# A Mathematical Model for an Inventory Management and Order Quantity Allocation Problem with Nonlinear Quantity Discounts and Nonlinear Price-Dependent Demand

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Description of the Problem and Its Analysis

_{i}), perfect rate (q

_{i}), and capacity (c

_{i}), ∀ i = 1, …, n. It is also essential to establish a minimum perfect rate (q

_{a}). Please refer to Table 1 for a summary of all the parameters and decision variables.

_{1}= 0.92, q

_{2}= 0.95, and q

_{3}= 0.98. Suppose the setup or order cost of $500 is dominant for purchases of a few items. In that case, the average cost can be reduced for purchases of larger quantities due to the volume discount offered by the supplier.

#### 3.1. The Previous Model

_{i}during the order cycle and the order size for each supplier Q

_{i}.

^{(1−e)}, minus the total cost per time unit. The profit function considers the ordering cost per time unit, holding cost per time unit, and purchasing cost per time unit.

#### 3.2. Reference Model’s Analysis: The Feasibility of the Solution Considering a New Focus on the Quality Parameters

- (i)
- The number of orders and the order quantity assigned to each supplier. The number of orders to supplier i is called j
_{i}; it is assumed that all orders assigned to a specific supplier are of the same size, and the size of the order assigned to supplier i is referred to as Q_{i}. - (ii)
- The order cycle period is in months. This, along with the number of orders and order quantity, determines how many items are purchased each month or each order cycle (the order cycle is expressed in months, it does not need to be an integer). In other words, this can be used to calculate the demand. Then, it is not necessary to provide the demand explicitly. Notice that the percentage of the demand covered by each supplier is not explicitly provided but can also be calculated.
- (iii)
- The selling price. This is also implicit since the demand can be calculated, and the selling price can be determined from Equation (1). Still, it is an important variable. In this work, we will consider it to be part of the solution to the problem.

## 4. The Proposed Model

#### 4.1. Model Analysis

_{c}:

_{a}.

_{i}and j

_{i}are positive numbers, no combination can satisfy Equation (26). Since this situation (low perfect rate situation) might be possible in real life, it is highly desirable that models can deal with it. An intuitive solution would be to shorten the order cycle to have more items to compensate for the number of imperfect items and satisfy the demand. Then, a new equation to calculate the order cycle may be required.

_{i}j

_{i}q

_{i}, and the total ordered quantity from supplier i as follows:

#### 4.2. Reformulated Model

#### 4.3. Particle Swarm Optimization (PSO)

Pseudocode. Particle Swarm Optimization |

P = P_{o}; /*Generating the initial population*/ |

t = 0; |

Repeat |

Update (${p}_{i}^{k}$ and ${g}^{k}$); |

Update velocity (${v}_{i}^{k+1}$); Update position (${x}_{i}^{k+1}$); |

t = t + 1 |

Until /*Stopping criteria*/ |

Output/*Best solution so far*/ |

#### 4.3.1. Initialization

#### 4.3.2. Velocity of Particles

_{1}) and social (c

_{2}) factors. The velocity ${v}_{i}^{k}$ of each particle or decision vector ${x}_{i}^{k}$ is updated by employing Equation (45):

#### 4.3.3. Movement of Particles

## 5. Solution of the Numerical Example

_{1}= 4, j

_{2}= 9, j

_{3}= 5, Q

_{1}= 608.68, Q

_{2}= 378.74, Q

_{3}= 486.95, and P = 15.84. This solution led to a monthly profit of $4179.91.

_{max}= 300). By balancing the number of individuals and iterations, we were able to conduct a thorough search for solutions within a reasonable timeframe. The problem involved nine decision variables, which means that it had nine dimensions (n = 9). Our cognitive and social factors were set at c

_{1}= 2 and c

_{2}= 2, respectively. To ensure accuracy and consistency, we repeated the optimization process 30 times for each metaheuristic algorithm. PSO yielded a total of 30 results, and we have provided a summary of the top 10 in Table 8.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters | |

c_{i} | Maximum monthly capacity of supplier i (given in units per month) |

k_{i} | Ordering or setup cost of supplier i (given in dollars per order), ∀ i = 1, …, n. |

r | Storage or holding cost rate (given in dollars per item per month) |

q_{i} | Perfect rate of supplier $i$, ∀ i = 1, …, n. |

q_{a} | Minimum required perfect rate |

v_{i} | Per unit cost (depends on the order quantity assigned Q_{i}) |

M | The maximum number of orders assigned per cycle |

Decision Variables | |

j_{i} | The number of orders assigned to supplier $i$ per order cycle (∑_{j} J_{ij}), ∀ i = 1, …, n. |

