# The Impacts of Payment Schemes and Carbon Emission Policies on Replenishment and Pricing Decisions for Perishable Products in a Supply Chain

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## Abstract

**:**

## 1. Introduction

## 2. Notations and Assumptions

${f}_{1}$ | the portion of the procurement cost that must be paid upfront, $0\le {f}_{1}\le 1$ |

${f}_{2}$ | the portion of the procurement cost payable upon delivery, $0\le {f}_{2}\le 1$ |

${f}_{3}$ | the portion of the procurement cost that allows for an acceptable delay from the supplier to the retailer, $0\le {f}_{3}\le 1$ and ${f}_{1}+{f}_{2}+{f}_{3}=1$ |

${k}_{l}$ | the duration of credit extended by the retailer to customers, i.e., the downstream credit period, ${k}_{l}\ge 0$ |

${k}_{u}$ | the duration of credit offered by the supplier to the retailer, i.e., the upstream credit period, ${k}_{u}\ge 0$ |

$\rho $ | the proportion of sales revenue that allows for an acceptable delay from the retailer to customers, $0\le \rho \le 1$ |

$\gamma $ | the annual compound interest paid per dollar |

${t}_{0}$ | the duration over years for which prepayments are made, ${t}_{0}$ > 0 |

$c$ | the procurement cost per unit in dollars, $c>0$ |

$h$ | the holding cost per unit per year in dollars, excluding interest charges, $h>0$ |

$b$ | carbon emissions generated during the purchase of a single unit of the product |

${h}_{e}$ | carbon emissions resulting from inventory holding per unit over a given timeframe |

${o}_{e}$ | carbon emissions generated by an order |

${c}_{e}$ | unit carbon trading emission price or unit carbon tax |

$B$ | carbon emission cap for the retailer |

${I}_{p}$ | the annual interest charged by the supplier per dollar |

${I}_{e}$ | the annual interest earned per dollar |

$x$ | the duration until the expiration date or shelf life, i.e., the stock age, measured in years, $x>0$ |

$o$ | the cost of ordering, expressed in dollars per order, $o>0$ |

$S$ | the selling price per unit in dollars, $S>c>0$ (decision variable) |

$t$ | the time in years, $t\ge 0$ |

$D\left(t,S\right)$ | the annual demand rate, $D\left(t,S\right)=f\left(S\right)\left(x-t\right)/x,f\left(S\right)0,{f}^{\prime}\left(S\right)0$ and ${f}^{\u2033}(S)>0$ |

$\theta \left(t\right)$ | the deteriorating rate at time $t,0\le \theta \left(t\right)\le 1,{\theta}^{\prime}\left(t\right)\ge 0$ and $\theta \left(x\right)=1$ |

$I\left(t\right)$ | the inventory level in units at time $t$ |

$Q$ | the order quantity |

$T$ | the duration of the cycle period, measured in years, $T\le x$ (decision variable) |

$PTP$ | the current value of the overall annual profit in dollars |

${IC}_{1}$ | the current value of interest incurred for advance payment per cycle period $T$ |

${IC}_{2}$ | the current value of interest incurred for cash payment per cycle period $T$ |

${IC}_{3}$ | the current value of interest incurred for credit payment per cycle period $T$ |

${IE}_{1}$ | the current value of interest accrued for credit payment per cycle period $T$ |

$CC$ | the current value of capital cost per cycle period $T$ |

- (1)
- Shortages are prohibited.
- (2)
- The ending inventory is zero.
- (3)
- The freshness and demand of perishable products are absolutely related to the stock age; hence, the demand rate is assumed as a function of selling price $S$ and stock age $x$, that is, $D\left(t,S\right)=f\left(S\right)\left(x-t\right)/x,0\le t\le x$, where $f\left(S\right)>0,{f}^{\prime}\left(S\right)0$ and ${f}^{\u2033}\left(S\right)>0$.
- (4)
- During the credit period ${k}_{u}$ provided by the upstream supplier, the unsettled account allows for the generated sales revenue to be deposited into an interest-bearing account. Once the permissible delay period elapses, the retailer settles the remaining procurement cost and proceeds to pay the accrued interest charges on the items in stock. The downstream credit period, ${k}_{l}$, is offered by the retailer. Consequently, the retailer can amass revenue in an account and acquire earned interest when ${k}_{u}\ge {k}_{l}$. There is no interest earned for the retailer when ${k}_{u}<{k}_{l}$. Because the product cannot be sold after the expiration date, it is assumed that ${k}_{l}\le x$ and ${k}_{u}\le x$.
- (5)
- The supplier asks that the retailer (i) prepay ${f}_{1}$ percentage of procurement cost in ${t}_{0}$ years prior to the time of delivery, (ii) pay another ${f}_{2}$ percentage of procurement cost at the time of delivery, and (iii) offer a credit period of ${k}_{u}$ years on the remaining ${f}_{3}$ percentage of procurement cost (i.e., $0\le {f}_{1},\hspace{0.17em}{f}_{2},\hspace{0.17em}{f}_{3}\le 1$, and ${f}_{1}+{f}_{2}+{f}_{3}=1$). Similarly, the retailer extends partial trade credit to customers, allowing them a credit period of ${k}_{l}$ years for $\rho $ proportion of sales, while the remaining portion (i.e., $1-\rho $ proportion of sales) is paid in cash.
- (6)
- Two main policy approaches apply to carbon emission reduction: the carbon cap-and-trade policy and the carbon tax policy. Within the framework of the carbon cap-and-trade policy, the retailer is assigned an initial cap for carbon emissions and has the flexibility to trade or exchange emission rights within this established cap. If the retailer exceeds its carbon cap $B$, it must purchase extra carbon allowances from the compliance carbon trading market to offset the difference between its actual emissions and the established cap. In the scenario where the retailer’s carbon emissions fall below its carbon cap, however, the retailer will have the opportunity to trade its surplus carbon allowances with other businesses on the compliance carbon trading market. In the carbon tax policy, the focus is solely on the amount of tax imposed on total carbon emissions, without considering other factors or mechanisms such as trading or allowances. The carbon emissions of the retailer primarily originate from various operational activities, encompassing tasks such as ordering, purchasing, and storage.

