Based on both the ordered quantity and lead time as decision variable, this article proposes a single item multi-supplier model that considers inventory problems with quantity discount and backorder. To complete our model, this paper regards crash cost as a function of both the order quantity and the shortened lead time, not as a function of the shortened lead time. In this article, the integral optimization approach is used to simplify the complex multiple-step process of the traditional quantitative methods. In addition, exerting piecewise linearization techniques solves non-linear problems to find an approximate global optimization. As a result, the proposed model not only can overcome the shortcoming of the traditional quantitative methods that can merely obtain the local optimal solutions, but can be easily implemented by the common commercial programs. In addition, the resource constraints can be easily added by decision-maker to suit real-world situations. Finally, several examples are included to demonstrate the usefulness and correctness of the proposed method.