Question

A grocery store manager must decide how to best present a limited supply of milk and cookies to its customers. Milk can be sold by itself for a profit of $1.50 per gallon. Cookies can likewise be sold at a profit of $2.50 per dozen. To increase appeal to customers, one gallon of milk and a dozen cookies can be packaged together and sold for a profit of $3.00 per bundle. The manager has at most 100 gallons of milk and 150 dozen cookies to make available each day. The manager has decided to stock at least 75 individual gallons of milk per day (excluding milk bundled with cookies). Demand for individual packages of cookies is at most 140 dozen per day (excluding cookies bundled with milk). The manager wishes to determine how much of each product to stock each day. Which of the following is the constraint that limits the amount of milk the store will use (both in bundles and sold separately) each day?

Answer 1

Solution:

Let the no. of gallons of milk that will be sold by itself be "x".

Let the no. of dozens of cookies that will be sold by itself be "y"

Let the gallons/dozens of packaged together milk and cookies that will be sold be "z"

Thus ,

Objective function = total profit

                               = 1.5x+2.5y+3z.

This has to be maximized.

Constraints:

1. x+z<=100 (as 100 gallons of milk is available per day)

2. y+z<=150 (as 150 dozen cookies is available each day)

3. x+z>=75 (stock will be at least 75 gallons of milk per day)

4. y+z = 140 (as demand for cookies is always 140 dozen per day)

5. x,y,z>=0 (non-negativity constraint)

Solving the above in solver the following solution is obtained:

X                     100    

Y                     140    

z                      0    

Total profit     500   Formula

                                                    1.5x+2.5y+3z

Constraints    

100          <=        100           x+z<=100

140           <=        150           y+z<=150

100          >=          75            x+z>=75

140           =          140            y+z = 140

Thus maximum daily profit is $500.

This is achieved by selling 100 gallons of milk by itself and 140 dozens of cookies by itself.