From this post:
This looks like an assignment problem with some side-constraints. The assignment part is boilerplate:I'm trying to solve a special assignment problem, but I cannot handle it.
The problem is to assign seats to audiences. Each audience has a ranking of the seats based on his/her own preference. I want to assign the seats to the audiences such that the overall satisfaction is maximized.
The above problem is nothing but an assignment problem, which can be solved by the Hungarian algorithm. Now, things becomes a little more complicated. The audiences are divided into male audiences and female audiences. When assigning the seats, there should be at least n empty seats between each male and female. I assume that there are enough seats to accommodate all audiences and that all seats are located in a row.
| \[\boxed{\begin{align}\max\>&\sum_{i,j}s_{i,j}x_{i,j}\\&\sum_j x_{i,j}=1 \>\forall i\\&\sum_i x_{i,j}\le 1\>\forall j\\&x_{i,j}\in \{0,1\}\end{align}}\] |
Here \(i\) indicates spectators and \(j\) refers to seats. Modeling the constraint that females cannot sit next to males is more challenging and interesting (from a strict modeling perspective that is: these seating arrangements do not look very attractive to me).
To experiment with some formulations I generate some random data for the preferences \(s_{i,j}\):
If we want to have \(n\) empty seats in between the males and females, then for \(m=m_F+m_M\) persons we need at least \(m+n\) seats.
Quadratic Model
Probably the easiest is to use a quadratic model. First we introduce a distance matrix \(d_{j,j’}\) indicating 1 + the number of seats between seats \(j\) and \(j’\). I.e. this matrix looks like:
Then we need to construct a set \(check_{i,i’}\) which has all unique female-male combinations we need to check. This set arranged as a matrix looks like:
With this data we can now write our side-constraint as:
| \[\sum_{j\ne j’} d_{j,j’} x_{i,j} x_{i’,j’} \ge n+1 \>\> \forall check(i,i’) \] |
This formulation is quadratic but unfortunately not convex (the Q matrix of quadratic coefficients is not negative semidefinite). We are lucky: some solvers like Cplex and Gurobi accept it nevertheless: they apply some reformulations behind the scenes.
The optimal assignment for \(n=1\) looks like:
Linearization
The multiplication of two binary variables can be easily linearized. A standard way to linearize \(z=x\cdot y\) with \(x,y \in \{0,1\}\) is:
| \[\boxed{\begin{align}&z \le x\\ &z \le y \\ & z\ge x+y-1\\& x,y,z \in \{0,1\}\end{align}}\] |
In our case we can simplify a bit:
| \[\boxed{\begin{align}&\sum_{j,j’} d_{j,j’} y_{i,j,i’,j’} \ge n+1\\ & y_{i,j,i’,j’} \le x_{i,j}\\ & y_{i,j,i’,j’} \le x_{i’,j’}\\ & y_{i,j,i’,j’} \in \{0,1\} \end{align}} \] |
This formulation leads to a very large model:
Although this linearized formulation finds the optimal solution, clearly, this is not the way to go.
Forbid patterns
A different formulation can be based on forbidding certain patterns in the seating. For \(n=1\) we do not allow F,M or M,F to appear.
The first thing we do is to introduce a variable that tells us if set \(j\) is occupied by a male or female (or empty). We can do this by using two binary variables \(m_j\) and \(f_j\), or we can use a single variable \(g_{j,fm} \in \{0,1\} \) where \(fm = \{f,m\}\) is a set indicating the gender.
To calculate \(g_{j,fm}\) we can use a simple summation over the females or males only, depending on what \(fm\) is:
| \[g_{j,fm}=\sum_{i|gender(i,fm)} x_{i,j} \] |
Variable \(g_{i,fm}\) will look like:
This is kind of an aggregation of the variable \(x_{i,j}\). Based on \(g_{j,fm}\) we can formulate a set of equations, again assuming \(n=1\):
| \[\boxed{\begin{align}&g_{seat1,female}+g_{seat2,male}\le1\\&g_{seat1,male}+g_{seat2,female}\le1\\&g_{seat2,female}+g_{seat3,male}\le1\\&g_{seat2,male}+g_{seat3,female}\le1\\&g_{seat3,female}+g_{seat4,male}\le1\\&g_{seat3,male}+g_{seat4,female}\le1\\&\dots\end{align}}\] |
This linear formulation has 83 variables and 180 equations and solves very fast. For \(n>1\) we can formulate generalizations of this scheme.