In this paper, we consider the min-max heterogeneous weighted delivery problem (the MMHWD problem). Specifically, given a weighted graph \(G=(V,E;w)\) with length function \(w:E\rightarrow {R}^+\) satisfying the triangle inequality, a fixed depot \(r\in V\) , m items, and k vehicles having nonuniform speeds \(\lambda _1\) , \(\lambda _2\) , \(\ldots \) , \(\lambda _k\) , each item is initially located at its source vertex \(s_j\) and it needs to be delivered to its target vertex \(t_j\) , \(j=1,2, \ldots , m\) , each vehicle can move along some edges of G and only deliver one item at a time and each item only can be continuously delivered by one vehicle, it is asked to find a set \(\mathcal {C}=\{C_1,C_2,\ldots , C_k\}\) of k tours for these k vehicles, each starting and ending at the same depot r, and collectively delivering all items, the objective is to minimize the maximum completion time of vehicles, where the completion time of a vehicle is its total length divided by its speed. We obtain the two main results. (1) Given any small constant \(\delta >0\) , we design an \(134.4434(1+\delta )\) -approximation algorithm to solve the MMHWD problem, its time complexity is bounded by a polynomial in the input size and \(\frac{1}{\delta }\) ; (2) We provide an \((\varphi +\frac{9}{5}-\frac{1}{k})\) -approximation algorithm to resolve the MMHWD problem, where \(\varphi \) is the ratio of the largest vehicle speed to the smallest one.