price competition, cost and demand disruptions and...

Price competition, cost and demand disruptions and coordination of a supply chain with one manufacturer

and two competing retailers

Tiaojun Xiao, Xiangtong QiSchool of Management Science and Engineering, Nanjing University, Nanjing, Jiangsu

210093, China

Department of Industrial Engineering and Logistics Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Published in OMEGA, November 2008

2018.01.16Liying Piao

Table of Contents• Introduction• Model• Coordination machanism• Conclusion

1 Introduction Background:

Generally, supply chain coordination management investigates a static

coordination mechanism under a deterministic environment such as a known market

demand and production cost. But the plan is often disrupted by some haphazard

events. So it is important to know how the supply chain can be still coordinated under

different disruptions.

Problem to deal with:How the supply chain which has 1 manufacture and 2 competing retailers can

be coordinated under the production cost disruption incurred to the manufacturer.

2.1 Static game with complete information based on Bertrand model. Price competition in static oligopoly model

– Market is dominated by small number of sellers → We called 2 sellers situation duopoly.

– The decisions of one seller are influenced by decisions of other sellers.

– Sellers are neither the subjectives nor objecctives of prices, but the seekers.

Bertrand model– Sellers compete in price → Choice variable is price.

– Method to get Nash Equilibrium:

• Each sellers must behave optimally assuming that rival behaves optimally.

• Mathematically, seller i's profit fuction is differentiable in , we drive each

seller's first-order condition with respenct to its own price for a given rival price and

solve the system of equations.

2 Model

ip

2.2 Basic model Parameters --- unit cost for manufacture produce goods --- unit cost for retailers add values

--- unit wholesale price for retailers

--- retail price for retailers --- demand of retailers

--- market scale --- substituatbility coefficient

Basic model

Linear market demand for retailer i(j ≠ i) is

The total profit of the supply chain is

Solving the first-order conditions we know that the optimal solution of

optimal quantity of retailer i in the centralized supply chain is

According to (1)(3), we got optimal quantity of retailer i in the centralized supply

chain is

2 Model

0c

）（ 2,1ii ic

ipa d

iq

2.3 Supply chain Centralized supply chainThe supply chain is managed by a central planner who is able to control all decisions.

The set of actions that can optimize the supply chain’s performance is called the

centralized optimal solution.

Decentralized supply chainMembers within a supply chain to be independent organizations that aim to maximize

their own objectives. The behavior of a decentralized supply chain can be characterized

using the Nash equilibrium concept

Coordinated supply chainA supply chain is coordinated if all supply chain members adopt the actions that

optimize the entire system’s performance.

2 Model

2.4 Centralized solution with cost disruption Parameters --- production cost with disruption, --- demand of retailer i with production cost disruption --- penalty cost for deviation production

Model of the centralized supply chain with cost disruption

we get optimal solution of (4),

Inserting the optimal retail prices into (1), we find that the optimal order quantities are

2 Model

0~00 ccc0

~c

iq~

0,0 su cc

0,max xx ）（

total profit of the supply chain without the deviation penalty

total penalty cost incurred by the increased production.

total penalty cost incurred by the decreased production.

0~

ip

2.4 Centralized solution with cost disruption Theorem 1

- We find that the optimal retail prices do not change when the absolute cost disruption is

sufficiently small.

- The central planner changes the retail prices if the amount of the unit cost is

sufficiently large.

- The larger the incremental unit cost the higher the optimal retail prices.

2 Model

2,1,

),(21

),(21

~

*

*

*

*

i

ccifccp

cccifp

ccifccp

p

ssi

sui

uui

i ，The optimal solution above is

c

c

3.1 Quantity discount

All-unit quantity discountIf order quantity is bigger than the breakpoint, the discount price would be operated for

all purchased units .

Two-part tariff based on incremental quantity discount- Two-part tariff: fixed fee & constant marginal cost

- Incremental quantity discount: charging different discount for different order quantity.

