Smart Production System with Random Imperfect Process, Partial Backordering, and Deterioration in an Inflationary Environment

Abstract

In today’s digital age, industrial methods are shifting away from humans and toward machines. We choose automated systems for various jobs related to production systems, such as screening, manufacturing. A smart manufacturing system is one in which machines take the place of humans. Under the influence of inflation, this study proposes a smart production-inventory model with partial backlogging, and an imperfect manufacturing process where the deterioration rate is constant. Every production system, in reality, has a random defect rate. A screening procedure is required due to the manufacture of some defective items, which is carried out by machine, i.e., by an automated system. Carbon is released during the manufacturing process due to actions such as holding deterioration. As a result, carbon emissions are taken into account in the current study. The goal of this study is to reduce total inventory costs as much as possible. To demonstrate the proposed model’s practical application, many numerical examples and sensitivity assessments with graphs are provided.

1. Introduction

Carbon emissions are a major contributor to global warming. Today, carbon emissions are a serious global issue, and many governments are taking considerable steps to reduce them. During the 1990s, several international meetings and conferences were held to try to discover a solution to this problem. In 1997, an international agreement known as the Kyoto Protocol was signed to reduce greenhouse gases and carbon dioxide in the atmosphere. The carbon Cap and Trade program, which is particularly useful in cutting carbon emissions, is one of the most effective methods of the Kyoto Protocol. This system sets a carbon emissions cap for all industries and allows them to buy and sell carbon emission rights within that cap. Industry (www.epa.gov/ghgemissions, accessed on 1 March 2022), transportation, and electricity production are the key sources of carbon emissions, according to the United States (US) Environmental Protection Agency. In 2019, the industry contributed to 23% of total greenhouse gas emissions, transportation accounted for 29%, and power generation accounted for 25% of all greenhouse gas emissions in the US (Figure 1).

Carbon emissions are caused by automobiles, industries, power generation, and so other industries. Industries are directly linked to various economies from an economic standpoint. As a result, while some international policies and agreements can help reduce carbon emissions, a complete removal is not conceivable. At this time and for profitability, a smart production system is required. Because of the high level of market competition nowadays, every firm strives to produce error-free products to improve their brand value, thus inspection is necessary for the production system. Automation is a smart method of maximizing the use of machine resources. Machine inspection, rather than human inspection, ensure errorless items and transforms the manufacturing system into a smart manufacturing system.

Because of increased concerns among stakeholders such as international organizations, governments, supply chain partners, and consumers about the environmental impact due to the manufacturing, shipping, storing, reworking, and disposal of products, many companies are investing in carbon emission reduction. To address these environmental concerns, businesses have implemented or are considering investments in carbon emissions reduction in a variety of ways, including the use of energy-efficient equipment and facilities, renewable energy resources, product and packaging redesign, and process optimization. All over the world, different nations opt for the carbon tax and cap mechanism to achieve this; the term cap is used to describe a limit on the total quantity of carbon that can be emitted by a firm in the system.

To achieve the task of meeting sustainable development goals, knowing the actual picture of the financial position of the business is very important. In developing countries such as Indonesia, India, Iraq, the role of inflation is very crucial. In general, inflation refers to a rise in the price level of an economy over time (or price inflation). When the price level rises, each unit of currency buys fewer goods and services. In other ways, it can says that inflation reduces the purchasing power of a currency. The following sections are included in the research work: Section 1 contains the introduction. Section 2 includes the literature survey and research gaps. Section 3 consists of the assumptions and notation, and in Section 4, the formulation of the mathematical model is described. The solution methodology is discussed in Section 5. Section 6 contains numerical analysis and Section 7 consists of marginal insight, and finally in Section 8, concluding remarks with future scope are given. The flow chart is given below in Figure 2.

2. Literature Review

2.1. Inventory Models with Carbon Emissions

Global warming is mostly caused by carbon emissions. Some governments are focusing their efforts on reducing carbon emissions right now. The main causes of greenhouse gas emissions that contribute to global warming are human activities such as burning fossil fuels for industrial purposes, generating electricity, and driving automobiles. From a global economic standpoint, the industry is intrinsically related to the economy. Hua et al. [1] examined carbon emissions and how corporations manage carbon emissions through carbon emission trading schemes. Carbon emission metrics were linked to several decision variables in a model proposed by Benjaafar et al. [2]. They looked at how emissions control policies affected prices and emissions. The carbon cap-and-trade mechanism was considered by Lou et al. [3] in the study of a two-stage supply chain policy, and in addition to this, investment was made for opting for green technology. In order to maximize profit and curb carbon emissions, Datta [4] designed an inventory model by considering green technology and carbon tax regulations. An economic production quantity (EPQ) model was designed by Daryanto et al. [5], incorporating deterioration as well as a percentage of defective items in the model. They used the concept of a secondary market for defective things with a discount price to see the environmental issues of carbon emission costs, and carbon cap and tax policies are included in its decision model. Jauhari et al. [6] investigated a system of a closed supply chain in which they took a single producer, single retailer, and single collector. Green technologies, faulty production, and remanufacturing procedures were all taken into consideration. Mishra et al. [7] explored a carbon pricing and cap mechanism for a sustainable supply chain management. They looked at energy efficient products with a green technology investment. Yadav et al. [8] examined the reduction of carbon emissions through the selection of items with cross-price elasticity of demand to form a sustainable supply chain. Sarkar et al. [9] discussed carbon tax policy in a sustainable managerial decision-making problem for a substitutable product in a dual channel.

2.2. Inventory Models with Backlogging

Backlog refers to the inability to meet demand quickly from stock. The consumer is expected to wait until the demand is met, which may take some time. In this period, some customers may choose other sources to fulfill demand, so there is a partial need for the present scenario. Considering time-varying backlogging, Sarkar et al. [10] calcalculated inventory within a smart production model considering Stackelrberg game policy. Wee et al. [11] considered two partial backordering costs. They considered linear and fixed backordering costs. Considering stock-and-price-dependent demand, Mishra et al. [12] investigated a controllable deterioration rate inventory model with shortages. Two cases of backordering were discussed: (i) partial backordering and (ii) full backordering. Preservation technology is also being considered. A supply chain model with reverse logistics was developed by Singh and Rani [13] with partially backordered items. They also considered inflation, defective production, and remanufacturing. A model in which demand is taken as stock-and-price-dependent with time-varying holding costs, partial backlog, and a quantity discount was described by Palanivel and Suganya [14]. Further, a concept of partial backlogging was introduced by Kumar et al. [15] in two warehouse inventory models with stock-dependent demand.

