Sustainable circular economy production system with emission control in LED bulb companies

Mother Earth has a completely sustainable circular life cycle pattern. In its life cycle, there is no harm created to any living creature or to the environment. In this paper, a sustainable circular economic production and consumption system for a LED bulb firm that follows the same cycle pattern as our planet is developed. The circular economy concept, green technology, and carbon cap-and-trade policy are introduced in this model to control the carbon emission rate and resource depletion in LED firms. The profit function is maximized by Lagrange’s multipliers method and Karush–Kuhn–Tucker (KKT) criteria. This paper determined the optimal production quantity and circularity index of the LED bulb for a wise manufacturing process. The concavity of the optimal profit function is proved by using the Hessian matrix method. Different linear and non-linear combinations of demand and profit functions were discussed. This article claims that the circularity level of LED bulbs has influenced their selling price, cost, and demand. Green technology and carbon cap-and-trade policies improved the sustainability of LED bulb companies. Numerical examples, results discussions, and an optimal solution table are provided to show the implication of this model for LED bulb companies. Sensitivity analysis is presented for key parameters. Managerial implication explained in terms of arrived results. Limitations and possible future extensions of this model are given in the conclusion section.


Parameters ω
The product (LED bulb) Circularity index Q Quantity of production (units/year) T Time period (days) λ(ω) Demand (units/year) λ 0 Demand for the product (LED bulb) when it is in linear version (units/year) c 0 The setup cost ($/setup) p(ω) Gross profit per unit ($/unit) p 0 The base unit gross profit ($/unit) c c The cost of holding a product (LED bulb) on a daily basis per unit c c ≠ 0 ($/unit/year) P Production rate of LED bulbs (P>D) (units/ year)

Introduction
The circular economy concept's implementation in the production industry and all other industries is increasing nowadays because of its sustainability. It is also the perfect replacement for the present linear economic model (See Fig. 2). Life revolves around a circular cycle. Everything on this planet grows and returns back to the earth, where it again flourishes. But we've managed to break the loop through our daily activities. Now the economy is exploding uncontrollably without any care for nature. Produce-Use-Dispose is a linear economic model that we are following currently (See Fig. 1). Every year, millions of tons of new resources are converted into new materials and consumed, then thrown away as waste. According to an international resource panel report, global resource consumption is expected to double by 2050 (Ekins and Hughes 2017). The Earth Overshoot Day was July 29, 2021, when all humans on the planet used up more natural resources than the planet could renew in that year. The assessment of the International Resource Panel (IRP, 2020) shows the necessity for drastic reductions in global resource consumption in order to accomplish both environmental and socioeconomic goals. The WWF issued the Living Planet Report 2020, which indicated a worldwide species loss of 68 percent in less than 50 years, a catastrophic loss never witnessed before (Almond et al. 2020). Therefore, a circular economy model is needed to solve these problems. Circular economy production model development is critically important for eliminating waste, pollution, climate change, biodiversity loss, new resource input, and saving the environment (EMF 2015). Ghisellini et al. (2016) collected EC related research articles from the last twenty years and gave an extensive review. They found the origins, basic concepts, advantages and drawbacks, design, and adoption of CE at various levels (micro, meso, and macro) worldwide. Velenturf and Purnell (2021) presented a value framework and ten principles designing, implementing, and evaluating a sustainable circular economy. Many circular economy literature reviews have been  done with the aim of finding the development of circular economy frameworks, practices, implementation, performance assessment, benefits, and limitations (Goyal et al. 2020;Kalmykova et al. 2018;Lieder and Rashid 2016;Merli et al. 2018;Prieto-Sandoval et al. 2018;Sassanelli et al. 2019;Suzanne et al. 2020). Recently, Geissdoerfer et al. (2020) did an excellent review on how the circular economy model minimizes waste, emissions, and resource depletion. They also discussed the historical development of circular economy models and innovations. Stahel (2016) emphasized the importance of globalizing the circular economy in order to conserve resources and energy. Awasthi et al. (2019) and Geng et al. (2019) discussed how the circular economy is addressing electronic waste while also reducing negative environmental and human health issues. The European Parliament presented its ideas to the European commission circular economic action plan (CEAP) on February 10, 2021, focusing on eight primary areas: batteries, construction and buildings, ICT, plastics, textiles, packaging, food, and water. Brydges (2021) investigated how circular economy implemented in fashion industry of Swedish.
However, most research works on creating sustainability in production and consumption processes have focused solely on reducing emissions through green technology and carbon cap-and-trade policies (Chen et al. 2013;Datta et al. 2019;Entezaminia et al. 2021;Lu et al. 2020;Mishra et al. 2020;Wang et al. 2018). But sustainability can be achieved only by reusing, recycling products for a long lifespan and reducing resource depletion (Jones et al. 2013;Rene et al. 2021;Singh and Ordoñez 2016;Troschinetz and Mihelcic 2009;Zaman and Lehmann 2013). As a result, the currently existing production and consumption models are not more effective for environmental sustainability given the current exponential growth of the human population and resource consumption to meet their needs. Given the lack of research into more efficient and sustainable production models in fast-growing industries, the goal of this study is to create a production model that incorporates a circular economy, green technology, and a carbon cap-and-trade policy.
The purpose of this study is to answer the following questions: 1. How can a sustainable production system be developed with the circular economy concept, green technology, and carbon cap-and-trade policy? 2. What is the optimal production quantity and circularity index of the product for maximum profit? How are the various demand and profit functions going to affect the profit? 3. How can this circular economy model be implemented in LED light companies?
This study will contribute more to eco-friendly production and consumption development by using circular economy systems in the current competitive economic world. This will help address the current shortage of resources and energy for human survival.
In Section "Literature review", a detailed literature review on related topics is provided. Notation and assumptions are given for model formulation in Section "SEPQ model with circularity index". The mathematical model formulation for finding the optimal production quantity, optimal circularity index of product, and optimal total profit function is derived in Section "The unit profit function and the demand function". Numerical examples are provided for five different models by considering LED company case study data: (a) linear demand with a linear profit function; (b) linear demand with an exponential profit function; (c) linear demand with logistic profit; (d) logistic demand with a linear profit; and (e) logarithmic demand with a linear profit. A concavity graph of total profit functions, an optimal solution table, special cases, and result discussion are provided in Section "Numerical examples and discussion". Sensitivity analysis and industry implications are presented in Section "Sensitivity analysis". Conclusion: future extension opportunities and limitations are given in the final section.

