Pricing and Retail Service Decisions in Fuzzy Uncertainty

Applied Mathematics and Computation 250 (2015) 580–592

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Applied Mathematics and Computation

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Pricing and retail service decisions in fuzzy uncertaintyenvironments

http://dx.doi.org/10.1016/j.amc.2014.11.0050096-3003/� 2014 Published by Elsevier Inc.

⇑ Corresponding author.E-mail address: [email protected] (J. Zhao).

Jing Zhao ⇑, Lisha WangSchool of Science, Tianjin Polytechnic University, Tianjin 300387, China

a r t i c l e i n f o

Keywords:PricingRetail serviceGame theoryFuzziness

a b s t r a c t

This paper studies the pricing and retail service decisions of a product in a supply chainwith one manufacturer and two retailers. It is assumed that the supply chain is operatedin fuzzy uncertainty environments. The fuzziness is associated with the customerdemands, manufacturing costs and service cost coefficients. Three different game struc-tures are considered, i.e., Manufacturer-leader Stackelberg, Retailer-leader Stackelberg,and Vertical Nash. Expected value models are developed to determine the optimal pricingand retail service strategies. The corresponding analytical equilibrium solutions areobtained by solving the models. Finally, numerical examples are presented to illustratethe effectiveness of the theoretical results, and to gain various marketing strategiesemployed under different situations.

� 2014 Published by Elsevier Inc.

1. Introduction

With the current dynamic and competitive environment, the market must compete with more complicated strategiesthan simply lowering the product’s price, because the consumers’ perception of value and their purchase decisions areinfluenced not exclusively by the product’s selling price, but also the amount of service that accompanies it. And servicecan help the customers obtain maximum value from their purchases [5]. Here, service is taken to broadly represent all formsof demand-enhancing effort, including customer service before and after the sale, product advertising, on-time productdelivery and product placement, and the overall quality of the shopping experience.

Early research focusing on attributes such as product quality and service can be found in the economics literature, e.g.,Spence [15], Dixit [3]. In marketing literature, Perry and Porter [14] focused on a type of service that has a positiveexternality effect across the retailers. Yan and Pei [22] considered retail services and firm profit in a dual-channel market,and suggested that the improved retail services effectively improve the supply chain performance in a competitive market.Lu et al. [12] studied three possible supply chain scenarios under manufacturer service and retail price, and obtained theresults and the modeling approach which are useful references for managerial decisions and administrations. Lederer andLi [10] considered a more general problem by considering multiple classes of customers with general service time that isclass-specific. Bernstein and Federgruen [2] developed a stochastic general equilibrium inventory model for an oligopolywith price and service competition. Ho and Zheng [7] considered a situation in which two competing service providers com-pete in terms of guaranteed delivery time and customer service level. The literature mentioned above studied the servicedecision with deterministic or random demand, whereas did not consider the fuzzy uncertainty of the supply chain.

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J. Zhao, L. Wang / Applied Mathematics and Computation 250 (2015) 580–592 581

In order to make supply chain management more effective, the fuzzy uncertainty that happens in the real world cannot beneglected. For example, it is difficult to provide exact estimates of the manufacturing cost (e.g. procurement costs may bevolatility), the customer demand (e.g. due to the innovation of products and market turbulence), the product supply (e.g.due to the weather conditions) and so on. However, the probability distribution may not be available in practice or maybe difficult to estimate from limited data points. Under these scenarios, the fuzzy theory provides a reasonable way to dealwith the possibility and linguistic expressions. The fuzzy theory provided by Zadeh [23] is an appropriate modeling toolwhen uncertain parameters cannot be described in stochastic distributions. There have been many researchers who adoptedfuzzy theory to depict uncertainty in supply chains. They mainly focus on coordination problem [20,21], inventory problem[4,6,17], supplier selection problem [1,8], contract problem [9,19], pricing problem [27,18]. We review the literature on pric-ing decision in fuzzy uncertainty environments as follows. Zhou et al. [27] focused on the pricing problem of a single productwith fuzzy customer demand. Wei and Zhao [18] considered the optimal pricing decision problem of a fuzzy closed-loopsupply chain with retail competition. Lin and Chang [11] presented a method for order selection and pricing of manufacturerusing a fuzzy approach. Zhao et al. [25,26] analyzed the pricing problem of substitutable products in a fuzzy supply chain.Zhao et al. [24] considered the optimal pricing and service decisions in a supply chain with two competing manufacturersand one common retailer. However, all of the above literature except [24] did not consider the service decision problem infuzzy uncertainty environments.

As far as we know, no research has considered the pricing and retail service decision problem in a two-echelon supplychain as used in this paper. Our research is most related to that of [24]. The major differences between this research andthe study of [24] are as follows: (1) This research considers a supply chain where one manufacturer produces a productand sells it to two retailers who are from two different areas with different economic development levels. However, Zhaoet al. [24] considered a supply chain with two competing manufacturers who produce two substitutable products and sellthem to one common retailer, respectively. (2) In this research, the retail service levels provided to the consumers are madeby the two retailers. However, in [24], the service levels provided to the consumers are directly made by the two competitivemanufacturers. (3) The manufacturer need decide the wholesale prices and the two retailers need decide the retail prices andservice levels in this study, however, the two competing manufacturer need decide the wholesale prices and service levelsand the common retailer only need decide the retail prices in [24].

In this paper, specifically, we consider a two-echelon supply chain where a monopolistic manufacturer sells a productthrough two retailers. The manufacturing cost, the customer demand and the retailer’s service cost coefficients are charac-terized as fuzzy variables. The manufacturer needs to decide their wholesale price, and the two retailers need to make theirretail price decisions and service level decisions. Our paper also focuses on the market power structures between the channelmembers. Three decentralized decision models are established, i.e., Manufacturer-leader Stackelberg (MS) game model,Retailer-leader Stackelberg (RS) game model and Vertical Nash (VN) game model. We assume that the two retailers haveequal market power, and they engage in Bertrand competition. The Manufacturer-leader Stackelberg game scenario repre-sents a market, in which there is a larger manufacturer and two relatively smaller retailers, and the market is controlled bythe manufacturer who plays the role of Stackelberg-leader with respect to the two retailers. The Retailer-leader Stackelberggame scenario arises in a market where the two retailers’ sizes are larger compared to the manufacturer’s, and the manu-facturer acts as the follower. If neither the manufacturer nor the retailers possess a larger bargaining power in negotiations,the supply chain interaction then follows a Vertical Nash game. Our main interest is to investigate how the monopolisticmanufacturer makes his wholesale pricing decisions, and how the two retailers make their retail pricing decisions and retailservice levels decisions when facing fuzzy uncertainty environments.

