Available via license: CC BY 4.0

Content may be subject to copyright.

Journal of Uncertainty

Analysis and Applications

Ke et al. Journal of Uncertainty Analysis and

Applications (2017) 5:5

DOI 10.1186/s40467-017-0059-2

RESEARCH Open Access

Pricing Decision in a Two-Echelon Supply

Chain with Competing Retailers Under

Uncertain Environment

Hua Ke1, Yong Wu

1, Hu Huang1* and Zhiyi Chen2

*Correspondence:

huanghu0213@163.com

1School of Economics and

Management, Tongji University,

Siping Road, Shanghai 200092,

China

Full list of author information is

available at the end of the article

Abstract

This paper explores a supply chain pricing competition problem in a two-echelon

supply chain with one manufacturer and two competing retailers. The manufacturing

costs, sales costs, and market bases are all characterized as uncertain variables whose

distributions are estimated by experts’ experienced data. In consideration of channel

members’ different market powers, three decentralized game models are employed to

explore the equilibrium behaviors in corresponding decision circumstances. How the

channel members should choose their most profitable pricing strategies in face of

uncertainties is derived from these models. Numerical experiments are conducted to

examine the effects of power structures and parameters’ uncertain degrees on the

pricing decisions of the members. The results show that the existence of dominant

powers in the supply chain will increase the sales prices and reduce the profit of the

whole supply chain. It is also found that the supply chain members may benefit from

higher uncertain degrees of their own costs while the other supply chain members will

gain less profits. Additionally, the results demonstrate that the uncertainty of the

supply chain will make end consumers pay more. Some other interesting managerial

highlights are also obtained in this paper.

Keywords: Two-echelon supply chain, Pricing competition, Competing retailers,

Uncertain variable, Stackelberg game

Introduction

In this paper, we explore a pricing decision problem in a decentralized supply chain with

competing retailers under uncertainty. In such supply chains, costs and demands may

be subject to some inherent indeterministic factors, such as material cost changes, labor

costs, technology improvements, and customer incomes. It is necessary to identify dis-

tributions of these parameters, which are essential for supply chain managers when they

make decisions, e.g., choosing prices, determining production quantities, choosing order-

ing quantities, and adding investment. Theoretically, we can collect enough samples to

estimate distributions of these parameters before we make decisions. However, products,

especially high-tech products, e.g., smartphones, PCs, and other digital devices, have

been rapidly updated recently. The costs and demands for these new products are usu-

ally the lack of historical data, or we can collect enough data, but these data may not be

applicable due to the dynamic environment. Practically, domain experts’ experience data

© The Author(s). 2017 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0

International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and

reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the

Creative Commons license, and indicate if changes were made.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 2 of 21

(belief degrees) are often applied to estimate the distributions in such cases. Nevertheless,

some surveys have shown that human beings (even the most experienced experts) usu-

ally estimate a much wider range of values than the object actually takes. It makes belief

degrees behave quite different from frequency, indicating that human belief degrees given

by managers and experts should not be treated as random variables or fuzzy variables [1].

Instead, based on normality, duality, subadditivity, and product axioms, uncertainty the-

ory, initiated by Liu [2] and refined by Liu [3], can be introduced to deal with parameters

estimated by human belief degrees.

The goal of this paper is to study supply chain pricing problems with uncertainty the-

ory. Our work mainly focuses on equilibrium behaviors of the decentralized supply chain

with one common manufacturer (upstream member) and duopoly retailers (downstream

members) under uncertainty. A supply chain with competing retailers is not uncommon

in the real industry world. For instance, smartphone producers distribute their products

through both traditional retailers and online retailers. How should the supply chain man-

agers choose the most profitable price policies with experts’ estimations? What effects

might the uncertain degrees of the parameters have on the supply chain members’ pricing

decisions and expected profits? Additionally, we assume that the manufacturer deter-

mines the wholesale prices and the two retailers compete for end consumers by choosing

their own sales prices. The decision sequence is mainly decided by the power structures

of the supply chain. In some supply chains, the manufacturers (like Intel and Microsoft)

often play a more powerful role with regard to the other channel members. In some other

chains, the retailers (like Warmart and Carrefour) who are much bigger than most man-

ufacturers often hold the dominant power. While in another type of supply chain, no

absolute dominance exists between different members and each member holds the same

power. How would these various power structures affect the performance of the supply

chain?

In order to answer these questions, three decentralized pricing models based on

uncertainty theory and game theory are built. Then, the corresponding closed-form

equilibrium solutions are derived from these models. Afterwards, numerical experi-

ments are conducted to analyze the equilibrium behaviors of the supply chain under

different power structures and uncertain settings. It is found that the manufac-

turer can benefit from a higher uncertain degree of the manufacturing cost, mean-

while, the retailers will gain less profits. Similarly, when the uncertain degree of

the sales costs increases, the retailers can make more profits while the manufacture

will gain less. Additionally, the uncertainty of the supply chain will make end con-

sumers pay more. Some other interesting managerial highlights are also obtained in

this paper.

The remainder of this paper is organized as follows: some related literatures are pre-

sented in the “Literature Review” section. Some basic concepts and properties with

respect to uncertain variable are reviewed in the “Preliminaries” section. Some use-

ful notations and necessary assumptions are discussed in the “Problem Description”

section. Three models are employed to derive the equilibria under three possible scenar-

ios in the “Models and Solution Approaches” section. In the “Numerical Experiments”

section, numerical experiments are applied to examine the equilibrium behaviors of the

supply chain members. Some conclusions and possible extensions are discussed in the

“Conclusions” section.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 3 of 21

Literature Review

By now, the pricing competition problem in decentralized supply chains, initiated in

1980s [4, 5], has been well studied by both scholars and practitioners. Most of these

researches on the pricing competition problem focus on the following four structures:

monopoly common retailer structure [6–8], chain-to-chain structure [4, 9, 10], dual chan-

nel structure [11–14], and monopoly common manufacturer structure, which is studied

in this paper. The research on the monopoly common manufacturer structure was ini-

tiated by Ingene and Parry [15], considering a coordination problem in a supply chain

where a manufacturer sells its products through competing retailers. Ingene and Parry

[16] demonstrated that the manufacturer can generally obtain more profits by setting a

unique two-part tariff wholesale pricing policy. For pricing competition, Yang et al.[17]

considered a pricing problem in a two-echelon supply chain with a manufacturer who

supplies a single product to two competing retailers. Assuming that the manufacturer

acts as a Stackelberg leader, they explored the effects of the retailers’ Bertrand and Col-

lusion behaviors on the performance of the supply chain. Wu et al.[18] investigated the

pricing decision in this monopoly common manufacturer channel structure and built six

noncooperative models where the two retailers play Stackelberg or Bertrand games under

three possible power structures. Besides, Huang et al.[19] considered a pricing compe-

tition and cooperation problem in a supply chain with one common manufacturer and

duopoly retailers and explored the conditions under which the retailers can benefit from

their collusion behaviors.

The work above typically focused on deterministic demands and costs. In fact, the real

world exists many indeterminate factors which cannot be ignored when making pricing

decisions. Thus, one important tendency is to explore the pricing competition prob-

lem under environments with indeterminate factors, namely randomness, fuzziness, or

uncertainty. For instance, Bernstein and Federgruen [20] investigated the equilibrium

behaviors of decentralized supply chains with competing retailers under random demand

and designed some contractual arrangements between the parties that allow the decen-

tralized chain to perform like a centralized one. Xiao and Yang [21] studied a price

and service competition of two supply chains with risk-averse retailers under stochastic

demand. They also analyzed the impacts of the retailer’s risk sensitivity on the man-

ufacturers’ equilibrium strategies. Wu et al.[9] considered a joint pricing and quantity

competition between two separate supply chains in random environment and explored

the effects of randomness on the equilibrium behaviors of the supply chain members.

