# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020028

## 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

Received  January 2019 Revised  August 2019 Published  February 2020

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.

Citation: Ali Naimi-Sadigh, S. Kamal Chaharsooghi, Marzieh Mozafari. Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020028
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The schematic view of SNG setting
The schematic view of SNGC
The effect of retail price elasticity
The effect of advertising expenditure elasticity
The effect of raw materials' price elasticity
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
 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
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
 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
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
 $R_1$ $R_2$ $R_3$ $M$ $S_1$ $S_2$ $S_3$ $S_4$ $SC$ 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
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
 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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