# An Algebraic Decision Support Model for Inventory Coordination in the Generalized n-Stage Non-Serial Supply Chain with Fixed and Linear Backorders Costs

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### Background

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Notation and Assumptions

- T = Basic inventory replenishment time duration for the retailers at the last chain tier.
- T
_{i}= Time duration for production/inventory replenishment time at the i-th chain tier. - S
_{ij}= Cost of setup for the j-th company at the i-th chain tier. - S
_{i}= The sum of setup total costs for all of the companies at the i-th chain tier. - K
_{i}= A whole number representing the cycle coordination multiplier at the i-th chain tier. - h
_{i}= Cost of holding inventory at the i-th chain tier. - n
_{i}= Total number of companies at the i-th chain tier. - D
_{ij}= Rate of demand for the j-th company at the i-th chain tier. - D = Rate of demand for the whole supply chain
- P
_{ij}= Rate of production at the j-th company at the i-th chain tier. - T
_{S}= the stock-out time at the last chain tier. - = Linear cost of backorder cost at the last chain tier.
- = Backorder cost per unit (fixed) for each retailer.
- A = The product of rates of production for all supply chain firms.
- B
_{ij}= A values obtained by multiplying all the rates of production for all companies in the supply chain, excluding the j-th firm at the i-th chain tier.

- (a)
- In this non-serial supply chain, a single product is produced.
- (b)
- Production rates for all firms at all stages are deterministic and uniform.
- (c)
- End consumers demand rate is deterministic and uniform.
- (d)
- Holding costs are equal for the partners sharing the same tiers.
- (e)
- Shipments are delivered in equal sizes between each two chain tiers.
- (f)
- Shortage is allowed at the last chain tier.
- (g)
- Replenishment time duration at each tier in the chain is an integer multiple of the time duration at the next chain tier.

#### 3.2. Formulation of the Supply Chain Model

_{i}

_{+1}units by the amount of T

_{i}

_{+1}D

_{ij}beginning with the amount of (T

_{i}− T

_{i}

_{+1})D

_{ij}as it is illustrated in Figure 2 (see also [40,41]). Consequently, the total cost for each company i at any stage j, excluding the final stage, is expressed by:

**Theorem**

**1.**

#### 3.3. Development of the Algebraic Optimization Algorithm

_{S}after allowing ${\left(T\sqrt{Y}-\sqrt{W}\right)}^{2}$ and allowing:

## 4. Results

#### 4.1. Solution Algorithm

_{1}thru K

_{n}

_{−1}. In step two, the algorithm calculates the values of Y and W that correspond to the optimal values of the basic inventory cycle time T and the optimal stock-out time T

_{S}. Finally, the optimal values of the basic inventory cycle time T, the optimal stock-out time T

_{S}and the total cost of the whole supply chain are computed in step three.

#### 4.2. Numerical Examples

**Example**

**1.**

**Example**

**2.**

_{0}= 0.1.

${\alpha}_{0}=$ | 1.02321 × 10^{+41} |

${\psi}_{0}=$ | 0 |

${\phi}_{0}=$ | 0 |

_{1}, K

_{2}, and K

_{3}. It first calculates K

_{1}the integer multiplier at the first stage. Second, it computes K

_{2}the integer multiplier at the second stage. Finally, the integer multiplier at the third stage K

_{3}is calculated.