Shi et al.[22] utilized a game-theory-based framework to formulate the power in a supply

chain and examined how power structure and demand indeterminacy affect the supply

chain members’ performances. Mahmoodi and Eshghi [23] studied this pricing problem

in duopoly supply chains with stochastic demand and explored the effects of competition

and demand indeterminacy intensity on the equilibria of the structures by a numerical

experiment. Li et al.[24] studied the pricing and remanufacturing decision problem with

random yield and demand.

More recently, fuzzy set theory has been introduced to the pricing decision problems.

Zhou et al.[25] considered the pricing decision problem in a supply chain which consists

of a manufacturer and a retailer in fuzzy environment. Zhao et al.[26] and Sang [27] stud-

ied the pricing problem on two substitutable products in a supply chain with different

structures in which the manufacturing costs and consumer demands were described by

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 4 of 21

fuzziness. Specially, Liu and Xu [28] and Ke et al.[29] studied the pricing problem in a

fuzzy supply chain consisting of one manufacturer and two competitive retailers.

To the best of our knowledge, there are little researches on the pricing decision problem

with competing retailers under uncertain settings. Differing from the literatures above,

this paper addresses the pricing equilibria under circumstances with only belief degrees

being available, which can be described by uncertainty theory. By far, the new theory has

been successfully applied to deal with many uncertain decision-making problems, e.g.,

option pricing [30, 31], facility location [32], portfolio selection [33], inventory problem

[34], project scheduling problem [35–37], and production control problem [38]. Recently,

Huang and Ke [8] applied uncertainty theory to a pricing decision problem in a sup-

ply chain with one common retailer. Besides, Chen et al.[39] studied an effort decision

problem in a supply chain with one manufacturer and one retailer under uncertain infor-

mation. However, both of them did not consider the pricing problem in a supply chain

with competing retailers.

This paper extends the above literatures on the pricing problems with competing

retailers [17–19, 28], by employing uncertainty theory to depict the costs and demands

estimated by experts’ experience data in practice. Three uncertain game models are

built to explore the equilibrium behaviors of the decentralized supply chain under differ-

ent power structures. Meanwhile, we consider the retailers’ sales costs which are often

ignored in most literatures. Then, we explore the effects of uncertain degrees of the

manufacturing costs and sales costs on the equilibrium prices determined by the supply

chain managers. Some interesting managerial highlights are gained from some numerical

experiments conducted in this paper.

Preliminaries

Let be a nonempty set and Laσ-algebra over .Eachelementin Lis called an event.

Definition 1 ([2]) The set function Mis called an uncertain measure if it satisfies:

Axiom 1 (Normality axiom) M{}=1.

Axiom 2 (Duality axiom) M{}+M{c}=1for any event .

Axiom 3 (Subadditivity axiom) For every countable sequence of events {i},i=1, 2, ···,

we have

M∞

i=1

i≤

∞

i=1

M{i}.

Besides, the product uncertain measure on the product σ-algebra Lwas defined by Liu

[40] as follows:

Axiom 4 (Product axiom) Let (k,Lk,Mk)be uncertainty spaces for k =1, 2, ··· The

product uncertain measure Mis an uncertain measure satisfying

M∞

k=1

k=

∞

k=1

Mk{k}

where kare arbitrarily chosen events from Lkfor k =1, 2, ···, respectively.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 5 of 21

Definition 2 ([2]) An uncertain variable is a measurable function ξfrom an uncertainty

space (,Ł,M)to the set of real numbers, i.e., for any Borel set B of real numbers, the set

{ξ∈B}={γ∈ξ(γ ) ∈B}

is an event.

Definition 3 ([40]) The uncertain variables ξ1,ξ2,··· ,ξnare said to be independent if

Mn

i=1

(ξi∈Bi)=

n

i=1

M{ξi∈Bi}

for any Borel sets B1,B2,··· ,Bnof real numbers.

Sometimes, we should know uncertainty distribution to model real-life uncertain

optimization problems.

Definition 4 ([2]) The uncertainty distribution of an uncertain variable ξis defined

by

(x)=M{ξ≤x}

for any real number x.

An uncertainty distribution is referred to be regular if its inverse function −1(α)

exists and is unique for each α∈[0,1].

Lemma 1 ([3])Letξ1,ξ2,··· ,ξnbe independent uncertain variables with regular

uncertainty distributions 1,2,··· ,n, respectively. If the function f (x1,x2,··· ,xn)is

strictly increasing with respect to x1,x2,··· ,xmand strictly decreasing with respect to

xm+1,xm+2,··· ,xn,then

ξ=f(ξ1,ξ2,··· ,ξn)

is an uncertain variable with inverse uncertainty distribution

−1(α) =f(−1

1(α),··· ,−1

m(α),−1

m+1(1−α),··· ,−1

n(1−α)).

Definition 5 ([2])Letξbe an uncertain variable. The expected value of ξis defined by

E[ξ]=+∞

0

M{ξ≥r}dr −0

−∞

M{ξ≤r}dr

provided that at least one of the above two integrals is finite.

Lemma 2 ([3])Letξbe an uncertain variable with uncertainty distribution .Ifthe

expected value exists, then

E[ξ]=+∞

0

(1−(x))dx −0

−∞

(x)dx.

Lemma 3 ([3])Letξbe an uncertain variable with regular uncertainty distribution .

If the expected value exists, then

E[ξ]=1

0

−1(α)dα.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 6 of 21

Lemma 4 ([41])Letξ1,ξ2,··· ,ξnbe independent uncertain variables with regular

uncertainty distributions 1,2,··· ,n, respectively. A function f (x1,x2,··· ,xn)is

strictly increasing with respect to x1,x2,··· ,xmand strictly decreasing with respect to

xm+1,xm+2,··· ,xn. Then, the expected value of ξ=f(ξ1,ξ2,··· ,ξn)is

E[ ξ]=1

0

f(−1

1(α),··· ,−1

m(α),−1

m+1(1−α),··· ,−1

n(1−α)))dα(1)

provided that the expected value E[ξ]exists.

Example 1 Let ξand ηbe two positive independent uncertain variables with regular

uncertainty distributions and , respectively. Then we have

Eξ

η=1

0

−1(α)

−1(1−α)dα.(2)

Problem Description

Consider a supply chain with one common manufacturer and two competing retailers:

the manufacturer supplies one common product to the two retailers, and in turn, the two

retailers compete to sell the product to end consumers in the same market.

Notations and Assumptions

It is assumed that the manufacturer determines the wholesale prices (w1,w2)to the two

retailers and then the two retailers choose their own markup policies r1and r2,respec-

tively. Note that the sales prices can be decided (pi=wi+ri,i=1, 2)when the

corresponding wholesale prices and retailer markup policies are determined. Consistent

with the extant literatures [6, 17–19, 42], we suppose that each retailer faces the following

linear demand function:

qi=di−βpi+γp3−i,i=1, 2

where direpresents the potential market size of retailer i,βis the self-price elasticity

coefficient, γis the cross-price elasticity coefficient, and piisthesalespriceofretaileri.

