The capacitated dispersion problem (CDP) presents a challenging optimization scenario where the objective is maximizing node dispersion while respecting a capacity constraint. This paper proposes a novel variant of CDP that incorporates a cost constraint and dynamic facility costs as well as a max-sum objective function. To address these complexities, a learnheuristic framework integrated with machine learning and a metaheuristic method is proposed. The operational cost of each facility can be affected by a Bernoulli distribution function, introducing the possibility that a facility might have zero operational cost. This uncertainty is addressed by a black-box mechanism that takes into account the utilization and energy consumption rate of each facility. The proposed methodology is evaluated using a set of benchmark instances. Results demonstrate the efficiency of our learnheuristic approach in achieving near-optimal solutions under dynamic cost conditions, specifying its potential for real-world applications in logistics, telecommunications, and beyond.