## 3. Mathematical Model Formulation

**Case 1:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**Case 1.1:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**${\mathit{k}}_{\mathit{u}}\le \mathit{T}\le \mathit{T}+{\mathit{k}}_{\mathit{l}}$

**Case 1.2:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**$\mathit{T}\le {\mathit{k}}_{\mathit{u}}\le \mathit{T}+{\mathit{k}}_{\mathit{l}}$

**Case 1.3:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**${\mathit{k}}_{\mathit{u}}\ge \mathit{T}+{\mathit{k}}_{\mathit{l}}$

**Case 2:**${\mathit{k}}_{\mathit{l}}\ge {\mathit{k}}_{\mathit{u}}$

## 4. Theoretical Findings

#### 4.1. Case 1: ${k}_{l}\le {k}_{u}$

**Case 1.1:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**${\mathit{k}}_{\mathit{u}}\le \mathit{T}\le \mathit{T}+{\mathit{k}}_{\mathit{l}}$

**Case 1.2:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**$\mathit{T}\le {\mathit{k}}_{\mathit{u}}\le \mathit{T}+{\mathit{k}}_{\mathit{l}}$

**Case 1.3:**${\mathit{k}}_{\mathit{l}}\le {\mathit{k}}_{\mathit{u}}$

**and**${\mathit{k}}_{\mathit{u}}\ge \mathit{T}+{\mathit{k}}_{\mathit{l}}$

#### 4.2. Case 2: ${k}_{l}\ge {k}_{u}$

## 5. Numerical Examples

**Example 1.**The annual demand rate is $f\left(S\right)=3000{e}^{-0.03S}$ and deteriorating rate is $\theta (t)=0.03$, given the other parameter values are ${f}_{1}=0.3$, ${f}_{2}=0.3$, ${f}_{3}=0.4$, $\rho =0.4,$ $\gamma =0.07$, ${k}_{l}=0.15$, ${k}_{u}=0.25$, $c=\mathrm{\$}30$, $h=\mathrm{\$}5$, ${t}_{0}=0.15$, $x=0.6$ years, $o=\mathrm{\$}250$, ${I}_{p}=0.07$, ${I}_{e}=0.05$, $b=5$, ${o}_{e}=400$, ${h}_{e}=3$, ${c}_{e}=0.2$, and $B=4000$.

**Example 2.**It is presumed that all parameters align with those in Example 1, except that the downstream credit period is ${k}_{l}=0.25$ years and the upstream credit period is ${k}_{u}=0.15$ years.

**Example 3.**It is presumed that all parameters align with those in Example 1, except that the unit carbon tax is ${c}_{e}=0.5$ and $B=0$. The solution process is the same as in Example 1. Because ${PTP}^{\ast}(S,T)=max\{PT{P}_{1}(S,T),PT{P}_{2}(S,T),PT{P}_{3}(S,T)\}=max\{\mathrm{8,583.12}$, $\mathrm{8,919.78},\mathrm{7,832.27}\}=\$\mathrm{8,919.78}$, the optimal solution to the problem is the optimal selling price of ${S}^{\ast}={S}_{2}=\$66.79$, the optimal replenishment time is ${T}^{\ast}={T}_{2}=0.18402$ years, the optimal current value of the total annual profit is ${PTP}^{\ast}(S,T)=PT{P}_{2}(S,T)=\$\mathrm{8,919.78}$, and the optimal order quantity is ${Q}^{\ast}=63.18$ units. In addition, the carbon tax per year is $C{E}^{\ast}/{T}^{\ast}=\$2000.92$. The Hessian matrix of $PT{P}^{\ast}(S,T)$ at the critical point $({S}^{\ast},{T}^{\ast})$ is derived as follows:

