Food X contains 6 units of vitamin A per gram and 7 units of vitamin B per gram and costs 12 paise per gram. Food Y contains 8 units of vitamin A per gram and 12 units of vitamin B per gram and costs 20 paise per gram. The daily minimum requirement of vitamins A & B is 100 units and 120 units respectively. Find the minimum cost of product. -

Mumbai University > MECH > Sem 7 > Operations Research

Marks: 10 M

Year: May 2014

We have to buy x quantity of food X and Y quantity of food Y to meet the daily requirement.

We have to minimize the cost of these foods.

Minimization function: Z = 12x + 20y

Converting to a maximization function: Z = –12x – 20y

Constraints:

Minimum 100 units of A: 6x + 8y ≥ 100

Minimum 120 units of B: 7x + 12y ≥ 120

Introducing surplus and artificial variables into the constraints:

$6x + 8y ≥ 100 → 6x + 8y – S/_1 + A_1 = 100$

$7x + 12y ≥ 120 → 7x + 12y – S_2 + A_2 = 120$

Using Big M method, maximization function becomes:$Z = –12x – 20y + 0S_1 + 0S_2 – MA_1 – MA_2$

Max. Z = 205

$\text{Therefore min. Z = 205 paise $\\$ x = 15 grams $\\$ y = 1.25 grams}$