Question

The Outhouse Plumbing Company sells commercial plumbing pipe in lengths of 4 feet, 8 feet, and 15 feet.  Their supplier can only ship pipes that are 30 feet long.  Outhouse needs to determine how to cut the 30-foot pipes to meet the customer demand given below for the various pipe lengths.  The lean manager wants the pipes to be cut so that the total remaining unusable pipe (waste after cutting) is minimized.  Determine the patterns and cutting plan for Outhouse.  Customer Demand for the customers' pipes are 40, 25, and 13.  (Hint: there are seven unique cutting patterns.)

Set up and solve the problem to minimize the unusable waste for the company.

The optimal objective function value should be 26 feet of unusable pipe.

Answer 1

The pipes can be cut in the following patterns:

In total, the pipe can be cut into the following seven patterns (the loss is minimized)

S. No.	Cut into	Number of pipes with pattern	Loss
1	4,4,4,4,4,4,4	x1	2
2	8,8,8	x2	6
3	15,15	x3	0
4	8,8,4,4,4	x4	2
5	15,4,4,4	x5	3
6	15,8	x6	7
7	15,8,4	x7	3

Say x1, x2,... x7 be the number of pipes that are cut with the patterns 1, 2, 3 ... 7 respectively. Now we can make the following linear equations:

(A) Demand for Pipe of length 4:

7*x1 + 3*x4 + 3*x5 + 1*x7 >= 40

(B) Demand for pipe of length 8:

3*x2 + 2*x4 + 1*x6 + 1*x7 >= 25

(C) Demand for pipe of length 15:

2*x3 + 1*x5 + 1*x6 + 1*x7 >= 13

Function to be minimized:

Loss = Z = 2*x1 + 6*x2 + 2*x4 + 3*x5 + 7*x6 + 3*x7 = 26

x1, x2 ... x7 >= 0

Let's input the same in Solver:

Now let's run the Solver, go to Data Analysis -> Solver:

Click on OK, the results are:

Hence, the lines are to be cut in the following patterns:

Pattern	Number of pipes to be cut
4,4,4,4,4,4,4,	1
15,15	7
8,8,4,4,4	12