In this paper, a period review inventory model with controllable lead time has been considered where shortages are partially backlogged. The backorder rate is dependent on the backorder discount and the length of the protection interval, which is sum of the review period and the lead time. Two cases have been discussed for protection interval demand which are (a) Demand distribution is known (Normal Distribution) (b) Demand distribution is unknown (Minimax distribution). Further, algorithms have been developed which jointly optimize the backorder discount, the review period and the lead time for each case. Numerical examples are also presented to illustrate the results.

This paper presents a new vendor-buyer system where there are different objectives for both sides. The proposed method of this paper is different from the other previously published works since it considers different objectives for both sides. In this paper, the vendor’s emphasis is on the crashing of the setup cost, which not only helps him compete in the market but also provides better services to his customers; and the buyer’s aim is to reduce the lead time, which not only facilitates the buyer to fulfill the customers’ demand on time but also enables him to earn a good reputation in the market or vice versa. In the light of the above stated facts, an integrated vendor-buyer stochastic inventory model is also developed. The propsed model considers two cases for demand during lead time: Case (i) Complete demand information, Case (ii) Partial demand information. The proposed model jointly optimizes the buyer’s ordered quantity and lead time along with vendor’s setup cost and the number of shipments. The results are demonstrated with the help of numerical examples.