2.3. Inventory Models with Random Imperfect Production Rate

It is common knowledge that any imperfect real-world production system has a random imperfect rate. This research takes a step in that approach by extending an inventory model to include random faulty rates. A supply chain with a random demand rate for different warehouse models were explored by Sarkar et al. [16], considering a single-stage production process. Jawla and Singh [17] gave a multi-item and imperfect-item economic production quantity model. They considered a multi-setup for production and rework, and also discussed a learning environment with preservation technology. Khara et al. [18] designed an inventory model considering the influence of imperfect manufacturing on cost. Moreover, demand was taken as reliability-dependent. Considering imperfect production systems, Al-Salamah [19] developed an EPQ model with adjustable rework rates that are both synchronous and asynchronous. In asynchronous production, defective things are held until the entire production lot is finished, but in synchronous production, as soon as the defective goods are generated, they are reworked. With a multiple transportation facility, Sarkar et al. [20] explored a supply chain model for biodegradable products. In a similar direction, Gupta et al. [21] discussed multiple distribution network with integrated policy. Lin [22] developed an EPQ model with uncertain demand, an imperfect rework process, and backlogging. He assumed that imperfect items were stocked separately and then reworked.

2.4. Inventory Models with Inflation

If money was fully neutral, inflation would have no influence on the real economy. Nevertheless, full neutrality is a concept that is not universally accepted, so consideration of inflation is necessary. The impacts of the time value of money and inflation on inventories were explored by Chandra and Michael [23]. For external and internal costs, they assumed two different inflation rates. The effect of inflation was explored by Yang et al. [24] on an inventory model where partial backlogging, stock-dependent demand, and deterioration were considered. Palanivel and Uthayakumar [25] investigated the influence of the time value of money and inflation on an EPQ model with probabilistic deterioration, partial backordering, and variable production. They talked about three different kinds of continuous probabilistic deterioration functions. A green supply chain (GSC) model was developed by Rani et al. [26] and explored the effect of inflation on it. They considered that for remanufacturing products, a secondary market is available. Huang et al. [27] created a model for perishable food with replenishment policies under inflation. Under inflation, an economic order quantity (EOQ) model was designed by Alamri et al. [28] for deteriorating items with carbon emissions. Padiyar et al. [29] discussed the inventory model in an inflationary environment with an imperfect production process and preservation technology (

Table 1. Summary of selected literature and the proposed model.

Author(s)	Model Type	Carbon Emission	Volume Flexibility	Inflation	Backorder	Imperfect Process
Hua et al. [1]	EOQ	√	-	-	-	-
Benjaafar et al. [2]	SCM	√	-	-	Full	-
Lou et al. [3]	EOQ	√
Datta [4]	EPQ	√	-	-	-	Other
Daryanto et al. [5]	SCM	√	-	-	-	Constant
Jauhari et al. [6]	SCM	√	-	-	-	Constant
Mishra at el. [7]	SCM	√	-	-	-	-
Yadav et al. [8]	SCM	√	√	-	Partial	Constant
Sarkar et al. [9]	SCM	√	-	-	-	-
Sarkar et al. [10]	SCM	-	-	-	-	Random
Wee et al. [11]	EPQ	-	-	-	Partial	-
Mishra et al. [12]	EOQ	-	-	-	Both	-
Singh and Rani [13]	EOQ	-	-	√	Partial	-
Kumar [15]	EOQ	-	-	-	Partial	-
Sarkar et al. [16]	SCM	-	-	-	-	-
Jawla and Singh [17]	EPQ	-	-	√	-	Constant
Khara [18]	EPQ	-	-	-	-	Constant
Al-Salamah [19]	EPQ	-	-	-	Full	Constant
Sarkar et al. [20]	SCM	√	-	-	-	-
Sarkar et al. [21]	EOQ	-	-	-	-	-
Lin [22]	EPQ	-	-	-	Full	Constant
Chandra and Michael [23]	EOQ	-	-	√	Full	-
Yang et al. [24]	EOQ	-	-	√	Partial	-
Palanivel and Uthayakumar [25]	EPQ	-	-	√	Partial	-
Rani and Ali [26]	SCM	-	-	√	-	-
Huang et al. [27]	EOQ	-	-	√	-	-
Alamri et al. [28]	EOQ	√	-	√	-	Random
Padiyar et al. [29]	SCM	-	-	√	-	Constant
Singh and Sharma [30]	SCM	-	-	√	Partially	Constant
This paper	EPQ	√	√	√	√	√

SCM: supply chain model; EPQ: economic production quantity; EOQ: economic order quantity.

).

2.5. Research Gap and Research Contributions

From the literature, it is observed that some researchers considered the reworking process, volume flexibility, shortages, inflation, random imperfect processes, or carbon emission independence, or considered two factors or three factors at the same time. As per our knowledge, no researcher has analyzed all these factors together while modelling the production-inventory model. The current work focuses on the following:

• To make goodwill in the market, screening items is necessary before satisfying the demand of customers. An attempt is made to develop a smart production system by considering that the screening process is performed with the help of an automated machine system.
• Due to the production process, due to storage, and other different operational activities, carbon emission is unavoidable. To make the model environment sustainable, carbon emissions due to different activities associated with manufacturing systems have been considered.
• Further, obtaining the goal of sustainability is achieved with help of reworking and proper waste management.
• The benefits of a flexible manufacturing system are explored by considering the production rate-dependent production cost.
• The fluctuating nature of the market is absorbed into the model by considering inflation.
• The model is analyzed under the effect of partial backlogging.

3. Assumptions, Notation, and Problem Description

For modelling purposes, the required assumptions and notation are presented in this section.

3.1. Assumptions

• Nowadays, production systems have gradually shifted from manual to the machine. Thus, in the current study, we consider a smart production system with an automated screening process. Moreover, we have considered a single-type product inventory.
• To avoid overstocking and understocking in the production system, it is very crucial to tune up the production rate with the demand for the product. Thus, the production rate p is variable (due to the smart production system).
• Demand rate d is constant and known.
• Due to the presence of inventory in the stock, deterioration is unavoidable. Thus, it is important to consider its effect on the model as it has some economical value. Thus, in the current study, the rate of deterioration of inventory is considered as the constant parameter θ, where $0<$ θ $<1$.
• Practically, it is observed that the production cost consists of different components such as raw material cost, tool cost, labor cost, energy cost, etc., and depends on the production rate. Thus, the production cost is considered as follows:

  $$c_p=C_0+C_{1}p+\left(C_2/p\right),$$

  where $C_0,C_1,C_2$ are raw material cost, labor/energy charges, and tool/die cost, respectively.
• Shortages are permitted and partially backlogged, i.e., a portion of the demand is backlogged and has been taken as B(t) = $e^{-\epsilon t}$, where t is the time of waiting and $\epsilon >0$ is the backlogging parameter.
• The rate of imperfect production is random and follows the known probability distribution.
• A portion of the defective items will be reworkable, while the remainder will be scrapped immediately if they cannot be reworked successfully.
• To absorb the market disturbance, the whole of the study is analyzed under the effect of inflation.
• Due to the different operational activities associated with the production system, the emission of carbon is unavoidable. Thus, to control carbon emissions, a regulatory mechanism is considered.

