Multiple Vehicle Routing Problem Research Articles

An approximation algorithm for vehicle routing with compatibility constraints

We study a multiple-vehicle routing problem with a minimum makespan objective and compatibility constraints. We provide an approximation algorithm and a nearly-matching hardness of approximation result. We also provide computational results on benchmark instances with diverse sizes showing that the proposed algorithm (i) has a good empirical approximation factor, (ii) runs in a short amount of time and (iii) produces solutions comparable to the best feasible solutions found by a direct integer program formulation.

Collaboration and transportation resource sharing in multiple centers vehicle routing optimization with delivery and pickup

The adoption of collaboration strategies among logistics facilities and the formation of one or multiple coalitions constitute a sustainable approach to vehicle routing network optimization. This paper introduces a collaborative multiple centers vehicle routing problem with simultaneous delivery and pickup (CMCVRPSDP) to minimize operating cost and the total number of vehicles in the network. Distribution and pickup centers are allowed to share vehicles and customers in order to increase the entire network's efficiency and maximize profit. To provide the coalition coordinators with good routing solutions, we propose a hybrid heuristic algorithm which properly combines k-means and Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Based on clustering solutions, the proposed Hybrid NSGA-II (HNSGA-II) first generates a real coded population to bind our mathematical model constraints and to obtain a large number of feasible solutions which converge to optimality. Chromosomes are divided for genetic operations with partial mapped crossover and swap mutation algorithms, before their recombination to ensure the quality of our results. Comparisons with the traditional NSGA-II and the Multi-Objective Particle Swarm Optimization (MOPSO) algorithm indicate better performances of HNSGA-II in terms of objective function values. We also apply Cost Gap Allocation method (CGA) and the strictly monotonic path selection principle to examine profit allocation schemes. Numerical analyses on part of Chongqing city's logistics network show the superiority of HNSGA-II over MOPSO and NSGA-II on the practical case study, as well that of CGA over the Minimum Costs-Remaining Savings (MCRS), Shapley and Game Quadratic Programming (GQP) methods. In addition, the proposed profit allocation approach has supported the establishment of a grand coalition instead of two sub-coalitions. CMCVRPSDP optimization reduces long-haul transportation, improves the vehicle loading rate and facilitates sustainable development. Through the rational allocation of profits, the proposed solution methodology assures the stability and fairness among coalition members. The implementation is also important to design sustainable urban transportation networks.

Economic and environmental evaluations in the two-echelon collaborative multiple centers vehicle routing optimization

Abstract The two-echelon collaborative multiple centers vehicle routing problem (2E-CMCVRP) integrates collaboration mechanism and the vehicle routing problem. Combining k-means clustering algorithm and an improved Non-dominated Sorting Genetic Algorithm-II (Im-NSGA-II), this paper proposes a three-phase approach to simultaneously minimize the aggregate operating cost and reduce carbon dioxide emission. To ensure the initial population's quality, the sweep algorithm is integrated as modification of the standard NSGA-II. The chromosome population consists of multiple depots and corresponding customer nodes independently assessed to find local solutions, and latterly combined to yield suboptimal routes. The nodes scan principle of the sweep algorithm is employed to enforce optimization constraints, and the non-dominated sorting of the population efficiently improves the solution search accuracy. Further, the Minimum Cost-Remaining Savings (MCRS) method is used to determine appropriate profit distribution schemes, and the selection of the optimal sequential coalition is executed on the basis of the strictly monotonic path principle. Computational comparisons on benchmark instances indicate the superiority of Im-NSGA-II over NSGA-II and MOGA, and an empirical study in Chongqing, China confirms the practicability of our solution approach. The evaluation of MCRS solution's stability displays outperformance over other methods including the Shapley value model, CGA and GQP, and suitable coalition sequences are selected and assessed to improve the efficiency of logistics network optimization as well as the achievement of environment-friendly objectives.

