Download Pay-Back-Revenue-Sharing Contract in Coordinating Supply

Pay-Back-Revenue-Sharing Contract in Coordinating Supply Chains with
Random Yield
Sammi Y. Tang
School of Business, University of Miami, Coral Gables, FL, [email protected]
Panos Kouvelis
Olin Business School, Washington University in St.Louis, St. Louis, MO, [email protected]
Abstract
We consider coordination issues in supply chains where supplier’s production process is
subject to random yield losses. For a simple supply chain with a single supplier and retailer
facing deterministic demand, a pay back contract which has the retailer paying a discount price
for the supplier’s excess units can provide the right incentive for the supplier to increase his
production size and achieve coordination. Building upon this result, we consider coordination
issues for two other supply chains: one with competing retailers, the other with stochastic
demand. When retailers compete for both demand and supply, they tend to over-order. We
show that a combination of a pay back and revenue sharing mechanism can coordinate the supply
chain, with the pay back mechanism correcting the supplier’s under-producing problem and the
revenue sharing mechanism correcting the retailers’ over-ordering problem. When demand is
stochastic, we consider a modified pay-back-revenue-sharing contract under which the retailer
agrees to not only purchase the supplier’s excess output (beyond the retailer’s order), but also
share with the supplier a portion of the revenue made from the sales of the excess output. We
show that this contract, by giving the supplier additional incentives in the form of revenue share,
can achieve coordination.
Key words: supply chain coordination; yield uncertainty; demand uncertainty;
1
1.
Introduction and Related Literature
In a decentralized supply chain, optimal supply chain performance may not be achieved because
supply chain members are primarily concerned with optimizing their own objectives, which creates
inefficiency in the system. It has been the focus of academic researchers as well as practitioners
to look for mechanisms that can eliminate such inefficiency. Though there exists a rich body of
work in the supply chain literature on coordinating contracts, the majority of it only considers
how to design such contracts in a decentralized supply chain facing stochastic demand but with no
supply uncertainty (see Cachon 2003 for a complete review). Our paper aims to address this gap
by explicitly studying coordination concerns in the presence of supply uncertainty.
In this paper, the uncertain supply takes the form of random yield. In many industries,
especially those adopting new technologies, the problem of random yield is common to the production processes. The output is often only a random fraction of the input. For example, in the
semiconductor industry, due to a complicated production process and strict product specifications,
the output often differs from the initial lot size. Yield uncertainty is the reality in the pharmaceutical industry as well. For example, the manufacturing process for influenza vaccine involves growing
the virus in embryonated eggs. Due to the inherent uncertainty regarding the growing characteristics of the viral strains, the quantity of vaccine that can be obtained per egg is uncertain (Deo
and Corbett 2009). Yield uncertainty is a primary planning concern in agribusiness environment
(soybeans, corn etc.), with unpredictable weather a main factor behind the yield unpredictability
in realized output from planted acres of land (see Jones et al. 2001). The problem of random yield
has been well studied in the context of single-item inventory models (see Yano and Lee 1995 for a
comprehensive review). In our paper, we adopt the stochastically proportional yield models commonly used in this literature. But our focus goes beyond single firm planning decisions to supply
chain coordination issues in a decentralized system.
Like a supply chain facing stochastic demand, a supply chain with random yield is also
subject to the double marginalization problem. In the presence of deterministic demand but random
yield, double marginalization leads to underproduction: the supplier does not produce enough from
the system’s point of view. Like a buy back contract (the supplier buys back excess inventory at
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a discount price) can coordinate a supply chain with stochastic demand, a pay back contract (the
retailer buys excess production output at a discount price, see Simchi-Levi et al. 2007, Chick et
al. 2008, Özer and Wei 2006) can achieve coordination for a supply chain with random yield. Under
a pay back contract, the retailer pays a discount price for supplier’s excess production output above
her order, so the supplier is given incentives to set higher production quantities, and at the right
incentive levels leads to the supply chain’s optimal performance.
Using this result, we search for coordination solutions for two other supply chains with
random yield. The first one is motivated by the semiconductor industry. As leading companies
such as Nvidia and AMD outsource their production of chips to foundries like TSMC, they may
suffer from TSMC’s low and unstable yield rate, especially when a new generation of chips first
comes out (Vilches 2009). In this situation, yield issues at TSMC lead to shortages at both Nvidia
and AMD, which affects the market share of these two major graphic card manufacturers. In
order to study if and how competition between buyers distorts the supply chain in the presence of
random yield, we consider a model with a single unreliable supplier and two competing retailers.
The retailers compete for the market demand of a known size. They also compete for the supplier’s
output when the supplier cannot fully satisfy retailers’ total order in the case of low yield realization.
The competition is captured using a proportional allocation model. We show that this supply chain
is distorted in two ways: first, the supplier’s production quantity is lower than the system optimal,
like the supply chain with a single retailer; second, the retailers’ order quantity may be higher than
the system optimal, especially when the wholesale price is low. To coordinate the supply chain,
we propose a contract combining two mechanisms: a pay back mechanism, which aligns supplier’s
production decision with the centralized system, and a revenue sharing mechanism, which aligns
retailers’ ordering decision with the centralized system.
In addition to address the coordination issue under buyer competition and supply uncertainty, another important issue we focus on is whether the presence of stochastic demand requires
a different coordination mechanism. For that, we consider a supply chain with a single unreliable
supplier and a single retailer facing stochastic demand. In this supply chain, when the supplier sets
the stocking quantity for the retailer (for example, supply chains with Vendor Managed Inventory
(VMI) agreements in place), coordination can be achieved by using the basic pay back contract.
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While the pay back contract can correct the distortion of the supplier’s under-producing problem,
another distortion is introduced when the retailer retains control over her inventory in the presence of uncertain demand. As the retailer only purchases up to the amount ordered, lost sales
might happen even when the supplier produces enough to satisfy the realized demand. To align
the decentralized system with the centralized one, we consider a modified pay-back-revenue-sharing
contract in which the supplier gets a share of the revenue made only from the sales of the output
in excess of the original retailer order. We show that this contract, again, by using a combination
of two mechanisms, can coordinate both the supplier and retailer’s behavior with the centralized
system. As the supplier’s production decision depends on both the yield and demand distribution,
the implementation of this contract requires information sharing between the two supply chain
partners.
From our discussions with procurement executives in the semiconductor and agribusiness
industries, both seriously affected by random yields, we were informed that contracts used in
these random yield environments have various complex provisions to deal with unpredictable yields
and erratic demand forecasts. Some provisions in currently used contracts actually reflect the
structure proposed in our paper. For example, Sun Edison, a company which produces custom
application wafers for various electronic manufacturers, often works with consigned inventories.
Their customers are willing to accept excess output in case of high yield realizations, but with the
understanding that quantities in excess of their formal forecasts (effectively their ordered quantity)
receive discounts. This is essentially the pay-back mechanism proposed in our paper. Similarly,
when customers require expedited use of excess output, often not in their consigned stocks, to
meet higher than expected demand, the supplier receives some extra compensation, very close to a
revenue share for such quantities. Similar observations were conveyed to us by production planning
managers in the soybeans and corn production divisions in agribusiness firms such as Monsanto
and Bunge.