3.2. Notation

Decision Variables

Parameters	Unit	Expression
$p$	units/unit time	Production rate
$t_1$	years	Time at which production starts
$t_3$	years	Time at which production stop

Model Parameters

$t_2$	years	Time at which backordered cleared partially
$t_4$	years	Rework process starting time
$t_5$	years	Rework process ending time
$t_6$	years	Whole length of a complete cycle
d	units/unit time	Demand rate
θ	units/unit time	Deterioration rate
$p_r$	units/unit time	Rework process rate
r	unit	Discount rate
i	unit	Rate of inflation
$\epsilon$	>0	Shape parameter of backorder rate
$s_p$	USD per setup	Setup cost for production
$s_r$	USD per setup	Setup cost for reworking station
$c_p$	USD/unit	Production cost
$c_{\theta }$	USD/unit	Deterioration cost
$c_r$	USD/unit	Cost of rework
$c_b$	USD/unit/unit time	Backorder cost
$c_s$	USD/unit	Scrapping cost
$c_L$	USD/unit	Lost sale cost
$C_0$	USD/unit	Raw material cost
$C_1$	USD/unit	Labor/energy charges
$C_2$	USD/unit	Tool/die cost
$h_s$	USD/unit/unit time	Serviceable items holding cost
$h_r$	USD/unit/unit time	Reworkable items holding cost
$e_p$	kg/unit	Carbon emissions by production
$e_{sp}$	kg/setup	Carbon emissions by setup process of production
$e_{sr}$	kg/setup	Carbon emissions by setup process of reworking station
$e_h$	kg/unit	Carbon emissions by holding the items in a warehouse
$e_r$	kg/unit	Carbon emissions due to reworking
$e_{\theta }$	kg/unit	Carbon emissions by deterioration
$e_s$	kg/unit	Carbon emissions by scrapping
$\delta$	USD/kg	Carbon tax per kg
Z	USD	Total cost
$Z_T$	USD/cycle	Total cost per cycle
$I_i\left(t\right)$	unit	Serviceable inventory level at time t, i = 1, 2, 3, 4, 5, 6
$I_j\left(t\right)$	unit	Reworkable inventory level at time t, j = 7, 8, 9
$x_p$	-	Defective proportion in production
$x_r$	-	Defective proportion in rework
$E\left[x_p\right]$	-	Expected value of defective proportion in production
$E\left[x_r\right]$	-	Expected value of defective proportion in rework

3.3. Problem Descriptions

In this section, a cleaner production inventory model is developed under the effect of inflation. Waste management and reworking are considered in the model to reduce the waste from the system as much as possible. Because of the environmentally friendly nature of the decision maker, government regulation is imposed on carbon emissions. The inventory situation of the problem is described in Figure 3 and Figure 4. Figure 3 represents the inventory situation of serviceable items, while Figure 4 represents the inventory situation of non-serviceable items. From Figure 3, it is observed that the production process started at the time $t_1$ and over the interval [$t_1,t_2$], backlogging was covered partially, and in the interval [$t_2,t_3$], the inventory level increased due to production, demand, and deterioration. The production process stopped at the time $t_3$, and in the interval [$t_3,t_4]$, the inventory level decreased due to demand and deterioration. Rework started at the time $t_4$, and in the interval [$t_4,t_5$], the inventory level increased due to rework, demand, and deterioration, and decreased in the interval due to demand and deterioration.

From Figure 4, it is observed that over the interval [$t_1,t_3$], the inventory level increased due to the random imperfectness of the production process, and in the interval [$t_3,t_4]$, the inventory level decreased due to the deterioration of imperfect items, and over the interval [$t_4,t_5]$, the inventory level decreased due to deterioration and rework, and reached zero at the time $t_5$.

4. Mathematical Model of Smart Production System

Initially, the inventory cycle starts with shortages in the interval [0, $t_1$], and at the time $t_1$, a fraction of the shortage is cleared by the backlogging rate B(t). In the interval [$t_1,t_2$], the inventory grows with (production, demand, and deterioration), in [$t_2,t_3$], the inventory decreases by (demand and deterioration), in [$t_3,t_4]$ with (rework, demand, and deterioration), and in the interval [$t_4,t_5$], the inventory decreases (demand and deterioration).

The inventory level declines due to the production, rework, demand, and deterioration rates. The inventory level of serviceable inventories can be represented mathematically with the help of the following differential equations:

$$\frac{d{I}_1\left(t\right)}{dt}=-dB\left(t_1-t\right)=-d{e}^{-\epsilon \left(t_1-t\right)},0\le t\le t_1;I_1\left(0\right)=0$$

(1)

$$\frac{d{I}_2\left(t\right)}{dt}=p\{1-E[x_p]\}-d,t_1\le t\le t_2,I_2\left(t_2\right)=0$$

(2)

$$\frac{d{I}_3\left(t\right)}{dt}=p\{1-E[x_p]\}-\theta I_3\left(t\right)-d,t_2\le t\le t_3,I_3\left(t_2\right)=0$$

(3)

$$\frac{d{I}_4\left(t\right)}{dt}=-d-\theta I_4\left(t\right),t_3\le t\le t_4,I_4\left(t_4\right)=0$$

(4)

$$\frac{d{I}_5\left(t\right)}{dt}=p_r\{1-E\left[x_r\right]\}-d-\theta I_5\left(t\right),t_4\le t\le t_5,I_5\left(t_4\right)=0$$

(5)

$$\frac{d{I}_6\left(t\right)}{dt}=-d-\theta I_6,t_5\le t\le t_6,I_6\left(t_6\right)=0.$$

(6)

The above-mentioned differential equations’ solutions are expressed as

$$I_1\left(t\right)=\frac{d}{\epsilon }{e}^{-\epsilon t_1}[1-e^{\epsilon t}],0\le t\le t_1$$

(7)

$$I_2\left(t\right)=[p\{1-E[x_p]\}-d]\left(t-t_2\right),t_1\le t\le t_2$$

(8)

$$I_3\left(t\right)=\frac{1}{\theta }[1-e^{-\theta \left(t-t_2\right)}][p\{1-E[x_p]\}-d],t_2\le t\le t_3,$$

(9)

$$I_4\left(t\right)=\frac{d}{\theta }[e^{\theta (t_4-t)}-1],t_3\le t\le t_4,$$

(10)

$$I_5\left(t\right)=\frac{1}{\theta }[1-e^{-\theta \left(t-t_4\right)}]\left[p_r\left\{1-E\left[x_r\right]\right\}-d\right],t_4\le t\le t_5,$$

(11)

$$I_6\left(t\right)=\frac{d}{\theta }[e^{\theta (t_6-t)}-1],t_5\le t\le t_6.$$

(12)

