Vehicle routing problems (VRPs) are a large class of well-studied and computationally hard combinatorial optimization problems. In the classical capacitated VRP, a fleet of homogeneous capacitated vehicles, that start and end their route at a depot, must satisfy customers’ demand at minimum cost. Many variants and extensions of this problem have been studied. In this paper, after discussing the capacitated VRP, we review the literature for the most-studied variants and extensions of the VRP and then focus on the most recent developments and trends.

The inventory routing problem (IRP) focuses on jointly optimizing inventory and distribution operations from a supplier to retailers over multiple days. Compared to other problems from the vehicle routing family, the interrelations between inventory and routing decisions render IRP optimization more challenging and call for advanced solution techniques. A few studies have focused on developing large neighborhood search approaches for this class of problems, but this remains a research area with vast possibilities due to the challenges related to the integration of inventory and routing decisions. In this study, we advance this research area by developing a new large neighborhood search operator tailored for the IRP. Specifically, the operator optimally removes and reinserts all visits to a specific retailer while minimizing routing and inventory costs. We propose an efficient tailored dynamic programming algorithm that exploits preprocessing and acceleration strategies. The operator is used to build an effective local search routine, and included in a state-of-the-art routing algorithm, i.e., Hybrid Genetic Search. Through extensive computational experiments, we demonstrate that the resulting heuristic algorithm leads to solutions of unmatched quality up to this date, especially on large-scale benchmark instances.

This paper investigates stochastic scheduling and routing problems in the online meal delivery (OMD) service. The huge increase in meal delivery demand requires the service providers to construct a highly efficient logistics network to deal with a large-volume of time-sensitive and fluctuating fulfillment, often using inhouse and crowdsourced drivers to secure the ambitious service quality. We aim to address the problem of developing an effective scheduling and routing policy that can handle real-life situations. To this end, we first model the dynamic problem as a Markov Decision Process (MDP) and analyze the structural properties of the optimal policy. Then we propose four integrated approaches to solve the operational level scheduling and routing problem. In addition, we provide a continuous approximation formula to estimate the bounds of required fleet size for the inhouse drivers. Numerical experiments based on a real dataset show the effectiveness of the proposed solution approaches. We also obtain several managerial insights that can help decision makers in solving similar resource allocation problems in real-time.

In the vehicle routing problem (VRP), a fleet of capacitated vehicles serves a set of customers with known demands at minimum cost. The Split Delivery VRP (SDVRP), a variant of VRP, permits customer demands to be served by different vehicles, and thereby results in a significant cost reduction. In this paper, we study the VRP with transfers (VRP-T), where, as in the SDVRP, customers can be visited by multiple vehicles. However, differently from the SDVRP, customer locations can be used as transfer locations to exchange load among vehicles. We find that the routing costs can potentially be reduced by at most 50% by allowing such transfers in the VRP. We develop a two-index mixed-integer programming (MIP) model and heuristic algorithm based on Multi-start local search (MSLS). The solutions obtained by the MSLS are used as warm-start for the MIP and the corresponding approach is tested. Computational results indicate substantial savings in the costs incurred and the number of vehicles required when transfers are allowed. Results also suggest that transfers are at least as effective in reducing the cost as split delivery.

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.

Challenges in last-mile delivery have encouraged innovative solutions like crowdsourced delivery, where online platforms leverage the services of drivers who occasionally perform delivery tasks for compensation. A key challenge is that occasional drivers' acceptance behavior towards offered tasks is uncertain and influenced by task properties and compensation. The current literature lacks formulations that fully address this challenge. Hence, we formulate an integrated problem that maximizes total expected cost savings by offering task bundles to occasional drivers. To this end, we simultaneously determine the optimal bundle set, their assignment to occasional drivers, and compensations for each pair while considering acceptance probabilities, which are captured via generic logistic functions. The vast number of potential bundles, combined with incorporating acceptance probabilities leads to a mixed-integer nonlinear program (MINLP) with exponentially many variables. Using mild assumptions, we address these complexities by exploiting properties of the problem, leading to an exact linearization of the MINLP which we solve via a tailored exact column generation algorithm. Our algorithm uses a variant of the elementary shortest path problem with resource constraints (ESPPRC) that features a non-linear and non-additive objective function as its subproblem, for which we develop tailored dominance and pruning strategies. We introduce several heuristic and exact variants and perform an extensive set of experiments evaluating the algorithm performances and solution structures. The results demonstrate the efficiency of the algorithms for instances with up to 120 tasks and 60 drivers and highlight the advantages of integrated decision-making over sequential approaches. The sensitivity analysis indicates that compensation is the most influential factor in shaping the bundle structure.

