We tackle three optimization problems in which a colored graph, where each node is assigned a color, must be partitioned into colorful connected components. A component is defined as colorful if each color appears at most once. The problems differ in the objective function, which determines which partition is the best one. These problems have applications in community detection, cybersecurity, and bioinformatics. We present integer nonlinear formulations, which are then linearized using standard techniques. To solve these formulations, we develop exact branch-and-cut algorithms, embedding various improving techniques, such as valid inequalities, bounds limiting the number of variables, and warm-start and preprocessing techniques. Extensive computational tests on benchmark instances demonstrate the effectiveness of the proposed procedures. The branch-and-cut algorithms can solve reasonably sized instances efficiently. To the best of our knowledge, we are the first to propose an exact algorithm for solving these problems. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of M. Cerulli was partially funded by project “SEcurity and RIghts in the CyberSpace” [Grant PE00000014] under the MUR National Recovery and Resilience Plan funded by the European Union - NextGenerationEU. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0927 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0927 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

ABSTRACT We introduce new matheuristic algorithms for the Inventory Routing Problem with unsplit and split deliveries for both Order‐Up‐to Level and Maximum Level replenishment policies. The first matheuristic is based on the Capacitated Concentrator Location problem. The second is a route‐based approach using routes found in other schemes as input, including the ones found in the first matheuristic. We carry out extensive experiments on benchmark instances to understand their effectiveness. The results show that they are effective and require a relatively short computational time.

ABSTRACT The Split Delivery Vehicle Routing Problem with Two‐dimensional Loading Constraints (2L‐SDVRP) integrates vehicle routing, split delivery, and two‐dimensional packing constraints. In the 2L‐SDVRP, customers can be served by multiple vehicles, and their demands consist of different two‐dimensional rectangular items that must be packed in the vehicles' bases. The problem involves determining the least‐cost routes that satisfy all customer demands while ensuring the feasible packing of items in each vehicle. We present tailored branch‐and‐cut (BC) methods for solving the 2L‐SDVRP. One of the methods is based on an effective, relaxed two‐index vehicle flow formulation that is newly introduced in this paper. To evaluate the performance of the BC methods, computational experiments were conducted using both benchmark instances and new realistic instances inspired by cases from Brazilian logistics companies. The results indicate the superior performance of the method based on the two‐index formulation, which obtained optimal solutions for 14 more instances than the other approach on the benchmark instances. This method also performed better on newly created instances, improving solutions by 5.6% on average.

Abstract The commodity constrained split delivery vehicle routing problem (C‐SDVRP) is a routing problem where customer demands are composed of multiple commodities. A fleet of capacitated vehicles must serve customer demands in a way that minimizes the total routing costs. Vehicles can transport any set of commodities and customers are allowed to be visited multiple times. However, the demand for a single commodity must be delivered by one vehicle only. In this work, we developed a heuristic with a performance guarantee to solve the C‐SDVRP. The proposed heuristic is based on a set covering formulation, where the exponentially‐many variables correspond to routes. First, a subset of the variables is obtained by solving the linear relaxation of the formulation by means of a column generation approach which embeds a new pricing heuristic aimed to reduce the computational time. Solving the linear relaxation gives a valid lower bound used as a performance guarantee for the heuristic. Then, we devise a restricted master heuristic to provide good upper bounds: the formulation is restricted to the subset of variables found so far and solved as an integer program with a commercial solver. A local search based on a mathematical programming operator is applied to improve the solution. We test the heuristic algorithm on benchmark instances from the literature. The comparison with the state‐of‐the‐art heuristics for solving the C‐SDVRP shows that our approach significantly improves the solution time, while keeping a comparable solution quality and improving some best‐known solutions. In addition, our approach is able to solve large instances with 100 customers and six commodities, and also provides very good quality lower bounds. Furthermore, an instance of the C‐SDVRP can be transformed into a CVRP instance by simply duplicating each customer as many times as the requested commodities and by assigning as demand the demand of the single commodity. Hence, we compare heuristics for the C‐SDVRP against the state‐of‐the‐art heuristic for the Capacitated Vehicle Routing Problem (CVRP). The latter approach revealed to have the best performance. However, our approach provides solutions of comparable quality and has the interest of providing a performance guarantee.

Abstract We study the benefit of introducing split deliveries in the inventory routing problem (IRP), both when the order‐up‐to level (OU) and the maximum level replenishment policies are applied. We first propose a mathematical formulation and solve it by implementing a branch‐and‐cut algorithm. Then, we carry out a worst‐case analysis to show the cost increase we have in the worst case by using unsplit deliveries instead of split deliveries, both for the OU and the maximum‐level replenishment policies. Extensive computational results on benchmark instances allow us to evaluate the benefit of introducing split deliveries. Finally, a sensitivity analysis on customer demands, initial inventory levels, maximum inventory levels and distance to the depot allows us to understand the instance features that make split deliveries effective in IRPs.

