# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2018172

## Designing a hub location and pricing network in a competitive environment

 1 Department of Industrial Engineering, Alzahra University, Tehran, Iran 2 PhD Student of Industrial Engineering, Alzahra University, Tehran, Iran

* Corresponding author:esmaeili_m@alzahra.ac.ir

Received  December 2016 Revised  June 2018 Published  October 2018

This paper models a novel mixed hub location and pricing problem in a network consists of two competitive firms with different economic positions (Stackelberg-game). The flow that reflects demand of each firm directly depends on its price (Bernard's model). The flow of each firm directly depends on both firms' prices simultaneously (Bernard's model). The firm with higher position (the leader) chooses its potential hubs while the firm in lower position (the follower) may choose either its own hub locations or the other firm's existing hub locations (the competitor's hub) through two real contracts; the airlines own and the long term usage contracts. Firms have to make decision on both the location-allocation and the price determination problems through maximizing their own profits. Moreover, firms make decisions for extending hub coverage through establishing new airline bands, gates and other infrastructures by considering extra cost. In order to evaluate the proposed model, an example derived from the CAB dataset has been solved using Imperialist Competitive Algorithm (ICA) and closed expression, respectively for the hub location-allocation and pricing decisions. Finally, a sensitivity analysis of the model is conducted to show the effect of each firm's share of fixed costs on the contract type selection.

Citation: Maryam Esmaeili, Samane Sedehzade. Designing a hub location and pricing network in a competitive environment. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018172
##### References:
 [1] S. AbbasiParizi, M. Aminnayeri and M. Bashri, Robust solution for a min-max regret hub location problem in a fuzzy stochastic environment, Journal of Industrial and Management Optimization, 14 (2018), 1271-1295. doi: 10.3934/jimo.2018083. [2] N. Adler and K. Smilowitz, Hub-and-spoke network alliances and mergers: Price-location competition in the airline industry, Transportation Research, 41 (2007), 394-409. doi: 10.1016/j.trb.2006.06.005. [3] S. Alumur and B. Y. Kara, Network hub location problems: The state of the art, European Journal of Operational Research, 190 (2008), 1-21. doi: 10.1016/j.ejor.2007.06.008. [4] E. Atashpas-Gargari and C. Lucas, Imperialist competitive algorithm: An algorithm for optimization inspired by imperialist competitive, Proceeding IEEE Congress on Evolutionary computation, (2007), 4661-4667. doi: 10.1109/CEC.2007.4425083. [5] C. Barbot, Vertical contracts between airports and airlines: Is there a trade-off between welfare and competitiveness?, Journal of Transport Economics and Policy, 45 (2011), 227-302. [6] J. F. Campbell, Integer programming formulations of discrete hub location problems, European Journal of Operational Research, 72 (1994), 387-405. doi: 10.1016/0377-2217(94)90318-2. [7] J. F. Campbell and M. O'Kelly, Twenty-five years of hub location research, Transportation Science, 46 (2012), 153-295. doi: 10.1287/trsc.1120.0410. [8] M. L. F. Cheong, R. Bhatnagar and S. C. Graves, Logistics network design with supplier consolidation hubs and multiple shipment options, Journal of Industrial and Management Optimization, 3 (2007), 51-69. doi: 10.3934/jimo.2007.3.51. [9] I. Correia, S. Nickel and F. S. Gama, A stochastic multi-period capacitated multiple allocation hub location problem: Formulation and inequalities, Omega, 74 (2017), 122-134. doi: 10.1016/j.omega.2017.01.011. [10] I. Correia, S. Nickel and F. Saldanha-da-Gama, The capacitated single-allocation hub location problem revisited: A note on a classical formulation, European Journal of Operation Research, 207 (2010), 92-96. doi: 10.1016/j.ejor.2010.04.015. [11] G. Dobson and P. J. Lederer, Airline scheduling and routing in a hub and spoke system, Transportation Science, 27 (1993), 209-312. doi: 