 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 287 https://internationalpubls.com Computational Analysis of the Framework Evaluation for Sustainable EOQ Considering Emission Tax with Capital Constraints Ajeet Baraskar1, Dr. Arun Thakare2, Dr. Yogesh Deshmukh3 1G H Raisoni University, Amravati-411701, India ajeetbaraskar@gmail.com 2G H Raisoni College of Engineering & Management, Pune-412207, India arun.thakare@raisoni.net 3Rajiv Gandhi Institute of Technology, Andheri-400053, India yogesh.deshmukh@mctrgit.ac.in Article History: Received: 15-03-2024 Revised: 14-05-2024 Accepted: 24-05-2024 Abstract: One of the key concerns in production planning has always been determining the economic lot size. Researchers have been studying this issue for a while, and numerous models have been created to meet objectives with the least amount of expense. This study's objective is to assess a more sophisticated approach to deciding lot size by taking into account the costs associated with emissions taxes (environmental impact) with capital constraints. It will be suggested to adopt a framework, called Sustainable Economic Order Quantity (SEOQ), for inventory control in the manufacturing industry. For the purpose of assisting decision-makers and policies on inventory issues, this study included a useful numerical analysis as well as a sensitivity analysis. Finally, the experimental findings demonstrated that the suggested models would solve the issues with the lowest possible overall inventory costs. Keywords: Sustainability, Inventory, emissions taxes, capital constraints, SEOQ. 1. INTRODUCTION In the era of industrial evolution and rapid technological advancements, the equilibrium between economic growth and environmental sustainability has become a paramount concern. Businesses are increasingly under pressure to optimize their operations not only for cost efficiency but also for environmental responsibility. This dual objective is particularly pertinent in inventory management, where the Economic Order Quantity (EOQ) model plays a critical role. Traditionally, EOQ models have focused on minimizing the total cost of inventory, including ordering and holding costs. However, in the context of sustainable development, it is imperative to incorporate environmental considerations into these models. The main factor that helps a business maintain its seamless functioning is its inventory. These days, environmental concerns are shared by all nations and businesses. Businesses must consider environmental factors, including carbon emissions and capital restrictions. The SEOQ model is a lot size strategy used to establish economic ordering when dealing with inventory by taking into account environmental factors [1]. The financial and environmental perspectives must be economically balanced within this framework in order for the business community to choose the best policy to promote sustainability. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 288 https://internationalpubls.com Numerous research, like Chen, et al. [2], Jaber, et al. [3], dan He, et al. [4], have examined inventory concerns that take carbon emissions into account. The studies often take effects of carbon emissions, order frequency, and storage volume into account [5]. The model was built by several researchers by including the cost of taxes on environmental consequences [6, 7]. In addition, the purchase inventory model was modified by some researchers to include environmental and tax costs [8]. Sustainable EOQ models were examined by Maulana et al. [9] while taking capital restrictions. In previous publications author Baraskar et. al. presented literature survey on inventory management [21] and studied SEOQ considering environmental factors [22]. This research paper presents a comprehensive computational analysis of a framework designed to evaluate the sustainable EOQ model considering emission taxes and capital