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doi: 10.3934/jimo.2018182

An iterated greedy algorithm with variable neighborhood descent for the planning of specialized diagnostic services in a segmented healthcare system

Rodolfo Mendoza-Gómez 1, , Roger Z. Ríos-Mercado 2,, and  Karla B. Valenzuela-Ocaña 1,

1. 

Tecnológico de Monterrey, Campus Toluca, Department of Industrial Engineering, Av. Eduardo Monroy Cárdenas 2000, San Antonio Buenavista, Toluca 50110, Mexico
2. 

Universidad Autónoma de Nuevo León (UANL), Graduate Program in Systems Engineering, Av. Universidad s/n, Cd. Universitaria, San Nicolás de los Garza, NL 66455, Mexico

* Corresponding author: R. Z. Ríos-Mercado

Received  February 2018 Revised  August 2018 Published  December 2018

Fund Project: The first author is supported by a scholarship for doctoral studies by the Mexican National Council for Science and Technology (CONACYT). The second author is supported by CONACYT grant CB2011-1-166397 and by UANL-PAICYT grant CE331-15

In this paper, a problem arising in the planning of specialized diagnostic services in a segmented public healthcare system is addressed. The problem consists of deciding which hospitals will provide the service and their capacity levels, the allocation of demand in each institution, and the reallocation of uncovered demand to other institutions or private providers, while minimizing the total equivalent annual cost of investment and operating cost required to satisfy all the demand. An associated mixed-integer linear programming model can be solved by conventional branch and bound for relatively small instances; however, for larger instances the problem becomes intractable. To effectively address larger instances, a hybrid metaheuristic framework combining iterated greedy (IGA) and variable neighborhood descent (VND) components for this problem is proposed. Two greedy construction heuristics are developed, one starting with an infeasible solution and iteratively adding capacity and the other starting with a feasible, but expensive, solution and iteratively decrease capacity. The iterated greedy algorithm includes destruction and reconstruction procedures. Four different neighborhood structures are designed and tested within a VND procedure. In addition, the computation of local search components benefit from an intelligent exploitation of problem structure since, when the first-level location variables (hospital location and capacity) are fixed, the remaining subproblem can be solved efficiently as it is very close to a transshipment problem. All components and different strategies were empirically assessed both individually and within the IGA-VND framework. The resulting metaheuristic is able to obtain near optimal solutions, within 3% of optimality, when tested over a data base of 60- to 300-hospital instances.

Keywords: Healthcare planning, integer programming, hybrid metaheuristics, iterated greedy algorithm, variable neighborhood descent.
Mathematics Subject Classification: Primary: 90C10, 90C59; Secondary: 90B80, 90C06, 90C90.
Citation: Rodolfo Mendoza-Gómez, Roger Z. Ríos-Mercado, Karla B. Valenzuela-Ocaña. An iterated greedy algorithm with variable neighborhood descent for the planning of specialized diagnostic services in a segmented healthcare system. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018182
References:
[1]

A. Ahmadi-Javid, P. Seyedi and S. S. Syam, A survey of healthcare facility location, Computers & Operations Research, 79 (2017), 223-226.  doi: 10.1016/j.cor.2016.05.018.  Google Scholar

[2]

N. Ayvaz and W. T. Huh, Allocation of hospital capacity to multiple types of patients, Journal of Revenue and Pricing Management, 9 (2010), 386-398.   Google Scholar

[3]

A. Chauhan and A. Singh, Healthcare waste management: A state-of-the-art literature review, International Journal of Environment and Waste Management, 18 (2016), 120-144.   Google Scholar

[4]

M. J. Coté, S. S. Syam, W. B. Vogel and D. C. Cowper, A mixed integer programming model to locate traumatic brain injury treatment units in the department of veterans affairs: A case study, Health Care Management Science, 10 (2007), 253-267.   Google Scholar

[5]

