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).     

Iterated local search and simulated annealing algorithms for the inventory routing problem

This paper addresses the inventory routing problem (IRP), which consists in defining the customer visit schedule, the delivery quantities, and the vehicle routing plan to meet the demands of a set of customers over a given time horizon. We consider the variant with a single item, a single supplier, multiple vehicles, and a finite multiperiod planning horizon, minimizing the sum of inventory and travel costs. In addition, we address an alternative objective function that minimizes the logistic ratio, defined as the total travel cost divided by the total quantity delivered to customers. This second objective function, while more realistic in some logistics settings, poses a challenge for integer programming models and exact methods because of its nonlinearity. To our knowledge, no heuristic method has been proposed to address this objective in the IRP variant addressed in this paper. To solve this problem with each of these objective functions, we propose effective metaheuristic algorithms based on iterated local search and simulated annealing. Computational experiments show that these algorithms provide reasonably high‐quality solutions in relatively short running times for both objective functions when compared to other methods for well‐known instances from the literature. Moreover, the algorithms produce new best solutions for some of these instances.     

The exact solution of several classes of inventory-routing problems

In order to be competitive companies need to take advantage of synergistic interactions between different decision areas. Two of these are related to the distribution and inventory management processes. Inventory-Routing Problems (IRPs) arise when inventory and routing decisions must be made simultaneously, which yields a difficult combinatorial optimization problem. In this paper, we propose a branch-and-cut algorithm for the exact solution of several classes of IRPs. Specifically, we solve the multi-vehicle IRP with a homogeneous and a heterogeneous fleet, the IRP with transshipment options, and the IRP with added consistency features. We perform an extensive computational analysis on benchmark instances.     

A maritime inventory routing problem: Practical approach

Despite of the practicality of the motivation of the inventory routing problem (IRP), there are few successful implementation stories of IRP based decision support systems which utilize optimization algorithms. Besides the fact that the IRP is an extremely challenging optimization problem, simplifications and assumptions made in the definition of typical IRP in the literature make it even more difficult to take advantage of the developed technologies for IRP in practice. This paper introduces a flexible modeling framework for IRP, which can accommodate various practical features. A simple algorithmic framework of an optimization based heuristic method is also proposed. A case study on a practical maritime inventory routing problem (MIRP) shows that the proposed modeling and algorithmic framework is flexible and effective enough to be a choice of model and solution method for practical inventory routing problems.     

The Flexible Periodic Vehicle Routing Problem

This paper introduces the Flexible Periodic Vehicle Routing Problem (FPVRP) where a carrier has to establish a distribution plan to serve his customers over a planning horizon. Each customer has a total demand that must be served within the horizon and a limit on the maximum quantity that can be delivered at each visit. A fleet of homogeneous capacitated vehicles is available to perform the services and the objective is to minimize the total routing cost. The FPVRP can be seen as a generalization of the Periodic Vehicle Routing Problem (PVRP) which instead has fixed service frequencies and schedules and where the quantity delivered at each visit is fixed. Moreover, the FPVRP shares some common characteristics with the Inventory Routing Problem (IRP) where inventory levels are considered at each time period and, typically, an inventory cost is involved in the objective function. We present a worst-case analysis which shows the advantages of the FPVRP with respect to both PVRP and IRP. Moreover, we propose a mathematical formulation for the problem, together with some valid inequalities. Computational results show that adding flexibility improves meaningfully the routing costs in comparison with both PVRP and IRP.     

A branch-and-price algorithm for an integrated production and inventory routing problem

