An Inventory Model in a Three-Echelon Supply Chain for Growing Items with Imperfect Quality, Mortality, and Shortages under Carbon Emissions When the Demand Is Price Sensitive

This research develops an optimization model for growing items in a supply chain with three stages: farmer, processor, and retailer while considering imperfect quality, mortality, shortages with full backordering, and carbon emissions. In the farmer stage, during the growing period, not all articles survive until the end of the period, so a density function of the probability of survival and death of the growing articles is taken into account. Moreover, it is considered imperfect quality in the retailer’s stage because as the supply chain goes down, there exists a greater probability of product defects. Here, the end customer (consumer) can detect poor-quality aspects such as poorly cut, poorly packed, expired products, etc. An inventory model that maximizes the expected total profit is formulated for a single type of growing items with price-dependent polynomial demand. An algorithm is developed to solve the optimization problem generating the optimal solution for order quantity, backordering quantity, selling price, and the number of shipments that maximizes the expected total profit per unit of time, and a numerical example is used to describe the applicability of the proposed inventory model. Finally, a sensitivity analysis has been carried out for all the input parameters of the inventory model, where the effect of each of the parameters on the decision variables is shown to extract some management knowledge. It was found that holding costs in the three stages of the supply chain have a substantial impact on the total profit per unit of time. In addition, as the demand scale parameter increases, the company must raise the selling price, which directly impacts the expected total profit per unit of time. This inventory model has the advantage that it can be applied to any growing item, including animals or plants, so it helps the owners of farms or crops to generate the most significant possible profit with their existing resources.