A manufacturer produces two items, bookcases and library tables. Each item requires processing in each of two departments. Department I has 40 hours available and department II has 36 hours available each week for production. To manufacture a bookcase requires 2 hours in department I and 4 hours in department II, while a library table requires 3 hours in department I and 2 hours in department II. Profits on the items are $6.00 for a bookcase and $7.00 for a library table.
Identify all constraints.
Identify all applicable corner points of the feasibility region.
If all units produced can be sold, how many of each should be made in order to maximize profits?
We are not going to do all your linear programming problems for you. I will rough out one of them. Then you try and post your work if you want feedback.
b = number of bookcases
t = number of tables
constraints:
Department 1:
2 b + 3 t </= 40
Department 2:
4 b + 2 t </= 36
Profit to maximize:
p = 6 b + 7 t
corner points: (b,t)
(0,0)
(0,13.33) because 13.33<36/2
(9,0) because 9<20
then the intersection
t = 11 and b = 3.5 or (3.5,11)
check corners for p =6 b + 7 t
(0,0) gives 0 profit
(0,13.33) gives 6*0+7*13.33=93.31
(9,0) gives 6*9 = 54
(3.5,11) gives 6*3.5+7*11 = 98 winner
I am assuming fractional bookcases and tables can be built because you presumably can spread this out over many weeks.
To identify the constraints, we need to consider the limitations given in the problem. These limitations are as follows:
1. Department I has 40 hours available per week.
2. Department II has 36 hours available per week.
3. Manufacturing a bookcase requires 2 hours in department I and 4 hours in department II.
4. Manufacturing a library table requires 3 hours in department I and 2 hours in department II.
Let's write these constraints in mathematical form:
1. Department I constraint: 2x + 3y ≤ 40, where x is the number of bookcases and y is the number of library tables.
2. Department II constraint: 4x + 2y ≤ 36, where x is the number of bookcases and y is the number of library tables.
To find the corner points of the feasibility region, we solve these equations simultaneously with the variables x and y:
2x + 3y ≤ 40
4x + 2y ≤ 36
Solving these equations will give us the values of x and y at each corner point.
To maximize profits, we need to determine the number of each item that should be made. Let's assign a profit function to be maximized: P = 6x + 7y, where P is the total profit, x is the number of bookcases, and y is the number of library tables.
Using the corner points obtained from solving the constraints, we substitute the values of x and y into the profit function to calculate the profit at each corner point. The combination that results in the highest profit is the optimal solution.
Now, let's calculate the corner points and determine the optimal solution.