EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6032
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Integrated Sustainable Inventory and Remanufacturing
Optimization in a Circular Economy: A Two-Echelon
Supply Chain Approach Under Carbon Regulations

R. Suvetha1, K. Rangarajan1, P. Rajadurai2,∗, M. Kaviyarasu3,∗,
Mohammad Alqahtani4,∗

1 Department of Mathematics, Saveetha School of Engineering, SIMATS Deemed University,
Chennai, Tamilnadu, India.
2 Department of Mathematics, Srinivasa Ramanujan Centre, SASTRA Deemed University,
Kumbakonam, Tamilnadu, India
3 Department of Mathematics, Vel Tech Rangarajan Dr. Saguthala R&D Institute of Science
and Technology, Chennai, Tamilnadu, India
4 Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic
University, Riyadh, Saudi Arabia

Abstract. This study presents an integrated sustainable inventory and remanufacturing model
within a two-echelon supply chain framework, comprising a manufacturer and a retailer. The model
captures the impact of carbon emissions regulations through both carbon tax and cap-and-trade
mechanisms, considering the dual nature of demand: price- and time-dependent for manufacturing
and circularity-index-dependent for remanufacturing. Unlike conventional multi-echelon models,
this study isolates the core dyadic relationship between the manufacturer and retailer to streamline
decision-making and emphasize emission and circularity optimization. Using analytical optimiza-
tion and numerical simulations in MATLAB R2024b, the study demonstrates that the cap-and-
trade mechanism offers slightly higher profitability while maintaining environmental targets. Key
contributions include the simultaneous integration of deteriorating items, dual demand functions
and policy-driven emissions constraints in a supply chain context.

2020 Mathematics Subject Classifications: 90B05, 90B50

Key Words and Phrases: Supply Chain, Circularity index, EPQ, Carbon tax & cap & trade,
Total profit, Two-echelon

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6032

Email addresses: psdurai17@gmail.com (P. Rajadurai),
drkaviyarasum@veltech.edu.in (M. Kaviyarasu), m.alqahtani@seu.edu.sa (M. Alqahtani)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



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1. Introduction

In today’s competitive business environment, the collaboration between retailers and
manufacturers is a critical component of effective supply chain management. This part-
nership enables manufacturers to utilize retailers market insights while ensuring a steady
supply of products, fostering mutual growth and operational efficiency. Such collaborations
are vital in addressing challenges like fluctuating consumer preferences, rapid innovation
and intense competition. Modern production systems often face imperfections due to ma-
chine breakdowns, human errors, and variability in raw materials. Addressing these chal-
lenges requires adaptive strategies, including quality improvement programs and flexible
production schedules, to ensure efficiency and product quality. Additionally, product de-
terioration, particularly in perishable goods and high-tech items, underscores the need for
effective inventory management and innovative solutions to minimize losses and maintain
customer satisfaction. This paper explores the retailer-manufacturer collaboration, with
a specific focus on the remanufacturing process under carbon-indicator-driven demand.
It also examines the role of carbon emissions in supply chain management, highlighting
the importance of sustainable practices. By addressing these interconnected elements,
the study aims to provide actionable insights for enhancing supply chain resilience and
sustainability.

Inventory system mathematical modeling is one of the many academic domains where
CE concepts have been used due to their significance for environmental sustainability.
Despite being relatively new in this field of study, CE applications have gained more
interest recently since Rabta’s (2020) [1] groundbreaking work was published. Rabta
(2020) [2] used the economic order quantity (EOQ) model theory, initially put out by
Harris (1913)[3], to build an economic order quantity extension for a product with a CE
indication. Carbon emissions indicator is only an index that ranges from 0 to 1, indicating
how much a product’s manufacturing and consumption adhere to the CE’s tenets. It
directly affects the product’s unit gross profit as well as the rate of demand for it. The
integration of sustainability and efficiency in supply chain systems has become increasingly
significant in addressing contemporary challenges such as environmental impact, product
quality, and resource optimization. Inventory systems, in particular, are at the forefront
of these efforts, where the interplay between deteriorating items, circular economy (CE)
principles, and carbon emissions policies necessitates innovative approaches. Rabta (2020)
[1] introduced a pivotal model that emphasized the dual optimization of order quantities
and CE indicators, paving the way for exploring critical facets such as item deterioration,
quality assurance, supply chain integration, and environmental regulations.

An illustrative example can be seen in the management of pharmaceutical products,
which often have strict expiration dates and require controlled environments to maintain
efficacy. Deteriorating items like these pose challenges for inventory control, necessitating
advanced strategies for monitoring quality, minimizing waste, and ensuring timely dis-
tribution. Moreover, recycling and reprocessing expired pharmaceuticals align with CE
principles, reducing environmental impact and conserving resources. These processes must
be balanced with stringent quality standards and compliance with regulatory frameworks.



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Simultaneously, effective supply chain integration is vital for leveraging core competencies
and achieving operational coherence among diverse stakeholders. As businesses increas-
ingly outsource non-core activities, fostering synergy within the supply chain becomes
imperative to ensure seamless functionality. Additionally, growing global awareness of en-
vironmental sustainability has intensified the need for compliance with carbon emissions
policies, including carbon taxes and cap-and-trade mechanisms.

The objective of this article is to create a CE-based integrated sustainable inventory
model for a two-tier supply chain that focuses on deteriorating and subpar products.
By examining various carbon emissions policies, the study seeks to optimize inventory
replenishment strategies and CE indicators while addressing key research questions:

• How can inventory levels of deteriorating items be optimized to reduce waste and
ensure quality?

• What degree of circularity should be incorporated into inventory systems for maxi-
mum sustainability?

• Which carbon emissions policy is most effective for balancing profitability and envi-
ronmental responsibility?

• How do different demand rates and profit structures impact the overall efficiency of
sustainable inventory models?

1.1. Orientation of the Manuscript

The structure of the paper is as follows: Section 1 provides the introduction. Section 2
reviews the existing literature relevant to the proposed model. In Section 3, the notations
and assumptions utilized in the study are introduced. Section 4 outlines the problem
description of the developed model. The mathematical formulation and corresponding
solution are presented in Section 5. Section 6 discusses the variations and extensions of
the proposed model. Section 7 describes the solution methodology and algorithm adopted.
Section 8 presents numerical examples, implemented using MATLAB R2024b. Section 9
includes a sensitivity analysis and graphical representation of the results. Managerial
insights derived from the model are discussed in Section 10. Finally, Section 11 concludes
the study with key observations and practical implications.

2. Literature Review

The proposed supply chain inventory system faces challenges related to the circularity
(CE) indicator of items, their potential for deterioration and the presence of substandard-
quality items. Additionally, the system operates under varying carbon emissions regula-
tions. As a result, inventory models that address carbon emissions policy, item deteriora-
tion, imperfect quality and CE indicators are the primary focus of the research study.



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2.1. Deteriorating Items in EPQ Models:

Deterioration is a significant factor affecting inventory decisions. The incorporation of
deterioration rates into EPQ models has been widely studied. Goyal and Giri (2001) [4]
analyzed replenishment policies for deteriorating items, laying the foundation for future
research. Later, Abad (2003) [5] integrated deterioration with variable production rates
in an EPQ context. In recent years, Sarkar et al. (2020) [6] introduced an EPQ model ad-
dressing perishability and environmental sustainability simultaneously. Dey et al. (2022)
[7] optimized the production schedule in conjunction with investments for automated in-
spection and green technology integration in a manufacturing-remanufacturing system for
assembled goods. In Sindhuja and Arathi (2023) [8], a preservative-based inventory model
for depreciating products with quality demand is investigated. This model incorporates
circularity index and price-sensitive demand dynamics. Employing constant deterioration
methods aims to enhance manufacturer profits through a new approach to defective items-
based inventory models, ultimately aiming for cost-effectiveness in future manufacturing
endeavors.

2.2. Circular Economy and EPQ Models:

Rabta (2020) [1] introduced an EOQ model integrating a product’s circularity indica-
tor to study its impact on inventory replenishment, modeling demand, and profit using
various functional forms and optimizing for profit with circularity and order quantity as
decision variables. Rabta (2020) [2] economic order quantity model has been executed
for multi-echelon supply chains and manufacturing systems. Incorporating CE principles
into EPQ models enhances resource efficiency and promotes sustainability. Kazancoglu
et al. (2020) [9] extended the EPQ framework to include remanufacturing and recycling
activities, highlighting the potential for reduced environmental impact. Singh et al. (2021)
[10] introduced a hybrid EPQ model combining CE principles with traditional inventory
management, demonstrating cost and waste reductions. John and Mishra (2023a) [11] and
Khan et al. (2023) [12] created EPQ extensions. John and Mishra included emissions,
carbon cap & trade policy and investment in green technology, while Khan et al. took
production, setup and storage of carbon emissions into account. Wani and Mishra (2022)
[13] and Thomas and Mishra (2022) [14] expanded two-echelon supply chain models, em-
phasizing sustainability investments and carbon emissions. John and Mishra (2023b) [15]
developed a three-echelon model for the textile industry, incorporating emissions, green
technologies, and textile waste.

2.3. Imperfect Quality in EPQ Models:

Imperfect production is a crucial aspect influencing inventory and production decisions.
Porteus (1986) [16] was among the first to study quality control within EPQ systems.
Salameh and Jaber’s (2000) [17] inventory model has garnered significant attention for its
applicability across various production systems. To find the ideal order amount, Goyal and
Cardenas-Barron (2002) [18] suggested a less complicated approach. Chang (2004) [19]



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extended the model to a fuzzy environment with fuzzy demand rates and quality fractions.
Wee et al. (2007) [20] introduced an executed allowing shortages with full backordering,
while Khan et al. (2011) [21] accounted for Type I and Type II errors in the screening
process, addressing the original model’s assumption of perfect screening. In order to
compare the performance of a three-tier supply chain with a Stackelberg game theory
approach for commodities of imperfect quality, Sana (2011) [22] constructed the model.
More recently, Khan and Qianli (2017) [23] integrated quality improvement into EPQ
models for sustainable supply chains. In related work, Sana et al. (2022) [24] proposed an
EPQ framework considering imperfect quality under carbon regulations, offering insights
into balancing quality and sustainability. Salameh and Jaber’s (2000) [17] model was
expanded by Gautam et al. (2022) [25] by adding reworking, price- and green-dependent
demand, and energy consumption. Additional noteworthy extensions include the use of
high-tech products (Ruidas et al., 2023b [26]), trade credit with two levels (Tiwari et
al., 2022 [27]), expanding products (Sebatjane & Adetunji, 2019, 2020, 2024 [28–30]) and
sophisticated algorithms such as outer approximated performance with the use of penalty
and equality relaxation (Gharaei et al., 2019 [29]).

