Branch-price-and-cut for the production routing problem with time windows

Multi-objective modeling of production and pollution routing problem with time window: A self-learning particle swarm optimization approach

Production and inventory routing problems consider a single-product supply chain operating under a vendor-managed inventory system. A plant creates a production plan and vehicle routes over a planning horizon to replenish its customers at minimum cost. In this paper, we present two- and three-index formulations, implement a branch-and-cut algorithm based on each formulation, and introduce a local search matheuristic-based algorithm to solve the problem. In order to combine all benefits of each algorithm, we design a parallel framework to integrate all three fronts, called the three-front parallel branch-and-cut algorithm (3FP-B&C). We assess the performance of our method on well-known benchmark instances of the inventory routing problem (IRP) and the production routing problem (PRP). The results show that our 3FP-B&C outperforms by far other approaches from the literature. For the 956 feasible small-size IRP instances, our method proves optimality for 746, being the first exact algorithm to solve all instances with up to two vehicles. 3FP-B&C finds 949 best known solutions (BKS) with 153 new BKS (NBKS). For the large-size set, our method provides two new optimal solutions (OPT), and finds 82% of BKS, being 70% of NBKS for instances with up to five vehicles. This result is more than twice the number of BKS considering all heuristic methods from the literature combined. Finally, our 3FP-B&C finds the best lower bounds (BLB) for 1,169/1,316 instances, outperforming all previous exact algorithms. On the PRP, our method obtained 278 OPT out of the 336 instances of benchmark set of small- and medium-size instances being 19 new ones in addition to 335 BKS (74 NBKS) and 313 BLB (52 new ones). On another set of PRP with medium- and large-size instances, our algorithm finds 1,105 BKS out of 1,440 instances with 584 NBKS. Besides that, our 3FP-B&C is the first exact algorithm to solve the instances with an unlimited fleet, providing the first lower bounds for this subset with an average optimality gap of 0.61%. We also address a very large-size instance set, the second exact algorithm to address this set, outperforming the previous approach by far. Finally, a comparative analysis of each front shows the gains of the integrated approach. History: This paper has been accepted for the Transportation Science Special Issue: DIMACS Implementation Challenge: Vehicle Routing. Funding: C. M. Schenekemberg was supported by the São Paulo Research Foundation (FAPESP) [Grant 2020/07145-8]. A. A. Chaves was supported by FAPESP [Grants 2018/15417-8 and 2016/01860-1] and Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 312747/2021-7 and 405702/2021-3]. L. C. Coelho was supported by the Canadian Natural Sciences and Engineering Research Council [Grant 2019-00094]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0261 .

Purpose The integrated production and routing problems in a supply chain, commonly referred to as the production routing problem (PRP), combine lot-sizing and vehicle routing problems. PRP involves determining production schedules, quantities to produce and distribution plans to efficiently meet customer demands. This research aims to analyze the research background, identify current trends, track evolutions and propose future research directions for PRP. Design/methodology/approach This study employs descriptive analysis and a systematic literature review (SLR) to explore PRP literature. Software such as R Studio and Excel was used for the analysis. An SLR of 90 articles (1994–2023) was conducted, classifying the literature into themes and sub-themes, providing an evolution of PRP research and introducing a diagnostic framework for a firm's readiness. Findings Existing PRP review papers cover only 25% of the literature. This study fills the gap by offering a comprehensive review, categorizing PRP research into themes such as nature of demand, problem structure, solution methodologies, mathematical formulations, software tools and applications. The review also identifies sub-themes, including vehicle routing problem variants, environmental and reverse logistics, inbound logistics, multiple objectives and multi-echelon. The study also traces the evolution of PRP literature and solution methodologies. Originality/value This study makes a significant contribution by identifying current trends, mapping the literature into various themes and sub-themes, introducing a diagnostic framework to assess firm readiness for PRP adoption and presenting PRP evolution from 1994 to 2023. It also outlines future research directions and theoretical and managerial implications for scholars and practitioners in the field of PRP.

