I seem to be having problems setting up Linear Programming questions.
For example:
Kustom Kars does van conversions. The Custom conversion required 15 hours of shop time, 8 hours of painting time, and 1 hour of inspection time. The Deluxe conversion requires 12 hours of conversion time, 12 hours of painting time, and 1.5 hours of inspection time. There are 90 hours of shop time, 72 hours of painting time, and 10 hours of inspection time in the coming two weeks. How many conversion of each type should Kustom Kars peform assuming that each Custom conversion results in $175 profit and each Deluxe conversion results in $225 profit? What's the Max profit?
I keep thinking that there are 3 variables (shop, paint, & inspection time) and am unable to graph any lines. So far this is how I've set this problem up:
X= number of Custom Conversions
Y= number of Deluxe Conversions
Max(P)=$175X + $225Y
X=15shop + 8paint + 1 inspection
Y=12shop + 12paint + 1.5 inspection
shop < 90
paint < 72
inspection < 10
What the heck am I doing wrong? I'd prefer a hint in the right direction over the solution.
Thanks,
Matt
There are two variables: deluxe, and custome
There are three constraints.
Plot on a graph the number of deluxe,custom you can make at the same time.
Start with the axis, and fill in the chart, it is only six points. Your max profit will be at one of the corners.
Hi Matt! It looks like you're on the right track, but there are a few key elements missing in your setup.
In linear programming problems, we want to optimize (maximize or minimize) an objective function while considering a set of constraints. In this case, the objective function is the profit, which is given by Max(P) = $175X + $225Y, where X is the number of Custom conversions and Y is the number of Deluxe conversions.
Now, let's consider the constraints. You correctly defined X and Y in terms of the hours required for each type of conversion. However, you need to remember that the number of hours available for each resource is limited. So the constraints should be:
1. Shop time constraint: 15X + 12Y ≤ 90 (since the total shop time available is 90 hours)
2. Painting time constraint: 8X + 12Y ≤ 72 (since the total painting time available is 72 hours)
3. Inspection time constraint: 1X + 1.5Y ≤ 10 (since the total inspection time available is 10 hours)
These constraints reflect the fact that the sum of hours spent on each type of conversion multiplied by the respective conversion durations must be less than or equal to the total available hours.
Now that you have your objective function and constraints, you can graph these inequalities on a coordinate plane to visualize the feasible region. The feasible region represents the set of all possible solutions that satisfy the constraints.
Remember that the objective is to maximize profit, so you need to find the corner point within the feasible region that will yield the highest profit. You can achieve this by applying graphical methods like graphing the lines, shading the feasible region, and determining the coordinates of the corner points.
I hope this helps you get started on solving your linear programming problem!