10.1287/trsc.27.3.281. [12] H. A. Eiselt and V. Marianov, A conditional p-hub location problem with attraction functions, Computers & Operations Research, 36 (2009), 3128-3135. doi: 10.1016/j.cor.2008.11.014. [13] M. Esmaeili, M. Aryanezhad and P. Zeephongsekul, A game theory approach in seller-buyer supply chain, European Journal of Operational Research, 195 (2009), 442-448. doi: 10.1016/j.ejor.2008.02.026. [14] J. M. Faulhaber, J. J. Schulthess, A. C. Eastmond, C., P. Lewis and R. W. Block, Airport/Airline Agreements Practices and Characteristics, Transportation Research Board, 2010. [15] S. Gelareh, S. Nickel and D. Pisinger, Liner shipping hub network design in a competitive environment, Transportation Research Part E, 46 (2010), 991-1004. doi: 10.1016/j.tre.2010.05.005. [16] A. Lüer-Villagra and V. Marianov, A competitive hub location and pricing problem, European Journal of Operational Research, 231 (2013), 734-744. doi: 10.1016/j.ejor.2013.06.006. [17] A. E. Mahmutogullari and B. Y. Kara, Hub location under competition, European Journal of Operational Research, 250 (2016), 214-225. doi: 10.1016/j.ejor.2015.09.008. [18] V. Marianov, D. Serra and C. ReVelle, Location of hubs in a competitive environment, European Journal of Operational Research, 114 (1999), 363-371. [19] M. Mohammadi, R. Tavakkoli-Moghaddam, A. Siadat and Y. Rahimi, A game-based meta-heuristic for a fuzzy bi-objective reliable hub location problem, Engineering Applications of Artificial Intelligence, 50 (2016), 1-19. doi: 10.1016/j.engappai.2015.12.009. [20] A. Niknamfar, S. T. Akhavan Niaki and S. A. Akhavan Niaki, Opposition-based learning for competitive hub location: A Bi-objective biogeography-based optimization algorithm, Knowledge-Based Systems, 128 (2017), 1-19. doi: 10.1016/j.knosys.2017.04.017. [21] M. E. O'Kelly, The location of interacting hub facilities, Transportation Science, 20 (1986), 65-141. doi: 10.1287/trsc.20.2.92. [22] M. E. O'Kelly, A quadratic integer program for the location of interacting hub facilities, European Journal of Operational Research, 32 (1987), 393-404. doi: 10.1016/S0377-2217(87)80007-3. [23] T. H. Oum and X. Fu, Impacts of airports on airline competition: Focus on airport performance and airport-airline vertical relations, JTRC Discussion paper, (2008), 2008-2017. [24] M. Sasaki and M. Fukushima, Stackelberg hub location problem, Journal of the Operations Research Society of Japan, 44 (2001), 390-402. doi: 10.15807/jorsj.44.390. [25] S. Sedehzadeh, R. Tavakkoli-Moghaddam, A. Baboli and M. Mohammadi, Optimization of a multi-modal tree hub location network with transportation energy consumption: A fuzzy approach, Journal of Intelligent & Fuzzy Systems, 30 (2016), 43-60. doi: 10.3233/IFS-151709. [26] S. Sedehzadeh, R. Tavakkoli-Moghaddam and F. Jolai, New Multi-Mode and Multi-Product Hub Covering Problem: A Priority M/M/c Queue Approach, International Journal of Industrial Mathematics, 72 (2015), 139-148. [27] A. S. Ta, L. T. An, D. Khadraoui and P. D. Tao, Solving Partitioning-Hub Location-Routing Problem using DCA, Journal of Industrial and Management Optimization, 8 (2012), 87-102. doi: 10.3934/jimo.2012.8.87. [28] B. Wagner, Model formulations for hub covering problems, The Journal of the Operational Research Society, 59 (2008), 932-938. doi: 10.1057/palgrave.jors.2602424. [29] B. Wagner, A note on "Location of hubs in a competitive environment", European Journal of Operational Research, 184 (2008), 57-62. doi: 10.1016/j.ejor.2006.10.057. [30] R. Zanjirani Farahani and M. Hekmatfar, Facility Location Concepts, Models, Algorithms and Case Studies, Chapter 11, 2009. [31] R. Zanjirani Farahani, M. Hekmatfar, A. Boloori Arabani and E. Nikbakhsh, Hub location problems: A review of models, classification, techniques and application, Computers & Industrial Engineering, 64 (2013), 1096-1109. doi: 10.1016/j.cie.2013.01.012. [32] M. Zhalechian, R. Tavakkoli-Moghaddam, Y. Rahimi and F. Jolai, An interactive possibilistic programming approach for a multi-objective hub location problem: Economic and environmental design, Applied Soft Computing, 52 (2017), 699-713. doi: 10.1016/j.asoc.2016.10.002.

