American Institute of Mathematical Sciences

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January  2019, 15(1): 343-364. doi: 10.3934/jimo.2018046

Pricing decisions for complementary products in a fuzzy dual-channel supply chain

Lisha Wang 1, , Huaming Song 1,, , Ding Zhang 2, and  Hui Yang 1,

1. 

School of Economic and Management, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China
2. 

School of Business, State University of New York at Oswego, New York 13126, USA

* Corresponding author: huaming@njust.edu.cn(Huaming Song)

Received  June 2017 Revised  November 2017 Published  January 2019 Early access  April 2018

Fund Project: The authors are supported by the National Natural Science Foundation of China 71571102 and the Key Scientific Research Projects of Colleges and Universities of Henan Province (No.18A630001).

The paper considers the pricing problem of complementary products in a fuzzy dual-channel supply chain environment where there are two manufacturers and one retailer. Four different decision models are established to study this problem: the centralized decision model, MS-Bertrand model, RS-Bertrand model and Nash game model, where the consumer demand and manufacturing cost for each product are characterized as fuzzy variables. A closed form solution has been obtained for each model by using game theory and fuzzy theory. Numerical examples are presented to compare the maximal expected profits and optimal pricing decisions, and to provide additional managerial insights. The finding shows that the decision makers are more likely to choose industries with higher self-price elastic coefficient and lower complementarity in the retail channel to cooperate. We can obtain that consumers can benefit from the cooperation of the two manufacturers because of lower prices. We can also find that it might not be bad for retailer because it can expand demand and obtain more maximal expected profits.

Keywords: Pricing decisions, complementary products, fuzzy dual-channel, game theory, supply chain.
Mathematics Subject Classification: Primary: 91A25, 91B24; Secondary: 91A40.
Citation: Lisha Wang, Huaming Song, Ding Zhang, Hui Yang. Pricing decisions for complementary products in a fuzzy dual-channel supply chain. Journal of Industrial & Management Optimization, 2019, 15 (1) : 343-364. doi: 10.3934/jimo.2018046
References:
[1]

P. Berger, J. Lee and B. Weinberg, Optimal cooperative advertising integration strategy for organizations adding a direct online channel, Journal of the Operational Research Society, 57 (2006), 920-927.   Google Scholar

[2]

G. Cai, Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.  doi: 10.1016/j.jretai.2009.11.002.  Google Scholar

[3]

E. Cao, Y. Ma and C. Wan, Contracting with asymmetric cost information in a dual-channel supply chain, Operation Research Letters, 41 (2013), 410-414.  doi: 10.1016/j.orl.2013.04.013.  Google Scholar

[4]

C. Cheng, Group opinion aggregation based on a grading process: A method for constructing triangular fuzzy numbers, Computers and Mathematics with Applications, 48 (2004), 1619-1632.  doi: 10.1016/j.camwa.2004.03.008.  Google Scholar

[5]

W. Chiang, D. Chhajed and J. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.  Google Scholar

[6]

S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.  doi: 10.1287/mksc.10.4.271.  Google Scholar

[7]

B. Dan, J. Xiao and X. Zhang, A collaboration strategy complementary products in dual-channel supply chains, Journal of Industrial Engineering and Engineering Management, 25 (2011), 162-166.   Google Scholar

[8]

Z. Ding and Y. Liu, Revenue sharing contract in dual channel supply chain in case of free riding, Journal of Systems Engineering, 28 (2013), 370-376.   Google Scholar

[9]

H. Kurata, D. Yao and J. Liu, Pricing policies under direct vs. indirect channel competition and nation vs. store brand competition, European Journal of Operational Research, 180 (2007), 262-281.   Google Scholar

[10]

N. Leila and S. Mehdi, Modeling and solving a pricing problem considering substitutable products in a dual supply chain with internet and traditional distribution channels, Cumhuriyet University Faculty of Science Science Journal, 36 (2015), 956-970.   Google Scholar

[11]

T. Li, X. Zhao and J. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499.   Google Scholar

[12]

Q. Li and B. Li, Dual-channel supply chain equilibrium problems regarding retail services and fairness concerns, Applied Mathematical Modelling, 40 (2016), 7349-7367.  doi: 10.1016/j.apm.2016.03.010.  Google Scholar

[13]

H. Lin and W. Chang, Order selection and pricing methods using flexible quantity and fuzzy approach for buyer evaluation, European Journal of Operational Research, 187 (2008), 415-428.  doi: 10.1016/j.ejor.2007.03.003.  Google Scholar