Capacitated and multiple cross-docked vehicle routing problem with pickup, delivery, and time windows

Abstract The vehicle routing problem (VRP) in cross-docked distribution networks is an important research topic in the supply chain management literature. This paper formulates a mixed integer linear programming model of a special type of capacitated and multiple cross-docked VRP with pickup, delivery, and time windows. The problem is to design a set of routes for vehicle fleets servicing pickup and delivery nodes with defined demands and time windows to achieve the minimum transportation and fixed costs. We propose a hybrid of the genetic algorithm and particle swarm optimization (HGP) to solve the formulated NP-hard problem. Small-size problems are solved by HGP and then compared with the exact method using CPLEX to validate the effectiveness of the proposed hybrid algorithm. Extensive experiments have been conducted for medium and large-size problems and the results show the proposed HGP provides better solutions in the allocated time compared to CPLEX.

Analysis of Mixed-Integer Linear Programming Formulations for a Fuel-Constrained Multiple Vehicle Routing Problem

This paper addresses a fuel-constrained, multiple vehicle routing problem (FCMVRP) in the presence of multiple refueling stations. We are given a set of targets, a set of refueling stations, and a depot where [Formula: see text] vehicles are stationed. The vehicles are allowed to refuel at any refueling station, and the objective of the problem is to determine a route for each vehicle starting and terminating at the depot, such that each target is visited by at least one vehicle, the vehicles never run out of fuel while traversing their routes, and the total travel cost of all the routes is a minimum. We present four new mixed-integer linear programming (MILP) formulations for the problem. These formulations are compared both analytically and empirically, and a branch-and-cut algorithm is developed to compute an optimal solution. Extensive computational results on a large class of test instances that corroborate the effectiveness of the algorithm are also presented.

Profit distribution in collaborative multiple centers vehicle routing problem

A collaborative multiple-center vehicle routing problem (CMCVRP) is a multi-constraint combinatorial and game optimization issue containing both vehicle routing optimization and profit distribution procedures. The CMCVRP is generally used to study the logistics network structure adjustment from a non-optimal network structure to a collaborative multiple DCs network optimization structure. The optimization of CMCVR can effectively improve vehicle loading rate and reduce the crisscross transportation phenomenon. Designing a reasonable profit distribution mechanism is a critical step of CMCVR optimization. Collaboration can be organized through a negotiation process by a logistics service provider. This paper establishes an integer-programming model that contains transportation costs among distribution centers (DCs) and vehicle routing costs in each DC to minimize the total costs of CMCVRP. A multi-phase hybrid approach with clustering, dynamic programming, and heuristic algorithm is presented to solve the model formulation. The clustering procedure increases the likelihood that the solution will converge to an optimal value, between-route operations (relocate, 2-opt* exchange, and swap move) in the heuristic algorithm will improve the initial solution, and the within-route (dynamic programming) procedure will calculate a good feasible solution for each vehicle route. Both between- and within-route operations are recursively executed to find the best solution. Profit distribution plans are then established using the improved Shapley value model. Optimal sequential coalitions are selected based on strictly monotonic path, cost reduction model, and best strategy of sequential coalition selection in cooperative game theory. An empirical study in Chongqing, China suggests that the proposed approach outperforms other algorithms, and the best sequential coalition can be selected and adjusted to increase the negotiation power for network optimization of logistics distribution.

The Multiple Trip Vehicle Routing Problem with Backhauls: Formulation and a Two-Level Variable Neighbourhood Search

In this paper a new VRP variant the Multiple Trip Vehicle Routing Problem with Backhauls (MT-VRPB) is investigated. The classical MT-VRP model is extended by including the backhauling aspect. An ILP formulation of the MT-VRPB is first presented and CPLEX results for small and medium size instances are reported. For large instances of the MT-VRPB a Two-Level VNS algorithm is developed. To gain a continuous balanced intensification and diversification during the search process VNS is embedded with the sequential VND and a multi-layer local search approach. The algorithm is tested on a set of new MT-VRPB data instances which we generated. Interesting computational results are presented. The Two-Level VNS produced excellent results when tested on the special variant of the VRPB.

