A fuzzy random integrated inventory model with imperfect production under optimal vendor investment

This article investigates an integrated single-vendor single-buyer production-inventory model with imperfect production under a mixed environment where fuzziness and randomness appear simultaneously. The paper focuses on representing the annual customer demand as a triangular fuzzy number together with an associated probability. A further assumption is that the production process is not perfect and goes ‘out-of-control’ with a certain probability. This causes the vendor, in particular, and the supply chain, in general, to incur an additional warranty cost and also leads to the production of larger batch sizes to compensate the imperfection. In order to avoid these extra costs, the vendor makes an investment to improve the production process quality and hence reduce the number of defective items produced. The expected annual integrated total cost is derived with these assumptions under the n-shipment policy. A methodology is proposed to minimize crisp equivalent of the expected annual integrated total cost so as to obtain the optimal values of the number of shipments, the shipment lot-size, the safety stock factor and the ‘out-of-control’ probability. A numerical example is given to illustrate this proposed methodology and to highlight the advantage of investing in reducing the probability of the production process going ‘out-of-control’.