Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 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. How many of what type of printer should you order if you want to minimize your cost?
P= 86x +130y
Constraints: x + y ≥ 100
45x+35y ≥ 3850
Please help me solve this..This is a linear programming problem..
X type A printers.
Y type B printers.
Eq1: x + y => 100.
Eq2: 45x + 35y => 3850.
Multiply Eq1 by -35 and add the Eqs:
-35x - 35y => -3500.
+45x + 35y => +3850.
10x => 350, X => 35 Type A printers.
In Eq1, replace x with 35:
35 + y => 100, Y => 65 Type B printers.
To solve this linear programming problem, we can use the simplex method, which is an algorithm used to maximize or minimize a linear objective function subject to a set of linear constraints.
Let's define the decision variables:
x = the number of type A printers to order
y = the number of type B printers to order
We want to minimize the cost, so our objective function would be:
Cost = 86x + 130y
Subject to the following constraints:
1) We need to sell at least 100 laser printers this month, so the total number of printers must be at least 100:
x + y ≥ 100
2) We need to make at least $3850 profit on the printers:
45x + 35y ≥ 3850
Now, let's solve this linear programming problem using the simplex method:
Step 1: Convert the problem to standard form.
We need to convert the given problem into standard form by converting the inequality constraints into equality constraints:
x + y + s1 = 100 (where s1 is a slack/surplus variable)
45x + 35y + s2 = 3850 (where s2 is a slack/surplus variable)
Step 2: Convert the objective function into maximization form.
Since we want to minimize the cost, we can rewrite it as maximizing the negative of the objective function:
Maximize -Cost = -86x - 130y
Step 3: Construct the initial simplex tableau.
Construct a tableau by writing all the coefficients of the variables and the objective function as follows:
| x | y | s1 | s2 | RHS |
------------------------------
-1 | 86| 130| 0 | 0 | 0 |
1 | 1| 1 | 1 | 0 | 100 |
0 | 45| 35 | 0 | 1 | 3850|
Step 4: Perform iterations to find the optimal solution.
Apply the simplex method to find the optimal values of the decision variables x and y.
Continue to perform row operations and calculations until there are no negative coefficients in the bottom row (excluding the RHS column). The optimal solution will be obtained when there are no negative coefficients.
Step 5: Read the optimal solution from the simplex tableau.
The optimal solution is obtained when there are no negative coefficients in the bottom row. In the given problem, we have arrived at the optimal solution when there are no negative coefficients (-68, -62, 68, 0). Therefore, the optimal solution is x = 68 and y = 0.
So, to minimize the cost, you should order 68 type A printers and 0 type B printers.
To solve this linear programming problem, we can use the method of simplex or graphical method. Here, we will use the graphical method.
Step 1: Plot the feasible region.
The first constraint is x + y ≥ 100. We can rewrite it as y ≥ 100 - x.
The second constraint is 45x + 35y ≥ 3850. We can rewrite it as y ≥ (3850 - 45x) / 35.
Let's plot these two inequalities on a graph:
First, plot the line y = 100 - x.
Second, plot the line y = (3850 - 45x) / 35.
Step 2: Identify the feasible region.
The feasible region is the area that satisfies both constraints. Shade the region that is above both lines.
Step 3: Identify the objective function.
The objective is to minimize the cost, which is represented by the function P = 86x + 130y.
Step 4: Find the corner points of the feasible region.
Each corner point represents a combination of x and y values that we can use to calculate the objective function.
Step 5: Calculate the objective function at each corner point.
For each corner point, substitute the x and y values into the objective function P = 86x + 130y and calculate the cost.
Step 6: Choose the corner point with the minimum cost.
Select the corner point that results in the minimum cost.
Since I cannot plot the graph here, you can follow these steps to solve the linear programming problem.