Carbon Reduction Technology Based on Imperfect Production System for Deteriorating Items with Warranty Periods and Greenness Dependent Demand

: In the current situation, environmental pollution is one of the vital issues affecting every country. In this research paper, we have developed a production inventory model based on carbon emissions, level of greenness, and the warranty of a product. However, very little research has reported on the topics mentioned above. To set up a sustainable imperfect production inventory model, the following suppositions are made: (a) that carbon is released during the production process and that it can be mitigated by imposing technology preventing carbon release directly into the environment; (b) that manufacturers provide a price discount on the selling price of the product to attract customers; and (c) that manufacturers also give attention to the warranty on the goods. This paper assesses the effects of the greenness, warranty of an item, and technology preventing carbon release into the environment on overall proﬁt to help decision-makers make more effective decisions about pricing and replenishment. Three decision variables will need to have their optimal values determined using an algorithm. To justify the proposed model, one numerical example is solved. Finally, a sensitivity analysis is performed to determine how various factors affect total proﬁt.


Introduction
One of the main problems that can harm the planet irreparably is environmental pollution, which affects the quality of the land, water, and air.Environmental pollution also contributes to global warming and climate change.In the last 20 years, there has been an increase in the production of greenhouse gases (GHG) such as carbon dioxide (CO2), methane (CH4), and nitrogen oxide (N2O) [1].Companies are becoming increasingly aware of the importance of investing in technologies that lower carbon emissions because carbon dioxide accounts for more than half of all emissions.Numerous conferences have been held since 1990 to discuss potential solutions to this issue.The "Kyoto Protocol", one of the most important outcomes of negotiations, was introduced in 1997.Developed nations put one of the Kyoto Protocol's effective measures into practice by regulating their programs to reduce carbon emissions [2].(The goal of regulations, however, is to encourage sustainability and help firms cut back on carbon emissions [3].A sustainability technique with a healthrelated focus, client emissions can be used in this context to gauge an item's freshness, and purchasers now heavily weigh the rate of degradation, which adds to the production system's costs, according to [4].The techniques being used to halt the deterioration process must be investigated in order to satisfy the demands of the market.In order to prevent products from degrading and becoming waste, preservation technologies are applied.It has been demonstrated that altering the humidity and temperature of storage areas affects how quickly products decay [5].Thus, the main goal of protection technologies is to slow down the rate of deterioration by keeping these parameters stable.
The prior production inventory models made the supposition that product manufacturing was flawless, despite the fact that this expectation is rarely realized in practice.The lengthy machine running times, human error, and comprehensive control over manufacturing supply chains make imperfect production impossible to avoid.As a first step of the suggested method, [6] created a defective item inventory model after inspecting everything to figure out the rate of imperfection.The impacts of lowering the supply of inferior goods while retaining quality were then studied by a large number of researchers.The goal of quality-improving technologies is to assist manufacturers in avoiding uncontrollable circumstances that result in the creation of unsatisfactory items [7].
The effect of emitting carbon into the environment on the substandard manufacturing of degrading items has not been covered very extensively, as far as we can tell from the literature.Given this gap, the following three research topics can be put forth: (1) What is the role of the greenness of the product?(2) What steps may be taken to reduce carbon emissions and advance imperishability?(3) How much of an effect does the warranty policy have throughout the business period?(4) How might this-improving facilitiesaffect a company's overall profit?To address these issues, and supply the appropriate information to the decision-makers to adopt the best replenishment and pricing methods, this paper focuses on combining carbon preventing technology, deterioration, greenness, and warranty of a product in an imperfect production system.We shall discuss the research gap that this study aims to fill and go into more detail regarding pertinent studies on the three issues we highlighted in the following section.
It is indicated that the remainder of our elaboration will come later.The thorough evaluation of the literature on articles focusing on carbon emissions, item deterioration, and flawed manufacturing in inventory models is covered in Section 2. In Section 3, symbols and presumptions linked to the mathematical model are discussed, along with a detailed overview of the model description and its improvement of the earlier efforts.In Section 4, the mathematical formulation is presented.An approach to a theoretical solution is also discussed in Section 4. The effects of some parameters are addressed using post optimality analysis, and one numerical illustration is shown in Sections 5 and 6 in order to demonstrate the validity of the models.Managerial insight is presented in Section 7. The results of this study are eventually summarized in Section 8.

