Optimal Inventory and Pricing Strategies for Integrated Supply Chains of Growing Items Under Carbon Emission Policies

Abstract

This study investigates inventory management and pricing techniques in a two-tier supply chain where newborn items are grown, slaughtered, and transported to retailers for consumer sale. This study assesses how certain carbon regulations can enhance or hinder profitability for suppliers and retailers, demonstrating the interdependence of their financial performance in connection to environmental regulations. A mathematical model considers demand as impacted by unit weight, selling price, and storage duration, with consumption patterns as a power function of these variables. This paper examines demand dynamics and proposes a solution for optimizing crucial factors such as the number of newborn items, the retailer’s selling price, and operating cycle time to increase profitability while maintaining excellent customer service.

1. Introduction

Collaborations between businesses and their partners are critical in today’s economy. Businesses comprehend that the performance of their partners heavily impacts the quality of their products and services. This interconnected network of partners is recognized as a supply chain (SC) in which numerous firms collaborate to provide goods or services to customers [1]. Effectively controlling the activity of these SC participants is critical for satisfying client expectations.

Supply chain management (SCM) refers to various operations involving various stakeholders, including raw material suppliers and end customers. Customer demand drives all SC activities. Thus, effective inventory management is critical for satisfying that demand and guaranteeing customer satisfaction. Proper inventory management can reduce costs throughout the system, leading to increased profitability [2]. Furthermore, a significant relationship exists between pricing and demand, i.e., lower prices usually result in increased demand. That is, improving pricing tactics can significantly boost profitability. Effective pricing, a crucial revenue management component, relies on efficient inventory management procedures. Consequently, combining inventory and revenue management methods is critical for improving SC performance.

In modern business transactions, a credit term is provided by suppliers, which aids inventory management by allowing retailers to settle outstanding accounts. Suppliers frequently provide a grace period during which retailers can postpone payment without incurring interest costs. This arrangement allows retailers to generate revenue from sales while possibly earning interest by investing those funds. However, suppliers might incur interest charges if payments are delayed beyond this grace period. Delayed payments can lower inventory holding costs. Trade-credit from suppliers motivates retailers to purchase more and acts as a marketing tool, luring new consumers who see it as an alternative to bulk discounts. As a result, suppliers and retailers rely largely on trade-credit.

The obligations of SCM must shift from traditional methods to incorporate novel concepts [3]. Inventory management often assumes that items remain unmodified throughout storage. However, this is not always the case. For example, the quality of inventory might decline or increase with time. While this idea is well explored in inventory management, the issue of growth—defined as increases in inventory levels owing to weight gain—receives far less scholarly attention. Farming, fishery, poultry, and livestock industries all experience growth as inventory goes through breeding and consumption. During breeding, inventory grows until slaughter, which is consumed to meet the demand. Managing the inventory of growing things is vital since they are susceptible to illnesses and quality loss. Poor assessments can lead to financial losses along with threats to consumer health. After slaughter, these goods become perishable. Optimizing inventory and pricing for such commodities can minimize waste and unnecessary expenditure.

The retailer’s activities produce large carbon emissions, necessitating taxes based on certain carbon restrictions by the authorities. This study answers numerous essential questions as follows: (i) How is the per unit weight price for a retailer affected by the carbon laws? (ii) How can the optimal permissible time be determined under carbon regulations? (iii) How is the integrated profitability affected by carbon regulations? The presented model primarily contributes to the research with integrated optimal pricing, permissible payment delay, and ordering policies for newborn growing items, all designed to maximize the overall profit while complying with carbon limits.

Section 2 provides a brief overview of the current literature in the remaining paper, while Section 3 outlines the notations and assumptions underpinning the model. The utilized mathematical framework is described in Section 4. The methodology for obtaining solutions is explained in Section 5. Section 6 examines the numerical results, while Section 7 delves into sensitivity analysis and its implications for management. Lastly, the work is concluded in Section 8, and several avenues are suggested for additional research.

2. Contextual Analysis of the Literature

The existing literature has been reviewed considering four aspects: (1) growing items inventory, (2) integrated SC system, (3) two-level permissible delay in payments (trade-credit), and (4) carbon emissions regulations.

Khalilpourazari and Pasandideh [4] developed a mathematical model to solve operational restrictions for businesses that manage various types of growing goods, utilizing hybrid meta-heuristic algorithms to identify the ideal number of newborn items to acquire. Nobil et al. [5] incorporated shortages into their economic order quantity (EOQ) model for poultry producers. Sarkar et al. [6] built an SCM model to reduce emissions from the system as well as emissions. Pourmohammad-Zia and Karimi [7] included degradation in their analysis, outlining the best feeding and consuming periods for slaughtered items. Alfares and Afzal [8] investigated growing products with defective quality and full backorder. Furthermore, Mittal and Sharma’s [9] study in 2021 advanced the knowledge of how financial regulations may strategically influence the purchase behavior in SCs with growing products, while Sharma and Mittal [10] examined its influence on selecting items of different quality.

Yu et al. [11] utilized profit-sharing contracts to improve coordination within an SC, especially when the supplier faces considerable upfront expenses. Moreover, revenue-sharing contracts and collaborative investment in inventory technology were implemented by Zhang et al. [12] to reduce degradation. These studies found that coordinating techniques boost system profitability. However, three-echelon supply networks were under-researched due to their complexity, as emphasized by Cai et al. [13] in their study, encompassing a supplier, distributor, and retailer. A three-tier SC was examined by Sebatjane and Adetunji [14] for growing commodities, taking the consumption rate to be both price- and freshness-sensitive, implementing an exponential function for price effect and a linear function for freshness over storage time. Sebatjane and Adetunji [15] then investigated a model emphasizing maximum lifespan-sensitive degradation rates, ignoring expiry date influences.

