Very large-scale vehicle routing: new test problems,algorithms, and results

The standard vehicle routing problem was introduced in the OR/MS literature about 45 years ago. Since then, the vehicle routing problem has attracted an enormous amount of research attention. In the late 1990s, large vehicle routing problem instances with nearly 500 customers were generated and solved using metaheuristics. In this paper, we focus on very large vehicle routing problems. Our contributions are threefold. First, we present problem instances with as many as 1200 customers along with estimated solutions. Second, we introduce the variable-length neighbor list as a tool to reduce the number of unproductive computations. Third, we apply record-to-record travel with a variable-length neighbor list to 32 problem instances and obtain high-quality solutions, very quickly.     

Solving the vehicle routing problem with adaptive memory programming methodology

In this paper we develop an adaptive memory programming method for solving the capacitated vehicle routing problem called Solutions’ Elite PArts Search (SEPAS). This iterative method, first generates initial solutions via a systematic diversification technique and stores their routes in an adaptive memory. Subsequently, a constructive heuristic merges route components (called elite parts) from those in the adaptive memory. Finally, a tabu search approach improves the heuristically constructed solution and the adaptive memory is appropriately updated. SEPAS has been tested on two benchmark data sets and provides high quality solutions in short computational times for all problem instances. The method reaches several new best solutions for benchmark instances with a large number of customers.     

A Multi-Trip Split-Delivery Vehicle Routing Problem with Time Windows for Inventory Replenishment Under Stochastic Travel Times

This study considers a multi-trip split-delivery vehicle routing problem with soft time windows for daily inventory replenishment under stochastic travel times. Considering uncertainty in travel times for vehicle routing problems is beneficial because more robust schedules can be generated and unanticipated consequences can be reduced when schedules are implemented in reality. However, uncertainties in model parameters have rarely been addressed for the problems in this category mainly due to the high problem complexity. In this study, an innovative and practical approach is proposed to consider stochastic travel times in the planning process. In the planning model, the possible outcomes of vehicle arrivals and product delivery at retailers are systematically categorized and their associated penalty and reward are estimated. Thus, unanticipated costs for every scheduling decision can be incorporated into the planning model to generate vehicle routing schedules that are more robust facing uncertain traffic conditions. To solve the model that is characterized as an NP-hard problem in a reasonable amount of time, a two-stage heuristic solution algorithm is proposed. Finally, the stochastic model is compared with the deterministic model in both planning and simulated operation stages using the data of a supply chain in Taiwan. The result confirms that the schedule generated by the stochastic model is more robust than the one created with the deterministic model because undesired outcomes such as unfulfilled demands are greatly reduced.     

A hybrid evolutionary algorithm for heterogeneous fleet vehicle routing problems with time windows

This paper presents a hybrid evolutionary algorithm (HEA) to solve heterogeneous fleet vehicle routing problems with time windows. There are two main types of such problems, namely the fleet size and mix vehicle routing problem with time windows (F) and the heterogeneous fixed fleet vehicle routing problem with time windows (H), where the latter, in contrast to the former, assumes a limited availability of vehicles. The main objective is to minimize the fixed vehicle cost and the distribution cost, where the latter can be defined with respect to en-route time (T) or distance (D). The proposed unified algorithm is able to solve the four variants of heterogeneous fleet routing problem, called FT, FD, HT and HD, where the last variant is new. The HEA successfully combines several metaheuristics and offers a number of new advanced efficient procedures tailored to handle the heterogeneous fleet dimension. Extensive computational experiments on benchmark instances have shown that the HEA is highly effective on FT, FD and HT. In particular, out of the 360 instances we obtained 75 new best solutions and matched 102 within reasonable computational times. New benchmark results on HD are also presented.     

A deterministic tabu search algorithm for the capacitated arc routing problem

The capacitated arc routing problem (CARP) is a difficult optimisation problem in vehicle routing with applications where a service must be provided by a set of vehicles on specified roads. A heuristic algorithm based on tabu search is proposed and tested on various sets of benchmark instances. The computational results show that the proposed algorithm produces high quality results within a reasonable computing time. Some new best solutions are reported for a set of test problems used in the literature.     