With globalization, the need to better integrate production and distribution decisions has become ever more pressing for manufacturers trying to streamline their supply chain. This paper investigates a previously developed mixed-integer programming (MIP) model aimed at minimizing production, inventory, and delivery costs across the various stages of the system. The problem being modeled includes a single production facility, a set of customers with time varying demand, a finite planning horizon, and a fleet of homogeneous vehicles. Demand can be satisfied from either inventory held at a customer site or from daily product distribution. Whether a customer is visited on a particular day is determined by an implicit tradeoff between inventory and distribution costs. Once the decision is made, a vehicle routing problem must be solved for those customers who are scheduled for a delivery.A hybrid methodology that combines exact and heuristic procedures within a branch-and-price framework is developed to solve the underlying MIP. The approach takes advantage of the efficiency of heuristics and the precision of branch and price. Implementation required devising a new branching strategy to accommodate the unique degeneracy characteristics of the master problem, and a new procedure for handling symmetry. A novel column generation heuristic and a rounding heuristic were also implemented to improve algorithmic efficiency. Computational testing on standard data sets showed that the hybrid scheme can solve instances with up to 50 customers and 8 time periods within 1 h. This level of performance could not be matched by either CPLEX or standard branch and price alone.     

A fast two-level variable neighborhood search for the clustered vehicle routing problem

In this paper, we present an improved two-level heuristic to solve the clustered vehicle routing problem (CluVRP). The CluVRP is a generalization of the classical capacitated vehicle routing problem (CVRP) in which customers are grouped into predefined clusters, and all customers in a cluster must be served consecutively by the same vehicle. This paper contributes to the literature in the following ways: (i) new upper bounds are presented for multiple benchmark instances, (ii) good heuristic solutions are provided in much smaller computing times than existing approaches, (iii) the CluVRP is reduced to its cluster level without assuming Euclidean coordinates or distances, and (iv) a new variant of the CluVRP, the CluVRP with weak cluster constraints, is introduced. In this variant, clusters are allocated to vehicles in their entirety, but all corresponding customers can be visited by the vehicle in any order.The proposed heuristic solves the CluVRP by combining two variable neighborhood search algorithms, that explore the solution space at the cluster level and the individual customer level respectively. The algorithm is tested on different benchmark instances from the literature with up to 484 nodes, obtaining high quality solutions while requiring only a limited calculation time.     

Heuristics for a multiperiod inventory routing problem with production decisions

Manufacturers who resupply a large number of retailers on a periodic basis continually struggle with the question of how to formulate a replenishment strategy. This paper presents a comparative analysis of a series of heuristics for an inventory routing problem (IRP) that arises in a manufacturing supply chain. The IRP is formulated as a mixed integer program with the objective of maximizing the net benefits associated with making deliveries in a specific time period to a widely dispersed set of customers. It is assumed that inventory can accumulate at the customer sites, but that all demand must be met without backlogging. Because optimal solutions were not within reach of exact methods, a two-step procedure was developed that first estimates daily delivery quantities and then solves a vehicle routing problem for each day of the planning horizon. As part of the methodology, a linear program is used to determine which days it is necessary to make at least some deliveries to avoid stockouts.The IRP is investigated in the context of an integrated production–inventory–distribution–routing problem (PIDRP). The full model takes into account production decisions and inventory flow balance in each period. For the computations, a previously developed branch-and-price algorithm is used that requires the solution of multiple IRPs (one in each period) to generate columns for the master problem. Testing showed that PIDRP instances with up to eight time periods and 50 customers can be solved within 1 h. This level of performance could not be matched by either CPLEX or an exact version of the branch-and-price algorithm.     

A two phase variable neighborhood search for the multi-product inventory routing problem

We address a multi-product inventory routing problem and propose a two-phase Variable Neighborhood Search (VNS) metaheuristic to solve it. In the first phase, VNS is used to solve a capacitated vehicle routing problem at each period to find an initial solution without taking into account the inventory. In the second phase, we iteratively improve the initial solution while minimizing both the transportation and inventory costs. For this, we propose two different algorithms, a Variable Neighborhood Descent and a Variable Neighborhood Search. We present an heuristic and a Linear Programming formulation, which are applied after each local search move, to determine the amount of products to collect from each supplier at each period. During the exploration, we use priority rules for suppliers and vehicles, based on the current delivery schedule over the planning horizon. Computational results show the efficiency of the proposed two-phase approach.     

