High transportation costs and poor quality of service are common vulnerabilities in various logistics networks, especially in food distribution. Here we propose a many-objective Customer-centric Perishable Food Distribution Problem that focuses on the cost, the quality of the product, and the service level improvement by considering not only time windows but also the customers’ target time and their priority. Recognizing the difficulty of solving such model, we propose a General Variable Neighborhood Search (GVNS) metaheuristic based approach that allows to efficiently solve a subproblem while allowing us to obtain a set of solutions. These solutions are evaluated over some non-optimized criteria and then ranked using an a posteriori approach that requires minimal information about decision maker preferences. The computational results show (a) GVNS achieved same quality solutions as an exact solver (CPLEX) in the subproblem; (b) GVNS can generate a wide number of candidate solutions, and (c) the use of the a posteriori approach makes easy to generate different decision maker profiles which in turn allows to obtain different rankings of the solutions.