Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 760 https://internationalpubls.com A Sustainable Supply Chain Model for Defective Items Under Fuzzy Environment Subhkaran1, Pushpendra Kumar1*, Soniya Gupta2, A. K. Malik3 1Department of Mathematics, Shri Khushal Das University, Hanumangarh (Rajasthan) 2Faculty of Engineering & Technology, Swami Vivekanand Subharti University, Meerut U. P. 3School of Sciences, UP Rajarshi Tandon Open University, Prayagraj (U.P.) Email: subhkaran@skduniversity.com; pushpendra.kumar@skduniversity.com; drsoniyainpg@gmail.com; profakmalik@uprtou.ac.in *Corresponding author Article History: Received: 14-02-2025 Revised: 15-03-2025 Accepted: 21-03-2025 Abstract These days it is very difficult to produce 100% good quality items. Hence, delivered lot may have some defective items in the lot and the process of inspection is must for the improvement of quality of items. We deal a sustainable supply chain model for defective items under centroid method defuzzification fuzzy environment by using of centroid method. Nowadays, the emission of carbon from the storage, various type of gases like carbon dioxide, sulphur dioxide and methane and transportations is more dangerous for the human life and also affects the activities of business. Due to carbon emissions some cities are in more and more problems like Delhi, Noida and Ghaziabad. These gases are more responsible for the damaging of the environment balancing and poisonous for the human style. In the Finally, we minimized the total inventory fuzzy cost with respect to the order quantity for the eco-friendly items and also presented numerical example for the justification of the proposed mode. The sensitivity observation has been presented for the ordering policies as well as decision makers and future scope of this scenario also presented. Keywords: EOQ, inspection, imperfect quality items, carbon emission. Abstract: 1. Introduction Inventory is the complete list of items such as goods in property. Inventory includes the raw material, partially manufactured items, prior to sale and departure from the manufacturing product. Inventory is widely used in public as well as private sector organization. Proposed strategy of Hill (1997) refers to the shipment needs to be delivered to retailer during the production segment of manufacturer cycle. In addition to Hill (1997) transportation cost and emission cost was added by Aljazzar et al. (2017). Where transportation cost & emission through transportation was defined in detail by Gurtu et al. (2015) and manufacturer emission was defined by Jaber et al. (2013). Whenever we purchase any item from the market, we must be inspecting the item because of many defects in between them (Jaggi et al., 2011). Also in mailto:subhkaran@skduniversity.com mailto:pushpendra.kumar@skduniversity.com mailto:drsoniyainpg@gmail.com mailto:profakmalik@uprtou.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 761 https://internationalpubls.com supply chain model it affects the demand when there are imperfect items available in the lot (Yadav et al. 2018). Huang (2007) research tasks have been modified by Teng and Chang (2009) with the help of a two-level credit financing scheme which is helpful for the consumers. Chen and Kang (2010) expand a model for an industrial system with the help of two-level credit policy with price dependent demand along with a satisfaction. Shah et at. (2010) has provided a large amount of articles for the inventory system with the help of a two-level credit financing scheme. That factor affected the cost of airplanes examined by Wright (1936). Mandal and Giri (2017) developed an imperfect quality based inventory model for two warehouse system under supply chain model where demand rate varies with respect to stock of the inventory. Sheikh and Patel (2017) improved a two-warehouse based model by using of the shortages concept for deteriorating items where the rate of demand varies with time. Some authors shown the idea of the fuzzy in the system of two warehouse. The basic idea of the fuzzy theory is suggested by Zadeh (1965). The learning effect on seller’s ordering policy for defective items with trade credit financing was suggested by Jayaswal (2019). The total manufacturing of profit-making models for items with defective quality subjected to the effect of learning described by Jaber et al. (2008). Kumar (2021) derived the formula for the EOQ under fuzzy environment. Malik and Garg (2021), Malik et al. (2017), Malik and Sharma (2011) and Vikram et al. (2016) generalized an inventory model for two-warehouse system under fuzzy environment. Yadav et al. (2022) developed a fuzzy based inventory model for deteriorating items under variable demand. Mandal (2023) proposed an inventory model with fuzzy environment for the deterioration demand under the variable demand with respect to the stock and also variable holding cost. The learning effect is a one type of mathematical too which helps to minimize the total fuzzy cost corresponding to the number of shipment. we included some authors who worked on the learning theory, Jayaswal et al. (2019) developed a learning based model with the effect of trade policy for imperfect quality items under inspection. Jayaswal et al. (2021) presented an imperfect based scenario with trade-credit policy for deteriorating items under inspection process. Vikram et al. (2016) and Singh and Malik (2010) and these authors gave an inventory model with different approach for deteriorating items under various. Jayaswal et al. (2021) assumed an inventory model with preservation technology for the perishable items under learning effect. Alamri et al. (2023) explained an inventory model with fuzzy environment for imperfect items under trade credit policy and some other authors gave an inventory model with different approach for deteriorating items under various. The research work of Goyal et al. (1985) has been generalized by Su et al. (2007) with the help of credit financing policy and payment rebate. Teng et al. (2006) arranged a two-level credit financing inventory model and in this model the total order is optimized below the permissible delay in amount. The analysis work of Huang (2003) has been explored by Huang (2006) and upgraded this model with the help of two-level credit financing as well as restricted storage space. Hung and Chung (2003) introduced an inventory model which is the enlarged form of Goyal (1985) describing how to reduce the total cost with the help of credit financing along with payment rebate scheme. Teng and Goyal (2007) introduced a stock model for the buyers when the buyer used the credit policy provided by the customer in the industrial sector. Huang (2007) stabilized the establishment of total Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 762 https://internationalpubls.com economic order which was increased with the form of Haung (2003) with the help of a two- level credit financing scheme. Huang and Hsu (2008) to expand a two-level credit financing scheme with the help of a partial credit scheme. An economic order quantity formulation has been presented by Jaggi et al. (2008) introducing an economic order quantity formulation with the help of two-level credit policy with credit dependent demand. In this paper we have studied the impact of incorporating the imperfect quality items in the lot to the total cost which includes setup cost, holding cost, transportation cost, production cost, interest earning & emission. If the total cost increases or decreases, it directly impacts the factors included in total cost. So here we have observed the changes in the total lot size and changes in total cost if there are some imperfect qualities in the lot delivered. 