Optimal scheduling of parallel machines with constrained resources

This paper analyzes a manufacturing system consisting of parallel machines, which produce one product-type with controllable production rates subject to continuously-divisible, time-dependent resources. The objective is to produce the required amount of product-type units by a due date while minimizing inventory, backlog and production related costs over a production horizon. With the aid of the maximum principle, a number of analytical rules of the optimal scheduling is derived whereby the continuous-time scheduling is reduced to discrete sequencing and timing. As a result, a polynomial-time algorithm is developed for solving the problem.