In this paper we consider a manufacturing firm with a capacity that can be used to produce several kinds of products. A mathematical model for this type of manufacturing firm is constructed and a new profit functional is introduced. Note that in this problem the derivatives of the profit integrand with respect to some state variables are only piecewise continuous. Thus, Pontryagin's Maximum Principle cannot be used for our problem. Therefore, a modified maximum principle of Pontryagin type is derived in § 5. Its proof follows the approach developed by Blatt and Gray (1976) and generalized by Moore and Teo (1976). This maximum principle, together with a basic theorem in linear programming problems, is used to obtain the optimal policy for the production rate and the advertising expenditure for each kind of product. For illustration, two numerical examples are given.