The key contribution of this paper lies in constructively combining two well-known coordination mechanisms to address supply chain coordination issues in environments with random yield
and competing retailers supplied by a common supplier or linear suply chains facing both yield and
demand uncertainties. The proposed contracts not only have some practical basis, but also have
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the potential to streamline rather complex provisions in the contracts currently used in random
yield environments.
To the best of our knowledge, there is scarce literature on supply chain coordination in the
presence of supply uncertainty. The most relevant work is a recent paper by Chick et al. (2008).
They study coordination issues in a specific industry, i.e., influenza vaccination manufacturing. In
their model, the buyer (government in their model, retailer in our setting) faces a deterministic
demand and a cost function depending on the desired demand target. The buyer’s objective is to
minimize the total cost (government’s cost for purchasing and administering vaccine and all relevant
costs for fighting influenza outbreaks). The supplier (vaccine retailer in their model), knowing the
demand target set by the buyer (government), makes the production decision. The basic model
presented in §2 in our paper closely resembles, but is not identical to, theirs (with piecewise-linear
number of infected individuals) in that yield is the only source of uncertainty in the supply chain.
Using our model, we show that a pay back contract can coordinate the supply chain when demand
is deterministic, while in the Chick et al. setting the pay back contract is non-coordinating. Our
paper builds upon the basic pay back contract and studies coordination issues in supply chains
with distortions caused by not only the unreliable supplier but other complicating factors such as
retailer competition and demand uncertainty.
Liu (2006) is the only work that studies coordinating contracts under both random supply
and stochastic demand. However, in Liu’s work there are two decisions to coordinate on: an input
level (initial lot size) and an output level (output sold to the market), while in our stochastic demand
model, we coordinate only the input level. Restricting the “output level” implies possible holding
back from the market, which is neither realistic nor efficient in most markets. Once production is
initiated, sales are determined by the minimum of the realized demand and output.
Gurnani and Gerchak (2007), Yan et al. (2010) study coordination in a decentralized assembly system with uncertain yield but deterministic demand. They show that a shortage penalty
contract penalizing the supplier with the worst delivery performance, or a surplus subsidy contract
rewarding the supplier with better-than-expected delivery performance, can achieve coordination.
Another related work is He (2005). He studies optimal behavior of firms under various risk sharing
contracts in the presence of random yield. For reasons of completeness in referencing, we point
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out work on coordination issues in the presence of random demand and other sources of supply
uncertainty (e.g., lead-time uncertainty, see Weng and McClurg 2003), but is of no relevance to our
yield uncertainty focused work.
The pay back mechanism studied in our paper can be thought of as a parallel to buy back
contracts for a newsvendor setting (e.g., see Pasternack 1985). Both allow the party which does
not directly face uncertainty to share the risk faced by its partner. The pay back contract also
has a similar flavor to quantity flexibility contracts (e.g., see Tsay and Lovejoy 1999, and Milner
and Rosenblatt 2002), in the sense that no firm commitment is made. Under a quantity flexibility
contract, a retailer could adjust her initial order within a certain range after demand realization.
In a pay back contract, a supplier may deliver a quantity different from the received order after
yield realization. However, in the former, the maximum allowed change is part of the contract
terms; while in the latter, the pay back price rather than the supplier’s delivery quantity is part of
the contract. The revenue sharing mechanism considered in §3 works in the same way as the one
commonly studied in the literature (e.g., see Cachon and Lariviere 2005). However, the revenue
sharing mechanism considered in §4.2 is different because it only allows the supplier to earn a
portion of the revenue made from sales in excess of the retailer’s original order. Also, it does not
require the supplier to sell below cost. Therefore, our study complements the current literature on
revenue sharing.
Our study is also related to the literature on Vendor Managed Inventory (VMI). Our description of the two decentralized systems follows the framework introduced in Aviv and Federgruen
(1998). Similar to other VMI literature (see Cetinkaya and Lee 2000, Cheung and Lee 1998, Lee et
al. 1997), they focus on inventory and distribution cost performance measures of the supply chain
in a multi-period model and study the value of information and policy coordination. Different from
these studies, our paper considers a single-period model, which is commonly used to study coordinating contract design (e.g., Cachon and Lariviere 2005, Chick et al. 2008). Most importantly,
we introduce yield uncertainty and try to devise coordinating contracts under both supply chain
partnership and collaboration as well as more traditional arms-length buyer-supplier relationships.
The rest of the paper is organized as follows. In §2 we introduce the notation and present
some preliminary results for the basic pay back contract in a supply chain with a single supplier and
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retailer with deterministic demand. In §3 we study pay-back-revenue-sharing contract in supply
chains with unreliable supplier and competing retailers. Then in §4 we consider a modified payback-revenue-sharing contract in coordinating supply chains with unreliable supplier and stochastic
demand. Finally, §5 summarizes the main results and concludes.
2.
Preliminaries
In this paper, we consider supply chains facing uncertain supply. In the supply chain, a retailer
orders a product from a supplier and then sell to the end market. We use the same model when a
manufacturer source components from a supplier and assemble a finished product to sell to an end
market.
The production process at the supplier is modeled as having a stochastic proportional yield
(see Yano and Lee 1995). For an input of size q, the output is qY , where the random yield rate Y is
distributed on [0,1] and has a continuous probability density function h(·), a cumulative distribution
function H(·), and a mean µ, all independent of the lot size. The production cost is c per unit.
Since usually yield can only be observed after the production process is finished, the supplier has
to pay the production cost for the whole lot even though he might only collect revenue on the
quality (meeting specification) output. The supplier gets paid for each quality unit delivered at a
wholesale price w. The product is sold to the end-market at a unit price p. We assume that due
to production and assembly lead-time and tight customer response windows, early test production
or a second production run are not possible. Also, without loss of generality and for presentation
convenience, we normalize the cost of any value-added activity performed by the retailer, and the
salvage value for unused units, and shortage penalty to zero (including these parameters in the
model will not affect the qualitative insights).
We assume that both the retailer and supplier are risk neutral and maximize their expected
profits. Throughout the paper, we assume that all parties have a reservation profit of zero and
suppress this condition in the optimization problem. We use Π(π) to denote the retailer(supplier)’s
expected profit, and subscript c to denote expressions related to the centralized system.
If a supply chain only faces supply uncertainty, i.e., the retailer faces deterministic demand
and the supplier is subject to random yield, one would expect that a wholesale price contract may
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lead to a “double marginalization” problem and cause some distortion, similar to supply chains
with stochastic demand. In fact, one can design coordination mechanisms for this supply chain
using a logic similar to the stochastic demand case. To set the stage for our later analysis, we next
formalize this conjecture as a preliminary result.
