How can we bound the optimal solution of the dual bin packing when we solve the knapsack problem for each bin?
The "dual bin packing problem" is more commonly referred to as the Multiple Knapsack problem. One can show that $ALG \ge (1-1/e) OPT_b$ assuming an optimal algorithm for the Knapsack problem is used in each bin. If the bins are of different sizes then one can show that $ALG \ge OPT_b/2$ irrespective of the order in which the bins are processed. See Section 4 in the paper below for these results. http://epubs.siam.org/doi/abs/10.1137/S0097539700382820