This work aims at formulating a shape optimization problem within a multiobjective optimization framework and approximating it by means of the so-called Multiple-Gradient Descent Algorithm (MGDA), a gradient-based strategy that extends classical Steepest-Descent Method to the case of the simultaneous optimization of several criteria. We describe several variants of MGDA and we apply them to a shape optimization problem in linear elasticity using a numerical solver based on IsoGeometric Analysis (IGA). In particular, we study a multiobjective gradient-based method that approximates the gradients of the functionals by means of the Finite Difference Method; kriging-assisted MGDA that couples a statistical model to predict the values of the objective functionals rather than actually computing them; a variant of MGDA based on the analytical gradients of the functionals extracted from the NURBS -based parametrization of the IGA solver. Some numerical simulations for a test case in computational mechanics are carried on to validate the methods and a comparative analysis of the results is presented.