Article History: Received: 12-01-2025 Revised: 14-02-2025 Accepted: 10-03-2025 Abstract: In today’s competitive scenario, many business organizations pro- vide their customers with personalization options, thereby increasing customer satisfaction through a wide range of choices. This strategy boosts profitability for various technologically advanced businesses. In this study, a dual-channel supply chain model with customization is developed to enhance firm profits. Dual-channel retailing delivers personalized products online, while standard products are available via traditional retail channels. This paper modifies the existing dual-channel model by consider- ing customer switching behavior between the two channels. It also introduces a pre-assigned threshold that signifies demand decline when the price difference between the online and offline channels exceeds a specific limit. Furthermore, the model incorporates de- mand uncertainty and fluctuation using a distribution-free approach. Demand price-sensitivity is also considered in constructing the cen- tralized dual-channel supply chain model. The study concludes that adopting a dual-channel policy leads to better predictability and per- formance compared to conventional single-channel systems. Ad- ditionally, customer switching depends on which channel offers a lower price, especially when the price gap surpasses the threshold. Keywords: supply chain management, max-min distribution-free method, dual-channel retailing, demand uncertainty, customization, SDG-12 1. INTRODUCTION The introduction of technology has made the retailing system more organized and competi- tive. E-commerce is increasing with the advancement of technology. The concept of smart manufacturing is evolving from simple digitization and utilizing the data collected from the Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) A Dual-Channel Supply Chain Model with Distribution-Free Approach using Customization Quazzafi Rabbani1, Krishna Kumar2, and Anjani Kumar Shukla3 1Department of Mathematics & Statistics, Integral University Lucknow, India 2Department of Mathematics & Statistics, Integral University Lucknow, India 3Dept. of Mathematics, Lovely Professional University, Phagwara, Jalandhar, Punjab, India https://internationalpubls.com 2232 customers to make decisions on the products.Sustainability in smart supply chain management also takes into account all three pillars of sustainability like environmental, social, and eco- nomic due to inflexible rules and regulations. In 2023, air pollution claimed approximately 7.1 million deaths (or approximately one in 10 deaths) globally, consequently, declaring it the fifth leading risk factor for mortality according to the State of global air/2019. Furthermore, the report discovered that air pollution claims more lives than traffic accidents or malaria. the ejection of CO2 from firms is manifested as a rate of manufacturing, speed of machinery, and consumption of energy function as all these contribute to the total greenhouse gases (Bazan et al., 2015). Moreover, government charges penalties to the firms once they cross the carbon emission ceiling. The European Union Emission Trading System (EU-ETS) works on a cap- and-trade policy where firms are penalized if they cross the carbon emission limit. If emissions are less than the limit firms can trade with another firm that surpasses the limit. Thus, many organizations are emphasizing on reducing greenhouse