Researchers have extensively investigated the best ordering procedures for businesses operating on fixed credit durations across various scenarios. Huang et al. [16] conducted research on collaborative inventory management between suppliers and customers to optimize order quantities and scheduling to improve SC efficiency. Mahata et al. [17] studied inventory models created particularly for items with a short shelf life, highlighting the importance of timely replenishment to avoid spoiling losses. Mahata [18] examined the issues that retailers confront when physical or operational constraints limit their capacity to hold or handle goods, requiring careful order quantity planning. Vandana and Sharma [19], for example, expanded the credit term to incorporate payment delays based on purchase quantity. In certain circumstances, if the retailer’s purchase fulfills a minimum amount, the supplier will provide a defined credit period; otherwise, immediate payment is needed. Furthermore, some works suggest partial payment delays to stimulate larger purchases, in which the retailer pays a certain percentage beforehand and the balance at the end of the credit term [18]. Seifert and Seifert [20] provided extensive assessments of trade-credit policies.

In the late 1960s, authorities began implementing pollution control rules to minimize human-caused greenhouse gas emissions. Product handling operations in logistics facilities account for about 13% of total emissions, as shown by the World Economic Forum in 2009. The product storage operations in UK warehouses produce around 10.2 million tons of CO₂ [21]. An effective strategy was employed by Hua et al. [22] to investigate how trading mechanisms affect firm expenses when certain carbon caps are imposed. Chen et al. [23] investigated how a carbon cap policy affects the optimum ordering policy, considering purchased items as an emissions source. Toptal et al. [24] examined purchase and investment strategies for lowering carbon emissions under three tax schemes. Hovelaque and Bironneau [25] investigated how carbon pricing influences consumer behavior in the context of carbon price legislation. Lin and Sarker [26] incorporated certain rules regarding carbon taxes to determine optimal ordering strategies for low-quality goods. Li and Hai [27] developed effective reorder regulations considering various retailers utilizing a single warehouse, factoring carbon tax into the overall costs of buying and maintaining inventory. Dhanda et al. [28] evaluated the implications of carbon emission regulations on fresh food SCs, emphasizing degrading products of varying quality.

Alamri [29], Chang and Tseng [30], and De-la-Cruz-Márquez et al. [31] analyzed sustainable SC models for growing and perishable commodities, considering trade-credit, price, carbon emissions, imperfect quality, and replenishment techniques to optimize inventory and financial decisions.

Research Gap

This research draws on these models, particularly the work by Nobil et al. [5] as a basis for incorporating growth dynamics and quality into inventory management in SCs with growing products. Building upon these insights, it addresses the complexities of a three-echelon SCM, particularly in managing growing products and integrating inventory, trade-credit, and carbon emissions considerations. This research further extends the previous studies by considering the dual challenge of carbon emissions and product degradation, particularly in growing SCs. The work by Hua et al. [22] and Dhanda et al. [28] is a foundation for understanding how carbon pricing can impact SC decisions, which is further integrated with trade-credit and growth dynamics.

Building on the previous major contributions, this study intends to support an integrated business-focused model by providing optimal integrated price, payment delay, and ordering policies for newborn growing goods. These strategies aim to optimize the integrated profit while considering power demand patterns and different carbon emission regulations. The gap in the emerging literature is addressed in the presented research by comparing its findings to previous research and offering contributions in

Table 1. Author contributions in the existing literature.

Authors	Imperfect Quality Items	Growing Items	Trade-Credit Policy	Demand	Carbon Emissions
Khalilpourazari et al. [4]		✓		Constant
Sebatjane and Adetunji [32]	✓	✓		Constant
Pourmohammad et al. [7]		✓		Constant
Mittal and Sharma [9]		✓	✓	Constant
Sarkar and Bhuniya [33]	✓			Variable
Sharma and Mittal [10]	✓	✓	✓	Constant
Alfares and Afzal [8]	✓	✓		Constant
Sebatjane and Adetunji [14]		✓		Price and freshness
Mahata et al. [18]			✓	Constant
Garai and Sarkar [34]				Fuzzy
Moon et al. [35]	✓			Variable	✓
Mahata et al. [17]			✓	Price sensitive
Mahapatra et al. [36]	✓			Selling and advertisement	✓
Hovelaque and Bironneau [25]				Carbon emissions	✓
Lin et al. [26]	✓			Constant	✓
Sarkar et al. [37]				Price	✓
Present study	✓	✓	✓	Price and time-dependent	✓

.

3. Nomenclature and Assumptions

This section provides notation used in the paper and associative assumptions to build the model.

3.1. Nomenclature

The following nomenclature is adopted in the presented model as given in

Table 2. Nomenclature.