2. Notations The following notations have been given under below with meaning description 𝑄 = Order lot (decesion variable) (units) 𝐷 = Retailerβ€²s demand rate (units/year) οΏ½ΜƒοΏ½ = Retailerβ€²s fuzzy demand rate (units/year 𝑃 = Manufacturer’s production ratewhere(𝐷 < 𝑃) π΄π‘Ÿ = Ordering cost for reailer ($/shipment) πΆπ‘Ÿ = Item unit cost for reailer ($/unit) β„Žπ‘Ÿ = Holding cost for reailer ($/unit/year) π‘ π‘Ÿ = Storage holding cost for for reailer ($/unit/year) π΄π‘š = Ordering cost for manufacturer ($/shipment) πΆπ‘š = Item unit cost for manufacturer (in $) β„Žπ‘š = Holding cost for manufacturer($/unit/year) π‘ π‘š = Storage holding cost for for reailer ($/unit/year) Ξ» = Shipment lot size multiplier in a vendor cycle Tm = Manufacturer’s cycle time = Ξ»Q/D (year) 𝛿 = Percentage of defective items Te = Transport emission tax (in $) Tci = Emission tax rate ($/Ton) Tcap = Truck capacity (in integer) Ξ· = Number of trucks per shipment (Q/Tcap) Em = Manufacturing emission (g/unit) 𝑑 = Distance traveled in one side (Km) 𝑇𝐴𝐢𝑅 =Rretailer’s total annual cost (in $) π‘‡π΄πΆπ‘š =Mmanufacturer’s total annual cost (in $) ( )Q = Integrated total annual cost (in $) under crisp model ( )Q ~ = Total integrated fuzzyfied cost (in $) under fuzzy environment ( )Q ~~ = Total integrated defuzzified cost (in $) under fuzzy environment Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 763 https://internationalpubls.com 3. Assumptions ➒ The rate of demand obeys the effect of fuzzy environment. ➒ The demand rate is taken as triangular fuzzy number ➒ Shortages are not permitted in this model. ➒ The rate of demand is less than the rate of production. ➒ The capacity of all trucks is same and uses same path for each truck during supply chain. ➒ Single type of items is used during this supply chain. ➒ The cost of holding units is included during non-production phase. ➒ Delivered lot has some defective items. ➒ Imperfect quality items are sold at low price in another market. ➒ Manufacturer emission cost will be fixed during supply chain ➒ The cost related to holding units is termed as financial cost which will be same during supply chain. ➒ The financial cost for retailer is greater than the financial cost of manufacturer at the retailer’s end. 4. Proposed Model Our proposed model follows the model mythology of Hill (1997) and its inventory representation has been depicted in figure 1. Manufacturer prepared Q units as per demanded retailer. It is suppose that 𝐷is the rate of demand for retailer. In this order, we studied the model of Jaber et. al. (2013) which presented for the manufacturer emissions which are equal to Em = aP2 – bP + c. Gurtu et al. (2015) explained the carbon emission cost associated with transportation and transport with the help of formula (2d eTci). Figure 1: Inventory process with time Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 764 https://internationalpubls.com 4.1 Retailer’s total annual cost In this section, we calculated the total annual cost for the retailer which is the sum of the setup cost (𝑆𝐢 = (π΄π‘Ÿ)𝐷 𝑄 ), item cost (𝐼𝐢 = πΆπ‘Ÿπ·), holding