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I'm helping organizing a one-day conference which will host 6 events and a few hundred people, and wouldn't mind some help myself. There are four locations we've staked out with varying capacities, and three time slots of equal length. Each event will be allotted two time slots, and four events will run concurrently during any given time slot at the different locations. Any given event can switch locations between their allotted time slots.

All of the people who will attend have ordered their four favorite events beforehand and they all must be assigned three events with disjoint time slots. (Since we don't know otherwise, the two events not rated by the participants are both rated 5th.) Ultimately, people need to be assigned to events and events need to be assigned places and times such that the average of the sum of the participants' ratings of their assigned events is minimized.

Attempts at a solution

I initially thought the assignment of times to events could be arbitrary so long as the timetable looked like this (each event is a represented by a color):

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which ensures there are no time conflicts. However I realized that if many people were assigned two events with the same set of time slots (e.g. red and green events), they would be forced to attend the third at the remaining time slot (e.g. orange event must be at t3). If there was a sufficiently popular triplet of event choices where two shared the same set of time slots, the third might have a capacity problem during the remaining time slot.

This being the case, my current approach has been reduced to brute force: assigning participants to their first three choices, testing out all combinations of time and place assignments for each event, and continually adjusting participants' and events' assignments as necessary until the program stumbles upon a viable solution. Aside from being terribly inefficient and susceptible to infinite loops, I doubt it'll produce the optimal solution. I must profess that I know little about formal optimization theory, so any help is greatly appreciated!

concat
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  • Can people choose to attend events from one location to the other location?. Or are they restricted on the location. 2) Could you explain to me what you mean by "a popular triplet of event choices ( in terms of your color in the example) where two shared the same set of time slots". I quite did not get you on that. Do you have the frequency of ratings of the six events by participants before the conference ( It does not matter now but will matter if you set it up as an optimization routine). Let me try to throw an attempt at it? – Satish Ramanathan Jan 19 '15 at 01:11
  • @satishramanathan There's no restriction on where the participants go, so long as they don't need to be in two places at once. My concern with that schedule is that not all assignments of time slots to events are optimized: if three events are colored red, green and orange and are often selected together, the orange event will be forced to host many people in the third time slot. Alternatively, they could be colored red, yellow and orange, allowing the orange event access to the first time slot. – concat Jan 19 '15 at 02:09
  • It is too late in the night, I will continue to make an attempt as soon as I get up from my bed. Till then, I wish you get a solution from somebody if it were urgent. – Satish Ramanathan Jan 19 '15 at 06:22

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