This study involves an application of linear programming. The model, though hypothetical, is constructed in terms of relevance to characteristic multi-product and process operations, such as those found in the oil industry. All important phases, production, manufacturing (refining), transportation and marketing, are covered. Through tax considerations, financial as well as the more usual price, cost, and technological conditions, are included. The dual theorem of linear programming is used in a manner which makes it relevant to sensitivity analysis involved in studying technological and structural changes. This suggests possible uses of the dual theorem to areas other than transportation and classical decentralization analysis which have, to date, received major attention in the programming literature. By means of the dual formulation and specified computational procedures detailed in this paper considerable simplification in sensitivity analysis is achieved. A general theorem is established (easily applied in this and similar cases) by means of which an optimal solution to the dual is obtained without the need for matrix inversion or calculating an optimal simplex tableau. Finally, the solution and analysis are carried out with respect to a "kinked" functional which is only piecewise linear. A prototype is thus provided for handling a certain class of nonlinear programming problems which are also relevant to important areas of economic analysis.