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

A matheuristic solution approach for the production routing problem with visit spacing policy

A progressive hedging-based matheuristic for the stochastic production routing problem with adaptive routing

The inventory routing problem (IRP) and the production routing problem (PRP) are two difficult problems arising in the planning of integrated supply chains. These problems are solved in an attempt to jointly optimize production, inventory, distribution, and routing decisions. Although several studies have proposed exact algorithms to solve the single-vehicle problems, the multivehicle aspect is often neglected because of its complexity. We introduce multivehicle PRP and IRP formulations, with and without a vehicle index, to solve the problems under both the maximum level (ML) and order-up-to level (OU) inventory replenishment policies. The vehicle index formulations are further improved using symmetry breaking constraints; the nonvehicle index formulations are strengthened by several cuts. A heuristic based on an adaptive large neighborhood search technique is also developed to determine initial solutions, and branch-and-cut algorithms are proposed to solve the different formulations. The results show that the vehicle index formulations are superior in finding optimal solutions, whereas the nonvehicle index formulations are generally better at providing good lower bounds on larger instances. IRP and PRP instances with up to 35 customers, three periods, and three vehicles can be solved to optimality within two hours for the ML policy. By using parallel computing, the algorithms could solve the instances for the same policy with up to 45 and 50 customers, three periods, and three vehicles for the IRP and PRP, respectively. For the more difficult IRP (PRP) under the OU policy, the algorithms could handle instances with up to 30 customers, three (six) periods, and three vehicles on a single core machine, and up to 45 (35) customers, three (six) periods, and three vehicles on a multicore machine.

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

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.

The Inventory Routing Problem (IRP) is an integrated logistic problem arising in several industries (e.g. petrochemical, grocery, soft drink and automotive). A vendor decides the optimal delivery strategy for a set of customers, taking into account their inventory policies and avoiding product stock-out in a finite and discrete time horizon. Delivery strategy includes the time and size of deliveries in order to minimize the total delivery cost. Most commonly studied are IRP real cases where a single homogeneous product with deterministic but time-varying demand is delivered over a finite time horizon. This paper is focused on an efficient methodology for industrial problems where a vendor resupplies a set of customers of heterogeneous products (as in the supermarket distribution industry). In this context, the paper reports on an effort facing the inventory routing problem for multi-category products per customer in conjunction with different inventory policies per category. The paper is motivated by real applications arising in the food engineering field. For instance, industries dealing with food’s distribution to stores located in a given geographic area. The planning strategy is formulated as a linear model. The core of the decision problem consists in determining both the delivery route and the corresponding day of activation along the time horizon. A decomposition of the problem into two phases has been proposed. A suitable penalty cost modeled by simulating the possibility of having an early or delayed product delivery on the delivery day returned from the inventory model (e.g. Economic Order Quantity) is the key feature of the first phase. In the second phase, deliveries are scheduled on a daily basis by taking into account the time windows associated to each customer. This is accomplished by using a constructive heuristic algorithm for the vehicle routing problem with time windows. Computational results on some realistic instances are presented and discussed.

In this thesis we study three combinatorial optimization problems belonging to the classes of Network Design and Vehicle Routing problems that are strongly linked in the context of the design and management of transportation networks: the Non-Bifurcated Capacitated Network Design Problem (NBP), the Period Vehicle Routing Problem (PVRP) and the Pickup and Delivery Problem with Time Windows (PDPTW). These problems are NP-hard and contain as special cases some well known difficult problems such as the Traveling Salesman Problem and the Steiner Tree Problem. Moreover, they model the core structure of many practical problems arising in logistics and telecommunications. The NBP is the problem of designing the optimum network to satisfy a given set of traffic demands. Given a set of nodes, a set of potential links and a set of point-to-point demands called commodities, the objective is to select the links to install and dimension their capacities so that all the demands can be routed between their respective endpoints, and the sum of link fixed costs and commodity routing costs is minimized. The problem is called non- bifurcated because the solution network must allow each demand to follow a single path, i.e., the flow of each demand cannot be splitted. Although this is the case in many real applications, the NBP has received significantly less attention in the literature than other capacitated network design problems that allow bifurcation. We describe an exact algorithm for the NBP that is based on solving by an integer programming solver a formulation of the problem strengthened by simple valid inequalities and four new heuristic algorithms. One of these heuristics is an adaptive memory metaheuristic, based on partial enumeration, that could be applied to a wider class of structured combinatorial optimization problems. In the PVRP a fleet of vehicles of identical capacity must be used to service a set of customers over a planning period of several days. Each customer specifies a service frequency, a set of allowable day-combinations and a quantity of product that the customer must receive every time he is visited. For example, a customer may require to be visited twice during a 5-day period imposing that these visits take place on Monday-Thursday or Monday-Friday or Tuesday-Friday. The problem consists in simultaneously assigning a day- combination to each customer and in designing the vehicle routes for each day so that each customer is visited the required number of times, the number of routes on each day does not exceed the number of vehicles available, and the total cost of the routes over the period is minimized. We also consider a tactical variant of this problem, called Tactical Planning Vehicle Routing Problem, where customers require to be visited on a specific day of the period but a penalty cost, called service cost, can be paid to postpone the visit to a later day than that required. At our knowledge all the algorithms proposed in the literature for the PVRP are heuristics. In this thesis we present for the first time an exact algorithm for the PVRP that is based on different relaxations of a set partitioning-like formulation. The effectiveness of the proposed algorithm is tested on a set of instances from the literature and on a new set of instances. Finally, the PDPTW is to service a set of transportation requests using a fleet of identical vehicles of limited capacity located at a central depot. Each request specifies a pickup location and a delivery location and requires that a given quantity of load is transported from the pickup location to the delivery location. Moreover, each location can be visited only within an associated time window. Each vehicle can perform at most one route and the problem is to satisfy all the requests using the available vehicles so that each request is serviced by a single vehicle, the load on each vehicle does not exceed the capacity, and all locations are visited according to their time window. We formulate the PDPTW as a set partitioning-like problem with additional cuts and we propose an exact algorithm based on different relaxations of the mathematical formulation and a branch-and-cut-and-price algorithm. The new algorithm is tested on two classes of problems from the literature and compared with a recent branch-and-cut-and-price algorithm from the literature.

