A Sustainable Production Scheduling with Backorders under Different Forms of Rework Process and Green Investment

: Rework is currently a necessity for businesses and commercial organizations across the world. It is only beneﬁcial in tackling climate change if the process emits less greenhouse gases than would otherwise be emitted. This study designs an optimal production scheduling model to reduce both carbon emissions during the processes of production, transport and storage, and setup cost by leveraging on green technology efforts in an imperfect production process where a fraction of items is erroneous so that the ﬁrm may run out of inventory. The producer implements a rework strategy to rectify the ﬂawed products, anda ﬂexible rework rate is offered since the rework might be executed on various schemes. The ﬂexible rework allows the producer to choose therework rate, which can differ from the production rate, as well as the rework process itself, which can be asynchronous or synchronous.The two forms of green investments: quadratic and exponential are considered in the study. The main point of the study is to derive a solution procedure of the various problem settings associated with the rework rate, rework process and green investment. The ﬁndings suggest that developing the optimal production schedule (lot-sizes, backorders, setup cost and green investment amount) can lower the manufacturing sector’s excessive ecological carbon emissions. The ﬁndings also support the idea that making green investments is the most cost-effective way to cut carbon emissions and setup cost simultaneously.


Introduction
The Economic Production Quantity (EPQ) inventory techniqueis one of the most vital approaches in the manufacturing system for managing production since it notifies the producer when to halt production and use the products in inventory to meet consumer demand.The EPQ is based on the assumption that the company will create its own quantity or that the components will be delivered to the company as they are built, allowing orders to be available or received incrementally while the products are being produced.A shortage occurs when demand for a product or service exceeds available supply.This is a transitory state since the item will be restocked, and the market will return to balance.Unfortunately, no factory's production is perfect.Product faults abound in the manufacturing industry.They are available in a range of shapes and sizes.Moreover, they are an issue that might have a significant influence on their bottom line as an importer.As a result, we expect to diminish the amount of faulty goods by modifying them.

Literature Review
Rosenblatt and Lee [1] invented an EPQ model for a flawed production procedure with a constant, linear, exponential, or multistate defective rate.Later, a number of scholars extended Rosenblatt and Lee's [1] work witha variety of hypotheses (see [2][3][4][5][6][7][8][9]), and Resource conservation and efficiency are ensured through sustainable industrialdevelopment.Producers must analyze how raw materials are mined, components are made, products are created, and return markets are structured in order to optimize the supply chain and increase resource productivity.Think about innovative business models that would give us more control over every aspect of our operations to ensure that we are practicing environmental safety.The reducedecological effect through pollution avoidance is one of the most crucial elements of sustainability.Waste produces pollution, which can be avoided, repurposed, or decreased to provide environmental protection.There areseveral financial advantages to sustainable industrial growth.The sector itself promotes the employment and revenue opportunities connected to lessening ecological impacts.Additionally, sustainable industrial growth may help firms cut operational expenses.Processes that are efficient and sustainable require less energy, water, and materials, which may save a lot of money.The reducedecological impact is possibly the most evident benefit of sustainable industrialization.Many industrial firms are moving toward ecologically friendly development in order to conserve their ethical agreement to guarantee a safer and cleaner ecosystem.Sustainable industrial development aims to reduce greenhouse gas CO 2 while conserving natural resources.

