A Supply Chain Inventory Model for a Deteriorating Material under a Finite Planning Horizon with the Carbon Tax and Shortage in All Cycles

In recent years, there has been a lot of discussion about inventory systems that emit carbon and degrade the environment. Carbon emissions have been triggering global warming for many years, which has garnered considerable attention. Carbon emission gases, such as carbon dioxide and methane, heat our planet and lead to global warming. To learn more about this, we can visit the page on Climate Change Indicators: Atmospheric Concentrations of Greenhouse Gases. Global warming produces considerable environmental harm because of its destructive, pervasive, and long-term consequences. It is quickly destroying our planet's biodiversity, ultimately causing the extinction of countless plant and animal species. Global warming also causes sea-level rise, ozone layer depletion, rising global temperatures, extreme weather conditions, drought, and flooding. Due to increasingly powerful and frequent severe weather events, global climate change and the greenhouse effect have received a lot of attention. The Paris Protocol, also known as Cap-and-Trade, was created to lessen the greenhouse effect. Under the carbon tax, companies are paid a fixed sum for every tonne of emissions they generate, while the cap-and-trade policy issues a specified number of emissions allowances per year under the cap-trade program [1, 2].

There is growing agreement that Carbon emissions from enterprise economic activities are increasingly being blamed for serious climate change and global warming. Furthermore, reducing a polluted environment is one of the most considerable economic benefits of lowering carbon emissions, and it is becoming a big worry for nations worldwide. Only a few of them consider environmental issues in supply chain management, which include reducing carbon emissions. To reduce carbon emissions, the government of any nation and several regulatory bodies have developed carbon emission programs. The primary regulatory policy is a carbon tax. A carbon tax is charged or imposed by various government agencies on commercial enterprises or businesses that create carbon dioxide throughout their production process and generate environmental destruction [3]. The primary goal of the government entities responsible for taxation is to prevent global warming and safeguard the environment. In other terms, the carbon tax is a cost levied on corporations that use harmful raw materials, such as fossil fuels, in their manufacturing process and transportation, and they are blamed for global warming. Also, a carbon tax can prevent environmental degradation and total cost of Inventory model [4].

Chen et al. investigated an emissions inventory issue using the EOQ model and carbon schemes such as carbon tax, cap, and cap-and-offset [4]. Analyzed the influence of the credit period and environmental policies on inventory management, whenever the credit period has an impact on the demand rate [5]. Determined the optimum lot size and emissions with two of the most commonly used carbon policies to reduce carbon emissions: cap-and-trade and carbon tax [6]. It is investigated that the combined price and production of various commodities in the context of the carbon tax and cap-and-trade legislation. This report considers the joint decisions of retailers concerning inventory supplies and the reducing carbon emission investments model, taking into consideration the three carbon regulatory policies. In particular, it expands the economic quantitative order model to take into account the supply of carbon decreasing expenditure, in addition to complying with the carbon cap, price, and trade policies [7]. Based entirely on the EOQ model, the researcher discusses production lot-sizing problems under the carbon tax, cap, and carbon trade norms. From the previous research study, it is known that there is very little study on carbon emission in finite planning. Therefore, I have decided to focus on it in my work.