Similar to the above, the inventory level of reworkable inventories is governed by the following differential equations.

$$\frac{d{I}_7\left(t\right)}{dt}=pE[x_p]-\theta I_7,t_1\le t\le t_3,I_7\left(t_1\right)=0$$

(13)

$$\frac{d{I}_8\left(t\right)}{dt}=-\theta I_8,t_3\le t\le t_4,I_7\left(t_3\right)=I_8\left(t_3\right)$$

(14)

$$\frac{d{I}_9\left(t\right)}{dt}=-p_r-\theta I_9,t_4\le t\le t_5{I}_9\left(t_5\right)=0.$$

(15)

The above-mentioned differential equations’ solutions are expressed as

$$I_7\left(t\right)=\frac{1}{\theta }[1-e^{-\theta \left(t-t_1\right)}\left]pE[x_p\right],t_1\le t\le t_3,$$

(16)

$$I_8\left(t\right)=\frac{1}{\theta }{e}^{-\theta t}[e^{\theta t_3}-e^{\theta t_1}\left]pE[x_p\right],t_3\le t\le t_4,$$

(17)

$$I_9\left(t\right)=\frac{p_r}{\theta }[e^{\theta (t_5-t)}-1],t_4\le t\le t_5.$$

(18)

Using continuity $I_1\left(t_1\right)$ = $I_2\left(t_1\right)$, $I_3(t_3$) = $I_4\left(t_3\right)$, $I_5\left(t_5\right)$ = $I_6\left(t_5\right)$, and $I_8\left(t_4\right)$ = $I_9\left(t_4\right)$ and Taylor’s series expansion and omitting higher-order terms, we obtain the values of $t_2$, $t_4$, $t_5$, $t_6$ in terms of $t_1$ and $t_3$.

$$\mathrm{Assume}a=p\{1-E[x_p]\}\mathrm{and}\mathrm{b}=pE[x_p]\mathrm{c}=p_r\left\{1-E\left[x_r\right]\right\}.\mathrm{Then},$$

$$t_2=t_1+\frac{d}{\left(a-d\right)\epsilon }\left[1-e^{-\epsilon t_1}\right]$$

(19)

$$t_4=t_3+\frac{a-d}{d}(t_3-t_2)[1-\frac{\theta }{2}\left(t_3-t_2\right)]$$

(20)

$$t_6=t_5+\frac{c-d}{d}(t_5-t_4)[1-\frac{\theta }{2}\left(t_5-t_4\right)]$$

(21)

$$t_5=t_4+\frac{b}{p_r}(t_3-t_1)[1+\frac{\theta }{2}\left(t_3+t_1-2t_4\right)].$$

(22)

Now, different costs can be evaluated step by step as follows:

$$\mathrm{Production}\mathrm{cost}\mathrm{per}\mathrm{cycle}(CP)=c_p{{\displaystyle \int }}_{t_1}^{t_2}p{e}^{-Rt}dt+c_p{{\displaystyle \int }}_{t_2}^{t_3}p{e}^{-Rt}dt=\frac{c_{p}p}{R}\left[e^{-R{t}_1}-e^{-R{t}_3}\right].$$

$$\mathrm{Rework}\mathrm{cost}\mathrm{per}\mathrm{cycle}(CR)=c_r{{\displaystyle \int }}_{t_4}^{t_5}{p}_r{e}^{-Rt}dt=\frac{c_r{p}_r}{R}\left[e^{-R{t}_4}-e^{-R{t}_5}\right].$$

$$\mathrm{Setup}\mathrm{cost}\mathrm{per}\mathrm{cycle}(C{S}_{pr})=s_p+s_r.$$

$$\begin{array}{ll}\mathrm{Backorder}\mathrm{cost}(CB)& =-c_b[{{\displaystyle \int }}_0^{t_1}{I}_1\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_1}^{t_2}{I}_2\left(t\right)e^{-Rt}dt].\\ & =-c_b[\frac{d}{\epsilon }{e}^{-\epsilon t_1}\{\frac{1}{R}(1-e^{-R{t}_1})+\frac{1}{\epsilon -R}(1-e^{\left(\epsilon -R\right)t_1})\}+[p\{1-E[x_p]\}-d].\\ & [\frac{1}{R}{e}^{-R{t}_1}\left(t_1-t_2\right)+\frac{1}{R^2}(e^{-R{t}_1}-e^{-R{t}_2})]].\end{array}$$

$$\mathrm{Lost}\mathrm{sale}\mathrm{cost}\mathrm{per}\mathrm{cycle}(CL)=c_L{{\displaystyle \int }}_0^{t_1}(1-e^{-\epsilon \left(t_1-t\right)})d{e}^{-Rt}dt=c_{L}d[\frac{1}{R}\left[(1-e^{-R{t}_1}\right)-\frac{e^{-\epsilon t_1}}{\epsilon -\mathrm{R}}(e^{t_1\left(\epsilon -R\right)}-1)].$$

The total deterioration cost is given by

$$\begin{array}{lll}C\theta & =& c_p{{\displaystyle \int }}_{t_2}^{t_3}p\{1-E[x_p]\}{e}^{-Rt}dt-c_p{{\displaystyle \int }}_{t_2}^{t_4}d{e}^{-Rt}dt+c_r{{\displaystyle \int }}_{t_4}^{t_5}{p}_r\left\{1-E\left[x_r\right]\right\}{e}^{-Rt}dt\\ & & -c_r{{\displaystyle \int }}_{t_4}^{t_6}d{e}^{-Rt}dt+c_p{{\displaystyle \int }}_{t_1}^{t_3}pE[x_p]e^{-Rt}dt-c_p{{\displaystyle \int }}_{t_4}^{t_5}{p}_r\left\{1-E\left[x_r\right]\right\}{e}^{-Rt}dt\\ & =& c_{p}p\{1-E[x_p]\}\frac{1}{R}\left[e^{-R{t}_2}-e^{-R{t}_3}\right]-c_{p}d\frac{1}{R}[e^{-R{t}_2}-e^{-R{t}_4}]+c_r{p}_r\left\{1-E\left[x_r\right]\right\}\\ & & \frac{1}{R}[e^{-R{t}_4}-e^{-R{t}_5}]-[c_{r}d\frac{1}{R}{e}^{-R{t}_4}-e^{-R{t}_6}]+c_{p}pE[x_p]\frac{1}{R}\left[e^{-R{t}_1}-e^{-R{t}_3}\right]\\ & & -c_p{p}_r\left\{1-E\left[x_r\right]\right\}\frac{1}{R}\left[e^{-R{t}_4}-e^{-R{t}_5}\right].\end{array}$$

Scrapping cost per cycle (CS) = $c_s{p}_{r}E{\left[x\right]}_r\left(t_5-t_4\right)e^{-R{t}_5}$.

