Optimization Problem Help ASAP... Thanks - XP Math - Forums

 XP Math - Forums Optimization Problem Help ASAP... Thanks

 10-17-2011 #1 franksurvived   Join Date: Oct 2011 Posts: 1 Optimization Problem Help ASAP... Thanks Calculus Optimization Help ASAP... Thanks in advance...? The state has created new legislation about “crunch-*‐time”, and as such requires us to adjust our projects for the next quarter. So for the next 13 weeks we will assume that there will be no scheduled overtime. Our adjusted budget for the quarter gives us 520 man-*‐hours per employee. If we have to schedule more hours, we will have to cut into the profits, which we don’t want to do. So, with these new adjustments here are our projected needs, based on previous projects: Note: To convert from man-*‐hours to number of employees, divide by 520 (which is the number of full time hours in the quarter) Problem 1 To create a console game, we will assume 10,920 man-*‐hours of development, 13,000 man-*‐hours on art, 3,120 man-*‐hours for design, and 2,080 man-*‐hours for production management. The profit margin for a console game is 80% higher; that is, about \$1.8 million for a console title and \$1 million for a handheld title. The work requirements for a handheld game quite a bit different: 7,280 man-*‐hours in development, 2,600 man-*‐hours art, 9,360 man-*‐hours in design, and 2,600 man-*‐hours in production management. Our current staff consists of 238 programmers, 225 artists, 180 designers, and 57 production managers. Figure out how many console and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded. Note: If a console game makes 1.8x more profit than a handheld game, a handheld game makes 1x profit. Problem 2 We have decided to expand and create a new PC games department. Out projection indicate a PC game makes 40% more profit; that is, about \$1.4 million in profit. The work requirements are 9,360 man-*‐hours for development, 8,840 man-*‐hours on artwork, 5,720 man-*‐hours for design, and 1,560 man-*‐ hours for production management. To help staff this department, we hired 44 more programmers for the development team, 58 more artists for the art team, and 2 more managers for the management team. Figure out how many console, PC, and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded. Thanks I although need to see the work being laid out so I can understand I'm having such a hard time with these two problems. That even includes Optimization as a whole thanks for your input and answers. 30 minutes ago - 4 days left to answer.
10-18-2011   #2
MAS1

Join Date: Dec 2008
Posts: 249

Quote:
 Originally Posted by franksurvived Calculus Optimization Help ASAP... Thanks in advance...? The state has created new legislation about “crunch-*‐time”, and as such requires us to adjust our projects for the next quarter. So for the next 13 weeks we will assume that there will be no scheduled overtime. Our adjusted budget for the quarter gives us 520 man-*‐hours per employee. If we have to schedule more hours, we will have to cut into the profits, which we don’t want to do. So, with these new adjustments here are our projected needs, based on previous projects: Note: To convert from man-*‐hours to number of employees, divide by 520 (which is the number of full time hours in the quarter) Problem 1 To create a console game, we will assume 10,920 man-*‐hours of development, 13,000 man-*‐hours on art, 3,120 man-*‐hours for design, and 2,080 man-*‐hours for production management. The profit margin for a console game is 80% higher; that is, about \$1.8 million for a console title and \$1 million for a handheld title. The work requirements for a handheld game quite a bit different: 7,280 man-*‐hours in development, 2,600 man-*‐hours art, 9,360 man-*‐hours in design, and 2,600 man-*‐hours in production management. Our current staff consists of 238 programmers, 225 artists, 180 designers, and 57 production managers. Figure out how many console and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded. Note: If a console game makes 1.8x more profit than a handheld game, a handheld game makes 1x profit. Problem 2 We have decided to expand and create a new PC games department. Out projection indicate a PC game makes 40% more profit; that is, about \$1.4 million in profit. The work requirements are 9,360 man-*‐hours for development, 8,840 man-*‐hours on artwork, 5,720 man-*‐hours for design, and 1,560 man-*‐ hours for production management. To help staff this department, we hired 44 more programmers for the development team, 58 more artists for the art team, and 2 more managers for the management team. Figure out how many console, PC, and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded. Thanks I although need to see the work being laid out so I can understand I'm having such a hard time with these two problems. That even includes Optimization as a whole thanks for your input and answers. 30 minutes ago - 4 days left to answer.
I'm not sure that you need to use calculus to solve these problems. They looked more like linear programming problems to me, so I used linear programming to solve problem 1.

d = development
a = art
s = design
p = production

Console (c= number of consoles)
----------------
d: 10920/520 = 21 people
a: 13000/520 = 25 people
s: 3120/520 = 6 people
p: 2080/520 = 4 people

Handheld (h = number of handhelds)
----------------------
d: 7280/520 = 14 people
a: 2600/520 = 5 people
s: 9360/520 = 18 people
p: 2600/520 = 5 people

So our limits are given by:

Eq. 1: 21c + 14h <= 238
Eq. 2: 25c + 5h <= 225
Eq. 3: 6c + 18h <= 180
Eq. 4: 4c + 5h <= 57

Since each of these are linear equations I assumed c was the x-axis and h was the y-axis.

