Abstract
We propose new exact solution methods for cyclic Fair Sequence Problem (cFSP). The cFSP is a kind of scheduling problem to allocate fractional executions of given jobs to the cyclic sequence of unit-time slots with the given periodical length, where respective jobs must be processed in specified number of slots per period, and with as equal intervals as possible. The measure of equal intervals is somewhat ambiguous. Corominas et al. presented exact solution method for cFSP with basic equality measure that minimizes the sum of square of differences between actual and ideal intervals. They reported that instances with less than or equal to 40 periodical time slots or so can be solved through their formulation by a general purpose mixed integer programming (MIP) solver. We propose three formulations to solve cFSP based on the partition of cyclic slots by pre-generated job allocation patterns: simple partitioning formulation, improved formulation with reduced patterns and Traveling Salesman problem (TSP) like formulation considering patterns as salesman's tours. These formulations can flexibly treat any measure of equality as far as it based on the difference between actual and ideal intervals. The results of numerical experiments to evaluate proposed methods show that TSP-like formulation outperforms existing ones and that it succeeds to solve 40% or so larger instances than the existing best.