A probabilistic approach to optimal quality usage

The production process of a certain item exhibits some quality characteristics governed by a probability measure μ(A). The consumption (or usage) of all items of this production is described by another probability measure ν(B), where A and B are elements in the product space of all quality characteristics of the totality of produced items. When a product with characteristics x ε{lunate} A is being used instead of a product with characteristics y ε{lunate} B, then a loss φ{symbol}(x,y) is incurred. Any ditribution plan θ(A,B) of the product for consumption produces a total expected loss τφ(θ) = E_φ(x, y). Using some general results in the theory of probability metrics, under given marginals, we develop models for finding the optimal distribution plan θ and the corresponding minimal total losses τ_φ and establish some particular forms and inequalities. A brief discussion of the results follows.