Per cycle holding cost is the sum of (i) per cycle holding cost for serviceable inventory and (ii) per cycle holding cost for reworkable inventory. Per cycle holding cost for serviceable inventory

$$\begin{array}{ll}H{C}_s& =h_s[{{\displaystyle \int }}_{t_2}^{t_3}{I}_3\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}{I}_4\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_4}^{t_5}{I}_5\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_5}^{t_6}{I}_6\left(t\right)e^{-Rt}dt]\\ & =h_s[\frac{\{p\{1-E[x_p]\}-d\}}{\theta }[\frac{1}{R}(e^{-R{t}_2}-e^{-R{t}_3})-\frac{e^{\theta t_2}}{\theta +R}(e^{-\left(\theta +R\right)t_2}-e^{-\left(\theta +R\right)t_3})]+\frac{d}{\theta }[\frac{e^{\theta t_4}}{\theta +R}(e^{-\left(\theta +R\right)t_3}\\ & -e^{-\left(\theta +R\right)t_4})+\frac{1}{R}(e^{-R{t}_4}-e^{-R{t}_3})]+\frac{\left\{p_r\left\{1-E\left[x_r\right]\right\}-d\right\}}{\theta }[\frac{1}{R}(e^{-R{t}_4}-e^{-R{t}_5})-\frac{e^{\theta t_4}}{\theta +R}(e^{-\left(\theta +R\right)t_4}-\\ & e^{-\left(\theta +R\right)t_5})]+\frac{d}{\theta }[\frac{e^{\theta t_6}}{\theta +R}(e^{-\left(\theta +R\right)t_5}-e^{-\left(\theta +R\right)t_6})+\frac{1}{R}(e^{-R{t}_6}-e^{-R{t}_5})]].\end{array}$$

Per cycle holding cost for recoverable inventory

$$\begin{array}{ll}H{C}_r& =h_r[{{\displaystyle \int }}_{t_1}^{t_3}{I}_7\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}{I}_8\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_4}^{t_5}{I}_9\left(t\right)e^{-Rt}dt]\\ & =h_r[\frac{pE[x_p]}{\theta }[\frac{1}{R}(e^{-R{t}_1}-e^{-R{t}_3})-\frac{e^{\theta t_1}}{\theta +R}(e^{-\left(\theta +R\right)t_1}-e^{-\left(\theta +R\right)t_3})]+\frac{1}{\theta }[e^{\theta t_3}-e^{\theta t_1}\left]pE[x_p\right]\\ & \frac{1}{\theta +R}(e^{-\left(\theta +R\right)t_3}-e^{-\left(\theta +R\right)t_4})+\frac{p_r}{\theta }[\frac{e^{\theta t_5}}{\theta +R}(e^{-\left(\theta +R\right)t_4}-e^{-\left(\theta +R\right)t_5})+\frac{1}{R}(e^{-R{t}_5}-e^{-R{t}_4})].\end{array}$$

Total inventory cost per cycle

$$T{C}_1=\frac{1}{t_6}(CP+CR+C{S}_{pr}+CB+CL+C\theta +CS+H{C}_s+H{C}_r).$$

(23)

Now, due to production, rework, and deterioration, the holding carbon emissions costs can be obtained step by step as follow.

• Carbon emissions cost due to setup of production and reworking station

  $$(CS{E}_{})=\frac{\delta }{t_6}\left[e_{sp}+e_{sr}\right]$$
• Per cycle carbon emissions cost due to production

  $$(C{E}_p)=\frac{\delta }{t_6}{{\displaystyle \int }}_{t_1}^{t_3}{e}_{p}p{e}^{-Rt}dt=\frac{\delta e_{p}p}{t_{6}R}[e^{-R{t}_1}-e^{-R{t}_3}]$$
• Per cycle carbon emissions cost due to rework

  $$(C{E}_r)=\frac{\delta }{t_6}{{\displaystyle \int }}_{t_4}^{t_5}{e}_r{p}_r{e}^{-Rt}dt=\frac{\delta e_r{p}_r}{t_{6}R}[e^{-R{t}_4}-e^{-R{t}_5}]$$
• Per cycle carbon emissions cost due to scrapping

  $$(C{E}_s)=\frac{\delta }{t_6}{e}_s{p}_{r}E\left[x_r\right]\left(t_5-t_4\right)e^{-R{t}_5}$$
• Per cycle carbon emissions cost due to holding inventory

  $$\begin{array}{ll}C{E}_h=\frac{\delta }{T}& e_h[{{\displaystyle \int }}_{t_2}^{t_3}{I}_3\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}{I}_4\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_4}^{t_5}{I}_5\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_5}^{t_6}{I}_6\left(t\right)e^{-Rt}dt+\\ & {{\displaystyle \int }}_{t_1}^{t_3}{I}_7\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}{I}_8\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_4}^{t_5}{I}_9\left(t\right)e^{-Rt}dt]\\ & =\frac{\delta }{t_6}{e}_h[\frac{\{p\{1-E[x_p]\}-d\}}{\theta }[\frac{1}{R}(e^{-R{t}_2}-e^{-R{t}_3})-\frac{e^{\theta t_2}}{\theta +R}(e^{-\left(\theta +R\right)t_2}-e^{-\left(\theta +R\right)t_3})]+\frac{d}{\theta }[\frac{e^{\theta t_4}}{\theta +R}(e^{-\left(\theta +R\right)t_3}\\ & -e^{-\left(\theta +R\right)t_4})+\frac{1}{R}(e^{-R{t}_4}-e^{-R{t}_3})]+\frac{\left\{{\mathrm{p}}_{\mathrm{r}}\left\{1-E\left[x_r\right]\right\}-d\right\}}{\theta }[\frac{1}{R}(e^{-R{t}_4}-e^{-R{t}_5})-\frac{e^{\theta t_4}}{\theta +R}(e^{-\left(\theta +R\right)t_4}-\\ & e^{-\left(\theta +R\right)t_5})]+\frac{d}{\theta }[\frac{e^{\theta t_6}}{\theta +R}(e^{-\left(\theta +R\right)t_5}-e^{-\left(\theta +R\right)t_6})+\frac{1}{R}(e^{-R{t}_6}-e^{-R{t}_5})+\frac{pE[x_p]}{\theta }\\ & [\frac{1}{R}\left((e^{-R{t}_1}-e^{-R{t}_3}\right)-\frac{e^{\theta t_1}}{\theta +R}(e^{-\left(\theta +R\right)t_1}-e^{-\left(\theta +R\right)t_3})]+\frac{1}{\theta }[e^{\theta t_3}-e^{\theta t_1}\left]pE[x_p\right]\\ & \frac{1}{\theta +R}(e^{-\left(\theta +R\right)t_3}-e^{-\left(\theta +R\right)t_4})+\frac{p_r}{\theta }[\frac{e^{\theta t_5}}{\theta +R}(e^{-\left(\theta +R\right)t_4}-e^{-\left(\theta +R\right)t_5})]-\frac{1}{R}(e^{-R{t}_4}-e^{-R{t}_5})].\end{array}$$

