# A Discount Technique-Based Inventory Management on Electronics Products Supply Chain

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^{2}

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## Abstract

**:**

## 1. Introduction and Literature Review

- It critically evaluates when one needs to impose a discount and when to not, especially when a bulk purchase has been made by a retailer with a huge investment in a limited storage shop in a highly expensive location.
- A synergy between stock, price, and time-dependent demand and implications of discount policy has been meticulously explained.
- A sensitivity analysis with some theoretical findings has been suggesting to achieve the maximum profit in the chain for the managers of the industry and shown a threshold point of discount offered time.

#### 1.1. Literature Review

#### 1.1.1. Influence of Stock Dependent Demand on Traditional Inventory Model

#### 1.1.2. Influence of Price Sensitive Demand on Tradition Inventory Model

#### 1.1.3. Influence of Sensitiveness of Time on Traditional Inventory Mode

#### 1.1.4. Impacts of Discount Policy on Electronic Products

## 2. Assumption and Notations

#### 2.1. Assumptions

- The replenishment rate is infinite and Lead-time is negligible.
- This model is for a single type of item.
- The planning horizon is considered infinite.
- In this paper, the demand function comprises price, time, and stock-dependence in the form of$$D=\{\begin{array}{l}\left(a-bp\left(1-\delta \right)\right)+\alpha t+\beta {t}^{2}+s{I}_{1}(t),\mathrm{when}\hspace{0.17em}0\le t\le {t}_{1}\text{}\left(\mathrm{discount}\text{}\mathrm{is}\text{}\mathrm{given}\text{}\mathrm{on}\text{}\mathrm{price}\right)\\ (a-bp)+\alpha t+\beta {t}^{2}+s{I}_{2}(t),\hspace{0.17em}\mathrm{when}\hspace{0.17em}{t}_{1}t\le T\hspace{0.17em}\left(\mathrm{without}\text{}\mathrm{discount}\right)\end{array}$$$a$ is the initial rate of demand$b$ is the rate decrease demand on prices$p$ is the product price$\delta $ is the discount rate on price of product$\alpha $ is the rate with which the demand rate increases on time$\beta $ is the rate of changes of rate on time in the demand rate itself$s$ is the rate depending on stock, $0<s\le 1$
- There are no shortages considered in this model.

