When　a　system　of　max-min　equations　is　inconsistent,　which　is　frequently　encountered　in　modelling　with　fuzzy　relations,　the　approximate　solutions　may　be　considered　instead　within　an　admissible　error　bound.　This　paper　tackles　the　linear　optimization　problem　over　the　approximate　solutions　with　respect　to　the　L∞　and　the　L1　residual　error,　respectively.　It　demonstrates　that　such　an　optimization　problem　can　be　reformulated　as　either　a　generalized　set　covering　problem　for　the　L∞　scenario　or　a　mixed　integer　linear　programming　problem　for　the　L1　scenario.　As　a　result,　it　may　be　solved　to　optimality　with　the　aid　of　a　state-of-the-art　optimization　solver　and　hence　does　not　demand　a　tailored　solving　method　except　for　very　large　instances.　©　2024　Elsevier　B.V.