The rest of this paper is organized as follows. Section 2 briefly introduces the notations and the problem formulation.Section 3 gives and analyzes the three decentralized decision models. In Section 4, numerical examples are presented toillustrate the effectiveness of the theoretical results, and to compare the analytical equilibrium solutions for prices, services,and the maximal expected demands and profits under three scenarios. Section 5 summarizes our results and presents severalextensions for future research.

2. Problem description

In our two-echelon supply chain structure, there are one monopolistic manufacturer and two retailers. The manufacturerproduces a product with a manufacturing cost c, and wholesales it to two retailers (indexed by 1;2) with wholesale prices w1

and w2, respectively. We assume the two retailers are from two different areas with different economic development levels,so the manufacturer adopts the differential wholesale prices. The retailer i sells the product to the end consumers with retailprice piði ¼ 1;2Þ and provides services directly to the consumers. We assume all activity occurs within a single period.Moreover, the manufacturer and the two retailers have perfect information of the demands and the cost structures of otherchannel members.

Consumer demand for the product is sensitive to two factors: retail prices and retail service levels. In defining the demandfunction, we follow the approach by McGuire and Staelin [13]. The demand function is decreasing in its own price, butincreasing in the opponent’s price. Moreover, similar to [16,12], the demand for a product is increasing in its own servicelevel and decreasing in the opponent’s service level. The customer demand faced by retailer i can be described as

Diðp1; p2; s1; s2Þ ¼ ai � bppi þ cppj þ bssi � cssj; i ¼ 1;2; j ¼ 3� i; ð1Þ

582 J. Zhao, L. Wang / Applied Mathematics and Computation 250 (2015) 580–592

where the parameter ai represents the market base of the product sold by the retailer i, the parameters bp and cp are priceelastic coefficients, which measure the effectiveness of the retail prices in stimulating or restraining product’s consumerdemand, and the parameters bs and cs are service elastic coefficients, which measure the effectiveness of the retail servicesin stimulating or restraining product’s consumer demand.

We assume that the cost of providing service has a decreasing return property, the next dollar invested would produceless unit of service than the last dollar, i.e., it becomes more expensive to provide the next unit of service. This diminishingreturn of service can be captured by the quadratic form of service cost. In our model, we assume that the cost of providing si

units of service is gis2i =2, where gi is the service cost coefficient of retailer i; ði ¼ 1;2Þ. This is similar to the function used in

[16,12].Due to the innovation of products or market turbulence, the history data of the customer’s demand are not always

available or reliable. Therefore, quantitative demand forecasts and the estimates of manufacturing cost and service costcoefficient based on decision makes judgements, intuitions and experience seem to be more appropriate. In this paper,we assume the parameters c;ai; bp; cp; bs; cs and giði ¼ 1;2Þ are nonnegative and independent fuzzy variables. The parametersbp; bs and cp; cs should satisfy E½bp� > E½cp� and E½bs� > E½cs�, because the expected demand should be more sensitive to thechanges in its own retail price and service than to the changes in the rival’s retail price and service.

The notations of this paper can be summarized as follows:

c: unit manufacturing cost of the product;pi: unit retail price of the retailer i, which is the retailer i’s decision variable, satisfying pi > 0; i ¼ 1;2;wi: unit wholesale price of the product selling to the retailer, which is the manufacturer’s decision variable, satisfyingpi > wi; i ¼ 1;2;si: the retailer i’s service level, which is the retailer’s decision variable, i ¼ 1;2;Di: consumer demand for the product selling by the retailer iði ¼ 1;2Þ, which is a function of p1; p2; s1 and s2;ai: market base of products selling by the retailer i; i ¼ 1;2;gi: the retailer i’s service cost coefficient, i ¼ 1;2;pm: the manufacturer’s profit, which is a function of w1 and w2;pri

: the retailer i’s profit, which is a function of pi and si; i ¼ 1;2.

Each retailer chooses his own retail price and service level, and the manufacturer determines the wholesale prices. Allchannel members try to maximize their own expected profit and behave as if they have perfect information of the demandand the cost structure of other channel members.

According to the problem descriptions and assumptions, the profits of the two retailers and the manufacturer can beexpressed as follows

priðpi; siÞ ¼ ðpi �wiÞDiðp1;p2; s1; s2Þ � gis

2i =2; i ¼ 1;2; ð2Þ

pmðw1;w2Þ ¼ ðw1 � cÞD1ðp1;p2; s1; s2Þ þ ðw2 � cÞD2ðp1;p2; s1; s2Þ: ð3Þ

To ensure that the various profit expressions will be well behaved and possess a unique optimum, similar to [16], weimpose the following condition on the parameters: 2E½bp�E½gi� � E½bs�

2> 0;C11 < 0;C11C22 � C12C21 > 0 and 4E½bp�ð2F11E½bs�

þE½gi�Þ � ð2E½bs� þ E½bp�F11Þ2 > 0, where i ¼ 1;2;C11;C22;C12;C21 and F11 are constants defined in Appendix A.

3. Main results

In this section, we follow a game-theoretical approach to analyze our model. Variation in the market power can createone of the following three scenarios. The first scenario is that there is a larger manufacturer and two relatively smaller retail-ers in the market, and the manufacturer acts as the Stackelberg leader. We establish the Manufacturer-leader Stackelberg(MS) game model in this case. The second scenario is Retailer-leader Stackelberg where the two retailers’ sizes are largercompared to the manufacturer, and the market is controlled by the retailers, we establish a Retailer-leader Stackelberg(RS) game model in this scenario. The Vertical Nash (VN) game model is established in the scenario where every part inthe system has an equal bargaining power, so they simultaneously make their decisions. We will discuss these models inmore detail in the following sections.

3.1. MS game model

The manufacturer first announces the wholesale prices of the product. After observing the wholesale prices, the tworetailers then set their own retail prices and service levels, simultaneously. The objective of each participant is to maximizehis own expected profit. The MS model can be formulated as follows:


maxðw1 ;w2Þ

E pmðw1;w2;p�1ðw1;w2Þ; p�2ðw1;w2Þ; s�1ðw1;w2Þ; s�2ðw1;w2ÞÞ� �

p�i ðw1;w2Þ; s�i ðw1;w2Þ; i ¼ 1;2 are derived from solving the problemmaxðpi ;siÞ

E½priðpi; siÞ� i ¼ 1;2:

8>>><>>>:

ð4Þ

The problem can be solved by backwards induction. We first derive the reaction functions of the two retailers. TheProposition 1 gives the results.