Symbol	Description
Parameters
$P_s$	Per unit supplier’s selling cost ($/kg)
$O_r$	Retailer’s fixed ordering cost ($)
$F_r$	Retailer’s fixed transportation cost ($)
$I_r$	Per unit retailer’s inspection cost ($)
$H_r$	Per unit retailer’s holding cost ($/kg/year)
$V$	Per unit selling price of defective items ($)
$\gamma$	Fraction of defective items
$\alpha$	Scale parameter of the time-sensitive demand where $\alpha$∈ (−∞,∞) (kg/day)
$\beta$	Index of power demand pattern in customers’ demand where $\beta$> 0
$u$	Highest potential customers’ demand rate without impacts of price and time (kg/day)
$v$	Price-sensitive parameter in customer’s demand ($v$> 0)
$w$	Price index based on customer’s demand ($w$≥ 1)
$s1$	Highest possible selling price of each growing item ($/kg)
${\mathsf{\phi }}_1$	Supplier’s inventory cycle length (years)
${\mathsf{\phi }}_2$	Screening period (years)
$U_i$	Weight of each newborn growing item initially (kg)
$U_t$	Weight of each newborn growing item at any time $t$(kg)
$O_{s }$	Supplier’s fixed ordering cost ($)
$P_c$	Per unit supplier’s purchasing cost ($/kg)
$I_s$	Per unit supplier’s opportunity cost ($/kg)
$F_s$	Per unit feeding cost of newborn items ($/kg)
$M$	Retailer’s trade-credit period offered by the supplier (years)
$n$	Customer’s trade-credit period offered by the retailer (years)
$I_E$	Retailer’s earned interest (%/year)
$I_P$	Retailer’s paid interest (%/year)
$b$	Constant of integration (>0)
$\eta$	Asymptotic weight of each growing item when the feeding period tends to infinity (>0) (kg)
$\theta$	Growth rate (>0)
$\widehat{P}$	Carbon emissions from purchasing growing items (ton/kg/year)
$\widehat{O}$	Carbon emissions from ordering growing items (ton/kg/year)
$\widehat{H}$	Carbon emissions from holding growing items (ton/kg/year)
$\widehat{T}$	Carbon emissions from transporting growing items (ton/kg/year)
$\widehat{c}$	Cap for carbon emissions (ton/kg/year)
$\widehat{z_1}$	Tax on carbon emissions ($\mathrm{\$}/\mathrm{t}\mathrm{o}\mathrm{n}/\mathrm{k}\mathrm{g}$)
$\widehat{z_2}$	Market price of carbon emissions ($\mathrm{\$}/\mathrm{t}\mathrm{o}\mathrm{n}/\mathrm{k}\mathrm{g}$)
Decision Variables
$X$	Number of newborn items purchased by the supplier (integer)
$s$	Selling price of good quality items ($/kg)
$\mathsf{\phi }$	Retailer’s inventory cycle length (years)

.

3.2. Assumptions

• The supplier’s inventory techniques do not involve stock-outs.
• Carbon emissions are accounted for in all inventory activities, resulting in higher expenses due to various carbon regulations enforced by authorities.
• The customer demand structure, $D\left(t,s\right)$, is a nonlinear function of both selling price per weight unit and storage time, stated in a separable and additive manner. The demand function is represented as $D\left(t,s\right)=u-v{s}^w+\alpha \beta {\left({\displaystyle \frac{t}{\mathsf{\phi }}}\right)}^{\beta -1}$, where $u>0, v>0$, $w\ge 1, -\infty <\alpha <\infty , \beta >0$, and $s\le s1$. The highest selling price possible for each weight unit is taken to be $s1={\left({\displaystyle \frac{u}{v}}\right)}^{{\displaystyle \frac{1}{w}}}$.
• The credit period enables retailers to collect revenue from sales while earning interest on their capital without paying any interest expenses. This structure gives them more financial freedom and improves their overall profitability.

4. Mathematical Model

Consider that the supplier orders $X$ newborn growing items of a specific type from an outside supplier, each having an initial weight of $U_{\mathit{i}}$. The total initial weight of these items upon delivery is $X{U}_i$. These items are then fed and nurtured throughout the growth time ${\mathsf{\phi }}_1$, using a three-parameter logistic growth function

$$U\left(t\right)={\displaystyle \frac{\eta }{1+b{e}^{-\theta t}}}$$

(1)

where $U\left(t\right)$ represents each item’s weight at time $t$, $\eta >0$ is the asymptotic weight, $b>0$ is the integration constant, and $\theta >0$ is the growth rate.

When each item achieves the required weight $U_t$ after feeding and growing, it is slaughtered, signaling the end of the growing phase ${\mathsf{\phi }}_1$, after which it is sold to the retailer for further inspection, as shown in Figure 1.

Hence, Equation (1) yields

$${\mathrm{U}}_t=U\left({\mathsf{\phi }}_1\right)={\displaystyle \frac{\eta }{1+b{e}^{-\theta {\mathsf{\phi }}_1}}}$$

(2)

which gives the length of the growing period as

$${\mathsf{\phi }}_1=-{\displaystyle \frac{1}{\theta }}Log\left({\displaystyle \frac{\eta -U_t}{b{U}_t}}\right).$$

(3)

Following slaughter, the retailer commences the inspection with the initial inventory $X{U}_t$ units of weight after which the items are sold to the consumers. The differential equation below describes the on-hand inventory weight, $I\left(t\right)$, throughout the time $t\in \left[0, \mathsf{\phi }\right]$.

$$I^{\prime }\left(t\right)=-D\left(t,s\right)=-\left\{u-v{s}^w+\alpha \beta {\left({\displaystyle \frac{t}{\mathsf{\phi }}}\right)}^{\beta -1}\right\}, 0<t<\phi$$

where the auxiliary conditions are $I\left(0\right)=X{U}_t$ and $I\left(\mathsf{\phi }\right)=0$.

