This paper investigates a dual-channel, two-echelon supply chain comprising a manufacturer and a retailer for a single perishable product. In today's global marketplace, a well-designed return policy is a crucial element that can lead to higher revenues, reduced expenses, improved profitability, and enhanced customer loyalty. Offering the option to return perishable goods can significantly increase consumer willingness to purchase them. The model incorporates a return policy, recognizing its importance in enhancing consumer acceptance and demand in both traditional retail and the manufacturer's e-tail channels. Analytical solutions are derived for centralized and decentralized systems to optimize decision variables that maximize total profit. Numerical examples illustrate the model's findings, revealing that a return policy for perishable goods significantly benefits the supply chain members by increasing profit, particularly with higher demand.

In real-world situations, the human-based inspection process often involves errors in detecting defective products, affecting the quality of products. When the production system enters an out-of-control state in a long-run process, it produces faulty products at a random rate. Based on this, the model introduces an autonomation inspection technology in an imperfect smart complex production system to separate faulty items using a machine-based inspection strategy. The manufacturer reworks the separated faulty items to control their quality. The manufacturer sells new quality and reworked quality products at variable prices and offers service for sold products. The model introduces variable demand and formulates it mathematically by incorporating space and budget constraints. The formulated nonlinear function is solved using the Karush-Kuhn-Tucker method. A numerical example and sensitivity analysis are provided to illustrate the model. Results show that autonomous inspection, quality, and service improve the profit of the proposed model.

Wastewater from a garment production system in a textile supply chain management is one of the main sources of water pollution. Wastewater is still an increasing problem for medium to large production industries as traditional wastewater treatments are not much effective, especially for removal of contaminated organic particles and heavy metals. Photocatalytic ultraviolet process is an eco-friendly and promising wastewater treatment since the last one decade with the removal of organic particles. This study examines a textile supply chain management, which uses the photocatalytic ultraviolet process for the wastewater treatment from the production system. The production system plays an important part to exclude wastewater from the system. The unnecessary production run-time not only produces excess garments but also produces excess wastewater, which is blended with dyestuff extracts. A smart production is able to control the garment production based on the necessity and the control of wastewater generation is possible. Despite of the amount of wastewater, the treatment of wastewater gradually becomes apart of the textile supply chain with economic valuation. The textile supply chain management is built by multi-supplier and multi-manufacture, where the garment production is controlled by a smart production system. The manufacturers provide a photocatalytic ultraviolet wastewater treatment and this wastewater treatment positively effects the market demand of garments. The global optimum solutions of the study prove that the use of smart production reduces the cost of the textile supply chain 18.55% followed by the PUV wastewater treatment 16.01%. The results reveal that an involvement of categorized labor for material handling in a textile supply chain management is cost effective rather than using uncategorized labor. Skilled labors are found to be the most sensitive fora textile supply chain management.

Nowadays, barter exchange has become growingly popular in the national and global industries as an alternative to excessive inventory transfers. Many companies exchange their surplus products on barter platforms for products they need without using money. In a traditional supply chain, the retailer holds unsold products that hinder the supply chain's profitability from reaching its maximum level. This paper solves this issue by proposing a stochastic model of two players (single manufacturer, single retailer) with trade-credit (delayed payment) and barter exchange policy under a supply chain management. In this work, to entice the retailer and increase sales, the manufacturer grants a credit payment facility to the retailer on the items ordered for a specified period, and the manufacturer does not charge any interest on the outstanding amount during this credit period. The concept of a credit period raises the possibility of default risk. In this case, some interest is charged. Several investments are made here to diminish setup and ordering costs and improve the product quality of the system. This work focuses on the flexible production to manage the demand uncertainty and the marginal reduction technology to lessen carbon emissions that occur during the production and inventory holding. In contrast, a retailer can exchange unsold items in the barter market for its required products, which is extensively discussed in this study. Finally, the maximum profit is assessed in terms of credit period, investments, quality improvement, and production rate. The result numerically and graphically proves a huge impact of the barter platform for any business industry on overstock transfers, conserving cash, managing unpredictable demand, and reaching the maximum profit. Moreover, the significant finding is observed in the proposed work that the idea of the flexible production, barter exchange policy, and several investments increase the system profit up to 50.55%.