[14]

B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag: Heidelberg, 2002. Google Scholar

[15]

B. Liu, Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Springer-Verlag: Berlin, 2004.  Google Scholar

[16]

S. Liu and Z. Xu, Stackelberg game models between two competitive retailers in fuzzy decision environment, Fuzzy Optimization and Decision Making, 13 (2014), 33-48.  doi: 10.1007/s10700-013-9165-x.  Google Scholar

[17]

S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1 (1978), 97-110.  doi: 10.1016/0165-0114(78)90011-8.  Google Scholar

[18]

J. Raju, R. Sethuraman and S. Dhar, The introduction and performance of store brands, Management Science, 41 (1995), 957-978.  doi: 10.1287/mnsc.41.6.957.  Google Scholar

[19]

S. Sang, Price competition of manufacturers in supply chain under a fuzzy decision environment, Fuzzy Optimization and Decision Making, 14 (2015), 335-363.  doi: 10.1007/s10700-014-9202-4.  Google Scholar

[20]

S. Sayman, S. Hoch and J. Raju, Positioning of store brands, Marketing Science, 21 (2002), 378-397.  doi: 10.1287/mksc.21.4.378.134.  Google Scholar

[21]

H. Shin and W. Benton, Quantity discount-based inventory coordination: Effectiveness and critical environmental factors, Production and Operations Management, 13 (2004), 63-76.  doi: 10.1111/j.1937-5956.2004.tb00145.x.  Google Scholar

[22]

W. Soon, A review of multi-product pricing models, Applied Mathematics and Computation, 217 (2011), 8149-8165.  doi: 10.1016/j.amc.2011.03.042.  Google Scholar

[23]

L. Wang, H. Song and Y. Wang, Pricing and service decisions of complementary products in a dual-channel supply chain, Computers and Industrial Engineering, 105 (2017), 223-233.  doi: 10.1016/j.cie.2016.12.034.  Google Scholar

[24]

J. Wei and J. Zhao, Reverse channel decisions for a fuzzy closed-loop supply chain, Applied Mathematical Modeling, 37 (2013), 1502-1513.  doi: 10.1016/j.apm.2012.04.003.  Google Scholar

[25]

E. Wolfstetter, Topics in Microeconomics: Industrial Organization, Auction, and Incentives, Cambridge, UK: Cambridge University Press, 1999. doi: 10.1017/CBO9780511625787.  Google Scholar

[26]

C. Wu, C. Chen and C. Hsieh, Competitive pricing decisions in a two-echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2011), 265-274.  doi: 10.1016/j.ijpe.2011.07.020.  Google Scholar

[27]

J. Xiao, B. Dan and X. Zhang, Study on cooperation strategy between electronic channels and retailers in dual-channel supply chain, Journal of Systems Engineering, 24 (2009), 673-679.   Google Scholar

[28]

D. Yang and T. Xiao, Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, Journal of Cleaner Production, 149 (2017), 1174-1187.  doi: 10.1016/j.jclepro.2017.02.138.  Google Scholar

[29]

X. Yue and J. Liu, Demand forecast sharing in a dual-channel supply chain, European Journal of Operational Research, 174 (2006), 646-667.  doi: 10.1016/j.ejor.2004.12.020.  Google Scholar

[30]

X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.  doi: 10.1016/j.jbusres.2005.06.005.  Google Scholar

[31]

J. Zhao, W. Liu and J. Wei, Competition under manufacturer service and price in fuzzy environments, Knowledge-Based Systems, 50 (2013), 121-133.  doi: 10.1016/j.knosys.2013.06.003.  Google Scholar

[32]

J. Zhao, W. Tang and J. Wei, Pricing decision for substitutable products with retail competition in a fuzzy environment, International Journal of Production Economics, 135 (2012), 144-153.  doi: 10.1016/j.ijpe.2010.12.024.  Google Scholar

[33]

J. Zhao and L. Wang, Pricing and retail service decisions in fuzzy uncertainty environments, Applied Mathematics and Computation, 250 (2015), 580-592.  doi: 10.1016/j.amc.2014.11.005.  Google Scholar

[34]

C. Zhou, R. Zhao and W. Tang, Two-echelon supply chain games in a fuzzy environment, Computers and Industrial Engineering, 55 (2008), 390-405.  doi: 10.1016/j.cie.2008.01.014.  Google Scholar

[35]