First, denote the deterministic demand by D0 . The centralized system solves the following
problem:
max EY
Qc
p min{D0 , Qc Y } − cQc .
(1)
Assuming that the expected profit margin is high enough so that the business is profitable, i.e.,
pµ > c, we can characterize the optimal production quantity as the following:
Lemma 1 When demand is a constant D0 , the optimal production quantity Q∗c for the centralized
system satisfies:
Z
D0
Q∗
c
yh(y) dy =
0
c
.
p
(2)
Define function:
Λ(x) :=
Z
x
yh(y) dy.
(3)
0
Notice that Λ is increasing in x. Since Λ(1) = µ > c/p by the assumption, and Λ(0) = 0, there
must exist a unique value of x ∈ (0, 1) that satisfies Λ(x) = c/p. So we see that the optimal
production quantity is a linear function of D0 : Q∗c = kc D0 , where kc = 1/Λ−1 (c/p) > 1 is a
constant independent of D0 (Λ−1 denotes the inverse of Λ). In other words, the optimal production
quantity is inflated above the demand. From (2) we can show that kc is decreasing in c, and
increasing in p. So the cheaper the production cost or the higher the sales price, the higher the
production quantity. Based on Lemma 1, the centralized system’s optimal expected profit can be
expressed as Π∗c = pH̄(1/kc )D0 .
Under a wholesale price contract, the supplier bears the supply risk alone and would set a
production quantity lower than Q∗c . So a coordinating contract should provide incentives to the
supplier to increase his production size. One such contract is the pay back contract which tries
to correct the problem of supplier under-producing by having the retailer purchase the supplier’s
excess output at a discount price (a price lower than the wholesale price). In this way, the retailer
shares the supplier’s risk of over-producing, i.e., the risk of ending up with more products than
8
needed in good yield realizations. (Note that execution of the contract may not require physical
delivery of the product if in excess of demand and may only involve transfer of money1 .)
Assume that the pay back price is wm , where wm < c/µ < w. The supplier’s optimal
production quantity q ∗ can be determined by solving the following problem:
max π = EY w min(D0 , qY ) + wm (qY − D0 )+ − cq .
q
So
q∗ =
Λ−1
D
0
c−wm µ
w−wm
.
(4)
From (2) and (4) we can determine the coordination condition:
Proposition 1 When demand is a constant D0 , then a pay back contract where w and wm satisfy
c/p = (c − wm µ)/(w − wm ) coordinates the decentralized supply chain. Furthermore, the supply
chain profit can be arbitrarily allocated between the two parties by varying wm within (0, c/µ).
The pay back contract is in essence similar to a buy back contract designed for a stochastic
demand world. Both contracts allow the firm which does not directly face uncertainty to share the
risk faced by its partner. In a buy back contract, the supplier pays the retailer for unsold units,
while in the pay back contract the retailer pays the supplier for excess production output.
Same analogy can be used to design other coordinating contract in a random yield world. For
example, like a revenue sharing contract, a cost sharing contract which has the retailer share part
of the supplier’s production cost (incurred for all units, not just the delivered units) can achieve
coordination. And similar to the equivalence established for the buy back and revenue sharing
contract (Cachon 2003), there is an equivalence relationship between the pay back and cost sharing
contract.
Based on the understanding of the pay back contract in coordinating a supply chain simply
facing yield uncertainty, we will proceed to study coordination issues in two other related supply
chains with more elements.
1
Also, it can be shown that in execution of the pay back contract with deterministic demand, it is optimal for the
retailer to truthfully share the demand information with the supplier.
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3.
Coordination with Retailer Competition
In this section, we will consider a supply chain with multiple retailers which compete with each
other. Specifically, we assume that two identical retailers, R1 and R2 , source a common product
from a single supplier S. The assumption for the supplier’s production process is similar to that
in §2. We continue to assume that the market demand for the product is a constant D0 . The
retailers compete with each other in two dimensions: first, when there is limited supply due to low
yield realization, the supply is rationed between the retailers proportional to their order quantity.
In other words, the retailer which orders more gets more units. Second, the constant demand is
divided between the retailers proportional to their stocking quantity.
The sequence of events goes as follows: first, the retailers places orders with the supplier
simultaneously, with retailer Ri ordering Qi . After the supplier receives the orders, he sets a
production quantity q. Then the yield realizes, and the supplier obtains an output of qy, where y
is the realized yield rate. Finally, retailers receive delivery and make sales. If the output is more
than the total order, i.e., when qy ≥ Q1 + Q2 , then each retailer gets her full order. So retailer Ri ’s
stocking quantity is simply Qi . According to the proportional allocation model, the demand faced
by retailer Ri is
Qi
Q1 +Q2 D0 ,
i
and hence the sales made by Ri is min{Qi , Q1Q+Q
D0 }2 . Obviously, in
2
this case retailers only compete for the demand but not for the supply. If the output is less than
the total order, i.e., when qy ≤ Q1 + Q2 , then retailer Ri receives a delivery proportional to her
order quantity. So retailer Ri ’s stocking quantity is
still
Qi
Q1 +Q2 D0 .
Qi
Q1 +Q2 qy.
The demand faced by retailer Ri is
i
i
So the sales made by Ri becomes min{ Q1Q+Q
qy, Q1Q+Q
D0 }.
2
2
Before analyzing the decentralized system, we first quickly look at the centralized system
which turns out to be very similar to the one studied in §2 Lemma 1. Since the two retailers are
symmetric, the central planner will demand
D0
2
from the supplier for each retailer. The supplier’s
production decision then can be characterized by Lemma 1.
In analyzing the decentralized system, we use backward induction, starting from the supplier’s problem first. Upon receiving orders Q1 and Q2 , the supplier sets the production quantity
2
The case for asymmetric retailers where retailer 1’s proportion is αQ1 /(αQ1 +Q2 ) and retailer 2’s is Q2 /(αQ1 +Q2 )
(α > 1) can be analyzed in a similar way and will lead to the same qualitative results. The coordination condition
will depend on α. The symmetric case with α = 1 is a special case.
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q to maximize his expected profit. Under a wholesale price contract with wholesale price w, the
supplier’s optimal production quantity q ∗ (Q1 , Q2 ) can be determined in a way similar to that used
for Lemma 1. So q ∗ (Q1 , Q2 ) satisfies:
Z
Q1 +Q2
q∗
yh(y) dy =
0
c
.
w
(5)
Under the assumption that w > c/µ, we see that q ∗ (Q1 , Q2 ) can be expressed as a linear function
of the total order Q1 + Q2 , i.e., q ∗ = k(Q1 + Q2 ), where k > 1.
Next, we look at the retailers’ problem. Retailer R1 solves the following profit maximization
problem (we suppress the dependence of q ∗ on Q1 and Q2 ):
max πR1
Q1
Q1 +Q2
q∗
o
n Q
Q1
Q1
1
∗
∗
=
q y,
D0 − w
q y h(y) dy
p min
Q1 + Q2
Q1 + Q2
Q1 + Q2
0
Z 1
o
n
Q1
+
D0 − wQ1 h(y) dy,
p min Q1 ,
Q1 +Q2
Q1 + Q2
∗
Z
q
where each integrand represents the total revenue net the procurement cost. The sales is determined
according to the proportional allocation model described earlier, and recall that the retailers only
need to pay for quality products up to their order quantity.
