The capacitated dispersion problem is a variant of the maximum diversity problem in which a set of elements in a network must be determined. These elements might represent, for instance, facilities in a logistics network or transmission devices in a telecommunication network. Usually, it is considered that each element is limited in its servicing capacity. Hence, given a set of possible locations, the capacitated dispersion problem consists of selecting a subset that maximizes the minimum distance between any pair of elements while reaching an aggregated servicing capacity. Since this servicing capacity is a highly usual constraint in real-world problems, the capacitated dispersion problem is often a more realistic approach than is the traditional maximum diversity problem. Given that the capacitated dispersion problem is an NP-hard problem, whenever large-sized instances are considered, we need to use heuristic-based algorithms to obtain high-quality solutions in reasonable computational times. Accordingly, this work proposes a multi-start biased-randomized algorithm to efficiently solve the capacitated dispersion problem. A series of computational experiments is conducted employing small-, medium-, and large-sized instances. Our results are compared with the best-known solutions reported in the literature, some of which have been proven to be optimal. Our proposed approach is proven to be highly competitive, as it achieves either optimal or near-optimal solutions and outperforms the non-optimal best-known solutions in many cases. Finally, a sensitive analysis considering different levels of the minimum aggregate capacity is performed as well to complete our study.

The capacitated dispersion problem, which is a variant of the maximum diversity problem, aims to determine a set of elements within a network. These elements could symbolize, for instance, facilities in a supply chain or transmission nodes in a telecommunication network. While each element typically has a bounded service capacity, in this research, we introduce a twist. The capacity of each node might be influenced by a random Bernoulli component, thereby rendering the possibility of a node having zero capacity, which is contingent upon a black box mechanism that accounts for environmental variables. Recognizing the inherent complexity and the NP-hard nature of the capacitated dispersion problem, heuristic algorithms have become indispensable for handling larger instances. In this paper, we introduce a novel approach by hybridizing a heuristic algorithm with reinforcement learning to address this intricate problem variant.

In an effort to balance the distribution of services across a given territory, dispersion and diversity models typically aim to maximize the minimum distance between any pair of facilities. Specifically, in the capacitated dispersion problem (CDP), each facility has an associated capacity or level of service, and the objective is to select a set of facilities so that the minimum distance between any pair of them (dispersion) is maximized, while ensuring a user-defined level of service. This problem can be formulated as a linear integer model, where the sum of the capacities of the selected facilities must match or exceed the total demand in the network. Real-life applications often necessitate considering the levels of uncertainty affecting the capacity of the nodes. Failure to account for this uncertainty could lead to low-quality or infeasible solutions in practical scenarios. However, research addressing the stochastic version of the CDP is scarce. This paper introduces two models for the CDP with stochastic capacities, incorporating soft constraints and penalty costs for violating the total capacity constraint. The first model includes a probabilistic constraint to ensure the required level of service with a certain probability, while the second model introduces a soft constraint with penalty costs for violations. To solve both variants of the model, a forward–backward simheuristic algorithm is proposed. Our approach combines a metaheuristic algorithm with Monte Carlo simulation, enabling the efficient handling of the random behavior of node capacities and obtaining reliable solutions regardless of their probability distribution.

The team orienteering problem (TOP) is a well-studied optimization challenge in the field of Operations Research, where multiple vehicles aim to maximize the total collected rewards within a given time limit by visiting a subset of nodes in a network. With the goal of including dynamic and uncertain conditions inherent in real-world transportation scenarios, we introduce a novel dynamic variant of the TOP that considers real-time changes in environmental conditions affecting reward acquisition at each node. Specifically, we model the dynamic nature of environmental factors—such as traffic congestion, weather conditions, and battery level of each vehicle—to reflect their impact on the probability of obtaining the reward when visiting each type of node in a heterogeneous network. To address this problem, a learnheuristic optimization framework is proposed. It combines a metaheuristic algorithm with Thompson sampling to make informed decisions in dynamic environments. Furthermore, we conduct empirical experiments to assess the impact of varying reward probabilities on resource allocation and route planning within the context of this dynamic TOP, where nodes might offer a different reward behavior depending upon the environmental conditions. Our numerical results indicate that the proposed learnheuristic algorithm outperforms static approaches, achieving up to $25\%$ better performance in highly dynamic scenarios. Our findings highlight the effectiveness of our approach in adapting to dynamic conditions and optimizing decision-making processes in transportation systems.

Given a network of nodes and a certain demand that needs to be satisfied, the capacitated dispersion problem (CDP) involves selecting a subset of nodes to maximize dispersion between them. In many practical instances, symmetry in the structure of the selected nodes (e.g., using nodes of the same type) can lead to synergies. Hence, this paper studies a bi-objective variant of the CDP to account for these symmetries. The first goal seeks to maximize the minimum distance between opened nodes, while the second goal accounts for symmetry by penalizing the use of nodes of different types (in our case, represented by different colors). We formalize the problem as a bi-objective mathematical program and address it through a classical multiobjective strategy, the $ϵ$-constraint method. Exact methods are used when the problem size allows, even though the problem is NP-hard. To tackle larger instances, we design a biased-randomized algorithm based on a constructive heuristic. Computational experiments show that our biased-randomized algorithm provides high-quality approximations of the Pareto frontier.

The periodic vehicle routing problem (PVRP) is a well-known challenge in real-life logistics, requiring the planning of vehicle routes over multiple days while enforcing visitation frequency constraints. Although numerous metaheuristic and exact methods have tackled various PVRP extensions, real-world settings call for additional features such as depot configurations, tight visitation frequency constraints, and heterogeneous fleets. In this paper, we present a two-phase biased–randomized algorithm that addresses these complexities. In the first phase, a round-robin assignment quickly generates feasible and promising solutions, ensuring each customer’s frequency requirement is met across the multi-day horizon. The second phase refines these assignments via an iterative search procedure, improving route efficiency and reducing total operational costs. Extensive experimentation on standard PVRP benchmarks shows that our approach is able to generate solutions of comparable quality to established state-of-the-art algorithms in relatively low computational times and stands out in many instances, making it a practical choice for real life multi-day vehicle routing applications.