Literature Review
Rosenblatt and Lee [1] invented an EPQ model for a flawed production procedure with a constant, linear, exponential, or multistate defective rate.Later, a number of scholars extended Rosenblatt and Lee's [1] work witha variety of hypotheses (see refs.[2][3][4][5][6][7][8][9]), and all of these models utilize a method for removing damaged products once they are detected.Rather than being discarded, broken products are recovered and used as raw materials in everyday production.In view of this, Liu and Yang's [10] EPQ model argues that a flawed manufacturing system can create damaged items that are both reworkable and non-reworkable.Hayek and Salameh [11] estimated the manufacturing lot-size when shortages are granted, and the portion of spoiled goods is a random variable.
Liao et al. [12,13] evaluated the EPQ and optimal preemptive upkeep schedule for inadequate production activity including the rework of damaged goods.Krishnamurthy et al. [14] extended an EPQ model withaproblematic manufacturing structure to include frequent manufacturing rework and sales returns.After production, defective items are detected and reworked.If manufacturing demands are unique, production planning may be a challenge.In the case of defective production, for example, requirements may vary with the amount of stock; this issue is examined and appraised in [15].Repairingdamaged items may be conductedin two ways: after-producing rework and during-producing rework.Rework of items and manufacturingare considered synchronous operations, but the rework of faulty goods after production is considered asynchronous.Nihar et al. [16] implied the requirement of taking the synchronous and asynchronous decision-making activities of diverse inventory systems.They studied how the various synchronous and asynchronous functions affect the system's actions.Al-Salamah [17] formed an EPQ inventory model with synchronous and asynchronous variable rework rates to account for an imperfect manufacturing process.He offered two configurations for the rework process.Imperfect components may only be modified utilizing the asynchronous rework option after the entire lot has been formed.Instead, with synchronous rework, damaged items may be repaired as soon as they are made.
Coates et al. [18] derived a method for lowering the cost of product setup in industries.Sarkar and Moon [19] created a quality improvement model with a variable setup cost and backorder rate using the concept of Porteus [20].Lung Hou [21] established an EPQ model that included capital expenditure which is a function of setup cost and process quality.For the EPQ model with flaws, Freimer et al. [22] calculated the worth of setup cost reduction optimization.To decrease the setup in production systems with work-in-process inventories, Nye et al. [23] adopted an optimum investment.Sarkar et al. [24] designed a setup cost reduction inventory model with quality upgrading.Then, Tiwari et al. [25] studied an integrated multi-echelon inventory system whose coordination is hampered by quality concerns and human error.By conductingan early investment in the vendor's manufacturing amenities, the buyer is prepared to minimize the vendor's set-up costs.
Different sustainable strategies to reduce CO 2 have been established by the carbon regulating bodies in many industrialized nations.The main sustainable approaches are limited CO 2 , carbon taxation, carbon cap and trade, and Green Lean Six Sigma (GLSS) which are often adopted by governments and private industries.In this connection, Bouchery et al. [26] explored traditional inventory procedures while analyzing the approach ofsustainability.They highlighted how CO 2 was slashed to a single goal function in terms of sustainable growth.Benjaafar et al. [27] created a model based on the cost function and CO 2 footprint by connecting CO 2 quantities to a variety of decision criteria.They were able to broaden their stance on CO 2 cut by making small operational changes, such as investing in green technologies.Toptal et al. [28] explored a joint inventory strategy with three unique CO 2 investment policies.Dye and Yang [29] investigated a trade-creditinventory system that included issues ondemand sustainability depending on credit terms.They discussed how credit duration and environmental restrictions influence the inventory model in the context of a CO 2 levy and cap system, with default risk rates.Qin et al. [30] developed a trade-credit inventory model for a CO 2 tax, a CO 2 cap, anda demand-based trade strategy under credit-period demand.Then, Datta [31] analyzed the effect of green investment to reduce CO 2 in an EPQ model.Following that, Huang et al. [32] derived a supply chain system that considered logistics, green investment, and various CO 2 norms.Mishra et al. [33] developed a long-term production-inventory model to reduce CO 2 when resources are scarce.Hasanet al. [34] figured out how to maximize inventory levels and technical investment withdifferent CO 2 strategies.We notice that the aforesaid papers considered the first three sustainable approaches.Despite rising curiosity about GLSS, only a small amount of research has been conductedon its use, and there has been no research conductedon the obstacles that prevent GLSS from being employed.The reduction in GLSS implementation hurdles in the industrial sector was examined by Kaswan et al. [35] based on their interaction with one another.Then, Kaswanet al. [36] proposed a GLSS implementation framework for enhanced organizational performance.The selection of the GLSS project for the industrial sector in the dynamic decision-making ecosystem is the focus of the study.Rathi et al. [37] also recently created a systematic GLSS framework for increasing operational effectivenesstogether with social and environmental sustainability.The framework, which covers the systematic application of numerous Green paradigm, Lean, and Six Sigma techniques from the identification and evaluation of the problem to the maintenance of the realized measures, was created with perceptions learned from the literature and industrial people.Mohan et al. [38] offered an analysis of GLSS research focused on a systematic literature study and expedited the organization's readiness to apply sustainable GLSS practice via deep knowledge of realization.

Research Gaps and Contributions
The majority of studies in the collection of imperfect production were designed with reworks, repairs, etc.Although synchronous and asynchronous rework processes were studied by a few scholars, sustainable EPQ CO 2 tax and cap models of optimizing setup cost and CO 2 simultaneously under bothaforementioned rework processes are notaccessible.We enhance Al Salamah's [17] approach in order to reduce setup costs and control CO 2 since the presence of CO 2 and cost reductions in setup make the model more realistic.The overview of the literature is given in Table 1.In comparison to earlier studies, our study made the following contributions: This research takes into account a flawed production system with two rework processes.Previously published studies avoided the availability of green technologies to manage CO 2 and setup costs at the same time.According to Porteus [20], a logarithmic expression may be utilized to lessen the setup cost, and two distinct types of CO 2 reduction functions for green technology are being investigated to reduce CO 2 .

Research Methodology
The models in this study are based on mathematically oriented inventory theory, and the methodology used is the quantitative method, which is based on the principles of operations research and management science.The schematic diagram of the methodology is shown in Figure 2. In this study, we develop mathematical models and use differential calculus optimization techniques to find the optimal solutions forthe models.The methodology followed in this research to find the optimal production scheduling (lot-sizes, backorders, setup cost and green investment amount) is listed below:

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Description of the problem  The rest of the study is designed in the same way: required notations and assumptions.Sections 3 and 4 form along with solution techniques.The sensitivity and num Section 5. Section 6 concludes the paper.

Descriptions of Problem
A producer creates inventory items in a flawed produ customer-ordered quantities.A 100% inspection is perfor parts, which are stored apart from faultless ones and rem rate is variable and different from the manufacturing rate synchronous or asynchronous.We examine CO2 and extre system's many industrial processes.The company intend duction system by investing in modern technology, ene costs, non-traditional energy, and other elements.The am vested appears to be limited.The producer's budget for th The rest of the study is designed in the same way: Section 2 shows the research's required notations and assumptions.Sections 3 and 4 formulate the mathematical models along with solution techniques.The sensitivity and numerical analysis are discussed in Section 5. Section 6 concludes the paper.

