Question

(Fertilizer Company, 10 points). Consider a fertiliser company producing three brands:

SulphurGrower (S)

EssarGrower (E)

NitroGrower (N).

The raw material for each of these are:

SulphurGrower EssarGrower NitroGrower

15% Sulphur 10% Sulphur 5% Sulphur

5% Nitrate 10% Nitrate 10% Nitrate

10% Potash 10% Potash 15% Potash

During a given period the company can obtain 1000 tons of Sulphur, 1600 tons Nitrate, and 2500 tons Potash. Other ingredients have no supply limit.

During the same period the company has contracts to supply at least 1000 tons of SulphurGrower (S), and 500 tons of EssarGrower (E). There is no limit on what the company can sell otherwise.

The profits for each of the products are:

SulphurGrower £3/ton

EssarGrower £5/ton

NitroGrower £5/ton

(a) Give the LP for maximising the profit

(b) Write it in standard form

Answer 1

We tabulate the data as shown below:

Minimum SulphurNitrate Potash Profits requirements Sulphur Grower (S Essar Grower (E Nitro Grower (N Capacity (Tons) 15% 10% 5% 1000 5% 10% 10% 1600 10% 10% 15% 2500 1000 £ 3 500 E 5

Let the amount of fertilizer sold to Sulphur Grower be S, Essar Grower be E and Nitro Grower be N

We have to maximize profit for total fertilizer sold

Hence, we get

Maximize z = 3S+5E+5N

Total sulphur produced = 0.15S + 0.1E + 0.05N

Total Nitrate produced = 0.05S + 0.1E + 0.1N

Total Potash produced = 0.1S + 0.1E + 0.15N

a) Hence, from above table as well as from above equations we get LP as follows:

Maximize z = 3S+5E+5N

Subject to

0.15S + 0.1E + 0.05N <= 1000

0.05S + 0.1E + 0.1N <= 1600

0.1S + 0.1E + 0.15N <= 2500

S >= 1000

E >= 500

b) In standard form, we get

Maximize z = 3S+5E+5N + 0S1 + 0S2 + 0S3 + 0S4 + 0S5

Subject to

0.15S + 0.1E + 0.05N + S1 + 0S2 + 0S3 + 0S4 + 0S5 = 1000

0.05S + 0.1E + 0.1N + 0S1 + S2 + 0S3 + 0S4 + 0S5 = 1600

0.1S + 0.1E + 0.15N + 0S1 + 0S2 + S3 + 0S4 +0S5 = 2500

S + 0S1 + 0S2 + 0S3 - S4 + 0S5 = 1000

E + 0S1 + 0S2 + 0S3 +0S4 - S5 = 500

S, E, N, S1, S2, S3, S4, S5 >= 0

S1, S2, S3 are slack variable and S4, S5 are surplus variables