To respond to the adoption of carbon pricing regulations, researchers and industry are developing low carbon inventory models that can meet emission reduction targets while maintaining company profits. The challenge is getting tougher when the company is still facing problems related to imperfect product quality. This research solves this problem by developing an economic order quantity (EOQ) model by considering several sources of carbon emissions, as well as the influence of the defective rates, different demand rates, selling price and holding cost for defective products, and shortages backorder. The objective function of the formulated mathematical model is to minimize the total costs which include the emission costs. A numerical example is developed to illustrate the model based on the previous data set. Sensitivity analysis is also carried out to validate the model and to learn more about the system characteristics. The total emissions are calculated and the affecting factors are identified.

In this study an integrated model is proposed for the location inventory routing problem under uncertainty. This problem involves determining the location of distribution centers (DCs) in a three echelon supply chain. The DCs receive orders from the customer and according to a continuous review inventory replenishment policy place orders to the supplier. The products are directly shipped from the supplier to the DCs. The vehicles start from the DCs to fulfill the demands of the customers. Determining the routing of the vehicles is one of the decisions involved in this problem. The demands of customers are stochastically distributed and the capacity of DCs are limited. If one of the DCs undergo a disruption and is unable to fulfill the demands of the customers, shortage may occur. Moreover in the proposed model the shortage is considered as partial backlogging. This means that if shortage occurs, some of the orders result in lost sales and other orders are fulfilled in the next period. In order to optimally solve the proposed model a nonlinear integer programming (INLP) model is developed. However, since the problem is NP-hard, the mathematical formulation cannot be efficiently solved for large sized instances of the problem. Therefore an outer approximation method is developed to solve the problem more efficiently. The computational results show the efficiency of the proposed method.