Note that γ/β denotes the substitutability of the two retailers.

Owing to the complicated and changeable environment, the manufacturing cost ˜c,

sales costs (˜s1,˜s2),marketsizes(˜

d1,˜

d2), and price elasticity coefficients ˜

β,˜γare usu-

ally unknown to the supply chain managers. Thus, belief degrees, given by experienced

experts, are often employed to estimate these parameters due to the lack of historical data.

To sum up, we give the notations shown in Table 1.

Table 1 Notations

˜

c: Unit manufacturing cost, an uncertain variable

˜

si: Unit sales cost of retailer i,anuncertainvariable

wi: Unit wholesale price to retailer i, the manufacturer’s decision variable

ri: Unit markup price of retailer i, the retailer’s decision variable

pi: Unit retail price of retailer i,wherepi=wi+ri

˜

di: The market base of retailer i, an uncertain variable

πm: The profit of the manufacturer: πm=2

i=1(wi−˜

c)(˜

di−˜

βpi+˜γp3−i)

πri: The profit of retailer i:πri=(ri−˜

si)(˜

di−˜

βpi+˜γp3−i)

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 7 of 21

Note that some basic concepts and theorems of uncertainty theory are provided in

the “Preliminaries” section. Interested readers can consult Liu [1] for more details of the

uncertainty theory. In order to attain closed-form solutions, some necessary assumptions

are made as follows:

•As each retailer’s demand should be more sensitive to changes of its own price than

to changes of the other retailer, it is assumed that the elasticity coefficients ˜

βand ˜γ

satisfy: E[ ˜

β]>E[ ˜γ]>0.

•All the uncertain coefficients are assumed nonnegative and mutually independent.

•All the supply chain members have access to the same information on the demands

and the costs.

•All the supply chain members are risk neutral.

•All the pricing activities are assumed to be completed in a cycle.

•The wholesale prices and markups should exceed the costs and the retailers’

demands are always positive:

M{wi−˜c≤0}=0, M{ri−˜si≤0}=0,

M{˜

di−˜

βpi+˜γp3−i≤0}=0, i=1, 2. (3)

Crisp Forms of Objective Functions

In order to derive the equilibrium solutions, we transform uncertain objective functions

to equivalent crisp forms first. Let ˜c,˜si,˜

di,˜

β,and ˜γbe positive independent uncertain

variables with regular uncertainty distributions c,si,di,β,andγ,i=1, 2.

As M{wi−˜c≤0}=0, M{ri−˜si≤0}=0, M{˜

di−˜

βpi+˜γp3−i≤0}=0, i=1, 2, one

can find that πmis monotone increasing function of ˜

di,˜γand monotone decreasing with

˜c,˜

β,and˜si. Thus, referring to Lemma 4, the expected object function of the manufacturer

can be attained as follows:

πm=

2

i=1

E(wi−˜ci)˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i)

=

2

i=11

0wi−−1

c(1−α)−1

di(α) −−1

β(1−α) (ri+wi)

+−1

γ(α) (r3−i+w3−i)dα.

(4)

Define

E[˜a1−α˜

bα]=1

0

−1

a(1−α)−1

b(α)dα,

E[˜a1−α˜

b1−α]=1

0

−1

a(1−α)−1

b(1−α)dα.

(5)

where −1

aand −1

bare the inverse distribution functions of ˜aand ˜

b,respectively.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 8 of 21

Then, the crisp form of πmcan be attained as follows:

πm=

2

i=11

0

[(wi−−1

c(1−α))(−1

di(α) −−1

β(1−α)(ri+wi)

+−1

γ(α)(r3−i+w3−i))]dα

=

2

i=1

{−E[ ˜

β]w2

i+E[ ˜γ]w3−iwi+(−E[ ˜

β]ri+E[ ˜γ]r3−i+E[˜

di]+E[˜c1−α

i˜

β1−α])wi

−E[˜c1−α˜

dα

i]+E[˜c1−α˜

β1−α]ri−E[˜c1−α˜γα](r3−i+w3−i)}.

(6)

Similarly,thecrispformsoftheexpectedprofitfunctionsoftheretailerscanalsobe

attained as follows:

πri=−E[˜

β]r2

i+E[˜γ]r3−iri+(−E[˜

β]wi+E[˜γ]w3−i+E[˜

di]+E[˜s1−α

i˜

β1−α])ri

−E[˜s1−α

i˜

dα

i]+E[˜s1−α

i˜

β1−α]wi−E[˜s1−α

i˜γα](r3−i+w3−i),i=1, 2.

(7)

Models and Solution Approaches

In this section, starting with the Manufacturer Stackelberg (MS) model, we present the

general formulations and solutions to three types of supply chain pricing models with

different power structures.

MS Model

In the first case, we assume that the manufacturer plays a dominant role and announces

its wholesale prices (w1,w2)to maximize its own profit allowing for the retailers’ reac-

tion functions. Observing the wholesale prices, the two retailers noncooperatively choose

their own markup pricing schemes rito maximize their own profits conditional on

the other retailer’s decision. Then, the retail prices are decided as well as the ordering

quantities. Thus, a Stackelberg-Nash game can be applied as follows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

max

w1,w2

πm=

2

i=1

E[(wi−˜c)( ˜

di−˜

β(r∗

i+wi)+˜γ(r∗

3−i+w3−i))]

subject to:

M{wi−˜c≤0}=0, i=1, 2

where (r∗

1,r∗

2)solves problems:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

max

r1

πr1=E[(r1−˜s1)(˜

d1−˜

β(r1+w1)+˜γ(r2+w2))]

max

r2

πr2=E[(r2−˜s2)(˜

d2−˜

β(r2+w2)+˜γ(r1+w1))]

subject to:

M{ri−˜si≤0}=0,

M{˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i)≤0}=0, i=1, 2.

(8)

To solve this Stackelberg-Nash game model, opposite to the decision sequence, we

should first derive the Nash equilibrium at the lower level for the given wholesale prices

w1and w2specified by the manufacturer in advance.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 9 of 21

Proposition 1 Given the wholesale prices (w1,w2), the Nash equilibrium in the lower

level can be obtained as follows:

r1(w1,w2)=−2E[˜

β]2+E[˜γ]2

4E[˜

β]2−E[˜γ]2w1+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2w2

+2E[˜

β](E[˜

d1]+E[˜

β1−α˜s1−α

1])+E[˜γ](E[˜

d2]+E[˜

β1−α˜s1−α

2])

4E[˜

β]2−E[˜γ]2,

r2(w1,w2)=−2E[˜

β]2+E[˜γ]2

4E[˜

β]2−E[˜γ]2w2+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2w1

+2E[˜

β](E[˜

d2]+E[˜

β1−α˜s1−α

2])+E[˜γ](E[(˜

d1]+E[˜

β1−α˜s1−α

1])

4E[˜

β]2−E[˜γ]2.

(9)

Proof Referring to the two retailers’ expected objective functions with the given whole-

sale prices, we can get

∂2πr1

∂r2

1

=−2E[˜

β]<0, ∂2πr2

∂r2

2

=−2E[˜

β]<0, (10)

with the assumption E[˜

β]>E[˜γ]>0. Hence, πr1and πr2are concave in r1and r2,

respectively. Setting the first-order derivatives equaling zero, we have

∂πr1

∂r1

=−2E[˜

β]r1+E[˜γ]r2−E[˜

β]w1+E[˜γ]w2+E[˜

d1]+E[˜s1−α

1˜

β1−α]=0,

∂πr2

∂r2

=−2E[˜

β]r2+E[˜γ]r1−E[˜

β]w2+E[˜γ]w1+E[˜

d2]+E[˜s1−α

2˜

β1−α]=0.