基于划分的蚁群算法求解货物权重车辆路径问题

考虑单产品分销网络中的车辆路径问题(VRP:vehicle routing problem).与以往诸多研究不同的是,建立了一种带货物载重量的VRP模型(weighted VRP),即车辆在两个顾客之间行驶时的载重量也作为影响运输费用的一个因素考虑.因此,需求量较大的顾客拥有较高的车辆运输优先权.在分析了问题性质的基础上,提出一种基于划分策略的蚁群算法PMMAS求解货物权重车辆路径问题,并与其他常用的启发式算法进行比较分析,表明了算法的有效性.     

The customer-centric,multi-commodity vehicle routing problem with split delivery

In this paper, we present the Customer-centric, Multi-commodity Vehicle Routing Problem with Split Delivery (CMVRPSD) whose objective is to minimize total waiting time of customers in distributing multiple types of commodities by multiple capacitated vehicles. It is assumed that a customer's demand can be fulfilled by more than one vehicle. Two classes of decisions are involved in this problem: routing vehicles to customers and quantifying commodities to load and unload. The CMVRPSD can be applied to distributing commodities in customer-oriented distribution problems for both peacetime and disaster situations. The problem is formulated in two Mixed-Integer Linear Programming (MILP) models, and a heuristic method is proposed by adapting and synthesizing Simulated Annealing (SA) and Variable Neighborhood Search (VNS) for large-scale problems. Experimental results show that the proposed hybrid algorithm outperforms other applicable algorithms such as SA, VNS, and Nearest Neighborhood heuristic.     

Formulations for an inventory routing problem

In this paper, we present and compare formulations for the inventory routing problem (IRP) where the demand of customers has to be served, over a discrete time horizon, by capacitated vehicles starting and ending their routes at a depot. The objective of the IRP is the minimization of the sum of inventory and transportation costs. The formulations include known and new mathematical programming formulations. Valid inequalities are also presented. The formulations are tested on a large set of benchmark instances. One of the most significant conclusions is that the formulations that use vehicle‐indexed variables are superior to the more compact, aggregate formulations.     

A heuristic method for the inventory routing and pricing problem in a supply chain

The inventory routing problem (IRP) in a supply chain (SC) is to determine delivery routes from suppliers to some geographically dispersed retailers and inventory policy for retailers. In the past, the pricing and demand decisions seem ignored and assumed known in most IRP researches. Since the pricing decision affects the demand decision and then both inventory and routing decisions, it should be considered in the IRP simultaneously to achieve the objective of maximal profit in the supply chain. In this paper, a mathematical model for the inventory routing and pricing problem (IRPP) is proposed. Since the solution for this model is an NP (non-polynomial) problem, a heuristic method, tabu search adopting different neighborhood search approaches, is used to obtain the optimal solution. The proposed heuristic method was compared with two other methods considering the IRPP separately. The experimental results indicate that the proposed method is better than the two other methods in terms of average profit.     

基于多代理系统方法的存贮路径问题研究

在分析现有研究成果的基础上,运用智能化的多代理技术建模方法,将问题抽象映射成为4类自治的Agent,构造存贮路径问题（Inventory Routing Problem,IRP）的多代理模型,通过Agent间的通信与协作完成问题的求解,为该问题的处理探索一条新的途径。     

The production routing problem: A review of formulations and solution algorithms

The production routing problem (PRP) combines the lot-sizing problem and the vehicle routing problem, two classical problems that have been extensively studied for more than half a century. The PRP is solved in an attempt to jointly optimize production, inventory, distribution and routing decisions and is thus a generalization of the inventory routing problem (IRP). Although the PRP has a complicated structure, there has been a growing interest in this problem during the past decade in both academia and industry. This article provides a comprehensive review of various solution techniques that have been proposed to solve the PRP. We attempt to provide an in-depth summary and discussion of different formulation schemes and of algorithmic and computational issues. Finally, we point out interesting research directions for further developments in production routing.     