2.4. Carbon Emissions in EPQ Models:

Environmental regulations have increasingly influenced EPQ-based research. Benjaa-
far et al. (2013) [31] explored the impact of carbon emissions policies on production
systems, emphasizing the importance of regulatory compliance. Studies by Govindan and
Hasanagic (2018) [32] further explored CE integration into EPQ models under carbon
constraints. Chen et al. (2020) [33] developed a green EPQ model incorporating carbon
taxes, demonstrating its effectiveness in achieving sustainability targets. The influence
of the reduction of emission techniques and take-back laws on manufacturing enterprises’
reproduction and carbon tax was examined by Ding et al. (2020) [34]. Yu et al. (2020)
[35] developed an inventory model for degradable products while accounting for carbon
policy investments in preservation technologies. They examined the best ordering choices
that retailers would make in the constraints of the tax and cap & trade. By controlling
product stocks and delivery, Wangsa et al. (2022) [36] were able to optimize expenditures
associated with purchasing, inspection, food waste, packing, cold storage, transportation,
and carbon emissions. In order to address a sustainable and traceable fish closed-loop
supply chain, Andi Purnomo et al. (2022) [37] created a mathematical model that took
carbon emissions from transportation, production, and warehousing into account. The
model sought to minimize overall costs across a number of time periods, taking into ac-
count production, inventory, transportation, traceability, and emission costs. It did this by
using mixed-integer linear programming. In a multi-echelon dairy supply chain, Vanany
et al. (2023) [38] concentrated on reducing expenses, carbon emissions, and food waste.
An method to multi-objective mixed-integer linear programming was utilized to optimize
a system that involved farmers, distributors, retailers, and a processing plant. Dey et
al. (2023) [39] examined three distinct policy types within their model to reduce carbon
emissions and demonstrated that constrained carbon regulation was the most effective



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approach. While previous literature surveys have made advancements in carbon emission-
based inventory models and strategies for cost reduction through emission control, this
paper takes a slightly different approach by incorporating the remanufacturing method.
This introduces a novel component with greater potential for cost reduction compared to
conventional methods.

2.5. Sustainability and Environmental Considerations in EPQ Models:

Recent advancements in Economic Production Quantity (EPQ) models have increas-
ingly focused on integrating sustainability and real-world complexities into supply chain
management. Karim and Nakade (2022) [40] reviewed EPQ frameworks, emphasizing the
incorporation of carbon emissions and product recycling. Sana (2022) [41] developed a
two-echelon supply chain model to optimize inventory strategies under structural con-
straints, while Sana (2023) [42] explored the impact of greenhouse gas emission costs on
pricing and lot-sizing in imperfect production systems. Sivashankari et al. (2024) [43]
analyzed how advertising and price-dependent demand influence production and pricing
in a sustainable system with imperfect quality. Barman et al. (2024) [44] studied pricing
policies in dual-channel supply chains, integrating green investment and sales efforts with
a revenue-sharing contract. Suvetha et al. (2024) [45] identified emerging trends, including
additive manufacturing and fuzzy logic applications, in EPQ research from 2000 to 2022.
Salas-Navarro et al. (2020) [46] developed an EPQ model that considers probabilistic de-
mand and defective items, while Salas-Navarro et al. (2020) [47] proposed a three-echelon
model factoring in marketing efforts on demand. Shaikh et al. (2018) [48] offered solutions
for EPQ models with exponentially deteriorating items under partial trade credit policies.
These studies collectively highlight the shift toward more sustainable, adaptable EPQ
models addressing environmental impact, demand variability and financial constraints.

2.6. Research Gap:

Despite increasing research on sustainable supply chains, several critical gaps remain:

(i) Supply Chain Structure: Most studies consider multi-echelon supply chains that
include distributors and other intermediaries (e.g., Dey et al., (2023) [39]; Ding et
al., (2020) [34]). In contrast, this study focuses on a simplified two-echelon structure
involving only a manufacturer and a retailer. This allows for direct coordination
and streamlined decision-making, enhancing analytical tractability and managerial
relevance.

(ii) Demand Modeling: Prior research in inventory and production systems primarily
incorporates price-dependent or time-dependent demand models (e.g., Abad, (2003)
[5]; Chen et al., (2020) [33]). However, circularity index-based demand, which re-
flects consumer preference for environmentally sustainable products, has not been
integrated into such models. This study fills this gap by incorporating both tradi-
tional and circularity-driven demand functions.



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(iii) Carbon Emission Regulations: Most existing works examine only one type of
carbon policy—either carbon tax or cap-and-trade (e.g., Benjaafar et al., (2013)
[31]; Sana & Chaudhuri, (2022) [24]). The current study addresses this limitation by
analyzing and comparing both policies within a unified framework, offering greater
insight into regulatory trade-offs.

(iv) Quality Imperfections and Deterioration: Studies have addressed production
imperfections (e.g., Salameh & Jaber, (2000) [17]; Chang, (2004) [19]) and product
deterioration (e.g., Goyal & Giri, (2001) [4]) separately. However, these two factors
often coexist in real-world scenarios. This work integrates both deterioration and
imperfect quality simultaneously, enhancing the model’s practical applicability to
sustainable supply chains.

2.7. Contribution of the Study:

To address these gaps, this study:

(i) Develops a two-echelon sustainable supply chain model that excludes distributors,
enhancing decision-making efficiency.

(ii) Integrates dual-demand structures, where manufacturing is influenced by price and
time, while remanufacturing depends on circularity index demand.

(iii) Analyzes and compares multiple carbon emissions regulations, extending prior re-
search that focused on singular policies.

(iv) Incorporates imperfect quality and deterioration factors, optimizing inventory re-
plenishment in remanufacturing systems.

This study introduces a sustainable production-inventory model for a two-echelon
manufacturer-retailer supply chain with remanufacturing under various carbon emissions
policies. The findings reveal that cap & trade policies yield the highest profitability, de-
spite a moderate circularity index. Compared to previous studies (Dey et al., (2023) [39];
Ding et al., (2020) [34]; Rabta, (2020) [1]), this model enhances decision making by in-
tegrating dual demand functions, multiple carbon regulations, and quality deterioration
factors. These insights contribute to the growing body of research on sustainable sup-
ply chains, helping policymakers and companies develop environmentally conscious and
cost-effective inventory strategies.

3. Notations and Assumptions

In this manuscript, the following notations, acronyms and presumptions are used.

3.1. Notations

Table 1 & 2, describe the notation for all parameters used in this manuscript.



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3.2. Assumptions

The model formulation is based on the following assumptions:

(i) The supply chain consists of a manufacturer and a retailer. The inventory system
manages a single type of deteriorating product with a circularity index (ϵ), ranging
from 0 to 1.

(ii) Inventory levels decline over time due to deterioration at a constant rate θ, both at
the manufacturer and the retailer levels.

(iii) The manufacturing process is imperfect, resulting in a fixed fraction Y of defective
units. These are detected through 100% quality inspection performed by the retailer.

(iv) All imperfect items are collected and remanufactured in batches by the manufacturer
after screening.

(v) The demand rate for remanufactured products depends on the circularity index,
expressed as D(ϵ) = D0 + A ϵ, while the unit gross profit is defined as G P(ϵ) =
G0 + Bϵ, in line with Rabta (2020) [1].

(vi) The manufacturing and remanufacturing rates, i.e., PM&PR exceed the maximum
demand rate D0 + A and the screening rate i.e., X is assumed to be significantly
higher than the demand rate to ensure immediate processing.

(vii) Both high-quality items and remanufactured products contribute to the total gross
profit, reflecting realistic cost recovery mechanisms.

(viii) Carbon emissions arise from both manufacturing and remanufacturing activities and
are regulated under two policy mechanisms: carbon tax and cap-and-trade.

(ix) All system parameters are deterministic and constant over time. Shortages are not
permitted in this model.

(x) Classical optimization techniques are employed to determine optimal production
cycle and circularity level. The Hessian matrix is used to confirm the concavity and
local optimality of the objective function.

4. Problem Description

This study investigates a sustainable inventory and remanufacturing problem in a two-
echelon supply chain consisting of a manufacturer and a retailer, operating under carbon
emissions regulations. The manufacturer produces new items that deteriorate over time
and are subject to imperfect quality, while the retailer handles quality inspection and
facilitates the return of defective units for remanufacturing. The demand for newly man-
ufactured products is dependent on both selling price and time, whereas the demand for
remanufactured products is driven by a circularity index, which reflects the product’s



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environmental performance. The supply chain operates under either a carbon tax or a
cap-and-trade policy and the objective is to determine optimal production and remanufac-
turing cycle times that maximize total profit while satisfying environmental constraints.
The model also considers the effects of deterioration, carbon emissions, imperfect qual-
ity, and circular economy dynamics on inventory decisions. A mathematical framework is
developed to analyze the system behavior under both regulatory regimes, and numerical
validation is conducted using MATLAB to assess the impact of key parameters on supply
chain performance. Figure 1, shows the interaction between the manufacturer and retailer,
supporting sustainable and cost-effective inventory and remanufacturing decisions under
carbon regulations.

Figure 1: Flowchart of the proposed model

5. Mathematical Model with Solution

This section details the mathematical model of the proposed framework, while Figures
2 to 6 illustrate the overall configuration of the inventory system.

Figure 2: Inventory system of manufacturer



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Figure 3: Inventory system of retailer

Figure 4: Inventory system of re-manufacturer

5.1. Case:1 Manufacturing inventory model for deteriorating items with
price & time dependent demand

5.1.1. Total cost and emission function of the manufacturer:

As shown in Figure 2, the inventory levels at any time during the period (0,tM ) are
governed by the following differential equations:

dIM (t)

dt
+ θIM (t) = PM , 0 ≤ t ≤ tM ; (1)

Figure 5: Inventory system of retailer in re-manufacturing process



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Figure 6: Inventory system of the model

With the given initial and boundary conditions, IM (0) = 0 at t = 0, the manufac-
turer’s stock level at any given time t can be determined as follows:

IM (t) =
PM

θ
[1− e−θt]. (2)

The quantity delivered by the manufacturer to the retailer, represented as QM , can be
obtained by utilizing the boundary condition IM (tM ) = QM at t = tM . By incorporating
this condition, the preceding equations can be solved as follows:

QM = IM (tM ) =
PM

θ
[1− e−θtM ]. (3)

As illustrated in Figure 2, the manufacturer produces QM goods during the manu-
facturing period tM and starts a current manufacturing cycle every T time units. Fixed
setup cost of SM incurred at the start of each cycle, the setup cost per unit time for the
manufacturer is,

S CM =
SM

T
. (4)

In this study, products are stored until their demand is met, extending up to time T .
During the interval from 0 to tM , the holding cost is calculated for all items produced and
retained until demand is satisfied. This cost is determined based on the fixed holding cost
HM , the inventory level IM (t) and the demand D(t).

