# Improving Supply Chain Profit through Reverse Factoring: A New Multi-Suppliers Single-Vendor Joint Economic Lot Size Model

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Supply Chain Finance and Reverse Factoring

#### 2.2. Joint Economic Lot Size Models

## 3. Model Development

j | part index, $j=1,2,\dots ,k$; |

${u}_{j}$ | number of units of part type $j$ that go into one unit of the finished product, $j=1,2,\dots ,k$ (unit); |

${A}_{v}$ | vendor’s fixed order cost ($/order); |

${a}_{v,j}$ | cost for placing a purchase order for the $j$th part ($/order); |

${c}_{0}$ | unit cost ($/unit); |

$D$ | annual demand rate (unit/year); |

${h}_{v,PF}$ | unit holding cost of finished product per year, consisting of two components, one physical (${h}_{v,PF,p}$) and the other financial (${h}_{v,PF,f}$) ($/unit/year); |

${h}_{v,j}$ | unit holding cost of part j at the vendor’s warehouse per year, consisting of two components, one physical (${h}_{v,j,p}$) and the other financial (${h}_{v,j,f}$) ($/unit/year); |

$k$ | number of part types in the finished product; |

${p}_{v}$ | product unit selling price ($/unit); |

${P}_{v}$ | vendor’s production rate (unit/year); |

$q$ | lot size quantity (unit); |

${\rho}_{v}$ | interest rate the bank offers to the vendor (%/year); |

$S$ | vendor’s setup cost ($/setup). |

s | supplier index, $s$ = 1, 2, …, $m$; |

$m$ | total number of suppliers; |

${A}_{s,j}$ | setup cost that supplier $s$ incurs when producing the $j$th part, $j=1,2,\dots ,k$ ($/setup); |

${c}_{s,j}$ | unit production cost of part $j$ for supplier $s$ ($/unit); |

${h}_{s,j,0}$ | unit holding cost per unit of time for part $j$ supplied by supplier $s$, when there is no coordination of the financial flow. It consists of two contributions, one physical (${h}_{s,j,p}$) and the other financial (${h}_{s,j,f,0}$) ($/unit/year); |

${h}_{s,j}$ | holding cost per unit of time for $j$th part supplied by supplier $s$, consisting of two contributions, one physical (${h}_{s,j,p}$) and the other financial (${h}_{s,j,f}$) ($/unit/year); |

${n}_{s,j}$ | number of shipments for part $j$ supplier $s$ sends to the vendor; |

${P}_{s,j}$ | production rate of supplier $s$ for part $j$ (unit/year); |

${p}_{s,j}$ | supplier $s$ unit selling price for part $j$ ($/unit); |

${\rho}_{s}$ | interest rate the bank offers to supplier s when there is collaboration of financial flows in a supply chain (%/year); |

${\rho}_{s,0}$ | interest rate the bank offers to supplier s when there is no financial collaboration (%/year); |

${Y}_{s,j}$ | binary parameter assuming value 1 if the part $j$ is supplied by supplier s; 0 otherwise. |

#### 3.1. Problem Description and Assumptions

- Deterministic demand and constant over time which is lower than the production rate of the vendor ${P}_{v}$;
- The final product requires $k$ different parts;
- Shortages are not allowed;
- Lead time is assumed to be zero;
- An infinite time horizon is considered.

#### 3.2. Model Development

#### 3.2.1. The Vendor’s Annual Profit Function

#### 3.2.2. The Suppliers’ Annual Profit Function

#### 3.2.3. The Supply Chain’s Annual Profit Function

#### 3.2.4. Solution Procedure

- ${N}_{0}=2\left(S+{A}_{v}+{\displaystyle \sum _{j=1}^{k}}{a}_{v,j}+{\displaystyle \sum _{s=1}^{m}\sum _{j=1}^{k}}\frac{{A}_{s,j}}{{n}_{s,j}}{Y}_{s,j}\right)D$
- ${N}_{1}={\displaystyle \sum _{j=1}^{k}}{h}_{v,j}\frac{{u}_{j}D}{{P}_{v}}+{h}_{v,PF}\left(1-\frac{D}{{P}_{v}}\right)$
- ${N}_{2}={h}_{v,PF}\left(1-\frac{D}{{P}_{v}}\right)+{\displaystyle \sum _{s=1}^{m}\sum _{j=1}^{k}}{h}_{s,j}{u}_{j}\left[\left(1-\frac{{u}_{j}D}{{P}_{s,j}}\right){n}_{s,j}+\frac{2{u}_{j}D}{{P}_{s,j}}-1\right]{Y}_{s,j}$
- ${N}_{3}=2{\displaystyle \sum _{s=1}^{m}\sum _{j=1}^{k}\sum _{i=1}^{{n}_{s,j}}}\frac{{p}_{j}D}{{n}_{s,j}}{\rho}_{v}\left[\frac{{u}_{j}}{{P}_{s,j}}+\left(i-1\right)\frac{1}{D}\right]{Y}_{s,j}$

#### 3.3. Reference Case without Reverse Factoring

## 4. Numerical Example

## 5. Summary and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Classification of the main supply finance solutions (Observatory for Supply Chain Finance 2016).