**Example 4.**It is presumed that all parameters align with those in Example 2, except the unit carbon tax is ${c}_{e}=0.5$ and $B=0$. The solution process is also the same. Here, the optimal selling price is ${S}^{\ast}=\$67.39$, the optimal replenishment time is ${T}^{\ast}=0.18830$ years, the optimal current value of the total annual profit is $PT{P}^{\ast}(S,T)=\$\mathrm{8,613.43}$, the optimal order quantity is ${Q}^{\ast}=63.24$ units, and the carbon tax per year is $C{E}^{\ast}/{T}^{\ast}=\mathrm{\$}1957.25$. The Hessian matrix of $PT{P}^{\ast}(S,T)$ at the critical point $({S}^{\ast},{T}^{\ast})$ is derived as follows:

## 6. Sensitivity Analysis

- (1)
- When any of the parameters $(o,c,\gamma ,{k}_{l},\rho ,{t}_{0},h,x,{I}_{p},b,{o}_{e},{h}_{e},\mathrm{o}\mathrm{r}{c}_{e})$ increases, the optimal selling price $\left({S}^{\ast}\right)$ also increases. Conversely, the optimal selling price $\left({S}^{\ast}\right)$ decreases when the parameters $\left({k}_{u}\mathrm{o}\mathrm{r}{I}_{e}\right)$ increase. Furthermore, the ordering cost $\left(o\right)$ and the procurement cost $\left(c\right)$ exert a notable positive influence on the optimal selling price $\left({S}^{\ast}\right)$. However, the impact of other parameters on the optimal selling price $\left({S}^{\ast}\right)$ is relatively limited.
- (2)
- The influence of parameters $(o,c,{k}_{l},\rho ,{t}_{0},x,b,{o}_{e},\mathrm{o}\mathrm{r}{c}_{e})$ on the optimal replenishment time $\left({T}^{\ast}\right)$ is positive, indicating that an increase in any of these parameters results in a longer optimal replenishment time $\left({T}^{\ast}\right)$. Conversely, the influence of parameters $\left(\gamma ,{k}_{u},h,{I}_{p},{I}_{e},\mathrm{o}\mathrm{r}{h}_{e}\right)$ on $\left({T}^{\ast}\right)$ is negative, suggesting that an increase in any of these parameters leads to a shorter optimal replenishment time $\left({T}^{\ast}\right)$. Also important to highlight is that the ordering cost $\left(o\right)$, procurement cost $\left(c\right)$, stock age $\left(x\right)$, carbon emissions from an order $\left({o}_{e}\right)$, and carbon trading price $\left({c}_{e}\right)$ all exert a notable positive effect on the optimal replenishment time $\left({T}^{\ast}\right)$. Conversely, the holding cost $\left(h\right)$ exhibits a substantial negative impact on the optimal replenishment time $\left({T}^{\ast}\right)$. The impact of other parameters on the optimal replenishment time $\left({T}^{\ast}\right)$ is relatively modest.
- (3)
- A higher value of parameters $\left({o,k}_{u},\rho ,{t}_{0},x,{o}_{e},\mathrm{o}\mathrm{r}{c}_{e}\right)$ results in a higher optimal order quantity $\left({Q}^{\ast}\right)$, while a higher value of parameters $\left(c,\gamma ,h,{I}_{p},b,{I}_{e},\mathrm{o}\mathrm{r}{h}_{e}\right)$ leads to a lower $\left({Q}^{\ast}\right)$. Moreover, the ordering cost $\left(o\right)$, stock age $\left(x\right)$, carbon emissions from an order $\left({o}_{e}\right)$, and carbon trading price $\left({c}_{e}\right)$ all have a considerable positive impact on the optimal order quantity $\left({Q}^{\ast}\right)$. Conversely, the procurement cost $\left(c\right)$ has a substantial negative effect on the optimal order quantity $\left({Q}^{\ast}\right)$. The influence of other parameters on the optimal order quantity $\left({Q}^{\ast}\right)$ is relatively minor.
- (4)
- The parameters $\left({k}_{u},x,\mathrm{o}\mathrm{r}{I}_{e}\right)$ positively influence the optimal current value of the total annual profit $\left({PTP}^{\ast}\right)$, whereas the parameters $(o,c,\gamma ,{k}_{l},\rho ,{t}_{0},h,{I}_{p},b,{o}_{e},{h}_{e},\mathrm{o}\mathrm{r}{c}_{e})$ negatively affect $\left({PTP}^{\ast}\right)$. Furthermore, the stock age $\left(x\right)$ exerts a notable positive impact on the optimal current value of the total annual profit $\left({PTP}^{\ast}\right)$. Conversely, both the ordering cost $\left(o\right)$ and procurement cost $\left(c\right)$ have a considerable negative effect on the optimal current value of the total annual profit $\left({PTP}^{\ast}\right)$. The impact of other parameters on the optimal current value of the total annual profit $\left({PTP}^{\ast}\right)$ is relatively minor.
- (5)
- The impact of parameters $\left(\gamma ,{k}_{u},h,{I}_{e},b,{o}_{e},\mathrm{o}\mathrm{r}{h}_{e}\right)$ on the optimal amount of total annual carbon emissions $\left({TCE}^{\ast}\right)$ is positive, indicating that an increase in any of these parameters results in a higher $\left({TCE}^{\ast}\right)$. Conversely, the effect of parameters $\left(o,c,{k}_{l},\rho ,{t}_{0},x,{I}_{p},\mathrm{o}\mathrm{r}{c}_{e}\right)$ on $\left({TCE}^{\ast}\right)$ is negative, suggesting that an increase in any of these parameters leads to a lower $\left({TCE}^{\ast}\right)$. It is worth noting that the carbon emissions from buying a product $\left(b\right)$ and the carbon emissions from an order $\left({o}_{e}\right)$ have a substantial positive impact on the optimal amount of total annual carbon emissions $\left({TCE}^{\ast}\right)$. Conversely, the ordering cost $\left(o\right)$, procurement cost $\left(c\right)$, stock age $\left(x\right)$, and carbon trading price $\left({c}_{e}\right)$ all exert a significant negative effect on the optimal amount of total annual carbon emissions $\left({TCE}^{\ast}\right)$. The influence of other parameters on the optimal $\left({TCE}^{\ast}\right)$ is relatively limited.

