This paper studies sequencing situations with non-linear cost functions. We show that the neighbor switching gains are now time-dependent, in contrast to the standard sequencing situations with linear cost functions, which complicate finding an optimal order and stable allocations. We derive conditions on the time-dependent neighbor switching gains in a (general) sequencing situation to guarantee convexity of the associated sequencing game. Moreover, we provide two procedures that uniquely specify a path from the initial order to an optimal order and we define two corresponding allocation rules that divide the neighbor switching gains equally in every step of the path. We show that the same conditions on the gains also guarantee stability for the allocations prescribed by these rules