cost ((β„Žπ‘Ÿ + π‘†π‘Ÿ) [ (1βˆ’π›Ώ)𝑄 2 + 𝛿𝑄2 17520 ]), and transportation cost (𝑇𝐢 = πœ‚π·π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 ) and emission tax (𝐸𝑇 = πœ‚π·π‘‡π‘’ 𝑄 ). The Retailer’s annual cost(𝑇𝐴𝐢𝑅)can be defined as given below; 𝑇𝐴𝐢𝑅 = 𝑆𝐢 + 𝐼𝐢 + 𝐼𝐻𝐢 + 𝑇𝐢 + 𝐸𝑇 𝑇𝐴𝐢𝑅 = (π΄π‘Ÿ)𝐷 𝑄 + πΆπ‘Ÿπ· + (β„Žπ‘Ÿ + π‘†π‘Ÿ) [ (1βˆ’π›Ώ)𝑄 2 + 𝛿𝑄2 17520 ] + πœ‚π·π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 + πœ‚π·π‘‡π‘’ 𝑄 …(1) 4.2 Manufacturer’s total annual cost In this section, we calculated the total annual cost for the retailer which is the sum of the setup cost(π‘†πΆπ‘š = (π΄π‘š)𝐷 πœ†π‘„ ), item cost (πΌπΆπ‘š = πΆπ‘šπ·), holding cost (πΌπ»πΆπ‘š = (β„Žπ‘š + π‘†π‘š) 𝑄 2 [ 2𝐷 𝑃 + πœ†(π‘ƒβˆ’π·) 𝑃 βˆ’ 1]), and transportation cost (π‘‡πΆπ‘š = πœ‚π·π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 ) and emission tax (𝐸𝑇𝑀 = πΈπ‘šπ·π‘‡π‘π‘–). The manufacturer’s annual cost(π‘‡π΄πΆπ‘š) can be defined as given below; π‘‡π΄πΆπ‘š = π‘†πΆπ‘š + πΌπΆπ‘š + πΌπ»πΆπ‘š + π‘‡πΆπ‘š + πΈπ‘‡π‘š π‘‡π΄πΆπ‘š = (π΄π‘š)𝐷 πœ†π‘„ + πΆπ‘šπ· + (β„Žπ‘š + π‘†π‘š) 𝑄 2 [ 2𝐷 𝑃 + πœ†(π‘ƒβˆ’π·) 𝑃 βˆ’ 1] + πœ‚π·π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 + πΈπ‘šπ·π‘‡π‘π‘– ….(2) 4.3 Integrated total annual cost for supply chain system In this section, we have calculated total joint cost for supply chain which is the sum of retailer’s annual cost and manufacturer’s cost. From the equations (1) and (2), we get ( )Q = (π΄π‘š+πœ†π΄π‘Ÿ)𝐷 πœ†π‘„ + 𝐷(πΆπ‘Ÿ + πΆπ‘š) + (β„Žπ‘Ÿ + π‘†π‘Ÿ) [ (πŸβˆ’πœΉ)𝑸 𝟐 + πœΉπ‘ΈπŸ πŸπŸ•πŸ“πŸπŸŽ ] + (β„Žπ‘š+π‘ π‘š)𝑄 2 { 2𝐷 𝑃 + πœ†(π‘ƒβˆ’π·) 𝑃 βˆ’ 1} + πœ‚π·π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 + πœ‚π·π‘‡π‘’ 𝑄 + πΈπ‘šπ·π‘‡π‘π‘– … (3) 4.4 Integrated total annual cost for supply chain system under fuzzy environment Let us suppose that the demand rate is taken as triangular fuzzy number is οΏ½ΜƒοΏ½ = (𝑑1, 𝑑2, 𝑑3) and its 𝛼 βˆ’ cut of οΏ½ΜƒοΏ½ is represented by 𝐷(𝛼) and the value of 𝐷(𝛼) can be defined as 𝐷(𝛼) = [𝐷𝐿(𝛼 ), π·π‘ˆ(𝛼 )], where𝐷𝐿(𝛼 ) = 𝑑1 + (𝑑2 βˆ’ 𝑑1)𝛼 and π·π‘ˆ(𝛼 ) = 𝑑3 βˆ’ (𝑑1 βˆ’ 𝑑2)𝛼. The equation (3) fuzzyfied the total integrated cost, we get Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 765 https://internationalpubls.com ( )Q ~ = (π΄π‘š+πœ†π΄π‘Ÿ)οΏ½ΜƒοΏ½ πœ†π‘„ + οΏ½ΜƒοΏ½(πΆπ‘Ÿ + πΆπ‘š) + (β„Žπ‘Ÿ + π‘†π‘Ÿ) [ (πŸβˆ’πœΉ)𝑸 𝟐 + πœΉπ‘ΈπŸ πŸπŸ•πŸ“πŸπŸŽ ] + (β„Žπ‘š+π‘ π‘š)𝑄 2 { 2οΏ½ΜƒοΏ½ 𝑃 + πœ†(π‘ƒβˆ’οΏ½ΜƒοΏ½) 𝑃 βˆ’ 1} + πœ‚οΏ½ΜƒοΏ½π΄0 𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 + πœ‚οΏ½ΜƒοΏ½π‘‡π‘’ 𝑄 + πΈπ‘šοΏ½ΜƒοΏ½π‘‡π‘π‘– …(4) Now, defuzzified the equation (4) by using of centroid method, we get Integrated total fuzzy cost for the supply chain, ( ( )Q ~~ , 0Μƒ)= (π΄π‘š+πœ†π΄π‘Ÿ)(𝑑1+𝑑2+𝑑3) 3πœ†π‘„ + (πΆπ‘Ÿ(𝑑1+𝑑2+𝑑3)+πΆπ‘š(𝑑1+𝑑2+𝑑3)) 3 + (β„Žπ‘Ÿ + π‘†π‘Ÿ) [ (πŸβˆ’πœΉ)𝑸 𝟐 + πœΉπ‘ΈπŸ πŸπŸ•πŸ“πŸπŸŽ ] + (β„Žπ‘š+π‘ π‘š)𝑄 2 { 2(𝑑1+𝑑2+𝑑3) 3𝑃 + πœ†(3π‘ƒβˆ’(𝑑1+𝑑2+𝑑3)) 3𝑃 βˆ’ 1} + (𝑑1+𝑑2+𝑑3)πœ‚ 𝐴0 3𝑄 + πœ‚π΄1(2πœ†π‘›βˆ’πœ†βˆ’1) 2 + πœ‚(𝑑1+𝑑2+𝑑3)𝑇𝑒 3𝑄 + πΈπ‘š(𝑑1+𝑑2+𝑑3)𝑇𝑐𝑖 3 …(5) 5. Solution procedure We adopt the property of maxima and minima property for the calculation of integrated fuzzy cost, if ( ) ,0 ~~ =  dQ Qd we calculated the value of lot size, ( )Q by using of Mathematica 9.0, and suppose it is ο€ͺ= QQ (consider) and now we calculates ( ) , ~~ dQ Qd and if we get ( ) ,0 ~~ ο‚³  dQ Qd then ο€ͺ= QQ is called optimal value of Q and denoted by ο€ͺQ . The convexity of joint total fuzzy cost for supply has been shown in the figure with help of Mathematica 9.0 software. 