Finding all additive orders
One approach is to recursively extend prefixes of the order. At each recursive call, we have a prefix $\phi$ of the order. We go over all remaining subsets, and for each subset $S$, we check whether the order $\phi &lt; S$ is consistent. For each consistent $S$, we make another recursive call.

How do we check whether $\phi &lt; S$ is consistent? This is a linear programming problem, whose solution we can speed up by starting the simplex algorithm from the "witness" showing that $\phi$ is feasible. There are two things to notice here: First, presumably you are assuming that all weights are positive. Second, we replace each constraints $A &gt; B$ by $A \geq B+1$ (since LPs don't have strict constraints).

There are many optimizations which can be done here. For example, $S$ is always inclusion-minimal among the remaining subsets. Another optimization is to assume that the weights are ordered in a specific way (in your example, $x &lt; y &lt; z$); if you are really interested in all possible orders, you can always go over all permutations later.