Obsolescence, damage, and depreciation can all cause on-hand inventory to deteriorate over time, resulting in lost sales, decreased earnings, and poorer customer satisfaction. The variable fraction value of on-hand inventory represents the proportion of total inventory that is prone to deterioration over time. Managing the supply chain for goods that deteriorate and emit carbon is difficult and risky because the utility of such products can decline due to spoilage, damage, or degradation during storage or transport. Carbon emissions and deterioration mostly occur due to global warming. Many countries are increasingly focusing on reducing greenhouse gas emissions and environmental devastation, employing pivotal methods such as carbon caps and taxes to accomplish this task. Researchers examined an integrated supply chain model for a single supplier and a single buyer manufacturing stock, taking into consideration imperfect output, including reworked products, and various governmental mechanisms for decarbonization, such as a carbon tax, carbon cap-and-trade, and item deterioration, all at the same time. They developed a multi-objective framework to minimize the overall cost and emissions simultaneously. The demand for a product is generally influenced by several uncontrollable factors such as pricing, season, time, accessibility, and so on. However, addressing a fuzzy demand rather than static demand is considered [8]. Through an EOQ model, we know that retailers or buyers pay a constant carbon tax, and buyers can use one of three forms of payment to pay: cash payment, advance payment, or credit payment [9]. Credit transactions seem to be the most economical and effective of the multiple payment alternatives for lowering carbon emissions and maintaining the environment. They also state that items deteriorate over time and thus can't be sold after their expiry. Researchers addressed a production planning problem for deteriorating items, considering a carbon taxation policy with preservation technology investment [10]. This study also focuses on two carbon pollution schemes: carbon cap policies and the cap-and-trade system. It provides a single-vendor and single-buyer integrated supply chain stock model with decaying goods of inferior quality, considering carbon emissions [11]. Researchers introduced microbes initially forming a film on materials such as metals, and organic biomaterials which leads to the final degradation of the compound. A hot and humid climate catalyzes the process [12]. Due to carbon emissions, the deterioration of anything is more obvious that can’t be ignored. This paper mainly focuses on materials getting deteriorated because of corrosion, rust, rotting along with carbon emission. corrosion or rust are directly related to deterioration but indirectly related to emission. When metal products are manufactured or transported, they often require energy to be produced or moved. Additionally, there is a limited amount of research that specifically examines the relationship between carbon emission policies and asset deterioration over time in inventory control and management, and nobody has considered carbon emission policies with deterioration in a finite planning horizon for unequal cycle length with carbon emission policies.

The main objective is to control carbon emissions and find the optimal solution by considering three models: the carbon cap and tax model, excluding shortage, with partial backlogging, and full backlogging. Carbon emissions and carbon cap and taxes are the fundamental points of this article [3]. Shortages or stock-out scenarios arise in any business or industrial company due to many factors, such as pricing and a high rate of deterioration [13]. In today's scenario, taking stock-out into account in any inventory model is necessary. It is anticipated that when one merchant receives the requested amount from the other merchant, they would satisfy the customers who have been waiting and then stock the remaining items for their regular demand. However, due to their impatience or other sources accessible in the region, not all buyers waiting in line can wait for the supply to arrive, resulting in a proportion of consumers waiting to lead to lost sales. Shortages are therefore defined as partially backlogged. A model permit shortage in all iterations of replenishment except for the final cycle. Each cycle during which shortages are created begins with replenishment and finishes with a backlogged shortage in the final cycle [14]. Furthermore, presented a novel restocking approach where each cycle starts with shortfalls and ends with a positive inventory. Most researchers have considered shortage for a single cycle [15-17] and there was no literature on a shortage of carbon emission and deterioration under finite planning horizons for unequal cycle length.

The significant result of using a time-dependent quadratic demand mechanism is that it accommodates all three forms of demand functions: increasing, falling, or constant, according to demand constants [18]. Researchers introduced a supplier-retailer EOQ stock supply model with quadratic demand dependent on time and stock, as well as partial backlog throughout all cycles during a finite planned period [19]. Some researchers included a quadratic time-dependent demand function for both finite and infinite horizons in their inventory management system [20]. Furthermore, employed the quadratic time demand function in their research study under finite planning of equal replenishment cycles [21]. It is noticeable from the previous analysis that the discussion on deteriorating goods in nature has not been addressed for carbon tax cost and shortage along with carbon-dependent, time-dependent quadratic, and inventory-dependent demand under finite planning horizons for unequal cycle lengths (Table 1).

