Managing High-Tech Capacity via Managing High-Tech Capacity via Reservation Contracts Reservation Contracts
S. David WuS. David Wu
Lee A. Iacocca Professor and ChairmanLee A. Iacocca Professor and Chairman
Department of Industrial and Systems EngineeringDepartment of Industrial and Systems Engineering
Lehigh UniversityLehigh University
Pennsylvania, U. S. A.Pennsylvania, U. S. A.
*Joint work with Murat Erkoc
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An U.S.Telecommunications An U.S.Telecommunications Component-ManufacturerComponent-Manufacturer
Client Systems
Enterprise/Metro/ Wireless Access
Metro/Regional Transport
Long-Haul Backbone
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High-Tech Manufacturing Supply ChainHigh-Tech Manufacturing Supply Chain
Rapid innovation & volatile demandRapid innovation & volatile demand
High obsolescence rate on technologyHigh obsolescence rate on technology
Short product life-cycle (avg. 18 months)Short product life-cycle (avg. 18 months)
Capital intensive facilities ($2 Billion +)Capital intensive facilities ($2 Billion +)
High capacity costsHigh capacity costs
Manufacturers are conservative in capacity Manufacturers are conservative in capacity expansion of any formexpansion of any form
Capacity reservation contractsCapacity reservation contracts
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Some BackgroundSome Background
Capacity Planning vs. Capacity ReservationCapacity Planning vs. Capacity ReservationOptimization vs. CharacterizationOptimization vs. CharacterizationSolution vs. ProcessSolution vs. Process
The main motivations? The main motivations? how to price capacity reservation? how to price capacity reservation? to go over the thought process analyticallyto go over the thought process analytically preparing for the “soft capacity” environmentpreparing for the “soft capacity” environment
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CM 1Buyer1
BuyerM
Buyer2
Spot Market
CM 2
CM n Supplier K
Supplier 2
Contract Manufacturers
High-Tech Supply Chain Building BlocksHigh-Tech Supply Chain Building Blocks
Capacity Capacity Reservation Reservation
ContractContract
Supplier 1
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Capacity ReservationCapacity Reservation::Settings and Assumptions Settings and Assumptions (Erkoc and Wu, 2002a)(Erkoc and Wu, 2002a)
Between the supplier and a major customerBetween the supplier and a major customer
Stochastic market demandStochastic market demand
Capacity must be built in advance (tactical)Capacity must be built in advance (tactical) outsourcing, work force expansion, procurement of raw outsourcing, work force expansion, procurement of raw
material, or pre-processing of semi-finished productsmaterial, or pre-processing of semi-finished products
Convex capacity cost (considers residual value)Convex capacity cost (considers residual value)
No inventory carry-over (single period)No inventory carry-over (single period)
Complete information (cost, demand distribution)Complete information (cost, demand distribution)
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Related Literature Related Literature
Buyer’s perspective: Silver and Jan (1994), Jan and Silver Buyer’s perspective: Silver and Jan (1994), Jan and Silver (1995), Brown and Lee (1998)(1995), Brown and Lee (1998)Supplier-Buyer perspectives: Cachon and Larivier (2001), Supplier-Buyer perspectives: Cachon and Larivier (2001), Tomlin (1999), Serel et. al. (2001)Tomlin (1999), Serel et. al. (2001)Options: Donohue (2000), Barnes-Schuster et. al. (2000) Options: Donohue (2000), Barnes-Schuster et. al. (2000) Main DistinctionsMain Distinctions to reserve or not to reserve to reserve or not to reserve wholesale price is not a contract parameterwholesale price is not a contract parameter linear vs. convex capacity costlinear vs. convex capacity cost forced vs. voluntary complianceforced vs. voluntary compliance buyer vs. supplier led channelbuyer vs. supplier led channel
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Integrated ChannelIntegrated Channel
Supplier
w p
c V(k)
Buyer Market
F(x)
k
F dxxFkkS0
)()(expected sales given capacity k
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Delegation of Control in the Supply ChainDelegation of Control in the Supply Chain
System’s Profit (concave)System’s Profit (concave)
Let Let kkoo be the system optimal solutionbe the system optimal solution
Supplier’s newsvendor profit functionSupplier’s newsvendor profit function
Let Let kk** be the newsvendor solutionbe the newsvendor solution
)()(*)( kVkScp FI
)()(*)(0 kVkScw Fs
)(*
1*
cwFk
)(1
cpFk
oo
okk *
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Double MarginalizationDouble Marginalization- - suboptimalsuboptimal
Supplier
w p
c V(k)
Buyer Market
F(x)
Due to Due to double marginalizationdouble marginalization the system optimal capacity is the system optimal capacity is always greater than the supplier’s always greater than the supplier’s newsvendor capacitynewsvendor capacity
Optimal
Competitive Outcome
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Reservation Contract with Deductible FeeReservation Contract with Deductible Fee
For each unit of capacity reserved, the buyer is charged a feeFor each unit of capacity reserved, the buyer is charged a fee
The fee is later deducted from the purchasing price if the reserved The fee is later deducted from the purchasing price if the reserved capacity is used by the intended customercapacity is used by the intended customer
If the buyer’s demand exceeds the reservation amount, the orders If the buyer’s demand exceeds the reservation amount, the orders are satisfied based on the availability of “extra capacity”are satisfied based on the availability of “extra capacity”
Customer reservations are locked in and some extra capacity may Customer reservations are locked in and some extra capacity may be built during expansionbe built during expansion
The wholesale price The wholesale price w w and product cost and product cost cc are set at the “design- are set at the “design-win” phase, which are not negotiablewin” phase, which are not negotiable
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The Sequence of EventsThe Sequence of Events
Supplier announces reservation fee (r) Buyer announces how
much to reserve (q)
Supplier decides how much capacity to build (k)
Demand is realized (x)
Buyer is penalized for unused reserved capacity
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Supplier’s Capacity DecisionSupplier’s Capacity Decision
If , then the constraint must be bindingIf , then the constraint must be binding
qk
xqErkVkScw Fk
s.t.
