At the manufacturing plant or while the products are being transferred from one supply layer to another, there is a considerable possibility of receiving damaged or faulty items mixed in with non-defective commodities. This research focuses on the non-defective and defective products that are shipped to retailers by their suppliers. The retailer reworks faulty items to make them non-defective, and the retailer receives a discount on the cost of purchasing defective items. The presented inventory system addresses the uncertainty in inventory costs and also considers the deterioration of items with prioritized maximum product life. In this study, our aim is to minimize the total inventory cost when demand rate as a function of quality and power pattern of time under crisp and generalized triangular neutrosophic environments.Based on the payment deal, interest charges are imposed only when the payment delay has passed a particular allowable time limit. The neutrosophic number, which provides three different types of membership functions representing truth, hesitation, and falseness, is used in the inventory model to handle the cost pattern's uncertainty. A particle swarm optimization approach is used to analyze the proposed inventory model, and the results are validated using a numerical example and sensitivity analysis for various parameters.

The potential to obtain defective or damaged items with non-defective commodities is common to experience at the production unit or when shipping products from one layer to another. This research focuses on the faulty things that retailers receive from suppliers. The retailer has set a restriction on the percentage of defective things, and the retailer receives a discount on the cost of purchasing defective items. The proposed inventory system handles the uncertainty in inventory costs and also considers the demand and deterioration of items with prioritized maximum product life. This work minimizes total inventory cost when demand rate as a function of reliability and power pattern of time under a crisp and triangular neutrosophic environment. The inventory system for degrading items considers the predictability and power pattern of time with a reasonable payment delay. The interest charges are applied only after a specific permissible time limit in the proposed inventory system. The neutrosophic number that defines three different kinds of membership functions representing the truth, hesitation, and falseness is applied in the inventory model in handling the uncertainty of the cost pattern. The proposed inventory model is investigated using a particle swarm optimization algorithm, and the results are validated using a numerical example and a sensitivity analysis for various parameters.

Environmentally friendly goods are market-oriented goods that create less environmental damage. Their manufacture is related to a product development process designed to consider the environmental consequences that might develop throughout their life cycle. In reality, the global demand for herbal goods is expanding since herbal products are manufactured from plant extracts such as leaves, roots, flowers, and seeds, among others, and cause less environmental destruction. This study introduces a novel, eco-friendly demand determined by the usage of herbal and chemical substances in products. In this context, companies producing these products are encouraged. Firms are interested in producing eco-friendly products while keeping an eye on carbon emissions. This paper presents a sustainable inventory model of non-instantaneous decaying items that follow this eco-friendly demand under partially backlogged shortages. In this study, emission releases due to inventory setup, degradation, and holding were estimated, as were carbon emissions under cap and tax policies. This approach invests in green and preservation technologies to reduce carbon emissions and deterioration. To address the imprecision of the model’s cost parameters, we converted them to Pythagorean fuzzy numbers. The optimum profit of the inventory model with carbon emissions is estimated by considering the time that the inventory level takes to reach zero and the replenishment time as decision variables. Numerical examples and a sensitivity analysis of significant parameters have been conducted to examine the effect of variation in the optimal inventory policy.

Global warming is mainly caused by carbon emissions. Currently, fewer countries are concentrating on reducing carbon emissions. The primary strategy utilized by numerous countries to achieve carbon emissions reduction is the carbon tax policy. With this in mind, a sustainable two-warehouse inventory model was taken carbon tax into account for a controllable carbon emissions rate by investing in green technology initiatives under uncertain emission and cost parameters. The globe is currently experiencing an eco-friendly period. Many individuals are interested in purchasing natural or herbal items since they are made from natural sources and do not affect the environment. The demand for products made with herbal or natural ingredients is considered eco-friendly demand. This study examines a two-warehouse inventory model of deteriorating commodities with price and marketing-dependent eco-friendly demand. The inventory system is presented to handle the inventory in the depository with last-in-first-out and first-in-first-out strategies. After comparing both the policies under deterioration rate and holding cost, this study recommended a suitable dispatch policy. Interval-valued numbers and fuzzy numbers are the mathematical techniques that deal with uncertainties, so this model’s emission and cost parameters are taken as interval-valued numbers, and the storage capacity of the owned warehouse is a Pythagorean fuzzy number. The optimal solution for the two-warehouse inventory system is evaluated by taking the parametric form of interval-valued cost parameters and the new concept of the ranking function of triangular Pythagorean fuzzy numbers. Numerical results prove that emissions are reduced by 87% under green technology investment in both policies. As a consequence, in the FIFO policy, the total cost of the two-warehouse inventory system decreases by 34.45% and cycle length increases by 5.72%, and in the LIFO policy, the total cost of the two-warehouse inventory system decreases by 34.42% and cycle length increases by 11.19%. Sensitivity analysis of the key parameters has been performed to study the effect of various parameters on the optimal solution.

At the manufacturing plant or while the products are being transferred from one supply layer to another, there is a considerable possibility of receiving damaged or faulty items mixed in with non-defective commodities. This research focuses on the non-defective and defective products that are shipped to retailers by their suppliers. The retailer reworks faulty items to make them non-defective, and the retailer receives a discount on the cost of purchasing defective items. The presented inventory system addresses the uncertainty in inventory costs and also considers the deterioration of items with prioritized maximum product life. In this study, our aim is to minimize the total inventory cost when demand rate as a function of quality and power pattern of time under crisp and generalized triangular neutrosophic environments. Based on the payment deal, interest charges are imposed only when the payment delay has passed a particular allowable time limit. The neutrosophic number, which provides three different types of membership functions representing truth, hesitation, and falseness, is used in the inventory model to handle the cost pattern's uncertainty. A particle swarm optimization approach is used to analyze the proposed inventory model, and the results are validated using a numerical example and sensitivity analysis for various parameters.