Reduction of carbon emissions under sustainable supply chain management with uncertain human learning

: Customers’ growing concern for environmentally friendly goods and services has created a competitive and environmentally responsible business scenario. This global awareness toward a green environment has motivated several researchers and companies to work on reducing carbon emissions along with sustainable supply chain (SSC) management. This study explores a sustainable supply chain system in the context of an imperfect flexible production system with a single manufacturer and multiple competitive retailers. It aims to reduce the carbon footprints of the developed system through uncertain human learning. Three carbon regulation policies are designed to control carbon emissions caused by various supply chain activities. Despite the retailers being competitive in nature, the smart production system with a sustainable supply chain and two-level screening was found to reduce carbon emissions effectively with maximum profit. The obtained results explore the significance of uncertain human learning because of this total profit of the system increased to 0.039% and 2.23%, respectively. A comparative study of the model under different carbon regulatory policies showed a successful reduction in carbon emissions (beyond 20%), which meets the motive of this research.


Introduction
In today's world, customers have added environmental expectations for the products they purchase for their healthy living.That is why the market for sustainable products is becoming competitive and growing fast.In 2018, Nielsen predicted sales of sustainable products in the U.S. to be up to 39.9% greater in 2021 than in 2014 [1].This creates new business opportunities for industrialists to step into this competitive market by promoting a sustainable and smart product.Although these products are green and sophisticated and make people's lives easier, the production of such progressive products releases greenhouse gases, mainly carbon dioxide (CO2), into the atmosphere, which is a threat to our earth.
The American space agency, NASA, has also observed that industrial activities have increased atmospheric CO2 levels from 280 parts per million to 400 parts per million in the last 150 years [2].However, industries could overcome this problem by designing strategies like a sustainable and smart production system, green product management, carbon capture, customer awareness and so on in a supply chain [3].
Parsaeifar et al. [4] examined the effect of pricing, advertising and green product management in a competitive supply chain.Tiwari et al. [5] worked on remanufacturing and repair of imperfect products in a green production system with trade credit.Xiao et al. [6] worked on a SSC where the producer motivates its suppliers to invest in sustainable technology through price and cost-sharing contracts.They found a positive effect of these contracts on the sustainable technology level and profit of all supply chain members.The pricing strategy of retailers is very effective in the purchasing trend of consumers.Nowadays, due to strict governmental laws and regulations against plastic, the demand for biodegradable disposable cutlery made of wood is high.Figure 1 presents a global market analysis of disposable cutlery.Therefore, the government has designed policies to introduce eco-friendly and sustainable products in the market.In India, the Bharat stage emission standards have been implemented by the government to control increasing air pollution.In 2016, the government of India urged the adoption of BS-VI norms for vehicular emissions by 2020, rather than the BS-V norms [7] All this compels the manufacturer to design their production strategies with an environmentally conscious approach and attract customers by spreading awareness toward eco-friendly products.Ecoware is India's largest sustainable food packaging industry and produces 100% natural and biodegradable products made from plant biomass, thereby keeping costs and carbon footprint low.Also, Magnus Eco Concepts is a leading producer of ecological/green commodities which are made of areca palm leaf.Environmental attributes of a product are sometimes very confusing for a consumer to assess and trust compared to other available products.In this direction, Singh et al. [8] studied the "Ecomark," a tag introduced by the Indian government for the identification of green products.Bai et al. [9] formulated an inventory system considering a single manufacturer and two competing retailers.They considered deteriorating products with vendor-managed inventory (VMI) and studied the model under carbon emission policy and green technology investment for centralized and decentralized systems.Li et al. [10] presented a sustainable design for a coal supply chain under four carbon regulation policies.
Carbon footprints are hypersensitive to the type of technology implemented in the production process.Therefore, manufacturers should produce smart products along with the smart production process for environmental sustainability.Sarkar et al. [11] presented a variable production system with an automation policy and material requirement planning to lower carbon emissions for saving the environment.They worked on four subsystems using net present value and solved using an integral transform.
Getting motivation from the above research studies and keeping the above-described market analysis with environmental conditions in mind, we proposed the following research in an inventory supply chain model.The purpose of the present study is to design a SSC (for one manufacturer and multiple competitive retailers) (i) to optimize the total profit along with optimizing the production rate of the manufacturer, order quantities and selling prices of retailers through an imperfect production system in a coordinated and competitive supply chain, (ii) to reduce rate of deterioration using preservation technology, (iii) to study the effect of learning in fuzziness and (iv) to reduce carbon emissions by applying three carbon regulation policies.We have tried to find the answers to the following research questions in this study: 1) How do the manufacturer and competitive retailers coordinate to optimize their production rate, competitive prices and order levels in a SSC? 2) How do human learning and two-stage screening enhance the overall performance of a smart production system? 3) How are awareness programs and competitive price-dependent demand along with preservation technology beneficial for smart production and sustainability?4) How does the concept of learning in fuzziness help to tackle the uncertainty and competitive market scenario?5) How are flexible production and carbon reduction policies effective to achieve sustainability goals?6) How is the competitive advantage of a carbon-efficient supply chain sustained?
To answer these questions, a supply chain network (one manufacturer and multiple retailers) with an imperfect production system is addressed.An attempt has been made to add the following three key areas of sustainability in the supply chain to enrich its significance.
• Environmental sustainability: This is maintained by manufacturing green products, continuous carbon emission monitoring at each stage (production, holding, waste disposal, transportation and deterioration of product) of the supply chain and employing carbon reduction policies to lower it.Also, the waste disposal setup is managed by the manufacturer to avoid unnecessary landfills for throwing away waste [12,13] • Economic sustainability: This is achieved by (i) a smart/flexible production system and maintaining cooperation among members of the supply network (besides, retailers have a competition on price), (ii) retailer strategies to raise customer awareness through awareness programs (that their products are eco-friendly) and competitive prices, (iii) variable rate of deterioration (a function of a maximum lifetime and the cost due to preservation policies) at the retailer end and (iv) a strong inspection process to maintain goodwill toward the product in the market.In this process, first the screening is carried out through a machine by the manufacturer.Second, it is inspected by the retailer manually, and defective products are returned to the manufacturer.Error in manual screening process can be reduced due to learning effect.Also, (v) products with low quality are sent to an alternate market for gaining profit.[11,14].
• Social sustainability: This is covered by adding (i) learning in fuzziness to reduce market ambiguity, (ii) human learning in screening to reduce the percentage of defectives, (iii) awareness programs of retailers to increase customer consciousness toward the purchase of an environmentally friendly product and (iv) implementation of carbon emission policies along with proper waste management setup to reduce the negative impact on ecology [15,16].

