Download Competitive pricing decision in a two

Competitive pricing decision in a two-echelon supply chain with competing
retailers under uncertainty
Hu Huang Hua Ke∗ Lei Wang
School of Economics and Management, Tongji University, Shanghai 200092, China
Abstract
This paper explores a supply chain pricing competition problem in a two-echelon supply chain with one manufacturer
and two competing retailers. Specially, the manufacturing costs, sales costs and market bases are all characterized as
uncertain variables whose distributions are estimated by experts’ experimental 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 should the channel members choose their most profitable pricing strategy
in face of uncertainties are 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 dominant powers in 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, whereas
the supply chain members in the other level will gain less profits. Additionally, the results show that the uncertainness
of the supply chain will make the end consumers pay more. Some other interesting managerial highlights are also
gained in this paper.
Keywords: two-echelon supply chain, pricing competition, competing retailers, uncertain variable, Stackelberg game
1. Introduction
In this paper, we explore a pricing decision problem in a decentralized supply chain with competing retailers under
uncertain settings. In such supply chain, the costs and demands are usually unavailable to the managers which may
be subject to some inherent indeterministic factors such as changes of material costs, workers’ expenses, technology
improvements and customer incomes. It is necessary to identify the distributions of these parameters which are
essential for the supply chain managers when they make decisions, e.g., choosing prices, determining the production
quantities, choosing the ordering quantities and adding investment. Theoretically, we can collect enough samples to
estimate the distributions of these parameters before we make decisions. However, the products, especially the hightech products, e.g. smartphone, PC and other digital devices, are rapidly updating recently. The costs and demands
of these new products are usually lack of historical samples; or even though we can collect enough data but may not
be applicable due to the changeable environment. Practically, experienced experts’ experience data (belief degrees)
are often used to estimate the distributions in cases without historical examples. Some surveys have shown that,
however, human beings (even the most experienced experts) usually estimate a much wider range of values than it
actually takes. It makes the belief degrees behave quite different from frequency, indicating that human belief degree
given by managers and experts should not be treated as random variable or fuzzy variable [1]. Instead, uncertainty
theory, initiated by Liu [2] and refined by Liu [3] based on normality, duality, subadditivity and product axioms, can
be introduced to deal with variables estimated by human belief degrees.
The goal of this paper is to study these uncertainties existing in supply chain pricing problems by using uncertainty
theory. Our work mainly focuses on the equilibrium behavior of the decentralized supply chain with one common
∗ Corresponding
author.
Email: [email protected] (Hua Ke)[email protected] (Hu Huang) [email protected](Lei Wang)
Preprint submitted to Submitted for publication
May 22, 2015
manufacturer (upstream member) and duopoly retailers (downstream members) under uncertain settings. This type of
supply chain is not uncommon in the real industry world, e.g., smartphone producers distribute their products through
both traditional retailer and online retailer per area. Specially, in consideration of the changeable and complicated
environments, the products’ demands and costs, lack of historical data and estimated by experienced experts in practice, are characterized as uncertain variables. How should the supply chain managers choose the most profitable price
policies by using experts’ estimations? What effects the uncertain degrees of the parameters might have on the supply chain members’ pricing decisions and expected profits? Additionally, we assume that the common manufacturer
determines the wholesale prices and the two retailers compete for end consumers by choosing their own sales prices
respectively. The decision sequence of the channel members are 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. While in some other chains, the retailers like Warmart and Carrefour who are much
bigger than most manufacturers often hold the dominant power. There are also some chains, however, no absolute
dominance existing and each member hold the same power. How should these various power structures affect the
performances of the supply chain?
In order to cover these problems, three decentralized pricing models based on uncertainty theory and game theory
are built. And then the corresponding closed-form equilibrium solutions are derived from these models. Afterwards,
numerical experiments are conducted to analyze the equilibrium behaviors of the supply chain under different power
structures and uncertain settings. It is found that regardless the power structurer, the manufacturer can benefit from
higher uncertain degree of its manufacturing cost , whereas the retailers will gain less profits. Correspondingly, when
the uncertain degree of the sales cost increase, both retailer can procure more while the manufacture will gain less.
Additionally, the uncertainness of the supply chain will make the the end consumers pay more. Some other interesting
managerial highlights are also gained in this paper.
The reminder of this paper is organized as follows: Some related researches are shown in Section 2. Following
that, some useful notations and necessary assumptions are discussed in Section 3. Three models are employed to
derive the equilibria under three possible scenarios in Section 4. Afterwards, in Section 5, numerical experiments are
applied to examine the equilibrium behaviors of the supply chain members. Some conclusions and possible extensions
about this paper are discussed in Section 6.
2. Literature Review
By now, the pricing competition problem in decentralized supply chains, initialed in 1980s[4, 5], has been well
studied by both scholars and practitioners. Most of these researches concerning about the pricing competition problem
are focused on the following four structures: monopoly common retailer structure [6, 7, 8, 9], chain-to-chain structure
[4, 10, 11], dual channel structure [12, 13, 14, 15]and monopoly common manufacture structure which is studied in
this paper. The researches of the monopoly common manufacture structure were initialed by Ingene and Parry [16]
who considered a coordination problem in supply chain where a manufacturer sells its products through competing
retailers. Meanwhile, Ingene and Parry [17] demonstrated that the manufacturer can generally obtain more profits by
setting a unique two-part tariff wholesale pricing policy. As for pricing competition, , Yang and Zhou [18] 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 Collusion behaviors on the supply chains’ performance. After that, Wu et al. [19] investigated the
pricing decisions in this monopoly common manufacture channel structure and they built six non-cooperative models
where the two retailers play as Stackelberg or Bertrand games under three possible power structures.
The work above has 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 of a most important tendency of
these researches is to explore the pricing competition problem under environments with indeterminate factors, namely randomness, fuzziness and uncertainty; and correspondingly probability theory, fuzzy set theory and uncertainty
theory can be employed to the corresponding indeterminate variables.
For instance, Bernstein and Federgruen [20] investigated the equilibrium behavior of decentralized supply chains
