This paper advances and evaluates a recently proposed progressively interactive evolutionary multi-objective optimization algorithm. The algorithm uses preference information from the decision maker during the intermediate generations of the EMO and produces the most preferred solution on the Pareto-optimal front. The progress towards the Pareto-optimal front is made by approximating decision maker's value function. In this paper, a generalized polynomial value function has been proposed and the procedure to fit the value function to the decision maker's preference information has been described. The generality of the procedure of fitting a value function to the decision maker's preferences has been shown by using other existing value functions from the literature. The proposed generic polynomial value function has been incorporated in the PI-EMO-VF algorithm to efficiently approximate the decision maker's value function. The paper then evaluates the performance of the PI-EMO-VF algorithm on three and five objective test problems with constraints. It also evaluates the efficacy of the procedure in producing the most preferred solution when the decision maker is unable to provide perfect information, i.e., the decision maker finds certain pairs of solutions in the objective space to be incomparable. Results have been presented for three and five objective constrained test problems using the procedure.