I'm trying to solve one interesting math task.

Let’s imagine we have a number of instructors with different timespans during the day in which they work or they are available. We need to display to students timeslots where they can book inside the available timespan of the instructor. Within this time period we need to optimize the appointments so that there is a minimum number of "idle" timeslots. Once an appointment has been set we cannot move the appointment. The negotiation of an appointment must happen without negotiating with other students. Appointments have variable and unknown lengths?

Any ideas people ?

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    $\begingroup$ There's a relatively large literature on appointment scheduling. I think most of it is focused on medical appointments but it's probably the same basic idea as what you're describing. $\endgroup$ – LarrySnyder610 Apr 2 '20 at 13:07
  • $\begingroup$ @LarrySnyder610 Can you please share something with me ? $\endgroup$ – BigGinDaHouse Apr 2 '20 at 13:10
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    $\begingroup$ Just a google search: scholar.google.com/… $\endgroup$ – LarrySnyder610 Apr 2 '20 at 13:54
  • $\begingroup$ Assuming a fixed number of instructors with fixed individual timespans, the total instructor capacity is constant. For a given demand, the aggregate empty time slots will be capacity - demand, regardless of how you schedule. So to minimize "idle" time, it must somehow be different from unused time. How? For instance, do instructors leave early if they have not more scheduled appointments? $\endgroup$ – prubin Apr 3 '20 at 21:38

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