with competing retailers under random demand and designed some contractual arrangements between the parties that
allow the decentralized chain to perform as well as 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
2
of the retailers risk sensitivity on the manufacturers’ equilibrium strategies. Wu et al. [10] considered a joint pricing
and quantity competition between two separate supply chains in random environment and explored the effect 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 examine how power structure and demand indeterminacy
affect supply chain members’ performances. Mahmoodi and Eshghi [23] studied this pricing problem in duopoly
supply chains with stochastic demand and explored the effect of competition and demand indeterminacy intensity
on the equilibrium of the structures by a numerical example. Li et al. [24] studied the pricing and remanufacturing
decisions problem with random yield and random demand.
More recently, fuzzy set theory, initiated by Zadeh [25], has been introduced to the pricing decision problem. Zhou
et al. [26] considered the pricing decision problem in supply chains consisting of a manufacturer and a retailer in fuzzy
environment. Zhao et al.[7, 27, 28] and Wei and Zhao [29] studied the pricing problem of two substitutable products
in supply chains with different structures in which the manufacturing costs and consumer demands are described by
fuzziness. Specially, , Wei et al. ([30] and Liu and Xu [31] studied the pricing problem in a fuzzy supply chain
consisting of one manufacturer and two competitive retailers.
However, to the best of our knowledge, there are little researches on the pricing decision problems with competing
retailers with uncertain settings. Differing from the literature above, this paper addresses the pricing equilibrium under
circumstances where only belief degree is available by using uncertainty theory. By far, the new theory has been
successfully applied to deal with many uncertain decision-making problems, e.g., option pricing [32, 33], portfolio
selection [34], facility location [35], production control problem [36], inventory problem [37], project scheduling
problem [38], assignment problem [39] and network problem [40]. Recently, Huang and Ke [8] applied uncertainty
theory to a pricing decision problem in supply chain with one common retailer, but did not consider the pricing
problem in supply chain with competing retailers.
This paper, extending the above literature [18, 19, 30, 31] about pricing problem with competing retailers, employs
uncertainty theory to depict the costs and demands estimated by experts experience data in practice. And three
uncertain game models are built to explore the equilibrium behavior of the decentralized supply chain under different
power structures. Meanwhile, we additionally consider the retailers’ costs which are often ignored in most literature.
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 the numerical
experiments conducted in this paper.
3. 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 a same market.
3.1. 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 (r1 , r2 ) respectively. Note that the sales prices can be decided (pi =
wi + ri , i = 1, 2) when the corresponding wholesale price and retailer markup policy are determined. Consistent with
the extant literature [6] [18] [11] [19] [41], we suppose that each retailer faces the following linear demand function:
qi = di − βpi + γp3−i , i = 1, 2.
where di represents the potential market size of retailer i, β is the self-price elastic coefficient, γ is the cross-price
elastic coefficient and pi is the sales prices of retailer i. Note that γ/β denote the substitutability of the two retailers.
Owing to the complicated and changeable environments, the manufacturing cost c̃, sales costs ( s̃1 , s̃2 ), market
sizes (d̃1 , d̃2 ) and price elastic coefficients β̃, γ̃, also subjected to some indeterministic factors, are usually unknown to
the supply chain managers when making price decisions. Thus, experts’ belief degrees are often employed to estimate
these parameters due to lacking history data. Accordingly, we assume Table 1.
In order to attain closed-form solutions, some necessary assumptions are made in this paper. Note that some basic
concepts and introductions about uncertain variable are provided in Appendix. Interested readers can consult [1] for
3
Table 1: Notations
c̃ :
s̃i :
wi :
ri :
pi :
d̃i :
πm :
πr :
unit manufacturing cost, which is an uncertain variable.
unit sales cost of retailer i, which is an uncertain variable.
unit wholesale price of product i, which is the manufacturer’s decision variable.
unit markup price of product i, which is the retailer’s decision variable.
unit retail price of product i, where pi = wi + ri .
the market base of product i, which is an uncertain variable.
P
the manufacturer’s profit: πm = 2i=1 (wi − c̃)(d̃i − β̃pi + γ̃p3−i ).
the profit of retailer i: πri = (ri − s̃i )(d̃i − β̃pi + γ̃p3−i ).
more details about uncertainty theory. A brief introduction about uncertain variable and uncertain programming is
presented in Preliminaries (Section 6).
• Because the 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 elastic coefficients β̃ and γ̃ satisfy: E[β̃] > E[γ̃] > 0.
• All the uncertain coefficients are assumed nonnegative and mutually independent.
• All the supply chain members have access to the same information about the demands and the costs.
• All the supply chain members are risk neutral.
• The wholesale prices and markups should exceed the costs and the retailers’ demands are always positive:
M{wi − c̃ ≤ 0} = 0, M{ri − s̃i ≤ 0} = 0,
(1)
M{d̃i − β̃pi + γ̃p3−i ≤ 0} = 0, i = 1, 2.
3.2. Crisp form of the Objective function
In order to derive the equilibrium solutions, we should transform these uncertain objective functions to equivalent crisp form first. Let c̃, s̃i , d̃i , β̃ and γ̃ be positive independent uncertain variables with the regular uncertainty
distributions Φc , Φ si ,Φdi , Φβ and Φγ , i = 1, 2.
As M{wi − c̃ ≤ 0} = 0, M{ri − s̃i ≤ 0} = 0 M{d̃i − β̃pi + γ̃p3−i ≤ 0} = 0, i = 1, 2, one can find that πm is monotone
increasing with d̃i , γ̃ and monotone decreasing with c̃, β̃, s̃i . Thus, referring to Lemma 4, the expected functions of
the manufacturer can be attained as follows:
πm =
2
X
E[(wi − c̃i )(d̃i − β̃(ri + wi ) + γ̃(r3−i + w3−i ))]
i=1
=
2 Z
X
i=1
(2)
1
[(wi −
Φ−1
c (1
0
−
α))(Φ−1
di (α)
−
Φ−1
β (1
− α)(ri + wi ) +
Φ−1
γ (α)(r3−i
+ w3−i ))]dα
By defining
E[ã1−α b̃α ] =
1
Z
0
1−α 1−α
E[ã
b̃
]=
−1
Φ−1
a (1 − α)Φb (1 − α)dα,
Z
1
(3)
Φ−1
a (1
0
−
α)Φ−1
b (1
− α)dα.
−1
where Φ−1
a and Φb is the inverse distribution function of ã and b̃ respectively.
4
Then the crisp form of πm can be attained as follows:
πm =
2 Z
X
i=1
=
2
X
0
1
−1
−1
−1
[(wi − Φ−1
c (1 − α))(Φdi (α) − Φβ (1 − α)(ri + wi ) + Φγ (α)(r3−i + w3−i ))]dα
{−E[β̃]w2i + E[γ̃]w3−i wi + (−E[β̃]ri + E[γ̃]r3−i + E[d̃i ] + E[c̃1−α
β̃1−α ])wi
i
(4)
i=1
− E[c̃1−α d̃iα ] + E[c̃1−α β̃1−α ]ri − E[c̃1−α γ̃α ](r3−i + w3−i )}.
In the similar way, the crisp forms of the expected profit functions of the two retailers can also be attained as
follows:
πri = − E[β̃]ri2 + E[γ̃]r3−i ri + (−E[β̃]wi + E[γ̃]w3−i + E[d̃i ] + E[ s̃1−α
β̃1−α ])ri
i
(5)
d̃iα ] + E[ s̃1−α
β̃1−α ]wi − E[ s̃1−α
γ̃α ](r3−i + w3−i ), i = 1, 2.
− E[ s̃1−α
i
i
i
4. Models and solution approaches
In this section, starting with the Manufacturer Stackelberg (MS) model, we present the general formulations and
solutions to the three supply chain pricing models with different power structures.
4.1. MS Model
In first case, we assume that the manufacturer plays a dominant role and announces its wholesale prices (w1 , w2 ) to
maximize its own profits allowing for the retailers’ react functions. Observing the wholesale prices, the two retailers
non-cooperatively choose their own markup pricing schemes ri to 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:

2
P


∗

max πm = E[(wi − c̃)(d̃i − β̃(ri∗ + wi ) + γ̃(r3−i
+ w3−i ))]



w
,w
1 2

i=1





subject to:






M{w1 − c̃ ≤ 0} = 0 M{w2 − c̃ ≤ 0} = 0






where (r1∗ , r2∗ ) solves problems:



 
(6)


max πr1 = E[(r1 − s˜1 )(d̃1 − β̃(r1 + w1 ) + γ̃(r2 + w2 ))]





r1










max πr2 = E[(r2 − s˜2 )(d̃2 − β̃(r2 + w2 ) + γ̃(r1 + w1 ))]







 r2






subject to:













M{r1 − s̃1 ≤ 0} = 0
M{r2 − s̃2 ≤ 0} = 0







 M{d̃ − β̃(r + w ) + γ̃(r + w ) ≤ 0} = 0, i = 1, 2.
 
i
i
i
3−i
3−i
To solve this Stackelberg-Nash game model, opposite to the decision sequence, we should derive the Nash equilibrium in the lower level for the given wholesale prices w1 and w2 specified by the manufacturers in advance.
Referring to the two retailers’ expected value functions with the given wholesale prices, we can get
∂2 π r1
∂2 πr2
= −2E[β̃] < 0,
= −2E[β̃] < 0,
2
∂r1
∂r22
(7)
with the assumption E[β̃] > E[γ̃] > 0. Hence, πr1 and πr2 are concave in r1 and r2 respectively. Set the first-order
derivatives equaling zero.
∂πr1
1−α
= − 2E[β̃]r1 + E[γ̃]r2 − E[β̃]w1 + E[γ̃]w2 + E[d̃1 ] + E[ s̃1−α
] = 0,
1 β̃
∂r1
∂πr2
1−α
= − 2E[β̃]r2 + E[γ̃]r1 − E[β̃]w2 + E[γ̃]w1 + E[d̃2 ] + E[ s̃1−α
] = 0.
2 β̃
∂r2
5
(8)
The followers’ optimal responses to (w1 , w2 ) can be easily obtained by solving the above two equations as follows:
−2E[β̃]2 + E[γ̃]2
E[β̃]E[γ̃]
w1 +
w2
2
2
4E[β̃] − E[γ̃]
4E[β̃]2 − E[γ̃]2
1−α 1−α
2E[β̃](E[d̃1 ] + E[β̃1−α s̃1−α
s̃2 ])
1 ]) + E[γ̃](E[d̃2 ] + E[β̃
,
+
4E[β̃]2 − E[γ̃]2
−2E[β̃]2 + E[γ̃]2
E[β̃]E[γ̃]
r2∗ (w1 , w2 ) =
w2 +
w1 +
4E[β̃]2 − E[γ̃]2
4E[β̃]2 − E[γ̃]2
2E[β̃](E[d̃2 ] + E[β̃1−α s̃21−α ]) + E[γ̃](E[(d̃1 ] + E[β̃1−α s̃1−α
1 ])
.
2
2
4E[β̃] − E[γ̃]
r1∗ (w1 , w2 ) =
(9)
Substituting the two retailer’s react functions Eqs (9) into the the manufacturer’s profit function, and assuming
E[β̃]2 E[γ̃]
,
4E[β̃]2 − E[γ̃]2
˜ γ̃α ])
(2E[β̃]2 + E[β̃]E[γ̃])(E[c̃1−α β̃1−α ] − E[c1−α
C=
,
4E[β̃]2 − E[γ̃]2
]) + E[γ̃](E[d̃3−i ] + E[β̃1−α s̃1−α
2E[β̃](E[d̃i ] + E[β̃1−α s̃1−α
i
3−i ])
, i = 1, 2,
Si =
2
2
4E[β̃] − E[γ̃]
A=
2E[β̃]3 − E[β̃]E[γ̃]2
,
4E[β̃]2 − E[γ̃]2
B=
we obtain
πm =
2
X
E[(wi − c̃)(d̃i − β̃(ri∗ + wi ) + γ̃(ri∗ + wi ))]
i=1
=
2
X
i=1
E[(wi − c̃)(d̃i − β̃(
2E[β̃]2
E[β̃]E[γ̃]
2E[β̃]2
E[β̃]E[γ̃]
w
+
w
+
S
)
+
γ̃(
w3−i +
wi + S 3−i ))]
i
3−i
i
2
2
2
2
2
2
4E[β̃] − E[γ̃]
4E[β̃] − E[γ̃]
4E[β̃] − E[γ̃]
4E[β̃]2 − E[γ̃]2
2
X
= {E[d̃i ]wi − Aw2i + Bw3−i wi − E[β̃]S i wi + E[γ̃]S 3−i wi − E[c̃1−α d̃iα ] + E[c̃1−α β̃1−α ]S i − E[c̃1−α γ̃α ]S 3−i
i=1
2E[c̃1−α β̃1−α ]E[β̃]2 − E[β̃]E[γ̃]E[c̃1−α γ̃α ]
E[β̃]E[γ̃]E[c̃1−α β̃1−α ] − 2E[β̃]2 E[c̃1−α γ̃α ]
+
wi +
w3−i }
2
2
4E[β̃] − E[γ̃]
4E[β̃]2 − E[γ̃]2
(10)
Referring to Eq (10), we can obtain the second-order derivative of the equivalent objective function πm as follows:
2
∂2 πm ∂ π2m
−2A 2B ∂w
∂w
∂w
1
2
1
H = ∂2 πm
∂2 πm = 2B
−2A
2 ∂w2 ∂w1
∂w2
With assumption that E[β̃] > E[β̃] > 0, then
A=
2E[β̃]3 − E[β̃]E[γ̃]2
E[β̃]3
E[β̃]2 E[γ̃]
>
>
=B
4E[β̃]2 − E[γ̃]2
4E[β̃]2 − E[γ̃]2 4E[β̃]2 − E[γ̃]2
In can be easily seen that H is negative definite and πm is jointly concave with respect to w1 and w2 . Therefore,
differentiating πm by (w1 , w2 ) and equating the expressions to zero
 ∂π (w ,w )


