We consider the homogeneous variant of linear programming problems in the setting of a module over a linearly ordered commutative associative ring. We formulate the optimality condition for the primal and for the dual problem; by combining the aforementioned results, we obtain the Strong Duality Theorem for the homogeneous linear programming problems. Albeit the results are based on a general discrete variant of Farkas’ Lemma, which has been published recently, further restrictive assumptions are necessary to prove the results. We also propose a simple application of the results – an extension of the FMEA (Failure Mode and Effects Analysis) method – in business decision making of small and medium-sized enterprises.