# American Institute of Mathematical Sciences

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

* 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.

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:
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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
Example of the allocation problem
Calibration of $\rho$ in the IGA
Interval plot with a confidence interval of 95% for the number of suggested iterations
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
 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
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
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
 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
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
 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
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
 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
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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