(11)

The followers’ optimal responses to (w1,w2)can be easily obtained by solving the above

two equations as follows:

r∗

1(w1,w2)=−2E[˜

β]2+E[˜γ]2

4E[˜

β]2−E[˜γ]2w1+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2w2

+2E[˜

β](E[˜

d1]+E[˜

β1−α˜s1−α

1])+E[˜γ](E[˜

d2]+E[˜

β1−α˜s1−α

2])

4E[˜

β]2−E[˜γ]2,

r∗

2(w1,w2)=−2E[˜

β]2+E[˜γ]2

4E[˜

β]2−E[˜γ]2w2+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2w1

+2E[˜

β](E[˜

d2]+E[˜

β1−α˜s1−α

2])+E[˜γ](E[(˜

d1]+E[˜

β1−α˜s1−α

1])

4E[˜

β]2−E[˜γ]2.

(12)

Thus, Proposition 1 is proved.

The manufacturer, given the optimal responses of the two retailers, will make decisions

to maximize its expected profit.

Proposition 2 Given the optimal responses of the two retailers, we can obtain the fol-

lowing:

(1) πmis jointly concave with respect to w1and w2;

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 10 of 21

(2) The optimal decisions of the manufacturer are as follows:

w∗

1=A(E[˜

d1]−E[˜

β]S1+E[˜γ]S2+C)+B(E[˜

d2]−E[˜

β]S2+E[˜γ]S1+C)

2(A2−B2),

w∗

2=A(E[˜

d2]−E[˜

β]S2+E[˜γ]S1+C)+B(E[˜

d1]−E[˜

β]S1+E[˜γ]S2+C)

2(A2−B2)

(13)

where

A=2E[˜

β]3−E[˜

β]E[˜γ]2

4E[˜

β]2−E[˜γ]2,B=E[˜

β]2E[˜γ]

4E[˜

β]2−E[˜γ]2,

C=(2E[˜

β]2+E[˜

β]E[˜γ])(E[˜c1−α˜

β1−α]−E[˜

c1−α˜γα])

4E[˜

β]2−E[˜γ]2,

Si=2E[˜

β](E[˜

di]+E[˜

β1−α˜s1−α

i])+E[˜γ](E[˜

d3−i]+E[˜

β1−α˜s1−α

3−i])

4E[˜

β]2−E[˜γ]2,i=1, 2.

Proof Substituting the two retailers’ reaction functions Eqs‘. (9) into the manufacturer’s

profit function, and we obtain

πm=

2

i=1

E[(wi−˜c)( ˜

di−˜

β(r∗

i+wi)+˜γ(r∗

i+wi))]

=

2

i=1

E[(wi−˜c)( ˜

di−˜

β( 2E[˜

β]2

4E[˜

β]2−E[˜γ]2wi+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2w3−i+Si)

+˜γ( 2E[˜

β]2

4E[˜

β]2−E[˜γ]2w3−i+E[˜

β]E[˜γ]

4E[˜

β]2−E[˜γ]2wi+S3−i))]

=

2

i=1

{E[˜

di]wi−Aw2

i+Bw3−iwi−E[˜

β]Siwi+E[˜γ]S3−iwi

−E[˜c1−α˜

dα

i]+E[˜c1−α˜

β1−α]Si−E[˜c1−α˜γα]S3−i

+2E[˜c1−α˜

β1−α]E[˜

β]2−E[˜

β]E[˜γ]E[˜c1−α˜γα]

4E[˜

β]2−E[˜γ]2wi

+E[˜

β]E[˜γ]E[˜c1−α˜

β1−α]−2E[˜

β]2E[˜c1−α˜γα]

4E[˜

β]2−E[˜γ]2w3−i}.

(14)

Referring to Eq. (14), we can obtain the Hessian matrix with the corresponding second-

order derivatives of the objective function πmas follows:

H=

∂2πm

∂w2

1

∂2πm

∂w1∂w2

∂2πm

∂w2∂w1

∂2πm

∂w2

2

=

−2A2B

2B−2A

With assumption E[˜

β]>E[˜γ]>0, we can obtain

A=2E[˜

β]3−E[˜

β]E[˜γ]2

4E[˜

β]2−E[˜γ]2>E[˜

β]3

4E[˜

β]2−E[˜γ]2>E[˜

β]2E[˜γ]

4E[˜

β]2−E[˜γ]2=B.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 11 of 21

It can be easily seen that His negative definite and πmis jointly concave with respect to

w1and w2. Therefore, differentiating πmby (w1,w2)and equating the expressions to zero,

we have

∂πm(w1,w2)

∂w1

=−2Aw1+2Bw2+E[˜

d1]−E[˜

β]S1+E[˜γ]S2+C=0,

∂πm(w1,w2)

∂w2

=−2Aw2+2Bw1+E[˜

d2]−E[˜

β]S2+E[˜γ]S1+C=0,

and the leader’s equilibrium prices (w1,w2)can be easily obtained by solving the above

two equations

w∗

1=A(E[˜

d1]−E[˜

β]S1+E[˜γ]S2+C)+B(E[˜

d2]−E[˜

β]S2+E[˜γ]S1+C)

2(A2−B2),

w∗

2=A(E[˜

d2]−E[˜

β]S2+E[˜γ]S1+C)+B(E[˜

d1]−E[˜

β]S1+E[˜γ]S2+C)

2(A2−B2).

(15)

Thus, Proposition 2 is proved.

Based on Propositions 1 and 2, we can get the equilibrium prices of the two retailers as

follows:

r∗

1=−2E˜

β2+E˜γ2w∗

1+E˜

βE˜γw∗

2+2E˜

βE˜

d1+E˜

β˜s1

4E˜

β2−E˜γ2

+

E˜γE˜

d2+E˜

β˜s2

4E˜

β2−E˜γ2,

r∗

2=−2E˜

β2+E˜γ2w∗

2+E˜

βE˜γw∗

1+2E˜

βE˜

d2+E˜

β˜s2

4E˜

β2−E˜γ2

+

E˜γE˜

d1+E˜

β˜s1

4E˜

β2−E˜γ2.

RS Model

The second possible structure is that the manufacturer distributes its products through

two supper retailers which are much bigger than the manufacturer. We assume that the

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 12 of 21

two retailers can dominate the sales prices by choosing their markup policies firstly. Then,

a Nash-Stackberg model can be applied as follows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

max

r1

πr1=E[(r1−˜s1)( ˜

d1−˜

β(r1+w∗

1)+˜γ(r2+w∗

2))]

max

r2

πr2=E[(r2−˜s2)( ˜

d2−˜

β(r2+w∗

2)+˜γ(r1+w∗

1))]

subject to:

M{ri−˜si≤0}=0, i=1, 2

where (w∗

1,w∗

2)solves problem:

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

max

w1,w2

πm=

2

i=1

E[(wi−˜c)( ˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i))]

subject to:

M{˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i)≤0}=0,

M{wi−˜c≤0}=0, i=1, 2.

(16)

Similarly, opposite to the decision sequence, we should derive the follower’s response

to the given retail markups r1and r2specified by the two retailers.

