# Two Level Trade Credit Policy Approach in Inventory Model with Expiration Rate and Stock Dependent Demand under Nonzero Inventory and Partial Backlogged Shortages

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Assumptions

#### Assumptions

- (i)
- The lot size ($Q$) in a single batch is delivered.
- (ii)
- The inventory planning horizon is infinite.
- (iii)
- Replenishments are instantaneous.
- (iv)
- The freshness of the product depends on several factors such as the temperature of the environment, conditions of the stocking place, and humidity, among others. For these causes the product deteriorates over time continuously and it has an expiration date. In the beginning, when the lot size arrives, the freshness of the product is assumed to equal 1 when the item is in stock, in the storeroom, and in its displayed place. After that, the freshness of the product is decreased, and finally it reaches the expiration date. Thereofore, the freshness function is mathematically expressed as follows:$$f(t)=\frac{\left(m-t\right)}{m}\hspace{0.17em}0\le t\le m$$

- (i)
- It is also considered that when the cycle length finishes before the end of the stock (i.e., without ending zero inventory) on that time the demand depends on the power function of the inventory level, as well as the expiration rate. Therefore, the demand is given by$$D(t)=\alpha {\left[I(t)\right]}^{\beta}\frac{m-t}{m},\hspace{0.17em}{t}_{1}<t\le T.$$$$D(t)=\left\{\begin{array}{ll}\alpha {\left[I(t)\right]}^{\beta}\frac{m-t}{m}& {t}_{1}<t\le {t}_{2}\\ \alpha & {t}_{2}<t\le T\end{array}\right.$$
- (ii)
- The deteriorated products are neither repaired nor refunded.
- (iii)
- Shortages are allowed with partial backlogging with a rate of $\delta $.
- (iv)
- The supplier gives the credit time (M) facility to his/her retailer. The retailer offers credit time (N) facility to his/her customers.
- (v)
- The retailer uses the sales revenue to obtain the interest with a rate I
_{e}according to the terms and conditions given by the agreement. When the cycle length finishes, the credit is settled, and then interest charges start to be paid by the retailer on the items in stock with a rate I_{p}.

## 3. Mathematical Formulation of the Inventory Model with Expiration Date, Shelf Space, and Stock-Dependent Demand under Two-Level Trade Credit Financing and Partial Backlogging

#### 3.1. Mathematical Derivation When B > 0

- (i)
- Ordering cost = ${c}_{o}$
- (ii)
- Holding cost = $chol=h\left\{{\displaystyle \underset{0}{\overset{{t}_{1}}{\int}}{I}_{1}(t)dt+{\displaystyle \underset{{t}_{1}}{\overset{{t}_{2}}{\int}}{I}_{2}(t)dt}}\right\}=h\left(cho{l}_{1}+cho{l}_{2}\right)$, where $cho{l}_{1}=\left[\frac{\alpha {W}^{\beta}}{6m}{t}_{1}^{3}-\frac{\alpha {W}^{\beta}}{2}{t}_{1}^{2}+Q{t}_{1}\right]$ and $cho{l}_{2}=\frac{\left(W+B\right)\left(T-{t}_{1}\right)}{2}$.

#### 3.2. Mathematical Derivation When B < 0

- (i)
- Ordering cost = ${c}_{o}$
- (ii)
- Holding cost = $chol=h\left\{{\displaystyle \underset{0}{\overset{{t}_{1}}{\int}}{I}_{1}(t)dt+{\displaystyle \underset{{t}_{1}}{\overset{{t}_{2}}{\int}}{I}_{2}(t)dt}}\right\}=h\left(cho{l}_{1}+cho{l}_{2}\right)$, where $cho{l}_{1}=\left[\frac{\alpha {W}^{\beta}}{6m}{t}_{1}^{3}-\frac{\alpha {W}^{\beta}}{2}{t}_{1}^{2}+Q{t}_{1}\right]$ and $cho{l}_{2}=\frac{W\left({t}_{2}-{t}_{1}\right)}{2}$
- (iii)
- Shortage cost = $csho={c}_{b}{\displaystyle \underset{{t}_{2}}{\overset{T}{\int}}\left[-I(t)\right]dt}={c}_{b}{\displaystyle \underset{{t}_{2}}{\overset{T}{\int}}\alpha \delta \left(t-{t}_{2}\right)dt=\frac{{c}_{b}{B}^{2}}{2\delta \alpha}}$
- (iv)
- Cost of lost sale = $ocls={c}_{l}\left(1-\delta \right)\alpha {\displaystyle \underset{{t}_{2}}{\overset{T}{\int}}dt}=\frac{{c}_{l}\left(1-\delta \right)B}{\delta}$