Proposition 1. In the MS model, given the wholesale prices w1 and w2 made earlier by the manufacturer, the two retailers’optimal retail prices and service levels are:

p�1ðw1;w2Þ ¼B10 þ B11w1 þ B12w2

B; ð5Þ

p�2ðw1;w2Þ ¼B20 þ B21w1 þ B22w2

B; ð6Þ

s�1ðw1;w2Þ ¼E½bs�

BE½g1�ðB10 þ ðB11 � BÞw1 þ B12w2Þ; ð7Þ

s�2ðw1;w2Þ ¼E½bs�

BE½g2�ðB20 þ B21w1 þ ðB21 � BÞw2Þ; ð8Þ

where B;B10;B11;B12; B20;B21 and B22 are defined in Appendix A.The proof of Proposition 1 as well as the proofs of the other propositions, are given in Appendix B.After knowing the retailers’ reaction functions, the two manufacturers would use them to maximize his own expected

profit by choosing the wholesale prices. The following proposition gives the closed form solution of manufacturer’s optimalwholesale prices.

Proposition 2. In the MS model, the manufacturer’s optimal wholesale prices, denoted as wM�1 and wM�

2 , are given as follows

wM�1 ¼

C12C23 � C22C13

C11C22 � C12C21; ð9Þ

wM�2 ¼

C13C21 � C23C11

C11C22 � C12C21; ð10Þ

where C11;C12;C13;C21;C22 and C23 are defined in Appendix A.

Proposition 3. In the MS model, the two retailers’ optimal retail prices and service levels are

pM�1 ¼

B10

Bþ B11ðC12C23 � C22C13Þ þ B12ðC13C21 � C23C11Þ

BðC11C22 � C12C21Þ; ð11Þ

pM�2 ¼

B20

Bþ B21ðC12C23 � C22C13Þ þ B22ðC13C21 � C23C11Þ

BðC11C22 � C12C21Þ; ð12Þ

sM�1 ¼

E½bs�BE½g1�

B10 þðB11 � BÞðC12C23 � C22C13Þ þ B12ðC13C21 � C23C11Þ

C11C22 � C12C21

� �; ð13Þ

sM�2 ¼

E½bs�BE½g2�

B20 þB21ðC12C23 � C22C13Þ þ ðB21 � BÞðC13C21 � C23C11Þ

C11C22 � C12C21

� �: ð14Þ

The maximal expected profits of the manufacturer and the two retailers are

pM�m ¼

X2

i¼1

wM�i E½ai� � E½bp�pM�

i þ E½cp�pM�3�i þ E½bs�sM�

i � E½cs�sM�3�i

� �þ E½cbp�pM�

i þ E½ccs�sM�3�i �

pM�3�i

2

Z 1

0cLac

Upa þ cU

a cLpa

� �da

�

� sM�i

2

Z 1

0cLab

Usa þ cU

a bLsa

da� 1

2

Z 1

0cLaa

Uia þ cU

aaLia

da�; ð15Þ

pM�ri¼ pM�

i �wM�i

E½ai� � E½bp�pM�

i þ E½cp�pM�3�i þ E½bs�sM�

i � E½cs�sM�3�i

� �� E½gi�

sM�i

2

2: ð16Þ


3.2. RS game model

In this case, the two retailers’ sizes are larger compared to the manufacturer’s. The two retailers act as the Stackelbergleaders and simultaneously decide the retail prices and service levels to maximize their own expected profit, and thenthe Stackelberg follower, i.e., the manufacturer, determines the wholesale prices to maximize his expected profit. The RSmodel is formulated as

maxðpi ;siÞ

E priðpi; si;w�1ðp1; p2; s1; s2Þ;w�2ðp1;p2; s1; s2ÞÞ

� �i ¼ 1;2

w�1ðp1;p2; s1; s2Þ;w�2ðp1;p2; s1; s2Þ are derived from solving the problemmaxðw1 ;w2Þ

E½pmðw1;w2Þ�

8>>><>>>:

ð17Þ

Similar game-theoretic framework as applied in the MS model is implemented to solve this problem. Given retail pricesand service levels, we first derive the manufacturer’s response functions.

Proposition 4. In the RS model, for given retail prices p1 and p2, service levels s1 and s2, the manufacturer’s response functions areformulated as

w�1ðp1; p2; s1; s2Þ ¼ �p1 þ F11s1 þ F12s2 þ F01; ð18Þ

w�2ðp1; p2; s1; s2Þ ¼ �p2 þ F12s1 þ F11s2 þ F02; ð19Þ

where F01; F02; F11 and F12 are defined in Appendix A.Having the information about the response functions of the manufacturer, the two retailers would use them to maximize

their expected profits. The optimal retail prices and service levels of the two retailers can be derived as follows.

Proposition 5. In the RS model, the two retailers’ optimal retail prices and service levels are

pR�1 ¼

G1

G0; ð20Þ

pR�2 ¼

G2

G0; ð21Þ

sR�1 ¼

G3

G0; ð22Þ

sR�2 ¼

G4

G0; ð23Þ

where G0;G1;G2;G3 and G4 are defined in Appendix A.

Proposition 6. In the RS model, the manufacturer’s optimal wholesale prices are

wR�1 ¼

�G1 þ F11G3 þ F12G4 þ F01G0

G0; ð24Þ

wR�2 ¼

�G2 þ F12G3 þ F11G4 þ F02G0

G0: ð25Þ

3.3. VN game model

This scenario arises in a market in which there are relatively small to medium-sized manufacturers and retailers. It isreasonable to assume that the manufacturer must condition his wholesale price decisions on the retail prices and the retailservices since he cannot dominate the market over the retailers. On the other hand, the retailer must also condition her retailprice and service decisions on the wholesale prices, since he cannot dominate the market over the manufacturer. In this case,every firm has equal bargaining power and thus makes his decisions simultaneously, so the VN model can be formulated asfollows:

maxðw1 ;w2Þ

E½pmðw1;w2Þ�

maxðpi ;siÞ

E½priðpi; siÞ�; i ¼ 1;2

8<: ð26Þ

Solving the VN model, we can obtain the following results.