## 7. Conclusions

- (1)
- When facing an increase in ordering or procurement costs, the retailer is required to elevate the selling price and prolong the replenishment time. Nevertheless, in this scenario, the optimal current values of total annual profit and the optimal amount of total annual carbon emissions both diminish.
- (2)
- As the stock age rises, the retailer needs to elongate the replenishment time and augment the order quantity. Consequently, the optimal current values of total annual profit are expected to rise, while the optimal amount of total annual carbon emissions is anticipated to decrease.
- (3)
- As carbon emissions increase, the retailer must raise the selling price and extend the replenishment time. Consequently, the optimal amount of total annual carbon emissions will increase, while the optimal current values of the total annual profit will decrease.
- (4)
- When the carbon trading price rises, the retailer needs to raise the selling price, extend the replenishment time, and increase the order quantity. Consequently, both the optimal amount of total annual carbon emissions and the optimal current values of the total annual profit will decline.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- An, S.; Li, B.; Song, D.; Chen, X. Green credit financing versus trade credit financing in a supply chain with carbon emission limits. Eur. J. Oper. Res.
**2021**, 292, 125–142. [Google Scholar] [CrossRef] - Dong, G.; Liang, L.; Wei, L.; Xie, J.; Yang, G. Optimization model of trade credit and asset-based securitization financing in carbon emission reduction supply. Ann. Oper. Res.
**2023**, 331, 35–84. [Google Scholar] [CrossRef] - Fu, K.; Li, Y.; Mao, H.; Miao, Z. Firms’ production and green technology strategies: The role of emission asymmetry and carbon taxes. Eur. J. Oper. Res.
**2023**, 305, 1100–1112. [Google Scholar] [CrossRef] - Hammami, R.; Nouiram, I.; Frein, Y. Carbon emissions in a multi-echelon production-inventory model with lead time constraints. Int. J. Prod. Econ.
**2015**, 164, 292–307. [Google Scholar] [CrossRef] - Lu, C.J.; Gu, M.; Lee, T.S.; Yang, C.T. Impact of carbon emission policy combinations on the optimal production-inventory decisions for deteriorating items. Expert Syst. Appl.
**2022**, 201, 117234. [Google Scholar] [CrossRef] - Ma, X.; Wang, J.; Bai, Q.; Wang, S. Optimization of a three-echelon cold chain considering freshness-keeping efforts under cap-and-trade regulation in Industry 4.0. Int. J. Prod. Econ.
**2020**, 220, 107457. [Google Scholar] [CrossRef] - Maulana, S.K.B.; Utama, D.M.; Asrofi, M.S.; Ningurm, I.S.; Alba, N.; Ahfa, H.A.; Zein, T.A. The Capacitated Sustainable EOQ Models: Models considering tax emissions. J. Tek. Ind.
**2020**, 21, 12–21. [Google Scholar] [CrossRef] - Mishra, U.; Wu, J.Z.; Tsao, Y.C.; Tseng, M.L. Sustainable inventory system with controllable non-instantaneous deterioration and environmental emission rate. J. Clean. Prod.
**2020**, 244, 118807. [Google Scholar] [CrossRef] - Mubin, A.M.; Syahril, F.; Rosiani, T.Y. Sustainable EOQ model with multi container transportation problems. J. Tek. Ind.
**2021**, 22, 236–244. [Google Scholar] [CrossRef] - Sabzevar, N.; Enns, S.T.; Bergerson, J.; Kettunen, J. Modeling competitive firms’ performance under price-sensitive demand and cap-and-trade emissions constraints. Int. J. Prod. Econ.
**2017**, 184, 193–209. [Google Scholar] [CrossRef] - Sarkar, B.; Sarkar, M.; Ganguly, B.; Cárdenas-Barrón, L.E. Combined effects of carbon emission and production quality improvement for fixed lifetime products in a sustainable supply chain management. Int. J. Prod. Econ.
**2021**, 231, 107867. [Google Scholar] [CrossRef] - Shi, Y.; Zhang, Z.; Chen, S.C.; Cárdenas-Barrón, L.E. Optimal replenishment decisions for perishable products under cash, advance, and credit payments considering carbon tax regulations. Int. J. Prod. Econ.
**2020**, 223, 107514. [Google Scholar] [CrossRef] - Dye, C.Y.; Yang, C.T. Sustainable trade credit and replenishment decision with credit-linked demand under carbon emission constraints. Eur. J. Oper. Res.
**2015**, 244, 187–200. [Google Scholar] [CrossRef] - Daryanto, Y.; Wee, H.M.; Astanti, R.D. Three-echelon supply chain mode considering carbon emission and item deterioration. Transp. Res. E Logist. Transp.
