A fast local search based memetic algorithm for the parallel row ordering problem

The parallel row ordering problem (PROP) is concerned with arranging two groups of facilities along two parallel lines with the goal of minimizing the sum of the flow cost-weighted distances between the pairs of facilities. As the main result of this paper, we show that the insertion neighborhood for the PROP can be explored in optimal time $\Theta \left(n^2\right)$ by providing an $O\left(n^2\right)$-time procedure for performing this task, where n is the number of facilities. As a case study, we incorporate this procedure in a memetic algorithm (MA) for solving the PROP. We report on numerical experiments that we conducted with MA on PROP instances with up to 500 facilities. The experimental results demonstrate that the MA is superior to the adaptive iterated local search algorithm and the parallel hyper heuristic method, which are state-of-the-art for the PROP. Remarkably, our algorithm improved best known solutions for six largest instances in the literature. We conjecture that the time complexity of exploring the interchange neighborhood for the PROP is $\Theta \left(n^2\right)$, exactly as in the case of insertion operation.