Capacitated Vehicle Routing Research Articles

Optimal Solution for the Integrated Model of the Problems in Capacitated Vehicle Routing and Vehicle Scheduling at the Multi-door Depot

To have an efficient Supply Chain (SC), the coordination and integration of the activities in the SC are mandatory. Routing vehicles is an optimization problem in the SC. When the routed vehicles return to the depot and the doors at the depot are busy, the returned vehicles must wait to unload the accumulated shipments. Therefore, properly scheduling these vehicles to those doors at the depot to minimize the waiting time is an optimization problem in the SC. Therefore, in this study, routing vehicles to collect the shipments from suppliers and scheduling vehicles to doors at the depot, based on a first-come-first-serve basis, are simultaneously solved. Hence, the objective of this integrated vehicle routing and scheduling problem (VR&SP) is to minimize the total cost which contains the following components: vehicle travelling cost between suppliers, loading cost at the suppliers, vehicle waiting cost, unloading cost at the depot and vehicle operations cost. A Mixed Integer Quadratic Programming (MIQP) model is developed to solve the integrated VR&SP. The Branch and Bound algorithm method is employed to obtain the exact optimal solution to this MIQP using LINGO optimization software. The compatibility of the developed MIQP model is verified by the randomly generated small-scale instances of VR&SP. Therefore, it can be concluded that, this model solves the vehicle routing to suppliers and vehicle scheduling to doors at the depot simultaneously. Since VR&SP is a NP-hard problem, heuristics or meta-heuristic methods are recommended to solve the large-scale instances of VR&SP. Moreover, it can be recommended for further studies to amend this model by incorporating additional constraints to make it more applicable to real-world scenarios.

PHH: Policy-Based Hyper-Heuristic With Reinforcement Learning

Hyper-heuristics have a high level of generality and adaptability, allowing them to effectively solve a wide range of complex optimization problems. With reinforcement learning, hyper-heuristics can use experience and knowledge gained to tackle unforeseen problems, allowing hyper-heuristics to adapt and improve over time. Our paper proposes a framework for using policy-based reinforcement learning to improve the performance of hyper-heuristics. The framework trains hyper-heuristic agents to select the best generalized constructive low-level heuristics to solve combinatorial optimization problems. The framework evaluation was performed using three benchmarking problems: traveling salesman, capacitated vehicle routing, and bin packing problems. The results showed that the proposed framework can outperform existing meta-heuristic and hyper-heuristic-based algorithms for all large problem instances in all problem domains. The proposed framework was also evaluated by applying it to a cost optimization problem for workflow scheduling on a hybrid cloud with a deadline constraint. Eight agents were trained on medium-sized workflows with two deadlines and tested against traditional meta-heuristic and hyper-heuristic methods to solve smaller and larger workflows with unforeseen deadlines. Four workflow applications, three workflow sizes, and three deadlines were used in the evaluation. The results showed that our proposed framework provided significantly better solutions of up to 98% for benchmarking problems, and up to 22% for cost optimization in workflow scheduling. Moreover, trained with small problem instances, the framework performed well for unforeseen larger problem instances implying its generalization. The proposed framework, thus, has the potential to improve both the generality and performance of solving large combinatorial optimization problems.

SOLO: Search Online, Learn Offline for Combinatorial Optimization Problems

We study combinatorial problems with real world applications such as machine scheduling, routing, and assignment. We propose a method that combines Reinforcement Learning (RL) and planning. This method can equally be applied to both the offline, as well as online, variants of the combinatorial problem, in which the problem components (e.g., jobs in scheduling problems) are not known in advance, but rather arrive during the decision-making process. Our solution is quite generic, scalable, and leverages distributional knowledge of the problem parameters. We frame the solution process as an MDP, and take a Deep Q-Learning approach wherein states are represented as graphs, thereby allowing our trained policies to deal with arbitrary changes in a principled manner. Though learned policies work well in expectation, small deviations can have substantial negative effects in combinatorial settings. We mitigate these drawbacks by employing our graph-convolutional policies as non-optimal heuristics in a compatible search algorithm, Monte Carlo Tree Search, to significantly improve overall performance. We demonstrate our method on two problems: Machine Scheduling and Capacitated Vehicle Routing. We show that our method outperforms custom-tailored mathematical solvers, state of the art learning-based algorithms, and common heuristics, both in computation time and performance.

