Optimizing Inventory for Imperfect and Gradually Deteriorating Items Under Multi-Level Trade Credit in a Sustainable Supply Chain

Abstract

Reducing carbon emissions is of immense interest to most modern organizations striving for sustainability. Effective inventory management is crucial for achieving resource optimization and minimizing environmental impact. Very little work has been conducted up to this point on slowly declining, low-quality products with multi-level trade credit rules under the influence of carbon emissions. In this study, an inventory model is tailored specifically for imperfect and gradually deteriorating products with a multi-level trade credit policy. Further, the impact of carbon emissions on the retailer’s ordering strategies is also considered. To determine the optimal policy for supply chain partners, three trade credit instances with seven subcases are taken into consideration. To choose the best scenario out of ten cases, an algorithm is also developed. The model’s validity is illustrated through a numerical experiment and sensitivity analysis. This study is an innovative approach to balancing economic trade credit policy in sustainable supply chain management.

1. Introduction

Balancing economic and environmental goals is a significant challenge in practice. Business organizations are continuously undertaking this balancing act to offer competitive products while ensuring lower carbon emissions. Incorporating carbon emission costs and trade credit policies in inventory management results in efficiency.

In the contemporary business landscape, stakeholders must collaborate to enhance processes to reduce carbon emissions while providing financial support to achieve environmental and economic sustainability. While holding lower inventory levels is often perceived as a strategy for economic sustainability, it can lead to stockouts and dissatisfied customers. Inventory management strategies need to carefully balance these dynamics, particularly when dealing with goods categorized by their shelf life. Extended usability allows non-perishable items to satisfy demand over longer periods of time without suffering appreciable quality deterioration. Perishable commodities, on the other hand, pose difficulties because their quality and value decline with time.

Perishable goods can be further classified into two categories: those with constant utility throughout their lifespan, such as blood (e.g., a static 21-day usability period) and certain medications, and those with utility that decreases exponentially, such as fruits and fish. This distinction is critical for devising effective inventory management strategies that align with sustainability goals while ensuring customer satisfaction [1]. As a result, research is shifting focus toward enhancing the inventory models for products with deteriorating quality [2,3].

Interestingly though, the collaborative environment of modern business is not conducive for it to approach sustainability as a single entity. Cooperation between different business players and their mutual financial understanding is essential for economic sustainability. Such collaboration can also improve profits and financial benefits. Marketing practices include suppliers granting retailers an acceptable payment delay and retailers granting customers an acceptable payment delay, along with the settlement of any outstanding balances within a predetermined time frame known as a trade credit period [4]. The ecological dimensions with returnable transport items and remanufacturing were evaluated by Ref. [5]. This policy ensures an efficient trade balance, maximizing profits for both trade credit policy partners. The trade credit also makes money in interest and revenue from items sold while being able to protect the buyer in case of insolvency from the retailer. It is in the seller’s best interest not to receive prepaid payments or to pay interest within a given period. The final flow of inventory payment can be illustrated as follows: the supplier receives full value for their products, for example, the supplier is a business that provides inventory to the buyer in exchange for cash.

The supplier frequently collects interest on their cash balance [6]. The supplier pays out some of their interest to intermediate sellers on their cash balance. The intermediate sellers then pay out. If a person only makes minimum payments on their balance and does not pay off the entire account balance every month, there will be interest charges for all unpaid interest. If a person pays off 90% of their balance before the expiration date and only owes 10% inside the boundaries of the allowable holdup period, then they will not be billed with any interest. An increasing focus on sustainable development and environmental concerns make inventory management a major research topic. In order to provide retailers with the best ordering methods under multi-level trade credit regulations while lowering the cost of carbon emissions, an inventory model is therefore suggested. With the use of this modeling, businesspeople can reduce their carbon footprint and increase their profits.

A model has been created that focuses on the inventory management of a retailer by optimizing the ordering quantity and time, enabling the retailer to smoothly run their business. This model focuses on trade credit policy between the supply chain partners. By this method, the total holding cost per unit item will get reduced; this will maximize the total profit of the retailers.

This study introduces a novel inventory model integrating multi-level trade credit policies for imperfect and deteriorating products to optimize profitability and sustainability. Unlike traditional models, it incorporates flexible credit structures based on product quality, improving cash flow and reducing financial risk. By minimizing excess inventory and waste, the model aligns economic and environmental goals, lowering carbon emissions. It provides a quantitative framework to balance trade credit, deterioration, and profitability. Validated through numerical experiments and sensitivity analysis, it offers a practical solution for sustainable inventory management. This innovative approach enhances decision making in modern supply chains.

All of this basic foundation is briefly covered in Section 1, and then the literature of the model based on the current model is presented in Section 2. Notations and assumptions that are used in the model are presented in Section 3. Section 4 completes the mathematical portion, and Section 5 deals with the solution procedure. Section 6 provides an explanation of the numerical examples used to support the correctness of this concept. A sensitivity analysis presented in Section 7, and Section 8 includes the final reflections.

2. Literature Review

The literature is related to the following points: carbon emission in the inventory mode trade credit effect on the inventory model and imperfect products in the inventory models.

