Temporal dependencies between customer visits, such as synchronization constraints, pose a fundamental challenge in vehicle routing. These dependencies, which arise in applications such as home healthcare routing, aircraft scheduling, and technician routing, introduce inter-route constraints that make the resulting problems significantly harder to solve. We present an exact solution method for vehicle routing problems with temporal dependencies capable of handling all types of temporal dependencies studied in the literature, unlike most existing approaches that target specific subclasses. Our approach is based on a fragment-based formulation in which routes are represented as sequences of a new type of fragment, designed to handle temporal dependency constraints. This formulation is solved via a price-cut-and-enumerate algorithm that computes a lower bound using alternating column-and-row generation, obtains an initial upper bound, and iteratively refines both bounds through fragment enumeration and branch-and-cut, supported by several new classes of valid inequalities. Computational experiments show that our method significantly outperforms state-of-the-art benchmark methods and is able to solve previously intractable instances while covering a wider range of temporal dependencies.

The single-source capacitated facility location problem with customer preferences (SSCFLPCP) is known to be strongly NP-hard. Computational tests imply that state-of-the-art solvers struggle with computing exact solutions. In this paper, we contribute two novel preprocessing methods which reduce the size of the considered integer programming formulation and introduce sets of valid inequalities which decrease the integrality gap. Each of the introduced results utilises structural synergies between capacity constraints and customer preferences in the SSCFLPCP. First, we derive two preprocessing methods where the first method fixes location variables and the second method fixes allocation variables. Afterwards, we study cover-based inequalities. Here, we first strengthen the well-known cover inequalities: when determining covers, we also consider demands of customers not in the cover that must be assigned to the covered facility if a customer in the cover is assigned to it. We further strengthen these inequalities by including information on the assignments of customers in a cover if they are not assigned to the covered facility. Afterwards, we derive a new family of valid inequalities, which expresses the relation of open facilities based on sets of customers covering a facility. We then discuss solution methods for the corresponding separation problems and, finally, test our results for two preference types in a computational study. Our results show a clear positive impact of the preprocessing methods and inequalities, in particular when preferences are defined by assignment costs.

This paper introduces the concept of task-splitting into home healthcare (HHC) routing and scheduling. It focuses on the design of routes and timetables for caregivers providing services at patients' homes. Task-splitting is the division of a (lengthy) patient visit into separate visits that can be performed by different caregivers at different times. The resulting split parts may have reduced caregiver qualification requirements, relaxed visiting time windows, or a shorter/longer combined duration. However, additional temporal dependencies can arise between them. To incorporate task-splitting decisions into the planning process, we introduce two different mixed integer linear programming formulations, a Miller-Tucker-Zemlin and a time-indexed variant. These formulations aim to minimize operational costs while simultaneously deciding which visits to split and imposing a potentially wide range of temporal dependencies. We also propose pre-processing routines for the time-indexed formulation and two heuristic procedures. These methods are embedded into the branch-and-bound approach as primal and improvement heuristics. The results of our computational study demonstrate the additional computational difficulty introduced by task-splitting possibilities and the associated additional synchronization, and the usefulness of the proposed heuristic procedures. From a planning perspective, our results indicate that integrating task-splitting decisions into the planning process reduces staff requirements, decreases HHC operational costs, and allows caregivers to spend relatively more time on tasks aligned with their qualifications.

Two-stage stochastic programs become computationally challenging when the number of scenarios representing parameter uncertainties grows. Motivated by this, we propose the TULIP-algorithm ("Two-step warm start method Used for solving Large-scale stochastic mixed-Integer Problems"), a two-step approach for solving two-stage stochastic (mixed) integer linear programs with an exponential number of constraints. In this approach, we first generate a reduced set of representative scenarios and solve the root node of the corresponding integer linear program using a cutting-plane method. The generated constraints are then used to accelerate solving the original problem with the full scenario set in the second phase. We demonstrate the generic effectiveness of TULIP on two benchmark problems: the Stochastic Capacitated Vehicle Routing Problem and the Two-Stage Stochastic Steiner Forest Problem. The results of our extensive numerical experiments show that TULIP yields significant computational gains compared to solving the problem directly with branch-and-cut.

