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doi: 10.3934/jimo.2020028

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

Ali Naimi-Sadigh 1,, , S. Kamal Chaharsooghi 2, and  Marzieh Mozafari 3,

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

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.

Keywords: Pricing, Advertising, Production-Inventory, Stackelberg-Nash Equilibrium, Oligopoly Competition.
Mathematics Subject Classification: Primary: 91B25, 91A12, 91A40; Secondary: 90B60, 90B30, 90B05.
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
References:
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[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. Google Scholar

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M. Bazzara, H. Sherali and C. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley and Sons Inc., New York, Third edition, 1993. Google Scholar

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F. Bernstein and A. Federgruen, Pricing and replenishment strategies in a distribution system with competing retailers, Operations Research, 51 (2003), 409-426.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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F. Cheng and S. Sethi, A periodic review inventory model with demand influenced by promotional decisions, Management Science, 45 (1999), 1510-1523.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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A. P. Jeuland and S. M. Shugan, Managing Channel Profits, Marketing Science, 2 (1983), 239-272.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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show all references

References:
[1]

A. Arreola-Risa, Integrated multi-item production-inventory systems, European Journal of Operational Research, 89 (1996), 326-340.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[4]

M. Bazzara, H. Sherali and C. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley and Sons Inc., New York, Third edition, 1993. Google Scholar

[5]

F. Bernstein and A. Federgruen, Pricing and replenishment strategies in a distribution system with competing retailers, Operations Research, 51 (2003), 409-426.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[12]

F. Cheng and S. Sethi, A periodic review inventory model with demand influenced by promotional decisions, Management Science, 45 (1999), 1510-1523.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Drud, CONOPT - a large-scale GRG code, INFORMS Journal on Computing, 6 (1992), 207-216.   Google Scholar

[17]

J. P. Dube, G. J. Hitch and D. Manchanda, An empirical model of advertising dynamics, Quantitative Marketing and Economics, 3 (2005), 107-144.   Google Scholar

[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.   Google Scholar

[19]

F. Facchinei and C. Kanzow, Generalized Nash equilibrium problems, 4OR, 5 (2007), 173-210.  doi: 10.1007/s10288-007-0054-4.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[26]

A. P. Jeuland and S. M. Shugan, Channel of distribution profits when channel members form conjectures, Marketing Science, 7 (1988), 202-210.   Google Scholar

[27]

A. P. Jeuland and S. M. Shugan, Managing Channel Profits, Marketing Science, 2 (1983), 239-272.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[30]

Z. Lin, Price promotion with reference price effects in supply chain, Transportation Research Part E, 85 (2016), 520-568.   Google Scholar

[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.  Google Scholar

[32]

T. Malone and K. Crowston, The interdisciplinary study of coordination, ACM Computing Surveys, 26 (1994), 87-119.   Google Scholar

[33]

K. S. Moorthy, Managing channel profits: Comment, Marketing Science, 6 (1987), 375-379.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[51]

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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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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
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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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
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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$ 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
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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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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