Table 1. For a review of the above literature

1.1 Research gap

So far, multiple researches on material deterioration and emission have already been conducted. Furthermore, neither of the research has considered an inventory management model that incorporates carbon emission measures including a carbon tax, covering degrading materials with emission-dependent, nonlinear time-dependent, and inventory-dependent demand, along with shortage. This is a significant and strong research gap, and the study is especially unique in that the issues highlighted are entirely economic and environmental in scope. Based on the aforementioned research, the authors claim that none of the other researchers has developed a model for managing deteriorating products, incorporating emission-dependent, time-dependent quadratic, and inventory-dependent demand over a finite horizon. Finite planning horizons refer to the period in which decisions about inventory replenishment are made. Longer planning horizons allow for more time to make decisions, potentially leading to longer replenishment cycles, while shorter planning horizons can result in shorter replenishment cycles.

The remaining sections of this article are organized as follows: Section 2 presents a list of all the assumptions and notations used in the study. In Section 3, a mathematical model is presented, and a solution to it is provided. The optimality requirements for cost equations are discussed in section 4, with the help of a theorem and the Mathematica software. Section 5 includes a numerical example, an algorithm for finding ti, si, and total cost in the Mathematica tool, as well as sensitivity analysis, and a comparison discussion with graphs and tables. In Section 6, managerial suggestions are provided. Finally, the conclusions of the proposed model are discussed.

As seen in the inventory Figure 1 for finite planning along with shortages. Differential equation of inventory level besides shortages given by Eq. (1) and Eq. (5) Boundary condition IL_i+1 (s_i+1)=0.

$\begin{aligned} \frac{d I L_{i+1}(t)}{d t}+ & \left(\theta_1+\theta_2\right) I L_{i+1}(t)=-\mathrm{D}(\mathrm{t}) \\ & t_i<\mathrm{t}<s_{i+1}\end{aligned}$      (1)

where, {i=1,2,3…………………. n₁}.

Taking θ₂=αt and the differential equation's solution is given by:

$\begin{aligned} \frac{d I L_{i+1}(t)}{d t} & =-\mathrm{D}(\mathrm{t})-\alpha t I L_{i+1}(t) \\ t_i & <\mathrm{t}<s_{i+1}\end{aligned}$      (2)

$I L_{i+1}(t)=e^{-\frac{\alpha t^2}{2}-t \theta_1} \int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\frac{\alpha u^2}{2}+t} \mathrm{du}$     (3)

$I L_{i+1}(t)=\int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du}$      (4)

$\begin{gathered}I L_{i+1}(t)=\left[\left(1+t \theta_1+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)\right. \left.-\frac{\alpha}{6}\left(t^3-t_i^3\right)\right] D(t)\end{gathered}$

The instantaneous level of shortage ILS_i+1(t) with the boundary condition ILS_i+1(s_i)=0, is given by the following differential equation:

$\frac{d I L S_{i+1}(t)}{d t}=D(t) \beta(t) \quad$ where $s_i<\mathrm{t}<t_i$      (5)

$\begin{gathered}\frac{d I L S_{i+1}(t)}{d t}=\frac{D(t)}{\delta\left(t_i-t\right)+1} \\ I L S_{i+1}(t)=\int_{s_i}^{t_i} \frac{D(t)}{\delta\left(t_i-t\right)+1} d t=\frac{D(t)\left(t_i-t\right)}{\delta\left(t_i-t\right)+1}\end{gathered}$      (6)

So total amount of inventory held during the interval held during the $\left[t_i, s_{i+1}\right]$

$R_{i+1}=\int_{t_i}^{s_{i+1}}\left\{\int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du}\right\} \mathrm{dt}$      (7)

We may express Eq. (7) as below by changing the position of integration and skipping the higher powers of α².