H. Zimmermann, Application-oriented view of modeling uncertainty, European Journal of Operational Research, 122 (2000), 190-198.  doi: 10.1016/S0377-2217(99)00228-3.  Google Scholar

show all references

References:
[1]

P. Berger, J. Lee and B. Weinberg, Optimal cooperative advertising integration strategy for organizations adding a direct online channel, Journal of the Operational Research Society, 57 (2006), 920-927.   Google Scholar

[2]

G. Cai, Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.  doi: 10.1016/j.jretai.2009.11.002.  Google Scholar

[3]

E. Cao, Y. Ma and C. Wan, Contracting with asymmetric cost information in a dual-channel supply chain, Operation Research Letters, 41 (2013), 410-414.  doi: 10.1016/j.orl.2013.04.013.  Google Scholar

[4]

C. Cheng, Group opinion aggregation based on a grading process: A method for constructing triangular fuzzy numbers, Computers and Mathematics with Applications, 48 (2004), 1619-1632.  doi: 10.1016/j.camwa.2004.03.008.  Google Scholar

[5]

W. Chiang, D. Chhajed and J. Hess, Direct marketing, indirect profits: A strategic analysis of dual-channel supply chain design, Management Science, 49 (2003), 1-20.  doi: 10.1287/mnsc.49.1.1.12749.  Google Scholar

[6]

S. Choi, Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.  doi: 10.1287/mksc.10.4.271.  Google Scholar

[7]

B. Dan, J. Xiao and X. Zhang, A collaboration strategy complementary products in dual-channel supply chains, Journal of Industrial Engineering and Engineering Management, 25 (2011), 162-166.   Google Scholar

[8]

Z. Ding and Y. Liu, Revenue sharing contract in dual channel supply chain in case of free riding, Journal of Systems Engineering, 28 (2013), 370-376.   Google Scholar

[9]

H. Kurata, D. Yao and J. Liu, Pricing policies under direct vs. indirect channel competition and nation vs. store brand competition, European Journal of Operational Research, 180 (2007), 262-281.   Google Scholar

[10]

N. Leila and S. Mehdi, Modeling and solving a pricing problem considering substitutable products in a dual supply chain with internet and traditional distribution channels, Cumhuriyet University Faculty of Science Science Journal, 36 (2015), 956-970.   Google Scholar

[11]

T. Li, X. Zhao and J. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499.   Google Scholar

[12]

Q. Li and B. Li, Dual-channel supply chain equilibrium problems regarding retail services and fairness concerns, Applied Mathematical Modelling, 40 (2016), 7349-7367.  doi: 10.1016/j.apm.2016.03.010.  Google Scholar

[13]

H. Lin and W. Chang, Order selection and pricing methods using flexible quantity and fuzzy approach for buyer evaluation, European Journal of Operational Research, 187 (2008), 415-428.  doi: 10.1016/j.ejor.2007.03.003.  Google Scholar

[14]

B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag: Heidelberg, 2002. Google Scholar

[15]

B. Liu, Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Springer-Verlag: Berlin, 2004.  Google Scholar

[16]

S. Liu and Z. Xu, Stackelberg game models between two competitive retailers in fuzzy decision environment, Fuzzy Optimization and Decision Making, 13 (2014), 33-48.  doi: 10.1007/s10700-013-9165-x.  Google Scholar

[17]

S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1 (1978), 97-110.  doi: 10.1016/0165-0114(78)90011-8.  Google Scholar

[18]

J. Raju, R. Sethuraman and S. Dhar, The introduction and performance of store brands, Management Science, 41 (1995), 957-978.  doi: 10.1287/mnsc.41.6.957.  Google Scholar

[19]

S. Sang, Price competition of manufacturers in supply chain under a fuzzy decision environment, Fuzzy Optimization and Decision Making, 14 (2015), 335-363.  doi: 10.1007/s10700-014-9202-4.  Google Scholar

[20]

S. Sayman, S. Hoch and J. Raju, Positioning of store brands, Marketing Science, 21 (2002), 378-397.  doi: 10.1287/mksc.21.4.378.134.  Google Scholar

[21]

H. Shin and W. Benton, Quantity discount-based inventory coordination: Effectiveness and critical environmental factors, Production and Operations Management, 13 (2004), 63-76.  doi: 10.1111/j.1937-5956.2004.tb00145.x.  Google Scholar

[22]

W. Soon, A review of multi-product pricing models, Applied Mathematics and Computation, 217 (2011), 8149-8165.  doi: 10.1016/j.amc.2011.03.042.  Google Scholar

