A Mechanism Implementing in Nash Equilibria the Optimal Centralized Solution of a Supply Chain Problem

Transcription

1 Introduction Model Decentralized mechanism Conclusion A Mechanism Implementing in ash Equilibria the Optimal Centralized Solution of a Supply Chain Problem Shruti Sharma Ph.D. candidate, Electrical Engineering and Computer Science Volodymyr Babich, Mark Van Oyen Industrial and Operations Engineering Demos Teneketzis Electrical Engineering and Computer Science University of Michigan, Ann Arbor IFORMS Annual Meeting 2008, October 14, Washington DC, USA Shrutivandana Sharma University of Michigan, Ann Arbor 1 / 34

2 Introduction Model Decentralized mechanism Conclusion Outline 1 Introduction 2 The supply chain model Supply chain problem 3 Decentralized mechanism Solution approach: Implementation theory framework A decentralized mechanism for supply chain Results 4 Conclusion Shrutivandana Sharma University of Michigan, Ann Arbor 2 / 34

3 Introduction Model Decentralized mechanism Conclusion Overview Set-up A supply chain coordination problem Arbitrary number of suppliers and manufacturers Decentralized and asymmetric information Competitive/selfish/strategic decision makers with no prior beliefs Our work Design of a decentralized negotiation mechanism that, preserves private information of the agents makes the agents willingly participate in the mechanism obtains optimal centralized transactions at ash equilibrium Shrutivandana Sharma University of Michigan, Ann Arbor 3 / 34

4 Introduction Model Decentralized mechanism Conclusion Literature survey Principal-Agent Models Contracting in Supply Chain Management: Review by Cachon (2003) Economics: Myerson (1981, 1982), Grossman and Hart (1983), Guesnerie Laffont (1984), McAfee and McMillan (1986), Maskin and Tirole (1990, 1992) Operations Management: Corbett and Tang (1999), Corbett and de Groote (2000), Cachon and Lariviere (2001), Iyer, Schwarz, and Zenios (2005), Yang, Aydin, Babich, and Beil (2008a, b) Coordination might not be attainable: Ha (2001), Cakanyildirim, Gan, and Sethi (2006) Coordination of cross-functional decisions within a firm: Porteus and Whang (1991), Kouvelis and Lariviere (2000) Shrutivandana Sharma University of Michigan, Ann Arbor 4 / 34

5 Introduction Model Decentralized mechanism Conclusion Contribution Generality of model Arbitrary number of suppliers and manufacturers Complete decentralization of information Competitive/selfish decision makers with no prior beliefs about other agents Developed decentralized negotiation mechanism that, preserves private information of the agents makes the agents willingly participate in the mechanism obtains optimal centralized transactions at all ash equilibria balances the flow of products and money between the suppliers and manufacturers at equilibria Presented a method to characterize all ash equilibria for a given system wide objective, and a given decentralized negotiation mechanism Shrutivandana Sharma University of Michigan, Ann Arbor 5 / 34

6 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model Shrutivandana Sharma University of Michigan, Ann Arbor 6 / 34

7 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers Suppliers 1 i S Shrutivandana Sharma University of Michigan, Ann Arbor 7 / 34

8 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 8 / 34

9 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 9 / 34

10 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Supply vector: x i = (x i1, x i2,..., x il ) 0 Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 10 / 34

11 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Supply vector: x i = (x i1, x i2,..., x il ) 0 Capacity constraint: A i x i b i Cost of production: c i (x i ) Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 11 / 34

12 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Supply vector: x i = (x i1, x i2,..., x il ) 0 Capacity constraint: A i x i b i Cost of production: c i (x i ) Purchase vector: y j = (y j1, y j2,..., y jl ) 0 Value of purchase: v j (y j ) Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 12 / 34

13 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Supply vector: x i = (x i1, x i2,..., x il ) 0 Capacity constraint: A i x i b i Cost of production: c i (x i ) Purchase vector: y j = (y j1, y j2,..., y jl ) 0 Value of purchase: v j (y j ) Payment by manufacturers: g j > 0 Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 13 / 34

14 Introduction Model Decentralized mechanism Conclusion Supply chain problem The supply chain model S suppliers M manufacturers L products Supply vector: x i = (x i1, x i2,..., x il ) 0 Capacity constraint: A i x i b i Cost of production: c i (x i ) Purchase vector: y j = (y j1, y j2,..., y jl ) 0 Value of purchase: v j (y j ) Payment by manufacturers: g j > 0 Payment to suppliers: r i > 0 Suppliers 1 i S 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 14 / 34

15 Introduction Model Decentralized mechanism Conclusion Supply chain problem Information available to the agents Suppliers: Utility of supplier i: ui S (r i, x i ) = r i c i (x i ) 1 I D i S ((r i, x i )) (1) I D S ((r i, x i )) i Suppliers 1 i S payment received cost of production c i is a convex function of x i with c i (0) = 0. Each supplier s production capability, production capacity, and the cost of production are its private information. Suppliers are self utility maximizers / behave strategically. 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 15 / 34