The multi-depot split delivery vehicle routing problem: An integer programming-based heuristic, new test problems, and computational results

The multi-depot split delivery vehicle routing problem combines the split delivery vehicle routing problem and the multiple depot vehicle routing problem. We define this new problem and develop an integer programming-based heuristic for it. We apply our heuristic to 30 instances to determine the reduction in distance traveled that can be achieved by allowing split deliveries among vehicles based at the same depot and vehicles based at different depots. We generate new test instances with high-quality, visually estimated solutions and report results on these instances.     

Full-shipload tramp ship routing and scheduling with variable speeds

This paper investigates the simultaneous optimization problem of routing and sailing speed in the context of full-shipload tramp shipping. In this problem, a set of cargoes can be transported from their load to discharge ports by a fleet of heterogeneous ships of different speed ranges and load-dependent fuel consumption. The objective is to determine which orders to serve and to find the optimal route for each ship and the optimal sailing speed on each leg of the route so that the total profit is maximized. The problem originated from a real-life challenge faced by a Danish tramp shipping company in the tanker business. To solve the problem, a three-index mixed integer linear programming formulation as well as a set packing formulation is presented. A novel Branch-and-Price algorithm with efficient data preprocessing and heuristic column generation is proposed. The computational results on the test instances generated from real-life data show that the heuristic provides optimal solutions for small test instances and near-optimal solutions for larger test instances in a short running time. The effects of speed optimization and the sensitivity of the solutions to the fuel price change are analyzed. It is shown that speed optimization can improve the total profit by 16% on average and the fuel price has a significant effect on the average sailing speed and total profit.     

A parallel simulated annealing method for the vehicle routing problem with simultaneous pickup–delivery and time windows

This paper addresses a variant of the vehicle routing problem in which customers require simultaneous pickup and delivery of goods during specific individual time windows (VRPSPDTW). A general mixed integer programming model is employed to minimize the routing cost due to: the cost of vehicles and the travel cost of vehicles. A parallel Simulated Annealing (p-SA) algorithm that includes a Residual Capacity and Radial Surcharge (RCRS) insertion-based heuristic is developed and applied to solve this NP-hard optimization problem. Computational results are reported for 65 test problems from Wang and Chen’s benchmark and compared with the results from a Genetic Algorithm (GA) that minimizes the number of vehicles (NV) as the primary objective. Experimental results demonstrate the effectiveness of the p-SA algorithm, which is able to achieve the same objective response as 100% of the Wang and Chen small-scale benchmarks (number of customers from 10 to 50). For the Wang and Chen medium-scale benchmarks (number of 100 customers), the p-SA algorithm obtains better NV solutions for 12 instances and the same NV solutions for the remaining 44 instances. For the 44 instances with the same NV solutions, a secondary objective, travel distance (TD), the p-SA provides better solutions than the GA for 16 instances, and equal solutions for 7 instances. In addition, solutions are found for 30 large-scale instances, with customers of 200, 400, 600, 800 and 1000, which may serve as a new benchmark for the VRPSPDTW problem.     

A multi-phase heuristic for the production routing problem

This study considers the production routing problem where a plant produces and distributes a single item to multiple retailers over a multi-period time horizon. The problem is to decide on when and how much to produce and stock at the plant, when and how much to serve and stock at each retailer, and vehicle routes for shipments such that the sum of fixed production setup cost, variable production cost, distribution cost, and inventory carrying cost at the plant and retailers is minimized. A multi-phase heuristic is proposed for the problem. The proposed heuristic is a mathematical programming-based heuristic that relies on formulating and solving restricted versions of the problem as mixed integer programs. The computational experiments on benchmark instances show favorable results with regard to the quality of the solutions found at the expense of higher computing times on large instances. In particular, the heuristic managed to find new best solutions for the 65% of benchmark instances.     

