A Novel Integer Linear Programming Formulation for Job-Shop Scheduling Problems

ob-shop scheduling is an important but difficult problem arising in low-volume high-variety manufacturing. It is usually solved at the beginning of each shift with strict computational time requirements. For fast resolution of the problem, a promising direction is to formulate it in an Integer Linear Programming (ILP) form so as to take advantages of widely available ILP methods such as Branch-and-Cut (B&C). Nevertheless, computational requirements on ILP methods for existing ILP formulations are high. In this paper, a novel ILP formulation is presented. In the formulation, a set of binary indicator variables indicating whether an operation begins at a time slot on a machine group or not is selected as decision variables, and all constraints are innovatively formulated based on this set of variables. For fast resolution of large problems, our recent decomposition-and-coordination method “Surrogate Absolute-Value Lagrangian Relaxation” (SAVLR) is enhanced by using a 3-segment piecewise linear penalty function, which more accurately approximates a quadratic penalty function as compared to an absolute-value function. Testing results demonstrate that our new formulation drastically reduces the computational requirements of B&C as compared to our previous formulation. For large problems where B&C has difficulties, near-optimal solutions are efficiently obtained by using the enhanced SAVLR under the new formulation.