Table allocation for meetings when the hall has fewer tables than proposals

The floor plan came back with 40 tables in the meeting zone, because that is what fits once you allow for aisles and the catering point. The matching engine produced 300 proposals worth sending. Somebody has done the multiplication and it works, so nobody is worried.

Table allocation for meetings is where that arithmetic quietly stops working, and it usually stops working about ten days before the show when the third round of proposals lands. The reason is that 40 tables over 8 slots is 320 places, and 320 is the ceiling on meetings held rather than a target you work up to.

The capacity arithmetic, done properly

Tables multiplied by slots gives places. Forty tables, eight slots, 320 places. Three hundred proposals against 320 places looks like 94 per cent utilisation with room to spare.

Now take the deductions, all of which are real and none of which appear in the first calculation.

Four tables are held for sponsor meetings that get booked on the day. That is 4 times 8, or 32 places gone, leaving 288.

Slot 4 is lunch for half the hall, so 20 tables are unusable in that slot. Another 20 places gone, leaving 268.

The first slot on day one starts 20 minutes after doors open and in practice loses about a quarter of its bookings to late arrivals. That is a completion problem rather than a capacity one, and it shows up further down the report. Leave those places in the count for now.

So the honest capacity is 268 places against 300 proposals. The floor is over-subscribed by 32 before anybody has failed to turn up, and the shortfall is 10.7 per cent of proposals.

That is the number to put in front of the show director, because it changes the conversation from scheduling to rationing. Somebody is going to be told no. The only open question is whether you choose who, or whether the placement loop chooses for you by running out of tables halfway down a sorted list.

Two ways to ration, and they produce different floors

The first way is a global capacity constraint. Let the scheduler place proposals by score and stop when the tables are gone. This maximises total match score subject to capacity, which is the textbook objective and has an obvious appeal.

The second way is a per-participant quota. Give every buyer a ceiling, say 4 meetings on a 268 place floor with 60 buyers, which is 240 places allocated and 28 held back. Then let the scheduler fill each buyer's quota from their own best proposals.

These produce genuinely different shows. Under the global constraint, a buyer with 11 strong proposals takes 11 places and a buyer with 3 weak ones takes none, so the score is higher and forty per cent of your hosted buyers go home with nothing. Under quotas, everybody gets four, the total score is lower, and nobody writes to the show director.

Gale and Shapley set out the structure of this in the American Mathematical Monthly in 1962, in the paper that opens with a college considering n applicants of which it can admit a quota of only q. Their point, which transfers directly, is that a quota changes the problem from ranking into assignment, and that the assignment has a stability property worth having: no buyer and exhibitor should both prefer each other to what they were given. Roth showed in the Journal of Political Economy in 1984, working through the history of the medical residency match, that mechanisms without that property get gamed by participants until they collapse, and mechanisms with it survive for decades.

Your meeting programme is a weaker version of the same market, because meetings are not exclusive and nobody signs a contract. The lesson that transfers is that participants notice the rule and respond to it. Run a pure global capacity constraint for one edition and your exhibitors will learn to submit forty proposals each, because volume is what wins under that rule.

What 300 proposals against 40 tables actually looks like

Take the numbers concretely. 300 proposals, 60 buyers, 90 exhibitors, 8 slots, 40 tables, 268 usable places after deductions.

Under a global capacity rule with proposals sorted by score, the first 268 proposals that fit get placed, subject to nobody being double booked. In practice you will not place 268, because clashes bite before capacity does. Suppose 231 get placed, with 37 failing on clashes and 32 never reached because capacity ran out. Utilisation is 231 over 268, which is 86.2 per cent, and 69 proposals are unfulfilled.

Distribution is where it gets uncomfortable. Those 231 meetings across 60 buyers average 3.85 each, and the spread will run from 8 or 9 for the buyers with dense proposal sets down to 0 for perhaps 6 or 7 buyers. On a hosted programme where each of those buyers cost you money to bring, six buyers with no meetings is a number that gets said out loud in the debrief.

Under a quota rule of 4 per buyer, you allocate 240 places, place perhaps 218 of them after clashes, and hold 50 places back. Utilisation drops to 81.3 per cent. Total match score falls by something like 6 to 9 per cent, because you are declining strong proposals from the busy buyers to protect places for the quiet ones. Nobody gets zero.

I would run quotas on a hosted programme and global capacity on an open floor, and the reason is who paid. If the organiser paid for the buyer to be in the room, the organiser owes them meetings. If the buyer bought their own ticket and is browsing, the strongest matches winning is a defensible outcome.

Should tables be modelled as a hard constraint?

Yes, and it is worth being firm about this because the temptation to soften it is strong.

A schedule that books 290 meetings into 268 places is not a schedule with a small penalty attached. It is a document that will be corrected in the hall at 09:15 by an operations lead moving people to standing tables, and every correction is a meeting that starts late or does not happen. The failure is physical and it lands on the people least able to absorb it.

Model capacity as a limit on meetings per slot that the scheduler cannot exceed, and make the number a real count of tables rather than an aspiration from the floor plan. If 6 of the 40 tables sit in a corridor where nobody can hear, they are not tables. Count 34.

There is a related trap with tables that belong to one exhibitor all day. A stand with a dedicated meeting table sits outside the shared pool, and if you count it in both places you will over-allocate by exactly the number of dedicated tables. On a floor with 12 dedicated tables and 40 shared ones, the pool is 40, and the 12 carry their own availability that belongs to their exhibitor.

The same care applies to slot boundaries. Eight slots on a floor plan usually means eight meeting windows plus turnaround, and if the turnaround is 5 minutes on a 20 minute slot then the day needs 200 minutes of meeting zone time to deliver 8 slots. Where the hall is shared with a keynote that overruns, slot 5 disappears and the capacity ceiling drops by 40 places without anybody updating the spreadsheet the proposal count was checked against.

What happens to the 69 proposals that did not fit?

They are not waste. They are the most useful list you will hold on the morning of the show, because they are ranked, both parties already said yes, and roughly one meeting in eight will free up as cancellations land.

Holding those proposals as a ranked standby list per slot is the backfill mechanism that turns a cancellation into a second chance, and it is worth building before you spend money on more tables. Forty tables in a hall cost floor space that could be sold. A standby queue costs a database index and a text message.

The order you place proposals in the first pass also determines which ones end up on that list, which is why the placement rule and the capacity rule have to be designed together instead of one being bolted onto the other.

Where this stops

The arithmetic above assumes a table is a table. On most floors it is not.

Meeting length varies by programme, and a hall running 20 minute slots for hosted buyers and 45 minute slots for a supplier forum does not have one slot grid. Capacity in mixed-length programmes has to be computed in minutes and then divided, and the answer will be lower than the naive count because the boundaries do not line up.

The second limit is that capacity and demand are not independent. Publish a 4 meeting quota and exhibitors submit fewer proposals, because they can see the ceiling. Run a global rule and they submit more. Any capacity plan built on last edition's proposal count under a different rule is measuring a response to the old rule.

The third is that none of this survives a room with poor acoustics or a bad location. A table in the far corner gets declined by both parties, shows up as a placed meeting that nobody attended, and quietly makes your utilisation figure a fiction. Walk the meeting zone before you trust the table count.

Take last edition's proposal file and your current floor plan, compute places as tables times slots, subtract the sponsor holds and the lunch losses, and put that single number next to the proposal count. If the gap is more than about 10 per cent, decide your rationing rule now, because your matchmaking programme has already made the decision by default and it will not be the one you would have chosen.