## 4. Solution Procedure

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**,**then the necessary and sufficient condition for the optimal cycle time ${T}^{*}$ is determined and this is given below:

Algorithm 1. Steps to determine the solution to the inventory model. |

Step 1. Input all inventory parameters values. Step 2. Select without ending zero inventory situation. Step 3. Set the following initial values for $W=1$, $B=1$, and $j=0$ into Equation (44) and state the accuracy as $\epsilon ={10}^{-5}$. Step 4. Solve the Equation (44) to compute the value of ${T}_{j}^{*}$. Step 5. Set $T={T}_{j}^{*}$ and B = 1 into Equation (45) to calculate the value of ${W}_{j}^{*}$. Step 6. Set $T={T}_{j}^{*}$ and $W={W}_{j}^{*}$ into Equation (46) to obtain the value of ${B}_{j}^{*}$. Step 7. Iterate Equations (44), (45), and (46) until the accuracy is satisfied $\left|{T}_{j+1}^{*}-{T}_{j}\right|<\epsilon $, $\left|{W}_{j+1}^{*}-{W}_{j}\right|<\epsilon $, $\left|{B}_{j+1}^{*}-{B}_{j}\right|<\epsilon $ and $\left|T{P}_{i,j+1}^{*}(.)-T{P}_{i,j}^{}(.)\right|<\epsilon $. If all the conditions are satisfied, then store ${T}_{j}={T}_{j+1}^{*}$, ${W}_{j}={W}_{j+1}^{*}$, ${B}_{j}={B}_{j+1}^{*}$ and $T{P}_{i,j}(.)=T{P}_{i,j+1}^{*}(.)$. Otherwise, put $j=1$ and repeat Step 4 to Step 6. Step 8. Compare the value of the objective function to select the best profit $TP({W}^{*},{B}^{*},{T}^{*})=\underset{i=1,\dots ,5}{Max}\left\{T{P}_{i}(W,B,T)\right\}$. Step 9. Select partial backlogged shortages situation. Step 10. Set the initial values for $W=1$, $B=1$,$j=0$ into Equation (48) and state the accuracy as $\epsilon ={10}^{-5}$. Step 11. Solve the Equation (48) to calculate the value of ${T}_{j}^{*}$. Step 12. Set $T={T}_{j}^{*}$ and B = 1 into Equation (49) to obtain the value of ${W}_{j}^{*}$. Step 13. Set $T={T}_{j}^{*}$ and $W={W}_{j}^{*}$ into Equation (50) to get the value of ${B}_{j}^{*}$. Step 14. Iterate Equations (48), (49), and (50) until the accuracy is satisfied $\left|{T}_{j+1}^{*}-{T}_{j}\right|<\epsilon $, $\left|{W}_{j+1}^{*}-{W}_{j}\right|<\epsilon $, $\left|{B}_{j+1}^{*}-{B}_{j}\right|<\epsilon $ and $\left|T{P}_{i,j+1}^{*}(.)-T{P}_{i,j}^{}(.)\right|<\epsilon $. If all the conditions are satisfied, then store ${T}_{j}={T}_{j+1}^{*}$, ${W}_{j}={W}_{j+1}^{*}$, ${B}_{j}={B}_{j+1}^{*}$ and $T{P}_{i,j}(.)=T{P}_{i,j+1}^{*}(.)$. Otherwise, put $j=1$ and repeat Step 11 to Step 13. Step 15. Compare the value of the objective function to select the best profit $TP({W}^{*},{B}^{*},{T}^{*})=\underset{i=6,\dots ,10}{Max}\left\{T{P}_{i}(W,B,T)\right\}$. Step 16. Report the solution and stop. |