H CM =
HM

T

[ ∫ tM
0 IM (t)dt

]
,

H CM =
PM HM

θ2T

[
θtM + e−θtM − 1

]
. (5)



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The deteriorated inventory of the manufacturer per cycle is given by PM tM )-QM ,

PM tM − QM = PM tM − PM

θ
[1− e−θtM ],

The manufacturer’s deterioration cost per unit time, DCM , is determined using the de-
preciated inventory per cycle, the deterioration cost per unit (DM ) and the cycle of re-
plenishment duration T can be used to compute as,

DCM =
DM

T

[
PM tM − PM

θ [1− e−θtM ]
]
. (6)

The overall cost to the manufacturer includes ordering, degradation, and holding
charges. Therefore, by summing equations (4), (5) and (6), the total cost per unit time
(T CM ) can be obtained as follows:

T CM =

[
SM
T +

[
PMHM

θ2T

[
θtM + e−θtM − 1

] ]
+
[

DM
T

[
PM tM − PM

θ [1− e−θtM ]
] ] ]

. (7)

The manufacturer’s inventory holding, production setup and depreciation processes gen-
erate carbon emissions amounting to ŜM units per cycle, ĤM units per unit time and
D̂M per units time. Consequently, the manufacturer’s overall carbon emissions per time
unit, T EM are calculated as follows:

T EM =

 ˆSM
T +

[
PM

ˆHM
θ2T

[
θtM + e−θtM − 1

] ]
+
[

ˆDM
T

[
PM tM − PM

θ [1− e−θtM ]
] ]  . (8)

5.1.2. Retailer’s overall manufacturing process costs and emissions

In Figure 3, the manufacturer receives an order of QM products to the store at the
beginning of each cycle, with a fraction Y of these items being of lower quality. The
retailer performs a 100% screening of the lot at a rate of X after receiving the order in
order to separate the good from the bad. The lower-quality items Y QR are separated
from the lot and kept as a single batch at time t = IR(0)

X after the screening process is
finished. Since each cycle extends for tR time units and the retailer is obligated to pay a
fixed order cost of OR at the beginning of each cycle, the ordering cost per unit time,

OCR =
OR

tR
(9)

Assuming that the quantity of high-quality items received by the retailer is at least equal
to the demand during the screening period, the following constraint is enforced on the
inventory system of the retailer, as derived from Lee and Kim (2014).

(1− Y )IR(0) ≥ D(t)
IR(0)

X
, (10)



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Because demand and deterioration cause the retailer’s inventory to deplete at rates of
D(t) and θ, respectively, and because lower-quality items are not used to satisfy the de-
mand D(t) for high-quality items, the following differential equation governs the retailer’s
inventory level over time.

dIR(t)

dt
+ θIR(t) = −(1− Y )D(t), 0 ≤ t ≤ tR ; (11)

The retailer’s inventory system is divided into two periods, the non-screening period,
which runs from t = IR(0)

X to t = tR , and the screening period, which runs from t = 0

to t = IR(0)
X . Taking into account the boundary constraint IR(tR) = 0, the following

differential equations depict the variations in the retailer’s inventory level over time during
the screening and non-screening periods, respectively.

IR(t) =
1

θ2
[
(αθP − β)e−θt − αθP − βθt+ β

]
, 0 ≤ t ≤ IR(0)

X
; (12)

IR(t) =
(1− Y )

θ2

[
eθ(tR−t)[αθP + βθtR − β]

−[αθP + βθt− β]

]
,

IR(0)

X
≤ t ≤ tR ; (13)

Similarly, the buyer’s ordered quantity is provided as follows:

QR = IR(0) = αPtR + βt2R . (14)

The retail replenishment cycle duration (tR) is divided by the annual cost of keeping one
unit of inventory in storage (HR), and the average inventory level per cycle is multiplied
by the retailer to determine the holding cost per unit time (H CR). Therefore, the holding
cost is as follows:

H CR =
HR

tR

[ ∫ IR(0)

X
0 IR(t)dt+

∫ tR
IR(0)

X

IR(t)dt

]
,

H CR =
HR

tR

 1
2θX 2

[
(−αβθ + 2β)(α2P2t2R + β2t4R + 2αβPt3R)

]
+ (1−Y )(αP+βtR)

2X

[
2tR(X tR − αPtR − βt2R)

− 1
X 2 (X tR − (αPtR + βt2R))2

]  . (15)

In the retailer’s cycle D(t)tR , the order quantity QR lowers the demand, which is the
amount of inventory that depreciates per cycle. The amount of deteriorating inventory
that the merchant has per cycle can be roughly estimated using the yields of Tiwari et al.
(2018).

QR − D(t)tR = βtR(tR − t),

The retailer’s deterioration cost per unit time DCR is determined as follows, considering
the deteriorated inventory per cycle, the deterioration cost per unit DR and the replen-
ishment cycle time tR .

DCR = DRβ(tR − t). (16)



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If the retailer completes a screening procedure (i.e., screens all QR units) and the cost
to screen each unit is ZR , then the screening costs per unit time Z CR is calculated as
follows.

Z CR =
ZRQR

tR
= ZR [αP + βtR ]. (17)

The retailer’s total cost function includes ordering, holding, deterioration, and screening
costs. Thus, the retailer’s total cost per unit time T CR is found by adding equations
(9), (15), (16), and(17), as shown below.

T CR =

 HR
tR

 1
2θX 2

[
(−αβθ + 2β)(α2P2t2R + β2t4R + 2αβPt3R)

]
+ (1−Y )(αP+βtR)

2X

[
2tR(X tR − αPtR − βt2R)

− 1
X 2 (X tR − (αPtR + βt2R))2

] 
+DRβ(tR − t) + ZR [αP + βtR ] + OR

tR

 . (18)

At the retail facility, various inventory management activities contribute to carbon dioxide
emissions. Specifically, these activities generate ÔR emissions units per cycle, ĤR emis-
sions units per unit per unit time from holding and D̂R emissions units per unit per unit
time from deterioration. As a result, the retailer’s total carbon emissions per unit time
T ER are calculated using the following formula:

T ER =


ĤR
tR

 1
2θX 2

[
(−αβθ + 2β)(α2P2t2R + β2t4R + 2αβPt3R)

]
+ (1−Y )(αP+βtR)

2X

[
2tR(X tR − αPtR − βt2R)

− 1
X 2 (X tR − (αPtR + βt2R))2

] 
+D̂Rβ(tR − t) + ÔR

tR

 . (19)

G P(t) is the unit gross profit, which is determined by subtracting the acquisition cost of
the items from the selling price. D(t) indicates the demand rate. The average total supply
chain profit, therefore, is D(t)G P(t) − (T CM + T CR). Thus, the following formula is
used to calculate the total supply chain profit per unit time T PS C :

T PS C =



D(t)G P(t)−

[
SM
T +

[
PMHM

θ2T

[
θtM + e−θtM − 1

] ]
+
[

DM
T

[
PM tM − PM

θ [1− e−θtM ]
] ] ]

−

 HR
tR

 1
2θX 2

[
(−αβθ + 2β)(α2P2t2R + β2t4R + 2αβPt3R)

]
+ (1−Y )(αP+βtR)

2X

[
2tR(X tR − αPtR − βt2R)

− 1
X 2 (X tR − (αPtR + βt2R))2

] 
+DRβ(tR − t) + ZR [αP + βtR ] + OR

tR




.

(20)

The supply chain’s total carbon emissions originate from both the retailer’s and the man-
ufacturer’s facilities. Therefore, the overall carbon emissions per unit time in the supply



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 15 of 53

chain are determined as follows:

T ES C =




ĤR
tR

 1
2θX 2

[
(−αβθ + 2β)(α2P2t2R + β2t4R + 2αβPt3R)

]
+ (1−Y )(αP+βtR)

2X

[
2tR(X tR − αPtR − βt2R)

− 1
X 2 (X tR − (αPtR + βt2R))2

] 
+D̂Rβ(tR − t) + ÔR

tR


+

 ˆSM
T +

[
PM

ˆHM
θ2T

[
θtM + e−θtM − 1

] ]
+
[

ˆDM
T

[
PM tM − PM

θ [1− e−θtM ]
] ] 


.

(21)

5.2. Case:2 Remanufacturing inventory model for deteriorating items
with circularity index demand

5.2.1. Remanufacturer’s total cost and emission functions:

The production process begins with a constant rate PM > D , producing defective items
at a rate of Y , resulting in θ1 = Y PM . Remanufacturing starts at a rate PR to
convert defective items into perfect ones, with scrap generated at a rate of Z , leading
to θ2 = Z PR . The perfect items are then prepared for retail sale within time tR′ . The
on-hand inventory of perfect items after remanufacturing is IM ′ . Inventory levels between
(tR ,tM ′) are described by the following differential equations,

dIM ′(t)

dt
+ θIM ′(t) = PR , tR ≤ t ≤ tM ′ ; (22)

With the boundary condition IM ′(t) = Y PM at t = tR , the remanufacturer’s stock level
at any given time t can be expressed as,

IM ′(t) = e−θt
[

PR
2eθt−Y PMθ2

θPR

]
(23)

The remanufacturer’s delivery quantity to the retailer QM ′ is obtained by applying the
initial condition IM ′(tM ′) = Q′

M at t = tM ′ . The equations can be solved as follows,

QM ′(t) = e−θtM′
[

PR
2eθtM′−Y PMθ2

θPR

]
(24)

As depicted in Figure 4, the remanufacturer produces Q′
M goods during the manufacturing

period tM ′ and begins a manufacturing cycle every T time units. The fixed setup cost
SM ′ is incurred at the beginning of every cycle, and the manufacturer’s setup cost per
time unit is calculated as follows,

S CM ′ =
SM ′

T
. (25)

Products are stored until their demand is met, extending up to time T . From tR to tM ′ ,
the holding costs include all items produced and retained until demand is satisfied. This



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 16 of 53

cost is calculated based on the fixed holding cost HM ′ , the inventory level IM ′(t) and
the product demand D(ϵ).

H CM ′ =
HM ′

T

[ ∫ tM′
tR

IM ′(t)dt
]
,

H CM ′ =
HM ′

T

[
PR
θ [tM ′ − tR ] + Y PM

PR
e−θ(tM′−tR)

]
(26)

The remanufacturer’s deteriorated inventory per cycle is determined by subtracting the
quantity of goods delivered to the retailer Q′

M from the remanufacturer’s production
quantity PRtM ′ .

PRtM ′ − QM ′ = PRtM ′ − e−θtM′
[

PR
2eθtM′−Y PMθ2

θPR

]
Deterioration cost per unit time for the remanufacturer the amount of deteriorated

inventory every cycle, the deterioration cost per unit DM ′ and the duration of the replen-
ishment cycle T can be used to determine DCM ′ as follows:

DCM ′ =
DM ′

T

[
PRtM ′ − e−θtM′

[
PR

2eθtM′−Y PMθ2

θPR

] ]
(27)

The remanufacturer’s total cost includes setup, holding, and deteriorating costs. There-
fore, summing equations (23), (24), and (25) gives the remanufacturer’s total cost per unit
time (T CM ′) as follows,

T CM ′ =

 SM′
T +

HM′
T

[
PR
θ [tM ′ − tR ] + Y PM

PR
e−θ(tM′−tR)

]
+

DM′
T

[
PRtM ′ − e−θtM′

[
PR

2eθtM′−Y PMθ2

θPR

] ]  . (28)

Carbon emissions are ˆSM ′ units per cycle, ˆHM ′ units per unit time, and ˆDM ′ units
per unit time, respectively, from the remanufacturer’s production setup, deterioration
and inventory holding activities. T EM ′ , the total carbon emissions per unit of time, is
determined by

T EM ′ =

 ˆSM′
T +

ˆHM′
T

[
PR
θ [tM ′ − tR ] + Y PM

PR
e−θ(tM′−tR)

]
+

ˆDM′
T

[
PRtM ′ − e−θtM′

[
PR

2eθtM′−Y PMθ2

θPR

] ]
 . (29)

5.2.2. Retailer’s total cost and emissions function of remanufacturing process

According to Figure 5, the retailer receives QM ′ items at the start of each cycle, with a
fraction Z being of imperfect quality. The retailer separates the high-quality items from
the lower-quality ones by performing a 100% screening at a pace of X . The lower-quality

items Z QR′ are salvaged in a single batch at t =
IR′ (0)

X at the conclusion of the screening
procedure.



