An EPQ Model with Deterioration, Partial Backlogging, and Carbon Emissions: A Comparison of Two Minimization Methods

Abstract

In 2025, Wahyudi et al. constructed an EPQ-type inventory model involving non-defective and defective products, a demand rate that depends quadratically on time, a holding-cost rate that depends linearly on time, deterioration, and carbon emissions from production and storage activities. In that study, the total cost minimization was conducted using a method involving Maclaurin polynomial approximation without an investigation of the result's accuracy. In the present study, we modify the inventory model constructed by Wahyudi et al. by removing the defective-product assumption and incorporating a partial backlogging mechanism. We divide each inventory cycle into three periods: the production-consumption period, the post-production consumption period, and the shortage period. We conduct the total cost minimization using not only the method involving Maclaurin polynomial approximation but also the Lagrange multiplier method. Across five numerical parameter sets, the two methods yield closely comparable minimizers, with relative differences in the minimum total cost of at most 0.00635%. Furthermore, a sensitivity analysis shows that the length of the production-consumption period, the length of the period after production ceases, the no-shortage time proportion, the total carbon emissions, the regular production quantity, and the shortage quantity depend most sensitively on the lost-sales cost, whereas the minimum total cost depends most sensitively on the initial demand rate.