Submitted to Production and Operations Management
manuscript
Managing Supply Uncertainty under Chain-to-Chain
Competition
Biying Shou
Department of Management Sciences, City University of Hong Kong, [email protected]
Erick Zhaolin Li
Faculty of Economics and Business, The University of Sydney, Australia, [email protected]
We study the competition between two supply chains that are subject to supply uncertainty. Each supply
chain consists of a supplier and a retailer. The retailers engage in a Cournot competition by determining
order quantities from their exclusive suppliers. The yields of the suppliers are random. We examine the
suppliers’ and the retailers’ decisions at three different levels. At the operational level, we demonstrate that a
retailer should order more if its competing retailer has a less reliable supply, and it should order less if its own
supply becomes less reliable. At the design level, we characterize the optimal parameters for a wholesale price
contract with linear penalty for any given supply uncertainty. At the strategic level, we show that supply
chain centralization is a dominant strategy, and it always makes the customers better off. Nevertheless, if the
supply risk is low, centralization could actually decrease the the supply chain profit, compared with the case
where both chains do not choose centralization. This results in a prisoner’s dilemma. However, if the supply
risk is high, centralization always increases supply chain profit. The benefit of supply chain centralization is
strengthened by high supply uncertainty.
Key words : Supply chain risk management, supply uncertainty, supply chain competition, game theory
History :
1.
Introduction
With advances in information technology and the increasingly tense global competition, firms are
cooperating more closely than ever to optimize the performance of their supply chains. As a result,
the traditional model of firm-to-firm competition is giving way to a new paradigm of chain-to-chain
competition (Barnes (2006), Ha and Tong (2008)). Moreover, supply uncertainty is becoming a
major concern for global supply chain management. An industrial survey conducted by Protiviti
and APICS (American Production and Inventory Control Society) showed that 66% of respondents
considered supply interruption one of their most significant concerns among all supply chain related
risks (O’Keeffe (2006)).
This paper offers a systematic examination of how to design and operate supply chains to effectively deal with supply uncertainty and chain-to-chain competition. In particular, we study the
interaction between two competing supply chains. Each chain consists of one retailer and its exclusive supplier. Both chains are subject to supply uncertainty, i.e., the supplier may not be able to
fulfill the retailer’s order due to external (e.g., snowstorm, customs delays) or internal causes (e.g.,
labor strike, machine breakdown).
We can find many real-life examples where our model of chain-to-chain competition with supply
uncertainty is applicable. For instance, in the automobile industry, competing car manufacturers often sell through independent and exclusive dealerships. Events such as labor strikes, parts
shortages, earthquakes, and fires may disrupt supply. For example, due to the recent earthquake
in Japan, Toyota was forced to cut production because of compulsory power cuts and problems
in getting the parts to factories. Akio Toyoda, president of Toyota, announced in May 2011: “In
Japan, we are currently operating at about 50%. We will mark that as the low point from which
we intend to increase production. As for production outside Japan, we are currently producing at
1
2
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
about 40%.” The supply shortage of Japanese cars have seen its impact on the car dealerships.
The BBC news reported (May 2011), “As car demand in the US, China and Asia continues to
recover, an increasing number of customers are stopping by at dealer showrooms only to find there
are not enough Japanese cars. Consequently, the Japanese car makers are losing market shares to
American, European and South Korean car companies.”
Another example to consider is from the pharmaceutical industry. Two Chinese companies
account for approximately 70% of the market for manufacturing Heparin Sodium, an anti-coagulant
extracted from healthy pig intestines and widely used in open-heart surgery. Currently, both companies source from independent suppliers. Because quality problems with pig intestines often cause
disruption to their normal supply, one of the companies is considering vertical integration. Similarly, some diary product manufacturers in China have their own farms while others procure from
independent farmers. Different supply risks and supply chain structures are likely result in different
competitive strategies.
We can also find chain-to-chain competition in many other contexts. For instance, Wu and Chen
(2005) discussed the chain-to-chain competition between the Chinese wireless service providers who
are served by exclusive equipment suppliers. McGuire and Staelin (1983) discussed competition
between fast-food chains. These supply chains also face challenges managing supply disruptions
due to equipment failure and quality problems.
In this paper, we aim at addressing the following research questions:
• How does supply uncertainty and chain-to-chain competition affect the ordering decisions?
• How to determine the wholesale price and shortage penalty in a decentralized supply chain?
• Does centralization provide a competitive advantage when dealing with competition and supply
uncertainty?
We consider three different market structures: i) both supply chains are centralized (referred to
as Centralized Competition); ii) one supply chain is centralized while the other is decentralized
(referred to as Hybrid Competition); and iii) both supply chains are decentralized (referred to as
Decentralized Competition). We derive the equilibrium decisions for each game in terms of the
retailer ordering quantity and (for a decentralized chain) the contract terms. By comparing the
three equilibria, we examine the impact of supply uncertainty and chain-to-chain competition on
the value of centralization and customer service.
The key results of this paper are summarized as follows:
• Operational level: We show that a retailer should order more if its competing retailer’s supply
becomes less reliable or if its own supply becomes more reliable. Retailer’s order sizes can be larger
or smaller compared to the benchmark setting where supply uncertainty is absent.
• Design level: At the design level, we characterize the optimal contract terms for wholesale
price with linear penalty for supply shortage.
• Strategic level: We show that supply chain centralization is the dominant strategy under supply
uncertainty and chain-to-chain competition. Moreover, in terms of expected market supply and
price, customers are always better off with centralization than without. Nevertheless, supply chain
centralization may not always result in positive gains for the supply chain itself. In fact, we show
that if supply risk is low, centralization could actually decrease supply chain profit comparing to
the case that both chains choose to be decentralized, which results in a prisoner’s dilemma. On the
other hand, if supply risk is high, centralization always increases the supply chain profit. In other
words, the benefit of supply chain centralization is strengthened by high supply uncertainty.
The remainder of this paper is organized as follows. Section 2 reviews the related literature.
Section 3 describes the model and introduces the notation. Section 4 studies the operational and
design level decisions under three different game settings: the centralized game, the hybrid game,
and the decentralized game. Section 5 compares the three equilibria from Section 4 and derives
strategic insights on supply chain profits and customer benefits. Finally, we conclude in Section 6.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
2.
3
Literature
Our study is closely related to three areas: supply uncertainty, supply chain contracting, and supply
chain competition.
The existing work on supply uncertainty can be divided into three categories: (i) the randomyield model, which models the uncertainty by assuming that the supply level is a random function
of the input level (e.g., Yano and Lee (1995), Gerchak and Parlar (1990), Parlar and Wang (1993),
Swaminathan and Shanthikumar (1999), Kazaz (2004), Babich et al. (2007), Federgruen and Yang
(2008), Wang et al. (2008), Kazaz (2008), Deo and Corbett (2009)), (ii) the stochastic lead-time
model, which models the lead-time as a random variable (see a comprehensive review in Zipkin
(2000)), and (iii) the supply disruption model, which typically models the uncertainty of a supplier
as one of two states: “up” or “down” (e.g., Arreola-Risa and DeCroix (1998), Gupta (1996), Meyer
et al. (1979), Parlar and Berkin (1991), Song and Zipkin (1996), Tomlin (2006), Snyder and Shen
(2006b), Snyder and Shen (2006a), Yang et al. (2009)). Specifically, for the supply disruption model,
the orders are fulfilled on time and in full when the supplier is “up,” and no order can be fulfilled
when the supplier is “down”. Our proposed research builds upon the random yield model.
Conflict of interest is ubiquitous in a supply chain, and contracts are widely used to resolve these
conflicts. The key task in supply contract design is to align the objectives of the various firms with
the supply chain’s overall objective, and thereby reduce the inefficiency of the supply chain. Cachon
(2003) provides an excellent review of this field. There are several papers addressing contracting and
competition simultaneously, e.g., Parlar (1988), Cachon (2001), Corbett and Karmarkar (2001),
Carr and Karmarkar (2005), Boyaci and Gallego (2004), and Ha and Tong (2008). In particular,
Carr and Karmarkar (2005) examined quantity competition under different supply chain structures
with deterministic demand. Boyaci and Gallego (2004) studied two competing supply chains where
the manufacturers and the retailers could act to affect service quality, and examined the value of
centralization given full information. They showed that centralization is the dominant strategy, but
it reduces supply chain profit, presenting a prisoner’s dilemma. Ha and Tong (2008) investigated two
competing supply chains where only the retailers could act to directly affect market competition.
They studied the value of information sharing and the corresponding contracting choices. However,
none of these papers considered supply risks in a competitive environment.
Very few papers have focused on the effects of supply uncertainty in a competitive environment.
Motivated by the case of the U.S. influenza vaccine market, Deo and Corbett (2009) examined
the interaction between supply uncertainty and a firm’s strategic decision regarding market entry.
They found that, in an equilibrium state, supply uncertainty can lead to a high number of firms
concentrating in an industry as well as a reduction in the industry output and the expected
consumer surplus. Babich et al. (2007) studied a supply chain where one retailer does business
with competing high-risk suppliers that may default during their production lead times. They show
that low supplier default correlations dampen competition between the suppliers and increase the
equilibrium wholesale prices. Therefore, the retailer prefers suppliers with highly correlated default
events, despite consequently losing the benefits of diversification. In contrast, the suppliers and
the channel prefer negatively correlated defaults. Hopp et al. (2008) investigated the impact of
regional supply disruption on competing supply chains where the firms’ strategies consist of two
stages: (i) preparation, investing in measures that facilitate quick detection of a problem prior
to a disruption, and (ii) response, post-disruption purchasing of backup capacity for components
with compromised availability. As a measure of risk, they used the expected loss of profit due to
lack of preparedness, and they found that the products that pose the greatest risk are those with
valuable market share, low customer loyalty, and relatively limited backup capacity. Furthermore,
they showed that a dominant firm in such a market should focus primarily on protecting its market
share, while a weaker firm should focus on being ready to take advantage of a supply disruption
to gain market share.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
4
Our paper differs from the above-mentioned papers in that we focus on the ordering decisions,
contract design, and centralization choices given supply uncertainty and chain-to-chain competition. We contribute to the literature by offering a systematic examination of how to design and
operate supply chains to effectively deal with supply risks and competition.
3.
The Model
To investigate the impact of supply uncertainty and chain-to-chain competition, we consider the
model presented in Figure 1.
Supplier 1
Retailer 1
Market
Supplier 2
Figure 1
Retailer 2
Model of Supply Chain Competition and Supply Uncertainty
• There are two competing supply chains that offer the same product in the market. Each
chain consists of one retailer and one supplier. All parties are risk neutral. The two supply chains,
suppliers, and retailers are indexed by i and j where i, j ∈ {1, 2}, i 6= j. The retailers compete on
quantity.
• A supplier provides the product to its retailer exclusively, and it may be subject to supply
uncertainty. If retailer i places an order of Qi , he receives αi Qi , where αi is a random variable
between 0 and 1 (inclusive). The mean and standard deviation of αi is µi and σi . We use coefficient
of variation (CV), δi = σi /µi , to measure level of uncertainty: the larger δi , the more uncertainty.
We consider the general case where the two suppliers may have different uncertainty levels.
• The product has a short life cycle, and thus the two supply chains only interact once. The
market price of the product is determined by a function p(x) = A − x, where x is the total amount
of inventory available in the market (i.e., the sum of the actual deliveries received by the two
retailers) and the positive constant A characterizes the maximum market demand. This pricing
function is widely used in quantity competition papers (e.g., in Ha and Tong (2008)). The function
p(x) = A − x reflects the fact that a larger supply leads to a lower market price.