Proposition 7. In the VN model, the optimal retail prices pN�1 and pN�

2 , the optimal service levels sN�1 and sN�

2 chosen by the retailers,and the optimal wholesale prices wN�

1 and wN�2 chosen by the manufacturer, are

Table 1Relation

Man

Mark

Mark

Price

Price

Serv

Serv

Serv

Serv

wN�1 ¼

I12I23 � I22I13

I11I22 � I12I21; ð27Þ

wN�2 ¼

I13I21 � I23I11

I11I22 � I12I21; ð28Þ

pN�1 ¼

B10

Bþ B11ðI12I23 � I22I13Þ þ B12ðI13I21 � I23I11Þ

BðI11I22 � I12I21Þð29Þ

pN�2 ¼

B20

Bþ B21ðI12I23 � I22I13Þ þ B22ðI13I21 � I23I11Þ

BðI11I22 � I12I21Þ; ð30Þ

sN�1 ¼

E½bs�BE½g1�

þ B10 þðB11 � BÞðI12I23 � I22I13Þ þ B12ðI13I21 � I23I11Þ

I11I22 � I12I21

� �; ð31Þ

sN�2 ¼

E½bs�BE½g2�

þ B10 þB21ðI12I23 � I22I13Þ þ ðB22 � BÞðI13I21 � I23I11Þ

I11I22 � I12I21

� �; ð32Þ

where I11; I12; I13; I21; I22 and I23 are defined in Appendix A.

4. Numerical studies

Because the optimal decisions obtained in this paper are in a very complicated form, we have to use numerical studies tocompare the results obtained from the above three different decision models and to explore the channel members behaviorsfacing changing environments. Furthermore, the effects of the fuzzy degrees of parameters on the optimal prices, optimalservice levels, maximal expected profits and maximal expected demands are analyzed.

The relationship between the linguistic expressions and triangular fuzzy variables for the manufacturing cost, marketbases, price elasticity coefficients, service elastic coefficients and service cost coefficients are often determined by experts’experiences (see Table 1).

4.1. Discussion 1

Consider the case where the manufacturing cost c is high (c is about 7), the market bases a1 and a2 are large (a1 is about3000, a2 is about 3500), service cost coefficients g1 and g2 are sensitive (g1 is about 160, g2 is about 120), price elastic

between linguistic expression and triangular fuzzy variable.

Linguistic expression Triangular fuzzy variable

ufacturing cost c Low (about 3) (2, 3, 4)Medium (about 5) (4, 5, 6)High (about 7) (6, 7, 8)

et base a1 Large (about 3000) (2960, 3000, 3040)Small (about 2000) (1960, 2000, 2040)

et base a2 Large (about 3500) (3460, 3500, 3540)Small (about 2500) (2460, 2500, 2540)

elasticity bp Very sensitive (about 300) (270, 300, 330)Sensitive (about 200) (170, 200, 230)

elasticity cp Very sensitive (about 250) (220, 250, 280)Sensitive (about 150) (120, 150, 180)

ice elasticity bs Very sensitive (about 250) (220, 250, 280)Sensitive (about 150) (120, 150, 180)

ice elasticity cs Very sensitive (about 200) (170, 200, 230)Sensitive (about 100) (70, 100, 130)

ice cost coefficient g1 Very sensitive (about 180) (175, 180, 185)Sensitive (about 160) (150, 160, 170)

ice cost coefficient g2 Very sensitive (about 150) (130, 150, 170)Sensitive (about 120) (105, 120, 135)

Table 3The optimal decisions of retail prices, service levels and wholesale prices.

Scenario p�1 p�2 s�1 s�2 w�1 w�2

MS 41.68 45.50 5.37 11.50 35.95 36.30RS 44.21 46.96 4.68 7.39 32.55 33.16VN 42.17 45.70 5.80 10.54 35.99 37.27

Table 2The maximal expected profits and maximal expected demands.

Scenario E½Pm� E½Pr1 � E½Pr2 � E½Pr1 � þ E½Pr2 � E½D1� E½D2�

MS 89155 4249 8986 13235 1144.8 1839.3RS 67971 11835 15759 27594 1165.8 1379.7VN 89010 4960 7553 12513 1236.8 1686.2


coefficients bp and cp are sensitive (bp is about 200, cp is about 150), service elastic coefficients bs and cs are sensitive (bs isabout 150, cs is about 100). Using Table 1, c ¼ ð6;7;8Þ; a1 ¼ ð2960;3000;3040Þ; a2 ¼ ð3460;3500;3540Þ; g1 ¼ ð150;160;170Þ; g2 ¼ ð105;120;135Þ; bp ¼ ð170;200;230Þ; cp ¼ ð120;150;180Þ; bs ¼ ð120;150;180Þ; cs ¼ ð70;100;130Þ.

Following from Definition 5 of [25], the a-optimistic values and a-pessimistic values of c, ai; bp; cp; bs; cs and gi (i ¼ 1;2)are as follows.

Table 4The ma

Scen

MSRSNG

cLa ¼ 6þ a; aL

1a ¼ 2960þ 40a; aL2a ¼ 3460þ 40a; bL

pa ¼ 170þ 30a; cLpa ¼ 120þ 30a;

bLsa ¼ 120þ 30a; cL

sa ¼ 70þ 30a; gL1a ¼ 150þ 10a; gL

2a ¼ 105þ 15a;

cUa ¼ 8� a; aU

1a ¼ 3040� 40a; aU2a ¼ 3540� 40a; bU

pa ¼ 230� 30a; cUpa ¼ 180� 30a;

bUsa ¼ 180� 30a; cU

sa ¼ 130� 30a; gU1a ¼ 170� 10a; gU

2a ¼ 135� 15a:

The expected values are E½a1� ¼ 2960þ2�3000þ30404 ¼ 3000; E½a2� ¼ 3460þ2�3500þ3540

4 ¼ 3500; E½bp� ¼ 170þ2�200þ2304 ¼ 200; E½cp� ¼

120þ2�150þ1804 ¼ 150; E½bs� ¼ 120þ2�150þ180

4 ¼ 150; E½cs� ¼ 70þ2�100þ1304 ¼ 100; E½g1� ¼ 150þ2�160þ170

4 ¼ 160; E½g2� ¼ 105þ2�120þ1354

¼ 120, E½cbp� ¼ 1410; E½ccs� ¼ 710.The corresponding results are shown as in Tables 2 and 3.From Tables 2 and 3, we can obtain the following insights.

