On the MAXGAP Problem for Cutting Stock Problems

The MAXGAP problem of a linear integer optimization problem P consists in determining the maximum diierence (gap) (P) between the optimal value z (E) of an instance E 2 P and the LP bound z c (E) with respect to all E 2 P. In the case of the one-dimensional cutting stock problem (1D CSP) it is known that (1D CSP) 1 + 5 132 but there is also a conjecture that (1D CSP) 2. In this paper we show that in the case of the restricted exact 2-stage two-dimensional cutting stock problem (RE2 CSP) the maximum gap increases at least linearly in the number m of pieces. It holds (RE2 CSP) > m 3. For the (non-restricted) exact 2-stage two-dimensional cutting stock problem (E2 CSP) an example is given such that (E2 CSP) 2(1 + 5 132) > 2:075 follows which yields also a lower bound for the general two-dimensional cutting stock problem. Furthermore some consequences for higher-dimensional cutting stock problems and with respect to applications are discussed.