# An EOQ Model with Carbon Emissions and Inflation for Deteriorating Imperfect Quality Items under Learning Effect

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Literature Review with Regard to Carbon Emissions and Inspection

#### 2.2. Literature Review with Regard to Defective Items, Inflation and Inspection

#### 2.3. Literature Review with Regard to Defective Items, Inspection, and Learning Effect

#### 2.4. Research Gap

#### 2.5. Contribution Concern with Proposed Model

## 3. Assumptions

- (i)
- CO
_{2}is directly emitted from electricity and fuels consumption in product storage Daryanto et al. [14]. - (ii)
- Waste due to the deterioration process which is dangerous for the climate is properly arranged by investing in the waste management process.
- (iii)
- The continuity of replacement is allowed.
- (iv)
- Shortages and lead time are not involved in this model.
- (v)
- The screening rate is greater than the demand rate [8].
- (vi)
- The time horizon plane has been taken as finite.
- (vii)
- Lots have some defective items as per consideration by [7].
- (viii)
- Defective quality items follow the S-shaped learning curve suggested by Jaber and Goyal [25].
- (ix)
- Imperfect items are sold at rebate prices.
- (x)
- Lots have a constant deterioration rate in the whole cycle length.
- (xi)
- The inflation rate is constant.
- (xii)
- A carbon tax is allowed.

## 4. Mathematical Model

## 5. Solution Method

#### Numerical Example

## 6. Sensitivity Analysis

#### 6.1. Observations and Managerial Insights

- From Table 4, it is seen that if the number of shipments and learning rate increase from top to bottom, then cycle length, lot size, and retailer’s total profit rapidly increase up to the 10th shipment level with different learning rates. After the 10th shipment, cycle length, lot size, and retailer’s total profit increase very slowly and approach the maturity phase up to the 16th, and this phase is called the learning phase. Finally, order size, cycle length, and buyer’s total profit remain constant on 17th shipments and reach maturity phase. It means that retailers obtain the optimal length of cycle, maximum lot size, and maximum profit when the shipment is the 17th one and the learning rate is 1.4. Hence, the retailer obtains more profit due to decreased carbon emissions. It suggests that retailers should be aware of new strategies in the form of learning to obtain more profit.
- From Table 5, we analyzed that if the deterioration rate increases, then lot size, length of cycle, and buyer’s profit reduce due to deterioration. Deterioration affects the cycle time and order quantity, as well as buyer’s total profit. It reflects that the retailer should be aware during the transaction of business when products are deteriorating items. When this order quantity decreases, then carbon emissions increase due to deterioration. Hence, the retailer obtains less profit due to increased carbon emissions.
- From Table 6, we studied that if the rate of inflation increases, then the length of cycle, lot size, and retailer’s profit decrease. Inflationary situations affect the lot size, cycle length, and buyer’s total profit. It reflects that retailers should be aware during the transaction of business when products are deteriorating items. When this order quantity decreases, then the carbon emissions increase. Hence, retailers obtain less profit due to decreases in cycle time and order quantity and increase in carbon emission.

#### 6.2. Discussion with Observations

## 7. Concluding Remarks and Future Extension

_{,}and buyer’s total profit $\left(\mathsf{\Psi}\left({T}_{n}^{\ast}\right)\right)$ follow the S-shaped learning curve and achieve the maturity phase with variable shipment and learning rate. Furthermore, the retailer’s cycle length $\left({T}_{n}^{\ast}\right)$, lot size $\left({Q}^{\ast}\right)$

_{,}and retailer’s total profit $\left(\mathsf{\Psi}\left({T}_{n}^{\ast}\right)\right)$ are affected by the inflation rate and deterioration rate under carbon emissions, which are discussed in the sensitivity analysis section. Our work is important for those who want to obtain an optimal lot size, optimal cycle length, and retailer’s total profit with the various carbon emissions regulations imposed by the government or regulatory authorities. The observations revealed that (a) when the lot has more defective items, then the buyer should be more vigilant while ordering, (b) the present model provided good results when the learning rate is 1.40 and number of shipments is 17, (c) for high deterioration rate, the buyer should order less quantity more frequently, (d) in the highly inflationary market, the buyer should order a large quantity to increase his profit, and (e) the issue of environmental sustainability is addressed due to storage and is important for long-time existence. The present model is beneficial for business managers who want to obtain an optimal ordering quantity and strictly comply with carbon emission regulations imposed by the regulatory authorities or government. This research work can be extended by considering investments in green technology and exploring other mechanisms to decrease carbon releases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notations