2.1. Carbon Emission in Inventory Models

Carbon emission happens because of the storage and transporting of items. Carbon emissions occur because of the process of storing and transporting deteriorating products. Emissions in the warehouse are based on the inventory, as well as how much energy is used per unit. The disposal of items that are no longer needed is also a major contributor to carbon emission. For this problem, an inventory model is generated to balance global effects to meet the requirements without affecting the environment. Refs. [7,8] generated a model in which the lead time is taken as a stochastic incorporation with the production of defective products with a credit financing scheme. Ref. [9] developed a sustainability issue with a trade credit policy in their model. They have also worked on minimizing the greenhouse gas emission. Ref. [10] collaborated upon the carbon emission of defective products in their inventory model. Ref. [11] optimized the shipment amount in their model. They have also worked on the defective production rate. Ref. [12] generated a multi-stage formation model for defective items with a reducing carbon emission. Ref. [13] worked on reducing carbon output with the impact of credit financing on a multi-echelon inventory model. Ref. [14] generated a model using environmental and social strategies for the consumption of future energy. A sustainable inventory model was generated by [15]. This model worked on reducing carbon emission and. This model is also worked on non-instantaneous deteriorating products governed by a multi-level credit term policy.

2.2. Non-Instantaneous Degrading Items in Inventory Management Models

In traditional inventory models, the instantaneous deterioration of products upon delivery is often assumed. However, real-world scenarios reveal a delay between receipt and decay. This lag, known as non-instantaneous deterioration, has gained attention since 2006. Ref. [16] designed the structure of the inventory system allowing delayed payments to be available to benefit such products. Ref. [17] proposed a price-dependent demand model under trade credit facilities for non-instantaneous decaying commodities. Recent studies expanded on this concept [18], creating a production model incorporating region-specific, population-driven, and price-based demand. Ref. [19] introduced a multilevel trade credit policy for non-instantaneous deterioration, considering promotional and selling price influences on demand. In the case of non-instantaneously degrading items with a cost and time demand based on unpredictable conditions, Ref. [20] developed a two-warehouse EOQ model.

2.3. Trade Credit in Inventory Models

Delayed payments serve companies with a better access to the capital, and in such a process, they can empower the suppliers and incentivize a greater order size. Therefore, in this way, this policy ensures continuous financing. The theory of credit financing is different from business to business.

Research on credit financing inventory models has evolved significantly. Ref. [21] laid the groundwork by developing a model that ignored the distinction between purchase cost and selling price. Subsequent studies expanded on this concept. Ref. [22], for example, presented a two-level credit financing strategy for non-immediately degrading commodities. A model for degrading items with a two-level credit timeframe was created by [23], taking into account finite time horizons and stock-dependent demand. Ref. [24] made further progress by establishing the best policies under allowable payment delays for degrading products with multivariate demand rates. Optimal trade credit policies for perishable goods were investigated by [25], taking into consideration an imperfect supply and demand that is dependent on stocks. Additionally, they presented the idea of a twofold wait time as a way to reward retailers. Ref. [26] looked at situations in which suppliers are paid a portion of the whole order amount right away and the remaining amount after a predetermined amount of time. This idea was expanded upon by Ref. [27] to include multi-level trade credit, in which suppliers provide merchants grace periods, which retailers subsequently give to customers. A model with variable degradation under credit financing was developed by Ref. [28]. The joint trade credit problem for deteriorating inventory systems presented by Ref. [29] and the model for determining the economic order quantity with flexible trade credit put forth by Ref. [30] are recent contributions. A model for degrading items using advance payment methods was proposed by Ref. [31]. A model for credit financing policy was created by Ref. [32] in a two-stockroom setting with an ambiguous deterioration and demand modeled by the Weibull distribution.

2.4. Imperfect Within Inventory Models

During the manufacturing process, most items are manufactured with certain imperfections. The retailer has to deliver a good quality of items; therefore, it is important to identify this imperfection via an inspection process. In fact, researchers should focus on the imperfect quality of items as it is not possible to produce all goods of a perfect quality. Furthermore, this problem has been focused upon by many academicians in the past years.

Ref. [33] laid the groundwork with an inventory system aimed at establishing optimal ordering guidelines for defective items. Imperfect production processes were later accounted for by Ref. [34] through a generated model. Ref. [35] addressed a problem where item demand depends on production units, incorporating a screening process to eliminate defective items. Ref. [36] modified this model to include shortage situations and backordering costs. As the area continues to evolve, Ref. [37] developed an inventory model that included learning effects for damaged products. Ref. [38] examined the usage of two storage facilities for damaged commodities after developing an inventory model for credit financing schemes in [39]. A multiple-production stock management model designed especially for damaged products was developed by Ref. [40]. Ref. [41] presented an inventory model for low-quality products that took shortages into consideration, leading to the creation of an inventory model for defective objects. A multi-repository model with shortages and discount plans for defective commodities was most recently introduced by Ref. [42].

3. Problem Description, Notation, and Assumptions

3.1. Problem Description

This research addresses the issue of multi-level trade credit policies in the context of non-instantaneously degrading products, with the goal of reducing the carbon emissions associated with storage. Recognizing the inherent challenges in achieving 100% defect-free production, the proposed model accommodates imperfect items and effectively explores strategies for managing such scenarios. The study investigates the interrelations among suppliers, retailers, and customers, aiming to optimize the retailer’s profit and order quantities. By integrating environmental considerations and operational efficiency, this research offers a comprehensive framework for sustainable inventory management in supply chains.

3.2. Notations

This section contains all the notations which are used in this model (

Table 1. Notations.

