The single-machine scheduling problems of due window assignment with position-dependent weights are investigated with the aim of minimizing the total weighted sum of the starting time, the size of due window and total tardiness of  due window in a just-in-time environment. The goal is to find the optimal job processing sequence and due window starting time $d_k^1$ (finishing time $d_k^2$). Under common, slack and different due window assignments, the corresponding optimal properties are obtained from the theoretical analysis, it is proved that the problems can be solved in polynomial-time. For the common and slack due window assignments, the time complexity is $O(n^2 \log n)$, while the different due window scheduling problem can be solved in $O(n \log n)$ time, where $n$ is the number of given jobs.