Enhancing Product Quality Incorporating Additive Manufacturing on Recycled Inventory System with Deterioration and Delayed Payment

1. Introduction

As the world grapples with the mounting issue of plastic waste, innovative approaches are essential to mitigate the environmental impact of plastic products. One such approach is to leverage additive manufacturing (AM) or 3D printing technology to create innovative and high-quality products. Use of AM not only reduces the need for new production but also supports the principles of the circular economy and reduces waste on the environment. This offers precise control over material properties and product design, enabling the creation of customized and high-performance products made from recycled materials over traditional methods. [1] analysed various AM technologies that offer a significant advantage in reducing product development timelines for intricate structures. Furthermore, researchers discussed layer-by-layer additive process ensures high precision, enhancing manufacturing quality and reliability ([2,3]) and design an optimized, sustainable reverse logistics model for PVC recycling that minimizes costs and environmental impacts while improving operational efficiency. As AM integrates into manufacturing, its efficiency, sustainability, and adaptability reshape industries, promising a future marked by reduced waste, optimized inventory, and elevated production standards.

In the inventory system, one of the important parameters known as demand must be considered when making decisions about the stock and output of the warehouses. Researchers are emphasizing pivotal factors in inventory systems, notably focusing on aspects such as degradation, demand dynamics, and the assurance of reliability and quality. One of the significant challenges confronting the community today is the recycling of plastics by developing innovative and sustainable recycling with minimal environmental impact. In most of the research, facts are needed to answer the specific queries on inventory regarding AM as follows:

• Need to answer approaches to improve the quality of recycled products using additive manufacturing technology.
• How can a sustainable recycled inventory system be created through the use of AM technology?
• Analysis on investment from retailers in 3D printers can support the production of recycled products.
• How can the reliability of recycled plastic products be ensured while maximizing overall profit?

Retailers aim to maintain high-quality and eco-friendly products to meet customer demands within their inventory systems, driving a growing interest in sustainable and innovative manufacturing practices. Recycling plastic products, although vital for environmental sustainability, often leads to reduced product quality, reliability, and profitability. This study developed an Economic Order Quantity (EOQ) model by investigating sustainable strategies to enhance the quality and reliability by mitigating the waste, emissions, quality and reliability issues associated with recycled plastics. Furthermore, this advancement has motivated retailers to invest in 3D printing technologies, fostering the broader adoption of AM for sustainable and efficient inventory systems.

2. Literature Review

Nowadays, the majority of buyers prefer the acquisition of high-quality and dependable. Different scenarios in production inventory models have been analyzed that focus on particular situations incorporating demand affected by both price and inventory levels, decay of products ([4]), stochastic demand ([5]), and product reliability ([6]). Researchers ([7,8,9]) explored preventive maintenance strategies for manufacturing systems deteriorating in reliability and quality, assuming constant demand or the demand rates that vary with both stock levels and time [10]. Research has been developed on an inventory framework that incorporates fuzzy demand subject to price discounts ([11]) and also the considers sales price of the product ([12,13,14,15]).

The degradation rate significantly influences the optimal policy ([16,17]). Plastics degrade over time due to factors such as temperature, humidity, light exposure, and chemical interactions with other substances. Many researchers ([18,19]) have been studying inventory models for decaying items in recent years on the smart manufacturing system of degrading products through the application of preservation technologies. Numerous researchers ([20,21]) have created inventory models for perishable items with varying decay rates instead of a steady deterioration rate.

In industry, payment delay refers to the time it takes a company to settle a bill after receiving it, whereas late payments refer to payments that are delayed beyond the agreed-upon date. Several findings suggest that policies allowing for payment delays can result in heightened demand ([22,23,24]), whereas prepayment policies tend to diminish demand, depending on the duration of the prepayment terms ([25,26]).

Inventory shortages occur when demand exceeds available stock, leading to lost sales, reduced customer satisfaction, and reputational risks. To mitigate these challenges, advanced inventory management uses data-driven strategies like demand forecasting, safety stock optimization, and real-time tracking to minimize stockouts and improve efficiency. [27] created a sustainable inventory model with complete backlogs. developed a bi-objective model to make supply chains sustainable by minimizing production waste and reducing environmental emissions.

The implementation of AM technologies in inventory models for enhancing defect-free production, life cycle, and cost-controlled inventory reduces the environmental impact ([28]) irrespective of the kind of material and production process. [29] created a life cycle inventory framework for stereolithography, a commonly employed AM method. [30] developed an environmental evaluation of an AM method that used a generic framework to examine the ecological impact of 3D concrete printing via a parametric approach. [31] developed a bi-objective sustainable supply chain for reduce waste generation and pollution through the integration of AM. [32] reviewed the use of recycled polymers in 3D printing, emphasizing recycling methods, reinforcements, and parameter optimization for sustainable production. AM technology has been designed for strategies in supply chain network ([33]), conventional warehouse inventory system ([34]), and also on the inventory model ([35]) so that businesses may enhance. Researchers investigated and compared the effects of the incorporation of AM technology into the supply chain networks ([36,37,38,39,40]).