Based on the supplier’s optimal solution, we can plug q ∗ = k(Q1 + Q2 ) in the objective
function. For any given D0 and Q2 , it can be shown that if Q1 ≤ D0 − Q2 , the objective function
is increasing in Q1 with slope (p − w)kΛ( k1 ) + (p − w)H̄( k1 ). So the best response Q∗1 must be no
less than D0 − Q2 , and in this case the objective function can be written as:
πR1 =
Z
D0
k(Q1 +Q2 )
(pkQ1 y − wkQ1 y)h(y) dy +
0
+
Z
1
1
k
Z
1
k
D0
k(Q1 +Q2 )
Q1
p
D0 − wkQ1 y h(y) dy
Q1 + Q2
(6)
Q1
D0 − wQ1 h(y) dy.
p
Q1 + Q2
It can be shown that πR1 is concave in Q1 in this range. Hence, the following first order condition
is sufficient to characterize the best response function Q∗1 (Q2 ):
Z
D0
k(Q∗ +Q2 )
1
0
pkyh(y) dy −
Z
1
k
0
1
D0
pQ2 D0
= 0.
H̄
−
w
H̄
wkyh(y) dy +
(Q∗1 + Q2 )2
k(Q∗1 + Q2 )
k
(7)
Retailer R2 ’s best response function can be derived in the same way. The equilibrium then
can be found at the intersection of the two best response functions. A quick check of (7) indicates
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that at equilibrium Q∗1 must be equal to Q∗2 , which is quite intuitive since retailers are identical.
Denote the equilibrium order quantity as Q∗ . Then Q∗ can be characterized by the following result:
Proposition 2 In a supply chain with two competing retailers as described above, the retailers’
equilibrium order quantity Q∗ is greater than D0 /2 and satisfies:
Z
D0
2kQ∗
0
pkyh(y) dy −
Z
1
k
0
1
D0
pD0
= 0,
H̄
−
w
H̄
wkyh(y) dy +
4Q∗
2kQ∗
k
(8)
when
Otherwise, Q∗ =
D0
2 .
1 p
1
(p − w)kΛ
+
− w H̄
≥ 0.
k
2
k
(9)
Proposition 2 implies that the retailers’ equilibrium order quantities may be greater than the system
optimal, D0 /2. This is because they compete for the demand and possibly for the limited supply.
The fact that both demand and supply are allocated proportionally encourages retailers to order
aggressively to guarantee their market share and supply.
When the yield is subject to a uniform distribution, closed-form solution can be obtained.
A direct application of Proposition 2 leads to the following result:
Corollary 1 When Y has a uniform distribution, the retailers’ equilibrium order quantity is Q∗ =
p c −1
pD0
when w ≤ w0 , and Q∗ = D0 /2 otherwise, where the threshold value w0 satisfies:
4w (1 −
2w )
2w0 1 −
r
c
2w0
= p.
(10)
The decentralized supply chain is distorted in two ways: one, the supplier which is subject
to random yield losses chooses a production quantity lower than the system optimal; second, the
competing retailers may order more than the system optimal when the wholesale price is low. The
preliminary result in §2 suggests that when only the first distortion exists, coordination can be
achieved by using a pay back contract. So we will focus on the case where the wholesale price is
low and both distortions are present. We will show next that the second distortion can be corrected
by using a revenue sharing contract. Therefore, a combination of pay back and revenue sharing
mechanism can coordinate the supply chain.
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3.1
Pay-back-revenue-sharing contract
Consider the case where the supplier offers a pay-back-revenue-sharing contract to each retailer.
For a given wholesale price w, a pay-back-revenue-sharing contract needs to specify two terms, a
pay back price wm and a revenue share parameter φ. The pay back price wm is defined in a similar
way as in §2 and represents the discount price the retailers pay for the supplier’s excess output
beyond their orders. Again we assume a proportional allocation model where the portion of the
excess output retailer Ri pays is
Qi
+
Q1 +Q2 (qy −Q1 −Q2 ) .
The revenue sharing parameter φ is defined
as the fraction of the revenue the supplier gets, and hence the retailer earns (1 − φ) portion (similar
to what has been commonly used in the literature e.g., Cachon and Lariviere 2005).
We first look at the supplier’s problem. The supplier’s expected profit can be expressed as
the following:
+
πS =EY w min{qY, Q1 + Q2 } + wm (qY − Q1 − Q2 ) + φp min min{qY, Q1 + Q2 }, D0 − cq
=
Z
D0
q
(wqy + φpqy)h(y) dy +
0
+
Z
1
Q1 +Q2
q
h
Z
Q1 +Q2
q
D0
q
(wqy + φpD0 )h(y) dy
i
w(Q1 + Q2 ) + wm (qy − Q1 − Q2 ) + φpD0 h(y) dy − cq.
(11)
Recall that we only focus on the case where retailers over-order, i.e., when Q1 + Q2 > D0 . In the
low yield realization case (the first two terms in the last expression), i.e., when qy ≤ Q1 + Q2 , the
supplier sells all output and gets his revenue share. In this case, the total revenue from the two
retailers is p min{qy, D0 }. In the high yield realization case (the third term in the last expression),
i.e., when qy ≥ Q1 + Q2 , the supplier delivers retailers’ full order and has an excess output of
(qy − Q1 − Q2 ). Besides the regular sales, the supplier also gets some payment for the excess output
and gets his share of the total revenue from the retailers. The objective function is concave in q,
so the first order condition is sufficient to characterize q ∗ .
Lemma 2 Under a pay-back-revenue-sharing contract, for given order quantities Q1 and Q2 , the
supplier’s optimal production quantity q ∗ (Q1 , Q2 ) satisfies:
(w − wm )
Z
Q1 +Q2
q∗
yh(y) dy + φp
0
13
Z
D0
q∗
0
yh(y) dy = c − wm µ.
(12)
To derive retailers’ equilibrium order quantity, we first write retailer R1 ’s expected profit
function:
πR1 =
D0
q∗
i
Q1
Q1
q∗y − w
q ∗ y h(y) dy
Q1 + Q2
Q1 + Q2
0
Z Q1 +Q
2 h
i
q∗
Q1
Q1
(1 − φ)p
+
D0 − w
q ∗ y h(y) dy
D0
Q1 + Q2
Q1 + Q2
q∗
Z 1
i
h
Q1
Q1
D0 − wQ1 − wm
(q ∗ y − Q1 − Q2 ) h(y) dy,
(1 − φ)p
+
Q1 +Q2
Q1 + Q2
Q1 + Q2
∗
Z
h
(1 − φ)p
(13)
q
where q ∗ is determined by (12). Again, in the case where Q1 +Q2 > D0 , retailers only receive partial
order when q ∗ y < Q1 + Q2 , and retailer R1 gets her share
Q1
∗
Q1 +Q2 q y
according to the proportional
1
1
allocation model. In this case, retailer R1 makes a sale of min{ Q1Q+Q
q ∗ y, Q1Q+Q
D0 } and retains
2
2
(1 − φ) portion of the revenue. When q ∗ y > Q1 + Q2 , retailers receive their full order and share the
market demand (since the total order is more than D0 ). In this case, since the supplier has excess
output, each retailer needs to pay the supplier for her share of the excess output at the pay back
price.