3 Coordination machanism

3.2 Coordination via all-unit quantity discount scheme Coordination under cost disruption- General situation

When retailer i gets a wholesale price , his profit function is

Solving the firstorder conditions of (6) with respect to retail prices, we obtain Nash

equilibrium prices

The supply chain is coordinated only when the Nash equilibrium retail prices for the

decentralized supply chain equal the optimal retail prices for the centralized supply chain,

respectively: ,so we got

Rewriting the profit function of retailer i we have


i~

2,1~~ * ippii

N ，

3.2 Coordination via all-unit quantity discount scheme Coordination under cost disruption- General situation

--- 2 breakpoints of all-unit quantity discount scheme,

--- coresponding unit whole sale price ,

> >


21~̂,

~̂qq 21

~̂~̂0 qq

210~̂~̂~̂ 21

~̂,

~̂

1~̂q 2

~̂q0

0~̂ 1

~̂ 2

~̂

3.2 Coordination via all-unit quantity discount scheme Coordination under cost disruption- Symmetric case:

• the supply chain with competing retailers can be coordinated by an all-unit quantity discount scheme with a unique breakpoint if two retailers are identical.


21 cc

3.2 Coordination via all-unit quantity discount scheme Coordination under cost disruption- Asymmetric case:


21 cc

3.2 Coordination via all-unit quantity discount scheme Coordination under cost & demand disruptions- demand disruption:

- In general, the larger the market scale, the higher the optimal quantity although the retail

prices increase.

- Two retailers evenly share the incremental demand dueto the symmetric position.

- Nash equilibrium retail prices of the decentralized supply chain are


0~~~ caaaa ，

3.2 Coordination via all-unit quantity discount scheme Coordination under cost & demand disruptions- Substituting for in

And similar with Theorem 2, we got coordination machanism as follow:

- Symmetric case:


21 cc

aa a

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- Two-part tariff:

---the optimal tariff for retailer i.

--- marginal price --- fixed fee

The profit of retailer i

Nash equilibrium prices

If the supply chain is coordinated,


)~,~( **ii T

)~,~(),~,~( *2

*2

*1

*1 TT

*~iT

*~i

2,1~~ * ipp iNi ，

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- Theorem 6

The marginal wholesale prices equal the optimal transfer prices in an

centralized supply chain. The central decision-maker takes the marginal wholesale

prices


*~i

*~i

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- Two-part tariff:

---the optimal tariff for retailer i.

--- marginal price --- fixed fee

The profit of retailer i Nash equilibrium prices

if bothretailers accept the optimal marginal wholesale prices:


)~,~( **ii T

)~,~(),~,~( *2

*2

*1

*1 TT

*~iT

*~i

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- Constrains of two-part tariff:

Incentive compatibility constraint: neither retailer has a motivation to unilaterally

deviate from the optimal tariffs of the supply chain.

Individual rationality (IR) constraint: retailers would like to order and sell the manufacturer’s products.

--- reservation utility of retailer i


0i

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- coordination machanism

According to (17)(18)(19)

we got

It follows from that the two sides of inequality are nonnegative.


*2

*1

~~

3.3 Coordination via incremental quantity discount scheme Coordination under cost disruption- Theorem 7

- For the symmetric case, marginal wholesale prices are equal if and only if the two

retailers are identical. When the retailers are identical, both sides of (22) equal zero.

→ , which means same tariffs .

- For the asymmetric case, the higher marginal wholesale price goes with the lower fixed

fee in the coordinating menu of two-part tariffs.


*2

*1

~~ TT

For the all-unit quantity discount scheme, the supply chain can be fully

coordinated when two retailers are identical.

For the coordinating menu of two-part tariffs, under the original mechanism, the

lower-cost retailer accepts the original wholesale price, but the higher-cost retailer

may have a motivation to order more products to accept the lower wholesale

price.

4 Conclusions

price competition, cost and demand disruptions and...

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