# Stochastic Review Inventory Systems with Deteriorating Items; A Steady-State Non-Linear Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Inventory Models with Deterioration

#### 2.2. Replenishment Policies

#### 2.3. Integrated Queuing-Inventory Models

## 3. Material and Methods

#### 3.1. Queuing Inventory Models

#### 3.1.1. Problem Assumptions

- The first-rate concerns customers who have already entered the system while the inventory is not empty upon their arrival, but the quantity in hand deteriorated during the waiting time and became less than what they requested, making the customer wait longer.
- The second rate concerns the arrival of customers when the inventory is empty. In this case, they will be blocked.

#### 3.1.2. Mathematical Formulation

#### 3.1.3. M/M/1/N-System with Inventory and Lost Sales

#### 3.1.4. Performance Measures

#### 3.1.5. The Cost Function

#### 3.2. Classical Inventory Queuing System with Deterioration

#### Mathematical Formulation

## 4. Results and Discussion

#### 4.1. Numerical Analysis

#### 4.1.1. The Effect of $\lambda $ on Performance Measures for Different Q Values

#### 4.1.2. The Effect of $\mu $ on Performance Measures for Different Q Values

#### 4.2. M/M/1/N-System with Deterioration under Deterministic and Uniformly Distributed Order Size

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

**Theorem**

**A2.**

## Appendix B

## References

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**Figure 3.**The effect of increasing demand ($\lambda $) for different $\gamma $ values under deterministic order size.

**Figure 4.**The effect of increasing demand ($\lambda $) for different $\gamma $ values under uniform order size.

**Table 1.**The list of variables introduced during the building of the balance equations and the investigated performance metrics.

Term | Definition |
---|---|

Q | The ordered quantity by the management |

µ | The number of customers served per unit time (the service rate) |

$\lambda up$ | The number demand arrived per unit of time (customers arrival rate) |

${p}_{k}$ | The probability of the replenishment order with size k |

${F}_{P}$ | The discrete distribution |

$\overline{{F}_{p}}$ | The tail distribution corresponding to probability p |

${q}_{k}$ | The probability of the replenishment order is at least k units |

$\upsilon $ | The average lead intensities/rates |

$\gamma $ | The average deterioration intensities/rates per unit |

$\overline{I}$ | The average inventory level |

$\overline{p}$ | The mean order size of new delivery |

${E}_{Z}$ | State space of stochastic process representing the current state of the inventory system = {(n,k):n $\in N,k\in \left\{0,\dots \dots ,M\right\}\}$ |

M | The maximal size of the storage space measured in units of inventory |

N | The maximal capacity of the number of customers in the system at any given time |

n | The current number of customers in the system |

k | The current number unites available in the system |

${\lambda}_{R}$ | The mean number of replenishments per time unit |

${\lambda}_{A}$ | The mean number of customers arriving per unit time |

$\overline{LS}$ | The average number of lost sales incurred per unit of time |

${\overline{LS}}_{c}$ | The expected number of lost sales per cycle |

${\overline{L}}_{0}$ | The mean number of customers in the system |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

15 | 11 | 3.6886 | 0.943 | 7.543 | 3.457 | 3.667 | 0.686 | 0.324 | 0.079 | 0.043 | 0.010 | 8.532 |

15 | 15 | 3.487 | 1.154 | 9.231 | 5.769 | 5.000 | 0.615 | 0.500 | 0.167 | 0.054 | 0.018 | 10.782 |

15 | 25 | 2.776 | 1.531 | 12.245 | 12.755 | 8.333 | 0.490 | 1.250 | 0.694 | 0.102 | 0.057 | 18.028 |

15 | 40 | 2.126 | 1.874 | 14.994 | 25.006 | 13.341 | 0.375 | 7.649 | 6.761 | 0.510 | 0.451 | 36.101 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

25 | 11 | 7.020 | 0.660 | 8.850 | 2.420 | 3.667 | 0.780 | 0.324 | 0.079 | 0.038 | 0.009 | 10.370 |

25 | 15 | 6.500 | 0.833 | 10.833 | 4.167 | 5.000 | 0.722 | 0.500 | 0.167 | 0.046 | 0.015 | 11.907 |

25 | 25 | 5.484 | 1.172 | 15.234 | 9.776 | 8.333 | 0.609 | 1.250 | 0.694 | 0.082 | 0.046 | 17.455 |

25 | 40 | 4.445 | 1.518 | 19.737 | 20.263 | 13.346 | 0.493 | 7.653 | 6.765 | 0.388 | 0.343 | 33.430 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

33 | 11 | 9.597 | 0.532 | 9.048 | 1.952 | 3.667 | 0.823 | 0.324 | 0.079 | 0.036 | 0.009 | 12.361 |

33 | 15 | 9.015 | 0.682 | 11.951 | 3.409 | 5.000 | 0.773 | 0.500 | 0.167 | 0.043 | 0.014 | 13.530 |

33 | 25 | 7.829 | 0.987 | 16.776 | 8.224 | 8.333 | 0.609 | 1.250 | 0.694 | 0.075 | 0.041 | 18.107 |

