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I am trying to figure out how to maximize the number of appointment blocks in a day given the current appointment blocks with their specific times:

  • Appointment Type A
    • 8-10:30am
    • 10:30-1pm
    • 1-3:30pm
    • 3:30-6pm
    • 6-8:30pm
  • Appointment Type B
    • 8-9:30am
    • 9:30-11am
    • 11-12:30pm
    • 12:30-2pm
    • 2-3:30pm
    • 3:30-5pm
    • 5-6:30pm
    • 6:30-8pm

Let's say also that I want to spend 60% of my time on Appointment Type A and 40% of my time on appointment type B. I am also constrained by the fact that I can only have appointments in a day totaling roughly 480 minutes.

What is the process for figuring out the maximum number of appointment blocks I can have in a day given these constraints?

Thank you for your help!

T.S.
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2 Answers2

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Looks like every type A is 2.5 hours, and every type B is 1.5 hours, which means that you need exactly the same number of type A as type B in order to get the 60-40% ratio you want.

Because of the 480 minute = 8 hour limit, we see that we cannot choose 3 type A and 3 type B: 3*2.5 + 3*1.5 = 12. But we can choose 2 type A and 2 type B: 2*2.5+2*1.5 = 8.

Now it looks like there's several ways to choose 2 of each, e.g. the first two type B and the last 2 type A.

Louis
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    ...realizing you probably were more interested in an algorithm than a solution. Googling "interval scheduling problem" turns up a number of related problems which may help to come up with a good general algorithm for this particular type of interval scheduling problem. – Louis Nov 15 '13 at 01:09
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First you need to turn your constraints into numbers which work.

$60\%$ of $480$ is $288$ but Type B appointments last $150$ minutes, so you probably want two. $40\%$ of $480$ is $192$ but Type B appointments last $90$ minutes, so you probably want two. These will add up to $480$ minutes with a $62.5\% , 37.5\%$ split. You cannot spend any more time on appointments and any other pattern will give worse percentages.

So choose two Type A appointment times (there are ten possible pairs), and see if there are at least two Type B appointment times which do not overlap either of them. This will produce a lot of possibilities, all of which seem to meet your constraints.

Henry
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