A frequently studied problem in the context of digital marketing for online social networks is the influence maximization problem that seeks for an initial seed set of influencers to trigger an information propagation cascade (in terms of active message forwarders) of expected maximum impact. Previously studied problems typically neglect that the probability that individuals passively view content without forwarding it is much higher than the probability that they forward content. Considering passive viewing enables us to maximize more natural (social media) marketing metrics, including (a) the expected organic reach, (b) the expected number of total impressions, or (c) the expected patronage, all of which are investigated in this paper for the first time in the context of influence maximization. We propose mathematical models to maximize these objectives, whereby the model for variant (c) includes individual’s resistances and uses a multinomial logit model to model customer behavior. We also show that these models can be easily adapted to a competitive setting in which the seed set of a competitor is known. In a computational study based on network graphs from Twitter (now X) and from the literature, we show that one can increase the expected patronage, organic reach, and number of total impressions by 36% on average (and up to 13 times in particular cases) compared with seed sets obtained from the classical maximization of message-forwarding users. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Federal Ministry of Education, Science and Research of Austria and by the Austrian Agency for International Mobility and Cooperation in Education, Science and Research [Reference ICM-2019-13384]. 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.2023.0047 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0047 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We study the problem of optimizing assortment decisions in the presence of product-specific costs when customers choose according to a multinomial logit model. This problem is NP-hard, and approximate solutions methods have been proposed in the literature to obtain both lower and upper bounds in a tractable manner. We propose the first exact solution method for this problem and show that provably optimal assortments of instances with up to 1,000 products can be found, on average, in about 2/10 of a second. In particular, we propose a bounding procedure to enhance an approximation method originally proposed by Feldman and Topaloglu and provide tight lower and upper bounds at a fraction of a second. We show how these bounds can be used to effectively identify an optimal assortment. We also describe how to adapt our approach to handle cardinality or space/resource capacity constraints on the assortment as well as assortment optimization under a mixed-multinomial logit model. In both cases, our solution method provides significant computational boosts compared with exact methods from the literature.

One of the recent innovations in urban distribution is crowdsourced delivery, where deliveries are made by occasional drivers who wish to utilize their surplus resources (unused transport capacity) by making deliveries in exchange for some compensation. The potential benefits of crowdsourced delivery include reduced delivery costs and increased flexibility (by scaling delivery capacity up and down as needed). The use of occasional drivers poses new challenges because (unlike traditional couriers) neither their availability nor their behavior in accepting delivery offers is certain. The relationship between the compensation offered to occasional drivers and the probability that they will accept a task has been largely neglected in the scientific literature. Therefore, we consider a setting in which compensation-dependent acceptance probabilities are explicitly considered in the process of assigning delivery tasks to occasional drivers. We propose a mixed-integer nonlinear model that minimizes the expected delivery costs while identifying optimal assignments of tasks to a mix of professional and occasional drivers and their compensation. We propose an exact two-stage solution algorithm that allows to decompose compensation and assignment decisions for generic acceptance probability functions and show that the runtime of this algorithm is polynomial under mild conditions. Finally, we also study a more general case of the considered problem setting, show that it is NP-hard and propose an approximate linearization scheme of our mixed-integer nonlinear model. The results of our computational study show clear advantages of our new approach over existing ones. They also indicate that these advantages remain in dynamic settings when tasks and drivers are revealed over time and in which case our method constitutes a fast, yet powerful heuristic.

This archive is distributed in association with the INFORMS Journal on Computing under the MIT License. This repository contains codes (written in Julia) and results that were used in the research reported on in the paper: Kahr Michael, Leitner Markus, Ljubic Ivana. The impact of passive social media viewers in influence maximization, 2024, (https://doi.org/10.1287/ijoc.2023.0047). Important: This repository will not receive any updates. Moreover, the provided code is unlikely to work on machines with up-to-date software, because it uses outdated Julia packages and CPLEX versions (see the requirements below). An updated version (and potential future updates) can be found here: https://github.com/m-kahr/mnlcimp.

