In Nash's (1950) seminal result, independence of irrelevant alternatives (IIA) plays a central role, but it has long been a subject of criticism in axiomatic bargaining theory. This paper examines the implication of a weak version of IIA in multi-valued bargaining solutions defined on non-convex bargaining problems. We show that if a solution satisfies weak IIA together with standard axioms, it can be represented, like the Nash solution, using weighted products of normalized utility levels. In this representation, the weight assigned to players for evaluating each agreement is determined endogenously through a two-stage optimization process. These solutions bridge the two dominant solution concepts, the Nash solution and the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 1975). Furthermore, we consider special cases of these solutions in the context of bargaining over linear production technologies.