$\begin{aligned} R_{i+1}=\int_{t_i}^{s_{i+1}}\{(1+ & \left.\theta_1 t+\frac{\alpha}{2} t^2\right)\left(t-t_i\right) \\ & -\frac{\theta_1}{2}\left(t^2-t_i{ }^2\right) \\ & \left.-\frac{\alpha}{6}\left(t^3-t_i{ }^3\right)\right\} D(t) d t\end{aligned}$      (8)

The total amount of that quantity for which customers are waiting i.e., the amount of shortage during the interval $\left[s_i, t_i\right]$. After rearranging the ordering, $S_{i+1}$ can be given as:

$S_{i+1}=\int_{s_i}^{t_i} I L S_{i+1}(t) d t$$=\int_{s_i}^{t_i}\left\{\int_{s_i}^{t_i} \frac{D(u)}{\delta\left(t_i-u\right)+1} d u\right\} d t$$=\int_{s_i}^{t_i} \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1} d t$      (9)

The total order quantity for a finite planning horizon.

$\mathrm{Q}=\sum_{i=1}^n Q_{i+1}=\sum_{i=1}^n\left\{R_{i+1}+S_{i+1}\right\}$

$Q_{i+1}=\int_{t_i}^{s_{i+1}}\left\{\left(1+\theta_1 t+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)\right.$$\left.-\frac{\alpha}{6}\left(t^3-t_i^3\right)\right\} D(t) d t$$+\int_{s_i}^{t_i} \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1} d t$

The total number of deteriorated components throughout each replenishment is as follows:

$D_{i+1}=\int_{s_i}^{t_i} \theta_2 I L_{i+1}(t) d t$

$\begin{gathered}=\alpha t\left\{\int_{t_i}^{s_i}\left(1+\theta_1 t+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)\right. \left.-\frac{\alpha}{6}\left(t^3-t_i^3\right) D(t) d t\right\}\end{gathered}$       (10)

In the cases of some materials, the Buyer can generally not wait for some product, so only a proportion β(φ) of the demand during the stock-out duration is backlogged, where φ is the amount of time the buyer waits for the quantity of negative inventory. Therefore, the leftover proportion (1-β(φ)) is lost. Sarkar et al. [18], the amount has become lost during the interval $\left[s_i, t_i\right]$ is given as:

$L_{i+1}=\int_{s_i}^{t_i}\{D(t)-D(t) \beta(\varphi)\} d t$

$\begin{aligned} & =\int_{s_i}^{t_i}\{(1-\beta(\varphi)) D(t)\} d t \\ & =\int_{s_i}^{t_i}\left\{\delta \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1}\right\} d t\end{aligned}$      (11)

Amount of Carbon emission cost during the interval [t_i, s_i+1] can be expressed as:

$C e=\sum_{i=0}^{n_1-1} \mathrm{c}^{\hat{}}+\hat{P}_r * R_{i+1}+\widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$\begin{aligned} & C e=\sum_{i=0}^{n_1-1} \mathrm{c}^{\hat{}}+\hat{P}_r * Q_{i+1} \\ &+\widehat{h_c} \int_{t_i}^{s_{i+1}}\left\{\int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du}\right\} d t\end{aligned}$

The amount of CO₂ emissions from refrigeration systems is influenced by factors like refrigerant type, energy efficiency, usage frequency, system size, and power source. Usage of fossil fuels leads to higher emissions while energy-efficient technologies and smaller systems result in lower CO₂ emissions. According to researchers, the total cost of Carbon emission cost/carbon tax during the interval $\left[t_i, s_{i+1}\right]$ can be expressed as [22]:

$\begin{aligned} & C e=\tau\left\{\sum_{i=0}^{n_1-1}{c}^{\hat{}}+\hat{P}_r \int_{t_i}^{s_{i+1}}\left\{\left(1+\theta_1 t+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)\right.\right. \\ & \left.-\frac{\alpha}{6}\left(t^3-t_i{ }^3\right)\right\} D(t) d t \\ & +\int_{s_i}^{t_i} \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1} d t \\ & \left.+\widehat{h_c} \int_{t_i}^{s_{i+1}}\left\{\int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du}\right\} d t\right\} \\ & \end{aligned}$     (12)

Total cost = Replenishment cost + Stock holding cost + purchasing cost + Deteriorating cost + Storage cost + Lost sale cost + Carbon emission cost+ Transportation cost.

