Production–Inventory Decisions Under a Carbon Cap-and-Tax Policy with Stock-Dependent Demand in an Autonomation-Based Inspection Setting: A Fixed-Point and Karush–Kuhn–Tucker Analysis

Background: Automation-based inspection removes classification errors from imperfect production, but production inventory models built on it rarely account for carbon regulation or for the effect of displayed stock on demand. Methods: This paper extends an automation-based economic production quantity model with price–quality and service-sensitive demand in two directions: a five-source carbon-emission function regulated by a carbon cap-and-tax policy and a stock-dependent demand. Results: In the numerical study, price settles at the market ceiling, quality at the minimum acceptable level and the rework rate at one; these corner solutions are explained analytically. Relative to a demand, emission-blind baseline, the carbon tax alone lowers profit by 2.01%, stock-dependent demand alone raises it by 7.16%, and the integrated model raises it by 5.03%; because the cycle-average stock approximation overstates the stock effect, this gain falls to 2.33%. Emissions respond weakly to taxes, while a binding cap drives abatement through lower service levels and reduced rework. Conclusions: For the manufacturer, the carbon tax acts as a levy on an almost unchanged operating plan, whereas the cap is the instrument that forces abatement.