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Q: offset lithography of different versions ( Answered ,   0 Comments )
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 Subject: offset lithography of different versions Category: Business and Money Asked by: norman9-ga List Price: \$25.00 Posted: 30 Dec 2004 15:31 PST Expires: 29 Jan 2005 15:31 PST Question ID: 449428
 ```An offset lithographer has to print differnt verions 20 thousand, 74 thousand, 145 thousand, 148 thousand, 278 thousand and 187 thousand pieces... He puts these six versions onto three form. He can put 12 items on a form. How many of each version does he put on each form and how many of each form does he print to make the exact count? Can you create a general solution?``` Request for Question Clarification by joey-ga on 30 Dec 2004 17:09 PST ```Hi there. Do you think you may be able to better explain the problem? Before I can figure out a mathematical solution, it would help to better understand what you mean by: * the numbers for each version (is this the # printed, or an item number?) * "form" (what is this?)``` Clarification of Question by norman9-ga on 31 Dec 2004 09:46 PST ```Let's say a lithographer can print 12 letterheads on a single sheet. Let's say his customer has 6 offices he wants various quantities of letterheads for. Let's say the lithographer does not want 6 different forms, but would like to reduce his setup costs by printing on 3 forms. How many sheets of each form would he print and how many up of each type of letterhead would he impose on each form? This is a really tough problem, at least for me it is.``` Request for Question Clarification by joey-ga on 31 Dec 2004 15:57 PST ```Do you want this for practical usage or for academic reasons? I used the Excel solver function to find a possible solution (there are a number of possible "reasonable" solutions). If this is for practical purposes, I can provide you with the solution it suggests. It gets some of the quantities exactly equal to those required for each office, but some are over by a few thousand. It has managed to do this while keeping the total number of 12-letterhead sheets printed to no more than 73,000. Please let me know if you want this practical result. As far as an academically "true" solution, I don't think there is a clear method for an exact answer. Any solution will involve a significant number of iterations through an extended process to narrow down options. This is what Excel did, and it took it several thousand iterations. --Joey``` Clarification of Question by norman9-ga on 02 Jan 2005 03:26 PST ```This is a business problem. I make envelopes for a living and we have a 28 x 40 5 color offset press. I do need exact or best answers as I can come up with "good" answers right now by fooling around with a spreadsheet.```
 ```As I mentioned above, this sort of "optimization" problem can't be reduced to a simple general solution -- instead, they can only be solved by multiple iterations through a solving process (a "heuristic"). Luckily computers make this relatively simple and quick. In this case, the "optimal" solution appears to be the following: Assign the six items (offices) letters A-F with the following requirements: A: 20,000 B: 74,000 C: 145,000 D: 148,000 E: 278,000 F: 187,000 The forms should assign the following number of spots to each of the items: Form 1: PRINT 27,177 copies C (1 spot) E (7 spots) F (4 spots) Form 2: PRINT 37,000 copies B (2 spots) C (3 spots) D (4 spots) E (2 spots) F (1 spot) Form 3: PRINT 6,882 copies A (3 spots) C (1 spot) E (2 spots) F (6 spots) This results in the following number of each item provided: A: 20646 (over by 646) B: 74000 C: 145059 (over by 59) D: 148000 E: 278003 (over by 3) F: 187000 It requires a total number of 71,059 sheets to be printed (exceeding the theoretical minimum of 71,000 sheets by only 59). ------ To achieve this result, I downloaded Lindo's free-trial What'sBest add-in for Excel. It works like Excel's Solver function to solve this type of problem, but is far faster and gives better results. I set up the spreadsheet as necessary and then let it run for about 40 minutes (in the background while I did other things), and it came to this solution. After about 5 minutes, it had come up with a good solution that required 71,300 sheets, but over the next 35 minutes it found ways to reduce that to the optimal values shown above. This trial version is free and will work forever, but it is limited to 30 integer variables at a time. This should suit you well, as it only took 21 integer variables to solve this problem. It can be downloaded from: http://www.lindo.com/cgi/frameset.cgi?leftwb.html;wbf.html ------ The spreadsheet I created and used with the What'sBest add-in can be found at the following URL: http://69.61.18.4/temp/norman-9/lithographers_solution.xls It will be available until Wednesday, January 5th at 5:00pm EST. Please download it before then if you would like it. If you need some assistance setting up a spreadsheet for use with the What'sBest software, please let me know, and I can walk you through how to do it with your next printing problem. If you have any questions or anything is unclear, please let me know. --Joey Strategy: My personal knowledge of Excel and optimization strategies Google: excel advanced solver freeware``` Request for Answer Clarification by norman9-ga on 06 Jan 2005 09:37 PST ```Joey Sorry about the delay in responding. The 100% answer is: 20 1 0 0 74 2 0 2 145 3 1 3 148 4 2 0 278 2 4 6 187 0 5 1 20 34 17 Your answer is very good, though. I tried Lindo/What's Best over ten years ago, but could not figure it out. It is now Thursday, so I may not be able to download your solution. I'll contact you one way or the other. Thanks, Norman``` Clarification of Answer by joey-ga on 06 Jan 2005 21:49 PST ```I'm glad I could help, though I'm sorry I didn't hit on the true optimal solution (I guess this was more of a test to see how it could be found). If I'd let the What'sBest add-in run longer, I imagine it eventually would have found it, but after forty minutes it wasn't improving much -- it just hadn't iterated to that point yet. The spreadsheet I created is still up if you haven't gotten it -- I'll leave it for a few days. And please let me know if you have trouble figuring out how to use the Lindo software -- I can walk you through it. Take care. --Joey```
 norman9-ga rated this answer: ```Joey solved a problem I've been screwing around with for about 20 years. Thank you very much.```