On solving the above equation, the on-hand inventory weight for time $t\in \left[0, \mathsf{\phi }\right]$ is given by

$$I\left(t\right)=\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left\{1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right\}, 0\le t\le \phi .$$

(4)

Hence, the initial inventory weight for consumption is given by

$$X{U}_t=I\left(0\right)={\int }_0^{\phi }D\left(t,s\right)dt=\left(\alpha +u-v{s}^w\right)\phi$$

(5)

which gives the retailer’s entire cycle length as

$$\phi ={\displaystyle \frac{\eta X}{\left(1+b{e}^{-\theta {\mathsf{\phi }}_1}\right)\left(\alpha +u-v{s}^w\right)}}$$

(6)

4.1. Supplier’s Profit Components

The supplier’s profit function components are calculated to understand how credit financing influences their overall profitability:

• Purchasing Cost: Since the supplier’s purchase cost is $P_c$ per unit weight of a newborn item, the total purchase cost is given by

$${PC}_s=P_{c}X{U}_i.$$

ii.

Ordering Cost: The supplier’s fixed ordering cost is given by

$${OC}_s=O_{s }.$$

iii.

Feeding cost: Since the supplier feeds and nurtures the newborn items for the time period $0$ to ${\phi }_1$, the total feeding cost is

$${FC}_s=F_{s}X{\int }_0^{{\phi }_1}U\left(t\right)dt=F_{s}X\left\{\eta {\phi }_1+{\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}.$$

iv.

Opportunity Cost: When a supplier offers a retailer a credit period, they face an opportunity cost in the form of lost interest income, that is, the supplier misses out on the opportunity to invest that money elsewhere, which is given by

$${OPC}_s=I_s{P}_c\phi M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right).$$

v.

Sales Revenue: Since the supplier earns $P_s$ per unit weight of a grown item, the total sales revenue is

$${SR}_s=P_s\left(\alpha +u-v{s}^w\right)\phi .$$

Hence, the overall profit of the supplier is given by

$${PR}_s={SR}_s-\left({PC}_s+{ OC}_s+{ FC}_s+{OPC}_s\right).$$

$$\begin{gathered} {PR}_s\left(X,s,\phi \right)=P_s\left(\alpha +u-v{s}^w\right)\phi -P_{c}X{U}_i-O_{s }-I_s{P}_c\phi M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right) \\ -{ F}_{s}X\left\{\eta {\phi }_1+{\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\} \end{gathered}$$

(7)

4.2. Retailer’s Profit Components

The retailer’s profit function components include revenue from good quality and imperfect items and the total costs associated with acquiring them.

• Purchasing Cost: Since the retailer’s purchase cost is $P_s$ per unit weight of a grown item, the total purchase cost is given by

$${PC}_R=P_s\left(\alpha +u-v{s}^w\right)\phi .$$

ii.

Ordering Cost: The retailer’s fixed ordering cost is given by

$${OC}_R=O_r.$$

iii.

Transportation Cost: Since the retailer has to transport the items from the supplier at the cost of $F_r$ per unit weight, the total transportation cost is given by

$${TC}_R=F_r\left(\alpha +u-v{s}^w\right)\phi .$$

iv.

Inspection Cost: The retailer starts inspecting the items as the order is received from the supplier at the rate of $I_r$ per unit weight; hence, the total inspection cost is

$${IC}_R=I_r\left(\alpha +u-v{s}^w\right)\phi .$$

v.

Holding Cost: The retailer holds the inventory at the rate of $H_r$ per unit weight, until inspection is completed, and items are sold to the customers; hence, the holding cost is

$${HC}_R=H_r{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt.$$

vi.

Carbon Emissions Cost: The retailer bears the cost of producing emissions through various activities involving purchasing ($\widehat{P}$), ordering ($\widehat{O}$), holding ($\widehat{H}$), and transporting ($\widehat{T}$) the items from the supplier. Hence, the total carbon emissions cost will be

$$CE=\widehat{P}\left(\alpha +u-v{s}^w\right)\phi +\widehat{O}+\widehat{H}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt+\widehat{T}\left(\alpha +u-v{s}^w\right)\phi .$$

vii.

Sales Revenue: The retailer makes revenue by selling good quality items at $s$ per unit item and imperfect quality items at a discounted price of $V$ per unit item. Hence, the total sales revenue is

$${SR}_R=\left(s\left(1-\gamma \right)+V\gamma \right){\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt.$$

Hence, the overall retailer’s profit is

$${PR}_R={SR}_R-\left({PC}_R+{OC}_R+{TC}_R+{IC}_R+{HC}_R+CE\right).$$

$$\begin{gathered} {PR}_R\left(s,\phi \right)=\left(s\left(1-\gamma \right)+V\gamma \right){\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)\phi -O_r- \\ {F}_r\left(\alpha +u-v{s}^w\right)\phi -I_r\left(\alpha +u-v{s}^w\right)\phi -H_r{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt- \\ (\widehat{P}\left(\alpha +u-v{s}^w\right)\phi +\widehat{O}+\widehat{H}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\widehat{T}\left(\alpha +u-v{s}^w\right)\phi ) \end{gathered}$$

(8)

4.3. Trade-Credit Policy

Considering that the supplier provides the retailer a credit period of $M$ years to settle payments, while the retailer further offers $n$ years of credit period to the customer, the following cases of trade-credit are taken as

• $0\le N\le {\phi }_2\le \phi \le M$
• $0\le {\phi }_2\le N\le \phi \le M$
• $0\le {\phi }_2\le \phi \le N\le M$
• $0\le N\le {\phi }_2\le M\le \phi$
• $0\le {\phi }_2\le N\le M\le \phi$
• $0\le N\le M\le {\phi }_2\le \phi$.

• Case 1: $0\le N\le {\phi }_2\le \phi \le M$

Figure 2 depicts how the buyer receives interest from $0$ to $M$ based on the sales revenue earned. However, in this case, the buyer does not pay any interest to the seller for unsold items since the cycle length is shorter than the credit period, and the account is settled by time $M$.