**Figure 4.**Schematic representation of a supply chain consisting of multi-suppliers, a vendor, and a bank (third party), who manages the financial flows.

**Figure 6.**Supply chain total annual profit as a function of the lot size for the cases with and without a reverse factoring agreement.

**Figure 7.**Optimal lot size, and total cost of the supply chain as a function of interest rates: (

**a**) of the vendor ${\rho}_{v}$, (

**b**) of the supplier with reverse factoring ${\rho}_{s}$, and (

**c**) of the supplier without reverse factoring ${\rho}_{s,0}$.

Supplier | Part | Units for One Part of Final Product (${\mathit{u}}_{\mathit{j}}$) | Order Cost (${\mathit{a}}_{\mathit{v}\mathbf{,}\mathit{j}}$) | Setup Cost (${\mathit{A}}_{\mathit{s}\mathbf{,}\mathit{j}}$) | Unit Production Cost (${\mathit{c}}_{\mathit{s}\mathbf{,}\mathit{j}}$) | Production Rate (${\mathit{P}}_{\mathit{s}\mathbf{,}\mathit{j}}$) | Selling Price (${\mathit{p}}_{\mathit{s}\mathbf{,}\mathit{j}}$) | Physical Holding Cost (${\mathit{h}}_{\mathit{s}\mathbf{,}\mathit{j}\mathbf{,}\mathit{p}}$) | ${\mathit{\rho}}_{\mathit{s}}$ | ${\mathit{\rho}}_{\mathit{s}\mathbf{,}\mathbf{0}}$ |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 5 | 10 | 400 | 10 | 5500 | 15 | 1 | 2% | 15% |

1 | 2 | 2 | 10 | 400 | 20 | 3500 | 25 | 2 | ||

2 | 3 | 1 | 5 | 300 | 30 | 1500 | 40 | 2 | 3% | 20% |

Case | q | n_{1,1} | n_{1,2} | n_{2,3} | TP_{V} | TP_{S}_{,1} | TP_{S}_{,2} | TP_{SC} | $\mathbf{\Delta}\mathit{T}{\mathit{P}}_{\mathit{S}\mathit{C}}$ |
---|---|---|---|---|---|---|---|---|---|

with RF | 219 | 5 | 3 | 3 | $17,602 | $32,438 | $9120 | $59,161 | +3.23% |

without RF | 202 | 4 | 2 | 2 | $17,849 | $31,015 | $8450 | $57,313 |

Vendor | Supplier 1 | Supplier 2 | ||||||
---|---|---|---|---|---|---|---|---|

Case | SETUP COST | Order Cost | Holding Cost | Interest to the Bank | Setup Cost | Holding Cost | Setup Cost | Holding Cost |

with RF | $915 | $572 | $585 | $327 | $976 | $1585 | $457 | $423 |

without RF | $992 | $620 | $539 | - | $1488 | $2498 | $744 | $807 |

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**MDPI and ACS Style**

Marchi, B.; Zanoni, S.; Jaber, M.Y.
Improving Supply Chain Profit through Reverse Factoring: A New Multi-Suppliers Single-Vendor Joint Economic Lot Size Model. *Int. J. Financial Stud.* **2020**, *8*, 23.
https://doi.org/10.3390/ijfs8020023

**AMA Style**

Marchi B, Zanoni S, Jaber MY.
Improving Supply Chain Profit through Reverse Factoring: A New Multi-Suppliers Single-Vendor Joint Economic Lot Size Model. *International Journal of Financial Studies*. 2020; 8(2):23.
https://doi.org/10.3390/ijfs8020023

**Chicago/Turabian Style**

Marchi, Beatrice, Simone Zanoni, and Mohamad Y. Jaber.
2020. "Improving Supply Chain Profit through Reverse Factoring: A New Multi-Suppliers Single-Vendor Joint Economic Lot Size Model" *International Journal of Financial Studies* 8, no. 2: 23.
https://doi.org/10.3390/ijfs8020023