American Institute of Mathematical Sciences

Advanced
May  2021, 17(3): 1423-1450. 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  May 2021 Early access  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 and Management Optimization, 2021, 17 (3) : 1423-1450. doi: 10.3934/jimo.2020028
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. 

show all references

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. 

Figure 1.  The schematic view of SNG setting
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The schematic view of SNGC
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The effect of retail price elasticity
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The effect of advertising expenditure elasticity
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The effect of raw materials' price elasticity
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

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
Table Options

Download as excel

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
Table Options

Download as excel

$ 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
Table Options

Download as excel

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
[1]

Sanjoy Kumar Paul, Ruhul Sarker, Daryl Essam. Managing risk and disruption in production-inventory and supply chain systems: A review. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1009-1029. doi: 10.3934/jimo.2016.12.1009
[2]

Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021044
[3]

Yeong-Cheng Liou, Siegfried Schaible, Jen-Chih Yao. Supply chain inventory management via a Stackelberg equilibrium. Journal of Industrial and Management Optimization, 2006, 2 (1) : 81-94. doi: 10.3934/jimo.2006.2.81
[4]

Chloe A. Fletcher, Jason S. Howell. Dynamic modeling of nontargeted and targeted advertising strategies in an oligopoly. Journal of Dynamics and Games, 2017, 4 (2) : 97-124. doi: 10.3934/jdg.2017007
[5]

Ali Naimi Sadigh, S. Kamal Chaharsooghi, Majid Sheikhmohammady. A game theoretic approach to coordination of pricing, advertising, and inventory decisions in a competitive supply chain. Journal of Industrial and Management Optimization, 2016, 12 (1) : 337-355. doi: 10.3934/jimo.2016.12.337
[6]

Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial and Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843
[7]

Kegui Chen, Xinyu Wang, Min Huang, Wai-Ki Ching. Compensation plan, pricing and production decisions with inventory-dependent salvage value, and asymmetric risk-averse sales agent. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1397-1422. doi: 10.3934/jimo.2018013
[8]

Nickolas J. Michelacakis. Strategic delegation effects on Cournot and Stackelberg competition. Journal of Dynamics and Games, 2018, 5 (3) : 231-242. doi: 10.3934/jdg.2018015
[9]

F. M. Bass, A. Krishnamoorthy, A. Prasad, Suresh P. Sethi. Advertising competition with market expansion for finite horizon firms. Journal of Industrial and Management Optimization, 2005, 1 (1) : 1-19. doi: 10.3934/jimo.2005.1.1
[10]

Ganfu Wang, Xingzheng Ai, Chen Zheng, Li Zhong. Strategic inventory under suppliers competition. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2159-2173. doi: 10.3934/jimo.2019048
[11]

Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics and Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[12]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[13]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[14]

Feng Tao, Hao Shao, KinKeung Lai. Pricing and modularity decisions under competition. Journal of Industrial and Management Optimization, 2020, 16 (1) : 289-307. doi: 10.3934/jimo.2018152
[15]

Chenyin Wang, Yaodong Ni, Xiangfeng Yang. The inventory replenishment policy in an uncertain production-inventory-routing system. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021196
[16]

Sahar Vatankhah, Reza Samizadeh. Determining optimal marketing and pricing policies by considering customer lifetime network value in oligopoly markets. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021112
[17]

Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics and Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1
[18]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[19]

Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060
[20]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (504)
  • HTML views (677)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy