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The present report covers the mathematical foundation for implementing a 'repair versus replacement' policy which minimizes the expected total over-all cost at the end of a given, stipulated time, T. The model of repair costs could be kept fairly general so that the procedure should be applicable for a very broad class of actual situations. Specifically, it was assumed that the expected cumulative cost is a convex function of time, differentiable almost everywhere in the domain of interest. A more detailed analysis was made for the specific case where the expected total cost is parabolic. The influence of the parameter T and the effect of errors in estimating this parameter were investigated. (Author)