The impact of lead time on the complex behavior of a single-echelon supply chain Chong Zhang Southeast University POMS 20th Annual Conference

Southeast University Conclusions Subsystem and stability analysis The model Introduction Complex behavior of supply chain Outline

Southeast University Introduction - Related past studies  Forrester(1960) firstly studied the dynamic behavior of supply chains: “Demand Amplification”  Three broad approaches  The control theory  The behavioral science approach  The practitioner approach

Southeast University  Control theory approach develop methodologies applying system control principles to dampen the dynamics within a supply chain AthorsYearTopic Disney,Towill2002The stability of a VMI supply chain, adopting APIBPCS, using the discrete linear contr ol theory Dejonckheere et al 2002,2003, 2004 The bullwhip effect, applying control engineering techniques: z-transform, transfer fu nction, frequency plot Lin2004Bullwhip effect, linear discrete system with lead time and operation constraint Introduction - Related past studies

Southeast University  Behavioral science approach how human decision making generates uncertainties AthorsYearTopic Thomsen et al 1992Realistic parameters support hyperchaotic, higher-order hyperchaotic behavior Larsen et al1999found unstable behaviors, such as chaos/hyperchaos, generate higher cost than stable /periodic behavior Introduction - Related past studies

Southeast University  Larsen, et al(1999) Introduction - Related past studies

Southeast University Introduction - Related past studies

Southeast University  Practitioner approach simulation models to study the dynamic behavior of supply chains AthorsYearTopic Parunak, VanderBok 1998Multi-agent models Angerhofer, Angelides 2000System dynamics models Dong, Chenl2001Petri-net models Introduction - Related past studies

Southeast University  Wilding(1998) observed the chaos and discusses the implication of chaos theory: sensitivity to initial conditions, island of stability, pattern, and invalidation of the reductionist view.  Hwaring at el (2008) analyzed supply chain dynamics from a chaos perspective, such as demand pattern, ordering policy, demand-information sharing, lead time, different options of levels. Introduction - Related past studies

Southeast University  The objective of the paper  To understand lead time affects the dynamic behavior of the single-echelon supply chain system using switched system theory and discrete system control theory  To investigate the stability of the subsystems and the supply chain system by methods of analytics and simulation  To study chaos behavior of the whole supply chain system Introduction

Southeast University  The beer distribution model The model - The beer distribution model

Southeast University The model

Southeast University Step 1Step 2 Step 3Step 4 Satisfy the demand according to the current inventory Observe the demand of the customer Receive the previous orders Observe the new inventory Place the new order Step 5 The model

Southeast University  Consider a retailer, one external supplier who has the sufficient inventory  Customer demand is exogenous and like a step function  four units per time period in the first four time period  eight per period from fifth period till the end of the experiment or simulation  Unsatisfied order is complete backlogging  Placed orders cannot be cancelled and shipments made cannot be returned supply chain system turns out to be an autonomous switched system The model -Assumption

Southeast University External demand at time period t Forecasted demand of the retailer at the end of time period t Net inventory of the retailer at time period t Inventory of the retailer at time period t Received goods of the retailer at the beginning of time period t The model - Notations Backorders of the retailer at the end of time period t, non-positive value Work in process of the retailer at time period t Shipments of the retailer at time period t Amount order quantity of the retailer at the end of time period t Indicated order quantity of the retailer at the end of time period t

Southeast University Inventory and order based production control system (IOBPCS ）

Southeast University  Lead time (L)  The retailer uses period review system  A fixed replenishment lead time between the time when the order is placed by the retailer and the time when the order is received from the supplier  Demand policy The model

Southeast University  the received goods  The backup order  The inventory  The shipments  The net inventory The model -Inventory policy

Southeast University  John et al(1994)showed that, mixing work-in-process or inventory-on-transit in ventory into the inventory control process can make the inventory stable, and fl eetly respond to the client demand The model -Work-in-process policy