$\operatorname{TCR}\left(t_i, s_i, n_1\right)=n_1 * O_r \mathrm{z}+\sum_{i=0}^{n_1-1} H \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} W_h * Q_{i+1}$

$+\sum_{i=0}^{n_1-1} D_t * \theta_2 \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} I L S_{i+1}(t) d t$

$+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}\left\{\delta \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1}\right\} d t+\sum_{i=0}^{n_1-1} \mathrm{c}^{\wedge}+\hat{P}_r Q_{i+1}$

$+\widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$\operatorname{TCR}\left(t_i, s_i, n_1\right)=n_1 * O_r$$+\sum_{i=0}^{n_1-1} H \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t+\sum_{i=0}^{n_1-1} W_h * Q_{i+1}$

$+\sum_{i=0}^{n_1-1} D_t * \alpha t \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$$+\sum_{i=0}^{n_1-1} s \int_{s_i}^{t_i} I L S_{i+1}(t) d t$

$+\sum_{i=0}^{n_1-1} l \int_{s_i}^{t_i}\left\{\delta \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1}\right\} d t$

$+\sum_{i=0}^{n_1-1}\mathrm{c}^{\hat{}}+\hat{P}_r\left\{R_{i+1}+\int_{s_i}^{t_i} I L S_{i+1}(t) d t\right\}$$+\widehat{h_c} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$\operatorname{TCR}\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left\{H+\tau \widehat{h_c}\right\} \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$+\sum_{i=0}^{n_1-1} D_t * \alpha t \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$$+\left\{W_h+\tau \hat{P}_r\right\} \sum_{i=0}^{n_1-1} Q_{i+1}$

$+\{s+l \delta\} \sum_{i=0}^{n_1-1} \int_{s_i}^{t_i} I L S_{i+1}(t) d t+{c}^{\hat{}}$

$\operatorname{TCR}\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left(H+\tau \widehat{h_c}\right) \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$+\sum_{i=0}^{n_1-1} D_t * \alpha t \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$+\left\{W_h+\tau \hat{P}_r\right\} \sum_{i=0}^{n_1-1}\left(R_{i+1}+S_{i+1}\right)$

$+\left(s+\tau \hat{P}_r+l \delta\right) \sum_{i=0}^{n_1-1} \int_{s_i}^{t_i} I L S_{i+1}(t) d t+\tau \mathrm{c}^{\hat{}}$

$\operatorname{TC}\left(t_i, s_i, n_1\right)=n_1 * O_r$$+\sum_{i=0}^{n_1-1}\left(H+\tau \widehat{h_c}\right) \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$

$+\sum_{i=0}^{n_1-1} D_t * \alpha t \int_{t_i}^{s_{i+1}} I L_{i+1}(t) d t$$+\left\{W_h+\tau \hat{P}_r\right\} \sum_{i=0}^{n_1-1} R_{i+1}$

$+\left(s+W_h+\tau \hat{P}_r+l \delta\right) \sum_{i=0}^{n_1-1} \int_{s_i}^{t_i} I L S_{i+1}(t) d t+\tau {c}^{\hat{}}$

$\mathrm{TC}\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left(H+\tau \widehat{h_c}\right) \int_{t_i}^{s_{i+1}} \int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du} d t$

$+\sum_{i=0}^{n_1-1} D_t * \alpha t \int_{t_i}^{s_{i+1}} \int_t^{s_{i+1}} \mathrm{D}(\mathrm{u}) e^{\theta_1(u-t)+\frac{\alpha\left(u^2-t^2\right)}{2}} \mathrm{du} d t$