Literature review
A well-organized literature review of previous works in this field is required to comprehend future work in this field. By searching on the web of science, 737 papers were collected, and 182 papers were chosen after the first review. Then, 78 papers were selected out of 182 after the final review.

Circular economy
The circular economy is gaining popularity as a viable alternative to the present financial model of take-make-usethrow. Because it is a closed waste-free loop, it saves energy, reduces new resource consumption, and is eco-friendlier and more sustainable (see Fig. 2). According to (Ellen MacArthur Foundation 2019), the three main principles of a circular economy are eliminating waste and pollution, circulating products and materials, and restoring nature. We have explained these three principles in detail to get a better understanding of the circular economy business approach.

Eliminate waste and pollution
Eliminating waste and pollution is the first principle of a circular economy. In today's economy, we take resources from the Earth, use them to manufacture things, and then discard them as garbage. This is how a linear economy system works (see Fig. 1). If we consider crisp pockets, due to their making process and design, they cannot be recycled or composted, so they end up as waste. The Zero-Circle Company solved the plastic pollution problem by making biodegradable plastic from seaweed that can be used to package food products (Zero-Circle 2020). The company Apeel minimizes food waste by covering fresh vegetables and fruits with an edible, plant-based coating that prevents oxidation and water loss, both of which lead to deterioration (James 2012). Dye-Coo, a textile company, has invented a process for dying fabrics without requiring water, thereby eliminating harmful waste water (Dyecoo 2008).

Circulate products and materials
Circulate products and materials is the second principle of circular economy. It aims to keep the product and materials in use as much as possible. The products are kept in circulation via two cycles: the biological cycle and the technological cycle. The technical cycle repairs, reuses, recycles, and remanufactures items. Composting biodegradable materials on the earth and improving soil quality are part of the biological cycle. The company Ecovative creates completely biodegradable packaging products from mushroom roots (Ecovative 2007). Resortecs has made a thread that dissolves at a different melting point, which makes it easier to disassemble clothes for the recycling process. They have reduced carbon emissions and textile waste by up to 80% (Resortecs 2017). The company Loop collects old packaging from customers and stores it, sets up deposit returns, sorts and stores it, and then sends it back to manufacturers clean and hygienic so they can refill it again (Loop 2019).

Regenerative nature
Connect the Dots is helping farmers in rural Sao Paulo, Brazil, use regenerative techniques to build a circular food economy (Connect 2016). Using regenerative techniques improves the health of the soil, slows down climate change, and reduces the need for synthetic pesticides and fertilizers. Natura is a cosmetic company that helps reduce Amazon deforestation and creates a regenerative economy (Natura 1969). The firm incorporates ucuuba seeds into Ekos, a moisturizer for the skin. Therefore, the farmers are preserving the trees, and they get $15 per year from each tree. Shellfish and seaweed are produced in combination using Green-Wave's polyculture ocean farming technique in a manner that benefits the environment (Greenwave 2014). Ocean farms for seaweed and shellfish restore natural systems by lowering ocean acidity and providing a home for a variety of marine creatures. Seaweeds are good CO 2 observers, which reduces global warming. Bioplastics are made from seaweed, which reduces plastic waste and creates jobs in the coastal area. Therefore, this ocean farming regenerates nature. The virgin resource consumption rate can be reduced by increasing the recycling level of products, as shown in Fig. 3.
The circular economy is a way to solve global problems like climate change, biodiversity loss, pollution, and waste. Till today, most of the researchers extended the inventory production model by adding green technology and a capand-trade policy to create sustainability (Datta et al. 2019;Lu et al. 2020;Mishra et al. 2020). However, using these two techniques is not very effective in reducing new resource input. This problem needs a better solution model. The circular economy model is much more promising for reducing new resource consumption because of its reuse and recycle process model. Few researchers have used circular economy models alone to create sustainability, but no one has combined circular economy, green technology, and cap-and-trade policy to create a new model for sustainability. In the initial stage of creating a circular economy model, business strategists and product designers are playing the main roles, so guiding them is important. Bocken et al. (2016) developed an excellent framework of strategies to guide them and also provided many product designs and business model strategies for slowing resource input. Identifying the degree of recycling level of a product is essential, or, in other words, finding the circularity level of the product. Lewandowski (2016) designed an EOQ inventory model with the circular economy concept and proved the product might have a varied level of circularity, which can be quantified by an index. A circularity indicator enables the consumer to make wise decisions while buying products. The product's demand, cost, and selling price are all affected by the circularity level, which suggests a variety of linear and nonlinear relationships. Because of this, Rabta (2020)   the aluminum sector, reduce environmental pressure, and support green development. China's biggest aluminum making company is taken as a case study. Martinho (2021) gave novel insights on the implementation of circular economics measures and how they relate to sustainable growth. Finding the best CE indicator plays a vital role, so Yadav et al. (2020) provided 31 CE indicators as well as a framework for improving the CE adoption process. Later, Velasco-Muñoz et al. (2021) studied a total of 41 circularity indicators for uses in agricultural systems were evaluated in depth to establish their strengths and shortcomings. The applicability of circularity indicators to agriculture is assessed. Rajput and Singh (2020) optimized product-machine allocation to achieve cleaner production and a circular economy. An Industry 4.0 MILP model was proposed that uses mixed integer linear programming (MILP).
In Fig. 3, the red curve represents the decreasing of virgin resources with respect to decreasing of circularity index value ω. The green curve represents the increasing of recycled resources for production with respect to increasing of circularity index value ω.