**2019**, 122, 368–383. [Google Scholar] [CrossRef] - Ma, X.; Xu, J.; Peng, W.; Wang, S. Optimal freshness and carbon abatement decisions in a two-echelon cold chain. Appl. Math. Model.
**2021**, 96, 834–859. [Google Scholar] [CrossRef] - Cheng, M.C.; Lo, H.C.; Yang, C.T. Optimizing pricing, pre-sale incentive, and inventory decisions with advance sales and trade credit under carbon tax policy. Mathematics
**2023**, 11, 2534. [Google Scholar] [CrossRef] - Jain, R.; Mittal, M.; Mangla, S.K.; Baraiya, R. Optimizing supply chain strategies for deteriorating items and imperfect manufacturing under carbon emission regulations. Comput. Ind. Eng.
**2023**, 182, 109350. [Google Scholar] [CrossRef] - Goyal, S.K. Economic order quantity under conditions of permissible delay in payments. J. Oper. Res. Soc.
**1985**, 36, 335–338. [Google Scholar] [CrossRef] - Teng, J.T. On economic order quantity under conditions of permissible delay in payments. J. Oper. Res. Soc.
**2002**, 53, 915–918. [Google Scholar] [CrossRef] - Chang, C.T.; Ouyang, L.Y.; Teng, J.T. An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Appl. Math. Model.
**2003**, 27, 983–996. [Google Scholar] [CrossRef] - Teng, J.T. Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. Int. J. Prod. Econ.
**2009**, 119, 415–423. [Google Scholar] [CrossRef] - Teng, J.T.; Krommyda, I.P.; Skouri, K.; Lou, K.R. A comprehensive extension of optimal ordering policy for stock-dependent demand under progressive payment scheme. Eur. J. Oper. Res.
**2011**, 215, 97–104. [Google Scholar] [CrossRef] - Teng, J.T.; Min, J.; Pan, Q. Economic order quantity model with trade credit financing for non-decreasing demand. Omega
**2012**, 40, 328–335. [Google Scholar] [CrossRef] - Ouyang, L.Y.; Chang, C.T. Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging. Int. J. Prod. Econ.
**2013**, 144, 610–617. [Google Scholar] [CrossRef] - Sarkar, B.; Gupta, H.; Chaudhuri, K.; Goyal, S.K. An integrated inventory model with variable lead time, defective units and delay in payments. Appl. Math. Comput.
**2014**, 237, 650–658. [Google Scholar] [CrossRef] - Liao, J.J.; Huang, K.N.; Ting, P.S. Optimal strategy of deteriorating items with capacity constraints under two-levels of trade credit policy. Appl. Math. Comput.
**2014**, 233, 647–658. [Google Scholar] [CrossRef] - Chang, C.T.; Soong, P.Y.; Cheng, M.C. The influences of defective items and trade credits on replenishment decision. J. Inf. Manag.
**2017**, 28, 113–132. [Google Scholar] [CrossRef] - Majumder, P.; Bera, U.K.; Maiti, M. An EPQ model of deteriorating substitute items under trade credit policy. Int. J. Oper. Res.
**2019**, 34, 162–212. [Google Scholar] [CrossRef] - Panda, G.C.; Khan, M.A.; Shaikh, A.A. A credit policy approach in a two-warehouse inventory model for deteriorating items with price- and stock-dependent demand under partial backlogging. J. Ind. Eng. Int.
**2019**, 15, 147–170. [Google Scholar] [CrossRef] - Zhang, A.X. Optimal advance payment scheme involving fixed per-payment costs. Omega
**1996**, 24, 577–582. [Google Scholar] [CrossRef] - Maiti, A.K.; Maiti, M.K.; Maiti, M. Inventory model with stochastic lead-time and price dependent demand incorporating advance payment. Appl. Math. Model.
**2009**, 33, 2433–2443. [Google Scholar] [CrossRef] - Taleizadeh, A.A.; Niaki, S.T.A.; Nikousokhan, R. Constraint multiproduct joint-replenishment inventory control problem using uncertain programming. Appl. Soft Comput.
**2011**, 11, 5143–5154. [Google Scholar] [CrossRef] - Thangam, A. Optimal price discounting and lot-sizing policies for perishable items in a supply chain under advance payment scheme and two-echelon trade credits. Int. J. Prod. Econ.
**2012**, 139, 459–472. [Google Scholar] [CrossRef] - Taleizadeh, A.A.; Pentico, D.W.; Jabalameli, M.S.; Aryanezhad, M. An economic order quantity model with multiple partial prepayments and partial backordering. Math. Comput. Model.
**2013**, 57, 311–323. [Google Scholar] [CrossRef] - Taleizadeh, A.A. An economic order quantity model for deteriorating item in a purchasing system with multiple prepayments. Appl. Math. Model.