constraints. The primary objective is to provide a decision-making tool that helps businesses optimize their inventory management practices in a manner that aligns with environmental regulations and financial limitations. By integrating these factors, the proposed framework aims to offer a more holistic approach to EOQ, promoting sustainable business practices without compromising economic viability. The significance of this study lies in its interdisciplinary approach, merging concepts from environmental economics, operations management, and financial analysis. Traditional EOQ models are typically rooted in operational efficiency, focusing on minimizing costs associated with ordering and holding inventory. However, the incorporation of emission taxes introduces a layer of complexity that necessitates a broader analytical perspective. Emission taxes directly impact the cost structure of inventory management, influencing both the frequency and quantity of orders. Therefore, the proposed framework not only evaluates the economic aspects but also addresses the environmental impacts of inventory decisions. Furthermore, capital constraints add another layer of complexity to the EOQ model. Limited financial resources can restrict a company's ability to purchase in bulk or invest in sustainable practices, such as adopting cleaner technologies or sourcing eco-friendly materials. This constraint must be carefully balanced with the goal of minimizing total inventory costs, including emission tax liabilities. The framework proposed in this study incorporates financial constraints into the EOQ model, ensuring that the recommendations are practical and applicable to businesses with varying financial capabilities. 2. FRAMEWORKS In the current study, author assessed the SEOQ framework, which took into account the expenses associated with the sustainable inventory, including order cost, purchase cost, holding cost, the fixed cost of an environmental effect (carbon emission tax cost) for each cycle, and capital constraint. This study's goal is to assess a more complex approach for solving the issues of calculating lot size by taking environmental concerns, capital constraints of raw materials and purchasing of emission tax into account. As a result, our research evaluated perspectives on inventories, particularly for SEOQ models with the capital of raw material purchase with (2) /without (1) emission tax, which is achieved by adding a constraint function i.e., capital constraint. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 289 https://internationalpubls.com M = (c + v) ∗ Q 1 M = (c + pv) ∗ Q 2 2.1 ASSUMPTIONS  Demand rate (λ) is predictable, constant, and uniformly dispersed over the course of the year.  Every demand is met on schedule.  All of the model's variables remain constant over time.  The impact on the environment is taken into account for all costs.  The planning horizon is infinitely long.  A fixed cost (k + f) is incurred each and every time an order is placed. Each unit of inventory has a holding cost in inventory of (h + g). And the cost per unit of purchase is (c + v).  Each model is applied to a single emission product, and the tax cost per item  Tax price is per unit of emissions  Capital used is the capital for the purchase of raw materials and purchasing of emission tax 2.2 FRAMEWORK 1: Sustainable EOQ without Emission Tax with Capital Constraint Evaluation of the SEOQ with capital constraint and without emission tax is our aim. Study evaluated the problems of determining the lot size by considering capital constraints for purchasing raw of materials. In this study Lagrange function is used to minimize the total inventory cost against the capital constraint. The model of total inventory cost as evaluated in previous study [22] is as shown in equation (3) is added to the constraint function in equation (1). Langrange function of the proposes SEOQ model without tax and with capital constraint is shown in equation (4) TIC = Ordering cost + Purchasing cost + Holding cost TIC(Q) = (k + f) ∗ λ Q + (c + v) ∗ λ + (h + g) ∗ Q 2 3 Applying Langrange function to minimize total inventory cost against the constraint by adding the total inventory cost with capital constraint. 