Q_{i} | Ordered quantity assigned to supplier i (in units), ∀ i = 1, …, n. |

P | Selling price |

Additional Variables | |

T_{c} | Order cycle period (in months) |

R_{i} | The total number of items ordered to supplier $i$ during the order cycle period (R_{i} = j_{i}Q_{i}), ∀ i = 1, …, n. |

d | Demand per time unit |

Supplier 1 | ||
---|---|---|

Interval | Bound (Units) | Unit Cost (USD) |

1 | 0–50 | v_{1}= 9 |

2 | 50–100 | v_{1} = 8.9 |

3 | 100–150 | v_{1} = 8.8 |

4 | 150–200 | v_{1} = 8.7 |

5 | 200–$\infty $ | v_{1} = 8.6 |

Supplier 2 | ||
---|---|---|

Interval | Bound (Units) | Unit Price (USD) |

1 | 0–75 | v_{2} = 9.8 |

2 | 75–150 | v_{2} = 9.6 |

3 | 150–225 | v_{2} = 9.4 |

4 | 225–$\infty $ | v_{2} = 9.2 |

Supplier 3 | ||
---|---|---|

Interval | Bound (Units) | Unit Price (USD) |

1 | 0–100 | v_{3} = 10.5 |

2 | 100–200 | v_{3} = 10.4 |

3 | 200–$\infty $ | v_{3} = 10.3 |

Reference | EPQ/EOQ | Discounts | Demand Depends on | Methodology | Considering Quality |
---|---|---|---|---|---|

Moon I. et al. [31] | EPQ | No | Production cost | Geometric programming | No |

Kugele ASH. et al. [32] | Smart | No | N.A. | Geometric programming | No |

Pando et al. [33] | EOQ | No | stock | Analytic | No |

Pando et al. [34] | EOQ | No | stock | Analytic | No |

Cárdenas-Barrón et al. [35] | EOQ | No | stock | Analytic | No |

Ventura et al. [9] | EOQ | No | price | Analytic | No |

Venegas et al. [2] | EOQ | All units | price | Game theoretic model | No |

Adeinat and Ventura [4] | EOQ | All units | price | Analytic | Yes |

This study | EOQ | All units | price | Metaheuristic | Yes |

Reference | Quality Parameters Are Considered as | Demand Depends on | Parameters of the Order Cycle |
---|---|---|---|

Mendoza A. et al. [3] | Quality constraint | Constant | Demand, order quantity, number of orders |

Subramanian P. et al. [36] | Quality constraint | Constant | Demand, order quantity, number of orders |

Mendoza A. et al. [37] | Quality constraint | Constant | Demand, order quantity, number of orders |

Adeinat and Ventura [4] | Quality constraint | Price | Demand, order quantity, number of orders |

This study | Part of the order cycle | Price | Demand, order quantity, number of orders, quality parameters |

Variable | Numerical Example Data |
---|---|

r | 3 suppliers |

r | 0.3 per month |

k_{i} | k_{1} = 500, k_{2} = 250, k_{3} = 450, dollars per order |

q_{i} | q_{1} = 0.92, q_{2} = 0.95, q_{3} = 0.98 |

q_{a} | 0.95 |

c_{i} | c_{1} = 300, c_{2} = 350, c_{3} = 250, units a month |

α | 3,375,000 |

e | 3 |

j_{1} | j_{2} | j_{3} | Q_{1} | Q_{2} | Q_{3} | P | Profit | T_{c} |
---|---|---|---|---|---|---|---|---|

