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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

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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

Email: [email protected]


Outline
Outline Optimization:

  • 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


Supply chain management scm
Supply Chain Management ( Optimization: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-widecostsare minimized and

  • Service level requirements are satisfied

  • (Simchi-Levi et al.,2003)


A four stage serial supply chain
A four-Stage Serial Supply chain Optimization:

Materials and Services Flow

Information and Cash Flow

Supplier

Manufacturer

Distributor

Retailer


A four stage non serial supply chain
A four-stage non-serial supply chain Optimization:

Materials and Services Flow

Information and Cash Flow

Suppliers

Manufacturers

Distributors

S2

S1

S3

Retailers

S4


Related work
Related work Optimization:

  • 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


The problem
The Problem Optimization:

  • 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


Notation
Notation Optimization:

  • 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.


Assumptions
Assumptions Optimization:

  • 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.


Retailer inventory without back orders
Retailer Inventory without back- orders Optimization:

TDn.j

T

T

T

Inventory holding per cycle =

Inventory holding per unit time =


Retailer inventory with back orders
Retailer Inventory with Optimization:back- orders

TDn.j

T-

Ts

Ts

Ts

T-

Ts

T

T


Model development
Model Development Optimization:

  • The time-weighted total cost for the jth retailer

T


Model development cont
Model Development Cont. Optimization:

  • The time-weighted total cost for all of the retailers together is :

T


Model development cont1

Inventory level Optimization:

Finished products

Incoming

materials

Ti+1

Ti+1

Ti+1

Production Portion

Non Production portion

Ti

Model Development Cont.

  • Raw materials and finished products at two consecutive stages:


Model development cont2
Model Development Cont. Optimization:

The total annual cost for any firm at any stage, except for the final stage:

T


Model development cont3
Model Development Cont. Optimization:

  • The total cost for the entire supply chain is

T


Model development cont4
Model Development Cont. Optimization:

  • The total cost for the entire supply chain can be represented in the following compact form


Model development cont5
Model Development Cont. Optimization:

  • Initial values



Algebraic optimization of tc cont
Algebraic Optimization of Optimization:TC cont.


Algebraic optimization of tc cont1
Algebraic Optimization of Optimization:TC cont.


Algebraic optimization of tc cont2
Algebraic Optimization of Optimization:TC cont.


Algebraic optimization of tc cont3
Algebraic Optimization of Optimization:TC cont.


Algebraic optimization of tc cont4
Algebraic Optimization of Optimization:TC cont.


The algorithm
The Algorithm Optimization:


References
References Optimization:

  • 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. 183:933-943.

  • MoutazKhouja. (2003) Optimizing inventory decisions in a multi-stage multi-customer supply chain. Transportation Research. Part E, Logistics & Transportation Review, Exeter. pp 193-208.

  • M. E. Seliaman and AbRahman 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.: 409-418.

  • M. E. Seliaman (2011)“ using complete squares methods...”, Advances in Decision Sciences, 2011


Than you
Than You Optimization:


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