• We include two marginal costs for each supply chain i (similar as in Deo and Corbett (2009)):
(i) C1,i per unit of production quantity, and (ii) C2,i per unit of successful delivery. If supplier i
produces an order of Qi and the yield is αi Qi , the expected total cost is E[C1,i Qi + C2,i αi Qi ] =
C1,i Qi + C2,i µi Qi . To simplify the notation, we can further define Ci = C1,i /µi + C2,i , and thus the
expected total cost is Ci Qi . To exclude the non-interesting case where neither party sells to the
market, we assume A > Ci . We consider the general case where the two chains may have different
costs.
• We assume that the retailers have no option for backup (contingent) sourcing, that the supply
uncertainty are independent of each other, and that all parties have full information about the
entire market.
The sequence of events is as follows, and both chains (i = 1, 2) act simultaneously at each stage.
1. Contract decision: supplier i offers a contract to retailer i.
2. Order decision: retailer i decides on the order quantity Qi from supplier i.
3. Uncertain supply: retailer i receives αi Qi from supplier i .
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
5
4. Quantity competition: the market is cleared based on the total realized products from both
supply chains and each party receives the payoff.
4.
Design and Operation Decisions
In this section we examine the optimal ordering and contract design decisions under three settings.
First, we derive the equilibrium for the centralized competition game between two centralized
chains. Second, we derive the equilibrium for the hybrid competition game between a centralized
chain and a decentralized chain. Finally, we derive the equilibrium for the decentralized competition
game between two decentralized chains. Comparisons between the three settings and insights into
the strategic decisions are provided in Section 5.
4.1 Centralized Competition Game
We first consider the case in which both supply chains are centralized, i.e., retailers and suppliers
are fully aligned to achieve the supply chain’s optimal performance. We first consider each chain as
a central planner (instead of considering the supplier and the retailer as two decision makers), and
analyze their quantity competition under supply disruptions. We seek to derive the order quantity,
Qi and Qj , for each supply chain at the market equilibrium.
The goal of the central planner i (i.e., the person who dictates supply chain i) is to maximize
the expected profit for supply chain i, which can be expressed as follows:
E(πi ) = E(pαi Qi − C1,i Qi − C2,i αi Qi ) = (A − Ci )µi Qi − Q2i (µ2i + σi2 ) − Qi Qj µi µj
(1)
The following theorem characterizes the unique pure strategy equilibrium and the corresponding
players’ payoffs for the Centralized Competition game. For notation convenience, we denote Ki =
A − Ci , Kj = A − Cj , and λ = Ki /Kj .
Theorem 1. The unique pure strategy Nash equilibrium of the Centralized Competition game
is (Q∗i , Q∗j ), where the optimal order quantities of supply chains are
1
2
(i) when 2(1+δ
2 ) < λ < 2(1 + δi ),
j
(ii) when λ ≤
Q∗i =
2Ki − Kj + 2δj2 Ki
,
µi (4δi2 + 4δj2 + 4δi2 δj2 + 3)
(2)
Q∗j =
2Kj − Ki + 2δi2 Kj
,
µj (4δi2 + 4δj2 + 4δi2 δj2 + 3)
(3)
1
,
2(1+δj2 )
Q∗i = 0,
(4)
Q∗j =
Kj
,
2µj (1 + δj2 )
(5)
Q∗i =
Ki
,
2µi (1 + δi2 )
(6)
Q∗j = 0.
(7)
(iii) when λ ≥ 2(1 + δi2 ),
Note that as we assume Ki > 0 and Kj > 0, we exclude the trivial case of Q∗i = Q∗j = 0.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
6
The proof is provided in the Appendix. Now consider the special case where there is no supply
uncertainty (µi = µj = 1, δi = δj = 0). First, if the costs are the same (Ci = Cj = C), the Centralized
Competition game degenerates into a standard Cournot duopoly game, and the optimal order
quantities become Q∗i = Q∗j = (A − C)/3 and the expected profits are equal to (A − C)2 /9. Second,
if the difference between the two supply chain’s costs is significant, one chain simply does not enter
the market and the other chain becomes the market monopoly. For example, if Ki ≥ 2Kj , chain i
becomes the monopoly player and its optimal order quantity becomes (A − Ci )/2.
Next, we examine the impact of supply uncertainty. In the special case when Ci = Cj and δi = δj ,
Theorem 1 echoes the findings in Lemma 1 of Deo and Corbett (2008) when there are two firms.
The order quantities of Chain i and j are identical and decrease in the yield uncertainty δ. Now
consider the more general setting where the costs and yield uncertainties of two chains are different,
we find: first, when the market has only one monopoly player (cases (ii) and (iii) in Theorem
1), the optimal order quantity and the corresponding expected profit decrease when the supply
uncertainty increases; second, when both chains sell to the market (case (i) in Theorem 1), we
show that the following is true.
1
2
Corollary 1. when both chains sell to the market (i.e., when 2(1+δ
2 < λ < 2(1 + δi )), the
j)
equilibrium order quantity and expected profit for supply chain i increase in δj and decrease in δi .
That is, supply chain i’s order quantity and expected profit increase when the supply uncertainty of
the competing chain increases, and decrease when its own supply uncertainty increases.
0.3
0.5
Expected Profit of Chain i
Expected Supply of Chain i
0.6
δi=0
0.4
δi=0.5
0.3
δi=1
0.2
0.1
0.25
δi=0
0.2
δi=0.5
0.15
δi=1
0.1
0.05
δi=2
δi=2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
0
Supply Uncertainty Level of Chain j (δj)
Figure 2
Expected supply
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
Supply Uncertainty Level of Chain j (δj)
Figure 3
Expected profit
Corollary 1 can be proven easily by examining the first order derivative. Figures 2 and 3 present
some numerical examples. For the purpose of illustration, we let Ci = Cj = C and normalize (A − C)
to 1. The horizontal axis in both figures is the uncertainty level of supply chain j, δj . In Figure 2 the
vertical axis is the expected order quantity of supply chain i, E(Qi ) = µi Qi , whereas in Figure 3 the
vertical axis is the expected profit of supply chain i. The four curves represent different uncertainty
levels for supply chain i, with δi = 0, 0.5, 1, 2, respectively. The observations are as follows:
• For a given uncertainty level δi , the expected order quantity and profit of supply chain i
increase as its competing supply chain’s uncertainty level δj increases. For example, if supply chain
i has perfect supply (i.e., δi = 0), while the uncertainty of supply chain j (δj ) increases from 1 to
2, supply chain i’s expected supply increases by 10% and the expected profit increases by 22%.
6.5
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
7
• Given a fixed uncertainty level for the competing supply chain, the expected order quantity
and profit of supply chain i decreases as its own uncertainty level increases. For example, assuming
that the uncertainty level of supply chain j is fixed at δj = 1, while the uncertainty of supply chain
i (δi ) increases from 0.5 to 1, supply chain i’s expected supply decreases by 40% and the expected
profit decreases by 42%.
• As a supply chain’s uncertainty level becomes very large, its expected supply becomes very
small, and the competing supply chain behaves more like a monopoly player.
• Q∗i ≤ (A − C)/2 = 0.5 and E(πi∗ ) ≤ (A − C)2 /4 = 0.25, i.e., the optimal order quantities and
the expected profits with supply uncertainty are upper-bounded by the order quantities and the
profits for the standard monopoly game without supply uncertainty.
The next corollary shows the impact of supply uncertainty on the expected total market supply.
Corollary 2. When both chains sell to the market, the expected total market supply at the
Nash equilibrium for the centralized Competition game is
E(αi Qi + αj Qj ) = µi Q∗i + µj Q∗j =
Ki + Kj + 2Ki δj2 + 2Kj δi2
4δi2 + 4δj2 + 4δi2 δj2 + 3
(8)
The expected total market supply decreases in δi and δj .
4.2 Hybrid Competition Game
In this section, we study the case where one supply chain is centralized while the other is not.
We are interested in discovering how the equilibrium order quantities and the profits change when
compared to the centralized game. In particular, does the centralized chain have a competitive
edge over the decentralized chain in the hybrid scenario?
Without loss of generality, we assume that supply chain i is centralized and that supply chain
j is decentralized. We use superscript “h” to denote the hybrid competition game. The expected
profit of supply chain i is
2
2
h h
E(πih ) = E(pαi Qhi − C1,i Qhi − C2,i αi Qhi ) = (A − Ci )µi Qhi − Qh2
i (µi + σi ) − Qi Qj µi µj .
(9)
For supply chain j, we assume that the supplier charges wholesale price ωj per unit successful
delivery, and pays a penalty sj per unit of shortage. This is a linear contract that is easy to
implement and is widely adopted in practice. The expected profit of retailer j is
h
E(πj,r
) = E[(p − wj )αj Qhj + sj (1 − αj )Qhj ]
2
2
h h
h
= (A − wj )µj Qhj − Qh2
j (µj + σj ) − Qi Qj µi µj + sj (1 − µj )Qj .
(10)
Define m = wj µj − sj (1 − µj ) as the supplier j’s expected profit margin for each unit ordered by the
retailer, i.e., with probability µj the supplier receives ωj for each unit of successful delivery to the
retailer and with probability 1 − µj the supplier pays sj for each unit of shortage. The expected
profit of supplier j becomes
h
E(πj,s
) = (m − Cj µj )Qhj .
(11)
The setup of the hybrid competition game is as follows.
Definition 1 (Hybrid Competition Game). A Hybrid Competition game is defined as
• Players: a set of three players: central planner i, retailer j, and supplier j,
• There are two stages in the game
— Stage 1: supplier j announces the contract term m to retailer j.
— Stage 2: central planner i and retailer j choose production quantities Qhi and Qhj simultaneously.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
8
h
h
• Payoff: the expected profits for the three players are E(πi,r
(Qhi , Qhj )), E πj,r
(Qhi , Qhj , m) , and
h
E πj,s
(Qhj , m) , respectively.
When a different contract term m is offered by supplier j in the first stage, central planner i and
retailer j play a different subgame at the second stage. For an extensive game, the most commonly
used solution concept is a subgame perfect Nash equilibrium (SPNE), which is a refinement of the
concept of a Nash equilibrium. An equilibrium is an SPNE if the players’ strategies constitute a
Nash equilibrium in every subgame.
The SPNE of the Hybrid Competition game can be characterized as follows.
Definition 2. A pure strategy SPNE (Qhi (m), Qhj (m), m∗ ) of the Hybrid Competition game
satisfies
1. Retailers compete with each other in quantities to maximize their profits for any given contract
m (not necessary optimal), i.e.,
h
h
h
Qh∗
i (m) ∈ arg max E(πi,r (Qi , Qj (m))),
Qh
i ≥0
h
h
h
Qh∗
j (m) ∈ arg max E(πj,r (Qi (m), Qj , m)).
Qh
j ≥0
2. Supplier j sets the contract term to maximize its expected profit given the choices of retailers,
i.e.,
h
(Qhj (m), m) .
m∗ ∈ arg max E πj,s
m≥0
The following lemma characterizes the equilibrium strategies for both retailers as a function of
contract term m. Similar as before, we let Ki = A − Ci , Kj = A − Cj , and λ = Ki /Kj .
Lemma 1. At a pure
Hybrid Competition game,
strategy SPNE of the
µ K
1. if Aµj − 2 δj2 + 1 µj Ki < m < Aµj − 2δj2 +2i ,
i
m − Aµj + 2Ki µj + 2Ki µj δj2
,
µi µj (4δi2 + 4δj2 + 4δi2 δj2 + 3)
2Aµj − 2m − Ki µj − 2mδi2 + 2Aδi2 µj
Qhj (m) =
.