[23]

L. Wang, H. Song and Y. Wang, Pricing and service decisions of complementary products in a dual-channel supply chain, Computers and Industrial Engineering, 105 (2017), 223-233.  doi: 10.1016/j.cie.2016.12.034.  Google Scholar

[24]

J. Wei and J. Zhao, Reverse channel decisions for a fuzzy closed-loop supply chain, Applied Mathematical Modeling, 37 (2013), 1502-1513.  doi: 10.1016/j.apm.2012.04.003.  Google Scholar

[25]

E. Wolfstetter, Topics in Microeconomics: Industrial Organization, Auction, and Incentives, Cambridge, UK: Cambridge University Press, 1999. doi: 10.1017/CBO9780511625787.  Google Scholar

[26]

C. Wu, C. Chen and C. Hsieh, Competitive pricing decisions in a two-echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2011), 265-274.  doi: 10.1016/j.ijpe.2011.07.020.  Google Scholar

[27]

J. Xiao, B. Dan and X. Zhang, Study on cooperation strategy between electronic channels and retailers in dual-channel supply chain, Journal of Systems Engineering, 24 (2009), 673-679.   Google Scholar

[28]

D. Yang and T. Xiao, Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, Journal of Cleaner Production, 149 (2017), 1174-1187.  doi: 10.1016/j.jclepro.2017.02.138.  Google Scholar

[29]

X. Yue and J. Liu, Demand forecast sharing in a dual-channel supply chain, European Journal of Operational Research, 174 (2006), 646-667.  doi: 10.1016/j.ejor.2004.12.020.  Google Scholar

[30]

X. Yue, S. Mukhopadhyay and X. Zhu, A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.  doi: 10.1016/j.jbusres.2005.06.005.  Google Scholar

[31]

J. Zhao, W. Liu and J. Wei, Competition under manufacturer service and price in fuzzy environments, Knowledge-Based Systems, 50 (2013), 121-133.  doi: 10.1016/j.knosys.2013.06.003.  Google Scholar

[32]

J. Zhao, W. Tang and J. Wei, Pricing decision for substitutable products with retail competition in a fuzzy environment, International Journal of Production Economics, 135 (2012), 144-153.  doi: 10.1016/j.ijpe.2010.12.024.  Google Scholar

[33]

J. Zhao and L. Wang, Pricing and retail service decisions in fuzzy uncertainty environments, Applied Mathematics and Computation, 250 (2015), 580-592.  doi: 10.1016/j.amc.2014.11.005.  Google Scholar

[34]

C. Zhou, R. Zhao and W. Tang, Two-echelon supply chain games in a fuzzy environment, Computers and Industrial Engineering, 55 (2008), 390-405.  doi: 10.1016/j.cie.2008.01.014.  Google Scholar

[35]

H. Zimmermann, Application-oriented view of modeling uncertainty, European Journal of Operational Research, 122 (2000), 190-198.  doi: 10.1016/S0377-2217(99)00228-3.  Google Scholar