Interest Earned: $\left(s\left(1-\gamma \right)+V\gamma \right)I_E{\int }_n^{\phi }{\int }_0^t(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1})dxdt+{sI}_E(M-\phi ){\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt$.

Interest Paid: $0$.

The combined overall profit per unit time is calculated as

$${TPU}_1(X,s,\phi )={\displaystyle \frac{{PR}_S(X,s,\phi )+{PR}_R(s,\phi )}{\phi }}.$$

$$\begin{gathered} {TPU}_1\left(X,s,\phi \right)=P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-{ I}_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right)dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt+\widehat{T}\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{I}_E{\int }_n^{\phi }{\int }_0^t\left(\left(u-v{s}^w\right)+ \\ \alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1}\right)dxdt+{\displaystyle \frac{{sI}_E\left(M-\phi \right)}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt \end{gathered}$$

(9)

• Case 2: $0\le {\phi }_2\le N\le \phi \le M$

Figure 3 depicts how the buyer receives interest from $0$ to $M$ based on the sales revenue earned. However, in this case, the buyer does not pay any interest to the seller for unsold items since the cycle length is shorter than the credit period, and the account is settled by time $M$.

Interest Earned: $\left(s\left(1-\gamma \right)+V\gamma \right)I_E{\int }_n^{\phi }{\int }_0^t(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1})dxdt+{sI}_E(M-\phi ){\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt$.

Interest Paid: $0$.

The combined overall profit per unit time is calculated as

$$\begin{gathered} {TPU}_2\left(X,s,\phi \right)={ P}_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-I_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\widehat{ T}\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{I}_E{\int }_n^{\phi }{\int }_0^t\left(\left(u-v{s}^w\right)+ \\ \alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1}\right)dxdt+{\displaystyle \frac{{sI}_E\left(M-\phi \right)}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt. \end{gathered}$$

(10)

• Case 3: $0\le {\phi }_2\le \phi \le N\le M$

Figure 4 depicts how the buyer receives interest from $0$ to $M$ based on the sales revenue earned. However, in this case, the buyer does not pay any interest to the seller for unsold items since the cycle length is shorter than the credit period, and the account is settled by time $M$.

Interest Earned: ${sI}_E(M-\phi ){\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta })dt$.

Interest Paid: $0$.

The combined overall profit per unit of time is calculated as

$$\begin{gathered} {TPU}_3\left(X,s,\phi \right)={ P}_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-I_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\widehat{ T}\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{{sI}_E\left(M-\phi \right)}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+ \\ \alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt. \end{gathered}$$

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• Case 4: $0\le N\le {\phi }_2\le M\le \phi$

Figure 5 shows that the buyer receives interest on the money received from selling the items between time $0$ and $M$. However, the buyer continues to possess some items after $M$ and must pay interest to the vendor on those items till $\phi$. Hence, the buyer must manage the interest earned simultaneously with the interest paid to achieve a balanced financial outcome.

Interest Earned: $\left(s\left(1-\gamma \right)+V\gamma \right)I_E{\int }_n^M{\int }_0^t(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1})dxdt$.

Interest Paid: $P_s{I}_P{\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt$.

The combined overall profit per unit of time is calculated as

$$\begin{gathered} {TPU}_4(X,s,\phi )={ P}_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-I_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{I}_E{\int }_n^M{\int }_0^t\left(\left(u-v{s}^w\right)+ \\ \alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1}\right)dxdt-{\displaystyle \frac{P_s{I}_P}{\phi }} {\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt. \end{gathered}$$

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• Case 5: $0\le {\phi }_2\le N\le M\le \phi$

Figure 6 depicts how the buyer earns interest on the money received from selling items between time $0$ and $M$. However, the buyer continues to retain certain products after $M$ and must pay interest to the vendor until $\phi$.

Interest Earned: $\left(s\left(1-\gamma \right)+V\gamma \right)I_E{\int }_n^M{\int }_0^t(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1})dxdt$.

Interest Paid: $P_s{I}_P{\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt$.

The combined overall profit per unit of time is calculated as

$$\begin{gathered} {TPU}_5(X,s,\phi )={ P}_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-I_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\widehat{ T}\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{I}_E{\int }_n^M{\int }_0^t\left(\left(u-v{s}^w\right)+ \\ \alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1}\right)dxdt-{\displaystyle \frac{P_s{I}_P}{\phi }} {\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt. \end{gathered}$$

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• Case 6: $0\le N\le M\le {\phi }_2\le \phi$

Figure 7 depicts how the buyer earns interest on the money received from selling items between time $0$ and $M$. However, the buyer continues to retain certain products after $M$ and must pay interest to the vendor until $\phi$.

Interest Earned: $\left(s\left(1-\gamma \right)+V\gamma \right)I_E{\int }_n^M{\int }_0^t(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1})dxdt$.

Interest Paid: $P_s{I}_P{\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt$.