Descriptions of Problem
A producer creates inventory items in a flawed production system in order to supply customer-ordered quantities.A 100% inspection is performed to classify the problematic parts, which are stored apart from faultless ones and remodeled separately.The rework rate is variable and different from the manufacturing rate, and the rework activity can be synchronous or asynchronous.We examine CO 2 and extreme setup costs as a result of the system's many industrial processes.The company intends to shift toward a greener production system by investing in modern technology, energy-efficient equipment, setup costs, non-traditional energy, and other elements.The amount of money that may be invested appears to be limited.The producer's budget for the green technology renovation venture is denoted by this ceiling.With the producer's approval, the back-ordering of shortage items is also feasible.The mathematical models were developed using the following assumptions and notations.

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Consumer requirement(demand) and production rate are constant.

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CO 2 are generated from the process of production, transportation, and storage.

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There are two primary forms that green technology might reduce CO 2 : , where α stands for the offsetting CO 2 reduction factor and β for the CO 2 reduction efficiency factor (Huang et al. [32]).
where m stands for the effectiveness of greener technology in decreasing CO 2 , ξ is a proportion of CO 2 after investment in green technology, and F is a fraction of average CO 2 reduction (Mishra et al. [33]).

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The relationship between setup cost reduction and capital investment may be defined using the logarithmic investment cost function.Therefore, S and the capital expenditure for S reduction (Π) may be recorded as Π(S) = Mln S 0 S for 0 < S ≤ S 0 where M = 1/δ,δ is the fractional cut in S\dollar rise in Π(S).

Production Scheduling with Asynchronous Rework
Due to the accumulation and rework of defective items occurring only after the manufacturing lot is ended, the production and rework processes are not synchronized.Due to the adaptability of rework and the potential for manufacturer-dependent variations, there are two options to take into account.The inventory curvature will have a positive slope if perfect inventory accumulates during T 2 as a result of P R being larger than D. However, if P R is less than D, the inventory of the perfect items constantly drops over T 2 , resulting in a negative slope on the inventory curve for the perfect goods.In the next subsections, we examine each circumstance separately and compute the optimal Q, S, B, and G for each type of green investment.

The P R Is Higher than D (P R > D)
The following can be calculated from Figure 3, which depicts the inventory curve of perfect items in a cycle with backorders.
there are two options to take into account.The inventory curvature will have a positive slope if perfect inventory accumulates during  as a result of  being larger than D. However, if  is less than D, the inventory of the perfect items constantly drops over  , resulting in a negative slope on the inventory curve for the perfect goods.In the next subsections, we examine each circumstance separately and compute the optimal , , , and  for each type of green investment.

The PR Is Higher Than D (PR > D)
The following can be calculated from Figure 3, which depicts the inventory curve of perfect items in a cycle with backorders.
During the production period  1 +  2 , the  units of products produced.That is, The inventory curve is  1 () =  1 () during  2 .Then the total amount of inven- tory during  2 is condcutedby  The curve of back-order is The total backorder quantities during T 1 is provided by During the production period T 1 + T 2 , the Q units of products produced.That is, Then the total amount of inventory during T 2 is condcutedby The period T 3 is the time of rework rQ items, and T 3 = rQ P R since the rework rate is P R .For the period T 3 , the inventory curve is Then the total inventory during T 3 is The function of backorder quantities is B 2 (t) = Dt with terminal value is B 2 (T 5 ) = B during T 5 = B D .Hence, during T 5 , the total backorder is D .Next, we evaluate the inventory, which is depicted in Figure 4 as a curve of flawed items with asynchronous rework.The following can be deduced from The function of backorder quantities is  () =  with terminal value is  ( ) =  during  = .Hence, during  , the total backorder is  () = .Next, we evaluate the inventory, which is depicted in Figure 4 as a curve of flawed items with asynchronous rework.The following can be deduced from Figure 4.During the period T 1 + T 2 , the inventory curve of flawed items is D 1 (t) = rPt with the terminal value The inventory curve of the flawed products during T 3 is D 2 (t) = P R t.Then the total inventory of the flawed products during T 3 = rQ P R is Sustainability 2022, 14, 16999 Now the CO 2 throughout production setup, manufacture and inspection, shipping, and inventory keeping for perfect and flawed items. where The average inventory total cost per cycle is the sum of the following costs: setup, production, rework, backorder per unit of time and backorder per item, holding cost of perfect and flawed items, the CO 2 tax, and the investment cost function to cut the setup cost.It is mathematically derived as The manufactureris willing to spend money on eco-friendly technology to cut CO 2 and pay a CO 2 tax.Here Thus, the manufacturer can profit by selling the permit.The manufacturer's CO 2 is greater than the CO 2 cap Z when As a result, the manufacturer mustobtaina permit, which incurs a cost.Hence, the average total cost when P R > D under a carbon cap and tax functions for a quadratic form of green investment case is where The above-mentioned problem looks to be constrained non-linear programming (NLP).We use a method that is comparable to that used in the majority of the NLP literature to solve this type of NLP.Initially, we briefly ignore the constraint 0 < S ≤ S 0 , then attempt to determine the optimal solution of TC q1 (Q, B, G, S) through the following theorems and results.We also propose the following Algorithm 1 to pick the optimal Q, B, G, and S in the given situation.
Theorem 1.For fixed B, S and G, TC Aq1 (Q, B, G, S) is convex in Q.

