Continuous review inventory model with lost sale reduction and ordering cost dependent on lead time for the mixtures of distributions

We consider a continuous review inventory system for inventory model involving lost sales reduction through capital investment cost function and the reduction of lead time further which reduces the ordering cost. To reduce the lost sales rate, two forms of capital investment cost function, viz. logarithmic and power are employed. Two relationships between ordering cost and lead time, viz. linear and logarithmic are considered. We develop four inventory models by taking different combinations of capital investment cost function and ordering cost lead time relationship. Objective of the study is to reduce the total related cost by simultaneously optimizing the order quantity, safety factor, fraction of the shortages during the stock-out period that will be lost and length of lead time. The lead time demand is assumed to follow a mixture of normal distributions. The optimal solution is derived by developing computer programs using the software MATLAB. We also provide four numerical examples to illustrate the models.