A joint inventory model with reliability, carbon emission, and inspection errors in a defective production system

Nowadays, environment is an important concern of industries parallel to the economy. In this direction, a joint vendor-buyer model is exhibited where the system reliability and inspection errors are discussed along with the carbon emission issue. The main goal of this model is to obtain the optimum investment, shipment size, reliability and lead time even the inspection errors present in the system. A reliability dependent unit production cost is utilized to raise the machinery system reliability. Transportation of products use the single-setup-multi-unequal-delivery (SSMUD) policy to reduce carbon emission. Mathematical problem is solved analytically and a quasi-closed-form solution is found. Total cost is minimized with the optimum level of decision variables. Globality of the decisions is proved by Hessian matrix. Results demonstrate that the total cost is minimized even though the optimum solutions are obtained in quasi-closed- form. Numerical example is elaborated to test the validity of the model and to clarify the comparison among SSSD, SSMD, and SSMUD policies.