33 | 40 | 6.541 | 1.318 | 22.405 | 17.595 | 13.350 | 0.560 | 7.655 | 6.767 | 0.342 | 0.302 | 32.720 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 11 | 15.154 | 0.377 | 9.617 | 1.383 | 3.667 | 0.874 | 0.324 | 0.079 | 0.034 | 0.008 | 17.207 |

50 | 15 | 14.492 | 0.492 | 12.541 | 2.459 | 5.000 | 0.836 | 0.500 | 0.167 | 0.040 | 0.013 | 17.888 |

50 | 25 | 13.064 | 0.739 | 18.842 | 6.158 | 8.333 | 0.754 | 1.250 | 0.694 | 0.066 | 0.037 | 21.074 |

50 | 40 | 11.386 | 1.029 | 26.249 | 13.751 | 13.359 | 0.656 | 7.658 | 6.770 | 0.292 | 0.258 | 33.520 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

15 | 10 | 4.126 | 0.816 | 6.524 | 2.476 | 3.035 | 0.728 | 6.444 | 5.548 | 0.988 | 0.850 | 13.617 |

15 | 16 | 4.121 | 0.818 | 6.545 | 2.455 | 3.000 | 0.727 | 4.745 | 4.182 | 0.725 | 0.639 | 11.985 |

15 | 20 | 4.121 | 0.818 | 6.545 | 2.455 | 3.000 | 0.727 | 4.617 | 4.167 | 0.705 | 0.637 | 11.888 |

15 | 35 | 4.121 | 0.818 | 6.545 | 2.455 | 3.000 | 0.727 | 4.488 | 4.231 | 0.686 | 0.646 | 11.812 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

25 | 10 | 7.319 | 0.560 | 7.286 | 1.714 | 3.058 | 0.813 | 8.911 | 8.015 | 1.223 | 1.100 | 18.337 |

25 | 16 | 7.313 | 0.562 | 7.312 | 1.688 | 3.000 | 0.813 | 7.741 | 7.179 | 1.059 | 0.982 | 17.198 |

25 | 20 | 7.313 | 0.562 | 7.312 | 1.688 | 3.000 | 0.813 | 7.653 | 7.203 | 1.047 | 0.985 | 17.132 |

25 | 35 | 7.313 | 0.563 | 7.313 | 1.688 | 3.000 | 0.813 | 7.565 | 7.308 | 1.035 | 0.999 | 17.079 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

33 | 10 | 9.923 | 0.448 | 7.621 | 1.379 | 3.075 | 0.851 | 11.197 | 10.301 | 1.496 | 1.352 | 22.813 |

33 | 16 | 9.917 | 0.450 | 7.650 | 1.350 | 3.000 | 0.850 | 10.260 | 9.697 | 1.341 | 1.268 | 21.892 |

33 | 20 | 9.917 | 0.450 | 7.650 | 1.350 | 3.000 | 0.850 | 10.189 | 9.739 | 1.332 | 1.273 | 21.839 |

33 | 35 | 9.917 | 0.450 | 7.650 | 1.350 | 3.000 | 0.850 | 10.119 | 9.861 | 1.323 | 1.289 | 21.797 |

Q | $\mathit{\lambda}$ | $\overline{\mathit{I}}$ | ${\mathit{\lambda}}_{\mathit{R}}$ | ${\mathit{\lambda}}_{\mathit{A}}$ | $\overline{{\mathit{L}}_{\mathit{S}}}$ | ${\overline{\mathit{L}\mathit{S}}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\overline{{\mathit{L}}_{\mathit{o}}}$ | $\overline{\mathit{L}}$ | $\overline{{\mathit{W}}_{\mathit{o}}}$ | $\overline{\mathit{W}}$ | F |
---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | 10 | 15.516 | 0.315 | 8.021 | 0.979 | 3.311 | 0.895 | 16.410 | 15.514 | 2.046 | 1.934 | 33.125 |

50 | 16 | 15.509 | 0.316 | 8.053 | 0.979 | 3.000 | 0.895 | 15.749 | 15.187 | 1.956 | 1.886 | 32.462 |

50 | 20 | 15.509 | 0.316 | 8.053 | 0.979 | 3.000 | 0.895 | 15.700 | 15.250 | 1.950 | 1.894 | 32.425 |

50 | 35 | 15.509 | 0.316 | 8.053 | 0.979 | 3.000 | 0.895 | 15.650 | 15.393 | 1.944 | 1.912 | 32.395 |

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**MDPI and ACS Style**

Alrasheedi, A.F.; Alnowibet, K.A.; Alotaibi, I.T.
Stochastic Review Inventory Systems with Deteriorating Items; A Steady-State Non-Linear Approach. *Processes* **2022**, *10*, 781.
https://doi.org/10.3390/pr10040781

**AMA Style**

Alrasheedi AF, Alnowibet KA, Alotaibi IT.
Stochastic Review Inventory Systems with Deteriorating Items; A Steady-State Non-Linear Approach. *Processes*. 2022; 10(4):781.
https://doi.org/10.3390/pr10040781

**Chicago/Turabian Style**

Alrasheedi, Adel F., Khalid A. Alnowibet, and Ibtisam T. Alotaibi.
2022. "Stochastic Review Inventory Systems with Deteriorating Items; A Steady-State Non-Linear Approach" *Processes* 10, no. 4: 781.
https://doi.org/10.3390/pr10040781