𝑇𝐼𝐶(𝑄, 𝛽) = TIC(Q) − 𝛽 ∗ 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 4 From equation (1), constraint function will be (c + v) ∗ Q − M = 0 5 𝐿(𝑄, 𝛽) = TIC(Q, β) = (k + f) ∗ λ Q + (c + v) ∗ λ + (h + g) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑣) − 𝑀) 6 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 290 https://internationalpubls.com Now differentiate partially to Q, β and equate first differentiation to zero to get minimum cost. Equation (6) is differentiated with respect to Q to obtain the optimal Q for the SEOQ model with capital constraint without tax, and the first derivative is then set equal to zero to obtain the minimal cost. d dQ [TIC(Q, β) ] = d dQ [ (k + f) ∗ λ Q + (c + v) ∗ λ + (h + g) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑣) − 𝑀) ] = 0 7 (k + f)λ d dQ [ 1 Q ] + d dQ [(c + v)λ] + (h + g) 2 d dQ [Q] − 𝛽(𝑐 + 𝑣) d dQ [𝑄] + d dQ [𝛽𝑀] = 0 8 −(𝑘 + 𝑓)λ d dQ [𝑄] Q𝑠𝑚 2 + 0 + (h + g) 2 ∗ 1 − 𝛽(𝑐 + 𝑣) ∗ 1 + 0 = 0 9 − (𝑘 + 𝑓)λ Q𝑠𝑚 2 + (h + g) 2 − 𝛽(𝑐 + 𝑣) = 0 10 (𝑘 + 𝑓)λ Q𝑠𝑚 2 = (h + g) 2 − 𝛽(𝑐 + 𝑣) 11 (𝑘 + 𝑓)λ Q𝑠𝑚 2 = (h + g) − 2𝛽(𝑐 + 𝑣) 2 12 2(𝑘 + 𝑓)λ Q𝑠𝑚 2 = (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 13 Q𝑠𝑚 2 = 2λ(𝑘 + 𝑓) (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 14 𝑄𝑠𝑚 = √ 2λ(𝑘 + 𝑓) (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 15 Now differentiate equation (6) partially with respect to β and equate first differentiation to zero to get minimum cost. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 291 https://internationalpubls.com d dβ [TIC(Q, β) ] = d dβ [ (k + f) ∗ λ Q + (c + v) ∗ λ + (h + g) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑣) − 𝑀) ] = 0 16 d dβ [ (k + f)λ Q ] + d dβ [(c + v)λ] + d dβ [ (h + g)𝑄 2 ] − 𝑄(𝑐 + 𝑣) d dβ [𝛽] + M d dβ [𝛽] = 0 17 0 + 0 + 0 − 𝑄𝑠𝑚(𝑐 + 𝑣) ∗ 1 + M ∗ 1 = 0 18 𝑀 − 𝑄𝑠𝑚(𝑐 + 𝑣) = 0 19 𝑀 = 𝑄𝑠𝑚(𝑐 + 𝑣) 20 𝑄𝑠𝑚 = 𝑀 (𝑐 + 𝑣) 21 Equating equation [15] and [21] 𝑄𝑠𝑚 = 𝑀 (𝑐 + 𝑣) = √ 2(𝑘 + 𝑓)λ (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 22 𝑀2 (𝑐 + 𝑣)2 = 2λ(𝑘 + 𝑓) (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 23 𝑀2((ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣)) = 2λ(𝑘 + 𝑓)(𝑐 + 𝑣)2 24 𝑀2(ℎ + 𝑔) − 2𝛽𝑀2(𝑐 + 𝑣) = 2λ(𝑘 + 𝑓)(𝑐 + 𝑣)2 25 2𝛽𝑀2(𝑐 + 𝑣) = 𝑀2(ℎ + 𝑔) − 2λ(𝑘 + 𝑓)(𝑐 + 𝑣)2 26 𝛽 = 𝑀2(ℎ + 𝑔) − 2λ(𝑘 + 𝑓)(𝑐 + 𝑣)2 2𝑀2(𝑐 + 𝑣) 27 Further, substitute equation (15) and (27) in equation (6) to calculate the total inventory cost with capital constraint without emission tax (TIC(Qsm)) From figure 2, At Optimal quantity ordering cost and holding cost are same. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 292 https://internationalpubls.com Figure 2. Ordering and holding costs as functions of the order quantity. So equation (6) becomes TIC(Qsm) = 2(k + f) ∗ λ Qsm + (c + v) ∗ λ − 𝛽(𝑄𝑠𝑚(𝑐 + 𝑣) − 𝑀) 28 Further substituting constraint equation (20) in equation (28), we will get TIC(Qsm) as TIC(Qsm) = 2(k + f) ∗ λ Qsm + (c + v) ∗ λ − 𝛽 ∗ 0 29 . TIC(Qsm) = λ(c + v) + 2λ(k + f) Qsm 30 Further substituting equation (15) in equation (30), we will get TIC(Qsm) as TIC(Qsm) = λ(c + v) + 2λ(k + f) √ 2λ(𝑘 + 𝑓) (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 31 TIC(Qsm) = λ(c + v) + 2λ(k + f)√ (ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣) 2λ(𝑘 + 𝑓) 32 TIC(Qsm) = λ(c + v) + √ (2λ(k + f)) 2 ((ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣)) 2λ(𝑘 + 𝑓) 33 TIC(Qsm) = λ(c + v) + √2λ(k + f)((ℎ + 𝑔) − 2𝛽(𝑐 + 𝑣)) 34 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 293 https://internationalpubls.com 2.3 FRAMEWORK 2: Sustainable EOQ with Emission Tax with Capital Constraints Evaluation of the SEOQ with capital constraint and