# Help me with a sliding scheduling problem?November 6, 2014 8:58 PM   Subscribe

Lennon manages projects. Each project takes 125 about working days to complete. Lennon must complete 39 projects in any given twelve month period. How many projects must Lennon manage simultaneously so that the boss could cut a twelve-month wide window in a piece of paper, slide it over the Gantt chart, and always see the finalisation of 39 projects?

For example, Lennon's boss places the window over September 2014 to August 2015, and can see 39 projects that ended during that period. The boss moves the window two months over to reveal the period November 2014 to October 2015, and can still see the finalisation of 39 projects. No matter where the boss moves the window - even starting and finishing partway through a month - the ends of 39 125-day-long projects can be seen.

(The boss is not super picky, and one project up or down doesn't matter, if that helps.)
posted by obiwanwasabi to Work & Money (7 answers total) 2 users marked this as a favorite

Response by poster: (125 working days. Call it six months even of total calendar time if that helps.)
posted by obiwanwasabi at 9:00 PM on November 6, 2014

Best answer: Well, if we call it six months, that's easier.

If Lennon was working on only one project at a time, his boss would see 2 projects completed in any 12-month period.

If Lennon worked on two projects at a time, he would complete 4 projects in 12 months.

If Lennon worked on 3 projects at a time, he would complete 6 projects in 12 months, etc.

In general, if Lennon works on x projects at a time, he will complete 2x projects in 12 months.

Lennon needs to complete 39 projects in 12 months, which gives us

2x = 39
x = 39/2 = 19.5 projects. Call it 20 projects if it has to be a whole number.

This is assuming that he spaces the projects evenly.
posted by mekily at 9:08 PM on November 6, 2014

Best answer: Slightly different math from mekily, to get the same answer, in case this math works better for you:
With 39 projects per year, evenly split, he must finish a project every 9.35 (real, not working) days (365/39). Per your update, each project takes 182.5 (again, real) days to complete. 182.5/9.35 = 19.5 projects at once. And since it has to be a whole number, round it up to 20.
posted by brainmouse at 9:15 PM on November 6, 2014 [1 favorite]

Response by poster: If it's OK I'll try phrasing it a little differently to see if the answer is the same:

What's the minimum number of projects Lennon needs to manage at once, assuming the projects must start and finish on different days? So this is out:
|--------------------|
|--------------------|
|--------------------|
But this is in:
|--------------------|
|--------------------|
|--------------------|
So no matter where the window is, you'll always see 39 'steps down'.
posted by obiwanwasabi at 9:21 PM on November 6, 2014

Response by poster: Actually, I think I get brainmouse's calcs. The 'steps' can't be more than 9.35 days apart, otherwise if you line the window up at the end of the top one the bottom one will fall outside the window. Yes?

(Why yes I'm not at all visual and so clumsily mix visual metaphors like windows sliding over steps all the time why do you ask?)
posted by obiwanwasabi at 9:31 PM on November 6, 2014

Best answer: The answer is the same (in fact, that's the only way the top diagram can work -- each project has to start and end the same distance apart). There is no minimum/maximum, there is only 1 right answer. He needs to start a project every 9.35 days, and finish a project every 9.35 days. He needs to have 19.5 projects going at once. It's the only way it works to always see exactly 39 at once. If more than 39 is ok (as opposed to exactly 39) then as long as he finishes one every 9.35 days or more often, there will always be at least 39 completed projects in the last 365 days If he finishes two projects closer together than that, it doesn't push the next deadline out, because if it did it would mess up your 39 projects a year later, so make sure each one is being turned in no more than 9.35 days from the last one. But your rephrasing doesn't change anything.
posted by brainmouse at 9:31 PM on November 6, 2014

Response by poster: Thanks. This explains why Lennon has such a hard time. Scheduling 20 projects for the first half of the year and 20 for the second isn't enough, and nor is saying 'I'll just schedule a few extra in case a couple fall behind'. The maximum gap between the projects is what's critical.
posted by obiwanwasabi at 9:39 PM on November 6, 2014 [1 favorite]

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