Summary: Let \(A\) be a finite multiset of integers. If \(B\) be a multiset such that \(A\) and \(B\) are \(t\)-complementing multisets of integers, then \(B\) is periodic. We obtain the Biró-type upper bound for the smallest such period of \(B\) [A. Biró, Acta Arith. 118, No. 2, 117–127 (2005; Zbl 1088.11016)]: Let \(\varepsilon>0\). We assume that \(\text{diam}(A)\geq n_{0}(\varepsilon)\) and that \(\sum_{a\in A}w_{A(a)}\leq (\text{diam}(A)+1)^{c}\), where \(c\) is any constant such that \(c< 100\log2-2\). Then \(B\) is periodic with period \(\log k\leq (\text{diam}(A)+1)^{\frac{1}{3}+\varepsilon}\).