Abstract 
: 
Recently there has been considerable research on simple mixedinteger sets, called mixing sets, and closely related sets arising in uncapacitated and constant capacity lot sizing. This in turn has led to study of more general sets, called networkdual sets, for which it is possible to derive extended formulations whose projection gives the convex hull of the networkdual set. Unfortunately this formulation cannot be used (in general) to optimize in polynomial time. Furthermore the inequalities definining the convex hull of a networkdual set in the original space of variables are known only for some special cases.
Here we study two new cases, in which the continuous variables of the networkdual set are linked by a bidirected path. In the first case, which is motivated by lotsizing problems with (lost) sales, we provide a description of the convex hull as the intersection of the convex hulls of 2n mixing sets, where n is the number of continuous variables of the set. However optimization is polynomial as only n + 1 of the sets are required for any given objective function. In the second case, generalizing single arc flow sets, we describe again the convex hull as an intersection of an exponential number of mixing sets and also give a combinatorial polynomialtime separation algorithm.
