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New methods for computing parametric local $b$-functions are introduced for$\mu$-constant deformations of semi-weighted homogeneous singularities. Thekeys of the methods are comprehensive Gr\"obner systems inPoincar\'e-Birkhoff-Witt algebra and holonomic ${\mathcal D}$-modules. It isshown that the use of semi-weighted homogeneity reduces the computationalcomplexity of $b$-functions associated with $\mu$-constant deformations. In thecase of inner modality 2, local $b$-functions associated with $\mu$-constantdeformations are obtained by the resulting method and given the list ofparametric local $b$-functions.

Methods for computing $b$-functions associated with $\mu$-constant deformations -- Case of inner modality 2 --