An efficient variable neighborhood search heuristic for very large scale vehicle routing problems

In this paper, we present an efficient variable neighborhood search heuristic for the capacitated vehicle routing problem. The objective is to design least cost routes for a fleet of identically capacitated vehicles to service geographically scattered customers with known demands. The variable neighborhood search procedure is used to guide a set of standard improvement heuristics. In addition, a strategy reminiscent of the guided local search metaheuristic is used to help escape local minima. The developed solution method is specifically aimed at solving very large scale real-life vehicle routing problems. To speed up the method and cut down memory usage, new implementation concepts are used. Computational experiments on 32 existing large scale benchmarks, as well as on 20 new very large scale problem instances, demonstrate that the proposed method is fast, competitive and able to find high-quality solutions for problem instances with up to 20,000 customers within reasonable CPU times.     

Multiscale production routing in multicommodity supply chains with complex production facilities

In this work, we introduce the multiscale production routing problem (MPRP), which considers the coordination of production, inventory, distribution, and routing decisions in multicommodity supply chains with complex continuous production facilities. We propose an MILP model involving two different time grids. While a detailed mode-based production scheduling model captures all critical operational constraints on the fine time grid, vehicle routing is considered in each time period of the coarse time grid. In order to solve large instances of the MPRP, we propose an iterative MILP-based heuristic approach that solves the MILP model with a restricted set of candidate routes at each iteration and dynamically updates the set of candidate routes for the next iteration. The results of an extensive computational study show that the proposed algorithm finds high-quality solutions in reasonable computation times, and in large instances, it significantly outperforms a standard two-phase heuristic approach and a solution strategy involving a one-time heuristic pre-generation of candidate routes. Similar results are achieved in an industrial case study, which considers a real-world industrial gas supply chain.     

An exact hybrid method for the vehicle routing problem with time windows and multiple deliverymen

The vehicle routing problem with time windows and multiple deliverymen (VRPTWMD) is a variant of the vehicle routing problem with time windows in which service times at customers depend on the number of deliverymen assigned to the route that serves them. In particular, a larger number of deliverymen in a route leads to shorter service times. Hence, in addition to the usual routing and scheduling decisions, the crew size for each route is also an endogenous decision. This problem is commonly faced by companies that deliver goods to customers located in busy urban areas, a situation that requires nearby customers to be grouped in advance so that the deliverymen can serve all customers in a group during one vehicle stop. Consequently, service times can be relatively long compared to travel times, which complicates serving all scheduled customers during regular work hours. In this paper, we propose a hybrid method for the VRPTWMD, combining a branch-price-and-cut (BPC) algorithm with two metaheuristic approaches. We present a wide variety of computational results showing that the proposed hybrid approach outperforms the BPC algorithm used as standalone method in terms of both solution quality and running time, in some classes of problem instances from the literature. These results indicate the advantages of using specific algorithms to generate good feasible solutions within an exact method.     

Heuristic algorithms for a three-dimensional loading capacitated vehicle routing problem in a carrier

In this paper, we present heuristic algorithms for a three-dimensional loading capacitated vehicle routing problem arising in a real-world situation. In this problem, customers make requests of goods, which are packed in a sortment of boxes. The objective is to find minimum cost delivery routes for a set of identical vehicles that, departing from a depot, visit all customers only once and return to the depot. Apart of the usual 3D container loading constraints which ensure that the boxes are packed completely inside the vehicles and that the boxes do not overlap each other in each vehicle, the problem also takes into account constraints related to the vertical stability of the cargo and multi-drop situations. The algorithms are based on the combination of classical heuristics from both vehicle routing and container loading literatures, as well as two metaheuristic strategies, and their use in more elaborate procedures. Although these approaches cannot assure optimal solutions for the respective problems, they are relatively simple, fast enough to solve real instances, flexible enough to include other practical considerations, and normally assure relatively good solutions in acceptable computational times in practice. The approaches are also sufficiently generic to be embedded with algorithms other than those considered in this study, as well as they can be easily adapted to consider other practical constraints, such as the load bearing strength of the boxes, time windows and pickups and deliveries. Computational tests were performed with these methods considering instances based on the vehicle routing literature and actual customers’ orders, as well as instances based on a real-world situation of a Brazilian carrier. The results show that the heuristics are able to produce relatively good solutions for real instances with hundreds of customers and thousands of boxes.     