Abstract This article studies the Hazardous Orienteering Problem (HOP), a variant of the more famous Orienteering Problem (OP). In the OP, a vehicle earns a profit for each customer it visits (e.g., to pick up a parcel) subject to an upper bound on the tour time. In the HOP, the parcels picked up at some customers have a probability of triggering a catastrophic event. The probability depends on how long the parcels travel on the vehicle. If any catastrophic event triggers, the entire collected profit is lost. The goal is to determine the tour that maximizes the expected profit. The problem has interesting applications in routing of hazardous material, cash‐in‐transit, and law enforcement. We propose a mixed‐integer nonlinear formulation and techniques both to obtain dual bounds and to produce primal solutions. Computational tests investigate the efficacy of the methods proposed and allow to gain insights into solution features.

Long-haul truck transportation is concerned with freight transportation from shipments' origins to destinations, with vehicle trips lasting from some hours to several days. Drivers performing long-haul transportation are subject to strict rules derived from Hours of Service (HoS) regulations. There exists a large body of literature integrating HoS regulations within long-haul transportation. The optimization problems in this context generally deal with routing and scheduling decisions aimed at determining where a driver should stop and how long a rest should be. However, the overwhelming majority of the literature on long-haul transportation ignores refueling decisions and treats fuel costs as proportional to the traveled distance. In this talk we analyze a long-haul truck scheduling problem where a path has to be determined for a vehicle traveling from a specified origin to a specified destination. We consider refueling decisions along the path while accounting for heterogeneous fuel prices in a road network. Furthermore, the path has to comply with Hours of Service (HOS) regulations. Therefore, a path is defined by the actual road trajectory traveled by the vehicle, as well as the locations where the vehicle stops due to refueling, compliance with HOS regulations, or a combination of the two. This setting is cast in a bi-objective optimization problem, considering the minimization of fuel cost and the minimization of path duration. An algorithm is proposed to solve the problem on a road network. The algorithm builds a set of non-dominated paths with respect to the two objectives. Given the enormous theoretical size of the road network, the algorithm follows an interactive path construction mechanism. Specifically, the algorithm dynamically interacts with a geographic information system to identify the relevant potential paths and stop locations. Computational tests are made on real-sized instances where the distance covered ranges from 500 to 1500 km. The algorithm is compared with solutions obtained from a policy mimicking the current practice of a logistics company. The results show that the non-dominated solutions produced by the algorithm significantly dominate the ones generated by the current practice, in terms of fuel costs, while achieving similar path durations. The average number of non-dominated paths is 2.7, which allows decision-makers to ultimately visually inspect the proposed alternatives.

Abstract In the e‐commerce era, vendors have to satisfy a large number of on‐line orders, mainly from private customers, with low weight and volume, reduced delivery time, and overlap of customers' time windows. Production is made available all day long. New strategies and new technologies are emerging for deliveries. The processing time of the orders is reduced. These new features generate interesting challenges in formulating and solving Routing and Inventory Routing problems. After discussing these features and the corresponding challenges, we recall the relevant literature in Routing and Inventory Routing and provide future research directions, mainly related to routing problems with release dates, routing problems with crowdshipping, and inventory routing problems in the e‐commerce era.

We consider the inventory routing problem, in which a supplier has to replenish a set of customers by means of a limited fleet of capacitated vehicles over a discrete time horizon. The goal is to minimize the total cost of the distribution that comprises the inventory cost at the supplier and at the customers and the routing cost. We present a matheuristic that combines a tabu search and mathematical programming formulations. When compared with two exact methods on 640 small instances, the matheuristic finds 192 (48%) optima over the 402 instances with known optima and improves 125 upper bounds. Tested on 240 large instances (with up to 200 customers) for which no optimal solutions are known, it improves the best solution for 220 (92%) of the 240 instances. The online supplement is available at https://doi.org/10.1287/ijoc.2016.0737 .

Many logistic problems arising in supply chain management, distribution and inventory management call for the integration of different components of the production/distribution system which have to be coordinated in such a way that a common objective, which can be the cost minimization or the revenue maximization, is optimized. Thus, in order to find the best management policy, one should be able to tackle the problem as a whole and to find an integrated policy that is aimed at optimizing the system behavior. However, known practices as well as the scientific literature have shown a major attitude in proposing strategies that are aimed at decomposing the system in parts and then proposing optimal policies for each single part. This clearly leads to a strategy that is far from being optimal for the global system. The aim of this work is to focus on the advantages that integrated policies can provide when used to handle production, inventory and distribution problems. We will present some cases that are mainly dealing with distribution problems and show the strategies proposed in the literature. We will also present a study on a distribution system where the inventory and the distribution costs have to be minimized.