**2014**, 38, 5357–5366. [Google Scholar] [CrossRef] - Zhang, Q.; Tsao, Y.C.; Chen, T.H. Economic order quantity under advance payment. Appl. Math. Model.
**2014**, 38, 5910–5921. [Google Scholar] [CrossRef] - Teng, J.T.; Cárdenas-Barrón, L.E.; Chang, H.J.; Wu, J.; Hu, Y. Inventory lot-size policies for deteriorating items with expiration dates and advance payments. Appl. Math. Model.
**2016**, 40, 8605–8616. [Google Scholar] [CrossRef] - Taleizadeh, A.A.; Tavakoli, S.; San-Jose, L.A. A lot sizing model with advance payment and planned backordering. Ann. Oper. Res.
**2018**, 271, 1001–1022. [Google Scholar] [CrossRef] - Feng, L.; Wang, W.C.; Teng, J.T.; Cárdenas-Barrón, L.E. Pricing and lot-sizing decision for fresh goods when demand depends on unit price, displaying stocks and product age under generalized payment. Eur. J. Oper. Res.
**2022**, 296, 940–952. [Google Scholar] [CrossRef] - Tsao, Y.C.; Pantisoontorn, A.; Vu, T.L.; Chen, T.H. Optimal production and predictive maintenance decisions for deteriorated products under advance-cash-credit payments. Int. J. Prod. Econ.
**2024**, 269, 109132. [Google Scholar] [CrossRef] - Li, R.; Chan, Y.L.; Chang, C.T.; Cárdenas-Barrón, L.E. Pricing and lot-sizing policies for perishable products with advance-cash-credit payments by a discounted cash-flow analysis. Int. J. Prod. Econ.
**2017**, 193, 578–589. [Google Scholar] [CrossRef] - Wu, J.; Teng, J.T.; Chan, Y.L. Inventory policies for perishable products with expiration dates and advance-instant-delayed payment schemes. Int. J. Syst. Sci. Oper. Logist.
**2018**, 5, 310–326. [Google Scholar] [CrossRef] - Chang, C.T.; Ouyang, L.Y.; Teng, J.T.; Lai, K.K.; Cárdenas-Barrón, L.E. Manufacturer’s pricing and lot-sizing decisions for perishable goods under various payment terms by a discounted cash flow analysis. Int. J. Prod. Econ.
**2019**, 218, 83–95. [Google Scholar] [CrossRef] - Taleizadeh, A.A.; Tavakoli, S.; Bhattacharya, A. Inventory ordering policies for mixed sale of products under inspection policy, multiple prepayment, partial trade credit, payments linked to order quantity and full backordering. Ann. Oper. Res.
**2020**, 287, 403–437. [Google Scholar] [CrossRef] - Shi, Y.; Zhang, Z.; Tiwari, S.; Tao, Z. Retailer’s optimal strategy for a perishable product with increasing demand under various payment schemes. Ann. Oper. Res.
**2022**, 315, 899–929. [Google Scholar] [CrossRef] - Wu, J.; Chang, C.T.; Cheng, M.C.; Teng, J.T.; Al-khateeb, F.B. Inventory management for fresh produce when the time-varying demand depends on product freshness, stock level and expiration date. Int. J. Syst. Sci. Oper. Logist.
**2016**, 3, 138–147. [Google Scholar] [CrossRef] - Chen, S.C.; Min, J.; Teng, J.T.; Li, F. Inventory and shelf-space management for fresh produce with freshness-and-stock dependent demand and expiration date. J. Oper. Res. Soc.
**2016**, 67, 884–896. [Google Scholar] [CrossRef] - Li, R.; Teng, J.T. Pricing and lot-sizing decisions for perishable goods when demand depends on selling price, reference price, product freshness, and displayed stocks. Eur. J. Oper. Res.
**2018**, 270, 1099–1108. [Google Scholar] [CrossRef] - Wang, W.C.; Teng, J.T.; Lou, K.R. Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. Eur. J. Oper. Res.
**2014**, 232, 315–321. [Google Scholar] [CrossRef] - Taleizadeh, A.A.; Pourmohammad-Zia, N.; Konstantaras, I. Partial linked-to-order delayed payment and life time effects on decaying items ordering. Oper. Res.
**2021**, 21, 2077–2099. [Google Scholar] [CrossRef] - Li, R.; Teng, J.T.; Chang, C.T. Lot-sizing and pricing decision for perishable products under three-echelon supply chain when demand depends on price and stock-age. Ann. Oper. Res.
**2021**, 307, 303–328. [Google Scholar] [CrossRef] [PubMed] - Li, R.; Liu, Y.; Teng, J.T.; Tsao, Y.C. Optimal pricing, lot-sizing and backordering decisions when a seller demands for an advance-cash-credit payment scheme. Eur. J. Oper. Res.
**2019**, 278, 283–295. [Google Scholar] [CrossRef] - Roy, M.D.; Sana, S.S. Production rate and lot size-dependent lead time reduction strategies in a supply chain model with stochastic demand, controllable setup cost and trade-credit financing. RAIRO Oper. Res.
**2021**, 55, S1469–S1485. [Google Scholar] [CrossRef]