#### 2.2. Notations

## 3. Mathematical Formulation for Proposed Electronics Product Inventory Model

#### 3.1. Solution of Differential Equations from (1) and (2)

#### 3.2. The Total Cost per Unit Time per Cycle

- (a)
- Ordering cost per cycle = $C$
- (b)
- Holding cost (HC) = ${C}_{h}\left[{\displaystyle \underset{0}{\overset{{t}_{1}}{\int}}{I}_{1}(t)\hspace{0.17em}dt}+{\displaystyle \underset{{t}_{1}}{\overset{T}{\int}}{I}_{2}(t)\hspace{0.17em}dt}\right]$i.e.,$${C}_{h}\left[\begin{array}{l}\left(\frac{bp-a}{s}+\frac{\alpha}{{s}^{2}}-\frac{2\beta}{{s}^{3}}\right)T+\left(\frac{2\beta}{{s}^{2}}-\frac{\alpha}{s}\right)\frac{{T}^{2}}{2}+\left(\frac{w}{s}+\frac{b\delta p-bp+a}{{s}^{2}}-\frac{\alpha}{{s}^{3}}+\frac{2\beta}{{s}^{4}}\right)\left(1-{e}^{s{t}_{1}}\right)+\\ \left(\frac{bp-a-\alpha T-\beta {T}^{2}}{{s}^{2}}+\frac{\alpha +2\beta T}{{s}^{3}}-\frac{2\beta}{{s}^{4}}\right)\left(1-{e}^{s\left(T-{t}_{1}\right)}\right)-\frac{b\delta p{t}_{1}}{s}-\frac{\beta {T}^{3}}{3s}\end{array}\right]$$
- (c)
- Purchase cost (PC) = ${C}_{p}{}^{*}w$$${C}_{p}\left[\frac{b\delta p}{s}{e}^{s{t}_{1}}-\left(\frac{bp-a-\alpha T-\beta {T}^{2}}{s}+\frac{\alpha +2\beta T}{{s}^{2}}-\frac{2\beta}{{s}^{3}}\right){e}^{sT}-\left(\frac{b\delta p-bp+a}{s}-\frac{\alpha}{{s}^{2}}+\frac{2\beta}{{s}^{3}}\right)\right]$$
- (d)
- Transportation cost (TC) = ${T}_{fc}+{T}_{vc}{}^{*}w$$${T}_{fc}+{T}_{vc}\left[\begin{array}{l}\frac{b\delta p}{s}{e}^{s{t}_{1}}-\left(\frac{bp-a-\alpha T-\beta {T}^{2}}{s}+\frac{\alpha +2\beta T}{{s}^{2}}-\frac{2\beta}{{s}^{3}}\right){e}^{sT}\\ -\left(\frac{b\delta p-bp+a}{s}-\frac{\alpha}{{s}^{2}}+\frac{2\beta}{{s}^{3}}\right)\end{array}\right]$$
- (e)
- Sales revenue (SR) = ${p}^{*}\left[{\displaystyle {\int}_{0}^{{t}_{1}}D\hspace{0.17em}dt+{\displaystyle \underset{{t}_{1}}{\overset{T}{\int}}D\hspace{0.17em}dt}}\right]$$$={p}^{*}\left[{\displaystyle {\int}_{0}^{{t}_{1}}\left(a-b\hspace{0.17em}p\hspace{0.17em}\left(1-\delta \right)+\alpha \hspace{0.17em}t+\beta \hspace{0.17em}{t}^{2}+s\hspace{0.17em}{I}_{1}(t)\right)\hspace{0.17em}dt+{\displaystyle \underset{{t}_{1}}{\overset{T}{\int}}\left(a-b\hspace{0.17em}p\hspace{0.17em}+\alpha \hspace{0.17em}t+\beta \hspace{0.17em}{t}^{2}+s\hspace{0.17em}{I}_{2}(t)\right)\hspace{0.17em}dt}}\right]$$$$\begin{array}{l}=\left(a-bp\right)pT+\frac{\alpha p{T}^{2}}{2}+\frac{\beta p{T}^{3}}{3}+b{p}^{2}\delta {t}_{1}+\\ sp\left[\begin{array}{l}\left(\frac{bp-a}{s}+\frac{\alpha}{{s}^{2}}-\frac{2\beta}{{s}^{3}}\right)T+\left(\frac{2\beta}{{s}^{2}}-\frac{\alpha}{s}\right)\frac{{T}^{2}}{2}+\left(\frac{w}{s}+\frac{b\delta p-bp+a}{{s}^{2}}-\frac{\alpha}{{s}^{3}}+\frac{2\beta}{{s}^{4}}\right)\left(1-{e}^{s{t}_{1}}\right)+\\ \left(\frac{bp-a-\alpha T-\beta {T}^{2}}{{s}^{2}}+\frac{\alpha +2\beta T}{{s}^{3}}-\frac{2\beta}{{s}^{4}}\right)\left(1-{e}^{s\left(T-{t}_{1}\right)}\right)-\frac{b\delta p{t}_{1}}{s}-\frac{\beta {T}^{3}}{3s}\end{array}\right]\end{array}$$

## 4. Theoretical Derivations

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 4.1. Algorithms

#### 4.1.1. Algorithm for Single Decision Variable

- Step 1.
- Input all the parameters value ($C,\hspace{0.17em}a,\hspace{0.17em}\alpha ,\hspace{0.17em}b,\hspace{0.17em}\beta ,\hspace{0.17em}\delta ,\hspace{0.17em}s,\hspace{0.17em}{C}_{h},\hspace{0.17em}{C}_{p},\hspace{0.17em}{T}_{fc},\hspace{0.17em}{t}_{1},T$).
- Step 2.
- Evaluate the value of ${p}^{*}$ from Equation (13).
- Step 3.
- Evaluate the value of ω from Equation (12) using all the parameters and the value of p
^{*}. - Step 4.
- Output the value of p
^{*}and ω. - Step 5.
- End.