Figure 1.  The graphic method
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Figure 2.  The five kinds of CPs which selected randomly from the space$^*$
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Table 1.  Summary of the major literature review. (M: manufacturer; R: retailer)
Paper Supply chain structure Product status Dual channel Fuzzy or not Decisions
Chiang et al.[5] one M single Yes No pricing policy
one R ordering policy
Berger and Weinberg[1] one firm single Yes No pricing policy
advertising
Dan et al.[7] one M single Yes No pricing policy
one R service decision
Zhao et al.[32] one M substitutable No Yes pricing policy
two Rs
Zhao and Wang[33] one M single No Yes pricing policy
two Rs service decision
Leila and Mehdi[10] two Ms substitutable Yes No pricing policy
one R
Sang[19] two Ms substitutable No Yes pricing policy
one R
Wang et al.[23] two Ms complementary Yes No pricing policy
one R service decision
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Paper Supply chain structure Product status Dual channel Fuzzy or not Decisions
Chiang et al.[5] one M single Yes No pricing policy
one R ordering policy
Berger and Weinberg[1] one firm single Yes No pricing policy
advertising
Dan et al.[7] one M single Yes No pricing policy
one R service decision
Zhao et al.[32] one M substitutable No Yes pricing policy
two Rs
Zhao and Wang[33] one M single No Yes pricing policy
two Rs service decision
Leila and Mehdi[10] two Ms substitutable Yes No pricing policy
one R
Sang[19] two Ms substitutable No Yes pricing policy
one R
Wang et al.[23] two Ms complementary Yes No pricing policy
one R service decision
Table 2.  Relation between linguistic expression and triangular fuzzy variable.
Linguistic expression Triangular fuzzy variable
$\Phi_0$ Large (about 500) (350,500,700)
Small (about 200) (150,200,300)
$\Phi_1$ Large (about 800) (600,800,900)
Small (about 300) (200,300,350)
$\Phi_2$ Large (about 1200) (800,1200,1500)
Small (about 600) (500,600,800)
$ k_0 $ Very sensitive(about 5) (4, 5, 6)
Sensitive (about 2) (1, 2, 2.5)
$k_1$ Very sensitive(about 8) (5, 8, 10)
Sensitive (about 4) (3, 4, 5)
$k_2$ Very sensitive(about 30) (25, 30, 35)
Sensitive (about 20) (10, 20, 25)
$\beta$ Very sensitive(about 0.5) (0.3, 0.5, 0.8)
Sensitive(about 0.2) (0.1, 0.2, 0.3)
$\gamma_1$ Very sensitive(about 0.4) (0.3, 0.4, 0.6)
Sensitive (about 0.2) (0.1, 0.2, 0.3)
$\gamma_2$ Very sensitive(about 0.3) (0.1, 0.3, 0.5)
Sensitive (about 0.1) (0.05, 0.1, 0.15)
$c_1$ High(about 30) (25, 30, 40)
Low(about 15) (10, 15, 20)
$c_2$ High (about 5) (4, 5, 6)
Low (about 2) (1, 2, 4)
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Linguistic expression Triangular fuzzy variable
$\Phi_0$ Large (about 500) (350,500,700)
Small (about 200) (150,200,300)
$\Phi_1$ Large (about 800) (600,800,900)
Small (about 300) (200,300,350)
$\Phi_2$ Large (about 1200) (800,1200,1500)
Small (about 600) (500,600,800)
$ k_0 $ Very sensitive(about 5) (4, 5, 6)
Sensitive (about 2) (1, 2, 2.5)
$k_1$ Very sensitive(about 8) (5, 8, 10)
Sensitive (about 4) (3, 4, 5)
$k_2$ Very sensitive(about 30) (25, 30, 35)
Sensitive (about 20) (10, 20, 25)
$\beta$ Very sensitive(about 0.5) (0.3, 0.5, 0.8)
Sensitive(about 0.2) (0.1, 0.2, 0.3)