For i = 1: | |

${\alpha}_{1}=$ | 2.01612 × 10^{41} |

${\psi}_{1}=$ | 1.02321× 10^{41} |

${\phi}_{1}=$ | 1000 |

${K}_{1}=$ | 2 |

For i = 2: | |

${\alpha}_{2}=$ | 4.01455× 10^{41} |

${\psi}_{2}=$ | 4.06254× 10^{41} |

${\phi}_{2}=$ | 900 |

${K}_{2}=$ | 2 |

For i = 3: | |

${\alpha}_{3}=$ | 2.83046× 10^{39} |

${\psi}_{3}=$ | 1.21396× 10^{42} |

${\phi}_{3}=$ | 650 |

${K}_{3}=$ | 1 |

_{S}as follows:

Y = | 846,756.926 |

W = | 631.212 |

_{S}and the total cost as follows:

T* = | 0.0273 |

T_{S}* = | 0.0055 |

TC* = | 51752.94 |

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

**IFT20149**. The authors also acknowledge the Deanship of Scientific Research at King Faisal University for the financial support Institutional Financing

**Track 2020**.

## Conflicts of Interest

## Appendix A

## References

- Mahdavi, I.; Mohebbi, S.; Cho, N.; Paydar, M.M.; Mahdavi-Amiri, N. Designing a dynamic buyer-supplier coordination model in electronic markets using stochastic Petri nets. In Innovations in Supply Chain Management for Information Systems: Novel Approaches; Wang, J., Ed.; IGI Global: Hershey, PA, USA, 2010; pp. 279–296. [Google Scholar] [CrossRef]
- Park, K. A Heuristic Simulation–Optimization Approach to Information Sharing in Supply Chains. Symmetry
**2020**, 12, 1319. [Google Scholar] [CrossRef] - Ma, K.; Pal, R.; Gustafsson, E. What modelling research on supply chain collaboration informs us? Identifying key themes and future directions through a literature review. Int. J. Prod. Res.
**2019**, 57, 2203–2225. [Google Scholar] [CrossRef] [Green Version] - Pan, J.L.; Chiu, C.Y.; Wu, K.S.; Yen, H.F.; Wang, Y.W. Sustainable Production–Inventory Model in Technical Cooperation on Investment to Reduce Carbon Emissions. Processes
**2020**, 8, 1438. [Google Scholar] [CrossRef] - Prajogoa, D.; Olhager, J. Supply chain integration and performance: The effects of long-term relationships, information technology and sharing, and logistics integration. Int. J. Prod. Econ.
**2012**, 135, 514–522. [Google Scholar] [CrossRef] - Devaraj, S.; Krajewski, L.; Wei, J. Impact of e-business technologies on operational performance: The role of production information integration in the supply chain. J. Oper. Manag.
**2007**, 25, 1199–1216. [Google Scholar] [CrossRef] - Liu, Y.; Li, Q.; Yang, Z. A new production and shipment policy for a coordinated single-vendor single-buyer system with deteriorating items. Comput. Ind. Eng.
**2019**, 128, 492–501. [Google Scholar] [CrossRef] - Sett, B.K.; Dey, B.K.; Sarkar, B. The Effect of O2O Retail Service Quality in Supply Chain Management. Mathematics
**2020**, 8, 1743. [Google Scholar] [CrossRef] - Kim, C.O.; Jun, J.; Baek, J.K.; Smith, R.L.; Kim, Y.D. Adaptive inventory control models for supply chain management. Int. J. Adv. Manuf. Technol.
**2005**, 26, 1184–1192. [Google Scholar] [CrossRef] [Green Version] - Carter, J.R.; Ferrin, B.G.; Carter, C.R. The effect of less-than-truckload rates on the purchase order lot size decision. Transp. J.
**1995**, 34, 35–45. [Google Scholar] - Yao, Y.; Evers, P.T.; Dresner, M.E. Supply chain integration in vendor-managed inventory. Decis. Support Syst.