gas emissions by investing in green technologies (Reddy et al., 2020). By prioritizing sustainability, the company is committed to driving positive change in the industry and making substantial contributions to the global effort to combat climate change. Also, a sustainable model is exemplified by including out-of-order products and manageable CO2 emissions from the firm (Mashud et al., 2020). A supply chain model is all about handling the complete manufacturing of items from raw materials to the final product. Table 1: Overview of Related Literature Author DCSC SP CP UCS SSMD MTO DU DFA CLT MM Takahashi et al. ✓ ✓ Zhang and Choi ✓ ✓ Chiang and Monahan ✓ ✓ Shao ✓ ✓ ✓ Batarfi et al. ✓ ✓ ✓ ✓ Yue and Liu ✓ ✓ ✓ Yan and Pei ✓ ✓ ✓ Sarkar et al. ✓ Chiang et al. ✓ ✓ Dan et al. ✓ ✓ Hua et al. ✓ ✓ ✓ Majumder et al. ✓ Kaya et al. ✓ ✓ ✓ Tsay and Agrawal ✓ ✓ Modak and Kelle ✓ ✓ Li et al. ✓ ✓ ✓ Jing et al. ✓ ✓ ✓ Zhou et al. ✓ ✓ Majumder et al. ✓ Proposed paper ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Abbreviations: DCSC: Dual-channel supply chain; SP: Standard product; CP: Customized product; UCS: Unequal customer shifting (online/offline); MTO: Make-to-order; DU: Demand uncertainty; DFA: Distribution-free approach; CLT: Controllable lead time; MM: Markup margin. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2233 Table 2: Key Decision Variables in the Model Decision Variable Description Qd Quantity of core products ordered for customization (units) k Safety factor (units) to account for demand variability L Retailer’s lead time (days) Qr Quantity of standard products ordered by the retailer (units) n Number of batches delivered from the manufacturer to the retailer in one production cycle (positive integer) Table 3: Parameters of the model. Parameters Description Pr Production rate for the standard product (Pr > a1) (positive number) β1 Price sensitivity in retail channel (customer/day) β2 Price sensitivity in online channel (customer/day) δ1 Number of customers switching from retail channel to online channel a2 Number of customers prefer online channel σ Standard deviation of demand per unit time Cp Production cost for standard product ($/unit) δ2 Number of customers switching from online channel to retail channel Dr Variable demand of retail channel (units/year) π Unit backlogging cost for the retailer ($/unit) pr Retailer’s selling price of the standard product (pr > Cp) ($/order) Dd Variable demand of online channel (units/year) ϕdi Percentage of core product stock used for customized product (i = 1, 2, . . . , N ) h1 Manufacturer’s holding cost including financial and storage cost ($/unit) Ar Ordering cost of the retailer per order ($/order) m Markup margin (percentage) Cdi Production cost for customized product (i = 1, 2, . . . , N ) ($/unit) pdi Manufacturer’s selling price of customized product i ($/order) rv Holding cost rate of manufacturer ($/unit/unit time) rb Holding cost rate of retailer ($/unit/unit time) Sd Manufacturer’s setup cost for core product customized product ($/setup) Sr Manufacturer’s setup cost for standard product ($/setup) Cb Unit production cost paid by retailer ($/unit) Cvr Unit production cost paid by manufacturer ($/unit) R Reorder point of the retailer (units) s Safety factor of the retailer (units) E(·) Mathematical expectation 2. MATHEMATICAL MODEL This part explains the retailing system’s total cost function, optimal decisions, and algorithm for evaluating the solution of this model. 