Symbols	Description
Parameters
$I_i\left(t\right), i=\mathrm{1,2},3$	Inventory level for time t
$y$	Order level per cycle (units)
$D$	Demand rate (unit/years)
$A$	Ordering cost per order (in dollars)
$A^{\prime }$	Cost of carbon emissions while placing an order
$c$	Purchase price (in dollars) of an item from the retailer
$c^{\prime }$	Cost of carbon emissions when a store purchases a unit
$v$	Pricing each item for sale (in dollars) v>c
$\beta$	$\mathrm{Deterioration} \mathrm{rate} (0\le \beta \le 1$) (units)
$h$	Holding costs, excluding interest charges, for objects held per cycle (per unit annually)
$h^{\prime }$	Cost of carbon emissions per item for each cycle of storing items
$I_e$	Interest earned (per dollar per year)
$I_p$	Interest charge (per dollar per year)
$s$	Inspection cost per unit (in dollars)
$p$	Percentage of defective items in y (units)
$x$	Inspection rate (per unit per unit time)
$M$	Delayed payment offered by the supplier
$N$	Delayed payment offered by the retailer
$t_1$	Screening time (in years)
$t_d$	No deterioration time (in years)
Functions:
$f\left(p\right)$	Probability density function of p
$TC\left(T\right)$	Total cost (in dollars)
$TP\left(T\right)$	Total profit (in dollars)
TPU(T)	Total profit per unit of time (in dollars)
E[TPU]	Expected total profit per unit of time (in dollars)
Decision variable:
$T$	Cycle time (in years)

).

3.3. Assumptions

This section contains all the assumption which are considered in the model.

i.

There are carbon emissions and energy consumption per unit associated with storing inventory in warehouses. As a result, $h^{\prime }$ represents the holding cost related to carbon emissions per unit item.

ii.

$D$ is the rate of annual demand, which is known, uniform, and constant.

iii.

The distribution of the proportion of items of defective quality (p) is uniform in [α, β] where [0 ≤ α ≤ β ≤ 1].

iv.

There is a carbon emission cost $c^{\prime }$ on purchasing units from the supplier, which is paid by the retailer, as per government policy.

v.

The cost of carbon emissions when an item is ordered is $A^{\prime }$.

vi.

Lead time is never changing. The replenishment rate occurs at an infinite rate.

vii.

A single item is used in the inventory model.

viii.

No shortages occur in the inventory model.

ix.

The supplier’s (M) credit period is always greater than or equal to the retailer’s (N) credit duration for consumers. This relationship ensures M ≥ N.

x.

During case 1: $T\le M$, there is no interest charged. But in case 2: $N<T<M$ and case 3: $T\ge M$, $I_p$, the retailer will charge for those items that are left in stock.

xi.

Between periods N and M, the retailer earns a profit $I_e$ through credit financing. Earnings begin when customers start making payments and continue until the designated payment period for the goods expires.

4. Model Description and Analysis

This research investigates a multi-level trade credit policy for inventory comprising defective and non-instantaneously deteriorating items. Initially, a quantity related to y units enters the system. Within the time frame $\left[0, t_1\right]$, the inventory diminishes due to demand and imperfect item screening. After the screening time, items are separated as defective and non-defective items. The screening time can be calculated by t₁(=y/x). Following the screening, the inventory decreases solely due to the demand between $\left[t_1,t_d\right]$. Subsequently, deterioration commences, and the inventory is rejected due to the demand and item degeneration from $\left[t_d, T\right]$, depleting entirely at t = T. Let $I_1\left(t\right)$ and $I_2\left(t\right)$ represent the inventory levels during $t\in \left[0, t_1\right]$ and $t\in \left[t_1,t_d\right]$, respectively. The equations from Figure 1, which is differential of inventory level at any time t, are mentioned below:

$${\displaystyle \frac{d{I}_1\left(t\right)}{dt}}=-D, \left(0\le t\le t_1\right)$$

(1)

$${\displaystyle \frac{d{I}_2\left(t\right)}{dt}}=-D, \left(t_1\le t\le t_d\right)$$

(2)

The solution of the differential equations using $I_1\left(t_1\right)=y-D{t}_1,I_2\left(t_d\right)=y-py-D{t}_1-D{t}_d$ is

$$I_1\left(t\right)=y-Dt$$

(3)

$$I_2\left(t\right)=\left(1-p\right)y-D{t}_1-Dt$$

(4)

Likewise, let $I_3\left(t\right)$ denote the inventory level at time $t\in \left[t_d, T\right]$

$${\displaystyle \frac{d{I}_3\left(t\right)}{dt}}+\beta t=-D, \left(t_d\le t\le T\right)$$

(5)

Using $I_3\left(T\right)=0$, the solution of the equation is given by

$$I_3\left(t\right)={\displaystyle \frac{D}{\beta }}\left(e^{\beta \left(T-t\right)}-1\right)$$

(6)

For continuity of I(t) at $t=t_d$, $I_2\left(t_d\right)=I_3\left(t_d\right)$

$$\left(1-p\right)y-D{t}_1-D{t}_d={\displaystyle \frac{D}{\beta }}\left(e^{\beta \left(T-t_d\right)}-1\right)$$

(7)

which implies that the green inventory per cycle is given by

$$y={\displaystyle \frac{D\left(e^{\beta \left(T-t_d\right)}+\beta t_d-1\right)}{\beta \left(1-p-\frac{D}{x}\right)}}$$

(8)

Further, to prevent shortages during the screening period ($t_1$), the effective percentage, p (a random variable uniformly distributed in the range [a, b], where 0 < a < b < 1), is limited to $\left(1-p\right)y\ge D{t}_1$.

