Calculus Optimization

[Mitch Excel]

I'm very new to calculus so please excuse me if this doesn't quite make sense but I would appreciate if you could help me with this problem.

Inputs
Variable quantity of rectangles (Range 1-50)
Variable width on each rectangle (Range 1-8)
Variable height on each rectangle (Range 1-30)

Desired Output
One larger rectangle that can hold all the input rectangles and meets the following criteria:
1. Output rectangle must have Width of 8, height can be infinite.
2. Width and height of input rectangles must coincide with width and height of output rectangle. They cannot be rotated.
Formula should find the smallest possible output rectangle that meets the two criteria above. (Basically solve for smallest possible height, since width is set at 8.)

Thanks so much for your help!

 

[Dr.Peterson]

This sounds more like a computer programming problem than calculus. It definitely will not be solved by a mere formula.

What is the context? (That is, why are you asking?)

 

[Mitch Excel]

Its ultimately going to be used to determine staffing levels for a factory. I was trying to keep it simple with the rectangle illustration but here's the real world translation:
Quantity of rectangles = Number of lines that are running
Width of rectangle = Hours that individual line is running
Height of rectangle = Staff needed to run the line

Heres some more background on it:

Complex Staffing Formula

I'm putting together a spreadsheet that determines how much staffing we need to bring in. Row 5 contains permanent data showing how many people are needed to run a certain line. Row 7 and below show how many hours the line will be running each shift. I'm using the following formula to determine...

www.mrexcel.com

Its frustrating because I can solve each instance manually but cant figure out how to automate that thought process into a formula.

 

[Mitch Excel]

Here's a picture to illustrate, Line 1-4 hours and staffing are the inputs. Goal is to minimize total staffing numbers.

 

[Dr.Peterson]

You're looking for a packing algorithm. Wikipedia refers to this page, which sounds like almost what you want, except that the container width is not fixed.

 

[Mitch Excel]

Perfect, that's just what I'm looking for! Thanks for the time and references : )