Trying to solve a task here using the simplex method.

A company is producing three types of fertilizer. Their chemical composition is as follows (per ton): Type I has 0.1 tonnes of nitrates, 0.1 tonnes of phosphates, 0.2 tonnes of carbonates and 0.6 tonnes of filler; Type II has 0.1 tonnes nitrates, 0.2 tonnes phosphates, 0.1 tonnes carbonates and 0.6 tonnes filler; type III has 0.2 tonnes nitrates, 0.1 tonnes phosphates, 0.1 tonnes carbonates and 0.6 tonnes filler.

In a month, the plant can make use of 1200 tonnes of nitrates (costing $1500), 2000 tonnes phosphates ($600), 2200 tonnes of carbonates ($1200) and an unlimited supply (!) of filler ($100).

The fertilizers are sold at a per ton price of $830 (I), $810 (II) and $810 (III). Production costs, except raw material, are estimated at $110 in the case of all three fertilizers.

Please help me put together a linear programming task to maximise profit. If I could just get the contraints and the optimization formula right, then I could get on with solving the simplex tableu itself.

Assuming the variables are x1 - type I production, x2 - type II production, x3 - type III production.

Constraints (non-canonical): 0.1x1 + 0.1x2 + 0.2x3 <= 1200 (for nitrates)

0.1x1 + 0.2x2 + 0.1x3 <= 2000 (phosphates)

0.2x1 + 0.1x2 + 0.1x3 <= 2200 (carbonates)

0.6x1 + 0.6x2 + 0.6x3 <= ...(?)... (filler)

Some help would be greatly appreciated!