Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures

Ali Naimi-Sadigh; S. Kamal Chaharsooghi; Marzieh Mozafari

doi:10.3934/jimo.2020028

Article Contents

2021, Volume 17, Issue 3: 1423-1450. Doi: 10.3934/jimo.2020028

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Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures

1.
Electronic Business Research Group, Information Technology Research Department, Iranian Research Institute for Information Science and Technology (IRANDOC), Tehran, Iran
2.
Faculty of Industrial and Systems Engineering, Tarbiat Modares University, Tehran, Iran
3.
Department of Industrial Engineering, Electronic Branch, Islamic Azad University, Tehran, Iran

^* Corresponding author: Ali Naimi-Sadigh
^* Corresponding author: Ali Naimi-Sadigh

Received: December 31, 2018

Revised: July 31, 2019

Early access: February 2020

Published: May 2021

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Abstract

This paper addresses the coordination of pricing, advertising, and production-inventory decisions in a multi-product three-echelon supply chain composed of multiple suppliers, single manufacturer, and multiple retailers. The demand of each product is considered to be non-linearly influenced by the retail price and advertising expenditure. Taking into account the dominant power of the manufacturer and the suppliers' oligopoly competition, this paper aims at obtaining the equilibrium prices at each level of the supply chain and comparing two different scenarios of competitions and cooperation: The former focuses on the situation where the single manufacturer has the dominant power in the supply chain and acts as the leader followed by the retailers and the suppliers simultaneously. The latter implies the situation in which the dominant manufacturer enters cooperation with each independent retailer to boost sales while the suppliers play the role of the followers simultaneously. We develop the Stackelberg-Nash game (SNG), and the Stackelberg-Nash game with cooperation (SNGC) formulations to model the two market structures. The equilibrium decisions are achieved through the optimization methods and the existence and uniqueness properties are explored. Finally, analytical and computational analyses are carried out through a numerical example, and a comprehensive sensitivity analysis is conducted to discuss some managerial insights such as increasing competition among suppliers leads to reducing retail prices.

Keywords:

Mathematics Subject Classification: Primary: 91B25, 91A12, 91A40; Secondary: 90B60, 90B30, 90B05.

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Figure 1. The schematic view of SNG setting

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Figure 2. The schematic view of SNGC

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Figure 3. The effect of retail price elasticity

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Figure 4. The effect of advertising expenditure elasticity

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Figure 5. The effect of raw materials' price elasticity

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Table 1. The retailers' equilibrium strategies

Variables	Retailer			Variables	Retailer
(SNG)	1	2	3	(SNGC)	1	2	3
$ p_{ir}^* $	358.1	443.1	413.7	$ p_{ir}^* $	232.7	288.7	268.9
	342.7	486.5	486.5		190.3	271.8	167.4
	330.2	518.7	373.6		205.4	321.4	234.0
	345.6	420.3	369.1		239.8	292.7	255.6
$ ad_{ir}^* $	29.8	52.1	25.9	$ ad_{ir}^* $	19.4	34.0	16.8
	72.2	91.2	35.7		40.1	51.0	19.7
	76.8	172.9	101.9		47.8	107.1	63.9
	44.3	84.1	41.0		30.8	58.5	28.4
$ D_{ir}^* $	31.6	55.9	72.0	$ D_{ir}^* $	64.3	106.3	137.5
	21.1	58.3	74.1		51.0	124.3	180.6
	14.8	29.3	15.8		32.4	57.3	33.4
	15.1	16.7	27.7		28.0	29.7	49.8

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Table 2. The suppliers' equilibrium strategies