Inventory production model with green technology investment and emission control
The EPQ (Economic Production Quantity) model has been frequently utilized in many companies for better production. Many researchers have extended this basic model by adding new costs like green technology investment costs and carbon emission costs in order to create sustainability. At first, Chen et al. (2013) developed an EOQ model with a carbon cap and tax policy to reduce carbon emissions. They have altered the order quantity to cut emissions. Later Hovelaque and Bironneau (2015) also reduced emissions by altering batch sizes or order quantity in the EOQ model and found that it was more effective. In manufacturing firms, only more greenhouse gases are released so Datta (2017) Implemented green technology and carbon tax schemes in the production inventory system. They have taken the rate of production as a decision variable that could be varied up to the limits of the machine. In a real-life situation, it is not possible to have constant demand all the time. Sometimes we have to deal with uncertain demands. Liao and Deng (2018) developed an EOQ model for uncertain demand with carbon constraints to address this issue. In most production processes, defective products are made, so there is a need to remanufacture those defective products and also reduce carbon emissions. Wang et al. (2018) came up with the best policy on carbon taxes by taking a two-period approach to production decision-making that makes a clear distinction between new and remanufactured products. The consequences of the emissions tax on the best production and remanufacturing procedures were investigated. Many researchers have tried to reduce emissions by implementing either a carbon tax policy or a carbon cap and trade policy in the production process. But combining both policies and implementing them in the production process will be more effective in reducing emissions, so Datta et al. (2019) introduced a hybrid carbon-regulatory scheme to reduce emissions, and GT investment was made separately for each emission section. Mishra et al. (2020) demonstrated a carbon cost and cap model for sustainable economic manufacturing quantity that has been considered for CO2 emissions to be kept under control by making an investment in GT. Mishra et al. (2017) created a controllable emission and deterioration rate reduction, an SEOQ model with two forms of demand that depend on price. This study took into account making investments in PT and GT in both backordered and non-backordered circumstances.

Sustainable production and consumption
The manufacturing industries like electronics, automobiles, petroleum, chemicals, plastics, transportation equipment, food, leather, metal, paper, wood, textiles, and clothing are consuming huge amounts of non-renewable natural resources like oil, natural gas, coal, metals, and groundwater (Bp 2022). Although we are all aware that these resources are limited and non-renewable, we still continue to follow an unsustainable approach. The electronic waste, textile waste, toxic wastewater, and plastic waste generated from these industries are leading to the unsustainability of the earth and putting the lives of future generations in danger. Global warming has increased as a result of the huge amounts of greenhouse gases released by these industries. So, there is an urgent need to transition from unsustainable industrial consumption practices to sustainable ones. New innovative production strategies and resource-saving technologies are necessary to achieve sustainability without negatively impacting the environment, humans, or animals.
As a first step, Heijungs et al. (2010) created the ISOframework for product life cycle assessment in order to achieve balanced social, environmental, and economic growth for sustainable development. Collecting different ideas from different researchers to create sustainability will be more effective and powerful. So, Almeida et al. (2013) analyzed forty-eight articles from fifteen nations that were presented at the third International Workshop Advances in Cleaner Production held in São Paulo, Brazil, in 2011. From this analysis they found a variety of innovative ideas to assist the industrial sectors in their pursuit of sustainability. Also, Pallaro et al. (2015) analyzed 42 relevant articles published in major academic journals between 2004 and 2014, included in the collaboration of business institutes' academic journals, and discovered that many people are just concerned with production rather than consumption. They indicated that future research on integrated consumption and production stages might aid the automotive industry in developing business models that are long-term and sustainable. Implementation of these ideas is not enough; evaluating whether they are having a positive impact is more important. Tseng et al. (2013) studied the management of a green supply chain, design, and practices, looked at the possibilities for sustainable consumption and manufacturing in Asia. The article examined green supply practices, the implications of lean manufacturing, green innovation, the management of a green supply chain, and methods of evaluation and implementation procedures. Also, Lee and Lee (2014) developed a design for an inventory that focuses on the evaluation of articles related to sustainable manufacturing. Sustainability cannot be achieved with innovative ideas alone it requires mathematical proof, statistical data, and real-life case studies. Therefore, Zadjafar and Gholamian (2018) constructed a sustainable inventory using a mathematical model, and the effects of environmental elements on social aspects were explored. And they tested the suggested model's utility in a real-world pulp & paper mill case study. Also, Tayyab et al.
(2020) established a foundation for the sustainable growth of a cleaner system of textile production (in the water-processing category). In a many-step system of production, the manufacturing model is assessed in terms of policies relating to the environment and the treatment of sewage water, and the ideal amount of a batch is established using a metaheuristic technique with reduced CO 2 . Green technology and cap-and-trade policy ideas are mostly used by researchers to create sustainability. Some of the recent research works, the first of which is Lu et al. (2020) investigated potential cooperative and competitive concerns of sustainable production models with joint green investment under various CO 2 reductions. A Stackelberg approach to game theory was used to find the optimal solution between the vendor and retailer. The second one Mishra et al. (2020) demonstrated a carbon cost and cap model for sustainable economic manufacturing. It has been considered that by investing in green technologies, emissions can be kept under control, with and without scarcity conditions. The third one Sepehri (2021) found that by using preservation and carbon emission reduction technologies in production models, they had a positive impact on the environment and a more sustainable model.

