Rosebud said:
Homework Statement
A small purse manufacturer has a single machine that makes the metal parts of a purse. This takes 2 minutes. Another single machine makes the cloth parts in 3 minutes. Then it takes a worker 4 minutes to sew the cloth and metal parts together. How long will it take to make 6 purses?
Homework Equations
The solution in the back of the book states that it will take 4.5 minutes to make each purse but I have no idea how they got this value. So, it will obviously take 6 x 4.5 = 27 minutes to make 6 purses. How did they get a time of 4.5 minutes?
The Attempt at a Solution
[/B]
Task Time Preceding Task
A | 2 | None
B | 3 | None
C | 4 | A, B
The following is incorrect but it is my attempt at solving the problem.
A to C = 2 + 4 = 6C
B to C = 3 + 4 = 7
So, B to C is the critical path which yields a time of 7 minutes.
6 x 7 = 42
So, it would take 42 minutes to make six purses.
This does not look anything like a small PERT/CPM problem to me, but it IS a large one.
Say we start producing at time t = 0. Let's measure all time in minutes, so we can dispense with units.
At time t = 2 machine A is done on part 1 and sends its output to the assembly stage C; but C waits for the first job to complete on machine B, which is at t = 3. So, at t = 3, C starts, and so completes part 1 at t = 3+4 = 7. Meanwhile, from t = 2 to t = 4, A is working on part 2, and from t = 3 to t = 6, B is working on part 2. Therefore, at t = 4, A sends its output to C, where it waits in a "metal" queue. At time t = 6, B sends its output to C where it, too, sits in a "cloth" queue. At t = 7, C withdraws one part from each queue and begins assembly of part 2, which completes at time t = 7 + 4 = 11. Meanwhile, from t = 4 to t = 6, A is working on part 3 and sends its output to C, which goes into the "metal parts" queue; since C is still working on part 1 then, there are two parts in the metal queue from t = 6 to t = 7. At t = 6, B starts working on part 3 and finishes at t = 9, etc, etc.
You see ... it gets complicated and keeping track of things can be tricky. In problems of this type it is often recommended that you make use of a so-called Gantt Chart; see, eg.,
https://en.wikipedia.org/wiki/Gantt_chart .
However, if you prefer you CAN represent the problem as a (large) PERT network, with tasks A1, A2, A3, ..., A6 (metal parts), B1, B2, ..., B6 (cloth) and C1, C2, ...,C6 (assembly). The precedence constraint are A1 ← A2 ← A3 ← ⋅ ⋅ ⋅ ← A6, B1 ← B2 ← ⋅ ⋅ ⋅ ← B6, and (A1, B1) ← C1, (A2, B2) ← C2, ⋅ ⋅ ⋅, (A6, B6) ← C6. If you apply standard CPM methodology, the time you want is the project completion time, when C6 is finished. Applying the standard Critical Path Method should not be too hard, but it will be tedious and possible error-prone. You can do it in Excel or a package such a Matlab, Maple or Mathematica. I did it in Maple and got the book's answer of 27 minutes. Of course, 27/6 = 4.5 is the average time per purse, but I'm not sure that means very much. (For example, the average time per purse for 100 purses would be 4.03, and for 1000 purses would be 4.003, etc.)