Research aperture on the topic: 

This study further contrasts traditional manufacturing methods with AM technology within the inventory framework. There is a notable gap in addressing the integrated sustainable inventory model formulated to enhance the quality and reliability of recycled products using AM technology. This innovative model incorporates realistic factors such as price, time, quality, reliability demand, delayed payment, investment in AM technology, etc., aiming to achieve sustainable development. The present work aims to address the stated research gap by pursuing the following objectives:

• To develop a sustainable EOQ model integrating AM (3D printing) for eco-friendly inventory management.
• o enhance product quality and reliability while utilizing recycled plastic materials in production.
• To evaluate the economic and environmental benefits of adopting 3D printing technology in retailer operations.

Table 1 highlights the research gap and emphasises that this article is a pioneering effort to address all of the factors discussed collectively. More specific contributions to this manuscript are as follows:

• Introduce the additive manufacturing technology investments to increase the quality, reliability, and profitability of recycled products.
• A sustainable inventory model using AM technology for emission reduction and waste minimization to enhance the quality of recycled plastic products.

Table 1. A contribution-oriented comparative analysis of the present study against earlier models.

Reference	Sustainable	EOQ	Price dependent demand	Time dependent demand	Quality dependent demand	Reliability dependent demand	AM Technology or 3D printing technology	Delay payment
[41]	×	×	×	×	×	×	✓	×
[42]	×	✓	✓	×	×	×	×	×
[43]	×	✓	✓	×	×	×	×	×
[44]	×	×	✓	×	×	×	×	×
[45]	✓	×	×	×	×	×	✓	×
[46]	×	✓	✓	✓	×	×	×	×
[47]	✓	×	×	×	×	×	✓	×
[48]	×	✓	✓	×	×	×	×	×
[49]	×	×	×	×	×	×	✓	×
[50]	✓	✓	✓	×	×	×	×	✓
[51]	✓	×	×	×	×	×	✓	×
[52]	✓	×	✓	×	×	×	×	×
[53]	×	✓	✓	✓	×	×	×	✓
[54]	✓	✓	×	×	×	✓	✓	×
[55]	×	✓	✓	×	×	×	×	✓
[56]	✓	✓	×	✓	×	×	×	✓
[57]	×	×	×	×	×	×	✓	×
[58]	×	×	×	✓	×	×	✓	×
[59]	✓	×	✓	×	×	×	×	×
[60]	×	✓	✓	×	×	×	×	×
This paper	✓	✓	✓	✓	✓	✓	✓	✓

Table 1. A contribution-oriented comparative analysis of the present study against earlier models.

Reference	Sustainable	EOQ	Price dependent demand	Time dependent demand	Quality dependent demand	Reliability dependent demand	AM Technology or 3D printing technology	Delay payment
[41]	×	×	×	×	×	×	✓	×
[42]	×	✓	✓	×	×	×	×	×
[43]	×	✓	✓	×	×	×	×	×
[44]	×	×	✓	×	×	×	×	×
[45]	✓	×	×	×	×	×	✓	×
[46]	×	✓	✓	✓	×	×	×	×
[47]	✓	×	×	×	×	×	✓	×
[48]	×	✓	✓	×	×	×	×	×
[49]	×	×	×	×	×	×	✓	×
[50]	✓	✓	✓	×	×	×	×	✓
[51]	✓	×	×	×	×	×	✓	×
[52]	✓	×	✓	×	×	×	×	×
[53]	×	✓	✓	✓	×	×	×	✓
[54]	✓	✓	×	×	×	✓	✓	×
[55]	×	✓	✓	×	×	×	×	✓
[56]	✓	✓	×	✓	×	×	×	✓
[57]	×	×	×	×	×	×	✓	×
[58]	×	×	×	✓	×	×	✓	×
[59]	✓	×	✓	×	×	×	×	×
[60]	×	✓	✓	×	×	×	×	×
This paper	✓	✓	✓	✓	✓	✓	✓	✓

3. Development of Sustainable Inventory System

This study is consistent with a negligible replenishment rate of the horizon for inventory planning that extends infinitely. The following considerations are essential and consistent with the review of this study.

• The demand rate for this model which depends on the price, reliability and quality and increases with time i.e., $R(p,r,q)=a+(xr+yq-bp)t,o\le t\le t_1$, here a is initial demand.
• Degradation initiates immediately upon lot reception, and its rate remains constant at deterioration rate denoted by $\theta$$(0<\theta <1)$.
• If the retailer settles the payment within the designated credit period D provided by the supplier, they will avoid any interest fees. Conversely, if the retailer makes the payment after the the credit term D, they will incur interest at the rate $I_p$.
• In case-I: When the stock duration is consistently positive and exceeds the credit term D (i.e., $D\le t_1$). The consumer seeks to generate interest at an annual rate of $I_e$. This situation arises when the credit duration D exceeds the duration of the stock. Using the sales revenue, calculate $[0,t_1]$. After the credit period concludes, the unsold inventory will be financed at an annual rate of $I_p$ once the credit time has expired.
  Case-II: If the amount D equals or exceeds the allowable payment delay $t_1$(i.e, $D\ge t_1$), the consumer is excused from paying interest and instead accumulates interest at a constant yearly rate within the range (0, D). This occurs if D is greater than $t_1$, which indicates that D exceeds the allowable payment delay.
• The rate at which this model is in demand relies on product reliability, expressed as $r\left(s\right)=(1-e^{-a_{0}s})$. The reliability of products offered by the retailer is influenced by the quality of goods provided by the manufacturer.
• Quality of items in the inventory system is $q\left(A\right)=\omega \left({\displaystyle \frac{eA}{1+eA}}\right)$ i.e $q\%$ of quality item’s present in the inventory. Here $\omega$ is the amount of recycling products when 3D technology is investmented, and e is the efficiency of 3D technology in recycling products. The quality q = 0, when A = 0, and tends to $\omega$ when $A\to \infty$. The cost function of investment, q(A) is possesses continuous derivatives $q^{{‌}^{\prime }}\left(A\right)>0,q^{{‌}^{\prime \prime }}\left(A\right)<0.$