(1.1) The firm who is the leader in supply chain has the advantage to get the higher expected profits. For example, themanufacturer’s expected profit under MS model is the highest, while the two retailers have their own maximalexpected profits under RS model.

(1.2) Under the three game models, the optimal wholesale price w�2 > w�1, the profit margin p�2 �w�2 > p�1 �w�1, and themaximal expected demands E½D2� > E½D1�. This is because E½a2� > E½a1�.

(1.3) Regardless of what kind of power structure, the retailer 2’s optimal retail price and service level are always higherthan retailer 1’s. This is because the retailer 2 has an advantage on the market base of the product and service costcoefficient (i.e., E½a2� > E½a1�; E½g1� > E½g2�). This results in the retailer 2 gets a higher maximal expected profit thanthe retailer 1.

4.2. Discussion 2

In this discussion, we assume that the two retailers do not provide the service to the end consumers,i.e.bs ¼ cs ¼ 0;g1 ¼ g2 ¼ 0 in the corresponding models. The computation results are shown as in Tables 4 and 5.

From Tables 4 and 5, we can see pM�i ¼ pR�

i > pN�i ;wM�

i > wN�i > wR�

i ; E½pm�M > E½pm�N > E½pm�R, E½pri�R > E½pri

�N > E½pri�M ,

E½Di�N > E½Di�M ¼ E½Di�Rði ¼ 1;2Þ. The results are consistent with what have been obtained by Zhao et al. [25].

ximal expected profits and maximal expected demands without retailers’ services.

ario E½Pm� E½Pr1 � E½Pr2 � E½Pr1 � þ E½Pr2 � E½D1� E½D2�

67309 6122 7170 13292 1106.5 1197.554018 12244 14339 26583 1106.5 1197.566585 7361 9068 16429 1213.3 1346.7

Table 7The change of optimal decisions with the fuzzy degree of bs .

Scenario bs p�1 p�2 w�1 w�2 s�1 s�2

MS (120,150,180) 41.6785 45.5009 35.9544 36.3044 5.3664 11.4957– (130,150,170) 41.6859 45.5069 35.9621 36.3148 5.3660 11.4901– (140,150,160) 41.6932 45.5128 35.9698 36.3252 5.3657 11.4845– (145,150,155) 41.6969 45.5158 35.9737 36.3304 5.3655 11.4817

RS (120,150,180) 44.2070 46.9582 32.5494 33.1608 4.6839 7.3915– (130,150,170) 44.2070 46.9582 32.5494 33.1608 4.6839 7.3915– (140,150,160) 44.2070 46.9582 32.5494 33.1608 4.6839 7.3915– (145,150,155) 44.2070 46.9582 32.5494 33.1608 4.6839 7.3915

Table 6The change of maximal expected profits and demands with the fuzzy degree of bs .

Scenario bs E½pm� E½pr1 � E½pr2 � E½D1� E½D2�

MS (120,150,180) 89155 4249.0 8986 1144.80 1839.3– (130,150,170) 89099 4248.8 8977 1144.75 1838.4– (140,150,160) 89043 4248.4 8968 1144.70 1837.5– (145,150,155) 89014 4248.0 8964 1144.60 1837.1

RS (120,150,180) 67971 11835 15759 1165.8 1379.7– (130,150,170) 67931 11835 15759 1165.8 1379.7– (140,150,160) 67891 11835 15759 1165.8 1379.7– (145,150,155) 67871 11835 15759 1165.8 1379.7

Table 5The optimal decisions of retail prices and wholesale prices without retailers’ services.

Scenario p�1 p�2 w�1 w�2

MS 41.38 42.54 35.84 36.23RS 41.38 42.54 30.31 30.57NG 38.88 39.92 32.81 33.19


Compared with Tables 2 and 3, we find the following results.

(2.1) Regardless of what kind of power structure, when the two retailers provide services for consumers, the optimal priceswill increase, and the maximal expected demands will increase. This results in the increase of the manufacturer’s max-imal expected profit.

(2.2) When the two retailers provide services for consumers, the maximal expected profit of retailer 1 will decrease,whereas the maximal expected profit of retailer 2 will increase. This is because the service cost coefficient of retailer1 is bigger than that of retailer 2.

4.3. Discussion 3

Now we change the fuzzy degree of parameter bs and analyze its effect on the optimal prices, optimal service levels, max-imal expected profits and maximal expected demands in MS and RS games. The other parameters are as follows:c ¼ ð6;7;8Þ;a1 ¼ ð2960;3000;3040Þ;a2 ¼ ð3460;3500;3540Þ; bp ¼ ð170;200;230Þ; cp ¼ ð120;150;180Þ; cs ¼ ð70;100;130Þ;g1 ¼ ð150;160;170Þ and g1 ¼ ð105;120;135Þ. The corresponding results are shown as in Tables 6 and 7.

From Tables 6 and 7, we can obtain the following results.

(3.1) In MS game scenario, with decreasing fuzzy degree of parameter bs, the optimal retail prices and the optimal whole-sale prices slightly increase, whereas the optimal service levels, the maximal expected demands and the maximalexpected profits slightly decrease.

(3.2) In RS game scenario, the maximal expected profit of the manufacturer decreases with decreasing fuzzy degree ofparameter bs, this is consistent with MS game case. Different from MS game case, with decreasing fuzzy degree ofparameter bs, the optimal prices, the optimal service levels, the maximal expected demands and the maximal expectedprofits of the two retailers are unchanged.Similarly, we can also discuss the effects of the other parameters’ fuzzy degrees on the equilibrium solutions, such asservice elastic coefficient cs, the market bases a1 and a2, and so on.


5. Conclusions

This paper studies the pricing and retailers’ service decisions in a fuzzy supply chain where two retailers buy a productfrom one monopolistic manufacturer, and in turn sell it to the end consumers. Unlike others studies, our work makes con-tribution on three aspects. Firstly, we consider the retailers’ service decision and service competition. Secondly, we considerthe fuzzy uncertainty environments and characterize the consumer demands, manufacturing costs and two retailers’ servicecost coefficients as fuzzy variables. Thirdly, we study three possible scenarios for the strategic interactions between onemanufacturer and two retailers. By using game-theoretic approach, the corresponding analytical equilibrium solutions forthree models (MS, RS, VN) are obtained. Through numerical analysis, we compare analytical equilibrium solutions for themaximal expected profits, the maximal expected demands, the optimal pricing and service level decisions under three deci-sion scenarios, and study the behavior of firms facing changing fuzzy uncertainty environments. We also discuss the effectsof the fuzzy degrees of parameters on the equilibrium solutions. There are some possible extensions to improve our models,for example, the supply chain coordination under the decentralized decision cases and the decision model with nonlineardemand can be studied in the future.