Carbon emissions cost per cycle due to deterioration

$$\begin{array}{l}C{E}_{\theta }=\frac{\delta }{t_6}{e}_{\theta }[{{\displaystyle \int }}_{t_2}^{t_3}p\{1-E[x_p]\}{e}^{-Rt}dt-{{\displaystyle \int }}_{t_2}^{t_4}d{e}^{-Rt}dt+{{\displaystyle \int }}_{t_4}^{t_5}{p}_r\left\{1-E\left[x_r\right]\right\}{e}^{-Rt}dt-{{\displaystyle \int }}_{t_4}^{t_6}d{e}^{-Rt}dt+{{\displaystyle \int }}_{t_1}^{t_3}\theta I_7\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}\theta I_8\left(t\right)e^{-Rt}dt+{{\displaystyle \int }}_{t_3}^{t_4}\theta I_9\left(t\right)e^{-Rt}dt]\\ =\frac{\delta }{t_6}{e}_{\theta }[p\{1-E[x_p]\}\frac{1}{R}\left[e^{-R{t}_2}-e^{-R{t}_3}\right]-d\frac{1}{R}\left[e^{-R{t}_2}-e^{-R{t}_4}\right]+p_r\left\{1-E\left[x_r\right]\right\}\frac{1}{R}[e^{-R{t}_4}-\\ e^{-R{t}_5}]-d\frac{1}{R}\left[e^{-R{t}_4}-e^{-R{t}_6}\right]+pE[x_p][\frac{1}{R}\left(e^{-R{t}_1}-e^{-R{t}_3}\right)+\frac{e^{\theta t_1}}{\theta +R}(e^{-\left(\theta +R\right)t_3}-e^{-\left(\theta +R\right)t_1})\\ +pE[x_p](e^{\theta t_3}-e^{\theta t_1})\frac{1}{\theta +R}(e^{-\left(\theta +R\right)t_3}-e^{-\left(\theta +R\right)t_4})+p_r[\frac{e^{\theta t_5}}{\theta +R}(e^{-\left(\theta +R\right)t_4}-e^{-\left(\theta +R\right)t_5})+\frac{1}{R}(e^{-R{t}_5}-e^{-R{t}_4})]].\end{array}$$

Total carbon emission cost

$$T{C}_2=CS{E}_{}+C{E}_p+C{E}_r+C{E}_s+C{E}_h+C{E}_{\theta }$$

(24)

$$\mathrm{Total}\mathrm{cost}TC=T{C}_1+T{C}_2.$$

(25)

Now, it is considered that the rate of random defectiveness during production and rework follows two kinds of distribution: (1) uniform distribution and (2) triangular distribution.

Case 1. Rates of random defectiveness follow a uniform distribution.

It is assumed that the rates of random defectiveness $x_p$, $x_r$ both follow a uniform distribution. Then, the expected values of $x_p$, $x_r$ are given by

$$E\left[x_p\right]=\frac{l_{1+}{m}_1}{2},l_1>0,m_1>0,l_1<m_1$$

$$E\left[x_r\right]=\frac{l_{2+}{m}_2}{2},l_2>0,m_2>0,l_2<m_2.$$

After putting the values of $E\left[x_p\right]$, $E\left[x_r\right]$ in Equation (25), we can obtain the total cost of the system.

Objective

$$\mathrm{Min}TC=TC(p,t_1,t_3)$$

(26)

$$\mathrm{Subjected}\mathrm{to}p>0,t_1>0,t_3>0.$$

Case 2. Rates of random defectiveness follow the triangular distribution.

It is assumed that the rates of random defectiveness $x_p$, $x_r$ both follow the triangular distribution. The expected values of $x_p$, $x_r$ are given by

$$E\left[x_p\right]=\frac{l_{1+}{m}_{1+}{n}_1}{3},l_1<m_1\mathrm{and}{l}_1\le n_1\le m_1$$

$$E\left[x_r\right]=\frac{l_{2+}{m}_2+n_2}{3},l_2<m_2,\mathrm{and}{l}_2\le n_2\le m_2.$$

Now, after putting the values of $E\left[x_p\right]$, $E\left[x_r\right]$ in Equation (25), we can obtain the total cost.

Objective

$$\mathrm{Min}TC=TC(p,t_1,t_3)$$

(27)

$$\mathrm{Subjected}\mathrm{to}p>0,t_1>0,t_3>0.$$

Theoretical Result: To derive the theoretical results, we apply some of the theorems of Cambini and Martein. According to them, any function can be written as

$$f(x)=\frac{F\left(x\right)}{G\left(x\right)}x\mathsf{ϵ}{R}^n,$$

f(x) is a (strictly) pseudo-convex function, if F(x) is convex and differentiable and G(x) is positive and affine.

Theorem 1.

Cost function TC ($p,t_1,t_3$) is a pseudo-convex function with respect to$p,t_1$, and$t_3$, and hence there exist unique$p,t_1$, and$t_3$that satisfy the following equations:

$$\frac{\partial \left(TC\right)}{\partial p}=0,\frac{\partial \left(TC\right)}{\partial t_1}=0,\frac{\partial \left(TC\right)}{\partial t_3}=0$$

such that TC($p,t_1,t_3$) is minimum.

Proof: 

See Appendix A. □

5. Solution Methodology

The objective functions given by Equations (26) and (27) are nonlinear functions of $p,t_1,\mathrm{and}{t}_3$. Thus, the closed-form solution is not an easy task. Thus, in order to obtain the optimal solution, the following solution methodology has been adopted.

