# Inventory Management of Perishable Goods with Overconfident Retailers

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## Abstract

**:**

## 1. Introduction

## 2. Model

- (1)
- When the retailer of a perishable agricultural product lacks the ability and experience to predict the market size of the product, it may overestimate it. When the product market size prediction goes too high, without deviation in the market fluctuation prediction, we derive the demand function as follows:

- (2)
- When the retailer of a perishable agricultural products lacks the ability and experience to observe fluctuation deviations of the random variable that reflects the market demand, it may overestimate its forecasting ability, falsely believing that the market demand fluctuates less than it actually does. Specifically, the retailer may think that the variance ratio of the random variable $\epsilon $ is smaller than that in the rational case. Using the guaranteed mean value transformation method of random variables (see [30]), we derive the market demand function perceived by the overconfident retailer as follows:

#### 2.1. Optimal Decision Making and Profit of the Retailer under the Rational Scenario

#### 2.2. Optimal Decision Making and Profit under Retailer’s Over-Estimation Scenario

#### 2.3. Optimal Decision Making and Profit under Retailer’s Over-Precision Scenario

## 3. Numerical Studies

_{1}and ρ

_{2}are, the more unfavorable the overconfidence behavior is to the retailer.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Relative earning loss of overconfident retailer when the unit cost of handling surplus changes.

**Figure 3.**Relative earning loss of the overconfident retailer when the unit-shortage penalty cost changes.

$m$ | Maximum market size |

$k$ | Customer price sensitivity factor |

$\epsilon $ | A random variable representing the uncertain demand function (We assume that ε follows the uniform distribution U$\left[-d,d\right]$, with distribution function F(·) and probability density function f(·).) |

t | Listing time of the perishable product |

a | Sensitive factor of freshness to time, $a\ge 0$ |

${b}_{1}$ | Over-estimation level, $0\le {b}_{1}\le 1$ |

${b}_{2}$ | Over-precision level, $0\le {b}_{2}\le 1$ |

c | Unit production cost |

h | Unit surplus cost |

s | Unit shortage cost |

q | Order quantity |

p | Retail price |

$h$ | −3 | −1 | 1 | 3 | 5 |

${p}_{0}$ (rational) | 25.15 | 24.94 | 24.65 | 24.31 | 23.92 |

${p}_{1}$(over-estimated) | 31.99 | 31.86 | 31.66 | 31.43 | 31.16 |

${p}_{2}$(over-precise) | 25.20 | 25.14 | 25.06 | 24.96 | 24.86 |

${q}_{0}$ (rational) | 45.88 | 43.22 | 39.05 | 36.28 | 33.84 |

${q}_{1}$ (over-estimated) | 53.57 | 50.54 | 47.87 | 45.49 | 43.35 |

${q}_{2}$ (over-precise) | 27.93 | 26.84 | 25.91 | 25.12 | 24.43 |

$E\left[{\pi}_{0}\left({p}_{0},{q}_{0}\right)\right]$ | 353.56 | 305.65 | 264.05 | 227.73 | 195.88 |

$E\left[{\pi}_{0}\left({p}_{1},{q}_{1}\right)\right]$ | 306.94 | 258.53 | 216.44 | 179.36 | 146.83 |

$E\left[{\pi}_{0}\left({p}_{2},{q}_{2}\right)\right]$ | 280.86 | 249.25 | 220.88 | 195.25 | 171.81 |

${\rho}_{1}$ (over-estimated) | 13.19 | 15.42 | 18.03 | 21.24 | 25.04 |

${\rho}_{2}$ (over-precise) | 20.56 | 18.45 | 16.35 | 14.26 | 12.29 |

$c$ | 1 | 3 | 5 | 7 | 9 |

${p}_{0}$ | 23.16 | 23.96 | 24.65 | 25.24 | 25.74 |

${p}_{1}$ | 30.00 | 30.87 | 31.66 | 32.39 | 33.06 |

${p}_{2}$ | 23.21 | 24.15 | 25.06 | 25.94 | 26.81 |

${q}_{0}$ | 48.17 | 43.48 | 39.05 | 34.85 | 30.81 |

${q}_{1}$ | 55.76 | 51.71 | 47.87 | 44.20 | 40.67 |

${q}_{2}$ | 30.01 | 27.91 | 25.91 | 24.00 | 22.15 |

$E\left[{\pi}_{0}\left({p}_{0},{q}_{0}\right)\right]$ | 438.16 | 364.55 | 264.05 | 190.17 | 124.54 |