Acknowledgments

The authors wish to express their sincerest thanks to the editors and anonymous referees for their constructive commentsand suggestions on the paper. We gratefully acknowledge the support of National Natural Science Foundation of China, Nos.71371186, 71301116, and 71001106.

Appendix A

E½bs�E½cs��

E½bs�E½cs��

E2½bs� !

E2½bs� !

B ¼ E½cp� � E½cp� � � � 2E½bp� � 2E½bp� ;
E½g1� E½g2� E½g1� E½g2�
B10 ¼ E½a1�E2½bs�E½g2�

� 2E½bp� !

� E½a2� E½cp� �E½bs�E½cs�

E½g2�

� ;

B11 ¼ E½bp� �E2½bs�E½g1�

!E2½bs�E½g2�

� 2E½bp� !

� E½bs�E½cs�E½g1�

E½cp� �E½bs�E½cs�

E½g2�

� ;

B12 ¼E½bs�E½cs�

E½g2�E2½bs�E½g2�

� 2E½bp� !

� E½bp� �E2½bs�E½g2�

!E½cp� �

E½bs�E½cs�E½g2�

� ;

B20 ¼ E½a2�E2½bs�E½g1�

� 2E½bp� !

� E½a1� E½cp� �E½bs�E½cs�

E½g1�

� ;

B21 ¼E½bs�E½cs�

E½g1�E2½bs�E½g1�

� 2E½bp� !

� E½bp� �E2½bs�E½g1�

!E½cp� �

E½bs�E½cs�E½g1�

� ;

B22 ¼ E½bp� �E2½bs�E½g2�

!E2½bs�E½g1�

� 2E½bp� !

� E½bs�E½cs�E½g2�

E½cp� �E½bs�E½cs�

E½g1�

� ;

C11 ¼ 2B21E½cp�

B�

B11E½bp�B

þ E2½bs�ðB11 � BÞBE½g1�

� E½bs�E½cs�B21

BE½g2�

!;

C12 ¼ C21 ¼ðB22 þ B11ÞE½cp�

B�ðB12 þ B21ÞE½bp�

Bþ E2½bs�

BB12

E½g1�þ B21

E½g2�

� � E½bs�E½cs�

BB22 � B

E½g2�þ B11 � B

E½g1�

� ;

C13 ¼ E½a1� þB20E½cp�

B�

B10E½bp�B

þ B10E2½bs�BE½g1�

� B20E½bs�E½cs�BE½g2�

þ E½bs�E½ccs�B

B11 � BE½g1�

þ B21

E½g2�

�

þB11 þ B21E½cbp�

B� B11 þ B21

2B

Z 1

0cLac

Upa þ cU

a cLpa

� �da� 1

2E½bs�ðB11 � BÞ

BE½g1�þ E½bs�B21

BE½g2�

� Z 1

0cLab

Usa þ cU

a bLsa

da;


C22 ¼ 2B12E½cp�

B�

B22E½bp�B

þ E2½bs�ðB22 � BÞBE½g2�

� E½bs�E½cs�B12

BE½g1�

!;

C23 ¼ E½a2� þB10E½cp�

B�

B20E½bp�B

þ B20E2½bs�BE½g2�

� B20E½bs�E½cs�BE½g1�

þ E½bs�E½ccs�B

B12

E½g1�þ B22 � B

E½g2�

�

þðB12 þ B22ÞE½cbp�

B� B12 þ B22

2B

Z 1

0cLac

Upa þ cU

a cLpa

� �da� 1

2E½bs�B12

BE½g1�þ E½bs�ðB22 � BÞ

BE½g2�

� Z 1

0cLab

Usa þ cU

abLsa

da;

F11 ¼E½bs�E½bp� � E½cs�E½cp�

E2½bp� � E2½cp�; F12 ¼

E½bs�E½cp� � E½cs�E½bp�E2½bp� � E2½cp�

;

F01 ¼E½a1�E½bp� þ E½a2�E½cp� þ ðE½cbp� � 1

2

R 10 ðcL

acUpa þ cU

a cLpaÞdaÞðE½bp� þ E½cp�Þ

E2½bp� � E2½cp�;

F02 ¼E½a1�E½cp� þ E½a2�E½bp� þ ðE½cbp� � 1

2

R 10 ðcL

acUpa þ cU

a cLpaÞdaÞðE½bp� þ E½cp�Þ

E2½bp� � E2½cp�;

G01 ¼ 2E½a1� þ F01E½bp�; G03 ¼ �F11E½a1� � F01E½bs�;

G02 ¼ 2E½a2� þ F02E½bp�; G04 ¼ �F11E½a2� � F02E½bs�;

G11 ¼ �4E½bp�; G13 ¼ 2E½bs� þ F11E½bp�;

G12 ¼ 2E½cp�; G14 ¼ F12E½bp� � 2E½cs�;

G22 ¼ �F11E½cp�; G23 ¼ �2F11E½bs� � E½g1�;

G24 ¼ F11E½cs� � F12E½bs�; G33 ¼ �2F11E½bs� � E½g2�;

G1 ¼

�G01 G12 G13 G14

�G02 G11 G14 G13

�G03 G22 G23 G24

�G04 G13 G24 G33

��

��; G2 ¼

G11 �G01 G13 G14

G12 �G02 G14 G13

G13 �G03 G23 G24

G22 �G04 G24 G33

��

��;

G3 ¼

G11 G12 �G01 G14

G12 G11 �G02 G13

G13 G22 �G03 G24

G22 G13 �G04 G33

��

��; G4 ¼

G11 G12 G13 �G01

G12 G11 G14 �G02

G13 G22 G23 �G03

G22 G13 G24 �G04

��

��;

G0 ¼

G11 G12 G13 G14

G12 G11 G14 G13

G13 G22 G23 G24

G22 G13 G24 G33

��

��;

I11 ¼F11E½bs�ðB11 � BÞ

E½g1�þ F12E½bs�B21

E½g2�� B� B11;