**2007**, 43, 663–674. [Google Scholar] [CrossRef] - Chan, F.T.S.; Zhang, T. The impact of collaborative transportation management on supply chain performance: A simulation approach. Expert Syst. Appl.
**2011**, 38, 2319–2329. [Google Scholar] [CrossRef] - Fang, D.; Ren, Q. Optimal decision in a dual-channel supply chain under potential information leakage. Symmetry
**2019**, 11, 308. [Google Scholar] [CrossRef] [Green Version] - Ji, S.F.; Peng, X.S.; Luo, R.J. An integrated model for the production-inventory-distribution problem in the Physical Internet. Int. J. Prod. Res.
**2019**, 57, 1000–1017. [Google Scholar] [CrossRef] - Sarkar, S.; Giri, B.C. Stochastic supply chain model with imperfect production and controllable defective rate. IJSS Oper. Logist.
**2020**, 7, 133–146. [Google Scholar] [CrossRef] - Chang, W.S.; Sanchez-Loor, D.A. Downstream Information Leaking and Information Sharing between Partially Informed Retailers. J. Ind. Compet. Trade
**2020**. [Google Scholar] [CrossRef] - Chung, K.J. The economic production quantity with rework process in supply chain management. Comput. Math. Appl.
**2011**, 62, 2547–2550. [Google Scholar] [CrossRef] [Green Version] - Beck, F.G.; Glock, C.H.; Kim, T. Coordination of a production network with a single buyer and multiple vendors with geometrically increasing batch shipments. Int. J. Prod. Econ.
**2017**, 193, 633–646. [Google Scholar] [CrossRef] - Ben-daya, M.; Al-Nassar, A. Integrated multi-stage multi-customer supply chain. Prod. Plan. Control.
**2008**, 19, 97–104. [Google Scholar] [CrossRef] - Khouja, M.; Goyal, S.K. A review of the joint replenishment problem literature: 1989–2005. Eur. J. Oper. Res.
**2008**, 186. [Google Scholar] [CrossRef] - Abdul-Jalbar, B.; Gutiérrez, J.; Puerto, J.; Sicilia, J. Policies for inventory/distribution systems: The effect of centralization vs. decentralization. Int. J. Prod. Econ.
**2003**, 81–82, 281–293. [Google Scholar] [CrossRef] [Green Version] - Gnoni, M.; Iavagnilio, R.; Mossa, G.; Mummolo, G.; Leva, A.D. Production planning of a multi-site manufacturing system by hybrid modeling: A case study from the automotive industry. Int. J. Prod. Econ.
**2003**, 85, 251–262. [Google Scholar] [CrossRef] - Chen, J.M.; Chen, T.H. The multi-item replenishment problem in a two-echelon supply chain: The effect of centralization versus decentralization. Comput. Oper. Res.
**2005**, 32, 3191–3207. [Google Scholar] [CrossRef] - Grubbström, R.W. Modelling production opportunities—An historical overview. Int. J. Prod. Econ.
**1995**, 41, 1–14. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E. The economic production quantity (EPQ) with shortage derived algebraically. Int. J. Prod. Econ.
**2001**, 70, 289–292. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E. Optimizing inventory decisions in a multi-stage multi-customer supply chain: A note. Transp. Res. Part E Logist. Transp. Rev.
**2007**, 43, 647–654. [Google Scholar] [CrossRef] - Chung, C.J.; Wee, H.M. Optimizing the economic lot size of a three-stage supply chain with backordering derived without derivatives. Eur. J. Oper. Res.
**2007**, 183, 933–943. [Google Scholar] [CrossRef] - Wee, H.M.; Chung, C.J. A note on the economic lot size of the integrated –buyer inventory system derived without derivatives. Eur. J. Oper. Res.
