You own a computer store selling two types of laser printers. The first type,A, costs $141 and you make a $45 profit on each one. The second type,B, costs $110 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month and you need to make at least $3850 profit on them. If you must order at least one of each type of printer, how many of each type of printer should you order if you want to minimize your cost?
Can someone please show me the steps?
You need to minimize
C = 141a + 110b
subject to
45a + 35b >= 3850
a+b >= 100
a >= 1
b >= 1
Graph those lines, and check the value of C at each vertex. You will see that
C(35,65) = 3850 gives the minimum cost.
To minimize your cost, you need to determine the number of each type of printer that you should order. Let's use the following variables:
Let x = the number of printers of type A to be ordered
Let y = the number of printers of type B to be ordered
We know that the cost of each printer of type A is $141 and the profit is $45. Therefore, the total cost for the type A printers would be 141x, and the total profit would be 45x.
Similarly, the cost of each printer of type B is $110, and the profit is $35. Therefore, the total cost for the type B printers would be 110y, and the total profit would be 35y.
Now, given the constraints of at least 100 printers sold and a minimum profit of $3850, we can set up the following equations:
Constraint 1: x + y ≥ 100 (to ensure at least 100 printers are sold)
Constraint 2: 45x + 35y ≥ 3850 (to ensure a minimum profit of $3850)
To minimize the cost, we need to minimize the total expense, which is given by: Expense = 141x + 110y.
We can now solve this problem using linear programming techniques. One approach is to graph the feasible region defined by the two constraints (x + y ≥ 100 and 45x + 35y ≥ 3850) and find the point where the total expense (141x + 110y) is minimized.
However, I will solve this problem using the Simplex method, which is a common optimization algorithm.
The objective function is Expense = 141x + 110y.
The constraints are:
1. x + y ≥ 100
2. 45x + 35y ≥ 3850
First, let's convert the inequalities to equalities by introducing slack variables:
Constraint 1 becomes: x + y + s₁ = 100
Constraint 2 becomes: 45x + 35y + s₂ = 3850
Now, let's create a table:
| Cj | x | y | s₁ | s₂ | RHS | Ratio |
------------------------------------------------
Z | 1 | 0 | 0 | 0 | 0 | 0 | - |
------------------------------------------------
s₁ | 0 | 1 | 1 | 1 | 0 | 100 | - |
------------------------------------------------
s₂ | 0 | 45| 35| 0 | 1 | 3850| - |
Now, perform the Simplex algorithm:
- Choose the most negative coefficient in the Z row, which is -1, and its corresponding column is 'x'.
- Perform the row operations to make the coefficient of 'x' equal to 1 in the Z row and 0 in all other rows.
| Cj | x | y | s₁ | s₂ | RHS | Ratio |
------------------------------------------------------
Z | 1 | 0 | 0 | 0 | 0 | 0 | - |
------------------------------------------------------
s₁ | 0 | 1 | 1 | 1 | 0 | 100 | - |
------------------------------------------------------
-1s₂ | 0 | 45 | 35 | 0 | 1 | 3850| - |
Next, perform the Simplex iterations:
- Choose the column with the smallest Ratio (RHS/Column Coefficient) to enter the basis. In this case, 's₁' has the smallest ratio, so it enters the basis.
- Perform the row operations to make the coefficient of 's₁' equal to 1 in the s₁ row and 0 in all other rows.
| Cj | x | y | s₁ | s₂ | RHS | Ratio |
--------------------------------------------------------
Z | 0 | -1 | -1 | 1 | -1/3| 9100 | - |
--------------------------------------------------------
s₁ | 1 | 1 | 1 | 1 | 0 | 100 | - |
--------------------------------------------------------
-1s₂ | 0 |-10/3| 0 | -1/3| 1 |4100/3| - |
- Repeat this process until the Z row has no negative coefficients or all the ratios in the RHS column are negative.
Continuing with the iteration:
- Choose the column with the smallest Ratio (RHS/Column Coefficient) to enter the basis. In this case, 'y' has the smallest ratio, so it enters the basis.
- Perform the row operations to make the coefficient of 'y' equal to 1 in the y row and 0 in all other rows.
| Cj | x | y | s₁ | s₂ | RHS | Ratio |
-----------------------------------------------------------
Z | 0 | 0 | -1/3| 4/3 | -5/3| 4400/3| - |
-----------------------------------------------------------
s₁ | 1 | 0 | 2/3 | -1/3 | 1 | 2500/3| - |
-----------------------------------------------------------
-1x | 0 | 1 | 0 | -1/3 | 3/7 |1500/7 | - |
- Repeat the process again until the Z row has no negative coefficients or all the ratios in the RHS column are negative.
Continuing with the iteration:
- Choose the column with the smallest Ratio (RHS/Column Coefficient) to enter the basis. In this case, 'x' has the smallest ratio, so it enters the basis.
- Perform the row operations to make the coefficient of 'x' equal to 1 in the x row and 0 in all other rows.
| Cj | x | y | s₁ | s₂ | RHS | Ratio |
--------------------------------------------------------------------------
Z | 0 | 0 | -1/6 | 3/6 | -4/6 | 4700/6 | - |
--------------------------------------------------------------------------
s₁ | 1 | 0 | 1/2 | -1/2 | 1 | 500/2 | - |
--------------------------------------------------------------------------
-5x | 0 | 1 | -3/10 | 1/10 | 3/10 | 150/10 | - |
Now we can read the solution from the Simplex tableau:
The optimal solution is:
x = 150/10 = 15
y = 500/2 = 250
Therefore, to minimize the cost, you should order 15 printers of type A and 250 printers of type B.