 m∂w11 2 = −2Aw1 + 2Bw2 + E[d̃1 ] − E[β̃]S 1 + E[γ̃]S 2 + C = 0


 ∂πm (w1 ,w2 ) = −2Aw2 + 2Bw1 + E[d̃2 ] − E[β̃]S 2 + E[γ̃]S 1 + C = 0,
∂w2
6
the leaders’ equilibrium prices (w1 , w2 ) can be easily obtained by solving the above two equations
A(E[d̃1 ] − E[β̃]S 1 + E[γ̃]S 2 + C) + B(E[d̃2 ] − E[β̃]S 2 + E[γ̃]S 1 + C)
,
2(A2 − B2 )
A(E[d̃2 ] − E[β̃]S 2 + E[γ̃]S 1 + C) + B(E[d̃1 ] − E[β̃]S 1 + E[γ̃]S 2 + C)
w∗2 =
.
2(A2 − B2 )
w∗1 =
(11)
Obviously, we can get the equilibrium prices of the two retailers as follows:
r1∗ =
r2∗ =
(−2E[β̃]2 + E[γ̃]2 )w∗1 + E[β̃]E[γ̃]w∗2 + 2E[β̃](E[d̃1 ] + E[β̃ s̃1 ]) + E[γ̃](E[(d̃2 ] + E[β̃ s̃2 ])
4E[β̃]2 − E[γ̃]2
2
2 ∗
∗
(−2E[β̃] + E[γ̃] )w2 + E[β̃]E[γ̃]w1 + 2E[β̃](E[d̃2 ] + E[β̃ s̃2 ]) + E[γ̃](E[(d̃1 ] + E[β̃ s̃1 ])
4E[β̃]2 − E[γ̃]2
,
.
4.2. 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 two retailer can dominate the sales prices by choosing
their markup policies firstly. Then a Nash-Stackberg model can be applied as follows:


max πr1 = E[(r1 − s̃1 )(d̃1 − β̃(r1 + w∗1 ) + γ̃(r2 + w∗2 ))]



r1





max
πr2 = E[(r2 − s̃2 )(d̃2 − β̃(r2 + w∗2 ) + γ̃(r1 + w∗1 ))]subject to:



r2





M{wi − c̃ ≤ 0} = 0








(12)



where (w∗1 , w∗2 ) solves problems:





2


P






max πm = E[(wi − c̃)(d̃i − β̃(ri + wi ) + γ̃(r3−i + w3−i ))]subject to:







i=1
w1 ,w2







M{r
−
s̃
M{r2 − s̃2 ≤ 0} = 0

1
1 ≤ 0} = 0







 
 M{d̃i − β̃(ri + wi ) + γ̃(r3−i + w3−i ) ≤ 0} = 0, i = 1, 2.
With the second condition function of πm
2
∂ π2m
∂w
H2 = ∂2 π1m
∂w2 ∂w1
∂2 πm ∂w1 ∂w2 ∂2 πm ∂w22 −2E[β̃] 2E[γ̃] =
2E[γ̃] −2E[β̃].
Obviously, the Hessian matrix is negative definite with the assumption E[β̃] > E[γ̃] ≥ 0. Then the optimal react
functions to (r1 , r2 ) can be derived from the first order conditions Eqs.(13).
∂πm
= − 2E[β̃]w1 + 2E[γ̃]w2 + E[d̃1 ] − E[β̃]r1 + E[γ̃]r2 + E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ] = 0
∂w1
∂πm
= − 2E[β̃]w2 + 2E[γ̃]w1 + E[d̃2 ] − E[β̃]r2 + E[γ̃]r1 + E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ] = 0
∂w2
(13)
By solving Eqs.(13), we obtain:
E[β̃]E[d˜1 ] + E[γ̃]E[d˜2 ] + (E[β̃] + E[γ̃])(E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ]) 1
− r1 ,
2
2(E[β̃]2 − E[γ̃]2 )
1−α 1−α
1−α α
˜
˜
E[β̃]E[d2 ] + E[γ̃]E[d1 ] + (E[β̃] + E[γ̃])(E[c̃ β̃ ] − E[c̃ γ̃ ]) 1
w∗2 =
− r2 .
2
2(E[β̃]2 − E[γ̃]2 )
w∗1 =
7
(14)
Substituting Eqs.(19) into the two retailers’ profit functions, we obtain:
1
1
1
1−α
πr1 = − E[β̃]r12 + E[γ̃]r2 r1 + (E[d˜1 ] − E[c̃1−α β̃1−α ] + E[c̃1−α γ̃α ] + E[s1−α
])r1
1 β̃
2
2
2
1
α
1−α 1−α
− E[ s̃1−α
]C1 − E[ s̃11−α γ̃α ](C2 + r2 )
1 d̃1 ] + E[ s̃1 β̃
2
1
1
1
2
1−α
1−α
1−α
πr2 = − E[β̃]r2 + E[γ̃]r1 r2 + (E[d˜2 ] − E[c̃ β̃ ] + E[c̃1−α γ̃α ] + E[s1−α
])r1
2 β̃
2
2
2
1
α
1−α 1−α
− E[ s̃1−α
]C2 − E[ s̃21−α γ̃α ](C1 + r1 )
2 d̃2 ] + E[ s̃2 β̃
2
where
Ci =
˜ ] + (E[β̃] + E[γ̃])(E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ])
E[β̃]E[d̃i ] + E[γ̃]E[d3−i
, i = 1, 2.
2(E[β̃]2 − E[γ̃]2 )
(15)
(16)
∂2 π
Because the second order condition: ∂r2ri = −E[β̃] < 0, the optimal decisions of the two retailers can be derived
i
from their first order conditions (Eqs.(17)).
∂πr1
= − E[β̃]r1 +
∂r1
∂πr2
= − E[β̃]r2 +
∂r2
1
E[γ̃]r2 +
2
1
E[γ̃]r1 +
2
1
1−α
]) = 0
(E[d˜1 ] − E[c̃1−α β̃1−α ] + E[c̃1−α γ̃α ] + E[s1−α
1 β̃
2
1
1−α
(E[d˜2 ] − E[c̃1−α β̃1−α ] + E[c̃1−α γ̃α ] + E[s1−α
]) = 0
2 β̃
2
(17)
By solving Eqs.(17), we have
r1∗ =
r2∗ =
1−α
1−α
2E[β̃]E[d˜1 ] + E[γ̃]E[d˜2 ] − (2E[β̃] + E[γ̃])(E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ] + 2E[β̃]E[ s̃1−α
] + E[γ̃]E[ s̃1−α
]
1 β̃
2 β̃
4E[β̃]2 − E[γ̃]2
1−α
1−α
2E[β̃]E[d˜2 ] + E[γ̃]E[d˜1 ] − (2E[β̃] + E[γ̃])(E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ] + 2E[β̃]E[ s̃1−α
] + E[γ̃]E[ s̃1−α
]
2 β̃
1 β̃
4E[β̃]2 − E[γ̃]2
,
.
(18)
Obviously, we can get the equilibrium prices of the manufacturer as follows:
E[β̃]E[d˜1 ] + E[γ̃]E[d˜2 ] + (E[β̃] + E[γ̃])(E[c̃1−α β̃1−α ] − E[c̃1−α γ̃α ]) 1 ∗
− r1 ,
2
2(E[β̃]2 − E[γ̃]2 )
1−α
1−α
1−α
α
E[β̃]E[d˜2 ] + E[γ̃]E[d˜1 ] + (E[β̃] + E[γ̃])(E[c̃ β̃ ] − E[c̃ γ̃ ]) 1 ∗
− r2 .
w∗2 =
2
2(E[β̃]2 − E[γ̃]2 )
w∗1 =
(19)
4.3. VN Model
In some supply chains, there is no obvious dominance exists. Therefore, we assume that the three players move
simultaneously. Then a three-player Nash game model can be built as follows:

2

P



max πm = E[(wi − c̃)(d̃i − β̃(ri + wi ) + γ̃(r3−i + w3−i ))]


w1 ,w2

i=1





max
π
=
E[(r1 − s̃1 )(d̃1 − β̃(r1 + w1 ) + γ̃(r2 + w2 ))]

r
1


r1



max π = E[(r − s̃ )(d̃ − β̃(r + w∗ ) + γ̃(r + w∗ ))]
r2
2
2
2
2
1
(20)
2
1


r2





subject to:






M{wi − c̃ ≤ 0} = 0 M{ri − s̃i ≤ 0} = 0





 M{d̃i − β̃(ri + wi ) + γ̃(r3−i + w3−i ) ≤ 0} = 0, i = 1, 2.
Note that the πm is joint concave in (w1 , w2 ) and πri is concave with respect to ri , i = 1, 2. Thus, the
8
Setting the first-order derivatives equaling zero as follows:
∂πm
= − 2E[β̃]w1 + 2E[γ̃]w2 − E[β̃]r1 + E[γ̃]r2 + E[d̃1 ] + E[c̃1−α β̃α ] − E[c̃1−α γ̃α ] = 0,
∂w1
∂πm
= − 2E[β̃]w2 + 2E[γ̃]w1 − E[β̃]r2 + E[γ̃]r1 + E[d̃2 ] + E[c̃1−α β̃α ] − E[c̃1−α γ̃α ] = 0,
∂w2
(21)
∂πr1
α
= − 2E[β̃]r1 + E[γ̃]r2 − E[β̃]w1 + E[γ̃]w2 + E[d̃1 ] + E[ s̃1−α
2 β̃ ] = 0,
∂r1
∂πr2
α
= − 2E[β̃]r2 + E[γ̃]r1 − E[β̃]w2 + E[γ̃]w1 + E[d̃2 ] + E[ s̃1−α
2 β̃ ] = 0.
∂r2
(22)
By solving Eqs. (21) and (22), the equilibrium solutions can be attained:
r1∗ =
r2∗ =
1−α
1−α
(−6E[β̃]2 + 2E[γ̃]2 )C1 + 4E[β̃]E[γ̃]C2 + 6E[β̃](E[d̃1 ] + E[ s̃1−α
]) + 2E[γ̃](E[d̃2 ] + E[ s̃1−α
])
1 β̃
2 β̃
9E[β̃]2 − E[γ̃]2
1−α
1−α
(−6E[β̃]2 + 2E[γ̃]2 )C2 + 4E[β̃]E[γ̃]C1 + 6E[β̃](E[d̃2 ] + E[ s̃1−α
]) + 2E[γ̃](E[d̃1 ] + E[ s̃1−α
])
2 β̃
1 β̃
w∗1 =
w∗2 =
9E[β̃]2 − E[γ̃]2
2
2
1−α
1−α
(12E[β̃] − 2E[γ̃] )C1 − 2E[β̃]E[γ̃]C2 − 3E[β̃](E[d̃1 ] + E[ s̃1−α
]) − E[γ̃](E[d̃2 ] + E[ s̃1−α
])
1 β̃
2 β̃
9E[β̃]2 − E[γ̃]2
1−α
1−α
(12E[β̃]2 − 2E[γ̃]2 )C2 − 2E[β̃]E[γ̃]C1 − 3E[β̃](E[d̃2 ] + E[ s̃1−α
]) − E[γ̃](E[d̃1 ] + E[ s̃1−α
])
2 β̃
1 β̃
9E[β̃]2 − E[γ̃]2
,
,
,
,
In the similar way, the sales prices, order quantities and expected profits can also be attained.
5. Numerical Example
By the results obtained from the above models, the expressions of each supply chain member’s optimal decision
and optimal expected profits 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 attain general conclusions.
Instead, we employed numerical approach to illustrate supply chain members’ different behaviors and performances
facing various decision environments with different power structures and uncertain degrees.
When deal 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 [1] (Chapter 16: uncertain statistics) to get more details about how to collect
experts’ experimental data and how to estimate empirical distribution of uncertain variable from the experimental
data. For simplicity, we just give these uncertain variables’ distributions in Table 2.
Table 2: Distributions of uncertain variables
Parameter
Distribution
Expected Value
c̃
L(9,11)
10
s̃1
L(5,7)
6
s̃2
L(4, 6)
5
d̃1
Z(2900, 3000, 3300)
3050
d̃2
Z(2800, 3000, 3100)
2975
β̃
L(90,100)
100
γ̃
L(40,60)
50
Additionally, this paper uses E[γ̃]/E[β̃] to represent the substitutability of the two retailers. By keeping β̃ unchanged and varying γ̃ in five levels, we obtain Table 3.
With Lemma 3, we can attain E[c̃] = (9 + 11)/2 = 10, E[ s̃1 ] = (5 + 7)/2 = 6, E[ s̃2 ] = (4 + 6)/2 = 5E[d̃1 ] =
(2900 + 2 ∗ 3000 + 3300)/4 = 1025, E[d̃2 ] = (2800 + 2 ∗ 3000 + 3100)/4 = 2975, etc.
9
Table 3: The Substitutability of the Two Retailers
Substitutability
very low
low
medium
high
very high
γ̃
L(0, 20)
L(15, 25)
L(40, 60)
L(65, 75)
L(80, 100)
Expected Value
10
50
100
150
50
E[γ̃]/E[β̃]
0.10
0.25
0.5
0.75
0.90
According to Lemma 4, then we have
E[c̃
1−α
α
d˜1 ] =
Z
0
=
Z
1
−1
[Φ−1
c (1 − α)Φd1 (α)]dα
0.5
[9(1 − α)) + 11α)(3000 ∗ (1 − 2(1 − α)) + 2 ∗ 3300 ∗ (1 − α))dα
Z 1
+
(9(1 − α) + 11α)(2900 ∗ (2 − 2(1 − α)) + 3000 ∗ (2(1 − α) − 1))dα
0
0.5
(23)
=3043.33
Z 1
−1
E[c̃1−α γ̃α ] =
[Φ−1
c (1 − α)Φγ (α)]dα
0
Z 1
=
[11(1 − α)) + 9α)(40(1 − α) + 60α)dα
0
=496.67
1−α
α
In the same way, we can get the values of E[c̃1−α β̃1−α ], E[c̃1−α γ̃α ], E[ s̃1−α
], E[ s̃1−α
r β̃
r γ̃ ], etc.
Substitute these values into the equilibrium solutions attained in Section 4, we can obtain:
Table 4: Effects of β̃’s uncertain degrees on the prices under the three structures
Sub
0.10
0.25
0.5
0.75
0.90
Str
MS
VN
RS
MS
VN
RS
MS
VN
RS
MS
VN
RS
MS
VN
RS
w1
18.9288
16.1958
14.7560
22.2667
18.9688
17.0794
32.3167
27.9219
24.9778
62.5238
56.7831
52.1614
153.1904
146.4859
140.3493
w2
19.0879
16.2984
14.8317
22.4667
19.0918
17.1683
32.5667
28.0648
25.0778
62.8095
56.9354
52.2653
153.4930
146.6411
140.4537
πm
8286.20
7563.81
4733.49
13748.63
12794.36
8250.51
34302.99
32983.47
23308.72
111801.44
110451.84
89945.35
374529.21
373693.16
341037.74
r1
10.2395
11.5326
14.4123
11.2540
12.6625
16.4413
13.4056
14.8562
20.7444
16.4291
17.5481
26.7915
18.9077
19.4756
31.7487
p1
29.1683
27.7284
29.1683
33.5206
31.6312
