# American Institute of Mathematical Sciences

July  2018, 14(3): 1271-1295. doi: 10.3934/jimo.2018083

## Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment

 1 Department of Industrial Engineering & Management Systems, Amirkabir University of Technology, Tehran, Iran 2 Department of Industrial Engineering, Shahed University, Tehran, Iran

* Corresponding author: saeedabasi@aut.ac.ir

Received  August 2015 Revised  September 2016 Published  June 2018

In the present paper, a robust approach is used to locate hub facilities considering network risks. An additional objective function, minimax regret, is added to the classical objective function in the hub location problem. In the proposed model, risk factors such as availability, security, delay time, environmental guidelines and regional air pollution are considered using triangular fuzzy-stochastic numbers. Then an equivalent crisp single objective model is proposed and solved by the Benders decomposition method. Finally, the results of both Benders decomposition and commercial optimization software are compared for different instances. Numerical instances were developed based on the well-known Civil Aeronautics Board (CAB) data set, considering different levels of uncertainty in parameters. The results show that the proposed model is capable of selecting nodes as sustainable hubs. Also, the results confirm that using Benders decomposition is more efficient than using classical solution methods for large-scale problems.

Citation: Saeid Abbasi-Parizi, Majid Aminnayeri, Mahdi Bashiri. Robust solution for a minimax regret hub location problem in a fuzzy-stochastic environment. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1271-1295. doi: 10.3934/jimo.2018083
##### References:

