  The Multi-Objective Mixed-Integer Programming (MOMIP) problem is one of the
most challenging. To derive its Pareto optimal solutions one can use the
well-known Chebyshev scalarization and Mixed-Integer Programming (MIP) solvers.
However, for a large-scale instance of the MOMIP problem, its scalarization may
not be solved to optimality, even by state-of-the-art optimization packages,
within the time limit imposed on the optimization. If a MIP solver cannot
derive the optimal solution within the assumed time limit, it provides the
optimality gap, which gauges the quality of the approximate solution. However,
for the MOMIP case, no information is provided on the lower and upper bounds of
the components of the Pareto optimal outcome. For the MOMIP problem with two
and three objective functions, an algorithm is proposed to provide the
so-called interval representation of the Pareto optimal outcome designated by
the weighting vector when there is a time limit on solving the Chebyshev
scalarization. Such interval representations can be used to navigate on the
Pareto front. The results of several numerical experiments on selected
large-scale instances of the multi-objective, multidimensional 0-1 knapsack
problem illustrate the proposed approach. The limitations and possible
enhancements of the proposed method are also discussed.
