Question

A company blends two materials: A and B to produce two types of fertilizers. Fertilizer 1 must be at least 50% of A and sells for $65 per kilo gram. Fertilizer 2 must be at least 70% of B and sells for $48 per kilogram. The price of martial A is $10 per 100 kilo grams and the price of martial B is $14 per 100 kilo grams if they purchased over 10,000 kilo gram the price will be reduced by 10%. Total budget of the company to spend on raw martial is $2000.
a) Write the linear optimization model for the company to make the best decision

Answer 1

The linear Optimization model is described below:

Decision variable

Let F1 contains x fraction of A and (1-x) of B

Let F2 contains y fraction of B and (1-y)% of A

Let f1 kg & f2 kg be the quantity of F1 & F2 produced respectively

Objective function

Maximize profit:-

[ Max(65f1+48f2 - [(x*f1*0.1)+((1-x)*f1*0.14)]-[y*f2*0.14+(1-y)*f2*0.1)] , if f1+f2 < 10,000 .... (x* f1 = amount of A in F1 while 0.1 is price/Kg of A, Similarly for B)

Max(65f1+48f2 - 0.9[(x*f1*0.1)+((1-x)*f1*0.14)]-0.9[y*f2*0.14+(1-y)*f2*0.1)] , if f1+f2 > 10,000 ] ..... (Accounting for 10% bulk purchase discount)

Constraints

x >= 0.5 (Atleast 50% of A in F1)

y >= 0.7 (Atleast 70% of B in F2)

[(x*f1*0.1)+((1-x)*f1*0.14)]-[y*f2*0.14+(1-y)*f2*0.1)] <2000 if f1 + f2 <10,000 [Budget constraint]

0.9[(x*f1*0.1)+((1-x)*f1*0.14)]-0.9[y*f2*0.14+(1-y)*f2*0.1)]< 2000 , if f1+f2 > 10,000   [Budget constraint accounted with 10% bulk purchase discount)