$+\left\{W_h+\tau \hat{P}_r\right\} \sum_{i=0}^{n_1-1} R_{i+1}$

$+\left(s+W_h+\tau \hat{P}_r+l \delta\right) \sum_{i=0}^{n_1-1} \int_{s_i}^{t_i} I L S_{i+1}(t) d t+{c}^{\hat{}}$

$\operatorname{TC}\left(t_i, s_i, n_1\right)=n_1 * O_r+\sum_{i=0}^{n_1-1}\left(H+\tau \widehat{h}_c+W_h+\tau \hat{P}_r\right) \int_{t_i}^{s_{i+1}}\left[\left(1+t \theta_1+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)-\frac{\alpha}{6}\left(t^3-t_i^3\right)\right] D(t) d t$

$+\sum_{i=0}^{n_1-1} D_t \int_{t_i}^{s_{i+1}} \alpha t\left\{\left(1+\theta_1 t+\frac{\alpha}{2} t^2\right)\left(t-t_i\right)-\frac{\theta_1}{2}\left(t^2-t_i^2\right)-\frac{\alpha}{6}\left(t^3-t_i^3\right)\right\} D(t) d t$

$+\left(s+W_h+\tau \hat{P}_r+l \delta\right) \sum_{i=0}^{n_1-1} \int_{s_i}^{t_i} \frac{\left(t_i-t\right) D(t)}{\delta\left(t_i-t\right)+1} d t+\tau {c}^{\hat{}}$

The objective is that the fundamental values of ti and si must be determined to reduce the total variable cost TC of the stock control and management. The requirements to find the values of to t_i and s_i are given below:

$\begin{aligned} & \frac{\partial T C\left(t_i, s_i, n_1\right)}{\partial t_i}=0 \\ & \frac{\partial T C\left(t_i, s_i, n_1\right)}{\partial s_i}=0\end{aligned}$

By neglecting $\alpha^2$ and the higher terms of $\alpha$, because $\alpha$ have a negligible value then we get:

$\begin{aligned} \frac{\partial T C\left(t_i, s_i, n_1\right)}{\partial s_i}=\left(H+\tau \widehat{h_c}\right. & \left.+W_h+\tau \hat{P}_r\right)\left[\left(s_i \theta_1+\frac{\alpha}{2} s_i{ }^2+1\right)\left(s_i-t_{i-1}\right)\right. \\ & \left.-\frac{\theta_1}{2}\left(s_i{ }^2-t_{i-1}^2\right)-\frac{\alpha}{6}\left(s_i{ }^3-t_{i-1}^3\right)\right] D\left(s_i\right) \\ & +D_t \alpha s_i\left[\left(s_i \theta_1+1\right)\left(s_i-t_{i-1}\right)\right. \\ & \left.-\frac{\theta_1}{2}\left(s_i{ }^2-t_{i-1}^2\right)\right] D\left(s_i\right) \\ & +\left(s+W_h+\tau \hat{P}_r+l \delta\right)\left[\frac{\left(t_i-s_i\right)}{\delta\left(t_i-s_i\right)+1}\right] D\left(s_i\right)\end{aligned}$      (15)

$\begin{aligned} \frac{\partial T C\left(t_i, s_i, n_1\right)}{\partial t_i}=\left(H+\tau \widehat{h_c}+W_h\right. & \left.+\tau \widehat{P}_r\right) \int_{t_i}^{s_{i+1}}\left[\theta_1\left(t_i-t\right)+\frac{\alpha}{2}\left(t_i^2-t^2\right)-1\right] D(t) d t \\ & +D_t \int_{t_i}^{s_{i+1}} \alpha t\left[\theta_1\left(t_i-t\right)-1\right] D(t) d t \\ & +\left(s+W_h+\tau \widehat{P}_r+l \delta\right)\left[\int_{s_i}^{t_i} \frac{D(t)}{\delta\left(t_i-t\right)^2+1} d t\right]\end{aligned}$       (16)