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##### References:
 [1] S. AbbasiParizi, M. Aminnayeri and M. Bashri, Robust solution for a min-max regret hub location problem in a fuzzy stochastic environment, Journal of Industrial and Management Optimization, 14 (2018), 1271-1295. doi: 10.3934/jimo.2018083. [2] N. Adler and K. Smilowitz, Hub-and-spoke network alliances and mergers: Price-location competition in the airline industry, Transportation Research, 41 (2007), 394-409. doi: 10.1016/j.trb.2006.06.005. [3] S. Alumur and B. Y. Kara, Network hub location problems: The state of the art, European Journal of Operational Research, 190 (2008), 1-21. doi: 10.1016/j.ejor.2007.06.008. [4] E. Atashpas-Gargari and C. Lucas, Imperialist competitive algorithm: An algorithm for optimization inspired by imperialist competitive, Proceeding IEEE Congress on Evolutionary computation, (2007), 4661-4667. doi: 10.1109/CEC.2007.4425083. [5] C. Barbot, Vertical contracts between airports and airlines: Is there a trade-off between welfare and competitiveness?, Journal of Transport Economics and Policy, 45 (2011), 227-302. [6] J. F. Campbell, Integer programming formulations of discrete hub location problems, European Journal of Operational Research, 72 (1994), 387-405. doi: 10.1016/0377-2217(94)90318-2. [7] J. F. Campbell and M. O'Kelly, Twenty-five years of hub location research, Transportation Science, 46 (2012), 153-295. doi: 10.1287/trsc.1120.0410. [8] M. L. F. Cheong, R. Bhatnagar and S. C. Graves, Logistics network design with supplier consolidation hubs and multiple shipment options, Journal of Industrial and Management Optimization, 3 (2007), 51-69. doi: 10.3934/jimo.2007.3.51. [9] I. Correia, S. Nickel and F. S. Gama, A stochastic multi-period capacitated multiple allocation hub location problem: Formulation and inequalities, Omega, 74 (2017), 122-134. doi: 10.1016/j.omega.2017.01.011. [10] I. Correia, S. Nickel and F. Saldanha-da-Gama, The capacitated single-allocation hub location problem revisited: A note on a classical formulation, European Journal of Operation Research, 207 (2010), 92-96. doi: 10.1016/j.ejor.2010.04.015. [11] G. Dobson and P. J. Lederer, Airline scheduling and routing in a hub and spoke system, Transportation Science, 27 (1993), 209-312. doi: 10.1287/trsc.27.3.281. [12] H. A. Eiselt and V. Marianov, A conditional p-hub location problem with attraction functions, Computers & Operations Research, 36 (2009), 3128-3135. doi: 10.1016/j.cor.2008.11.014. [13] M. Esmaeili, M. Aryanezhad and P. Zeephongsekul, A game theory approach in seller-buyer supply chain, European Journal of Operational Research, 195 (2009), 442-448. doi: 10.1016/j.ejor.2008.02.026. [14] J. M. Faulhaber, J. J. Schulthess, A. C. Eastmond, C., P. Lewis and R. W. Block, Airport/Airline Agreements Practices and Characteristics, Transportation Research Board, 2010. [15] S. Gelareh, S. Nickel and D. Pisinger, Liner shipping hub network design in a competitive environment, Transportation Research Part E, 46 (2010), 991-1004. doi: 10.1016/j.tre.2010.05.005. [16] A. Lüer-Villagra and V. Marianov, A competitive hub location and pricing problem, European Journal of Operational Research, 231 (2013), 734-744. doi: 