**2007**, 177, 1289–1293. [Google Scholar] [CrossRef] - Seliaman, M.E. Using complete squares method to optimize replenishment policies in a four-stage supply chain with planned backorders. Adv. Decis. Sci.
**2011**, 745896. [Google Scholar] [CrossRef] - Chi, S.W. Production lot size problem with failure in repair and backlogging derived without derivatives. Eur. J. Oper. Res.
**2008**, 188, 610–615. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E. Optimal manufacturing batch size with rework in a single-stage production system—A simple derivation. Comput. Ind. Eng.
**2008**, 55, 758–765. [Google Scholar] [CrossRef] - Ben-Daya, M.; As’ad, R.; Seliaman, M. An integrated production inventory model with raw material replenishment considerations in a three layer supply chain. Int. J. Prod. Econ.
**2013**, 143, 53–61. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E.; Teng, J.T.; Treviño-Garza, G.; Wee, H.M.; Lou, K.R. An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain. Int. J. Prod. Econ.
**2012**, 136, 384–388. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E. The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra. Appl. Math. Model.
**2011**, 35, 2394–2407. [Google Scholar] [CrossRef] [Green Version] - Teng, J.T.; Cárdenas-Barrón, L.E.; Lou, K.R. The economic lot size of the integrated vendor–buyer inventory system derived without derivatives: A simple derivation. Appl. Math. Comput.
**2011**, 217, 5972–5977. [Google Scholar] [CrossRef] - Cárdenas-Barrón, L.E. An easy method to derive EOQ and EPQ inventory models with backorders. Comput. Math. Appl.
**2010**, 59, 948–952. [Google Scholar] [CrossRef] [Green Version] - Cárdenas-Barrón, L.E. A simple method to compute economic order quantities: Some observations. Appl. Math. Model.
**2010**, 34, 1684–1688. [Google Scholar] [CrossRef] [Green Version] - Sana, S.S. A production-inventory model of imperfect quality products in a three-layer supply chain. Decis. Support Syst.
**2011**, 50, 539–547. [Google Scholar] [CrossRef] - Chung, K.J.; Cárdenas-Barrón, L.E. The complete solution procedure for the EOQ and EPQ inventory models with linear and fixed backorder costs. Math. Comput. Model.
**2012**, 55, 2151–2156. [Google Scholar] [CrossRef] - Lin, S.S.C. Note on “The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra”. Appl. Math. Model.
**2019**, 73, 378–386. [Google Scholar] [CrossRef] - Seliaman, M.E.; Ahmad, A. A generalized algebraic model for optimizing inventory decisions in a multi-stage complex supply chain. Transp. Res. Part E Logist. Transp. Rev.
**2009**, 45, 409–418. [Google Scholar] [CrossRef] - Khouja, M. Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transp. Res. Part E Logist. Transp. Rev.
**2003**, 39, 193–208. [Google Scholar] [CrossRef] - García-Laguna, J.; San-José, L.A.; Cárdenas-Barrón, L.E.; Sicilia, J. The integrality of the lot size in the basic EOQ and EPQ models: Applications to other production-inventory models. Appl. Math. Comput.
**2010**, 216, 1660–1672. [Google Scholar] [CrossRef] - Khan, M.; Jaber, M.Y. Optimal inventory cycle in a two-stage supply chain incorporating imperfect items from suppliers. Int. J. Oper. Res.
**2011**, 10, 442–457. [Google Scholar] [CrossRef]