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##### References:
The reliable HLP vs. the classical HLP
Different sources of risk in network design
Convergence of lower and upper bounds in the BD algorithm
The overall performance of the BD method
Comparison of output for stochastic, fuzzy and fuzzy-stochastic problems
Risk sensitivity factor chart
A brief review on recent researches of multi-objective HLP considering uncertainty
 Reference Year Model Number of objectives MODM1 approach Risk assessment approach Risk factors Objective functions Uncertainty Solution approach Fuzzy Stochastic Parameter Approach Parameter Approach Chen et al. [12] 2011 - - - ANF Environmental risks - - - - - - Snyder et al. [48] 2006 Supply chain network Single - Expected value, Worst-case value Supply risks Cost - - Cost Scenario-based - Jabbarzadeh et al. [29] 2012 Supply chain network Single - Expected value Environmental and operational risks Profit - - Cost Scenario-based Lagrangian relaxation, GA Atoei et al. [3] 2013 Supply chain network Multiple ε-constraint method Expected value Environmental, macro and supply risks Cost/ Reliability - - Reliability/ Capacity Scenario-based NSGA-Ⅱ Garcia-Herreros et al. [22] 2014 Supply chain network Single - Expected value Operational risks Cost - - The availabilityof DC Scenario-based Accelerated BD Pishvaee et al. [39] 2014 Supply chain network Multiple WSM2 possibility Environmental, macro and Cost/ Environmental impact/ Social responsibility Social impact Credibility - - Accelerated BD Marianov et al. [33] 2003 HLF Single - - - Cost - - Number of plane Probability TS Snyder et al. [49] 2007 Location model Single - - - Cost - - Cost Scenario-based Lagrangian relaxation Camargo et al. [9] 2008 HLP Single - - - Cost - - - - BD Costa et al. [26] 2008 HLF Multiple WSM/WCD3 - - Cost/Tiine - - - - An interactive decision-aid approach Sim et al. [47] 2009 P-hub center problem Single - - - Time - - Time Chance constraint Radial heuristic & Teitz-Bart heuristic Yang [54] 2009 HLP Single - - Cost - - Demand Scenario-based Heuristic methods Vasconcelos et al. [53] 2011 HLP Single - - Cost - - - Probability - Camargo et al. [7] 2011 HLP Single - - Cost - - Demand/ Cost Probability BD & OA Contreras et al. [15] 2011 HLP Single - - Cost - - Demand/ Cost Stochastic BD Mohainmadi et al. [37] 2011 Hub covering location Multiple WSM - Cost/Time - - - - ICA Alumur et al. [2] 2012 HLP Single - - Max cost - - Demand/ Cost Scenario-based - Taghipourian et al. [50] 2012 Dynamic virtual hub location Single _ - Cost Flow and capacity Fuzzy numbers - - - Zhai et al. [57] 2012 HLF Single - Probability function Macro risks Services level - - Demand Chance constraint B& B Bashiri et al. [4] 2013 F-hub center problem Single - - - Max time Qualitative parameters Fuzzy VIKOR method - - GA Davari et al. [17] 2011 maximal covering problem Single - - - Time Time Credibility - - SA4 Yang et al. [55] 2013 P-hub center problem Single - - - Time Travel time Credibility - - PSO5 Mahaminadi et al. [38] 2013 Hub covering problem Multiple MOICA6 Reliability constraints Environmental and operational risks Cost/Max time - - Time Probability MOICA Eghhali et al. [19] 2013 Hub covering problem Multiple - - Cost/ Intermediate links _ _ Reliability of path Reliability NSGA-Ⅱ Proposed model 2018 HLP Multiple WSM Regret/ Expected value Environmental, operational, macro, security and supply risks Cost/ Risk Risk factors Fuzzy numbers Cost/ Risk factors Scenario-based BD
