Consider the following linear programming problem.

Question

Consider the following linear programming problem. A farmer plans to use two types of food to make a mix of low-cost feed for the animals on his farm. A bag of food A costs $16 and contains 24 units of proteins, 1 unit of minerals and 8 units of vitamins. A bag of food B costs $6.80 and contains 7 units of proteins, 1 unit of minerals and 15 units of vitamins. How many bags of food A and B should be consumed by the animals each day in order to meet the daily requirements of at least 280 units of proteins, 23 units of minerals and 240 units of vitamins at a minimum cost?

a) Define your variables using full sentences.

b) Write the objective (Minimize or Maximize) and the objective function.

c) State the constraints. Do NOT find the feasible set.

d) For this linear programming problem, the corner points of the feasible set and the values of the objective function at these points are given below.

Corner Points: #1 (0,40), #2 (7,16), #3(15,8), #4 (30,0)

Value of objective function on corner point: #1 272, #2 220.8, #3 294.8, #4 480.

The ordered pairs in the table above are in the form (food A, food B).

How many bags of food A and B should be bought in order to meet the nutritional requirements at a minimum cost? What is the minimum cost possible to satisfy the nutritional requirements?

Answer 1

a) A is the number of bags of Food A per days. B is the number of bags of Food B used per day.

b) Our objective is to minimize the total cost of food per day, so we can write the equation

C = 24A + 6.80B

c) Our constraints can be categorized into three types: protein, minerals, and vitamins

Protein: 24A + 7B ≥ 280

Minerals: 1A + 1 B ≥ 23

Vitamins: 8A + 15B ≥ 240

d) We were given the corner points for this problem as well as the value of the objective function at those points. The points are written in the format ( A , B ).

Corner Points: #1 (0,40), #2 (7,16), #3(15,8), #4 (30,0)

Value of objective function on corner point: #1 272, #2 220.8, #3 294.8, #4 480

If we want to minimize cost, we're looking for the smallest value of the objective function which is 220.8 and occurs at point #3. This means we want to buy 15 bags of Food A and 8 bags of Food B to meet the nutritional requirements.. This will cost us $220.80.