$Q$ | Order quantity has been taken as decision variable (units). |

$D$ | Rate of demand (units/year). |

${C}_{h}$ | Holding cost (USD/unit). |

${t}_{x}$ | The imposed unit tax for emissions (USD/ton). |

${F}_{c}$ | The carbon emission factor for fuels (tones/gallon). |

${E}_{c}$ | The carbon emission factor for electricity (tones/kWh). |

${C}_{p}$ | Unit purchasing cost (USD/unit). |

$p$ | Unit selling cost for perfect items (USD/units). |

$P$ | Percentage defective items are presents in $Q$. |

$P\left(n\right)$ | Imperfect quality items are following the S-shaped learning curve. |

${c}_{s}$ | Unit selling price for imperfect items, ${c}_{s}<p$ (USD/unit). |

${C}_{s}$ | Screening cost (USD/units). |

$\theta $ | Deterioration rate (per year). |

${e}_{c}$ | The variable amount of electricity utilized to store one unit of goods per time unit (KWh/year). |

${C}_{h}=h+{t}_{x}{E}_{c}{e}_{c}$ | Holding cost due carbon emission from variable electricity (USD/unit/year). |

${\tilde{C}}_{h}={e}_{c}{F}_{c}$ | Holding cost due carbon emission from generator fuels (USD/unit/year). |

${C}_{d}$ | Deterioration cost (USD/unit). |

${C}_{w}$ | Cost of waste management due to deterioration (USD/unit). |

$\lambda $ | Screening rate, $\lambda >D$ (USD/unit/year). |

${t}_{n}$ | Inspection time (year). |

${T}_{n}$ | Cycle length (year). |

${I}_{1}\left(t\right)$ | Inventory at $t\in \left[0,{t}_{n}\right]$. |

${I}_{2}\left(t\right)$ | Inventory at $t\in \left[{t}_{n},{T}_{n}\right]$. |

$SR$ | Total sales revenue (USD). |

$TC$ | Total buyer’s cost (USD). |

$\mathsf{\Psi}\left({T}_{n}\right)$ | Total buyer’s whole profit (USD). |

$r$ | Discount rate at $i$ inflation rate. |

$R$ | $r-i$, Inflation due to discount rate. |

## Appendix A

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Author(s) | Impact of Learning | Inspection | Carbon Emissions | Deterioration | Defective Items | Inflation |
---|---|---|---|---|---|---|

Wright [9] | ✔ | |||||

Salameh and Jaber [7] | ✔ | ✔ | ||||

Jaber et al. [25] | ✔ | ✔ | ✔ | |||

Khan et al. [27] | ✔ | ✔ | ✔ | |||

Jaggi and Khanna [37] | ✔ | ✔ | ✔ | ✔ | ||

Jaggi et al. [23] | ✔ | ✔ | ✔ | ✔ | ||

Jaggi et al. [8] | ✔ | ✔ | ||||

Jaggi et al. [38] | ✔ | ✔ | ✔ | ✔ | ||

Patro et al. [39] | ✔ | ✔ | ✔ | ✔ | ||

Daryanto et al. [14] | ✔ | ✔ | ✔ | ✔ | ||

Liao et al. [40] | ✔ | ✔ | ||||

Daryanto and Christata [41] | ✔ | ✔ | ✔ | |||

Barman et al. [42] | ✔ | ✔ | ||||

Jayaswal et al. [32] | ✔ | ✔ | ||||

Jayaswal et al. [43] | ✔ | ✔ | ✔ | ✔ | ||

Mashud et al. [44] | ✔ | ✔ | ✔ | |||

This paper | ✔ | ✔ | ✔ | ✔ | ✔ |

**Table 2.**Comparison of cycle length, lot size, and buyer’s whole profit with and without learning rate.

Model with No Learning Effect | Model with Learning Effect | ||||
---|---|---|---|---|---|

Cycle Length ${\mathit{T}}_{\mathit{n}}^{\ast}$ (Year) | Lot Size ${\mathit{Q}}^{\ast}$ (Units) | Buyer’s Total Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)$ (Dollars) | Cycle Length ${\mathit{T}}_{\mathit{n}}^{\ast}$ (Year) | Lot Size ${\mathit{Q}}^{\ast}$ (Units) | Buyer’s Total Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)$ (Dollars) |

