In　this　paper,　we　provide　a　streaming　algorithm　for　the　problem　of　maximizing　the　sum　of　a　supermodular　function　and　a　nonnegative　monotone　diminishing　return　submodular　(MDR-submodular)　function　with　a　knapsack　constraint　on　the　integer　lattice.　Inspired　by　the　SIEVE-STREAMING　algorithm,　we　present　a　two-pass　streaming　algorithm　by　using　the　threshold　technique.　Then,　we　improve　the　two-pass　streaming　algorithm　to　one-pass　and　further　reduce　its　space　complexity.　The　proposed　algorithms　are　proved　to　have　polynomial　time　and　space　complexity,　and　a　performance　guarantee　dependent　on　the　curvature　of　the　supermodular　function.　Finally,　we　carry　out　numerical　experiments　to　verify　the　performance　of　the　algorithm.