#### 4.1.2. Algorithm for Double Decision Variable

- Step 1.
- Declare $F(p,T)=\frac{\partial \omega}{\partial p}$ and $G(p,T)=\frac{\partial \phi}{\partial p}$ from Equations (18) and (19).
- Step 2.
- Input all the parameters value ($C,\hspace{0.17em}a,\hspace{0.17em}\alpha ,\hspace{0.17em}b,\hspace{0.17em}\beta ,\hspace{0.17em}\delta ,\hspace{0.17em}s,\hspace{0.17em}{C}_{h},\hspace{0.17em}{C}_{p},\hspace{0.17em}{T}_{fc},\hspace{0.17em}{t}_{1}$).
- Step 3.
- Take ${p}_{0},\hspace{0.17em}{T}_{0}$ where $\left({p}_{0}>0,{T}_{0}>0\right)$ and iterative variable $i=0$.
- Step 4.
- Find $D=\left|\begin{array}{cc}{\left[\frac{\partial F}{\partial p}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[\frac{\partial F}{\partial T}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\\ {\left[\frac{\partial G}{\partial p}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[\frac{\partial G}{\partial T}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\end{array}\right|$
- Step 5.
- IF $D=0$ and $i=0$, Go to Setp 3. And IF $D=0$ and $i\ne 0$ Go to Step 10.
- Step 6.
- Find $h=\frac{1}{D}\left|\begin{array}{cc}{\left[F\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[\frac{\partial F}{\partial T}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\\ {\left[G\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[\frac{\partial G}{\partial T}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\end{array}\right|$ and $k=\frac{1}{D}\left|\begin{array}{cc}{\left[\frac{\partial F}{\partial p}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[F\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\\ {\left[\frac{\partial G}{\partial p}\right]}_{\left(p={p}_{0},T={T}_{0}\right)}& {\left[G\right]}_{\left(p={p}_{0},T={T}_{0}\right)}\end{array}\right|$
- Step 7.
- Set ${p}_{1}={p}_{0}-h$ and ${T}_{1}={T}_{0}-k$.
- Step 8.
- If $\left|{p}_{1}-{p}_{0}\right|<\epsilon $ and $\left|{T}_{1}-{T}_{0}\right|<\epsilon $, Go to Step 10 ($\epsilon $ is small value).
- Step 9.
- Update ${p}_{0}={p}_{1},\hspace{0.17em}{T}_{0}={T}_{1}$ and $i=i+1$. Go to Step 4.
- Step 10.
- Evaluate $\omega ({p}_{1},\hspace{0.17em}{T}_{1})$ from Equation (12).
- Step 11.
- Output the value of ${p}_{1},{T}_{1}$ and $\omega $.
- Step 12.
- End.

#### 4.2. Case Study

#### 4.3. Numerical Illustration:

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

#### 4.4. Sensitivity Analysis

- When the ordering cost (C) of the system increased, the selling price (p) and as well as the cycle length (T) of the chain were raised. This happens because a higher ordering cost brings a more considerable lot and intensifies the total cost of the business. As a result, the retailer will need to sell his products at a comparatively higher selling price, and as the lot is massive so it is challenging to sell the products quickly. However, the profits without discount and with discount were increased.
- With the intensification of purchase costs, the profit was decreasing. However, the selling price and total cycle length also increased. If a retailer purchased any item at a high price to maintain the profit margin, he needs to sell it at a high price. Moreover, an increase in stock provides fluctuations in the profit and selling price of the system.
- The profit becomes lower with the upsurge of the per-unit holding cost of the item. Moreover, it increases the selling price (p) and cycle length (T) of the system. Furthermore, the increase in initial demand parameter (a) provides a more significant profit than usual. In contrast, an increase in another parameter (b) will give a decrease in profit.
- The increase of the rate of change of demand rate ($\alpha $) provides a lower profit for the system while it is vice versa for the increasing rate of demand parameter ($\beta $). The profit of the chain decreased with the increase of the period (${t}_{1}$). However, the rate depending on stock (s) when increased the system’s profit has been reduced. A significant change in profit has been noticed with variable transportation (${T}_{vc}$) and fixed transportation (${T}_{fc}$). However, for both costs, the retailer’s profit margin slightly drops due to the excessive expenses in the transportation system.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**An electronics shop. (Source: https://lh3.ggpht.com/p/AF1QipMf1FVqbSnGOjLJswWWY_TNg_rlQlbQ4OIXt4mb=s0). (accessed on 15 July 2021).

