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

Previous Article
On a modified extragradient method for variational inequality problem with application to industrial electricity production
JIMO Home
This Issue
Next Article
Effect of Bitcoin fee on transaction-confirmation process

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 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

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The five kinds of CPs which selected randomly from the space$^*$

Figure Options

Download full-size image

Download as PowerPoint slide

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

Table Options

Download as excel


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)

Table Options

Download as excel

	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

Table Options

Download as excel

$scenario$	$w_1^{*}$	$w_2^{*}$	$p_0^{*}$	$p_1^{*}$	$p_2^{*}$	$p_1^{}-w_1^{}$	$p_2^{}-w_2^{}$
$CD$	$-$	$-$	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

Table Options

Download as excel

$scenario$	$E[D_0^{*}]$	$E[D_1^{*}]$	$E[D_2^{*}]$	$E[\Pi_{m_1}^{*}]$	$E[\Pi_{m_2}^{*}]$	$E[\Pi_r^{*}]$	$E[\Pi_c^{*}]$
$CD$	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)

Table Options

Download as excel

	$\Phi_2$	$k_2$	$\gamma_1$
$CP_1$	(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

Table Options

Download as excel

	$CP_1$	$CP_2$	$CP_3$	$CP_4$	$CP_5$
$p_0^{N*}-p_0^{Nocp}$	-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

Table Options

Download as excel

$scenario$	$\beta$	$E[D_0^{*}]$	$E[D_1^{*}]$	$E[D_2^{*}]$	$E[\Pi_{m_1}^{*}]$	$E[\Pi_{m_2}^{*}]$	$E[\Pi_r^{*}]$	$E[\Pi_c^{*}]$
$MS-Bertrand$	(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

Table Options

Download as excel

$scenario$	$\beta$	$w_1^{*}$	$w_2^{*}$	$p_0^{*}$	$p_1^{*}$	$p_2^{*}$	$p_1^{}-w_1^{}$	$p_2^{}-w_2^{}$
$MS-Bertrand$	(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

Table Options

Download as excel

$scenario$	$k_2$	$E[D_0^{*}]$	$E[D_1^{*}]$	$E[D_2^{*}]$	$E[\Pi_{m_1}^{*}]$	$E[\Pi_{m_2}^{*}]$	$E[\Pi_r^{*}]$	$E[\Pi_c^{*}]$
$MS-Bertrand$	(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

Table Options

Download as excel

$scenario$	$k_2$	$w_1^{*}$	$w_2^{*}$	$p_0^{*}$	$p_1^{*}$	$p_2^{*}$	$p_1^{}-w_1^{}$	$p_2^{}-w_2^{}$
$MS-Bertrand$	(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

[1]	Mingyong Lai, Hongzhao Yang, Erbao Cao, Duo Qiu, Jing Qiu. Optimal decisions for a dual-channel supply chain under information asymmetry. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1023-1040. doi: 10.3934/jimo.2017088
[2]	Mitali Sarkar, Young Hae Lee. Optimum pricing strategy for complementary products with reservation price in a supply chain model. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1553-1586. doi: 10.3934/jimo.2017007
[3]	Lei Yang, Jingna Ji, Kebing Chen. Advertising games on national brand and store brand in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2018, 14 (1) : 105-134. doi: 10.3934/jimo.2017039
[4]	Tinggui Chen, Yanhui Jiang. Research on operating mechanism for creative products supply chain based on game theory. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1103-1112. doi: 10.3934/dcdss.2015.8.1103
[5]	Ali Naimi Sadigh, S. Kamal Chaharsooghi, Majid Sheikhmohammady. A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain. Journal of Industrial & Management Optimization, 2016, 12 (1) : 337-355. doi: 10.3934/jimo.2016.12.337
[6]	Claude Carlet, Sylvain Guilley. Complementary dual codes for counter-measures to side-channel attacks. Advances in Mathematics of Communications, 2016, 10 (1) : 131-150. doi: 10.3934/amc.2016.10.131
[7]	Jing Zhao, Jie Wei, Yongjian Li. Pricing and remanufacturing decisions for two substitutable products with a common retailer. Journal of Industrial & Management Optimization, 2017, 13 (2) : 1125-1147. doi: 10.3934/jimo.2016065
[8]	Gang Xie, Wuyi Yue, Shouyang Wang. Optimal selection of cleaner products in a green supply chain with risk aversion. Journal of Industrial & Management Optimization, 2015, 11 (2) : 515-528. doi: 10.3934/jimo.2015.11.515
[9]	Jingming Pan, Wenqing Shi, Xiaowo Tang. Pricing and ordering strategies of supply chain with selling gift cards. Journal of Industrial & Management Optimization, 2018, 14 (1) : 349-369. doi: 10.3934/jimo.2017050
[10]	Qiang Lin, Ying Peng, Ying Hu. Supplier financing service decisions for a capital-constrained supply chain: Trade credit vs. combined credit financing. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2019026
[11]	Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[12]	Jun Li, Hairong Feng, Kun-Jen Chung. Using the algebraic approach to determine the replenishment optimal policy with defective products, backlog and delay of payments in the supply chain management. Journal of Industrial & Management Optimization, 2012, 8 (1) : 263-269. doi: 10.3934/jimo.2012.8.263
[13]	Katherinne Salas Navarro, Jaime Acevedo Chedid, Whady F. Florez, Holman Ospina Mateus, Leopoldo Eduardo Cárdenas-Barrón, Shib Sankar Sana. A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2019020
[14]	Feng Tao, Hao Shao, KinKeung Lai. Pricing and modularity decisions under competition. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2018152
[15]	Xiaohong Chen, Kui Li, Fuqiang Wang, Xihua Li. Optimal production, pricing and government subsidy policies for a closed loop supply chain with uncertain returns. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-26. doi: 10.3934/jimo.2019008
[16]	Jiuping Xu, Pei Wei. Production-distribution planning of construction supply chain management under fuzzy random environment for large-scale construction projects. Journal of Industrial & Management Optimization, 2013, 9 (1) : 31-56. doi: 10.3934/jimo.2013.9.31
[17]	Mehmet Önal, H. Edwin Romeijn. Two-echelon requirements planning with pricing decisions. Journal of Industrial & Management Optimization, 2009, 5 (4) : 767-781. doi: 10.3934/jimo.2009.5.767
[18]	Cuilian You, Le Bo. Option pricing formulas for generalized fuzzy stock model. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-10. doi: 10.3934/jimo.2018158
[19]	Juliang Zhang, Jian Chen. Information sharing in a make-to-stock supply chain. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1169-1189. doi: 10.3934/jimo.2014.10.1169
[20]	Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial & Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81

2018 Impact Factor: 1.025

Tools

Metrics

Tools

Article outline

Figures and Tables

2018 Impact Factor: 1.025

Tools

Figures and Tables

2018 Impact Factor: 1.025

American Institute of Mathematical Sciences