An semidefinite programming-based $\varepsilon$-constraint method for the bi-objective single-row facility layout problem

In this work, we introduce a multi-objective version of the well-known single-row facility layout problem (SRFLP). In the SRFLP, a set of one-dimensional facilities should be placed along a single line such that the weighted sum of the center-to-center distances of each pair of facilities is minimized. In our multi-objective extension, there are multiple such weighted-sum objectives which we consider under the concept of Pareto optimality.
We develop a solution algorithm based on the $\varepsilon$-constraint method to solve the bi-objective SRFLP. Many existing works on the $\varepsilon$-constraint method use integer linear programming (ILP) solvers in a black-box fashion for solving the problems at the individual iterations of the method. In contrast to that, we use our own branch-and-bound procedure based on semidefinite programming (SDP), as SDP relaxations are known to be more effective for solving the SRFLP than linear programming relaxations of ILPs. This allows us to propose several enhancements procedures for our $\varepsilon$-constraint approach, such as non-binary branching and reusing of nodes within the branch-and-bound trees, which are usually not possible when using black-box solvers. We present a computational study to demonstrate the effectiveness of our solution approach and its enhancements.