Xerox Reduction Problem A copy machine supports 7%, 10%, and 37% reductions. In 2 copies, how many different reductions can a user achieve? In 3 copies? Is there an easy formula for 5 copies? Your an engineer at the copy machine company. Sales wants to add a 20% button, Marketing wants to add a 3% button, and your vice president thinks a 70% button should be added. Are any of these buttons good for the environment? How can you determine which is best? ANSWER Create a table consisting of all the different combinations of 2 copies, don't forget that only making 1 copy is a subset of 2 copies. You can skip copies that are symmetric (7% then 10% is the same as 10% then 7%), but my listing doesn't because I generated it with a computer program. [ 0.070 ] => 0.07000 round -> 0.07 [ 0.100 ] => 0.10000 round -> 0.10 [ 0.370 ] => 0.37000 round -> 0.37 [ 0.070 0.070 ] => 0.00490 round -> 0.00 (ignore) [ 0.070 0.100 ] => 0.00700 round -> 0.01 [ 0.070 0.370 ] => 0.02590 round -> 0.03 [ 0.100 0.070 ] => 0.00700 round -> 0.01 dup! [ 0.100 0.100 ] => 0.01000 round -> 0.01 dup! [ 0.100 0.370 ] => 0.03700 round -> 0.04 [ 0.370 0.070 ] => 0.02590 round -> 0.03 dup! [ 0.370 0.100 ] => 0.03700 round -> 0.04 dup! [ 0.370 0.370 ] => 0.13690 round -> 0.14 I round to a whole percentage because I doubt that users could actually see a fraction of a percent difference. I don't think 0 should be counted as a reduction, because that is the same as not making a copy at all. There is a number of duplicates, so you only get 7 total reductions possible in 2 or less copies. I use the unrounded reductions from 2 copies above and calculate all the possiblilties after a 3rd copy. The users may not be able to tell the difference in fractional percents, but the optics in the copy machine probably can, which is why I choose to use the unrounded values from 'round 2'. [ 0.070 0.070 0.070 ] => 0.00034 round -> 0.00 [ 0.070 0.070 0.100 ] => 0.00049 round -> 0.00 [ 0.070 0.070 0.370 ] => 0.00181 round -> 0.00 [ 0.070 0.100 0.070 ] => 0.00049 round -> 0.00 [ 0.070 0.100 0.100 ] => 0.00070 round -> 0.00 [ 0.070 0.100 0.370 ] => 0.00259 round -> 0.00 [ 0.070 0.370 0.070 ] => 0.00181 round -> 0.00 [ 0.070 0.370 0.100 ] => 0.00259 round -> 0.00 [ 0.070 0.370 0.370 ] => 0.00958 round -> 0.01 [ 0.100 0.070 0.070 ] => 0.00049 round -> 0.00 [ 0.100 0.070 0.100 ] => 0.00070 round -> 0.00 [ 0.100 0.070 0.370 ] => 0.00259 round -> 0.00 [ 0.100 0.100 0.070 ] => 0.00070 round -> 0.00 [ 0.100 0.100 0.100 ] => 0.00100 round -> 0.00 [ 0.100 0.100 0.370 ] => 0.00370 round -> 0.00 [ 0.100 0.370 0.070 ] => 0.00259 round -> 0.00 [ 0.100 0.370 0.100 ] => 0.00370 round -> 0.00 [ 0.100 0.370 0.370 ] => 0.01369 round -> 0.01 [ 0.370 0.070 0.070 ] => 0.00181 round -> 0.00 [ 0.370 0.070 0.100 ] => 0.00259 round -> 0.00 [ 0.370 0.070 0.370 ] => 0.00958 round -> 0.01 [ 0.370 0.100 0.070 ] => 0.00259 round -> 0.00 [ 0.370 0.100 0.100 ] => 0.00370 round -> 0.00 [ 0.370 0.100 0.370 ] => 0.01369 round -> 0.01 [ 0.370 0.370 0.070 ] => 0.00958 round -> 0.01 [ 0.370 0.370 0.100 ] => 0.01369 round -> 0.01 [ 0.370 0.370 0.370 ] => 0.05065 round -> 0.05 A third stage of copying only gives you 1 more new reduction value, 0.05. No, there is no easy formula for determining all the possible reductions after 5 copies. In fact, the 2 and 3 copy calculations probably aren't alot of fun; this is why I'm using a computer! Of the buttons that your company wants to add to the xerox machine (20%, 3%, and 70%) I think the only one you can draw an immediate conclusion on is 3%. In three copies we never generated a 20% or 70% reduction, so although adding either of these buttons would (probably) increase the total number of reductions we can get (after 3 copies), they may not save a user from making an unneeded copy. However, we did generate a 3% reduction by doing 2 copies at 7% then 37%. By adding a 3% button we would definitely save one copy for anyone wanting to make a 3% reduction. So, 3% is certainly a good button for the environment, evaluation of 20% and 70% requires a deeper analysis. The real question is: "What is a good metric for determining 'good' for the environment?" Since every copy that you make is bad for the environment (uses paper + toner + electricity, none of which are 'environmentally' free), a good metric may be the average number of copies required per reduction. We can then judge 'good' by being the smallest average number of copies per reduction. Here is a table of the current copier reductions: Redux Sequence: [ 0.070 ] => 0.070000 Redux Sequence: [ 0.100 ] => 0.100000 Redux Sequence: [ 0.370 ] => 0.370000 Redux Sequence: [ 0.070 0.370 ] => 0.030000 Redux Sequence: [ 0.070 0.100 ] => 0.010000 Redux Sequence: [ 0.370 0.370 ] => 0.140000 Redux Sequence: [ 0.100 0.370 ] => 0.040000 Redux Sequence: [ 0.370 0.370 0.370 ] => 0.050000 The average number of copies required per reduction is then: [ 1(3) + 2(4) + 3(1) ] / 8 [ 3 + 8 + 3 ] / 8 14/8 1.750 copies/reduction By generating 3 similar calculations for a hypothetical copy machines with 4 buttons, then you can compare average number of copies per reduction for each candidate machine. The candidate machine with the smallest number could be the 'best' for the environment. WHEW! I've provided four files: CurrentButtons, SalesButtons, MarketingButtons, and VpButtons. These contain an analysis as I propose above (except that I calculate 4 COPIES). The answer to the question is still not straightforward. If we stick to our original metric, then the 20% button would win, but the 70% button has a 2.18 average (close to the others) and provides almost twice as many reduction values as the alternatives. Some good follow-up questions to this problem would be: Q: Should you have felt 'optimistic' about the 70% button before seeing the results of the analysis? A: Probably. Note how many reductions were 'driven down' to zero after just 3 copies. Adding 70% into the mix would reduce this effect. Q: There is an important piece of information missing, and I suspect no company would make a decision based on an analysis like the above. What are we missing? A: We should know what the 'popular' or most used reductions are in practice. Right now our analyis gives equal weight to 50% reductions and 1% reductions. People probably don't make many reductions down to 1%.