µ2j (4δi2 + 4δj2 + 4δi2 δj2 + 3)
2. if m ≤ Aµj − 2 δj2 + 1 µj Ki ,
Qhi (m) =
(12)
(13)
Qhi (m) = 0,
(14)
Qhj (m)
(15)
Aµj − m
.
=
2 δj2 + 1 µ2j
µ K
3. if Aµj − 2δj2 +2i ≤ m ≤ Aµj ,
i
Ki
,
2µi (δi2 + 1)
h
Qj (m) = 0.
Qhi (m) =
(16)
(17)
The following theorem characterizes the equilibrium contract terms m∗ and order quantities.
Theorem 2. At an SPNE of the Hybrid Competition game, supplier j’s contract term m∗ ,
h∗
supply chain i’s order quantity Qh∗
i , and retailer j’s order quantity Qj satisfy
2
∗
i) if λ > 2δi + 2, m can be any value in [Cj µj , Aµj ],
Ki
,
2µi (1 + δi2 )
= 0.
Qh∗
i =
Qh∗
j
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
ii) if
2+2δi2
8δi2 +8δj2 +8δi2 δj2 +7
9
< λ ≤ 2δi2 + 2,
(Ki + 2Kj + 2Kj δi2 ) µj
,
4 (δi2 + 1)
2
2
7Ki − 2Kj + 8Ki δi + 8Ki δj − 2Kj δi2 + 8Ki δi2 δj2
,
=
4 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1) µi
2Kj + 2δi2 Kj − Ki
=
.
2µj (4δi2 + 4δj2 + 4δi2 δj2 + 3)
m∗ = Aµj −
Qh∗
i
Qh∗
j
iii) if
1
4δj2 +4
<λ≤
2+2δi2
2
8δi +8δj2 +8δi2 δj2 +7
,
m∗ = Aµj − 2 δj2 + 1 µj Ki ,
Qh∗
i = 0,
Ki
h∗
.
Qj =
µj
iv) if λ ≤
1
,
4δj2 +4
m∗ = Aµj −
Qh∗
i = 0,
Qh∗
j =
4
δj2
Kj µ j
,
2
Kj
.
+ 1 µj
h∗
Note that as we assume Ki > 0 and Kj > 0 we exclude the case where Qh∗
i = Qj = 0.
Similar to the Centralized Competition game, we obtain the following comparative statics.
Corollary 3. The equilibrium order quantity and expected profit for supply chain i increase
in δj and decrease in δi . That is, one supply chain’s order quantity and expected profit increase
as its competing supply chain’s uncertainty increases and decrease as its own supply uncertainty
increases.
Corollary 3 can be easily proven by checking the first order derivative. Figures 4 and 5 present
some numerical examples. For illustration purposes, we let Ci = Cj = C and normalize (A − C)
to 1. The horizontal axis in both figures represents the uncertainty level of supply chainj, δj . In
Figure 4 the vertical axis is the expected order quantity of supply chain i, whereas in Figure 5 the
vertical axis is the expected order quantity of supply chain j. The four curves represent different
uncertainty levels of supply chain i, with δi = 0, 0.5, 1, 2, respectively.
Corollary 4. At the SPNE of the Hybrid Competition game, the expected market supply
decreases in δi and δj . As a result, the expected market price increases in δi and δj .
Next we compare the equilibrium order quantities under the Hybrid Competition Game with
∗
h∗
∗
those in the Centralized Competition Game. We find that Qh∗
i > Qi and Qj < Qj . This indicates
that under the same uncertainty level, in the Hybrid Competition game the order quantity for
the centralized supply chain i increases while that for the decentralized supply chain j decreases,
when comparing with the Centralized Competition game. Figure 6 provides a numerical example.
Here we let Ci = Cj = C and δi = δj = δ and normalize (A − C) to 1. The horizontal axis is the
uncertainty level of supply chains δ, and the vertical axis is the expected order quantity of each
supply chain. The middle curve represents the order quantity of chain i and j in the Centralized
Competition game, whereas the top and bottom curves represent the order quantity of chain i
(centralized) and chain j (decentralized) in the Hybrid Competition game.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
0.6
0.5
Expected Supply of Chain j
Expected Supply of Chain i
10
δi=0
0.4
δi=0.5
0.3
δi=1
0.2
0.1
0
0
δi=2
0.25
0.2
0.15
0.1
0.05
0
0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
Supply Uncertainty Level of Chain j (δj)
Figure 4
δi=1
δi=0
δi=2
δi=0.5
1
2
3
4
5
Supply Uncertainty Level of Chain j (δj)
Expected supply of chain i
Figure 5
Expected supply of chain j
0.45
Expected Supply
0.4
Hybrid Competition, Chain i
0.35
0.3
Centralized Competition
0.25
0.2
0.15
0.1
0.05
0
0
Hybrid Competition, Chain j
0.5
1
1.5
2
2.5
3
3.5
Level of Supply Uncertainty
Figure 6
Expected supply of chain i
We have similar results for the expected profits of supply chains. More detailed discussions on
the comparison of expected market supplies and profits of different games are provided in Section
5.
4.3 Decentralized Competition Game
Assume that supply chain i ∈ {1, 2} adopts a linear contract with a wholesale price ωi and a penalty
si for failed delivery. We use superscript “u” to indicate that both chains are decentralized. Define
mi = µi ωi − (1 − µi )si .
Definition 3 (Decentralized Competition Game). A Decentralized Competition game is
defined as
• Players: a set of four players: retailer i, retailer j, supplier i, and supplier j.
• There are two stages in the game
— Stage 1: two suppliers announce the contract term mi and mj simultaneously.
— Stage 2: two retailers choose production quantity Qui and Quj simultaneously.
u
u
• Payoff: theexpected revenues for the four players are E(πi,r
(Qui , Quj , mi )), E πj,r
(Qui , Quj , mj ) ,
u
u
E πi,s
(Qui , mi ) , and E πj,s
(Quj , mj ) , respectively. The payoff functions are defined similarly as
in eqs. (10) and (11).
6.5
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
11
The SPNE of the Decentralized Competition game can be characterized as follows.
Definition 4. A pure strategy SPNE (Qui (mi , mj ), Quj (mi , mj ), m∗i , m∗j ) of the game between
two decentralized supply chains satisfies:
1. Retailers compete with each other in quantities to maximize their profits for any given contracts (mi , mj ) (not necessary optimal), i.e.,
u
u
Qui (mi , mj ) ∈ arg max
E
π
Q
,
Q
(m
,
m
),
m
,
i,r
i
j
i
i
j
u
Qi ≥0
2. Suppliers compete with each other in contract terms to maximize their profits given the
choices of retailers, i.e.,
m∗i ∈ arg max mi Qui (mi , m∗j ),
mi ≥0
The following results characterize the equilibrium strategies of the retailers and the suppliers.
For expositional simplicity, we let Bi = Aµi − mi and Bj = Aµj − mj .
Lemma 2. At a pure strategy SPNE of the Hybrid Competition game,
2
2Bi (µ2
B µi µj
j +σj )
1. if 2(µi2 +σ
,
2 ) < Bj <
µi µj
i
i
2Bi σj2 − Bj µi µj + 2Bi µ2j
4σi2 σj2 + 4σi2 µ2j + 4σj2 µ2i + 3µ2i µ2j
2Bj σi2 − Bi µi µj + 2Bj µ2i
Quj =
4σi2 σj2 + 4σi2 µ2j + 4σj2 µ2i + 3µ2i µ2j
Qui =
2. if Bj ≥
3. if Bj ≤
2
2Bi (µ2
j +σj )
µi µj
(18)
(19)
,
Qui = 0
(20)
Quj
Bj
=
2
2(µj + σj2 )
(21)
Bi
+ σi2 )
(22)
Bi µi µj
2 ,
2(µ2
i +σi )
Qui =
Quj
2(µ2i
=0
(23)
Similar as before, we define Ki = A − Ci , Kj = A − Cj , and λ = Ki /Kj .
Theorem 3. At the SPNE of the Decentralized Competition game, supplier i (i = 1, 2) chooses
the contract terms m∗i (equivalently Bi∗ ) and order quantity Qu∗
i as follows:
i) if
2+2δi2
2
8δi +8δj2 +8δi2 δj2 +7
<λ<
8δi2 +8δj2 +8δi2 δj2 +7
2δj2 +2
,
8Ki µi + 2Kj µi + 8Ki µi δi2 + 8Ki µi δj2 + 2Kj µi δi2 + 8Ki µi δi2 δj2
,
16δi2 + 16δj2 + 16δi2 δj2 + 15
2Ki µj + 8Kj µj + 2Ki µj δj2 + 8Kj δi2 µj + 8Kj µj δj2 + 8Kj δi2 µj δj2
Bj∗ =
,
16δi2 + 16δj2 + 16δi2 δj2 + 15
2 7Ki − 2Kj + 8Ki δi2 + 8Ki δj2 − 2Kj δi2 + 8Ki δi2 δj2 δj2 + 1
,
Qu∗
=
i
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3 µi
2 7Kj − 2Ki − 2Ki δj2 + 8Kj δi2 + 8Kj δj2 + 8Kj δi2 δj2 (δi2 + 1)
Qu∗
=
.
j
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3 µj
Bi∗ =
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
12
ii) if λ <
1
, Bi∗
4(1+δj2 )
can be any value in (0, Ki µi ],
Kj µ j
Bj∗ =
,
2
u∗
Qi = 0,
Kj
.
Qu∗
j =
4µj (1 + δj2 )
iii) if
1
4(1+δj2 )
≤λ≤
2δi2 +2
,
8δi2 +8δj2 +8δi2 δj2 +7
Bi∗ = Ki µi ,
Bj∗ = 2Ki µj + 2Ki µj δj2 ,
Qu∗
i = 0,
Ki
.
Qu∗
j =
µj
iv) if
8δi2 +8δj2 +8δi2 δj2 +7
2+2δj2
≤ λ ≤ 4(1 + δi2 ),
Bi∗ = 2Kj µi + 2Kj µi δi2 ,
Bj∗ = Kj µj ,
Kj
Qu∗
,
i =
µi
u∗
Qj = 0.
v) if λ > 4(1 + δi2 ), Bj∗ can be any value in (0, Kj µj ],
Ki µ i
,
2
Ki
=
,
4µi (1 + δi2 )
= 0.
Bi∗ =
Qu∗
i
Qu∗
j
u∗
Note that as we assume Ki > 0 and Kj > 0 we exclude the case where Qu∗
i = Qj = 0.
Similar to the Centralized Competition and Hybrid Competition games, we obtain the following
comparative statics.
Corollary 5. At the SPNE of the decentralized Competition game when both retailers sell to
the market, the equilibrium order quantity and expected profit for supply chain i increase with δj
and decreases with δi . That is, one supply chain’s order quantity and expected profit increase as its
competing supply chain’s uncertainty increases, and its order quantity and expected profit decrease
as its own supply uncertainty increases.
Corollary 6. At the SPNE of the Decentralized Competition game when both retailers sell to
the market,, the expected market supply increases in δi and δj . As a result, the expected market
price decreases in δi and δj .
5.
Strategic Decisions
In the previous section, we derive the optimal ordering decision and linear contract design (for
decentralized chain) for the Centralized, Hybrid and Decentralized Competition games. In this
section we consider the strategic-level question: with supply uncertainty and chain-to-chain competition, is there any incentive for a supply chain to become centralized? What is the impact of
supply uncertainty on the benefit/loss from supply chain centralization?
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
13
First, we examine whether a supply chain has an incentive to become centralized when both
chains sell to the market. Based on the equilibrium order quantities we have derived under each
game when both chains sell to the market, we can calculate the corresponding expected pay-offs
as follows.
