We consider a multicriteria integer linear programming problem with a targeting set of optimal solutions given by the set of all individual criterion minimizers (extrema). In this study, the lower and upper attainable bounds on the quasistability radius of the set of extremum solutions are obtained when solution and criterion spaces are endowed with different Hlder’s norms. As a corollary, an analytical formula for the quasistability radius is obtained for the case where criterion space is endowed with Chebyshev’s norm. Some computational challenges are also discussed.