Literature Review
In this section, the review of literature based on the contribution of the study is presented.

Competitive sustainable supply chain (SSC) and carbon footprints
Nowadays, industries focus on carbon-efficient supply chains to take the competitive advantage of the market.Usually, supply chain members design some pricing strategies that attract customers, to overcome market competitiveness.Figure 2 shows the annual total carbon-dioxide emissions variations across the world.From 1950 to 2018, emissions increased rapidly.From Fig. 2, it is clear that carbon emissions in India crosses 15 billion tons.In this direction, Wee and Chung [14] worked on an integrated buyer and supplier system consisting of decaying and green component (computer power-supply) production with just-in-time deliveries and remanufacturing.In this model, a new approach of coordinating between the time of storage and the supplier's production level is applied.Bonney and Jaber [17] investigated the effects of inventory planning on environmentally responsible models in detail.They included non-traditional costs related to packaging, waste management, and transportation for promoting green production.Also, Zanoni et al. [18] developed an integrated twostage model with price and environmentally sensitive demand.Mukhopadhyay and Goswami [20] presented an economical manufacturing system containing defective products under two cases.In the first case the pollution cost is constant, while in another case it is variable.Dye and Yang [21] worked on developing a sustainable model by adding commercial borrowing with environmental regulations.The demand was assumed to depend on the borrowing period and carbon cap and trade regulation.Further, Sarkar et al. [22] derived a supplier-manufacturerretailer model under carbon emissions and variable shipping costs using an algebraic method.Sarkar et al. [13] extended the Sarkar et al. [23] model, including a multi-level trade credit policy and a delivery strategy to reduce carbon emissions.A non-calculus method for determining the optimal policy was applied.Tiwari et al. [24] analyzed a coordinated vendor-buyer manufacturing process model which includes production of defectives along with perfect items.Kundu et al. [25] examined a manufacturing model, including remanufacturing with waste disposal, and investigated the impact of carbon emission policies (carbon cap and trade, carbon tax and strict carbon cap) on optimal results and amount of carbon emissions.It was assumed that the company sells its product in two markets and the return rate of the used items depends on the buyback price of the company.Jamali and Rasti-Barzoki [26] worked on a competitive scenario of pricing for green and non-green item producers to maintain economic and environmental sustainability in a supply chain using game theory.The model under integrated and non-integrated scenarios was analyzed, and it was found that the integrated policy is beneficial.Daryanto et al. [27] incorporated product deterioration with carbon footprints in an integrated network.It was observed from the obtained results that integration in the supply network decreases the cost as well as carbon emissions.Olatunji et al. [3] highlighted the importance of competitiveness and environmental consciousness for sustainability and carbon reduction in a production system.It was analyzed that the end customers are responsible for competitive awareness in the production supply chain.Hosseini-Motlagh et al. [28] designed an acquisition price strategy for producers to enhance the collection of used products in a sustainable closed-loop supply chain (CLSC).This strategy increased the market demand as well as a collection of used items.Seyedhosseini et al. [29] introduced a new price-dependent demand for retailers in a two-level (competitive) supply network.Ranjbar et al. [30] introduced two competitive recycling channels in a CLSC between retailer collecting and third-party collector.Wu and Kung [16] addressed SSC with competitive prices to reduce carbon emissions.It was concluded that financial risk is a major factor in controlling emissions which should be balanced by government initiatives.Huang et al. and Manupati et al. [31,32] studied various carbon reduction policies in SSC.Mishra et al. [33] examined green technology investment, preservation technology to control emissions and deterioration in a supply chain with trade credit.The model was studied under full backorder, partial backorder and no backorder.Sarkar et al. [34] introduced a closed-loop supply chain model assuming price and quality-based demand and learning in production.The significant effects of human learning in production were examined in the model.Recently, Alamri et al. [35] developed an economic order quantity model with learning effect, carbon emission and inflation.Wang et al. [36] studied global value chains and carbon reduction in developing countries.They explored a value-added accounting method under a new trade accounting framework to calculate the real emissions embodied in trade using the fuzzy C-means clustering method.Sun and Zhong [37] developed a low-carbon supply chain to study the effects of fairness concerns on optimal policy and utility.Kang and Tan [38] investigated a sustainable supply chain game model under capand-trade policy to study the investment decisions of manufacturers and suppliers.It suggested investing in decarbonization technologies to reduce carbon emissions.
Learning-based supply chains under inflation are rarely studied.It is a clear research gap that should be covered.Although the models presented above introduced low-carbon practices in SSC, no one had worked on implementing the government policies to curb emissions such as carbon caps and carbon trade policies in competitive SSC.

