=1000

*Due October 1, 1996*

**Aluminum 'R Us**

**INTER-OFFICE CORRESPONDENCE**

I've decided to give you a chance to prove your ideas about replacing the retiring Joe Beenherelong with a computer system. As you know, we have relied on Joe for thirty years to plan the cutting patterns that divide aluminum profiles. I find it hard to believe that you can do a better job than Joe, but I'll give you a chance.

Attached is a description of the problem and a sample of the problems Joe solves. I would like a report back from you in two weeks giving the following:

- An outline of your so-called ``variable generation'' approach to this problem. The most important difficulty seems to be in handling some of the more idiosyncratic constraints that Joe faces, so be sure the talk about them. In particular, any constraints that you cannot handle, or can handle in only a limited way are very important to identify.
- A comparison of the results of your method versus Joe's result on the sample problem. I know that there is not time to get a fully automated system, so some portions of your method should be done by hand.
- A description of how your method can help in answering some of Sale's questions regarding costs.
- A brief description of the hardware you believe will be necessary to run problems like the sample problem and the time required to get good solutions.

**Aluminum 'R Us**

**INTER-OFFICE CORRESPONDENCE**

T. Boss asked me to describe the profile division problem to you. I don't know why I need to waste my time when I have work to do, but here goes.

As you know, we manufacture aluminum profiles, which is a fancy word for a long stick of aluminum. We then divide the profile into smaller pieces to meet customer orders. We have a number of machines that do this division. It is the scheduling of these machines that is the hard part. Due to limitations on the type of profile each machine can handle, however, I generally need only solve single machine problems.

I generally do a production plan once per two week period for each machine. I generally know the following pieces of information:

- The lengths of the profiles that I can choose from. Most types of profile can be made in two to four lengths, though up to ten is possible.
- Production limitations on each profile length over this planning period. Most profiles have no limit, but some do.
- The cost of using each profile.
- Order quantities required for this production period. For any type of aluminum, there will be from three to thirty required lengths. You get these lengths by cutting up profiles. For each length, you have a quantity required.
- Limitations on the number of pieces that can come from each
profile. Due to technological reasons, it is not possible to divide a
profile arbitrarily. Normally, there is a limit on the number of
pieces that can come out of a profile. For instance, on a 600 cm long
profile on our X3000 machines, no more than 8 pieces can be cut. If
you cut 30 cm pieces, then you can get no more than 8, even though the
profile is plenty long. This limit does not include any scrap or
wasteage: it is a limit on usable pieces.

That's about it. The problem is working it all together. Here is a problem that I know you won't do better than me on.

Profiles: 1000 cm and 600 cm

Limitations: No more than 50,000 of the 600 cm profiles; unlimited on the 1000 cm profiles.

Cost: $1.10 per 600 cm profile; $1.90 per 1000 cm profile.

Orders:

Limitations on pieces: No more than 8 pieces from each profile. Even a pattern with no waste can have no more than 8 usable pieces.

I don't mind telling you that my production plan cost only $200,000. I won't tell you how I got that because otherwise you would cheat me. See if you can get anywhere near that!

**Aluminum 'R Us**

**INTER-OFFICE CORRESPONDENCE**

Glad to see that y'all are working to replace Joe Beenherelong. That guy was absolutely useless to us. Maybe he could do some planning (though some say his last good year was in 1971).

One problem we have always had is in estimating a price for the customers. Normally what we do is to take the longest applicable profile, determine the fraction of a profile a single piece takes and then multiply by the number of pieces and the cost of a profile to get the total cost of production. We then apply all sorts of other things. We would like to know if this is at all accurate. I know Joe gave you a sample problem. Even knowing whether we do fine for this sample would be good.

**Aluminum 'R Us**

**INTER-OFFICE CORRESPONDENCE**

My numbers were all messed up last month. I think it was because Joe messed up on a critical decision (would not be the first time!). On the problem that he sent to you, we had a chance to buy a lot more of the 600 cm profiles at a cost just a nickle more than normal. Silly guy said he had plenty and so we didn't buy. Can you help us on this?

Tue Sep 24 10:39:24 EDT 1996