6. Numerical example The input parameters have been taken from the contribution of Jaber et al. (2013) and Gurtu et al. (2015) and presented in the Table 1. Table 1 Input parameters. Variable Value Variable Value (𝑑1, 𝑑2, 𝑑3) (995,1000,1020) units per year Tcap 80 P 3200 units per year 𝜹 0.02 Am $200 per order Tci 30 Cm $15 per unit Te (2)(d)(0.00077344)(30) hm $ 3 per unit per year a 0.0000003 sm $ 9 per unit per year b 0.0012 km 0.001 c 1.4 Ar $ 30 per order Em aP2-bP +c Cr $ 20 per unit 𝜼 3 hr $ 4 per unit per year π‘„βˆ— 227 units sr $ 12 per unit per year ( )Q ~~ 60407 ($) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 766 https://internationalpubls.com kr 0.002 T 5 d 500 𝝀 2 For the calculation, we used Mathematica software for the optimization of order quantity and total integrated cost for supply chain. The optimal order quantity of this proposed model is π‘„βˆ— = 227 units and the optimal integrated cost with respect to this optimal order quantity is ( )= *Q 60407 $. The present study is also compared with previous study and has been shown in the Table 2. Table 2: Comparison of base model with present model Model Percentage defective item Order quantity Integrated total fuzzy cost Base Paper 0.02 224 60517. Proposed Model 0.02 227 60407 Figure 2: Graphical representation of present model with base model under integrated cost 60360 60380 60400 60420 60440 60460 60480 60500 60520 Base paper Present paper Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 767 https://internationalpubls.com Figure 3: Graphical representation of present model with base model under lot size 7. Sensitivity analysis Sensitive analysis is a vital constituent of research which analyses the decision which is variable with respect to the model parameters which have been given below: Table 3: Variable effect of percentage defective items on the lot size and integrated total fuzzy cost Percentage defective items Order quantity Integrated total fuzzy cost 0.01 218 60535 0.02 227 60407 0.03 229 59671 0.04 232 59542 0.05 242 59429 0.08 252 59342 Table 4: Variable effect of fuzzy demand rate on the order quantity and integrated total fuzzy cost Fuzzy Demand rate Production rate Order quantity Integrated total fuzzy cost (500,1000,1500) 3200 227 60407 (1500,2000,2500) 4200 432 297654 (2500,3000,3500) 7200 586 627654 222.5 223 223.5 224 224.5 225 225.5 226 226.5 227 Base model Present model Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 768 https://internationalpubls.com Table 5: Variable effect of number of truck on the order quantity and integrated total fuzzy cost Number of truck Order quantity Integrated total fuzzy cost 2 181 59765 3 227 60407 4 264 64321 5 273 65876 Managerial insights 1. From Table-3, we can conclude that the percentage of defective items increase then optimal shipment lot size increase while joint total cost decreases. 2. From Table-4, it can easily be analysed that as the demand rate and production rate increase, initially the order quantity and joint total cost increases. 3. From Table-5, it can easily be analysed that as the number of truck increases with keeping other model parameter constant then order quantity and Joint total cost increases. 8. Conclusion The purpose of this paper is to analyse the impact of incorporating the imperfect quality items in the lot & we observed that by adding imperfect items it helps to increase the lot size also the total cost is decreasing which includes holding cost, setup cost, item cost, transportation cost, emission tax cost. Findings together with mathematical analysis clearly suggested that the presence carbon emissions and imperfect items had an affirmative impact on retailer ordering policy. 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