at a specified origin and ending at a specified destination that maximizes the total profit while observing time window constraints on all nodes and not exceeding a designated time limit. The problem is classified as NP-hard and, thus, an exact algorithm that executes in reasonable computational time is unlikely to exist. Since the problem is highly-constrained, we were able to develop a heuristic (referred to as the ‘tree’ heuristic) based upon an exhaustive search of the feasible solution space. The tree heuristic systematically generates a list of feasible paths and then selects the most profitable path from the list. In comparison with an insertion heuristic, the tree heuristic was found to produce improved values of total profit for heavily-constrained, modest-sized problems with reasonable computational effort.     

A priori orienteering with time windows and stochastic wait times at customers

In the pharmaceutical industry, sales representatives visit doctors to inform them of their products and encourage them to become an active prescriber. On a daily basis, pharmaceutical sales representatives must decide which doctors to visit and the order to visit them. This situation motivates a problem we more generally refer to as a stochastic orienteering problem with time windows (SOPTW), in which a time window is associated with each customer and an uncertain wait time at a customer results from a queue of competing sales representatives. We develop a priori routes with the objective of maximizing expected sales. We operationalize the sales representative’s execution of the a priori route with relevant recourse actions and derive an analytical formula to compute the expected sales from an a priori tour. We tailor a variable neighborhood search heuristic to solve the problem. We demonstrate the value of modeling uncertainty by comparing the solutions to our model to solutions of a deterministic version using expected values of the associated random variables. We also compute an empirical upper bound on our solutions by solving deterministic instances corresponding to perfect information.     

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.     

Heuristic procedures for a stochastic batch service problem

This paper considers a multi-class batch service problem that involves a class-dependent waiting cost and a service cost in determining customer batch sizes. Unlike a fixed service cost used widely in standard models, the service cost considered in this work is incurred only if the total service time is over the capacity. We formulate this problem as an infinite horizon Markov decision process, and exploit its structural properties to establish theoretical results, including bounds on the optimal action space. We use the results to improve the value iteration procedure. Furthermore, we design heuristic algorithms for large problems. The numerical experiments demonstrate that the class-dependent waiting cost has a considerable influence on the optimal customer batch size. Finally, we evaluate the efficiency of the proposed value iteration procedure and the quality of the heuristic solutions.