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Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels
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 |
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.
Mathematics Subject Classification:
Primary: 90B80, 62G35; Secondary: 83C15.
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 and Management Optimization,
2018, 14
(3)
: 1271-1295.
doi: 10.3934/jimo.2018083
![]()
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Partitioning procedures for solving mixed-variables programming problems, Numer. Math., 4 (1962), 238-252.
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A hybrid outer-approximation/benders decomposition algorithm for the single allocation hub location problem under congestion, Oper. Res. Lett., 39 (2011), 329-337.
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Designing a road network for hazardous materials shipments, Comp. Oper. Res., 34 (2007), 1389-1405.
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Decomposition methods for network design, Procedia-Social and Behavioral Sciences, 20 (2011), 31-37.
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A. M. Geoffrion,
Generalized Benders decomposition, J. Optimization Theory Appl., 10 (1972), 237-260.
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M. da Graça Costa, M. E. Captivo and J. Clímaco,
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|
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I. Heckmann, T. Comes and S. Nickel,
A critical review on supply chain risk-Definition, measure and modeling, Omega, 52 (2015), 119-132.
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Multiple Attributes Decision Making. Methods and Applications, Springer-Verlag, Berlin-New York, 1981. |
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A. Jabbarzadeh, S. G. Jalali Naini, H. Davoudpour and N. Azad, Designing a supply chain network under the risk of disruptions,
Math. Probl. Eng., (2012), Art. ID 234324, 23 pp.
doi: 10.1155/2012/234324. |
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B. Y. Kara,
Modeling and Analysis of Issues in Hub Location Problems, Doctor of Philosophy Thesis, Bilkent University, Ankara, Turkey, 1999. |
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H. E. Lee, K. H. Park and Z. Z. Bien,
Iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data, Fuzzy Systems, IEEE Transactions on, 16 (2008), 263-277.
|
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J. Li, G. H. Huang, G. Zeng, I. Maqsood and Y. Huang,
An integrated fuzzy-stochastic modeling approach for risk assessment of groundwater contamination, Jour. Environ. Manag., 82 (2007), 173-188.
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Location models for airline hubs behaving as M/D/c queues, Comp. Oper. Res., 30 (2003), 983-1003.
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Robust intermodal hub location under polyhedral demand uncertainty, Transportation Research Part B: Methodological, 86 (2016), 66-85.
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Organizational risk perception and reduction: A literature review, British Journal of Management, 6 (1995), 115-133.
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A multi-objective imperialist competitive algorithm for a capacitated hub covering location problem, Int. J. Indust. Eng. Comp., 2 (2011), 671-688.
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M. Mohammadi, F. Jolai and R. Tavakkoli-Moghaddam,
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M. S. Pishvaee, J. Razmi and S. A. Torabi,
An accelerated Benders decomposition algorithm for sustainable supply chain network design under uncertainty: A case study of medical needle and syringe supply chain, Trans. Res. Part E: Log. Trans. Rev., 67 (2014), 14-38.
|
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E. M. de Sá, R. Morabito and R. S. de Camargo,
Benders decomposition applied to a robust multiple allocation incomplete hub location problem, Computers & Operations Research, 89 (2018), 31-50.
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E. M. de Sá, R. Morabito and R. S. de Camargo,
Efficient Benders decomposition algorithms for the robust multiple allocation incomplete hub location problem with service time requirements, Expert Systems with Applications, 93 (2018), 50-61.
|
| [43] |
E. M. de Sá, R. S. de Camargo and G. de Miranda,
An improved Benders decomposition algorithm for the tree of hubs location problem, European J. Oper. Res., 226 (2013), 185-202.
doi: 10.1016/j.ejor.2012.10.051. |
| [44] |
F. S. Salman and E. Yücel,
Emergency facility location under random network damage: Insights from the Istanbul case, Comput. Oper. Res., 62 (2015), 266-281.
doi: 10.1016/j.cor.2014.07.015. |
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T. Santoso, S. Ahmed, M. Goetschalckx and A. Shapiro,
A stochastic programming approach for supply chain network design under uncertainty, Eur. J. Oper. Res., 167 (2005), 96-115.