References | Demand Function | Payment Type | Carbon Policy |
---|---|---|---|

Chang et al. [43] | Price | ACC | X |

Cheng et al. [16] | Price | Advance-Credit | Carbon tax |

Feng et al. [39] | Price, Stock Age, and Level | ACC | X |

Jain et al. [17] | Constant | X | Cap-and-trade, Carbon tax, and Carbon offset |

Li and Teng [48] | Price, Stock Age, and Level | Cash | X |

Li et al. [51] | Price and Stock Age | ACC | X |

Li et al. [52] | Price | ACC | X |

Ma et al. [6] | Price and Freshness | X | Cap-and-trade |

Mishra et al. [8] | Price and Stock | X | Cap-and-trade |

Roy and Sana [53]) | Stochastic | Credit | X |

Sabzevar et al. [10] | Price | X | Cap-and-trade |

Shi et al. [12] | Constant | ACC | Carbon tax |

Shi et al. [45] | Time | ACC | X |

Taleizadeh et al. [44] | Constant | Advance-Credit | X |

Tsao et al. [40] | Constant | ACC | X |

Wu et al. [42] | Constant | ACC | X |

This paper | Price and Stock Age | ACC | Cap-and-trade and Carbon tax |

**Table 2.**Analysis of sensitivity and the trends in each parameter’s impact on the decision variable.

Decision | ${\mathit{S}}^{\mathit{*}}$ | ${\mathit{T}}^{\mathit{*}}$ | ${\mathit{Q}}^{\mathit{*}}$ | ${\mathit{P}\mathit{T}\mathit{P}}^{\mathit{*}}$ | ${\mathit{T}\mathit{C}\mathit{E}}^{\mathit{*}}$ | |
---|---|---|---|---|---|---|

Parameter | ||||||

$o=200$ | 65.01 | 0.14111 | 53.24 | 11,343.70 | 4797.41 | |

$o=250$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$o=300$ | 65.12 | 0.16538 | 60.78 | 10,684.10 | 4342.41 | |

trend | ↗ | ↗ | ↗ | ↘ | ↘ | |

$c=25$ | 59.97 | 0.14276 | 62.56 | 13,038.40 | 5082.61 | |

$c=30$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$c=35$ | 70.18 | 0.16551 | 52.25 | 9271.16 | 4069.54 | |

trend | ↗ | ↗ | ↘ | ↘ | ↘ | |

$\gamma =0.05$ | 65.02 | 0.15432 | 57.49 | 11,059.10 | 4536.50 | |

$\gamma =0.07$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$\gamma =0.09$ | 65.12 | 0.15303 | 56.91 | 10,942.80 | 4554.47 | |

trend | ↗ | ↘ | ↘ | ↘ | ↗ | |

${k}_{u}=0.20$ | 65.14 | 0.15393 | 57.16 | 10,965.60 | 4536.65 | |

${k}_{u}=0.25$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${k}_{u}=0.30$ | 65.00 | 0.15338 | 57.23 | 11,036.90 | 4554.96 | |