The expenses spent in the inventory model, which are used to compute the overall profit, are as follows:

(a)

Sale revenue

$$\left(SR\right)=vDT$$

(9)

(b)

The cost of order cost

$$\left(OC\right)=A+A^{\prime }$$

(10)

(c)

The cost of purchase

$$\left(PC\right)=\left(c+c^{\prime }\right)y$$

(d)

The cost to hold the inventory

$$HC=\left(h+h^{\prime }\right)\left[{\int }_0^{t_1}{I}_1\left(t\right)dt+{\int }_{t_1}^{t_d}{I}_2\left(t\right)dt+{\int }_{t_d}^T{I}_3\left(t\right)dt\right]$$

(11)

$$=\left(h+h^{\prime }\right)\left[\left(1-p\right)y{t}_d+{\displaystyle \frac{p{y}^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)\right]$$

(12)

(e)

The cost of deterioration

$$\left(DC\right)=\left(c+c^{\prime }\right)\left(\left(1-p\right)y-DT\right)$$

(13)

(f)

Screening cost

$$\left(SC\right)=sy$$

(14)

(g)

Further, on the basis of the credit periods M and N, the interest paid ($IP$) and interest earned (IE) will be calculated as per the cases and subcases.

From the above costs, the total annual profit (TP) can be determined by total profit:

$$(TP)={\displaystyle \frac{1}{T}}\left[SR-OC-PC-HC-DC-SC-IP-IE\right]$$

Additionally, three situations emerge based on the credit periods M and N, which are given below:

(i)

$T\ge M$

(ii)

$N\le T\le M$

(iii)

$T\le M$

Further, subcases also arise, which are given below in

Table 2. Cases and subcases that arise for I_p and I_e.

Case I: $T\ge M$	Case II: $N\le T\le M$	Case III: $T\le M$
(i) ${0\le t}_1\le t_d\le N\le M\le T$	$\left(\mathrm{i}\right) {0\le t}_1\le t_d\le N\le T\le M$	$\left(\mathrm{i}\right) {0\le t}_1\le t_d\le T\le N\le M$
(ii) ${0\le t}_1\le N\le t_d\le M\le T$	(ii) ${0\le t}_1\le N\le t_d\le T\le M$
(iii) ${0\le N\le t}_1\le t_d\le M\le T$	(iii) ${0\le N\le t}_1\le t_d\le T\le M$
(iv) ${0\le t}_1\le N\le M\le t_d\le T$
(v) ${0\le N\le t}_1\le M\le t_d\le T$
(vi) ${0\le N\le M\le t}_1\le t_d\le T$

.

Case I: $T\ge M$.

There are six subcases that arise based on the values of $t_1, t_d, N, M,$ and $T$:

$$\begin{gathered} \left(1.1\right) {0\le t}_1\le t_d\le N\le M\le T, \left(1.2\right) {0\le t}_1\le N\le t_d\le M\le T, \\ \left(1.3\right) {0\le N\le t}_1\le t_d\le M\le T, \left(1.4\right) {0\le t}_1\le N\le M\le t_d\le T, \\ \left(1.5\right) {0\le N\le t}_1\le M\le t_d\le T, \left(1.6\right) {0\le N\le M\le t}_1\le t_d\le T \end{gathered}$$

The formulation of these cases is discussed below:

Subcase I:

$${0\le t}_1\le t_d\le N\le M\le T$$

The retailer must pay interest $I_p$ for the left stock in $M\le T$ shown in Figure 2. The interest paid per cycle is as follows:

$$\begin{gathered} IP=c{I}_p{\int }_M^T{I}_3\left(t\right)dt \\ =c{I}_p{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-M\right)}-\left(T-M\right)\beta -\beta \right) \end{gathered}$$

(15)

During the time period N to M, the retailer can use the sale revenue $I_e$ shown in Figure 2. The interest earned is as follows:

$$IE = v{I}_e{\int }_N^{M}Dtdt=v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right)$$

(16)

Subcase II:

$${0\le t}_1\le N\le t_d\le M\le T$$

The retailer has some inventory left on hand as $M\le T$. As a result, for the remaining stock depicted in Figure 3, the merchant must pay interest at the rate of $I_p$. The amount of interest paid is as follows:

$$IP=c{I}_p{\int }_M^T{I}_3\left(t\right)dt=c{I}_p{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-M\right)}-\left(T-M\right)\beta -\beta \right)$$

(17)

The retailer can use selling income, as depicted in Figure 3, to earn interest at the rate of $I_e$ from N to M. The interest received per cycle is as follows:

$$IE=p{I}_e{\int }_N^{M}Dtdt=p{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right)$$

(18)

Subcase III:

$${0\le N\le t}_1\le t_d\le M\le T$$

The retailer has some inventory left on hand as $M\le T$. As a result, for the left stock depicted in Figure 4, the merchant must pay interest at the rate of $I_p$ (Figure 4). The amount of interest each cycle paid is as follows:

$$\begin{gathered} IP = c{I}_p{\int }_M^T{I}_3\left(t\right)dt \\ =c{I}_p{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-M\right)}-\left(T-M\right)\beta -\beta \right) \end{gathered}$$

(19)

The retailer can use selling income, as depicted in Figure 4, to earn interest at the rate of $I_e$ from N to M. The interest received per cycle is as follows:

$$\begin{gathered} IE = v{I}_e{\int }_N^{M}Dtdt \\ =v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

(20)

Subcase IV:

$${0\le t}_1\le N\le M\le t_d\le T$$

Some inventory on hand is left by the retailer as $M\le T$. As a result, for the left stock depicted in Figure 5, the merchant must pay interest at the rate of $I_p$. The amount of interest each cycle paid is as follows:

$$\begin{gathered} IP = c{I}_p\left({\int }_M^{t_d}{I}_2\left(t\right)dt+{\int }_{t_d}^T{I}_3\left(t\right)dt\right) \\ =c{I}_p\left({\displaystyle \frac{D}{2}}\left(M^2-t_d^2\right)+\left(1-p\right)y\left(t_d-M\right)+{\displaystyle \frac{Dy}{x}}\left(M-t_d\right)+{\displaystyle \frac{D}{\beta }}\left(t_d-T\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)\right) \end{gathered}$$