I12 ¼F11E½bs�B12

E½g1�þ F12E½bs�ðB22 � BÞ

E½g2�� B12;

I13 ¼F11E½bs�B10


E½g2�þ BF01 � B10;

I21 ¼F12E½bs�ðB11 � BÞ


E½g2�� B21;

I22 ¼F12E½bs�B12

E½g1�þ F11E½bs�ðB22 � BÞ

E½g2�� B� B22;

I23 ¼F12E½bs�B10


E½g2�þ BF02 � B20:


Appendix B

Proof of Proposition 1. It follows from Eqs. (1) and (2), together with Lemmas 3–5 of [25], that

E½priðpi; siÞ� ¼ ðpi �wiÞðE½ai� � E½bp�pi þ E½cp�pj þ E½bs�si � E½cs�sjÞ � E½gi�

s2i

2; i ¼ 1;2; j ¼ 3� i: ð33Þ

The first-order and second-order partial derivatives of with respect to pi and si can be shown as

@E½priðp1;p2; s1; s2Þ�@pi

¼ E½ai� � 2E½bp�pi þ E½cp�pj þ E½bs�si � E½cs�sj þwiE½bp� ð34Þ

@E½priðpi; siÞ�@si

¼ ðpi �wiÞE½bs� � E½gi�si; ð35Þ

@2E½priðpi; siÞ�

@p2i

¼ �2E½bp�; ð36Þ


@pi@si¼ @

2E½priðp1;p2; s1; s2Þ�@si@pi

¼ E½bs�; ð37Þ


@s2i

¼ �E½gi�: ð38Þ� �
It follows from Eqs. (36)–(38) and 2E½bp�E½gi� � E½bs�
2> 0 that the Hessian matrix H ¼ �2E½bp� E½bs�

E½bs� �E½gi�is negative definite.

Thus, the expected profit E½priðpi; siÞ� of retailer i is concave to pi and si. Setting Eqs. (34) and (35) to zero and solving them

simultaneously, we obtain Eqs. (5)–(8). Thus Proposition 1 holds.

Proof of Proposition 2. By substituting Eqs. (5)–(8) into Eq. (3), the expected profit E½pm� (as a shorthand forE½pmðw1;w2; p�1ðw1;w2Þ; p�2ðw1;w2Þ; s�1ðw1;w2Þ�) can be expressed as:

E½pm� ¼C11

2w2

1 þ C12w1w2 þC22

2w2

2 þ C13w1 þ C23w2 þðB10 þ B20ÞE½cbp�

Bþ E½bs�E½ccs�

BB10

E½g1�þ B20

E½g2�

�

� 12

Z 1

0cLaðaU

1a þ aU2aÞ þ cU

a aL1a þ aL

2a

� �da� B10 þ B20

2B

Z 1

0cLac

Upa þ cU

a cLpa

� �da

� 12

B10E½bs�BE½g1�

þ B20E½bs�BE½g2�

� Z 1

0cLab

Usa þ cU

a bLsa

da; ð39Þ

The first-order partial derivatives of E½pm� with respect to w1 and w2 can be shown as

@E½pm�@w1

¼ C11w1 þ C12w2 þ C13; ð40Þ

@E½pm�@w2

¼ C21w1 þ C22w2 þ C23: ð41Þ

It follows from Eqs. (40) and (41), and assumptions C11 < 0;C11C22 � C12C21 > 0, that the expected profit E½pm� is a con-cave function of w1 and w2. By setting Eqs. (40) and (41) to zero and solving for w1 and w2, the optimal wholesale prices wM�

1

and wM�2 can be obtained.

Proof of Proposition 3. By substituting wM�1 and wM�

2 into Eqs. (2), (3) and (6)–(9), Proposition 3 can be obtained.

Proof of Proposition 4. By using Eqs. (1) and (3), together with Lemma 3–5 of [25], E½pmðw1;w2Þ� can be expressed as

E½pmðw1;w2Þ� ¼ E½a1� � E½bp�p1 þ E½cp�p2 þ E½bs�s1 � E½cs�s2

� �w1 þ E½a2� � E½bp�p2 þ E½cp�p1 þ E½bs�s1 � E½cs�s2

� �w2

þ E½cbp�ðp1 þ p2Þ þ E½ccs�ðs1 þ s2Þ

� 12

Z 1

0cLa aU

1a þ aU2a þ cU

paðp1 þ p2Þ þ bUsaðs1 þ s2Þ

� �þ cU

a aL1a þ aL

2a þ cLpaðp1 þ p2Þ þ bL

saðs1 þ s2Þ� �h i

da:

ð42Þ


Letting and substituting pi ¼ wi þ ti; ti > 0 into Eq. (42), the first-order partial derivatives are given as

@E½pmðw1;w2Þ�@w1

¼ E½a1� � E½bp�p1 þ E½cp�p2 þ E½bs�s1 � E½cs�s2 � E½bp�w1 þ E½cp�w2 þ E½cbp� �12

Z 1

0cLac

Upa þ cU

a cLpa

� �da;

ð43Þ

@E½pmðw1;w2Þ�@w2

¼ E½a2� � E½bp�p2 þ E½cp�p1 þ E½bs�s2 � E½cs�s1 � E½bp�w2 þ E½cp�w1 þ E½cbp� �12

Z 1

0cLac

Upa þ cU

a cLpa

� �da:

ð44Þ� �
It follows from Eqs. (43) and (44) that the Hessian matrix H ¼ �2E½bp� 2E½cp�
2E½cp� �2E½bp�. It is negative definite since E½bp� > 0 and

E½bp� > E½cp�. So, E½pmðw1;w2Þ� is concave with respect to w1 and w2. By setting Eqs. (43) and (44) to zero and solving themsimultaneously, Eqs. (18) and (19) can be obtained.