T. G. Crainic, M. Gendreau, P. Hansen, N. Hoeb and N. Mladenović, Cooperative parallel variable neighborhood search for the p-median, Journal of Heuristics, 10 (2004), 293-314.   Google Scholar

[6]

M. S. Daskin and L. K. Dean, Location of health care facilities, in Operations Research and Health Care: A Handbook of Methods and Applications (eds. M. L. Brandeau, F. Sainfort and W. P. Pierskalla), Springer, New York, 2005, chapter 3, 43-76. Google Scholar

[7]

Z. Diakova and Y. Kochetov, A double VNS heuristic for the facility location and pricing problem, Electronic Notes in Discrete Mathematics, 39 (2012), 29-34.  doi: 10.1016/j.endm.2012.10.005.  Google Scholar

[8]

F. García-López, B. Melián-Batista, J. A. Moreno-Pérez and J. M. Moreno-Vega, The parallel variable neighborhood search for the p-median problem, Journal of Heuristics, 8 (2002), 375-388.   Google Scholar

[9]

P. Hansen and N. Mladenović, Variable neighborhood search for the p-median, Location Science, 5 (1997), 207-226.   Google Scholar

[10]

P. Hansen and N. Mladenović, Variable neighborhood decomposition search, Journal of Heuristics, 7 (2001), 335-350.   Google Scholar

[11]

H. H. Hoos and T. Stützle, Stochastic Local Search: Foundations and Applications, Morgan Kaufmann, San Francisco, 2004. Google Scholar

[12]

I. Ljubić, A hybrid VNS for connected facility location, in Hybrid Metaheuristics (eds. T. Bartz-Beielstein, M. J. Blesa Aguilera, C. Blum, B. Naujoks, A. Roli, G. Rudolph and M. Sampels), vol. 4771 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, 2007,157-169. Google Scholar

[13]

H. R. Lourenço, O. C. Martin and T. Stützle, Iterated local search, in Handbook of Metaheuristics (eds. F. Glover and G. A. Kochenberger), Springer, Boston, 2003, chapter 11,320-353. Google Scholar

[14]

S. Mahar, K. M. Bretthauer and P. A. Salzarulo, Locating specialized service capacity in a multi-hospital network, European Journal of Operational Research, 212 (2011), 596-605.   Google Scholar

[15]

M. Marić, Z. Stanimirović and S. Božović, Hybrid metaheuristic method for determining locations for long-term health care facilities, Annals of Operations Research, 227 (2013), 3-23.  doi: 10.1007/s10479-013-1313-8.  Google Scholar

[16]

S. McLafferty and D. Broe, Patient outcomes and regional planning of coronary care services: A location-allocation approach., Social Science and Medicine, 30 (1990), 297-305.   Google Scholar

[17]

R. Mendoza-Gómez, R. Z. Ríos-Mercado and K. B. Valenzuela, Efficient Planning of Specialized Diagnostic Services in a Segmented Healthcare System, Technical report PISIS-2016-01, Graduate Program in Systems Engineering, Universidad Autónoma de Nuevo León, 2016. Google Scholar

[18]

A. M. Mestre, M. D. Oliveira and A. P. Barbosa-Póvoa, Location-allocation approaches for hospital network planning under uncertainty, European Journal of Operational Research, 240 (2015), 791-806.   Google Scholar

[19]

N. Mladenović and P. Hansen, Variable neighborhood search, Computers and Operations Research, 24 (1997), 1097-1100.  doi: 10.1016/S0305-0548(97)00031-2.  Google Scholar

[20]

N. Mladenović and P. Hansen, Solving the p-center problem by tabu search and variable neighborhood search, Networks, 42 (2003), 48-64.  doi: 10.1002/net.10081.  Google Scholar

[21]

Q. K. Pan and R. Ruiz, An effective iterated greedy algorithm for the mixed no-idle permutation flowshop scheduling problem, Omega, 44 (2014), 41-50.   Google Scholar