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 17 of 53

At the beginning of each cycle, which lasts tR′ time units, the retailer incurs a fixed
order cost OR′ . The following formula is used to determine the ordering cost per unit of
time:

OCR′ =
OR′

tR′
(30)

The following restriction, derived from Lee and Kim (2014) [16], applies to the retailer’s
inventory system when the quantity of high-quality products received is sufficient to meet
demand during the screening period:

(1− Z )IR′(0) ≥ D(ϵ)
IR′(0)

X
, (31)

Assuming poorer quality items are excluded from fulfilling the demand rate D(ϵ) for good
quality goods, and considering inventory depletion due to demand D(ϵ) and deteriora-
tion at rate θ, the retailer’s inventory changes over time are governed by the following
differential equation,

dIR′(t)

dt
+ θIR′(t) = −

[
(1− Z )D(ϵ) + Z D(ϵ)

]
, 0 ≤ t ≤ tR′ ; (32)

The two periods in the retailer’s inventory system are the screening period [t = 0 to

t =
IR′ (0)

X ] and the and, according to Tiwari et al. (2018) [36], the non-screening period

[t =
IR′ (0)

X to t = tR′ ]. Inventory changes throughout these periods are described by the
following differential equations with the boundary condition IR(tR′) = 0.

IR′(t) =
D(ϵ)

θ

[
e−θt − 1

]
, 0 ≤ t ≤ IR′(0)

X
; (33)

IR′(t) =
(1− Z )D(ϵ)

θ

[
eθ(tR′−t) − 1

]
,

IR′(0)

X
≤ t ≤ tR′ ; (34)

The buyer’s purchase quantity is determined by

QR′ = IR′(0) =
D(ϵ)

θ

[
eθtR′ − 1

]
. (35)

The retailer’s holding cost per unit time H CR′ is determined by dividing the length of
the replenishment cycle tR by the average stock level per cycle and the yearly holding cost
per unit HR .

H CR′ =
HR′

tR′

[ ∫ IR′ (0)
X

0 IR′(t)dt+
∫ tR′

IR′ (0)
X

IR′(t)dt

]
,



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 18 of 53

H CR′ =
HR′

tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]− (D(ϵ))2tR′

2θX 2 [θD(ϵ)tR′ + 2X ]

]
. (36)

The retailer’s deteriorated inventory per cycle is the difference between the order quantity
QR′ and the demand during the period D(ϵ)tR′ . As per this quantity can be approximated
as follows,

QR′ − D(ϵ)tR′ = θD(ϵ)t2R′ ,

The retailer’s deterioration cost per unit of time DCR′ is calculated by considering the
deteriorated inventory each cycle, the deterioration cost per unit DR′ and the replenishing
cycle time tR′ .

DCR′ = DR′θD(ϵ)tR′ . (37)

Since the retailer performs 100% screening of the entire lot QR′ at a cost of ZR′ per unit,
the retailer’s screening cost per unit time Z CR′ is calculated as follows,

Z CR′ =
ZR′QR′

tR′
= θD(ϵ)ZR′tR′ . (38)

DPCR′ = θZ PR . (39)

The retailer’s total cost function includes ordering, holding, deteriorating, screening, and
disposal costs. Therefore, summing equations (28), (34), (35), (36) and (37) gives the re-
tailer’s total cost per unit time T CR′ as follows,

T CR′ =

 HR′
tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]− (D(ϵ))2tR′

2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

OR′
tR′

+ DR′θD(ϵ)tR′ + θD(ϵ)ZR′tR′ + θZ PR

 . (40)

Carbon emissions are produced by the retail facility’s inventory management operations,
such as ordering, holding, and deterioration. These actions are specifically related to
ÔR′ . Each cycle’s emissions, ĤR′ and D̂R′ units of emissions per time units, respectively.
Consequently, T ER′ , the retailer’s total emissions per units of time,

T ER′ =

[
ˆHR′

tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]

− (D(ϵ))2tR′
2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

ˆOR′
tR′

+ D̂R′θD(ϵ)tR′

]
. (41)

G P(ϵ) is the unit gross profit, which is achieved by subtracting the acquisition cost from
the selling price. D(ϵ) is the demand rate. D(ϵ)G P(ϵ) less the sum of T CM ′ and T CR′)



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 19 of 53

yields the average total supply chain profit. Therefore, T PS C ′ , the total supply chain
profit per unit time, is

T PS C ′ =


D(ϵ)G P(ϵ)−


HR′
tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]

− (D(ϵ))2tR′
2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

OR′
tR′

+ DR′θD(ϵ)tR′

+θD(ϵ)ZR′tR′ + θZ PR


−

 SM′
T +

HM′
T

[
PR
θ [tM ′ − tR ] + Y PM

PR
e−θ(tM′−tR)

]
+

DM′
T

[
PRtM ′ − e−θtM′

[
PR

2eθtM′−Y PMθ2

θPR

] ] 


. (42)

The total carbon emissions in the supply chain originate from both the retailer’s and
the manufacturer’s resources. Consequently, the overall carbon emissions per unit time
for the supply chain are calculated as follows:

T ES C ′ =



[
ˆHR′

tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]

− (D(ϵ))2tR′
2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

ˆOR′
tR′

+ D̂R′θD(ϵ)tR′

]

+

 ˆSM′
T +

ˆHM′
T

[
PR
θ [tM ′ − tR ] + Y PM

PR
e−θ(tM′−tR)

]
+

ˆDM′
T

[
PRtM ′ − e−θtM′

[
PR

2eθtM′−Y PMθ2

θPR

] ]


 . (43)

6. Carbon emissions policies and solution approaches

This section examines two carbon emissions policies: carbon tax and cap-and-trade
regulations. Each strategy in the study has two mathematical formulations and additional
analysis is done using various combinations of the unit profit and demand functions.

6.1. Mathematical models and solution methods for carbon emissions
policies

Carbon tax and cap & trade emissions policies are commonly used by governments
to motivate firms to reduce their carbon emissions. Both of these policies are used to
assess the suggested supply chain model, and mathematical formulations are created for
situations in which the unit profit and demand functions are both linear. Due to the com-
plexity of these formulations, analytical solutions are not feasible, so heuristic algorithms
are suggested as solution methods.

6.1.1. Carbon tax regulation

The supply chain must pay an extra tax under a carbon tax policy, which is determined by
the quantity of carbon emissions generated. The monetary tax levied per unit of carbon
emissions per unit of time is denoted by τ . The following is a mathematical formulation



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 20 of 53

of the problem:
Total profit of manufacturing process:

T PS C 1(tR , tM , T ) =



D(t)G P(t)

−


SM+τ ˆSM

T

+
[

PM (HM+τ ˆHM )
θ2T

[
θtM + e−θtM − 1

] ]
+
[

(DM+τ ˆDM )
T

[
PM tM − PM

θ [1− e−θtM ]
] ]



−


(HR+τĤR)

tR


1

2θX 2

[
(−αβθ + 2β)(α2P2t2R
+β2t4R + 2αβPt3R)

]

+ (1−Y )(αP+βtR)
2X


2tR(X tR

−αPtR − βt2R)
− 1

X 2 (X tR
−(αPtR + βt2R))2




+(DR + τD̂R)β(tR − t) + ZR [αP + βtR ] + (OR+τ ÔR)

tR





.

(44)

Total profit of remanufacturing process:

T PS C ′2(tR′ , tM ′ , ϵ) =



D(ϵ)G P(ϵ)

−


(HR′+τ ˆHR′ )

tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]

− (D(ϵ))2tR′
2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

(OR′+τ ˆOR′ )
tR′

+ (DR′ + τD̂R′)θD(ϵ)tR′

+θD(ϵ)ZR′tR′ + θZ PR



−



(SM′+τ ˆSM′ )
T

+
(HM′+τ ˆHM′ )

T

[
PR
θ [tM ′ − tR ]

+Y PM
PR

e−θ(tM′−tR)

]

+
(DM′+τ ˆDM′ )

T

[
PRtM ′

−e−θtM′
[

PR
2eθtM′−Y PMθ2

θPR

] ]




.

(45)

D(ϵ) = D0 + A ϵ, G P(ϵ) = G0 + Bϵ.

6.1.2. Carbon cap & trade regulation

In a carbon cap & trade regime, the supply chain is subject to a carbon emissions cap,
and excess emissions credits can be bought and sold through an emissions trading market.
The supply chain can sell excess credits at market value if emissions are below the cap. On
the other hand, going over the cap requires buying more credits at market value, which
adds to the expenses. Let δ be the market price for purchasing and disposing of emissions
credits, and let ρ be the emissions cap per unit of time. The following is a mathematical



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 21 of 53

formulation of the problem:
Total profit of manufacturing process:

T PS C 3(tR , tM , T ) =



D(t)G P(t) + ρδ

−


SM+δ ˆSM

T

+
[

PM (HM+δ ˆHM )
θ2T

[
θtM + e−θtM − 1

] ]
+
[

(DM+δ ˆDM )
T

[
PM tM − PM

θ [1− e−θtM ]
] ]



−


(HR+δĤR)

tR


1

2θX 2

[
(−αβθ + 2β)(α2P2t2R
+β2t4R + 2αβPt3R)

]

+ (1−Y )(αP+βtR)
2X


2tR(X tR

−αPtR − βt2R)
− 1

X 2 (X tR
−(αPtR + βt2R))2




+(DR + δD̂R)β(tR − T ) + ZR [αP + βtR ] + (OR+δÔR)

tR





.

(46)

Total profit of remanufacturing process:

T PS C ′4(tR′ , tM ′ , ϵ) =



D(ϵ)G P(ϵ) + ρδ

−


(HR′+δ ˆHR′ )

tR′

[
(1−Z )D(ϵ)t2

R′
2X 2 [X 2 − (D(ϵ))2]

− (D(ϵ))2tR′
2θX 2 [θD(ϵ)tR′ + 2X ]

]
+

(OR′+δ ˆOR′ )
tR′

+ (DR′ + δD̂R′)θD(ϵ)tR′

+θD(ϵ)ZR′tR′ + θZ PR



−



(SM′+δ ˆSM′ )
T

+
(HM′+δ ˆHM′ )

T

[
PR
θ [tM ′ − tR ]

+Y PM
PR

e−θ(tM′−tR)

]

+
(DM′+δ ˆDM′ )

T

[
PRtM ′

−e−θtM′
[

PR
2eθtM′−Y PMθ2

θPR

] ]




.

(47)

D(ϵ) = D0 + A ϵ, G P(ϵ) = G0 + Bϵ.