Let $Q_1$ is the starting point for the inventory level. As a result of demand and deterioration, the inventory level drops. Figure 1 shows two subperiods of T, when the inventory level drops to zero at time $t_1$, there are shortages in the fully backlogged period $(t_1,T)$.

$$\begin{array}{c}\hfill {\displaystyle \frac{dI\left(t\right)}{dt}}+\theta I\left(t\right)=-R(p,s,q),0\le t\le t_1.\end{array}$$

(1)

$$\begin{array}{c}\hfill {\displaystyle \frac{dI\left(t\right)}{dt}}=-R(p,s,q),t_1\le t\le T.\end{array}$$

(2)

with the condition $I\left(0\right)=Q_1,I\left(t_1\right)=0,I\left(T\right)=-Q_2$ and $I\left(t\right)$ is continuous at $t=t_1$

Resolving Eq.-(1) utilizing the provided condition $I\left(t_1\right)=0$, one has

$$\begin{array}{c}\hfill I\left(t\right)={\displaystyle \frac{1}{{\theta }^2}}\left((e^{\theta (t_1-t)}-1)(a\theta -(xr+yq-bp))+\theta (-bp)\left(e^{\theta (t_1-t)}{t}_1-t\right)\right),where0\le t\le t_1\end{array}$$

(3)

Once more, leveraging the boundary condition $I\left(0\right)=Q_1$, we can find

$$\begin{array}{c}\hfill Q_1={\displaystyle \frac{1}{{\theta }^2}}\left((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))+\theta t_1(xr+yq-bp)e^{\theta t_1}\right).\end{array}$$

(4)

Addressing the Eq.-(2) under the specified condition $I\left(t_1\right)=0$, yields

$$\begin{array}{c}\hfill I\left(t\right)=a(t_1-t)+{\displaystyle \frac{(xr+yq-bp)}{2}}(t_1^2-t^2),where{t}_1\le t\le T\end{array}$$

(5)

once again, utilizing the auxiliary condition $I\left(T\right)=-Q_2$, one can derive

$$\begin{array}{c}\hfill Q_2=a(T-t_1)+{\displaystyle \frac{(xr+yq-bp)}{2}}(T^2-t_1^2).\end{array}$$

(6)

The total number of the ordered amount is: $Q=Q_1+Q_2$

$$\begin{array}{ccc}\hfill Q& =& {\displaystyle \frac{1}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & +\theta t_1(xr+yq-bp)e^{\theta t_1})+\left(a(T-t_1)+{\displaystyle \frac{(xr+yq-bp)}{2}}(T^2-t_1^2)\right).\hfill \end{array}$$

(7)

4. Derivation of Costs of Inventory System

The cost interpretation of a business framework is vital to deciding the system’s endeavour for an inventory model with a supplier or manufacturer.

Ordering cost $\left(C_0\right)$: A constant expense associated with processing each order cycle, as given by

$$\begin{array}{c}\hfill OC=C_o.\end{array}$$

(8)

Holding cost ($C_h$ unit per unit time): It refers to the costs or losses incurred by a business for maintaining inventory over a specified duration. Using expressions for $I\left(t\right)$ appropriately by referring to Eq.-(3), the cost of holding $HC=C_h{\int }_0^{t_1}I\left(t\right)dt$ for the sellable inventory is expressible as

$$\begin{array}{ccc}\hfill HC& =& {\displaystyle \frac{C_h}{{\theta }^3}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))-\theta t_1(a\theta -(xr+yq-bp)e^{\theta t_1})\hfill \\ & & -{\displaystyle \frac{{\theta }^2{t}_1^2}{2}}(xr+yq-bp)).\hfill \end{array}$$

(9)

An organization’s purchasing cost ($C_p$ per unit time) is the amount it spends on purchasing the products and services it needs to operate or produce a product.

Purchasing cost ($PC$): The total cost expended by the merchant throughout the whole planning cycle for acquiring the requisite quantity of product in the form of purchase amounts is

$$\begin{array}{c}\hfill PC={\displaystyle \frac{C_p}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -xr-yq+bp)+t_1\theta (xr+yq-bp)e^{\theta t_1}+{\theta }^{2}a(T-t_1)\\ \hfill +{\theta }^2{\displaystyle \frac{(xr+yq-bp)}{2}}(T^2-t_1^2)).\end{array}$$

(10)