[22]

D. R. Quevedo-Orozco and R. Z. Ríos-Mercado, Improving the quality of heuristic solutions for the capacitated vertex $ p $-center problem through iterated greedy local search and variable neighborhood descent, Computers and Operations Research, 62 (2015), 133-144.  doi: 10.1016/j.cor.2014.12.013.  Google Scholar

[23]

A. Rais and A. Viana, Operations research in healthcare: A survey, International Transactions in Operational Research, 18 (2011), 1-31.  doi: 10.1111/j.1475-3995.2010.00767.x.  Google Scholar

[24]

N. Rego and J. P. de Sousa, Supply chain coordination in hospitals, in Leveraging Knowledge for Innovation in Collaborative Networks (eds. L. M. Camarinha-Matos, I. Paraskakis and H. Afsarmanesh), vol. 307 of IFIP Advances in Information and Communication Technology (IFIPAICT), Springer, Berlin, Germany, 2009,117-127. Google Scholar

[25]

I. Ribas, R. Companys and X. Tort-Martorell, An iterated greedy algorithm for the flowshop scheduling problem with blocking, Omega, 39 (2011), 293-301.   Google Scholar

[26]

R. Ruiz and T. Stützle, A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem, European Journal of Operational Research, 177 (2006), 2033-2049.   Google Scholar

[27]

R. Ruiz and T. Stützle, An iterated greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives, European Journal of Operational Research, 187 (2008), 1143-1159.   Google Scholar

[28]

R. J. Ruth, A mixed integer programming model for regional planning of a hospital inpatient service, Management Science, 27 (1981), 521-533.   Google Scholar

[29]

C. Stummer, K. Doerner, A. Focke and K. Heidenberger, Determining location and size of medical departments in a hospital network: A multiobjective decision support approach, Health Care Management Science, 7 (2004), 63-71.   Google Scholar

[30]

S. S. Syam and M. J. Coté, A location-allocation model for service providers with application to not-for-profit health care organizations, Omega, 38 (2010), 157-166.   Google Scholar

[31]

S. S. Syam and M. J. Coté, A comprehensive location-allocation method for specialized healthcare services, Operations Research for Health Care, 1 (2012), 73-83.   Google Scholar

[32]

H. Tlahig, A. J. H. Bouchriha and P. Ladet, Centralized versus distributed sterilization service: A location-allocation decision model, Operations Research for Health Care, 2 (2013), 75-85.   Google Scholar

[33]

Z. Yuan, A. Fügenschuh, H. Homfeld, P. Balaprakash, T. Stützle and M. Schoch, Iterated greedy algorithms for a real-world cyclic train scheduling problem, in Hybrid Metaheuristics (eds. M. J. Blesa, C. Blum, C. Cotta, A. J. Fernández, J. E. Gallardo, A. Roli and M. Sampels), vol. 5296 of Lecture Notes in Computer Science, Springer, 2008,102-116. Google Scholar

[34]

N. Zarrinpoor, M. S. Fallahnezhad and M. S. Pishvaee, Design of a reliable hierarchical location-allocation model under disruptions for health service networks: A two-stage robust approach, Computers & Industrial Engineering, 109 (2017), 130-150.   Google Scholar

show all references

References:
[1]

A. Ahmadi-Javid, P. Seyedi and S. S. Syam, A survey of healthcare facility location, Computers & Operations Research, 79 (2017), 223-226.  doi: 10.1016/j.cor.2016.05.018.  Google Scholar

[2]

N. Ayvaz and W. T. Huh, Allocation of hospital capacity to multiple types of patients, Journal of Revenue and Pricing Management, 9 (2010), 386-398.   Google Scholar

[3]

A. Chauhan and A. Singh, Healthcare waste management: A state-of-the-art literature review, International Journal of Environment and Waste Management, 18 (2016), 120-144.   Google Scholar

[4]