2.1. FORMULATION OF TOTAL COST The equations of the total cost of manufacturer and retailer for core and personalized items are derived in this segment. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2234 2.1.1 Cost parameter for Retailer The costs associated with the core item, bore by the retailer are given as follows. Ordering cost The cost bear by the retailer for placing an order is given as OC1 = Ord1 q1 (1) Holding cost Retailer’s inventory follows an “Economic Order Quantity (EOQ)” model. The retailer places the order of q1 quantity whenever the inventory measure sinks to R reorder point. Thus, inven- tory measure sinks to R− d1l before the order is received. Moreover, q1 +R− d1l depicts the expected level of inventory after receiving quantity q1. Over a cycle, the average inventory is represented as q1 2 +R− d1l. Therefore, the retailer’s inventory cost of holding is HC3 = hrCb (q1 2 +R− d1l ) (2) Shortage cost If ρ is the stochastic demand of lead time and R is the reorder point then, the expected shortage at the end of the cycle is given as E(ρ−R)+ and the shortage cost is Shortcost = νd1 q1 E(ρ−R)+ (3) ≤ νd1 q1  √ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) 2  (4) Lead time crashing cost The lead time ‘l’ has ‘m’ collectively independent elements. For the ‘jth’ element, aj =minimum time span, bj = normal time span, and cj = crashing cost per unit time. Practically, c1 < c2 < . . . < cm. Additionally, l0 = ∑n j=1 and li be the duration of the lead time with elements 1, 2, ..., i sinks to their least span of time, then Li can be represented as li = l0 − ∑n j=1(bj − aj), i = 1, 2, . . . , n. The lead time crashing cost per cycle CL is manifested as CL = ci(li−1 − l) + ∑i−1 j=1 cj(bj − aj) (Sarkar et al., 2018). Therefore, the cost of lead time crashing is CLL = d1CL q1 (5) Total cost The retailer’s aggregate cost per unit of time by selling the core item is TC1 = Cost of ordering + Cost of holding + Cost of shortage + Cost of lead time crashing Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2235 TC1 = Ord1 q1 + hrCb (q1 2 +R− d1l ) + νd1 q1 E(ρ−R)+ + d1CL q1 (6) 2.1.2 Manufacturer’s cost parameters for standard product The costs associated with the standard product, bore by the manufacturer are given as follows. Setup cost The manufacturer’s setup cost is equal to Sm c d1 ηq1 that is derived from machines and cutting tools i.e., the setup of resources. Moreover, as q1 is the quantity ordered by the retailer, ηq1 quantity is manufactured by the producer where η is a positive integer. Therefore, the setup orders of the producer are d1 ηq1 and the cost is Sm c d1 ηq1 . Henceforth, the setup cost for the manufacturer is given as SC1 = Sm c d1 ηq1 (7) Holding cost Holding cost is induced when the producer stores the produced product for some time. The producer’s average inventory is evaluated as follows. Whereas, the membership functions d1 ( ηq1 ( q1 P1 + (η−1)q1d1 η ) − η2q21 2P1 − ( q21 d1 (1 + 2 + . . .