Support　Vector　Machines　(SVMs),　originally　proposed　for　classifications　of　two　classes,　have　become　a　very　popular　technique　in　the　machine　learning　field.　For　multi-class　classifications,　various　single-objective　models　and　multi-objective　ones　have　been　proposed.　However,in　most　single-objective　models,　neither　the　different　costs　of　different　misclassifications　nor　the　users＇　preferences　were　considered.　This　drawback　has　been　taken　into　account　in　multi-objective　models.　In　these　models,　large　and　hard　second-order　cone　programs(SOCPs)　were　constructed　ane　weakly　Pareto-optimal　solutions　were　offered.　In　this　paper,　we　propose　a　Projected　Multi-objective　SVM　(PM),　which　is　a　multi-objective　technique　that　works　in　a　higher　dimensional　space　than　the　object　space.　For　PM,　we　can　characterize　the　associated　Pareto-optimal　solutions.　Additionally,　it　significantly　alleviates　the　computational　bottlenecks　for　classifications　with　large　numbers　of　classes.　From　our　experimental　results,　we　can　see　PM　outperforms　the　single-objective　multi-class　SVMs　(based　on　an　all-together　method,　one-against-all　method　and　one-against-one　method)　and　other　multi-objective　SVMs.　Compared　to　the　single-objective　multi-class　SVMs,　PM　provides　a　wider　set　of　options　designed　for　different　misclassifications,　without　sacrificing　training　time.　Compared　to　other　multi-objective　methods,　PM　promises　the　out-of-sample　quality　of　the　approximation　of　the　Pareto　frontier,　with　a　considerable　reduction　of　the　computational　burden.