Scope and Purpose--It is well known by managers and researchers that early completion, and/or tardy completion of orders, is undesirable. Each order, ideally, should be completed exactly at its due date, which gives good services to customers and reduces inventory costs. This is the philosophy of Just-in-time (JIT) production In this paper, we examine a one-machine scheduling problem where the objective is to minimize the weighted earliness subject to a maximum tardiness constraint for each job. This is consistent with the ideas of JIT and pull systems. The exact solution to the problem is usually found by using a branch and bound algorithm or a dynamic programming algorithm. On the other hand, the problem is NP-complete. so it is necessary to develop effective heuristic algorithm. A heuristic algorithm and a branch and bound algorithm are proposed in this paper.