Question 1119561: A graphic designer can design a magazine cover or a logo. Her company makes a profit of $800 for each magazine cover and $500 for each logo. She estimates that it takes her 4 hours of brainstorming for a magazine cover and 2 hours of brainstorming for a logo. She'd like to keep the total brainstorming time under 24 hours a week. Further, she estimates that it takes her 2 hours to lay out a magazine cover and 0.5 hours to sketch up a logo, and she must fit this into 10 hours a week. Her boss requires her to design no more than 4 logos for each magazine cover she designs.
How many of each should she design in order to maximize the company's profits?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52787)   (Show Source):
You can put this solution on YOUR website! .
Let X = the number of the magazine covers,
Y = the number of logos.
Then the profit function is
P(X,Y) = 800*X + 500*Y (1) dollars.
The restrictions are
4*X + 2*Y <= 24 (2) (hours per week)
2*X + 0.5*Y <= 10 (3) (hours per week)
Y <= 4X (4) ("no more than 4 logos for each magazine cover")
Other restrictions are non-negativity
X >= 0; Y >= 0.
The feasible domain is shown in the plot below.
Plot 4*X + 2*Y = 24 (red); 2*X + 0.5*Y = 10 (green); and Y = 4X (blue)
It is the quadrilateral in QI, adjacent to x-axis and bounded by the blue, red and green lines.
It has vertices
P1 = (2,8)
P2 = (4,4) and
P3 = (5,0) (the green line x-intercept).
According to the Linear Programming method, we should calculate and compare the values of the profit function at these three points
at P1: P(2,8) = 800*2 + 500*8 = 5560 dollars;
at P2: P(4,4) = 800*4 + 500*4 = 5200 dollars.
at P3: P(5,0) = 800*5 + 500*0 = 4000 dollars.
The maximum value of the profit function is at P1.
It is the optimal solution.
Answer. Optimal solution to the problem is 4 magazine covers and 8 logos.
It satisfies the restrictions and gives maximal profit of 5560 dollars.
Solved.
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To see other similar problems solved by the Linear Programming method, look into the lesson
- Solving minimax problems by the Linear Programming method
in this site.
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
Tutor @ikleyn has provided a thorough solution along with a clear explanation of how to solve the problem using the standard linear programming method.
However, you can refine the standard method to allow you to reach the final answer with less work.
The three constraint lines determine the boundaries of the feasibility region. The maximum value of the objective function has to occur at a corner of the feasibility region, or perhaps at any point along one edge of the feasibility region.
The standard solution process says you should evaluate the objective function at all corners of the feasibility region to determine the maximum value of the objective function. In fact, that is not true.
Furthermore, you don't even have to find all of the intersection points of the constraint lines; you can tell where the maximum value of the objective function is going to be by comparing the slope of the objective function to the slopes of the three constraint lines.
The slopes of the three constraint lines in this problem are 4, -2, and -4; the slope of the objective function is -8/5. Since -8/5 is between 4 and -2, the maximum value of the objective function is going to be where the constraint lines with slopes 4 and -2 intersect.
So after you have the slopes of the three constraint lines and the objective function, you only need to find one corner of the feasibility region and evaluate the objective function at that corner.
To understand why this shortcut works, look at the graph in tutor @ikleyn's response and imagine various lines with the slope of the objective function, -8/5. You want a line with slope -8/5 that just touches a corner of the feasibility region; that will be at the corner where the slope of -8/5 is between the slopes of the two intersecting constraint lines.
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