+ (η − 1)) )) ηq1 = q1 2 ( η ( 1− d1 P1 ) − 1 + 2d1 P1 ) The manufacturer’s cost of holding is hmcpq1 2 ( η ( 1− d1 P1 ) − 1 + 2d1 P1 ) . . Therefore, the ex- pected cost of holding per unit time per unit item is HC1 = hmcpq1 2 ( η ( 1− d1 P1 ) − 1 + 2d1 P1 ) Manufacturer’s production cost For manufacturing the finished product, The manufacture has to invest in the form of capital, energy, and labors which is termed as cost of production. Thus, the cost of production is PC1 = Cvrd1 (8) Cost of imperfect items In the course of production of ηq1 lost size, nearly ηq1ζ 2 defective items are believed to produce. Thus, the year-long cost of imperfect items is believed to be IC1 = sd1ηq1ζ 2 (9) Investment in the quality improvement of the product Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2236 The capital expenditure Iζ is presumed for reducing the probability of out-of-control ζ . There- fore, Iζ can be exhibited as Iζ = b ln ( ζ0 ζ ) for 0 < ζ ≤ ζ0, i.e., Iζ = b(ln ζ0 − ln ζ), where as initial probability is given by ζ0, for which the manufacturing can go out-of-control. And b = 1 ∆ , where ∆ depicts the shrink percentage in ζ per dollar rise in Iζ . Therefore, the financing done for improving the quality of the product is given as I = αb(ln ζ0 − ln ζ) (10) Where the capital investment’s fraction of annual cost is α. Total cost The aggregate cost of selling the core item through the retail channel per unit of time is TC2 = Cost of setup + Cost of holding + Cost of production + Cost of imperfect items + Investment in the quality improvement of the product TC2 = Sm c d1 ηq1 + hmcpq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] +Cvrd1+ sd1ηq1ζ 2 +αb(ln ζ0− ln ζ) (11) 2.1.3 Manufacturer’s cost parameters for customized product The costs associated with the customized product, bore by the manufacturer are given as fol- lows. Setup cost Setup cost includes the charge of supplies and materials, i.e., cutting tools, materials, and ma- chines. Thus the setup cost for the manufacturer is given as SC2 = Sm p d2 q2 (12) Holding cost The manufacturer’s average inventory is evaluated as Average inventory = q2 2 ( 1− d2 P2 ) The cost of holding item by the manufacturer is hq2 2 ( 1− d2 P2 ) . Therefore, the expected cost of holding per unit time per unit item is HC2 = hq2 2 ( 1− d2 P2 ) Manufacturing cost Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2237 For manufacturing the customized product, the manufacturer has to invest separately in the form of workers, energy, and capital. Thus, the production cost is given as MC2 = K∑ j=1 cjψid2 (13) Total cost In e-commerce, the aggregate cost of production per unit of time by selling the personalized item from the online route is TC3 = Setup cost + Holding cost + Manufacturing cost TC3 = Sm p d2 q2 + ( hq2 2 )( 1− d2 P2 ) + K∑ j=1 cjψid2 (14) Total cost TCSof the retailing system following a single route is derived by summing TC1 and TC2 which is TCS = TC1 + TC2 TCS = Sm c d1 ηq1 + hmcpq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] + Cvrd1 + sd1ηq1ζ 2 + αb(ln ζ0 − ln ζ) + Ord1 q1 + hrCb (q1 2 +R− d1l ) + νd1 q1 E(ρ−R)+ + d1CL q1 (15) Thus, for retailing system having centralized strategy with single-route, the expected aggregate cost TCS is given by summing TC1 and TC2 that is TCS = TC1 + TC2 = Ord1 q1 + hrCb (q1 2 + δσ √ l ) + d1CL q1 + νd1 2q1 [√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) ] + Sm c d1 q1η + hmcpq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] + Cvrd1 + sd1ηq1ζ 2 + αb(ln ζ0 − ln ζ) (16) And for the retailing system having a centralized strategy with dual-route, the expected aggre- gate cost TCD is given by summing TC1, TC2, and TC3 is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2238 TCD = TC1 + TC2 + TC3 = Ord1 q1 + hrCb (q1 2 + δσ √ l ) + d1CL q1 + νd1 2q1 [√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) ] + Sm c d1 q1η + hmcpq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] + Cvrd1 + sd1ηq1ζ 2 + αb(ln ζ0 − ln ζ) + Sm p d2 q2 + ( hq2 2 )( 1− d2 P2 ) + K∑ j=1 cjψid2 (17) 2.2. OUTCOME OF THE RETAILING SYSTEM Since, Equation (19) is non-linear in nature therefore for constant positive integer µ, partial derivative of the cost with respect to q2, q1, δ, & ζ is taken and then equated to zero for obtaining the optimal solution. Henceforth, we get q∗2 = √√√√ Sm p d2 h 2 ( 1− d2 P2 ) (18) q∗1 = √√√√√−d1CL+ Sm c d1 +Ord1 + νd1 2 √ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + d1l − δσ √ l hrCb + hm 2 ( η ( 1− d1 P1 ) − 1 + 2d1 P1 ) + sd1ηζ 2 (19) δ∗ = (−σ √ q1Cbhr(−q1Cbhr+νd1)) −q1Cbhr+νd1 + d1νσ √ q1Cbhr(−q1Cbhr+νd1)l 2q1Cbhr(νd1−q1Cbhr) + ld1 √ lσ (20) ζ∗ = αb sd1ηq1 (21) Optimal solution for q∗1 , q∗2 , δ∗, ζ∗ and so obtained dependents upon each other. Moreover, it is difficult to evaluate the closed-form expression for the centralized total cost function. There- fore, a numerical procedure is required for evaluating these optimal values. Along with the following algorithm an iteration method is utilized to find the managerial decisions. 2.3. ANALYZING DIFFERENT PILLARS OF SUSTAINABILITY IN SINGLE CHAN- NEL 2.3.1 Environmental pillar 1. CO2 emissions throughout the production Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2239 The aggregate of CO2 emitted (ton/unit) in the production process is: E = E(P1) = x1P 2 1 − x2P1 + x3 (22) where x1, x2, and x3 can be experimentally verified from Bazan et al. (2015). The experiment gives a way to understand how operating a machine tool adds a carbon emis- sion burden and gave a quadratic Equation (24) reflecting the equivalent CO2 emissions. Therefore, the cost of the carbon exude because of production is given as EC1 = Ed1C ec (23) 2. Penalty because of excess of CO2 emission When the carbon burden from the firm surpasses the predetermined ceiling, then the penalty cost is collected from it. Thus, the penalty cost as a consequence of carbon emission is given as EC2 = l∑ i=1 YiCep,i (24) where Yi = { 1 Ed1 > Eli (i = 1, 2, ..., l) 0 otherwise (25) 2.3.2 Social pillar Cumulative social cost as a result of hard labor, medical maintenance, welfare, and social consciousness (Khan et al., 2016) for single channel supply chain model, is SC1 = ηS1q1 (26) Therefore, for the supply chain having a centralized policy and following a single route, the expected aggregate cost TCS is TCS = Ord1 q1 + hrCb (q1 2 + δσ √ l ) + d1CL q1 + νd1 2q1 [√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) ] + Sm c d1 q1η + hmq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] + Cvrd1 + sd1ηq1ζ 2 + αb(ln ζ0 − ln ζ) + Ed1Cec + l∑ i=1 YiCep,i + ηS1q1 (27) 2.4. ANALYZING DIFFERENT PILLARS OF SUSTAINABILITY IN DUAL CHANNEL 2.4.1 Environmental pillar 1. CO2 emissions throughout the production. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2240 The aggregate of CO2 emitted (ton/unit) in the production process is: E ′ = E(P ) = x1P 2 − x2P + x3 (28) where x1, x2, and x3 can be experimentally verified from Bazan et al. (2015). The experiment gives a way to understand how operating a machine tool adds a carbon emis- sion burden and gave a quadratic Equation (30) reflecting the equivalent CO2 emissions. Moreover, it also reflects that increased cutting speed converts tool wears into consider- able lofty which shortens its life span and elevates CO2 emissions. Further, there is also a trade relation with the cutting speed as carbon stress build-up by electricity utilization