Proposition 3 Observing the two retailers’ retail markups r1and r2, the manufacturer’s

optimal response is:

w1(r1,r2)=E[˜

β]E[˜

d1]+E[˜γ]E[˜

d2]+(E[˜

β]+E[˜γ])(E[˜c1−α˜

β1−α]−E[˜c1−α˜γα])

2(E[˜

β]2−E[˜γ]2)−1

2r1,

w2(r1,r2)=E[˜

β]E[˜

d2]+E[˜γ]E[˜

d1]+(E[˜

β]+E[˜γ])(E[˜c1−α˜

β1−α]−E[˜c1−α˜γα])

2(E[˜

β]2−E[˜γ]2)−1

2r2.

(17)

Proof Referring to Eq.(6), the Hessian matrix with the corresponding second-order

derivatives of the equivalent objective function πmcan be attained:

H2=

∂2πm

∂w2

1

∂2πm

∂w1∂w2

∂2πm

∂w2∂w1

∂2πm

∂w2

2

=

−2E[˜

β]2E[˜γ]

2E[˜γ]−2E[˜

β]

Obviously, the Hessian matrix is negative definite with the assumption E[˜

β]>E[˜γ]>

0. Then, the optimal reaction functions to (r1,r2)can be derived from the first-order

conditions.

∂πm

∂w1

=−2E˜

βw1+2E˜γw2+E˜

d1−E˜

βr1+E˜γr2+E˜c1−α˜

β1−α

−E˜c1−α˜γα=0,

∂πm

∂w2

=−2E˜

βw2+2E˜γw1+E˜

d2−E˜

βr2+E˜γr1+E˜c1−α˜

β1−α

−E˜c1−α˜γα=0.

(18)

By solving Eqs. (18), we can obtain:

w1(r1,r2)=E[˜

β]E[˜

d1]+E[˜γ]E[˜

d2]+(E[˜

β]+E[˜γ])(E[˜c1−α˜

β1−α]−E[˜c1−α˜γα])

2(E[˜

β]2−E[˜γ]2)−1

2r1,

w2(r1,r2)=E[˜

β]E[˜

d2]+E[˜γ]E[˜

d1]+(E[˜

β]+E[˜γ])(E[˜c1−α˜

β1−α]−E[˜c1−α˜γα])

2(E[˜

β]2−E[˜γ]2)−1

2r2.

(19)

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 13 of 21

Thus, Proposition 3 is proved.

Given the manufacturer’s response, the two retailers compete to choose their own sales

markups.

Proposition 4 The Nash equilibrium between the two retailers is as follows:

r∗

1=

2E˜

βE˜

d1+E˜γE˜

d2−2E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

4E˜

β2−E˜γ2

+

2E˜

βE˜s1−α

1˜

β1−α+E˜γE˜s1−α

2˜

β1−α

4E˜

β2−E˜γ2,

r∗

2=

2E˜

βE˜

d2+E˜γE˜

d1−2E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

4E˜

β2−E˜γ2

+

2E˜

βE˜s1−α

2˜

β1−α+E˜γE˜s1−α

1˜

β1−α

4E˜

β2−E˜γ2.

(20)

Proof Substituting Eqs. (17) into the two retailers’ profit functions, we obtain:

πr1=− 1

2E˜

βr2

1+1

2E˜γr2r1+1

2E˜

d1−E˜c1−α˜

β1−α+E˜c1−α˜γα

+Es1−α

1˜

β1−αr1−E˜s1−α

1˜

dα

1+E˜s1−α

1˜

β1−αD1−E˜s1−α

1˜γαD2+1

2r2,

πr2=− 1

2E˜

βr2

2+1

2E˜γr1r2+1

2E˜

d2−E˜c1−α˜

β1−α+E˜c1−α˜γα

+Es1−α

2˜

β1−αr1−E˜s1−α

2˜

dα

2+E˜s1−α

2˜

β1−αD2−E˜s1−α

2˜γαD1+1

2r1

(21)

where

Di=

E˜

βE˜

di+E˜γE˜

d3−i+E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

2E˜

β2−E˜γ2,i=1, 2.

(22)

With the second-order condition ∂2πri

∂r2

i

=−E[˜

β]<0, the first-order conditions can be

shown as follows:

∂πr1

∂r1

=−E˜

βr1+1

2E˜γr2+1

2E˜

d1−E˜c1−α˜

β1−α+E˜c1−α˜γα+Es1−α

1˜

β1−α=0,

∂πr2

∂r2

=−E˜

βr2+1

2E˜γr1+1

2E˜

d2−E˜c1−α˜

β1−α+E˜c1−α˜γα+Es1−α

2˜

β1−α=0.

(23)

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 14 of 21

By solving Eq. (23), the optimal decisions of the two retailers can be derived:

r∗

1=

2E˜

βE˜

d1+E˜γE˜

d2−2E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

4E˜

β2−E˜γ2

+

2E˜

βE˜s1−α

1˜

β1−α+E˜γE˜s1−α

2˜

β1−α

4E˜

β2−E˜γ2,

r∗

2=

2E˜

βE˜

d2+E˜γE˜

d1−2E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

4E˜

β2−E˜γ2

+

2E˜

βE˜s1−α

2˜

β1−α+E˜γE˜s1−α

1˜

β1−α

4E˜

β2−E˜γ2.

(24)

Thus, Proposition 4 is proved.

Based on Propositions 3 and 4, we can get the equilibrium prices of the manufacturer

as follows:

w∗

1=

E˜

βE˜

d1+E˜γE˜

d2+E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

2E˜

β2−E˜γ2−1

2r∗

1,

w∗

2=

E˜

βE˜

d2+E˜γE˜

d1+E˜

β+E˜γE˜c1−α˜

β1−α−E˜c1−α˜γα

2E˜

β2−E˜γ2−1

2r∗

2.

(25)

VN Model

In some supply chains, there is no obvious dominance between chain members. In this

case, we assume that the three players move simultaneously. Then, a three-player Nash

game model can be built as follows:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

max

w1,w2

πm=

2

i=1

E(wi−˜c)˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i)

max

r1

πr1=E(r1−˜s1)˜

d1−˜

β(r1+w1)+˜γ(r2+w2)

max

r2

πr2=E(r2−˜s2)˜

d2−˜

β(r2+w2)+˜γ(r1+w1)

subject to:

M{wi−˜c≤0}=0, M{ri−˜si≤0}=0,

M˜

di−˜

β(ri+wi)+˜γ(r3−i+w3−i)≤0=0, i=1, 2.

(26)

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 15 of 21

Proposition 5 In the Nash equilibrium of the three-player game, the equilibrium prices

are obtained as follows:

r∗

1=−6E˜

β2+2E˜γ2D1+4E˜

βE˜γD2+6E˜

βE˜

d1+E˜s1−α

1˜

β1−α

9E˜

β2−E˜γ2

+

2E˜γE˜

d2+E˜s1−α

2˜

β1−α

9E˜

β2−E˜γ2,

r∗

2=−6E˜

β2+2E˜γ2D2+4E˜

βE˜γD1+6E˜

βE˜

d2+E˜s1−α

2˜

β1−α

9E˜

β2−E˜γ2

+

2E˜γE˜

d1+E˜s1−α

1˜

β1−α

9E˜

β2−E˜γ2,

w∗

1=12E˜

β2−2E˜γ2D1−2E˜

βE˜γD2−3E˜

βE˜

d1+E˜s1−α

1˜

β1−α

9E˜

β2−E˜γ2

−

E˜γE˜

d2+E˜s1−α

2˜

β1−α

9E˜

β2−E˜γ2,

w∗

2=12E˜

β2−2E˜γ2D2−2E˜

βE˜γD1−3E˜

βE˜

d2+E˜s1−α

2˜

β1−α

9E˜

β2−E˜γ2

−

E˜γE˜

d1+E˜s1−α

1˜

β1−α

9E˜

β2−E˜γ2.