Variables	Raw Material				Variables	Raw Material
(SNG)	1	2	3	4	(SNGC)	1	2	3	4
$ F_{js}^* $	8.1	8.6	10.8	7.5	$ F_{js}^* $	15.6	16.4	20.6	14.4
	7.2	7.3	7.6	7.1		15.5	15.8	16.4	15.4
	11.6	7.6	12.6	8.1		22.9	15.0	24.7	15.9
	6.9	9.0	10.9	8.3		14.1	18.5	22.3	16.9
	4.0	4.1	4.2	3.5		8.4	8.6	9.0	7.5
$ v_{js}^* $	43.7	43.7	61.8	41.5	$ v_{js}^* $	82.3	85.7	117.4	76.4
	39.3	46.2	49.9	42.2		86.5	97.2	105.3	90.9
	61.4	29.0	62.8	35.8		119.6	56.9	123.3	70.1
	23.3	45.7	55.0	38.0		47.4	92.1	111.2	77.9
	29.7	27.0	33.4	27.9		62.0	59.9	69.6	55.5

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Table 3. Comparisons among different game settings

	$ R_1 $	$ R_2 $	$ R_3 $	$ M $	$ S_1 $	$ S_2 $	$ S_3 $	$ S_4 $	$ SC $
Nash	19449.7	48501.5	50858.4	15378.8	6214.9	5542.3	10155.1	4804.2	163613.9
SNG	14880.6	42892.2	40079.2	37803.0	1548.4	1363.0	2459.6	1238.7	142264.7
SNGC	150431.6				6388.9	5818.4	10079.9	5195.4	177914.2

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Table 4. Sensitivity of the whole supply chain benefit with respect to the main parameters

Para-	SC	Test problem No.
meter	benefit	1	2	3	4	5	6	7	8
$ \alpha $	Nash	644055	360043	251351	163614	107806	71085	46138	28418
	SNG	553476	304006	213389	142265	98018	69366	50194	37006
	SNGC	650427	370022	263496	177914	123688	88119	64095	47456
	SNGC to	$ 1\% $	$ 2.8\% $	$ 4.8\% $	$ 8.7\% $	$ 14.7\% $	$ 24\% $	$ 38.9\% $	$ 67\% $
	Nash
	improvement
	SNGC to	$ 17.5\% $	$ 21.7\% $	$ 23.5\% $	$ 25.1\% $	$ 26.2\% $	$ 27\% $	$ 27.7\% $	$ 28.2\% $
	SNG
	improvement
$ \beta $	Nash	146552	151699	157361	163614	170547	178262	186900	196622
	SNG	127737	132110	146931	142265	148192	154815	162263	170702
	SNGC	160470	165743	171534	177914	184970	192807	201560	211394
	SNGC to	$ 9.5\% $	$ 9.3\% $	$ 9\% $	$ 8.7\% $	$ 8.5\% $	$ 8.2\% $	$ 7.8\% $	$ 7.5\% $
	Nash
	improvement
	SNGC to	$ 25.6\% $	$ 25.5\% $	$ 25.3\% $	$ 25.1\% $	$ 24.8\% $	$ 24.5\% $	$ 24.2\% $	$ 23.8\% $
	SNG
	improvement
$ \eta $	Nash	151410	155902	159941	163614	166990	170113	173021	175744
	SNG	133047	136441	139491	142265	144811	147168	149363	151418
	SNGC	166417	170645	174450	177914	181098	184047	186796	189372
	SNGC to	$ 9.9\% $	$ 9.5\% $	$ 9.1\% $	$ 8.7\% $	$ 8.5\% $	$ 8.2\% $	$ 8\% $	$ 7.8\% $
	Nash
	improvement
	SNGC to	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $	$ 25.1\% $
	SNG
	improvement

