Reducing carbon emissions is crucial in the development of a green supply chain. Greenhouse firms regulate carbon emission constraints to achieve sustainable supply chains. Deterioration can be controlled by utilizing a preservation technology investment in most of greenhouse firms as they have a specific (summer) time. Despite the importance of these concerns, few researches have been done on the effect of carbon emissions on the lot sizing sustainable inventory management. To fill this gap, this study develops a carbon cap and tax-regulated sustainable inventory management for a buyer utilizing a linear and non-linear price-dependent demand. This is the first study to explore sustainable inventory management under controllable carbon emissions from a greenhouse farm. An investment in preservation and green technology efforts under different backorder situations is needed. This model furnishes both theoretical and numerical solutions to determine optimal strategies relating to preservation technology (PRT) and green technology (GRT) investment, selling price, cycle time, and the fraction period length for sustainable inventory management. Several numerical examples are presented to validate all sustainable economic order quantity models. This problem is solved by theoretical and analytical methods. The results of this paper, in a controllable sustainable inventory management partial backordering case, exhibit a justifiable amount of profit compared with other backordering cases. The results of this paper are sustainable inventory management in the carbon tax and cap partial backlogging case has a better justifiable profit with the highest cycle time and the lowest value of the fraction length period as well as the lower green technology investment cost compared with other models. Finally, sensitivity analysis is discussed in relation to the managerial implications of the optimal feasible solution with respect to key parameters. © 2020 Elsevier Ltd