0.9032 | 46,694 | 1,472,210 | 1.0049 | 48,225 | 1,662,440 |

$D$ | $50,000\text{}\mathrm{units}/\mathrm{year}$ | ${F}_{e}$ | $0.0026\text{}\mathrm{ton}\text{}{\mathrm{CO}}_{2}/\mathrm{L}$ | $b$ | $1.40$ |

$\lambda $ | $175,000\text{}\mathrm{unit}/\mathrm{year}$ | ${E}_{e}$ | $0.0005\text{}\mathrm{ton}\text{}{\mathrm{CO}}_{2}/\mathrm{kWh},\text{}$ | $\theta $ | $0.1/\mathrm{year}$ |

$p$ | $\mathrm{USD}50/\mathrm{unit}$ | $R$ | $0.08$ | ${n}^{\ast}$ | $17$ |

${C}_{s}$ | $\mathrm{USD}0.5/\mathrm{unit}$ | ${C}_{k}$ | $\mathrm{USD}2000/\mathrm{order}$ | $p\left(17\right)$ | $2.36887\times {10}^{-8}$ |

${C}_{p}$ | $\mathrm{USD}25/\mathrm{unit}$ | ${C}_{d}$ | $\mathrm{USD}600/\mathrm{unit}$ | ${T}_{n}^{\ast}$ | $1.00941\text{}\mathrm{year}$ |

$h$ | $\mathrm{USD}60/\mathrm{unit}$ | $a$ | $40$ | ${Q}^{\ast}$ | $48,225\text{}\mathrm{units},$ |

${e}_{c}$ | $1.44\mathrm{kWh}/\mathrm{unit}/\mathrm{year}$ | $g$ | $999$ | ${t}_{n}^{\ast}$ | $0.2752\text{}\mathrm{year}$ |

${T}_{x}$ | $\mathrm{USD}75/{\mathrm{tonCO}}_{2}$ | ${C}_{h}$ | $\mathrm{USD}2/\mathrm{unit}$ | $\mathsf{\Psi}\left({T}_{n}^{\ast}\right)$ | $\mathrm{USD}1,662,440$ |

**Table 4.**Effect of learning and shipments on the cycle length, order size, and buyer’s total profit with carbon emissions.

Number of Shipment $\left(\mathit{n}\right)$ | Rate of Learning | ||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{b}=1.00$ | $\mathit{b}=1.2\text{}{0}_{\text{}}$ | $\mathit{b}=1.40$ | |||||||

Cycle Time ${{\mathit{T}}_{\mathit{n}}}_{\text{}}$ | Lot Size $\mathit{Q}$ | Retailer’s Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)\left(\mathbf{USD}\right)$ | Cycle Time ${\mathit{T}}_{\mathit{n}}$ | Lot Size $\mathit{Q}$ | Retailer’s Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)\left(\mathbf{USD}\right)$ | Cycle Length ${{\mathit{T}}_{\mathit{n}}}_{\text{}}$ | Lot Size $\mathit{Q}$ | Retailer’s Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)\left(\mathbf{USD}\right)$ | |

1 | 0.9034 | 46,669 | 1,472,500 | 0.9034 | 46,697 | 1,472,600 | 0.9035 | 46,698 | 1,472,720 |

2 | 0.9038 | 46,706 | 1,473,230 | 0.9040 | 46,709 | 1,473,750 | 0.9044 | 46,713 | 1,474,510 |

3 | 0.9047 | 46,720 | 1,475,010 | 0.9059 | 46,742 | 1,477,140 | 0.9078 | 46,755 | 1,480,720 |

4 | 0.9087 | 46,850 | 1,482,380 | 0.9109 | 46,824 | 1,486,610 | 0.9181 | 46,924 | 1,500,070 |

5 | 0.9124 | 46,852 | 1,489,280 | 0.9234 | 47,031 | 1,510,020 | 0.9422 | 47,331 | 1,545,150 |

6 | 0.9234 | 47,031 | 1,510,020 | 0.9467 | 47,400 | 1,553,570 | 0.9736 | 47,799 | 1,603,860 |

7 | 0.9422 | 47,331 | 1,545,150 | 0.9583 | 47,575 | 1,575,280 | 0.9937 | 48,075 | 1,641,570 |

8 | 0.9652 | 47,680 | 1,588,050 | 0.9918 | 48,052 | 1,637,910 | 1.0015 | 48,182 | 1,656,160 |

