$w_i$ of the follower and weights $v_i$ of the leader are integers randomly distributed in $[1,100]$: 10 instances are generated for each value of $n$. The follower’s capacity $C_l$ is set to $\lceil (INS/11)\sum_i^{n}w_i\rceil$ where $INS$ $(=1, \dots, 10)$ denotes the instance identifier. The leader’s capacity is randomly selected in the interval $[C_l - 10; C_l + 10]$.

We first tested our approach on these 50 benchmark instances. After some preliminary computational tests, we chose the following parameter entries for our approach: $\alpha = 100 $, $\beta = 100$, $\delta = 10$, $\mu = 150$, $\gamma = 2$. The corresponding results are presented in Table \[tab:CPUTime1\]. For each instance, we report the optimal solution value, the CPU time to obtain an optimal solution and the number of subproblems explored in Step 2. The last column also reports the number of times model $CRIT_2(c)$ is solved along the two steps.

Algorithm CCLW in [@CaCarLoWo16] solves all instances with 50 items within a CPU time limit of 3600 seconds but runs out of time limit in instances 55-3, 55-4. Algorithm in [@FiLjMoSi18] solves all benchmark instances, requiring at most a computation time of about 85 seconds for solving instance 55-3. As the 