Sustainability inventory management model with warm-up process and shortage

Fast-paced markets require complex interactions from all supply-chain agents to satisfy customer demands and needs. The manufacturing industries face some difficulties in terms of production amounts and smooth delivery rates. Technical experts found that a warm-up period before a production run helps address those challenges and improves the workability of machine tools in the manufacturing process. The use of a warm-up process causes a reduction of faulty products (an adverse production outcome) and improves operational efficiency. Also, a shortage in the supply of commodities creates difficult conditions for inventory management decisions, posing the same production problems as mentioned above. Consideration of the warm-up process has recently been included in the scope of operations research, but it is necessary to study its interaction with the presence of shortage. This study presents a system where a manufacturing environment utilizes the warm-up process in its initial phase and shortages are allowed during the production period, in addition, the study takes into account carbon emissions during manufacturing to integrate environmental concerns. We assume that the company has the capability to trade the surplus carbon capacity it hasn't produced. This study offers a comprehensive framework that incorporates former research that addresses warm-up process, carbon emissions, shortages, and defective items. To solve the proposed non-linear programming problem with inequality constraints, we employ the Karush-Kuhn-Tucker (KKT) conditions method to determine the optimal solutions. Managerial insights are derived, and sensitivity analysis highlights the effects of the system parameters on the decision variables. The sensitivity analysis results indicate that the carbon trading cost has a significant impact on the overall cost, and subsequently, the company's profit.