Bipolar N-soft set theory with applications

In this paper, the notion of bipolar N-soft set, which is the bipolar extension of N-soft set, and its fundamental properties are introduced. This new idea is illustrated with real-life examples. Moreover, some useful operations and products on the bipolar N-soft sets are derived. We thoroughly discuss the idempotent, commutative, associative, and distributive laws for these emerging operations and products. Also, we set forth two outstanding algorithms to handle the decision-making problems under bipolar N-soft set environments. We give potential applications and comparison analysis to demonstrate the efficiency and advantages of algorithms.