Proof. See Appendix A.
Result 1.By equating Equation (A1) to zero, the optimal Q Aq1 as (2) Proof.See Appendix B.
Result 2. By equating Equation (A2) to zero, the optimal B Aq1 as Proof.See Appendix C.
Result 3. By equating Equation (A3) to zero, the optimal S Aq1 is Theorem 4. For fixed Q,B and S,TC Aq1 (Q, B, G, S) is convex in G.
Proof.See Appendix D.
Result 4. By equating Equation (A4) to zero, the optimal G Aq1 is Algorithm 1. Optimal Solution for the Quadratic Case.

Carbon Tax with Exponential form of Green Investment Function
In this case, we take into account green investment as an exponential function.The average total cost of the proposed problem for this case when P R > D under a CO 2 cap and tax functions is designed by Here, . Similar to the case ofaquadratic form, the average total cost for the current case is written as where The solution approach for Problem (7) is similar to that of the previous case 3.1.1.The same solution procedures are omitted in this theoretical derivation to avoid redundancy.
Result 5.The optimal Q AE1 as Result 6.The optimal G AE1 as Result 7. The optimal B AE1 as Remark 1. Equation ( 4) is still valid for finding the optimal value of S AE1 in the exponential green investment case as it does not change by any assumption about green investment.We present Algorithm 2 to find the optimal Q , B, G and S for the current case.(1.2) Substituting B 1 , S 1 and S 1 into Equation ( 8) evaluates Q 1 .
Step 2. Compare S with S 0 (i) If S < S 0 , go to step (4).

The P R Is Lower than D(P R < D)
If P R < D, excellent items are retrieved from inventory at a faster rate than purchasing, resulting in a drop in inventory during the rework phase and a negative slope on the inventory curve.The inventory curves in this situation are depicted in Figure 5.The inventory curves for flawed products retain the same shape as in Figure 4. where Π (, ) =  − − − .The average total cost when  <  and quadratic form of investment for the p The total inventory and backorder for the perfect items in the period T 1 , T 2 , T 4 , T 5 and the inventory of flawed items for the period T 3 defined in Section 3.1 (P R > D) are the same in the case that P R < D as any assumption regarding P R has no effect on these values.
The inventory rate is decreasing during T 3 , so the inventory curve during T 3 altered by , the total inventory of perfect items during T 3 is Hence, the average total cost per cycle and CO 2 for this case P R < D is where

Carbon Tax with Quadratic Form of Green Investment
The average total cost when P R < D and quadratic form of investment for the present scenario is Theorem 5.For fixed B, S and G ,TC Aq2 (Q, B, S, G) is convex in Q.
Proof.See Appendix E.
Result 8.By equating Equation (A5) to zero, the optimal Q Aq2 as Remark 2. Equations ( 3)-( 5) are still applicable to obtain the optimal B Aq2 , S Aq2 and G Aq2 , respectively, under the case P R < D since any assumption regarding P R has no effect on these values.Moreover, we may utilize the same Algorithm 1 approach that was generated in the earlier part to obtain the optimal values in the present scenario.

Carbon Tax with Exponential form of Green Investment function
The total cost of the current scenario when P R < D per cycle is That is, The solution approach for problem ( 13) is similar to that of previous case Section 3.2.1.The same solution procedures are omitted in this theoretical derivation to avoid redundancy.Result 9.The optimal Q AE2 as Result 10.The optimal G AE2 as Remark 3. Equations ( 4) and (10) are still applicable to determine the optimal values of B AE2 and S AE2 , respectively, under the case P R < D since any assumption regarding P R has no effect on these values.Moreover, in the current scenario, we may utilize the same Algorithm 2 method that was generated in the preceding case to find the optimal values.

Production Scheduling with Synchronous Rework
The concept of a manufacturing process with synchronous rework offers the advantage of permitting faulty inventory items to be removed and backorders to be filled more quickly.There are two cases that must be investigated, and they are as follows: P R > D and P R < D. The total inventory of flawed items may be calculated as follows: For time  +  , the total inventory of flawed items is  .
During  , the total inventory of flawed items is ( )  .Proof.See Appendix F. □ The inventory curve has a slope ((1 − r)P + P R − D) throughout the production period T 1 + T 2 , since perfect items emerge from the rework process at a rate of P R .Additionally, it is assumed that P R < rP to prevent disruption in the rework process. During D .The total inventory of flawed items may be calculated as follows: For time T 1 + T 2 , the total inventory of flawed items is 1   2  rP−P R P 2 Q 2 .During T 3 , the total inventory of flawed items is 1 2 Then the average total cost per cycle for thecurrent case when P R > D is Then, the CO 2 is given by 4.1.1.Carbon Tax with Quadratic form of Green Investment Function The average total cost per cycle with variable green investment is Theorem 6.For fixed B, S and G,TC Sq1 (Q, B, S, G) is convex in Q.
Proof.See Appendix F.
Result 11.By equating Equation (A6) to zero, the optimal Q Sq1 as Theorem 7.For fixed Q, S and G, TC Sq1 (Q, B, S, G) is convex in B.
Proof.See Appendix G.
Result 12.By equating Equation (A7) to zero, the optimal B Sq1 as Remark 4. Equations ( 4) and ( 5) are still applicable to obtain the optimal S Sq1 and G Sq1 , respectively, under the case of quadratic green investment since any assumption regarding synchronous rework has no effect on these values.Moreover, we can utilize the same Algorithm 1 from Section 3 to obtain the optimal values for the current situation.

