Before 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.
After revision:  We prove that for additive valuations, an allocation that is both Pareto optimal and strongly WEF1 always exists and can be computed in pseudo-polynomial time; moreover , an allocation that maximizes the weighted Nash social welfare may not be strongly WEF1 but always satisfies the weak version of the property.