## 5. Numerical Examples

**Example**

**1.**

**Example**

**2.**

## 6. Sensitivity Analysis

- The total profit is highly sensitive to all parameters except to the ordering cost parameter ${c}_{o}$ and it is infeasible when the value of $\beta $ is +20%.
- The initial stock level (Q) is highly sensitive to all parameters except to the ordering cost parameter ${c}_{o}$ and it is infeasible when the value of $\beta $ is +20%.
- The ending inventory level (B) is highly sensitive to all parameters except to the ordering cost parameter ${c}_{o}$ and it is infeasible when the value of $\beta $ is +20%.
- The shelf-space (W) is highly sensitive to all parameters except the ordering cost parameter ${c}_{o}$ and it is infeasible the value of $\beta $ is +20%.
- The time of the displayed units $({t}_{1})$ is insensible to the parameters $\alpha $ and ${c}_{o}$; and it is moderately sensitive to the parameter $h$, and highly sensitive to the rest parameters, and it is infeasible when the value of $\beta $ is +20%.
- The cycle length $(T)$ is insensible to the parameters $\alpha $ and ${c}_{o}$. It is moderately sensitive to the parameters $h,p,s,$ and $u$. It is highly sensible to the rest parameters, and it is infeasible the value of $\beta $ is +20%.

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

Parameter | Description |

$p$ | Selling price per unit ($/unit) |

$S$ | Salvage price per unit ($/unit) |

${c}_{o}$ | Ordering cost per order ($/order) |

$c$ | Purchase cost per unit ($/unit) |

$h$ | Holding cost per unit per time unit ($/unit/time unit) |

${c}_{b}$ | Shortage cost per unit per time unit ($/unit/time unit) |

${c}_{l}$ | Unit opportunity cost due to lost sale per unit per time unit ($/unit/time unit) |

$u$ | Shelf space cost ($/unit) |

$\delta $ | Backlogging parameter |

$m$ | The expiration time of the product (time unit) |

$M$ | Credit period offered by the supplier to his/her retailer (time unit) |

N | Credit period offered by the retailer to the customer (time unit) |

${I}_{e}$ | Interest earned by the retailer (%/time unit) |

${I}_{p}$ | Interest charged by the suppliers to the retailers (%/time unit) |

$I\left(t\right)$ | Inventory level at time t (units) |

$T{P}_{i}{}^{(.)}$ | The total profit ($/time unit) |

Dependent decision variables | |

${t}_{1}$ | Time of the displayed units (time unit) |

${t}_{2}$ | Time when inventory level reaches zero (time unit) |

$Q$ | The stock level at the beginning of the period (units) |

Decision variables | |

$T$ | Cycle length (time unit) |

$W$ | The number of units displayed in shelf space (units). |

$B$ | Ending inventory level at time T (units) |

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Author (s) | Type of Model | Product Lifetime/Expiration Date | Demand Dependent on | Partial/Complete Backlogging | Trade Credit Policy |
---|---|---|---|---|---|

Chen and Teng [2] | Purchase | √ | Constant | √ | |

Wu et al. [3] | Purchase | √ | Expiration rate | √ | |

Teng et al. [4] | Purchase | √ | Constant | √ | |

Sarkar [5] | Supply chain | √ | Constant | √ | |

Tiwari et al. [6] | Purchase | √ | Selling price | √ | √ |

Uthayakumar and Tharani [7] | Purchase | √ | Constant | √ | √ |

Liao et al. [8] | Purchase | Constant | √ | ||

Banu et al. [21] | Production | Display stock and credit period | √ | ||

Min et al. [22] | Purchase | Display stock | √ | ||

Panda et al. [23] | Purchase | Display stock | √ | √ | |

Jaggi et al. [41] | Purchase | Selling price | √ | ||

Mohanty et al. [42] | Purchase | Time | √ | ||

Aliabadi et al. [43] | Purchase | Credit period and selling price | √ | √ | |

Shaikh et al. [45] | Production | Selling price | √ | ||

Das et al. [46] | Purchase | Selling price | √ | ||

Chakraborty et al. [47] | Purchase | Display stock | √ | ||

Manna et al. [48] | Purchase | Time | √ | ||

Barman et al. [49] | Supply chain | Green level and selling price | |||

Li and Teng [50] | Purchase | Display shelf space, stock dependent (power function), and expiration rate | |||