The shortage cost ($C_b$) represents the expenditures or losses incurred by an organization due to insufficient inventory or product availability for customer demand, resulting in missed sales opportunities or customer dissatisfaction, which is evaluated as

$$\begin{array}{ccc}\hfill SC& =& -C_b{\int }_{t_1}^{T}I\left(t\right)dt\hfill \\ & =& -C_b\left(a{t}_1(T-t_1)-{\displaystyle \frac{a}{2}}(T^2-t_1^2)+{\displaystyle \frac{xr+yq-bp}{6}}\left(3t_1^2(T-t_1)-(T^3-t_1^3)\right)\right).\hfill \end{array}$$

(11)

Sales revenue: The total amount of money that the inventory system will make from the sale of goods throughout the planning cycle is provided by

$$\begin{array}{c}\hfill SR=p\left({\int }_0^{t_1}R(p,r,q)dt+Q_2\right)=p\left(aT+{\displaystyle \frac{(xr+yq-bp)}{2}}{T}^2\right).\end{array}$$

(12)

Deterioration cost ($C_d$ of deterioration per unit over time): From the stock that the retailer receives in excellent condition, a certain percentage of the items begin to deteriorate over time; as a result, the cost of deterioration is calculated as:

$$\begin{array}{ccc}\hfill DC& =& C_d\left(Q_1-{\int }_0^{t_1}R(p,s,q)dt\right)\hfill \\ & =& {\displaystyle \frac{C_d}{{\theta }^2}}\{\left((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))+\theta t_1(xr+yq-bp)e^{\theta t_1}\right)\hfill \\ & & -{\theta }^2\left(a{t}_1+{\displaystyle \frac{(xr+yq-bp)t_1^2}{2}}\right)\}.\hfill \end{array}$$

(13)

The AM technology investment cost per cycle is given by

$$\begin{array}{ccc}\hfill AIC& =& AT.\hfill \end{array}$$

(14)

Delay Based Payment Frameworks 

There are two possible results from this inventory system: either the buyer’s conditional payment delays are larger or shorter than the cycle time. In order to determine the overall inventory costs, we evaluate the interest paid in both situations due to payment delays.

Case I: $(D<t_1)$ payment delay is less than the cycle time:

The interest earned from 0 to $t_1$ is provided by

$$\begin{array}{c}\hfill I{E}_1=C_p{I}_e{\int }_0^{t_1}tR(p,s,q)dt=C_p{I}_e\left({\displaystyle \frac{a{t}_1^2}{2}}+(xr+yq-bp){\displaystyle \frac{t_1^3}{3}}\right).\end{array}$$

(15)

The retailer’s cycle-specific interest rate is calculated by

$$\begin{array}{ccc}\hfill I{P}_1& =& C_p{I}_p{\int }_D^{T}I\left(t\right)dt=C_p{I}_p\left({\int }_D^{t_1}I\left(t\right)dt+{\int }_{t_1}^{T}I\left(t\right)dt\right)\hfill \\ \hfill ‌& =& {\displaystyle \frac{C_p{I}_p}{{\theta }^3}}\left[\right((e^{\theta (t_1-D)}-1)\left(a\theta -(xr+yq-bp)\right)+\theta (xr+yq-bp)t_1)\hfill \\ & & -\theta (a\theta -(xs+yq-bp))(t_1-M)-{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(t_1^2-D^2)\hfill \\ & & +{\displaystyle \frac{{\theta }^3(T-t_1)}{6}}\left(3a(t_1-T)+(xr+yq-bp)(2t_1^2-T^2-T{t}_1)\right)].\hfill \end{array}$$

(16)

In case $(D<t_1)$, the total inventory cost per unit of time $T{P}_1(t_1,T)$ is calculated as ${\displaystyle \frac{1}{T}}\left(SR-HC-PC-OC-DC-SC-I{P}_1-+I{E}_1-AIC\right)$.

$$\begin{array}{ccc}\hfill ‌& & T{P}_1(t_1,T)={\displaystyle \frac{1}{T}}[p\left(aT+{\displaystyle \frac{(xr+yq-bp)}{2}}{T}^2\right)-{\displaystyle \frac{C_h}{{\theta }^3}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1(a\theta -(xr+yq-bp)e^{\theta t_1})-{\displaystyle \frac{{\theta }^2{t}_1^2}{2}}(xr+yq-bp))-{\displaystyle \frac{C_p}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & +t_1\theta (xr+yq-bp)e^{\theta t_1}+a{\theta }^2(T-t_1)+{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(T^2-t_1^2))-C_0\hfill \\ & & -{\displaystyle \frac{C_d}{{\theta }^2}}\left((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))-\theta t_1(xr+yq-bp)e^{\theta t_1}-{\theta }^2\left(a{t}_1+{\displaystyle \frac{(xr+yq-bp)t_1^2}{2}}\right)\right)\hfill \\ & & +C_b\left(a{t}_1(T-t_1)-{\displaystyle \frac{a}{2}}(T^2-t_1^2)+{\displaystyle \frac{(xr+yq-bp)}{6}}\left(3t_1^2(T-t_1)-(T^3-t_1^3)\right)\right)\hfill \\ & & -{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}((e^{(t_1-M)}-1)(a\theta -(xr+yq-bp))+\theta (xr+yq-bp)t_1\hfill \\ & & -\theta (a\theta -(xr+yq-bp))(t_1-M)-{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(t_1^2-M^2)\hfill \\ & & +{\displaystyle \frac{{\theta }^3(T-t_1)}{6}}\left(3a(t_1-T)+(xr+yq-bp)r(2t_1^2-T^2-T{t}_1)\right))\hfill \\ & & +C_p{I}_e\left({\displaystyle \frac{a{t}_1^2}{2}}+(xr+yq-bp){\displaystyle \frac{t_1^3}{3}}\right)-AT].\hfill \end{array}$$