Abstract In this article, we study the capacitated team orienteering problem where split deliveries are allowed. A set of potential customers is given, each associated with a demand and a profit. The set of customers to be served by a fleet of capacitated vehicles has to be identified in such a way that the profit collected is maximized, while satisfying constraints on the maximum time duration of each route and the vehicle capacity constraints. When split deliveries are allowed, each customer may be served by more than one vehicle. We show that the profit collected by allowing split deliveries may be as large as twice the profit collected under the constraint that each customer has to be served by one vehicle at most. We then present a branch‐and‐price exact algorithm and a hybrid heuristic. We show the effectiveness of the proposed approaches on benchmark instances and on a new set of instances that allow us to computationally evaluate the impact of split deliveries. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(1), 16–33 2014

Abstract In this article, we study a variant of the capacitated team orienteering problem, that is the problem where a fleet of vehicles, each with a constraint on the time available, is given to serve profitable customers with the objective of maximizing the collected profit. We study the variant where customers may be only partially served (incomplete service) and, if beneficial, also by more than one vehicle (split deliveries). We will analyze the maximum theoretical increase of the profit due to the incomplete service and to the split deliveries. We also computationally measure such increase on a set of instances, by means of an exact algorithm on small/medium size instances and of two heuristics on instances of larger size. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 63(2), 135–145 2014

We consider an inventory routing problem in discrete time where a supplier has to serve a set of customers over a multiperiod horizon. A capacity constraint for the inventory is given for each customer, and the service cannot cause any stockout situation. Two different replenishment policies are considered: the order-up-to-level and the maximum-level policies. A single vehicle with a given capacity is available. The transportation cost is proportional to the distance traveled, whereas the inventory holding cost is proportional to the level of the inventory at the customers and at the supplier. The objective is the minimization of the sum of the inventory and transportation costs. We present a heuristic that combines a tabu search scheme with ad hoc designed mixed-integer programming models. The effectiveness of the heuristic is proved over a set of benchmark instances for which the optimal solution is known.

Abstract This article studies the pickup and delivery traveling salesman problem with multiple stacks. The vehicle contains a number of (horizontal) stacks of finite capacity for loading items from the rear of the vehicle. Each stack must satisfy the last‐in‐first‐out constraint that states that any new item must be loaded on top of a stack and any unloaded item must be on top of its stack. A branch‐and‐cut algorithm is proposed for solving this problem. Computational results are reported on different types of randomly generated instances as well as on classical instances for some well‐known special cases of the problem. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

Abstract In this article we present a branch‐and‐price‐and‐cut method for the solution of the split delivery vehicle routing problem (SDVRP). The SDVRP is the problem to serve customers with a fleet of capacitated vehicles at minimum traveling cost. With respect to the classical vehicle routing problem, where each customer is visited exactly once, in the SDVRP a customer may be visited any number of times. The exact method we propose is based on a decomposition of the problem where the possible routes, with the delivery quantities, are generated in the subproblem. The generated routes are also used to find a heuristic solution to the problem. We consider both the case where the fleet of vehicles is unlimited and the case where the fleet is limited to the minimum possible number of vehicles. We solve to optimality instances with larger size with respect to previous approaches, find new best solutions to several benchmark instances and reduce the optimality gap on most of the benchmark instances. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(4), 241–254 2011

Abstract In this article we present mathematical programming formulations and solution approaches for the optimal solution of the Double Travelling Salesman Problem with Multiple Stacks (DTSPMS). A set of orders is given, each one requiring transportation of one item from a customer in a pickup region to a customer in a delivery region. The vehicle available for the transportation in each region carries a container. The container is organized in rows of given length. Each row is handled independently from the others according to a Last In First Out stack policy. The DTSPMS problem consists of determining the pickup tour, the loading plan of the container and the delivery tour in such a way that the total length of the two tours is minimized. The formulations are based on different modeling ideas and each formulation gives rise to a specific solution approach. We present computational results on a set of benchmark instances that compare the different approaches and show that the most successful one is a decomposition approach applied to a new model. © 2010 Wiley Periodicals, Inc. NETWORKS, 2010

Abstract In this paper, we study the reoptimization problems which arise when a new node is added to an optimal solution of a traveling salesman problem (TSP) instance or when a node is removed. We show that both reoptimization problems are NP‐hard. Moreover, we show that, while the cheapest insertion heuristic has a tight worst‐case ratio equal to 2 when applied to a TSP instance, it guarantees, in linear time, a tight worst‐case ratio equal to 3/2 when used to add the new node and that also the simplest heuristic to remove a node from the optimal tour guarantees a tight ratio equal to 3/2 in constant time. © 2003 Wiley Periodicals, Inc.