Giving formal roles to elevators for breaking symmetry in static elevator operation problems

In this paper, we propose a technique to decrease computational times in solving an integer linear programming (ILP) model for the static elevator operation problem (SEOP). The SEOP is a problem to optimally operate elevators on such assumption that all information on passengers who use an elevator system is known beforehand. In planning, there is a symmetry on elevators that exchanging 2 elevators does not affect the value of the objective function, if initial states of those elevators are identical. Such symmetry requires much computational times for problems to be solved, since there are at least 2 optimal solutions which differ only in allocations of elevators and partial solutions for those solutions can not be bound. That symmetry is resolved by giving different roles to elevators, and those roles are assignment pattern numbers (APNs) in the proposed technique. An APN of an elevator is a decimal number which is calculated from a binary vector which represents assignments of passengers to that elevator. The proposed technique deploys such a straightforward fact that all elevators have different APNs, and enfoces an elevator with a smaller index to have a smaller APN than other elevators with larger indexes. The effectiveness of that technique is numerically examined by applying a mathematical solver to ILP equations generated from some problem instances.