A fertilizer producing company purchases nitrates, phosphates, potash, and an inert chalk base and produces four different fertilizers A, B,C, and D. The cost of these nitrates, phosphates, potash, and an inert chalk base are $\$ 1600, \$ 550, \$ 1100$, and $\$ 110$ per ton, respectively. The cost of production, selling price, and composition of the four fertilizers are given in the following table.

Table 2.2

Cost of production, selling price, and composition of fertilizers

Fertilizer	Production cost ($/ton)	Selling price ($/ton)	Percentage composition by weight
			Nitrates	Phosphates	Potash	Inert chalk base
A	98	340	4.5	10	5.5	80
B	155	560	6	15	9	70
C	260	720	14	6	15	65
D	195	440	10	18	10	62

The supply of nitrates, phosphates, and potash is limited, no more than 1100 tons of nitrate, 2200 tons of phosphates, and 1600 tons of potash will be available for a week. The company is required to supply to its customers a minimum of 5200 tons of fertilizer $A$ and 4100 tons of fertilizer $D$ per week; however, it is otherwise free to produce the fertilizers in any quantities it satisfies. Formulate the problem (objective function and constrained functions) to find the quantity of each fertilizer to be produced by the company that maximize its profit.