Table A
                                      Operations
Items	Std Setup Times   Op A      Op B        Op C    Op D    Op E
                          MacA1	 MacB1 MacB2    MacC1   MacD1   MacE1
Item X	20 Mins            100	  50	75	 75	 150	 200
Item Y	50 Min	            75	  75	85	 X	 150	 100
Item Z	120 Mins	    50	 100    105	 125	  X	  50

Legend,
Mac: Machine
X:   Operation not applicable
Op:  Operation
Std: Standard
----------------------------------------------------

Table B
Weekly production targets(Completed Items)
Items	Week 1	Week 2	Week 3	Week 4
Item X	500	700	300	200
Item Y	400	300	600	300
Item Z	200	200	200	400

-------------------------------------------------

Our problem is related to the auto parts manufacturing industry where
different auto parts are manufactured through the application of
different processes on a single piece of raw material.

The items being produced may be different in form but many of them go
through the same processes, hence different parts may be in contention
for the same operation at a time. Different operations may have more
than one machines with different efficiency levels (New Machines may
have better production capacity per hour whereas old ones may have
less, Table A). There is also a setup time for each item for each
machine i.e if a machine has to switch over to producing a different
item, there may be a setup time of say 20 minutes to 2 hours.

Items are completed only when all operations have been performed. Not
all items require all operations (Refer to table A)

Lets assume that our daily production shift duration is of 8 hours. We
have to find the optimal production schedule, operation and machine
wise to achieve our targets. (Table B),
We require the complete LP Model including the objective function and
the constraints along with the solution as soon

as possible. 

Please contact incase of further verifications. Thanks.

---

Clarification of Question by wasiq-ga on 26 Aug 2004 11:11 PDT

Thanks for your interest in this problem. I shall try to clarify all
of the issues that you have mentioned in your comment in the order
they were posted.

1. Normally the factory works in two shifts per day, but during the
peak production days, the factory may be required to do a third shift
as well. At this moment we shall consider 2 shifts only.

2. Again, the working week is of 5 days, but on peak production, the
factory may be required to operate all seven days. But right now we
are assuming a 5 day week.

3. Machine operation duration is in minutes beacause of the fact that
even the smallest of the operation takes more than a minute. Even your
consideration of seconds as the duration would pose no problems as I
can convert that on my own, but please be careful that whatever unit
of duration you choose should be consistent throughtout the solution
of the problem.

Units can be items processed per hour. Again I can convert that to
other measurements on my own if the need arises.

4. There is no travel time between the machines, they are located
pretty close to each other.

5. Yes, initial start state is a clear line.

6. I really did'nt get your question this time around. What do you
mean by initial configuration of machines?

7. Yes, it can be either, depending on the availability. As a rule of
thumb, if all machines for an operation are available then the most
efficient one should be used.

8. Yes, all machines for an operation can be used at the same time.

9. Yes, its OK to complete a shift mid cycle with no down time.

10. No there is no limit to the number of items that can queued. This
is actually a major concern for me as some process being slow becomes
the bottleneck operation, therefore the number of completed items
cannot be greater than the bottlneck operation.

Thanks again for your interest. I am available for for queries.

Regards
Wasiq

---

Clarification of Question by wasiq-ga on 03 Sep 2004 05:08 PDT

Hi manuka-ga 

I think that you've misunderstood the data presented in Table-A. There
is Standard Setup Times columns which states the Minutes required to
setup a machine to produce a particular item i.e when switching
between one item and another.

The other columns contain the Operations and their machines. For
example In   OP-B, there are two machines i.e MacB1 and MacB2. The
value 50 for MacB1 and 75 for MacB2 represent the number of operations
that can be performed for a particular item and these do not represent
Minute values!!

I hope that answers your concern. There are a $100 up for grabs for
anyone who solves this problem whether or not he is a researcher or
not!

Thanks 
Wasiq

This looks like an interesting problem that I'd like to have a go at
but please clarify:

1. Are you assuming continuous 24 hour per day operation. i.e. 3 x 8 hour shifts?
2. Is your work week 5 days or 7 days? How many 8 hour shifts = 1 week?
3.Are the machine operation durations in minutes or seconds? (I assume seconds)
4. I assume no travel time for items between the various machines in
the production line.
5. Assume initial start state is no items part assembled, i.e. a clear line?
6. Assume initial configuration of machines is for the item Z?

I'm thinking of setting this up in Excel and using the Solver function
with constraints as a possible route to a solution. Please note, I'm
not a researcher so this is just of academic interest to me!

Some more clarifications!

7. For operation B can it be either Machine B1 OR Machine B2 that is in use?
8. Can both Operation B machines be in use at the same time?
9. Assume that it's ok to complete the shift mid cycle on an operation
and start the next shift at the same point with no down time?

thanks.

3. Units are items processed per hour?

sorry, one final clarification:

10. Is there a limit of the number of items that can be queuing at
each machine? e.g. for item Z, machine C1 can release finished items
to machine E1 even though machine E1 is busy with previous item?
Another way of looking at this would be to say that each machine has a
storage bin on it's input side waiting to be processed from the
previous machine?

I'm on UK time so I won't get back to these messages until tomorrow.

regards

It looks to me like you have some very serious scale issues here. If I
am interpreting the question correctly, in Week 1 you want to produce
500 X, 400 Y and 200 Z, requiring a minimum of (500*100) + (400*75) +
(200*50) = 90,000 minutes on Machine A alone. This equates to 1500
hours, i.e. 62.5 24-hour days or 93.75 2-shift days, which is a little
hard to cram into one week! Or do you have *lots* of machines?