For each equation I found the x and y intercepts by setting c equal to 0 and solving for h, then setting h equal to 0 and solving for c.

c | h
-----------------
Eq. 1: 0 | 17
Eq. 1: 34/3 | 0
Eq. 2: 0 | 45
Eq. 2: 9 | 0
Eq. 3: 0 | 10
Eq. 3: 30 | 0
Eq. 4: 0 | 11.4
Eq. 4: 14.25 | 0

I used the intercepts to find the slope of the line for each equation.

m1 = (17 - 0)/(0 - 34/3) = -1.5
m2 = (45 - 0)/(0 - 9) = -5
m3 = (10 - 0)/(0 - 30) = -1/3
m4 = (11.4 - 0)/(0 - 14.25) = -0.8

Rewriting the equations gives:
Eq. 1: h = -1.5c + 17
Eq. 2: h = -5c + 45
Eq. 3: h = -c/3 + 10
Eq. 4: h = -0.8c + 11.4

I then graphed them to see where they intersected each other and the x and y axes. There were 4 points which are easy to see if you graph them.

Point1: (0,10)
Point2: (3,9)
Point3: (8,5)
Point4: (9,0)

Three of the lines (1, 2, and 4) intersect at (8,5).

Then plug in the values for c and h into the equation

Profit = 1.8c + 1h

And pick the largest to find the maximum profit.

Profit1 = 1.8(0) + 10 = 10 million
Profit2 = 1.8(3) + 9 = 14.4 million
Profit3 = 1.8(8) + 5 = 19.4 million
Profit4 = 1.8(9) + 0 = 16.2 million

So the max profit occurs when 8 consoles and 5 handhelds are produced

 06-17-2012 #3 HelpxMe   Join Date: Jun 2012 Posts: 1 HELP!! Homework due tomorrow night!! Hey all, I need help on this last question.. Its due tomorrow at 11:59pm and I don't understand how to complete this problem!! HELP PLEASE.. Problem 2 We have decided to expand and create a new PC games department. Out projection indicate a PC game makes 40% more profit; that is, about \$1.4 million in profit. The work requirements are 9,360 man-*‐hours for development, 8,840 man-*‐hours on artwork, 5,720 man-*‐hours for design, and 1,560 man-*‐ hours for production management. To help staff this department, we hired 44 more programmers for the development team, 58 more artists for the art team, and 2 more managers for the management team. Figure out how many console, PC, and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded.
06-21-2012   #4
MAS1

Join Date: Dec 2008
Posts: 249

Quote:
 Originally Posted by HelpxMe Hey all, I need help on this last question.. Its due tomorrow at 11:59pm and I don't understand how to complete this problem!! HELP PLEASE.. Problem 2 We have decided to expand and create a new PC games department. Out projection indicate a PC game makes 40% more profit; that is, about \$1.4 million in profit. The work requirements are 9,360 man-*‐hours for development, 8,840 man-*‐hours on artwork, 5,720 man-*‐hours for design, and 1,560 man-*‐ hours for production management. To help staff this department, we hired 44 more programmers for the development team, 58 more artists for the art team, and 2 more managers for the management team. Figure out how many console, PC, and handheld games can be made this quarter to maximize our profit. In addition, report what pools (development, artists, designers, and managers) have some unutilized employees, and which pools need to be expanded.
I used some of my answer to problem 1 for this problem.

d = development
a = art
s = design
p = production

Console (c= number of consoles)
----------------
d: 10920/520 = 21 people
a: 13000/520 = 25 people
s: 3120/520 = 6 people
p: 2080/520 = 4 people

Handheld (h = number of handhelds)
----------------------
d: 7280/520 = 14 people
a: 2600/520 = 5 people
s: 9360/520 = 18 people
p: 2600/520 = 5 people

PC (p = number of PCs)
----------------------
d: 9360/520 = 18 people
a: 8840/520 = 17 people
s: 5720/520 = 11 people
p: 1560/520 = 3 people

238 + 44 = 282 programmers for development
225 + 58 = 283 artists
180 designers
57 + 2 = 59 managers for production

So our limits are given by:

Eq. 1: 21c + 14h + 18p <= 282
Eq. 2: 25c + 5h + 17p <= 283
Eq. 3: 6c + 18h + 11p <= 180
Eq. 4: 4c + 5h + 3p <= 59

Now we have 4 equations with three variables. We have to use combinations of 4 equations taken three at a time (4 combinations) to find the vertices. Using an online matrix calculator to solve systems of 3 simultaneous equations gives:

Eq.1, Eq. 2, Eq. 3: c = -29.835, h = -20.774, p = 66.632
Eq.1, Eq. 2, Eq. 4: c = 8, h = 3, p = 4
Eq.1, Eq. 3, Eq. 4: c = 4.048, h = 3.741, p = 8.034
Eq.2, Eq. 3, Eq. 4: c = 4.092, h = 2.609, p = 9.862

We can ignore the results of Eq.1, 2, and 3 because we get negative values for c, and h.

Plugging the other results into our profit equation and looking for the maximum gives:

Profit = 1.8c + 1h + 1.4p

Since we cannot make fractions of a console, handheld, or pc game then round the values to give:

Profit = 1.8(8) + 1(3) + 1.4(4) = 23 million dollars
Profit = 1.8(4) + 1(3) + 1.4(8) = 21.4 million dollars
Profit = 1.8(4) + 1(2) + 1.4(9) = 21.8 million dollars

So max profit comes from making 8 consoles, 3 handhelds, and 4 pc games.

Last edited by MAS1; 06-29-2012 at 01:03 PM.. Reason: New idea for solving the problem

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