16 Introduction Model Decentralized mechanism Conclusion Supply chain problem Information available to the agents Manufacturers: Utility of manufacturer j: uj M (g j, y j ) 1 I D M (y j ) = g j +v j (y j ) j (2) I D M (y j ) j Suppliers 1 i S payment made + value of purchase v j is concave in y j with v j (0) = 0. v j increases in each element of y j. Each manufacturer s purchase value is its private information. Manufacturers are self utility maximizers / behave strategically. 1 j M Manufacturers Shrutivandana Sharma University of Michigan, Ann Arbor 16 / 34

17 Introduction Model Decentralized mechanism Conclusion Supply chain problem The centralized supply chain problem Problem (P SC ) max (g,r,x,y) S ui S (r i, x i ) + i=1 M j=1 u M j (g j, y j ) (3) s.t. and S M x i = y j (4) i=1 j=1 M S g j = r i (5) j=1 i=1 (P SC ) obtains a transaction that is balanced in product and money transfers and maximizes the sum of utilities of suppliers and manufacturers. Solution of Problem (P SC ) = Ideal transaction Shrutivandana Sharma University of Michigan, Ann Arbor 17 / 34

18 Introduction Model Decentralized mechanism Conclusion Supply chain problem How to obtain centralized solution Characteristics of the supply chain model Decentralized information: obody has complete system information. Strategic agents: The suppliers and manufacturers are selfish. Solution approach: Implementation theory Provides guidelines for: how the agents should communicate with one another, and how the information communicated by the agents should be used to determine the transactions so as to induce the selfish agents to communicate information that results in an optimal centralized transaction. Reference: Implementation theory Maskin (1985), Jackson (2001), Palfrey (2002), Stoenescu and Teneketzis (2005) Shrutivandana Sharma University of Michigan, Ann Arbor 18 / 34

19 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Shrutivandana Sharma University of Michigan, Ann Arbor 19 / 34

20 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Shrutivandana Sharma University of Michigan, Ann Arbor 20 / 34

21 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Allocation space Feasible product and money transactions Shrutivandana Sharma University of Michigan, Ann Arbor 21 / 34

22 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Goal correspondence Solution set of Problem ( P SC ) Allocation space Feasible product and money transactions Shrutivandana Sharma University of Michigan, Ann Arbor 22 / 34

23 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Goal correspondence Solution set of Problem ( P SC ) Allocation space Feasible product and money transactions Message space Μ Shrutivandana Sharma University of Michigan, Ann Arbor 23 / 34

24 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Goal correspondence Solution set of Problem ( P SC ) Allocation space Feasible product and money transactions Message space Μ Outcome function f Decentralized mechanism Game form: (M, f ) Shrutivandana Sharma University of Michigan, Ann Arbor 24 / 34

25 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Goal correspondence Solution set of Problem ( P SC ) Allocation space Feasible product and money transactions Agent's strategy induced by ( M, f ) and equilibrium concept Message space Μ Outcome function f Decentralized mechanism Game form: (M, f ) Induced game: (M, f, {u S i } S i=1, {um j } M j=1 ) Shrutivandana Sharma University of Michigan, Ann Arbor 25 / 34

26 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Supply chain problem in implementation theory framework Environment space Agent's utility, private and common information Goal correspondence Solution set of Problem ( P SC ) Allocation space Feasible product and money transactions Agent's strategy induced by ( M, f ) and equilibrium concept Message space Μ Outcome function f ash equilibrium: A message profile m is a E if, u S i (f (m )) u S i (f ((m S i, m /i))), m S i M S i, i {1, 2,..., S } u M j (f (m )) u M j (f ((m M j, m /j))), m M j M M j, j {1, 2,..., M } Shrutivandana Sharma University of Michigan, Ann Arbor 26 / 34

27 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Interpretation of ash equilibria Traditional definition of ash equilibria for games of complete information Difference in supply chain model Supply chain model does not result in game of complete information Agents utilities are private information Suppliers and manufacturers are involved in a message exchange process Interpretation The stationary points of the message exchange process should have properties of ash equilibria. Shrutivandana Sharma University of Michigan, Ann Arbor 27 / 34

28 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Desirable properties of a decentralized mechanism Implementation in ash equilibria: A game form (M, f ) fully implements the goal correspondence π in ash equilibria if, for all problem environments, Set of transactions at all ash equilibria = Set of optimal centralized transactions Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

29 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Desirable properties of a decentralized mechanism Implementation in ash equilibria: A game form (M, f ) fully implements the goal correspondence π in ash equilibria if, for all problem environments, Set of transactions at all ash equilibria = Set of optimal centralized transactions Individual rationality: A game form (M, f ) is individually rational if, for all suppliers and manufacturers, Utility at all ash equilibria Utility before/without participating in the negotiation process specified by the game form Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

30 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Desirable properties of a decentralized mechanism Implementation in ash equilibria: A game form (M, f ) fully implements the goal correspondence π in ash equilibria if, for all problem environments, Set of transactions at all ash equilibria = Set of optimal centralized transactions Individual rationality: A game form (M, f ) is individually rational if, for all suppliers and manufacturers, Utility at all ash equilibria Utility before/without participating in the negotiation process specified by the game form Goal: To design an individually rational game form (M, f ) for the Supply Chain problem that implements in ash Equilibria the goal correspondence π corresponding to (P SC ). Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