Adaptation to LED bulb and carbon emission reduction
Adaptation to LED lighting has created a huge change in the lighting industry with a positive impact. The light-emitting diode is the most energy-saving and newly emerging lighting technology. LED light bulbs of higher quality last longer, endure longer, and deliver light quality that is comparable to or better than incandescent and fluorescent lighting (Byun et al. 2013;Khorasanizadeh et al. 2015;Pattison et al. 2018;Reineke et al. 2009;Singh et al. 2015). The benefits of LED's are their long lifespan, energy efficiency, performance improvement in the environment, ability to operate at low voltage with no UV emissions and heat release, flexibility in design, and eco-friendliness. LED companies can easily adopt the circular economy model. The transformation of the linear economic production and consumption model of the LED bulb company into a sustainable circular production and consumption model is shown in Fig. 4. Additionally, by adopting the circular economy concept in LED bulb production and lighting systems, we can reuse, reducing resource input, carbon emissions, and environmental damage (Balaram 2019; Buchert et al. 2012;Mills and Jacobson 2011;Rahman et al. 2021;Wehbie and Semetey 2022;Zamprogno Rebello et al. 2020). Adaptation of LED lights is necessary for developing countries due to an increase in power needs. For example, India's lighting power consumption is expected to climb to 120,000 GWh/year by 2030, up from the present 55,000 GWh/year. In order to have a comprehensive knowledge of led light adaptation benefits, it is necessary to review the research works of led light adaptations by various nations in various sectors up until today. So here we have chronologically listed the LED light adaptation with carbon emission reduction research works. The first one is (1) Sangwan et al. (2014) examined the ecological consequences of four types of lights in India: incandescent, fluorescent, CFL, and LED lamps, over the course of their lives. (2) Every house can be lighted with LED bulbs because they are easy to adopt and money-saving. Khorasanizadeh et al. (2015) replacing incandescent lights with LED bulbs, each Malaysian household can reduce yearly power use, electrical energy expenditures, CO 2 and other GHG emissions in terms of illumination. (3) Park et al. (2015) studied the economic advantages of switching squid-jigging vessels' fishing lights from metal halide lights to light-emitting diode lights in order to minimize fuel expenditures and pollution. (4) Beu et al. (2018) the LED lights have significantly revolutionized the lighting business, with more than four times the performance of fluorescent lights and new possibilities for advanced control systems. The necessity of CE cannot be overstated. Retrofitting present luminaires and installing contemporary management systems should be the solution to further lowering carbon emissions. (5) Nakano et al. (2018) in Bogor, Indonesia, researchers looked into what factors influenced people's willingness to buy home energy-saving technologies. And the energy efficiency labeling scheme has been found to have a favorable impact on LED lighting purchases. (6) Booysen et al. (2021) fluorescent lights are being replaced with LED bulbs in South African schools. Showed that they saved more energy and reduced electricity expenditure and carbon emissions. (7) Kamat et al. (2020) studied India's lighting market's recent, quick, and ongoing transition to LED technology, from a minimal market share to LEDs becoming the leading lighting products within 5 years. In addition, between 2014 and 2018, annual LED bulb sales increased more than 130 times, resulting in an estimated annual energy savings of over 30 billion kWh, which led to low carbon emissions. (8) Nguyen et al. (2021) demonstrated in the Vietnamese purse seine fishery that employing LED lighting can boost profitability while reducing fuel use and CO 2 emissions. A comparative study of this paper and existing works is presented in Table 1.

SEPQ model with circularity index
The currently existing production models are only focused on minimizing the cost and maximizing the profit of the company without any concern for the environment. We have updated the assumptions of the production model by taking into account the circularity index of the LED bulb products and green technology investment with a cap-and-trade policy to reduce CO 2 emissions. Consider one type of LED bulb product, the one-location inventory model. The manager produces a predetermined quantity Q at the start of each period to ensure that demand is met precisely in the next time period. The LED bulb can be manufactured in either a normal or a circular fashion (more recycling efficiency), where the circularity of the LED bulb is determined by an ω(0 ≤ ω ≤ 1) index. The unit gross profit is assumed to be a function of the LED bulb circularity level. The graphical representation of LED bulb Production system with two different circularity index value of ω is shown in Fig. 5.

Assumptions
• The demand for LED bulb is a function of the circularity index (ω). So, it can be written as λ(ω)=λ 0 +αω where 0 ≤ ω≤ 1 and α represents maximum additional demand factor and λ 0 is the demand for the product (LED bulb) when it is in linear version. (Rabta 2020;Thomas and Mishra 2022). • The price index of LED bulb is a function of circularity index (ω). So, it can be written as p(ω)=p 0 +bω where 0 ≤ ω≤ 1 and b represents unit premium factor and p 0 is the base unit gross profit. The circularity index means that how much percentage of the original LED bulbs can be recycled and used it again. (Rabta 2020; Thomas and Mishra 2022). • The demand rate λ(ω)=λ 0 +αω is deterministic and constant for a fixed ω (Rabta 2020). • An amount of carbon emission reduction due to green technology (G) is ζ (1-e -ψG ) because ζ is the reduced amount of carbon emission after the green technology investment and ψ measures the efficiency of green technologies (Mashud et al. 2021). • Lead time is zero. • The inventory is replenished as soon as the level of the inventory reaches to zero. Thus, shortages are not allowed. • Quantity discounts are not allowed. • Production is done in batches or lots. • Units are produced and used /or sold simultaneously.