Smart production system
Many researchers in the literature have considered the production rate as constant.Most of the carbon emissions are caused during the manufacturing process, holding of the inventories, solid waste disposal, deterioration and damage in the transportation of the product.Therefore, nowadays, manufacturers prefer smart production systems to curb overall carbon footprints.Generally, smart production is referred to as a controllable/variable production rate.The basic traditional production models could be transformed into smart production systems by utilizing a flexible rate of production, strong screening and emission reduction.Sana et al. [39] incorporated volume flexibility in a deficient manufacturing process.In the model, the manufacturing cost varies with the rate of production, and profit is optimized using the penalty function method.Glock [40] examined flexible production on total cost by reducing production at different time intervals in a two-level inventory system.It was concluded that reducing the production rate resulted in a lesser overall cost.Glock [41] extended the Glock [40] model into a multi-level system with varying production rates and made a comparison with a constant rate of production.Singhal and Singh [42] worked on a volume-flexible inventory process with machine breakdown under uncertainty and shortages.Singhal and Singh [43] extended the Singhal and Singh [42] model assuming damageable items with partially fulfilled shortages and the backorder rate as random.In addition, the uncertainty of the market was covered by the concept of randomness and the learning effect.Sarkar et al. [23] examined a system considering a fixed lifetime of decaying items and a variable backorder rate.Tayal et al. [44] applied preservation technology for dropping the rate of deterioration of the product in a two-stage coordinated production model with shortages and delays in deficits.In addition, a Stackelberg game method was applied to find the solution of the problem.Manna et al. [45] worked on a defective manufacturing system assuming variable demand with screening.It was considered that the production rate is variable, but they did not consider shortages in their model.Sarkar and Chung [46] designed a supply chain network with a flexible production system in which the production rate lies within a prescribed interval.Gautam et al. [12] emphasized reducing defects and carbon emissions in a production system of a two-level supply network.The study involved a strong (multiple) inspection process to decrease waste, and they constructed two different models, by using the integrated problem-solving approach and Stackelberg policy, respectively.The results showed that rather than the Stackelberg policy, the integrated policy is beneficial for reducing emissions without affecting the profit of green SSC.Dey et al. [47] proved that autonomation policy can converge over human inspection error through the automated inspection process, but the investment in the automated inspection system may not be supported by all industries.Thus, the industry may have human inspection if autonomation is not utilized.Sarkar and Sarkar [48] developed a production model of pure biofuel ensuring minimum energy consumption and carbon emission.They applied a controllable production rate to minimize the impurities, and impure fuel is again reworked to produce pure fuel.Sarkar et al. [35] discussed a sustainable supply chain considering the combined effects of improved quality of products and carbon emission reduction through controlling the production process.Recently, Mridha et al. [49] showed combined effects of improved quality of biofuel and carbon emission control with flexible production rate in a sustainable supply chain management.The models presented above did not introduce emission reduction policies for smart production and focused neither on reducing deterioration nor on implementing preservation technology, which is a clear research gap.

