We investigate set theoretic properties that characterize the collection of connected coalitions in cycle-free undirected graphs among the accessible union stable network structures. It turns out that the only additional requirement is that every non-unitary feasible coalition can be written in a unique way as a union of non-unitary supports. As a consequence, a fairness axiom for solutions for cooperative games on cycle-free accessible union stable network structures can be defined that, together with the well-known component efficiency, characterizes the Shapley value on the class of cycle-free accessible union stable network structures. Since this fairness axiom combines ideas behind the traditional fairness axiom and balanced contributions, we refer to it as balanced fairness.

The research field of Attended Home Delivery (AHD) and Attended Home Service (AHS) problems has experienced fast growing interest in the last two decades, with the rapid growth of online platforms and e-commerce transactions. The COVID-19 pandemic has reinforced that interest, raising further challenges and opportunities that have to be tackled by innovative methodologies and policies. The aim of this work is to provide an extensive literature review on the state of the art for AHD and AHS problems, with a particular focus on real-world applications. A discussion of promising future research directions is also provided.