33.5206
45.7222
42.7781
45.7222
78.9529
74.3312
78.9529
172.0980
165.9615
172.0980
πr1
2124.87
3365.45
3866.11
3124.97
4775.00
5815.78
5956.74
8276.47
11342.67
11661.49
14067.03
22399.47
18360.75
19793.74
34849.92
r2
9.3228
10.6455
13.5789
10.3651
11.8163
15.6635
12.5556
14.0705
20.0444
15.6109
16.8148
26.1552
18.1060
18.7705
31.1452
p2
28.4107
26.9440
28.4107
32.8317
30.9082
32.8317
45.1222
42.1352
45.1222
78.4204
73.7503
78.4204
171.5989
165.4115
171.5989
πr2
2595.67
3905.69
4407.14
3656.11
5416.60
6463.42
6613.49
9124.66
12221.84
12492.06
15179.93
23610.17
19332.69
21100.95
36335.00
πt
13006.74
14834.95
13006.74
20529.71
22985.96
20529.71
46873.23
50384.61
46873.23
135954.99
139698.79
135954.99
412222.66
414587.85
412222.66
Referring to Table 4, one can find that
• Regardless of the leadership, the sales prices and the expected profits of the total system in MS and RS structures
are the same, indicating that who holds the power has no influence on the total supply chain. In consideration
of the individual firms, however, the firm as a leader will gain more profit than as a follower.
10
• However, the dominant power structure will increase the sales prices and lower the profit of the whole supply
chain as the sales prices in VN case is lowest.
Next, we analyze the effects of the uncertain degrees of manufacturing costs c and sales costs s1 , s2 on optimal
pricing decisions under the three possible structures. The uncertain degree of a parameter mainly depend on its
own inherent variance and experts’ personal knowledge. If more informations about a parameter are available, more
accurate estimations the experts can make; as a result, the uncertain degree of the parameter will become lower. Of
course, if we have enough information (history data) about these parameters 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 with the deterministic assumption if we assume
the supply chain members are risk neutral.
By varying the uncertain degrees of these parameters and keeping the other parameters remaining as shown in
Table 2, the changes of optimal prices are shown in Tables 5, 6 and 7.
Table 5: Effects of manufacturing cost’s uncertain degree on the prices and profits
Str
MS
VN
RS
c̃
10
L(9, 11)
L(8, 12)
L(7, 13)
10
L(9, 11)
L(8, 12)
L(7, 13)
10
L(9, 11)
L(8, 12)
L(7, 13)
w1
32.2167
32.3167
32.4167
32.5167
27.8019
27.9219
28.0419
28.1619
24.8444
24.9778
25.1111
25.2444
w2
32.4667
32.5667
32.6667
32.7667
27.9448
28.0648
28.1848
28.3048
24.9444
25.0778
25.2111
25.3444
πm
33278.55
34302.99
35328.77
36355.88
31947.14
32983.47
34021.08
35059.98
22185.20
23308.72
24433.12
25558.42
r1
13.4389
13.4056
13.3722
13.3389
14.8962
14.8562
14.8162
14.7762
20.8111
20.7444
20.6778
20.6111
p1
45.6556
45.7222
45.7889
45.8556
42.6981
42.7781
42.8581
42.9381
45.6556
45.7222
45.7889
45.8556
πr1
6005.34
5956.74
5908.37
5860.23
8346.42
8276.47
8206.85
8137.55
11440.30
11342.67
11245.49
11148.75
r2
12.5889
12.5556
12.5222
12.4889
14.1105
14.0705
14.0305
13.9905
20.1111
20.0444
19.9778
19.9111
p2
45.0556
45.1222
45.1889
45.2556
42.0552
42.1352
42.2152
42.2952
45.0556
45.1222
45.1889
45.2556
πr2
6663.09
6613.49
6564.12
6514.98
9196.32
9124.66
9053.33
8982.31
12321.47
12221.84
12122.65
12023.91
πt
45946.97
46873.23
47801.27
48731.08
49489.87
50384.61
51281.26
52179.83
45946.97
46873.23
47801.27
48731.08
Referring to Table 5:
• 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.
• Even though the wholesale supply chain will gain more, but the consumer will have to pay more when the
uncertain degree of the manufacturing cost increases.
Referring to Tables 6 and 7:
• The retailer will be charged a lower wholesale price when the uncertain degree of its sales cost become higher.
While the wholesale price for the other retailer will keep the same in MS structure and drop slightly in RS and
VN structure.
• Both retailers can charge a higher markup prices when the uncertain degree of some of the sales cost increases.
• Contrary to the case of the manufacturing cost, the retailers can gain higher profits while the common manufacturer will be suffer from lower profits.
• In the same way, when the uncertain degree of the sales cost increase, the consumer will be afford to higher
prices while the whole supply chain will gain more.
In the similar way, more experiments can be conducted to explore the effects of other parameters such as the two
price elastic coefficients or the market sizes of the two retailers.
11
Table 6: Effects of retailer 1’s sales cost’s uncertain degree on the prices and profits
Str
MS
VN
RS
s̃1
6