Learning in fuzziness
Learning in fuzziness has broad applications.It is a concept that handles the uncertainty level of the existing information base, which is usually gained by time or the number of times work is done.It is a mixture of fuzzy systems and learning specialties.Although many researchers have worked on learning in fuzziness, still it is a new concept for the majority.Bera et al. [50] incorporated the effect of learning in setup cost of every production cycle in a deteriorating model.Glock et al. [51] introduced a learning effect in fuzzy demand in an Economic order quantity (EOQ) model.It was observed that the fuzziness of the information decreased as the learning rate decreased.Pathak et al. [52] studied the learning and forgetting effects in a production process with shortages.It included two models with fixed and fuzzy costs assuming variable demand and decay rates.The two models were analyzed with the help of three examples.Yadav et al. [53] studied the effects of human learning in an inventory model of imperfect production with fuzzy demand and error in screening.Kumar and Goswami [54] added a learning effect in a production model in an imprecise environment.In this model, the faulty items were reworked, and shortages were partially fulfilled.Kazemi et al. [55] developed a fuzzy model considering learning and backorders.It was found that learning in fuzziness reduces the cost and increases the performance of the system.Shekarian et al. [56] extended an imperfect quality model with two different holding costs under learning in an imprecise environment.In the model, the nature of the demand parameter is fuzzy and a function of marketing cost.Soni et al. [57] investigated an imprecise inventory problem and applied the learning phenomenon to control the fuzziness, which results in reduced costs and fluctuations.A comparison of the developed model with or without the learning effect was done, and it was analyzed that the impact of learning in fuzziness lowers the uncertainty of information collected and is helpful in decision making.Giri and Masanta [58] examined the effect of learning in the manufacturing process of a closed-loop supply chain.Saha and Chakrabarti [59] studied the effect of learning on the production cost in a supply chain with a return policy.Huang and Wang [60] analyzed the coordination between information sharing and the effect of learning on pricing decisions in a closed-loop supply chain.Recently, Jayaswal et al. [61] developed an inventory model with trade credit and backorders.In the model, there were effects of learning and trade credit financing with fuzzy and fuzzy learning scenarios.Poursoltan et al. [62] studied the impact of human learning in a vendor-managed closed-loop supply network.Alsaedi et al. [63] developed a sustainable green supply chain model with carbon emissions and two-stage inspection under learning in a fuzzy environment.Supply chain models that implement two stages of human learning are rarely studied.In the studies presented above, it is observed that no one focused on SSC with learning in fuzziness.It is a major research gap in the existing literature.From the above literature survey and gap analysis from Table 1, it is observed that there is a clear research gap in introducing learning in fuzziness in an environmentally sensitive and imperfect production system with two-level screening and variable deterioration rate along with awareness program and price-sensitive demand.Hence, attempting to cover this research gap, the present study was done.To the best of our knowledge, no research has yet used the idea of learning in fuzziness in a SSC.The present study has not been previously labeled.

Problem description
The current study is related to a supply chain network containing a single manufacturer and multiple competitive retailers.During production, the screening process is carried out by the manufacturer with the help of the machine.The screening process segregates the produced lot into the following three categories: (i) perfect products, which are delivered to multiple retailers; (ii) products with some design flaws, which are sold by the manufacturer in an alternate market with low price; and (iii) waste products, which are disposed of by the manufacturer at some disposal cost.Figure 3 illustrates the variations of product level in the described supply network.
When retailers receive the products, they also perform manual screening.Due to manual selection, error in screening is evident, which can be reduced by introducing the learning effect.After screening, retailers use the perfect products for satisfying their market demands and send the waste products to the manufacturer, as the manufacturer has waste management set up.First, a basic (crisp) mathematical structure for manufacturer and retailers is established.Then, it is formulated with a coordinated supply chain model including carbon emissions.Further, the model is fuzzified by using a triangular fuzzy number and defuzzified with the help of the signed distance method.After defuzzification, the obtained fuzzy model is extended to the model with learning in fuzziness.In the model with learning in fuzziness, three carbon emission policies are designed to check carbon emissions.

Notation
The following notation is used to design the mathematical model.Unit selling cost of carbon emission credit   Total amount of carbon emissions generated

