Linear programming-based algorithms for the minimum makespan high multiplicity jobshop problem

We study a generalized version of the minimum makespan jobshop problem in which multiple instances of each job are to be processed. The system starts with specified inventory levels in all buffers and finishes with some desired inventory levels of the buffers at the end of the planning horizon. A schedule that minimizes the completion time of all the operations is sought. We develop a polynomial time asymptotic approximation procedure for the problem. That is, the ratio between the value of the delivered solution and the optimal one converge into one, as the multiplicity of the problem increases. Our algorithm uses the solution of the linear relaxation of a time-indexed Mixed-Integer formulation of the problem. In addition, a heuristic method inspired by this approximation algorithm is presented and is numerically shown to out-perform known methods for a large set of standard test problems of moderate job multiplicity.