trend | ↘ | ↘ | ↗ | ↗ | ↗ | |

${k}_{l}=0.10$ | 64.96 | 0.15034 | 57.19 | 11,061.90 | 4563.57 | |

${k}_{l}=0.15$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${k}_{l}=0.20$ | 65.18 | 0.15426 | 57.20 | 10,942.60 | 4528.62 | |

trend | ↗ | ↗ | ↗ | ↘ | ↘ | |

$\rho =0.3$ | 64.99 | 0.15313 | 57.16 | 11,058.50 | 4560.09 | |

$\rho =0.4$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$\rho =0.5$ | 65.15 | 0.15422 | 57.24 | 10,943.40 | 4531.09 | |

trend | ↗ | ↗ | ↗ | ↘ | ↘ | |

${t}_{0}=0.10$ | 65.00 | 0.15325 | 57.18 | 11,032.20 | 4557.21 | |

${t}_{0}=0.15$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${t}_{0}=0.20$ | 65.13 | 0.15410 | 57.22 | 10,969.60 | 4534.00 | |

trend | ↗ | ↗ | ↗ | ↘ | ↘ | |

$h=4$ | 65.00 | 0.15444 | 57.56 | 11,028.10 | 4535.59 | |

$h=5$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$h=6$ | 65.14 | 0.15291 | 56.84 | 10,973.90 | 4555.57 | |

trend | ↗ | ↘ | ↘ | ↘ | ↗ | |

$x=0.5$ | 65.04 | 0.14261 | 52.23 | 10,628.10 | 4710.01 | |

$x=0.6$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$x=0.7$ | 65.09 | 0.16331 | 61.56 | 11,292.90 | 4422.22 | |

trend | ↗ | ↗ | ↗ | ↗ | ↘ | |

${I}_{p}=0.06$ | 65.02 | 0.15368 | 57.29 | 11,019.40 | 4548.37 | |

${I}_{p}=0.07$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${I}_{p}=0.08$ | 65.12 | 0.15366 | 57.11 | 10,982.40 | 4542.85 | |

trend | ↗ | ↘ | ↘ | ↘ | ↘ | |

${I}_{e}=0.04$ | 65.13 | 0.15427 | 57.29 | 10,959.70 | 4531.37 | |

${I}_{e}=0.05$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${I}_{e}=0.06$ | 65.01 | 0.15308 | 57.11 | 11,042.20 | 4559.84 | |

trend | ↘ | ↘ | ↘ | ↗ | ↗ | |

$b=4$ | 64.87 | 0.15322 | 57.40 | 11,075.60 | 4191.14 | |

$b=5$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

$b=6$ | 65.27 | 0.15413 | 57.00 | 10,926.70 | 4895.31 | |

trend | ↗ | ↗ | ↘ | ↘ | ↗ | |

${o}_{e}=350$ | 65.06 | 0.15123 | 56.44 | 11,067.20 | 4260.84 | |

${o}_{e}=400$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${o}_{e}=450$ | 65.08 | 0.15607 | 57.94 | 10,935.60 | 4822.01 | |

trend | ↗ | ↗ | ↗ | ↘ | ↗ | |

${h}_{e}=2$ | 65.05 | 0.15382 | 57.27 | 11,006.30 | 4516.39 | |

${h}_{e}=3$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${h}_{e}=4$ | 65.08 | 0.15352 | 57.13 | 10,995.50 | 4574.74 | |

trend | ↗ | ↘ | ↘ | ↘ | ↗ | |

${c}_{e}=0.1$ | 64.50 | 0.14286 | 54.65 | 11,070.20 | 4790.71 | |

${c}_{e}=0.2$ | 65.07 | 0.15367 | 57.20 | 11,000.90 | 4545.60 | |

${c}_{e}=0.3$ | 65.64 | 0.16408 | 59.44 | 10,954.50 | 4333.59 | |

trend | ↗ | ↗ | ↗ | ↘ | ↘ |

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## Share and Cite

**MDPI and ACS Style**

Chang, C.-T.; Tseng, Y.-T.
The Impacts of Payment Schemes and Carbon Emission Policies on Replenishment and Pricing Decisions for Perishable Products in a Supply Chain. *Mathematics* **2024**, *12*, 1033.
https://doi.org/10.3390/math12071033

**AMA Style**

Chang C-T, Tseng Y-T.
The Impacts of Payment Schemes and Carbon Emission Policies on Replenishment and Pricing Decisions for Perishable Products in a Supply Chain. *Mathematics*. 2024; 12(7):1033.
https://doi.org/10.3390/math12071033

**Chicago/Turabian Style**

Chang, Chun-Tao, and Yao-Ting Tseng.
2024. "The Impacts of Payment Schemes and Carbon Emission Policies on Replenishment and Pricing Decisions for Perishable Products in a Supply Chain" *Mathematics* 12, no. 7: 1033.
https://doi.org/10.3390/math12071033