The unit profit function and the demand function
Assuming a linear relationship between demand and unit profit is the simplest way to describe the impact of circular labelling on demand and unit profit λ(ω)= λ 0 +α ω>0 and p(ω)=p 0 +bω>0 where α and b are constants. However, the linearity assumption of the two components of the model may be simplistic and unrealistic. A measure of circularity can take into account the non-linear relationship of product (LED bulb) demand to better represent its impact on cost and price. For instance, some features of circularity may be straightforward to execute to some extent, but later advancements become more difficult and expensive. This can be written as an exponential cost function with the same impact on unit profit.
Let p 0 , δ b and b are parameters.
It's also possible to write the demand function in a nonlinear way. Consider the demand function which is logarithmic.
Let λ 0 , γ and α are to be constant parameters. Also, the demand function which is logistic Let λ 0 , α and γ are constant.

The profit functions
Taking into account the impact of the circularity of LED light on sales prices, demand, and costs, we will continue to maximize profits instead of minimizing costs like in the basic EPQ inventory. The main goal is to figure out what the best production quantity Q * also to find ω * in [0, 1] the optimal circularity level of LED light which increase the profit. Let p (ω) and λ(ω) are monotone and continuously differentiable twice. Suppose product's level of circularity has no effect on unit profit or on demand (e.g.λ(ω) and p(ω)=p 0 ) then our model corresponds to the traditional production model. If omit this circumstance then the following cases are raised.
Case a Depending on the value of ω the unit gross profit and demand are rising (or one remains unchanged while the other one is increasing). Clearly, the best result is ω * =1 and.

Case b
Depending on the value of ω the unit gross profit and demand are decreasing (or one remains unchanged while the other one is decreasing). clearly, the best result is ω * =0 and.
Case c either the unit gross profit function p(ω) is increasing in ω or the demand function λ(ω) is decreasing in ω. In this situation, the model's optimum parameters may vary from the above trivial results. The average profit is.
Our aim is to increase the average profit subject to 0 ≤ω The Lagrangian function is The optimality of solution is given by the Karush-Kuhn-Tucker (KKT) criteria are Based on conditions (10) and (11), there are three possible cases that let us find solutions to the maximization problem. When ω = 0. then, when μ 1 =0 then, when μ 2 =0 and ω-1 = 0. Thus, ω=1, Q = .
The answers we obtained should fulfill all of feasibility requirements. (The value of ω and Q should be positive also Eq. (13) and constraints should satisfy). If the objective function is concave, then KKT criteria are enough for the optimality of a solution. Anyway, the attributes of the unit profit function p(ω) and demand function λ(ω) influence the concavity objective function. In the coming sections, (10) 1 ( − 1) = 0, we examine at the linear type of the demand and unit profit functions, as well as some nonlinear cases.
Proof See appendix 1.
Proof See appendix 2. Proof See appendix 5.

Algorithm
In this section, we have provided an algorithm for finding optimal total profit value with optimal production quantity, .
. an optimal circularity index, and optimal green technology investment. The process of finding optimal total profit is given step by step in the algorithm (See Fig. 6). The optimum solution can be found by using the algorithm to validate the problem numerically.

Numerical examples and discussion
This section provides five SEPQ inventory models. Each of these models has different demand and profit functions. They are: a) linear demand and linear profit function; b) linear demand and exponential profit function; c) linear demand and logical profit function; d) logical demand and linear profit function; and e) logarithmic demand and linear profit function. In order to show the applicability of these sustainable production models in real-life situations, the Electric Lamp and Component Manufacturers Association of India (ELCOMA) Company has the same product inventory system. According to the Green Tech Solution Company, 95% of LED and CFL bulbs can be recycled. So, this company can easily adopt a circular economic production model for producing green LED and CFL light bulbs. Converting to LED made a good impact on the environment. For example, in India, the Domestic Efficient Lighting Program (DELP) was launched on 1 May 2015. As of August 28, 2021, India had saved 21, 937,5747 kWh of energy per day, INR 745,740,000 in cost per day, and 59 MW of avoided peak demand per day. In addition, using LED bulbs reduces emissions by 179,888 tons per day. The company can determine the optimal production quantity Q*, the optimal circularity index of the product ω*, and the total profit of the company separately from each of these five different production models by using algorithms and the Mathematica 9 software version.

Example 1: Linear demand and linear profit functio
Using Mathematic 9.0 software and Algorithm (See Fig. 6) with considering initial parameters.α = 150,b = -0.25, P = 30, p 0 =2, c o = 16, c c = 0.2, λ 0 = 10,K =4, η = 1, ζ = 0.2, ψ = 0.4, J =1000 The algorithm was used to find out the maximum total profit value with the optimal production quantity, optimal circularity index, and green technology investment in the case of a linear demand and profit function model. We have obtained the optimal total profit TP=969.002 with green technology investment G=2, optimal production quantity Q*=8.02 and the optimal circularity index of the LED bulb ω*=0.03611. The optimal solution is highlighted in Table 2. The graphical representation of total profit is shown in Fig. 7. The contour plot of the total profit function w.r.t ω and G is shown in Fig. 7(a), and the concavity of the total profit function w.r.t ω and G is shown in Fig. 7(b).