(17)

Case II:($t_1<D$) payment delay is greater than the cycle time:

Here $I_e$ is the retailer interest earned per cycle at the return rate. If $t_1<D$, per cycle’s annual interest is given by

$$\begin{array}{ccc}\hfill I{E}_2& =& C_p{I}_e\left({\int }_0^{t_1}tR(p,s,q)dt+(M-t_1){\int }_0^{t_1}R(p,r,q)dt\right)\hfill \\ & =& C_p{I}_e\left(3a{t}_1(2M-t_1)+(xr+yq-bp)t_1^2(3M-t_1)\right).\hfill \end{array}$$

(18)

In case ($t_1<D$), the total inventory cost per unit of time $T{P}_2(t_1,T)$ is calculated as ${\displaystyle \frac{1}{T}}\left(SR-HC-PC-OC-DC-SC-I{P}_2+I{E}_2-AIC\right)$

$$\begin{array}{ccc}\hfill ‌& & T{P}_2(t_1,T)={\displaystyle \frac{1}{T}}[p\left(aT+{\displaystyle \frac{(xr+yq-bp)}{2}}{T}^2\right)-{\displaystyle \frac{C_h}{{\theta }^3}}((e^{\theta t_1}-1)(a\theta -(rs+yq-bp))\hfill \\ & & -\theta t_1\left(a\theta -(xs+yq-bp)e^{\theta t_1}\right)-{\displaystyle \frac{{\theta }^2{t}_1^2}{2}}(xr+yq-bp))\hfill \\ & & -{\displaystyle \frac{C_p}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))+t_1\theta (xr+yq-bp)e^{\theta t_1}+a{\theta }^2(T-t_1)\hfill \\ & & +{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(T^2-t_1^2))-C_0-{\displaystyle \frac{C_d}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1(xr+yq-bp)e^{\theta t_1}-{\theta }^2\left(a{t}_1+{\displaystyle \frac{(xr+yq-bp)t_1^2}{2}}\right))\hfill \\ & & +C_b\left(a{t}_1(T-t_1)-{\displaystyle \frac{a}{2}}(T^2-t_1^2)+{\displaystyle \frac{(xr+yq-bp)}{6}}\left(3t_1^2(T-t_1)-(T^3-t_1^3)\right)\right)\hfill \\ & & +C_p{I}_e\left(3a{t}_1(2M-t_1)+(xr+yq-bp)t_1^2(3M-t_1)\right)-AT].\hfill \end{array}$$

(19)

The total profit of the inventory system assessed per unit of time throughout the planning cycle for both the payment delays channel in AM technology expenditure.

5. Methodology

This section examines the optimal solution for sustainable inventory management, focusing on investments in additive manufacturing technology and credit options.

A function denoted as $\mathsf{\Gamma }\left(t\right)={\displaystyle \frac{G\left(t\right)}{H\left(t\right)}}$, $t_1\in R^n$ is strictly pseudo-concave under the condition that not only is the function I(t) differentiable but also strictly concave. Additionally, J(t) is infinite and strictly adheres to this characteristic. The outcome is derived from the analysis of the function $\mathsf{\Gamma }\left(t\right)$.

Theorem 1. 

If $\left(e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)+(xr+yq-bp)e^{\theta t_1}(t_1\theta +2)({\displaystyle \frac{C_h}{\theta }}-C_p-C_d)+{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}\left((t_1-M)e^{(t_1-M)}(a\theta -(xr+yq-bp))\right)+a(C_b-C_p{I}_p-C_p{I}_e)+(xr+yq-bp)((T-2t_1)(C_p{I}_p-C_p)-C_p{I}_e{e}^{\theta t_1})\right)\left(a(C_b-C_p{I}_p)+(xr+yq-bp)(T(C_b-C_p{I}_p)+C_p-S)\right)>{\left((C_b-C_p{I}_p)(a+t_1(xr+yp-bp))\right)}^2$. The Hessian matrix pertaining to $T{P}_1(t_1,T)$ demonstrates negative definiteness, thereby establishing that $T{P}_1(t_1,T)$ achieves a global maximum at the coordinates $(t_1^{*},T^{*})$. Importantly, the coordinates $(t_1^{*},T^{*})$ represent a singular and unique point of attainment.

Proof. 

