A vendor–buyer integrated production–inventory model with normal distribution of lead time

By relaxing the long-term assumption of the deterministic lead time, recently three coordinated vendor–buyer models with exponential distribution of lead time in a two-stage supply chain were presented. The vendor produces a product at a finite rate and delivers the lot to the buyer with a number of equal-sized batches (sub-lots) to meet the deterministic demand. The next batch is ordered when the previous one drops to a reorder point. Shortages were allowed and completely backordered. However, in exponential distribution of lead time, the probability of arrival of a batch earlier is higher than the probability of arrival of a batch late or in the mean lead time. But usually, probability of arrival of a batch earlier or late appears to be smaller than the probability of arrival of a batch in the mean lead time. Thus normal distribution of lead time seems to be a better fit to the problem. Hence their models seem unfit to the concerned problem in practice. Based on this notion, we develop a vendor–buyer integrated production–inventory model following normal distribution of lead time but retaining their other assumptions. To make the model more realistic, set up time per set up of a machine, the highest limit on the capacity of the transport vehicle and the transportation cost and time per batch are imposed. Then we derive an optimal solution technique to the model to obtain minimum expected joint total cost that follows development of the solution algorithm. Extensive comparative studies on the results of some numerical problems are carried out to highlight the potential significance of the present method. Sensitivity analysis to the solutions with variations of some parameter values are also carried out.