Development and application of an iterative heuristic for roadway snow and ice control

Many states in the U.S. have experienced increased demand for roadway snow and ice control (RSIC) operations due to an increase in extreme winter weather. As the number and severity of extreme weather events increases, the costs associated with winter roadway maintenance materials, plow operator time, equipment maintenance and replacement, and fuel use will also increase. In this paper, we introduce a unique heuristic procedure which we combine with real-world operational constraints to advance both the modeling and practical application of RSIC operations by incorporating a continuous measure of priority into a sequenced, iterative heuristic for network clustering, vehicle allocation, and capacitated vehicle routing. The heuristic balances the competing objectives of minimizing the total vehicle hours traveled for the fleet and minimizing the total time required to service the most critical links in the roadway network, while ensuring that the entire fleet is put to use. We also introduce a new measure of route-system performance, which is based on an effective link length (adjusted for how critical the link is to the performance of the entire system) and the time it takes to service the most critical links. We demonstrate the approach in practice by running five different applications of the heuristic on the statewide roadway network in Vermont. We demonstrate conclusively that our heuristic is effective for servicing the most critical links in the network in the least amount of time. We show that our more advanced vehicle allocation methods result in more effective RSIC service operations than adding vehicles to the fleet.

Algorithm for directing cooperative vehicles of a vehicle routing problem for improving fault-tolerance

This paper presents a new algorithm for solving capacitated single depot vehicle routing problems without time windows in a fault-tolerant manner where the distances between points are symmetrical. The sudden breakdown of a vehicle on its pre-planned route is considered as a fault. In this case one of the other cooperating vehicles will take over the remaining tasks and perform them in the most effective way. We introduce the concept of “route change cost” as the difference between the length of the modified and the original routes of the helper vehicle, and our goal is to minimize this value. Starting from an arbitrary VRP solution, we determine the optimal direction, in terms of the route change cost, in which each route should be performed by a vehicle. A simulation with case studies was used to test the algorithm. We found that applying the proposed method, the execution of the original tasks requires routes with lengths 4.1–8.6% shorter on average than using the non-directed routes of the solution of the original vehicle routing problem in arbitrary directions.

A unified framework for routing problems with a fixed fleet size

Vehicle routing problems constitute the core of many operations research efforts. Early studies introduced tailor-made solutions for each variant of a vehicle routing problem, but unified frameworks have emerged more recently. These approaches typically generalise across many vehicle routing problems and implement a method for tackling the generalised problem. In line with this, this research proposes a generic method for solving several fixed fleet vehicle routing problems - capacitated vehicle routing, open vehicle routing, vehicle routing with soft and hard time windows, open vehicle routing with soft and hard time windows, and time-dependent vehicle routing with soft and hard time windows - by transforming them into a time-dependent vehicle routing problem with soft time windows, solved by a variable neighbourhood search using a unique parameter setting, regardless of the original problem. Computational tests using standard benchmark instances from prior literature show that genericity does not come at the expense of solution quality. Moreover, the algorithm yields competitive results and some new best known solutions are obtained for the vehicle routing problem with soft time windows, the open vehicle routing problem with hard time windows, and time-dependent vehicle routing problem with hard time windows.

A unified framework for routing problems with a fixed fleet size

Vehicle routing problems constitute the core of many operations research efforts. Early studies introduced tailor-made solutions for each variant of a vehicle routing problem, but unified frameworks have emerged more recently. These approaches typically generalise across many vehicle routing problems and implement a method for tackling the generalised problem. In line with this, this research proposes a generic method for solving several fixed fleet vehicle routing problems - capacitated vehicle routing, open vehicle routing, vehicle routing with soft and hard time windows, open vehicle routing with soft and hard time windows, and time-dependent vehicle routing with soft and hard time windows - by transforming them into a time-dependent vehicle routing problem with soft time windows, solved by a variable neighbourhood search using a unique parameter setting, regardless of the original problem. Computational tests using standard benchmark instances from prior literature show that genericity does not come at the expense of solution quality. Moreover, the algorithm yields competitive results and some new best known solutions are obtained for the vehicle routing problem with soft time windows, the open vehicle routing problem with hard time windows, and time-dependent vehicle routing problem with hard time windows.

