Solution of a Tinned Iron Purchasing Problem using Lagrangean Relaxation

The following problem is considered. A tin factory obtains its material from steel works. It consists of rectangular sheets of tinned iron of various sizes and thickness of tinfoil, with several other specifications, like aftertreatment, temper, etc. The prices per unit of volume of the sheets vary with thickness, thickness of tinfoil, and width. They also depend on discounts, which are given for large quantities of the same sheets. As a consequence, it is often profitable to the factory to order sheets of larger sizes and/or thicker tin layers than needed. This is allowed, provided that sheets of a certain size can only be replaced by sheets of one size. Leftover pieces can be sold as scrap.