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References

[1]	A. Arreola-Risa, Integrated multi-item production-inventory systems, European Journal of Operational Research, 89 (1996), 326-340.
[2]	K. Arshinder, A. Kanda and S. G. Deshmukh, A Review on Supply Chain Coordination: Coordination Mechanisms, Managing Uncertainty and Research Directions, In T.-M. Choi and T. E. Cheng (Eds.), Supply Chain Coordination under Uncertainty. Berlin, Heidelberg: Springer Verlag, 2011.
[3]	G. Aust and U. Buscher, Cooperative advertising models in supply chain management: A review, European Journal of Operation Research, 234 (2014), 1-14. doi: 10.1016/j.ejor.2013.08.010.
[4]	M. Bazzara, H. Sherali and C. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley and Sons Inc., New York, Third edition, 1993.
[5]	F. Bernstein and A. Federgruen, Pricing and replenishment strategies in a distribution system with competing retailers, Operations Research, 51 (2003), 409-426.
[6]	T. Boyaci and G. Gallego, Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers, International Journal of Production Economics, 77 (2002), 95-111.
[7]	G. G. Cai, W. C. Chiang and X. Chen, Game theoretic pricing and ordering decisions with partial lost sales in two-stage supply chains, International Journal of Production Economics, 130 (2011), 175-185.
[8]	L. E. Cardenas-Barron and S. S. Sana, A production-inventory model for a two-echelon supply chain when demand is dependent on sales teams' initiatives, International Journal of Production Economics, 155 (2014), 249-258.
[9]	F. R. Chen, A. Federgruen and Y. S. Zheng, Coordination mechanisms for a distribution system with one supplier and multiple retailers, Management Science, 47 (2001), 693-708.
[10]	T. H. Chen and H. M. Chang, Optimal ordering and pricing policies for deteriorating items in one-vendor multi-retailer supply chain, The International Journal of Advanced Manufacturing Technology, 49 (2010), 341-355.
[11]	X. Chen, Z. Pang and L. Pan, Coordinating inventory control and pricing strategies for perishable products, Operations Research, 62 (2014), 284-300. doi: 10.1287/opre.2014.1261.
[12]	F. Cheng and S. Sethi, A periodic review inventory model with demand influenced by promotional decisions, Management Science, 45 (1999), 1510-1523.
[13]	W. Chung, S. Talluri and R. Narasimhan, Price markdown scheme in a multi echelon supply chain in a high-tech industry, European Journal of Operational Research, 215 (2011), 581-589. doi: 10.1016/j.ejor.2011.07.002.
[14]	S. Dempe, Bilevel programmin - a survey, Essays and Surveys in Global Optimization, 7 (2005), 165-193. doi: 10.1007/0-387-25570-2_6.
[15]	Z. S. Dong, W. Chen, Q. Zhao and J. Li, Optimal pricing and inventory strategies for introducing a new product based on demand substitution effects, Journal of Industrial and Management Optimization, (2019). doi: 10.3934/jimo.2018175.
[16]	A. Drud, CONOPT - a large-scale GRG code, INFORMS Journal on Computing, 6 (1992), 207-216.
[17]	J. P. Dube, G. J. Hitch and D. Manchanda, An empirical model of advertising dynamics, Quantitative Marketing and Economics, 3 (2005), 107-144.
[18]	W. Elmaghraby and P. Keskinocak, Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions, Management Science, 49 (2003), 1287-1309.
[19]	F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, 5 (2007), 173-210. doi: 10.1007/s10288-007-0054-4.
[20]	Y. Ghiami and T. Williams, A two-echelon production-inventory model for deteriorating items with multiple buyers, International Journal of Production Economics, 159 (2016), 233-240.
[21]	X. Hong, L. Xu, P. Du and W. Wang, Joint advertising, pricing and collection decisions in a closed-loop supply chain, International Journal of Production Economics, 167 (2015), 12-22.
[22]	H. Huang, H. Ke and L. Wang, Equilibrium analysis of pricing competition and cooperation in supply chain with one common manufacturer and duopoly retailers, International Journal of Production Economics, 178 (2016), 12-21.
