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

Saeid Abbasi-Parizi; Majid Aminnayeri; Mahdi Bashiri

doi:10.3934/jimo.2018083

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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

Saeid Abbasi-Parizi ^1,, , Majid Aminnayeri ^1, and Mahdi Bashiri ^2,

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.

Keywords: Hub location, robust optimization, minimax regret, Benders decomposition algorithm.

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

References:

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[5]	J. F. Benders, Partitioning procedures for solving mixed-variables programming problems, Numer. Math., 4 (1962), 238-252. doi: 10.1007/BF01386316. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	G. Chen and T. T. Pham, Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems, CRC Press, 2000. doi: 10.1201/9781420039818. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	K. Deb, Multi-objective Optimization Using Evolutionary Algorithms, Wiley-Interscience Series in Systems and Optimization, John Wiley & Sons, Ltd., Chichester, 2001. Google Scholar
[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. Google Scholar
[20]	E. Erkut and O. Alp, Designing a road network for hazardous materials shipments, Comp. Oper. Res., 34 (2007), 1389-1405. Google Scholar
[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). Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	A. M. Geoffrion, Generalized Benders decomposition, J. Optimization Theory Appl., 10 (1972), 237-260. doi: 10.1007/BF00934810. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	C. L. Hwang and K. Yoon, Multiple Attributes Decision Making. Methods and Applications, Springer-Verlag, Berlin-New York, 1981. Google Scholar
[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. Google Scholar
[30]	B. Y. Kara, Modeling and Analysis of Issues in Hub Location Problems, Doctor of Philosophy Thesis, Bilkent University, Ankara, Turkey, 1999. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	V. Marianov and D. Serra, Location models for airline hubs behaving as M/D/c queues, Comp. Oper. Res., 30 (2003), 983-1003. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[46]	E. S. Sheppard, A conceptual framework for dynamic location-Allocation analysis, Environment and Planning A, 6 (1974), 547-564. doi: 10.1068/a060547. Google Scholar
[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. Google Scholar
[48]	L. V. Snyder, M. S. Daskin and C. P. Teo, The stochastic location model with risk pooling, Eur. J. Oper. Res., 179 (2007), 1221-1238. doi: 10.1016/j.ejor.2005.03.076. Google Scholar
[49]	L. V. Snyder, M. P. Scaparra, M. S. Daskin and R. L. Church, Planning for disruptions in supply chain networks, Tutorials in Operations Research, (2006), 234-257. doi: 10.1287/educ.1063.0025. Google Scholar
[50]	F. Taghipourian, I. Mahdavi, N. Mahdavi-Amiri and A. Makui, A fuzzy programming approach for dynamic virtual hub location problem, Appl. Math. Model., 36 (2012), 3257-3270. doi: 10.1016/j.apm.2011.10.016. Google Scholar
[51]	A. Tajbakhsh, H. Haleh and J. Razmi, A multi-objective model to single-allocation ordered hub location problems by genetic algorithm, Int. J. Acad. Res. Bus. Soc. Sci., 3 (2013). Available from: http://hrmars.com/admin/pics/1640.pdf. Google Scholar
[52]	S. A. Torabi and E. Hassini, An interactive possibilistic programming approach for multiple objective supply chain master planning, Fuzzy Sets and Systems, 159 (2008), 193-214. doi: 10.1016/j.fss.2007.08.010. Google Scholar
[53]	A. D. Vasconcelos, C. D. Nassi and L. A. Lopes, The uncapacitated hub location problem in networks under decentralized management, Comput. Oper. Res., 38 (2011), 1656-1666. doi: 10.1016/j.cor.2011.03.004. Google Scholar
[54]	T.-H. Yang, Stochastic air freight hub location and flight routes planning, Appl. Math. Model., 33 (2009), 4424-4430. doi: 10.1016/j.apm.2009.03.018. Google Scholar
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Figure 1. The reliable HLP vs. the classical HLP

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Figure 2. Different sources of risk in network design

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Figure 3. Convergence of lower and upper bounds in the BD algorithm

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Figure 4. The overall performance of the BD method

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Figure 5. Comparison of output for stochastic, fuzzy and fuzzy-stochastic problems

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Figure 6. Risk sensitivity factor chart

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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

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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
Problem	No. of constraints	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} $

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Problem	No. of constraints	No. of variables
Problem	No. of constraints	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} $

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
\|N\|\|H\|		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

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\|N\|\|H\|		No of variables		No of constraints		B&B solution			BD solution			No of iterations	%Gap
\|N\|\|H\|		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

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
\|N\|\|H\|		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

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\|N\|\|H\|		No of variables		No of constraints		B&B solution			BD solution			No of iterations	%Gap
\|N\|\|H\|		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

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
\|N\|\|H\|		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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\|N\|\|H\|		No of variables		No of constraints		CM solution		RHLP-FSE solution
\|N\|\|H\|		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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2019 Impact Factor: 1.366

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2019 Impact Factor: 1.366

American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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