(21)

The retailer can use selling income, as depicted in Figure 5, to earn interest at the rate of $I_e$ from N to M. The interest received per cycle is as follows:

$$\begin{gathered} IE=v{I}_e{\int }_N^{M}Dtdt \\ =v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

(22)

Subcase V:

$${0\le N\le t}_1\le M\le t_d\le T$$

Some inventory on hand is left by the retailer as $M\le T$. As a result, for the left stock depicted in Figure 6, the merchant must pay interest at the rate of $I_p$. The amount of interest each cycle paid is as follows:

$$\begin{gathered} IP=c{I}_p\left({\int }_M^{t_d}{I}_2\left(t\right)dt+{\int }_{t_d}^T{I}_3\left(t\right)dt\right) \\ =c{I}_p\left({\displaystyle \frac{D}{2}}\left(M^2-t_d^2\right)+\left(1-p\right)y\left(t_d-M\right)+{\displaystyle \frac{Dy}{x}}\left(M-t_d\right)+{\displaystyle \frac{D}{\beta }}\left(t_d-T\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)\right) \end{gathered}$$

(23)

The retailer can use selling income, as depicted in Figure 6, to earn interest at the rate of $I_e$ from N to M. The interest received per cycle is as follows:

$$\begin{gathered} IE=v{I}_e{\int }_N^{M}Dtdt \\ =v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

(24)

Subcase VI:

$${0\le N\le M\le t}_1\le t_d\le T$$

Some inventory on hand is left by the retailer as $M\le T$. As a result, for the left stock depicted in Figure 7, the merchant must pay interest at the rate of $I_p$. The amount of interest each cycle cleared is as follows:

$$\begin{gathered} IP=c{I}_p\left({\int }_M^{t_1}{I}_1\left(t\right)dt+{\int }_{t_1}^{t_d}{I}_2\left(t\right)dt+{\int }_{t_d}^T{I}_3\left(t\right)dt\right) \\ =c{I}_p\left({\displaystyle \frac{D{M}^2}{2}}+y\left({\displaystyle \frac{y}{x}}-M\right)+{\displaystyle \frac{D}{2}}\left({\displaystyle \frac{y^2}{x^2}}-M\right)-{\displaystyle \frac{D{y}^2}{2x^2}}+\left(1-p\right)y\left(t_d-{\displaystyle \frac{y}{x}}\right)+{\displaystyle \frac{D\ast y}{x}}\left({\displaystyle \frac{y}{x}}-t_d\right)+{\displaystyle \frac{D}{\beta }}\left(t_d-T\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)\right) \end{gathered}$$

(25)

The retailer can use selling income, as depicted in Figure 7, to earn interest at the rate of $I_e$ from N to M. The interest received per cycle is as follows:

$$\begin{gathered} IE=v{I}_e{\int }_N^{M}Dtdt \\ =vD\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

(26)

Case II:

$$N\le T\le M$$

Based on the values of $t_1, t_d, N, M,$ and $T$, there are three subcases:

(1.1)

${0\le t}_1\le t_d\le N\le T\le M$

(1.2)

${ 0\le t}_1\le N\le t_d\le T\le M$

(1.3)

${0\le N\le t}_1\le t_d\le T\le M$

The formulation of these cases is discussed below:

Subcase I:

$${0\le t}_1\le t_d\le N\le T\le M$$

Figure 8 illustrates how the store generates income and interest (at rate $I_e$) from the inventory sales from N to T. Every revenue is subject to interest from T to M, accumulating interest from N to M each cycle.

$$\begin{gathered} IE=v{I}_e\left({\int }_N^{T}Dtdt+DT\left(M-T\right)\right) \\ =v{I}_{e}D\left(TM-{\displaystyle \frac{N^2}{2}}-{\displaystyle \frac{T^2}{2}}\right) \end{gathered}$$

(27)

Subcase II:

$${0\le t}_1\le N\le t_d\le T\le M$$

Figure 9 illustrates how the store generates income and interest (at rate $I_e$) from inventory sales from N to T. Every revenue is subject to interest from T to M, accumulating interest from N to M each cycle.

$$\begin{gathered} IE=v{I}_e\left({\int }_N^{T}Dtdt+DT\left(M-T\right)\right) \\ =v{I}_{e}D\left(TM-{\displaystyle \frac{T^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

(28)

Subcase III:

$${0\le N\le t}_1\le t_d\le T\le M$$

Figure 10 illustrates how the store generates income and interest (at rate $I_e$) from inventory sales from N to T. Every revenue is subject to interest from T to M, accumulating interest from N to M each cycle.

$$\begin{gathered} IE=v{I}_e\left({\int }_N^{T}Dtdt+DT\left(M-T\right)\right) \\ =v{I}_{e}D\left(TM-{\displaystyle \frac{T^2}{2}}-{\displaystyle \frac{N^2}{2}}\right) \end{gathered}$$

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Case III:

$$T\le M$$

Based on the values of $t_1,t_d,N,M,$ and $T,$ only ${0\le t}_1\le t_d\le T\le N\le M$ is the subcase. The formulation of this subcase is discussed below.

