Formulate the following linear programming problem. DO NOT SOLVE THE SYSTEM: A f

Question

Formulate the following linear programming problem. DO NOT SOLVE THE SYSTEM: A fruit grower can use two different fertilizer mixes in his apple orchard, brand X or brand Y. One bag of brand X contains 4 pounds of phosphoric acid, 4 pounds of chloride, and 6 pounds of nitrogen. One bag of brand Y contains 6 pounds of phosphoric acid, 2 pound of chloride, and 5 pounds of nitrogen. Tests indicate that the orchard needs at least 2400 pounds of phosphoric acid and no more 1600 pounds of chloride. If the grower wants to maximize the amount of nitrogen added to the orchard, how many bags of each mix should be used? Clearly define the unknowns, and then give the objective function and the constraints which are needed to determine how many bags of each fertilizer mix should be used to maximize nitrogen additive. Do not graph. DO NOT SOLVE.

Explanation / Answer

Let the bags of fertilizer X required be = a

and the bags of fertilizer Y required be = b

Fertilizer X composition for 1 bag is

P (Phosphoric acid) = 4 lbs

C (Chlorine) = 4 lbs

N (Nitrogen) = 6 lbs

Therefore, 'a' bags of X would contain

P = 4a lbs

C = 4a lbs

N = 6a lbs

Similarily, Fertilizer Y composition for 1 bag is

P (Phosphoric acid) = 6 lbs

C (Chlorine) = 2 lbs

N (Nitrogen) = 5 lbs

Therefore, 'b' bags of Y would contain

P = 6b lbs

C = 2b lbs

N = 5b lbs

The mixture of (a+b) bags will have the following composition

P = 4a + 6b

C = 4a + 2b

N = 6a + 5b

Given, the conditions that when 'a' bags of X and 'b' bags of Y are mixed,

P>= 2400 lbs and C<=1600 lbs

we have to maximize the quantity of N in the mixture

P>=2400

therefore,

4a + 6b >= 2400          

C<= 1600

therefore,

4a + 2b <= 1600           

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