Literature Review
The technique outlined of the present work mainly consists of three steps: (1) a thorough keyword search of the relevant literature; (2) a thorough study of the published manuscripts and summarizing them; and (3) a discussion of the contributions of their study to this research.Web of Science served as the database for scientific research, and manuscripts from renowned journals such as Elsevier and Springer were examined in order to cover as much literature as possible.To compile and assess the relevant papers, two thorough literature reviews were also used: [8,9].

Carbon Emissions Related Inventory Model
The authors' investigation of challenges with operations management is motivated by the expanding awareness of how greenhouse gases damage the environment.Management of inventory is one of the businesses that has a long-term benefits for emissions from manufacturers/suppliers/retailers.In order to comply with government regulation, the authors investigated how to manage carbon dioxide emissions from various operations.Ref. [10] proposed carbon footprint related work, which is considered the frontier of the earliest research.It is claimed to have suggested a carbon cap-and-trade system for situations where ordering and warehousing procedures generate emissions.The work was then expanded upon in a number of different ways.Ref. [11] were the first to propose the concept of carbon discharge control in an inventory analysis, and looked at the application of a carbon price to reduce emissions.This article investigated a vendor-buyer supply chain for carbon discharge protection, in order to lower emissions arising in the environment.More articles looked into the presumptions they had made with their models and concluded that other related activities were the primary reason for emissions in the following articles.By way of illustration, [12] studied the emissions caused by throwing away outdated products in the year of 2014.Depending on the method of transportation in this investigation, the replenishment quantity can change.Ref. [13] evaluated the implications of cap-and-trade and tax policies, and developed a model that contrasted different emission regimes.In a published article, [14] listed remanufacturing as one of the primary causes of emissions in the year of 2015.With consideration for the quantity of newly created and previously manufactured batches, as well as the size of each batch, the overall cost can be reduced.Ref. [15] considered a supportable model for damaged goods and created a carbon price policy without considering backlogged situations.Ref. [16] broadened [14] framework for the stochastic demand case in the year 2015.This study examined how onward and reverse logistics should be coordinated to produce the maximum environmental benefit while avoiding environmental pollution.Unlike coordinated and uncoordinated sustainable inventory models, [17] developed a vendor-buyer model.Their essay considers the effects of different carbon emission regulations.Ref. [18] proposed a sustainable inventory model for non-instantaneous deteriorating items under a carbon emission investment and trade credit facility.Ref. [19], take into account the concept of lead time depending on transportation activities such as unloading and loading.They investigated a sustainable inventory model with a tolerable shortfall and demand, which is stochastic in nature, in order to calculate the appropriate lead time, order quantity, safety factor, numbers of shipments, and emissions.Another contribution to our research comes from the [1], who published a manuscript which established carbon tax and carbon-cap as controlled when setup, manufacturing, warehousing, remanufacturing, and disposing of old materials are the major contributors to carbon discharge to the environment.Ref. [20] proposed an imperfect interval production system using carbon reduction technology.Ref. [21] studied a carbon emission related production inventory model in an interval environment, and they have used soft computing techniques for solving said problem.

Deteriorating Inventory Model
Ref. [22] studied a non-instantaneous deteriorating inventory model under stock dependent demand in a two-warehouse system.Due to nonlinearity of the objective function, they have used soft computing techniques for solving this type of problem.A significant study that was conducted prior to research recommended by [23], created an inventory model for decaying goods that concurrently took into account carbon emissions and poor manufacture (2018).They examined an integrated method for a two-echelon inventory model where carbon is released as a result of storing, transporting, and discarding damaged products.In another article that examined inventory models with a sustainable nature for deteriorating goods, [24] created a trade-and-cap strategy for emissions produced by production and storage operations where green technology can reduce emissions.Finding the pricing and optimal replenishment in both centralized and decentralized situations is the major goal of this effort.According to [4], different payment options including cash, credit, and prepayment have an impact on the model of depreciating inventory in the year of 2020.They have studied a strategy on carbon tax for emissions produced by manufacturing processes where it is feasible to invest in green technologies.Ref. [25] studied a supportable inventory problem with a focus on funding technologies that cut down on deterioration, ordering costs, and carbon emissions simultaneously.When demand is based on both price and hybrid stock, this study takes a continuous emission into account.Ref. [26] demonstrated a nonlinear stock dependent inventory model with nonlinear holding costs for deteriorating items.Ref. [27] suggested a deteriorating inventory model with order size dependent trade credit and completely backlogged situations.This manuscript's primary objective is to explore the results of green investing technologies to cut emissions in compliance with cap-and-trade and its regulations.The study provided by [25] that examined how to control carbon emissions and product deterioration simultaneously in a greenhouse farm addressed other shortages.Linear and nonlinear price dependent demand functions are contrasted in this essay.Ref. [28] introduced a price discount inventory model based on advanced and delayed payment for deteriorating inventory model under shortages.Ref. [29] proposed a non-instantaneous deteriorating inventory model with hybrid payment of cash and advance with time variations, holding cost-and time-dependent demand.Ref. [30] investigated a deteriorating inventory model with discount facility in an interval environment.They have used the parametric approach to intervals as a new technique to tackle the interval uncertainty.