7. Solution Methodology

To determine the optimal values in the proposed two-echelon supply chain model,
we employed classical optimization techniques suitable for nonlinear objective functions.
These methods rely on iterative procedures that use gradient and Hessian information to
navigate the solution space. Approaches such as the steepest descent and conjugate gradi-
ent methods follow the direction of the negative gradient to reach a local optimum, while
Newton’s method utilizes second-order derivatives via the Hessian matrix to accelerate



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 22 of 53

convergence. Given the highly nonlinear nature of the objective function—which incor-
porates cost, deterioration, circularity and carbon emissions—numerical optimization was
performed using a structured algorithm. Since classical methods generally converge to
local optima, we evaluated the Hessian matrix to assess concavity. Where the Hessian is
negative definite, we confirm the existence of a local maximum. To enhance confidence in
the results, multiple initial values were used to explore the solution space and reduce the
risk of convergence to suboptimal points. This approach ensures robustness in deriving
economically and environmentally optimal decisions under the model’s constraints.

7.1. Solution Algorithm

This section presents a solution algorithm designed to obtain the optimal results for the
proposed study. Figure 7 provides a visual depiction of the computational steps involved
in the model.

Figure 7: Algorithm of the proposed model

Step 1: Consider the following parameters values: OR , ÔR ,HR , ĤR ,
DR , D̂R ,SM ,SM ,HM , ĤM ,DM ,PM , τ, ρ, and δ.

Step 2: From the equation (44), to find ∂TP
∂tR

, ∂TP
∂tM

and ∂TP
∂T . Detailed solutions are

given.
Step 3: To determine the values of tR , tM , and T using Step 2, utilize the values from

Step 1.
Step 4: Substituting, tR , tM and T in equations (44) and you will get the values of

total profit.
Step 5: Evaluate distinct prominent minor of the Hessian matrix,

∂2(TP )
∂t2M

∂2(TP )
∂tM∂tR

∂2(TP )
∂tM∂T

∂2(TP )
∂tM∂tR

∂2(TP )
∂t2R

∂2(TP )
∂tR∂T

∂2(TP )
∂tM∂T

∂2(TP )
∂tR∂T

∂2(TP )
∂T 2


at the point tM , tR and T .

Step 6: This provides the global optimal solution if the hessian matrix is negative
definite at the points (tM , tR , and T ).

Step 7: Determine TP (tM , tR and T ), the system’s optimal profit.



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 23 of 53

8. Numerical Analysis

This section provides numerical examples to demonstrate the applicability and validate
the findings of the study in real-world scenarios. The effectiveness of green technology in-
vestment, outsourcing, deterioration, circularity index and flexibility in the manufacturing
and remanufacturing process are discussed using numerical data. The models at best fit
of Rabta et al. (2020) [1], Dey et al. (2022) [7] and Sebatjane et al. (2024) [30] are used to
determine the parametric values for various parameters. MATLAB R2024b was used on
a Windows 10 PC with 16 GB of RAM and 128 GB of SSD to determine the best values
for the choice criteria.

The manufacturing and remanufacturing process (with and without circularity index)
are the two types of models we looked at because the deterioration rate is random. We
determined the overall profit for each model by applying the profit formula found in
the Sebatjane [28]. The best results with green technology investment are shown in the
following sections, which also examine a number of unusual circumstances.

8.1. Numerical Assessment for Manufacturing and Remanufacturing Pro-
cess

In this portion, we explore the best outcomes for this investigation concerning price
& time dependent and circularity index demand in manufacturing & remanufacturing
process. The parameter values are associated with the retailer’s fixed screening cost
(ZR&Z ′

R)= 0.2 $/unit, retailer’s fixed ordering cost (OR&O ′
R)= 50 ($/unit), retailer’s

fixed deteriorating cost (DR&D ′
R)= 0.75 ($/unit/yr), retailer’s fixed holding cost (HR&H ′

R)=
0.075 ($/unit/yr), retailer’s carbon emissions as a result of the expense of ordering mer-
chandise (ÔR&ÔR′)= 15 (lb of Co2), retailer’s carbon emissions as a result of the ex-
pense of keeping inventory (ĤR&ĤR′)= 0.0481 (lb of Co2/unit/yr), retailer’s carbon
emissions as a result of declining inventory prices (D̂R&D̂R′)= 0.15 (lb of Co2/unit/yr),
manufacturer’s fixed setup cost (SM&S ′

M )= 2,500 ($/unit), manufacturer’s fixed holding
cost (HM&H ′

M )= 0.05 ($/unit/yr), manufacturer’s fixed deteriorating cost (DM&D ′
M )=

0.375 ($/unit/yr),Carbon emissions from the producer as a result of inventory setup ex-
penses (ŜM& ˆSM ′)= 500 (lb of Co2), manufacturer’s carbon emissions as a result of the
expense of keeping inventory (ĤM& ˆHM ′)= 0.01875 (lb of Co2), manufacturer’s carbon
emission due to inventory deteriorating cost (D̂M& ˆDM ′)= 0.0927 (lb of Co2), fixed de-
terioration rate (θ)= 0.25, fraction of poorer items in manufacturing process (Y )= 0.5,
fixed screening rate (X )= 1,75,000 ($/units/yr), production rate in manufacturing period
(PM )= 1,23,000 (units/yr), production rate in remanufacturing period (PR)= 1,00,000
(units/yr), demand rate(D0)= 65,200 (units/yr), gross profit (go)= 2($/unit), demand
rate including circularity index (A )= 9,800 (units/yr), gross profit including circularity
index (B)= 0.145 ($/unit), carbon emissions tax (τ)= 1.25 ($ /lb of Co2), carbon emis-
sions cap (ρ)= 8000 (lb of Co2 /yr), market price of carbon emissions (δ)= 2.5 ($ /lb of
Co2).



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The principal minors of the Hessian matrix at the optimal decision variable values are
as follows: | H11 |= −3.8235 < 0, | H22 |= 1.1673 > 0 and | H33 |= −2.0131 < 0. When all
primary minors have alternating signs, the generated profit reaches its global maximum,
indicating that the revenue function is negative definite at the optimal decision variable
values.

Table 3, presents the optimal results of the production system under two different
carbon emission policies: carbon tax and carbon cap-and-trade, based on the specified
parameter values. The table evaluates the impact of these policies on key decision variables
and total profit in both manufacturing and remanufacturing processes. Given the influence
of carbon regulations on production efficiency, we determine the optimal values for cycle
times and total profit under each policy. The results indicate that the cap-and-trade policy
leads to higher profitability in both processes compared to a carbon tax.

Table 3, shows that under the carbon cap & trade policy, the total profit reaches
$92, 393 per cycle for the manufacturing process and $99, 058 per cycle for the remanu-
facturing process. In comparison, under the carbon tax policy, the total profit is slightly
lower, at $90, 271 and $97, 692, respectively. This suggests that a cap & trade system pro-
vides more economic benefits by allowing flexibility in managing emissions. The optimal
cycle times for manufacturing and remanufacturing are also affected by the policy choice.
In the manufacturing process, the cycle time (tM ) is slightly higher under cap & trade
(0.1392) than under a carbon tax (0.1304), while the remanufacturing cycle time (tR) is
lower under cap & trade (1.2931 vs. 1.5498 under tax). Similarly, for the remanufacturing
process, the manufacturing cycle time is 0.1405 under cap & trade and 0.1472 under car-
bon tax, while the remanufacturing cycle time remains slightly lower under cap & trade
(1.2169 vs. 1.2324). Overall, the results indicate that cap-and-trade enhances profitabil-
ity by 2.35% in manufacturing and 1.40% in remanufacturing compared to a carbon tax.
Furthermore, the total cycle time (T) remains relatively stable across policies, ensuring
that production efficiency is not significantly disrupted.

9. Sensitivity Analysis

In this part, a sensitivity analysis of the parameters is reviewed. Through individually
modifying each parameter within a range of 50%, while maintaining the other parameters
constant, we conduct a comprehensive sensitivity analysis.

9.1. Sensitivity Analysis for Case 1:

Tables 4 and 5 present the results of the sensitivity analysis in manufacturing process.

9.1.1. Sensitivity Analysis for Case 1:

Tables 6, 7 and 8 present the results of the sensitivity analysis in remanufacturing process.



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9.2. Sensitivity Analysis for Case 2:

Tables 9, 10 and 11 present the results of the sensitivity analysis in manufacturing
process.

9.2.1. Sensitivity Analysis for Case 2:

Tables 12, 13 and 14 present the results of the sensitivity analysis in remanufacturing
process.

9.3. Sensitivity analysis of demand related parameters:

Demand plays a critical role in both manufacturing and remanufacturing systems, sig-
nificantly impacting overall supply chain profitability. This study examines the influence
of demand on total profit, highlighting the importance of strategic decision-making for
industry managers. By effectively managing demand through optimal pricing strategies,
managers can enhance revenue and profitability. This study considers two distinct types of
demand: (i) price- and time-dependent demand, which is associated with the manufactur-
ing process and (ii) circularity index-based demand, which pertains to the remanufacturing
process. Demand varies across different parameters, ranging from minimum to maximum
values, except for D0 and G0. The findings indicate that an increase in demand for these
parameters significantly enhances revenue generation. Sensitivity analysis of demand-
related parameters reveals that emission-related parameters contribute to a profit increase
of approximately 50% to 69%, whereas other parameters result in a profit increase of ap-
proximately 25% to 45%. These scaling parameters, particularly those associated with
demand-driven remanufacturing, exhibit high sensitivity. Figures 8 to 12 provide a visual
representation of total profit and its correlation with demand-related parameters.

Figure 8: Comparison of the overall profit in parameters D0

9.4. Sensitivity analysis of cost related parameters:

The cost function is essential for optimizing manufacturing and remanufacturing pro-
cesses by minimizing expenses and maximizing profitability. Key cost components include



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Figure 9: Comparison of the overall profit in parameters G0

Figure 10: Comparison of the overall profit in parameters A

ordering cost, setup cost, holding cost, production cost, screening cost, and deteriorating
cost, all of which influence overall operational efficiency. In manufacturing, it helps man-
age raw material procurement, labor, and energy consumption, while in remanufacturing,
it assesses refurbishment, recycling, and reverse logistics costs. An efficient cost func-
tion aids in resource allocation, waste reduction, and pricing strategies, enhancing supply
chain performance. Moreover, it supports sustainable practices by balancing economic and
environmental objectives in industrial systems. The total profit of the remanufacturing
process varies between 97,337 and 99,990, influenced by fluctuations in the parameters
OR , ÔR , ĤR , HM and ĤM within a range of -50% to +50%. Additionally, the setup

Figure 11: Comparison of the overall profit in parameters B



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Figure 12: Comparison of the overall profit in parameters θ

cost parameters SM and ŜM exhibit an increasing trend from top to bottom. Figures 13
to 24 provide a visual representation of total profit variations in response to changes in
cost parameters.

