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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M. da Graça CostaM. E. Captivo and J. Clímaco, Capacitated single allocation hub location problem-A bi-criteria approach, Comp. Oper. Res., 35 (2008), 3671-3695.   Google Scholar

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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. AlumurS. 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. AtoeiE. 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. BashiriM. 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. CaballeroM. GonzálezF. M. GuerreroJ. 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 CamargoG. de MirandaJr. 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 CamargoG. de MirandaJr.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 CamargoG. de MirandaJr. 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 CamargoG. de MirandaJr. 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. ChenH. LiH. RenQ. 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. ContrerasJ.-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. ContrerasJ.-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. ContrerasJ.-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. DavariM. 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. EghbaliM. 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-HerrerosJ. 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 CostaM. 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. HeckmannT. 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. LeeK. 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. LiG. H. HuangG. ZengI. 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. MohammadiR. 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. MohammadiF. 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. PishvaeeJ. 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 SáR. 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 SáR. 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 SáR. 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. SantosoS. AhmedM. 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. SimT. 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. SnyderM. 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]

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Figure 1.  The reliable HLP vs. the classical HLP
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 $3n^{4} ks-n^{3} ks+5n^{2} ks-nks$ $n$ $n^{4} ks+3n^{2} ks$
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} $
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 ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ 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
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 ${{\Omega }^*}_{B\&B}$ ${z^*}_{B\&B}$ 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
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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