 Reference Year Model Number of objectives MODM1 approach Risk assessment approach Risk factors Objective functions Uncertainty Solution approach Fuzzy Stochastic Parameter Approach Parameter Approach Chen et al. [12] 2011 - - - ANF Environmental risks - - - - - - Snyder et al. [48] 2006 Supply chain network Single - Expected value, Worst-case value Supply risks Cost - - Cost Scenario-based - Jabbarzadeh et al. [29] 2012 Supply chain network Single - Expected value Environmental and operational risks Profit - - Cost Scenario-based Lagrangian relaxation, GA Atoei et al. [3] 2013 Supply chain network Multiple ε-constraint method Expected value Environmental, macro and supply risks Cost/ Reliability - - Reliability/ Capacity Scenario-based NSGA-Ⅱ Garcia-Herreros et al. [22] 2014 Supply chain network Single - Expected value Operational risks Cost - - The availabilityof DC Scenario-based Accelerated BD Pishvaee et al. [39] 2014 Supply chain network Multiple WSM2 possibility Environmental, macro and Cost/ Environmental impact/ Social responsibility Social impact Credibility - - Accelerated BD Marianov et al. [33] 2003 HLF Single - - - Cost - - Number of plane Probability TS Snyder et al. [49] 2007 Location model Single - - - Cost - - Cost Scenario-based Lagrangian relaxation Camargo et al. [9] 2008 HLP Single - - - Cost - - - - BD Costa et al. [26] 2008 HLF Multiple WSM/WCD3 - - Cost/Tiine - - - - An interactive decision-aid approach Sim et al. [47] 2009 P-hub center problem Single - - - Time - - Time Chance constraint Radial heuristic & Teitz-Bart heuristic Yang [54] 2009 HLP Single - - Cost - - Demand Scenario-based Heuristic methods Vasconcelos et al. [53] 2011 HLP Single - - Cost - - - Probability - Camargo et al. [7] 2011 HLP Single - - Cost - - Demand/ Cost Probability BD & OA Contreras et al. [15] 2011 HLP Single - - Cost - - Demand/ Cost Stochastic BD Mohainmadi et al. [37] 2011 Hub covering location Multiple WSM - Cost/Time - - - - ICA Alumur et al. [2] 2012 HLP Single - - Max cost - - Demand/ Cost Scenario-based - Taghipourian et al. [50] 2012 Dynamic virtual hub location Single _ - Cost Flow and capacity Fuzzy numbers - - - Zhai et al. [57] 2012 HLF Single - Probability function Macro risks Services level - - Demand Chance constraint B& B Bashiri et al. [4] 2013 F-hub center problem Single - - - Max time Qualitative parameters Fuzzy VIKOR method - - GA Davari et al. [17] 2011 maximal covering problem Single - - - Time Time Credibility - - SA4 Yang et al. [55] 2013 P-hub center problem Single - - - Time Travel time Credibility - - PSO5 Mahaminadi et al. [38] 2013 Hub covering problem Multiple MOICA6 Reliability constraints Environmental and operational risks Cost/Max time - - Time Probability MOICA Eghhali et al. [19] 2013 Hub covering problem Multiple - - Cost/ Intermediate links _ _ Reliability of path Reliability NSGA-Ⅱ Proposed model 2018 HLP Multiple WSM Regret/ Expected value Environmental, operational, macro, security and supply risks Cost/ Risk Risk factors Fuzzy numbers Cost/ Risk factors Scenario-based BD
Models complexity comparison
 Problem No. of constraints No. of variables Binary Continues RHLP-FSE $3n^{4} ks-n^{3} ks+5n^{2} ks-nks$ $n$ $n^{4} ks+3n^{2} ks$ Classical HLP $2n^{4} +n^{2}$ $n$ $n^{4}$
 Problem No. of constraints No. of variables Binary Continues RHLP-FSE $3n^{4} ks-n^{3} ks+5n^{2} ks-nks$ $n$ $n^{4} ks+3n^{2} ks$ Classical HLP $2n^{4} +n^{2}$ $n$ $n^{4}$
Results of different-sized instances for class Ⅰ risk factors
 |N||H| No of variables No of constraints B&B solution BD solution No of iterations %Gap Continues Binary Equality Inequality ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ Time (s) ${{\Omega }^*}_{BD}$ ${z^*}_{BD}$ Time (s) 4 9 21736 9 245 65520 0.3214 4 92.9 0.3214 4 153.2 2 0 6 7 28456 7 545 91665 0.3255 6 98.3 0.3255 6 194.5 7 0 6 9 46575 9 540 147830 0.3410 3, 4 108.9 0.3410 3, 4 178.7 16 0 8 7 49456 7 965 160125 0.3289 6 265.2 0.3289 6 234.3 8 0 10 7 76336 7 1505 247905 n.a. n.a. _1 0.2979 2, 4 478.3 11 0 12 7 109095 7 2160 355010 n.a. n.a. _1 0.2390 3, 5, 7 763.2 11 10-6 15 7 169261 7 3380 551880 n.a. n.a. _2 0.2881 2, 5, 7 1101.1 8 10-5 16 5 98775 5 3840 332330 n.a. n.a. _2 0.2671 5 1371.6 7 10-4 18 5 124575 5 4860 419630 n.a. n.a. _2 0.2648 5 1427.4 10 10-4 20 5 298936 5 6005 976605 n.a. n.a. _2 0.2754 4, 18 1872.2 9 10-3 B&B: B&B result, BD: Benders decomposition result, −1 : Time limitation, −2: Lack of memory
