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doi: 10.3934/jimo.2022076
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## Location and capacity planning for preventive healthcare facilities with congestion effects

 1 School of Economics and Management, Southeast University, Nanjing, China 2 Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China 3 Key Laboratory of Road and Traffic Engineering of Ministry of Education, School of Transportation Engineering, Tongji University, Shanghai, China

*Corresponding author: Min Xu

Received  February 2022 Early access May 2022

A painful lesson got from pandemic COVID-19 is that preventive healthcare service is of utmost importance to governments since it can make massive savings on healthcare expenditure and promote the welfare of the society. Recognizing the importance of preventive healthcare, this research aims to present a methodology for designing a network of preventive healthcare facilities in order to prevent diseases early. The problem is formulated as a bilevel non-linear integer programming model. The upper level is a facility location and capacity planning problem under a limited budget, while the lower level is a user choice problem that determines the allocation of clients to facilities. A genetic algorithm (GA) is developed to solve the upper level problem and a method of successive averages (MSA) is adopted to solve the lower level problem. The model and algorithm is applied to analyze an illustrative case in the Sioux Falls transport network and a number of interesting results and managerial insights are provided. It shows that solutions to medium-scale instances can be obtained in a reasonable time and the marginal benefit of investment is decreasing.

Citation: Hongzhi Lin, Min Xu, Chi Xie. Location and capacity planning for preventive healthcare facilities with congestion effects. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022076
##### References:
 [1] S. Davari, The incremental cooperative design of preventive healthcare networks, Ann. Oper. Res., 272 (2019), 445-492.  doi: 10.1007/s10479-017-2569-1. [2] S. Davari, K. Kilic and G. Ertek, Fuzzy bi-objective preventive health care network design, Health Care Management Science, 18 (2015), 303-317. [3] S. Davari, K. Kilic and S. Naderi, A heuristic approach to solve the preventive health care problem with budget and congestion constraints, Appl. Math. Comput., 276 (2016), 442-453.  doi: 10.1016/j.amc.2015.11.073. [4] K. Dogan, M. Karatas and E. Yakici, A model for locating preventive health care facilities, Cent. Eur. J. Oper. Res., 28 (2020), 1091-1121.  doi: 10.1007/s10100-019-00621-4. [5] M. M. Ershadi and H. S Shemirani, Using mathematical modeling for analysis of the impact of client choice on preventive healthcare facility network design, International J. Healthcare Management, 14 (2021), 588-602. [6] W. Gu, X. Wang and S. E. McGregor, Optimization of preventive health care facility locations, International J. Health Geographics, 9 (2010), 17. [7] K. Haase and S. Müller, Insights into clients' choice in preventive health care facility location planning, OR Spectrum, 37 (2015), 273-291.  doi: 10.1007/s00291-014-0367-6. [8] M. Hamzeei and J. Luedtke, Service network design with equilibrium-driven demands, IISE Transactions, 50 (2018), 959-969. [9] S. Javanmardi, H. Hosseini-nasab, A. Mostafaeipour, M. Fakhrzad and H. Khademizare, Developing a new algorithm for a utility-based network design problem with elastic demand, International J. Engineering, Transactions B: Appl., 30 (2017), 758-767. [10] R. Krohn, S. Müller and K. Haase, Preventive healthcare facility location planning with quality-conscious clients, OR Spectrum, 43 (2021), 59-87.  doi: 10.1007/s00291-020-00605-w. [11] B. Kucukyazici, Y. Zhang, A. Ardestani-Jaafari and L. Song, Incorporating patient preferences in the design and operation of cancer screening facility networks, European J. Oper. Res., 278 (2020), 616-632.  doi: 10.1016/j.ejor.2020.03.082. [12] V. Marianov, M. Ríos and M. J. Icaza, Facility location for market capture when users rank facilities by shorter travel and waiting times, European J. Oper. Res., 191 (2008), 32-44.  doi: 10.1016/j.ejor.2007.07.025. [13] M. Ndiaye and H. Alfares, Modeling health care facility location for moving population groups, Computers and Operations Research, 35 (2008), 2154-2161. [14] S. Risanger, B. Singh, D. Morton and L. Meyers, Selecting pharmacies for COVID-19 testing to ensure access, Health Care Management Science, 24 (2021), 330-338. [15] J. F. Shortle, J. M. Thompson, D. Gross ans C. M. Harris, Fundamentals of Queueing Theory, 5$^{th}$ edition, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2018. [16] V. Verter and S. D. Lapierre, Location of preventive health care facilities, Ann. Oper. Res., 110 (2002), 123-132.  doi: 10.1023/A:1020767501233. [17] N. Vidyarthi and O. Kuzgunkaya, The impact of directed choice on the design of preventive healthcare facility network under congestion, Health Care Management Science, 18 (2015), 459-474. [18] Y. Zhang and D. Atkins, Medical facility network design: User-choice and system-optimal models, European J. Oper. Res., 273 (2019), 305-319.  doi: 10.1016/j.ejor.2018.08.008. [19] Y. Zhang, O. Berman, P. Marcotte and V. Verter, A bilevel model for preventive healthcare facility network design with congestion, IIE Transactions, 42 (2010), 865-880. [20] Y. Zhang, O. Berman and V. Verter, The impact of client choice on preventive healthcare facility network design, OR Spectrum, 34 (2012), 349-370.  doi: 10.1007/s00291-011-0280-1. [21] Y. Zhang, O. Berman and V. Verter, Incorporating congestion in preventive healthcare facility network design, European J. Oper. Res., 198 (2009), 922-935.