Under the Centralized Competition game:
2
2 2
+
1)
K
−
2K
−
2K
δ
(δ
j
i
i
i
j
E(πi∗ ) =
(24)
2
4δi2 + 4δj2 + 4δi2 δj2 + 3
2
δj2 + 1 (Ki − 2Kj − 2Kj δi2 )
∗
E(πj ) =
(25)
2
4δi2 + 4δj2 + 4δi2 δj2 + 3
Under the Decentralized Competition game:
2
2 δj2 + 1 6δi2 + 6δj2 + 6δi2 δj2 + 5 2Kj − 7Ki − 8Ki δi2 − 8Ki δj2 + 2Kj δi2 − 8Ki δi2 δj2
u
E(πi ) =
(26)
2
2
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3
2
2 (δi2 + 1) 6δi2 + 6δj2 + 6δi2 δj2 + 5 2Ki − 7Kj − 8Kj δj2 − 8Kj δi2 + 2Ki δj2 − 8Kj δi2 δj2
u
E(πj ) =
(27)
2
2
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3
Under the Hybrid Competition game where chain i is centralized and chain j is decentralized:
2
2Kj − 7Ki − 8Ki δi2 − 8Ki δj2 + 2Kj δi2 − 8Ki δi2 δj2
h
(28)
E(πi ) =
2
16 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
2
6δi2 + 6δj2 + 6δi2 δj2 + 5 (2Kj − Ki + 2Kj δi2 )
h
(29)
E(πj ) =
2
8 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
Under the Hybrid Competition game where chain j is centralized and chain i is decentralized:
2
6δi2 + 6δj2 + 6δi2 δj2 + 5 2Ki − Kj + 2Ki δj2
h0
E(πi ) =
(30)
2
8 4δi2 + 4δj2 + 4δi2 δj2 + 3 δj2 + 1
2
2
2
2 2 2
2K
−
7K
−
8K
δ
−
8K
δ
+
2K
δ
−
8K
δ
δ
0
i
j
j
j
i
j
j
i
j
i
j
(31)
E(πjh ) =
2
16 4δi2 + 4δj2 + 4δi2 δj2 + 3 δj2 + 1
Table 1
Payoff Table for Strategic Choice of Integration
Supply Chain j
centralized
decentralized
Supply centralized
E(πi∗ ), E(πj∗ ) E(πih ), E(πjh )
0
0
Chain i decentralized E(πih ), E(πjh ) E(πiu ), E(πju )
Table 1 summarizes these results. We compare these expected profits under different competition
games and obtain the following theorem.
0
Theorem 4. When both chains sell to the market, we find E(πih ) > E(πiu ), E(πi∗ ) > E(πih ),
0
E(πjh ) > E(πju ), and E(πj∗ ) > E(πjh ). Hence, given the competing supply chain’s fixed strategic
decision (to centralize or not), a supply chain is always better off by choosing to centralize. That is,
centralization is the dominant strategy for both supply chains. However, centralization may or may
not result in positive gains for each supply chain, as the payoff difference between the centralized
and decentralized games, E(πi∗ ) − E(πiu ), can be either negative or positive.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
14
The proof is provided in the Appendix. If we consider the choice of integration (i.e., centralization) as a strategic level game, Theorem 4 implies that integration would be a unique Nash
Equilibrium. However, a prisoners’ dilemma may occur.
To further understand the implications of Theorem 4, let us consider the special case with
symmetric supply chains, i.e., δi = δj = δ and Ci = Cj = C. Let K = A − C. It can be verified that
the expected profits for supply chain i and j under the Centralized Competition game become
E(πi∗ ) = E(πj∗ ) =
(δ 2 + 1) K 2
2 , R.
(2δ 2 + 3)
To simplify exposition, we define it as “R”. The expected supply chain profits under the Decentralized Competition game become
E(πiu ) = E(πju ) =
2K 2 (12δ 2 + 6δ 4 + 5) (δ 2 + 1) 2 (12δ 2 + 6δ 4 + 5)
=
R.
2
2
2
(4δ 2 + 3) (2δ 2 + 3)
(4δ 2 + 3)
The expected supply chain profits under the Hybrid Competition game are as follows.
2
2
E(πih )
=
0
E(πjh )
0
(4δ 2 + 5)
(4δ 2 + 5) K 2
=
=
2
2 R,
16 (2δ 2 + 3) (δ 2 + 1) 16 (δ 2 + 1)
E(πjh ) = E(πih ) =
(12δ 2 + 6δ 4 + 5) K 2 (12δ 2 + 6δ 4 + 5)
=
R.
2
2
8 (2δ 2 + 3) (δ 2 + 1)
8 (δ 2 + 1)
Here “h” denotes the case when chain i is centralized and chain j is not, whereas “h0 ” denotes the
case when chain j is centralized and chain i is not.
These results are summarized in Table 2. Figure 7 presents some numerical examples, where the
x axis is level of supply uncertainty δ, the y axis is expected supply chain profit, and the three
curves represent the Centralized, Hybrid and Decentralized Competition, respectively. Note that
chain i is the centralized one in the Hybrid Competition game.
We can see that a decentralized supply chain i is better off by moving toward centralization
2
+6δ 4 +5)
if supply chain j is centralized, since (12δ
< 1. Furthermore, a decentralized supply chain
8(δ 2 +1)2
i is better off by moving toward centralization if the supply chain j remains decentralized, since
2(12δ 2 +6δ 4 +5)
4(δ 2 +5)2
>
. Hence, centralization is the dominant strategy for supply chain i. A similar
2
2
2
16(δ +1)
(4δ2 +3)
argument is true for supply chain j, in other words, centralization is the dominant strategy for
both supply chains.
Table 2
Payoff Table for Strategic Choice of Integration with Symmetric Chains
Supply Chain j
centralized
decentralized
2
(4δ +5)
(12δ2 +6δ4 +5)
Supply centralized
R, R
R
2 R,
2
2
16(δ +1)
8(δ 2 +1)
2
2(12δ 2 +6δ 4 +5)
2(12δ 2 +6δ 4 +5)
(12δ2 +6δ4 +5)
(4δ2 +5)
Chain i decentralized
R,
R
R,
R
2
2
2
2
8(δ 2 +1)
16(δ 2 +1)
(4δ2 +3)
(4δ2 +3)
2
√
Furthermore, we can show that when δ < 22 a chain receives less profit in the centralized game
compared with the decentralized game. This is the case of prisoner’s dilemma, i.e., both supply
chains are worse off under the centralized scenario although centralization is the dominated strategy. Boyaci and Gallego (2004) first discovered the phenomenon of the prisoner’s dilemma in supply
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
15
chain competition, assuming that there is no supply uncertainty. Here we generalize√ their findings and show that it still exists with low levels of supply risk. However, when δ > 22 , we have
2(12δ 2 +6δ 4 +5)
< 1, i.e., the profit for each chain in the centralized competition game is higher than
2
(4δ2 +3)
that in the decentralized competition game. Hence, if the level of supply uncertainty is high, it
is better for the supply chain to centralize since it aligns the intra-chain objectives and results
in higher gain. In other words, the benefit of supply chain centralization is strengthened by high
supply risks.
0.18
Expected Profit
0.16
Hybrid Competition, Chain i
Decentralized Competition
0.12
Centralized Competition
0.08
0.04
Hybrid Competition, Chain j
0
0
0.5
1
1.5
2
2.5
3
3.5
Level of Supply Uncertainty
Figure 7
Comparison of the expected supply chain profits
Now a related question is: do individual retailer and supplier in a decentralized chain have the
right incentives to integrate?
We may consider a revenue sharing contract where the supplier’s income consists of three parts:
ω for each unit ordered by the retailer, a percentage of retail revenue for each successfully delivered
order, and a penalty for each unit that the supplier fails to deliver. Let φ ∈ [0, 1] be the fraction
of supply chain revenue the retailer keeps, so the supplier earns the remaining (1 − φ). The profit
functions for the retailer, the supplier, and the central planner are
E(πr ) = E[(φp − ω)Qα + SQ(1 − α)],
E(πs ) = E {[(1 − φ)p + ω]Qα − SQ(1 − α)]},
E(π) = E(πr ) + E(πs ) = E(pQα).
Here α is a random variable that represents yield uncertainty. Assume that α is generally distributed
in (0,1] with average value of µ. When ωµ − (1 − µ)S = 0, we have
E(πr ) = E[φpQI] = φE(π),
i.e., the profit of the retailer is an affine function of the central planner. Hence, such a revenue
sharing contract coordinates the supply chain. Based on this finding, we can show that the following
is true.
Lemma 3. For any given linear price and penalty contract, there always exists a coordinating
revenue-sharing contract that makes both the supplier and retailer better off.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
16
The proof for Lemma 3 is provided in the Appendix. Next, we examine customer service in terms
of expected total market supply and price under the Centralized, the Hybrid, and the Decentralized
Competition games. From the customers’ point of view, a higher total expected market supply
means a lower expected market price, and thus a better customer service.
Theorem 5. When both chains sell to the market, the centralized competition game provides
the best service to customers in terms of total expected market supply, the hybrid competition game
provides the second, and the decentralized competition game provides the worst service, i.e.,
µi Q∗i + µj Q∗j > µi Qhi + µj Qhj > µi Quj + µj Quj .
(32)
The proof is provided in the Appendix. Theorem 5 indicates that customers always benefit from
supply chain centralization with more supplies and a lower price. Figure 8 provides a numerical
example, where we assume that the two supply chains have the same costs and the same levels of
supply uncertainty. The horizontal axis is the level of supply uncertainty. The three curves represent
the expected total market supply under the Centralized, Hybrid, and Decentralized Competition
games, respectively.
Expected Total Market Supply
0.7
0.6
Centralized Competition
0.5
0.4
Hybrid Competition
0.3
0.2
0.1
0
0
Decentralized Competition
0.5
1
1.5
2
2.5
3
3.5
Level of Supply Uncertainty
Figure 8
6.
Comparison of the expected total market supplies
Conclusion
In this paper, we provide a systematic examination of how to design and operate supply chains to
effectively deal with supply uncertainty and competition for short life-cycle products. We derive
the equilibrium decisions for ordering decisions and contract terms for the Centralized, Hybrid,
and Decentralized competition games. By comparing these three scenarios, we obtain the following
managerial insights on supply chain management with the presence of competition and supply
uncertainty.
First, at the operational level we show that a retailer should order more if its competing retailer’s
supply becomes less reliable, and order less if its own supply becomes less reliable. Retailer’s order
sizes can be larger or smaller compared to the case where supply uncertainty is absent.
Second, at the design level we characterize the optimal terms for a wholesale price contract
with linear penalty for any given level of supply uncertainty. We also show that, for such a linear
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
17
contract there exists a coordinating revenue management contract which can make both supplier
and retailer better off.
Third, at the strategic level we show that supply chain centralization is always the dominant
strategy under supply uncertainty and chain-to-chain competition. Nevertheless, depending on the
level of supply uncertainty, centralization efforts may or may not result in positive gains for each
supply chain. For example, given competition between two symmetric supply chains, centralization
efforts guarantee a positive gain for each supply chain when the supply is highly uncertain. However,
when the supply is sufficiently reliable, centralization reduces the profit for each supply chain,
which results in a prisoner’s dilemma. In other words, the benefit of supply chain centralization
is strengthened by high supply uncertainty. Furthermore, centralization always benefits customers
by giving them a lower expected market price.
