Pay-Back-Revenue-Sharing Contract in Coordinating Supply Chains with

Random Yield

Sammi Y. Tang

School of Business, University of Miami, Coral Gables, FL, ytang@miami.edu

Panos Kouvelis

Olin Business School, Washington University in St.Louis, St. Louis, MO, kouvelis@wustl.edu

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 pro-

duction 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 pharmaceuti-

cal 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 character-

istics 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 com-

monly 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

2

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, Ozer 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 uncer-

tainty, 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 pres-

ence 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 coordi-

nation 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

4

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 as-

sembly 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 de-

scription 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 coor-

dinating 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

6

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 pay-

back-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

7

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:

maxQc

EY

[

pmin{D0, QcY } − 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:∫

D0Q∗

c

0yh(y) dy =

c

p. (2)

Define function:

Λ(x) :=

∫ x

0yh(y) dy. (3)

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 = kcD0, 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 = pH(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:

maxq

π = EY

[

wmin(D0, qY ) + wm(qY −D0)+ − cq

]

.

So

q∗ =D0

Λ−1(

c−wmµw−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 − wmµ)/(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.

1Also, 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 isQi

Q1+Q2D0, and hence the sales made by Ri is min{Qi,

Qi

Q1+Q2D0}

2. Obviously, in

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 Qi

Q1+Q2qy. The demand faced by retailer Ri is

still Qi

Q1+Q2D0. So the sales made by Ri becomes min{ Qi

Q1+Q2qy, Qi

Q1+Q2D0}.

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 D02 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 sup-

plier’s problem first. Upon receiving orders Q1 and Q2, the supplier sets the production quantity

2The 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:

∫

Q1+Q2q∗

0yh(y) dy =

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):

maxQ1

πR1 =

∫Q1+Q2

q∗

0

{

pmin{ Q1

Q1 +Q2q∗y,

Q1

Q1 +Q2D0

}

− wQ1

Q1 +Q2q∗y

}

h(y) dy

+

∫ 1

Q1+Q2q∗

{

pmin{

Q1,Q1

Q1 +Q2D0

}

− wQ1

}

h(y) dy,

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( 1k ) + (p − w)H( 1k ). 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 =

∫

D0k(Q1+Q2)

0(pkQ1y − wkQ1y)h(y) dy +

∫ 1k

D0k(Q1+Q2)

(

pQ1

Q1 +Q2D0 − wkQ1y

)

h(y) dy

+

∫ 1

1k

(

pQ1

Q1 +Q2D0 − wQ1

)

h(y) dy.

(6)

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):

∫

D0k(Q∗

1+Q2)

0pkyh(y) dy −

∫ 1k

0wkyh(y) dy +

pQ2D0

(Q∗

1 +Q2)2H

(

D0

k(Q∗

1 +Q2)

)

− wH(1

k

)

= 0. (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:

∫D0

2kQ∗

0pkyh(y) dy −

∫ 1k

0wkyh(y) dy +

pD0

4Q∗H

(

D0

2kQ∗

)

− wH(1

k

)

= 0, (8)

when

(p −w)kΛ( 1

k

)

+(p

2− w

)

H(1

k

)

≥ 0. (9)

Otherwise, Q∗ = D02 .

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∗ =

pD0

4w (1−√

c2w )

−1 when w ≤ w0, and Q∗ = D0/2 otherwise, where the threshold value w0 satisfies:

2w0

(

1−

√

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 isQi

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

[

wmin{qY,Q1 +Q2}+ wm(qY −Q1 −Q2)+ + φpmin

{

min{qY,Q1 +Q2},D0

}

]

− cq

=

∫D0q

0(wqy + φpqy)h(y) dy +

∫Q1+Q2

q

D0q

(wqy + φpD0)h(y) dy

+

∫ 1

Q1+Q2q

[

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 pmin{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)

∫Q1+Q2

q∗

0yh(y) dy + φp

∫D0q∗

0yh(y) dy = c− wmµ. (12)

13

To derive retailers’ equilibrium order quantity, we first write retailer R1’s expected profit

function:

πR1 =

∫

D0q∗

0

[

(1− φ)pQ1

Q1 +Q2q∗y − w

Q1

Q1 +Q2q∗y

]

h(y) dy

+

∫

Q1+Q2q∗

D0q∗

[

(1− φ)pQ1

Q1 +Q2D0 − w

Q1

Q1 +Q2q∗y

]

h(y) dy

+

∫ 1

Q1+Q2q∗

[

(1− φ)pQ1

Q1 +Q2D0 − wQ1 − wm

Q1

Q1 +Q2(q∗y −Q1 −Q2)

]

h(y) dy,

(13)

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+Q2q∗y according to the proportional

allocation model. In this case, retailer R1 makes a sale of min{ Q1

Q1+Q2q∗y, Q1

Q1+Q2D0} and retains

(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-revenue-

sharing contract can coordinate the decentralized supply chain if the contract terms satisfy: φ =

1− 2wµpµ+c and wm = c

pµ+cw.