Imperfect Quality Inventory Model
According to [31] Hou et al., first research among the related research looking at quality upgradation in a sustainable production model should be written.However, this paper's addition to prior work is minor because no methods for cutting emissions are given.Ref. [12] suggested the study that was expanded by [32], as a cap-and-trade method for defective items in a production firm where carbon is released due to the production, transportation activities and remanufacturing.They concluded that implementing carbon emission laws reduces the cost and emissions associated with remanufacturing activities.Later, [33] investigated a model for which carbon is released as a result of procedures such as scrapping and warehousing.The primary new component of this paper is the investigation of how inspection impacts long-term inventory management.Ref. [34] suggested an imperfect quality item model by taking carbon emission investment in the area of inventory management.One of the articles that helped with our study was the [35] publication, which built an integrated production model taking into account objects degrading when faulty manufacturing is accessible.The major goal of this research has been to analyze how each carbon emission technique affects the overall revenue produced by the inventory system.However, no one has yet addressed how to lower carbon emissions, the deterioration of goods, or uneven manufacturing.Ref. [7] proposed a coordinated three-echelon supply chain with expiration date-related deterioration.By utilizing quality improvement techniques and technology, this study seeks to decrease the occurrence of poor goods.Ref. [36] studied an imperfect production model in a fuzzy environment.They have used fuzzy differential and fuzzy integral methods to represent this model.

Main Contribution
The papers cited above demonstrate historical research on the three mentioned topics.There have been more articles published in this field of study in recent years.However, it is evident that articles rarely examined the junction of the three domains.This discussion analyzes the works of [6,10], two of the core studies that gained a lot of interest, in order to fill up the remarked research gap and develop a sustainable manufacturing model incorporating price, greenness, and warranty dependent demand.The possibility for studying in technology for quality enhancement, carbon reduction, and green investment is also taken into consideration when considering specific modifications of these two publications [7,25,34].This paper's main objective is to explain how to decision-makers might change their preferences and take into account a traditional production model.The major contribution of this work is given below: During the production process, manufacturing firms eject some carbon directly to the environment.Carbon emission reduction technology is used in order to control the ejection of carbon directly to the environment.
(i) Demand of the product is considered here according to linear price, nonlinear level of greenness, and nonlinear warranty of the product.
(ii) Discount is an important factor to attract more customers.In this context, a certain percentage of fixed discounts on the purchase amount is taken into consideration for developing this model.
Currently, no one has reported on the imperfect production model in the existing literature by combining all of the points mentioned above for developing the model.It is the main contribution of the proposed model that combining all the factors together formulates an imperfect production model.

Problem Description and Notation
To fill the gap, developed a manufacturing model for a system taking into account into a single production system and single customer is created the model examined based on [35].To meet the retailer's demand, the producer produces the goods at a steady rate.In actuality, the production of faulty (lower quality) goods due to the equipment malfunctions or human error is not negligible.As a result, a certain number of defective products (ξ) remain, and the defective items have undergone reworking.Even if the renovated goods are thought to be perfect, some of them are still unrepairable and will be discarded.The manufacturer invests in green technology to lower emissions in accordance with modifying sustainability of the production system [11].In addition, the effects of carbon preventing investments, warranty of the product, and green investment technology are covered in together.

Notation
The terminology and assumptions that were utilized are provided below to help with accurate model analysis (Table 1).

