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Quantity Discounts in Supply Chain Coordination under Multi-level Information Asymmetry
Zissis D.1, Ioannou G.1, Burnetas A.2
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

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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 node’s 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
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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

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Bibliography
Corbett and De Groote (2000): A Supplier’s 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
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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)
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Corbett and De Groote (2000)
2-node supply chain, in which a single product is traded
Asymmetric Information:
The supplier does not know the retailer’s 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

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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

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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, HΗ}
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
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Without Quantity Discounts
Supplier’s cost function: CS(Q) = KSD/Q
Retailer’s cost function: CR(Q) = KRD/Q + HRQ/2
Decision maker only Retailer
According to EOQ model Q*R = (2KRD/HR)1/2
Supplier’s cost: CS(Q*R)
Retailer’s cost: CR(Q*R)

10. Reservation Levels They refer to the worst case scenario payoffs
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Reservation Levels
They refer to the worst case scenario payoffs
C+R,L := (2KRDHL)1/2 if Q*R,L = (2KRD/HL)1/2
Retailer’s :
C+R,H := (2KRDHH)1/2 if Q*R,H = (2KRD/HH)1/2
Supplier’s : CS(Q*R,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)

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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
Supplier’s cost function: TCS(Q) = KSD/Q + P(Q)
Retailer’s cost function: TCR(Q) = KRD/Q + HRQ/2 - P(Q)

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Complete Information
Supplier knows the real value of retailer’s holing cost
It is known that the chain could be coordinated
Supplier provides in the order quantity Q*J = {2(KS + KR)D/HR}1/ the discount Y * = KRD/Q*J + HRQ*J /2 –(2KRDHR)1/2
ΤCS(Q*J ) = (2DHR)1/2 {(KS + KR)1/2 - KR1/2} < CS(Q*R)
ΤCR(Q*J) = (2KRDHR)1/2 = CR(Q*R)
Coordination
Supplier takes all the profits which arise from the coordination

13. Asymmetry Information
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Asymmetry Information
According to Mechanism Design, the Supplier provides the quantity-price pair discount: m ={P(QL) = YL, P(QH) = YH}
without discount, Q*R (i.e., Q*R,L, Q*R,H)
Retailer’s 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
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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

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Constraints:
I.R. - constraints CR,L(XL) ≤ C+R,L (1) CR,H(XH) ≤ C+R,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:
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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:
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Solution
We distinguish 3 cases according to the parameters’ values:
Case A:
2 < {(1-p)(1+ KS/KR)}1/2 {(HL)1/2+(HH)1/2 } / (HH - pHL)1/2
Case B:
{(1-p)(1+KS/KR)}1/2 {(HL)1/2+(HH)1/2 }/(HH-pHL)1/2≤ 2<(1+KS/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: Solution of Case B: Solution of Case C:
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Solution
Solution of Case A:
Q*L= {2(KS + KR)D/HL}1/ Y*L=CR,L(Q*L)-CR,L(Q*H)+CR,H(Q*H)-CR,L(Q*R)
Q*H = {2(1-p)(KS+KR)D/(HH-pHL)}1/ Y*H = CR,H(Q*H)-CR,L(Q*R)
Solution of Case B:
Q*L= {2(KS + KR)D/ HL}1/ Y*L = CR,L(Q*L) - CR,L(Q*R)
Q*H = 2(2KRD)1/2/{(HL)1/2+(HR)1/2 } Y*H = CR,H(Q*H)- CR,H(Q*R)
Solution of Case C:
Q*L= {2(KS + KR)D/HL}1/ Y*L = CR,L(Q*L) - CR,L(Q*R)
Q*H = {2(KS +KR)D/HH}1/ Y*H = CR,H(Q*H)- CR,H(Q*R)
In Case C, we have perfect coordination Q*L= Q*J,L and Q*H = Q*J,H
In all cases, we have perfect coordination for low holding cost, Q*L= Q*J,L

19. Numerical Experiments
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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] %
D ∈ (1000, 10000], HH/HL ∈ (1, 3], KS/KR ∈ (0,10], p ∈ (0,1] %
D ∈ (1000, 10000], HH/HL ∈ (1, 2], KS/KR ∈ (0,10], p ∈ (0,1] %

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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

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Differences
The Retailer knows the real value of holding cost, HR ={HL, HΜ, HΗ}
The Supplier assumes distribution {p, q, 1-p-q}
C+R,L := (2KRDHL)1/2 if Q*R,L = (2KRD/HL)1/2
Retailer’s: C+R,M := (2KRDHM)1/2 if Q*R,M = (2KRD/HM)1/2
C+R,H := (2KRDHH)1/2 if Q*R,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. CR,L(XL) ≤ C+R,L (1) CR,M(XM) ≤ C+R,M (2) CR,H(XH) ≤ C+R,H (3)
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I.R.-constraints:
CR,L(XL) ≤ C+R,L (1) CR,M(XM) ≤ C+R,M (2) CR,H(XH) ≤ C+R,H (3)

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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:
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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
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Questions
Examine if the coordination is always feasible
Find the appropriate indexes which evaluate the improvement, comparatively to the solution without discounts
Information rent (Retailer’s gains from the coordination)

26. Extensions – Future Research
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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)

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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.