and the cooling liquid is comparable with time. The quadratic Equation (30) manifests the behavior of the corresponding carbon ejection. Therefore, the cost of the carbon exude because of production is given as EC ′ 1 = E ′dCec (29) 2. Penalty on firm because of excess of CO2 emission When the carbon burden from the firm surpasses the predetermined ceiling then the penalty cost is collected from it. Thus, the penalty cost as a consequence of carbon emission is given as EC ′ 2 = l∑ i=1 YiCep,i (30) where Yi = { 1 E ′d > Eli (i = 1, 2, ..., l) 0 otherwise (31) 2.4.2 Social pillar Cumulative social cost as a result of hard labor, medical maintenance, welfare, and social consciousness (Khan et al., 2016) for supply chain model following dual route, is SC1 = η (S1q1 + S2q2) (32) Therefore, for the supply chain having a centralized policy and following a dual route, the expected aggregate cost TCD is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2241 TCD = Sm c d1 q1η + hmq1 2 [ η ( 1− d1 P1 ) − 1 + 2d1 P1 ] + Cvrd1 + sd1ηq1ζ 2 + αb(ln ζ0 − ln ζ) + Ord1 q1 + hrCb (q1 2 + δσ √ l ) + d1CL q1 + νd1 2q1 [√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) ] + Sm p d2 q2 + ( hq2 2 )( 1− d2 P2 ) + K∑ j=1 cjψid2 + E ′dCec + l∑ i=1 YiCep,i + η(S1q1 + S2q2) (33) 2.5. SOLUTION ALGORITHM To solve the current model succeeding ARV algorithm is applied. 1. Assign all parameters values as defined in Table 2. 2. Set η = 1. 3. Perform the underneath steps for all the values of lj; j = 1, 2, . . . (a) From Equation (20), evaluate q2. (b) From Equation (21), evaluate q1. (c) From Equation (22), evaluate δ. (d) From Equation (23), evaluate ζ . (e) Redo the steps from 3(a) to 3(d) unless there is no variation in the values of q1, q2, δ, and ζ until a specified accuracy level. 4. Evaluate E1d and E ′d using the following steps: (a) If E1d < 220 then Cep,i = 0 and Yi = 0 else go to step 4(b) and if E ′d < 220 then Cep,i = 0 and Yi = 0 else go to step 4(b). (b) If 220 < E1d < 330 then Cep,i = 1000 and Yi = 1 else go to step 4(c) and if 220 < E ′d < 330 then Cep,i = 1000 and Yi = 1 else go to step 4(c). (c) If 330 < E1d < 440 then Cep,i = 2000 and Yi = 1 else go to step 4(d) and if 330 < E ′d < 440 then Cep,i = 2000 and Yi = 1 else go to step 4(d). (d) If 440 < E1d < 550 then Cep,i = 3000 and Yi = 1 else go to step 4(e) and if 440 < E ′d < 550 then Cep,i = 3000 and Yi = 1 else go to step 4(e). (e) If 550 < E1d < 660 then Cep,i = 4000 and Yi = 1 else go to step 4(f) and if 550 < E ′d < 660 then Cep,i = 4000 and Yi = 1 else go to step 4(f). (f) If E1d > 660 then Cep,i = 4000 and Yi = 1 and if E ′d > 660 then Cep,i = 4000 and Yi = 1. 5. Obtain the value of EC1 and EC ′ 1 from Equations (25) and (31). 6. Obtain the value of EC2 and EC ′ 2 from Equations (26) and (32). 7. Obtain the value of SC1 and SC2 from Equations (28) and (34). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2242 8. Use the values of q1, ζ , δ, Edr, EC1, EC2, and SC1 to obtain TCS from Equation (29). 9. Use the values of q2, q1, ζ , δ, E ′ d, EC ′ 1, EC ′ 2, and SC2 to obtain TCD from Equation (35). 10. Set η = η + 1 and repeat steps 3 to 11. 11. If TCD(η) > TCD(η + 1) then redo the steps from 2 to 6 or else end the algorithm. 