Analysis of the single-vehicle cyclic inventory routing problem

The single-vehicle cyclic inventory routing problem (SV-CIRP) consists of a repetitive distribution of a product from a single depot to a selected subset of customers. For each customer, selected for replenishments, the supplier collects a corresponding fixed reward. The objective is to determine the subset of customers to replenish, the quantity of the product to be delivered to each and to design the vehicle route so that the resulting profit (difference between the total reward and the total logistical cost) is maximised while preventing stockouts at each of the selected customers. This problem appears often as a sub-problem in many logistical problems. In this article, the SV-CIRP is formulated as a mixed-integer program with a nonlinear objective function. After a thorough analysis of the structure of the problem and its features, an exact algorithm for its solution is proposed. This exact algorithm requires only solutions of linear mixed-integer programs. Values of a savings-based heuristic for this problem are compared to the optimal values obtained for a set of some test problems. In general, the gap may get as large as 25%, which justifies the effort to continue exploring and developing exact and approximation algorithms for the SV-CIRP.     

A two-stage tabu search algorithm with enhanced packing heuristics for the 3L-CVRP and M3L-CVRP

The Three-Dimensional Loading Capacitated Vehicle Routing Problem (3L-CVRP) addresses practical constraints frequently encountered in the freight transportation industry. In this problem, the task is to serve all customers using a homogeneous fleet of vehicles at minimum traveling cost. The constraints imposed by the three-dimensional shape of the goods, the unloading order, item fragility, and the stability of the loading plan of each vehicle are explicitly considered. We improved two well-known packing heuristics, namely the Deepest-Bottom-Left-Fill heuristic and the Maximum Touching Area heuristic, for the three-dimensional loading sub-problem and provided efficient implementations. Based on these two new heuristics, an effective tabu search algorithm is given to address the overall problem. Computational experiments on publicly available test instances show our new approach outperforms the current best algorithms for 20 out of 27 instances. Our approach is also superior to the existing algorithm on benchmark data for the closely related problem variant M3L-CVRP (which uses a slightly different unloading order constraint compared to 3L-CVRP).     

A tabu search procedure for coordinating production,inventory and distribution routing problems

This paper addresses the problem of optimally coordinating a production‐distribution system over a multi‐period finite horizon, where a facility production produces several items that are distributed to a set of customers by a fleet of homogeneous vehicles. The demand for each item at each customer is known over the horizon. The production planning determines how much to produce of each item in every period, while the distribution planning defines when customers should be visited, the amount of each item that should be delivered to customers and the vehicle routes. The objective is to minimize the sum of production and inventory costs at the facility, inventory costs at the customers and distribution costs. We also consider a related problem of inventory routing, where a supplier receives or produces known quantities of items in each period and has to solve the distribution problem. We propose a tabu search procedure for solving such problems, and this approach is compared with vendor managed policies proposed in the literature, in which the facility knows the inventory levels of the customers and determines the replenishment policies.     

Exact and heuristic algorithms for the vehicle routing problem with multiple interdependent time windows

In this paper a model and several solution procedures for a novel type of vehicle routing problems where time windows for the pickup of perishable goods depend on the dispatching policy used in the solution process are presented. This problem is referred to as Vehicle Routing Problem with multiple interdependent time windows (VRPmiTW) and is motivated by a project carried out with the Austrian Red Cross blood program to assist their logistics department. Several variants of a heuristic constructive procedure as well as a branch-and-bound based algorithm for this problem were developed and implemented. Besides finding the expected reduction in costs when compared with the current procedures of the Austrian Red Cross, the results show that the heuristic algorithms find solutions reasonably close to the optimum in fractions of a second. Another important finding is that increasing the number of pickups at selected customers beyond the theoretical minimum number of pickups yields significantly greater potential for cost reductions.     

An iterated local search heuristic for the split delivery vehicle routing problem

This paper concerns the Split Delivery Vehicle Routing Problem (SDVRP). This problem is a relaxation of the Capacitated Vehicle Routing Problem (CVRP) since the customers׳ demands are allowed to be split. We deal with the cases where the fleet is unlimited (SDVRP-UF) and limited (SDVRP-LF). In order to solve them, we implemented a multi-start Iterated Local Search (ILS) based heuristic that includes a novel perturbation mechanism. Extensive computational experiments were carried out on benchmark instances available in the literature. The results obtained are highly competitive, more precisely, 55 best known solutions were equaled and new improved solutions were found for 243 out of 324 instances, with an average and maximum improvement of 1.15% and 2.81%, respectively.