solve this linear programming problem;
A chain saw requires 4 hours of assembly and a wood chipper 6 hours. A maximum of 48 hours of assembly time is available. The profit is $150 on a chain saw and $220 on a chipper. How many of each should be assembled for maximum profit??
Thanks in advance guys!! =]
See:
http://www.jiskha.com/display.cgi?id=1302108683
maximum profit is $1800
To solve this linear programming problem, we'll use a graphical method called the graphical solution technique.
Step 1: Define the decision variables:
Let's denote the number of chain saws to be assembled as 'x' and the number of wood chippers to be assembled as 'y'.
Step 2: Write down the objective function:
The objective is to maximize profit, which can be represented by the following equation:
Profit = 150x + 220y
Step 3: Write down the constraints:
1. Assembly time constraint: 4x + 6y ≤ 48 (since a maximum of 48 hours of assembly time is available)
2. Non-negativity constraint: x ≥ 0 and y ≥ 0 (since the number of items cannot be negative)
Step 4: Graph the feasible region:
To graph the feasible region, we'll plot the equations on a Cartesian plane and shade the region that satisfies all the constraints.
First, rewrite the assembly time constraint in slope-intercept form:
6y ≤ 48 - 4x
y ≤ (48 - 4x) / 6
Now, create a table of values for this inequality:
x | y
0 | 8
12 | 0
Plot these points on a graph and draw a line passing through them.
Step 5: Identify the feasible region:
Shade the region below the line since it satisfies the inequality.
Step 6: Optimal solution:
Next, we need to find the corner points of the feasible region. These points will be tested in the objective function to find the point that maximizes the profit.
The corner points for this problem are:
Point A: (0, 8)
Point B: (12, 0)
Step 7: Evaluate the objective function:
Substitute the coordinates of each corner point into the objective function and calculate the profit for each point.
Profit at point A = 150(0) + 220(8) = $1760
Profit at point B = 150(12) + 220(0) = $1800
Step 8: Determine the maximum profit:
Since the profit at point B (12, 0) is higher than the profit at point A (0, 8), the maximum profit is $1800.
Step 9: Find the number of each item to be assembled:
From the optimal solution, we can see that to maximize the profit of $1800, 12 chain saws and 0 wood chippers should be assembled.
Therefore, to maximize the profit, you should assemble 12 chain saws and 0 wood chippers.