The supply chain is coordinated when the retailers’ equilibrium order quantities Q∗1 and
Q∗2 equal to the system optimal D0 /2, and the supplier’s production quantity is aligned with the
centralized system. Using the optimality condition for the centralized system given in (2) with
retailers’ total order quantity equal to D0 , and the first order condition for the decentralized
system, we can obtain the coordination condition as the following.
Proposition 3 In a supply chain with two competing retailers described above, a pay-back-revenuesharing contract can coordinate the decentralized supply chain if the contract terms satisfy: φ =
1−
2wµ
pµ+c
and wm =
c
pµ+c w.
The revenue sharing mechanism here is different from its counterpart in a stochastic demand
world in several ways. First, in the latter (see Cachon and Lariviere 2005 and §4.2 in this paper)
revenue sharing contract is used to correct buyer’s under-ordering problem. Here, this mechanism
is also able to correct buyers’ over-ordering problem. The difference is that with stochastic demand,
revenue sharing allows the supplier to share some of the buyer’s demand risk by letting the buyer
pay a lower wholesale price initially and compensate the supplier with a portion of sales revenue
later. In our model with deterministic demand and buyer competition, revenue sharing is not used
14
to reallocate the demand risk but reallocate the buyers’ revenue so their benefit from over-ordering
is dampened.
Second, the coordination condition in Proposition 3 shows that the fraction of revenue the
supplier gets decreases in the wholesale price w. This is in contrast to the revenue sharing contract
for a stochastic demand world. In the decentralized supply chain considered in this section, the
retailers order more than the system optimal when the wholesale price is low. From Corollary 1,
we see that the more retailers order (as w decreases) and the further the decentralized system’s
performance deviates from the centralized one. The revenue sharing mechanism, in an effort to
correct the resulting distortion dampens the retailers’ over-ordering behavior by offering them a
small share (i.e., (1-φ) is smaller when w is lower).
Proposition 3 also shows that the pay back price wm increases in the wholesale price w. This
is in contrast to the basic pay back contract studied in §2. In simple supply chains considered there,
the pay back mechanism corrects the problem of supplier under-producing by giving the supplier
incentives in the form of payment for excess output. The higher the wholesale price the supplier
charges, less such incentive is needed. However, in the supply chain considered in this section,
the pay back mechanism is coupled with the revenue sharing mechanism to restore supply chain
coordination. At a higher wholesale price, the supplier gets a smaller share of the revenue (as φ
decreases in w). Also, retailers’ order quantities are lower at a higher wholesale price level (e.g.,
see Corollary 1 for the case of uniform yield distribution). Both effects discourage the supplier to
set a high production quantity. Therefore, a higher pay back price is needed to induce the supplier
to inflate the production at the same rate as the centralized system.
By varying the wholesale price w, and setting the two contract terms wm and φ according
to the coordination condition, the supply chain profit can be allocated between the supplier and
the retailers in many different ways. However, unlike the basic pay back contract, not all possible
profit splits are achievable, as shown in the following result:
Corollary 2 For a uniform yield distribution, by varying the wholesale price w between c/µ and
min{w0 , (pµ + c)/2µ}, the supplier’s expected profit is γΠ∗c where Π∗c is the supply chain’s optimal
2w0 µ
pµ−c expected profit, and γ ∈ max{ 12 , 1 − pµ+c
}, pµ+c
.
Therefore, as long as each retailer’s expected profit under a wholesale price contract (or other
15
non-coordinating contracts) is lower than (1 − γ)Π∗c /2, one can always find a coordinating payback-revenue-sharing contract that makes all parties better off. Notice that w, wm and φ need
to be negotiated at the same time to achieve different profit allocations. From the supplier’s
optimal expected profit under a coordinating contract, Π∗s =
(1+φ)
2 pD0 H̄(1/kc ),
we can see that the
supplier(retailer) gets a larger portion of supply chain profit when φ is big(small), and wm , w are
small(big).
4.
Coordination under Both Yield and Demand Uncertainty
In this section, we consider supply chains with a single supplier and retailer but there are uncertainties on both the demand and supply side. The assumptions for the random yield are the same
as in §2. The demand is assumed to be distributed on (0, ∞) with a pdf f (ξ), CDF F (ξ) and mean
µD . We first set the stage by analyzing the centralized system. To differentiate from the centralized
system facing only yield uncertainty studied in §2, here we use a “hat” (ˆ·) to denote corresponding
variables. The centralized system’s problem can be expressed as:
max Π̂c =ED,Y p min{D, Q̂c Y } − cQ̂c
Q̂c
=
Z
1
y=0
Z
Q̂c y
pξf (ξ) dξ +
ξ=0
Z
∞
ξ=Q̂c y
pQ̂c yf (ξ) dξ h(y) dy − cQ̂c .
(14)
Unlike the deterministic demand model, now the sales is determined by both the realized demand
and the available output. The optimal production quantity can be characterized as follows:
Lemma 3 When demand and supply are both stochastic, the optimal production quantity Q̂∗c for
the centralized system satisfies:
Z
1
0
y F̄ (Q̂∗c y)h(y) dy =
c
.
p
The optimal expected profit for the centralized system is:
Z Q̂∗c
ξ ∗
ξ H̄
Π̂c = p
f (ξ) dξ.
Q̂∗c
0
(15)
(16)
With uncertainty on both the demand and supply side, one might expect that two coordination mechanisms are needed, one for each uncertainty. As we show next, this is not necessary if
demand information is shared between the two parties and the supplier manages the inventory at
the retailer’s site.
16
4.1
Coordination under a VMI program
When the retailer has joined the supplier’s VMI program, the supplier makes the production decision based on the demand forecast shared by the retailer as well as the nature of the production
process. After production is completed, the supplier makes the delivery based on the realized demand. Next we show that the pay back contract studied in §2 can also achieve coordination under
both demand and yield uncertainty. Under the pay back contract, the supplier’s problem is:
h
i
max π = ED,Y w min{D, qY } + wm (qY − D)+ − cq .
q
The profit function can be shown to be concave in q, so the supplier’s optimal production quantity
can be characterized by the following first order condition:
Z
1
y F̄ (q ∗ y)h(y) dy =
0
c − wm µ
.
w − wm
(17)
From (15) and (17), we see that the coordination condition is the same as the deterministic
demand case, i.e., c/p = (c−wm µ)/(w−wm ). Furthermore, the supply chain profit can be arbitrarily
allocated between the two parties by varying wm within (0, c/µ). It can be also shown that supply
chain coordination can be achieved using a cost sharing contract, similar to the deterministic
demand case.
