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

Job-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. To obtain near-optimal solutions with quantifiable quality within strict time limits, a direction is to formulate them 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 for ILP methods on existing ILP formulations are high. In this paper, a novel ILP formulation for minimizing total weighted tardiness is presented. The new formulation has much fewer decision variables and constraints, and is proven to be tighter as compared to our previous formulation. 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.