The combined overall profit per unit time is calculated as

$$\begin{gathered} {TPU}_6(X,s,\phi )={ P}_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{P_{c}X{U}_i}{\phi }}-{\displaystyle \frac{O_{s }}{\phi }}-I_s{P}_{c}M\left(1-\gamma \right)\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{{ F}_{s}X}{\phi }}\left\{\eta {\phi }_1+ \\ {\displaystyle \frac{\eta }{\theta }}\left(Log\left[1+b{e}^{-\theta {\phi }_1}\right]-Log\left[1+b\right]\right)\right\}+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{\int }_0^{\phi }\left(\left(u-v{s}^w\right)+\alpha \beta {\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta -1}\right)dt-P_s\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{O_{r }}{\phi }}- \\ {F}_r\left(\alpha +u-v{s}^w\right)-I_r\left(\alpha +u-v{s}^w\right)-{\displaystyle \frac{H_{r }}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi \left(1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }\right))dt-(\widehat{P}\left(\alpha +u-v{s}^w\right)+ \\ {\displaystyle \frac{\widehat{O}}{\phi }}+{\displaystyle \frac{\widehat{H}}{\phi }}{\int }_0^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt+\widehat{ T}\left(\alpha +u-v{s}^w\right))+{\displaystyle \frac{\left(s\left(1-\gamma \right)+V\gamma \right)}{\phi }}{I}_E{\int }_n^M{\int }_0^t\left(\left(u-v{s}^w\right)+ \\ \alpha \beta {\left({\displaystyle \frac{x}{\phi }}\right)}^{\beta -1}\right)dxdt-{\displaystyle \frac{P_s{I}_P}{\phi }} {\int }_M^{\phi }(\left(u-v{s}^w\right)\left(\phi -t\right)+\alpha \phi (1-{\left({\displaystyle \frac{t}{\phi }}\right)}^{\beta }))dt. \end{gathered}$$

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4.4. Implementing Carbon Emissions Policies

• A carbon tax policy imposes additional expenditures on the SC by imposing taxes based on the carbon emissions produced. By representing the problem numerically, where $\widehat{z_1}$ represents the tax amount levied for each unit of carbon released per unit of time, companies can better understand the financial implications and develop strategies to mitigate these costs while promoting sustainable practices within their operations. Hence,

$${ TP}_i\left(X,s,\phi \right)={\displaystyle \frac{{PR}_S\left(X,s,\phi \right)+{PR}_R\left(s,\phi \right)+{IE}_i-{IP}_i-\widehat{z_1}CE}{\phi }} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} 1\le i\le 6.$$

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ii.

A carbon cap policy imposes critical restrictions on SCs by limiting total carbon emissions. The produced emissions throughout the SC must stay below this limit. Using $\widehat{c}$ as the maximum permissible carbon emissions per unit of time, ${TP}_i\left(X,s,\phi \right)$ may be represented analytically as

$${TP}_i\left(X,s,\phi \right)={\displaystyle \frac{{PR}_S\left(X,s,\phi \right)+{PR}_R\left(s,\phi \right)+{IE}_i-{IP}_i}{\phi }} \mathrm{where} 1\le i\le 6.$$

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iii.

A carbon cap-and-trade policy effectively allows companies to trade carbon credits, encouraging businesses to innovate and find ways to lower their emissions. Companies that reduce their emissions below the cap can sell their extra credits to other businesses failing to meet their targets. This allows additional money because these credits may be sold at current market pricing. If a company exceeds its emissions limit, it must purchase more credits from the market to offset the excess emissions. By defining $\widehat{c}$ as the emissions cap per unit time and $\widehat{z_2}$ as the market price for selling carbon emissions, $TP\left(X,s,\phi \right)$ may be mathematically expressed as

$${TP}_i\left(X,s,\phi \right)={\displaystyle \frac{{PR}_S\left(X,s,\phi \right)+{PR}_R\left(s,\phi \right)+{IE}_i-{IP}_i-\widehat{z_2}CE+\widehat{z_2}\widehat{c}}{\phi }} \mathrm{where} 1\le i\le 6.$$

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5. Solution Procedure

The necessary conditions required for the maximization of ${TPU}_i\left(X,s,\phi \right)$ in each case is defined by ${\displaystyle \frac{\mathsf{\partial }\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }X}}=0, {\displaystyle \frac{\mathsf{\partial }\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }s}}=0,$ and ${\displaystyle \frac{\mathsf{\partial }\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }\phi }}=0$, which provide the optimal values for $X,$ $s$, and $\phi$. This objective is achieved by following a specific methodology from Mahato et al. [38].

• Differentiate ${TPU}_i\left(X,s,\phi \right) \left(1\le i\le 6\right)$ partially with respect to $s$ and $\phi$ for any fixed $X$, and then put it equal to $0$ to optimize the variables to maximize ${TPU}_i$.
• However, it is impossible to analytically find the values of $s$ and $\phi$. Therefore, the values are calculated in the following section through a numerical example.
• The second optimality condition to demonstrate the concavity of ${TPU}_i\left(X,s,\phi \right)$ in each case is given by ${\displaystyle \frac{{\mathsf{\partial }}^2\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }{X}^2}}\le 0, {\displaystyle \frac{{\mathsf{\partial }}^2\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }{s}^2}}\le 0,$ and ${\displaystyle \frac{{\mathsf{\partial }}^2\left[{TPU}_i\left(X,s,\phi \right)\right]}{\mathsf{\partial }{\phi }^2}}\le 0$.

(Refer to Appendix A.)

6. Numerical Example

This example applies the parametric values from Mahato et al. [38] and Sebatjane [39] and the extra needed data in order to comply with the given model: F_r = $1, V = 40, η = 6.87 kg, b = 120, U_i = 0.064 kg, U_t = 1.5 kg, X = 148, θ = 0.12, I_s = $0.25 kg, u = 243 kg/day, v = 1, w = 1.25, α = 10 kg/day, β = 2, γ = 0.02, P_s = $24/kg, H_r = $2.5/kg/year, O_r = $300, $\widehat{P}$ = 0.5 ton/kg/year, $\widehat{O}$ = 0.5 ton/kg/year, $\widehat{H}$ = 0.4 ton/kg/year, $\widehat{T}$ = 0.04 ton/kg/year, I_E = 0.05, I_P = 0.08, P_c = $12/kg, O_s = $1000, F_s = $0.5, $\widehat{z1}$ = $1.25 ton/kg, $\widehat{c}$ = 2000 ton/kg/year, $\widehat{z2}$ = $2.5 ton/kg.