Fig. 6
Optimal total profit finding algorithm  . 7 The red mark represents the local maximum feasible point in the contour plot of TP with respect to ω and G, as illustrated in Fig. 7a. Total profit Concavity with a circularity index and investment in green technologies in the linear profit function case, red spot indicates the optimal value of profit in Fig. 7b 1 3

Example 2: linear demand and exponential profit function
Using Mathematic 9.0 software and Algorithm (See Fig. 6) with considering initial parameters.α =600,b = -0.25, P = 40, p 0 = 4,c o = 40, c c = 0.2, λ 0 = 10,K =8, η = 1, ζ =0.2, δ = 8, ψ = 0.4, J =1000. The algorithm was used to find out the maximum total profit value with the optimal production quantity, optimal circularity index, and green technology investment in the case of a linear demand and exponential profit function model. We have obtained the optimal total profit TP=923.078 with green technology investment G=5, optimal production quantity Q*=9.44295 and the optimal circularity index of the LED bulb ω*=0.01013. The optimal solution is highlighted in Table 2. The graphical representation of total profit is shown in Fig. 8. The contour plot of the total profit function w.r.t ω and G is shown in Fig. 8(a), and the concavity of the total profit function w.r.t ω and G is shown in Fig. 8(b).

Example 3: linear demand and logistic profit function
By using Mathematic 9.0 software and Algorithm (See Fig. 6) with considering initial parameters.α = 600,b = -0.25, P = 40, p 0 = 2,c o = 20, c c = 0.2, λ 0 = 20,K =6, η = 1, ζ =0.2, δ =8, ω 0 = 0.3, ψ = 0.4, J =1000. The algorithm was used to find out the maximum total profit value with the optimal production quantity, optimal circularity index, and green technology investment in the case of a linear demand and logistic profit function model. We have obtained the optimal total profit TP=944.648 with green technology investment G=5, optimal production quantity Q*=10.0204 and the optimal circularity index of the LED bulb ω*=0.00916. The optimal solution is highlighted in Table 2. The graphical representation of total profit is shown in Fig. 9. The contour plot of the total profit function w.r.t ω and G is shown in Fig. 9(a), and the concavity of the total profit function w.r.t ω and G is shown in Fig. 9(b).

Example 4: logistic demand and linear profit function
By using Mathematic 9.0 software and Algorithm (See The algorithm was used to find out the maximum total profit value with the optimal production quantity, optimal circularity index, and green technology investment in the case of a logistic demand function and linear profit function model. We have obtained the optimal total profit TP=5908.79 with green technology investment G=4, optimal production quantity Q*=10.6422 and the optimal circularity index of the LED bulb ω*=0.06168. The optimal Fig. 8 The red mark represents the local maximum feasible point in the contour plot of TP with respect to ω and G, as illustrated in Fig. 8a. Total profit Concavity with a circularity index and investment in green technologies in the exponential profit function case, red spot indicates the optimal value of profit in Fig. 8b 1 3 solution is highlighted in Table 2. The graphical representation of total profit is shown in Fig. 10. The contour plot of the total profit function w.r.t ω and G is shown in Fig. 10(a), and the concavity of the total profit function w.r.t ω and G is shown in Fig. 10(b).

Example 5: Logarithmic demand and linear profit function
By using Mathematic 9.0 software and Algorithm (See Fig. 6) with considering initial parameters.α = 400,b =

Fig. 9
The red mark represents the local maximum feasible point in the contour plot of TP with respect to ω and G, as illustrated in Fig. 9a. Total profit Concavity with a circularity index and investment in green technologies in the logistic profit function case, red spot indicates the optimal value of profit in Fig. 9b Fig . 10 The red mark represents the local maximum feasible point in the contour plot of TP with respect to ω and G, as illustrated in Fig. 10a. Total profit Concavity with a circularity index and invest-ment in green technologies in the logistic demand function case, red spot indicates the optimal value of profit in Fig. 10b -0.25, P = 350, p 0 = 1,c o = 6, c c = 4, λ 0 = 5, K =6, η = 1, ζ =0.5, δ = 8, ψ=0.4, = 0.1, J =1000. The algorithm was used to find out the maximum total profit value with the optimal production quantity, optimal circularity index, and green technology investment in the case of a logarithmic demand and linear profit function model. We have obtained the optimal total profit TP=962.507 with green technology investment G=5, optimal production quantity Q*=6.3075 and the optimal circularity index of the LED bulb ω*=0.1532. The optimal solution is highlighted in Table 2. The graphical representation of total profit is shown in Fig. 11. The contour plot of the total profit function w.r.t ω and G is shown in Fig. 11(a), and the concavity of the total profit function w.r.t ω and G is shown in Fig. 11(b).

Linear
With no green technology G=0 and with circularity index ω=0 we get the optimum production quantity Q*=6.2725 and TP=968.984. Now with green technology G=2 and with circularity index ω=0 we get the optimum production quantity Q*=6.6426 and TP=969.826. And with no green technology G=0 and with circularity index ω=1 we get the optimum production quantity Q*=26791 and TP=1088.89.now with green G=6 and with circularity index ω=1 we get the optimum production quantity Q*=30.0275 and TP=1103.49. The linear profit function special case results are shown in Table 3.

Exponential
With no green technology G=0 and with circularity index ω=0 we get the optimum production quantity Q*=6.76123 and TP=921.678. Now with green technology G=4 and with circularity index ω=0 we get the optimum production quantity Q*=7.31613 and TP=926.652. And with no green technology G=0 and with circularity index ω=1 we get the optimum production quantity Q*=28170.2 and TP= 179350.now with green G=30 and with circularity index ω=1 we get the optimum production quantity Q*=38728.3 and TP=307274. The exponential profit function special case results are shown in Table 3.