Assumptions
The model presented is designed according to these assumptions: 1) In this study, the proposed supply chain considers one manufacturer and multiple retailers over an infinite planning horizon.2) The manufacturer produces eco-friendly products, and the rate of production is the decision variable, i.e., volume flexibility is considered.
3) The manufacturer adopts the lot for lot policy during each production cycle for delivering finished products to retailers.4) During production, 100% of the screening process through the machine is carried out at the manufacturer end.The screening process ends as the production cycle completes.Based on screening, products are segregated into three different categories.
5) Due to transportation, wear and tear is unavoidable, so a screening process is also carried out manually by the retailers with high screening rate.6) Lead time from the ordering of products to supply of products to the retailers is negligible.7) On increasing number of shipments, percentage (%) of defective products is defined as   () =  +  , where b, g are model parameters, c is the learning exponent, and m is the cumulative number of shipments.8) With the help of advertisement policy, retailers can make the customers aware that their products are eco-friendly to get a competitive edge in the market.As retailers are considered as competitors of each other, the pricing policy of one will influence the market of the other.So, a retailer's demand depends on the frequency of advertisement, selling price proposed by him/her and the selling price fixed by other retailers.
Where N is the number of advertisements,   (> 0) is the market base,   (> 0) is the elasticity of demand regarding selling price, and   (> 0) and   (> 0) are the effects of competitor's selling price and advertisements on demand, respectively.9) Rate of deterioration   () = 1 1+  − is considered as the function of maximum lifetime.Here,   is lifetime (maximum) of product for the  ℎ retailer, and lim →    () = 1.
10) Lifetime (maximum) of a product can be improved by adopting different policies of preservation.So, the rate of deterioration at the retailer's end is considered as the function of maximum lifetime and cost due to the adaptation of preservation policies.So, the resultant deterioration rate is 11) Effect of preservation technology cost is defined as   () =   +    , where (  > 0) and (  > 0) are model parameters.12) Partial backlogging is considered here at the retailer's end.13) Carbon emission costs are considered for manufacturing, transportation, waste disposal, inventory holding and keeping the deteriorating items.

Formulation of basic mathematical model for manufacturer and retailers
Here, a basic model for manufacturer and retailers is presented.After that, supply chain models have been developed in crisp, fuzzy, and fuzzy learning environment considering without any carbon regulatory authority.Further, it is extended with some carbon regulatory mechanisms.The inventory level at the manufacturer end follows the pattern depicted in Figure 4.During production, the inventory size of the manufacturer increases (time   ) continually up to time T. In [  , T] , the change in inventory size can be written as

Formulation of manufacturer inventory model
Using initial condition   (  ) = 0, the solution of Eq 1 is given by Total products manufactured per cycle The screening process also completes as soon as the production completes.It separates the manufactured products into three categories: (i) perfect products with probability  1 , (ii) products with design flaws with probability  2 and (iii) waste or scrap products with probability  3 .Out of the total of manufactured products,  1  are delivered to retailers with zero lead time.
The total cost of manufacture is the summation of total holding cost, set up price, manufacturing cost, screening cost, waste disposal cost and transportation cost.For detailed calculation of these costs, see Appendix 1.

𝑇𝑇𝐶𝐶𝑇𝑇
Total profit of manufacturer considering carbon emissions is as follows:

Formulation of retailer inventory model
The inventory level of the  ℎ retailer is shown in Figure 5.When the  ℎ retailer receives   products at time T, some of the products in the lot are found to be damaged.Damage of products may be caused by many reasons, mainly due to combined pressure of piled stocks during transportation.So, the retailer needs to perform manual screening to sort the defective products.This screening process is carried out with high screening rate by the retailer manually.In the proposed model, the retailer's screening process is assumed to be error-prone while screening the products.That is, some of the useful items will be categorized as defective with a probability (1 −   ()), whereas some faulty items will be classified as non-defective, with a probability   ().
When screening ends,   (1 −   ()) units are found to be perfect quality products, which retailers use to fulfill their demand, and     () units of product are found defective.For the  ℎ retailer, the stock at T is   .In [,  1 ] , the inventory level of the  ℎ retailer continuously decreases due to demand and deterioration.
Using initial condition   1 () =   , the solution of Eq 8 is given by At time  1 , the screening process ends, and inventory level decreases by     () units.In interval [ 1 ,  2 ] , the inventory level of the  ℎ retailer is Using the condition   2 ( 2 ) = 0, Eq 10 gives the solution as Using the condition   1 ( 1 ) =     () +   2 ( 1 ),  1 is obtained as In [ 2 , 2], a shortage occurs, from which some are backlogged with backlogging rate   .
Using the condition   3 ( 2 ) = 0, Eq 13 gives the solution as Using the condition   3 (2) = −  ,  2 is obtained as Total cost for the  ℎ retailer can be obtained by summation of ordering cost, buying cost, preservation technology cost, total holding cost of perfect products and waste products, screening cost, backlogging cost, lost sale cost and advertisement cost.For detailed calculation of these costs, see Appendix 2. Total cost of the  ℎ retailer Carbon emissions in inventory holding of the  ℎ retailer Carbon emission from deteriorating products of the  ℎ retailer Total carbon emission for  ℎ retailer considering carbon emissions

Formulation of supply chain inventory model in absence of regulatory authority
The different costs are imprecise in nature, and due to learning, impreciseness decreases.So, it is important to study models under three scenarios: the crisp case, the fuzzy case and learning in fuzziness.Therefore, in this sub-section, we develop three different models for a centralized system by assuming that there is no regulatory body controlling the carbon emissions.These models are as follows: Model 1: Centralized supply chain model including carbon emission without any carbon control mechanism (crisp case).
Model 2: Centralized supply chain model including carbon emission without any carbon control mechanism (fuzzy case).
Model 3: Centralized supply chain model including carbon emission without any carbon control mechanism (learning in fuzziness).