emission tax is our aim. Study evaluated the problems of determining the lot size by considering sustainability and capital constraints for purchasing raw of materials. In this study Lagrange function is used to minimize the total inventory cost against the capital constraint. The model of total inventory cost as evaluated in previous study [22] is as shown in equation (35) is added to the constraint function in equation (2). Langrange function of the proposes SEOQ model with tax and capital constraint is shown in equation (36) TIC = Ordering cost + Purchasing cost + Holding cost TIC(Q) = (k + pf) ∗ λ Q + (c + pv) ∗ λ + (h + pg) ∗ Q 2 35 Applying Langrange function to minimize total inventory cost against the constraint by adding the total inventory cost with capital constraint. 𝑇𝐼𝐶(𝑄, 𝛽) = TIC(Q) − 𝛽 ∗ 𝑐𝑜𝑛𝑠𝑡𝑟𝑎𝑖𝑛𝑡 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 36 From equation (2), constraint function will be (c + pv) ∗ Q − M = 0 37 𝐿(𝑄, 𝛽) = TIC(Q, β) = (k + pf) ∗ λ Q + (c + pv) ∗ λ + (h + pg) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑝𝑣) − 𝑀) 38 Now differentiate partially to Q, β and equate first differentiation to zero to get minimum cost. Equation (38) is differentiated with respect to Q to obtain the optimal Q for the SEOQ model with capital constraint with tax, and the first derivative is then set equal to zero to obtain the minimal cost. d dQ [TIC(Q, β) ] = d dQ [ (k + pf) ∗ λ Q + (c + pv) ∗ λ + (h + pg) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑝𝑣) − 𝑀) ] = 0 39 (k + pf)λ d dQ [ 1 Q ] + d dQ [(c + pv)λ] + (h + pg) 2 d dQ [Q] − 𝛽(𝑐 + 𝑝𝑣) d dQ [𝑄] + d dQ [𝛽𝑀] = 0 40 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 294 https://internationalpubls.com −(𝑘 + 𝑝𝑓)λ d dQ [𝑄] Q𝑠𝑝𝑚 2 + 0 + (h + pg) 2 ∗ 1 − 𝛽(𝑐 + 𝑝𝑣) ∗ 1 + 0 = 0 41 − (𝑘 + 𝑝𝑓)λ Q𝑠𝑝𝑚 2 + (h + pg) 2 − 𝛽(𝑐 + 𝑝𝑣) = 0 42 (𝑘 + 𝑝𝑓)λ Q𝑠𝑝𝑚 2 = (h + pg) 2 − 𝛽(𝑐 + 𝑝𝑣) 43 (𝑘 + 𝑝𝑓)λ Q𝑠𝑝𝑚 2 = (h + pg) − 2𝛽(𝑐 + 𝑝𝑣) 2 44 2(𝑘 + 𝑝𝑓)λ Q𝑠𝑝𝑚 2 = (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 45 Q𝑠𝑝𝑚 2 = 2λ(𝑘 + 𝑝𝑓) (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 46 𝑄𝑠𝑝𝑚 = √ 2λ(𝑘 + 𝑝𝑓) (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 47 Now differentiate equation (38) partially with respect to β and equate first differentiation to zero to get minimum cost. d dβ [TIC(Q, β) ] = d dβ [ (k + pf) ∗ λ Q + (c + pv) ∗ λ + (h + pg) ∗ Q 2 − 𝛽(𝑄(𝑐 + 𝑝𝑣) − 𝑀) ] = 0 48 d dβ [ (k + pf)λ Q ] + d dβ [(c + pv)λ] + d dβ [ (h + pg)𝑄 2 ] − 𝑄(𝑐 + 𝑝𝑣) d dβ [𝛽] + M d dβ [𝛽] = 0 49 0 + 0 + 0 − 𝑄𝑠𝑝𝑚(𝑐 + 𝑝𝑣) ∗ 1 + M ∗ 1 = 0 50 𝑀 − 𝑄𝑠𝑝𝑚(𝑐 + 𝑝𝑣) = 0 51 𝑀 = 𝑄𝑠𝑝𝑚(𝑐 + 𝑝𝑣) 52 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 295 https://internationalpubls.com 𝑄𝑠𝑝𝑚 = 𝑀 (𝑐 + 𝑝𝑣) 53 Equating equation [47] and [53] 𝑄𝑠𝑝𝑚 = 𝑀 (𝑐 + 𝑝𝑣) = √ 2(𝑘 + 𝑝𝑓)λ (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 54 𝑀2 (𝑐 + 𝑝𝑣)2 = 2λ(𝑘 + 𝑝𝑓) (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 55 𝑀2((ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣)) = 2λ(𝑘 + 𝑝𝑓)(𝑐 + 𝑝𝑣)2 56 𝑀2(ℎ + 𝑝𝑔) − 2𝛽𝑀2(𝑐 + 𝑝𝑣) = 2λ(𝑘 + 𝑝𝑓)(𝑐 + 𝑝𝑣)2 57 2𝛽𝑀2(𝑐 + 𝑝𝑣) = 𝑀2(ℎ + 𝑝𝑔) − 2λ(𝑘 + 𝑝𝑓)(𝑐 + 𝑝𝑣)2 58 𝛽 = 𝑀2(ℎ + 𝑝𝑔) − 2λ(𝑘 + 𝑝𝑓)(𝑐 + 𝑝𝑣)2 2𝑀2(𝑐 + 𝑝𝑣) 59 Further, substitute equation (47) and (59) in equation (38) to calculate the total inventory cost with capital constraint with emission tax (TIC(Qspm)) At Optimal quantity ordering cost and holding cost are same. So equation (38) becomes TIC(Qspm) = 2(k + pf) ∗ λ Qspm + (c + pv) ∗ λ − 𝛽(𝑄𝑠𝑝𝑚(𝑐 + 𝑝𝑣) − 𝑀) 60 Further substituting constraint equation (52) in equation (60), we will get TIC(Qspm) as TIC(Qspm) = 2(k + pf) ∗ λ Qspm + (c + pv) ∗ λ − 𝛽 ∗ 0 61 . TIC(Qspm) = λ(c + pv) + 2λ(k + pf) Qspm 62 Further substituting equation (47) in equation (62), we will get TIC(Qspm) as TIC(Qspm) = λ(c + pv) + 2λ(k + pf) √ 2λ(𝑘 + 𝑝𝑓) (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 63 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 296 https://internationalpubls.com TIC(Qspm) = λ(c + pv) + 