Considerasystemcontainingatypeofndevices.Eachdeviceconsistsofmparts.Givenmnpartsforndeviceswithdifferentworkingprobabilities,howdowegroupthemintondevicessuchthatthesystemismostreliable.Amonotonicgroupingmeansthatthebestpartsareputtogether,thenthebestremainingpartsareputtogether,etc..Asufficientandnecessaryconditionforsuchgroupingbeingoptimalwillbegiveninthispaper.Asanapplication,weshowthatthemonotonicgroupingisoptimalforthetwo-stagek-out-of-nsystem.Whenm=2,wealsostudythegroupingthatthebestpairstotheworst,thesecondbestpairstothesecondworst,etc.,associatedwitharedundantdeviceproblem.