Logistic
With no green technology G=0 and with circularity index ω=0 we get the optimum production quantity Q*=8. 13116 and TP=941.194. Now with green technology G=4 and with circularity index ω=0 we get the optimum production quantity Q*=8.86286 and TP=945.32. And with no green technology G=0 and with circularity index ω=1 we get the optimum production quantity Q*=52.2042 and the TP=1610.51.now with green G=8 and with circularity index ω=1 we get the optimum Fig. 11 The red mark represents the local maximum feasible point in the contour plot of TP with respect to ω and G, as illustrated in Fig. 11a. Total profit Concavity with a circularity index and invest-ment in green technologies in the logarithmic demand function case, red spot indicates the optimal value of profit in Fig. 11b production quantity Q*=60.4005 and TP=1666.98. The logistic profit function special case results are shown in Table 3.

Logistic demand
With no green technology G=0 and with circularity index ω=0 we get the optimum production quantity Q*=8.5586 and TP=5900.86. Now with green technology G=4 and with circularity index ω=0 we get the optimum production quantity Q*=10.66247 and TP=5908.74. And with no green technology G=0 and with circularity index ω=1 we get the optimum production quantity Q*=8.26351 and TP=5889.62 now with green G=4 and with circularity index ω=1 we get the optimum production quantity Q*=10.9052 and TP=5897.81. The logistic demand function special case results are shown in Table 3.

Logarithmic function
With no green technology G=0 and with circularity index ω=0 we get the optimum production quantity Q*=1.93996 and TP=947.072. Now with green technology G=2 and with circularity index ω=0 we get the optimum production quantity Q*=2.17882 and TP=975.462. And with no green technology G=0 and with circularity index ω=1 we get the optimum production quantity Q*=11.9329 and TP=957.263.now with green G=7 and with circularity index ω=1 we get the optimum production quantity Q*=15.4048 and TP=998.155. Suppose there is no carbon tax, no cost of green technology investment, and no product circularity index in this study. The study then switches back to the classic EPQ paradigm. The logarithm demand function special case results are shown in Table 3.

Discussion
All of the numerical example results are addressed in this section. The optimal solution table clearly shows that when the circularity index value of LED bulbs and green technology increases, then the profit also increases. The circularity index value of an LED bulb represents the degree of recycling capacity of an LED bulb, and green technology investment costs are used for installing solar panels and wind turbines to access renewable energy resources and reduce emissions. From Table 2, we can observe that the linear demand and linear profit function, the linear demand and exponential profit function, the linear demand and logistic profit function, the logistic demand and linear profit function, and the logarithmic demand and linear profit function production model cases with GT investment obtained the maximum total optimum profit value compared to those without GT investment. In linear demand and linear profit function, the total profit TP=969.002 with G=2 and optimal circularity index of the LED bulb ω*=0.03611. In linear demand and exponential profit function, the total profit TP=923.078 with G=5 and the optimal circularity index of the LED bulb ω*=0.01013. In linear demand and logistic profit function, the total profit TP=944.648 with G=5 and the optimal circularity index of the LED bulb ω*=0.00916. In logistic demand and linear profit function, the total profit TP=5908.79 with G=4 and the optimal circularity index of the LED bulb ω*=0.06168. In logarithmic demand and linear profit function, the total profit TP=962.507 with G=5 and the optimal circularity index of the LED bulb ω*=0.1532. From these five cases, the company can adopt any one of the production models' cases for themselves in order to increase their company's profit and decrease carbon emissions and the use of virgin resources in the production process. In the logistic demand function case, more profit is obtained compared to other cases.

Sensitivity analysis
Sensitive analysis is required to find the impact on total profit when we make small changes to key parameter values of α, b, P, K, c 0 , and c c . Let us examine the impact of minor parameter modifications on all five cases of optimal profit. To accomplish this, we vary the value of the key parameters by a relatively small amount while maintaining all other parameters are held constant and analyze the changes in the optimum revenue. The firm management can choose a bit more distinct value than the maximum one because, in all five examples, small modifications to one of those variables do not result in a significant change in total revenue. The firm management can choose a bit more distinct value than the maximum because, in all five examples, small modifications to one of those variables do not result in a significant change in total revenue. For example, in the case of logistic demand and a linear profit function, when we consider the parameter value a = 1999.8, we get a total profit of 5908.27 and when we consider the optimal parameter value a = 2000, we get a total profit of 5908.79. So, likewise, the circularity index and order quantity can be changed without affecting the total profit. So, from this, we can clearly understand that sometimes, adopting a more circular product can be done for absolutely zero additional expense. The results of the sensitivity analysis are presented in Table 4.

