In logistics, customer clusters are often predefined based on geographic locations to optimize resource management and workload distribution. However, this approach frequently leads to significant cluster imbalance, particularly when not all customers require daily service or when operational constraints restrict the possibility of reallocating customers across clusters. These imbalances may affect workload equity, routing efficiency, and overall service performance. This paper addresses the problem of balancing customer clusters under relocation constraints by introducing two alternative criteria for measuring imbalance and enforcing them within a constrained reassignment framework. Our goal is to redistribute the set of customers so as to construct new clusters whose cardinalities are as balanced as possible while fully respecting the reassignment limitations. After formulating both nonlinear optimization models, we propose an exact algorithm that systematically reallocates customers through sequences of feasible moves, thereby ensuring progress toward a balanced configuration. We provide formal proofs showing that the algorithm always converges to an optimal solution for both imbalance criteria after a finite number of iterations. From these results, it follows that both models yield the same optimal solution. In addition, computational experiments demonstrate the efficiency of the method, showing that optimal solutions are obtained with very low running times for large-scale instances.
This paper presents the Hazardous Team Orienteering Problem (HTOP), an extension of the Hazardous Orienteering Problem designed for multiple vehicles. In HTOP, visiting certain customers exposes the vehicle to a time-dependent risk of catastrophic loss. Each hazardous customer has an associated exponential failure rate, meaning that if an explosion occurs, the entire profit earned from that route is lost. The goal is to maximize the total expected profit collected by a fleet of vehicles while adhering to time constraints and ensuring customer exclusivity. We introduce two solution approaches. First, we develop an exact Branch-and-Price algorithm based on a set-partitioning reformulation, where each column corresponds to a feasible hazardous route valued by its expected profit. Second, we create an Adaptive Large Neighborhood Search (ALNS) metaheuristic tailored to the nonlinear expected-profit structure of the problem. Computational experiments on benchmark instances derived from the existing literature on the Hazardous Orienteering Problem demonstrate that the pricing phase is the main computational bottleneck of the exact approach. Nevertheless, our method successfully solves small and medium-sized instances to optimality. The ALNS consistently generates high-quality solutions in a short amount of time and matches the optimal solutions for single-vehicle benchmarks while showing strong scalability in multi-vehicle scenarios.