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E. S. Sheppard,
A conceptual framework for dynamic location-Allocation analysis, Environment and Planning A, 6 (1974), 547-564.
doi: 10.1068/a060547. |
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T. Sim, T. J. Lowe and B. W. Thomas,
The stochastic p-hub center problem with service-level constraints, Comput. Oper. Res., 36 (2009), 3166-3177.
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show all references
References:
| [1] |
S. Alumur and B. Y. Kara,
A new model for the hazardous waste location-routing problem, Comp. Oper. Res., 34 (2007), 1406-1423.
doi: 10.1016/j.cor.2005.06.012. |
| [2] |
S. A. Alumur, S. Nickel and F. Saldanha-da-Gama,
Hub location under uncertainty, Trans. Res. Part B: Method., 46 (2012), 529-543.
doi: 10.1016/j.trb.2011.11.006. |
| [3] |
F. Atoei, E. Teimory and A. Amiri,
Designing reliable supply chain network with disruption risk, Int. J. Indust. Eng. Comp., 4 (2013), 111-126.
|
| [4] |
M. Bashiri, M. Mirzaei and M. Randall,
Modeling fuzzy capacitated p-hub center problem and a genetic algorithm solution, Appl. Math. Model., 37 (2013), 3513-3525.
doi: 10.1016/j.apm.2012.07.018. |
| [5] |
J. F. Benders,
Partitioning procedures for solving mixed-variables programming problems, Numer. Math., 4 (1962), 238-252.
doi: 10.1007/BF01386316. |
| [6] |
R. Caballero, M. González, F. M. Guerrero, J. Molina and C. Paralera,
Solving a multiobjective location routing problem with a metaheuristic based on tabu search. Application to a real case in Andalusia, European Journal of Operational Research, 177 (2007), 1751-1763.
doi: 10.1016/j.ejor.2005.10.017. |
| [7] |
R. S. de Camargo, G. de Miranda, Jr. and R. P. M. Ferreira,
A hybrid outer-approximation/benders decomposition algorithm for the single allocation hub location problem under congestion, Oper. Res. Lett., 39 (2011), 329-337.
doi: 10.1016/j.orl.2011.06.015. |
| [8] |
R. S. de Camargo, G. de Miranda, Jr., R. P. M. Ferreira and H. P. Luna,
Multiple allocation hub-and-spoke network design under hub congestion, Comp. Oper. Res., 36 (2009), 3097-3106.
doi: 10.1016/j.cor.2008.10.004. |
| [9] |
R. S. de Camargo, G. de Miranda, Jr. and H. P. Luna,
Benders decomposition for the uncapacitated multiple allocation hub location problem, Comp. Oper. Res., 35 (2008), 1047-1064.
doi: 10.1016/j.cor.2006.07.002. |
| [10] |
R. S. de Camargo, G. de Miranda, Jr. and H. L. P. Luna,
Benders decomposition for hub location problems with economies of scale, Transportation Science, 43 (2008), 86-97.
doi: 10.1287/trsc.1080.0233. |
| [11] |
J. F. Campbell,
Integer programming formulations of discrete hub location problems, Eur. J. Oper. R., 72 (1994), 387-405.
doi: 10.1016/0377-2217(94)90318-2. |
| [12] |
Z. Chen, H. Li, H. Ren, Q. Xu and J. Hong,
A total environmental risk assessment model for international hub airports, Int. J. Proj. Man., 29 (2011), 856-866.
doi: 10.1016/j.ijproman.2011.03.004. |
| [13] |
G. Chen and T. T. Pham, Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems, CRC Press, 2000.
doi: 10.1201/9781420039818.
|
| [14] |
I. Contreras, J.-F. Cordeau and G. Laporte,
Benders decomposition for large-scale uncapacitated hub location, Operations Research, 59 (2011), 1477-1490.
doi: 10.1287/opre.1110.0965. |
| [15] |
I. Contreras, J.-F. Cordeau and G. Laporte,
Stochastic uncapacitated hub location, Eur. J. Oper. Res., 212 (2011), 518-528.
doi: 10.1016/j.ejor.2011.02.018. |
| [16] |
I. Contreras, J.-F. Cordeau and G. Laporte,
Exact solution of large-scale hub location problems with multiple capacity levels, Transportation Science, 46 (2012), 439-459.
