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This paper addresses the optimization of process models that involve discontinuous investment cost functions with fixed charges and are defined over several regions for the sizes. These discontinuous cost functions are naturally expressed by disjunctions. Conventional modeling and solution techniques for the optimization and synthesis of process systems with these cost models include a direct NLP approach, the use of smoothing functions and MINLP models with big-M constraints. In this paper we propose the application of disjunctive programming approaches based on the convex hull formulation of disjunctions, and disjunctive branch and bound with surrogates or linear underestimators. Theoretical comparison between the convex hull formulation, the big-M model, and the model with linear underestimators is presented. It is proved that the convex hull formulation of disjunctions gives the tightest relaxation among the alternative modeling techniques. The proposed solution techniques are tested on heat exchanger networks, process flowsheet design, and the synthesis of a large petrochemical complex. It is shown that the convex hull formulation of disjunctions gives the overall best performance among the proposed techniques.