Farkas Pricing doesnt lead to feasibility

I am currently trying to initialize my column generation in the root node with Farkas Pricing. I am starting with a completely empty model. My model will look like this. Where \(\Lambda_{ij}\) is the binary variable that indicates whether the column \(i\) is used in the iteration \(j\). The SP passes the column consisting of the \(x_{iks}\), i.e. whether person \(i\) is lying in bed \(k\) on day \(s\). The length of stay \(P_t\) is also transferred.
The MP then looks like this:
\begin{align}
\min \sum_{i,j}\Lambda_{ij}\cdot F_i^j\\
s.t.\\\
\sum_{i,j}\Lambda_{ij}\cdot x_{iks}^j\leq M_{ks}~~~\forall k,s\\\
\sum_j\Lambda_{ij}=1~~\forall i\\
\Lambda_{ij}\in \{0;1\}
\end{align}
The empty then looks like this:
\begin{align}
\min 0\\
s.t.\\
0\leq Max_{ks}~~~\forall k,s\\
0=1~~\forall i
\end{align}
This model is of course infeasible, which is why I optimize the SP with Farka's duals and a cost coefficient of 0. Now assume \(I=3\), \(K=2\) and \(S=4\). Here \(M_{11}=2, M_{12}=2, M_{13}=2, M_{14}=2, M_{21}=0, M_{22}=1, M_{23}=1, M_{24}=1\).
Then the empty LP. It looks like this:

    \ Model MasterProblem
    \ LP format - for model browsing. Use MPS format to capture full model detail.
    Minimize
      0 lmbda[1,1] + 0 lmbda[2,1] + 0 lmbda[3,1] + 2 lambda[1,2] + 4 lambda[2,3] + lambda[3,4]
    Subject To
     lmbda(1): = 1
     lmbda(2): = 1
     lmbda(3): = 1
     max(1,1): <= 2
     max(1,2): <= 2
     max(1,3): <= 2
     max(1,4): <= 2
     max(2,1): <= 0
     max(2,2): <= 1
     max(2,3): <= 1
     max(2,4): <= 1
    Bounds
    Generals

    End

Now I run four iterations and there is a column for each \(i\), so the convexity constraint is satisfied for all \(i\)'s. Unfortunately, it looks different for the other constraints, where variables are added, but the RHS is 0, which is why the MP remains infeasible. 

    \ Model MasterProblem
    \ LP format - for model browsing. Use MPS format to capture full model detail.
    Minimize
      0 lmbda[1,1] + 0 lmbda[2,1] + 0 lmbda[3,1] + 2 lambda[1,2] + 4 lambda[2,3] + lambda[3,4] + 2 lambda[1,5]
       + 4 lambda[2,5] + lambda[3,5]
    Subject To
     lmbda(1): lambda[1,2] + lambda[1,5] = 1
     lmbda(2): lambda[2,3] + lambda[2,5] = 1
     lmbda(3): lambda[3,4] + lambda[3,5] = 1
     max(1,1): <= 2
     max(1,2): <= 2
     max(1,3): <= 2
     max(1,4): <= 2
     max(2,1): lambda[2,3] + lambda[2,5] <= 0
     max(2,2): lambda[1,2] + lambda[1,5] <= 1
     max(2,3): lambda[1,2] + lambda[2,3] + lambda[3,4] + lambda[1,5]
       + lambda[2,5] + lambda[3,5] <= 1
     max(2,4): lambda[2,3] + lambda[2,5] <= 1
    Bounds
    Generals
     lambda[1,5] lambda[2,5] lambda[3,5]
    End

But now I have the problem that if I run the whole thing for more iterations, then the Farkas Duals always remain the same and therefore the same columns are always found and it always remains infeasible. What could be the reason for this? The duals look like this:

    {(1, 1): 0.0, (1, 2): 0.0, (1, 3): 0.0, (1, 4): 0.0, (2, 1): 0.0, (2, 2): 0.0, (2, 3): 1.0, (2, 4): 0.0}, {1: 1.0, 2: 1.0, 3: 1.0}

EDIT: Upon further inspection i may found the reason behind the 'same' column, which however doesnt fix the non-feasibility. The objective function in the SP looks like this and I have the termination criterion that columns are added as long as no more reduced costs \(<0\) are found for any subproblem. 
\begin{align}
\min (SP_i)~~~ 0 - \sum_{k,s}x_{iks}\cdot\pi_{ks} - \mu_i
\end{align}
Since \(\pi_{ks}\) are the duals of the first constraint and \(\mu_i\) those of the convexity constraint. For \(i=3\) and taking into account the Farkas duals, the objective function reduces to
 \begin{align}
\min (SP_3)~~~ 0 - x_{322}\cdot1 - 1
\end{align}
As the objective function is minimized, \(x_{322}=1\) is always reassigned and thus the same column is produced again.

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