A fortification game (FG) is a three-level, two-player hierarchical game, also known as defender-attacker -defender game, in which at the uppermost level, the defender selects some assets to be protected from potential malicious attacks. At the middle level, the attacker solves an interdiction game by depreciating unprotected assets, i.e., reducing the values of such assets for the defender, while at the innermost level the defender solves a recourse problem over the surviving or partially damaged assets. Fortification games have applications in various important areas, such as military operations, design of survivable networks, protection of facilities or power grid protection. In this work, we present an exact solution algorithm for FGs, in which the recourse problems correspond to (possibly NP-hard) combinatorial optimization prob-lems. The algorithm is based on a new generic mixed-integer linear programming reformulation in the natural space of fortification variables. Our new model makes use of fortification cuts that measure the contribution of a given fortification strategy to the objective function value. These cuts are generated when needed by solving separation problems, which correspond to (modified) middle-level interdiction games. We design a branch-and-cut-based solution algorithm based on fortification cuts, their strength-ened versions and other speed-up techniques. We present a computational study using the knapsack for-tification game and the shortest path fortification game. For the latter one, we include a comparison with a state-of-the-art solution method from the literature. Our algorithm outperforms this method and allows us to solve previously unsolved instances with up to 330 386 nodes and 1 202 458 arcs to optimality.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Concerns about greenhouse gas emissions and government regulations foster the use of electric vehicles. Several recently published articles study the use of electric vehicles (EVs) in node-routing problems. In contrast, this article considers EVs in the context of arc routing while also addressing practically relevant aspects that have not been addressed sufficiently so far. These include dynamic charging of EVs while driving, speed-dependent energy consumption, and nonlinear charging functions that depend on the battery???s state of charge and the charging time. A generic way of dealing with these aspects is introduced through the concept of an energy-indexed graph, which is used to derive an integer linear programming formulation and a solution framework based on branch and cut. Efficient construction heuristics and a local search for approximately solving large-scale instances are proposed. A computational study is performed on realistic problem instances. Besides analyzing the performance of all proposed methods, the obtained results also provide insights into strategic decisions related to the battery size and the amount of charging facilities.

We study the multi-depot split-delivery vehicle routing problem (MDSDVRP) that combines the advantages and potential cost savings of multiple depots and split deliveries and develop the first exact algorithm for this problem. We propose an integer programming formulation using a small number of decision variables and several sets of valid inequalities. These inequalities focus on ensuring the vehicles’ capacity limits and that vehicles return to their initial depot. As we show that the new constraints do not guarantee these aspects, our branch-and-cut framework also includes an efficient feasibility check for candidate solutions and explicit feasibility cuts. The algorithm that also uses a comparably simple, yet effective heuristic to compute high-quality initial solutions is tested on the MDSDVRP and two well-known special cases, the split-delivery vehicle routing problem (SDVRP) and the multi-depot traveling salesman problem (MDTSP). The results show that the new inequalities tighten the linear programming relaxation, increase the performance of the branch-and-cut algorithm, and reduce the number of required feasibility cuts. We report the first proven optimal results for the MDSDVRP and show that our algorithm significantly outperforms the state-of-the-art for the MDTSP while being competitive on the SDVRP. For the latter, 20 instances are solved for the first time, and new best primal and dual bounds are found for others. Funding: This work was supported by the Fundação para a Ciência e a Tecnologia [Project UIDB/04561/2020]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.1179 .

We consider the robust standard quadratic optimization problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is optimized over the standard simplex. Following most approaches, we model the uncertainty sets by balls, polyhedra, or spectrahedra, more generally, by ellipsoids or order intervals intersected with subcones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of the aforementioned uncertainty sets and present conditions under which the RStQP reduces to the standard quadratic optimization problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.

A fortification game (FG) is a three-level, two-player Stackelberg game, also known as defender-attacker-defender game, in which at the uppermost level, the defender selects some assets to be protected from potential malicious attacks. At the middle level, the attacker solves an interdiction game by depreciating unprotected assets, i.e., reducing the values of such assets for the defender, while at the innermost level the defender solves a recourse problem over the surviving or partially damaged assets. Fortification games have applications in various important areas, such as military operations, design of survivable networks, protection of facilities, or power grid protection. In this work, we present an exact solution algorithm for FGs, in which the recourse problems correspond to (possibly NP-hard) combinatorial optimization problems. The algorithm is based on a new generic mixed-integer linear programming reformulation in the natural space of fortification variables. Our new model makes use of fortification cuts that measure the contribution of a given fortification strategy to the objective function value. These cuts are generated on-the-fly by solving separation problems, which correspond to (modified) middle-level interdiction games. We design a branch-and-cut-based solution algorithm based on fortification cuts, their lifted versions, and other speed-up techniques. We present a computational study using the knapsack fortification game and the shortest path fortification game. For the latter one, we include a comparison with a state-of-the-art solution method from the literature. Our algorithm outperforms this method and allows us to solve previously unsolved instances to optimality.

