1. International Conference on Numerical Analysis & Optimization:
Theory and Applications
December 18-19, 2011, Dhahran, Saudi Arabia.
Algebraic Optimization Algorithm for Inventory Coordination in the N-Stage Non-Serial Supply Chain with Planned Back-Orders
Mohammed E. Seliaman
Department of Information Systems
College of Computer Science and Information Technology,
King Faisal University, KSA

2. Outline Supply Chain Management (SCM) The Serial and non-serial SC
Related work
The problem definition
Model Development
Algebraic method for optimization
The algorithm
Conclusion , any remarks or questions

3. Supply Chain Management (SCM)
A set of approaches used to efficiently integrate
Suppliers Manufacturers Warehouses Distributers
So that the product is produced and distributed
In the right quantities
To the right locations
And at the right time
System-wide costs are minimized and
Service level requirements are satisfied
(Simchi-Levi et al.,2003)

4. A four-Stage Serial Supply chain
Information and Cash Flow
Supplier
Manufacturer
Distributor
Retailer
Materials and Services Flow

5. A four-stage non-serial supply chain
Information and Cash Flow
Suppliers
Manufacturers
Distributors S1 S2 S3
Retailers
Materials and Services Flow S4

6. Related work
Cárdenas-Barrón (2007) solved the n-stage- multi-customer supply chain inventory model with the equal cycle inventory coordination.
Seliaman and Ahmad (2009) solved the same model considering integer multipliers coordination without backorders.
Chung and Wee, (2007) solved a three-stage serial SC model with backorders
Seliaman(2011) extended this model by adding a fourth stage and using integer multipliers

7. The Problem
We develop an algebraic optimization algorithm for the generalized n-stage, multi-customer, non-serial supply chain inventory problem.
We consider the integer multiplier inventory coordination mechanism with planned backorders and linear and fixed penalties

8. Notation T =Basic cycle time, cycle time at the end retailers
Ti =Cycle time at stage i
Sij = Setup cost for firm j at stage i
Si =Total setup cost for all firms at stage i
Ki =Integer multiplier at stage i
hi =Inventory holding cost at stage i
ni =Number of firms at stage i
Dij =The demand rate of firm j at stage i
D=The demand rate of the entire supply chain
Pij =Production rate of firm j at stage i
= Backordering cost per unit per unit time
=per unit backorder cost(fixed)
A=The product of all production rates for all the companies in the supply chain.
Bij= The product of all production rates for all the companies in the supply chain, except for the company j in stage i.

9. Assumptions
A single product is produced and distributed through a multi- stage, multi-customer, non-serial, supply chain.
Production rates and Demand rates are deterministic and uniform.
Ordering /setup costs are the same for firms at the same stage.
Holding costs cost are the same for firms at the same stage.
A lot produced at stage is sent in equal shipments to the downstream stage.
The supply chain is vertically integrated and the entire supply chain optimization is acceptable for all partners in the chain.
Shortages are allowed for the end retailers.
Cycle time at each stage is an integer multiplier of the cycle time used at the adjacent downstream stage.

10. Retailer Inventory without back- orders
TDn.j T T T
Inventory holding per cycle =
Inventory holding per unit time =

11. Retailer Inventory with back- orders
TDn.j T- Ts Ts T- Ts Ts T T

12. Model Development
The time-weighted total cost for the jth retailer T

13. Model Development Cont.
The time-weighted total cost for all of the retailers together is : T

14. Model Development Cont.
Raw materials and finished products at two consecutive stages:
Ti+1 Ti
Production Portion
Non Production portion
Finished products
Incoming
materials
Inv ent ory lev el

15. Model Development Cont.
The total annual cost for any firm at any stage, except for the final stage: T

16. Model Development Cont.
The total cost for the entire supply chain is T

17. Model Development Cont.
The total cost for the entire supply chain can be represented in the following compact form

18. Model Development Cont.
Initial values

19. Algebraic Optimization of TC

20. Algebraic Optimization of TC cont.

21. Algebraic Optimization of TC cont.

22. Algebraic Optimization of TC cont.

23. Algebraic Optimization of TC cont.

24. Algebraic Optimization of TC cont.

25. The Algorithm

26. References
Cárdenas-Barrón, L. E. (2006). Optimizing Inventory Decisions in a Multi-stage Multi-Customer Supply Chain: A Note. Transportation Research Part E: Logistics and Transportation Review. In Press.
Chung, C. J. and Wee, H. M. (2007). Optimizing the Economic Lot Size of a Three-Stage Supply Chain with Backordering Derived without Derivatives. European Journal of Operational Research. 183:
Moutaz Khouja. (2003) Optimizing inventory decisions in a multi- stage multi-customer supply chain. Transportation Research. Part E, Logistics & Transportation Review, Exeter. pp
M. E. Seliaman and Ab Rahman Ahmad, “A Generalized algebraic Model for Optimizing Inventory Decisions in a Multi-Stage Complex Supply Chain”, Transportation Research Part E: Logistics and Transportation Review, Elsevier Inc. 45(3), 2009, pp.:
M. E. Seliaman (2011)“ using complete squares methods...”, Advances in Decision Sciences, 2011

27. Than You