  We consider two models of fair division with indivisible items: one for goods
and one for bads. For goods, we study two generalized envy freeness proxies
(EF1 and EFX for goods) and three common welfare (utilitarian, egalitarian and
Nash) efficiency notions. For bads, we study two generalized envy freeness
proxies (1EF and XEF for goods) and two less common diswelfare (egalitarian and
Nash) efficiency notions. Some existing algorithms for goods do not work for
bads. We thus propose several new algorithms for the model with bads. Our new
algorithms exhibit many nice properties. For example, with additive identical
valuations, an allocation that maximizes the egalitarian diswelfare or Nash
diswelfare is XEF and PE. Finally, we also give simple and tractable cases when
these envy freeness proxies and welfare efficiency are attainable in
combination (e.g. binary valuations, house allocations).