31 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results A game form for the supply chain problem Message space: Suppliers: m S i := (x i, p S i ); x i R L, p S i R L + (6) (Supply vector, Price vector) proposal for L products Manufacturers: m M j := (y j, p M j ); y j R L, p M j R L + (7) (Purchase vector, Price vector) proposal for L products (8) Shrutivandana Sharma University of Michigan, Ann Arbor 29 / 34

32 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results A game form for the supply chain problem Message space: Suppliers: m S i := (x i, p S i ); x i R L, p S i R L + (6) (Supply vector, Price vector) proposal for L products Manufacturers: m M j := (y j, p M j ); y j R L, p M j R L + (7) (Purchase vector, Price vector) proposal for L products (8) Inspiring paper: Outcome functions yielding Walrasian and Lindahl allocations at ash equilibrium points, Hurwicz, 1979 Shrutivandana Sharma University of Michigan, Ann Arbor 29 / 34

33 Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results A game form for the supply chain problem (cont ) Outcome function: ˆx i (m) = 1 x i ( S 1) k S k i x k + 1 S j M y j (9) Deviation from average supply + Average demand ŷ j (m) = 1 y j ( M 1) k M k j y k + 1 M i S ˆx i (m) (10) Deviation from average demand + Average supply ˆr i (m) = p S i (m)t ˆx i ( p S i p S i (m)) T ( p S i p S i (m)) (11) where p S i (m) := 1 ( p S k ( S 1 + M ) + ) p M j (12) k j S M k i ĝ j (m) = p M j (m)t ŷ j + ( p M j p M j (m)) T ( p M j p M j (m)) (13) where p M j (m) := 1 ( p M k ( M 1 + S ) + ) p S i (14) k i M S k j Equilibrium price does not depend on agent s own message (15) Quadratic penalty term forces the agents to agree on one price (16) Shrutivandana Sharma University of Michigan, Ann Arbor 30 / 34

34 Results Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results Theorem 1: Let m be a ash equilibria of the game specified by the game form and the agents utility functions. Let (ĝ(m ), ˆr(m ), ˆx(m ), ŷ(m )) =: (ĝ, ˆr, ˆx, ŷ ) be the transaction at m determined by the game form. Then, (a) (ĝ, ˆr, ˆx, ŷ ) is individually rational, and (b) (ĝ, ˆr, ˆx, ŷ ) is an optimal solution of Problem (P SC ). Theorem 2: Given the optimum supply and purchase vector (ˆx, ŷ ) of Problem (P SC ), there exists at least one ash equilibria m of the game corresponding to the proposed game form and the agents utility functions such that, (ˆx(m ), ŷ(m )) = (ˆx, ŷ ). Furthermore, given (ˆx, ŷ ), the set of all ash equilibriathat result in (ˆx, ŷ ) can be characterized. Shrutivandana Sharma University of Michigan, Ann Arbor 31 / 34

35 Introduction Model Decentralized mechanism Conclusion Conclusion Conclusion Studied a decentralized supply chain coordination problem with arbitrary number of suppliers and manufacturers under competitive set up. Developed a decentralized negotiation mechanism that obtains optimal centralized transactions at all ash equilibria. Presented a method to characterize all ash equilibria corresponding to the decentralized mechanism. Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

36 Introduction Model Decentralized mechanism Conclusion Conclusion Conclusion Studied a decentralized supply chain coordination problem with arbitrary number of suppliers and manufacturers under competitive set up. Developed a decentralized negotiation mechanism that obtains optimal centralized transactions at all ash equilibria. Presented a method to characterize all ash equilibria corresponding to the decentralized mechanism. Future scope We have a constructive proof for the existence of ash equilibria. We do not have an algorithm to show how to converge to the ash equilibria. Orthogonal/greedy search is not guaranteed to converge because the resulting game is not supermodular. Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

37 Introduction Model Decentralized mechanism Conclusion Conclusion Conclusion Studied a decentralized supply chain coordination problem with arbitrary number of suppliers and manufacturers under competitive set up. Developed a decentralized negotiation mechanism that obtains optimal centralized transactions at all ash equilibria. Presented a method to characterize all ash equilibria corresponding to the decentralized mechanism. Future scope We have a constructive proof for the existence of ash equilibria. We do not have an algorithm to show how to converge to the ash equilibria. Orthogonal/greedy search is not guaranteed to converge because the resulting game is not supermodular. Developing algorithms or supermodular games that lead to the optimum centralized transactions. Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

38 Introduction Model Decentralized mechanism Conclusion Thank You! Shrutivandana Sharma University of Michigan, Ann Arbor 33 / 34

39 Introduction Model Decentralized mechanism Conclusion Questions? Contact: Shrutivandana Sharma web: svandana Volodymyr Babich web: babich Demosthenis Teneketzis web: teneket Mark P. Van Oyen web: vanoyen Shrutivandana Sharma University of Michigan, Ann Arbor 34 / 34