Carbon Tax with Exponential form of Green Investment Function
With exponential green investment, the average total cost per cycle is That is, The solution approach for Problem ( 19) is similar to that of previous case Section 4.1.1.The same solution procedures are omitted in this theoretical derivation to avoid redundancy.
Result 13.The optimal Q SE1 as Result 14.The optimal B SE1 as Result 15.The optimal G SE1 as Remark 5. Equation ( 4) is still valid to determine the optimal S SE1 under the exponential green investment case since this value does not change by any assumption about synchronous rework.Furthermore, we may use Algorithm 1 from Section 3 to obtain the optimal values for the current scenario.

The P R Is Lower than D (P R < D)
Figure 8 depicts the inventory curve for perfect items.When P R > D, as shown in Figure 7, the inventory curves of flawed items have the same functional forms as flawed items.The inventory curve of the perfect products during During T 3 = rQ P R − Q P , the total inventory is Inventory cost per cycle for the current scenario when P R < D is where Then the CO 2 is given by

Carbon Tax with Quadratic form of Green Investment Function
The average total cost when P R < D with the quadratic form of investment is Theorem 8.For fixed B, S and G,TC Sq2 (Q, B, S, G) is convex in Q.
Proof.See Appendix H.
Result 16.By setting Equation (A8) to zero, the optimal Q Sq2 as Remark 6. Equations ( 4), ( 5) and ( 18) are still applicable to determine the optimal values of B Sq2 , S Sq2 and G Sq2 , respectively, under the case P R < D since any assumption regarding P R has no effect on these values.Moreover, in the present scenario, we may utilize the same Algorithm 1 method that was generated in the preceding case to find the optimal values.

Carbon Tax with Exponential form of Green Investment Function
The average total cost with exponential green investment is That is, The solution approach for problem (25) is similar to that of previous case Section 4.2.1.The same solution procedures are omitted in this theoretical derivation to avoid redundancy.
Result 17.The optimal Q SE2 as Result 18.The optimal G SE2 as Remark 7. Equations ( 4) and ( 21) are still valid to obtain the optimal B SE2 and S SE2 , respectively, under the case P R < D since any assumption regarding P R has no effect on these values.Furthermore, we may utilize the same Algorithm 2 approach that was generated in the preceding case to obtain the optimal values in the present scenario.
We study the variations in optimal solutions subject to two main parameters r and P R for both quadratic and exponential cases.All the parameters are retained constant in the initial event, with the exclusion of r, which is altered to see how it affects decision variables for both quadratic (Q Aqi , B Aqi ,G Aqi , S Aqi ) and exponential (Q AEi , B AEi ,G AEi , S AEi ) cases, i = 1, 2. Similarly, the rework rate P R is examined in the second event to see how the values of decision variables for both cases vary for low and high rework rates.
Table 2 reveals the optimal Q Aq1 , B Aq1 , G Aq1 and S Aq1 for a range of values of r when P R = 40,000 items/year.The result is compared to the total cost with and without the green investment, which is also included in Table 2, to show how reducing setup costs and CO 2 affect each other.

Synchronous Rework
We will utilize the same firm as in the preceding section, but this time we will assume that products with flaws are fixed as soon as they are made.Once the manufacturing lot is complete, each defective item that was not corrected during production is individually remade.The model must meet the assumptions that P R < rP and D < P R , according to Al-Salamah [17], which states that D = 190 items per year and P R = 200 items per year.In Table 3, the results are compared to the total cost with and without the green investment.The visual comparison of CO 2 and tax with and without green investment vs.r when P R > D is shown in Figures 11 and 12.

Discussion and Comparison of Findings
This study explores the connection between green investments and CO 2 .Using the quadratic and exponential forms of CO 2 reduction functions offered by Huang et al. [32] and Mishra et al. [33], this study examines the dependence structure between green technology and CO 2 .

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According to Tables 2-5 and Figures 13 and 14, Al-Salamah's [17] model performs similarly to ours, with the exception that the optimum lotsizes (Q increase and backorders (B Aq1 , B AE1, B Sq2 , B SE2 ) decrease more quickly.It should be noted that Al-Salamah's [17] model disregards green investments as a means of reducing CO 2 and setup costs.Moreover, if green technology is not employed to lower setup costs and CO 2 , costs and CO 2 increase.A firm may save between 8.4% and 25.5% in costs when it invests in green technology to lower setup and CO 2 emissions.Green technology therebydecreases the system's overall cost of production and cuts CO 2 .
We will utilize the same firm as in the preceding section, but this time we wi that products with flaws are fixed as soon as they are made.Once the manufac is complete, each defective item that was not corrected during production is ind remade.The model must meet the assumptions that  <  and  <  , acc Al-Salamah [17], which states that D = 190 items per year and  = 200 items In Table 3, the results are compared to the total cost with and without the gree ment.The visual comparison of CO2 and tax with and without green investm when  >  is shown in Figures 11 and 12.