Proof Note that πmis jointly concave in (w1,w2)and πriis concave with respect to

ri,i=1, 2. Set the first-order derivatives equaling zero as follows:

∂πm

∂w1

=−2E˜

βw1+2E˜γw2−E˜

βr1+E˜γr2+E˜

d1+E˜c1−α˜

βα

−E˜c1−α˜γα=0,

∂πm

∂w2

=−2E˜

βw2+2E˜γw1−E˜

βr2+E˜γr1+E˜

d2+E˜c1−α˜

βα

−E˜c1−α˜γα=0,

∂πr1

∂r1

=−2E˜

βr1+E˜γr2−E˜

βw1+E˜γw2+E˜

d1+E˜s1−α

2˜

βα=0,

∂πr2

∂r2

=−2E˜

βr2+E˜γr1−E˜

βw2+E˜γw1+E˜

d2+E˜s1−α

2˜

βα=0.

(27)

By solving Eqs.(27), the equilibrium solutions of Eqs.(26) can be attained.

Similarly, the sales prices, ordering quantities, and expected profits can also be attained.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 16 of 21

Numerical Experiments

By the results obtained from the above models, the expressions of each supply chain mem-

ber’s optimal decision and optimal expected profit can be easily attained. Because of the

complicated forms of the equilibrium prices and expected profits, it is quite difficult (if

possible) to conduct analytical comparisons to obtain general conclusions. Instead, we

employ numerical approach to illustrate supply chain members’ different behaviors and

performances facing various decision environments with different power structures and

uncertain degrees.

When dealing with indeterminate quantities (e.g., demands and costs) without enough

historical data (samples), experienced experts are usually employed to estimate these

quantities in practice and hence uncertainty theory can be applied. Interested readers can

consult Liu [1] (Chapter 16: Uncertain Statistics) to get more details on how to collect

experts’ experimental data and how to estimate empirical distribution of uncertain vari-

able from the experimental data. For simplicity, we just list uncertainty distributions of

the uncertain parameters in Table 2.

Additionally, this paper uses E[˜γ]/E[˜

β] to represent the substitutability of the two

retailers. Let ˜

βremain constant and vary ˜γat three levels, we show the data in Table 3.

With Lemma 3, we can attain E[˜c]=(9+11)/2=10, E[˜s1]=(5+7)/2=6, E[˜s2]=

(4+6)/2=5, E[˜

d1]=(2900 +2×3000 +3300)/4=3050, E[˜

d2]=(2800 +2×3000 +

3100)/4=2975, etc.

According to Lemma 4, we have

E˜c1−α˜

d1

α=1

0−1

c(1−α)−1

d1(α)dα

=0.5

0

(11(1−α)) +9α)(2900 ×(1−2α) +2×3000α)dα

+1

0.5

(11(1−α) +9α)(3000 ×(2−2α) +3300 ×(2α−1))dα

=3043.33,

E˜c1−α˜γα=1

0−1

c(1−α)−1

γ(α)dα

=1

0

(11(1−α)) +9α)(40(1−α) +60α)dα

=496.67.

(28)

Table 2 Distributions of uncertain variables

Parameter Distribution Expected value

˜

cL(9, 11) 10

˜

s1L(5, 7) 6

˜

s2L(4, 6)5

˜

d1Z(2900, 3000, 3300)3050

˜

d2Z(2800, 3000, 3100)2975

˜

βL(80, 120) 100

˜γL(40, 60) 50

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 17 of 21

Table 3 Substitutability of the two retailers

Substitutability ˜γExpected value E[˜γ]/E[˜

β]

Low L(15, 35)25 0.25

Medium L(40, 60)50 0.50

High L(70, 80)75 0.75

Inthesameway,wecangetthevaluesofE[˜c1−α˜

β1−α], E[˜c1−α˜γα], E[˜s1−α

r˜

β1−α],

E[˜s1−α

r˜γα], etc.

Substituting these values into the equilibrium solutions attained in the “Models and

Solution Approaches” section, we can obtain Table 4.

Referring to Table 4, one can find the following:

•Regardless of the leadership, the sales prices and the expected profits of the total

system in the MS and RS structures are the same, indicating that no matter who holds

the power has no influence on the total supply chain. However, in consideration of

the individual firms, the firm acts as a leader will gain more profit than as a follower.

•The dominant power structure will increase the sales prices and lower the profits of

the whole supply chain as the sales prices in the VN case are the lowest.

Next,weanalyzetheeffectsoftheuncertaindegreesofthemanufacturingcostsand

sales costs on the optimal pricing decisions under the three possible structures. The

uncertain degree of a parameter mainly depends on its own inherent variance and experts’

personal knowledge. If more information on a parameter is available, more accurate esti-

mations the experts can make. Then, as a result, the uncertain degree of the parameter

will decrease. Of course, if we have enough information (historical data) on these param-

eters and their distributions can be precisely estimated, then we can apply probability

theory rather than uncertainty theory. Note that the forms of the equilibrium solutions

of these models under stochastic environment are the same to those under deterministic

environment with the assumption of risk neutral members.

By varying the uncertain degrees of these parameters and keeping the other parameters

unchanged as shown in Table 2, the changes of optimal prices are shown in Tables 5, 6,

and 7.

Referring to Table 5, we obtain the following results:

Table 4 The equilibrium results of the three structures under different substitutability

Sub. Str. w1w2πmr1p1πr1r2p2πr2πt

0.25 MS 22.2667 22.4667 13,748.63 11.2540 33.5206 3124.97 10.3651 32.8317 3656.11 20,529.71

VN 18.9688 19.0918 12,794.36 12.6625 31.6312 4775.00 11.8163 30.9082 5416.60 22,985.96

RS 17.0794 17.1683 8250.51 16.4413 33.5206 5815.78 15.6635 32.8317 6463.42 20,529.71

0.5 MS 32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23

VN 27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50384.61

RS 24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23

0.75 MS 62.5238 62.8095 111,801.44 16.4291 78.9529 11,661.49 15.6109 78.4204 12,492.06 135,954.99

VN 56.7831 56.9354 110,451.84 17.5481 74.3312 14,067.03 16.8148 73.7503 15,179.93 139,698.79

RS 52.1614 52.2653 89,945.35 26.7915 78.9529 22,399.47 26.1552 78.4204 23,610.17 135,954.99

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 18 of 21

Table 5 Effects of the manufacturing cost’s uncertain degree on the prices and profits

˜

cw

1w2πmr1p1πr1r2p2πr2πt

MS 10 32.2167 32.4667 33,278.55 13.4389 45.6556 6005.34 12.5889 45.0556 6663.09 45,946.97

L(9, 11)32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23

L(8, 12)32.4167 32.6667 35,328.77 13.3722 45.7889 5908.37 12.5222 45.1889 6564.12 47,801.27

L(7, 13)32.5167 32.7667 36,355.88 13.3389 45.8556 5860.23 12.4889 45.2556 6514.98 48,731.08

VN 10 27.8019 27.9448 31,947.14 14.8962 42.6981 8346.42 14.1105 42.0552 9196.32 49,489.87

L(9, 11)27.9219 28.0648 32983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61

L(8, 12)28.0419 28.1848 34,021.08 14.8162 42.8581 8206.85 14.0305 42.2152 9053.33 51,281.26

L(7, 13)28.1619 28.3048 35,059.98 14.7762 42.9381 8137.55 13.9905 42.2952 8982.31 52,179.83

RS 10 24.8444 24.9444 22,185.20 20.8111 45.6556 11,440.30 20.1111 45.0556 12,321.47 45,946.97

L(9, 11)24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23

L(8, 12)25.1111 25.2111 24,433.12 20.6778 45.7889 11,245.49 19.9778 45.1889 12,122.65 47,801.27

L(7, 13)25.2444 25.3444 25,558.42 20.6111 45.8556 11,148.75 19.9111 45.2556 12,023.91 48,731.08

•The wholesale prices will increase while the markup prices of the two retailers will

drop when the uncertain degree of the manufacturing cost increases in the three

structures.

