Three Plants P1, P2, and P3 of a Gas Corporation

Essay

Three plants P1, P2, and P3 of a gas corporation supply gasoline to three of their distributors located in the city at three different locations A, B, and C. The plants' daily capacities are 4500, 3000, and 5000, gallons respectively, while the distributors' daily requirements are 5500, 2500, and 4200 gallons. The per-gallon transportation costs (in $) are provided in the table below.
​ $$\begin{array}{l}\begin{array} { l c c c } &\text { Distributor }\\\hline \text { Plant } & \text { A } & \text { B } & \text { C } \\\hline \text { P1 } & 0.8 & 0.5 & 1 \\\text { P2 } & 0.7 & 0.65 & 0.8 \\\text { P3 } & 0.5 & 0.45 & 0.7\end{array}\end{array}$$ ​
Because of a failure of expected supply earlier, the distributors - A, B, and C this time have decided to charge a penalty of $0.45, $0.55, and $0.5 per gallon, respectively, to avoid any further delays.
Now, determine the optimum supply of gasoline to the distributors, in order to minimize the total transportation cost as well as the charges payable as penalty.