Solving the truck and trailer routing problem based on a simulated annealing heuristic

In this study, we consider the application of a simulated annealing (SA) heuristic to the truck and trailer routing problem (TTRP), a variant of the vehicle routing problem (VRP). In the TTRP, some customers can be serviced by either a complete vehicle (that is, a truck pulling a trailer) or a single truck, while others can only be serviced by a single truck for various reasons. SA has seen widespread applications to various combinatorial optimization problems, including the VRP. However, to our best knowledge, it has not been applied to the TTRP. So far, all the best known results for benchmark TTRP instances were obtained using tabu search (TS). We applied SA to the TTRP and obtained 17 best solutions to the 21 benchmark TTRP benchmark problems, including 11 new best solutions. Moreover, the computational time required by the proposed SA heuristic is less than those reported in prior studies. The results suggest that SA is competitive with TS on solving the TTRP.     

Hyperpath-based vehicle routing and scheduling method in time-varying networks for airport shuttle service

It is very significant for a reasonable vehicle routing and scheduling in city airport shuttle service to decrease operational costs and increase passenger satisfaction. Most of the existing reports for such problems assumed that the travel time was invariable. However, the ever-increasing traffic congestion often makes it variable. In this study, considering the time-varying networks, a vehicle routing and scheduling method is proposed, where the time-varying feature enables the traveler to select a direction among all the Pareto-optimal paths at each node in response to the knowledge of the time window demands. Such Pareto-optimal paths are referred to hyperpaths herein. To obtain the hyperpaths, an exact algorithm is designed in this study for addressing the bi-criteria shortest paths problem, where the travel time comes to be discontinuous time-varying. Given the techniques that generate all Pareto-optimal solutions exhibiting exponential worst-case computational complexity, embedded in the exact algorithm, a computationally efficient bound strategy is reported on the basis of passenger locations, pickup time windows and arrival time windows. As such, the vehicle routing and scheduling problem viewed as an arc selection model can be solved by a proposed heuristic algorithm combined with a dynamic programming method. A series of experiments by using the practical pickup data indicate that the proposed methods can obtain cost-saving schedules under the condition of time-varying travel times.

     

A dynamic vehicle routing problem with time-dependent travel times

In this paper we present a formulation for the dynamic vehicle routing problem with time-dependent travel times. We also present a genetic algorithm to solve the problem. The problem is a pick-up or delivery vehicle routing problem with soft time windows in which we consider multiple vehicles with different capacities, real-time service requests, and real-time variations in travel times between demand nodes.The performance of the genetic algorithm is evaluated by comparing its results with exact solutions and lower bounds for randomly generated test problems. For small size problems with up to 10 demands, the genetic algorithm provides almost the same results as the exact solutions, while its computation time is less than 10% of the time required to produce the exact solutions. For the problems with 30 demand nodes, the genetic algorithm results have less than 8% gap with lower bounds.This research also shows that as the uncertainty in the travel time information increases, a dynamic routing strategy that takes the real-time traffic information into account becomes increasingly superior to a static one. This is clear when we compare the static and dynamic routing strategies in problem scenarios that have different levels of uncertainty in travel time information. In additional tests on a simulated network, the proposed algorithm works well in dealing with situations in which accidents cause significant congestion in some part of the transportation network.     

A genetic and set partitioning two-phase approach for the vehicle routing problem with time windows

