Computing Fair Solutions in Single Machine Scheduling

We address a scheduling problem arising when two agents, each with a set of jobs, compete to perform their respective jobs on a common processing resource. Each agent wants to minimize the total completion times of its jobs only and, associated with one agent's objective value, a certain utility can be derived for each agent. On the other hand, we adopt as an index of collective satisfaction (system utility) the sum of the agents’ utilities. A system optimum is any solution maximizing system utility. However, such a solution may well be highly unbalanced and therefore possibly unacceptable by the worse-off agent. Hence, we are interested in a solution that incorporates some criterion of equity for the agents and, to this purpose, we make use of a concept of fairness—namely the Kalai-Smorodinsky solution—which is standard in game theory. We propose different MIP models and a heuristic algorithm to tackle the problem of determining a schedule which is fair to both agents. These approaches are then tested to assess their performance. Finally, an empirical evaluation of the amount of system utility that must be traded to reach a fair solution is given.