A　graph　G　=　(V,　E)　is　(k,　K′)-total　weight　choosable　if　the　following　is　true:　For　any　(k,　K′)-total　list　assignment　L　that　assigns　to　each　vertex　v　a　set　L(v)　of　k　real　numbers　as　permissible　weights,　and　assigns　to　each　edge　e　a　set　L(e)　of　k′　real　numbers　as　permissible　weights,　there　is　a　proper　L-total　weighting,　i.e.,　a　mapping　f:　V　∪　E　→　such　that　f(y)　∈　L(y)　for　each　y　∈　V　∪　E,　and　for　any　two　adjacent　vertices　u　and　u,　Σe∈E(u)　f(e)+f(u)　≠　Σe∈E(v)/(e)　+　f(v)-　This　Paper　introduces　a　method,　the　max-min　weighting　method,　for　finding　proper　L-total　weightings　of　graphs.　Using　this　method,　we　prove　that　complete　multipartite　graphs　of　the　form　K　n,m,1,1,.,1　are　(2,　2)-total　weight　choosable　and　complete　bipartite　graphs　other　than　K2　are　(1,　2)-total　weight　choosable.