$E\left[{\pi}_{0}\left({p}_{1},{q}_{1}\right)\right]$ | 391.52 | 299.49 | 216.44 | 141.72 | 75.02 |

$E\left[{\pi}_{0}\left({p}_{2},{q}_{2}\right)\right]$ | 358.22 | 286.69 | 220.88 | 160.47 | 105.10 |

${\rho}_{1}$ | 10.64 | 13.58 | 18.03 | 25.48 | 39.76 |

${\rho}_{2}$ | 18.24 | 17.27 | 16.35 | 15.62 | 15.61 |

$s$ | 1 | 3 | 5 | 7 | 9 |

${p}_{0}$ | 24.46 | 24.57 | 24.65 | 24.72 | 24.78 |

${p}_{1}$ | 31.57 | 31.62 | 31.66 | 31.70 | 31.74 |

${p}_{2}$ | 25.01 | 25.04 | 25.06 | 25.08 | 25.10 |

${q}_{0}$ | 37.38 | 38.28 | 39.05 | 39.73 | 40.32 |

${q}_{1}$ | 46.80 | 47.36 | 47.87 | 48.32 | 48.73 |

${q}_{2}$ | 25.44 | 25.69 | 25.91 | 26.11 | 26.27 |

$E\left[{\pi}_{0}\left({p}_{0},{q}_{0}\right)\right]$ | 269.37 | 266.51 | 264.05 | 261.89 | 259.99 |

$E\left[{\pi}_{0}\left({p}_{1},{q}_{1}\right)\right]$ | 221.08 | 218.60 | 216.44 | 214.45 | 212.60 |

$E\left[{\pi}_{0}\left({p}_{2},{q}_{2}\right)\right]$ | 238.42 | 229.55 | 220.88 | 212.44 | 204.00 |

${\rho}_{1}$ | 17.93 | 17.98 | 18.03 | 18.11 | 18.23 |

${\rho}_{2}$ | 11.49 | 13.87 | 16.35 | 18.88 | 21.54 |

${b}_{1}$ | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 |

${p}_{0}$ | 24.65 | 24.65 | 24.65 | 24.65 | 24.65 |

${p}_{1}$ | 27.00 | 29.34 | 31.66 | 33.98 | 36.29 |

${q}_{0}$ | 39.05 | 39.05 | 39.05 | 39.05 | 39.05 |

${q}_{1}$ | 42.09 | 45.02 | 47.87 | 50.66 | 53.38 |

$E\left[{\pi}_{0}\left({p}_{0},{q}_{0}\right)\right]$ | 264.05 | 264.05 | 264.05 | 264.05 | 264.05 |

$E\left[{\pi}_{0}\left({p}_{1},{q}_{1}\right)\right]$ | 258.73 | 242.79 | 216.44 | 179.55 | 132.30 |

${\rho}_{1}$ | 2.01 | 8.05 | 18.03 | 32.00 | 49.90 |

${b}_{2}$ | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 |

${p}_{0}$ | 24.65 | 24.65 | 24.65 | 24.65 | 24.65 |

${p}_{2}$ | 24.83 | 24.89 | 24.95 | 25.00 | 25.04 |

${q}_{0}$ | 39.05 | 39.05 | 39.05 | 39.05 | 39.05 |

${q}_{2}$ | 33.54 | 31.57 | 29.68 | 27.80 | 25.91 |

$E\left[{\pi}_{0}\left({p}_{0},{q}_{0}\right)\right]$ | 264.05 | 264.05 | 264.05 | 264.05 | 264.05 |

$E\left[{\pi}_{0}\left({p}_{2},{q}_{2}\right)\right]$ | 256.26 | 250.14 | 242.19 | 232.46 | 220.88 |

${\rho}_{2}$ | 2.95 | 5.27 | 8.28 | 11.96 | 16.35 |

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**MDPI and ACS Style**

Zhang, M.; Yang, X.; Cheng, T.E.; Chang, C. Inventory Management of Perishable Goods with Overconfident Retailers. *Mathematics* **2022**, *10*, 1716.
https://doi.org/10.3390/math10101716

**AMA Style**

Zhang M, Yang X, Cheng TE, Chang C. Inventory Management of Perishable Goods with Overconfident Retailers. *Mathematics*. 2022; 10(10):1716.
https://doi.org/10.3390/math10101716

**Chicago/Turabian Style**

Zhang, Mingyang, Xufeng Yang, Taichiu Edwin Cheng, and Chen Chang. 2022. "Inventory Management of Perishable Goods with Overconfident Retailers" *Mathematics* 10, no. 10: 1716.
https://doi.org/10.3390/math10101716