Question

In: Economics

Consider the following linear program where X1, X2, and X3represent the number of gourmet cookies, brownies,...

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.

• They are limited to 500 pounds of flour

• They must use at least 450 grams of sugar (or else it will go bad)

• They cannot use more than 400 oz of oil

• They need to produce at least 40 batches of cookies for charity

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?

Solutions

Expert Solution

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

Rahul Sunny answered 1 year ago

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