2λ(k + pf)√ (ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣) 2λ(𝑘 + 𝑝𝑓) 64 TIC(Qspm) = λ(c + pv) + √ (2λ(k + pf)) 2 ((ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣)) 2λ(𝑘 + 𝑝𝑓) 65 TIC(Qspm) = λ(c + pv) + √2λ(k + pf)((ℎ + 𝑝𝑔) − 2𝛽(𝑐 + 𝑝𝑣)) 66 2.4 NUMERICAL EXAMPLE This section shows the numerical experiment procedure on the proposed SEOQ models. The experiment was carried out to test the sensitivity of the proposed models. A case study data is presented in Table 1. Table 1. Experimental Data Variables Data Variables Unit Value Demand 𝜆 qty 50 Cost per order k $/order 40 Purchasing cost per unit c $/qty 20 Emission tax cost p $/qty 2 Holding cost per unit h $/qty 10 Total emissions from ordering f $/qty CO2 60 Total emissions from purchasing v $/qty CO2 5 Total emissions from holding g $/qty CO2 1 Capital M $ 1000 2.5 RESULTS AND DISCUSSION The jamovi project (2022). jamovi. (Version 2.3) Software is used for solving frameworks and checking numerical sensitivity and sustainability. 2.5.1 Sustainable EOQ without Emission Tax with Capital Constraints Figure 3. Ordering and holding costs as functions of the order quantity. 0 500 1000 1500 2000 2500 3000 0 20 40 60 80 100 C O ST Q OC PC HC TIC(TMQsm) TIC(Qsm) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 297 https://internationalpubls.com Figure 4. Effect of % Change in k, c, h on Qsm and TIC(Qsm) along with standard error (Std. Err.). 2.5.2 Sustainable EOQ with Emission Tax with Capital Constraints Figure 5. Ordering and holding costs as functions of the order quantity. Figure 6. Effect of % Change in k, c, h on Qspm and TIC(Qspm) along with standard error (Std. Err.). 0 500 1000 1500 2000 2500 3000 3500 4000 0 10 20 30 40 50 60 70 80 90 C O ST Q Ocp PCp HCp TIC(TMQspm) TIC(Qspm) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 298 https://internationalpubls.com 2.6 SENSITIVITY ANALYSIS 2.6.1 Sensitivity Table 2: Change in Qsm, Qspm, TIC(Qsm) and TIC(Qspm) due to change in cost per order(k). % Change k f 𝜆 c v h g p M 𝛽 Qsm TIC(Qsm) B Qspm TIC(Qspm) -50 20 60 50 20 5 10 1 2 1000 0.120 40 1450 -0.010 54 1920 -40 24 60 50 20 5 10 1 2 1000 0.115 40 1460 -0.016 53 1932 -30 28 60 50 20 5 10 1 2 1000 0.110 40 1470 -0.022 52 1944 -20 32 60 50 20 5 10 1 2 1000 0.105 40 1480 -0.028 52 1956 -10 36 60 50 20 5 10 1 2 1000 0.100 40 1490 -0.034 51 1968 0 40 60 50 20 5 10 1 2 1000 0.095 40 1500 -0.040 50 1980 10 44 60 50 20 5 10 1 2 1000 0.090 40 1510 -0.046 50 1992 20 48 60 50 20 5 10 1 2 1000 0.085 40 1520 -0.052 49 2004 30 52 60 50 20 5 10 1 2 1000 0.080 40 1530 -0.058 49 2016 40 56 60 50 20 5 10 1 2 1000 0.075 40 1540 -0.064 48 2028 50 60 60 50 20 5 10 1 2 1000 0.070 40 1550 -0.070 48 2040 Table 3: Change in Qsm, Qspm, TIC(Qsm) and TIC(Qspm) due to change in purchase cost per unit(c). % Change k f 𝜆 c v h g p M 𝛽 Qsm TIC(Qsm) B Qspm TIC(Qspm) -50 40 60 50 10 5 10 1 2 1000 0.292 67 900 0.140 219 1320 -40 40 60 50 12 5 10 1 2 1000 0.239 59 1020 0.097 103 1452 -30 40 60 50 14 5 10 1 2 1000 0.194 53 1140 0.058 77 1584 -20 40 60 50 16 5 10 1 2 1000 0.157 48 1260 0.023 65 1716 -10 40 60 50 18 5 10 1 2 1000 0.124 43 1380 -0.010 56 1848 0 40 60 50 20 5 10 1 2 1000 0.095 40 1500 -0.040 50 1980 10 40 60 50 22 5 10 1 2 1000 0.069 37 1620 -0.069 46 2112 20 40 60 50 24 5 10 1 2 1000 0.045 34 1740 -0.096 42 2244 30 40 60 50 26 5 10 1 2 1000 0.022 32 1860 -0.121 39 2376 40 40 60 50 28 5 10 1 2 1000 0.002 30 1980 -0.146 37 2508 50 40 60 50 30 5 10 1 2 1000 -0.018 29 2100 -0.170 35 2640 Table 4: Change in Qsm, Qspm, TIC(Qsm) and TIC(Qspm) due to change in holding cost per unit(h). % Change k f 𝜆 c v h g p M 𝛽 Qsm TIC(Qsm) B Qspm TIC(Qspm) -50 40 60 50 20 5 5 1 2 1000 -0.005 40 1500 -0.123 47 1980 -40 40 60 50 20 5 6 1 2 1000 0.015 40 1500 -0.107 47 1980 -30 40 60 50 20 5 7 1 2 1000 0.035 40 1500 -0.090 48 1980 -20 40 60 50 20 5 8 1 2 1000 0.055 40 1500 -0.073 49 1980 -10 40 60 50 20 5 9 1 2 1000 0.075 40 1500 -0.057 50 1980 