An industrial recycling centre uses two scrap aluminium metals, A and B$,$ to produce a special alloy. Scrap A contains $60\%$ aluminium, $3\%$ silicon, and $4\%$ carbon. Scrap B has $30\%$ aluminium, $6\%$ silicon, and $3\%$ carbon. The costs per ton for scraps A and B are $$100$ and $$80,$ respectively. The specifications of the special alloy require that $(1)$ the aluminium content must be at least $30\%$ and at most $60\%,(2)$ the silicon content must lie between $3\%$ and $5\%,$ and $(3)$ the carbon content must be between $3\%$ and $7\%.$ Determine the optimum mix of the scraps that should be used in producing $1000$ tons of the alloy.