This paper proposes a modification of the traditional water network design model, whose single objective is to minimize the total cost of the network. To relax the model, node pressures are allowed to deviate from a previously established minimum value. By adding a penalty for these deviations in the objective function, the model becomes multi-objective. In order to test the validity of this new approach, the model is particularized for a water distribution networks from the scientific literature, concretely the Two-loop network. Moreover, the results of the model are analysed and discussed under different scenarios, i.e., given different weights to the two components of the objective function, costs and pressure deviation. Since the model is nonlinear, due to the hydraulic constraints governing the behaviour of water networks, a genetic algorithm is used to solve it. Results indicate that this new approach significantly reduces total design costs by slightly lowering the pressure in certain nodes. However, a post-processing step is necessary to ensure that no node exhibits a deviation greater than a defined threshold. In many of the simulations performed using the genetic algorithm, a solution is obtained that improves upon the best result reported in the literature by 3,000 m.u. (reducing the cost from 419,000 to 416,000 m.u.). This reduction is achieved because the pressure at one of the nodes is 28.99 instead of 30 m.w.c., a deviation that appears acceptable and results in significant savings.