This is in contrast to the stochastic demand (perfect yield) literature. Although the wellknown revenue sharing and buy back contract are designed to increase a retailer’s order to the
system optimal quantity, as mentioned in Wang et al. (2004) and Gong (2008), perfect coordination
cannot be achieved if a VMI program is in place. However, in our model the pay back and cost
sharing contract can achieve coordination even under a VMI system. This is because in a supply
chain facing only demand uncertainty, it is the retailer whose action causes the distortion. The
revenue sharing and buy back contract are used to correct this distortion at the retailer’s site.
Therefore, if the supplier becomes the decision maker under VMI, these contracts may fail to
coordinate the supplier’s action. On the contrary, in our model, it is the supplier’s production
decision that needs to be coordinated. This is why the pay back and cost sharing contract designed
to correct the supplier’s action continue to work when the supply chain faces demand uncertainty
and a VMI program is in place.
17
Since VMI requires only one coordination mechanism to deal with both demand and supply
uncertainty, it greatly facilitates supply chain coordination. However, in reality it is not always
feasible to structure one. Despite all the benefits of VMI, there are disadvantages or obstacles to
implementing it. For suppliers, an obvious disadvantage is related to increased costs to manage
inventory and retain their access to market information. VMI cannot be successfully implemented
if suppliers do not have the capability to monitor and manage inventory at retailers’ site. Retailers
also can be reluctant to share the information or infrastructure with suppliers with the concerns
that they may lose their proprietary information and control over their own inventory. As we
will show next, when this is the case, a modified version of the pay-back-revenue-sharing contract
studied in §3 can coordinate the supply chain.
4.2
Coordination with retailer controlling own inventory
In this section, we consider supply chains where the retailer shares demand forecast with the
supplier, but unlike a VMI system, the retailer manages her own inventory. As a result, the retailer
still needs to initiate orders. It can be shown (details included in Appendix A.2) that when only a
pay back mechanism is used, the supplier’s production quantity is still below the system optimal
one because the retailer does not place a big enough order. This suggests that incentives should be
provided to the retailer to increase her order size. Revenue sharing is a well known mechanism that
is able to do so. Research (e.g., Cachon and Lariviere 2005) has shown that this mechanism can
correct buyer’s under-ordering problem only when supplier sells below cost initially and profits from
the shared revenue later. We observe the same finding in our model. The pay-back-revenue-sharing
contract in §3 can lead to coordination with the requirement that w < c/µ. In fact, it can be shown
(details included in Appendix A.2) that when w > c/µ this contract fails to coordinate because
the supplier is given generous incentives and as a result his production quantity is higher than
the system optimal. Since in some situations pricing below cost is not feasible because of either
the accounting pressure from inside the firm or some industry norm, in this section we consider a
modified version of the pay-back-revenue-sharing contract studied in §3. Under this contract the
retailer not only subsidizes the supplier for all excess output, but also shares with the supplier a
portion of the revenue made from the sales in excess of the original order. The difference from
18
the contract studied in §3 is that here not all revenues are shared, but only the sales beyond the
retailer’s original order. Notice that since the supplier’s revenue is now dependent not only on the
retailer’s order, but also on the end-market demand, it is only possible to arrange such a contract
when demand information is shared with the supplier.
We continue to use wm and φ to denote the contract terms, where wm is the pay back
price and φ is the revenue share the supplier gets. The sequence of events goes as follows. First,
the retailer and supplier negotiate and agree on the contract terms. Then the retailer submits an
order of size Q to the supplier, and the supplier proceeds to plan a production quantity q. After
production is completed, the retailer makes a first payment to the supplier based on the delivered
quantity and pay back units, if any. She pays the regular wholesale price w for output up to Q units
and the pay back price wm for any excess output. Finally demand gets realized and the retailer
may make a second payment to the supplier for sales in excess of his original order. For each unit
sold in excess of the original order, the retailer shares a portion φ of the sales revenue with the
supplier and retains (1 − φ) portion of it.
We first look at the supplier’s problem. Given an order size Q, the supplier solves the
following problem:
maxq π = ED,Y
+
+
wmin(Q, qY ) + wm (qY − Q) + φpmin (D − Q) , (qY − Q) − cq ,
+
where the third term is the shared revenue from the retailer’s excess sales. As the expression
indicates, the excess sales amount is determined by the minimum of excess demand (D − Q)+ and
the excess supply (qy − Q)+ .
We assume w > c/µ for the reason described above. Under this condition, the optimal
production size is no less than the received order Q, and the supplier’s expected profit can be
expressed as the following:
Z 1 Z
π=
wQ + wm (qy − Q) +
y= Q
q
+
Z
Q
q
qy
φp(ξ − Q)f (ξ) dξ +
Z
∞
ξ=qy
ξ=Q
φp(qy − Q)f (ξ) dξ h(y) dy
wqyh(y) dy − cq.
y=0
The following result characterizes the supplier’s optimal production quantity q ∗ (Q):
19
(18)
Lemma 4 Under a modified pay-back-revenue-sharing contract, for a given order size Q, if w −
wm ≥ φpF̄ (Q), then the supplier’s optimal production quantity q ∗ (Q) satisfies:
Z 1
Q
y F̄ q ∗ y h(y) dy = c.
(w − wm )Λ ∗ + wm µ + φp
Q
q
∗
(19)
q
By comparing q ∗ (Q) with the supplier’s production quantity under a pay back contract in (4) (with
D0 replaced by Q), we see that he is willing to set a higher production quantity under the modified
pay-back-revenue-sharing contract. As the supplier expects potential revenue share from his excess
output, he is willing to set a higher production quantity. The supplier’s optimal expected profit
can be expressed as:
h
Z 1 Z q∗ y
i
Q
∗
w − wm − φpF̄ (Q) H̄ ∗ Q + φp
π =
ξf (ξ) dξ h(y) dy.
q
y= Q∗ ξ=Q
(20)
q
Now we study the retailer’s problem. Under the pay back mechanism, the retailer has access
to the supplier’s entire output. As a result, the retailer’s expected sales are not directly determined
by her order size, but by the realized demand and supplier’s realized output. The expected profit
function when she places an order of size Q can be expressed as the following:
o
n
∗
∗
+
+
∗
+
,
Π = ED,Y p min{D, q Y } − wmin Q, q Y − wm (qY − Q) − φpmin (D − Q) , (q Y − Q)
where the supplier’s optimal production quantity q ∗ is a function of the retailer’s order Q according
to (19).
In order to coordinate the supply chain, we need to align the supplier’s production quantity
with the centralized system, i.e., q ∗ = Q̂∗c . Using the optimality condition for the centralized system
(15) and the corresponding ones for the supplier and retailer in the decentralized system, we see
that coordination requires the following two conditions to hold simultaneously:



 w = wm + φpF̄ (Q∗ ),
∗
R


 (w − wm )Λ Q∗ + wm µ + φp Q̂1 ∗ y F̄ Q̂∗c y h(y) dy = c.