5 | 10 | 0 | 625.2024 | 357.5136 | 0 | 17.48964 | $4195.2628 | 10.4658 |

5 | 10 | 0 | 460.4654 | 263.152 | 0 | 17.48778 | $4195.5702 | 7.7031 |

10 | 10 | 5 | 447.4173 | 507.6502 | 399.8 | 16.40596 | $4199.647 | 15.0089 |

4 | 9 | 0 | 618.4156 | 317.4855 | 0 | 17.47399 | $4200.1496 | 8.3043 |

9 | 10 | 0 | 395.3861 | 415.1201 | 0 | 17.419 | $4200.6401 | 11.8975 |

7 | 10 | 0 | 481.8841 | 397.9198 | 0 | 17.46638 | $4211.5892 | 11.3869 |

6 | 10 | 0 | 517.472 | 372.3754 | 0 | 17.48854 | $4212.0876 | 10.6669 |

7 | 10 | 0 | 438.2362 | 361.4035 | 0 | 17.46361 | $4213.3668 | 10.3913 |

5 | 10 | 2 | 668.2426 | 387.1903 | 409.3942 | 16.84653 | $4219.4276 | 11.2654 |

6 | 10 | 0 | 491.0763 | 346.7467 | 0 | 17.43933 | $4223.2641 | 9.9333 |

3 | 5 | 1 | 566.374 | 394.1614 | 352.0102 | 16.89048 | $4236.1505 | 5.6816 |

Method | j_{1} | j_{2} | j_{3} | Q_{1} | Q_{2} | Q_{3} | P | Profit | T_{c} |
---|---|---|---|---|---|---|---|---|---|

Proposed | 6 | 10 | 0 | 491.0763 | 346.7467 | 0 | 17.43933 | $4223.26 | 9.93 |

Proposed | 3 | 5 | 1 | 566.374 | 394.1614 | 352.0102 | 16.89048 | $4236.1505 | 5.68 |

Reference | 4 | 9 | 5 | 608.6800 | 378.7400 | 486.9500 | 15.8400 | $4179.91 | 9.74 |

j_{1} | j_{2} | j_{3} | Q_{1} | Q_{2} | Q_{3} | P | Profit | T_{c} |
---|---|---|---|---|---|---|---|---|

3 | 10 | 2 | 611.43482 | 302.2221 | 1000 | 16.422054 | $3991.35 | 8.9973 |

2 | 4 | 2 | 603.3485 | 549.5823 | 631.8927 | 17.07176 | $3998.97 | 6.8828 |

2 | 5 | 2 | 447.7809 | 252.1053 | 547.7207 | 16.58173 | $4016.19 | 4.3924 |

3 | 10 | 5 | 1000 | 397.1731 | 603.42911 | 16.259653 | $4045.13 | 12.7226 |

4 | 10 | 10 | 1000 | 598.94822 | 401.12309 | 16.079424 | $4046.64 | 17.246 |

4 | 10 | 10 | 781.919 | 500.231 | 357.0862 | 16.27514 | $4066.50 | 14.9458 |

4 | 9 | 5 | 703.4059 | 377.2035 | 570.511 | 16.5024 | $4070.85 | 12.0655 |

6 | 9 | 7 | 446.8211 | 507.0697 | 464.7826 | 16.34089 | $4085.31 | 13.5725 |

6 | 10 | 10 | 599.3858 | 521.8324 | 370.7003 | 15.91724 | $4133.20 | 14.9621 |

6 | 10 | 8 | 462.4226 | 416.3043 | 368.7471 | 15.98633 | $4134.55 | 11.9691 |

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**MDPI and ACS Style**

Alejo-Reyes, A.; Mendoza, A.; Cuevas, E.; Alcaraz-Rivera, M.
A Mathematical Model for an Inventory Management and Order Quantity Allocation Problem with Nonlinear Quantity Discounts and Nonlinear Price-Dependent Demand. *Axioms* **2023**, *12*, 547.
https://doi.org/10.3390/axioms12060547

**AMA Style**

Alejo-Reyes A, Mendoza A, Cuevas E, Alcaraz-Rivera M.
A Mathematical Model for an Inventory Management and Order Quantity Allocation Problem with Nonlinear Quantity Discounts and Nonlinear Price-Dependent Demand. *Axioms*. 2023; 12(6):547.
https://doi.org/10.3390/axioms12060547

**Chicago/Turabian Style**

Alejo-Reyes, Avelina, Abraham Mendoza, Erik Cuevas, and Miguel Alcaraz-Rivera.
2023. "A Mathematical Model for an Inventory Management and Order Quantity Allocation Problem with Nonlinear Quantity Discounts and Nonlinear Price-Dependent Demand" *Axioms* 12, no. 6: 547.
https://doi.org/10.3390/axioms12060547