Discussion and Comparison of Findings
This study explores the connection between green investments and CO2.U that products with flaws are fixed as soon as they are made.Once the manufa is complete, each defective item that was not corrected during production is in remade.The model must meet the assumptions that  <  and  <  , ac Al-Salamah [17], which states that D = 190 items per year and  = 200 item In Table 3, the results are compared to the total cost with and without the gre ment.The visual comparison of CO2 and tax with and without green invest when  >  is shown in Figures 11 and 12.

Discussion and Comparison of Findings
This study explores the connection between green investments and CO2.quadratic and exponential forms of CO2 reduction functions offered by Huang      • Since the optimal lot size raises as the percentage of flawed rises, Figures 15a, 16a, 17a, 18a, 19a, 20a, 21a and 22a explore the combined effects of both r and P R on lot-sizes For large values of r, the optimum lot sizes (Q Aq1 , Q AE1, Q Sq2 , Q SE2 ) are more sensitive to changes in the P R for high values of r than for small values of r < 0.1, as seen in the picture.As a result, when r > 0.1, it is claimed that lot sizes (Q

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The lot-sizes (Q Aq1 , Q AE1 ) and green investments (G Aq1 , G AE1 ) under the asynchronous rework model are slightly lower than the lot sizes (Q Aq2 , Q AE2 ) and G Aq2 , G AE2 under the synchronous rework model for the range of r values indicated in Tables 2-5.
When the rework is asynchronous, the backorder is much higher than when it is synchronous for most values of r; though the differences between the backorders are minor, and certain backorders are almost equal for r = 0.4.

Insights and Implications for the Industry
The financial industry has been a significant pillar of human progress s mencement of the industrial revolution.The global financial sector's funda tion is to make optimal use of global savings.Investments that are used w prove people's quality of life.People have invested their resources in ecologi ous initiatives, particularly those that worsen human-induced climate chang of the banking system's collapse.Despite the fact that finance plays a critic anthropogenic (i.e., human effect on the environment), nothing has been per tegrate environmental problems into finance.Green investments have obt attention in the financial industry in recent years, which has helped to adva ble growth.Green investment is an intersection between environmentally fri ior and the financial and business world.On the basis of the results, the m sights can be derived as follows: Making decisions to improve the sustainability of the inventory system timizing payout backorder and lot size, may assist the green inventory mode will be better able to concentrate on reducing the overall inventory in stora CO2 costs are included in the model.This will help to reduce the price of CO2 Firms must focus on transportation if they want to reduce overall costs.
This paper demonstrates that shifting to sustainable invention signifi the inventory system.Producers can use green technology to reduce CO2 fr turing, transport, and storage to abide by CO2 price rules.Green technology cycling technology, eco-friendly polymers, green chemical processes, and r ergy (solar, wind, hydro).Policymakers must thus sensibly prefer the opt green technology.They must take into account additional factors in addition

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Our research found that increasing C t lowers CO 2 levels.The findings of Dwicahyani et al. [40] and Hasanov et al. [41], who found that tariffs had a beneficial effect on CO 2 reduction, are consistent with this conclusion.The firm has new options for lowering CO 2 produced by industrial operations with the use of green technology.The firm will gain from less CO 2 even though green technology has higher direct costs.Studies, including Bai et al. [42] and others, have produced results that are similar.

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According to the findings (Tables 2-5), the optimal Q Aq1 ,G Aq1 and S Aq1 grow continuously as r increases, whereas B Aq1 progressively decreases as the fraction of defectives rises.The model of Al-Salamah [17] shows a similar pattern, with the exception that the optimal lot size grows faster, and the backorder decreases more slowly than ours.It is worth noting that Al-Salamah's [17] approach ignores green investment in terms of CO 2 and setup costs.In addition, CO 2 and total cost increase when green technology is not used for both CO 2 and setup costs.

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We may look at Table 4 and Figures 18 and 24 to see how the optimal solutions react when P R assumptions fluctuate.When r is increased, it is shown that Q Aq2 for P R < D rises more quickly than Q Aq1 for P R > D does if P R = 2500 units/year.The optimal backorders respond in a number of ways when r's value rises.B Aq1 declines when r rises, as was previously discovered.On the other hand, raising the value of r causes B Aq2 to rise.Additionally, Figures 12 and 24 provide a visual comparison of tax and CO 2 with and without green investment vs.r when P R < D.

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Table 5 and Figures 22 and 26 show how the optimal solutions change when the P R assumption changes.When P R = 2500 units/year, it is seen that, similar to the asynchronous situation when r is increased, Q Aq2 for P R < D raises faster than Q Aq1 for P R > D. The optimal backorders react in a number of ways as the value of r increases.As previously discovered, B Aq1 lessens as r rises.In contrast, increasing the value of r results in an increase in B Aq2 .Besides, Figures 12 and 26 depict a visual contrast of CO 2 and taxes with and without green investment vs.r when P R < D.

