 Research
 Open Access
Pricing Decision in a TwoEchelon Supply Chain with Competing Retailers Under Uncertain Environment
 Hua Ke^{1},
 Yong Wu^{1},
 Hu Huang^{1}Email authorView ORCID ID profile and
 Zhiyi Chen^{2}
https://doi.org/10.1186/s4046701700592
© The Author(s) 2017
 Received: 5 February 2017
 Accepted: 5 March 2017
 Published: 20 March 2017
Abstract
This paper explores a supply chain pricing competition problem in a twoechelon 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
 Twoechelon 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 distributions of these parameters, which are essential for supply chain managers when they make decisions, e.g., choosing prices, determining production quantities, choosing ordering quantities, and adding investment. Theoretically, we can collect enough samples to estimate distributions of these parameters before we make decisions. However, products, especially hightech products, e.g., smartphones, PCs, and other digital devices, have been rapidly updated recently. The costs and demands for these new products are usually 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 (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) usually 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 theory, 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 theory. 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 managers 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 determines 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 manufacturers 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 closedform 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 the manufacturer can benefit from a higher uncertain degree of the manufacturing cost, meanwhile, 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 consumers 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 presented in the “Literature Review” section. Some basic concepts and properties with respect to uncertain variable are reviewed in the “Preliminaries” section. Some useful notations and necessary assumptions are discussed in the “Problem Description” section. Three models are employed to derive the equilibria under three possible scenarios 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.
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], chaintochain structure [4, 9, 10], dual channel structure [11–14], and monopoly common manufacturer structure, which is studied in this paper. The research on the monopoly common manufacturer structure was initiated 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 twopart tariff wholesale pricing policy. For pricing competition, Yang et al.[17] considered a pricing problem in a twoechelon 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 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 competition 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 problem 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 decentralized chain to perform like a centralized one. Xiao and Yang [21] studied a price and service competition of two supply chains with riskaverse retailers under stochastic demand. They also analyzed the impacts of the retailer’s risk sensitivity on the manufacturers’ 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 gametheorybased 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] studied 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 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 decisionmaking 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 supply 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 information. 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 different 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 Γ be a nonempty set and a σalgebra over Γ. Each element Λ in is called an event.
Definition 1
([ 2 ]) The set function is called an uncertain measure if it satisfies:
Besides, the product uncertain measure on the product σalgebra was defined by Liu [40] as follows:
Axiom 4
(Product axiom) Let be uncertainty spaces for k=1,2,⋯ The product uncertain measure is an uncertain measure satisfying where Λ _{ k } are arbitrarily chosen events from for k=1,2,⋯, respectively.
Definition 2
Definition 3
([ 40 ]) The uncertain variables ξ _{1},ξ _{2},⋯,ξ _{ n } are said to be independent if for any Borel sets B _{1},B _{2},⋯,B _{ n } of real numbers.
Sometimes, we should know uncertainty distribution to model reallife uncertain optimization problems.
Definition 4
([ 2 ]) The uncertainty distribution Φ of an uncertain variable ξ is defined by 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
Definition 5
([ 2 ]) Let ξ be an uncertain variable. The expected value of ξ is defined by provided that at least one of the above two integrals is finite.
Lemma 2
Lemma 3
Lemma 4
provided that the expected value E[ξ] exists.
Example 1
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
Notations
\(\tilde {c}:\)  Unit manufacturing cost, an uncertain variable 
\(\tilde {s}_{i}:\)  Unit sales cost of retailer i, an uncertain variable 
w _{ i }:  Unit wholesale price to retailer i, the manufacturer’s decision variable 
r _{ i }:  Unit markup price of retailer i, the retailer’s decision variable 
p _{ i }:  Unit retail price of retailer i, where p _{ i }=w _{ i }+r _{ i } 
\(\tilde {d}_{i}:\)  The market base of retailer i, an uncertain variable 
π _{ m }:  The profit of the manufacturer: \(\pi _{m}=\sum _{i=1}^{2}(w_{i}\tilde {c})(\tilde {d}_{i}\tilde {\beta } p_{i}+\tilde {\gamma } p_{3i})\) 
\(\pi _{r_{i}}\):  The profit of retailer i: \(\pi _{r_{i}}=(r_{i}\tilde {s}_{i})(\tilde {d}_{i}\tilde {\beta } p_{i}+\tilde {\gamma } p_{3i})\) 

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 \(\tilde {\beta }\) and \(\tilde {\gamma }\) satisfy: \(\mathrm {E}[\tilde {\beta }]>\mathrm {E}[\tilde {\gamma }]>0\).

