Operation reasearch-feed mix problem

[Jammey]

An agricultural mill produces a different meal for cattles, sheep and chicken by mixing the following raw ingredients: corn, limestone, soyabeans and fishmeal. This ingredients contain the following nutrients: vitamins, protein, calcium and crude fat in the following quantities:

Nutrients k

Ingredients: Vitamin Protein Calcium Crude fat

Corn 8 10 6 8

Limestone 6 5 10 6

Soyabeans 10 12 6 6

Fish meals 4 18 6 9

Let a_ik= quantity of nutirnt k per kg of ingredient i

CONSTRAINTS: The meal has contract for the following:

Cattle= 10000 kg

demand d_j Sheep= 6000 kg

Chicken= 8000kg

There are limited availabilities of the raw ingredients:

Corn= 6000kg

Supply S_i Limestone= 10000kg

Soyabeans= 4000kg

Fishmeal= 5000kg

the different feeds have quality bound per kg

Vitamin Protein Calcium Crude fat

min max min max min max min max

Cattle 6 - 6 - 7 - 4 8

Sheep 6 - 6 - 6 - 4 8

chicken 4 6 6 - 6 - 4 8

The above value represent bounds: L_{jk and} U_jk

Cost per kg of ingredients is as follows:

Corn = N20

Limestone= N12

Soyabeans= N24

Fishmeal= N12

Formulate problem has a linear programming whose solution yields desired feed production levels at minimum cost.

DATA

d_j = demand for product j (kg)

S_i = supply of ingredient i (kg)

L_{jk =} lower bound on number of nutrients of type k per kg of product j

U_jk = Upper bound on number of nutrient of type k per kg of product j

C_i = cost per kg of ingredient i

a_ik = number of nutrients k per kg of ingredients i

DECISION VARIABLES

X_ij = amount (kg) of ingredient i used in producing product j

demand for product j (kg)