The store sells inventory from N to M and earns interest at a rate of I_e on the total sales income, as shown in Figure 11, which causes interest to accumulate throughout this time in each cycle.

$$IE=pvDT\left(M-N\right)$$

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From the above cases, since p is a random variable with a known probability density function f(p), then the expected total profit per unit of time, ${E\left[TPU\right]}_{i.j}, where i=1, 2, 3 and j=1, 2, 3, 4, 5, 6,$ can be expressed as follows:

Case I: When $T\ge M$ then

$${E\left[TPU\right]}_{i.j}=\left\{\begin{array}{c}{E\left[TPU\right]}_{1.1}={E\left[TPU\right]}_{1.2}={E\left[TPU\right]}_{1.3}, 0\le t_d\le N\le M\le T\\ { E\left[TPU\right]}_{1.4}={E\left[TPU\right]}_{1.5},{ 0\le t}_1\le N\le M\le t_d\le T \\ {E\left[TPU\right]}_{1.6},{ 0\le N\le M\le t}_1\le t_d\le T\end{array}\right.$$

(31)

Case II: When $N\le T\le M$ then

$${E\left[TPU\right]}_{i.j}={E\left[TPU\right]}_{2.1}={E\left[TPU\right]}_{2.2}\left(T\right)={E\left[TPU\right]}_{2.3,} 0 \le t_d\le N\le T\le M$$

(32)

Case III: When $T\ge M$ then

$${E\left[TPU\right]}_{i.j}={E\left[TPU\right]}_{3.1}, {0\le t}_1\le t_d\le T\le N\le M$$

(33)

where

$$\begin{array}{ll}{E\left[TPU\right]}_{1.1}={\displaystyle \frac{1}{T}}[& vDT-\left(A+A^{\prime }\right)-\left(c+c^{\prime }\right)y\\ & -(h+h^{\prime })(\left(1-E\left[p\right]\right)y{t}_d+{\displaystyle \frac{E\left[p\right]y^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)\\ & +{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right))-\left(c+c^{\prime }\right)\left(\left(1-E\left[p\right]\right)y-DT\right)-sy-c{I}_p{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-M\right)}-\left(T-M\right)\beta -\beta \right)\\ & +v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right)]={E\left[TPU\right]}_{1.2}={E\left[TPU\right]}_{1.3}\end{array}$$

(34)

$$\begin{array}{ll}{E\left[TPU\right]}_{1.4}={\displaystyle \frac{1}{T}}[& vDT-\left(A+A^{\prime }\right)-\left(c+c^{\prime }\right)y\\ & -\left(h+h^{\prime }\right)(\left(1-E\left[p\right]\right)y{t}_d+{\displaystyle \frac{E\left[p\right]y^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)\\ & +{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right))-\left(c+c^{\prime }\right)\left(\left(1-E\left[p\right]\right)y-DT\right)-sy\\ & -c{I}_p\left({\displaystyle \frac{D}{2}}\left(M^2-t_d^2\right)+\left(1-E\left[p\right]\right)y\left(t_d-M\right)+{\displaystyle \frac{Dy}{x}}\left(M-t_d\right)+{\displaystyle \frac{D}{\beta }}\left(t_d-T\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)\right)\\ & +v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right)]={E\left[TPU\right]}_{1.5}\end{array}$$

(35)

$$\begin{array}{ll}{E\left[TPU\right]}_{1.6}={\displaystyle \frac{1}{T}}[& vDT-\left(A+A^{\prime }\right)-\left(c+c^{\prime }\right)y\\ & -\left(h+h^{\prime }\right)(\left(1-E\left[p\right]\right)y{t}_d+{\displaystyle \frac{E\left[p\right]y^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)\\ & +{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right))-\left(c+c^{\prime }\right)\left(\left(1-E\left[p\right]\right)y-DT\right)-sy\\ & -c{I}_p({\displaystyle \frac{D{M}^2}{2}}+y\left({\displaystyle \frac{y}{x}}-M\right)+{\displaystyle \frac{D}{2}}\left({\displaystyle \frac{y^2}{x^2}}-M\right)-{\displaystyle \frac{D{y}^2}{2x^2}}+\left(1-E\left[p\right]\right)y\left(t_d-{\displaystyle \frac{y}{x}}\right)+{\displaystyle \frac{Dy}{x}}\left({\displaystyle \frac{y}{x}}-t_d\right)\\ & +{\displaystyle \frac{D}{\beta }}\left(t_d-T\right)+{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right)]+v{I}_{e}D\left({\displaystyle \frac{M^2}{2}}-{\displaystyle \frac{N^2}{2}}\right)]\end{array}$$

(36)

$$\begin{array}{ll}{E\left[TPU\right]}_{2.1}={\displaystyle \frac{1}{T}}[& vDT-\left(A+A^{\prime }\right)-\left(c+c^{\prime }\right)y\\ & -\left(h+h^{\prime }\right)(\left(1-E\left[p\right]\right)y{t}_d+{\displaystyle \frac{E\left[p\right]y^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)\\ & +{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right))-\left(c+c^{\prime }\right)\left(\left(1-E\left[p\right]\right)y-DT\right)-sy+v{I}_{e}D\left(TM-{\displaystyle \frac{N^2}{2}}-{\displaystyle \frac{T^2}{2}}\right)]\end{array}$$

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$$\begin{array}{ll}{E\left[TPU\right]}_{3.1}={\displaystyle \frac{1}{T}}[& vDT-\left(A+A^{\prime }\right)-\left(c+c^{\prime }\right)y\\ & -\left(h+h^{\prime }\right)(\left(1-E\left[p\right]\right)y{t}_d+{\displaystyle \frac{E\left[p\right]y^2}{x}}-{\displaystyle \frac{Dy{t}_d}{x}}+{\displaystyle \frac{D{y}^2}{x^2}}-{\displaystyle \frac{D{t_d}^2}{2}}-{\displaystyle \frac{D}{\beta }}\left(T-t_d\right)\\ & +{\displaystyle \frac{D}{{\beta }^2}}\left(e^{\beta \left(T-t_d\right)}-1\right))-\left(c+c^{\prime }\right)\left(\left(1-E\left[p\right]\right)y-DT\right)-sy+v{I}_{e}DT\left(M-N\right)]\end{array}$$

(38)

5. Solution Procedure

Furthermore, concavity is extremely difficult to show theoretically because the derivatives of the expected total profit functions are complex. Additionally, Figure 12 illustrates the nature of the functions for each of the ten cases, as explained below:

The necessary condition required for maximization of total profit is defined by ${\displaystyle \frac{d E\left[TPU\right]}{dT}}=0$ , which provides the optimal values for $T$. This study aims to optimize the cycle time to maximize total profit, $E\left[TPU\right]$. Hence, the profit function’s behaviour is analyzed for all cases using the derivative method.