3. MATHEMATICAL SOLUTION OF FIELD EQUATION E(ρ−R)+ = |ρ−R|+ ρ−R 2 E(ρ−R)+ ≤ √ E(ρ−R)2 + E(ρ−R) 2 Considering, ρ = d1 √ L+X summation of variability and randomness. E(ρ−R)+ ≤ √ E(d1 √ l + E(X −R))2 + E(d1 √ l +X −R) 2 where, R = d1l + δσ √ l is a safety factor E(ρ−R)+ ≤ √ E(d1l +X − d1l + δσ √ l)2 + E(d1l +X − d1l + δσ √ l) 2 In this current model, the worst practicable distribution of random variable d1 with mean d1l and standard deviation σ √ l. Thus we obtain, E(ρ−R)+ = √ E(X + δσ √ l)2 + E((X + δσ √ l)) 2 = √ E(X2 + δ2σ2l + 2Xδσ √ l) + E(X + δσ √ l) 2 =  √ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l + (d1l − δσ √ l) 2  COMPUTATION OF PROPOSITION 1 Underneath mentioned is the Hessian matrix H for the retailing system with dual-route. The Hessian matrix H for retailing system having dual-route is given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2243 (HTCD )1 =  ∂2TCD ∂q21 ∂2TCD ∂q1∂q2 ∂2TCD ∂q1∂δ ∂2TCD ∂q1∂ζ ∂2TCD ∂q2∂q1 ∂2TCD ∂q22 ∂2TCD ∂q2∂δ ∂2TCD ∂q2∂ζ ∂2TCD ∂q1∂δ ∂2TCD ∂q2∂δ ∂2TCD ∂δ2 ∂2TCD ∂δ∂ζ ∂2TCD ∂q1∂ζ ∂2TCD ∂q2∂ζ ∂2TCD ∂δ∂ζ ∂2TCD ∂ζ2  where, ∂2TCD ∂q22 = 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )] , ∂2TCD ∂q1∂q2 = ∂2TCD ∂q2∂δ = ∂2TCD ∂q2∂ζ = ∂2TCD ∂δ∂ζ = 0. ∂2TCD ∂q1∂δ = −νd1 2q21 ( δσ2l−d1lσ √ l√ σ2l+(d1l)2+δ2σ2l−2d1lδσ √ l − σ √ l ) , ∂2TCD ∂q1∂ζ = sd1η 2, ∂2TCD ∂q22 = 2Sm p d2 q32 . ∂2TCD ∂δ2 = νd1 2q1 ( σ4l2√ σ2l+(d1l)2+δ2σ2l−2d1lδσ √ l ) , ∂2TCD ∂ζ2 = αb ζ2 . The Hessian matrix’s |(HTCD )1|, principal minor of the order 1× 1 is |(HTCD )1,1| = ∣∣∣∣∂2TCD ∂q21 ∣∣∣∣ (q∗1 ,q ∗ 2 ,δ ∗) = 2 q31 [ Ord1 + Sm c d1 η + d1CL + νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l ) > 0. The Hessian matrix’s |(HTCD )1|, principal minor of the order 2× 2 is |((HTCD )1)2,2|(q∗1 ,q∗2 ,δ∗,ζ∗) = ∣∣∣∣∣ ∂2TCD ∂q21 ∂2TCD ∂q1∂q2 ∂2TCD ∂q2∂q1 ∂2TCD ∂q22 ∣∣∣∣∣ (q∗1 ,q ∗ 2 ,δ ∗,ζ∗) |((HTCD )1)2,2|(q∗1 ,q∗2 ,δ∗,ζ∗) = ( ∂2TCD ∂q21 )( ∂2TCD ∂q22 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2244 = ( 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )])( 2Sm p d2 q32 ) > 0 The Hessian matrix’s |(HTCD )1|, principal minor of the order 3× 3 is |((HTCD )1)3,3|(q∗1 ,q∗2 ,δ∗,ζ∗) = ∣∣∣∣∣∣∣ ∂2TCD ∂q22 ∂2TCD ∂q2∂δ ∂2TCD ∂q2∂ζ ∂2TCD ∂δ∂q2 ∂2TCD ∂δ2 ∂2TCD ∂δ∂ζ ∂2TCD ∂ζ∂q2 ∂2TCD ∂ζ∂δ ∂2TCD ∂ζ2 ∣∣∣∣∣∣∣ (q∗1 ,q ∗ 2 ,δ ∗,ζ∗) |((HTCD )1)3,3|(q∗1 ,q∗2 ,δ∗,ζ∗) = ( ∂2TCD ∂q22 )( ∂2TCD ∂δ2 )( ∂2TCD ∂ζ2 ) = ( 2Sm p d2 q32 )νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l ( αb ζ2 ) > 0 The Hessian matrix’s |(HTCD )1|, principal minor of the order 4× 4 is |((HTCD )1)4,4|(q∗1 ,q∗2 ,δ∗,ζ∗) = ∣∣∣∣∣∣∣∣∣∣ ∂2TCD ∂q21 ∂2TCD ∂q1∂q2 ∂2TCD ∂q1∂δ ∂2TCD ∂q1∂ζ ∂2TCD ∂q2∂q1 ∂2TCD ∂q22 ∂2TCD ∂q2∂δ ∂2TCD ∂q2∂ζ ∂2TCD ∂δ∂q1 ∂2TCD ∂δ∂q2 ∂2TCD ∂δ2 ∂2TCD ∂δ∂ζ ∂2TCD ∂ζ∂q1 ∂2TCD ∂ζ∂q2 ∂2TCD ∂ζ∂δ ∂2TCD ∂ζ2 ∣∣∣∣∣∣∣∣∣∣ (q∗1 ,q ∗ 2 ,δ ∗,ζ∗) = ( ∂2TCD ∂q21 )( ∂2TCD ∂q22 )( ∂2TCD ∂δ2 )( ∂2TCD ∂ζ2 ) − ( ∂2TCD ∂q22 )( ∂2TCD ∂ζ2 )( ∂2TCD ∂δ∂q1 )2 |(HTCS )1,1| (q∗1, δ∗, ζ∗) = ∣∣∣∣∂2TCS ∂q21 ∣∣∣∣ (q∗1 ,δ ∗,ζ∗) = 2 q31 [ Ord1 + Sm c d1 η + d1CL + νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )] > 0 νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l − ( 2Sm p d2 q32 )( αb ζ2 ) −νd1 2q21  δσ2l − d1lσ √ l√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l − σ √ l 2 > 0 =⇒ ( 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )]) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2245 νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l  > −νd1 2q21  δσ2l − d1lσ √ l√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l − σ √ l 2 Since the Hessian matrix’s every single principal minor is positive, therefore, at (q∗1, q ∗ 2, δ ∗, ζ∗), (HTCD )1 is positive definite. Moreover, at the same point, the total cost of the firm offering a product online along with a traditional