),0max()()()( Max
*kq
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Buyer’s ProfitBuyer’s Profit
Buyer profit function is conditional due to the fact that he Buyer profit function is conditional due to the fact that he never reserves less than never reserves less than k*k*
wo
kqB
/)S(kw)(p
ifdxF(x)rS(q)w)(p
*
*q
0
0
• In optimality
• Reservation quantity q decreases in reservation feerwp
wpqF
)( *
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Buyer’s Incentive to ReserveBuyer’s Incentive to ReserveBuyer never reserves less than Buyer never reserves less than k*k*. In fact, she reserves . In fact, she reserves only if the reservation fee is below a certain threshold, only if the reservation fee is below a certain threshold, rrtt. . The reservation quantity will be at least The reservation quantity will be at least qqtt , , wherewhere
qqtt is a function ofis a function of ww and does not depend on and does not depend on pp
The buyer reserves iff The buyer reserves iff r r rrtt
)()( *0 kq Bt
B
)(
)()(
t
tt
qF
qFwpr Threshold fee
Threshold Reservation Qty.
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Supplier’s ProfitSupplier’s Profit
The The reservation reservation quantityquantity determines the final determines the final capacity, so we may rewrite the profit function ascapacity, so we may rewrite the profit function as
Supplier’s profit function is unimodal if the Supplier’s profit function is unimodal if the demand distribution has increasing failure rate demand distribution has increasing failure rate (IFR)(IFR)The function is strictly decreasing in The function is strictly decreasing in [k[koo,,))
),max(max wt qqq
q
sqq
dxxFqF
qFwpqVqScw
0
)()(
)()()()()(Max
max
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The Supplier’s Faces Three Profit Scenarios in The Supplier’s Faces Three Profit Scenarios in sequence as the sequence as the Buyer’s MarginBuyer’s Margin Decreases Decreases
)()(
)()(*
*
kSqS
kVqVcp
t
tt
)()( *kq It
I
rminr*
Reservation Fee
Exp
ec
ted
Pro
fit
Reservation FeeE
xpe
cte
d P
rofi
t
Reservation Fee
Exp
ec
ted
Pro
fit
rmin
rmin
Scenario 1: High Scenario 2: Medium
Scenario 3: Low
The buyer’s margin must be sufficiently high to justify capacity reservation
Theorem: It is individually rational for the supplier and the buyer to enter the reservation contract if the buyer revenue margin p is no less than the threshold pt as follows:
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Impact of Market Size and Variability: Impact of Market Size and Variability: Shifted Family DistributionsShifted Family Distributions
If the distribution is from a “shifted” familyIf the distribution is from a “shifted” family
the standard deviation remains the same as the market size increasesthe standard deviation remains the same as the market size increasesWith With linearlinear capacity cost capacity cost
the threshold reservation fee the threshold reservation fee rrtt does not change in market size does not change in market size the threshold reservation quantity the threshold reservation quantity qqtt does does increase in increase in (literature)(literature)
With With convexconvex capacity cost, capacity cost, bothboth the threshold fee the threshold fee rrtt and quantity and quantity qqtt increase in market size increase in market size there exists a threshold there exists a threshold tt over which reservation is not “win-win” over which reservation is not “win-win” the decrease in the coefficient of variation due to market size may not sufficient the decrease in the coefficient of variation due to market size may not sufficient
to offset the risk of expansion.to offset the risk of expansion.