Unreliability of the manufacturer is a challenging issue for a retailer in order to provide service to consumers and meet the market demand. Due to the unreliability of the manufacturer, the lead time increases, causing shortages. In turn, the retailer faces huge shortages and losses. The lead time can be minimized by reducing the flow time during work-in-process. To reduce the holding cost of the retailer under an increasing demand, the single-setup-multi-unequal-increasing-delivery is introduced by the unreliable manufacturer. But delivered products to the retailer variable demand are lower in volume than the ordered products. Due to the variable demand that is selling price and service dependent, the number of shipments during transportation increases for the single-setup-multi-unequal-increasing-delivery policy. The main goal of this research is to manage unequal shipments from the unreliable manufacturer for gaining more profit. The stochastic optimization approach is considered for the analytical solution. The quasi-closed-form solution is determined for the decision variables of the model. The study is illustrated both numerically and graphically. Results prove that the retailer can still control the profit if the manufacturer can reduce the flow time of the production and maintain a perfect retailing strategy. The research shows that the single-setup-multi-unequal-increasing-delivery policy is 1.14% more profitable than the single-setup-multi-delivery policy, and 8.53% more profitable than the single-setup-single-delivery policy.

Currently, apart from manufacturing processes, the remanufacturing of products is considerably important. Appropriate remanufacturing requires the operation of long-run manufacturing systems. However, in long-run processes, the production system may convert to an out-of-control state due to machine breakdowns. Then, defective products are frequently produced; this increases wastage and disrupts environmental sustainability. In this model, a smart autonomation policy is deliberated for an error-free inspection in separating defective products during production. The autonomation policy facilitates waste reduction through remanufacturing. This paper concentrates on customer awareness and service-dependent demand, which directly improves the overall profitability of the system. A discrete investment to reduce setup cost, continuous investment to collect used goods, and cap-and-trade strategy to limit carbon emission are considered to obtain a more realistic model. Classical optimization method is applied for global maximum profit test of the profit function with respect to cycle length, customer awareness, service investment, discrete investment to reduce setup cost, number of shipments, and container capacities. Numerical testing, sensitivity to total profit in different cost parameters, and comparisons with previous research are explained. Some special scenarios including graphical representations are discussed to prove that a large investment is more beneficial than the cost of specific setup and collection.

In recent times, environmental responsibility is an important factor that determines the success of a Supply Chain. In this study, we have considered green production in the light of various co-ordinations and contracts. This is a two-echelon Supply chain consisting of one manufacturer who designs and develops a green product and the retailer sells it to the environmentally aware customers and the awareness is converted to actual purchasing behaviour by the retailers marketing strategy and the manufacturer’s product design and development which includes technology usage to develop the greenness,packaging and several other factors which were not studied earlier. All these factors are involved in our demand function which is distinct from the existing literature. The model is developed under three contracts, Price-only, green marketing cost sharing and two-part tariff contracts. This is an well-established fact that co-ordination enhances the economic benefits to every tier member of a chain. Our findings also establish that co-ordination and co-operation among members will enhance their environmental sustainability. In this way they can carry out their social responsibilities towards our environment. It is also noticed that as the environmental consciousness of the consumers increases, the cost sharing contract is more profitable for manufacturer than that for retailer whereas the price only contract is profitable for the retailer.

The corona virus pandemic situation has changed customers’ purchasing habits and this change can be lasting. As a consequence, new marketing patterns need to be identified and appropriate strategies need to be adopted to reach the majority of consumers. The current paper explores a dual channel supply chain considering the impact of customers’ channel preferences under a stochastic demand environment. The influences of retailers’ cooperative advertising and manufacturers’ direct online services on the channel’s best decisions and coordination have been investigated. The model analyzes the best decisions and benefits of single and dual-channel strategies under centralized and decentralized decisions. Based on customers’ channel preferences, the proposed model identifies some regions and the corresponding best-profit channel strategies that are preferred by each member of the chain and the whole chain. The numerical simulation of the proposed model confirms that in dual-channel marketing, cooperative advertising and online services are beneficial to the manufacturer and the whole supply chain but detrimental to the traditional retailer. The scope for dual-channel use becomes wider when the manufacturer offers a return policy to the seller. The decentralized situation of the proposed dual-channel model has been coordinated by sharing cooperative advertising costs and adopting a return policy. Finally, the proposed paper analyzes the sensitivity of key parameters and provides some managerial insights for choosing an optimal channel strategy.

Smart production plays a significant role to maintain good business terms among supply chain players in different situations. Adjustment in production uptime is possible because of the smart production system. The management may need to reduce production uptime to deliver products ontime. But, a decrement in production uptime reduces the projected production quantity. Then, the management uses a limited investment for pursuing possible alternatives to maintain production schedules and the quality of products. This present study develops a mathematical model for a smart production system with partial outsourcing and reworking. The market demand for the product is price dependent. The study aims to maximize the total profit of the production system. Even in a smart production system, defective production rate may be less but unavoidable. Those defective products are repairable. The model is solved by classical optimization. Results show that the application of a variable production rate of the smart production for variable market demand has a higher profit than a traditional production (52.65%) and constant demand (12.45%).