0 40 60 50 20 5 10 1 2 1000 0.095 40 1500 -0.040 50 1980 10 40 60 50 20 5 11 1 2 1000 0.115 40 1500 -0.023 51 1980 20 40 60 50 20 5 12 1 2 1000 0.135 40 1500 -0.007 52 1980 30 40 60 50 20 5 13 1 2 1000 0.155 40 1500 0.010 53 1980 40 40 60 50 20 5 14 1 2 1000 0.175 40 1500 0.027 54 1980 50 40 60 50 20 5 15 1 2 1000 0.195 40 1500 0.043 55 1980 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 299 https://internationalpubls.com Table 5: Change in Qsm, Qspm, TIC(Qsm) and TIC(Qspm) due to change in Capital(M). % Change k f 𝜆 c v h g p M 𝛽 Qsm TIC(Qsm) B Qspm TIC(Qspm) -50 40 60 50 20 5 10 1 2 500 -0.280 20 1750 -0.760 24 2460 -40 40 60 50 20 5 10 1 2 600 -0.127 24 1667 -0.467 29 2300 -30 40 60 50 20 5 10 1 2 700 -0.035 28 1607 -0.290 34 2186 -20 40 60 50 20 5 10 1 2 800 0.025 32 1563 -0.175 39 2100 -10 40 60 50 20 5 10 1 2 900 0.066 36 1528 -0.096 45 2033 0 40 60 50 20 5 10 1 2 1000 0.095 40 1500 -0.040 50 1980 10 40 60 50 20 5 10 1 2 1100 0.117 44 1477 0.002 57 1936 20 40 60 50 20 5 10 1 2 1200 0.133 48 1458 0.033 63 1900 30 40 60 50 20 5 10 1 2 1300 0.146 52 1442 0.058 70 1869 40 40 60 50 20 5 10 1 2 1400 0.156 56 1429 0.078 78 1843 50 40 60 50 20 5 10 1 2 1500 0.164 60 1417 0.093 87 1820 Figure 7. Change in TIC(Qsm) due to change in cost per order(k), purchase(c), holding(h) cost per unit and capital(M). Figure 8. Change in TIC(Qspm) due to change in cost per order(k), purchase(c) and holding(h) cost per unit and capital(M). 0 500 1000 1500 2000 2500 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 TI C (Q SM ) % CHANGE Sensitivity of cost per order(k) to TIC(Qsm) Sensitivity of purchase cost per unit(c) to TIC(Qsm) Sensitivity of holding cost per unit(h) to TIC(Qsm) Sensitivity of Capital to TIC(Qsm) 0 500 1000 1500 2000 2500 3000 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 TI C Q SP M % CHANGE Sensitivity of cost per order(k) to TIC(Qspm) Sensitivity of purchase cost per unit(c) to TIC(Qspm) Sensitivity of holding cost per unit(h) to TIC(Qspm) Sensitivity of Capital to TIC(Qspm) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 300 https://internationalpubls.com Figure 9. Change in Qsm due to change in cost per order, purchase and holding cost per unit and capital. Figure 10. Change in Qspm due to change in cost per order, purchase and holding cost per unit and capital. 2.6.2 ANOVA Table 6: ANOVA. ANOVA – TIC(Qspm) Sum of Squares df Mean Square F p Overall Model 0 30 0 1.98 0.096 K 0 NaN c 0 0 h 0 0 k✻c 0 0 k✻h 0 0 c✻h 0 0 k✻c✻h 0 0 Residuals 427398 13 32877 0 10 20 30 40 50 60 70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Q SM % CHANGE Sensitivity of cost per order(k) to EOQ (Qsm) Sensitivity of purchase cost per unit(c) to EOQ (Qsm) Sensitivity of holding cost per unit(h) to EOQ (Qsm) Sensitivity of Capital to EOQ (Qsm) 0 50 100 150 200 250 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 Q SP M % CHANGE Sensitivity of cost per order(k) to EOQ (Qspm) Sensitivity of purchase cost per unit(c) to EOQ (Qspm) Sensitivity of holding cost per unit(h) to EOQ (Qspm) Sensitivity of Capital to EOQ (Qspm) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 301 https://internationalpubls.com 2.6.3 Assumptions Checks Table 7: Homogeneity of Variances & Normality Test Homogeneity of Variances Test (Levene's) F df1 df2 p 0.444 30 13 0.967 Normality Test (Shapiro-Wilk) Static P 0.642 <0.001 Figure 11. Q-Q Plot. 2.6.4 Correlation Matrix Table 8: Correlation Matrix Qsm Qspm TIC(Qsm) TIC(Qspm) k c h M Qsm Pearson's r p-value 95% CI Upper 95% CI Lower N - - - - - Qspm Pearson's r p-value 95% CI Upper 95% CI Lower N 0.817 <0 .001 0.897 0.687 44 - - - - - TIC(Qsm) Pearson's r p-value 95% CI Upper 95% CI Lower N -0.811 <0 .001 -0.677 -0.893 44 -0.746 <0 .001 -0.576 -0.854 44 - - - - - TIC(Qspm) Pearson's r p-value 95% CI Upper 95% CI Lower N -0.883 <0 .001 -0.795 -0.935 44 -0.758 <0 .001 -0.594 -0.861 44 0.984 <0.001 0.991 0.971 44 - - - - - Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 302 https://internationalpubls.com k Pearson's r p-value 95% CI Upper 95% CI Lower N 0.000 1.000 0.297 -0.297 44 -0.033 0.834 0.267 -0.326 44 0.080 0.605 0.368 -0.222 44 0.082 0.599 0.370 -0.221 44 - - - - - c Pearson's r p-value 95% CI Upper 95% CI Lower N -0.656 <0 .001 -0.446 -0.797 44 -0.683 <0 .001 -0.484 -0.815 44 0.961 <0.001 0.978 0.929 44 0.897 <0.001 0.943 0.819 44 0.000 1.000 0.297 -0.297 44 - - - - - h Pearson's r p-value 95% CI Upper 95% CI Lower N 0.000 1.000 0.297 -0.297 44 0.045 0.773 0.337 -0.256 44 0.000 1.000 0.297 -0.297 44 0.000 1.000 0.297 -0.297 44 0.000 1.000 0.297 -0.297 44 0.000 1.000 0.297 -0.297 44 - - - - - M Pearson's r p-value 95% CI Upper 95% CI Lower N 0.724 <0.001 0.840 0.544 44 0.344 0.022 0.581 0.052 44 -0.245 0.108 0.055 -0.505 44 -0.400 0.007 -0.117 -0.623 44 0.000 1.000 0.297 -0.297 44 0.000 1.000 0.297 -0.297 44 0.000 1.000 0.297 -0.297 44 - - - - - Figure 12. Correlation Matrix Plot. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 303 https://internationalpubls.com Table 9: Descriptives k c h M Qsm TIC(Qsm) Qspm TIC (Qspm) N 44 44 44 44 44 44 44 44 Mean 40.0 20.0 10.0 1000 40.7 1508 55.9 1995 Median 40.0 20.0 10.0 1000 40.0 1500 50.4 1980 Standard Deviation 6.40 3.20 1.60 160 8.83 200 28.9 235 Minimum 20 10 5 500 20.0 900 23.6 1320 Maximum 60 30 15 1500 66.7 2100 219 2640 Figure 13. Correlation Heatmap. 2.7 CONCLUSION In the current study, SEOQ framework is evaluated, which took into account the expenses associated with the sustainable inventory, including order cost, purchase cost, holding cost, fixed cost of an environmental effect (carbon emission tax cost) and capital required for raw materials and purchasing of emission tax into account. As a result, research produced perspectives on inventories, particularly for SEOQ models with cost of the emission tax and capital constraints. Further research can be done on study of different use cases SEOQ for different industries. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 304 https://internationalpubls.com REFERENCES: [1] Battini, D., Persona, A., & Sgarbossa, F. (2014). A sustainable EOQ model: Theoretical formulation and applications. International Journal of Production Economics, 149, 145–153. doi: 10.1016/j.ijpe.2013.06.026. [2] X. Chen, S. Benjaafar, & A. Elomri (2013). The carbon-constrained EOQ. Operations Research Letters, vol. 41, pp. 172-179. https://doi.org/10.1016/j.orl.2012.12.003. [3] M. Y. Jaber, M. Bonney, & H. Jawad (2017). Comparison between economic order/manufacture quantity and just-in- time models from a thermodynamics point of view. Computers & Industrial Engineering, vol. 112, pp. 503-510. https://doi.org/10.1016/j.cie.2016.08.023. [4] P. He, W. Zhang, X. Xu, and Y. Bian (2015). Production lot-sizing and carbon emissions under cap-and-trade and carbon tax regulations. Journal of Cleaner Production, vol. 103, pp. 241-248. https://doi.org/10.1016/j.jclepro.2014.08.102. [5] A. Gurtu, M. Y. Jaber, and C. Searcy (2015), Impact of fuel price and emissions on inventory policies. Applied Mathematical Modelling, vol. 39, pp. 1202-1216. https://doi.org/10.1016/j.apm.2014.08.001. [6] Arslan, M. C., & Turkay, M. (2013). EOQ Revisited with Sustainability Considerations. Foundations of Computing and Decision Sciences, 38(4), 223–249. doi:10.2478/fcds-2013-0011. [7] M. Bonney and M. Y. Jaber (2011), Environmentally responsible inventory models: Non-classical models for a non- classical era. International Journal of Production Economics, vol. 133, pp. 43-53, https://doi.org/10.1016/j.ijpe.2009.10.033. [8] D.