W.R. Grace10 strip mines phosphates in strata numbered from i = 1 at the top to i = n at the deepest level. Each stratum must be removed before the next can be mined, but only some of the layers contain enough suitable minerals to justify processing into the company’s three products: pebble, concentrate, and flotation feed 1j = 1, 2, 32.

The company can estimate from drill samples the quantity ai,j of product j available in each stratum i, the fraction bi,j of BPL (a measure of phosphate content) in the part of i suitable for j, and the corresponding fraction pi,j of pollutant chemicals. They wish to choose a mining plan that maximizes the product output while keeping the average fraction BPL of material processed for each product j at least bj and the average pollution fraction at most pj. Formulate an ILP model of this mining problem using the decision variables 1i = 1,

c, n2 xi ! e 1 if stratum i is removed 0 otherwise yi ! e 1 if stratum i is processed 0 otherwise