$\begin{aligned} \frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial s_i^2}=\frac{1}{6}\left\{\left(H+\tau \widehat{h}_c\right.\right. & \left.+W_h+\tau \widehat{P}_r\right)\left(s_i-t_{i-1}\right)\left[6\left(\theta_1+\alpha s_i\right)\left(a+b s_i+c s_i^2\right)\right. \\ & +\left(3\left(2 s_i \theta_1+\alpha s_i^2+2\right)-3 \theta_1\left(s_i+t_i-1\right)-\alpha\left(s_i^2+t_{i-1}^2\right.\right. \\ & \left.\left.\left.+s_i t_{i-1}\right)\right)\left(a+b s_i+c s_i^2\right)\right] \\ & +3 D_t \alpha\left(s_i-t_{i-1}\right)\left[\left(2-\theta_1 t_{i-1}+\theta_1 s_i\right)\left(b+2 c s_i\right)\right. \\ & \left.+\left(2-\theta_1 t_{i-1}+\theta_1 s_i\right)\left(a+b s_i+c s_i^2\right)\right] \\ & +6\left(s_h+W_h+\tau P_r^{\wedge}+I \delta\right)\left[\frac{\left(t_i-s_i\right)}{1+\delta\left(t_i-s_i\right)}\left(b+2 c s_i\right)\right. \\ & \left.\left.-\frac{1}{1+\delta\left(t_i-s_i\right)^2}\left(a+b s_i+c s_i^2\right)\right]\right\}\end{aligned}$       (17)

$\begin{aligned} \frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial t_i^2}=\left(H+\tau \widehat{h_c}\right. & \left.+W_h+\tau \widehat{P}_r\right) \int_{t_i}^{s_{i+1}}\left(\theta_1+\alpha t_i\right)\left(a+b t+c t^2\right) d t \\ & +\left(H+\tau \widehat{h}_c+W_h+\tau \widehat{P}_r\right)\left(a+b t_i+c t_i^2\right) \\ & +D_t \int_{t_i}^{s_{i+1}} \alpha t \theta_1\left(a+b t+c t^2\right) d t \\ & +D_t \alpha t_i\left(a+b t_i+c t_i^2\right) \\ & -2\left(s+W_h+\tau \hat{P}_r+l \delta\right)\left[\int_{s_i}^{t_i} \frac{\left(a+b t+c t^2\right)}{\left(\delta\left(t_i-t\right)^2+1\right)^2} d t\right]\end{aligned}$       (18)

The sufficient requirement, the Hessian matrix with TC has to be positive definite for TC to be minimum for a fixed n₁. Moreover, Theorem proves that TC is positive. Therefore, By the iterative method and using Mathematica software the optimal value of t_i and s_i for a given positive integer n₁ may be calculated from the above Eqns. (16) and (17).

Theorem: -

If t_i and s_i satisfy the inequality, (i) $\frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial t_i^2} \geq 0$, (ii) $\frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial s_i^2} \geq 0,(i i i) \frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial t_i^2}-\left|\frac{\partial T c\left(t_i, s_i, n_1\right)}{\partial t_i \partial s_i}\right| \geq 0$ and (iv)$\frac{\partial^2 T C\left(t_i, s_i, n_1\right)}{\partial s_i{ }^2}-\left|\frac{\partial T C\left(t_i, s_i, n_1\right)}{\partial s_i \partial t_i}\right| \geq 0$ for all i = 1, 2,..., n then TC will be positive definite.

The sufficient condition for TC to be minimum for a fixed n1 is that the Hessian matrix with TC has to be positive definite.

Theorem proves that TC is positive. Hence, by utilizing the iterative method and Mathematica software, the optimal values of t_i and s_i for a given positive integer n₁ may be calculated from the above Eqns. (16) and (17).

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