Develop a linear programming model that can be used to determine the plan that will minimize total distribution costs. Let x$\_\{11\}=$ amount shipped from Southern Gas to Hamilton x$\_\{12\}=$ amount shipped from Southern Gas to Butler x$\_\{13\}=$ amount shipped from Southern Gas to Clermont x$\_\{21\}=$ amount shipped from Northwest Gas to Hamilton x$\_\{22\}=$ amount shipped from Northwest Gas to Butler x$\_\{23\}=$ amount shipped from Northwest Gas to Clermont Min boxed boxed $10$ x $11+$ boxed $20$ x $12+$ boxed $15$ x $13+$ boxed $12|$x $21+$ boxed $15|$x $22+$ boxed $18|$x $2$ s$.$t$.$ x$\_\{11\}+$ x$\_\{12\}+$ x$\_\{13\}$ S$500$ X$23400$ X$21400200$ X$13300$ x $11,$x $12,$x $13,$x $21$ prime $,$x $23,$x $33>=0$