This paper provides a tutorial on Branch-Price-and-Cut (BPC) algorithms for a generic class of problems whose objective is to find a set of feasible paths in a graph while optimising a given objective function. The tutorial is split into two main parts. First, we describe the building blocks of a BPC algorithm: the Branch-and-Bound algorithm, the column generation procedure and the Branch-and-Cut algorithm. Then, we focus on the description of a BPC algorithm for the class of problems we consider. Precisely, we present the classical and advanced techniques that should be embedded in an efficient algorithm. Particular attention is devoted to the solution of the pricing problem in the case where it is formulated as an Elementary Shortest Path Problem with Resource Constraints. The aim of the tutorial is pedagogical. Hence, its intended reader is someone facing the first implementation of a BPC algorithm. Implementation tips and examples accompany the techniques and concepts to ease their comprehension. Precisely, the examples are based on the Capacitated Vehicle Routing Problem, which is a well-known problem belonging to the class we consider.

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.

We tackle three optimization problems in which a colored graph must be partitioned into connected colorful components. A component is colorful if all its vertices have different colors. We consider three different objectives, formulate the resulting problems as integer nonlinear programs and linearize them through standard linearization techniques. Then, we develop exact branch-and-cut algorithms, including families of valid inequalities for each problem, bounds reducing the size of the models, warm-start and preprocessing techniques.

Recent advancements in deep reinforcement learning have sparked a growing interest in the application of this approach to solve combinatorial optimization (CO) problems. This paper presents neural combinatorial optimization (NCO) as a framework for constructing functions that work as heuristics for CO problems. Given the rapid expansion of the field and the increasing interest in the topic, this tutorial introduces the main techniques utilized in NCO and explores the current open issues in the field.We define key terms and concepts related to NCO and present the latest developments, using the Knapsack Problem as a running example to complement theoretical explanations. Finally, we analyze prominent works in the field of NCO, with a focus on their application to the Traveling Salesman Problem, which serves as the most extensively studied problem in this domain.

We tackle the problems of workforce sizing and shift scheduling of a logistic operator delivering parcels in the last-mile segment of the supply chain. Our working hypothesis is that the relevant decisions are affected by two main trade-offs: workforce size and shift stability. A large workforce can deal with demand fluctuations but incurs higher fixed costs; by contrast, a small workforce might require excessive outsourcing to third-party logistic providers. Stable shifts, i.e., with predictable start times and lengths, improve worker satisfaction and reduce turnover; at the same time, they might be less able to adapt to an unsteady demand. We test these assumptions through an extensive computational campaign based on a novel mathematical formulation. We find that extreme shift stability is, indeed, unsuitable for last-mile operations. At the same time, introducing a very limited amount of flexibility achieves similar effects as moving to a completely flexible system while ensuring a better work-life balance for the workers. Several recent studies in the social sciences have warned about the consequences of precarious working conditions for couriers and retail workers and have recommended—among other things—stable work schedules. Our work shows that it is possible to offer better working conditions in terms of shift stability without sacrificing the company’s bottom line. Thus, companies prioritising profitability (as is often the case) can improve workers’ well-being and increase retention with a negligible cost impact.

We tackle the problem of coordinating a three-echelon last-mile delivery system. In the first echelon, trucks transport parcels from distribution centres outside the city to public transport stops. In the second echelon, parcels move on public transport and reach the city centre. In the third echelon, zero-emission vehicles pick the parcels at public transport stops and deliver them to customers. We introduce two extended formulations this problem. The first has two exponential sets of variables, while the second has one. We propose column generation algorithms and compare several methods to solve the pricing problems on specially constructed graphs. We also devise dual bounds, which we can compute even when the graphs are so large that not single pricing round completes within the time limit. Compared to previous formulations, our models find new best known solutions out of an existing dataset of 24 instances from the literature.

This paper introduces the Green Commodity constrained Split Delivery Vehicle Routing Problem (GC-SDVRP), which involves designing efficient and environmentally friendly delivery routes that reduce the CO2 emissions associated with transporting multiple commodities. In this problem, different commodities demanded by a customer can be delivered by one or more vehicles, if beneficial, which poses additional modeling and solution challenges. We propose a relaxed formulation that provides a lower bound on the optimal value of the GC-SDVRP, and adapt two other formulations from the literature to address this problem. Additionally, we develop a branch-and-cut (BC) method based on two of these formulations, and introduce a procedure for deriving feasible solutions to the GC-SDVRP from solutions obtained with the relaxed formulations. The results of computational experiments performed on benchmark instances indicate the superior performance of the BC method based on the proposed formulation. Furthermore, they show that, contrary to the traditional objective of minimizing distance, the GC-SDVRP is significantly easier to solve to optimality and can reduce CO2 emissions by 2.59% compared to the problem that minimizes total travel distance. Our investigation also reveals that increasing vehicle capacity improves solution quality in the GC-SDVRP, while split delivery can enable further reductions in CO2 emissions. Finally, although increasing the number of commodities imposes challenges in solving the problem, the possibility of split delivery mitigates its impact on the value of the final solution, indicating that an increase in the number of commodities does not necessarily result in higher CO2 emissions.