Computation of Lower Bounds for a Multiple Depot, Multiple Vehicle Routing Problem With Motion Constraints

This paper considers the problem of planning paths for a collection of identical vehicles visiting a given set of targets, such that the total lengths of their paths are minimum. Each vehicle starts at a specified location (called a depot) and it is required that each target to be on the path of at least one vehicle. The path of every vehicle must satisfy the motion constraints of every vehicle. In this paper, we develop a method to compute lower bound to the minimum total path lengths by relaxing some of the constraints and posing it as a standard multiple traveling salesmen problem (MTSP). A lower bound is often important to ascertain suboptimality bounds for heuristics and for developing stopping criterion for algorithms computing an optimal solution. Simulation results are presented to show that the proposed method can be used to improve the lower bounds of instances with four vehicles and 40 targets by approximately 39%.

Liquefied Natural Gas Ship Route Planning Model Considering Market Trend Change

We consider a new biannual liquefied natural gas (LNG) ship routing and scheduling problem and a stochastic extension under boil-off gas (BOG) uncertainty while serving geographically dispersed multiple customers using a fleet of heterogeneous vessels. We are motivated not only by contract trend changes to shorter ones but also by technological advancesin LNG vessel design. The mutual coincidence of both transitions enables developing a new LNG shipping strategy to keep up with emerging market trend. We fi rst propose a deterministic LNG scheduling model formulated as a multiple vehicle routing problem (VRP). The model is then extended to consider BOG using a two-stage stochastic modeling approach in which BOG is a random variable. Since the VRP is typically a combinatorial optimization problem, its stochastic extension is much harder to solve. In order to overcome this computational burden, a Monte Carlo sampling optimization is used to reduce the number of scenarios in the stochastic model while ensuring ggood quality of solutions. The solutions are evaluated using expected value of perfect information (EVPI) and value of stochastic solution (VSS). The result shows that our proposed model yields more stable solutions than the deterministic model. The study was made possible by the NPRP award [NPRP 4-1249-2-492] from the Qatar National Research Fund (a member of the Qatar Foundation).

A cloud theory-based particle swarm optimization for multiple decision maker vehicle routing problems with fuzzy random time windows

This article puts forward a cloud theory-based particle swarm optimization (CTPSO) algorithm for solving a variant of the vehicle routing problem, namely a multiple decision maker vehicle routing problem with fuzzy random time windows (MDVRPFRTW). A new mathematical model is developed for the proposed problem in which fuzzy random theory is used to describe the time windows and bi-level programming is applied to describe the relationship between the multiple decision makers. To solve the problem, a cloud theory-based particle swarm optimization (CTPSO) is proposed. More specifically, this approach makes improvements in initialization, inertia weight and particle updates to overcome the shortcomings of the basic particle swarm optimization (PSO). Parameter tests and results analysis are presented to highlight the performance of the optimization method, and comparison of the algorithm with the basic PSO and the genetic algorithm demonstrates its efficiency.

Heuristics for Routing Heterogeneous Unmanned Vehicles with Fuel Constraints

This paper addresses a multiple depot, multiple unmanned vehicle routing problem with fuel constraints. The objective of the problem is to find a tour for each vehicle such that all the specified targets are visited at least once by some vehicle, the tours satisfy the fuel constraints, and the total travel cost of the vehicles is a minimum. We consider a scenario where the vehicles are allowed to refuel by visiting any of the depots or fuel stations. This is a difficult optimization problem that involves partitioning the targets among the vehicles and finding a feasible tour for each vehicle. The focus of this paper is on developing fast variable neighborhood descent (VND) and variable neighborhood search (VNS) heuristics for finding good feasible solutions for large instances of the vehicle routing problem. Simulation results are presented to corroborate the performance of the proposed heuristics on a set of 23 large instances obtained from a standard library. These results show that the proposed VND heuristic, on an average, performed better than the proposed VNS heuristic for the tested instances.