Additionally, the following adequate conditions must be met for the predicted total profit $E[T{PU]}_{i,j}$ to be concave: $\left({\displaystyle \frac{d^{2}E[T{PU]}_{i,j}}{{dT}^2}}\right)\le 0$ (where i = 1, 2, 3 and j = 1, 2, 3, 4, 5, 6).

Appendix A contains the first and second derivatives of the expected total profit function, which are quite big equations.

In order to find the optimal value T*, the following Algorithm 1 is proposed:

Algorithm 1. Finding the optimal value T*

Step 1: Put values of all the parameters in Equation (A1) Step 2: Determine the value of cycle length, T. Now, using the value of T, calculate the values of y from Equation (8). $\mathrm{If} (T\ge M \&$ ${0\le t}_1\le t_d\le N\le M\le T$ (case 1: subcase 1.1 satisfy))     then the cycle length will be T and the value of expected total profit,   E[TPU], can be obtained by putting T in Equation (A1),    else     this case is not feasible and set E[TPU] = 0. $\mathrm{If} ({0\le t}_1\le N\le t_d\le M\le T)$     then find T and E[TPU] for this case from Equation (A1),    else     case is not possible, set E[TPU] = 0. If (${0\le N\le t}_1\le t_d\le M\le T$)     Then determine T and E[TPU] from Equation (A1),    else     put E[TPU] = 0, in case of not feasible $\mathrm{If} ({0\le t}_1\le N\le M\le t_d\le T$)     then calculate T and E[TPU] from Equation (A3),    else     E[TPU] = 0. $\mathrm{If} ({0\le N\le t}_1\le M\le t_d\le T)$     then find T and predicted total profit E[TPU] from (A3)    else     put E[TPU] = 0. $\mathrm{If} ({0\le N\le M\le t}_1\le t_d\le T)$     then determine T and E[TPU] from Equation (A5)    else     place E[TPU] = 0. $\mathrm{If} (N\le T\le M \& {0\le t}_1\le t_d\le N\le T\le M$)     then compute T and E[TPU] from (A6)    else     set E[TPU] = 0. $\mathrm{If} (N\le T\le M \& {0\le t}_1\le N\le t_d\le T\le M$)     then evaluate T and E[TPU] from (A6)    else     put E[TPU]= 0. $\mathrm{If} (N\le T\le M \& {0\le N\le t}_1\le t_d\le T\le M$)     then determine T and E[TPU] from Equation (A6)    else     put E[TPU] = 0. $\mathrm{If} (T\le M \& {0\le t}_1\le t_d\le T\le N\le M$)     then evaluate T and E[TPU] from (A8)    else     in the case of not feasible, put E[TPU] = 0. Step 3. Compare the expected total profit for all the 10 cases, i.e.,     Max {for all cases E[TPU]i.j, (i = 1, 2, 3 and j = 1, 2, 3, 4, 5, 6}

Choose the ideal scenario, which has the optimal value of T and the largest predicted total profit, E[TPU].

6. Numerical Example

This section presents a numerical example to illustrate and validate the theoretical findings detailed in the examples. Most of the parameter values are considered from Ref. [38].

Example 1. 

$D=7000 units/year, A=100 \$/order, h=5 \$/unit/year, \beta =0.06, t_d=0.04 year, c=25\$, s=0.3\$/unit , A^{\prime }=1\$/order, h^{\prime }=0.1\$/unit/year, c^{\prime }=1\$, x=175200 units /year, v=50\$/units, I_e=0.08 \$/year, I_p=0.12\$/year$. N and M values are considered as per the cases 1 to 10. $f\left(p\right)=\left\{\begin{array}{c}25, 0\le p\le 0.04\\ 0 otherwise \end{array}\right.$, $then the expected value of defective items will be E\left[p\right]=0.02$.

From

Table 3. Optimal values of cycle length and expected total profit for all the cases.

Cases	$\mathit{T}$(Yr)	E [TPU] ($)
1	$0.0590$	$163195$
2	$0.0576$	$163276$
3	$0.0573$	$163291$
4	$0.0581$	$163287$
5	$0.0577$	$163311$
6	$0.0593$	$162408$
7	$0.0667$	$163444$
8	$0.0619$	$163754$
9	$0.0561$	$164134$
10	$0.0743$	$163350$

, The first five cases have relatively consistent values of T around 0.0573 yr to 0.0581 yr, showing minimal variation. A noticeable increase occurs in cases 7 (0.0667 yr) and 10 (0.0743 yr), indicating some deviation in the values. The $\mathrm{E}\left[\mathrm{T}\mathrm{P}\mathrm{U}\right]$ values show a stable pattern, though case 6 has a notably lower $\mathrm{E}\left[\mathrm{T}\mathrm{P}\mathrm{U}\right]$ of 162408$. Case 9 has the highest total cost at 164134$, and the optimal order quantity (y) is 418 units, implying better performance or increased computational efficiency.