M. J. Coté, S. S. Syam, W. B. Vogel and D. C. Cowper, A mixed integer programming model to locate traumatic brain injury treatment units in the department of veterans affairs: A case study, Health Care Management Science, 10 (2007), 253-267.   Google Scholar

[5]

T. G. Crainic, M. Gendreau, P. Hansen, N. Hoeb and N. Mladenović, Cooperative parallel variable neighborhood search for the p-median, Journal of Heuristics, 10 (2004), 293-314.   Google Scholar

[6]

M. S. Daskin and L. K. Dean, Location of health care facilities, in Operations Research and Health Care: A Handbook of Methods and Applications (eds. M. L. Brandeau, F. Sainfort and W. P. Pierskalla), Springer, New York, 2005, chapter 3, 43-76. Google Scholar

[7]

Z. Diakova and Y. Kochetov, A double VNS heuristic for the facility location and pricing problem, Electronic Notes in Discrete Mathematics, 39 (2012), 29-34.  doi: 10.1016/j.endm.2012.10.005.  Google Scholar

[8]

F. García-López, B. Melián-Batista, J. A. Moreno-Pérez and J. M. Moreno-Vega, The parallel variable neighborhood search for the p-median problem, Journal of Heuristics, 8 (2002), 375-388.   Google Scholar

[9]

P. Hansen and N. Mladenović, Variable neighborhood search for the p-median, Location Science, 5 (1997), 207-226.   Google Scholar

[10]

P. Hansen and N. Mladenović, Variable neighborhood decomposition search, Journal of Heuristics, 7 (2001), 335-350.   Google Scholar

[11]

H. H. Hoos and T. Stützle, Stochastic Local Search: Foundations and Applications, Morgan Kaufmann, San Francisco, 2004. Google Scholar

[12]

I. Ljubić, A hybrid VNS for connected facility location, in Hybrid Metaheuristics (eds. T. Bartz-Beielstein, M. J. Blesa Aguilera, C. Blum, B. Naujoks, A. Roli, G. Rudolph and M. Sampels), vol. 4771 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, Germany, 2007,157-169. Google Scholar

[13]

H. R. Lourenço, O. C. Martin and T. Stützle, Iterated local search, in Handbook of Metaheuristics (eds. F. Glover and G. A. Kochenberger), Springer, Boston, 2003, chapter 11,320-353. Google Scholar

[14]

S. Mahar, K. M. Bretthauer and P. A. Salzarulo, Locating specialized service capacity in a multi-hospital network, European Journal of Operational Research, 212 (2011), 596-605.   Google Scholar

[15]

M. Marić, Z. Stanimirović and S. Božović, Hybrid metaheuristic method for determining locations for long-term health care facilities, Annals of Operations Research, 227 (2013), 3-23.  doi: 10.1007/s10479-013-1313-8.  Google Scholar

[16]

S. McLafferty and D. Broe, Patient outcomes and regional planning of coronary care services: A location-allocation approach., Social Science and Medicine, 30 (1990), 297-305.   Google Scholar

[17]

R. Mendoza-Gómez, R. Z. Ríos-Mercado and K. B. Valenzuela, Efficient Planning of Specialized Diagnostic Services in a Segmented Healthcare System, Technical report PISIS-2016-01, Graduate Program in Systems Engineering, Universidad Autónoma de Nuevo León, 2016. Google Scholar

[18]

A. M. Mestre, M. D. Oliveira and A. P. Barbosa-Póvoa, Location-allocation approaches for hospital network planning under uncertainty, European Journal of Operational Research, 240 (2015), 791-806.   Google Scholar

[19]

N. Mladenović and P. Hansen, Variable neighborhood search, Computers and Operations Research, 24 (1997), 1097-1100.  doi: 10.1016/S0305-0548(97)00031-2.  Google Scholar

[20]

N. Mladenović and P. Hansen, Solving the p-center problem by tabu search and variable neighborhood search, Networks, 42 (2003), 48-64.  doi: 10.1002/net.10081.  Google Scholar