Heuristics to Solve a Real-world Asymmetric Vehicle Routing Problem with Side Constraints

To maintain patients at home as most as possible, healthcare services are nowadays quite diversified. We present the case of a public medical clinic offering activities, mostly to the elderly, at a daycare center. Users are brought into the daycare center by bus or by taxi. The global problem consists in defining routes to pick up users while assigning them to time slots in the week. At first sight this problem could be viewed as an asymmetric multiple vehicle routing problem. However many additional constraints must be considered. In this paper, we propose a metaheuristic including two phases to solve the problem. In the first phase, the initial solution is determined using one of two proposed constructed algorithms and is improved using tabu search in the second phase. These algorithms are tested on 10 problem instances. Experimental results are presented.

Optimization of Logistics Distribution Route Based on the Save Mileage Method and the Ant Colony Algorithm

This paper established the mathematical model of Logistics distribution routing problem,and departed the multiple vehicle routing problem into a single vehicle distribution problem with the save mileage method ,then the ant colony algorithm were used to optimize the single vehicle distribution problem. Finally through a case shows that the model for the practical application of logistics distribution has certain reference significance.

Evolutionary algorithm with fuzzy numbers for planning active distribution network

Modern approach to distribution systems planning requires developing tools able to interpret and analyze the stochastic nature of large power system problems. At the same time, it has to be able to optimize multiple, conflicting goals that appear in today’s open market environment. The presented evolutionary algorithm is based on a Multiple Vehicle Routing Problem adjusted for solving active electric distribution networks. An original approach to fuzzy number modeling is used for active power sources, consumption substations and distributed generation. This model, besides shorter computational time and less memory usage, is applicable to any form of fuzzy number shape required by the planner in modeling the stochastic nature of elements. An idea of defuzzification using pessimism is presented and compared to the solutions obtained without pessimism included. Application of the original universal fuzzy number modeling, using FER-fuzzy modeling, is demonstrated on multi-objective evolutionary algorithm and applied to real Croatian distribution network.

A web spatial decision support system for vehicle routing using Google Maps

This article presents a user-friendly web-based spatial decision support system (wSDSS) aimed at generating optimized vehicle routes for multiple vehicle routing problems that involve serving the demand located along arcs of a transportation network. The wSDSS incorporates Google Maps ™ (cartography and network data), a database, a heuristic and an ant-colony meta-heuristic developed by the authors to generate routes and detailed individual vehicle route maps. It accommodates realistic system specifics, such as vehicle capacity and shift time constraints, as well as network constraints such as one-way streets and prohibited turns. The wSDSS can be used for “what-if” analysis related to possible changes to input parameters such as vehicle capacity, maximum driving shift time, seasonal variations of demand, network modifications, and imposed arc orientations. Since just a web browser is needed, it can be easily adapted to be widely used in many real-world situations. The system was tested for urban trash collection in Coimbra, Portugal.

An adaptive guidance approach for the heuristic solution of a minimum multiple trip vehicle routing problem

One of the most important problems in combinatorial optimization is the well-known vehicle routing problem (VRP), which calls for the determination of the optimal routes to be performed by a fleet of vehicles to serve a given set of customers. Recently, there has been an increasing interest towards extensions of VRP arising from real-world applications. In this paper we consider a variant in which time windows for service at the customers are given, and vehicles may perform more than one route within a working shift. We call the resulting problem the minimum multiple trip VRP (MMTVRP), where a “multiple trip” is a sequence of routes corresponding to a working shift for a vehicle. The problem objective is to minimize the overall number of the multiple trips (hence the size of the required fleet), breaking ties in favor of the minimum routing cost. We propose an iterative solution approach based on the decomposition of the problem into simpler ones, each solved by specific heuristics that are suitably combined to produce feasible MMTVRP solutions. An adaptive guidance mechanism is used to guide the heuristics to possibly improve the current solution. Computational experiments have been performed on a set of real-world instances arising from a multi-regional scale distribution problem. The obtained results show that the proposed adaptive guidance mechanism is considerably effective, being able to reduce the overall number of required vehicles within a limited computing time.