$\gamma_1$ Very sensitive(about 0.4) (0.3, 0.4, 0.6)
Sensitive (about 0.2) (0.1, 0.2, 0.3)
$\gamma_2$ Very sensitive(about 0.3) (0.1, 0.3, 0.5)
Sensitive (about 0.1) (0.05, 0.1, 0.15)
$c_1$ High(about 30) (25, 30, 40)
Low(about 15) (10, 15, 20)
$c_2$ High (about 5) (4, 5, 6)
Low (about 2) (1, 2, 4)
Table 3.  The optimal decisions of wholesale prices, direct sale price and retail prices
$scenario$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$CD$ $-$ $-$ 63.0097 60.8519 20.8009 $-$ $-$
$MS-Bertrand$ 61.17 21.36 63.3476 82.0076 29.2987 20.8376 7.9387
$RS-Bertrand$ 40.02 13.20 63.3317 82.0065 29.3002 41.9865 16.0902
$NG$ 47.14 15.97 63.4526 74.9997 26.6042 27.8597 10.6342
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$scenario$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$CD$ $-$ $-$ 63.0097 60.8519 20.8009 $-$ $-$
$MS-Bertrand$ 61.17 21.36 63.3476 82.0076 29.2987 20.8376 7.9387
$RS-Bertrand$ 40.02 13.20 63.3317 82.0065 29.3002 41.9865 16.0902
$NG$ 47.14 15.97 63.4526 74.9997 26.6042 27.8597 10.6342
Table 4.  Maximal expected profits and demands in different models
$scenario$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$CD$ 220.56 330.2375 505.94 $-$ $-$ $-$ 37076
$MS-Bertrand$ 226.36 163.9087 244.52 19578 4173 5358 29109
$RS-Bertrand$ 226.44 163.9089 244.48 16111 2180 10818 29109
$NG$ 223.31 219.0836 327.41 18895 3761 9585 32240
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$scenario$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$CD$ 220.56 330.2375 505.94 $-$ $-$ $-$ 37076
$MS-Bertrand$ 226.36 163.9087 244.52 19578 4173 5358 29109
$RS-Bertrand$ 226.44 163.9089 244.48 16111 2180 10818 29109
$NG$ 223.31 219.0836 327.41 18895 3761 9585 32240
Table 5.  The corresponding triangular fuzzy variables of the CPs
$\Phi_2$ $k_2$ $\gamma_1$
$CP_1$ (1800,2000,2200) (3, 4, 5) (0.4, 0.5, 0.6)
$CP_2$ (3201,3302,3403) (6, 8, 10) (0.1, 0.3, 0.5)
$CP_3$ (800,1200,1500) (15, 20, 25) (0.1, 0.2, 0.3)
$CP_4$ (500,600,700) (25, 30, 35) (0.3, 0.4, 0.6)
$CP_5$ (1000,1156,1312) (15, 20, 25) (0.4, 0.5, 0.6)
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$\Phi_2$ $k_2$ $\gamma_1$
$CP_1$ (1800,2000,2200) (3, 4, 5) (0.4, 0.5, 0.6)
$CP_2$ (3201,3302,3403) (6, 8, 10) (0.1, 0.3, 0.5)
$CP_3$ (800,1200,1500) (15, 20, 25) (0.1, 0.2, 0.3)
$CP_4$ (500,600,700) (25, 30, 35) (0.3, 0.4, 0.6)
$CP_5$ (1000,1156,1312) (15, 20, 25) (0.4, 0.5, 0.6)
Table 6.  Comparison of the optimal decisions with and without CPs
$CP_1$ $CP_2$ $CP_3$ $CP_4$ $CP_5$
$p_0^{N*}-p_0^{Nocp}$ -36.3415 -28.0346 -19.6351 -20.7345 -19.284
$p_1^{N*}-p_1^{Nocp}$ -22.3776 -18.0985 -2.4605 -2.5192 -0.827
$p_2^{N*}$ 322.6005 271.3165 38.6035 37.2342 13.1502
$w_1^{N*}-w_1^{Nocp}$ 12.2558 14.0035 18.0239 17.8967 18.3916
$E[D_0^{N*}]-E[D_0^{Nocp}]$ 8.6757 49.2929 89.1796 83.7494 90.7422
$E[D_1^{N*}]-E[D_1^{Nocp}]$ -26.3475 -23.7178 -12.0458 -11.9774 -10.7595
$E[D_2^{NG*}]$ 640.6542 1070.9 342.9444 319.2612 129.3525
$E[\Pi_{m_1}^{N*}]-E[\Pi_{m_1}^{Nocp}]$ -3141 0 5177 4690 5528
$E[\Pi_{m_2}^{N*}]$ 102830 143660 6082 5501 611.5339
$E[\Pi_r^{N*}]-E[\Pi_{r}^{Nocp}]$ 92082 133632 417 0 -4373
$E[\Pi_{m_1+r}^{N*}]-E[\Pi_{c}^{Nocp}]$ 88941 133632 5594 4690 155
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$CP_1$ $CP_2$ $CP_3$ $CP_4$ $CP_5$