Table 3. Comparative analysis of decision variables and TPU.

Cases	${\mathit{s}}^{\mathit{*}}$ ($/kg)	${\mathit{\phi }}^{\mathit{*}}$ (years)	${\mathit{T}\mathit{P}}_{\mathit{R}}$ ($/cycle)	${\mathit{T}\mathit{P}}_{\mathit{S}}$ ($/cycle)	$\mathit{T}\mathit{P}\mathit{U}$ ($/cycle)
1	66	3.6	9475.36	2865.67	3428.06
2	47	3.03	8168.02	6285.25	4770.06
3	46.5	2.21	8352.2	7885.03	4689.39
4	46	2.8	6226.67	6280.85	4466.97
5	47	2.9	6329.27	6367.68	4378.26
6	46	2.85	5715.67	6587.15	4316.76

presents an assessment of all six cases, highlighting how $s^{*}$ and ${\phi }^{*}$ affect $TPU$. By understanding the correlation between these variables and integrated profit, SC managers can make educated decisions that improve their efficiency and profitability. It can be seen from Figure 8 and Figure 9 that Case 2 is the most profitable, where $TPU$ reaches its maximum value ($4770.06/year) when the optimal values of $s^{*}=\$47$/kg and ${\phi }^{*}=3.03$ years.

Table 4. Integrated profit under different carbon emission policies.

Policy	$\widehat{{\mathit{z}}_1}$ $(\mathbf{\$}/\mathbf{t}\mathbf{o}\mathbf{n}/\mathbf{k}\mathbf{g})$	$\widehat{\mathit{c}}$ $(\mathbf{t}\mathbf{o}\mathbf{n}/\mathbf{k}\mathbf{g}/\mathbf{y}\mathbf{e}\mathbf{a}\mathbf{r}$)	$\widehat{{\mathit{z}}_2}$ $(\mathbf{\$}/\mathbf{t}\mathbf{o}\mathbf{n}/\mathbf{k}\mathbf{g}$)	$\mathit{T}\mathit{P}\mathit{U}$$(\mathbf{\$}$/cycle)
Carbon tax	1.25	0	0	4732.28
Carbon cap	0	2000	0	4921.15
Carbon cap and trade	0	2000	2.5	6193.58

indicates that combining a cap with the flexibility of trading licenses yields the most profitable or efficient results. Imposing a strict cap on emissions, even without trading, is more successful than directly taxing emissions. It can be said that charging emissions without imposing restrictions or allowing for market-based flexibility is less effective at maximizing profit.

Table 5. Comparison of integrated profit ($\mathrm{\$}$/cycle) for all cases with carbon emissions policies.

Policy	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
Carbon tax	3406.99	4732.28	4656.68	4429.83	4341.35	4279.3
Carbon cap	3512.35	4921.15	4820.29	4615.54	4525.9	4466.71
Carbon cap and trade	4690.52	6193.58	5492.71	6029.83	5880.94	5846.27

represents different cases under which various restrictions on carbon emissions are evaluated. The carbon cap-and-trade policy provides the highest values across all scenarios, with a particularly noticeable rise in Case 2 ($\mathrm{\$}$6193.58/cycle). Even the smallest value for this policy ($\mathrm{\$}$5846.27/cycle in Case 6) exceeds the greatest values of the other two policies, implying that it is the most resilient and robust policy option among those analyzed in this study. For all three policies, Case 2 appears to provide the best results, which might imply that the conditions in Case 2 are particularly favorable for these policies.

7. Sensitivity Investigation

A sensitive investigation is undertaken based on the above numerical example to study the impact of certain parameters ($\gamma , \alpha , \beta , \widehat{H}, \widehat{O}, \widehat{P}, \widehat{T}, I_e$) on the selling price per unit weight $\left(s^{*}\right)$, cycle length (${\phi }^{*}$), and $TPU.$

Table 6. Impact of changes in key parameters on $TPU$.

Parameters	% of Change	${\mathit{s}}^{\mathit{*}}$	${\mathit{\phi }}^{\mathit{*}}$	$\mathit{T}\mathit{P}\mathit{U}$
$\gamma$	−40	47.2366	3.04211	6034.30
	−20	47.1804	3.04149	6032.24
	+20	47.059	3.03654	6030.03
	+40	46.9939	3.03472	6029.01
$\alpha$	−40	46.4436	3.07327	6009.49
	−20	46.7709	3.04818	6103.38
	+20	47.3542	3.01862	6281.29
	+40	47.6131	2.98854	6380.35
$\beta$	−40	47.1967	3.06737	6187.41
	−20	47.1371	3.05138	6188.05
	+20	47.1018	3.02744	6190.86
	+40	47.0878	3.0188	6192.02
$\widehat{H}$	−40	47.0408	3.08359	6244.86
	−20	47.0788	3.06071	6216.89
	+20	47.1553	3.05991	6136.41
	+40	47.1938	2.9952	6135.44
$\widehat{O}$	−40	47.1181	3.03826	6189.61
	−20	47.1181	3.0383	6189.47
	20	47.1182	3.0384	6189.25
	40	47.1182	3.0385	6189.11
$\widehat{P}$	−40	47.0338	3.03662	6254.83
	−20	47.0760	3.03749	6222.06
	+20	47.1604	3.03923	6156.70
	+40	47.2025	3.0401	6124.10
$\widehat{T}$	−40	47.1114	3.03822	6194.58
	−20	47.1148	3.03829	6191.97
	+20	47.1215	3.03843	6186.74
	+40	47.1249	3.0385	6184.12
$I_e$	−40	47.3184	3.18697	5853.21
	−20	47.2115	3.10584	6023.54
	+20	47.0356	2.98132	6351.47
	+40	46.9618	2.93247	6510.53

offers an insightful overview of the analysis results.

