Transcript Newsvendor Problem - Department of Industrial Engineering

Supply Chain Contracts
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
A contract provides the
parameters within which a
retailer places orders and the
supplier fulfills them.
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.
E[retailer profit] =

p[m   (X  q )f (X )dX ]  wq
q
2.
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.
E[retailer profit] =

p[m   (X  q )f (X)dX ]  wq + bE[overstock]
q
2.
E[supplier profit] =
q(w-c) – bE[overstock]
3.
E[overstock] =

q  m   (X  q )f (X)dX
q
Expected Results from Buy-back
Contracts for Ski Example
Price w
$100
$100
$100
$110
$110
$110
$120
$120
$120
Price b Order Size
$
1000
$
30
1067
$
60
1170
$
962
$
78
1191
$
105
1486
$
924
$
96
1221
$
116
1501
Profit
$ 76,063
$ 80,154
$ 85,724
$ 66,252
$ 78,074
$ 86,938
$ 56,819
$ 70,508
$ 77,500
Returns
120
156
223
102
239
493
80
261
506
Profit
Chain Profits
$ 90,000 $ 166,063
$ 91,338 $ 171,492
$ 91,886 $ 177,610
$ 96,230 $ 162,482
$ 100,480 $ 178,555
$ 96,872 $ 183,810
$ 101,640 $ 158,459
$ 109,225 $ 179,733
$ 106,310 $ 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+fp(q-E[overstock])
•
E[retailer profit]=
(1-f)p(q-E[overstock])+v E[overstock]-wq
Expected results from revenuesharing contracts for ski example
Wholesale
Price w
Optimal
Order Size
Expected
Overstock
Retail
Expected
Profit
$10
0.3
1440
449
$124,273 $ 59,429 $183,702
0.5
1384
399
$ 84,735 $ 98,580 $183,315
0.7
1290
317
$ 45,503 $136,278 $181,781
0.9
1000
120
$ 7,606
$158,457 $166,063
0.3
1320
342
$110,523
$ 71,886 $182,409
0.5
1252
286
$ 71,601 $109,176 $180,777
0.7
1129
195
$ 33,455 $142,051 $175,506
$10
$10
$10
$20
$20
$20
Supplier.
Expected
Profit
Expected
Supply
Chain
Profit
Revenuesharing
Fraction, f
“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
dq units
Quantity-flexibility Contract for
Ski Example
d
0
0.2
0.4
0
0.15
0.42
0
0.2
0.5
Price w
$100
$100
$100
$110
$110
$110
$120
$120
$120
Order Size Purchase
1000
1000
1050
1024
1070
1011
962
962
1014
1009
1048
1007
924
924
1000
1000
1040
1005
Sales
880
968
994
860
945
993
838
955
994
Profit
$ 76,063
$ 91,167
$ 97,689
$ 66,252
$ 78,153
$ 87,932
$ 56,819
$ 70,933
$ 78,171
$
$
$
$
$
$
$
$
$
Profit
90,000
89,830
86,122
96,200
99,282
95,879
101,640
108,000
105,640
Chain Profits
$ 166,063
$ 180,997
$ 183,811
$ 162,452
$ 177,435
$ 183,811
$ 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:
Accept
Contract?
N
Y
Q
Production
End
Product Delivery
Demand
Recognition
Transfer
payments
Assumptions
•
Risk neutral
•
Full information
•
Forced compliance
Profit Equations
p= price per unit sold
pr = pS(q) – T
S(q)= expected sales
c= production cost
ps = T – cq
P(q) = pS(q) – cq = pr +ps
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
Newsvendor Problem
Wholesale Price Contract
Decide on q, w
Let w be what the supplier
charges the retailer
per unit purchased
Tw(q,w)=wq
Retailer’s profit function
pr= pS(q)-T
Supplier’s Profit Function
ps= (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: pr = pS(q) – T
pr = pS(q) – wq – b(q – S(q))
= (p – b)S(q) – (w – b)q
= P(q)
Recall: ps = T - cq
ps = wq – b(q – S(q)) – cq
= bS(q) + (w – b)q – cq
= (1  )P(q)
Results
Since q0 maximizes p(q),
q0 is the optimal quantity for both
pr and ps
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
Tr= wq + (1-f) pS(q)
Retailer’s Profit
pr= f pS(q)- T
•
For  Є (0,1], let
fp= p
w= c
pr= P(q)
Similar to Buy-Back
From Previous Slide:
pr(q,wr,f)= P(q)
Recall from Buy-Back:
pr(q,wr,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 dq
unsold units i.e. min{I,dq}
Expected # units retailer gets
compensated for is Ir
q
Ir =
 F(x )dx
(1 d ) q
Proof:
Retailer’s profit function
q
pr = pS(q) – wq + w
 F( x )dx
(1 d ) q
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
q
ps = wq – w
 F( x )dx
(1 d ) q
What is supplier’s optimal q?
Key result
•
The supply chain is not coordinated if
(1 – d)2f((1 – d)q0) > f(q0)
q0 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 q0,
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 2nd 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