Solution of a tinned iron purchasing problem by Lagrangean relaxation

A tin factory obtains its material from steel works. This consists of sheets of tinned iron which may have very diverging specifications with respect to length, width, thickness, and thicknesses of tinfoils. Prices per unit of volume vary with width and thickness. For large quantities of the same size discounts are given. As a consequence of the price structure it is often advantageous for the factory to order sheets of larger sizes than needed and to resell the leftover pieces as scrap. The question is which sizes and quantities one should order if one wishes to minimize total purchase cost. This problem is formulated as a combinatorial optimization problem that is solved by Lagrangean relaxation and subgradient techniques.