Indifference sets of reference points in multi-objective integer linear programming

Reference point approaches for multi-objective problems rely on the definition of an achievement scalarizing function that projects reference points onto the non-dominated solution set. In this paper, we investigate the behaviour of reference points using a Tchebycheff metric-based scalarizing function in multi-objective pure integer linear programming (MOILP). Since the non-dominated solutions are discrete in MOILP, there are multiple reference points that lead to the same solution, i.e. there are indifference sets on the reference point space. We investigate some properties of the reference points in MOILP and also the graphical representation of indifference sets of tri-objective problems. We further investigate properties of the reference points when additional limitations on the objective function values are introduced.