An Algebraic Approach for the Search Space of Permutations with Repetition

We present an algebraic approach for dealing with combinatorial optimization problems based on permutations with repetition. The approach is an extension of an algebraic framework defined for combinatorial search spaces which can be represented by a group (in the algebraic sense). Since permutations with repetition does not have the group structure, in this work we derive some definitions and we devise discrete operators that allow to design algebraic evolutionary algorithms whose search behavior is in line with the algebraic framework. In particular, a discrete Differential Evolution algorithm which directly works on the space of permutations with repetition is defined and analyzed. As a case of study, an implementation of this algorithm is provided for the Job Shop Scheduling Problem. Experiments have been held on commonly adopted benchmark suites, and they show that the proposed approach obtains competitive results compared to the known optimal objective values.