Tuesday, October 4, 2016

Optimal Opera Seating: a MIP formulation

Here is an interesting problem that is somehow not so easy to model as a Mixed Integer Programming (MIP) problem:

So i have an opera houses with 10 rows and 10 columns of seats (Total :100). Each seat is allocated a preference value Aij. The preference value is halved if the group do not get seats in same row. Eg: If a reservation in opera house is for 5 people and only 2 can be accommodated in top row and 3 in next row, the preference value is actually halved for each seat. There are total of n reservations with 'n' > 100 seats. What will the best way to maximize the customer preference (n *Aij).If it can be done by linear programming, how should the equation look like.


Interior of Teatro de Romea in Murcia, Spain (link)

The data

I assume we have a set of groups \(g\) and their sizes: \(size_g\). A couple would be a group of size 2. Further use the rule that only complete groups can be seated (i.e. no partial groups).

The seats are organized as a grid \((i,j)\) where \(i\) is the row and \(j\) is the column. Each seat has a “preference value” \(a_{i,j}\).

A simpler case: assignment

Let’s first ignore the difficulty about single or multiple row assignments. Without this complication we are essentially dealing with an assignment problem:

\[\boxed{\begin{align}
\max\>&\sum_{i,j,g} a_{i,j} x_{i,j,g}\\
&\sum_g x_{i,j,g} \le 1 \>\>\forall i,j\\
&\sum_{i,j} x_{i,j,g} = \mathit{place}_g \cdot \mathit{size}_g \>\>\forall g\\
&x_{i,j,g} \in \{0,1\}\\
&\mathit{place}_g \in \{0,1\}
\end{align}}\]

Here \(x_{i,j,g}=1\) if a member of group \(g\) is seated in location \((i,j)\). The binary variable \(\mathit{place}_g\) indicates whether we accommodate group \(g\) (and thus seat all its members).

This model works both for the case where \(n=\sum_g \mathit{size}_g\) exceeds the number of available seats or when \(n\) is a smaller number.

This is quite easy. However using random \(a_{i,j}\) we see groups scattered around the rows:

image

The numbers in each cell are the group IDs.

The tough case: penalize multiple row assignments

I am not sure if this is the best or even a good formulation. What I did is introduce a new index \(r = \{\text{OneRow},\text{MultipleRows}\}\). Then my main variables are \(x_{i,j,g,r} \in \{0,1\}\). Here is the complete model:

image

The parameter pref2 was populated as:

image

Admittedly: this model is not very straightforward and obvious. I am curious if there are better formulations.

The results look ok: there is a heavy preference for placing groups in the same row:

image

I would guess a constraint Programming (CP) approach may make it easier to model this problem.