Maximizing income

Question

a bakery makes jumbo biscuits and regular biscuits. The bakery is limited to:

a. 5the oven can bake at most 120 biscuits per day. 

B. Each jumbo biscuis requires 3oz of flour and each regular biscuit requires 1oz of flour. The bakery has 180oz of flour available per day. 

C. Due to cooking time constraints, the total number of regular biscuits cannot exceed 100. 

D. Due to cooking time constraints, the total number of jumbo biscuits cannot exceed 50. 

 

The revenue from each jumbo biscuit sold is $1, and the revenue from each regular biscuit sold is $0.65. The cost to make each jumbo biscuit is $0.20, and the cost to make each regular biscuit is $0.15. 

1. Let x be the number of regular biscuits and y the number of jumbo biscuits. Write an inequality for the limitations a, b, c, and d. 

 

Answer 1

This is called linear programming.

a.  x + y <= 120

b.  3y + x <= 180

c.  x <= 100

d. y <= 50

We add two constraints for sanity:

e.  x >= 0

f.  y >= 0 

(you can't make a negative number of biscuits.)

 

You can plot the region determined by a. - f. as follows: translate each into the bounding equality

a.  x + y = 120

b.  3y + x = 180

c .  x = 100

d.  y = 50

e.  x = 0

f.  y = 0

Each of these is linear, hence a line (and hence the term linear programming). Shade the region bounded by the lines so that it matches up with the direction of the original inequalities. In other words, shade every part of the x,y plane that is feasible as solutions to the system of linear inequalities. 

 

Finally, we write down the formula to maximize:

R = 0.65x + 0.20y

 

It is a theorem of linear programming that any maximum or minimum is attained at a vertex of the polygon shaded in the first part of this solution. So we find each of the vertices (a,b) and plug them into the formula for R. Whichever vertex gives the maximum value is the solution. If the constraints did not give a complete polygon, but give a region that goes off to infinity, then we check two points along a boundary line that goes off to infinity and verify that it is going in the opposite direction of what we are trying to find. That is, since we are finding a maximum, we verify that going off to infinity would only decrease the values. It is a theorem that simply verifying two points confirms the overall tendency of going further in that direction.

 

This problem cannot be explained much more clearly without drawing on an x,y plane.