Southeast University  APIOBPCS (Automatic Pipeline Inventory and Order Based Production Contr ol System) policy -- extended forms of the Anchoring and Adjustment Heuristic(Sterman1989) is target inventory level is target work-in-process are the rate of the discrepancy The model -Order policy

Southeast University  the return policy is not permitted, just as the constraint of the order quantity in beer game  Net inventory  Work-in-process The model

Southeast University System state space equation  system input  system output  state variables of the supply chain system  Denote The state space of supply chain system is

Southeast University Subsystems and stability analysis  Stability criterion of the subsystems Case when lead time is equal to 1

Southeast University Subsystems and stability analysis  Stability criterion of the subsystems Case when lead time is equal to 1

Southeast University Case when lead time is equal to 1  Case when lead time is equal to 1 That is, there is no work-in-process inventory Indicated order quantity simplify as Switch rule is subsystem 11 subsystem 12

Southeast University  Subsystem 11 Case when lead time is equal to 1 State space model State variable Characteristic equation The eigenvalues According to theorem 1, this subsystem 11 is Lyapunov stable.

Southeast University  Subsystem 12 Case when lead time is equal to 1 State space model Characteristic equation The eigenvalues According to theorem 1, this subsystem12 is Lyapunov stable.

Southeast University Case when lead time is greater than or equal to 2  Case when lead time is equal to 1 Switch rule is subsystem 22 subsystem 21 subsystem 23 subsystem 24

Southeast University  Subsystem 21 State space model State variable Characteristic equation The eigenvalues According to theorem 2, this subsystem 21 is non-stable. Case when lead time is greater than or equal to 2

Southeast University  Subsystem 22 State space model The eigenvalues According to theorem 2, this subsystem 22 is Lyapunov stable. Case when lead time is greater than or equal to 2

Southeast University  Subsystem 23 State space model Two of the eigenvalues Characteristic equation We only analyze the equation Case when lead time is greater than or equal to 2

Southeast University Case when lead time is greater than or equal to 2 Fig.2 All the roots in a unit circle as lead time increases According to theorem 2, this subsystem 23 is not stable.

Southeast University  Subsystem 24 State space model Two of the eigenvalues Characteristic equation We only analyze the equation Case when lead time is greater than or equal to 2

Southeast University Case when lead time is greater than or equal to 2 Fig.3 All the roots in a unit circle as lead time increases subsystem 24 is stable. subsystem 24 is not stable. Hoberg(2007) obtained the same conclusions by simulation analysis

Southeast University Case when lead time is greater than or equal to 2

Southeast University Complex behavior of supply chain  The stability of the supply chain system Fig.4 The inventory, indicated order and evolving process of supply chain system

Southeast University Complex behavior of supply chain Fig.5 The inventory, indicated order and evolving process of supply chain system

Southeast University Complex behavior of supply chain Fig.6 The inventory, indicated order and evolving process of supply chain system

Southeast University Chaos behavior in the supply chain system

Southeast University  Calculate the Largest Lyapunov Exponents (LLE)  Insights can be obtained:  the degree of system chaos increases as the adjustment parameters enlarge, as to make inventory run up and down (imply: both of the inventory cost and out-of-stock co st will be higher)  lead time takes a very important role in the effect on the system chaos  to indicates that target inventory and work-in-process should be cautiously. Chaos behavior in the supply chain system

Southeast University Conclusions  To study the stability and dynamic behavior exhibited at a single-echelon supply chain u nder the influence of lead time  The stability of the subsystems of the supply chain is analyzed by means of linear discret e system control theory and switched system theory.  The behavior of supply chain system becomes non-stable or chaos as lead time increases

Southeast University Conclusions  Future research Assumptions:  Unsatisfied ordering of the retailer is given up  Unsatisfied order is partial backlogging  Return policy is permitted The total objective:  To increase the service level  To study the model of supply chain system on basis of cost minimization

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