: Let us consider

$$\begin{array}{ccc}\hfill G(t_1,T)& =& {\displaystyle \frac{1}{T}}[p\left(aT+{\displaystyle \frac{(xr+yq-bp)}{2}}{T}^2\right)-{\displaystyle \frac{C_h}{{\theta }^3}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1(a\theta -(xr+yq-bp)e^{\theta t_1})-{\displaystyle \frac{{\theta }^2{t}_1^2}{2}}(xr+yq-bp))\hfill \\ & & -{\displaystyle \frac{C_p}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))+t_1\theta (xr+yq-bp)e^{\theta t_1}+a{\theta }^2(T-t_1)\hfill \\ & & +{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(T^2-t_1^2))-C_0-{\displaystyle \frac{C_d}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1(xr+yq-bp)e^{\theta t_1}-{\theta }^2\left(a{t}_1+{\displaystyle \frac{(xr+yq-bp)t_1^2}{2}}\right))\hfill \\ & & +C_b\left(a{t}_1(T-t_1)-{\displaystyle \frac{a}{2}}(T^2-t_1^2)+{\displaystyle \frac{(xr+yq-bp)}{6}}\left(3t_1^2(T-t_1)-(T^3-t_1^3)\right)\right)\hfill \\ & & -{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}((e^{(t_1-M)}-1)(a\theta -(xr+yq-bp))+\theta (xr+yq-bp)t_1\hfill \\ & & -\theta (a\theta -(xr+yq-bp))(t_1-M)-{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(t_1^2-M^2)\hfill \\ & & +{\displaystyle \frac{{\theta }^3(T-t_1)}{6}}\left(3a(t_1-T)+(xr+yq-bp)(2t_1^2-T^2-T{t}_1)\right))\hfill \\ & & +C_p{I}_e\left({\displaystyle \frac{a{t}_1^2}{2}}+(xr+yq-bp){\displaystyle \frac{t_1^3}{3}}\right)-AT].\hfill \end{array}$$

(20)

and

$H(t_1,T)=T$

Partial differentiation with respect to both $t_1$ and T of the Eq.-(20)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial G(t_1,T)}{\partial t_1}}& =& [-{\displaystyle \frac{C_h}{{\theta }^3}}\left(\theta e^{\theta t_1}(a\theta -(xr+yq-bp))-\theta (a\theta -(xr+yq-bp)(t_1\theta e^{\theta t_1}+e^{\theta t_1})\right)\hfill \\ & & -{\theta }^2(xr+yq-bp)t_1)-{\displaystyle \frac{C_p}{{\theta }^2}}(\theta e^{\theta t_1}(a\theta -(xr+yq-bp))+\theta (xr+yq-bp)\hfill \\ & & (t_1\theta e^{t_1}+e^{\theta t_1})-a{\theta }^2-{\theta }^2(xr+yq-bp)t_1-{\displaystyle \frac{C_d}{{\theta }^2}}(\theta e^{\theta t_1}(a\theta -(xr+yq-bp))\hfill \\ & & -\theta (xr+yq-bp)(t_1\theta e^{t_1}+e^{\theta t_1})-{\theta }^{2}a-(xr+yq-bp)t_1)+C_b(aT-a{t}_1\hfill \\ & & -{\displaystyle \frac{xr+yq-bp)}{6}}(6T{t}_1-6t_1^2))-{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}((t_1-M)e^{(t_1-M)}(a\theta -(xr+yq-bp))\hfill \\ & & +\theta (xr+yq-bp)+\theta (a\theta -(xr+yq-bp))-{\theta }^2(xr+yq-bp)\hfill \\ & & +{\displaystyle \frac{{\theta }^3}{6}}\left((6aT-6a{t}_1)+(xr+yq-bp)(6T{t}_1-6t_1^2))\right)+C_p{I}_e(a{t}_1+t_1^2(xr+yq-bp))].\hfill \end{array}$$

(21)

Again, partial differentiation concerning both $t_1$ and T of Eq.-(21), we can derive

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1^2}}& =& [-e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)-(xr+yq-bp)e^{\theta t_1}\hfill \\ & & (t_1\theta +2)({\displaystyle \frac{C_h}{\theta }}+C_p-C_d)-(xr+yq-bp)({\displaystyle \frac{C_h}{\theta }}-C_p-C_d)\hfill \\ & & -{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}\left((t_1-M)e^{t_1-M}(a\theta -(xr+yq-bp))\right)-a(C_b-C_p{I}_p-C_p{I}_e)\hfill \\ & & -(xr+yq-bp)\left((T-2t_1)(C_p{I}_p-C_b)-C_p{I}_{e}2t_1\right)].\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T\partial t_1}}& =& (C_b-C_p{I}_p)(a+t_1(xr+yp-bp)).\hfill \end{array}$$

(23)

By taking the partial derivative of Eq.-(19) with respect to T, we can deduce

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial G(t_1,T)}{\partial T}}& =& [s(a+(xr+yq-bp)T)-C_p(a+(xr+yq-bp)T)+C_b(a{t}_1-aT)\hfill \\ & & +{\displaystyle \frac{(xr+yq-bp)}{6}}(3t_1^2-3T^2)-{\displaystyle \frac{C_p{I}_p}{6}}(6a{t}_1-6aT+(xr+yq-bp)(3t_1^2-3T^2)].\hfill \end{array}$$

(24)

Taking the partial derivative of Eq.-(24) with respect to T allows for the derivation of

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T^2}}& =& -a(C_b-C_p{I}_p)-(xr+yq-bp)((C_p-s)+T(C_b-C_p{I}_p)).\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1\partial T}}& =& (C_b-C_p{I}_p)(a+t_1(xr+yp-bp)).\hfill \end{array}$$

(26)

