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

2. 2 2015-6-28 Southeast University Conclusions Subsystem and stability analysis The model Introduction Complex behavior of supply chain Outline

3. 3 2015-6-28 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

4. 4 2015-6-28 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

5. 5 2015-6-28 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

6. 6 2015-6-28 Southeast University  Larsen, et al(1999) Introduction - Related past studies

7. 7 2015-6-28 Southeast University Introduction - Related past studies

8. 8 2015-6-28 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

9. 9 2015-6-28 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

10. 10 2015-6-28 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

11. 11 2015-6-28 Southeast University  The beer distribution model The model - The beer distribution model

12. 12 2015-6-28 Southeast University The model

13. 13 2015-6-28 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

14. 14 2015-6-28 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

15. 15 2015-6-28 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

16. 16 2015-6-28 Southeast University Inventory and order based production control system (IOBPCS ）

17. 17 2015-6-28 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

18. 18 2015-6-28 Southeast University  the received goods  The backup order  The inventory  The shipments  The net inventory The model -Inventory policy

19. 19 2015-6-28 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

20. 20 2015-6-28 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

21. 21 2015-6-28 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

22. 22 2015-6-28 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

23. 23 2015-6-28 Southeast University Subsystems and stability analysis  Stability criterion of the subsystems Case when lead time is equal to 1

24. 24 2015-6-28 Southeast University Subsystems and stability analysis  Stability criterion of the subsystems Case when lead time is equal to 1

25. 25 2015-6-28 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

26. 26 2015-6-28 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.

27. 27 2015-6-28 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.

28. 28 2015-6-28 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

29. 29 2015-6-28 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

30. 30 2015-6-28 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

31. 31 2015-6-28 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

32. 32 2015-6-28 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.

33. 33 2015-6-28 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

34. 34 2015-6-28 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

35. 35 2015-6-28 Southeast University Case when lead time is greater than or equal to 2

36. 36 2015-6-28 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

37. 37 2015-6-28 Southeast University Complex behavior of supply chain Fig.5 The inventory, indicated order and evolving process of supply chain system

38. 38 2015-6-28 Southeast University Complex behavior of supply chain Fig.6 The inventory, indicated order and evolving process of supply chain system

39. 39 2015-6-28 Southeast University Chaos behavior in the supply chain system

40. 40 2015-6-28 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

41. 41 2015-6-28 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

42. 42 2015-6-28 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

43. Thank you