Model Construction

[mitchfel]

Maria Eagle is a Native American artisan. She works part time making bowls and mugs by hand from special pottery clay and then sells her items to the Beaver Creek Pottery Company, a Native American crafts guild. She has 60 hours available each month to make bowls and mugs, and it takes her 12 hours to make a bowl and 15 hours to make a mug. She uses 9 pounds of special clay to make a bowl, and she needs 5 pounds to make a mug; Maria has 30 pounds of clay available each month. She makes a profit of $300 for each bowl she delivers, and she makes $250 for each mug. Determine all the possible combinations of bowls and mugs Maria can make each month, given her limited resources, and select the most profitable combination of bowls and mugs Maria should make each month. Find the solution.

 

[skeeter]

[MATH]b[/MATH] = number of bowls
[MATH]m[/MATH] = number of mugs

She has 60 hours available each month to make bowls and mugs, and it takes her 12 hours to make a bowl and 15 hours to make a mug.

[MATH]12b+15m = 60[/MATH]

She uses 9 pounds of special clay to make a bowl, and she needs 5 pounds to make a mug; Maria has 30 pounds of clay available each month.

[MATH]9b+5m = 30[/MATH]

She makes a profit of $300 for each bowl she delivers, and she makes $250 for each mug.

let [MATH]P[/MATH] = profit in dollars

[MATH]P = 300b + 250m[/MATH]

I'm assuming you've studied something about linear optimization ... what now?

 

[HallsofIvy]

I would have said that 12b+ 15m ≤ 60 and 9b+5m ≤ 30 but since the maximum of a linear function will be on the maxima, $12b+ 15m= 60$ and $9b+5m= 30$ are good.

Mitchfel, the basic "rule" of linear programming is that in a region bounded by straight lines, a linear function will take on maximum or minimum values at a "corner" or "vertex' of the region. The lines $12b+ 15m= 60$ and $9m+ 5m=30$ intersect in a single point. What is that point?

 

[JeffM]

I might approach this problem differently because it is an integer programming problem rather than a linear programming problem.

What is the maximum number of mugs that can be made?

What is the maximum number of bowls that can be made?

So what possible combinations of mugs and bowls can be made subject to the following constraints?

[MATH]b \text { and } m \text { must be non-negative integers.}[/MATH]
[MATH]0 \le 12b + 15m \le 60.[/MATH]
[MATH]0 \le 9b+ 5m \le 30.[/MATH]

 

[Deleted member 4993]

[MATH]b[/MATH] = number of bowls
[MATH]m[/MATH] = number of mugs
[MATH]12b+15m = 60[/MATH][MATH]9b+5m = 30[/MATH]let [MATH]P[/MATH] = profit in dollars
[MATH]P = 300b + 250m[/MATH]I'm assuming you've studied something about linear optimization ... what now?

I would change some of those "=" signs as follows:

12b+15m=60

12b+15m ≤ 60

9b+5m=30

9b+5m ≤ 30 and

b ≥ 0 and m≥0