Gabriela Contreras
Wendy O’Donnell
April 8, 2005

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems and open questions

Example: Music store
•
•
•
•
Supplier’s cost c=$1.00/unit
Supplier’s revenue w=$4.00/unit
Retail price p=$10.00/unit
Retailer’s service level
CSL*=0.5

Question
What is the highest service level both the supplier and retailer can hope to achieve?

Example: Music store (continued)
•
•
•
•
Supplier’s cost c=$1.00/unit
Supplier’s revenue w=$4.00/unit
Retail price p=$10.00/unit
Supplier & retailer’s service level
CSL*=0.9

Characteristics of an Effective
Contract:
• Replacement of traditional strategies
• No room for improvement
• Risk sharing
• Flexibility
• Ease of implementation

Why?
Sharing risk increase in order quantity increases supply chain profit

Types of Contracts:
• Wholesale price contracts
• Buyback contracts
• Revenue-sharing contracts
• Quantity flexibility contracts

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems & open questions

Example: Ski Jacket Supplier
•
•
•
•
Supplier cost c = $10/unit
Supplier revenue w = $100/unit
Retail price p = $200/unit
Assume:
– Demand is normal( m=1000,s=300)
– No salvage value

Formulas for General Case
1.
2.
E[retailer profit] =
 p [ m 
( X
 q ) f ( X ) dX ] q
  wq
E[supplier profit] = q(w-c)
3.
E[supply chain profit] =
E[retailer profit] + E[supplier profit]

Results:
Optimal order quantity for retailer = 1,000
Retail profit = $76,063
Supplier profit = $90,000
Total supply chain profit = $166,063
Loss on unsold jackets:
– For retailer = $100/unit
– For supply chain = $10/unit

Optimal Quantities for Supply Chain:
•
•
•
•
When we use cost = $10/unit, supply chain makes $190/unit
Optimal order quantity for retailer =
1,493
Supply chain profit = $183,812
Difference in supply chain profits =
$17,749

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Buy-Back Contracts
Supplier agrees to buy back all unsold goods for agreed upon price
$b/unit

Change in Formulas:
1.
2.
E[retailer profit] =
 p [ m  q

( X
 q ) f ( X ) dX ]
 wq
E[supplier profit] = q(w-c)
– bE[overstock]
+ bE[overstock]
3.
E[overstock] =
 q
 m  
( X
 q ) q f ( X ) dX

Expected Results from Buy-back
Contracts for Ski Example
Price w Price b Order Size Profit Returns
$100
$100
$ -
$ 30
1000
1067
$
$
76,063
80,154
120
156
$100
$110
$110
$110
$
$
$
$
-
60
78
105
1170
962
1191
1486
$
$
$
$
85,724
66,252
78,074
86,938
223
102
239
493
$120
$120
$120
$ -
$ 96
$ 116
924
1221
1501
$ 56,819
$ 70,508
$ 77,500
80
261
506
Profit Chain Profits
$ 90,000
$ 91,338
$
$
166,063
171,492
$ 91,886
$ 96,230
$ 100,480
$ 96,872
$
$
$
$
177,610
162,482
178,555
183,810
$ 101,640
$ 109,225
$ 106,310
$ 158,459
$ 179,733
$ 183,810

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Revenue-sharing Contracts
Seller agrees to reduce the wholesale price and shares a fraction f of the revenue

Change in formulas
• E[supplier profit]=
(w-c)q+ f p(q-E[overstock])
• E[retailer profit]=
(1f
)p(q-E[overstock])+v E[overstock]-wq

Expected results from revenuesharing contracts for ski example
Wholesale
Price w
Revenuesharing
Fraction, f
Optimal
Order Size
Expected
Overstock
Retail
Expected
Profit
Supplier.
Expected
Profit
Expected
Supply
Chain
Profit
$10
$10
0.3
1440 449 $124,273 $ 59,429 $183,702
0.5
1384 399 $ 84,735 $ 98,580 $183,315
$10
0.7
1290 317 $ 45,503 $136,278 $181,781
$10
0.9
1000 120 $ 7,606 $158,457 $166,063
$20
0.3
1320 342 $110,523 $ 71,886 $182,409
$20
0.5
1252 286 $ 71,601 $109,176 $180,777
$20 0.7
1129 195 $ 33,455 $142,051 $175,506

“Go Away Happy”
“Guaranteed to be There”

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Quantity-flexibility Contracts
• Retailer can change order quantity after observing demand
• Supplier agrees to a full refund of d q units

Quantity-flexibility Contract for
Ski Example d
Price w Order Size Purchase Sales
0 $100 1000 1000 880
0.2
$100 1050 1024 968
0.4
$100
0 $110
0.15
$110
0.42
$110
1070
962
1014
1048
1011
962
1009
1007
994
860
945
993
0 $120
0.2
$120
0.5
$120
924
1000
1040
924
1000
1005
838
955
994
Profit
$ 76,063
$ 91,167
$ 97,689
$ 66,252
$ 78,153
$ 87,932
$ 56,819
$ 70,933
$ 78,171
Profit Chain Profits
$ 90,000 $ 166,063
$ 89,830 $ 180,997
$ 86,122
$ 96,200
$ 99,282
$ 95,879
$
$
$
$
183,811
162,452
177,435
183,811
$ 101,640
$ 108,000
$ 105,640
$ 158,459
$ 178,933
$ 183,811