The Vehicle Routing Problem with Time Windows (VRPTW) is a well-known and complex combinatorial problem, which has received considerable attention in recent years. This problem has been addressed using many different techniques including both exact and heuristic methods. The VRPTW benchmark problems of Solomon [Algorithms for the vehicle routing and scheduling problems with time window constraints, Operations Research 1987; 35(2): 254–65] have been most commonly chosen to evaluate and compare all algorithms. Results from exact methods have been improved considerably because of parallel implementations and modern branch-and-cut techniques. However, 24 out of the 56 high order instances from Solomon's original test set still remain unsolved. Additionally, in many cases a prohibitive time is needed to find the exact solution. Many of the heuristic methods developed have proved to be efficient in identifying good solutions in reasonable amounts of time. Unfortunately, whilst the research efforts based on exact methods have been focused on the total travel distance, the focus of almost all heuristic attempts has been on the number of vehicles. Consequently, it is more difficult to compare and take advantage of the strong points from each approach. This paper proposes a robust heuristic approach for the VRPTW using travel distance as the main objective through an efficient genetic algorithm and a set partitioning formulation. The tests were produced using real numbers and truncated data type, allowing a direct comparison of its results against previously published heuristic and exact methods. Furthermore, computational results show that the proposed heuristic approach outperforms all previously known and published heuristic methods in terms of the minimal travel distance.     

The robust vehicle routing problem with time windows

This paper addresses the robust vehicle routing problem with time windows. We are motivated by a problem that arises in maritime transportation where delays are frequent and should be taken into account. Our model only allows routes that are feasible for all values of the travel times in a predetermined uncertainty polytope, which yields a robust optimization problem. We propose two new formulations for the robust problem, each based on a different robust approach. The first formulation extends the well-known resource inequalities formulation by employing adjustable robust optimization. We propose two techniques, which, using the structure of the problem, allow to reduce significantly the number of extreme points of the uncertainty polytope. The second formulation generalizes a path inequalities formulation to the uncertain context. The uncertainty appears implicitly in this formulation, so that we develop a new cutting plane technique for robust combinatorial optimization problems with complicated constraints. In particular, efficient separation procedures are discussed. We compare the two formulations on a test bed composed of maritime transportation instances. These results show that the solution times are similar for both formulations while being significantly faster than the solutions times of a layered formulation recently proposed for the problem.     

A note on the truck and trailer routing problem

This study considers the relaxed truck and trailer routing problem (RTTRP), a relaxation of the truck and trailer routing problem (TTRP). TTRP is a variant of the well studied vehicle routing problem (VRP). In TTRP, a fleet of trucks and trailers are used to service a set of customers with known demands. Some customers may be serviced by a truck pulling a trailer, while the others may only be serviced by a single truck. This is the main difference between TTRP and VRP. The number of available trucks and available trailers is limited in the original TTRP but there are no fixed costs associated with the use of trucks or trailers. Therefore, it is reasonable to relax this fleet size constraint to see if it is possible to further reduce the total routing cost (distance). In addition, the resulting RTTRP can also be used to determine a better fleet mix. We developed a simulated annealing heuristic for solving RTTRP and tested it on 21 existing TTRP benchmark problems and 36 newly generated TTRP instances. Computational results indicate that the solutions for RTTRP are generally better than the best solutions in the literature for TTRP. The proposed SA heuristic is able to find better solutions to 18 of the 21 existing benchmark TTRP instances. The solutions for the remaining three problems are tied with the best so far solutions in the literature. For the 36 newly generated problems, the average percentage improvement of RTTRP solutions over TTRP solutions is about 5%. Considering the ever rising crude oil price, even small reduction in the route length is significant.     

Variable Neighborhood Search heuristic for the Inventory Routing Problem in fuel delivery

In this paper we observe the extension of the vehicle routing problem (VRP) in fuel delivery that includes petrol stations inventory management and which can be classified as the Inventory Routing Problem (IRP) in fuel delivery. The objective of the IRP is to minimize the total cost of vehicle routing and inventory management. We developed a Variable Neighborhood Search (VNS) heuristic for solving a multi-product multi-period IRP in fuel delivery with multi-compartment homogeneous vehicles, and deterministic consumption that varies with each petrol station and each fuel type. The stochastic VNS heuristic is compared to a Mixed Integer Linear Programming (MILP) model and the deterministic “compartment transfer” (CT) heuristic. For three different scale problems, with different vehicle types, the developed VNS heuristic outperforms the deterministic CT heuristic. Also, for the smallest scale problem instances, the developed VNS was capable of obtaining the near optimal and optimal solutions (the MILP model was able to solve only the smallest scale problem instances).