Managerial Implications

• Figure 10 shows that, as $\gamma$ changes, the relationship between selling price and cycle length with $TPU$ becomes nonlinear. Increasing the selling price usually raises $TPU$, but the impact reduces as prices rise, demonstrating an optimal pricing range. Similarly, while a longer cycle length initially increases $TPU$, it eventually decreases $TPU$, implying that there is an optimal shorter cycle length for maximizing profits.
• Figure 11 shows that as $\alpha$ increases from −40% to 40%, increasing the selling price usually raises $TPU$, but the impact reduces as prices rise. Similarly, while a longer cycle length initially increases $TPU$, it eventually decreases $TPU$, implying that there is an optimal shorter cycle length for maximizing profits.
• Figure 12 shows that as $\beta$ increases from −40% to 40%, increasing the selling price decreases $TPU$, but the impact reduces as prices reduce. Similarly, while a longer cycle length initially decreases $TPU$, it eventually increases $TPU$, implying that there is an optimal shorter cycle length for maximizing profits.
• Figure 13 shows that as $\widehat{\mathrm{H}}$ increases from −40% to 40%, increasing the selling price decreases $TPU$, while decreasing the cycle length reduces $TPU$.
• Figure 14, Figure 15 and Figure 16 show that as there is a change in $\widehat{\mathrm{O}}, \widehat{\mathrm{P}},$ and $\widehat{\mathrm{T}}$ from −40% to 40%, both the selling price and cycle length have a negative impact on $TPU$. However, as the selling price and cycle length increase, $TPU$ eventually decreases.
• Figure 17 shows that both the selling price and cycle length have a negative impact on $TPU$ as $I_e$ increases, which implies that a decrease in the selling price and cycle length leads to an increase in $TPU$.

8. Conclusions

This study examined the interactions between suppliers and retailers, specifically how their relationships impact the management of stocks along with pricing decisions. This study intended to provide insights into how varying levels of trade-credit might influence a company’s financial dynamics and operational strategies. The emphasis on items that grow and are then processed highlights the complexity of managing perishable or time-sensitive inventories, which necessitates careful planning and forecasting. This paper looked at how retailers manage their stock levels while balancing customer demand and supplier restrictions, especially in a volatile market environment, and the influence of various carbon restrictions on their integrated profit. The mathematical model highlighted how two primary factors, the per unit weight selling price and the duration for which items are stored, impact product demand. This approach recognized that pricing strategies and storage times substantially impact customer behavior and purchase decisions. The model employed a power function to describe how demand fluctuates in response to the selling price and storage duration. This implied that minor changes in either aspect might result in varied amounts of demand, providing flexibility in predicting and planning. The model might simulate various customer responses by incorporating different scenarios within the demand function. This agility was critical for organizations operating in volatile market situations. Based on the analytical results, an algorithm was presented to balance the optimal values of the number of newborn items that are brought to the market based on the expected demand, the best price for each unit weight of the items, and the time between product releases or restocking periods to optimize sales while ensuring that clients have timely access to the items.

The algorithm’s effectiveness was shown using a numerical example, and various insights are presented into how optimal techniques change with variations in system parameters. As $\gamma$ increased from −40 to 40%, it led to a slight decrease in $TPU$, $s^{*}$, and ${\phi }^{*}$. However, an increase in $\alpha$ led to a considerable increase in $TPU$ from $\mathrm{\$}$6009.49/cycle to $\mathrm{\$}$6380.35/cycle, while ${\phi }^{*}$ decreased and $s^{*}$ increased. Changes in $\beta$ had a negligible effect on $TPU$, remaining relatively stable around $\mathrm{\$}$6190/cycle, whereas $s^{*}$ and ${\phi }^{*}$ exhibited a modest decrease. $TPU$ dropped from $\mathrm{\$}$6244.86/cycle to $\mathrm{\$}$6135.44/cycle as $\widehat{H}$ increases, while it remained stable across different levels of $\widehat{O}$. $TPU$ declined from $\mathrm{\$}$6254.83/cycle to $\mathrm{\$}$6124.10/cycle as $\widehat{P}$ increased, while $s^{*}$ increased slightly. $s^{*}$ and ${\phi }^{*}$ varied minimally as $\widehat{T}$ changed, while leading to a decrease in $TPU$. As $I_e$ increased by 40%, $s^{*}$ and ${\phi }^{*}$ decreased while $TPU$ rose significantly from $\mathrm{\$}$5853.21/cycle to $\mathrm{\$}$6510.53/cycle. Overall, parameters $\alpha$ and $I_e$ had the biggest impact on total profit per unit time ($TPU$), with larger values resulting in significant increases in profitability.

Future Implication

While this research focuses on a three-echelon SCM, future studies can extend this model to multi-echelon systems, considering additional layers such as wholesalers or transportation providers. Future research can incorporate dynamic and uncertain demand models, especially considering the volatility in the pricing and demand for growing products. Future research may look into distinct growth curves, feeding functions, deterioration patterns, and cash flow [40]. Furthermore, adding backordering and unpredictable demand into the assumptions may be beneficial. Advertisement policies [41] for deteriorating nature-type products [42] can be more effective for further studies, along with managing any defective products within the system using additional investment [6]. Another potential area of research is the study of growth within a three-level SCM [43].