Abstract: This paper discusses the problem that $n$ jobs with their own deadlines and precedence constraints have to be processed on a single machine considering the idle insert. The objective is to find a preemptive job sequence and determine jobs' starting times so as to minimize the maximum earliness cost. Firstly, we consider two special cases: (1) common deadline and precedence constraints; (2) arbitrary deadlines and no precedence constraint. The $O(n^2)$ algorithms are provided respectively. Then the general case of arbitrary deadlines and precedence constraints is considered based on the two special cases, and an $O(n^2)$ algorithm is developed too. Since tardy jobs are prohibited, it's possible that there is no feasible sequence for the problem. We consider the feasibility primarily.