L(5, 7)
L(4, 8)
L(3, 9)
6
L(5, 7)
L(4, 8)
L(3, 9)
6
L(5, 7)
L(4, 8)
L(3, 9)
w1
32.3500
32.3167
32.2833
32.2500
27.9448
27.9219
27.8990
27.8762
24.9956
24.9778
24.9600
24.9422
w2
32.5667
32.5667
32.5667
32.5667
28.0686
28.0648
28.0610
28.0571
25.0822
25.0778
25.0733
25.0689
πm
34351.97
34302.99
34254.12
34205.35
33030.55
32983.47
32936.47
32889.57
23341.49
23308.72
23275.99
23243.32
r1
13.3544
13.4056
13.4567
13.5078
14.8105
14.8562
14.9019
14.9476
20.7089
20.7444
20.7800
20.8156
p1
45.7044
45.7222
45.7400
45.7578
42.7552
42.7781
42.8010
42.8238
45.7044
45.7222
45.7400
45.7578
πr1
5408.79
5956.74
6504.81
7052.98
7762.45
8276.47
8790.64
9304.94
10817.57
11342.67
11867.93
12393.35
r2
12.5511
12.5556
12.5600
12.5644
14.0629
14.0705
14.0781
14.0857
20.0356
20.0444
20.0533
20.0622
p2
45.1178
45.1222
45.1267
45.1311
42.1314
42.1352
42.1390
42.1429
45.1178
45.1222
45.1267
45.1311
πr2
6596.68
6613.49
6630.32
6647.14
9096.45
9124.66
9152.89
9181.13
12198.37
12221.84
12245.32
12268.81
πt
46357.43
46873.23
47389.24
47905.47
49889.45
50384.61
50880.00
51375.64
46357.43
46873.23
47389.24
47905.47
Table 7: Effects of retailer 2’s sales cost’s uncertain degree on the prices and profits
Str
MS
VN
RS
s̃2
5
L(4.5, 5.5)
L(4, 6)
L(3.5, 6.5)
5
L(4.5, 5.5)
L(4, 6)
L(3.5, 6.5)
5
L(4.5, 5.5)
L(4, 6)
L(3.5, 6.5)
w1
32.3167
32.3167
32.3167
32.3167
27.9257
27.9238
27.9219
27.9200
24.9822
24.9800
24.9778
24.9756
w2
32.6000
32.5833
32.5667
32.5500
28.0876
28.0762
28.0648
28.0533
25.0956
25.0867
25.0778
25.0689
πm
34352.97
34327.97
34302.99
34278.04
33031.37
33007.41
32983.47
32959.55
23341.89
23325.30
23308.72
23292.15
r1
13.4011
13.4033
13.4056
13.4078
14.8486
14.8524
14.8562
14.8600
20.7356
20.7400
20.7444
20.7489
12
p1
45.7178
45.7200
45.7222
45.7244
42.7743
42.7762
42.7781
42.7800
45.7178
45.7200
45.7222
45.7244
πr1
5950.10
5953.42
5956.74
5960.07
8262.93
8269.70
8276.47
8283.25
11329.51
11336.09
11342.67
11349.26
r2
12.5044
12.5300
12.5556
12.5811
14.0248
14.0476
14.0705
14.0933
20.0089
20.0267
20.0444
20.0622
p2
45.1044
45.1133
45.1222
45.1311
42.1124
42.1238
42.1352
42.1467
45.1044
45.1133
45.1222
45.1311
πr2
6091.67
6352.58
6613.49
6874.41
8641.06
8882.86
9124.66
9366.47
11723.34
11972.58
12221.84
12471.12
πt
46394.74
46633.97
46873.23
47112.52
49935.37
50159.97
50384.61
50609.27
46394.74
46633.97
46873.23
47112.52
6. Conclusion
In this paper, we considered a pricing competing problem in supply chains with competing retailers. Specially,
manufacturing cost, 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 behavior of the supply chain under three possible power structures are derived from these
models. Afterwards, numerical experiments were also given to explore the impact of uncertain degrees of the parameters on the pricing decisions. The results showed that the dominant power in supply chain will increase the sales
prices and lower the profit of the whole supply chain. The results also demonstrated that the supply chain uncertain
factors have great influences on the decision makers. It was found that the supply chain members may benefit from
higher uncertain degrees of their own costs, whereas the supply chain members in the other level will gain less profits.
Additionally, the result showed that the uncertainness of the supply chain will make the the end consumers pay more.
As this study was based on some assumptions, future researches can be focused on some more general problems. For instance, this paper only considered one type of indeterminacy, while the real world might behave more
complicated in which randomness and uncertainties might co-exist. Therefore, one possible extension of this paper
is to study the pricing problem with twofold indeterminacy and in which uncertain random variable can be applied.
Besides, this paper assumed that all the participants are risk neutral 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.
Acknowledgments
This work was supported by National Natural Science Foundation of China (Nos.71371141, 71001080).
Preliminaries
In this section, we will introduce some important concepts and theorems of uncertainty theory for modeling the
pricing decision problem with human belief degree.
Let Γ be a nonempty set and L a σ-algebra over Γ. Each element Λ in L is called an event.
Definition 1. ([2]) The set function M is called an uncertain measure if it satisfies:
Axiom 1. (Normality Axiom) M{Γ} = 1.
Axiom 2. (Duality Axiom) M{Λ} + M{Λc } = 1 for any event Λ
Axiom 3. (Subadditivity Axiom) For every countable sequence of events {Λi }, i = 1, 2, · · · , we have
∞ 
∞