9 | 0.9841 | 47,954 | 1,623,590 | 1.0014 | 48,163 | 1,653,510 | 1.0036 | 48,195 | 1,660,660 |

10 | 0.9954 | 48,098 | 1,644,740 | 1.0032 | 48,204 | 1,659,370 | 1.0046 | 48,222 | 1,661,940 |

11 | 1.0091 | 48,586 | 1,654,940 | 1.0043 | 48,218 | 1,661,410 | 1.0048 | 48,222 | 1,662,300 |

12 | 1.0032 | 48,204 | 1,659,370 | 1.0047 | 48,223 | 1,662,100 | 1.0048 | 48,225 | 1,662,400 |

13 | 1.0042 | 48,217 | 1,661,200 | 1.0048 | 48,225 | 1,662,330 | 1.0049 | 48,225 | 1,662,430 |

14 | 1.0046 | 48,222 | 1,661,940 | 1.0048 | 48,223 | 1,662,400 | 1.0049 | 48,225 | 1,662,440 |

15 | 1.0048 | 48,224 | 1,662,240 | 1.0049 | 48,225 | 1,662,430 | 1.0049 | 48,225 | 1,662,440 |

16 | 1.0048 | 48,224 | 1,662,360 | 1.0049 | 48,225 | 1,662,420 | 1.0049 | 48,225 | 1,662,440 |

17 | 1.00491 | 48,225 | 1,662,440 | 1.00491 | 48,225 | 1,662,440 | 1.00491 | 48,225 | 1,662,440 |

18 | 1.00491 | 48,225 | 1,662,440 | 1.00491 | 48,225 | 1,662,440 | 1.00491 | 48,225 | 1,662,440 |

**Table 5.**Effect of deterioration rate on lot size, cycle length, and buyer’s total profit with carbon emissions and fixed learning rate $\left(b=1.4\right)$ and no. of shipments $\left(n=17\right)$.

Deterioration Rate $\mathit{\theta}$ | Cycle Length ${\mathit{T}}_{\mathit{n}}$ (Year) | Lot Size $\mathit{Q}$ (Units) | Buyer’s Total Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)$ (USD) |
---|---|---|---|

0.10 | 1.00491 | 48,225 | 1,662,440 |

0.15 | 0.6998 | 33,653 | 1,148,080 |

0.20 | 0.5365 | 25,828 | 876,186 |

0.25 | 0.4349 | 20,951 | 708,152 |

**Table 6.**Impact of inflation rate on cycle length, order size, and buyer’s total profit with carbon emissions and fixed learning rate $\left(b=1.4\right)$ and no. of shipments $\left({n}^{\ast}=17\right)$.

Inflation Rate $\mathit{R}$ | Cycle Length ${\mathit{T}}_{\mathit{n}}$ (Year) | Lot Size $\mathit{Q}$ (Units) | Buyer’s Total Profit $\mathsf{\Psi}\left({\mathit{T}}_{\mathit{n}}\right)$ (USD) |
---|---|---|---|

0.02 | 1.0349 | 49,735 | 1,710,570 |

0.04 | 1.0195 | 48,959 | 1,685,940 |

0.06 | 1.0049 | 48,225 | 1,662,440 |

0.08 | 0.9910 | 47,531 | 1,639,990 |

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**MDPI and ACS Style**

Alamri, O.A.; Jayaswal, M.K.; Khan, F.A.; Mittal, M.
An EOQ Model with Carbon Emissions and Inflation for Deteriorating Imperfect Quality Items under Learning Effect. *Sustainability* **2022**, *14*, 1365.
https://doi.org/10.3390/su14031365

**AMA Style**

Alamri OA, Jayaswal MK, Khan FA, Mittal M.
An EOQ Model with Carbon Emissions and Inflation for Deteriorating Imperfect Quality Items under Learning Effect. *Sustainability*. 2022; 14(3):1365.
https://doi.org/10.3390/su14031365

**Chicago/Turabian Style**

Alamri, Osama Abdulaziz, Mahesh Kumar Jayaswal, Faizan Ahmad Khan, and Mandeep Mittal.
2022. "An EOQ Model with Carbon Emissions and Inflation for Deteriorating Imperfect Quality Items under Learning Effect" *Sustainability* 14, no. 3: 1365.
https://doi.org/10.3390/su14031365