$p_0^{N*}-p_0^{Nocp}$ -36.3415 -28.0346 -19.6351 -20.7345 -19.284
$p_1^{N*}-p_1^{Nocp}$ -22.3776 -18.0985 -2.4605 -2.5192 -0.827
$p_2^{N*}$ 322.6005 271.3165 38.6035 37.2342 13.1502
$w_1^{N*}-w_1^{Nocp}$ 12.2558 14.0035 18.0239 17.8967 18.3916
$E[D_0^{N*}]-E[D_0^{Nocp}]$ 8.6757 49.2929 89.1796 83.7494 90.7422
$E[D_1^{N*}]-E[D_1^{Nocp}]$ -26.3475 -23.7178 -12.0458 -11.9774 -10.7595
$E[D_2^{NG*}]$ 640.6542 1070.9 342.9444 319.2612 129.3525
$E[\Pi_{m_1}^{N*}]-E[\Pi_{m_1}^{Nocp}]$ -3141 0 5177 4690 5528
$E[\Pi_{m_2}^{N*}]$ 102830 143660 6082 5501 611.5339
$E[\Pi_r^{N*}]-E[\Pi_{r}^{Nocp}]$ 92082 133632 417 0 -4373
$E[\Pi_{m_1+r}^{N*}]-E[\Pi_{c}^{Nocp}]$ 88941 133632 5594 4690 155
Table 7.  The change of maximal expected demands and profits with the fuzzy degree of $\beta$
$scenario$ $\beta$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$MS-Bertrand$ (0.2, 0.5, 0.8) 225.6026 163.7939 244.5593 19452 4175.0 5354.1 28981
(0.3, 0.5, 0.7) 225.6886 163.8355 244.5621 19427 4175.1 5355.9 28958
(0.4, 0.5, 0.6) 225.7746 163.8772 244.5649 19403 4175.2 5357.6 28936
(0.45, 0.5, 0.55) 225.8176 163.8980 244.5663 19391 4175.3 5358.5 28925
$RS-Bertrand$ (0.2, 0.5, 0.8) 225.6799 163.7941 244.5216 15990 2180.7 10810 28980
(0.3, 0.5, 0.7) 225.7658 163.8357 244.5244 15964 2180.8 10813 28958
(0.4, 0.5, 0.6) 225.8518 163.8774 244.5272 15938 2180.9 10817 28936
(0.45, 0.5, 0.55) 225.8948 163.8982 244.5286 15925 2181.0 10818 28924
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$scenario$ $\beta$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$MS-Bertrand$ (0.2, 0.5, 0.8) 225.6026 163.7939 244.5593 19452 4175.0 5354.1 28981
(0.3, 0.5, 0.7) 225.6886 163.8355 244.5621 19427 4175.1 5355.9 28958
(0.4, 0.5, 0.6) 225.7746 163.8772 244.5649 19403 4175.2 5357.6 28936
(0.45, 0.5, 0.55) 225.8176 163.8980 244.5663 19391 4175.3 5358.5 28925
$RS-Bertrand$ (0.2, 0.5, 0.8) 225.6799 163.7941 244.5216 15990 2180.7 10810 28980
(0.3, 0.5, 0.7) 225.7658 163.8357 244.5244 15964 2180.8 10813 28958
(0.4, 0.5, 0.6) 225.8518 163.8774 244.5272 15938 2180.9 10817 28936
(0.45, 0.5, 0.55) 225.8948 163.8982 244.5286 15925 2181.0 10818 28924
Table 8.  The change of optimal prices with the fuzzy degree of $\beta$
$scenario$ $\beta$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$MS-Bertrand$ (0.2, 0.5, 0.8) 60.9727 21.3595 63.0687 81.7999 29.3032 20.8272 7.9437
(0.3, 0.5, 0.7) 60.9608 21.3597 63.0508 81.7934 29.3034 20.8326 7.9437
(0.4, 0.5, 0.6) 60.9489 21.3599 63.0329 81.7869 29.3037 20.8380 7.9438
(0.45, 0.5, 0.55) 60.9430 21.3600 63.0240 81.7836 29.3038 20.8406 7.9438
$RS-Bertrand$ (0.2, 0.5, 0.8) 39.8371 13.2063 63.0530 81.7988 29.3047 41.9617 16.0984
(0.3, 0.5, 0.7) 39.8198 13.2064 63.0351 81.7923 29.3049 41.9725 16.0985
(0.4, 0.5, 0.6) 39.8025 13.2065 63.0173 81.7858 29.3052 41.9833 16.0987
(0.45, 0.5, 0.55) 39.7939 13.2066 63.0083 81.7825 29.3053 41.9886 16.0987
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$scenario$ $\beta$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$MS-Bertrand$ (0.2, 0.5, 0.8) 60.9727 21.3595 63.0687 81.7999 29.3032 20.8272 7.9437
(0.3, 0.5, 0.7) 60.9608 21.3597 63.0508 81.7934 29.3034 20.8326 7.9437
(0.4, 0.5, 0.6) 60.9489 21.3599 63.0329 81.7869 29.3037 20.8380 7.9438
(0.45, 0.5, 0.55) 60.9430 21.3600 63.0240 81.7836 29.3038 20.8406 7.9438
