A variety of practical situations involve supplying a particular commodity by some locations to satisfy the demand at others. If the demands and the costs of producing varying amounts of commodity at each location are known, then the question is how much commodity should be supplied by each...

The use of least concave utility functions describing a given concavifiable preference relation is suggested for determining the complementary vis-a-vis substitute nature of a pair of commodities.

Spingarn and Rockafellar [13] showed in a program (Q_{v}^{u}) minimize {f(x) - u^{T}x subject to g(x) <= b + v} where f : R^{n} approaches R, g : R^{n} approaches R^{m}; f in C^{2}, g in C^{n}; u in R^{n}, v in R^{m}; n >= m; that at any local minimum point x of (Q_{v}^{u}) the Jacobian matrix of g at x has full rank, strict complementary slackness holds and the second order sufficiency conditions hold, for...</=>