Linear Algebra w/ Objective Function & Constraints

Question

Quality Security runs a state prison. Part of their contract requires them to provide each inmate with at least the minimum nutritional requirements. After the inmate’s daily meals, it has been determined that each inmate still needs 120 units of vitamin A, 50 units of B-complex, and 20 units of iron. There are four different supplements that they are considering.

Vitamin A B-Complex Iron Cost per ounce of supplement

Supplement I 20 6 0 .01

Supplement II 11 12 3 .02

Supplement III 0 22 6 .03

Supplement IV 28 0 10 .04

How much of each supplement should a prisoner be given so that the minimum requirements are met at the lowest cost?”

(a)   Find the objective function

(b)  Find the constraints.

Answer 1

They are already hinting at what should the objective function should be since they want to minimize cost

Let me define the following

a = (20, 11, 0, 28) vector with the amount of vitamin A in each supplement

b = (6, 12, 22, 0) vector with the amount of vitamin B in each supplement

f = (0, 3, 6, 10) vector with the amount of Iron in each supplement

c = (0.01, 0.02, 0.03, 0.04) cost of each suplement

s = (s1, s2, s3, s4) vector with the amount of each supplement to buy

(a) Give the definitions below the total cost is the scalar product

T = c*s

This is our objective function

(b) Our constrains are the requirement each prisoner needs, in this case

a*s = 120

b*s = 50

f*s = 20