Insights and Implications for the Industry
The financial industry has been a significant pillar of human progress since the commencement of the industrial revolution.The global financial sector's fundamental function is to make optimal use of global savings.Investments that are used wisely can improve people's quality of life.People have invested their resources in ecologically hazardous initiatives, particularly those that worsen human-induced climate change, as a result of the banking system's collapse.Despite the fact that finance plays a critical part in the anthropogenic (i.e., human effect on the environment), nothing has been performedto integrate environmental problems into finance.Green investments have obtaineda lot of attention in the financial industry in recent years, which has helped to advance sustainable growth.Green investment is an intersection between environmentally friendly behavior and the financial and business world.On the basis of the results, the managerial insights can be derived as follows: Making decisions to improve the sustainability of the inventory system, such asoptimizing payout backorder and lot size, may assist the green inventory model.Businesses will be better able to concentrate on reducing the overall inventory in storage facilities if CO 2 costs are included in the model.This will help to reduce the price of CO 2 from storage.Firms must focus on transportation if they want to reduce overall costs.
This paper demonstrates that shifting to sustainable invention significantly affects the inventory system.Producers can use green technology to reduce CO 2 from manufacturing, transport, and storage to abide by CO 2 price rules.Green technology includes recycling technology, eco-friendly polymers, green chemical processes, and renewable energy (solar, wind, hydro).Policymakers must thus sensibly prefer the optimal kind of green technology.They must take into account additional factors in addition to economic aspects when choosing the exact technology, such as the technology's ability to reduce pollution and compatibility with machines.
Managers can adjust the production rate using the suggested approach by controlling production allocation.A strategy of production rate adjustment is essential when the rate of production has a substantial impact on the volume of CO 2 generated.According to our research, the system can profit from a decrease in production rate by balancing supply and demand and reducing CO 2 .Unfortunately, this benefit was not accessible since earlier inventory models did not take these limitations into account.Our research shows that the decision-making criteria and ultimate cost are affected by concerns about CO 2 .The study's conclusions also provide a roadmap that inventory decision-makers may use to achieve successful long-term inventory management.

Conclusions
Today an increasing number of businesses have made sustainability a top priority in their strategy and operations to boost growth and global competitiveness.This movement currently encompasses several well-known businesses from a wide range of industries, considerably beyond the small number of firms thatpreviously positioned themselves as green.This study offers an extension to anearlier study that intends at cutting the CO 2 and setup cost simultaneously in a backorder situation.This research considers an imperfect production process where a fraction of the items is faulty, and the firm employs a rework approach to rectify the faulty items under two realistic scenarios: asynchronous and synchronous.We used twoforms of green investment to attain the lowest cost in terms of optimal lot size, backorder and decreased setup cost while reducing CO 2 .We have formulated eight mathematical models under various problem settings.Iterativesolution approaches are derived and proved analytically and numerically for all models.The examples show how the lot size grows and the backorder reduces as the fraction of defects for asynchronous rework with a rework rate greater than the demand rises.
The proposed model can be expanded upon in future research because this study has some limitations.The failure of this research to demonstrate the impact of COVID-19 on the company's transportation system is a limitation.The pandemic may, therefore, affect customer demand, leading to a fluctuating demand that changes over time.This type of work could be a great extension of this study.We also missedincluding the effect of learning onquality.If so, a more intriguing extension would be to look into whether investing in screening-related learning is worthwhile.Moreover, the limitations of utilizing a CO 2 reduction process, such as Green Lean Six Sigma, cap-and-trade and carbon offsets, were lacking inthis research.

Figure 2 .
Figure 2. Schematic diagram of the research methodology.

Figure 2 .
Figure 2. Schematic diagram of the research methodology.

Figure 3 .
Figure 3. Inventory curves of perfect items when P R > D (asynchronous rework).Bule represents available stock, and purple represents out of stock.
r)P−D rQ P R .The inventory curve is F 3 (t) = Dt with the end value F 3 (T 4 ) = F 2 (T 3 ) = (P R − D)T 3 + ((1 − r)P − D)T 2 during T 4 .The T 4 can be derived as T 4 = (P R −D)T 3 +((1−r)P−D)T 2 D = Q D − B D − Q P − rQ P R from the terminal value.The total inventory during T 4 is

Figure 3 .
Figure 3. Inventory curves of perfect items when  >  (asynchronous rework).Bule represents available stock, and purple represents out of stock.The inventory curve is F 3 (t) = Dt with the end value F (T ) = F (T ) = (P − D)T + (1 − r)P − D T during T .The T can be derived as T = ( ) ( )

Figure 4 .
Figure 4. Inventory curves of the flawed items when P >  (asynchronous rework).

Figure 4 .
Figure 4. Inventory curves of the flawed items when P R > D (asynchronous rework).

Step 4 .( 2 . 1 )
Loop step (2.1) to (2.3) until the values Q and B have converged, and the solutions denote by ( Let S = S 0 and B 1 = DC b /b (2.2) Substitute B 1 in Equation (2) (switch S by S 0 ) to obtain the new Q 1.

Algorithm 2 .
Optimal Solution for the Exponential Case.Step 1. Do step (1.1)-(1.3)until the values Q, B, G and S have converged, and the solutions represented by Q, B, G, S .(1.1) Start with B 1 = DC b /b , G 1 = lnξ and S 1 = S 0 .

Step 3 .
Do step (2.1)-(2.3)until the values Q, B and G have converged, and the solutions represented by . 1) Let S = S 0 , B 1 = DC b /b and G 1 = lnξ.(2.2) Substitute B 1 and G 1 in Equation (8) (replace S by S 0 ) to obtain the new Q 1.

Figure 5 .
Figure 5. Inventory curves of perfect items with asynchronous rework and  < .Grey represe available stock, and purple represents out of stock.

Figure 5 .
Figure 5. Inventory curves of perfect items with asynchronous rework and P R < D. Grey represents available stock, and purple represents out of stock.

