Once the configuration of molds is fixed, the planning of production of aluminum ingots18 reduces to allocating the time of furnaces j = 1,c, n among alloys i = 1,c, m and ingot sizes s = 1,c, p. Yields aj, s of ingots of size s producible from furnace j during the entire planning period can be estimated. Planning should meet demands di, s for ingots of alloy i and size s, but this may require misapplying some ingots (i.e., trimming some larger ingots sizes s=

greater than s to meet demands for size s of the same alloy). Misapplications result in a trim-loss cost ci, s=, s for each ingot cut down from size s=

to s. Managers want to find a feasible plan that minimizes the total cost of these misapplications.

Formulate an LP model of this ingot production planning problem using decision variables 1i = 1,c, m; j = 1,c, n; s = 1,c, p, s= 7 s2 xi, j, s ! fraction of time on furnace j dedicated to making ingots of alloy i, size s yi, x=, s ! number of ingots of alloy i, size s=, misapplied to meet demand for size s