Quantity Discounts in Supply Chain Coordination under Multi-level Information Asymmetry

1

• Quantity Discounts in Supply Chain Coordination
  under Multi-level Information Asymmetry
• Zissis D.1, Ioannou G.1, Burnetas A.2
• dzisis_at_aueb.gr, ioannou_at_aueb.gr,
  aburnetas_at_math.uoa.gr
• 1 Department of Management Science Technology
• Athens University of Economics Business
• 2 Department of Mathematics, University of Athens
  
• 10th Conference on SMMSO 2015
• Volos, June 2015

2
Motivation

• Investigate the feasibility of coordination in
  2-node supply chains
• Assume private information
• Means to achieve coordination Quantity discounts
• To approach the problem, we use
• Tools of Game Theory (every nodes decisions
  affect the other nodes decisions and all the
  payoffs)
• Tools of Supply Chain Management (storage and
  transportation policies, as well as production
  rules affect individual strategies for increasing
  profits)

3
Bibliography

• Economic Order Quantity
• Harris (1913) How many parts to make at once.
  The Magazine of Management
• Quantity Discounts
• Monahan (1984) A quantity discount pricing model
  to increase vendor profits. Management Science
• Weng (1995) Channel coordination and quantity
  discounts. Management Science

4
Bibliography

• Corbett and De Groote (2000) A Suppliers
  Optimal Quantity Discount Policy under Asymmetric
  Information. Management Science
• Chen et al. (2001) Coordination Mechanism for a
  Distribution System with One Supplier and
  Multiple Retailers. Management Science
• Cakanyildirim et al. (2012) Contracting and
  Coordination under Asymmetric Production Cost
  Information. Production and Operations Management

5
Bibliography

• Revelation Principle
• Gibbard (1973) Manipulation of Voting Schemes A
  General Result. Econometrica
• Myerson (1979) Incentive-Compatibility and the
  Bargaining Problem. Econometrica
• Myerson (1982) Optimal Coordination Mechanisms
  in Generalized Principal Agent Problems.
  Journal of Mathematical Economics

6
Corbett and De Groote (2000)

• 2-node supply chain, in which a single product is
  traded
• Asymmetric Information
• The supplier does not know the retailers holding
  cost (cost function), but assumes a prior
  distribution over a continuous range
• In our work, we consider a model with discrete
  values of holding cost in order to
• gain insights on the effect of discrete inventory
  holding cost values
• further investigate a case applicable in practice

7
Model A

• 2-nodes supply chain (Supplier-Retailer), in
  which a single product is traded
• Retailer (R)
• Setup cost KR
• HL, with probability p
• Holding cost HR
• HH, with probability 1-p
• Policy Economic Order Quantity Q
• Supplier (S)
• Setup cost KS
• Policy lot for lot

8
Assumptions

• D Demand is assumed constant, independent of the
  product price
• Shortages and backorders are not allowed
• The Retailer knows the real value of holding
  cost, HR HL, H?
• The Supplier assumes distribution p, 1-p
• rational (minimize its own cost function)
• Nodes are
• risk neutral (due to asymmetric information)
  expected cost

9
Without Quantity Discounts

• Suppliers cost function CS(Q) KSD/Q
• Retailers cost function CR(Q) KRD/Q HRQ/2
• Decision maker only Retailer
• According to EOQ model QR (2KRD/HR)1/2
• Suppliers cost CS(QR)
• Retailers cost CR(QR)

10
Reservation Levels

• They refer to the worst case scenario payoffs
• CR,L (2KRDHL)1/2 if QR,L
  (2KRD/HL)1/2
• Retailers
• CR,H (2KRDHH)1/2 if QR,H (2KRD/HH)1/2
• Suppliers CS(QR,H) KS(DHH)1/2/(2KS)1/2
• Any solution, even one with quantity discounts,
  must conform to the reservation levels of both
  nodes (i.e., have costs up to the reservation
  levels)

11
Quantity Discounts

• The issue is whether an efficient node
  coordination (i.e., lower expected costs -
  individual/total) can be achieved without node
  coalitions or contracts
• Mean Quantity Discount from the Supplier
• Supplier the discount, P(Q), which he will
  offer to Retailer
• Decisions
• Retailer the order quantity (Q)
• Stackelberg game, with Supplier as the leader and
  Retailer as the follower
• Suppliers cost function TCS(Q) KSD/Q P(Q)
• Retailers cost function TCR(Q) KRD/Q HRQ/2
  - P(Q)

12
Complete Information

• Supplier knows the real value of retailers
  holing cost
• It is known that the chain could be coordinated
• Supplier provides in the order quantity QJ
  2(KS KR)D/HR1/2 the discount
  Y KRD/QJ HRQJ /2 (2KRDHR)1/2
• ?CS(QJ ) (2DHR)1/2 (KS KR)1/2 - KR1/2 lt
  CS(QR)
• ?CR(QJ) (2KRDHR)1/2 CR(QR)
• Coordination
• Supplier takes all the profits which arise from
  the coordination

13
Asymmetry Information

• According to Mechanism Design, the Supplier
  provides the quantity-price pair discount m
  P(QL) YL, P(QH) YH
• without
  discount, QR (i.e., QR,L, QR,H)
• Retailers options discount corresponding
  to actual holding cost value
• discount
  corresponding to the other holding cost value
• Thus, we have to determine QL, QH, YL, YH

14
Asymmetry Information

• The Revelation Principle states that for the
  Supplier, designing a mechanism in a way which
  the Retailer reveals his actual holding cost
  value, is an equilibrium strategy
• 
  incentive-compatibility (I.C.) constraints
• P(Q) must conform
• 
  individual-rationality (I.R.) constraints