Q̂
c
(21)
Q∗
c
It can be shown that conditions (21) also ensure that Q∗ is indeed a local maximum point for Π
by verifying that ∂ 2 Π/∂Q2 < 0 at Q∗ .
When the supply chain is coordinated under such a contract, the supplier’s profit (20)
becomes:
∗
π = φp
Z
1
∗
y= Q∗
Q̂c
Z
Q̂∗c y
ξf (ξ) dξ h(y) dy.
ξ=Q∗
20
(22)
Notice that from the first equation in (21), Q∗ can be expressed as a function of wm and φ. Similar to
what has been shown for the basic pay back contract, by varying w, and setting wm and φ according
to the coordination condition, the supply chain profit can be arbitrarily allocated between the two
firms. The following result formally characterizes the coordinating contract:
Proposition 4 When both demand and yield are stochastic, a modified pay-back-revenue-sharing
contract can coordinate the decentralized supply chain if the contract terms satisfy:
Z 1
v(w , φ) m
(w − wm )Λ
y F̄ (Q̂∗c y)h(y) dy = c − wm µ,
+ φp
v(wm ,φ)
Q̂∗c
Q̂∗
(23)
c
where v(wm , φ) := F̄ −1 ((w − wm )/(φp)) and F̄ −1 represents the inverse of CCDF for the demand
distribution. Furthermore, the supply chain profit can be arbitrarily split between the two parties.
Next, we describe some properties of the coordinating contract, which are useful in designing
or negotiating such contracts in practice. Define:
η :=
Z
1
0
y F̄ (Q∗c y)h(y) dy.
Corollary 3 For a given wholesale price w:
wµ−c
, 1].
(i) The feasible range for the revenue share φ is [ p(µ−η)
(ii) The feasible range for the pay back price wm is [w − pF̄ (Qmax ), c−wη
µ−η ], where Qmax is determined by the equation:
Z
1
h
i
wµ − c
y F̄ (Qmax ) − F̄ (Q̂∗c y) h(y) dy =
.
p
Qmax /Q̂∗c
(24)
(iii) The supplier’s expected profit is increasing in φ and decreasing in wm .
(iv) The feasible range for the supplier’s expected profit is [π,π̄], where
Z Z ∗
wµ − c 1 Q̂c y
π=
ξf (ξ) dξh(y) dy, and
µ−η 0 0
Z 1
Z Q̂∗c y
π̄ =
pξf (ξ) dξh(y) dy,
Qmax /Q̂∗c
Qmax
and Qmax is determined by (24).
Corollary 3 part (iii) states that even for a given wholesale price, the supply chain profit can still
be allocated between the two parties in different ways by varying the contract terms wm and φ. It
21
also implies that the supplier is better off if he earns a larger portion of the retailer’s excess sales,
as expected. However, the supplier is worse off at a higher pay back price. The latter result is
because of the fact that a higher pay back price wm leads to a lower revenue share φ and a lower
retailer’s order Q∗ as a result of the coordination condition. Since both the wholesale and retail
price are fixed, the supplier obtains lower profit on both the regular delivery and shared revenue.
Despite of a higher revenue from the excess output at a higher pay back price wm , it cannot offset
the revenue loss and the supplier finds itself worse off.
When the wholesale price is negotiated along with the other contract terms (wm , φ), designing a coordinating contract turns out to be more convenient. We first state some properties which
will be useful later:
Corollary 4 For a given revenue share φ:
(i) The feasible range for the pay back price wm is [(c − φpη)/µ, c/µ − φpF̄ (Q̂∗c )].
(ii) The supplier’s expected profit is decreasing in the pay back price wm , and increasing in w.
Corollary 4 part (ii) states that when the retailer agrees to share a fixed portion of her revenue
from excess sales, the supply chain profit can be allocated between the two parties in different ways,
with larger portion allocated to the supplier by increasing w and decreasing wm while satisfying the
coordination condition. This implies that the supplier is worse off if the retailer agrees to buy the
excess supply at a higher pay back price. To explain this, notice that the supplier’s expected profit
(22) is decreasing in the retailer’s order Q∗ for a given φ. Since coordination results in supplier
setting his production quantity always to be Q̂∗c , the higher the pay back price is, the more the
retailer orders from the supplier so that the cost of paying for excess supply is less.
We summarize the steps in designing a coordinating pay-back-revenue-sharing contract as
follows (assuming w is not fixed; one can easily modify it for the case where w is given):
Step 1: First, obtain the centralized system’s solution (Q̂∗c , Π̂∗c ) from (15) and (16). The two
supply chain partners negotiate an allocation (α, 1 − α) of the system profit, where α
denotes the supplier’s portion. Denote the supplier’s expected profit as π = αΠ̂∗c .
c
Step 2: Next, choose a revenue sharing parameter φ from the range (0, pη
).
Step 3: Then, calculate Q∗ from the supplier’s optimal profit function (22) using π determined
22
in Step 1 and φ in Step 2.
Step 4: Then, determine wm from the following equation:
Z 1
1
Q∗
wm =
c − φp F̄ (Q∗ )Λ( ) + ∗ y F̄ (Q̂∗c y)h(y) dy .
Q
µ
Q̂∗c
∗
Q̂c
Step 5: Finally, determine w from the first condition in (21).
The upper bound of φ in Step 2 is chosen to ensure that the pay back price is positive based on
Corollary 4 part (i). The expression for wm in Step 4 is obtained by writing the coordination
conditions (21) as an equation of φ, Q∗ and wm (i.e., independent of w).
4.3
A numerical example
We illustrate the modified pay-back-revenue-sharing contract using an example in this section. Let
demand be distributed according to Gamma(2, 50), yield be distributed with Beta(3, 1), p = 10,
and c = 3. The system optimal production quantity Q̂∗c = 129.03 and expected profit Π̂∗c = 310.43.
Figure 1 shows the two contract parameters (wm , φ) for eight different coordinating contracts, with
the left vertical axis corresponding to the pay back price, and the right vertical axis corresponding
to the revenue share proportion. For example, the coordinating contract at the wholesale price
wm
w = 5.24 has wm = 1.65 and φ = 0.6.
w
Figure 1: Illustration of eight different coordinating (modified) pay-back-revenue-sharing contracts
Figure 2 shows each supply chain party and the total supply chain’s profit under both the
above eight coordinating contracts as well as the wholesale price contract. The solid lines repre23
sent the profits under coordinating pay-back-revenue-sharing contract, and dashed lines represent
those under wholesale price contract. Notice that since the former contracts fully coordinate the
supply chain, the resulting total supply chain profit is the same as the centralized system, which
is represented by the horizontal line in Figure 2. Clearly, we see that each coordinating contract
leads to a different allocation of the system profit between the two firms. For example, the contract
at w = 5.24 with (wm , φ) = (1.65, 0.6) leads to a supplier profit of 94.42 and retailer profit of
216.01 (or 30% and 70% of the system profit respectively), while the contract at w = 7.18 with
(wm , φ) = (0.81, 0.8) leads to a supplier profit of 206.97 and retailer profit of 103.46 (or 67% and
33% of the system profit respectively). Figure 2 also shows that for the same wholesale price w,
both the supplier and retailer are better off under a coordinating pay-back-revenue-sharing contract
than under a wholesale price contract. Again, to achieve different allocation of the supply chain
profit, both w and the other contract terms (wm , φ) need to be negotiated at the same time.