Model 1: Centralized supply chain model including carbon emissions without any carbon control mechanism (crisp case)
For the integrated model, the manufacturer and n retailers work as team members and find the optimal values of P,   and   to optimize the total profit of a system in each cycle.The emissions caused during manufacturing, warehousing, deterioration, waste disposal and transportation activities are investigated throughout the supply chain.The total carbon emission per cycle of the system is The total profit of the system in each cycle is Where   (,   ) and    (  ,   ) are defined by Eq 6 and Eq 18.So, the objective function in Model 1 is defined as Total fuzzy profit of  ℎ retailer using Eq 18: Now, to defuzzify   � (,   ) and    � (  ), the signed distance method is applied.The signed distance of   � (,   ) to 0 � is as follows: The signed distance of    � (  ) to 0 � is as follows: For the defuzzification process, the signed distance method is applied.For this, see Appendix 4.
Substituting the above values in Eq 22 and Eq 23, crisp functions for total fuzzy costs of the manufacturer and  ℎ retailer are obtained.
The total fuzzy profit of the centralized system in each cycle is Where   (  (,   )) and   (   (  ,   )) are defined by Eq 26 and Eq 27.So, the objective function of Model 2 is defined as

Model 3: Centralized supply chain model including carbon emission without any carbon control mechanism (learning in fuzziness)
In this sub-section, the decision maker's learning in estimating the fuzziness values has been used, provided that build-up of knowledge occurs with the number of shipments.It is assumed that decisionmakers learn with time and use their expertise to reduce the fuzziness of the parameters while giving a fuzziness value for the parameters.The learning curve follows Wright's (1936) [67] power learning curve.If learning affects the fuzzy parameters and if their value changes according to the number of shipments, then for  =   ,   ,   ,   ,   ,   ,    ,    ,    ,    ,    ,   ,    ,    ,    and , the values of the  ℎ upper and lower fuzziness parameters at the time of the  ℎ shipment will be The total fuzzy profit functions using Eq 6 and Eq 18 with learning for the  ℎ shipment ( ≥ 1) of the manufacturer and the  ℎ retailer are given as The total fuzzy profit of the centralized system in each cycle is Where   (  (,   )) and   (   (  ,   )) are defined by Eq 32 and Eq 33.So, the objective function of Model 2 is defined as Where total emissions per cycle of the system are given by the Eq 19.

Formulation of supply chain inventory model under the restriction of regulatory authority
In this section, we extend Model 3 by adopting the policies given by the regulatory authority to reduce the carbon footprint in the system.These three models are as follows: Model 4: Centralized supply chain model with carbon tax policy under the effect of learning in fuzziness Model 5: Centralized supply chain model with carbon cap policy under the effect of learning in fuzziness Model 6: Centralized supply chain model with carbon cap and trade policy under the effect of learning in fuzziness 4.2.1.
Model 4: Centralized supply chain model with carbon tax policy under the effect of learning in fuzziness According to this policy, the supply chain manager (company) has to pay a tax on the quantity of carbon emitted in various processes.Suppose  is the per unit carbon tax.Then, the optimization model is represented as Where   �  (,   ,   )� and   are defined by Eq 35 and Eq 19.

Model 5: Centralized supply chain model with carbon cap policy under the effect of learning in fuzziness
According to this regulation, the supply chain manager (company) has an essential restriction, i.e., cap, on the quantity of carbon emitted by them.Suppose   is the carbon cap.So, the optimization model can be written as Where   �  (,   ,   )� and   are defined by Eq 37 and Eq 19.

