Approximate Dynamic Programming by Linear Programming for Stochastic Scheduling

In stochastic scheduling, we want to allocate a limited amount of resources to a set of jobs that need to be serviced. Unlike in deterministic scheduling, however, the parameters of the system may be stochastic. For example, the time it takes to process a job may be subject to random fluctuations. Stochastic scheduling problems occur in a variety of practical situations, such as manufacturing, construction, and compiler optimization. As in deterministic scheduling, the set of stochastic scheduling problems is incredibly large and diverse. One important class of models involves scheduling a fixed number of jobs on a fixed number of identical parallel machines while minimizing a given performance measure. The processing times of the jobs are assumed to follow some joint probability distribution. In addition, there may be precedence constraints, or interdependencies between the jobs that require certain jobs not be scheduled until others are completed. The deterministic counterpart to this class of problems has been studied extensively [KSW98]. A näıve approach to these problems would be to take the expected processing times and use the algorithms for the deterministic problems. Unfortunately, it is easy to construct examples that shows that this approach can produce solutions that are arbitrarily bad. Fortunately, however, this class of stochastic scheduling problems can be easily cast as a Markov decision process (MDP) and therefore can be attacked by dynamic programming methods. Results for this class of stochastic scheduling problems are somewhat scattered, and have been obtained using a variety of methods. Rothkopf [R66] showed that for one machine without precedence constraints, an index rule minimizes the expected sum of weighted completion times for arbitrary processing time probability distributions. Möhring, Radermacher and Weiss [MRW84, MRW85] study the analytic properties of various classes of scheduling policies, and determine optimal policies for special cases. Möhring, Schulz, and Uetz [MSU99] developed approximation algorithms for a variety of stochastic