- max PΣ kiui u,v i=' Σ1 - in Σ=1Vj≤Cout vjaijui = 0 U1,..., um ≥0 V1, Vn ≥ 0, 2 where m and n are the numbers of crude materials and end products, respec- tively, ki is the unit purchase cost of crude i, p; is the unit market price of product j, aij is the yield from crude i to product j, and Cin and Cout are the input and output capacities, respectively. The decision variables u; (i = 1,..., m) and v; (j = 1,...,n) represent supply of crude i and production of product j, respectively.

- max PΣ kiui u,v i=' Σ1 - in Σ=1Vj≤Cout vjaijui = 0 U1,..., um ≥0 V1, Vn ≥ 0, 2 where m and n are the numbers of crude materials and end products, respec- tively, ki is the unit purchase cost of crude i, p; is the unit market price of product j, aij is the yield from crude i to product j, and Cin and Cout are the input and output capacities, respectively. The decision variables u; (i = 1,..., m) and v; (j = 1,...,n) represent supply of crude i and production of product j, respectively.