•The manufacturer can benefit from the vagueness of the manufacturing costs, while

the other channel members, namely the two retailers, will suffer from less profits.

•When the uncertain degree of the manufacturing cost increases, the whole supply

chain will gain more, meanwhile, consumers have to pay more.

Referring to Tables 6 and 7, we obtain the following results:

•When the uncertain degree of its sales cost becomes higher, each retailer will be

charged a lower wholesale price, while the wholesale price for the other retailer will

keep the same in the MS structure and drop slightly in the RS and VN structures.

•Both the retailers can charge higher markup prices when the uncertain degree of

either of the sales costs increases.

•Contrary to the effect of the manufacturing cost’s uncertain degree, the retailers can

gain higher profits while the common manufacturer will suffer from lower profit with

the increase of the uncertain degree of the sales costs.

Table 6 Effects of the 1st retailer’s sales cost’s uncertain degree on the prices and profits

˜

s1w1w2πmr1p1πr1r2p2πr2πt

MS 6 32.3500 32.5667 34,351.97 13.3544 45.7044 5408.79 12.5511 45.1178 6596.68 46,357.43

L(5, 7)32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23

L(4, 8)32.2833 32.5667 34,254.12 13.4567 45.7400 6504.81 12.5600 45.1267 6630.32 47,389.24

L(3, 9)32.2500 32.5667 34,205.35 13.5078 45.7578 7052.98 12.5644 45.1311 6647.14 47,905.47

VN 6 27.9448 28.0686 33,030.55 14.8105 42.7552 7762.45 14.0629 42.1314 9096.45 49,889.45

L(5, 7)27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61

L(4, 8)27.8990 28.0610 32,936.47 14.9019 42.8010 8790.64 14.0781 42.1390 9152.89 50,880.00

L(3, 9)27.8762 28.0571 32,889.57 14.9476 42.8238 9304.94 14.0857 42.1429 9181.13 51,375.64

RS 6 24.9956 25.0822 23,341.49 20.7089 45.7044 10,817.57 20.0356 45.1178 12,198.37 46,357.43

L(5, 7)24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23

L(4, 8)24.9600 25.0733 23,275.99 20.7800 45.7400 11,867.93 20.0533 45.1267 12,245.32 47,389.24

L(3, 9)24.9422 25.0689 23,243.32 20.8156 45.7578 12,393.35 20.0622 45.1311 12,268.81 47,905.47

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 19 of 21

Table 7 Effects of the 2nd retailer’s sales cost’s uncertain degree on the prices and profits

˜

s2w1w2πmr1p1πr1r2p2πr2πt

MS 5 32.3167 32.6000 34,352.97 13.4011 45.7178 5950.10 12.5044 45.1044 6091.67 46,394.74

L(4.5, 5.5)32.3167 32.5833 34,327.97 13.4033 45.7200 5953.42 12.5300 45.1133 6352.58 46,633.97

L(4, 6)32.3167 32.5667 34,302.99 13.4056 45.7222 5956.74 12.5556 45.1222 6613.49 46,873.23

L(3.5, 6.5)32.3167 32.5500 34,278.04 13.4078 45.7244 5960.07 12.5811 45.1311 6874.41 47,112.52

VN 5 27.9257 28.0876 33,031.37 14.8486 42.7743 8262.93 14.0248 42.1124 8641.06 49,935.37

L(4.5, 5.5)27.9238 28.0762 33,007.41 14.8524 42.7762 8269.70 14.0476 42.1238 8882.86 50,159.97

L(4, 6)27.9219 28.0648 32,983.47 14.8562 42.7781 8276.47 14.0705 42.1352 9124.66 50,384.61

L(3.5, 6.5)27.9200 28.0533 32,959.55 14.8600 42.7800 8283.25 14.0933 42.1467 9366.47 50,609.27

RS 5 24.9822 25.0956 23,341.89 20.7356 45.7178 11,329.51 20.0089 45.1044 11,723.34 46,394.74

L(4.5, 5.5)24.9800 25.0867 23,325.30 20.7400 45.7200 11,336.09 20.0267 45.1133 11,972.58 46,633.97

L(4, 6)24.9778 25.0778 23,308.72 20.7444 45.7222 11,342.67 20.0444 45.1222 12,221.84 46,873.23

L(3.5, 6.5)24.9756 25.0689 23,292.15 20.7489 45.7244 11,349.26 20.0622 45.1311 12,471.12 47,112.52

•In the same way, when the uncertain degree of the sales costs increases, end

consumers will afford higher prices while the whole supply chain will gain more.

Similarly, more experiments can be conducted to explore the effects of the other

parameters, such as the two price elastic coefficients or the market sizes of the two

retailers.

Conclusions

In this paper, we considered a pricing competing problem in supply chain with com-

peting retailers. Specially, the manufacturing costs, sales costs, and consumer demands

were characterized as uncertain variables. Meanwhile, three decentralized models based

on uncertainty theory and game theory were built to formulate the pricing decision

problems. The equilibrium behaviors of the supply chain under three possible power

structures were derived from these models. Afterwards, numerical experiments were also

given to explore the impacts of uncertain degrees of the parameters on the pricing deci-

sions. The results showed that the existence of dominant power in a supply chain will

increase the sales prices and lower the profits of the whole supply chain. It was also found

that the supply chain members may benefit from higher uncertain degrees of their own

costs, whereas the supply chain members at the other level will gain less profits. Addi-

tionally, the results demonstrated that the uncertainty of the supply chain will make end

consumers pay more.

As this study was based on some assumptions, future researches can focus on some

more general problems. For instance, this paper only considered one type of indetermi-

nacy, while the real world might behave more complicated in which randomness and

uncertainty might coexist. Therefore, one possible extension of this paper is to study

the pricing problem with twofold indeterminacy, in which uncertain random variable

can be applied. Besides, this paper assumed that all the participants are risk neu-

tral while the decision makers may be risk sensitive in the real world. The research

can be more applicable if the equilibrium behaviors with risk-sensitive members are

considered.

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 20 of 21

Acknowledgements

This work was supported by National Natural Science Foundation of China (No. 71371141) and the Fundamental

Research Funds for the Central Universities.

Authors’ Contributions

YW and HK carried out the study in the paper and drafted the first version of the manuscript. HH designed the framework

of the paper and with ZC carried out the numerical experiments. All authors read and approved the final manuscript.

Competing Interests

The authors declare that they have no competing interests.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Author details

1School of Economics and Management, Tongji University, Siping Road, Shanghai 200092, China. 2Department of

Geotechnical Engineering, Tongji University, Siping Road, Shanghai 200092, China.