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:
Crisp Forms of Objective Functions
In order to derive the equilibrium solutions, we transform uncertain objective functions to equivalent crisp forms first. Let \(\tilde {c}, \tilde {s}_{i}, \tilde {d}_{i}, \tilde {\beta }\), and \(\tilde {\gamma }\) be positive independent uncertain variables with regular uncertainty distributions Φ _{ c }, \(\Phi _{s_{i}},\Phi _{d_{i}}\), Φ _{ β }, and Φ _{ γ }, i=1,2.
where \(\Phi _{a}^{1}\) and \(\Phi _{b}^{1}\) are the inverse distribution functions of \(\tilde {a}\) and \(\tilde {b}\), respectively.
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 (w _{1},w _{2}) to maximize its own profit allowing for the retailers’ reaction functions. Observing the wholesale prices, the two retailers noncooperatively choose their own markup pricing schemes r _{ i } 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 StackelbergNash game can be applied as follows:
To solve this StackelbergNash game model, opposite to the decision sequence, we should first derive the Nash equilibrium at the lower level for the given wholesale prices w _{1} and w _{2} specified by the manufacturer in advance.
Proposition 1
Proof
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
Proof
Thus, Proposition 2 is proved. □
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 retailers can dominate the sales prices by choosing their markup policies firstly. Then, a NashStackberg model can be applied as follows:
Similarly, opposite to the decision sequence, we should derive the follower’s response to the given retail markups r _{1} and r _{2} specified by the two retailers.
Proposition 3
Proof
Thus, Proposition 3 is proved. □
Given the manufacturer’s response, the two retailers compete to choose their own sales markups.
Proposition 4
Proof
Thus, Proposition 4 is proved. □
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 threeplayer Nash game model can be built as follows:
Proposition 5
Proof
By solving Eqs.(23), the equilibrium solutions of Eqs.(26) can be attained. □
Similarly, the sales prices, ordering quantities, and expected profits can also be attained.
Numerical Experiments
By the results obtained from the above models, the expressions of each supply chain member’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.
Distributions of uncertain variables
With Lemma 3, we can attain \(E[\tilde {c}]=(9+11)/2=10, \ E[\tilde {s}_{1}]=(5+7)/2=6\), \(E[\tilde {s}_{2}]=(4+6)/2=5\), \(E[\tilde {d}_{1}]=(2900+2\times 3000+3300)/4=3050\), \(E[\tilde {d}_{2}]=(2800+2\times 3000+3100)/4=2975\), etc.
In the same way, we can get the values of \(E[\tilde {c}^{1\alpha }\tilde {\beta }^{1\alpha }]\), \(E[\tilde {c}^{1\alpha }\tilde {\gamma }^{\alpha }]\), \(E[\tilde {s}_{r}^{1\alpha }\tilde {\beta }^{1\alpha }]\), \(E[\tilde {s}_{r}^{1\alpha }\tilde {\gamma }^{\alpha }]\), etc.
The equilibrium results of the three structures under different substitutability
Sub.  Str.  w _{1}  w _{2}  π _{ m }  r _{1}  p _{1}  \(\pi _{r_{1}}\)  r _{2}  p _{2}  \(\pi _{r_{2}}\)  π _{ t } 

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 

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, we analyze the effects of the uncertain degrees of the manufacturing costs and 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 estimations 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 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 to those under deterministic environment with the assumption of risk neutral members.
Effects of the manufacturing cost’s uncertain degree on the prices and profits
\(\tilde {c}\)  w _{1}  w _{2}  π _{ m }  r _{1}  p _{1}  \(\pi _{r_{1}}\)  r _{2}  p _{2}  \(\pi _{r_{2}}\)  π _{ t }  