doi: 10.1287/trsc.1110.0398. |
| [17] |
S. Davari, M. H. Fazel Zarandi and A. Hemmati,
Maximal covering location problem (MCLP) with fuzzy travel times, Expert Systems with Applications, 38 (2011), 14535-14541.
doi: 10.1016/j.eswa.2011.05.031. |
| [18] |
K. Deb,
Multi-objective Optimization Using Evolutionary Algorithms, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 2001. |
| [19] |
M. Eghbali, M. Abedzadeh and M. Setak,
Multi-objective reliable hub covering location considering customer convenience using NSGA-Ⅱ, Int. J. Syst. Assur. Eng. Manag., 5 (2014), 450-460.
doi: 10.1007/s13198-013-0189-y. |
| [20] |
E. Erkut and O. Alp,
Designing a road network for hazardous materials shipments, Comp. Oper. Res., 34 (2007), 1389-1405.
|
| [21] |
A. Eydi and A. Mirakhorli, An extended model for the uncapacitated single allocation hub covering problem in a fuzzy environment, in Proceedings of the International Multiconference of Engineers and Computer Scientists, International Association of Engineers, Hong Kong, Vol Ⅱ, (2012). |
| [22] |
P. Garcia-Herreros, J. M. Wassick and I. E. Grossmann,
Design of resilient supply chains with risk of facility disruptions, Indust. Eng. Chem. Res., 53 (2014), 17240-17251.
doi: 10.1021/ie5004174. |
| [23] |
S. Gelareh and S. Nickel, A benders decomposition for hub location problems arising in public
transport, in Operations Research Proceedings 2007 Part Ⅵ, Springer Berlin Heidelberg,
2008,129-134. |
| [24] |
B. Gendron,
Decomposition methods for network design, Procedia-Social and Behavioral Sciences, 20 (2011), 31-37.
doi: 10.1016/j.sbspro.2011.08.006. |
| [25] |
A. M. Geoffrion,
Generalized Benders decomposition, J. Optimization Theory Appl., 10 (1972), 237-260.
doi: 10.1007/BF00934810. |
| [26] |
M. da Graça Costa, M. E. Captivo and J. Clímaco,
Capacitated single allocation hub location problem-A bi-criteria approach, Comp. Oper. Res., 35 (2008), 3671-3695.
|
| [27] |
I. Heckmann, T. Comes and S. Nickel,
A critical review on supply chain risk-Definition, measure and modeling, Omega, 52 (2015), 119-132.
doi: 10.1016/j.omega.2014.10.004. |
| [28] |
C. L. Hwang and K. Yoon,
Multiple Attributes Decision Making. Methods and Applications, Springer-Verlag, Berlin-New York, 1981. |
| [29] |
A. Jabbarzadeh, S. G. Jalali Naini, H. Davoudpour and N. Azad, Designing a supply chain network under the risk of disruptions,
Math. Probl. Eng., (2012), Art. ID 234324, 23 pp.
doi: 10.1155/2012/234324. |
| [30] |
B. Y. Kara,
Modeling and Analysis of Issues in Hub Location Problems, Doctor of Philosophy Thesis, Bilkent University, Ankara, Turkey, 1999. |
| [31] |
H. E. Lee, K. H. Park and Z. Z. Bien,
Iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data, Fuzzy Systems, IEEE Transactions on, 16 (2008), 263-277.
|
| [32] |
J. Li, G. H. Huang, G. Zeng, I. Maqsood and Y. Huang,
An integrated fuzzy-stochastic modeling approach for risk assessment of groundwater contamination, Jour. Environ. Manag., 82 (2007), 173-188.
doi: 10.1016/j.jenvman.2005.12.018. |
| [33] |
V. Marianov and D. Serra,
Location models for airline hubs behaving as M/D/c queues, Comp. Oper. Res., 30 (2003), 983-1003.
|
| [34] |
M. Merakl and H. Yaman,
Robust intermodal hub location under polyhedral demand uncertainty, Transportation Research Part B: Methodological, 86 (2016), 66-85.