**Figure 3.**The effect of changing the linear backorder cost on the percent savings resulting from the integer multiple cycle time as compared to the equal cycle time.

**Figure 4.**The effect of changing the fixed backorder cost on the percent savings resulting from the integer multiple cycle time as compared to the equal cycle time.

Stage i | Party Index j | Parent (i,j) | Inventory Cost Incoming h_{i} | Annual Production P_{ij} | Annual Demand D_{ij} | Setup Cost Si | Backordering Costs $\mathit{\pi}$ | $\widehat{\mathit{\pi}}$ |
---|---|---|---|---|---|---|---|---|

Supplier | 1 | _ | 0.8 | 399,000 | 133,000 | 800 | - | - |

Manufactures | 1 | 1 | 2 | 140,000 | 70,000 | 200 | - | - |

2 | 1 | 2 | 108,000 | 36,000 | 200 | - | - | |

3 | 1 | 2 | 108,000 | 27,000 | 200 | - | - | |

Retailers | 1 | 1 | 5 | - | 10,000 | 50 | 0.1 | 9.5 |

2 | 1 | 5 | - | 20,000 | 50 | 0.1 | 9.5 | |

3 | 1 | 5 | - | 40,000 | 50 | 0.1 | 9.5 | |

4 | 2 | 5 | - | 12,000 | 50 | 0.1 | 9.5 | |

5 | 2 | 5 | - | 24,000 | 50 | 0.1 | 9.5 | |

6 | 3 | 5 | - | 9000 | 50 | 0.1 | 9.5 | |

7 | 3 | 5 | - | 18,000 | 50 | 0.1 | 9.5 |

Stage | Multiplier | Cycle Time | Shortage Duration | Cost |
---|---|---|---|---|

Supplier | 2 | 0.095 | - | $12,788.18 |

Manufactures | 1 | 0.048 | - | $16,176.75 |

Retailers | - | 0.048 | 0.0095 | $30,355.05 |

Entire Supply chain | - | - | - | $59,319.99 |

Tier i | Member j | Index (i,j) | H_{i} | P_{ij} | D_{ij} | S_{i} | Backordering Costs $\mathit{\pi}$ | $\widehat{\mathit{\pi}}$ |
---|---|---|---|---|---|---|---|---|

Supplier | 1 | _ | 0.8 | 396,000 | 130,000 | 1000 | - | - |

Manufactures | 1 | 1 | 2 | 180,000 | 60,000 | 200 | - | - |

2 | 1 | 2 | 200,000 | 70,000 | 200 | - | - | |

Distributors | 1 | 1 | 4 | 90,000 | 35,000 | 50 | - | - |

2 | 1 | 4 | 70,000 | 25,000 | 50 | - | - | |

3 | 2 | 4 | 80,000 | 30,000 | 50 | - | - | |

4 | 2 | 4 | 100,000 | 40,000 | 50 | - | - | |

Retailers | 1 | 1 | 7 | - | 20,000 | 10 | 0.1 | 9.5 |

2 | 1 | 7 | - | 15,000 | 10 | 0.1 | 9.5 | |

3 | 2 | 7 | - | 25,000 | 10 | 0.1 | 9.5 | |

4 | 3 | 7 | - | 30,000 | 10 | 0.1 | 9.5 | |

5 | 4 | 7 | - | 15,000 | 10 | 0.1 | 9.5 | |

6 | 4 | 7 | - | 25,000 | 10 | 0.1 | 9.5 |

Tier | Multiplier K | T | T_{S} | Cost |
---|---|---|---|---|

Supplier | 2 | 0.1092 | - | 14,093.40 |

Producers | 2 | 0.0546 | - | 14,276.55 |

Wholesale Distributor | 1 | 0.0273 | - | 12,518.31 |

Retailers | - | 0.0273 | 0.0055 | 10,864.68 |

Entire Supply chain | - | - | 51,752.94 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Seliaman, M.; Cárdenas-Barrón, L.; Rushd, S.
An Algebraic Decision Support Model for Inventory Coordination in the Generalized *n*-Stage Non-Serial Supply Chain with Fixed and Linear Backorders Costs. *Symmetry* **2020**, *12*, 1998.
https://doi.org/10.3390/sym12121998

**AMA Style**

Seliaman M, Cárdenas-Barrón L, Rushd S.
An Algebraic Decision Support Model for Inventory Coordination in the Generalized *n*-Stage Non-Serial Supply Chain with Fixed and Linear Backorders Costs. *Symmetry*. 2020; 12(12):1998.
https://doi.org/10.3390/sym12121998

**Chicago/Turabian Style**

Seliaman, Mohamed, Leopoldo Cárdenas-Barrón, and Sayeed Rushd.
2020. "An Algebraic Decision Support Model for Inventory Coordination in the Generalized *n*-Stage Non-Serial Supply Chain with Fixed and Linear Backorders Costs" *Symmetry* 12, no. 12: 1998.
https://doi.org/10.3390/sym12121998