We first consider the assortment optimization problem with fixed product costs under the Mixtures of Multinomials (MMNL) Model. The problem is NP-hard even under the Multinomial Logit Model and the existing literature focuses on developing heuristics and bounds. We develop a conic integer programming formulation for the problem and valid inequalities to strengthen the formulation. We show that this approach can be used to solve instances that are very large { sizes beyond which it would be very difficult to accurately estimate parameters of the choice model { in a short amount of time, eliminating the need to develop and implement specialized algorithms for the problem. We also study the assortment planning problem where the inventory and replenishment costs are considered using the Economic Order Quantity model and the customers' choice is governed by the MMNL model. We show that the problem is NP-hard and propose a conic integer program for this problem. Our numerical experiments show that moderately sized instances can be solved in reasonable times and McCormick inequalities are effective in tightening the formulation.