**Figure 3.**Concave graph of ${\psi}^{*}$ regarding decision variables selling price (p) and cycle length (T).

Notations | Units | Description |
---|---|---|

$C$ | $/Cycle | Ordering cost per cycle |

${C}_{p}$ | $/Unit | Purchasing cost per unit |

${C}_{h}$ | $/Unit | Holding cost per unit per unit time |

$w$ | Units/Cycle | Ordering quantity per cycle |

${T}_{fc}$ | $/Cycle | Fixed transportation cost |

${T}_{vc}$ | $/unit | Variable transportation cost |

${t}_{1}$ | Months | Discount time from the beginning of cycle |

$D$ | Units | Demand function |

${I}_{i}\left(t\right)$ | Units | inventory level at any time t where $0\le t\le {t}_{1}$ when i = 1, ${t}_{1}\le t\le T$ when i = 2 |

$\delta $ | Constant | Discount rate on price of product |

$\omega \left(p,T\right)$ | $/Month | Total profit per unit time |

Decision variables | ||

$p$ | $/Unit | Selling price per unit of product |

$T$ | Months | replenishment time. |

$\mathit{\delta}$ | Optimal Solutions | ||
---|---|---|---|

${\mathit{p}}^{\mathbf{*}}$ | ${\mathit{T}}^{\mathbf{*}}$ | ${\mathit{\psi}}^{\mathbf{*}}$ | |

0 | 369.559 | 18.027 | 14,462.310 |

0.5 | 373.951 | 18.737 | 11,531.170 |

1 | 380.394 | 19.984 | 8643.954 |

1.5 | 392.878 | 23.020 | 5872.356 |

2 | 847.031 | 90.767 | 5190.119 |

2.5 | 648.980 | 70.028 | 2489.949 |

3 | 575.141 | 60.376 | 217.017 |

3.5 | … | … | … |

N.B. (…) means infeasible solution |

Parameter | % Change | With Discount | Without Discount | $\mathbf{\left(}{\mathit{\psi}}^{\mathbf{*}}\mathbf{-}{\mathit{\psi}}^{\mathbf{*}\mathbf{*}}\mathbf{\right)}\mathbf{\%}$ | ||
---|---|---|---|---|---|---|

${\mathit{p}}^{\mathbf{*}}$ | ${\mathit{T}}^{\mathbf{*}}$ | ${\mathit{\psi}}^{\mathbf{*}}$ | ${\mathit{\psi}}^{\mathbf{*}\mathbf{*}}$ | |||

