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

Rodolfo Mendoza-Gómez; Roger Z. Ríos-Mercado; Karla B. Valenzuela-Ocaña

doi:10.3934/jimo.2018182

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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

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 March 2020 Early access 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, 2020, 16 (2) : 857-885. 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
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[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%

	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

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	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 (%)
($\mathcal{N}$ Order)	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

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VND	Final	Imp	Ave	Max	Average final gap (%)
($\mathcal{N}$ Order)	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	$\rho$	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

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			Average relative gap for NS (%)						Ave
Exp	$\rho$	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	$+$0.06	1,781	$+$ 38
M3	CM1	0.83	1.38	1.67	2.21	3.61	1.94	$+$0.29	1,907	$+$164
M4	VND1	1.13	1.42	1.98	2.67	3.66	2.17	$+$0.52	603	$-$1,140
M5	IGA2	0.96	1.72	2.10	2.62	3.54	2.19	$+$0.54	46	$-$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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	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	$+$0.06	1,781	$+$ 38
M3	CM1	0.83	1.38	1.67	2.21	3.61	1.94	$+$0.29	1,907	$+$164
M4	VND1	1.13	1.42	1.98	2.67	3.66	2.17	$+$0.52	603	$-$1,140
M5	IGA2	0.96	1.72	2.10	2.62	3.54	2.19	$+$0.54	46	$-$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 (%)
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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	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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American Institute of Mathematical Sciences

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

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American Institute of Mathematical Sciences

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

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