Competitive and Cooperative Inventory Policies in a Two-Stage Supply-Chain

Transcription

1 Compeiive and Cooperaive Invenory Policies in a Two-Sage Supply-Chain (G. P. Cachon and P. H. Zipkin) Presened by Shruivandana Sharma IOE 64, Supply Chain Managemen, Winer 2009 Universiy of Michigan, Ann Arbor

2 Ouline Inroducion Model Cenralized opimizaion problem Invenory games Nash equilibrium oucomes vs. cenralized soluion Opimal linear conracs Conclusion Fuure scope

3 Overview Two sage serial supply chain Saionary sochasic demand Fixed ransporaion ime Single produc Invenory holding coss a each sage Consumer backorder penaly a each sage

4 Moivaion Reailers Kroger JCPenny Bes Buy Suppliers Kellogg Nike Apple

5 Conribuion Models compeiive behavior of agens Game heoreic analysis Each agen has equal posiion in he game Analysis of Nash equilibria Sudy of wo differen games Echelon invenory racking Local invenory racking Design of linear ransfer paymens ha help minimize sysem cos a Nash equilibrium

6 The Model Time L L + L Time is sloed

7 The Model Time L L + L Source Supplier Reailer Time is sloed Flow of produc: Source Supplier Reailer

8 The Model Order quaniy Time L L + L Source Supplier Reailer Time is sloed Flow of produc: Source Supplier Reailer Supplier and reailer submi he orders

9 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Time is sloed Flow of produc: Source Supplier Reailer Supplier and reailer submi he orders Shipmens are immediaely released Lead ime: Source o Supplier ( ), Supplier o Reailer ( L ) L2

10 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Demand: τ D random oal demand over τ periods τ Saionary disribuion: densiy φ, disribuion Demand is a coninuous random variable Posiive demand occurs in each period τ Φ

11 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Invenory levels of ineres: IT i In ransi invenory:, Supplier ( = 2), Reailer IL i i ( i =) Echelon invenory level:, all invenory a sage i or lower in he sysem minus consumer backorders Local invenory level: ILi, invenory a sage i minus backorders a sage i

12 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Invenory levels of ineres: Echelon invenory posiion: Local invenory posiion:

13 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

14 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

15 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

16 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Holding coss: Supplier: h 2 per period for each uni in is sock or en roue o he reailer h + Reailer: per period for each uni in is sock Assumpion: h 2 h2 > 0, h 0

17 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

18 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

19 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer

20 The Model Order quaniy Time L L + L L2 L Source Supplier Reailer Backorder coss: Sysem backorder cos: p per uni backorder Supplier: Reailer: αp Assumpion: ( α) p 0 α

21 Cos Funcions Reailer: Cos in period : Expeced cos in period + L : depends on supplier s order and demand up o ime If If Toal expeced cos

22 Cos Funcions Supplier: Backorder cos in period : Expeced backorder cos in period + L : Le Toal expeced cos

23 Sysem opimal soluion A sysem opimal soluion minimizes he oal average cos per period. ), ( ), ( arg min ), ( ), ( 2 2 s s H s s H s s s s o o + =

24 Echelon Invenory (EI) Game Players: i =,2 Sraegies:, i =,2 Payoffs: -, i =,2 Pure sraegy Nash equilibrium such ha, Game is common knowledge

25 Local Invenory (LI) Game Players: i =,2 Sraegies:, i =,2 Payoffs: -, i =,2 Pure sraegy Nash equilibrium such ha, Game is common knowledge

26 Nash equilibria Theorem 4: For EI game has a unique Nash equilibrium. Theorem 8: For LI game has a unique Nash equilibrium.

27 Nash equilibria (con ) Theorem 9: For EI game has he following Nash equilibria. Theorem 0: For LI game has a unique Nash equilibrium. Theorem : For boh EI and LI games have unique Nash equilibrium and hey are idenical.

28 Comparing Nash equilibria Theorem 2: For he base sock levels for boh firms are higher in he LI game equilibrium han in he EI game equilibrium, i.e. Theorem 3: For he supplier s cos in he LI game equilibrium is lower han is cos in he EI game equilibrium.

29 Nash equilibria and Opimal Soluion Theorem 4: In an EI game equilibrium, he reailer s base sock level is lower han in he opimal soluion. Theorem 5 & 6: For α, he supplier s base sock level in boh he LI and he EI equilibria is lower han in he sysem opimal soluion. Theorem 7: For he sysem opimal soluion is no a Nash equilibrium in eiher game. Theorem 8: For he sysem opimal soluion is a Nash equilibrium in he LI game only when

30 Linear Conracs Period ransfer paymen from supplier o reailer Expeced ransfer paymen in period + L due o reailer invenory and backorders Expeced per period ransfer paymen from supplier o reailer

31 Sysem wih modified coss Coss accouning for ransfer paymens Objecive: To deermine he se of conracs, such ha is a Nash equilibrium for he cos funcions where and

32 Finding opimal linear conrac Assuming o be sricly concave in find he conracs saisfying and a sysems opimal Ou of his se of conracs, selec he subse of conracs ha make he cos funcions sricly concave.

33 Se of opimal linear conracs Theorem 9: When he firms choose a conrac ha saisfies he following properies, hen he opimal policy is a Nash equilibrium.

34 Conclusion When boh players care abou consumer backorders, here is a unique Nash equilibrium in EI game as well as LI game, and hese equilibria differ. The Nash equilibrium of such EI and LI game does no provide opimal soluion of supply chain. Compeiion lowers he supply chain invenory relaive o he opimal soluion. Appropriae linear conracs can help achieving opimal supply chain soluion a some Nash equilibrium.

35 Fuure Scope Muli produc supply chains where demands of differen producs are correlaed and are saionary, can be sudied similarly by considering join disribuion of demands of hese producs. Oher conracs can be invesigaed which ensure ha all Nash equilibria provide opimal supply chain soluion. The work can be exended o incorporae processing imes of orders.

36 Thank you!