 |N||H| No of variables No of constraints B&B solution BD solution No of iterations %Gap Continues Binary Equality Inequality ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ Time (s) ${{\Omega }^*}_{BD}$ ${z^*}_{BD}$ Time (s) 4 9 21736 9 245 65520 0.3214 4 92.9 0.3214 4 153.2 2 0 6 7 28456 7 545 91665 0.3255 6 98.3 0.3255 6 194.5 7 0 6 9 46575 9 540 147830 0.3410 3, 4 108.9 0.3410 3, 4 178.7 16 0 8 7 49456 7 965 160125 0.3289 6 265.2 0.3289 6 234.3 8 0 10 7 76336 7 1505 247905 n.a. n.a. _1 0.2979 2, 4 478.3 11 0 12 7 109095 7 2160 355010 n.a. n.a. _1 0.2390 3, 5, 7 763.2 11 10-6 15 7 169261 7 3380 551880 n.a. n.a. _2 0.2881 2, 5, 7 1101.1 8 10-5 16 5 98775 5 3840 332330 n.a. n.a. _2 0.2671 5 1371.6 7 10-4 18 5 124575 5 4860 419630 n.a. n.a. _2 0.2648 5 1427.4 10 10-4 20 5 298936 5 6005 976605 n.a. n.a. _2 0.2754 4, 18 1872.2 9 10-3 B&B: B&B result, BD: Benders decomposition result, −1 : Time limitation, −2: Lack of memory
Results of different-sized instances for class Ⅱ risk factors
 |N||H| No of variables No of constraints B&B solution BD solution No of iterations %Gap Continues Binary Equality Inequality ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ Time (s) ${{\Omega }^*}_{BD}$ ${z^*}_{BD}$ Time (s) 4 9 21736 9 245 65520 0.2670 4 87.4 0.2670 4 133.4 2 0 6 7 28456 7 545 91665 0.2487 1 201.4 0.2487 1 198.9 4 0 6 9 46575 9 540 147830 0.2500 6 214.4 0.2500 6 220.0 5 0 8 7 49456 7 965 160125 0.2384 6 264.2 0.2384 6 248.3 3 0 10 7 76336 7 1505 247905 n.a. n.a. _1 0.2425 5 348.5 5 0 12 7 109095 7 2160 355010 n.a. n.a. _1 0.3131 4, 7 654.2 7 10-6 15 7 169261 7 3380 551880 n.a. n.a. _2 0.2657 2, 5 985.3 6 10-5 16 5 98775 5 3840 332330 n.a. n.a. _2 0.3117 5 1321.6 6 10-4 18 5 124575 5 4860 419630 n.a. n.a. _2 0.3736 2 1563.2 12 10-4 20 5 298936 5 6005 976605 n.a. n.a. _2 0.2576 20 1983.1 7 10-3
 |N||H| No of variables No of constraints B&B solution BD solution No of iterations %Gap Continues Binary Equality Inequality ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ Time (s) ${{\Omega }^*}_{BD}$ ${z^*}_{BD}$ Time (s) 4 9 21736 9 245 65520 0.2670 4 87.4 0.2670 4 133.4 2 0 6 7 28456 7 545 91665 0.2487 1 201.4 0.2487 1 198.9 4 0 6 9 46575 9 540 147830 0.2500 6 214.4 0.2500 6 220.0 5 0 8 7 49456 7 965 160125 0.2384 6 264.2 0.2384 6 248.3 3 0 10 7 76336 7 1505 247905 n.a. n.a. _1 0.2425 5 348.5 5 0 12 7 109095 7 2160 355010 n.a. n.a. _1 0.3131 4, 7 654.2 7 10-6 15 7 169261 7 3380 551880 n.a. n.a. _2 0.2657 2, 5 985.3 6 10-5 16 5 98775 5 3840 332330 n.a. n.a. _2 0.3117 5 1321.6 6 10-4 18 5 124575 5 4860 419630 n.a. n.a. _2 0.3736 2 1563.2 12 10-4 20 5 298936 5 6005 976605 n.a. n.a. _2 0.2576 20 1983.1 7 10-3
Results of different sized instances for the RHLP-FSE model and the CM model
 |N||H| No of variables No of constraints CM solution RHLP-FSE solution Continues Binary Equality Inequality z*CM Lost flows z*RHLP −FSE Lost flows 4 5 6976 5 245 22725 1, 3, 4 7543 3, 4 6060 4 7 13336 7 245 42525 1, 2, 3, 4 12110 4 9599 6 5 14776 5 545 48825 1, 3, 4 3714 2 4892 6 7 28456 7 545 91665 1, 3, 4, 6 8825 2, 4 5982 8 5 25576 5 965 85125 3, 5 9691 6 598 8 7 49456 7 965 160125 1, 3, 4, 6, 7 9687 6, 7 1446
 |N||H| No of variables No of constraints CM solution RHLP-FSE solution Continues Binary Equality Inequality z*CM Lost flows z*RHLP −FSE Lost flows 4 5 6976 5 245 22725 1, 3, 4 7543 3, 4 6060 4 7 13336 7 245 42525 1, 2, 3, 4 12110 4 9599 6 5 14776 5 545 48825 1, 3, 4 3714 2 4892 6 7 28456 7 545 91665 1, 3, 4, 6 8825 2, 4 5982 8 5 25576 5 965 85125 3, 5 9691 6 598 8 7 49456 7 965 160125 1, 3, 4, 6, 7 9687 6, 7 1446
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