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

•
•
Contracts and the
Newsvendor Problem
One supplier, one retailer
Game description:
Y
Accept
Contract?
N
End
Q
Production
Product Delivery
Demand
Recognition
Transfer payments

Assumptions
• Risk neutral
• Full information
• Forced compliance

Profit Equations p= price per unit sold
S(q)= expected sales c= production cost p p r
= pS(q) – T s
= T – cq
P( q
)
= pS(q) – cq = p r
+ p s
Proof:

Transfer Payment
What the retailer pays the supplier after demand is recognized
T = wq w = what the supplier charges the retailer per unit purchased

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Wholesale Price Contract
Decide on q, w

Let w be what the supplier charges the retailer per unit purchased
T w
(q,w)=wq

Retailer’s profit function p r
= pS(q)-T

Supplier’s Profit Function p s
= (w-c)q

Results:
• Commonly used
• Does not coordinate the supply chain
• Simpler to administer

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Buy-back Contracts
• Decide on q,w,b
• Transfer payment
T = wq – bI(q)
= wq – b(q – S(q))

Claim
A contract coordinates retailer’s and supplier’s action when each firm’s profit with the contract equals a constant fraction of the supply chain profit. i.e. a Nash equilibrium is a profit sharing contract

Buy-back contracts coordinate if w & b are chosen such that:
 
( 0 , 1 ] p
 b
=  p w b
 b
=  c

Recall: p r
= pS(q) – T p r
= pS(q) – wq – b(q – S(q))
= (p – b)S(q) – (w – b)q
=
P
(q)

Recall: p s p s
= T - cq
= wq – b(q – S(q)) – cq
= bS(q) + (w – b)q – cq
= (1  )P( q)

Results
Since q 0 maximizes p
(q), q 0 is the optimal quantity for both p r and p s
And both players receive a fraction of the supply chain profit

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Newsvendor Problem
Revenue-Sharing Contracts
Decide on q, w, f

Transfer Payment
T r
= wq + pS(q)

Retailer’s Profit p r
= pS(q)- T
• For
 Є (0,1], let f p=
 p w=
 c p r=
P
(q)

Similar to Buy-Back
From Previous Slide: p r
(q,w r
, f
)=
P
(q)
Recall from Buy-Back: p r
(q,w r
,b)=
P
(q)

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems

Quantity-flexibility Contracts
• Decide on q,w, d
Supplier gives full refund on d q unsold units i.e. min{I, d q}

Expected # units retailer gets compensated for is I r
I r
=
( 1
 d
 q
) q
F ( x ) dx
Proof:

Retailer’s profit function p r
= pS(q) – wq + w
( 1
 d
 q
) q
F ( x ) dx

Optimal q satisfies: w = p(1 – F(q))
1 – F(q) + F((1 – d
)q)(1 – d
)
If supplier plays this w, will the retailer play this q?

Only if retailer’s profit function is concave
As long as w < p and w > 0

Supplier’s profit function p s
= wq – w
( 1
 d
 q
)
F ( q x ) dx
What is supplier’s optimal q?

Key result
• The supply chain is not coordinated if
(1 – d
) 2 f((1 – d
)q 0 ) > f(q 0 ) q 0 is the minimum

Result
• Supply chain coordination is not guaranteed with a quantityflexibility contract
•
•
Even if optimal w(q) is chosen
It depends on d
& f(q)

Summary
You can coordinate the supply chain by designing a contract that encourages both players to always want to play q 0 , the optimal supply chain order quantity

Outline
•
•
•
•
Introducing Contracts
Example: ski jackets
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Newsvendor Problem
– Wholesale
– Buy-back
– Revenue-sharing
– Quantity-flexibility
Results for other problems and open questions

Newsvendor with Price Dependent
Demand
• Retailer chooses his price and stocking level
• Price reflects demand conditions
• Can contracts that coordinate the retailer’s order quantity also coordinate the retailer’s pricing?
• Revenue-sharing coordinates

Multiple Newsvendors
•
•
•
•
One supplier, multiple competing retailers
Fixed retail price
Demand is allocated among retailers proportionally to their inventory level
Buy-back permits the supplier to coordinate the S.C.

Competing Newsvendors with
Market Clearing Prices
• Market price depends on the realization of demand (high or low) & amount of inventory purchased
• Retailers order inventory before demand occurs
• After demand occurs, the market clearing price is determined
• Buy-back coordinates the S.C.

Two-stage Newsvendor
• Retailer has a 2 nd opportunity to place an order
• Buy-back
•
Supplier’s margin with later production < margin with early production

Open Questions
• Current contracting models assume on single shot contracting.
• Multiple suppliers competing for the affection of multiple retailers
• Eliminate risk neutrality assumption
• Non-competing heterogeneous retailers