[23]	J. Huang, M. Leng and M. Parlar, Demand functions in decision modeling: A comprehensive survey and research directions, Decision Sciences, 44 (2013), 557-609.
[24]	Y. Huang, G. Q. Huang and S. T. Newman, Coordinating pricing and inventory decisions in a multi-level supply chain: A game-theoretic approach, Transportation Research Part E, 47 (2011), 115-129.
[25]	H. B. Hwarng, C. S. P. Chong, N. Xie and T. F. Burgess, Modelling a complex supply chain: Understanding the effect of simplified assumptions, International Journal of Production Research, 43 (2005), 2829-2872.
[26]	A. P. Jeuland and S. M. Shugan, Channel of distribution profits when channel members form conjectures, Marketing Science, 7 (1988), 202-210.
[27]	A. P. Jeuland and S. M. Shugan, Managing Channel Profits, Marketing Science, 2 (1983), 239-272.
[28]	M. Karakul, Joint pricing and procurement of fashion products in the existence of clearance markets, International Journal of Production Economics, 114 (2008), 487-506.
[29]	T. Kim, Y. Hong and J. Lee, Joint economic production allocation and ordering policies in a supply chain consisting of multiple plants and a single retailer, International Journal of Production Research, 43 (2006), 3619-3632.
[30]	Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E, 85 (2016), 520-568.
[31]	J. D. C. Little, BRANDAID: A marketing-mix model, Parts 1 and 2, Operations Research, 23 (1975), 628-655. doi: 10.1287/opre.23.4.628.
[32]	T. Malone and K. Crowston, The interdisciplinary study of coordination, ACM Computing Surveys, 26 (1994), 87-119.
[33]	K. S. Moorthy, Managing channel profits: Comment, Marketing Science, 6 (1987), 375-379.
[34]	S. Mukhopadhyay, R. N. Mukherjee and K. S. Chaudhuri, Joint pricing and ordering policy for a deteriorating inventory, Computers & Industrial Engineering, 47 (2004), 339-349.
[35]	A. Naimi Sadigh, S. K. Chaharsooghi and M. Sheikhmohammady, A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain, Journal of Industrial and Management Optimization, 12 (2016), 337-355. doi: 10.3934/jimo.2016.12.337.
[36]	A. Naimi Sadigh, S. K. Chaharsooghi and M. Sheikhmohammady, Game-theoretic analysis of coordinating pricing and marketing decisions in a multi-product multi-echelon supply chain, Scientia Iranica E, 23 (2016b), 1459-1473.
[37]	A. Naimi Sadigh, B. Karimi and R. Zanjirani Farhani, A game theoretic approach for two echelon supply chains with continuous depletion, International Journal of Management Science and Engineering Management, 6 (2011), 408-412. doi: 10.1080/17509653.2011.10671190.
[38]	K. S. Navarro, J. A. Chedid, W. F. Florez, H. O. Mateus, L. E. Cardenas-Barron and S. S. 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 and Management Optimization, (2019). doi: 10.3934/jimo.2019020.
[39]	S. Parsaeifar, A. Bozorgi-Amiri, A. Naimi-Sadigh and M. S. Sangari, A game theoretical for coordination of pricing, recycling, and green product decisions in the supply chain, Journal of Cleaner Production, 226 (2019), 37-49.
[40]	Y. Qi, W. Ni and K. Shi, Game theoretic analysis of one manufacturer two retailer supply chain with customer market search, International Journal of Production Economics, 164 (2015), 57-64.
[41]	S. Rezapour, J. K. Allen and F. Mistree, Reliable flow in forward and after-sales supply chains considering propagated uncertainty, Transportation Research Part E, 93 (2016), 409-436.
[42]	S. Saha and S. K. Goyal, Supply chain coordination contracts with inventory level and retail price dependent demand, International Journal of Production Economics, 161 (2015), 140-152.
[43]	M. S. Sajadieh and M. R. Akbari-Jokar, Optimizing shipment, ordering and pricing policies in a two-stage supply chain with price-sensitive demand, Transportation Research Part E: Logistics and Transportation Review, 45 (2009), 564-571.
[44]	A. Sajedinejad and S. K. Chaharsooghi, Multi-criteria supplier selection decisions in supply chain networks: A multi-objective optimization approach, Industrial Engineering & Management Systems, 17 (2018), 392-406.
[45]	S. S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support Systems, 50 (2011), 539-547.