Assumption
(i) A single producer provides products for a single retailer under a single-item model, which is defined by the instantaneous replenishment of inventory.(ii) Both lead times and shortages are neglected.Ref. [35] claim that the rate of degeneration is constant for all items (δ).(iii) Demand of an item is taken as the combination of price, greenness, and warranty period of the product.Mathematically, it can be represented as D(g, p, w) = a + cg α − bp + βw γ where 0 < α, γ ≤ 1 (iv) Imperfect production is taken with the rate (ξ) and the rate of production (P) is considered as constant [11,12,37].

Mathematical Formulation
Let us consider a manufacturing firm produces a single item with the rate P.During the production process they will also produce some imperfect item with the rate ξ.All the items are stored in a stocking point in order to satisfy the retailers' demand.This production process is continued up to the time period t = t 1 .In this instance, the customer purchases the goods by paying in advance the total purchase cost before receiving a large quantity.Prior to receiving goods from a seller who gives an early payment discount, a customer first purchases items, or Q units, by paying the full purchase price.Due to the combined effects of the demand D(g, p, w) and the deterioration, the inventory level decreased, and at time t = T the stock level was zero.As a result, the suggested model followed the given differential equation.
with the boundary conditions q(0) = Q,q(T) = 0 and q(t) is continuous at t = t 1 .
Solving the Equations ( 1) and ( 2) with the help of boundary conditions q(0) = Q and q(T) = 0, we have Now, using the condition of continuity the maximum stock can be written as: The associated costs of the inventory system are given below: Hence, the profit of the system is as follows: Therefore, average profit is given by: π(g, p, T) = TP T . i.e., − hD(g,p,w) Now, we have to optimize the objective function (6) in terms of the decision variables p, g, and T. In the next section, we are going to discuss the theoretical derivations of the objective function (6).

Theoretical Derivations
This section discusses the concavity of the average profit function; [44] Cambini and Martein's results were used to analyze the concavity of the model ( 2009).Theorems 3.2.9 and 3.2.10 from [44] determine the form's function as: It is (strictly) pseudo-concave if f (t) is a negative, differentiable, and (strictly) concave function yet g(t) is a positive, differentiable, and concave function.Using this method, one can easily demonstrate that profit per unit function ( 6) is a pseudo-concave function with respect to the decision variables g * , p * , and T * which will be the maximums of the average profit function (6).
In this section, we are going to examine the concavity for the said profit function π(g, p, T) with respect to g, p and T, where: and: g(g, p, T) = T In accordance with [44] Theorems 3.2.9 and 3.2.10,we must demonstrate that f (g, p, T) is a (strictly) joint concave, differentiable function, and negative definite with respect to p, g and T.
Next, differentiate Equation ( 7) with regard to p, g and T, and we have: Now again differentiate Equation ( 9) with respect to p, g and T, we get: Again, differentiate Equation ( 10) with respect to g, we get: Taking derivatives of Equation ( 11) with respect to T, we get: Now calculating the Hessian matrix of the problem we get: If this Hessian matrix is a negative definite then the objective function is pseudoconcave and the objective functions attain the maximum value with respect to the decision variable.
According to the necessary condition to find the optimal solution of the problem: ∂π(.) ∂p = 0, ∂π(.) ∂g = 0, ∂π(.) ∂T = 0. To find the optimal values of the decision variables, we have used the following algorithms.

Algorithm
Step 1: Input all the values of the inventory parameters.
Step 3: Set g i = 1 and T k = 1 and solve the equation ∂π (.)   ∂p j = 0. Restore the obtained value of p as p * j .
Step 4: Take the restored value of p * j and T k = 1 and solve the equation ∂π (.)   ∂g i = 0. Restore the obtained value of g as g * i Step 5: Take the restored value of p * j and g * i and solve the equation < ε then the optimality criterion is satisfied, so store the values of p * ,g * and T * .
Step 7: Find the value of the objective function using the optimal values of p * ,g * and T * .

Numerical Illustration
A single numerical instance is looked at to validate the proposed model.The values of the parameters are taken hypothetically.We did not perform any case studies in order to obtain the values for the parameters.However, the values seem to be realistic.The system parameter values are as follows:

Sensitivity Analyses
To demonstrate the effect of the different production system parameters on total cycle duration (T), starting stock level, maximum greenness, and profit per unit 'π', sensitivity studies were conducted relative to numerical Example 1 by varying the parameter values from −20 percent to +20 percent.Figures 4-10 are graphical representations of the ideal findings of these investigations.