The above findings are derived through closed-form analysis, facilitated by several simplifying
assumptions. We can extend our analysis in several directions. First, we can consider contingent
sourcing, i.e., after a retailer observes his supplier’s status, he may purchase from another reliable
source or from the spot market. Second, we may explore the effect of correlated supply disruptions,
i.e., a severe crisis could disrupt both chains simultaneously. Third, suppliers and retailers may
have private information about demand and reliability, and level of supply uncertainty could be
an endogenous decision. Incorporating these factors will add extra complexity to the analysis and
make the closed-form results very tedious and difficult to obtain. Finally, we may study different
types of games (e.g., Bertrand competition) and different type of contracts (e.g. with nonlinear
price and penalty cost).
Appendix
Proof of Theorem 1. Under the Centralized Competition Game, the expected profits for chain i (i = 1, 2;
j 6= i) are:
E(πi ) = E(pαi Qi − C1i Qi − C2i αi Qi ) = (A − Ci )µi Qi − Q2i (µ2i + σi2 ) − Qi Qj µi µj ,
which is a concave function in Qi . The first-order conditions for each chain are:
(A − Ci )µi − 2Qi (µ2i + σi2 ) − Qj µi µj = 0
(A − Cj )µj − 2Qj (µ2j + σj2 ) − Qi µi µj = 0
Solving these equations, we obtain the equilibrium production quantities as shown in Theorem 1.
Proof of Lemma 1. Under the Hybrid Competition game, the expected profit for chain i (centralized)
becomes:
E(πih ) = Aµi Qhi − Qhi 2 (µ2i + σi2 ) − Qhi Qhj µi µj − Ci Qhi .
The expected profits for retailer and supplier in chain j (decentralized) are respectively:
h
E(πj,r
) = Aµj Qhj − Qhj 2 (µ2j + σj2 ) − Qhi Qhj µi µj − mQhj ,
E(πj,s ) = (m − Cj µj )Qhj ,
Now consider the ordering decisions for any given m. The first-order conditions are:
Aµi − 2Qhi (µ2i + σi2 ) − Qhj µi µj − Ci µi = 0
Aµj − 2Qhj (µ2j + σj2 ) − Qhi µi µj − m = 0
Solving these equations, we find the equilibrium order quantities for any given m, as shown in Lemma 1.
Proof of Theorem 2. Supplier j’s optimal choice of m is to maximize E(πj,s ) = (m − Cj µj )Qj for
Cj µj ≤ m ≤ Aµj . Let Ki = A − Ci , Kj = A − Cj , λ = Ki /Kj , Gi = Aµi − Ci µi , Gj = Aµj − m. Because Gj
and m are one-to-one matching, we seek the optimal Gj in [0, Kj µj ] that maximizes the supplier’s profit
E(πj,s ) = (Aµj − Gj − Cj µj )Qj .
Based on the results of Lemma 1, we discuss the following three cases to derive supplier j’s optimal decision,
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
18
G µ µ
Case A: if Kj µj ≤ 2(σi2 +i µj2 ) , i.e., λ ≥ 2δi2 + 2, we only need to consider scenario 3) in the first stage (Lemma
i
i
1). Gj can be any value in (0, Kj µj ], Qi = 2(µ2G+i σ2 ) , Qj = 0.
Gi µi µj
Case B: if
2(σi2 +µ2
)
i
≤ Kj µj <
2
2Gi (µ2
j +σj )
µi µj
i
i
,i.e.,
1
2δj2 +2
≤ λ < 2δi2 + 2, we need to compare supplier’s expected
G µ µ
profit under scenarios 1) and 3) in the first stage. We first consider scenario 1) where 2(σi2 +i µj2 ) < Gj < Kj µj .
i
i
It is easy to show that E(πj,s ) is a concave function w.r.t Gj . Solving the first order condition, we obtain
(Ki + 2Kj + 2Kj δi2 ) µj
.
4(δi2 + 1)
Gj =
When λ < 2δi2 + 2, it satisfies
Gi µi µj
2(σi2 +µ2
)
i
< Gj < Kj µj . Acoordingly,
Qj =
2Kj + 2δi2 Kj − Ki
,
2µj (4δi2 + 4δj2 + 4δi2 δj2 + 3)
− 1 2
− 1
2
1
4δi2 + 4δj2 + 4δi2 δj2 + 3
δi + 1
2Kj − Ki + 2Kj δi2 .
8
Because the expected profit in scenario 3) is 0, obviously scenario 1) is a better choice. This means that if
−1
1
≤ λ < 2δi2 + 2, Gj = 14 (δi2 + 1) (Ki + 2Kj + 2Kj δi2 ) µj .
2δ 2 +2
E(πj,s ) =
j
Case C: Kj µj ≥
2
2Gi (µ2
j +σj )
µi µj
, i.e., λ ≤
1
,
2δj2 +2
we need to compare supplier’s expected profit under scenarios
2G (µ2 +σ 2 )
G
1), 2) and 3) in the first stage. First consider case 2) where i µijµj j ≤ Gj ≤ Kj µj , Qi = 0, Qj = 2(µ2 +j σ2 )
j
j
.It’s straightforward to show that E(πj,s ) is a concave function w.r.t Gj . Solving the first order condition,
we get
Kj µj
Gj =
.
2
There are two cases to discuss:
Kj2
2G (µ2 +σ 2 )
(A−Cj )µj
(K µ
Aµ −C µ
.
• If λ ≤ 4δ21+4 , then
≥ i µijµj j . Hence Gj = 2j j , Qj = 4σj2 +4jµ2j , E(πj,s ) = 8δ2 +8
2
j
2G (µ2 +σ 2 )
(A−C )µ
1
j
i
j
j
j
j
< λ ≤ 2δ21+2 then
<
. Hence Gj =
2
µi µj
j
2
Kj − 2Ki − 2Ki δj Ki .
Next consider scenario 1). Based on the first-order condition, we have
• If
Gj =
j
2
2Gi (µ2
j +σj )
4δj2 +4
µi µj
j
, Qj =
Gi
µi µj
, E(πj,s ) =
−1
1 2
δi + 1
Ki + 2Kj + 2Kj δi2 µj
4
G µ µ
2G (µ2 +σ 2 )
We need to check whether the solution falls into the feasible region 2(σi2 +i µj2 ) ≤ Gj ≤ i µijµj j . There are
i
i
two cases to discuss:
2
2+2δ
• if 8δ2 +8δ2 +8iδ2 δ2 +7 ≤ λ ≤ 2δ21+2 , then the solution falls into the feasible region, hence Gj =
j
i j
i
j
−1 2
2
−1
−1
1
1
1 −Ki +2Kj +2δi Kj
2
2
2
2
2 2
(δi + 1)
+
1)
(K
+
2K
+
2K
δ
)
µ
(δ
j , Qj = 2µ 4δ 2 +4δ 2 +4δ 2 δ 2 +3 , E(πj,s ) = 8 4δi + 4δj + 4δi δj + 3
i
j
j i
i
4
j
i
2
j
i j
(2Kj − Ki + 2Kj δi2 ) .
2+2δ 2
• if λ ≤ 8δ2 +8δ2 +8iδ2 δ2 +7 , then the solution is smaller than the lower bound of the feasible region, hence
j
i j
i
2
2Gi (µ2
j +σj )
Gj =
,Qj = µGi µij , E(πj,s ) = Kj − 2Ki − 2Ki δj2 Ki .
µi µj
Because the expected profit in scenario 3) is 0, we only need to compare scenarios 1) and 2). We find
• When λ ≤ 4δ21+4 ,the expected profits under scenario 2 is larger than that under scenario 1. Thus Gj =
j
(A−Cj )µj
2
, Qj =
1
4δj2 +4
2
2
2Gi (µj +σj )
• When
Gj =
Aµj −Cj µj
4σj2 +4µ2
j
µi µj
• When
Thus,
.
≤λ≤
, Qj =
2+2δi2
,the
8δi2 +8δj2 +8δi2 δj2 +7
Gi
µi µj
optimal solution under scenarios 1 and 2 are the same, thus
.
2+2δi2
8δi2 +8δj2 +8δi2 δj2 +7
<λ≤
1
,
2δj2 +2
Gj =
the expected profit in scenario 1 is larger than that in scenario 2.
−1
1 2
δ +1
Ki + 2Kj + 2Kj δi2 µj ,
4 i
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
Qi =
19
7Ki − 2Kj + 8δi2 Ki − 2δi2 Kj + 8δj2 Ki + 8δi2 δj2 Ki
,
16µi δi4 δj2 + 16µi δi4 + 32µi δi2 δj2 + 28µi δi2 + 16µi δj2 + 12µi
Qj =
2Kj + 2δi2 Kj − Ki
.
2µj (4δi2 + 4δj2 + 4δi2 δj2 + 3)
Summarizing cases A, B and C, we derive the SPNE of Gj , Qi and Qj , as shown in Thoerem 2.
Proof of Lemma 2. Under the Decentralized Competition game, the expected profit of retailer i(i = 1, 2)
is:
E(πi,r ) = Aµi Qui − Qui 2 (µ2i + σi2 ) − Qui Quj µi µj − mi Qui .
For any given mi , retailer i wants to find the optimal ordering quantity to maximize its expected profit. It’s
easy to show that the expected profit function is concave in Qi . The first-order condition leads to:
Aµi − 2Qui (µ2i + σi2 ) − Quj µi µj − mi = 0.
For exposition simplicity, we define Bi = Aµi − mi , where i = 1, 2. Thus,
Bi − 2Qui (µ2i + σi2 ) − Quj µi µj = 0.
Solving the two first-order conditions, we obtain the equilibrium ordering quantities Quj and Quj for any given
Bj and Bj , as shown in Lemma 2.
Proof of Theorem 3. Consider the suppliers’ decision for decentralized game. Supplier i’s optimal choice
of mi to maximize E(πi,s ) = (mi − Ci µi )Qi , or equivalently, to maximize E(πi,s ) = (Aµi − Bi − Ci µi )Qi for
0 ≤ Bi ≤ (A − Ci )µi . Based on the results of Lemma 2, we charaterize the best-response functions of Bj and
Bj and then compute SPNE.
Best-response function of Bj :
we first derive the best-response function of Bj for any given Bi . There are three possible cases.
µ Bi
B µ µ
, we only need to consider scenario 3) in the first stage. Bj can be
Case A: if Kj µj ≤ 2(σi2 +i µj2 ) = 2µ jδ2 +1
)
i( i
i
i
Bi
any value in (0, Kj µj ], Qi = 2(µ2 +σ2 ) , Qj = 0.
i
i
2
2(δj2 +1)µj Bi
2Bi (µ2
B µ µ
µ Bi
j +σj )
Case B: if 2(σi2 +i µj2 ) ≤ Kj µj ≤
≤ K j µj ≤
, i.e., 2µ jδ2 +1
, we need to compare the
µ
µ
µi
i
j
)
i( i
i
i
expected profits under scenarios 1) and 3) in the first stage to decide the the optimal choice of Bj . We
µ Bi
< Bj < Kj µj . It is easy to show that E(πj,s ) is a
first consider scenario 1) in the first stage where 2µ jδ2 +1
)
i( i
concave function w.r.t Bj . The first-order condition leads to:
(Aµj − Bj − Cj µj )(
2Bj σ 2 − Bi µi µj + 2Bj µ2i
2σi2 + 2µ2i
)− 2 2 i 2 2
= 0.
2 2
2 2
2 2
4σi σj + 4σi µj + 4σj µi + 3µi µj
4σi σj + 4σi µj + 4σj2 µ2i + 3µ2i µ2j
2
2
The solution is
Bj =
We can show that this solution satisfies
Qj =
(Bi + 2Kj µi + 2Kj µi δi2 ) µj
.