15
Constraints

• I.R. - constraints
• CR,L(XL) CR,L (1)
• CR,H(XH) CR,H (2)
• I.C. - constraints
• CR,L(XL) -YL CR,L(XH) YH (3)
• CR,H(XH) -YH CR,H(XL) YL (4)

16
Solution

• The Supplier has to solve the following
  optimization problem
• (Expected Supplier's costs)
• p (CS(XL) YL) (1-p) (CS(XH) YH)
• s.t. (1) - (4)

17
Solution

• We distinguish 3 cases according to the
  parameters values
• Case A
• 2 lt (1-p)(1 KS/KR)1/2 (HL)1/2(HH)1/2 / (HH
  - pHL)1/2
• Case B
• (1-p)(1KS/KR)1/2 (HL)1/2(HH)1/2
  /(HH-pHL)1/2 2lt(1KS/KR)1/2 1(HH/HL)1/2
• Case C
• (1 KS/KR)1/2 1(HH/HL)1/2 2

18
Solution

• Solution of Case A
• QL 2(KS KR)D/HL1/2
  YLCR,L(QL)-CR,L(QH)CR,H(QH)-CR,L(QR)
• QH 2(1-p)(KSKR)D/(HH-pHL)1/2
  YH CR,H(QH)-CR,L(QR)
• Solution of Case B
• QL 2(KS KR)D/ HL1/2
  YL CR,L(QL) - CR,L(QR)
• QH 2(2KRD)1/2/(HL)1/2(HR)1/2
  YH CR,H(QH)- CR,H(QR)
• Solution of Case C
• QL 2(KS KR)D/HL1/2
  YL CR,L(QL) - CR,L(QR)
• QH 2(KS KR)D/HH1/2
  YH CR,H(QH)- CR,H(QR)
• In Case C, we have perfect coordination QL
  QJ,L and QH QJ,H
• In all cases, we have perfect coordination for
  low holding cost, QL QJ,L

19
Numerical Experiments

• Zissis, Ioannou, Burnetas, 2015, Omega
• 100 values for each of the parameters (D, HH/HL,
  KS/KR, p) i.e., 108 Scenarios
• Data Sets D ? (1000, 10000, HH/HL ? (1, 5,
  KS/KR ? (0, 10, p ? (0, 1
• Maximum divergence from the whole supply chain
  cost under perfect coordination (coordination
  cost) is less that 11 (HH/HL 5, KS/KR 10, p
  0.85)
• D ? (1000, 10000, HH/HL ? (1, 5, KS/KR ? (0,
  7, p ? (0,1 9.5966
• D ? (1000, 10000, HH/HL ? (1, 3, KS/KR ?
  (0,10, p ? (0,1 8.1471
• D ? (1000, 10000, HH/HL ? (1, 2, KS/KR ?
  (0,10, p ? (0,1 5.4178

20
Model B

• 2-nodes supply chain (Supplier-Retailer), in
  which a single product is traded
• Retailer (R)
• Setup cost KR
• HL, with probability p
• Holding cost HR HM, with probability q
• HH, with probability 1-p-q
• Policy Economic Order Quantity Q

21
Differences

• The Retailer knows the real value of holding
  cost, HR HL, H?, H?
• The Supplier assumes distribution p, q, 1-p-q
• CR,L (2KRDHL)1/2 if QR,L
  (2KRD/HL)1/2
• Retailers CR,M (2KRDHM)1/2 if QR,M
  (2KRD/HM)1/2
• CR,H (2KRDHH)1/2 if QR,H (2KRD/HH)1/2
• According to Mechanism Design, the Supplier
  provides the quantity-price pair discount m
  P(QL) YL, P(QM) YM, P(QH) YH
• Thus, we have to determine QL, QM, QH, YL, YM,
  YH

22
I.R.-constraints

• CR,L(XL) CR,L (1)
• CR,M(XM) CR,M (2)
• CR,H(XH) CR,H (3)

23
I.C.-constraints

• CR,L(XL) -YL CR,L(XM) -YM (4)
• CR,L(XL) -YL CR,L(XH) YH (5)
• CR,M(XM) -YM CR,M(XL) YL (6)
• CR,M(XM) -YM CR,M(XH) YH (7)
• CR,H(XH) -YH CR,H(XL) YL (8)
• CR,H(XH) -YH CR,H(XM) YM (9)

24
Solution

• The Supplier has to solve the following
  optimization problem
• (Expected Supplier's costs)
• p(CS(XL) YL) q(CS(XM) YM)(1-p-q)(CS(XH)
  YH)
• s.t. (1)-(9)

25
Questions

• Examine if the coordination is always feasible
• Find the appropriate indexes which evaluate the
  improvement, comparatively to the solution
  without discounts
• Information rent (Retailers gains from the
  coordination)

26
Extensions Future Research

• Examine applicability of vendor-managed inventory
  (VMI) policies
• Investigate the feasibility of perfect
  coordination in supply chains with more than two
  nodes, both in sequence (e.g., three sequential
  nodes manufacturer, distributor, retailer) as
  well as in parallel (e.g., two or more retailers
  served by a single manufacturer)

27

• Thank you!
• This research has been co-financed by the
  European Union (European Social Fund ESF) and
  Greek national funds through the Operational
  Program "Education and Lifelong Learning" of the
  National Strategic Reference Framework (NSRF) -
  Research Funding Program THALES. Investing in
  knowledge society through the European Social
  Fund.