• Step 1: The values of all parameters are put in the objective functions in Equations (26) and (27).
• Step 2: Different derivatives of objective functions $p$, $t_1$, and $t_3$ are evaluated, and then each expression is put to zero to obtain the stationary points:

  $$\frac{\partial \left(TC\right)}{\partial p}=0$$

  (28)

  $$\frac{\partial \left(TC\right)}{\partial t_1}=0$$

  (29)

  $$\frac{\partial \left(TC\right)}{\partial t_3}=0.$$

  (30)

  On solving the above system of equations, we obtain the values $\left(p,t_1,t_3\right)$.
• Step 3: The Hessian matrix is obtained as follows to check the convexity of the objective function.

  $$H=\left[\begin{array}{c}\frac{{\partial }^{2}TC}{\partial {\left(p\right)}^2}\frac{{\partial }^{2}TC}{\partial p\partial t_1}\frac{{\partial }^{2}TC}{\partial p\partial t_3}\\ \frac{{\partial }^{2}TC}{\partial t_1\partial p}\frac{{\partial }^{2}TC}{\partial {\left(t_1\right)}^2}\frac{{\partial }^{2}TC}{\partial t_1\partial t_3}\\ \frac{{\partial }^{2}TC}{\partial t_3\partial p}\frac{{\partial }^{2}TC}{\partial t_3\partial t_1}\frac{{\partial }^{2}TC}{\partial {\left(t_3\right)}^2}\end{array}\right].$$

After that, we calculate the different principal minors of the Hessian matrix as follows:

$$D_{11}={\left(\frac{{\partial }^{2}TC}{\partial {\left(p\right)}^2}\right)}_{\left(p,t_1,t_3\right)},D_{22}={\left|\begin{array}{cc}\frac{{\partial }^{2}TC}{\partial {\left(p\right)}^2}& \frac{{\partial }^{2}TC}{\partial p\partial t_1}\\ \frac{{\partial }^{2}TC}{\partial t_1\partial p}& \frac{{\partial }^{2}TC}{\partial {\left(t_1\right)}^2}\end{array}\right|}_{\left(p,t_1,t_3\right)},D_{33}={\left|\begin{array}{c}\frac{{\partial }^{2}TC}{\partial {\left(p\right)}^2}\frac{{\partial }^{2}TC}{\partial p\partial t_1}\frac{{\partial }^{2}TC}{\partial p\partial t_3}\\ \frac{{\partial }^{2}TC}{\partial t_1\partial p}\frac{{\partial }^{2}TC}{\partial {\left(t_1\right)}^2}\frac{{\partial }^{2}TC}{\partial t_1\partial t_3}\\ \frac{{\partial }^{2}TC}{\partial t_3\partial p}\frac{{\partial }^{2}TC}{\partial t_3\partial t_1}\frac{{\partial }^{2}TC}{\partial {\left(t_3\right)}^2}\end{array}\right|}_{\left(p,t_1,t_3\right)}.$$

If D₁₁ > 0; D₂₂ > 0; and D₃₃ > 0, i.e., the Hessian matrix is positive-definite, then at the obtained point $\left(p,t_1,t_3\right)$, the objective function is minimum.

• Step 4: Thus, $\left(p^{*},t_1^{*},t_3^{*}\right)$ is an extreme point and $TC\left(p^{*},t_1^{*},t_3^{*}\right)$ is the optimal value of the objective function.

6. Numerical Analysis

The suggested model is illustrated with numerical examples in this section. For this purpose, data have been taken from the literature with appropriate modifications.

Example 1.

When$x_p$,$x_r$follow a uniform distribution.

$$\begin{array}{l}d=1200\mathrm{unit}/\mathrm{unit}\mathrm{time},p_r=0.05\mathrm{unit}/\mathrm{unit}\mathrm{time},c_{\theta }=10\\ \mathrm{USD}/\mathrm{unit},c_r=4\mathrm{USD}/\mathrm{unit};c_b=32\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time};c_L=27\mathrm{USD}/\mathrm{unit},c_s=3\\ \mathrm{USD}/\mathrm{unit},c=6,s_p=250\mathrm{USD}/\mathrm{setup},s=100\mathrm{USD}/\mathrm{setup},{h_s}_{}=5\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time},h_r\\ =6\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time},=0.08,e_p=5\mathrm{kg}/\mathrm{unit},e=2\mathrm{kg}/\mathrm{unit},e_{\mathrm{sp}}=100\mathrm{kg}/\mathrm{setup},e_{\mathit{sr}}=\\ 50\mathrm{kg}/\mathrm{setup},e_h=1\mathrm{kg}/\mathrm{unit},e=10\mathrm{kg}/\mathrm{unit},e_{\theta }=3\mathrm{kg}/\mathrm{unit},\delta =5\mathrm{USD}/\mathrm{kg},l_1=0.09,m_1=\\ 0.11,l_2=0.07,m_2=0.08,C_0=14\mathrm{USD}/\mathrm{unit},C_1=0.005\mathrm{USD}/\mathrm{unit};C_2=0.5\mathrm{USD}/\mathrm{unit};\epsilon =\\ 0.3.\end{array}$$

Now, on applying the solution methodology described in Section 4, the solution of Equations (28)–(30) is as follows:

$$p=2050\mathrm{unit}/\mathrm{unit}\mathrm{time},t_1=0.47\mathrm{years},t_3=1.48\mathrm{years}$$

Now, we evaluate the different principal minors of the Hessian matrix as follows:

D₁₁ = 0.0114417 > 0, D₁₂ = 27.1434 > 0, D₁₃ = 21,897.9 > 0,

The above result shows that the objective function is convex. Hence, the optimal solution is as follows:

$$p^{*}=2050\mathrm{unit}/\mathrm{unit}\mathrm{time},t_1^{*}=0.47\mathrm{years};t_3^{*}=1.48\mathrm{years};{\mathrm{TC}}^{*}=\mathrm{85,103.2}\mathrm{USD}/\mathrm{cycle}.$$

Example 2.

When$x_p$,$x_r$follow the triangular distribution.

$$\begin{array}{l}d=1200\mathrm{unit}/\mathrm{unit}\mathrm{time},p_r=0.05\mathrm{unit}/\mathrm{unit}\mathrm{time},c_{\theta }=10\\ \mathrm{USD}/\mathrm{unit},c_r=4\mathrm{USD}/\mathrm{unit};c_b=32\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time};c_L=27\mathrm{USD}/\mathrm{unit},c_s=3\\ \mathrm{USD}/\mathrm{unit},c=6,s_p=250\mathrm{USD}/\mathrm{setup},s=100\mathrm{USD}/\mathrm{setup},{h_s}_{}=5\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time},h_r\\ =6\mathrm{USD}/\mathrm{unit}/\mathrm{unit}\mathrm{time},=0.08,e_p=5\mathrm{kg}/\mathrm{unit},e=2\mathrm{kg}/\mathrm{unit},e_{\mathrm{sp}}=100\mathrm{kg}/\mathrm{setup},e_{\mathit{sr}}=\\ 50\mathrm{kg}/\mathrm{setup},e_h=1\mathrm{kg}/\mathrm{unit},e=10\mathrm{kg}/\mathrm{unit},e_{\theta }=3\mathrm{kg}/\mathrm{unit},\delta =5\mathrm{USD}/\mathrm{kg},l_1=0.11,m_1=\\ 0.09,l_2=0.080,m_2=0.070,C_0=14\mathrm{USD}/\mathrm{unit},C_1=0.005\mathrm{USD}/\mathrm{unit};C_2=0.5\mathrm{USD}/\mathrm{unit};\epsilon =\\ 0.3.\end{array}$$

Now, on applying the solution methodology described in Section 4, the solution of Equations (28)–(30) is as follows:

$$p=2068\mathrm{unit}/\mathrm{unit}\mathrm{time},t_1=\mathrm{years},t_3\mathrm{years}.$$

Now, we evaluate different principal minors of the Hessian matrix as follows:

D₁₁ = 0.0114417 > 0, D₁₂ = 27.1434 > 0, D₁₃ = 21,897.9 > 0.