4 (δi2 + 1) µi
µj Bi
2µi (δi2 +1)
(33)
< Bj < Kj µj . Accordingly,
−1
1 −1
1
4δi2 + 4δj2 + 4δi2 δj2 + 3
2Kj µi − Bi + 2Kj µi δi2 µ−
i µj ,
2
2 2
−1 2
−1
1
4δi2 + 4δj2 + 4δi2 δj2 + 3
δi + 1
2Kj µi − Bi + 2Kj µi δi2 µ−
i .
8
Because the expected profit for supplier j in case 3) is 0, obviously case 1) is better, which means that if
2(δj2 +1)µj Bi
(Bi +2Kj µi +2Kj µi δi2 )µj
µj Bi
≤
K
.
,
B
j µj ≤
j =
2
µ
2µi (δi +1)
4(δi2 +1)µi
i
2
2
2
δ
+1
µ
B
2Bi (µ2
+
σ
)
(
)
j
i
j
j
j
=
, we need to compare the expected profits of supplier j under
Case C: Kj µj ≥
µi µj
µi
scenarios 1), 2) and 3) in the first stage to decide the optimal choice of Bj . First consider scenario 2) where
2(δj2 +1)µj Bi
B
≤ Bj ≤ Kj µj , Qi = 0, Qj = 2(µ2 +j σ2 ) . It’s straightforward to show that E(πj,s ) is a concave function
µi
E(πj,s ) =
j
w.r.t Bj . Let
d
dBj
E(πj,s ) = 0. We get Bj =
j
Kj µj
2
. We need to consider two cases:
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
20
Kj µj
2
2Bi (µ2
j +σj )
Kj µj
K2
K µ
j j
j
, Qj = 4σ2 +4
, E(πj,s ) = 8δ2 +8
.
µ2
j
j
j
2
2
2
+1
µ
B
2
δ
2
B
(
µ
+
σ
)
) j i
(
K µ
2
, Qj = µBi µi j , E(πj,s ) = µ−
• If j2 j < i µijµj j ,then Bj = j µi
Kj µi − 2Bi − 2Bi δj2 Bi .
i
Next we consider scenario 1) in the first stage. We need to check whether the FOC solution expressed by
2B (µ2 +σ 2 )
(Bi +2Kj µi +2Kj µi δi2 )µj
B µ µ
B µ µ
Eq.(33), Bj =
, satisfies 2(µi2 +i σj2 ) < Bj < i µijµj j . Because 2(µi2 +i σj2 ) < Kj µj , we find
4(δi2 +1)µi
i
i
i
i
1
1
B
µ
µ
−
1
i i j
Bi µi µj + 2Kj σi2 µj + 2Kj µ2i µj −
=
σi2 + µ2i
2Kj σi2 − Bi µi + 2Kj µ2i µj > 0.
2
2
2
2
4σi + 4µi
2(µi + σi ) 4
• If
2
≥
µi µj
However, to have
, then Bj =
2
2Bi (µ2j + σj2 )
1
2
2
B
µ
µ
+
2K
σ
µ
+
2K
µ
µ
−
> 0,
i
i
j
j
j
j
j
i
i
4σi2 + 4µ2i
µi µj
it should satisfy
Bi >
2Kj µ3i µ2j + 2Kj σi2 µi µ2j
.
8σi σj2 + 8σi2 µ2j + 8σj2 µ2i + 7µ2i µ2j
2
Hence we need3 to
consider
two cases:
2
2
2Kj µi µ2
K µ µ2
j +2Kj σi µi µj
1
• If 8σ2 σ2 +8
< Bi < 2(µj2 +i σj2 ) , then Bj = 4σ2 +4
(Bi µi µj + 2Kj σi2 µj + 2Kj µ2i µj ) , E(πj,s ) =
2
2
2
2
σi µj +8σj µi +7µ2
µ2
µ2
i j
i j
j
j
i
i
−1
−1
2
2
1
4δi2 + 4δj2 + 4δi2 δj2 + 3
(δi2 + 1) (2Kj µi − Bi + 2Kj µi δi2 ) µ−
i .
8
2
2
2
2
2
2
B
(
µ
+
σ
)
2Kj µ3
µ
+2
K
σ
µ
µ
i j
2
, then Bj = i µijµj j ,Qj = µBi µi j , E(πj,s ) = µ−
Kj µi − 2Bi − 2Bi δj2 Bi .
• If Bi < 8σ2 σ2 +8σi 2 µj 2 +8σj2 µi2 +7
i
µ2
µ2
i j
i j
j i
i j
Because the expected profit of supplier j under scenario 3) is always 0 and that under scenarios 1 and 2
are positive, we only need to compare scenarios 1 and 2. We find:
2B (µ2 +σ 2 )
K µ µ2
(A−Cj )µj
• when
≥ i µijµj j , i.e., Bi ≤ 4(µj2 +i σj2 ) , the expected profit of supplier j under scenario 2) is
2
j
j
larger than that under scenario 1), therefore the optimal decision is: Bj =
• when
Kj µi µ2
j
+σj2 )
4(µ2
j
≤ Bi <
Kj µj
2
, Qj =
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
2 µ2 +7µ2 µ2 , the
+8
σ
8σi2 σj2 +8σi2 µ2
i j
j i
j
2
2Bi (µ2
Bi
j +σj )
and 2) are the same, thus Bj =
µi µj
2K µ3 µ2 +2K σ 2 µ µ2
Kj µj
4σj2 +4µ2
j
, E(πj,s ) =
Kj2
8δj2 +8
.
expected profit and choice of Bj under scenarios 1)
2
, Qj = µi µj , E(πj,s ) = µ−
Kj µi − 2Bi − 2Bi δj2 Bi .
i
2(σ 2 +µ2 )K
j i j
j i i j
≤ Bi ≤ i µi i j , the expected profit in scenario 1 is larger than that
• When 8σ2 σ2 +8
µ2
+7µ2
+8σj2 µ2
σi2 µ2
i j
i
j
i j
in scenario 2. Thus,
− 1
1
1 2
δ +1
Bi + 2Kj µi + 2Kj µi δi2 µ−
Bj =
i µj .
4 i
Summarizing cases A, B and C, the best response of Bj for any given Bi is:
2(σ 2 +µ2 )K
i) if Bi > i µi i j , Bj can be any value in (0, Kj µj ], Qi = 2(µ2B+i σ2 ) , Qj = 0, E(πj,s ) = 0.
ii) if
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
2 µ2 +7µ2 µ2
+8
σ
8σi2 σj2 +8σi2 µ2
i j
j i
j
< Bi ≤
2(σi2 +µ2
i )Kj
µi
i
i
,
−1
1
1 2
δ +1
Bi + 2Kj µi + 2Kj µi δi2 µ−
i µj
4 i
−1 −2
4δi2 + 4δj2 + 4δi2 δj2 + 3
µi 7Bi − 2Kj µi + 8Bi δi2 + 8Bi δj2 − 2Kj µi δi2 + 8Bi δi2 δj2
Bj =
Qi =
−1
1
δi2 + 1
4
Qj =
iii) if
Kj µi µ2
j
4(µ2
+σj2 )
j
iv) if Bi ≤
< Bi ≤
Kj µi µ2
j
4(µ2
+σj2 )
j
−1
1 −1
1
4δi2 + 4δj2 + 4δi2 δj2 + 3
2Kj µi − Bi + 2Kj µi δi2 µ−
i µj .
2
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
, Bj
8σi2 σj2 +8σi2 µ2
+8σj2 µ2
+7µ2
µ2
j
i
i j
, Bj =
Kj µj
2
, Qi = 0, Qj =
=
Kj µj
4σj2 +4µ2
j
2
2Bi (µ2
j +σj )
µi µj
Bi
µi µj
, Qj =
, E(πj,s ) =
Kj2
8δj2 +8
2
, E(πj,s ) = µ−
Kj µi − 2Bi − 2Bi δj2 Bi .
i
.
Best-response function of Bi :
Similarily, we can obtain the best-response function of Bi for any given Bj as follows.
2(σ 2 +µ2 )K
B
i) if Bj > j µj j i , Bi can be any value in (0, Ki µi ], Qj = 2(µ2 +j σ2 ) , Qi = 0, E(πi,s ) = 0.
ii) if
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
8σj2 σi2 +8σj2 µ2
+8
σ
i
i j
j i
< Bj ≤
Bi =
2(σj2 +µ2
j )Ki
µj
j
j
,
−1
1
1 2
δ +1
Bj + 2Ki µj + 2Ki µj δj2 µ−
j µi
4 j
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
21
−1
−1 −2
1
Qj =
δj2 + 1
4δj2 + 4δi2 + 4δj2 δi2 + 3
µj 7Bj − 2Ki µj + 8Bj δj2 + 8Bj δi2 − 2Ki µj δj2 + 8Bj δj2 δi2
4
1 −1
−1
1
4δj2 + 4δi2 + 4δj2 δi2 + 3
2Ki µj − Bj + 2Ki µj δj2 µ−
Qi =
j µi .
2
iii) if
Ki µj µ2
i
< Bj ≤
4(µ2
+σi2 )
i
iv) if Bj ≤
Ki µj µ2
i
4(µ2
+σi2 )
i
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
, Bi
8σj2 σi2 +8σj2 µ2
+8σi2 µ2
+7µ2
µ2
i
j
j i
, Bi =
Ki µi
2
, Qj = 0, Qi =
=
Ki µi
4σi2 +4µ2
i
2
2Bj (µ2
i +σi )
µj µi
Bj
µj µi
, Qi =
Ki2
8δi2 +8
, E(πi,s ) =
2
2
, E(πi,s ) = µ−
j (Ki µj − 2Bj − 2Bj δi ) Bj .
.
Subgame Perfect Nash Equilibriums:
Based on the best response functions, we analyze the following possible scenarios and compute the Subgame
Perfect Nash Equilibriums.
2(σj2 +µ2
2(σi2 +µ2
j )Ki
i )Kj
and Bi >
, Bi can be any value in (0, Ki µi ] and Bj can be any value
1. if Bj >
µj
µi
in (0, Kj µj ]. Because Bj >
1
≤ 12 .
2(δj2 +1)
2(σi2 +µ2
i)
2(σj2 +µ2
j )Ki
and Bj ≤ Kj µj , Kj µj >
µj
However, because Bi >
2(σi2 +µ2
i )Kj
µi
2(σj2 +µ2
j )Ki
µj
, which leads to λ <
2(σi2 +µ2
i )Kj
and Bi ≤ Ki µi , Ki µi >
µi
µ2
j
2(σj2 +µ2
)
j
=
, which leads to λ >
= 2(δi2 + 1) ≥ 2. Contradiction.
µ2
i
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
8σj2 σi2 +8σj2 µ2
+8
σ
i
i j
j i
2. if
< Bj ≤
2(σj2 +µ2
j )Ki
µj
2K µ3 µ2 +2K σ 2 µ µ2
j i i j
j i j
< Bi ≤
, 8σ2 σ2 +8
σ 2 µ2 +8σ 2 µ2 +7µ2 µ2
2(σi2 +µ2
i )Kj
µi
, that is,
i j
i j
j i
i j
2+2δ 2
< Bj ≤ 2µj Ki δj2 + 1 , 8δ2 +8δ2 +8iδ2 δ2 +7 µi Kj < Bi ≤ 2µi Kj (δi2 + 1) , we have
2+2δj2
Kµ
8δi2 +8δj2 +8δi2 δj2 +7 i j
i
j
i j
−1
1
1 2
δ +1
Bj + 2Ki µj + 2Ki µj δj2 µ−
j µi
4 j
−1
1 2
1
δ +1
Bj =
Bi + 2Kj µi + 2Kj µi δi2 µ−
i µj .