4. 1 .Figure 6
Figure6depicts the inventory curve of perfect items under the premise of synchronous rework, whereas Figure7depicts the inventory curve of flawed items.

Figure 6 .Figure 7 . 16 ) 6 .
Figure 6.Inventory curves of perfect items with synchronous rework.Blue represents available stock, and purple represents out of stock.

Figure 7 .
Figure 7. Inventory curves of flawed items with synchronous rework.

Figure 8 .
Figure 8. Inventory curves of perfect products with synchronous rework when P R < D. Blue represents available stock, and purple represents out of stock.

Figure 9 .
Figure 9.Comparison of the CO2 when  >  with and without green investment vs

Figure 9 .
Figure 9.Comparison of the CO 2 when P R > D with and without green investment vs.r.

Figure 9 .Figure 10 .
Figure 9.Comparison of the CO2 when  >  with and without green investment v

Figure 10 .
Figure 10.Comparison of the CO 2 when P R < D with and without green investment vs.r.

Figure 11 .
Figure 11.Comparison of CO2 with and without green investment vs.r when  > .

Figure 12 .
Figure 12.CO2 comparison with and without green investment vs.r.

Figure 11 .
Figure 11.Comparison of CO 2 with and without green investment vs.r when P R > D.

Figure 11 .
Figure 11.Comparison of CO2 with and without green investment vs.r when  > .

Figure 12 .
Figure 12.CO2 comparison with and without green investment vs.r.

Figure 12 .
Figure 12.CO 2 comparison with and without green investment vs.r.
on the other hand, are less sensitive to changes in the P R for large values of r than when the percentage is small (r < 0.1), as shown in Figures15c, 16c, 17c, 18c, 19c, 20c, 21c and 22c.Backorder size behavior leads to a similar conclusion.Figure15b, Figure16b, Figure17b, Figure18b, Figure19b, Figure 20b, Figure 21b, and Figure 22b indicate that a rise in P R induces a big fall in (B Aq1 , B AE1, B Sq2 , B SE2 ) for values of r > 0.1.• Figures 23-26 show the CO 2 reduces due to the increase in C t with r = 0.1.

Figure 16 .Figure 16 .Figure 17 .
Figure 16.How the Q AE1 , B AE1 , G AE1 , S AE1 changes with the rate of asynchronous rework P R > D (Exponential case).(a) Quantity lot size Q AE1 variations P R ; (b) Backorder B AE1 variations with P R ; (c) Green investment G AE1 variations with P R ; (d) Setup cost S AE1 variations with P R .

Figure 17 .Figure 18 .
Figure 17.How the Q Aq2 , B Aq2 , G Aq2 , S Aq2 changes with the rate of asynchronous rework P R < D (Quadratic case).(a) Quantity lot size Q Aq2 variations with P R ; (b) Backorder B Aq2 variations with P R ; (c) Green investment G Aq2 variations with P R ; (d) Setup cost S Aq2 variations with P R .Sustainability 2022, 14, x FOR PEER REVIEW 29 of 39

Figure 18 .Figure 18 .Figure 19 .Figure 20 .
Figure 18.How the Q AE2 , B AE2 , G AE2 , S AE2 changes with the rate of asynchronous rework P R < D (Exponential case).(a) Quantity lot size Q AE2 variations with P R ; (b) Backorder B AE2 variations with P R ; (c) Green investment G AE2 variations with P R ; (d) Setup cost S AE2 variations with P R .

Figure 19 .Figure 19 .Figure 20 .
Figure 19.How the Q Sq1 , B Sq1 , G Sq1 , S Sq1 changes with the rate of synchronous rework P R (Quadratic case).(a) Quantity lot size Q Sq1 variations with P R ; (b) Backorder B Sq1 variations with P R ; (c) Green investment G Sq1 variations with P R ; (d) Setup cost S Sq1 variations with P R .

Figure 20 .
Figure 20.How the Q SE1 , B SE1 , G SE1 , S SE1 changes with the rate of synchronous rework P R (Exponential case).(a) Quantity lot size Q SE1 variations with P R ; (b) Backorder B SE1 variations with P R ; (c) Green investment G SE1 variations with P R ; (d) Setup cost S SE1 variations with P R .

Figure 22 .Figure 22 .
Figure 22.How the Q SE2 , B SE2 , G SE2 , S SE2 changes with the rate of synchronous rework P R (Exponential case).(a) Quantity lot size Q SE2 variations with P R ; (b) Backorder B SE2 variations with P R ; (c) Green investment G SE2 variations with P R ; (d) Setup cost S SE2 variations with P R .

Figure 23 .Figure 23 .
Figure 23.CO 2 for various C t with r = 0.1 when P R > D.

Figure 24 .Figure 25 .
Figure 24.CO 2 for various C t with r = 0.1 when P R < D. ustainability 2022, 14, x FOR PEER REVIEW

Figure 25 .
Figure 25.CO 2 for various C t with r = 0.1 when P R > D.

Figure 26 .
Figure 26.CO 2 for various C t with r = 0.1.

Table 2 .
Changeable r and asynchronous rework with  > .

Table 2 .
Changeable r and asynchronous rework with P R > D.

Table 3 .
Changeable r and synchronous rework with P R > D.

Table 4 .
Changeable r and asynchronous rework with P R < D.

Table 5 .
Changeable r and synchronous rework with P R < D.