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: January 2019

Revised: August 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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