10.1016/j.ejor.2013.06.006. [17] A. E. Mahmutogullari and B. Y. Kara, Hub location under competition, European Journal of Operational Research, 250 (2016), 214-225. doi: 10.1016/j.ejor.2015.09.008. [18] V. Marianov, D. Serra and C. ReVelle, Location of hubs in a competitive environment, European Journal of Operational Research, 114 (1999), 363-371. [19] M. Mohammadi, R. Tavakkoli-Moghaddam, A. Siadat and Y. Rahimi, A game-based meta-heuristic for a fuzzy bi-objective reliable hub location problem, Engineering Applications of Artificial Intelligence, 50 (2016), 1-19. doi: 10.1016/j.engappai.2015.12.009. [20] A. Niknamfar, S. T. Akhavan Niaki and S. A. Akhavan Niaki, Opposition-based learning for competitive hub location: A Bi-objective biogeography-based optimization algorithm, Knowledge-Based Systems, 128 (2017), 1-19. doi: 10.1016/j.knosys.2017.04.017. [21] M. E. O'Kelly, The location of interacting hub facilities, Transportation Science, 20 (1986), 65-141. doi: 10.1287/trsc.20.2.92. [22] M. E. O'Kelly, A quadratic integer program for the location of interacting hub facilities, European Journal of Operational Research, 32 (1987), 393-404. doi: 10.1016/S0377-2217(87)80007-3. [23] T. H. Oum and X. Fu, Impacts of airports on airline competition: Focus on airport performance and airport-airline vertical relations, JTRC Discussion paper, (2008), 2008-2017. [24] M. Sasaki and M. Fukushima, Stackelberg hub location problem, Journal of the Operations Research Society of Japan, 44 (2001), 390-402. doi: 10.15807/jorsj.44.390. [25] S. Sedehzadeh, R. Tavakkoli-Moghaddam, A. Baboli and M. Mohammadi, Optimization of a multi-modal tree hub location network with transportation energy consumption: A fuzzy approach, Journal of Intelligent & Fuzzy Systems, 30 (2016), 43-60. doi: 10.3233/IFS-151709. [26] S. Sedehzadeh, R. Tavakkoli-Moghaddam and F. Jolai, New Multi-Mode and Multi-Product Hub Covering Problem: A Priority M/M/c Queue Approach, International Journal of Industrial Mathematics, 72 (2015), 139-148. [27] A. S. Ta, L. T. An, D. Khadraoui and P. D. Tao, Solving Partitioning-Hub Location-Routing Problem using DCA, Journal of Industrial and Management Optimization, 8 (2012), 87-102. doi: 10.3934/jimo.2012.8.87. [28] B. Wagner, Model formulations for hub covering problems, The Journal of the Operational Research Society, 59 (2008), 932-938. doi: 10.1057/palgrave.jors.2602424. [29] B. Wagner, A note on "Location of hubs in a competitive environment", European Journal of Operational Research, 184 (2008), 57-62. doi: 10.1016/j.ejor.2006.10.057. [30] R. Zanjirani Farahani and M. Hekmatfar, Facility Location Concepts, Models, Algorithms and Case Studies, Chapter 11, 2009. [31] R. Zanjirani Farahani, M. Hekmatfar, A. Boloori Arabani and E. Nikbakhsh, Hub location problems: A review of models, classification, techniques and application, Computers & Industrial Engineering, 64 (2013), 1096-1109. doi: 10.1016/j.cie.2013.01.012. [32] M. Zhalechian, R. Tavakkoli-Moghaddam, Y. Rahimi and F. Jolai, An interactive possibilistic programming approach for a multi-objective hub location problem: Economic and environmental design, Applied Soft Computing, 52 (2017), 699-713. doi: 10.1016/j.asoc.2016.10.002.