The above result shows that the objective function is convex. Hence, the optimal solution is as follows:

$$p^{*}=2068\mathrm{unit}/\mathrm{unit}\mathrm{time},t_1^{*}=0.39\mathrm{years};t_3^{*}=1.43\mathrm{years};{\mathrm{TC}}^{*}=109855\mathrm{USD}/\mathrm{cycle}.$$

7. Sensitivity Analysis

Sensitivity analysis with respect to important parameters provides the decision maker with fundamental information about the model’s behavior (

Table 2. Sensitive analysis with respect to key parameters.

Parameters	% Change	TC	% Change in TC
Serviceable items holding cost ($h_s$)	+20	85,114.9	+0.014
	+10	85,109	+0.007
	−10	85,097.3	−0.007
	−20	85,091.5	−0.014
Reworkable items holding cost ($h_r$)	+20	90,048.8	+5.81
	+10	87,576	+2.91
	−10	82,630.4	−2.91
	−20	80,157.6	−5.81
Scrapping cost ($c_s$)	+20	85,108.4	+0.006
	+10	85,105.8	+0.003
	−10	85,100.6	−0.003
	−20	85,098	−0.006
Setup cost for production (sp)	+20	85,132	+0.034
	+10	85,117.6	+0.017
	−10	85,088.8	−0.017
	−20	85,074.4	−0.034
Setup cost for reworking station (sr)	+20	85,114.7	+0.014
	+10	85,108.9	+0.007
	−10	85,097.4	−0.007
	−20	85,091.7	−0.014
Demand rate ($d$)	+20	103,356	+21.45
	+10	93,765.1	+10.18
	−10	76,909.6	−9.63
	−20	68,973.7	−18.95
Backorder cost ($c_b$)	+20	85,536.5	+0.51
	+10	85,319.9	+0.26
	−10	84,886	−0.26
	−20	84,669.9	−0.51
Carbon tax per cycle ($\delta$)	+20	91,084.2	+7.03
	+10	88,093.7	+3.51
	−10	82,112.7	−3.51
	−20	79,122.2	−7.03
Carbon emission by production ($e_p$)	+20	90,612.4	+6.47
	+10	87,857.8	+3.24
	−10	82,348.6	−3.24
	−20	79,594	−6.47
Carbon emission due to re-working ($e_r$)	+20	85,305.8	+0.24
	+10	85,204.5	+0.12
	−10	85,001.9	−0.12
	−20	84,900.6	−0.24
Carbon emission by holding the items in warehouse ($e_h$)	+20	85,185.1	+0.096
	+10	85,144.1	+0.048
	−10	85,062.3	−0.048
	−20	85,021.3	−0.096
Carbon emission by deterioration ($e_{\theta }$)	+20	85,117.7	+0.017
	+10	85,110.4	+0.008
	−10	85,095.9	−0.009
	−20	85,088.7	−0.012
Carbon emission by scrapping ($e_s$)	+20	85,189.8	+0.10
	+10	85,146.5	+0.05
	−10	85,059.9	−0.05
	−20	85,016.6	−0.10

). This section discusses sensitivity to key parameters and their implications for the model.

The effect of the holding cost for non-serviceable and serviceable items, cost of scrapping, backorder cost, demand, setup cost for both the reworking station and manufacturing station, carbon tax, carbon emissions by production and rework, carbon emission by deterioration, and carbon emission by holding have been observed. The following inferences have been obtained.

(i)

The total cost of the system changes only slightly due to the rise in the holding cost for serviceable products, but there is a significant rise due to a rise in the holding cost for reworkable items.

(ii)

A minor change in the total inventory cost in the system is detected due to increases in the scrapping cost, setup cost for the reworking station, and setup cost for serviceable products.

(iii)

A high inclination in the total inventory cost of the system is observed due to the increase in the customer base.

(iv)

A high change in the total cost of the system is observed due to the change in emission by production, while a slight change is observed due to the change in emission by rework.

(v)

It is observed that holding cost, reworkable cost, scrapping cost, backorder cost, lost-sale cost, and carbon emissions costs are all positively correlated with total cost.

8. Managerial Insights

The current study’s goal is to deliver some constructive and economically useful insights to industry executives. The following are some of the most relevant managerial implications of the current study.

• To lessen the influence of human mistakes in the screening process, the current study advises that automated screening procedures be established.
• According to the study, the cost of carbon emissions related to production has a significant impact on the system’s total cost. As a result, managers must consider measures to reduce carbon emissions from manufacturing from both an environmental and economic standpoint.
• Managers will have an understanding of how rework benefits the environment and how it can be used to meet customer requests as a result of this research.
• Inflation is extremely sensitive in developing economies such as India, Bangladesh, and others. As a result, managers must consider this element when designing the optimal strategies for the system.

9. Concluding Remarks and Future Extension

Today’s competitive business environment had compelled sectors to make fact-based decisions to survive. Furthermore, organizations suffered financial losses as a result of carbon emissions and human errors. Furthermore, environmental challenges were linked to waste generation and carbon emissions. Taking cognizance of all these, a smart production-inventory model was developed for deteriorating items with an automated screening process. From the environmental point of view, costs due to carbon emissions, reworking, and waste management were considered in the model. Here, randomness was considered in proportion to defectiveness. The developed models were illustrated with numerical examples in both cases of uniform and triangular distributions, demonstrating which was better from an economic standpoint. Total inventory costs were determined to be 29% greater in the case of triangular distribution versus uniform distribution. The cost of carbon emissions had a significant negative influence on overall inventory cost, with an approximately 7% change in inventory cost seen as a result of this cost change. The findings suggested that managers should look for ways to expand their customer base, as the change in demand caused a 21% increase in inventory expenses. According to one study, carbon emissions have a negative impact on both the environment and the economy. Carbon emissions in the manufacturing process raise inventory costs by about 6%. As a result, managers must investigate the possibility of greenness in the manufacturing process. In this modern environment, we prefer machine systems over traditional human systems, and when it comes to reworking systems, we do not require a secondary market to sell damaged products at a discount. As a result, from an industrial standpoint, this study is relevant when considering the reworking process, backlog, and automation policy in screening. In the future, this model can be expanded to include selling price and advertisement-dependent demand to build a customer base, as well as time-varying deterioration, rework [31], and a carbon cap and trade policy for environmental sustainability. Preservation, green technologies, and outsourcing [32] can be utilized to expand this research and limit the number of imperfect objects. This study will aid in the reduction of carbon emissions and economic development analysis [33] by applying green technology within a complex production system [34].