The present paper analyzes a dual channel two-echelon supply chain model consisting of one manufacturer and one retailer for a single product. The retailer uses traditional retail channel whereas the manufacturer uses e-tail channel as well as retail channel to boost the sales of the product. Customer feedback is an important factor in present global business environment because from the customer feedback, the manufacturer can reveal the level of satisfaction of the customers and this can help the to improve the product quality. Good feedback from the customers also attracts other new customers to buy the products. A company can never know whether customers are getting value out of their product without customer feedback. The effects of getting customer feedback from the customers and the promotional effort by advertising or by sales team’s initiatives are taken into account in the present model. The model is solved analytically to determine the decision variables which maximize the total profit in case of decentralized as well as centralized systems and the results have been illustrated by numerical examples. The study of this paper reveals that the profit margins of the individual channel members increase considerably if the feedback of the customers on the product is considered.

Utility of efficient automobiles has compelled us to explore various aspect of its promotion. Our manuscript analyzes a monopolistic automobile manufacturer trying to produce efficient vehicles along with traditional vehicles. A utility-based dynamic production design has been studied under price-dependent subsidy (PDS) model to explore the effects of government subsidy on manufacturing of Battery-operated vehicles (BOV) and Diesel vehicles (DV). We have compared the PDS mechanism under two sustainable and green strategies namely, BOV socially responsible (BSR) strategy and DV energy reduction (DER) strategy to bring out the best possible channel dynamics that can optimize its gross profit margins. We speculated that an increase in threshold price has positive impacts under BSR and DER strategies. However we found numerically and analytically that, even under subsidized BOV's, energy efficient DV's are still in much demand. Our study compels us to explore three different cost cases and the best possible optimal variant has been numerically established. We note that BSR strategy holds good in some circumstances while DER strategy rules in other. Our findings have been illustrated graphically by using real-life scenarios to validate our models.

Production of defective products is a very general phenomenon. But backorder and short-ages occur due to this defective product, and it hampers the manufacturer's reputation along with customer satisfaction. That is why, these outsourced products supply, a portion of required products for in-line production. This study develops a flexible production model that reworks repairable defec-tive products and outsources products to prevent backlogging. A percentage of total in-line production is defective products, which is random, and those defective products are repairable. A green invest-ment helps the reworking process, which has a direct impact on the market demand for products. A classical optimization solves the profit maximization model, and a numerical method proves the global optimal solutions. Sensitivity analysis, managerial insights, and discussions provide the highlights and decision-making strategies for the applicability of this model.

The paper describes an integrated/centralised supply chain model consisting of one supplier, one manufacturer and one retailer within a finite time horizon. The manufacturer produces, at a finite rate, in each lot. The lot production rate in a batch increases with a rate λ in successive batch and the produced items are supplied to the retailer. The objective of the proposed model is to optimize the average total profit under the consideration of the proportional increase in the size of successive shipments within a batch production run and the production time of the supplier. The corresponding average profits of the supplier, the manufacturer and the retailer and the average total profit of integrated model are obtained. The results obtained in the numerical examples clearly establish that it is always beneficial in terms of profit when the size of the successive shipment is a variable. Therefore, size of the successive shipment should be variable in order to get more profit. A sensitivity analysis of the optimal solution with respect to changes of the parameter values is also carried out to strengthen the proposed model.

We introduce the definition of intuitionistic fuzzy pseudo-norm and study some properties of convergence and [Formula: see text]-convergence in intuitionistic fuzzy pseudo-normed linear spaces.

The proposed study described the application of innovative technology to solve the issues in a supply chain model due to the players' unreliability. The unreliable manufacturer delivers a percentage of the ordered quantity to the retailer, which causes shortages. At the same time, the retailer provides wrong information regarding the amount of the sales of the product. Besides intelligent technology, a single setup multiple unequal increasing delivery transportation policy is applied in this study to reduce the holding cost of the retailer. A consumed fuel and electricity-dependent carbon emission cost are used for environmental sustainability. Since the industries face problems with smooth functioning in each of its steps for unreliable players, the study is proposed to solve the unpredictable player problem in the supply chain. The robust distribution approach is utilized to overcome the situation of unknown lead time demand. Two metaheuristic optimization techniques, genetic algorithm (GA) and particle swarm optimization (PSO) are used to optimize the total cost. From the numerical section, it is clear the PSO is $ 0.32 $ % more beneficial than GA to obtain the minimum total cost of the supply chain. The discussed case studies show that the applied single-setup-multi-unequal-increasing delivery policy is $ 0.62 $ % beneficial compared to the single-setup-single-delivery policy and $ 0.35 $ % beneficial compared to the single-setup-multi-delivery policy. The sensitivity analysis with graphical representation is provided to explain the result clearly.