Model 6: Centralized supply chain model with carbon cap and trade policy under the effect of learning in fuzziness
This policy provides the supply chain manager (company) with an option to purchase an emission limit.According to this policy, a fixed carbon emission limit is provided to the company.Also, extra emission limits can be purchased (if needed).Suppose ∈ 1 and ∈ 2 are the purchasing and selling prices per unit carbon emission.So, the optimization model can be written as Where   �  (,   ,   )� and   are defined by Eq 37 and Eq 19.
2) The optimal production rate decreases in the fuzzy and fuzzy-learning models by 6.45% and 2.03%, respectively, i.e., more in the fuzzy model.
3) The optimal order quantities increase by 104.54% and 82.82% for both retailers in the fuzzy model, which shows an increase in market demand for the product.4) Selling prices of both retailers decrease by 6.26% and 6.01% in the fuzzy model and by 1.75% and 0.55% in the fuzzy-learning model, i.e., more in the fuzzy case.This motivates the customer to buy more.5) Carbon emissions decrease by 4.41% and 1.35% in the fuzzy and fuzzy learning models, respectively.
The results reveal that the models with fuzziness and learning in fuzziness both increase the profit of the system, without increasing the production rate.The fuzzy model generates more profit and less emissions in comparison to the fuzzy-learning model, whereas market uncertainty could be best handled through the fuzzy-learning model.It is also observed that the results obtained in the learning in fuzziness model are closer to crisp models, which shows the significance of learning in fuzziness over fuzziness and proves that learning in fuzziness is an appropriate tool to reduce the cloudiness.Hence, learning in fuzziness Model 3 is recommended as an optimal strategy for decision-makers.From the results of Table 3, the following are observed on applying carbon regulation policies in Model 3: 1) Optimal production rate decreases by 20.65%, 26.96% and 20.65%, corresponding to all the three policies; but for the carbon cap policy (Model 5), it decreases the most.2) Optimal order quantity for retailer 1 increased by 117.91%, 193.28% and 117.91%, and for retailer 2, it increased by 24.34%, 60.18% and 23.89%, respective to all three policies applied, with the maximum with the carbon cap policy (Model 5).
3) The selling prices increase in the carbon cap policy (Model 5) for retailer 1 by 19.19% and for retailer 2 by 20.45%.4) The total profit of the system decreases due to carbon emission cost for all the three policies by 0.32%, 0.13% and 0.31%, respectively.5) Carbon emissions are reduced in all the policies by 15.24%, 20.36% and 15.24%.6) Comparing among all policies discussed, the carbon cap policy (Model 5) shows a maximum drop in carbon emission and production rate, but the total profit of the system corresponding to this policy is minimum in comparison with others.7) The carbon cap and trade policy (Model 6) shows effective drops in production rate and in carbon footprints along with little decrements in the total profit of the system.8) The selection of a strategy for controlling carbon footprints should be customized to the requirements of the supply network in order to make a perfect balance between the needs of a successful SSC and carbon control goals for a cleaner production system.9) It is concluded that among all the three policies, carbon cap and trade policy, Model 6, is best for environmental and economical sustainability both, as it reduces emissions, rate of production and competitive prices efficiently and enhances the profitability of the system.

Sensitivity analysis
In this section, the optimal results obtained for Model 3, Model 4, Model 5 and Model 6 are examined concerning all essential parameters of the system.

Sensitivity analysis of Model 3
For sensitivity analysis, important parameters of Model 3 are increased or decreased by 20%, and the results are presented in Table 4. Based on Table 4, the following conclusions are drawn: 1) On increasing traditional cost parameters (setup cost, holding cost, deterioration cost) of manufacturer and retailers, total profit and production rate decrease, but competitive prices increase.2) On increasing the selling price of the product by the manufacturer in an alternate market, total profit and production rate increase, but competitive prices decrease.
3) On increasing the maximum lifetime of the product, the total profit increases, and production rate decreases.Meanwhile, on increasing preservation technology cost, total profit and production rate both drops.4) Since this is the base model for other models defined, these parameters behave the same in all models.Table 5 shows variations in the total profit of the coordinated supply chain under the competitive prices of two retailers.If the selling price of one retailer decreases from 77 to 74, while keeping other retailer's selling price fixed, the total profit increases.If the selling prices of both retailers decrease together, the total profit again increases.This implies that the manufacturer could convince retailers to reduce their competitive prices to gain profit and increase market demand for a green product.In this way, the competitive advantage of the supply chain could be sustained.Table 6 shows changes in optimal profit when learning rate changes from 0.862 to 0.074 and for different numbers of shipments from m = 25 to m = 100.On increasing the rate of learning, the profitability of the company also increases.Figure 6 shows the improvement due to learning.On the other hand, the optimal profit decreases with an increase in the frequency of shipments (with a fixed learning rate).However, whenever shipments increase, the learning rate usually increases (resulting in increasing total profit).In this way, learning in fuzziness removes the illusion about optimal profit and helps the decision maker to make an appropriate decision.From the results of Table 7, it is interesting that on increasing carbon tax from 1.2 to 1.9 in Model 4, carbon emissions are reduced from 14.90 to 14.74, and total profit is also reduced due to carbon tax.Change in total gain with carbon tax can also be studied in Figure 7.

Sensitivity analysis of Model 5
A carbon cap is a critical parameter in this model.Increasing the carbon cap, the total profit of the system changes, while other variables are almost insensitive to changes in carbon cap. Figure 8 shows change in total profit with change in carbon cap.The above Table 8 shows higher price decreases the total profits.