),min(* wrr t
)0|()|( xFxF
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the standard deviation increases with the market size, thus, the standard deviation increases with the market size, thus, the threshold reservation fee the threshold reservation fee rrtt increasesincreases in market size in market size ..
even with linear capacity cost, the increase in reservation even with linear capacity cost, the increase in reservation payment (due to market size) is not sufficient to balance payment (due to market size) is not sufficient to balance the increase in riskthe increase in risk
Impact of Market Size and Variability: Impact of Market Size and Variability: Scaled Family DistributionsScaled Family Distributions
)1|()|(
xFxF
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Capacity reservation may create surplus for the Capacity reservation may create surplus for the channelchannel
However, except for a few special cases the However, except for a few special cases the surplus is sub-optimalsurplus is sub-optimal
We design two additional coordination We design two additional coordination mechanisms to achieve optimalitymechanisms to achieve optimality Partial-deduction (PD) contractPartial-deduction (PD) contract Cost-sharing (Options) contractCost-sharing (Options) contract
System Optimality and ReservationsSystem Optimality and Reservations
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Capacity Reservation with Partially Deductible FeesCapacity Reservation with Partially Deductible Fees
Suppose that the supplier offers a contract Suppose that the supplier offers a contract (r, r(r, r22)) where the reservation where the reservation fee is fee is r r and the deductible portion is and the deductible portion is rr22
TheoremTheorem. Under PD contract the buyer will reserve the channel optimal . Under PD contract the buyer will reserve the channel optimal capacity, i.e., capacity, i.e., q(r) = kq(r) = k00. Moreover,. Moreover,
(1) The buyer's profit is increasing in (1) The buyer's profit is increasing in rr,,(2) The supplier's profit is decreasing in (2) The supplier's profit is decreasing in rr,,(3) The supplier would only offer such contract if (3) The supplier would only offer such contract if vv0 0 2w 2wcc
thus,0but ,)( 22
rrwprv
cpr
o
cp
vwpr
vcp
vwp
o
o
o
)()(
marginal capacity cost at k0
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Cost-Sharing Contracts: Cost-Sharing Contracts: Capacity OptionsCapacity Options
Suppose the supplier proposes a contract where the buyer Suppose the supplier proposes a contract where the buyer pays pays V(Q) V(Q) for reserving for reserving Q Q units (options) whereas the units (options) whereas the supplier returns supplier returns rr22 for each exercised reservation quantity. for each exercised reservation quantity.
The channel is coordinated if The channel is coordinated if
Contract setting is independent of the demand distributionContract setting is independent of the demand distribution
*
0*
*
0
2 and )()(I
SI
I
Bwpcpr
*)1( getsSupplier I * getsBuyer I
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Capacity Reservation with Buyer CompetitionCapacity Reservation with Buyer Competition(Erkoc and Wu, 2002b)(Erkoc and Wu, 2002b)
Introducing horizontal buyer competitionIntroducing horizontal buyer competition
Still a supplier-lead single-period modelStill a supplier-lead single-period model
Two identical buyers from independent markets Two identical buyers from independent markets competing for the capacity competing for the capacity
The demand distribution for buyer The demand distribution for buyer ii is is GGii(x)(x)
The supplier offers a reservation fee, The supplier offers a reservation fee, rr, for each unit of , for each unit of capacity that is uniform across buyerscapacity that is uniform across buyers
Excess capacity, if any exists, is distributed Excess capacity, if any exists, is distributed proportionalproportional to the buyers’ reservation quantitiesto the buyers’ reservation quantities
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Integrated ChannelIntegrated Channel
Supplier
w p
c V(k)
Buyer i Market
Gi(x)
p
Buyer -i Market
G-i(x)
w
The buyers’ competition will be
•a Fixed Capacity Reservation (FCR) Game, if (total reservation amount) k*
•a Variable Capacity Reservation (VCR) Game, otherwise
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Fixed Capacity Reservation Game (FCR)Fixed Capacity Reservation Game (FCR)
Capacity is fixed at Capacity is fixed at k* k*
A zero-sum gameA zero-sum game
At equilibrium the buyers are compelled to reserveAt equilibrium the buyers are compelled to reserve
Capacity allocated to buyer Capacity allocated to buyer i:i:
With identical buyersWith identical buyers
*kqq
qh
ji
ii
)2/()(4
)( *2* kGkqGq
wpr
kk
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Capacity AllocationCapacity Allocation
q-i qi
Buyer -i demand, x-i Buyer i demand, xi
ii hx
iiii hkxhx *,
iiii hkxhx *,
iiii hxhx ,
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Variable Capacity Reservation (VCR) GameVariable Capacity Reservation (VCR) Game
Capacity is determined by total reservations: Capacity is determined by total reservations: k = qk = qi i + q+ q-i-i
Buyers’ profit function:Buyers’ profit function:
Optimal buyer profits decrease in reservation feeOptimal buyer profits decrease in reservation fee
At any point, each buyer wishes the other to At any point, each buyer wishes the other to reserve morereserve more
iq
iiqB dxxGrqqSwp0
)(),()(
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Game within a Game:FCR or VCR?Game within a Game:FCR or VCR?
