Maximal Material Yield in Gemstone Cutting

The fabrication of faceted gems from rough gems, when viewed mathematically, is a problem of maximal material yield. Here, one or more jewels must be embedded within an irregularly shaped rough stone such that the total value of the jewels is maximal. Both the position and the shape of the embedded stones can vary, which distinguishes this problem from the cutting and packing problems known in the literature. On the modeling side, there are three extremely interesting challenges: separating the continuous parameters, significant for yield maximization, from the discrete parameters, describing the facets, finding a mathematically tractable description of the esthetic requirements of the faceted stones, and reformulating the containment and/or non-overlapping conditions into tractable constraints. The methods of general semi-infinite optimization lend themselves to this last-named challenge. The numerical solution of such optimization problems having a practical nature is difficult and, in the mathematical literature, one frequently finds only conceptual solution approaches. Among other things, this chapter describes a novel approach and shows how it can be successfully applied to the problem.