$C$ | −20 | 372.922 | 18.560 | 12,122.700 | 14,470.070 | −16.22% |

−10 | 372.937 | 18.563 | 12,118.930 | 14,466.190 | −16.23% | |

10 | 372.966 | 18.568 | 12,111.390 | 14,458.430 | −16.23% | |

20 | 372.980 | 18.571 | 12,107.620 | 14,454.540 | −16.24% | |

$W$ | −20 | 370.786 | 17.408 | 9576.597 | 12,056.410 | −20.57% |

−10 | 371.828 | 18.004 | 10,858.480 | 13,266.230 | −18.15% | |

10 | 374.111 | 19.095 | 13,351.800 | 15,647.190 | −14.67% | |

20 | 372.951 | 18.566 | 12,115.160 | 16,822.770 | −27.98% | |

${C}_{p}$ | −20 | 352.280 | 15.439 | 18,783.750 | 21,230.690 | −11.53% |

−10 | 362.195 | 16.765 | 15,297.790 | 17,712.490 | −13.63% | |

10 | 386.700 | 21.553 | 9294.282 | 11,516.450 | −19.30% | |

20 | … | … | … | … | … | |

${C}_{h}$ | −20 | 372.902 | 18.556 | 12,145.900 | 14,496.270 | −16.21% |

−10 | 372.927 | 18.561 | 12,130.530 | 14,479.290 | −16.22% | |

10 | 372.976 | 18.570 | 12,099.800 | 14,445.330 | −16.24% | |

20 | 373.000 | 18.575 | 12,084.430 | 14,428.360 | −16.25% | |

$a$ | −20 | … | … | … | … | … |

−10 | 356.797 | 23.530 | 8425.492 | 10,169.700 | −17.15% | |

10 | 396.470 | 16.294 | 16,417.390 | 19,396.440 | −15.36% | |

20 | 421.316 | 14.743 | 21,174.930 | 24,849.270 | −14.79% | |

$b$ | −20 | 440.256 | 15.430 | 23,542.580 | 26,605.330 | −11.51% |

−10 | 402.391 | 16.758 | 17,024.210 | 19,709.090 | −13.62% | |

10 | 351.622 | 21.574 | 8430.589 | 10,448.790 | −19.32% | |

20 | … | … | … | … | … | |

$\alpha $ | −20 | 372.562 | 18.559 | 12,069.910 | 14,412.980 | −16.26% |

−10 | 372.757 | 18.562 | 12,092.530 | 14,437.640 | −16.24% | |

10 | 373.146 | 18.569 | 12,137.790 | 14,486.980 | −16.22% | |

20 | 373.341 | 18.573 | 12,160.430 | 14,511.660 | −16.20% | |

$\beta $ | −20 | 368.712 | 18.256 | 11,801.260 | 14,129.450 | −16.48% |

−10 | 370.784 | 18.405 | 11,957.220 | 14,294.920 | −16.35% | |

10 | 375.226 | 18.739 | 12,275.230 | 14,631.730 | −16.11% | |

20 | 377.627 | 18.928 | 12,437.600 | 14,803.340 | −15.98% | |

$s$ | −20 | 372.370 | 20.753 | 12,731.030 | 14,804.060 | −14.00% |

−10 | 372.435 | 19.538 | 12,496.160 | 14,709.430 | −15.05% | |

10 | 373.801 | 17.786 | 11,596.440 | 14,070.370 | −17.58% | |

20 | 374.922 | 17.169 | 10,948.390 | 13,540.900 | −19.15% | |

${t}_{1}$ | −20 | 365.411 | 17.276 | 14,929.950 | 16,843.630 | −11.36% |

−10 | 368.994 | 17.861 | 13,503.730 | 15,639.110 | −13.65% | |

10 | 377.446 | 19.446 | 10,770.670 | 13,315.880 | −19.11% | |

20 | 382.828 | 20.617 | 9479.562 | 12,203.190 | −22.32% | |

${T}_{fc}$ | −20 | 372.948 | 18.565 | 12,116.020 | 14,463.200 | −16.23% |

−10 | 372.950 | 18.565 | 12,115.590 | 14,462.750 | −16.23% | |

10 | 372.953 | 18.566 | 12,114.730 | 14,461.860 | −16.23% | |

20 | 372.955 | 18.566 | 12,114.300 | 14,461.420 | −16.23% | |

${T}_{vc}$ | −20 | 372.928 | 18.561 | 12,121.190 | 14,468.520 | −16.22% |

−10 | 372.940 | 18.564 | 12,118.180 | 14,465.410 | −16.23% | |

10 | 372.963 | 18.568 | 12,112.150 | 14,459.200 | −16.23% | |

20 | 372.974 | 18.570 | 12,109.130 | 14,456.100 | −16.24% | |

N.B. (…) means infeasible solution |

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## Share and Cite

**MDPI and ACS Style**

Miah, M.S.; Islam, M.M.; Hasan, M.; Mashud, A.H.M.; Roy, D.; Sana, S.S. A Discount Technique-Based Inventory Management on Electronics Products Supply Chain. *J. Risk Financial Manag.* **2021**, *14*, 398.
https://doi.org/10.3390/jrfm14090398

**AMA Style**

Miah MS, Islam MM, Hasan M, Mashud AHM, Roy D, Sana SS. A Discount Technique-Based Inventory Management on Electronics Products Supply Chain. *Journal of Risk and Financial Management*. 2021; 14(9):398.
https://doi.org/10.3390/jrfm14090398

**Chicago/Turabian Style**

Miah, Md. Sujan, Md. Mominul Islam, Mahmudul Hasan, Abu Hashan Md. Mashud, Dipa Roy, and Shib Sankar Sana. 2021. "A Discount Technique-Based Inventory Management on Electronics Products Supply Chain" *Journal of Risk and Financial Management* 14, no. 9: 398.
https://doi.org/10.3390/jrfm14090398