doi: 10.1016/j.trb.2016.01.010. |
| [35] |
A. Mirakhorli, Capacitated single-assignment hub covering location problem under fuzzy environment, in Proceedings of the World Congress on Engineering and Computer Science, 2
(2010), 20-22. |
| [36] |
V.-W. Mitchell,
Organizational risk perception and reduction: A literature review, British Journal of Management, 6 (1995), 115-133.
doi: 10.1111/j.1467-8551.1995.tb00089.x. |
| [37] |
M. Mohammadi, R. Tavakkoli-Moghaddam and R. Rostami,
A multi-objective imperialist competitive algorithm for a capacitated hub covering location problem, Int. J. Indust. Eng. Comp., 2 (2011), 671-688.
doi: 10.5267/j.ijiec.2010.08.003. |
| [38] |
M. Mohammadi, F. Jolai and R. Tavakkoli-Moghaddam,
Solving a new stochastic multi-mode p-hub covering location problem considering risk by a novel multi-objective algorithm, Appl. Math. Model., 37 (2013), 10053-10073.
doi: 10.1016/j.apm.2013.05.063. |
| [39] |
M. S. Pishvaee, J. Razmi and S. A. Torabi,
An accelerated Benders decomposition algorithm for sustainable supply chain network design under uncertainty: A case study of medical needle and syringe supply chain, Trans. Res. Part E: Log. Trans. Rev., 67 (2014), 14-38.
|
| [40] |
T. K. Roy and M. Maiti,
A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity, Eur. J. Oper. Res., 99 (1997), 425-432.
doi: 10.1016/S0377-2217(96)00163-4. |
| [41] |
E. M. de Sá, R. Morabito and R. S. de Camargo,
Benders decomposition applied to a robust multiple allocation incomplete hub location problem, Computers & Operations Research, 89 (2018), 31-50.
doi: 10.1016/j.cor.2017.08.001. |
| [42] |
E. M. de Sá, R. Morabito and R. S. de Camargo,
Efficient Benders decomposition algorithms for the robust multiple allocation incomplete hub location problem with service time requirements, Expert Systems with Applications, 93 (2018), 50-61.
|
| [43] |
E. M. de Sá, R. S. de Camargo and G. de Miranda,
An improved Benders decomposition algorithm for the tree of hubs location problem, European J. Oper. Res., 226 (2013), 185-202.
doi: 10.1016/j.ejor.2012.10.051. |
| [44] |
F. S. Salman and E. Yücel,
Emergency facility location under random network damage: Insights from the Istanbul case, Comput. Oper. Res., 62 (2015), 266-281.
doi: 10.1016/j.cor.2014.07.015. |
| [45] |
T. Santoso, S. Ahmed, M. Goetschalckx and A. Shapiro,
A stochastic programming approach for supply chain network design under uncertainty, Eur. J. Oper. Res., 167 (2005), 96-115.
doi: 10.1016/j.ejor.2004.01.046. |
| [46] |
E. S. Sheppard,
A conceptual framework for dynamic location-Allocation analysis, Environment and Planning A, 6 (1974), 547-564.
doi: 10.1068/a060547. |
| [47] |
T. Sim, T. J. Lowe and B. W. Thomas,
The stochastic p-hub center problem with service-level constraints, Comput. Oper. Res., 36 (2009), 3166-3177.
doi: 10.1016/j.cor.2008.11.020. |
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L. V. Snyder, M. S. Daskin and C. P. Teo,
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Figure 2.
Different sources of risk in network design
Figure 3.
Convergence of lower and upper bounds in the BD algorithm
Figure 4.
The overall performance of the BD method
Figure 5.
Comparison of output for stochastic, fuzzy and fuzzy-stochastic problems
Figure 6.
Risk sensitivity factor chart
Table 1.
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 |
Table 2.
Models complexity comparison
| Problem | No. of constraints | No. of variables | |
| Binary | Continues | ||
| RHLP-FSE | |||
| Classical HLP | |||
| Problem | No. of constraints | No. of variables | |
| Binary | Continues | ||
| RHLP-FSE | |||
| Classical HLP | |||
Table 3.
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 | Time (s) | 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 | Time (s) | 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 | |||||||||||||
Table 4.
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 | Time (s) | 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 | Time (s) | 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 |
Table 5.
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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