In: Economics
Consider the following linear program where X1, X2, and X3represent the number of gourmet cookies, brownies, and cakes sold by Litton’s Bakery on a weekly basis. They profit $25 from cookies, $15 from brownies, and $35 per cake. The bakery has the following constraints.
Using the output below, answer the questions on the next page
MAX 25X1+15X2+35X3
S.T. 1) 4X1+2X2+4X3<500
2) 3X1+3X2+5X3>450
3) 2X1+1X2+3X3<400
4) 1X1>40
OPTIMAL SOLUTION
Objective Function Value = 3975.000
Variable Value Reduced Costs
-------------- --------------- ------------------
X1 40.000 0.000
X2 0.000 2.500
X3 85.000 0.000
Constraint Slack/Surplus Dual Prices
-------------- --------------- ------------------
1 0.000 8.750
2 95.000 0.000
3 65.000 0.000
4 0.000 -10.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
X1 No Lower Limit 25.000 35.000
X2 No Lower Limit 15.000 17.500
X3 30.000 35.000 No Upper Limit
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
1 424.000 500.000 586.667
2 No Lower Limit 450.000 545.000
3 335.000 400.000 No Upper Limit
4 0.000 40.000 87.500
How many cookies, brownies, and cakes should Litton’s bakeeach week?
How much profit will Litton’s make?
How many grams of sugar will be used?
If 20 extra pounds of flour were available, what impact what that have on Litton’s profit?
If Litton’s charged $30 per cake instead of $35, how many cakes should they plan to bake?
Q. How many cookies, brownies, and cakes should Litton’s bakeeach week?
ANS:
According to output
X1 = 40, X2 = 0, X3 = 85
Litton’s should bake each week
Units of Cookies = 40 units
Units of brownies = 0 units
Units of Cakes = 85 units
Q. How much profit will Litton’s make?
ANS: The objective function value is given as 3975.00
Thus, with optimal product mix, total profit Litton can make is $3,975
Q: How many grams of sugar will be used?
ANS:
According to the report, the slack/surplus of the constraint 2 is +95. Thus there are 95 units surplus according to optimal solution.
Amount of sugar used = min. limit + slack = 450 + 95 = 545 grams
Q. If 20 extra pounds of flour were available, what impact what that have on Litton’s profit?
ANS:
The Flour availability constraint is represented y constraint (1). According to Dual price report, the dual price of constraint (1) is 8.750
It means that if additional unit of the constraint is made available, the profit will increase by $8.75 per additional unit.
Thus, with additional 20 pounds of flour will increase the profit by 20*$8.75 = $175
But the product mix will remain same.
Thus, the profit will increase by $175 by making 20 extra pounds of flour available.
Q.If Litton’s charged $30 per cake instead of $35, how many cakes should they plan to bake?
ANS:
The lower limit of the cake is 30, thus if the profit of cake is reduced from 35 to 30, the product mix will not change, the unit of cakes produced will be same.
Cake produced = 85 units