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We describe the primitive middle-dimensional cohomology $\mathbb{H}$ of acompact simplicial toric complete intersection variety in terms of a twisted deRham complex. Then this enables us to construct a concrete algorithm of formalflat $F$-manifold structures on $\mathbb{H}$ in the Calabi-Yau case by usingthe techniques of \cite{Park23}, which turn the twisted de Rham complex into aquantization dGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra andseek for an algorithmic solution to an associated \textit{weak primitive form.}

Twisted de Rham complex for toric Calabi-Yau complete intersections and flat $F$-manifold structures