[21]

Q. K. Pan and R. Ruiz, An effective iterated greedy algorithm for the mixed no-idle permutation flowshop scheduling problem, Omega, 44 (2014), 41-50.   Google Scholar

[22]

D. R. Quevedo-Orozco and R. Z. Ríos-Mercado, Improving the quality of heuristic solutions for the capacitated vertex $ p $-center problem through iterated greedy local search and variable neighborhood descent, Computers and Operations Research, 62 (2015), 133-144.  doi: 10.1016/j.cor.2014.12.013.  Google Scholar

[23]

A. Rais and A. Viana, Operations research in healthcare: A survey, International Transactions in Operational Research, 18 (2011), 1-31.  doi: 10.1111/j.1475-3995.2010.00767.x.  Google Scholar

[24]

N. Rego and J. P. de Sousa, Supply chain coordination in hospitals, in Leveraging Knowledge for Innovation in Collaborative Networks (eds. L. M. Camarinha-Matos, I. Paraskakis and H. Afsarmanesh), vol. 307 of IFIP Advances in Information and Communication Technology (IFIPAICT), Springer, Berlin, Germany, 2009,117-127. Google Scholar

[25]

I. Ribas, R. Companys and X. Tort-Martorell, An iterated greedy algorithm for the flowshop scheduling problem with blocking, Omega, 39 (2011), 293-301.   Google Scholar

[26]

R. Ruiz and T. Stützle, A simple and effective iterated greedy algorithm for the permutation flowshop scheduling problem, European Journal of Operational Research, 177 (2006), 2033-2049.   Google Scholar

[27]

R. Ruiz and T. Stützle, An iterated greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives, European Journal of Operational Research, 187 (2008), 1143-1159.   Google Scholar

[28]

R. J. Ruth, A mixed integer programming model for regional planning of a hospital inpatient service, Management Science, 27 (1981), 521-533.   Google Scholar

[29]

C. Stummer, K. Doerner, A. Focke and K. Heidenberger, Determining location and size of medical departments in a hospital network: A multiobjective decision support approach, Health Care Management Science, 7 (2004), 63-71.   Google Scholar

[30]

S. S. Syam and M. J. Coté, A location-allocation model for service providers with application to not-for-profit health care organizations, Omega, 38 (2010), 157-166.   Google Scholar

[31]

S. S. Syam and M. J. Coté, A comprehensive location-allocation method for specialized healthcare services, Operations Research for Health Care, 1 (2012), 73-83.   Google Scholar

[32]

H. Tlahig, A. J. H. Bouchriha and P. Ladet, Centralized versus distributed sterilization service: A location-allocation decision model, Operations Research for Health Care, 2 (2013), 75-85.   Google Scholar

[33]

Z. Yuan, A. Fügenschuh, H. Homfeld, P. Balaprakash, T. Stützle and M. Schoch, Iterated greedy algorithms for a real-world cyclic train scheduling problem, in Hybrid Metaheuristics (eds. M. J. Blesa, C. Blum, C. Cotta, A. J. Fernández, J. E. Gallardo, A. Roli and M. Sampels), vol. 5296 of Lecture Notes in Computer Science, Springer, 2008,102-116. Google Scholar

[34]

N. Zarrinpoor, M. S. Fallahnezhad and M. S. Pishvaee, Design of a reliable hierarchical location-allocation model under disruptions for health service networks: A two-stage robust approach, Computers & Industrial Engineering, 109 (2017), 130-150.   Google Scholar

