A paper mill received three orders: one for 100ft of 5ft wide paper, one for 300

Question

A paper mill received three orders: one for 100ft of 5ft wide paper, one for 300ft of 7ft wide paper and one for 200ft of 9ft wide paper. The orders will be filled by cutting from either a 10ft or a 20ft very long roll of paper using some combination of the patterns listed below:

The patterns having a '1' as their first digit (e.g. x12) are cut from 10ft wide rolls. The patterns having 2 as their first digit are cut from 20ft wide rolls. Set up and run the LP to minimize trim waste. For each size required (5,7, and 9 feet) compute how man more feet were cut than needed.

Pattern Designation 5' 7' 9' x11 2 0 0 x12 0 1 0 x13 0 0 1 x21 4 0 0 x22 2 1 0 x23 2 0 1 x24 1 2 0 x25 0 1 1 x26 0 0 2

Explanation / Answer

total no. of

5ft wide rolls = 100 = 2x11+4x21+2x22+2x23+x24

7''                 = 300 = x12+x22+2x24+x25

9''                 = 200 = x13+x23+x25+2x26

total wastage = 3x12+x13+3x22+x23+x24+4x25+2x26 = W

We have to minimize wastage given the above constraints. Solve using simplex method on page

http://www.zweigmedia.com/RealWorld/simplex.html

Type

Minimize p = 3b+c+3e+f+g+4h+2i subject to
c+f+h+i=200
b+e+2g+h=300
2a+4d+2e+2f+g=100
a>=0
b>=0
c>=0
d>=0
e>=0
f>=0
g>=0
h>=0
i>=0

to get W min = 600 , x12=100, x13=200, x24=100

ie cut 10 ft roll into 7 ft roll - W=3*100=300 to obtain 100 7ft rolls

(200 more 7ft left, 100 more 5ft left, 200 more 9ft left)

cut 10 ft roll into 9 ft roll - W=200*1=200 to obtain 200 9 ft rolls

(9ft done, 200 more 7ft left, 100 more 5ft left)

cut 20 ft roll into one 5 ft and two 7 ft - W=100*1=100 to obtain 100 5ft and 200 7ft

(everything done)

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