[ 
 X
M
Λi 
≤
M{Λi }.




i=1
i=1
Besides, the product uncertain measure on the product σ-algebra L was defined by Liu [42] as follows:
Axiom 4. (Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · The product uncertain measure M
is an uncertain measure satisfying
∞

∞



Y 
 ^
M
A
=
Mk {Ak }

k



k=1
k=1
where Ak are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively.
Definition 2. ([2]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of
real numbers, i.e., for any Borel set B of real numbers, the set
{ξ ∈ B} = {γ ∈ Γ ξ(γ) ∈ B}
is an event.
13
Definition 3. ([42]) The uncertain variables ξ1 , ξ2 , · · · , ξn are said to be independent if

 n
n




 ^
\
=
M{ξi ∈ Bi }
(ξ
∈
B
)
M
i
i 




i=1
i=1
for any Borel sets B1 , B2 , · · · , Bn .
Sometimes, we should know uncertainty distribution to model real-life uncertain optimization problems.
Definition 4. ([2]) The uncertainty distribution Φ of an uncertain variable ξ is defined by
Φ(x) = M{ξ ≤ x}
for any real number x.
An uncertainty distribution Φ is referred to be regular if its inverse function Φ−1 (α) exists and is unique for each
α ∈ [0, 1]
Lemma 1. ([3]) Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables with regular uncertainty distributions Φ1 , Φ2 , · · · , Φn ,
respectively. If the function f (x1 , x2 , · · · , xn ) is strictly increasing with respect to x1 , x2 , · · · , xm and 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
−1
−1
Φ−1 (α) = f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α)).
Definition 5. ([2]) Let ξ be an uncertain variable. The expected value of ξ is defined by
E[ξ] =
Z
+∞
M{ξ ≥ r}dr −
Z
0
M{ξ ≤ r}dr
−∞
0
provided that at least one of the above two integrals is finite.
Lemma 2. ([3]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then
E[ξ] =
+∞
Z
(1 − Φ(x))dx −
Z
0
Φ(x)dx.
−∞
0
Lemma 3. ([3]) Let ξ be an uncertain variable with regular uncertainty distribution Φ. If the expected value exists,
then
Z
1
E[ξ] =
Φ−1 (α)dα.
0
Example 1. Let ξ = L(a, b) be an linear uncertain variable, then



0, x < a;




Φ(x) = 
(x − a)/(b − a) a ≤ x ≤ b




0, x > b
(24)
and its inverse uncertainty distribution Φ−1 (α) = a + (b − a)α, The expected value can be attained
E[ξ] =
Z
1
a + (b − a)αdα =
0
14
a+b
.
2
(25)
Example 2. The zigzag uncertain variable ξ = Z(a, b, c) whose uncertainty distribution is



0, i f x < a;






(x − a)/2(b − a), if a ≤ x ≤ b
Φ(x) = 


(x + c − 2b)/2(c − b), if b < x ≤ c





0, i f x > c.
(26)
and its inverse uncertainty distribution



(1 − 2α)a + 2αb, if α < 0.5
φ (α) = 

(2 − 2α)b + (2α − 1)c, if α ≥ 0.5.
−1
(27)
Thus, its expected value is as follows:
E[ξ] =
Z
0.5
((1 − 2α)a + 2αb)dα +
0
Z
1
((2 − 2α)b + (1 − 2α)c)dα =
0.5
a + 2b + c
.
4
(28)
Lemma 4. ([43]) Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables with regular uncertainty distributions Φ1 , Φ2 , · · · , Φn ,
respectively. A function f (x1 , x2 , · · · , xn ) is strictly increasing with respect to x1 , x2 , · · · , xm and strictly decreasing
with respect to xm+1 , xm+2 , · · · , xn . Then the expected value of ξ = f (ξ1 , ξ2 , · · · , ξn ) is
E[ξ] =
Z
0
1
−1
−1
−1
f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α)))dα
(29)
provided that the expected value E[ξ] exists.
Example 3. Let ξ and η be two positive independent uncertain variables with regular uncertainty distributions Φ and
Ψ, respectively. Then we have
" # Z 1
ξ
Φ−1 (α)
(30)
E
=
dα.
−1
η
0 Ψ (1 − α)
Uncertain programming, as a type of mathematical programming involving uncertain variables, was initiated by [44].



max E[ f (x, ξ)]




(31)
subject to:





 M{gi (x, ξ) ≤ 0} ≥ αi , i = 1, 2, · · · , p
where x is a decision vector, ξ is an uncertain vector of parameters, E[ f (x, ξ)] means the expected value of the
objective function while M{gi (x, ξ) ≤ 0} ≥ αi , i = 1, 2, · · · , p, is a set of chance constraints. With the above concepts
and theorems, we can model the pricing decision problem in uncertain environments.
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16