This study proposes a multi-objective location-routing problem considering the capacity of vehicles to decline the system's costs. The model considers probabilistic times of traveling, service, and waiting by vehicles while guaranteeing the least probability which the cumulative values of these parameters are less than a pre-determined value when minimization of this value is considered an objective function. To cope with uncertainty, fuzzy numbers for important parameters of customer demand, vehicle capacity, variable and fixed transportation costs, and depot opening costs are used. Moreover, the nonlinear constraints are linearized to reduce computational time. We also use a fuzzy ranking method to transform the presented model into an equivalent auxiliary crisp model. As the model is NP-hard, we introduce a novel Multi-Objective Imperialist Competitive Algorithm (MOICA) to address the issue. The efficacy of the presented MOICA is evaluated by comparing its performance against two well-established multi-objective metaheuristics, Pareto Archived Evolution Strategy (PAES), and Non-Dominated Sorting Genetic Algorithm-II (NSGA-II). Leveraging Response Surface Methodology (RSM), the mutation and crossover operators employed by each algorithm were meticulously tuned. Subsequently, the performance of all three algorithms was examined using four benchmark comparison metrics across a range of established benchmark examples. The results demonstrably substantiate the superiority of the proposed MOICA in achieving optimal solutions.
