Effects of discrete objective functions with different granularities on the search behavior of EMO algorithms

Objective functions in combinatorial optimization are discrete. The number of possible values of a discrete objective function is totally different from problem to problem. Optimization of a discrete objective function is often very difficult. In the case of multiobjective optimization, a different objective function has a different number of possible values. This means that each axis of the objective space has a different granularity. Some axes may have fine granularities while others are coarse. In this paper, we examine the effect of discrete objective functions with different granularities on the search behavior of EMO (evolutionary multiobjective optimization) algorithms through computational experiments. Experimental results show that a discrete objective function with a coarse granularity slows down the search of EMO algorithms along that objective. An interesting observation is that such a slow-down along one objective often leads to the speed-up of the search along other objectives. We also examine the effect of adding a small random noise to each discrete objective function in order to increase the number of possible objective values.