Industry implications and managerial insights
This paper suggests some insights for supply chain managers and production managers of industries.
For maintaining sustainability, the production manager should first focus on decisions related to production rate.The controllable production rate applied in this study is a very effective strategy for the production of a green and innovative product.It not only reduces the cost due to overproduction or underproduction but also lowers industrial waste and extra energy consumption.Three different strategies, (i) a strong (two-level) inspection with human learning to reduce inspection error, (ii) an alternate market for selling defective products and (iii) a waste management setup to dispose of overall waste, help the decision maker of the production system, where the production process is not perfect, to reduce waste.Hence, the policy for smart production considered in this paper would optimize the sustainability goals of production managers.
Results of the current study give the direction to the supply chain managers that they should motivate their members to set their price competitively, work with team spirit and enhance customer awareness towards green purchases through promotional activities.
For the sustainability of the supply chain, the results of this paper suggest that managers should continuously monitor emissions generated at each stage of the supply chain and apply emissionreducing policies to minimize them.The optimal solutions under different schemes show that each reduces emissions, but carbon cap and trade policy is optimal for environmental and economic sustainability.Therefore, the carbon cap and trade policy with learning in fuzziness should be the most favorable policy for decision-makers to gain profit and minimize carbon footprints.
The sensitivity analysis of different inventory parameters advises inventory planners to take appropriate values of the highly sensitive inventory parameters like learning rates, maximum lifetime of product and preservation technology cost to enhance the gain of this centralized system, along with respective parameters of carbon control policy adopted for overall sustainability.

Conclusions
The carbon regulation policies and learning in fuzziness are the two practical tools to handle the present competitive market situations.In this study, a two-echelon competitive supply network is presented in the shape of a (single manufacturer and multiple retailers) flexible production model for deteriorating products under learning in fuzziness.A manufacturer produces green products, which undergo two-stage screening before dispatching in the market.It is presumed that the deterioration rate depends on the maximum lifetime and preservation technology costs.Retailers promote the product to increase its market demand.The model for a coordinated supply network is investigated under three scenarios: crisp, fuzzy and fuzzy-learning.Further, the model is extended by implementing different carbon regulation policies.The main findings of this study are summarized as follows: • Fuzziness and learning in fuzziness result in enhancing the total profit by 2.23% and 0.04%, along with decreasing the carbon footprint of the system by 4.41% and 1.35%, respectively.Results confirmed that human learning has an effect on maintaining the SSC management for the smart product subject to the reduction of carbon footprints.
• Retailer awareness programs and competitive demand attract customers to buy more, so industrial managers should motivate their supply team members correspondingly.
• The production process, transportation and deterioration are the main contributors to carbon footprints in the system.The applications of three carbon control policies, carbon tax, carbon cap and carbon cap and trade, show (i) decreases in total profit by 0.32%, 0.13% and 0.31%; (ii) drops in carbon footprints by 15.24%, 20.36% and 15.24%; and (iii) increases in order quantity of retailer 1 by 117.91%, 193.28% and 117.91% and of retailer 2 by 24.34%, 60.18% and 23.89%, respectively.
• The results suggest optimal planning of the SSC under learning in fuzziness along with controlling carbon footprints through cap and trade policy.
• Although the implementation of a carbon regulation policy reduces emissions, it increases the financial liabilities of firms.Further, the execution of human learning does not need a large amount of investment for industries.Instead, it reduces carbon emissions, market ambiguity and defective products effectively.Hence, learning in fuzziness and human learning both are important tools to maintain sustainability.
• The present study can be helpful for inventory managers in decision making to gain profit, reduce waste by human learning and decrease vagueness through the fuzzy-learning effect along with an efficient carbon management.In this way, it serves all the three expectations, i.e., economic, social and environmental, for a SSC and leads the research to move a step forward toward a cleaner and safer planet.
• The past research done in this field has not touched these critical areas together.This study has narrowed this gap.
This study has significant applicability to give a new direction to research, and its numerical results are appealing.Still, the effects of competitive prices and advertisement-based demand could be better demonstrated if they were studied in centralized and decentralized scenarios, both using a gametheoretical approach.Therefore, in future research, this study could be considered accordingly.
Further, it can be extended into a closed-loop structure under reverse logistics.A following study could be done with random production, rework, different demand patterns and shortages.Moreover, another attractive extension could be done by adding different profit-sharing contracts along with some government schemes to motivate eco-friendly production.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Figure 1 .
Figure 1.Global market interpretations for global disposable cutlery market.

Figure 3 .
Figure 3. Flow of products in single manufacturer and multi-retailer supply chain.

Figure 4 .
Figure 4. Inventory level of the manufacturer.

Figure 6 .
Figure 6.Variations in total profit with variations in human learning.

Figure 7 .
Figure 7. Change in total profit for a carbon tax for Model 4.

Figure 8 .
Figure 8. Change in total profit for carbon cap for Model 5.

Table 1 .
Literature review with gap analysis.

Table 2 .
Optimal results for various presented models.

Table 3 .
Optimal results corresponding to various policies.

Table 4 .
Results of sensitivity of parameters corresponding to Model 3.

Table 5 .
Sensitivity of competitive selling prices of retailers with total profit in Model 3.

Table 6 .
The sensitivity of learning rate and number of shipments with total profit in Model 3.

Table 7 .
Sensitivity of carbon tax on optimal policy of Model 4.

Table 8 .
The sensitivity of purchasing price on the optimal policy of Model 6.