This paper studies Pickup and Delivery Problems on Rings (PDP-R), i.e., a circular network with m stations. A set of transportation requests has to be served, where each request is associated with an origin, a destination and a quantity. A fleet of homogeneous capacitated vehicles is available at the depot. The ring can be traversed in one direction only, i.e, clockwise. The objective is to assign requests to vehicles and define the service sequence for each vehicle, while minimizing the total completion time of requests, i.e, the sum of the time at which each request is served. We prove that the problem is NP-hard. We propose an ILP formulation and we show its effectiveness through exhaustive computational tests on synthetic instances.

In this paper, we study a Kidney Exchange Problem (KEP) with altruistic donors and incompatible patient-donor pairs. Kidney exchanges can be modelled in a directed graph as circuits starting and ending in an incompatible pair or as paths starting at an altruistic donor. For medical reasons, circuits and paths are of limited length and are associated with a medical benefit to performing the transplants. The aim of the KEP is to determine a set of disjoint kidney exchanges of maximal medical benefit or maximal cardinality. Several solution methods are available in the literature to solve the KEP. However, they are usually most able to efficiently address instances with specific characteristics or of limited size. In this work, we propose an efficient method that stands out from the literature as it is able to report solutions of very good quality on large-size instances regardless of the characteristics (the type of objective function and the maximum length of circuits or paths). To do so, we consider a set-packing formulation for the KEP with exponentially many variables associated with circuits and paths, and develop a Branch-Price-and-Cut (BPC) algorithm to solve it. As a methodological contribution, the BPC algorithm features two novel heuristics to separate a well-known family of nonrobust inequalities, namely, the subset-row inequalities. The heuristics aim to detect such inequalities via an original transformation from violated clique and odd-hole inequalities. Extensive computational experiments have been performed on three sets of instances from the literature and on a newly generated set of challenging instances. On the easiest instances, the BPC algorithm yields results comparable with the literature, whereas on the other sets it clearly outperforms the previous approaches.

Same-day delivery (SDD) has become a new standard to satisfy the “instant gratification” of online customers. Despite existing powerful technologies deployed in last-mile delivery, SDD services face new decision-making challenges on the tradeoff between delivery costs and time. In addition, new concerns on environmental issues, customer satisfaction, and fairness arise. Researchers have explored various approaches to face these challenges in SDD, where data uncertainty plays a fundamental role. In this paper, we carefully review the emerging routing problems and solutions proposed in the existing literature for SDD services. We survey papers on how to manage dynamic order arrivals, how to allocate time slots for deliveries, how to select the right delivery options, how to design pickup and delivery routes, and how to partition delivery areas and decide the composition of the fleet. We also propose mathematical formulations for representative problems. Finally, we sketch managerial insights and identify future research directions.

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 non-linear 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.

This study explores the potential of using public transportation systems for freight delivery, where we intend to utilize the spare capacities of public vehicles like buses, trams, metros, and trains, particularly during off-peak hours, to transport packages within the city instead of using dedicated delivery vehicles. The study contributes to the growing literature on innovative strategies for performing sustainable last mile deliveries. We study an operational level problem called the Three-Tier Delivery Problem on Public Transportation, where packages are first transported from the Consolidation and Distribution Center (CDC) to nearby public vehicle stations by delivery trucks, comprising the first tier of the problem. In the second tier, the public vehicles pick them up from the stops and transport them into the city area. The last leg, or the third tier of the delivery, is performed to deliver the packages to their respective customers using green vehicles or eco-friendly systems. We propose mixed-integer linear programming formulations to study the transport of packages from the CDC to the customers and employ decomposition-based matheuristics to solve them. We have three decomposition approaches based on the order of solving the tiers, resulting from the tier we start solving the problem from. We use a heuristic methodology to link the tiers by coordinating the flow of packages between them, and utilize CPLEX to solve the individual tiers. We provide numerical experiments to demonstrate the efficiency and effectiveness of the system. Our results show that this system has the potential to reduce the length of trips performed by traditional delivery trucks by 85.91

This paper investigates a variant of an inventory-routing problem (IRP) that enforces two conditions on the structure of the solution: time-invariant routes, and a fixed, injective (i.e., one-to-one) assignment of routes to vehicles. The practical benefits of concurrent route invariance and driver assignments are numerous. Fixed routes reduce the solution space of the problem and improve its tractability; they simplify operations; and they increase the viability of newer delivery technologies like drones and autonomous vehicles. Consistency between driver and customer is linked to improved service, driver job satisfaction and delivery efficiency, and is also an important consideration in certain contexts like home healthcare. After formulating the problem a mixed integer-linear program, we recast it as a set partitioning problem whose linear relaxation is solved via column generation. Due to the prohibitively expensive nature of the pricing problem that generates new columns, we present a novel column generation-based heuristic for it that relies on decoupling routing and inventory management decisions. We demonstrate the effectiveness of the proposed method via a numerical study.