Supply chain integrated inventory model with present value and dependent crashing cost is polynomial

Due to today's highly competitive environment, both reducing lead time and the associated inventory cost are critically important issues in supply chain. And, the consideration of time value effect is lacked in most previous researches. However, the effect of inflation is too critical to ignore. Therefore, we develop an integrated inventory model with crashing cost which was determined by the length of lead time is polynomial to recover the real inventory problems. The objective of this research is to minimize present value of the joint expected total cost over infinite time horizon. Then, we provide a solution algorithm to determine the optimal order quantity, the length of lead time and the number of lots which are delivered from the vendor to the buyer in the solution procedure. Numerical example is provided here to illustrate the solution procedure.