# 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
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##### References:
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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