A robust optimization approach for the production-inventory-routing problem with simultaneous pickup and delivery

In today’s complex and dynamic transportation networks, increasing energy costs and adverse environmental impacts necessitate the efficient transport of goods or raw materials across a network to minimize all related costs through vehicle assignment and routing decisions. Vehicle routing problems under dynamic and stochastic conditions are known to be very challenging in both mathematical modeling and computational complexity. In this study, a special variant of the full-truckload vehicle assignment and routing problem was investigated. First, a detailed analysis of the processes in a liquid transportation logistics firm with a large fleet of tanker trucks was conducted. Then, a new original problem with distinctive features compared with similar studies in the literature was formulated, including pickup/delivery time windows, nodes with different functions (pickup/delivery, washing facilities, and parking), a heterogeneous truck fleet, multiple trips per truck, multiple trailer types, multiple freight types, and setup times between changing freight types. This dynamic optimization problem was solved using an intelligent multi-agent model with agent designs that run on vehicle assignment and routing algorithms. To assess the performance of the proposed approach under varying environmental conditions (e.g., congestion factors and the ratio of orders with multiple trips) and different algorithmic parameter levels (e.g., the latest response time to orders and activating the interchange of trip assignments between vehicles), a detailed scenario analysis was conducted based on a set of designed simulation experiments. The simulation results indicate that the proposed dynamic approach is capable of providing good and efficient solutions in response to dynamic conditions. Furthermore, using longer latest response times and activating the interchange mechanism have significant positive impacts on the relevant costs, profitability, ratios of loaded trips over the total distance traveled, and the acceptance ratios of customer orders.

Vehicle Routing Problem (VRP) is addressed to a class of problems for determining a set of vehicle routes, in which each vehicle departs from a given depot, serves a given set of customers, and returns back to the same depot. On the other hand, simultaneous delivery and pickup problems have drawn much attention in the past few years due to its high usage in real world cases. This study, therefore, considered a Vehicle Routing Problem with Time Windows and Simultaneous Delivery and Pickup (VRPTWSDP) and formulated it into a mixed binary integer programming. Due to the NP-hard nature of this problem, we proposed a variant of Particle Swarm Optimization (PSO) to solve VRPTWSDP. Moreover, in this paper we improve the basic PSO approach to solve the several variants of VRP including Vehicle Routing Problem with Time Windows and Simultaneous Delivery and Pickup (VRPTWSDP), Vehicle Routing Problem with Time Windows (VRPTW), Capacitated Vehicle Routing Problem (CVRP) as well as Open Vehicle Routing Problem (OVRP). In proposed algorithm, called Improved Particle Swarm Optimization (IPSO), we use some removal and insertion techniques and also combine PSO with Simulated Annealing (SA) to improve the searching ability of PSO and maintain the diversity of solutions. It is worth mentioning that these algorithms help to achieve a trade-off between exploration and exploitation abilities and converge to the global solution. Finally, for evaluating and analyzing the proposed solution algorithm, extensive computational tests on a class of popular benchmark instances, clearly show the high effectiveness of the proposed solution algorithm.

The two-echelon production routing problem with cross-docking satellites