Received: 5 February 2017 Accepted: 5 March 2017

References

1. Liu, B: Uncertainty Theory, 4th Edition. Springer, Berlin (2015)

2. Liu, B: Uncertainty Theory, 2nd Edition. Springer, Berlin (2007)

3. Liu, B: Uncertainty Theory: a Branch of Mathematics for Modeling Human Uncertainty. Springer, Berlin (2010)

4. McGuire, TW, Staelin, R: An industry equilibrium analysis of downstream vertical integration. Mark. Sci. 2(2), 161–191

(1983)

5. Coughlan, AT: Competition and cooperation in marketing channel choice: theory and application. Mark. Sci. 4(2),

110–129 (1985)

6. Choi, SC: Price competition in a channel structure with a common retailer. Mark. Sci. 10(4), 271–296 (1991)

7. Zhao, J, Tang, W, Zhao, R, Wei, J: Pricing decisions for substitutable products with a common retailer in fuzzy

environments. Eur. J. Oper. Res. 216(2), 409–419 (2012)

8. Huang, H, Ke, H: Pricing decision problem for substitutable products based on uncertainty theory. J. Intell. Manuf

(2014). doi:10.1007/s10845-014-0991-7

9. Wu, D, Baron, O, Berman, O: Bargaining in competing supply chains with uncertainty. Eur. J. Oper. Res. 197(2),

548–556 (2009)

10. Anderson, EJ, Bao, Y: Price competition with integrated and decentralized supply chains. Eur. J. Oper. Res. 200(1),

227–234 (2010)

11. Webb, KL: Managing channels of distribution in the age of electronic commerce. Ind. Mark. Manag. 31(2), 95–102

(2002)

12. Dumrongsiri, A, Fan, M, Jain, A, Moinzadeh, K: A supply chain model with direct and retail channels. Eur. J. Oper. Res.

187(3), 691–718 (2008)

13. Hu, W, Li, Y: Retail service for mixed retail and e-tail channels. Ann. Oper. Res. 192(1), 151–171 (2012)

14. Soleimani, F: Optimal pricing decisions in a fuzzy dual-channel supply chain. Soft Comput. 20(2), 689–696 (2016)

15. Ingene, CA, Parry, ME: Channel coordination when retailers compete. Mark. Sci. 14(4), 360–377 (1995)

16. Ingene, CA, Parry, ME: Coordination and manufacturer profit maximization: the multiple retailer channel. J. Retail.

71(2), 129–151 (1995)

17. Yang, S-L, Zhou, Y-W: Two-echelon supply chain models: considering duopolistic retailers’ different competitive

behaviors. Int. J. Prod. Econ. 103(1), 104–116 (2006)

18. Wu, C-H, Chen, C-W, Hsieh, C-C: Competitive pricing decisions in a two-echelon supply chain with horizontal and

vertical competition. Int. J. Prod. Econ. 135(1), 265–274 (2012)

19. Huang, H, Ke, H, Wang, L: Equilibrium analysis of pricing competition and cooperation in supply chain with one

common manufacturer and duopoly retailers. Int. J. Prod. Econ. 178, 12–21 (2016)

20. Bernstein, F, Federgruen, A: Decentralized supply chains with competing retailers under demand uncertainty.

Manag. Sci. 51(1), 18–29 (2005)

21. Xiao, T, Yang, D: Price and service competition of supply chains with risk-averse retailers under demand uncertainty.

Int. J. Prod. Econ. 114(1), 187–200 (2008)

22. Shi, R, Zhang, J, Ru, J: Impacts of power structure on supply chains with uncertain demand. Prod. Oper. Manag. 22(5),

1232–1249 (2013)

23. Mahmoodi, A, Eshghi, K: Price competition in duopoly supply chains with stochastic demand. J. Manuf. Syst. 33(4),

604–612 (2014)

24. Li, X, Li, Y, Cai, X: Remanufacturing and pricing decisions with random yield and random demand. Comput. Oper.

Res. 54, 195–203 (2015)

25. Zhou, C, Zhao, R, Tang, W: Two-echelon supply chain games in a fuzzy environment. Comput. Ind. Eng. 55(2),

390–405 (2008)

26. Zhao, J, Liu, W, Wei, J: Competition under manufacturer service and price in fuzzy environments. Knowledge-Based

Syst. 50, 121–133 (2013)

27. Sang, S: Price competition of manufacturers in supply chain under a fuzzy decision environment. Fuzzy Optim.

Decis. Making. 14(3), 335–363 (2015)

28. Liu, S, Xu, Z: Stackelberg game models between two competitive retailers in fuzzy decision environment. Fuzzy

Optim. Decis. Making. 13(1), 33–48 (2014)

Ke et al. Journal of Uncertainty Analysis and Applications (2017) 5:5 Page 21 of 21

29. Ke, H, Huang, H, Ralescu, DA, Wang, L: Fuzzy bilevel programming with multiple non-cooperative followers: model,

algorithm and application. Int. J. Gen. Syst. 45(3), 336–351 (2016)

30. Chen, X: American option pricing formula for uncertain financial market. Int. J. Oper. Res. 8(2), 32–37 (2011)

31. Sun, J, Chen, X: Asian option pricing formula for uncertain financial market. J. Uncertain. Anal. Appl. 3(1), 11 (2015)

32. Gao, Y: Uncertain models for single facility location problems on networks. Appl. Math. Model. 36(6), 2592–2599

(2012)

33. Ning, Y, Ke, H, Fu, Z: Triangular entropy of uncertain variables with application to portfolio selection. Soft Comput.

19(8), 2203–2209 (2015)

34. Ding, S: Uncertain multi-product newsboy problem with chance constraint. Appl. Math. Comput. 223, 139–146

(2013)

35. Ke, H: A genetic algorithm-based optimizing approach for project time-cost trade-off with uncertain measure. J.

Uncertain. Anal. Appl. 2(1), 8 (2014)

36. Wang, L, Huang, H, Ke, H: Chance-constrained model for rcpsp with uncertain durations. J. Uncertain. Anal. Appl.

3(1), 12 (2015)

37. Ma, W, Che, Y, Huang, H, Ke, H: Resource-constrained project scheduling problem with uncertain durations and

renewable resources. Int. J. Mach. Learn. Cybernet. 7(4), 613–621 (2016)

38. Ke, H, Su, T, Ni, Y: Uncertain random multilevel programming with application to production control problem. Soft

Comput. 19(6), 1739–1746 (2015)

39. Chen, L, Peng, J, Liu, Z, Zhao, R: Pricing and effort decisions for a supply chain with uncertain information. Int. J. Prod.

Res (2016). doi:10.1080/00207543.2016.1204475

40. Liu, B: Some research problems in uncertainty theory. J. Uncertain Syst. 3(1), 3–10 (2009)

41. Liu, Y, Ha, M: Expected value of function of uncertain variables. J. Uncertain Syst. 4(3), 181–186 (2010)

42. Wei, J, Govindan, K, Li, Y, Zhao, J: Pricing and collecting decisions in a closed-loop supply chain with symmetric and

asymmetric information. Comput. Oper. Res. 54, 257–265 (2015)

Submit your manuscript to a

journal and beneﬁ t from:

7 Convenient online submission

7 Rigorous peer review

7 Immediate publication on acceptance

7 Open access: articles freely available online

7 High visibility within the ﬁ eld

7 Retaining the copyright to your article

Submit your next manuscript at 7 springeropen.com