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

expected profit, and γ ∈[

max{12 , 1−

2w0µpµ+c},

pµ−cpµ+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 pay-

back-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 pD0H(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 uncer-

tainties 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” (·) to denote corresponding

variables. The centralized system’s problem can be expressed as:

maxQc

Πc =ED,Y

[

p min{D, QcY } − cQc

]

=

∫ 1

y=0

{∫ Qcy

ξ=0pξf(ξ) dξ +

∫

∞

ξ=QcypQcyf(ξ) dξ

}

h(y) dy − cQc.

(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 Q∗

c for

the centralized system satisfies:∫ 1

0yF (Q∗

cy)h(y) dy =c

p. (15)

The optimal expected profit for the centralized system is:

Π∗

c = p

∫ Q∗

c

0ξH

( ξ

Q∗

c

)

f(ξ) dξ. (16)

With uncertainty on both the demand and supply side, one might expect that two coordi-

nation 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 de-

cision 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 de-

mand. 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:

maxq

π = ED,Y

[

wmin{D, qY }+wm(qY −D)+ − cq]

.

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:

∫ 1

0yF (q∗y)h(y) dy =

c− wmµ

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−wmµ)/(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 well-

known 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:

π =

∫ 1

y=Q

q

{

wQ+ wm(qy −Q) +

∫ qy

ξ=Qφp(ξ −Q)f(ξ) dξ +

∫

∞

ξ=qyφp(qy −Q)f(ξ) dξ

}

h(y) dy

+

∫Q

q

y=0wqyh(y) dy − cq.

(18)

The following result characterizes the supplier’s optimal production quantity q∗(Q):

19

Lemma 4 Under a modified pay-back-revenue-sharing contract, for a given order size Q, if w −

wm ≥ φpF (Q), then the supplier’s optimal production quantity q∗(Q) satisfies:

(w − wm)Λ(Q

q∗

)

+ wmµ+ φp

∫ 1

Q

q∗

yF(

q∗y)

h(y) dy = c. (19)

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:

π∗ =

{

[

w − wm − φpF (Q)]

H(Q

q∗)

}

Q+ φp

∫ 1

y= Qq∗

∫ q∗y

ξ=Qξf(ξ) dξ h(y) dy. (20)

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:

Π = ED,Y

[

pmin{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∗ = Q∗

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 + φpF (Q∗),

(w − wm)Λ(

Q∗

Q∗

c

)

+ wmµ+ φp∫ 1

Q∗

Q∗

c

yF(

Q∗

cy)

h(y) dy = c.

(21)

It can be shown that conditions (21) also ensure that Q∗ is indeed a local maximum point for Π

by verifying that ∂2Π/∂Q2 < 0 at Q∗.

When the supply chain is coordinated under such a contract, the supplier’s profit (20)

becomes:

π∗ = φp

∫ 1

y=Q∗

Q∗

c

∫ Q∗

cy

ξ=Q∗

ξf(ξ) dξ h(y) dy. (22)

20

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:

(w − wm)Λ(v(wm, φ)

Q∗

c

)

+ φp

∫ 1

v(wm,φ)

Q∗

c

yF (Q∗

cy)h(y) dy = c− wmµ, (23)

where v(wm, φ) := F−1((w − wm)/(φp)) and F−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 :=

∫ 1

0yF (Q∗

cy)h(y) dy.

Corollary 3 For a given wholesale price w:

(i) The feasible range for the revenue share φ is [ wµ−cp(µ−η) , 1].

(ii) The feasible range for the pay back price wm is [w − pF (Qmax),c−wηµ−η ], where Qmax is deter-

mined by the equation:

∫ 1

Qmax/Q∗

c

y[

F (Qmax)− F (Q∗

cy)]

h(y) dy =wµ− c

p. (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,π], where

π =wµ− c

µ− η

∫ 1

0

∫ Q∗

cy

0ξf(ξ) dξh(y) dy, and

π =

∫ 1

Qmax/Q∗

c

∫ Q∗

cy

Qmax

pξf(ξ) dξh(y) dy,

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), design-

ing 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 − φpF (Q∗

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 Q∗

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 (Q∗

c , Π∗

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 = αΠ∗

c .

Step 2: Next, choose a revenue sharing parameter φ from the range (0, cpη ).

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:

wm =1

µ

[

c− φp(

F (Q∗)Λ(Q∗

Q∗

c

) +

∫ 1

Q∗

Q∗

c

yF (Q∗

cy)h(y) dy)

]

.

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 Q∗

c = 129.03 and expected profit Π∗

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

w = 5.24 has wm = 1.65 and φ = 0.6.

wm

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 repre-

23

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) pay-

back-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.

(QF - WP) (PBRS - WP)WP WP

4.08 310.36 310.38 0.004% 310.43 0.02%

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%

Table 1: Comparison between quantity flexibility and modified pay-back-revenue-sharing contract

Wholesale

price

contract

(WP)

w

Quantity flexibility

contract (QF)

Modified PB-RS

contract (PBRS)

Profit Profit

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-revenue-

sharing 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 replenish-

ment, 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 under-

produces. 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 ran-

dom 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 im-

plement. 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 set-

tings. Our paper provides a building block for models addressing these remaining interesting yet

challenging problems for supply chains with random yields.

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