Social activities, economic benefits, and environmental friendly approach are very much essential for a sustainable production system. This is widely observed during the Covid-19 pandemic situation. The demand for essential goods in the business sector is always changing due to different unavoidable situations. The proposed study introduces a variable demand for controlling the fluctuating demand. However, a reworking of produced imperfect products makes the production model more profitable. Partial outsourcing of the good quality products has made the production system more popular and profitable. Separate holding cost for the reworked and produced products are very helpful idea for the proposed model. Moreover, consumption of energy during various purpose are considered. Separate green investment make the model more sustainable and eco-friendly. The main focus of the model is to find the maximum profit through considering optimum value of lot size quantity, average selling price, and green investment. The classical optimization technique is utilized here for optimizing the solution theoretically. The use of concave 3D graphs, different examples, and sensitivity analyses are considered here. Furthermore, managerial insights from this study can be used for industry improvement.

In this paper, focus is on the study of spectrum and the spectral properties of bounded linear operators in intuitionistic fuzzy pseudo normed linear spaces(IFPNLS). It is done by studying regular value, resolvent set, spectrum of a linear operator in IFPNLS. Also, some properties of spectrum and resolvent of strongly intuitionistic fuzzy bounded(IFB) linear operators in IFPNLS are being developed. It is observed that, for a linear operator P in an IFPNLS, the resolvent set rho(P) and spectrum sigma(P) are nonempty, rho(P) is open and sigma(P) is closed set.

While developing supply chain models, many researchers have shown great interest on how to reduce the consumption of non-renewable sources of energy, as non-renewable sources of energy is limited. The purpose of this paper is to formulate a three echelon supply chain model when the demand of items is assumed to be stochastically dependent on price, quality and reduction of energy. In the centralized model, suppler, manufacturer and retailer are the three members of the supply chain. The model is solved analytically to obtain optimal values of order quantity, unit price, promotional effort and amount of energy consumption which maximizes the profit function of the supply chain. Two decentralized models namely MR-Nash and MS-Nash have also been considered in a separate section. These two models have also been solved analytically to obtain the optimal solution of the decision variables. Three proposed models have been illustrated with a numerical example by considering exponential distribution of customer's demand. The sensitivity of the optimal solution revealed the appropriate channel strategy in case of decentralized scenario. It is speculated that when the manufacturer and the supplier collaborates, the profit difference is reduced by \begin{document}$ 39 \% $\end{document} than that of the MR-Nash.

Currently, most countries are moving towards digitalization, and their energy consumption is increasing daily. Thus, power networks face major challenges in controlling energy consumption and supplying huge amounts of electricity. Again, using excessive power reduces the stored fossil fuels and affects the environment in terms of $ {\rm CO_{2}} $ emissions. Keep these issues in mind; this study focuses on energy-efficient products in an energy supply chain management model under credit sales, variable production, and stochastic demand. Here, the manufacturer grants a credit period for the retailer to get more orders; thus, the order quantity is related to the credit period envisaged in this model. Considering such components, supply chain members can reduce negative environmental impacts and significant energy consumption, achieve optimal results and avoid drastic financial losses. Additionally, including a credit period increases the possibility of default risk, for which a certain interest is charged. The marginal reduction cost for limiting carbon emissions, flexible production to meet fluctuating demand, and continuous investment to improve product quality are considered here. The global optimality of system profit function and decision variables (credit period, quality improvement, and production rate) is ensured through the classical optimization method. Interpretive sensitivity analyses and numerical investigations are performed to validate the proposed model. The results demonstrate that the idea of credit sales, flexible production, and quality improvement increases total system profit by $ 28.64\% $ and marginal reduction technology reduces $ {\rm CO_{2}} $ emissions up to $ 4.01\% $.

The waste of energy in the present era is a dangerous signal for the future. All categories of consumers should come forward to moderate energy use and prevent wastage. This study focuses on a controllable energy consumption-based sustainable inventory model incorporating variable production rates, improved service, partial outsourcing planning, defective production, restoring reworkable items, disposing of non-reworkable items, and energy-saving steps. Reducing unusual energy consumption in production systems reduces carbon emissions and maximizes the system’s profit. An improved service level attracts customers, increases demand, and improves product reputation. Separate holding costs of reworked, defective, and perfect-quality items are considered for every lot delivered and reworked. The demand in the market is related to price and service. A traditional optimization technique examines the global optimization for the profit function and decision variables. Numerical illustrations as well as concave 3D graphs validate the analytical results and provide a sensitivity analysis for different parameters. The model is validated through special cases and comparison graphs.