Industry implication in LED Bulb Company
1. In the case of linear demand and linear profit function, an increase in ordering cost parameter c 0 results in a decreased profit. Small changes in the parameters α, b, P, K and c c resulted in no significant increase in profit. From a managerial perspective, if the LED bulb company manager takes the minimum order cost, the company will get the maximum profit. Reducing the ordering cost leads to a reduction in the total cost of the company. Since small changes in the other parameters do not affect the total profit, the LED bulb company can obtain more profit by increasing production and reducing carrying costs. 2. In the case of linear demand and an exponential profit function, increasing parameter K reduces profit. Small changes in the parameters α, b, P, c 0 and c c lead to a slight change in profit. From a managerial perspective, an increase in K that is the amount of carbon emissions dependent on per unit produced reduces the total profit, so the LED bulb company can make a small reduction in production and also make small adjustments in other parameters to increase the profit of the LED bulb company. Because small changes in the other parameters lead to only a slight change in profit. 3. In the case of linear demand and logistic profit function, an increase in parameter K results in a decrease in total profit, and an increase in ordering cost parameter c 0 and carrying cost parameter c c results in a profit decrease. Small changes in the parameters α, b and P did not lead to a big change in profit. From a managerial perspective, if c 0 and c c decrease, then the total cost of the model decreases, which means the total profit will increase for the LED bulb company and they can also make small adjustment in other parameter to maximize the company's profit. 4. In the case of logistic demand and linear demand functions, an increase in ordering cost parameter c 0 and carrying cost parameter c c results in a profit decrease. Small adjustment in the parameters α, b, P and K did not lead to a big change in profit. From a managerial standpoint, if c 0 and c c decrease, the total cost of the model reduces, implying that the LED bulb company's total profit will grow, and they can also make tiny modifications to other parameters to maximize the company's profit. 5. In the case of logarithmic demand and linear profit function, an increase in impact factor parameter K results in a decrease in profit, and an increase in ordering cost parameter c 0 and carrying cost parameter c c results in a profit decrease. Small adjustment in the parameters α, b and P did not lead to a big change in profit. From a managerial standpoint, in this case, reducing ordering costs and carrying costs leads to a reduction in the total cost of the model, which means increasing the LED Company's profit. They can also make minor changes to other parameters to increase the company's profit. 6. In all five cases, if ψ increases, then total profit increases.
From a managerial standpoint, an increase in ψ values results increase in profit due to less carbon release. Therefore, the company should apply this to get more profit in their business. If ψ value decreases, then profit decreases, so the company manager should not apply it in their business. They can also make minor changes to other parameters to increase the company's profit.

Industrial prospective
New resource input can be avoided by implementing the circular economy concept in LED bulb production companies.

Table 4
Sensitivity analysis for all five cases with GT investment LED bulbs lead to more electricity savings, which means less fossil fuel burning and carbon dioxide emissions. This has a good influence on nature and decreases global warming and unexpected natural disasters. The circular economy concept can be introduced to many more production industries so that they can increase their profits as well as create a path to a sustainable production environment.

Conclusion
The modern business sector prefers circular economy policies rather than conventional economic policies. The circularity of the product is very relevant in circular economic policies. Companies are seeking methods to evaluate the circularity level of their products. In this context, in this research article, a sustainable circular economy production inventory model with circularity index under controllable emission rate by investing in green technology was presented. The circularity index is used to measure the percentage of recycled content in a LED bulb. The optimal circularity index of LED bulbs and optimal profit were calculated for distinct combinations of the non-linear and linear functions of demand and unit profit. The Lagrange multiplier method is used to maximize the total profit with optimal strategies under controllable emissions. It was observed that the circularity level of LED bulbs had impacted the cost, demand, and sales price of the product and explained various non-linear and linear relationships. It was observed that increased circularity index value and green technology investment helped to increase profit and reduced carbon release and environmental damage. This model could be adopted by any LED and CFL bulb production company in order to reduce resource input and carbon release. It is discovered that the total profit has increased in all five cases, and in the managerial implication section, profit-increasing strategies are found. The limitations of the model if the circularity index of the product is very small, then this model will not be more effective. This model is not more effective for products that are not recyclable. This idea can be extended to a sustainable circular economy production model with shortage situations, as well as to other inventory models with shortage situations under controllable emissions. This circular idea is possible to implement in a single-period inventory model. Also, it may be extended to an integrated production retailer model, and more emission parameters may be included.

Appendix 1
Let us assume the linear relationship ( ) = 0 + a > 0 and p( ) = p 0 + b > 0 where b and a are constants. Prove the objective function concavity by Hessian matrix.

The Hessian matrix is H
To assure concavity, it must be negative semi-definite. It is observed that.
Since ab < 0, 0 + a > − (2aPc0+aQ 2 c c ) 2 16P 2 Qc 0 ab . By using above condition, one of the possible solutions given below is optimal and feasible. We continue to work on solving our optimization problem. The value can be found from the latter simplification of the equation (A.1).

Appendix 2
Assume the relationship ( ) = 0 + a > 0 and p( ) = p 0 + be ( −1) > 0 where , b and a are constants. The objective function is Our aim is to increase the average profit subject to 0 ≤ ≤ 1 or l ≤ ≤ u ; For The Lagrangian function is The optimality of solution is given by the Karush-Kuhn-Tucker (KKT) criteria are.

The Hessian matrix is
To assure concavity, it should be negative semi-definite. Unfortunately, we only get it for certain parameter values a , b and . If b > 0, a > 0 and > 0 then we can use a simple condition to verify the profit function's concavity. In this situation, the h 11 and h 22 elements Hessian matrix are having negative values and (B.1) By using the above condition, we continue to work on solving our optimization problem and.
These are the possible cases we get. .
. This solution is feasible and optimal if and only if ) .

This solution is feasible (and optimal) if and only if
.
To assure concavity, it should be negative semi-definite. Unfortunately, we only get it for certain parameter values a , b and . If b > 0, a > 0 and > 0 then we can use a simple condition to verify the profit function's concavity. In this situation, the h 11 and h 22 elements Hessian matrix are having negative values and The Lagrangian function The optimality of solution is given by the Karush-Kuhn-Tucker (KKT) criteria are.

The Hessian matrix is
To assure concavity, it should be negative semi-definite. Unfortunately, we only get it for certain parameter values a , In this instance, with the aim to solve our optimizing problem, we must consider the following three possibilities. The value can be found from the latter simplification of the Eq. (26). To be optimal, the related solution must meet the feasibility requirement.
The value can be found from the latter simplification of the Eq. (30). To be optimal, the related solution must meet the feasibility requirement.

Declarations
Ethics approval Not applicable.