MS  10  32.2167  32.4667  33,278.55  13.4389  45.6556  6005.34  12.5889  45.0556  6663.09  45,946.97 
32.3167  32.5667  34,302.99  13.4056  45.7222  5956.74  12.5556  45.1222  6613.49  46,873.23  
32.4167  32.6667  35,328.77  13.3722  45.7889  5908.37  12.5222  45.1889  6564.12  47,801.27  
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 
27.9219  28.0648  32983.47  14.8562  42.7781  8276.47  14.0705  42.1352  9124.66  50,384.61  
28.0419  28.1848  34,021.08  14.8162  42.8581  8206.85  14.0305  42.2152  9053.33  51,281.26  
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 
24.9778  25.0778  23,308.72  20.7444  45.7222  11,342.67  20.0444  45.1222  12,221.84  46,873.23  
25.1111  25.2111  24,433.12  20.6778  45.7889  11,245.49  19.9778  45.1889  12,122.65  47,801.27  
25.2444  25.3444  25,558.42  20.6111  45.8556  11,148.75  19.9111  45.2556  12,023.91  48,731.08 
Effects of the 1st retailer’s sales cost’s uncertain degree on the prices and profits
\(\tilde {s}_{1}\)  w _{1}  w _{2}  π _{ m }  r _{1}  p _{1}  \(\pi _{r_{1}}\)  r _{2}  p _{2}  \(\pi _{r_{2}}\)  π _{ t }  

MS  6  32.3500  32.5667  34,351.97  13.3544  45.7044  5408.79  12.5511  45.1178  6596.68  46,357.43 
32.3167  32.5667  34,302.99  13.4056  45.7222  5956.74  12.5556  45.1222  6613.49  46,873.23  
32.2833  32.5667  34,254.12  13.4567  45.7400  6504.81  12.5600  45.1267  6630.32  47,389.24  
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 
27.9219  28.0648  32,983.47  14.8562  42.7781  8276.47  14.0705  42.1352  9124.66  50,384.61  
27.8990  28.0610  32,936.47  14.9019  42.8010  8790.64  14.0781  42.1390  9152.89  50,880.00  
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 
24.9778  25.0778  23,308.72  20.7444  45.7222  11,342.67  20.0444  45.1222  12,221.84  46,873.23  
24.9600  25.0733  23,275.99  20.7800  45.7400  11,867.93  20.0533  45.1267  12,245.32  47,389.24  
24.9422  25.0689  23,243.32  20.8156  45.7578  12,393.35  20.0622  45.1311  12,268.81  47,905.47 
Effects of the 2nd retailer’s sales cost’s uncertain degree on the prices and profits
\(\tilde {s}_{2}\)  w _{1}  w _{2}  π _{ m }  r _{1}  p _{1}  \(\pi _{r_{1}}\)  r _{2}  p _{2}  \(\pi _{r_{2}}\)  π _{ t }  

MS  5  32.3167  32.6000  34,352.97  13.4011  45.7178  5950.10  12.5044  45.1044  6091.67  46,394.74 
32.3167  32.5833  34,327.97  13.4033  45.7200  5953.42  12.5300  45.1133  6352.58  46,633.97  
32.3167  32.5667  34,302.99  13.4056  45.7222  5956.74  12.5556  45.1222  6613.49  46,873.23  
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 
27.9238  28.0762  33,007.41  14.8524  42.7762  8269.70  14.0476  42.1238  8882.86  50,159.97  
27.9219  28.0648  32,983.47  14.8562  42.7781  8276.47  14.0705  42.1352  9124.66  50,384.61  
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 
24.9800  25.0867  23,325.30  20.7400  45.7200  11,336.09  20.0267  45.1133  11,972.58  46,633.97  
24.9778  25.0778  23,308.72  20.7444  45.7222  11,342.67  20.0444  45.1222  12,221.84  46,873.23  
24.9756  25.0689  23,292.15  20.7489  45.7244  11,349.26  20.0622  45.1311  12,471.12  47,112.52 

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.

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.

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 competing 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 decisions. 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. Additionally, 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 indeterminacy, 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 neutral while the decision makers may be risk sensitive in the real world. The research can be more applicable if the equilibrium behaviors with risksensitive members are considered.
Declarations
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.
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Authors’ Affiliations
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