An integrated production‐distribution inventory system involving probabilistic defective and errors in quality inspection under variable setup cost

This paper presents a probabilistic defective vendor–buyer integrated inventory model with the consideration of quality inspection errors at the buyer's end and setup cost as function of capital investment. An integrated inventory model is established to find the optimal solutions of lot size, setup cost, and the total number of shipments from the vendor to the buyer in one production run, so that the joint expected total cost incurred has the minimum value. We consider three types of continuous probabilistic defective function to find the associated cost of the system. The expected total cost function is derived for each of these three distributions, its convexity is proved via differential calculus. An efficient iterative algorithm is designed to obtain the optimal solution of the model. The computational effort and time are small for the proposed algorithm and it is simple to implement. Numerical examples and sensitivity analysis are used to demonstrate the application and the performance of the proposed methodology. The computational results indicate that if we make decisions with the capital investment in reducing setup cost, it will help to lower the system cost, and we obtain a significant amount of savings to increase the competitive edge in business.