the items $j: j \geq c$ are added to the solution. Correspondingly, the initial profit and weight of the split solution are modified by subtracting the profits and the weights of the removed items and by summing up the profits and the weights of the added items.

Then, for any given subset $\tau$ of the items set $c-\delta,...,c+\delta$, let $p^{\tau}$ and $w^{\tau}$ be the overall profit (namely the value of the improvement upon the split solution) and weight contributions of the items in $\tau$, namely: $$\begin{aligned}
p^{\tau} = -\sum\limits_{i \in \tau: i < c} p_i + \sum\limits_{j \in \tau: j \geq c} p_j; \label{pT}\\
w^{\tau} = -\sum\limits_{i \in \tau: i < c} w_i + \sum\limits_{j \in \tau: j \geq c} w_j. \label{wT}\end{aligned}$$

A subset $\tau$ with $p^\tau \leq 0$ is not considered since it does not improve upon the split solution. Instead, an improving subset with $p^\tau > 0$ is feasible only if $w^{\tau} \leq w_c$ and all items in $\tau$ are not interdicted by the leader. In that case, by keeping the notation of model $CRIT_1(c)$, an improvement $\pi$ can be determined if the following constraint is added:

$$\label{TupleCons}
\pi \geq p^\tau(\sum\limits_{j = \max\{1;w^\tau\}}^{w_c} k_j - \sum\limits_{i \in \tau} 