Flowchart of the imperialist competitive algorithm
Continuous solution encoding for HLP
Links of firms 1 and 2 on CAB dataset for different Beta
Firm 1's profit during two contracts for different Beta
Firm 2's profit during two contracts for different Beta
Value and distribution of input parameters
 Value & Distribution Fk(hundred  $) Kij hundred ($ ) C ${\rm{\tilde U}}$(100, 1000) ${\rm{\tilde U}}$(10, 50) $0.01 \times d$ Q (hundred  $) R (K.M)$\beta$50 600 0.5  Value & Distribution Fk(hundred$ ) Kij hundred ( $) C${\rm{\tilde U}}$(100, 1000)${\rm{\tilde U}}$(10, 50)$0.01 \times d$Q (hundred$ ) R (K.M) $\beta$ 50 600 0.5
The optimal rout between nodes (1-10) and nodes (8-25)
 Route Cost Price Q Contract 1 - Beta 0.5 Example 1 (1-10) Firm 1 1-5-5-10 12.53 57.16 24.18 Firm 2 1-20-20-10 16.52 51.48 34.96 Example 2 (8-25) Firm 1 8-5-5-25 14.80 55.76 24.29 Firm 2 8-20-20-25 14.96 50.08 35.12 Contract 1 - Beta 1 Example 1 (1-10) Firm 1 1-5-5-10 12.53 75.93 22.63 Firm 2 1-22-22-10 37.91 70.62 32.72 Example 2 (8-25) Firm 1 8-5-5-25 14.80 56.33 24.24 Firm 2 8-22-5-25 15.59 50.64 35.08 Contract 1 - Beta 1.5 Example 1 (1-10) Firm 1 1-5-5-10 12.53 78.44 22.42 Firm 2 1-23-23-10 40.76 73.17 32.42 Example 2 (8-25) Firm 1 8-5-5-25 14.80 72.03 22.95 Firm 2 8-23-23-25 33.46 66.65 33.18 Contract 2 Example 1 (1-10) Firm 1 1-6-6-10 16.64 73.31 22.84 Firm 2 1-23-6-10 34.92 67.95 33.08 Example 2 (8-25) Firm 1 8-6-6-25 15.16 73.03 22.95 Firm 2 8-23-23-25 33.46 66.65 33.18
 Route Cost Price Q Contract 1 - Beta 0.5 Example 1 (1-10) Firm 1 1-5-5-10 12.53 57.16 24.18 Firm 2 1-20-20-10 16.52 51.48 34.96 Example 2 (8-25) Firm 1 8-5-5-25 14.80 55.76 24.29 Firm 2 8-20-20-25 14.96 50.08 35.12 Contract 1 - Beta 1 Example 1 (1-10) Firm 1 1-5-5-10 12.53 75.93 22.63 Firm 2 1-22-22-10 37.91 70.62 32.72 Example 2 (8-25) Firm 1 8-5-5-25 14.80 56.33 24.24 Firm 2 8-22-5-25 15.59 50.64 35.08 Contract 1 - Beta 1.5 Example 1 (1-10) Firm 1 1-5-5-10 12.53 78.44 22.42 Firm 2 1-23-23-10 40.76 73.17 32.42 Example 2 (8-25) Firm 1 8-5-5-25 14.80 72.03 22.95 Firm 2 8-23-23-25 33.46 66.65 33.18 Contract 2 Example 1 (1-10) Firm 1 1-6-6-10 16.64 73.31 22.84 Firm 2 1-23-6-10 34.92 67.95 33.08 Example 2 (8-25) Firm 1 8-6-6-25 15.16 73.03 22.95 Firm 2 8-23-23-25 33.46 66.65 33.18
Result of CAB data set for the proposed model
 Contract 1 Contract 2 Beta = 0.5 Beta = 1 Beta = 1.5 Firm 1 #Hub 3 1 1 1 Hubs 5, 20, 21 5 5 6 Increase Cover radius 1436, 0,181 1436 1436 1565 Cost 328185 340926 341911 343243 Income 676086 791716 794212 778227 Profit 347901 450790 452301 434984 Firm 2 459283 #Hub 1 2 2 2 Hubs 20 5, 22 21, 23 6, 23 Increase Cover radius 0 0, 0 0, 0 1565, 0 Cost 1680 92644 428533 346233 Income 635061 656739 656096 656286 Profit 633381 564095 227562 310053
 Contract 1 Contract 2 Beta = 0.5 Beta = 1 Beta = 1.5 Firm 1 #Hub 3 1 1 1 Hubs 5, 20, 21 5 5 6 Increase Cover radius 1436, 0,181 1436 1436 1565 Cost 328185 340926 341911 343243 Income 676086 791716 794212 778227 Profit 347901 450790 452301 434984 Firm 2 459283 #Hub 1 2 2 2 Hubs 20 5, 22 21, 23 6, 23 Increase Cover radius 0 0, 0 0, 0 1565, 0 Cost 1680 92644 428533 346233 Income 635061 656739 656096 656286 Profit 633381 564095 227562 310053
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