[46]	S. S. Sana, J. A. Chedid and K. S. Navarro, A three layer supply chain model with multiple suppliers, manufacturers and retailers for multiple items, Applied Mathematics and Computations, 229 (2014), 139-150. doi: 10.1016/j.amc.2013.12.006.
[47]	S. F. Sethi and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems, Birkha user Boston, Cambridge, MA, 1994. doi: 10.1007/978-1-4612-0285-1.
[48]	A. G. Sogomonian and C. S. Tang, A modeling framework for coordinating promotion and production decisions within a firm, Management Science, 39 (1993), 191-203.
[49]	F. Soleimani, A. Khamesh and B. Naderi, Optimal decisions in a dual-channel supply chain under simultaneous demand and production cost disruptions, Annals of Operations Research, 243 (2016), 301-321. doi: 10.1007/s10479-014-1675-6.
[50]	J. G. Szmerekovsky and J. Zhang, Pricing and two-tier advertising with one manufacturer and one retailer, European Journal of Operational Research, 192 (2009), 904-917. doi: 10.1016/j.ejor.2007.10.005.
[51]	A. A. Taleizadeh and M. Noori-daryan, Pricing, inventory and production policies in a supply chain of pharmacological products with rework process: A game theoretic approach, Operational Research, 16 (2015), 89-115.
[52]	Y. C. Tsao and G. J. Sheen, Dynamic pricing, promotion and replenishment policies for a deteriorating item under permissible delay in payments, Computers & Operations Research, 35 (2008), 3562-3580.
[53]	Y. C. Tsao and J. C. Lu, Trade promotion policies in manufacturer-retailer supply Chains, Transportation Research Part E, 96 (2016), 20-39.
[54]	Y. C. Tsao and G. J. Sheen, Effects of promotion cost sharing policy with the sales learning curve on supply chain coordination, Computers and Operations Research, 39 (2012), 1872-1878. doi: 10.1016/j.cor.2011.07.009.
[55]	J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, NJ, US, 1944.
[56]	C. Wang, R. Huang and Q. Wei, Integrated pricing and lot-sizing decision in a two-echelon supply chain with a finite production rate, International Journal of Production Economics, 161 (2015), 44-53.
[57]	S. Webster and Z. K. Weng, Ordering and pricing policies in a manufacturing and distribution supply chain for fashion products, International Journal of Production Economics, 114 (2008), 476-486.
[58]	T. M. Whitin, Inventory control and price theory, Management Science, 2 (1955), 61-80.
[59]	J. Xie and A. Neyret, Co-op advertising and pricing models in manufacturer-retailer supply chain, Computers & Industrial Engineering, 56 (2009), 1375-1385.
[60]	J. Xie and J. C. Wei, Coordinating advertising and pricing in a manufacturer-retailer channel, European Journal of Operational Research, 197 (2009), 785-791. doi: 10.1016/j.ejor.2008.07.014.
[61]	P. C. Yang and H. M. Wee, A single-vendor and multiple buyers production-inventory policy for deteriorating item, European Journal of Operational Research, 143 (2002), 570-581. doi: 10.1016/S0377-2217(01)00345-9.
[62]	C. A. Yano and S. M. Gilbert, Coordinated pricing and production/ procurement decisions: A review, In: A. Chakravarty, A., Eliashberg, J. (Eds.), Managing Business Interfaces: Marketing, Engineering and Manufacturing Perspectives. Kluwer Academic Publishers, Dordrecht, 2003.
[63]	S. Yin, T. Nishi and I. E. Grossman, Optimal quantity discount coordination for supply chain optimization with one manufacturer and multiple suppliers under demand uncertainty, The International Journal of Advanced Manufacturing Technology, 76 (2015), 1173-1184.
[64]	Y. Yu, G. Q. Huang and L. Liang, Stackelberg game-theoretic model for optimizing advertising, pricing and inventory policies in vendor managed inventory VMI production supply chains, Computers & Industrial Engineering, 57 (2009), 368-382.
[65]	J. L. Zhang, J. Chen and C. Y. Lee, Joint optimization on pricing, promotion and inventory control with stochastic demand, International Journal of Production Economics, 116 (2008), 190-198.
[66]	W. Zhao and Y. Wang, Coordination of joint pricing-production decisions in a supply chain, IIE Transactions, 34 (2002), 701-715.