Figure 1.
Example of the allocation problem
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Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization
March
2020, 16(2): 857-885.
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
| 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 |
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 and Management Optimization,
2020, 16
(2)
: 857-885.
doi: 10.3934/jimo.2018182
![]()
References:
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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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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.
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| [9] |
P. Hansen and N. Mladenović,
Variable neighborhood search for the p-median, Location Science, 5 (1997), 207-226.
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P. Hansen and N. Mladenović,
Variable neighborhood decomposition search, Journal of Heuristics, 7 (2001), 335-350.
|
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H. H. Hoos and T. Stützle,
Stochastic Local Search: Foundations and Applications, Morgan Kaufmann, San Francisco, 2004. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [28] |
R. J. Ruth,
A mixed integer programming model for regional planning of a hospital inpatient service, Management Science, 27 (1981), 521-533.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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.
|
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. |
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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. |
| [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.
|
| [9] |
P. Hansen and N. Mladenović,
Variable neighborhood search for the p-median, Location Science, 5 (1997), 207-226.
|
| [10] |
P. Hansen and N. Mladenović,
Variable neighborhood decomposition search, Journal of Heuristics, 7 (2001), 335-350.
|
| [11] |
H. H. Hoos and T. Stützle,
Stochastic Local Search: Foundations and Applications, Morgan Kaufmann, San Francisco, 2004. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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.
|
| [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.
|
| [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.
|
| [28] |
R. J. Ruth,
A mixed integer programming model for regional planning of a hospital inpatient service, Management Science, 27 (1981), 521-533.
|
| [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.
|
| [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.
|
| [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.
|
| [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.
|
| [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. |
| [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.
|
Figure 2.
Calibration of $ \rho $ in the IGA
Figure 3.
Interval plot with a confidence interval of 95% for the number of suggested iterations
Table 1.
Comparison of constructive methods and B & B
| Method | 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 |
| Method | 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%
| Final | Imp | Ave | Max | Average final gap (%) | |||||
| gap (%) | (%) | time (s) | time (s) | 60 | 120 | 180 | 240 | 300 | |
| LS1 | 3.30 | 12.28 | 0.9 | 15.0 | 1.93 | 2.62 | 3.12 | 3.92 | 4.93 |
| LS2 | 3.61 | 4.17 | 0.4 | 6.0 | 1.70 | 2.96 | 3.49 | 4.48 | 5.42 |
| LS3 | 2.85 | 24.40 | 18.1 | 198.0 | 1.61 | 2.32 | 2.69 | 3.43 | 4.18 |
| LS4 | 2.47 | 34.29 | 184.1 | 1,578.4 | 1.43 | 1.99 | 2.40 | 2.88 | 3.66 |
| Final | Imp | Ave | Max | Average final gap (%) | |||||
| gap (%) | (%) | time (s) | time (s) | 60 | 120 | 180 | 240 | 300 | |
| LS1 | 3.30 | 12.28 | 0.9 | 15.0 | 1.93 | 2.62 | 3.12 | 3.92 | 4.93 |
| LS2 | 3.61 | 4.17 | 0.4 | 6.0 | 1.70 | 2.96 | 3.49 | 4.48 | 5.42 |
| LS3 | 2.85 | 24.40 | 18.1 | 198.0 | 1.61 | 2.32 | 2.69 | 3.43 | 4.18 |
| LS4 | 2.47 | 34.29 | 184.1 | 1,578.4 | 1.43 | 1.99 | 2.40 | 2.88 | 3.66 |
Table 3.
VND evaluation, initial relative gap = 3.77%
| VND | Final | Imp | Ave | Max | Average final gap (%) | ||||
| ( | gap (%) | (%) | time (s) | time (s) | 60 | 120 | 180 | 240 | 300 |
| VND1(1-2-3) | 2.30 | 38.90 | 21.8 | 218.8 | 1.04 | 1.86 | 2.23 | 2.71 | 3.68 |
| VND2(1-2-3-4) | 2.04 | 48.87 | 263.5 | 2,660.5 | 0.91 | 1.62 | 1.89 | 2.44 | 3.34 |
| VND3(2-1-3) | 2.31 | 38.74 | 53.1 | 420.5 | 1.05 | 1.88 | 2.21 | 2.72 | 3.68 |
| VND4(2-1-3-4) | 2.05 | 45.70 | 257.4 | 2,139.6 | 0.91 | 1.60 | 1.88 | 2.45 | 3.39 |
| VND5(2-3) | 2.45 | 35.07 | 17.1 | 178.8 | 1.45 | 1.97 | 2.29 | 2.81 | 3.73 |
| VND | Final | Imp | Ave | Max | Average final gap (%) | ||||
| ( | gap (%) | (%) | time (s) | time (s) | 60 | 120 | 180 | 240 | 300 |
| VND1(1-2-3) | 2.30 | 38.90 | 21.8 | 218.8 | 1.04 | 1.86 | 2.23 | 2.71 | 3.68 |
| VND2(1-2-3-4) | 2.04 | 48.87 | 263.5 | 2,660.5 | 0.91 | 1.62 | 1.89 | 2.44 | 3.34 |
| VND3(2-1-3) | 2.31 | 38.74 | 53.1 | 420.5 | 1.05 | 1.88 | 2.21 | 2.72 | 3.68 |
| VND4(2-1-3-4) | 2.05 | 45.70 | 257.4 | 2,139.6 | 0.91 | 1.60 | 1.88 | 2.45 | 3.39 |
| VND5(2-3) | 2.45 | 35.07 | 17.1 | 178.8 | 1.45 | 1.97 | 2.29 | 2.81 | 3.73 |
Table 4.
