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

Lisha Wang; Huaming Song; Ding Zhang; Hui Yang

doi:10.3934/jimo.2018046

Article Contents

2019, Volume 15, Issue 1: 343-364. Doi: 10.3934/jimo.2018046

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

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

Received: June 2017

Revised: November 2017

Early access: April 2018

Published: January 2019

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 91A25, 91B24; Secondary: 91A40.

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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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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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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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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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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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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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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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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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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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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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