For the sale of a single object through an auction, we assume discrete type space for agents and make use of linear programming to find optimal mechanism design for a risk-neutral seller. First, we show that the celebrated incentive compatible mechanism, second price auction, is not optimal. We find a slightly different optimal mechanism referred to as “discrete second price auction”. Second we consider the problem of allocation with costly inspection. We obtain the optimal solution in the form of a favored-agent mechanism by the Greedy Algorithm. Moreover, we relax the common prior assumption and maximize the worst-case utility of an ambiguity averse seller for the two problems mentioned above. While the problem does not yield a useful optimal mechanism in general, optimal solutions for some special cases are obtained.