Figure 13: Comparison of the overall profit in parameters ˆDM

9.5. Sensitivity analysis of emissions related parameters:

In this study, two environmental policies carbon tax and carbon cap & trade are
considered to assess their impact on manufacturing and remanufacturing systems. The
carbon tax directly penalizes emissions, encouraging firms to adopt cleaner technologies
and optimize resource utilization. Meanwhile, the cap-and-trade system sets an emission
limit while allowing firms to trade allowances, promoting cost-effective emissions reduc-
tion. Both policies play a crucial role in enhancing sustainability, reducing environmental
impact, and improving long-term profitability by balancing economic and ecological objec-
tives in industrial operations. The parameters ρ and δ exhibit the highest recorded values
among all considered parameters, with values of 99,199 and 99,671, respectively. Similarly,
the carbon tax parameter τ demonstrates a high value of 97,797. Figures 25 to 29 provide
a visual representation of total profit variations corresponding to these parameters.



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Figure 14: Comparison of the overall profit in parameters DM

Figure 15: Comparison of the overall profit in parameters D̂R

10. Managerial Insights

This study makes meaningful theoretical and practical contributions, providing impor-
tant guidance for businesses and policymakers. The findings underscore the significance
of managing inventory efficiently while addressing carbon emissions, item deterioration,
and fluctuating demand patterns.

Figure 16: Comparison of the overall profit in parameters DR



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Figure 17: Comparison of the overall profit in parameters ĤM

Figure 18: Comparison of the overall profit in parameters HM

10.1. Theoretical Implementation

The findings of this study provide valuable implications for supply chain managers
aiming to balance economic performance with environmental compliance. Coordinated
decision-making between the manufacturer and retailer, as modeled in the two-echelon
framework, leads to more efficient inventory and remanufacturing strategies under carbon
regulations. The model demonstrates that cap-and-trade policies offer greater flexibility
and slightly higher profitability than carbon taxation, suggesting that firms can benefit
from strategic participation in carbon credit markets. Furthermore, the use of a circularity

Figure 19: Comparison of the overall profit in parameters HR



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Figure 20: Comparison of the overall profit in parameters ÔR

Figure 21: Comparison of the overall profit in parameters OR

index as a demand driver emphasizes the operational and market advantages of promoting
product circularity, particularly in remanufacturing contexts. Effective quality inspection
at the retail level supports value recovery from imperfect items, aligning with circular
economy goals. The model also highlights the importance of minimizing deterioration
losses through optimized cycle times, which is especially relevant for semi-durable goods.
Finally, the integration of analytical tools such as MATLAB enables managers to simu-
late real-world scenarios and optimize decisions across cost, environmental, and service
objectives. These insights are particularly useful for small and mid-sized enterprises that

Figure 22: Comparison of the overall profit in parameters ŜM



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Figure 23: Comparison of the overall profit in parameters SM

Figure 24: Comparison of the overall profit in parameters ZR

operate in simplified supply chain structures yet seek to improve sustainability outcomes.

10.2. Practical Implementation

The proposed two-echelon sustainable supply chain model has strong practical rele-
vance across several industrial sectors where product deterioration, remanufacturing po-
tential, and environmental constraints intersect. The dual-demand framework and integra-
tion of carbon policies provide a strategic decision-making tool for real-world operations.
Key application domains include:

Figure 25: Comparison of the overall profit in parameters PM



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Figure 26: Comparison of the overall profit in parameters PR

Figure 27: Comparison of the overall profit in parameters ρ

(i) Electronics and Consumer Appliances: Manufacturers of high-value consumer
goods—such as smartphones, laptops, and televisions—routinely manage defective
units through repair, refurbishing, or resale channels. Remanufacturing practices in
this sector align with circular economy goals and are influenced by extended producer
responsibility (EPR) policies. The model aids in determining optimal production and
remanufacturing cycles by considering deterioration of unsold items, emission costs,
and profitability. Notable examples include companies like Apple and Dell, which
have established certified refurbishing programs under sustainability commitments.

Figure 28: Comparison of the overall profit in parameters τ



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Figure 29: Comparison of the overall profit in parameters δ

(ii) Automotive Components and Heavy Machinery: Remanufacturing is well-
established in automotive supply chains, particularly for engines, gearboxes, alter-
nators, and turbochargers. Organizations such as Bosch, Caterpillar, and Cummins
manage large-scale remanufacturing operations that reduce raw material consump-
tion and emissions. The proposed model is suitable for coordinating replenishment
and remanufacturing decisions for deteriorating parts, optimizing profit under emis-
sions caps or taxes. The circularity index can quantify the reused material content
or lifecycle extension of remanufactured components.

(iii) Medical Equipment and Diagnostic Devices: Hospitals and manufacturers fre-
quently remanufacture or refurbish medical imaging equipment (e.g., MRI, CT scan-
ners) and surgical tools due to their high cost and regulatory lifespan. The model
is particularly applicable in optimizing inventory management for returned or used
medical devices. It supports decisions based on equipment deterioration, environ-
mental compliance, and cost-effectiveness, in accordance with healthcare regulatory
frameworks (e.g., FDA or EU MDR guidelines).

(iv) Battery Recycling and Energy Storage Systems: In the context of electric
vehicles (EVs) and renewable energy, remanufacturing and recycling of lithium-ion
batteries have become critical. Companies like Redwood Materials and Tesla engage
in battery recovery to extract valuable materials such as lithium, cobalt, and nickel.
The proposed model facilitates optimization of reverse logistics, refurbishment cycles,
and emission controls in remanufacturing processes, with circularity index reflecting
the proportion of recovered energy materials.

(v) Industrial Equipment and Construction Tools : Industries involved in mining,
construction, or manufacturing often refurbish tools and machinery to extend service
life and reduce replacement costs. The model can be applied to plan maintenance
and remanufacturing cycles for deteriorating mechanical parts under fluctuating de-
mand and carbon regulations. Companies like Komatsu and Volvo Construction
Equipment employ similar practices under sustainability initiatives.



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10.3. Policy & Industry Recommendations

This study offers key recommendations for promoting sustainable production and cir-
cular economy practices. Policymakers are encouraged to enhance cap-and-trade systems
and include circularity metrics in reporting standards. Industry practitioners should align
operations with environmental regulations, improve quality inspections, and manage de-
terioration through forecasting tools. Using decision-support systems like MATLAB can
further support data-driven, low-carbon supply chain strategies that balance profitability
with sustainability.

11. Conclusion

This paper presents a comprehensive two-echelon supply chain model that integrates
sustainable inventory and remanufacturing decisions under carbon regulation. Unlike tra-
ditional models that focus solely on production or inventory, this framework emphasizes
the interaction between the manufacturer and retailer, optimizing operations based on
deteriorating items, imperfect quality, and environmentally influenced demand. Circular-
ity index demand introduces a novel dimension in remanufacturing, reflecting real-world
shifts toward circular economy practices. Our findings indicate that cap-and-trade policies
are slightly more effective in balancing profitability and environmental goals compared to
carbon tax. The sensitivity analysis highlights the importance of circularity-based de-
mand parameters and carbon cost variations. This model is best suited for semi-durable
goods, such as electronics or medical devices, rather than highly perishable items. Future
extensions may include stochastic elements, traceability technologies and multiple product
types to further enhance realism and applicability.

Future studies can extend this work by incorporating uncertainty in demand, carbon
credit pricing, and return rates, reflecting more realistic supply chain environments. The
model may be expanded to include stochastic deterioration, multiple product types, and
capacity constraints in remanufacturing. Additionally, integrating blockchain or IoT-based
traceability mechanisms could enhance circularity tracking and compliance in closed-loop
systems. Exploring decentralized decision-making frameworks and contractual coordina-
tion between supply chain tiers may also provide deeper insights into collaborative sustain-
ability strategies. Finally, empirical validation using industry-specific data would further
strengthen the practical applicability of the model across diverse sectors.

While the proposed model offers valuable insights into sustainable inventory and re-
manufacturing planning within a two-echelon supply chain, certain limitations must be
acknowledged. The model assumes deterministic parameters, including demand rates, de-
terioration and return quantities, which may not fully capture real-world uncertainties.
Additionally, the system considers perfect quality screening and instantaneous remanu-
facturing, whereas practical operations may involve delays and classification errors. The
current model also excludes capacity constraints, lead times and stochastic emissions,



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 35 of 53

which can influence supply chain performance in practice. Addressing these limitations in
future studies by incorporating uncertainty modeling, stochastic demand, dynamic carbon
pricing and real-time remanufacturing logistics would enhance the model’s robustness and
applicability.

Acknowledgements

The authors are thankful to the editor and learned referees for their worthy suggestions
which helped in improving the paper.

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Appendix



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Table 1: Notations of the proposed model.

Input Parameters

θ Deterioration rate of items
Y &Z The proportion of inferior products in the retailer’s order

amount while it is being manufactured and remanufactured
X Screening rate at the retailer (units / yr)

ZR&ZR′ Retailer’s fixed screening cost in
manufacturing & re-manufacturing period ($ / unit)

OR&OR′ Retailer’s fixed ordering cost in
manufacturing & re-manufacturing period ($)

DR&DR′ Retailer’s fixed deteriorating cost in
manufacturing & re-manufacturing period ($ / unit)

HR&HR′ Retailer’s fixed holding cost in
manufacturing & re-manufacturing period ($ / unit)

T CR&T CR′ Retailer’s total cost in
manufacturing & re-manufacturing period ($ / unit)

T ER&T ER′ Retailer’s total carbon emission cost in
manufacturing & re-manufacturing period ($ / unit)

ÔR&ÔR′ Carbon emissions from inventory
ordering by retailers ( lb of CO2)

D̂R&D̂R′ Carbon emissions from the retailer’s
declining inventory (lb of CO2 /units / yr)

ĤR&ĤR′ Carbon emissions of retailers
inventory holdings (lb of CO2 /units / yr)

PM&PR Production rate in Manufacturing and
Re-manufacturing period (units / yr)

SM&SM ′ Manufacturer’s fixed setup cost in
manufacturing & re-manufacturing period ($)

DM&DM ′ Manufacturer’s fixed deteriorating cost in
manufacturing & re-manufacturing period ($ / unit)

HM&HM ′ Manufacturer’s fixed holding cost in
manufacturing & re-manufacturing period ($ / unit)

T CM&T CM ′ Manufacturer’s total cost in
manufacturing & re-manufacturing period ($ / unit)

T EM&T EM ′ Manufacturer’s total carbon emission cost in
manufacturing & re-manufacturing period ($ / unit)

ŜM& ˆSM ′ Carbon emissions from the manufacturer’s
production setup (lb of CO2)

D̂M& ˆDM ′ Carbon emissions from the manufacturer’s
declining inventory (lb of CO2 /units / yr)

ĤM& ˆHM ′ Manufacturer’s carbon emission from
inventory holding (lb of CO2 /units / yr)



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 41 of 53

Table 2: Notations of the proposed model.

