An oil refinery has three process plants, and four grades of motor oil have been produced from these plants. The refinery is liable to meet the demand of customers. The refinery incurs a penalty for failing to meet the demand for any particular grade of motor oil. The capacities of the various plants, the costs of production, the demands of motor oil of the different grades, and the penalties have shown in the following table:

Table 2.4

The details of various plants

Process plant	Plant capacity (kgal/day)	Cost of production ($/day)
		$\mathbf{1}$	$\mathbf{2}$	$\mathbf{3}$	$\mathbf{4}$
1	100	800	900	1000	1200
2	150	850	950	1150	1400
3	200	900	1000	1250	1600
Demand (kgal/day)		45	140	100	70
Penalty (for shortage of each kilogallon)		$$ 9$	$$ 12$	$$ 15$	$$ 21$

Formulate the optimization problem as an LPP for minimizing the overall cost.