Beyond Smith's Rule: An Optimal Dynamic Index, Rule for Single Machine Stochastic Scheduling with Convex Holding Costs

Abstract

We study a model for scheduling 'n' classes of stochastic jobs on a single machine, with the objective of minimizing the total expected holding cost (discounted or undiscounted). We allow general holding cost rates that are separable, nondecreasing and convex on the number of jobs in each class. We formulate the problem as a linear program over a certain greedoid polytope, and establish that it is solved optimally by a dynamic (priority) index rule, which extends the classical Smith's rule (1956) for the linear case.

Keywords: Stochastic scheduling, dynamic index rule, decomposition, convex holding costs, conservation laws, achievable region, linear programming relaxation, polyhedral methods.

JEL Classification: C60, C61