Implementing a multi-vehicle multi-route spatial decision support system for efficient trash collection in Portugal

More efficient vehicle routing can improve a firm’s competitive advantage or increase the efficiency by which governmental agencies supply public services. More efficient routing can also reduce traffic congestion and air pollution which are growing problems in many urban areas. Unfortunately, the identification of the optimal solution to most vehicle routing problems is computationally intractable. This article presents a user-friendly spatial decision support system (SDSS) to generate vehicle routes for multiple-vehicle routing problems that serve demand located along arcs and at nodes of the transportation network. The SDSS incorporates a geographical information system (GIS) and heuristic solution procedures to generate routes, system-wide data, and maps, as well as individual vehicle route maps, directions, and data quickly. It accommodates realistic system specifics such as vehicle capacity and time constraints and network constraints such as one-way streets, and prohibited turns. The system was tested for trash collection in Coimbra, Portugal. In addition, the SDSS can be used for “what-if” analysis related to possible changes to input parameters such as vehicle capacity and maximum driving time.

A shortest path approach to the multiple-vehicle routing problem with split pick-ups

We consider a multiple-vehicle routing problem with split pick-ups (mVRPSP). This problem involves multiple suppliers, a single depot, and a fleet of identical capacity trucks responsible for delivering supplies from the suppliers to the depot. Any supplier may be visited by more than one truck, thus allowing split pick-ups. The problem is to determine, for each truck, which suppliers to visit and the size of loads to pick up so as to minimize the total transportation cost for the fleet, which depends on the number of trucks used and their routes. We develop a fundamentally new model for the mVRPSP, a deterministic dynamic program (DP). Although the most natural DP formulation results in a DP with uncountably-infinite state and action spaces, an optimality-invariance condition we establish leads to an equivalent DP with finite state and action spaces. This DP formulation leads to a new exact algorithm for solving the mVRPSP, based on a shortest path search algorithm, which is conceptually simple and easy to implement.

Scenario-Based Planning for Partially Dynamic Vehicle Routing with Stochastic Customers

The multiple vehicle routing problem with time windows (VRPTW) is a hard and extensively studied combinatorial optimization problem. This paper considers a dynamic VRPTW with stochastic customers, where the goal is to maximize the number of serviced customers. It presents a multiple scenario approach (MSA) that continuously generates routing plans for scenarios including known and future requests. Decisions during execution use a distinguished plan chosen, at each decision, by a consensus function. The approach was evaluated on vehicle routing problems adapted from the Solomon benchmarks with a degree of dynamism varying between 30% and 80%. They indicate that MSA exhibits dramatic improvements over approaches not exploiting stochastic information, that the use of consensus function improves the quality of the solutions significantly, and that the benefits of MSA increase with the (effective) degree of dynamism.

A PROPOSITION FOR A DISTANCE MEASURE IN NEIGHBOURHOOD SEARCH FOR SCHEDULING PROBLEMS

ABSTRACT Meta-heuristics use iterations in which moves towards neighbourhood solutions are performed. Their power in escaping from local minima lies in accepting, according to some mechanism, solutions which worsen the current solution. It is generally accepted, that perturbations of solutions should be small. A solution for a scheduling problem is represented many times as a string of characters. The characters identify jobs to be scheduled. The use of a distance measure can enhance the search mechanism in a set of potential solutions. The notion of distance between two strings can be applied to patterns of production sequences. A distance measure is computed between permutations of the same string, which represent various single machine schedules. If cyclic permutations of the symbol sequence have distance zero, it is applicable to the Travelling Salesman Problem, where the symbol sequence represents the sequence of cities in the tour. Sometimes symbol sequences are divided into sub-strings called ‘words’ where the sequence of the words has no importance. This example can serve as a scheduling problem on parallel processors or as a multiple vehicle routing problem. The application of the distance measure to the single machine scheduling problem and the travelling salesman problem is shown. Also some implementation issues are discussed if the concept is applied within a genetic algorithm approach.