This paper | Purchase | √ | Display shelf space, stock dependent (power function), and expiration rate | Partial backlogging | Two-level trade credit policy approach |

Q | B | $\mathit{W}$ | ${\mathit{t}}_{1}$ | $\mathit{T}$ | Total Profit | Case |
---|---|---|---|---|---|---|

2590.934 | 964.5861 | 1873.553 | 0.08180 | 0.3057132 | 35357.71 | 1 |

2591.885 | 965.3788 | 1871.911 | 0.08219178 | 0.3057701 | 35357.59 | 2 |

2414.413 | 850.9161 | 2013.839 | 0.04109 | 0.3011034 | 33605.92 | 3 |

145.3784 | 58.75093 | 106.8986 | 0.030395 | 0.08219 | 6911.664 | 4 |

18.23070 | 7.478258 | 13.44068 | 0.015854 | 0.041095 | 1514.526 | 5 |

Q | B | $\mathit{W}$ | ${\mathit{t}}_{1}$ | $\mathit{T}$ | Total Profit | Case |
---|---|---|---|---|---|---|

30.73261 | 0.0000 | 16.72789 | 0.041095 | 0.3318969 | 1521.353 | 1 |

33.28520 | 0.0000 | 23.52837 | 0.021995 | 0.3269761 | 1576.334 | 2 |

33.29935 | 0.0000 | 23.58922 | 0.021846 | 0.3269347 | 1577.467 | 3 |

1.351514 | 0.0000 | 0.9899749 | 0.0073494 | 0.082191 | 144.7568 | 4 |

0.1615969 | 0.0000 | 0.1189071 | 0.0038087 | 0.041095 | −178.9718 | 5 |

Q | B | $\mathit{W}$ | ${\mathit{t}}_{1}$ | ${\mathit{t}}_{2}$ | $\mathit{T}$ | Total Profit | Case |
---|---|---|---|---|---|---|---|

3.924188 | 0.5730634 | 0.5730634 | 0.04109 | 0.1947118 | 0.1994873 | 141.2968 | 6 |

3.924255 | 0.5730742 | 0.5730742 | 0.041095 | 0.1947162 | 0.1994919 | 140.8073 | 7 |

1.563487 | 0.6658734 | 0.6658734 | 0.0312564 | 0.0901456 | 0.0942564 | −5.830897 | 8 |

0.8069315 | 0.7086727 | 0.7086727 | 0.0008951 | 0.082191 | 0.088097 | −33.83189 | 9 |

0.002679 | 0.0026796 | 0.0026796 | 0.000006 | 0.041095 | 0.0411182 | −223.2291 | 10 |

Parameter | % Change of Parameter | % Change in | |||||
---|---|---|---|---|---|---|---|

Total Profit | $\mathit{Q}$ | $\mathit{B}$ | $\mathit{W}$ | ${\mathit{t}}_{1}$ | $\mathit{T}$ | ||

