Abstract
Job shops are an important production environment for low-volume
high-variety manufacturing. Its scheduling has recently been
formulated as an Integer Linear Programming (ILP) problem to take
advantages of popular Mixed-Integer Linear Programming (MILP) methods,
e.g., branch-and-cut. When considering a large number of parts, MILP
methods may experience difficulties. To address this, a critical but
much overlooked issue is formulation tightening. The idea is that if
problem constraints can be transformed to directly delineate the problem
convex hull in the data pre-processing stage, then a solution can be
obtained by using linear programming methods without much difficulty.
The tightening process, however, is NP hard because of the existence of
integer variables. In this paper, an innovative and systematic approach
is established for the first time to tighten the formulations of
individual parts, each with multiple operations, in the data
pre-processing stage. It is a major extension from our previous work on
problems with binary and continuous variables to integer variables. The
idea is to first link integer variables to binary variables by
innovatively combining constraints so that the integer variables are
uniquely determined by binary variables. With binary variables and
continuous only, the vertices of the convex hull can be obtained based
on the vertices of the linear problem after relaxing binary requirements
with proved tightness. These vertices are then converted to tight
constraints for general use. This approach significantly improves and
extends our previous results on tightening single-operation parts
without actually achieving tightness. Numerical results demonstrate
significant benefits on solution quality and computational efficiency.
This approach also applies to other ILP problems with similar
characteristics and fundamentally changes the way how such problems are
formulated and solved.