The Hessian matrix for $G(t_1,T)$ is given by $\left[\begin{array}{cc}{\displaystyle \frac{\partial G(t_1,T)}{\partial t_1^2}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial t_1\partial T}}\\ {\displaystyle \frac{\partial G(t_1,T)}{\partial T\partial t_1}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial T^2}}\end{array}\right]$. The first principal minor is:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T^2}}& =& -a(C_b-C_p{I}_p)-(xr+yq-bp)((C_p-s)+T(C_b-C_p{I}_p))<0.\hfill \end{array}$$

(27)

Even more, the second principal minor is : $\left|\begin{array}{cc}{\displaystyle \frac{\partial G(t_1,T)}{\partial t_1^2}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial t_1\partial T}}\\ {\displaystyle \frac{\partial G(t_1,T)}{\partial T\partial t_1}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial T^2}}\end{array}\right|$=${\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1^2}}{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T^2}}-{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1\partial T}}{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T\partial t_1}}$

$=\left(e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)+(xr+yq-bp)e^{\theta t_1}(t_1\theta +2)({\displaystyle \frac{C_h}{\theta }}-C_p-C_d)+{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}\left((t_1-M)e^{(t_1-M)}(a\theta -(xr+yq-bp))\right)+a(C_b-C_p{I}_p-C_p{I}_e)+(xr+yq-bp)((T-2t_1)(C_p{I}_p-C_p)-C_p{I}_e{e}^{\theta t_1})\right)\left(a(C_b-C_p{I}_p)+(xr+yq-bp)(T(C_b-C_p{I}_p)+C_p-S)\right)-{\left((C_b-C_p{I}_p)(a+t_1(xr+yq-bp))\right)}^2$.

The second principal minor is >0 only if $\left(e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)+(xr+yq-bp)e^{\theta t_1}(t_1\theta +2)({\displaystyle \frac{C_h}{\theta }}-C_p-C_d)+{\displaystyle \frac{C_p{I}_p}{{\theta }^3}}\left((t_1-M)e^{(t_1-M)}(a\theta -(xr+yq-bp))\right)+a(C_b-C_p{I}_p-C_p{I}_e)+(xr+yq-bp)((T-2t_1)(C_p{I}_p-C_p)-C_p{I}_e{e}^{\theta t_1})\right)\left(a(C_b-C_p{I}_p)+(xr+yq-bp)(T(C_b-C_p{I}_p)+C_p-S)\right)>{\left((C_b-C_p{I}_p)(a+t_1(xr+yp-bp))\right)}^2$.

Given that ${\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1^2}}<0$ and $H_{2,2}>0$, the function $G(t_1,T)$ exhibits strict concavity and differentiability. Moreover, $H(t_1,T)=T$ represents a strictly positive affine function. According to theorem 3.2.5 from [61], $T{P}_1(t_1,T)$ attains the global maximum value at a specific point under the specified conditions, denoted as $(t_1^{*},T^{*}).$ This concludes the proof. □

Theorem 2. 

If $\left(e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)+e^{\theta t_1}(t_1\theta +2)(xr+yq-bp)({\displaystyle \frac{c_h}{\theta }}+C_p-C_d)+(xr+yp-bp)(-{\displaystyle \frac{C_h}{\theta }}-C_p-C_d)+a(C_b+6C_p{I}_e)+(xr+yq-bp)(C_b(T-2t_1)+6C_p{I}_p(M-t_1)\right)\left(C_{b}a+(xr+yq-bp)(S-C_p-C_{b}T)\right)>{\left(C_b(a+(xr+yq-bp)t_1)\right)}^2$. The Hessian matrix associated with $T{P}_2(t_1,T)$ consistently exhibits negative definiteness, signifying that it attains its global maximum at the singular point $(t_1^{*},T^{*})$, and this extremum is unique.

Proof. 

 : Let us consider

$$\begin{array}{ccc}\hfill G(t_1,T)& =& {\displaystyle \frac{1}{T}}[p\left(aT+{\displaystyle \frac{(xr+yq-bp)}{2}}{T}^2\right)-{\displaystyle \frac{C_h}{{\theta }^3}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1\left(a\theta -(xr+yq-bp)e^{\theta t_1}\right)-{\displaystyle \frac{{\theta }^2{t}_1^2}{2}}(xr+yq-bp))\hfill \\ & & -{\displaystyle \frac{C_p}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))+t_1\theta (xr+yq-bp)e^{\theta t_1}+a{\theta }^2(T-t_1)\hfill \\ & & +{\displaystyle \frac{{\theta }^2(xr+yq-bp)}{2}}(T^2-t_1^2))-C_0-{\displaystyle \frac{C_d}{{\theta }^2}}((e^{\theta t_1}-1)(a\theta -(xr+yq-bp))\hfill \\ & & -\theta t_1(xr+yq-bp)e^{\theta t_1}-{\theta }^2\left(a{t}_1+{\displaystyle \frac{(xr+yq-bp)t_1^2}{2}}\right))\hfill \\ & & +C_b\left(a{t}_1(T-t_1)-{\displaystyle \frac{a}{2}}(T^2-t_1^2)+{\displaystyle \frac{(xr+yq-bp)}{6}}\left(3t_1^2(T-t_1)-(T^3-t_1^3)\right)\right)\hfill \\ & & +C_p{I}_e\left(3a{t}_1(2M-t_1)+(xr+yq-bp)t_1^2(3M-t_1)\right)-AT].\hfill \end{array}$$