platform obtains its global minimum. COMPUTATION OF PROPOSITION 2 Underneath mentioned is the Hessian matrix H for the retailing system with single-route. The Hessian matrix H for retailing system having single-channel is given as (HTCS )2 =  ∂2TCS ∂q21 ∂2TCS ∂q1∂δ ∂2TCS ∂q1∂ζ ∂2TCS ∂δ∂q1 ∂2TCS ∂δ2 ∂2TCS ∂δ∂ζ ∂2TCS ∂ζ∂q1 ∂2TCS ∂ζ∂δ ∂2TCS ∂ζ2  where, ∂2TCS ∂q21 = 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )] , ∂2TCS ∂q1∂δ = −νd1 2q21  δσ2l − d1lσ √ l√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l − σ √ l  , ∂2TCS ∂q1∂ζ = sd1η 2 , ∂2TCS ∂δ2 = νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l  , ∂2TCS ∂δ∂ζ = ∂2TCS ∂ζ∂δ = 0, ∂2TCS ∂ζ2 = αb ζ2 . The Hessian matrix’s |(HTCS )2|, principal minor of the order 1× 1 is |(HTCS )1,1| (q∗1, δ∗, ζ∗) = ∣∣∣∣∂2TCS ∂q21 ∣∣∣∣ (q∗1 ,δ ∗,ζ∗) = 2 q31 [ Ord1 + Sm c d1 η + d1CL + νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l ) > 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2246 The Hessian matrix’s |(HTCS )2|, principal minor of the order 2× 2 is |(HTCS )2,2| (q∗1, δ∗, ζ∗) = ∣∣∣∣∣ ∂2TCS ∂δ2 ∂2TCS ∂δ∂ζ ∂2TCS ∂ζ∂δ ∂2TCS ∂ζ2 ∣∣∣∣∣ (q∗1 ,δ ∗,ζ∗) = ( ∂2TCS ∂δ2 )( ∂2TCS ∂ζ2 ) − ( ∂2TCS ∂δ∂ζ )( ∂2TCS ∂ζ∂δ ) = νd1αb 2q1ζ2  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l  > 0 The Hessian matrix’s ∣∣(HTCS)2 ∣∣, principal minor of the order 3× 3 is |(HTCS )3,3|(q∗1 ,δ∗,ζ∗) = ∣∣∣∣∣∣∣ ∂2TC ∂q21 ∂2TC ∂q1∂δ ∂2TC ∂q1∂ζ ∂2TC ∂δ∂q1 ∂2TC ∂δ2 ∂2TC ∂δ∂ζ ∂2TC ∂ζ∂q1 ∂2TC ∂ζ∂δ ∂2TC ∂ζ2 ∣∣∣∣∣∣∣ = ( ∂2TC ∂q21 )( ∂2TC ∂δ2 )( ∂2TC ∂ζ2 ) − ( ∂2TC ∂q1∂δ )( ∂2TC ∂δ∂q1 )( ∂2TC ∂ζ2 ) + ( ∂2TC ∂q1∂ζ )( −∂ 2TC ∂ζ∂q1 )( ∂2TC ∂δ2 ) 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )]( νd1 2q1 )( αb ζ2 )  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l  − ( νd1 2q21 )2  δσ2l − d1lσ √ l√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l − σ √ l 2( αb ζ2 ) + ( sd1η 2 )2 νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l  > 0 2 q31 [ Ord1 + Sm c d1 η + d1CL+ νd1 (√ σ2l + (d1l)2 + δ2σ2l − 2d1lδσ √ l )]( νd1 2q1 )( αb ζ2 ) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s(2025) https://internationalpubls.com 2247  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l  + ( sd1η 2 )2 νd1 2q1  σ4l2√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l  > ( νd1 2q21 )2  δσ2l − d1lσ √ l√ σ2l + (d1l)2 + δ2σ2l − 2d1δlσ √ l − σ √ l 2( αb ζ2 ) Since the Hessian matrix’s every single principal minor is positive, therefore, at (q∗1, q ∗ 2, δ ∗, ζ∗), (HTCS )1 is positive definite. Moreover, at the same point, the total cost of the firm offering a product on a traditional platform obtains its global minimum. 4. CONCLUSION Recently, many industries accept integrating dual-channel retailing systems into their business model. This is evident from the results of this study that traditional-online retailing increases the profitability of the company. Customized products have given the provision to the cus- tomer of choosing merchandise of their own choice. Moreover, this paper assumed a threshold amount and unequal shifting of customers between the channels. These two assumptions turned the model into a more practical scenario. The study also incorporated the policy of extensive quality checks in the production process as the customized product needs to be analyzed prop- erly before delivery. The considerations of sustainable pillars have shown how the company could improve its profit while achieving sustainable goals. 5. 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