Question

. Ms. Lee is famous for her parties in the Scoul area. In each four-year administration parties, set m tables at each party and seats n people at e people at the start of the four year period and invites them to every party. Her ta each other (teak, oak, cherry, etc.) and km. It is rumored that her parties are clever seating arrangements. She makes sure that each of her guests sits at each she believes very strongly that the atmosphere remains lively if each g his/her table at each party. So, every first time, she awards herself an imaginary penalty of c units, whe hrows k she t ach table. She makes a list of mn influential bles are all distinct from famous because of hor uest meets different people at time any pair of guests finds themselves at the same table after the integers 2 2 and k S m. formulate the problem of determining the seating arrangement o at the tables at the various parties, subject to the constraint that no guest sits at a table more than o and the overall penalty is as small as possible, as an integer program. 9 nce that a zero penalty seating arrangement ty seating arrangement using your If km, what condition should m,n satisfy to guarantee exists? When k mn 2, show that there is no zero penalty seating arrangement using y formulation above.

Answer 1

SOLUTION

First let's define our seating variables: (1 if the Ith person sits at table i in party j 0 if they don't This variable allows us to control which guest sits where and when. Using this we can already formulate the first constraint -since we can't seat the same person twice at the same table then it must be the case that if af, = 1 implies that a = 0 (for so our constraint looks like so: = l Now to find the cost function we have to define another variable to control how many times a pair of guests will meet at a table. For this we define as |1 if guest IAND guest ' sit at table i in party j |0 if they don't Since this only happens when both a, = 1 and af, = 1 then we can more accurately define b as ia)) Now we define Su as the number of times guests l and ' meet at a table after the first time -() () Su (1) 1 ij

However, we cannot define the cost function yet since in Su we are counting guests who never meet. To fix this, we add another variable: fiif guests I and never meet 0 if they do Which, in terms of the a's is given by: |0 otherwise Also notice that in (1) we have that b =1 for every partyj so from the final cost function we must eliminate these cases. This final cost function is given: mn mn mine ΣΣ(ΣΣ+)-