Overall assessment of IGA-VND strategies
| Average relative gap for NS (%) | Ave | ||||||||
| Exp | | Iter | 60 | 120 | 180 | 240 | 300 | Global | time (s) |
| E1 | 0.20 | 50 | 0.87 | 1.44 | 1.64 | 2.26 | 3.63 | 1.97 | 1,902 |
| E2 | 0.20 | 50 | 0.83 | 1.38 | 1.67 | 2.21 | 3.61 | 1.94 | 1,907 |
| E3 | 0.20 | 50 | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 | 1.65 | 1,743 |
| E4 | 0.20 | 100 | 0.67 | 0.97 | 1.45 | 2.02 | 3.06 | 1.64 | 2,495 |
| E5 | 0.10 | 50 | 0.67 | 1.05 | 1.58 | 2.04 | 3.11 | 1.69 | 1,550 |
| Method: | |||||||||
| E1 = IGA2_VND1 | |||||||||
| E2 = IGA2_VND1_LS4 | |||||||||
| E3 = CM1_IGA2_VND1_LS4 | |||||||||
| E4 = CM1_IGA2_VND1_LS4 | |||||||||
| E5 = CM1_IGA2_VND1_LS4 | |||||||||
| Average relative gap for NS (%) | Ave | ||||||||
| Exp | | Iter | 60 | 120 | 180 | 240 | 300 | Global | time (s) |
| E1 | 0.20 | 50 | 0.87 | 1.44 | 1.64 | 2.26 | 3.63 | 1.97 | 1,902 |
| E2 | 0.20 | 50 | 0.83 | 1.38 | 1.67 | 2.21 | 3.61 | 1.94 | 1,907 |
| E3 | 0.20 | 50 | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 | 1.65 | 1,743 |
| E4 | 0.20 | 100 | 0.67 | 0.97 | 1.45 | 2.02 | 3.06 | 1.64 | 2,495 |
| E5 | 0.10 | 50 | 0.67 | 1.05 | 1.58 | 2.04 | 3.11 | 1.69 | 1,550 |
| 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
| Omitted | Average relative gap for NS (%) | Ave. time (s) | ||||||||
| M | component | 60 | 120 | 180 | 240 | 300 | Global | Shift | Global | Shift |
| M1 | Neither | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 | 1.65 | 1,743 | ||
| M2 | LS4 | 0.73 | 1.06 | 1.54 | 2.08 | 3.12 | 1.71 | 1,781 | | |
| M3 | CM1 | 0.83 | 1.38 | 1.67 | 2.21 | 3.61 | 1.94 | 1,907 | | |
| M4 | VND1 | 1.13 | 1.42 | 1.98 | 2.67 | 3.66 | 2.17 | 603 | | |
| M5 | IGA2 | 0.96 | 1.72 | 2.10 | 2.62 | 3.54 | 2.19 | 46 | | |
| Method: | ||||||||||
| M1 = CM1_IGA2_VND1_LS4 | ||||||||||
| M2 = CM1_IGA2_VND1 | ||||||||||
| M3 = IGA2_VND1_LS4 | ||||||||||
| M4 = CM1_IGA2_LS4 | ||||||||||
| M5 = CM1_VND1_LS4 | ||||||||||
| Omitted | Average relative gap for NS (%) | Ave. time (s) | ||||||||
| M | component | 60 | 120 | 180 | 240 | 300 | Global | Shift | Global | Shift |
| M1 | Neither | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 | 1.65 | 1,743 | ||
| M2 | LS4 | 0.73 | 1.06 | 1.54 | 2.08 | 3.12 | 1.71 | 1,781 | | |
| M3 | CM1 | 0.83 | 1.38 | 1.67 | 2.21 | 3.61 | 1.94 | 1,907 | | |
| M4 | VND1 | 1.13 | 1.42 | 1.98 | 2.67 | 3.66 | 2.17 | 603 | | |
| M5 | IGA2 | 0.96 | 1.72 | 2.10 | 2.62 | 3.54 | 2.19 | 46 | | |
| 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 (%) | |||||
| Method | 60 | 120 | 180 | 240 | 300 |
| B&B | 0.32 | 2.24 | 6.98 | 20.94 | 43.96 |
| CM1_IGA2_VND1_LS4 | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 |
| Average run-time for NS (s) | |||||
| Method | 60 | 120 | 180 | 240 | 300 |
| B&B | 6,755 | 10,566 | 10,654 | 10,693 | 10,761 |
| CM1_IGA2_VND1_LS4 | 105 | 562 | 1,720 | 3,529 | 6,542 |
| Average relative gap for NS (%) | |||||
| Method | 60 | 120 | 180 | 240 | 300 |
| B&B | 0.32 | 2.24 | 6.98 | 20.94 | 43.96 |
| CM1_IGA2_VND1_LS4 | 0.68 | 0.97 | 1.49 | 2.03 | 3.06 |
| Average run-time for NS (s) | |||||
| Method | 60 | 120 | 180 | 240 | 300 |
| B&B | 6,755 | 10,566 | 10,654 | 10,693 | 10,761 |
| CM1_IGA2_VND1_LS4 | 105 | 562 | 1,720 | 3,529 | 6,542 |
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