(28)

and $H(t_1,T)=T.$

Using partial differentiation with regard to $t_1$ and T of Eq.-(28), one can derive

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial G(t_1,T)}{\partial t_1}}& =& [-{\displaystyle \frac{C_h}{{\theta }^3}}(\theta e^{\theta t_1}(a\theta -(xr+yq-bp))-\theta (a\theta +(xr+yq-bp)\hfill \\ & & (t_1\theta e^{\theta t_1}+e^{\theta t_1})-{\theta }^2(xr+yq-bp)t_1)-{\displaystyle \frac{C_p}{{\theta }^2}}(\theta e^{\theta t_1}(a\theta -(xr+yq-bp))\hfill \\ & & +\theta (xr+yq-bp)(t_1\theta e^{\theta t_1}+e^{\theta t_1})+a{\theta }^2-{\theta }^2(xr+yq-bp)t_1)\hfill \\ & & -{\displaystyle \frac{C_d}{{\theta }^2}}(e^{\theta t_1}(a\theta -(xr+yq-bp))-\theta (xr+yq-bp)(t_1\theta e^{\theta t_1}+e^{\theta t_1})\hfill \\ & & -{\theta }^2(a+(xr+yq-bp)t_1))+C_b\left(aT-a{t}_1+(xr+yq-bp)(T{t}_1-t_1^2)\right)\hfill \\ & & +C_p{I}_e\left(6Ma-6a{t}_1+(xr+yq-bp)(6M{t}_1-6t_1^2)\right)].\hfill \end{array}$$

(29)

Using partial differentiation with regard to both $t_1$ and T of Eq.-(29), we can find

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1^2}}& =& [-e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)-e^{\theta t_1}(t_1\theta +2)\hfill \\ & & (xr+yq-bp)({\displaystyle \frac{C_h}{\theta }}+C_p-C_d)r-(xr+yq-bp)({\displaystyle \frac{-C_h}{\theta }}-C_p-C_d)\hfill \\ & & -a(C_b+C_p+6C_p{I}_e)-(xr+yq-bp)(C_b(T-2t_1)-6C_p{I}_e(6M-6t_1)].\hfill \end{array}$$

(30)

and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T\partial t_1}}& =& C_b\left(a+(xr+yq-bp)t_1\right).\hfill \end{array}$$

(31)

By partially differentiating Eq.-(28) with regard to T, we can obtain

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial G(t_1,T)}{\partial T}}& =& S(a+(xr+yq-bp)T)+C_b\left(a{t}_1-aT+{\displaystyle \frac{(xr+yq-bp)}{6}}(3t_1^2-3T^2)\right)\hfill \\ & & -C_p(xr+yq-bp).\hfill \end{array}$$

(32)

Taking the partial derivative of Eq.-(32) with respect to T , can derive

$$\begin{array}{ccc}\hfill {\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T^2}}& =& -\left(C_{b}a+(xr+yq-bp)(S-C_p-C_{b}T)\right).\hfill \end{array}$$

(33)

Now the Hessian matrix of $G(t_1,T)$ is given by $\left[\begin{array}{cc}{\displaystyle \frac{\partial G(t_1,T)}{\partial t_1^2}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial t_1\partial T}}\\ {\displaystyle \frac{\partial G(t_1,T)}{\partial T\partial t_1}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial T^2}}\end{array}\right]$. The first principal minor is:

$\left|\begin{array}{cc}{\displaystyle \frac{\partial G(t_1,T)}{\partial t_1^2}}& {\displaystyle \frac{\partial F(t_1,T)}{\partial t_1\partial T}}\\ {\displaystyle \frac{\partial G(t_1,T)}{\partial T\partial t_1}}& {\displaystyle \frac{\partial G(t_1,T)}{\partial T^2}}\end{array}\right|$=${\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1^2}}{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T^2}}-{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial t_1\partial T}}{\displaystyle \frac{{\partial }^{2}G(t_1,T)}{\partial T\partial t_1}}$

The second principal minor is positive >0 only if

$\left(e^{\theta t_1}(a\theta -(xr+yq-bp))({\displaystyle \frac{C_h}{\theta }}+C_p+C_d)+e^{\theta t_1}(t_1\theta +2)(xr+yq-bp)({\displaystyle \frac{c_h}{\theta }}+C_p-C_d)+(xr+yp-bp)(-{\displaystyle \frac{C_h}{\theta }}-C_p-C_d)+a(C_b+6C_p{I}_e)+(xr+yq-bp)(C_b(T-2t_1)+6C_p{I}_p(M-t_1)\right)\left(C_{b}a+(xr+yq-bp)(S-C_p-C_{b}T)\right)-{\left(C_b(a+(xr+yq-bp)t_1)\right)}^2>0$.

Since ${\displaystyle \frac{{\partial }^{2}T{P}_2(t_1,T)}{\partial t_1^2}}<0$ and $H_{2,2}>0$, the function $G(t_1,T)$ is both strictly concave and differentiable.

In addition, the function $H(t_1,T)=T$ is strictly positive and affine. Based on theorem 3.2.5 of [61], $T{P}_2(t_1,T)$ attains its global maximum at the unique point determined by the necessary conditions, specifically, it is $(t_1^{*},T^{*}).$ This concludes the proof. □