Figure 1.  Example of the allocation problem
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Figure 2.  Calibration of $ \rho $ in the IGA
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Figure 3.  Interval plot with a confidence interval of 95% for the number of suggested iterations
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Table 1.  Comparison of constructive methods and B & B
Method $ n $ Ave
gap (%)		 Min
gap (%)		 Max
gap (%)		 Ave
gap (s) 		Min
gap (s) 		Max
gap (s)
60 0.32 0.00 2.31 6,755 18.5 10,800
120 2.24 0.00 10.62 10,566 3,568 10,800
B & B 180 6.98 0.81 23.91 10,654 8,244 10,800
240 20.94 3.11 59.43 10,693 7,557 10,800
300 43.96 6.45 76.65 10,762 9,367 10,800
60 2.11 0.26 7.91 0.7 0.2 1.7
120 3.06 0.66 16.87 3.2 0.9 8.1
CM1 180 3.58 1.09 8.86 8.4 2.1 24.7
240 4.59 0.98 12.12 14.7 3.7 36.3
300 5.49 1.75 10.52 27.2 5.8 59.0
60 2.04 0.0 6.67 0.5 0.2 1.1
120 2.46 0.81 6.50 1.8 0.7 3.2
CM2 180 3.38 0.56 10.07 5.0 1.3 10.9
240 3.36 0.70 9.43 8.5 2.6 18.5
300 5.22 1.08 10.03 16.0 4.7 38.9
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Method $ n $ Ave
gap (%)		 Min
gap (%)		 Max
gap (%)		 Ave
gap (s) 		Min
gap (s) 		Max
gap (s)
60 0.32 0.00 2.31 6,755 18.5 10,800
120 2.24 0.00 10.62 10,566 3,568 10,800
B & B 180 6.98 0.81 23.91 10,654 8,244 10,800
240 20.94 3.11 59.43 10,693 7,557 10,800
300 43.96 6.45 76.65 10,762 9,367 10,800
60 2.11 0.26 7.91 0.7 0.2 1.7
120 3.06 0.66 16.87 3.2 0.9 8.1
CM1 180 3.58 1.09 8.86 8.4 2.1 24.7
240 4.59 0.98 12.12 14.7 3.7 36.3
300 5.49 1.75 10.52 27.2 5.8 59.0
60 2.04 0.0 6.67 0.5 0.2 1.1
120 2.46 0.81 6.50 1.8 0.7 3.2
CM2 180 3.38 0.56 10.07 5.0 1.3 10.9
240 3.36 0.70 9.43 8.5 2.6 18.5
300 5.22 1.08 10.03 16.0 4.7 38.9
Table 2.  Individual neighborhood evaluation, initial relative gap = 3.77%
FinalImpAveMaxAverage final gap (%)
gap (%)(%)time (s)time (s)60120180240300
LS13.3012.280.915.01.932.623.123.924.93
LS23.614.170.46.01.702.963.494.485.42
LS32.8524.4018.1198.01.612.322.693.434.18
LS42.4734.29184.11,578.41.431.992.402.883.66
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FinalImpAveMaxAverage final gap (%)
gap (%)(%)time (s)time (s)60120180240300
LS13.3012.280.915.01.932.623.123.924.93
LS23.614.170.46.01.702.963.494.485.42
LS32.8524.4018.1198.01.612.322.693.434.18
LS42.4734.29184.11,578.41.431.992.402.883.66
Table 3.  VND evaluation, initial relative gap = 3.77%
VNDFinalImpAveMaxAverage final gap (%)
($\mathcal{N}$ Order)gap (%)(%)time (s)time (s)60120180240300
VND1(1-2-3)2.3038.9021.8218.81.041.862.232.713.68
VND2(1-2-3-4)2.0448.87263.52,660.50.911.621.892.443.34
VND3(2-1-3)2.3138.7453.1420.51.051.882.212.723.68
VND4(2-1-3-4)2.0545.70257.42,139.60.911.601.882.453.39
VND5(2-3)2.4535.0717.1178.81.451.972.292.813.73
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VNDFinalImpAveMaxAverage final gap (%)