Input Parameters

D0 Rate of base demand without implementing
the circularity index(units / yr)

G0 Gross profit per base unit less the
circularity index factor ($ / unit)

A Demand function sensitivity to the
circularity index (units / yr)

B Unit gross profit function sensitivity
to the circularity index ($ /unit)

ρ Carbon emissions cap (lb of CO2 / yr)
τ Carbon tax on emissions ($/ lb of CO2 )
δ Market price for carbon emissions credits ($ / lb of CO2)

Explicit Decision Variables

tR&tR′ Retailer’s cycle time in manufacturing
and re-manufacturing period (yr)

tM&tM ′ Manufacturer’s cycle time in manufacturing
and re-manufacturing period (yr)

ξ Circularity index ranging from 0 to 1

Implicit Decision Variables

T Inventory cycle time
QR&Q′

R Retailer’s order quantity in manufacturing
and re-manufacturing period (units)

QM&Q′
M Deliveries made by the manufacturer to the retailer

during the production and remanufacturing phases. (units)

Functions

IR(t)&I ′
R(t) Inventory levels of retailers in manufacturing

and remanufacturing at any given time t (units)
G P(ξ) Gross profit per unit as a function of circularity ($ /unit)

IM (t)&I ′
M (t) Inventory levels of manufacturers in production and

remanufacturing processes at any given time t (units)
D(ξ) Demand function for remanufactured items; defined as D0 + A ξ (units / yr)

T PS C Total profit of the supply chain ($ / yr)
T ES C Total carbon emissions from the supply chain (lb of CO2/ yr)

Abbreviations

AGI Annual Green technology Investment
AHC Annual Holding Cost
TP Total Profit
APC Annual Purchasing Cost
TCE Total Carbon Emission
ASC Annual Setup Cost
NAR Net Annual Revenue



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 42 of 53

Table 3: Optimum results under carbon tax and carbon cap and trade policy.

Manufacturing Process Remanufacturing Process

Decision Variables Carbon tax Carbon cap & trade Carbon tax Carbon cap & trade

tM 0.1304 0.1392 0.1472 0.1405
tR 1.5498 1.2931 1.2324 1.2169
T 2.0283 2.0157 2.0217 2.0043

Total Profit ($/cycle) 90,271 92,393 97,692 99,058



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 43 of 53

Table 4: Manufacturing process concerning the parameters OR, ÔR,HR, ĤR,DR, D̂R,SM , ŜM and HM

under carbon tax policy.

Parameter Percentage tM tR T T PS C 1

5* OR -50% 0.1127 1.4302 2.1996 85,239
-25% 0.1291 1.4999 2.1001 88,111
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1405 1.6027 2.0037 90,768
+50% 0.1493 1.6831 1.9380 91,170

5* ÔR -50% 0.1302 1.5496 2.0281 90,269
-25% 0.1303 1.5497 2.0282 90,270
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1305 1.5499 2.0284 90,272
+50% 0.1306 1.5500 2.0285 90,273

5* HR -50% 0.1304 1.2317 1.9371 90,094
-25% 0.1304 1.4052 2.0035 90,125
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1304 1.5931 2.0394 90,299
+50% 0.1304 1.6021 2.0501 90,310

5* ĤR -50% 0.0972 1.5488 2.0281 76,313
-25% 0.1089 1.5493 2.0282 82,106
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1315 1.5504 2.0284 91,534
+50% 0.1377 1.5511 2.0285 92,079

5* DR -50% 0.2947 1.7832 2.1988 88,394
-25% 0.2039 1.6549 2.1220 89,530
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1211 1.4311 2.0030 91,927
+50% 0.1021 1.3927 1.9900 92,105

5* D̂R -50% 0.1101 1.3271 2.9314 77,184
-25% 0.1203 1.4750 2.5177 89,200
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1409 1.6111 1.9117 92,824
+50% 0.1507 1.7004 1.2349 99,732

5* SM -50% 0.0751 1.5494 2.0095 74,359
-25% 0.0922 1.5496 2.0139 89,205
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1718 1.5500 2.0314 96,593
+50% 0.2349 1.5502 2.0406 99,217

5* ŜM -50% 0.1302 0.7632 1.0937 69,344
-25% 0.1303 0.9315 1.8754 80,077
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1305 1.5901 2.0976 99,745
+50% 0.1306 1.6537 2.1085 1,01,007

5* HM -50% 0.1304 1.5498 2.0283 90,271
-25% 0.1304 1.5498 2.0283 90,271
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1304 1.5498 2.0283 90,271
+50% 0.1304 1.5498 2.0283 90,271



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Table 5: Manufacturing process concerning the parameters ĤM ,DM , ˆDM , θ,ZR, Y,X,PM and τ under
carbon tax policy.

Parameter Percentage tM tR T T PS C 1

5* ĤM -50% 0.1291 1.3287 1.9317 90,114
-25% 0.1297 1.4155 1.9924 90,203
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1307 1.6022 2.0954 90,308
+50% 0.1312 1.6384 2.1036 90,379

5* DM -50% 0.1283 1.4322 2.0134 89,333
-25% 0.1291 1.4905 2.0199 89,925
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1311 1.5807 2.0307 90,846
+50% 0.1357 1.5994 2.0358 91,320

5* D̂M -50% 0.1304 1.5498 2.0283 90,271
-25% 0.1304 1.5498 2.0283 90,271
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1307 1.5531 2.1377 95,890
+50% 0.1307 1.5531 2.1377 95,890

5* θ -50% 0.1304 1.5498 2.0281 90,268
-25% 0.1304 1.5498 2.0282 90,270
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1304 1.5498 2.0284 90,272
+50% 0.1304 1.5498 2.0285 90,274

5* ZR -50% 0.1452 1.5271 2.0179 90,195
-25% 0.1375 1.5335 2.0211 90,218
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1286 1.5584 2.0314 90,306
+50% 0.1208 1.5679 2.0396 90,353

5* Y -50% 0.1304 1.5498 2.0283 90,271
-25% 0.1304 1.5498 2.0283 90,271
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1304 1.5498 2.0283 90,271
+50% 0.1304 1.5498 2.0283 90,271

5* X -50% 0.1302 1.5496 2.0281 90,269
-25% 0.1303 1.5497 2.0282 90,270
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1305 1.5499 2.0284 90,272
+50% 0.1306 1.5500 2.0285 90,273

5* PM -50% 0.1304 1.5431 2.0032 86,425
-25% 0.1304 1.5477 2.0157 88,607
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1306 1.5539 2.0311 92,380
+50% 0.1306 1.5590 2.0399 95,491

5* τ -50% 0.1175 1.7901 2.1189 81,394
-25% 0.1293 1.6324 2.0547 86,736
100% 0.1304 1.5498 2.0283 90,271
+25% 0.1389 1.4730 2.0025 93,115
+50% 0.1472 1.3956 1.9621 99,218



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Table 6: Remanufacturing process concerning the parameters OR′ , ˆOR′ ,HR′ , ĤR′ ,DR′ , ˆDR′ ,SM′ , ˆSM′ and
HM′ under carbon tax policy.

Parameter Percentage tM ′ tR′ T T PS C ′2

5* OR′ -50% 0.1230 1.2322 2.0193 95,311
-25% 0.1317 1.2323 2.0205 96,178
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1511 1.2325 2.0224 98,314
+50% 0.1590 1.2326 2.0236 98,729

5* ÔR′ -50% 0.1734 1.2321 2.0535 97,714
-25% 0.1502 1.2323 2.0411 97,707
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1236 1.2325 2.0010 97,653
+50% 0.1098 1.2325 1.9832 97,650

5* HR′ -50% 0.1221 1.2301 2.0204 97,591
-25% 0.1305 1.2319 2.0211 97,630
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1509 1.2328 2.0223 97,717
+50% 0.1637 1.2335 2.0229 97,778

5* ĤR′ -50% 0.1531 1.2917 2.0534 98,337
-25% 0.1509 1.2697 2.0319 97,920
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1458 1.2016 2.0092 97,219
+50% 0.1420 1.1995 1.9971 97,034

5* DR′ -50% 0.1294 1.1936 2.0138 97,337
-25% 0.1313 1.2172 2.0190 97,483
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1509 1.2506 2.0254 97,821
+50% 0.1581 1.2781 2.0279 97,908

5* D̂R′ -50% 0.1301 1.2101 2.0134 97,525
-25% 0.1396 1.2297 2.0182 97,607
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1499 1.2358 2.0256 97,713
+50% 0.1584 1.2373 2.0290 97,789

5* SM ′ -50% 0.1470 1.2320 2.0215 97,690
-25% 0.1471 1.2322 2.0216 97,691
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1473 1.2326 2.0218 97,693
+50% 0.1474 1.2328 2.0219 97,694

5* ˆSM ′ -50% 0.1472 1.2324 2.0217 97,692
-25% 0.1472 1.2324 2.0217 97,692
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1472 1.2324 2.0217 97,692
+50% 0.1472 1.2324 2.0217 97,692

5* HM ′ -50% 0.1593 1.1938 2.0211 97,683
-25% 0.1511 1.2102 2.0214 97,689
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1439 1.2575 2.0221 97,696
+50% 0.1397 1.2781 2.0225 97,701



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 46 of 53

Table 7: Remanufacturing process concerning the parameters ˆHM′ ,DM′ , ˆDM′ , θ,ZR′ ,PR,D0,A and g0
under carbon tax policy.

Parameter Percentage tM ′ tR′ T T PS C ′2

5* ˆHM ′ -50% 0.1298 1.2311 2.0183 97,611
-25% 0.1354 1.2319 2.0201 97,677
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1581 1.2332 2.0265 97,705
+50% 0.1606 1.2338 2.0290 97,719

5* DM ′ -50% 0.1938 1.3711 2.0591 98,031
-25% 0.1765 1.3097 2.0428 97,754
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1281 1.2009 2.0053 97,320
+50% 0.1054 1.1976 1.1926 96,917

5* ˆDM ′ -50% 0.1359 1.1983 2.0191 97,520
-25% 0.1413 1.2052 2.0209 97,605
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1508 1.2461 2.0228 97,713
+50% 0.1544 1.2599 2.0236 97,788

5* θ -50% 0.1470 1.2320 1.7281 97,234
-25% 0.1471 1.2322 1.9901 97,410
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1473 1.2326 2.0493 97,803
+50% 0.1474 1.2328 2.0668 97,994

5* ZR′ -50% 0.1423 1.2322 2.0203 97,678
-25% 0.1456 1.2323 2.0211 97,683
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1497 1.2325 2.0224 97,704
+50% 0.1505 1.2326 2.0238 97,721

5* PR -50% 0.1334 1.2322 2.0180 97,604
-25% 0.1390 1.2323 2.0204 97,657
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1499 1.2325 2.0249 97,723
+50% 0.1523 1.2326 2.0271 97,780

5* D0 -50% 0.1453 1.2310 2.0211 97,674
-25% 0.1461 1.2317 2.0215 97,681
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1485 1.2339 2.0220 97,714
+50% 0.1496 1.2358 2.0224 97,738

5* A -50% 0.1359 1.2301 2.0215 97,512
-25% 0.1423 1.2311 2.0216 97,620
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1519 1.2347 2.0218 97,749
+50% 0.1587 1.2356 2.0219 97,800

5* g0 -50% 0.1468 1.2320 2.0211 97,686
-25% 0.1470 1.2322 2.0214 97,689
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1474 1.2326 2.0220 97,696
+50% 0.1476 1.2328 2.0223 97,700



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Table 8: Remanufacturing process concerning the parameters B and τ under carbon tax policy.