Figure 2: Comparison of supplier, retailer, supply chain profit under coordinating (modified) payback-revenue-sharing (PB-RS) and wholesale price contract
Finally, we briefly discuss the comparison with other coordinating contracts, e.g., quantity
flexibility contract. A quantity flexibility contract allows the retailer to adjust her original order
size Q within a range [(1 − β)Q, (1 + α)Q] after uncertainty realizes. In a supply chain with both
stochastic demand and random yield, this contract, like the one studied in this section, allows
24
demand and supply risk to be shared between the two parties. Therefore, it may also improve the
supply chain performance. In Table 1, we compare the performance of the coordinating (modified)
pay-back-revenue-sharing contract to quantity flexibility contract, with wholesale price contract as
a benchmark. We see that although a quantity flexibility contract does better than a wholesale
price contract, it cannot fully restore system efficiency like the modified pay-back-revenue-sharing
contract does. Despite the fact that the allowed change can be negotiated between the two parties,
there still exist situations where supplier’s output cannot be used to satisfy the realized demand.
w
Wholesale
price
contract
(WP)
Quantity flexibility
contract (QF)
Modified PB-RS
contract (PBRS)
4.08
310.36
(QF - WP)
WP
310.38 0.004%
4.42
308.58
308.88
0.10%
310.43
0.60%
4.67
305.93
306.58
0.21%
310.43
1.47%
4.94
301.95
302.97
0.34%
310.43
2.81%
5.24
296.34
299.29
1.00%
310.43
4.76%
6.42
263.90
287.44
8.92%
310.43
17.63%
7.18
235.15
293.61
24.86%
310.43
32.01%
9.18
155.73
309.30
98.61%
310.43
99.34%
Profit
(PBRS - WP)
WP
310.43
0.02%
Profit
Table 1: Comparison between quantity flexibility and modified pay-back-revenue-sharing contract
5.
Conclusion
In this paper, we studied coordination issues in supply chains where supplier’s production process is
subject to random yield. We modeled yield randomness via a stochastic proportional yield model.
We first pointed out some similarities in coordination challenges between a supply chain facing only
yield uncertainty (hence with deterministic demand) and the widely studied case of supply chains
facing only demand uncertainty. In both situations, under a wholesale price contract the “double
marginalization” problem causes the party which faces uncertainty to deviate its action from the
system optimal one: when it is the retailer which faces demand uncertainty, she orders less from
the supplier than a centralized system does; when it is the supplier which faces yield uncertainty,
he sets a production quantity lower than a centralized system does. In a supply chain with a
25
single unreliable supplier and a single retailer, since the supplier is exposed to the risk of ending up
with output beyond the received order, he under-produces in the decentralized system. Therefore,
coordinating contracts should aim for the retailer to share this risk with the supplier and induce
him to set a higher production quantity. Like a buy back contract gives the retailer incentives to
order more in a stochastic demand world, a pay back contract has the retailer pay a discount price
for the supplier’s excess output in a random yield world and thus providing the right incentives to
the supplier.
Building upon this result, we considered coordination issues for two other supply chains with
random yield. The first one is motivated by the semiconductor industry and looks at competing
retailers for constant market demand while being constrained by the common supplier’s limited
output in low yield realizations. We considered a proportional allocation model where each retailer’s
share of the market demand and share of the limited supply is proportional to her order quantity.
We showed that in this decentralized supply chain there are two distortions: one, the retailers tend
to over-order as a result of their competition, especially when the wholesale price is low; second, the
supplier tends to under-produce due to double marginalization. We proposed a pay-back-revenuesharing contract under which each retailer pays a discount price for her share of the excess output
and shares a portion of her revenue with the supplier. We proved that this contract can achieve
coordination and can allocate the supply chain profit in many different ways between the supplier
and retailers. We found that in the coordinating contract, both the retailer’s revenue share and the
pay back price increase in the wholesale price, which contrasts with the revenue sharing contract
for stochastic demand supply chains and the basic pay back contract for random yield deterministic
demand environments.
The second supply chain we considered is one with both demand and yield uncertainty.
One may expect that a combination of the two coordination mechanisms for demand (revenue
sharing) and yield (pay back) is required for coordination in this environment. We showed that
this is not necessary if the supplier manages inventory at the retailer’s site, for example, under a
VMI program. A pay back contract can be designed using the same coordination condition as the
deterministic demand case. This is in contrast to the buy back contract, which fails to achieve
perfect coordination if a VMI program is in place in the presence of stochastic demand. Since in
26
a VMI system, the supplier makes the production and/or inventory decision for the whole supply
chain, contracts designed to correct his action rather than the retailer’s prove to be more useful in
coordination.
When it is not possible to structure a VMI program, often due to retailer’s reluctance in
delegating inventory responsibilities to supplier or supplier’s lack of capabilities in timely replenishment, and the retailer retains the control over her own inventory, another coordination distortion
exists besides supplier under-producing: lost sales might occur when both the realized output and
realized demand are more than the retailer’s order quantity. The pay back contract alone cannot
coordinate in this setting, since the retailer tends to under-order and as a result the supplier underproduces. The pay-back-revenue-sharing contract can coordinate only when the supplier sells below
cost. When the wholesale price is above cost, this contract seems to overdo it in supplier incentives
and as a result the supplier over-produces. For this situation we considered a modified version
of the pay-back-revenue-sharing contract in which the supplier only gets a portion of the revenue
made from sales of the supplied output in excess of the retailer’s original order. We proved that this
contract can achieve coordination. Interestingly, we found that under such a coordinating contract
and for a given wholesale price, the supplier’s expected profit is decreasing in the pay back price
because coordination calls for a smaller revenue share and a smaller retailer order.
Our study has important contributions to the supply chain coordination literature with random yield. Building upon the basic pay back contract for a simple supply chain with a single
unreliable supplier and a single retailer, we studied how to coordinate a supply chain with random
yield and competing retailers supplied by a common supplier or linear supply chains facing both
yield and demand uncertainties. The proposed pay-back-revenue-sharing contract constructively
combines two well-known coordination mechanisms for simpler supply chains and is easy to implement. Future research may consider other contract forms such as two-part tariffs, competing
suppliers, or risk-averse parties. Also, one may consider information sharing issues in these settings. Our paper provides a building block for models addressing these remaining interesting yet
challenging problems for supply chains with random yields.
27
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