($\mathcal{N}$ Order)gap (%)(%)time (s)time (s)60120180240300
VND1(1-2-3)2.3038.9021.8218.81.041.862.232.713.68
VND2(1-2-3-4)2.0448.87263.52,660.50.911.621.892.443.34
VND3(2-1-3)2.3138.7453.1420.51.051.882.212.723.68
VND4(2-1-3-4)2.0545.70257.42,139.60.911.601.882.453.39
VND5(2-3)2.4535.0717.1178.81.451.972.292.813.73
Table 4.  Overall assessment of IGA-VND strategies
Average relative gap for NS (%)Ave
Exp $\rho$Iter60120180240300Globaltime (s)
E10.20500.871.441.642.263.631.971,902
E20.20500.831.381.672.213.611.941,907
E30.20500.680.971.492.033.061.651,743
E40.201000.670.971.452.023.061.642,495
E50.10500.671.051.582.043.111.691,550
Method:
E1 = IGA2_VND1
E2 = IGA2_VND1_LS4
E3 = CM1_IGA2_VND1_LS4
E4 = CM1_IGA2_VND1_LS4
E5 = CM1_IGA2_VND1_LS4
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Average relative gap for NS (%)Ave
Exp $\rho$Iter60120180240300Globaltime (s)
E10.20500.871.441.642.263.631.971,902
E20.20500.831.381.672.213.611.941,907
E30.20500.680.971.492.033.061.651,743
E40.201000.670.971.452.023.061.642,495
E50.10500.671.051.582.043.111.691,550
Method:
E1 = IGA2_VND1
E2 = IGA2_VND1_LS4
E3 = CM1_IGA2_VND1_LS4
E4 = CM1_IGA2_VND1_LS4
E5 = CM1_IGA2_VND1_LS4
Table 5.  Assessment of individual components
OmittedAverage relative gap for NS (%)Ave. time (s)
Mcomponent60120180240300GlobalShiftGlobalShift
M1Neither0.680.971.492.033.061.651,743
M2LS40.731.061.542.083.121.71$+$0.061,781 $+$ 38
M3CM10.831.381.672.213.611.94$+$0.291,907 $+$164
M4VND11.131.421.982.673.662.17$+$0.52603 $-$1,140
M5IGA20.961.722.102.623.542.19$+$0.5446 $-$1,697
Method:
M1 = CM1_IGA2_VND1_LS4
M2 = CM1_IGA2_VND1
M3 = IGA2_VND1_LS4
M4 = CM1_IGA2_LS4
M5 = CM1_VND1_LS4
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OmittedAverage relative gap for NS (%)Ave. time (s)
Mcomponent60120180240300GlobalShiftGlobalShift
M1Neither0.680.971.492.033.061.651,743
M2LS40.731.061.542.083.121.71$+$0.061,781 $+$ 38
M3CM10.831.381.672.213.611.94$+$0.291,907 $+$164
M4VND11.131.421.982.673.662.17$+$0.52603 $-$1,140
M5IGA20.961.722.102.623.542.19$+$0.5446 $-$1,697
Method:
M1 = CM1_IGA2_VND1_LS4
M2 = CM1_IGA2_VND1
M3 = IGA2_VND1_LS4
M4 = CM1_IGA2_LS4
M5 = CM1_VND1_LS4
Table 6.  Comparison between IGA-VND and B&B
Average relative gap for NS (%)
Method60120180240300
B&B0.322.246.9820.9443.96
CM1_IGA2_VND1_LS40.680.971.492.033.06
Average run-time for NS (s)
Method60120180240300
B&B6,75510,56610,65410,69310,761
CM1_IGA2_VND1_LS41055621,7203,5296,542
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Average relative gap for NS (%)
Method60120180240300
B&B0.322.246.9820.9443.96
CM1_IGA2_VND1_LS40.680.971.492.033.06
Average run-time for NS (s)
Method60120180240300
B&B6,75510,56610,65410,69310,761
CM1_IGA2_VND1_LS41055621,7203,5296,542
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