$\alpha $ | −20 | −52.52 | −52.46 | −52.46 | −52.46 | 0.01 | 0.03 |

−10 | −29.64 | −29.61 | −29.61 | −29.61 | 0.01 | 0.01 | |

10 | 37.43 | 37.39 | 37.39 | 37.39 | 0.00 | −0.01 | |

20 | 83.70 | 83.61 | 83.61 | 83.61 | −0.01 | −0.01 | |

$\beta $ | −20 | −89.24 | −94.15 | −95.29 | −94.50 | −1.60 | −12.35 |

−10 | −74.25 | −81.29 | −83.02 | −81.80 | −0.66 | −6.01 | |

10 | 175.79 | 185.74 | 195.19 | 136.66 | 0.38 | 5.62 | |

20 | -- | -- | -- | -- | -- | -- | |

$c$ | −20 | 179.27 | 211.44 | 329.40 | 227.76 | 19.80 | −1.61 |

−10 | 64.41 | 71.22 | 102.69 | 75.99 | 7.84 | −0.66 | |

10 | −38.14 | −39.06 | −49.55 | −41.01 | −4.98 | 0.46 | |

20 | −61.61 | −61.76 | −74.46 | −64.41 | −7.75 | 0.77 | |

$h$ | −20 | 3.80 | 5.43 | 6.94 | 5.69 | 0.88 | 0.87 |

−10 | 1.87 | 2.66 | 3.39 | 2.79 | 0.43 | 0.43 | |

10 | −1.82 | −2.57 | −3.25 | −2.68 | −0.43 | −0.43 | |

20 | −3.59 | −5.04 | −6.36 | −5.27 | −0.85 | −0.85 | |

$m$ | −20 | −36.98 | −48.24 | −47.05 | −48.04 | −18.85 | −18.85 |

−10 | −19.47 | −26.58 | −25.74 | −26.44 | −9.36 | −9.36 | |

10 | 21.34 | 31.82 | 30.35 | 31.57 | 9.24 | 9.24 | |

20 | 44.46 | 69.14 | 65.42 | 68.50 | 18.36 | 18.36 | |

${c}_{o}$ | −20 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 | −0.01 |

−10 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

10 | −0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | |

20 | −0.02 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | |

$p$ | −20 | −78.15 | −71.21 | −82.64 | −75.78 | 11.28 | −0.26 |

−10 | −48.75 | −42.83 | −53.21 | −46.59 | 4.43 | −0.09 | |

10 | 73.64 | 60.69 | 85.17 | 68.56 | −3.11 | 0.04 | |

20 | 178.04 | 142.81 | 211.04 | 163.91 | −5.42 | 0.07 | |

$S$ | −20 | −15.18 | −20.4 | −27.23 | −20.49 | −7.01 | 0.75 |

−10 | −8.21 | −11.19 | −15.14 | −11.24 | −3.72 | 0.40 | |

10 | 9.77 | 13.76 | 19.25 | 13.83 | 4.23 | −0.45 | |

20 | 21.56 | 30.97 | 44.14 | 31.14 | 9.09 | −0.97 | |

$u$ | −20 | 19.72 | 21.94 | 28.90 | 28.55 | −13.51 | −0.51 |

−10 | 9.23 | 10.21 | 13.26 | 13.11 | −6.57 | −0.25 | |

10 | −8.16 | −8.92 | −11.32 | −11.21 | 6.22 | 0.23 | |

20 | −15.4 | −16.76 | −21.03 | −20.84 | 12.14 | 0.44 |

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## Share and Cite

**MDPI and ACS Style**

Shaikh, A.A.; Cárdenas-Barrón, L.E.; Manna, A.K.; Céspedes-Mota, A.; Treviño-Garza, G.
Two Level Trade Credit Policy Approach in Inventory Model with Expiration Rate and Stock Dependent Demand under Nonzero Inventory and Partial Backlogged Shortages. *Sustainability* **2021**, *13*, 13493.
https://doi.org/10.3390/su132313493

**AMA Style**

Shaikh AA, Cárdenas-Barrón LE, Manna AK, Céspedes-Mota A, Treviño-Garza G.
Two Level Trade Credit Policy Approach in Inventory Model with Expiration Rate and Stock Dependent Demand under Nonzero Inventory and Partial Backlogged Shortages. *Sustainability*. 2021; 13(23):13493.
https://doi.org/10.3390/su132313493

**Chicago/Turabian Style**

Shaikh, Ali Akbar, Leopoldo Eduardo Cárdenas-Barrón, Amalesh Kumar Manna, Armando Céspedes-Mota, and Gerardo Treviño-Garza.
2021. "Two Level Trade Credit Policy Approach in Inventory Model with Expiration Rate and Stock Dependent Demand under Nonzero Inventory and Partial Backlogged Shortages" *Sustainability* 13, no. 23: 13493.
https://doi.org/10.3390/su132313493