Parameter Percentage tM ′ tR′ T T PS C ′2

5* B -50% 0.1329 1.8921 2.0212 97,621
-25% 0.1390 1.5512 2.0215 97,664
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1501 1.0732 2.0220 97,718
+50% 0.1584 0.9321 2.0226 97,779

5* τ -50% 0.1465 1.2109 2.0198 97,631
-25% 0.1468 1.2272 2.0209 97,654
100% 0.1472 1.2324 2.0217 97,692
+25% 0.1479 1.2390 2.0224 97,739
+50% 0.1488 1.2416 2.0238 97,797



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 48 of 53

Table 9: Manufacturing process concerning the parameters OR, ÔR,HR, ĤR,DR, D̂R,SM , ŜM and HM

under carbon cap and trade policy.

Parameter Percentage tM tR T T PS C 3

5* OR -50% 0.1390 1.2720 1.0908 92,190
-25% 0.1391 1.2809 1.0999 92,289
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1393 1.3019 2.0194 92,478
+50% 0.1394 1.3679 2.0236 92,542

5* ÔR -50% 0.1277 1.2291 2.0122 92,137
-25% 0.1329 1.2635 2.0139 92,260
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1401 1.3107 2.0171 92,439
+50% 0.1473 1.3776 2.0190 92,492

5* HR -50% 0.1713 1.3509 2.1057 92,826
-25% 0.1586 1.3182 2.0685 92,521
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1100 1.2830 1.9801 92,111
+50% 0.0935 1.2497 1.8054 91,909

5* ĤR -50% 0.1156 1.2317 1.8507 91,973
-25% 0.1209 1.2677 1.9912 92,011
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1470 1.3351 2.0391 92,467
+50% 0.1514 1.3602 2.0509 92,616

5* DR -50% 0.1387 1.2922 2.0141 92,371
-25% 0.1388 1.2927 2.0150 92,384
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1396 1.2938 2.0166 92,412
+50% 0.1405 1.2949 2.0174 92,469

5* D̂R -50% 0.1373 1.2888 2.0111 92,325
-25% 0.1384 1.2911 2.0135 92,364
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1405 1.2960 2.0178 92,439
+50% 0.1438 1.2993 2.0213 92,490

5* SM -50% 0.1279 1.2900 2.0011 92,112
-25% 0.1303 1.2919 2.0099 92,275
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1486 1.2964 2.0192 92,436
+50% 0.1590 1.2995 2.0278 92,505

5* ŜM -50% 0.1376 1.2726 2.0062 91,990
-25% 0.1384 1.2860 2.0109 92,121
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1409 1.3015 2.0236 92,514
+50% 0.1427 1.3742 2.0390 92,782

5* HM -50% 0.1390 1.2921 2.0157 92,382
-25% 0.1391 1.2927 2.0157 92,389
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1393 1.2938 2.0157 92,400
+50% 0.1394 1.2949 2.0157 92,451



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 49 of 53

Table 10: Manufacturing process concerning the parameters ĤM ,DM , ˆDM , θ,ZR, Y,X,PM , ρ and δ under
carbon cap and trade policy.

Parameter Percentage tM tR T T PS C 3

5* ĤM -50% 0.1300 1.2929 2.0152 92,374
-25% 0.1347 1.2930 2.0155 92,386
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1405 1.2932 2.0160 92,405
+50% 0.1469 1.2933 2.0166 92,421

5* DM -50% 0.1247 1.2929 2.0132 92,371
-25% 0.1309 1.2929 2.0141 92,384
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1437 1.2934 2.0169 92,402
+50% 0.1490 1.2935 2.0178 92,437

5* D̂M -50% 0.1373 1.2903 2.0120 92,371
-25% 0.1380 1.2915 2.0139 92,385
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1409 1.2949 2.0171 92,407
+50% 0.1427 1.2964 2.0194 92,419

5* θ -50% 0.1607 1.3674 2.0139 92,311
-25% 0.1538 1.3199 2.0144 92,364
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1284 1.2290 2.0163 92,449
+50% 0.1105 1.2068 2.0178 92,482

5* ZR -50% 0.1691 1.2380 2.0133 92,681
-25% 0.1583 1.2611 2.0144 92,537
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1194 1.3277 2.0169 92,118
+50% 0.0917 1.3609 2.0176 91,840

5* Y -50% 0.1390 1.2939 2.0155 92,371
-25% 0.1391 1.2935 2.0156 92,384
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1393 1.2927 2.0158 92,416
+50% 0.1394 1.2918 2.0159 92,439

5* X -50% 0.1379 1.2714 2.1511 93,610
-25% 0.1386 1.2859 2.0805 92,901
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1401 1.3017 2.0003 91,717
+50% 0.1435 1.3290 1.9452 91,022

5* PM -50% 0.1317 1.3217 2.0111 91,908
-25% 0.1351 1.3108 2.0139 92,099
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1415 1.2629 2.0182 92,426
+50% 0.1478 1.2394 2.0208 92,497



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 50 of 53

Table 11: Manufacturing process concerning the parameters ρ, α, β, t, b and δ under carbon cap and trade
policy.

Parameter Percentage tM tR T T PS C 3

5* ρ -50% 0.1391 1.2912 2.0132 92,377
-25% 0.1391 1.2926 2.0145 92,384
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1394 1.2938 2.0163 92,401
+50% 0.1396 1.2947 2.0177 92,416

5* δ -50% 0.1390 1.2929 2.0157 92,386
-25% 0.1391 1.2930 2.0157 92,390
100% 0.1392 1.2931 2.0157 92,393
+25% 0.1393 1.2932 2.0157 92,397
+50% 0.1394 1.2933 2.0157 92,402



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 51 of 53

Table 12: Remanufacturing process concerning the parameters OR′ , ˆOR′ ,HR′ , ĤR′ ,DR′ , ˆDR′ ,SM′ , ˆSM′

and HM′ under carbon cap and trade policy.

Parameter Percentage tM ′ tR′ T T PS C ′4

5* OR′ -50% 0.1381 1.2026 1.1739 98,617
-25% 0.1393 1.2104 1.1907 98,980
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1429 1.2192 2.0110 99,137
+50% 0.1450 1.2211 2.0155 99,544

5* ÔR′ -50% 0.1332 1.2074 1.9901 97,111
-25% 0.1379 1.2111 1.9987 98,703
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1458 1.2205 2.0101 99,511
+50% 0.1490 1.2923 2.0456 99,990

5* HR′ -50% 0.1403 1.2167 2.0041 99,056
-25% 0.1404 1.2168 2.0042 99,057
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1406 1.2170 2.0044 99,059
+50% 0.1407 1.2171 2.0045 99,060

5* ĤR′ -50% 0.1400 1.2168 2.0041 99,032
-25% 0.1403 1.2169 2.0042 99,046
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1408 1.2170 2.0044 99,065
+50% 0.1411 1.2170 2.0045 99,077

5* DR′ -50% 0.1891 1.4444 2.9600 99,995
-25% 0.1633 1.3805 2.4802 99,512
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1299 1.1573 1.8999 98,768
+50% 0.9767 1.1057 1.5077 98,057

5* D̂R′ -50% 0.1913 1.2430 2.0043 99,313
-25% 0.1639 1.2327 2.0043 99,280
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1210 1.2102 2.0043 98,900
+50% 0.9084 1.2006 2.0043 98,845

5* SM ′ -50% 0.1492 1.2155 2.0041 99,317
-25% 0.1451 1.2162 2.0042 99,193
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1383 1.2177 2.0044 98,904
+50% 0.1317 1.2184 2.0045 98,799

5* ˆSM ′ -50% 0.1405 1.2167 2.0042 99,056
-25% 0.1405 1.2168 2.0042 99,057
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1405 1.2170 2.0043 99,059
+50% 0.1405 1.2171 2.0044 99,060

5* HM ′ -50% 0.1400 1.2165 2.0037 99,012
-25% 0.1403 1.2167 2.0041 99,033
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1407 1.2172 2.0045 99,076
+50% 0.1409 1.2175 2.0047 99,119



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 52 of 53

Table 13: Remanufacturing process concerning the parameters ˆHM′ ,DM′ , ˆDM′ , θ,ZR′ ,PR,D0,A and g0
under carbon cap and trade policy.

5* ˆHM ′ -50% 0.1511 1.2190 2.0088 99,382
-25% 0.1470 1.2178 2.0061 99,200
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1383 1.2151 2.0029 99,001
+50% 0.1327 1.2137 2.0017 98,947

5* DM ′ -50% 0.1403 1.2169 2.0043 99,056
-25% 0.1404 1.2169 2.0043 99,057
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1406 1.2169 2.0043 99,058
+50% 0.1407 1.2170 2.0044 99,061

5* ˆDM ′ -50% 0.1404 1.2167 2.0039 99,091
-25% 0.1404 1.2168 2.0041 99,070
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1405 1.2170 2.0045 99,039
+50% 0.1405 1.2171 2.0047 99,005

5* θ -50% 0.1490 1.2169 2.0000 99,711
-25% 0.1451 1.2169 2.0027 99,396
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1379 1.2169 2.0069 98,707
+50% 0.1336 1.2169 2.0085 98,599

5* ZR′ -50% 0.1400 1.2111 2.0043 99,010
-25% 0.1403 1.2147 2.0043 99,037
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1407 1.2183 2.0043 99,076
+50% 0.1409 1.2205 2.0043 99,099

5* PR -50% 0.1478 1.2244 2.0099 99,132
-25% 0.1439 1.2205 2.0061 99,085
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1381 1.2116 2.0027 99,039
+50% 0.1347 1.2078 2.0004 99,006

5* D0 -50% 0.1496 1.2318 2.0123 99,112
-25% 0.1449 1.2271 2.0095 99,076
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1371 1.2085 2.0007 99,021
+50% 0.1348 1.1734 1.9711 99,009

5* A -50% 0.1390 1.2099 1.9045 98,887
-25% 0.1397 1.2138 1.9650 98,966
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1414 1.2190 2.0081 99,139
+50% 0.1462 1.2237 2.0194 99,210

5* g0 -50% 0.1473 1.2192 2.0071 99,111
-25% 0.1451 1.2176 2.0058 99,083
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1377 1.2137 2.0027 99,017
+50% 0.1328 1.2102 2.0005 98,915



R. Suvetha et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6032 53 of 53

Table 14: Remanufacturing process concerning the parameters B, ρ and δ under carbon cap and trade policy.

Parameter Percentage tM ′ tR′ T T PS C ′4

5* B -50% 0.1401 1.2163 2.0037 99,053
-25% 0.1403 1.2166 2.0040 99,056
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1407 1.2172 2.0046 99,061
+50% 0.1409 1.2175 2.0049 99,063

5* ρ -50% 0.1476 1.2193 2.0158 99,671
-25% 0.1431 1.2174 2.0102 99,312
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1383 1.2152 2.0009 98,965
+50% 0.1329 1.2131 1.9904 98,903

5* δ -50% 0.1391 1.2087 2.0038 98,913
-25% 0.1397 1.2112 2.0040 98,987
100% 0.1405 1.2169 2.0043 99,058
+25% 0.1413 1.2191 2.0045 99,173
+50% 0.1426 1.2234 2.0048 99,199