Bi-criteria multiple knapsack problem with grouped items

The multiple knapsack problem with grouped items aims to maximize rewards by assigning groups of among knapsacks, without exceeding capacities. Either all in a group are assigned or none at all. We study the bi-criteria variation problem, where capacities can be exceeded and second objective is minimize maximum capacity. propose approximation algorithms that run pseudo-polynomial time guarantee not less than optimal solution capacity-feasible bound on have different factors, no capacity more 2, 1, $$1/2$$ times improved $$1/3$$ when equal. also prove for certain cases, solutions obtained always optimal—they never exceed To obtain solutions, we binary-search heuristic combined algorithms. test performance heuristics an extensive set experiments randomly generated instances show they efficient effective, i.e., reasonably fast generate good quality solutions.