Alternative solution on an assigment problem using PuLP

Question

I'm creating a program to solve assignment problems and so far so good, it works properly and in the end, it solves the problem. I'm using the PuLP library. But I just realized that when the problem has alternative solutions (instead of the unique optimal basic feasible solution) it only shows me the optimal one. Is there any function inside PuLP or any other library that can give me the number of total solutions (for example) or the objective function value for those said alt solutions?

    prob.solve()
    print(" ");
    print("* Status: ", LpStatus[prob.status]);
    print(" -------");
    print(" ");

    for v in prob.variables():
        if v.varValue > 0:
            print("------>", v.name,"=", v.varValue);
    
    print("Total = ", value(prob.objective))

Status: Optimal

    ------> Asignación:  Agente 1 Tarea 2 = 1.0
    ------> Asignación:  Agente 2 Tarea 4 = 1.0
    ------> Asignación:  Agente 3 Tarea 3 = 1.0
    ------> Asignación:  Agente 4 Tarea 1 = 1.0

Total = 15.0

Erwin Kalvelagen · Accepted Answer · 2022-05-01T16:55:40.290

The basic assignment problem looks like:

 min sum((i,j), c[i,j]*x[i,j])
 subject to
     sum(j, x[i,j]) = 1  ∀i
     sum(i, x[i,j]) = 1  ∀j
     x[i,j] ∈ {0,1}

If you solve this, you get one optimal solution x*[i,j]. Let's record this solution in a[i,j,k] with initially k=0. To find a next best solution we can add the constraint:

     sum((i,j), a[i,j,k]*x[i,j]) ≤ n-1  ∀k

where n is the size of set i (and j) or the number of ones in the solution. This constraint will precisely cut off previous solutions. The algorithm becomes:

solve original assignment problem
set k := 0
store solution in a[i,j,k]
solve problem with additional constraint
if infeasible: STOP
if no more solutions needed: STOP
k := k + 1
go to step 3.

An alternative would be to use a solver that has a so-called solution pool. That will do all of this in just one solve.

For an example see: https://yetanothermathprogrammingconsultant.blogspot.com/2018/04/k-best-solutions-for-assignment-problem.html

Alternative solution on an assigment problem using PuLP

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