4 i
Solving these two equations, we obtain
Bi =
8Ki µi + 2Kj µi + 8Ki µi δi2 + 8Ki µi δj2 + 2Kj µi δi2 + 8Ki µi δi2 δj2
,
16δi2 + 16δj2 + 16δi2 δj2 + 15
2Ki µj + 8Kj µj + 2Ki µj δj2 + 8Kj δi2 µj + 8Kj µj δj2 + 8Kj δi2 µj δj2
Bj =
.
16δi2 + 16δj2 + 16δi2 δj2 + 15
Bi =
To satisfy
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
+8
σ
8σj2 σi2 +8σj2 µ2
j i
i j
i
show that we need
Ki µj µ2
i
+σi2 )
4(µ2
i
3. if
2K µ3 µ2 +2K σ 2 µ µ2
j i j
j i i j
< Bi ≤
, 8σ2 σ2 +8
σ 2 µ2 +8σ 2 µ2 +7µ2 µ2
i j
8δi2 +8δj2 +8δi2 δj2 +7
2+2δi2
2
2
2
2
2
8δi +8δj +8δi δj +7
2δj +2
2
2
2
Kj µi µ2
2Ki µ3
j
j µi +2Ki σj µj µi
2
2
2
2
2
2
2
2
j
+σj2 )
8σj σi +8σj µi +8σi µj +7µj µi 4(µ2
j
2
2
2
2Bi (µj +σj )
2(µi +σi2 )
Bi
Ki µj µ2
i
4(µ2
+σi2 )
i
<λ<
i
j
j
i
i
2(σi2 +µ2
i )Kj
µi
j
< Bi ≤
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
2
2
2
2
2
2
µ2
8σi σj +8σi µj +8σj µi +7µ2
i j
,we have Bi =
2(µ2 +σ 2 )
µi µj
+σj2 )
2(µ2
j
Bi
Bj
, Bi =
Ki µi
2
2(σi2 +µ2
i )Kj
µi
, Bi ≤
Kj µi µ2
j
4(µ2
+σj2 )
j
, Bj =
Kj µj
2
, we have
Ki µi
2
≤
Kj µi µ2
j
= 2(δj2 + 1)Ki µj , Bi can be any value in (0, Ki µi ] and
µj
and
4(µ2
+σj2 )
j
2+2δ 2
j
2
i j
2Kj µi +2Kj µi δi2
8δi2 +8δj2 +8δi2 δj2 +7
Ki µj µ2
i
4(µ2
+σi2 )
i
, that
(Bi + 2Kj µi + 2Kj µi δi )
which contradicts with
2+2δi2
K µ < Bi ≤ 2µi Kj (δi2 + 1) .Similarly, we can derive contradiction when Bi
8δi2 +8δj2 +8δi2 δj2 +7 j i
2+2δ 2
and 8δ2 +8δ2 +8jδ2 δ2 +7 Ki µj < Bj ≤ 2µj Ki δj2 + 1 .
i
j
i j
2
2
2
2(σ 2 +µ2 )K
K µ µ2
2Kj µ3
i µj +2Kj σi µi µj
6. if Bj > j µj j i and 4(µj2 +i σj2 ) < Bi ≤ 8σ2 σ2 +8
2 µ2 +8σ 2 µ2 +7µ2 µ2 , Bi can be any value
σ
j
j
i j
i j
j i
i j
2
2
2Bi (µ2
2Bi (µ2
2(σj2 +µ2
j +σj )
j +σj )
j )Ki
µi µj
≤
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
<
8σi2 σj2 +8σi2 µ2
+8σj2 µ2
+7µ2
µ2
j
i
i j
−1
2
( 8δ2 +8δ2 +8iδ2 δ2 +7 Kj µi < Bi ≤ 2µi Kj (δi2 + 1)).Since Bj = 41 (δi2 + 1)
i
Kj µj
+ 1).Contradiction.
1
2
2
µ−
i µj > 2(δj + 1)Ki µj ≥ 2(δj + 1)Bi µj /µi , we get Bi ≥
Bj =
, it’s easy to
.
,
1
and λ ≥ 2(δi2
2(δj2 +1)
2(σj2 +µ2
j )Ki
5. if Bj >
Bi ≤
µj
and Bj =
,which means Bj = µj µi and
=
. However, since µij µi i −
µi µj
−
1
1 −1
µ−
4σi2 σj2 + 4σi2 µ2j + 4σj2 µ2i + 3µ2i µ2j > 0. This leads to contradiction.
= 12 σj2 + µ2j
i µj
4. if Bj ≤
is, λ ≤
2(σj2 +µ2
j )Ki
<B ≤
2
2Bj (µ2
i +σi )
µj µi
µi µj
2(µ2
+σj2 )
j
< Bj ≤
. Since Bj =
µi µj
can derive contradiction when Bi >
>
2(σi2 +µ2
i )Kj
µi
µj
and
> 2(δi2 + 1)Kj µi
in (0, Ki µi ] and
,which means Bi > Ki µi . Contradiction. Similarly, we
Ki µj µ2
i
4(µ2
+σi2 )
i
< Bj ≤
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2
2
2
2
2
2
8σi σj +8σi µj +8σj µi +7µ2
µ2
i j
.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
22
7. if Bj >
Bj >
2(σj2 +µ2
j )Ki
and Bi ≤
µj
2(σj2 +µ2
j )Ki
and Bi ≤
µj
Kj µi µ2
j
4(µ2
+σj2 )
j
Kj µi µ2
j
4(µ2
+σj2 )
j
Kj µj
, we need
>
2
2(σj2 +µ2
j )Ki
µj
,i.e., λ <
1
.
4(1+δj2 )
8. if
< Bj ≤
2(σj2 +µ2
j )Ki
µj
,
Kj µi µ2
j
4(µ2
+σj2 )
j
< Bi ≤
2
. To satisfy
Similarly, we can show that
if λ > 4(1 + δi2 ),there is an equilibrium where Bj can be any value in (0, Kj µj ] and Bi =
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
8σj2 σi2 +8σj2 µ2
+8
σ
i
i j
j i
Kj µj
, Bi can be any value in (0, Ki µi ] and Bj =
Ki µi
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
2 µ2 +7µ2 µ2
8σi2 σj2 +8σi2 µ2
+8
σ
j
j i
i j
.
2
,
−1
1
1 2
δj + 1
Bj + 2Ki µj + 2Ki µj δj2 µ−
j µi
4
2
2Bi µj (1 + δj )
Bj =
µi
Bi =
Solving these equations, we obtain Bi = Ki µi , Bj = 2Ki µj + 2Ki µj δj2 . In order to satisfy
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
8σj2 σi2 +8σj2 µ2
+8
σ
i
i j
j i
2Ki µj + 2Ki µj δj2 and
< Bj ≤
Kj µi
4(1+δj2 )
2(σj2 +µ2
j )Ki
µj
< K i µi ≤
K µ µ2
, 4(µj2 +i σj2 ) < Bi ≤
j
j
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
2 µ2 +7µ2 µ2 ,
8σi2 σj2 +8σi2 µ2
+8
σ
j
j i
i j
2+2δi2
Kµ.
8δi2 +8δj2 +8δi2 δj2 +7 j i
we need
and the latter is eqivalent to
<λ≤
Similarly, we can show that if
2
λ < 4(1 + δi ), there is an equilibrium where Bj = Kj µj , Bi = 2Kj µi + 2Kj µi δi2 .
2
2
2
2(σ 2 +µ2 )K
K µ µ2
2Ki µ3
j µi +2Ki σj µj µi
< Bj ≤ j µj j i , Bi ≤ 4(µj2 +i σj2 ) , we have
9. if 8σ2 σ2 +8
σ 2 µ2 +8σ 2 µ2 +7µ2 µ2
j
i
j
i
i
j
j
i
j
<
It’s easy to show that the former is satisfied
2+2δi2
.
8δi2 +8δj2 +8δi2 δj2 +7
1
4(1+δj2 )
2+2δj2
Kµ
8δi2 +8δj2 +8δi2 δj2 +7 i j
8δi2 +8δj2 +8δi2 δj2 +7
2+2δj2
≤
j
−1
1
1 2
δ +1
Bj + 2Ki µj + 2Ki µj δj2 µ−
j µi
4 j
K j µj
.
Bj =
2
Solving these equations, we obtain Bi = 8δ21+8 4Ki µi + Kj µi + 4Ki µi δj2 , Bj =
Bi =
satisfy
2
2
2
2Ki µ3
j µi +2Ki σj µj µi
2 µ2 +7µ2 µ2
+8
σ
8σj2 σi2 +8σj2 µ2
j i
i j
i
< Bj ≤
j
2(σj2 +µ2
j )Ki
, we need
µj
8δi2 +8δj2 +8δi2 δj2 +7
4+4δi2
2+2δj2
8δi2 +8δj2 +8δi2 δj2 +7
1
Kµ.
2 j j
Ki µj <
In order to
1
Kµ
2 j j
≤ 2Ki µj +
Kj µi µ2
j
+σj2 )
4(µ2
j
1
, we need
≤λ<
. In order to satisfy Bi ≤
4(1+δj2 )
Kj µi
1
1
1
2
4Ki µi + Kj µi + 4Ki µi δj ≤ 4(1+δ2 ) , which is equal to λ ≤ 4(1+δ2 ) . Thus, we need λ = 4(1+
. Note
8δj2 +8
δj2 )
j
j
Kj µj
1
1
2
2
that when λ = 4(1+δ2 ) , Bi = 8δ2 +8 4Ki µi + Kj µi + 4Ki µi δj = Ki µi , Bj = 2 = 2Ki µj + 2Ki µj δj . Simij
j
2
2
2Ki µj δj2 , which equals to
larly, we can show that if λ = 4(1 + δi ), Bj = Kj µj , Bi = 2Kj µi (1 + δi ).
2
2
2
K µ µ2
2Ki µ3
K µ µ2
j µi +2Ki σj µj µi
and Bi ≤ 4(µj2 +i σj2 ) , Bi =
10. if 4(µi2 +j σi2 ) < Bj ≤ 8σ2 σ2 +8
σ 2 µ2 +8σ 2 µ2 +7µ2 µ2
i
Bi =
i
2
2Bj (µ2
i +σi )
µj µi
j
=
j
i
2
2(µ2
i +σi ) Kj µj
µj µi
2
i
i
j
j
i
j
j
2
2
2Bj (µ2
i +σi )
µj µi
= Kj µi (1 + δi ) ≥ Kj µi , this contradicts with Bi ≤
Kj µi µ2
j
2
2
2
2Kj µ3
i µj +2Kj σi µi µj
and
µ2
+7µ2
+8σj2 µ2
8σi2 σj2 +8σi2 µ2
i j
i
j
, Bj =
Kj µi
4(1+δj2 )
≤
Kj µj
2
Kj µi
4
. Because
. Similarly,
Ki µj µ2
i
+σi2 )
4(µ2
i
Bj ≤
we can derive contradiction if 4(µ2 +σ2 ) < Bi ≤
.
j
j
Summarizing all the cases discussed above, we derive the Subgame Perfect Nash Equilibriums as shown
in Theorem 3.
Proof of Theorem 4.