-H. Lee, M. Dong, & W. Bian (2010). The design of sustainable logistics network under uncertainty. International Journal of Production Economics, vol. 128, pp. 159-166, 2010. https://doi.org/10.1016/j.ijpe.2010.06.009. [9] Maulana, S. K. D. B., Utama, D. M., Asrofi, M. S., Ningrum, I. S., Alba, N., Ahfa, H. A., & Zein, T. A. (2019). The Capacitated Sustainable EOQ Models: Models Considering Tax Emissions. Jurnal Teknik Industri, 21(1), 12-21. https://doi.org/10.22219/JTIUMM.Vol21.No1.12-21. [10] Hovelaque, V., & Bironneau, L. (2015). The carbon-constrained EOQ model with carbon emission dependent demand. International Journal of Production Economics, 164, 285–291. doi: 10.1016/j.ijpe.2014.11.022. [11] Hua, G., Cheng, T. C. E., & Wang, S. (2011). Managing carbon footprints in inventory management. International Journal of Production Economics, 132(2), 178–185. doi: 10.1016/j.ijpe.2011.03.024. [12] Paul, S., Wahab, M. I. M., & Cao, X. F. (2014). Supply chain coordination with energy price uncertainty, carbon emission cost, and product return. In Handbook of EOQ Inventory Problems (pp. 179-199). Springer, Boston, MA. [13] S. Ruidas, M. R. Seikh, P.K. Nayak (2021). A production inventory model with interval-valued carbon emission parameters under price-sensitive demand. Computer & Industrial Engineering 107154. https://doi.org/10.1016/j.cie.2021.107154. [14] X. Ma, P. Ji, W. Ho, & C. H. Yang (2018). Optimal procurement decision with a carbon tax for the manufacturing industry. Computers & Operations Research, vol. 89, pp. 360-368. https://doi.org/10.1016/j.cor.2016.02.017. [15] N. Absi, S. Dauzère-Pérès, S. Kedad-Sidhoum, B. Penz, & C. Rapine (2016). The single-item green lot-sizing problem with fixed carbon emissions. European Journal of Operational Research, vol. 248, pp. 849- 855. https://doi.org/10.1016/j.ejor.2015.07.052. [16] The jamovi project (2022). jamovi. (Version 2.3) [Computer Software]. Retrieved from https://www.jamovi.org. [17] R Core Team (2021). R: A Language and environment for statistical computing. (Version 4.1) [Computer software]. Retrieved from https://cran.r-project.org. (R packages retrieved from MRAN snapshot 2022-01-01). [18] R Core Team (2018). A Language and Envionment for Statistical Computing. [Computer software]. Retrieved from https://cran.r-project.org/. [19] Fox, J., & Weisberg, S. (2020). car: Companion to Applied Regression. [R package]. Retrieved from https://cran.rproject.org/package=car. [20] Revelle, W. (2019). psych: Procedures for Psychological, Psychometric, and Personality Research. [R package]. Retrieved from https://cran.r-project.org/package=psych. [21] A. Baraskar, Y. Deshmukh (2023). A literature survey on inventory management in manufacturing industry. Journal Of Aeronautical Materials. Vol. 43, Issue-1. Pp. 203-234. https://www.hkclxb.cn/article/view/2023/203.html. https://doi.org/10.1016/j.cie.2016.08.023 https://doi.org/10.1016/j.jclepro.2014.08.102 https://doi.org/10.1016/j.cie.2021.107154 https://doi.org/10.1016/j.cor.2016.02.017 https://doi.org/10.1016/j.ejor.2015.07.052 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 305 https://internationalpubls.com [22] A. Baraskar, Y. Deshmukh, A. Thakare (2023). A framework Evolution for Sustainable EOQ Considering Environmental Factors. Journal Of Harbin Engineering University. Vol. 44, Issue-8. Pp. 56-68. https://harbinengineeringjournal.com/index.php/journal/article/view/826. THE NOTATIONS: : Demand k : Cost per order c : Purchasing cost per unit h : Holding cost per unit g : Total emissions from holding v : Total emissions from purchasing f : Total emissions from ordering p : Emission tax cost β : Lagrange multipliers M : capital Q : Number of orders Qs : Optimal sustainable number of orders without tax Qsp : Optimal sustainable number of orders with tax Qsm : Optimal sustainable number of orders without tax with capital constraint Qspm : Optimal sustainable number of orders with tax and capital constraint TIC : Total inventory cost TIC(Qs) : Optimum total inventory cost without tax TIC(Qsp) : Optimum total inventory cost with tax TIC(Qsm) : Optimum total inventory cost without tax with capital constraints TIC(Qspm) : Optimum total inventory cost with tax and capital constraint