Conservative scales in packing problems

Packing problems (sometimes also called cutting problems) are combinatorial optimization problems concerned with placement of objects (items) in one or several containers. Some packing problems are special cases of several other problems such as resource-constrained scheduling, capacitated vehicle routing, etc. In this paper we consider a bounding technique for one- and higher-dimensional orthogonal packing problems, called conservative scales (CS) (in the scheduling terminology, redundant resources). CS are related to the possible structure of resource consumption: filling of a bin, distribution of the resource to the jobs, etc. In terms of packing, CS are modified item sizes such that the set of feasible packings is not reduced. In fact, every CS represents a valid inequality for a certain binary knapsack polyhedron. CS correspond to dual variables of the set-partitioning model of a special 1D cutting-stock problem. Some CS can be constructed by (data-dependent) dual-feasible functions ((D)DFFs). We discuss the relation of CS to DFFs: CS assume that at most one copy of each object can appear in a combination, whereas DFFs allow several copies. The literature has investigated the so-called extremal maximal DFFs (EMDFFs) which should provide very strong CS. Analogously, we introduce the notions of maximal CS (MCS) and extremal maximal CS (EMCS) and show that EMDFFs do not necessarily produce (E)MCS. We propose fast greedy methods to maximize a given CS. Using the fact that EMCS define facets of the binary knapsack polyhedron, we use lifted cover inequalities as EMCS. For higher-dimensional orthogonal packing, we propose a Sequential LP (SLP) method over the set of CS and investigate its convergence. Numerical results are presented.

Capacitated-Vehicle Routing Problem with Backhauls on Trees

The capacitated-vehicle routing problem (CVRP) is a classical problem that has been well studied by the transportation science community. A special case of the CVRP in which the network is constrained to have a tree structure (TCVRP) is studied here. Such tree networks arise when the cost of constructing and maintaining roads is much more than the routing cost—or when the transportation network consists of a main highway (e.g., Interstate system) and the customer locations are located off the highway. A new variant of the TCVRP, in which the customers are divided into two subsets—line haul and backhaul customers (TCVRPB)—is introduced in this paper. Line haul customers require delivery from the depot, and backhaul customers have supply that needs to be delivered to the depot. In any vehicle tour, line haul customers must be serviced before backhaul customers. The TCVRPB's relationship to the two-dimensional bin-packing problem is studied and, with the use of that problem, conditions for the lower bound on the problem are derived. An integer-programming (IP) formulation of the problem is presented. A two-approximation algorithm, which gives a feasible set of vehicle routes and customer assignments, is also presented. The problem is tested on two real-world treelike networks of varying sizes and on a randomly generated test network. The IP formulation works exceptionally well on small and medium-sized networks, and the gap between the approximation algorithm and the IP solution did not exceed 2% for nine of the 12 problem instances tested.

Economic and Risk Implications in the Distribution of Petrol Products to Service Stations under Retailer Managed and Vendor Managed Inventories

ABSTRACT Retailer inventory management is the common approach that is used by oil companies in the distribution of petrol products to service stations. Service station retailers send orders to the oil company vendor according to their own perception of inventory and of the future demand. The oil vendor consequently optimizes the deliveries by solving a related capacitated-vehicle routing problem. Nowadays, the technological possibility to monitor remotely the level of the underground storage tanks in the service stations allows a new formulation of the problem, planning deliveries based on the actual level of the inventory and on a prediction of future demand, that is, shifting also the inventory management on the vendor side. From a sustainable viewpoint, the centralization of the management of the retailer inventories has an economic impact, that in the case of petrol products is also related to risk impact, that has not yet been evaluated. Under this perspective, a comparison between the two approaches is described, where the vendor inventory management is proposed according to an original formulation of the problem integrating both the inventory and the routing costs in the objective function of the related mathematical programming formulation. This formulation has been implemented, tested and compared with respect to a simple simulated retailer inventory procedure in a series of random trials. In addition, the evaluation has been applied to an actual case study, related to the Nice (France) district, to evaluate the improvements of the performance of the distribution of petrol products by tank trucks when the central depot can receive the levels of the underground storage tanks in real-time. The results show that the proposed approach would have allowed a reduction in the number of deliveries and in the overall amount of kilometers necessary for the deliveries themselves, that is with a certain related economic impact. As a consequence, it is also shown that risk in the transportation of petrol products also decreases in the territory affected by this transportation, thus improving both the economic and the risk sustainability of this transport.

Comparison of bundle and classical column generation

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

Metaheuristic algorithms for combinatorial optimization problems

We present the main results in the author’s Ph.D. thesis (Iori 2004), defended at the University of Bologna in April 2004 and supervised by S. Martello. The thesis is written in English and is available from the author upon request. It proposes exact and metaheuristic algorithms for solving some relevant combinatorial optimization problems, with particular emphasis on scheduling, two-dimensional cutting and packing and capacitated vehicle routing. The performance of each algorithm is tested through extensive computational experiments and comparison with other approaches in the literature.

Two exact algorithms for the vehicle routing problem on trees

This article describes a heuristic and two exact algorithms for several classes of vehicle routing problems defined on tree networks. These include capacitated and time-constrained vehicle routing problems. One of the exact algorithms is based on the computation of bin packing lower bounds. The other uses column generation. The first algorithm performs better on problems containing small customer demands and in which all vehicles are identical. Otherwise, the second algorithm is more powerful and more versatile. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 75–89, 1999