rk rqrt
Bu
ye
r P
rofi
t
Reservation Fee
FCR
VCR
0.7
0.8
0.9
1
1.1
1.2
1 2 3 4 5 6 7 8 9 10
Reservation Fee
Res
erv
atio
n A
mo
un
t
Equilibrium Reservations, p=15,w=10,V(k)=5k,c=3,D~N(1,1/4)
FCR
VCR
In equilibrium, the buyers would prefer the VCR game so long as the reservation fee is below a certain threshold rt
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Supplier’s ProfitSupplier’s Profit
If the total reservation amount is If the total reservation amount is less less than than k*k*, the , the supplier’s optimal capacity will be supplier’s optimal capacity will be k* k* andand
Otherwise, the total reservation amount determines the Otherwise, the total reservation amount determines the final capacity andfinal capacity and
eqe
kFCRs dxxGqrkVkScw
0
** )()(2)()()(
eqeVCReeVCR
s dxxGqrqVqScw0
)()(2)2()2()(
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Supplier’s First Stage ModelSupplier’s First Stage Model
1,0
)(
0)1(
s.t.
)1( Max VCRs
FCRs
y
wqq
qyq
yy
t
qy=0 iff the surplus is created at the threshold reservation fee, i.e.,
)()( * tII qk
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TheThe N- N-Buyer CaseBuyer Case
Proportional allocation of excess capacityProportional allocation of excess capacity Becomes intractable Becomes intractable Existence of equilibrium can be established for a subset Existence of equilibrium can be established for a subset
of of uplifting contractsuplifting contracts
Uniform allocation of excess capacityUniform allocation of excess capacity
Equilibrium can be established via supermodularityEquilibrium can be established via supermodularity The equilibrium is uniqueThe equilibrium is unique
)),q(()q,( iiiiGiiVCRi xZqSqS
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ConclusionsConclusions
Impact of Impact of capacity reservationscapacity reservations on supply chain efficiency on supply chain efficiency (when are they beneficial?)(when are they beneficial?)
Convex capacity cost Convex capacity cost
Effects of market size and demand variabilityEffects of market size and demand variability
Horizontal buyer competitionHorizontal buyer competition
Proposed Contracts for channel coordinationProposed Contracts for channel coordination
Other ConsiderationsOther Considerations Voluntary ComplianceVoluntary Compliance Partial demand informationPartial demand information
Working papers Working papers (Erkoc and Wu, 2002 a,b)(Erkoc and Wu, 2002 a,b)
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Future ConsiderationsFuture Considerations
Multiple periods with inventory considerationsMultiple periods with inventory considerationsCorrelated demands and/or intra-market competitionCorrelated demands and/or intra-market competitionSequential or repeated negotiationsSequential or repeated negotiationsMultiple-supplier modelMultiple-supplier modelIn house production v.s. outsourcingIn house production v.s. outsourcingCapacity Reservation from the Capacity Reservation from the Contract ManufacturersContract Manufacturers PerspectivePerspective
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Buyer’s Profit and EquilibriumBuyer’s Profit and Equilibrium
i
i
i
i
h
hk
h
h
iik dydxxgygyKdxxKGxxgdxxxgqqS0
**
0)()()()()()(),(
*
* *
0
* )()()()()(K
h
xK
iiii
hGhGhdydxxgygyK
At equilibrium both buyers reserve the same quantity, qk yielding
0
0
** )()2/,2/()( B
q
keB
k
dxxGrKKSwp )2/(
4
)()( *2* KGK
r
wpqGq kk
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The Uplifting Contract (Spot Market Pricing)The Uplifting Contract (Spot Market Pricing)The supplier charges an extra The supplier charges an extra for any purchase order beyond the reservation for any purchase order beyond the reservation amount (spot market price)amount (spot market price)
The supplier offers a contract The supplier offers a contract ((,r,r))The contract coordinates the channel if The contract coordinates the channel if b,d b,d satisfy the following:satisfy the following:
With With dd the supplier dictates if the buyers choose to play the VCR game the supplier dictates if the buyers choose to play the VCR gameBy adjusting the reservation fee, the supplier may dictate which game (FCR or By adjusting the reservation fee, the supplier may dictate which game (FCR or VCR) the buyers playVCR) the buyers play
2/
0
)()(
)()()(2
1)2/(
)(ok
o
ooo
dxxkGxg
kFkGwpkrGrb
o/w
2/ if *
b
kqdb
)()2/(
)2/()(
*
*
rbkG
kGrrd