$RS-Bertrand$ (0.2, 0.5, 0.8) 39.8371 13.2063 63.0530 81.7988 29.3047 41.9617 16.0984
(0.3, 0.5, 0.7) 39.8198 13.2064 63.0351 81.7923 29.3049 41.9725 16.0985
(0.4, 0.5, 0.6) 39.8025 13.2065 63.0173 81.7858 29.3052 41.9833 16.0987
(0.45, 0.5, 0.55) 39.7939 13.2066 63.0083 81.7825 29.3053 41.9886 16.0987
Table 9.  The change of maximal expected demands and profits with the fuzzy degree of $k_2$
$scenario$ $k_2$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$MS-Bertrand$ (10, 30, 50) 226.3554 163.9024 243.2656 19576 4320.2 5338.1 29234
(15, 30, 45) 226.3583 163.9045 243.6821 19577 4271.3 5344.8 29192
(20, 30, 40) 226.3613 163.9066 244.0987 19577 4222.4 5351.5 29151
(25, 30, 35) 226.3642 163.9087 244.5152 19578 4173.6 5358.2 29109
$RS-Bertrand$ (10, 30, 50) 226.4335 163.9026 243.2279 16110 2347.0 10777 29234
(15, 30, 45) 226.4364 163.9047 243.6443 16110 2291.3 10791 29192
(20, 30, 40) 226.4394 163.9068 244.0608 16111 2235.7 10804 29151
(25, 30, 35) 226.4423 163.9089 244.4773 16111 2180.0 10818 29109
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$scenario$ $k_2$ $E[D_0^{*}]$ $E[D_1^{*}]$ $E[D_2^{*}]$ $E[\Pi_{m_1}^{*}]$ $E[\Pi_{m_2}^{*}]$ $E[\Pi_r^{*}]$ $E[\Pi_c^{*}]$
$MS-Bertrand$ (10, 30, 50) 226.3554 163.9024 243.2656 19576 4320.2 5338.1 29234
(15, 30, 45) 226.3583 163.9045 243.6821 19577 4271.3 5344.8 29192
(20, 30, 40) 226.3613 163.9066 244.0987 19577 4222.4 5351.5 29151
(25, 30, 35) 226.3642 163.9087 244.5152 19578 4173.6 5358.2 29109
$RS-Bertrand$ (10, 30, 50) 226.4335 163.9026 243.2279 16110 2347.0 10777 29234
(15, 30, 45) 226.4364 163.9047 243.6443 16110 2291.3 10791 29192
(20, 30, 40) 226.4394 163.9068 244.0608 16111 2235.7 10804 29151
(25, 30, 35) 226.4423 163.9089 244.4773 16111 2180.0 10818 29109
Table 10.  The change of optimal prices with the fuzzy degree of $k_2$
$scenario$ $k_2$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$MS-Bertrand$ (10, 30, 50) 61.1638 21.4399 63.3457 82.0067 29.3404 20.8429 7.9005
(15, 30, 45) 61.1644 21.4121 63.3463 82.0070 29.3265 20.8426 7.9144
(20, 30, 40) 61.1650 21.3844 63.3469 82.0073 29.3126 20.8423 7.9282
(25, 30, 35) 61.1656 21.3566 63.3476 82.0076 29.2987 20.8420 7.9421
$RS-Bertrand$ (10, 30, 50) 40.0141 13.3298 63.3298 82.0055 29.3418 41.9914 16.0120
(15, 30, 45) 40.0144 13.2881 63.3305 82.0059 29.3279 41.9915 16.0398
(20, 30, 40) 40.0147 13.2465 63.3311 82.0062 29.3141 41.9915 16.0676
(25, 30, 35) 40.0150 13.2048 63.3317 82.0065 29.3002 41.9915 16.0954
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$scenario$ $k_2$ $w_1^{*}$ $w_2^{*}$ $p_0^{*}$ $p_1^{*}$ $p_2^{*}$ $p_1^{*}-w_1^{*}$ $p_2^{*}-w_2^{*}$
$MS-Bertrand$ (10, 30, 50) 61.1638 21.4399 63.3457 82.0067 29.3404 20.8429 7.9005
(15, 30, 45) 61.1644 21.4121 63.3463 82.0070 29.3265 20.8426 7.9144
(20, 30, 40) 61.1650 21.3844 63.3469 82.0073 29.3126 20.8423 7.9282
(25, 30, 35) 61.1656 21.3566 63.3476 82.0076 29.2987 20.8420 7.9421
$RS-Bertrand$ (10, 30, 50) 40.0141 13.3298 63.3298 82.0055 29.3418 41.9914 16.0120
(15, 30, 45) 40.0144 13.2881 63.3305 82.0059 29.3279 41.9915 16.0398
(20, 30, 40) 40.0147 13.2465 63.3311 82.0062 29.3141 41.9915 16.0676
(25, 30, 35) 40.0150 13.2048 63.3317 82.0065 29.3002 41.9915 16.0954
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