7. Sensitivity Analysis

To determine the robustness of the model, a sensitivity analysis is carried out by adjusting a number of effective parameters (

Table 4. Effect of changes in $E\left[\mathit{p}\right]$ on $E\mathit{T}\mathit{P}\mathit{U}$.

E[p]	$\mathit{T}$(Yr)	E [TPU] ($)
0.01	0.056147	163863
0.02	0.056133	164134
0.03	0.056118	164410
0.04	0.056103	164692
0.05	0.056087	164980

,

Table 5. Effect of changes in ${\mathbf{I}}_{\mathit{e}}$ on $\mathit{T}\mathit{P}$.

${\mathbf{I}}_{\mathit{e}}$	$\mathit{T}$(Yr)	E[TPU] ($)
0.096	0.053861	164372
0.088	0.054961	164252
0.080	0.056133	164134
0.072	0.057382	164017
0.064	0.058718	163902

,

Table 6. Effect of changes in $\mathit{h}$ on $E\left[\mathit{T}\mathit{P}\mathit{U}\right]$.

$\mathit{h}$	$\mathbf{T}$(Yr)	E[TPU] ($)
6.0	0.053319	163941
5.5	0.054671	164036
5.0	0.056133	164134
4.5	0.057718	164234
4.0	0.059447	164337

and

Table 7. Effect of changes in $\beta$ on $E\left[TPU\right]$.

$\mathit{\beta }$	$\mathit{T}$(Yr)	E[TPU] ($)
0.02	0.056197	164135
0.04	0.056165	164134
0.06	0.056133	164134
0.08	0.056100	164133
0.10	0.056068	164132

and Figure 13, Figure 14 and Figure 15). The effect of the effective parameters is considered in the best case, i.e., 9.

• As ${\mathrm{I}}_e$ decreases, the processing time T gradually increases. This implies that higher interest rates may incentivize quicker or more efficient operations, likely due to better resource utilization.
• $\mathrm{E}\left[\mathrm{T}\mathrm{P}\mathrm{U}\right]$ steadily declines as ${\mathrm{I}}_e$ decreases. This decline indicates that lower interest earnings directly affect profitability, possibly due to reduced capital reinvestment benefits.

As the holding cost decreases from 6 to 4, the $\mathrm{E}\left[\mathbf{T}\mathbf{P}\mathrm{U}\right]$ steadily increases. This suggests that lower holding costs may introduce inefficiencies or require more time to process or execute certain operations. Higher $\mathrm{E}\left[\mathbf{T}\mathbf{P}\mathrm{U}\right]$ values typically indicate better or more efficient system output, suggesting that lower holding costs may promote higher performance despite the increased time.

Table 7 shows that the overall profit would decline as the rate of deterioration rises. An increase in the rate of deterioration will raise the order quantity to make up for the loss from deterioration. As the deterioration rate rises, the cycle length decreases as well, slowing down the deterioration process and raising overall profit.

8. Managerial Implications

The results derived from the numerical examples highlight the strategic approaches inventory managers should adopt when handling inventories under a multi-level trade credit program that incorporates carbon emissions. The study examines three scenarios based on multi-level trade credit. The most advantageous case occurs when the retailer’s and customer’s credit periods fall between the screening time and the non-deteriorating time duration. This scenario enables inventory managers to achieve higher profitability. Managerial decisions must account for ordering costs, holding costs, interest payments, and carbon emissions, emphasizing the importance of a holistic approach to inventory management. Inventory managers must carefully evaluate all options before procurement, as decisions impact both overall profitability and the quantity of orders placed. Larger order quantities increase holding costs and carbon emission costs, highlighting the need for managers to optimize total order quantities. Incorporating carbon emissions into inventory management strategies helps minimize order quantities, directly reducing holding- and emission-related costs. Furthermore, profitability is enhanced when managers focus on trade credit terms and the management of imperfect-quality items. Managers achieve higher profits when the interest earned exceeds the interest paid, underlining the critical role of financial strategies in inventory management. This research aims to address carbon emissions within a multi-level trade credit framework for imperfect-quality items. By integrating carbon emission considerations, the proposed model supports more sustainable practices, balancing environmental and economic objectives while maximizing profitability. By effectively managing a perishable and deteriorating inventory, this model can help industries including FMCG, pharmaceuticals, and food and beverage. It can be used to optimize stock levels and reduce waste in the chemical and electronic industries. Through better trade credit regulations, the model aids in balancing sustainability and profitability.

9. Concluding Remarks and Future Scope

This study aims to reduce carbon emissions by implementing multi-level trade credit policies. An inventory model has been developed for items with non-instantaneous deterioration. In practical production scenarios, achieving a perfect quality rate for all units is unrealistic, and this issue is addressed within the study. The research focuses on items with a gradual deterioration rate, such as food products, electronic components, and fashion items, where defects during production can be addressed through an inspection process. The model examines imperfect-quality items within the framework of a multi-level trade credit policy, integrating an economic order quantity (EOQ) model. This approach emphasizes the optimization of the retailer’s total profit while accounting for items with production defects that are rectifiable through inspection. Multi-level trade credit policies emerge as the most effective strategy for maximizing profits in this context. The study explores various scenarios involving suppliers, retailers, and customers, using numerical examples to validate the results. The optimal scenario for retailers occurs when the credit periods for both retailers and customers fall between the screening time and the non-instantaneous deterioration period. This configuration ensures enhanced profitability. The model’s applicability can be further extended by incorporating finite replenishment rates, variable demand rates, or other dynamic parameters, offering a robust foundation for addressing complex inventory management challenges. Additionally, this model can be expanded to include closed-loop supply chains and warehouses.