Proof of Proposition 5. It follows from Eqs. (1), (2), (18) and (19) that

E pripi;si;w�1ðp1;p2;s1;s2Þ;w�2ðp1;p2;s1;s2Þ � �

¼ð2pi�F11si�F12sj�F0iÞ E½ai��E½bp�piþE½cp�pjþE½bs�si�E½cs�sj

� ��E½gi�

s2i

2:

ð45Þ

From Eq. (45), the first-order partial derivatives of E½priðpi; si;w�1ðp1; p2; s1; s2Þ;w�2ðp1; p2; s1; s2ÞÞ� with respect to pi and si

abbreviated as@E½pri

�@pi

andE½pri

�@si

� �can be shown as

@E½pri�

@pi¼ �4E½bp�pi þ 2E½cp�pj þ ð2E½bs� þ E½bp�F11Þsi þ ðE½bp�F12 � 2E½cs�Þsj þ ð2E½ai� þ F0iE½bp�Þ; ð46Þ

@E½pri�

@si¼ ð2E½bs� þ E½bp�F11Þpi � F11E½cp�pj � ð2F11E½bs� þ E½gi�Þsi þ ðF11E½cs� � F12E½bs�Þsj � ðF11E½ai� þ F0iE½bs�Þ: ð47Þ

From Eqs. (46) and (47) and the assumption 4E½bp�ð2F11E½bs� þ E½gi�Þ � ð2E½bs� þ E½bp�F11Þ2 > 0, we can prove that theexpected profit E½pri

ðpi; si;w�1ðp1; p2; s1; s2Þ;w�2ðp1; p2; s1; s2ÞÞ� is a concave function of pi and si. By setting Eqs. (46) and (47)to zero and solving for pi and si, the optimal retail price pR�

i and the optimal service level sR�i can be obtained.

Proof of Proposition 6. By Propositions 4 and 5, we can easily see that Proposition 6 holds.

Proof of Proposition 7. Consider that the decisions for the two retailers and the manufacturer are already derived in the MSmodel and the RS model respectively. The two retailers’ reaction functions for given wholesale prices are Eqs. (5)–(8), and themanufacturer reaction functions for given retail prices and service levels are Eqs. (18) and (19).

Solving these equations simultaneously yields the Vertical Nash equilibrium solutions. The equilibrium solutions can bederived as Eqs. (27)–(32).

References

[1] A. Amid, S. Ghodsypour, C. O’Brien, Fuzzy multiobjective linear model for supplier selection in a supply chain, Int. J. Prod. Econ. 104 (2) (2006) 394–407.[2] F. Bernstein, A. Federgruen, Dynamic inventory and pricing models for competing retailers, Nav. Res. Logist. 51 (2) (2004) 258–274.[3] A. Dixit, Quality and quantity competition, Rev. Econ. Stud. 46 (1979) 587–599.[4] I. Giannaoccaro, P. Pontrandolfo, B. Scozzi, A fuzzy echelon approach for inventory management, Eur. J. Oper. Res. 149 (2003) 185–196.[5] K. Goffin, Customer support: a cross-industry study of distribution channels and strategies, Int. J. Phys. Distrib. Logist. Manage. 29 (6) (1999) 374–397.[6] A. Gumus, A. Guneri, A multi-echelon inventory management framework for stochastic and fuzzy supply chains, Exp. Syst. Appl. 36 (2009) 5565–5575.[7] T. Ho, Y. Zheng, Setting customer expectation in service delivery: an integrated marketing-operations perspective, Manage. Sci. 50 (4) (2004) 479–488.[8] M. Kumar, P. Vrat, R. Shankar, A fuzzy programming approach for vendor selection problem in a supply chain, Int. J. Prod. Econ. 101 (2006) 273–285.[9] Y. Lan, R. Zhao, W. Tang, A fuzzy supply chain contract problem with pricing and warranty, J. Intell. Fuzzy Syst. 26 (3) (2014) 1527–1538.

[10] P. Lederer, L. Li, Pricing, production, scheduling, and delivery-time competition, Oper. Res. 45 (3) (1997) 407–420.[11] H. Lin, W. Chang, Order selection and pricing methods using flexible quantity and fuzzy approach for buyer evaluation, Eur. J. Oper. Res. 187 (2) (2008)

415–428.[12] J. Lu, Y. Tsao, C. Charoensiriwath, Competition under manufacturer service and retail price, Econ. Model. 28 (13) (2011) 1256–1264.[13] T. McGuire, R. Staelin, An industry equilibrium analysis of downstream vertical integration, Market. Sci. 2 (1983) 161–192.[14] M. Perry, R. Porter, Can resale price maintenance and franchise fees correct sub-optimal levels of retail service, Int. J. Ind. Organ. 8 (1990) 115–141.[15] A. Spence, Monopoly, quality, and regulation, Bell J. Econ. 6 (1975) 417–429.[16] A. Tsay, N. Agrawal, Channel dynamics under price and service competition, Manuf. Serv. Oper. Manage. 2 (4) (2000) 372–391.[17] J. Wang, A supply chain application of fuzzy set theory to inventory control models-DRP system analysis, Exp. Syst. Appl. 36 (2009) 9229–9239.[18] J. Wei, J. Zhao, Pricing decisions with retail competition in a fuzzy closed loop supply chain, Exp. Syst. Appl. 38 (2011) 11209–11216.[19] X. Wu, R. Zhao, W. Tang, Optimal contracts for the agency problem with multiple uncertain information, Knowl. Based Syst. 59 (2014) 161–172.[20] Y. Xie, D. Petrovic, K. Burnham, A heuristic procedure for the two-level control of serial supply chains under fuzzy customer demand, Int. J. Prod. Econ.

102 (2006) 37–50.

http://refhub.elsevier.com/S0096-3003(14)01510-0/h0005























[21] R. Xu, X. Zhai, Manufacturers coordination mechanism for single-period supply chain problems with fuzzy demand, Math. Comput. Model. 51 (5–6)(2010) 693–699.

[22] R. Yan, Z. Pei, Retail services and firm profit in a dual-channel market, J. Retail. Consum. Serv. 4 (16) (2009) 306–314.[23] L. Zadeh, Fuzzy sets, Inf. Cont. 8 (1965) 338–353.[24] J. Zhao, W. Liu, J. Wei, Competition under manufacturer service and price in fuzzy environments, Knowl. Based Syst. 50 (2013) 121–133.[25] J. Zhao, W. Tang, J. Wei, Pricing decision for substitutable products with retail competition in a fuzzy environment, Int. J. Prod. Econ. 135 (2012) 144–

153.[26] J. Zhao, W. Tang, R. Zhao, J. Wei, Pricing decisions for substitutable products with a common retailer in fuzzy environments, Eur. J. Oper. Res. 216

(2012) 409–419.[27] C. Zhou, R. Zhao, W. Tang, Two-echlon supply chain games in a fuzzy environment, Comput. Ind. Eng. 55 (2) (2008) 390–405.











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Pricing and Retail Service Decisions in Fuzzy Uncertainty