Sensitivity Analyses
To demonstrate the effect of the different production system parameters on tota cle duration (T), starting stock level, maximum greenness, and profit per unit ' π ', s                         According to Figures 4-10, the profit per unit (π) is quite sensible to changes in the price of selling (p), demand location parameter 'a', and parameter scaling 'b'; conversely, (π) has the opposite impact on 'b'.In contrast to the average profit (π), which is insensitive to changes in either direction for ordering costs 'K o ', 'µ', and 'c h ', the profit per unit (π) is less sensitive to changes in either direction for costs 'δ' and 'c'.show that the business length (T) significantly affects whether the demand parameter 'b'changes in a positive or negative way.The business period (T) is often equally sensible to changes in the ordering cost 'A' and the location parameter of demand 'a', which might be positive or negative.The business period (T) is also less sensitive to variations in the price of selling (p), constant rate of deterioration (δ), and holding cost (c h ), but the opposite is true for (δ) and (c h ).
The beginning stock level (Q) is often equally sensible to changes in ordering cost 'A', whether they are positive or negative.The starting inventory level (Q) is less vulnerable to negative or positive changes in 'δ', 'c h ', and 'a' respectively, whereas 'δ' and 'c h ' are adversely affected.With negative or positive modifications of 'α', 'b', and 'µ' respectively, the original stock level (Q) is insensitive.

Managerial Implications
Based on the experimental results, the following implications may be suggested:

•
Price of the product (p) and demand of a product have a great influence on the profit of the system.In this context, the decision-maker should think about the appropriate selling price for satisfying the customers' demand.In addition, the organizer/decisionmaker must think about the warranty of the product.

•
It is also noted that discount rate has great influence on the demand of the product.Manufactures should consider the rate of discounts on purchased amount in order to increase the demand of the product.

•
Currently, customers are very much aware of their health.Generally, they want to use green products.So, producers should consider how to increase the appeal of the product in this category in order to increase their demand, as well as increase their total profit.

•
Warranty of an item is an important issue for purchasing a product.Thus, the decisionmaker must consider increasing the warranty period of the product.

•
The business terms could be considered by the decision-maker.As the business term gets longer, maintenance costs will increase, which will have a direct effect on the average profit of the system.

Figure 1 .
Figure 1.Graphic demonstration of concavity of profit function for two variables p and g.

Figure 1 .Figure 2 .
Figure 1.Graphic demonstration of concavity of profit function for two variables p and g.

Figure 3 .
Figure 3. Graphic demonstration of concavity of profit function for two variables p and T.

Figure 2 .Figure 2 .
Figure 2. Graphic demonstration of concavity of profit function for two variables g and T.

Figure 3 .
Figure 3. Graphic demonstration of concavity of profit function for two variables p and T.

Figure 3 .
Figure 3. Graphic demonstration of concavity of profit function for two variables p and T.
tivity studies were conducted relative to numerical Example 1 by varying the param values from −20 percent to +20 percent.Figures 4-10 are graphical representations o ideal findings of these investigations.

Figure 4 .
Figure 4. Post optimality analysis of a.

Figure 5 .
Figure 5. Post optimality analysis of b.

Figure 4 .
Figure 4. Post optimality analysis of a.

Figure 4 .
Figure 4. Post optimality analysis of a.

Figure 5 .
Figure 5. Post optimality analysis of b.

Figure 5 .
Figure 5. Post optimality analysis of b.

Figure 4 .
Figure 4. Post optimality analysis of a.

Figure 5 .
Figure 5. Post optimality analysis of b.

Figure 8 .
Figure 8. Post optimality analysis of h.

Figure 9 .
Figure 9. Post optimality analysis of p c .

Figure 8 .
Figure 8. Post optimality analysis of h.

Figure 9 .
Figure 9. Post optimality analysis of p c .

Figure 8 .
Figure 8. Post optimality analysis of h.

Figure 9 .
Figure 9. Post optimality analysis of p c .

Figure 9 .
Figure 9. Post optimality analysis of c p .

Figure 9 .
Figure 9. Post optimality analysis of p c .

Figure 10 .
Figure 10.Post optimality analysis of A.

Figure 10 .
Figure 10.Post optimality analysis of A.

Table 2 .
Optimal solutions obtained for solving the objective function.

Table 2 .
Optimal solutions obtained for solving the objective function.