Influence maximization aims at identifying a limited set of key individuals in a (social) network which spreads information based on some propagation model and maximizes the number of individuals reached. We show that influence maximization based on the probabilistic independent cascade model can be modeled as a stochastic maximal covering location problem. A reformulation based on Benders decomposition is proposed and a relation between obtained Benders optimality cuts and submodular cuts for correspondingly defined subsets is established. We introduce preprocessing tests, which allow us to remove variables from the model and develop efficient algorithms for the separation of Benders cuts. Both aspects are shown to be crucial ingredients of the developed branch-and-cut algorithm since reallife social network instances may be very large. In a computational study, the considered variants of this branch-and-cut algorithm outperform the state-of-the-art approach for influence maximization by orders of magnitude. (C) 2020 Elsevier B.V. All rights reserved.

This paper provides a first polyhedral study of the diameter constrained minimum spanning tree problem (DMSTP). We introduce a new set of inequalities, the circular-jump inequalities which strengthen the well-known jump inequalities. These inequalities are further generalized in two ways: either by increasing the number of partitions defining a jump, or by combining jumps with cutsets. Most of the proposed new inequalities are shown to define facets of the DMSTP polytope under very mild conditions. Currently best known lower bounds for the DMSTP are obtained from an extended formulation on a layered graph using the concept of central nodes/edges. A subset of the new families of inequalities is shown to be not implied by this layered graph formulation.

Electric vehicles are prime candidates for use within urban car sharing systems, both from economic and environmental perspectives. However, their relatively short range necessitates frequent and rather time-consuming recharging throughout the day. Thus, charging stations must be built throughout the system’s operational area where cars can be charged between uses. In this work, we introduce and study an optimization problem that models the task of finding optimal locations and sizes for charging stations, using the number of expected trips that can be accepted (or their resulting revenue) as a gauge of quality. Integer linear programming formulations and construction heuristics are introduced, and the resulting algorithms are tested on grid-graph-based instances, as well as on real-world instances from Vienna. The results of our computational study show that the best-performing exact algorithm solves most of the benchmark instances to optimality and usually provides small optimality gaps for the remaining ones, whereas our heuristics provide high-quality solutions very quickly. Our algorithms also provide better solutions than a sequential approach that considers strategic and operational decisions separately. A cross-validation study analyzes the algorithms’ performance in cases where demand is uncertain and shows the advantage of combining individual solutions into a single consensus solution, and a simulation study investigates their behavior in car sharing systems that provide their customers with more flexibility regarding vehicle selection.

In a personal communication with Ruslan Sadykov from Inria, France, we found an implementation error in our code for importing the benchmark instance set MUT leading to wrong numerical results in our original article (Gouveia, Leitner, and Ruthmair, 2017). In this corrigendum we provide corrected results for all experiments on instance set MUT. The general findings and conclusions drawn from the results however do not change.

Online social networks have become crucial to propagate information. Prominent use cases include marketing campaigns for products or political candidates in which maximizing the expected number of reached individuals is a common objective. The latter can be achieved by incentivizing an appropriately selected seed set of influencers that trigger an influence cascade with expected maximum impact. In real-world settings, competing influence spreads need to be considered frequently. These may, for in stance, stem from marketing activities for a substitute product of a different com pany or bad actors that spread (mis-)information about a political candidate. This article focuses on competitive settings in which the seed individuals of one entity are already known. Another entity wants to choose its seed set of individuals that triggers an influence cascade of maximum impact. The propagation process is modeled by a variant of the probabilistic independent cascade model. An algorithmic framework based on a Benders decomposition is developed that also employs preprocessing and initial heuristics. This framework is used within a sample average approximation scheme that allows to approximate the exact objective function value. The algorithms are tested on real-world instances from the literature and newly-obtained ones from Twitter. A computational study reports on the algorithms' performance next to providing further insights. The latter are based on analyzing expected losses that are caused by competition, the gain from solving subproblems to optimality using the Benders decomposition based algorithm, and the influence of different seed set choices of the first entity. (c) 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)

In this article we consider the network design problem with relays (NDPR), which gives answers to some important strategic design questions in telecommunication network design. Given a family of origin-destination pairs and a set of existing links these questions are as follows: (1) What are the optimal locations for signal regeneration devices (relays) and how many of them are needed? (2) Could the available infrastructure be enhanced by installing additional links in order to reduce the travel distance and therefore reduce the number of necessary relays?In contrast to previous work on the NDPR, which mainly focused on heuristic approaches, we discuss exact methods based on different mixed-integer linear programming formulations for the problem. We develop branch-and-price and branch-price-and-cut algorithms that build upon models with an exponential number of variables (and constraints). In an extensive computational study, we analyze the performance of these approaches for instances that reflect different real-world settings. Finally, we also point out the relevance of the NDPR in the context of electric mobility.