A dietician wishes to mix two types of food in such a way that the vitamin contends of the mixture contains at least 8 units of vitamin A and 10 units of vitamin B. Food I contains 2 units per Kg of vitamin A and 1 unit per Kg of vitamin B while food II contains 1 unit per Kg of vitamin A and 2 units of vitamin B. It costs 5 birr per Kg to purchase food I and birr 8 per Kg to purchase food II. Prepare the linear programming model for the problem
Let:
x = number of kilograms of food I to be mixed
y = number of kilograms of food II to be mixed
The objective is to minimize the cost, which is given by the cost of food I multiplied by x and the cost of food II multiplied by y.
Objective function: Cost = 5x + 8y
The constraints are:
1. The vitamin A content in the mixture should be at least 8 units:
Vitamin A constraint: 2x + y ≥ 8
This inequality represents the requirement for vitamin A.
2. The vitamin B content in the mixture should be at least 10 units:
Vitamin B constraint: x + 2y ≥ 10
This inequality represents the requirement for vitamin B.
3. Non-negativity constraint:
x ≥ 0, y ≥ 0
The number of kilograms of food cannot be negative.
Therefore, the complete linear programming model is as follows:
Minimize: Cost = 5x + 8y
Subject to:
2x + y ≥ 8 (Vitamin A constraint)
x + 2y ≥ 10 (Vitamin B constraint)
x ≥ 0
y ≥ 0