Linear and nonlinear multiplexer based physical unclonable functions (MUX PUFs) are used for authentication of devices. The responses to some input challenges can be unreliable or unstable due to inherent hardware variations. Thus, the server needs to guarantee that a challenge is stable before it is issued. This can be done by computing a metric called total delay-difference and then thresholding it to decide whether a challenge is stable or not. A straightforward approach is to discard and issue a new random challenge for authentication if the current challenge is unstable. This paper presents a novel bit-flipping approach where flipping few bits of the original unstable challenge can convert it to a stable one. We compare the computation complexities for the straightforward and proposed approaches. In terms of number of addition operations, the proposed approach has comparable average-case performance but much better worst-case performance than the straightforward approach.