2
2Kj − 7Ki − 8Ki δi2 − 8Ki δj2 + 2Kj δi2 − 8Ki δi2 δj2
=
2
16 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
2
2 δj2 + 1 6δi2 + 6δj2 + 6δi2 δj2 + 5 2Kj − 7Ki − 8Ki δi2 − 8Ki δj2 + 2Kj δi2 − 8Ki δi2 δj2
−
2
2
16δi2 + 16δj2 + 16δi2 δj2 + 15
4δi2 + 4δj2 + 4δi2 δj2 + 3
2
2Kj − 7Ki − 8Ki δi2 − 8Ki δj2 + 2Kj δi2 − 8Ki δi2 δj2
128δi2 + 128δj2 + 64δi4 + 64δj4 + 256δi2 δj2 + 128δi2 δj4 + 128δi4 δj2 + 64δi4 δj4 + 65
=
2
2
16 16δi2 + 16δj2 + 16δi2 δj2 + 15
4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
>0
E(πih ) − E(πiu )
E(πj∗ ) − E(πjh )
2
2
6δi2 + 6δj2 + 6δi2 δj2 + 5 2Kj − Ki + 2Kj δi2
Ki − 2Kj − 2Kj δi2
=
−
2
2
4δi2 + 4δj2 + 4δi2 δj2 + 3
8 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
2
2Kj − Ki + 2Kj δi2
2δi2 + 2δj2 + 2δi2 δj2 + 3
=
2
8 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
>0
δj2 + 1
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
0
E(πi∗ ) − E(πih )
23
2
2
Kj − 2Ki − 2Ki δj2
6δi2 + 6δj2 + 6δi2 δj2 + 5 2Ki − Kj + 2Ki δj2
=
−
2 2
2
8 4δi2 + 4δj2 + 4δi2 δj2 + 3
4δi2 + 4δj2 + 4δi2 δj2 + 3
δj + 1
2
Kj − 2Ki − 2Ki δj2
2δi2 + 2δj2 + 2δi2 δj2 + 3
=
2 2
8 4δi2 + 4δj2 + 4δi2 δj2 + 3
δj + 1
>0
δi2 + 1
2
2Ki − 7Kj − 8Kj δj2 − 8Kj δi2 + 2Ki δj2 − 8Kj δi2 δj2
=
2 2
16 4δi2 + 4δj2 + 4δi2 δj2 + 3
δj + 1
2
2 δi2 + 1 6δi2 + 6δj2 + 6δi2 δj2 + 5 2Ki − 7Kj − 8Kj δj2 − 8Kj δi2 + 2Ki δj2 − 8Kj δi2 δj2
−
2
2
16δi2 + 16δj2 + 16δi2 δj2 + 15
4δi2 + 4δj2 + 4δi2 δj2 + 3
2
7Kj − 2Ki − 2Ki δj2 + 8Kj δi2 + 8Kj δj2 + 8Kj δi2 δj2
128δi2 + 128δj2 + 64δi4 + 64δj4 + 256δi2 δj2 + 128δi2 δj4 + 128δi4 δj2 + 64δi4 δj4 + 65
=
2
2
16 4δi2 + 4δj2 + 4δi2 δj2 + 3
16δi2 + 16δj2 + 16δi2 δj2 + 15
δj2 + 1
> 0.
0
E(πjh ) − E(πju )
This implies that a decentralized supply chain is better off by centralization regardless whether the other
supply chain is centralized or not. Hence, centralization is the dominant strategy.
Proof of Lemma 3. We examine the centralization decision for a supply chain in two cases: (i) when the
other supply chain is centralized; and (ii) when the other supply chain is decentralized.
Case (i) when the other supply chain is centralized: Without loss of generality, we assume that supply
chain j uses a linear contract and supply chain i is centralized. As Theorem 4 suggested, E(πjh ) < E(πj∗ ),
meaning that the entire supply chain j would be better off if it centralizes. Suppose that under a given linear
h
h
). Consider when supplier
) and supplier j earns a profit of E(πj,s
contract, retailer j earns a profit of E(πj,r
h
)/E(πjh ). It is clear
j and retailer j engage in a revenue-sharing contract with profit split ratio φi = E(πj,r
that retailer j is better off since
φi E(πj∗ ) > φi E(πjh ) =
h
)
E(πj,r
h
E(πjh ) = E(πj,r
).
h
E(πj )
and so is supplier j since
h
(1 − φi )E(πj∗ ) > (1 − φi )E(πjh ) = E(πj,s
).
Case (ii) when the other supply chain is decentralized: Without loss of generality, we assume that supply
chain j uses a linear contract and supply chain i can choose either to coordinate or stick with a given linear
u
contract. As Theorem 4 suggested, E(πiu ) < E(πih ). We define φi = E(πi,r
)/E(πiu ) and let φi be the profit
split ratio of a revenue-sharing contract, we see that both the retailer and supplier are better off under this
contract.
In conclusion, there exists a coordinating contract that achieves Pareto improvement in a supply chain
regardless whether the other supply chain is centralized or not.
Proof of Theorem 5. Based on the results in Theorems 1, 2 and 3, we find
µi Q∗i + µj Q∗j =
2Ki − Kj + 2Ki δj2
Ki + Kj + 2Ki δj2 + 2Kj δi2
2Kj − Ki + 2Kj δi2
+
=
,
2
2
2 2
2
2
2 2
(4δi + 4δj + 4δi δj + 3) (4δi + 4δj + 4δi δj + 3)
4δi2 + 4δj2 + 4δi2 δj2 + 3
7Ki − 2Kj + 8Ki δi2 + 8Ki δj2 − 2Kj δi2 + 8Ki δi2 δj2
2Kj − Ki + 2Kj δi2
2
+
2
2
2 2
4 4δi + 4δj + 4δi δj + 3 (δi + 1)
2 4δi2 + 4δj2 + 4δi2 δj2 + 3
5Ki + 2Kj + 6Ki δi2 + 8Ki δj2 + 6Kj δi2 + 4Kj δi4 + 8Ki δi2 δj2
=
,
4 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
h∗
µi Qh∗
=
i + µj Qj
u∗
i
µi Q
u∗
j
+ µj Q
2 δj2 + 1 7Ki − 2Kj + 8Ki δi2 + 8Ki δj2 − 2Kj δi2 + 8Ki δi2 δj2
=
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3
2 (δi2 + 1) 7Kj − 2Ki − 2Ki δj2 + 8Kj δi2 + 8Kj δj2 + 8Kj δi2 δj2
.
+
16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
24
It can be verified that
h∗
µi Q∗i + µj Q∗j − (µi Qh∗
i + µj Qj ) =
(2Kj − Ki + 2Kj δi2 ) (2δi2 + 1)
> 0,
4 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
(7Ki − 2Kj + 8Ki δi2 + 8Ki δj2 − 2Kj δi2 + 8Ki δi2 δj2 ) 6δi2 + 8δj2 + 8δi2 δj2 + 5
>0
µi Q +µj Q −(µi Q +µj Q ) =
4 16δi2 + 16δj2 + 16δi2 δj2 + 15 4δi2 + 4δj2 + 4δi2 δj2 + 3 (δi2 + 1)
h∗
i
h∗
j
u∗
i
u∗
j
h∗
u∗
u∗
Hence, µi Q∗i + µj Q∗j > µi Qh∗
i + µj Qj > µi Qj + µj Qj .
References
Arreola-Risa, A., G.A. DeCroix. 1998. Inventory management under random supply disruptions and partial
backorders. Naval Research Logistics 45(7) 687–703.
Babich, V., P. H. Ritchken, A. Burnetas. 2007. Competition and Diversification Effects in Supply Chains
with Supplier Default Risk. Manufacturing & Service Operations Management 9 (2) 123 – 146.
Barnes, Dave. 2006. Competing supply chains are the future. Financial Times .
Boyaci, T., G. Gallego. 2004. Supply Chain Coordination in a Market with Customer Service Competition.
Production and Operations Management 13(1) 3–22.
Cachon, G.P. 2001. Stock Wars: Inventory Competition in a Two-Echelon Supply Chain with Multiple
Retailers. Operations Research 49(5) 658–674.
Cachon, G.P. 2003. Supply Chain Coordination with Contracts. Handbooks in Operations Research and
Management Science 11 229–340.
Carr, S.M., U.S. Karmarkar. 2005. Competition in Multiechelon Assembly Supply Chains. Management
Science 51(1) 45.
Corbett, C.J., U.S. Karmarkar. 2001. Competition and Structure in Serial Supply Chains with Deterministic
Demand. Management Science 47(7) 966.
Deo, S., C.J. Corbett. 2009. Cournot competition under yield uncertainty: The case of the us influenza
vaccine market. Manufacturing & Service Operations Management 11(4) 563–576.
Federgruen, A., N. Yang. 2008. Selecting a Portfolio of Suppliers Under Demand and Supply Risks. Operations
Research 56(4) 916.
Gerchak, Y., M. Parlar. 1990. Yield Randomness/Cost Tradeoffs and Diversification in the EOQ Model.
Naval Research Logistics 37(3) 341–354.
Gupta, D. 1996. The (Q, r) Inventory System with an Unreliable Supplier. INFOR 34 59–76.
Ha, A. Y., S. Tong. 2008. Contracting and information sharing under supply chain competition. Management
Science 54 701–715.
Hopp, W. J., S. M. R. Iravani, Z. Liu. 2008. Strategic risk from supply chain disruptions. Working paper.
Kazaz, B. 2004. Production Planning Under Yield and Demand Uncertainty with Yield-Dependent Cost
and Price. Manufacturing & Service Operations Management 6(3) 209.
Kazaz, B. 2008. Pricing and production planning under supply uncertainty. Working paper.
McGuire, T.W., R. Staelin. 1983. An industry equilibrium analysis of downstream vertical integration.
Marketing Science 161–191.
Meyer, R.R., M.H. Rothkopf, S.A. Smith. 1979. Reliability and inventory in a production-storage system.
Management Science 25(8) 799–807.
O’Keeffe, P. 2006. Understanding supply chain risk areas, solutions, and plans: a five-paper series. APICS,
The Association of Operations Management, Alexandria, VA. .
Parlar, M. 1988. Game theoretic analysis of the substitutable product inventory problem with random
demands. Naval Research Logistics 35 397–409.
Parlar, M., D. Berkin. 1991. Future supply uncertainty in EOQ models. Naval Research Logistics 38(1)
107–121.
Author: Article Short Title
Article submitted to Production and Operations Management; manuscript no.
25
Parlar, M., D. Wang. 1993. Diversification under yield randomness in inventory models. European Journal
of Operational Research 66(1) 52–64.
Snyder, L. V., Z.-J. M. Shen. 2006a. Supply and demand uncertainty in multi-echelon supply chains. Under
revision, Naval Research Logistics.
Snyder, L.V., Z.-J. M. Shen. 2006b. Supply chain managing under the threat of disruptions. The Bridge
(National Academy of Engineering) 36(4) 39.
Song, J.S., PH Zipkin. 1996. Inventory control with information about supply conditions. Management
science 42(10) 1409–1419.
Swaminathan, J.M., J.G. Shanthikumar. 1999. Supplier diversification: effect of discrete demand. Operations
Research Letters 24(5) 213–221.
Tomlin, B. 2006. On the Value of Mitigation and Contingency Strategies for Managing Supply Chain
Disruption Risks. Management Science 52(5) 639.
Wang, Y., W. Gilland, B. Tomlin. 2008. Mitigating supply risk: Dual sourcing or process improvement?
Working paper.
Wu, O., H. Chen. 2005. Chain-to-chain competition under demand uncertainty. Working paper, University
of British Columbia, Canada .
Yang, Z., G. Aydin, V. Babich, D. R. Beil. 2009. Supply disruptions, asymmetric information, and a backup
production option. Management Science 55(2) 192–209.
Yano, C.A., H.L. Lee. 1995. Lot-sizing with random yields: a review. Operations Research 43(3) 311–334.
Zipkin, P.H. 2000. Foundations of inventory management. McGraw-Hill.