Before revision:  We prove that for additive valuations, a strongly weighted envy-free allocation up to one item always existsand can be efficiently computed by means of a weight-based picking sequence . For two agents, we can also efficiently achieve strong weighted envy-freeness up to one item in conjunction with Pareto optimality using a weighted version of the classic adjusted winner algorithm.
After revision:  We prove that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists; however, an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1 but always satisfies the weak version of the property. Moreover, we establish that a generalization of the round-robin picking sequence produces in polynomial time a strongly WEF1 allocation for an arbitrary number of agents; for two agents, we can also efficiently achieve strong weighted envy-freeness up to one item in conjunction with Pareto optimality using a weighted version of the classic adjusted winner algorithm.