show all references

##### References:
 [1] S. Davari, The incremental cooperative design of preventive healthcare networks, Ann. Oper. Res., 272 (2019), 445-492.  doi: 10.1007/s10479-017-2569-1. [2] S. Davari, K. Kilic and G. Ertek, Fuzzy bi-objective preventive health care network design, Health Care Management Science, 18 (2015), 303-317. [3] S. Davari, K. Kilic and S. Naderi, A heuristic approach to solve the preventive health care problem with budget and congestion constraints, Appl. Math. Comput., 276 (2016), 442-453.  doi: 10.1016/j.amc.2015.11.073. [4] K. Dogan, M. Karatas and E. Yakici, A model for locating preventive health care facilities, Cent. Eur. J. Oper. Res., 28 (2020), 1091-1121.  doi: 10.1007/s10100-019-00621-4. [5] M. M. Ershadi and H. S Shemirani, Using mathematical modeling for analysis of the impact of client choice on preventive healthcare facility network design, International J. Healthcare Management, 14 (2021), 588-602. [6] W. Gu, X. Wang and S. E. McGregor, Optimization of preventive health care facility locations, International J. Health Geographics, 9 (2010), 17. [7] K. Haase and S. Müller, Insights into clients' choice in preventive health care facility location planning, OR Spectrum, 37 (2015), 273-291.  doi: 10.1007/s00291-014-0367-6. [8] M. Hamzeei and J. Luedtke, Service network design with equilibrium-driven demands, IISE Transactions, 50 (2018), 959-969. [9] S. Javanmardi, H. Hosseini-nasab, A. Mostafaeipour, M. Fakhrzad and H. Khademizare, Developing a new algorithm for a utility-based network design problem with elastic demand, International J. Engineering, Transactions B: Appl., 30 (2017), 758-767. [10] R. Krohn, S. Müller and K. Haase, Preventive healthcare facility location planning with quality-conscious clients, OR Spectrum, 43 (2021), 59-87.  doi: 10.1007/s00291-020-00605-w. [11] B. Kucukyazici, Y. Zhang, A. Ardestani-Jaafari and L. Song, Incorporating patient preferences in the design and operation of cancer screening facility networks, European J. Oper. Res., 278 (2020), 616-632.  doi: 10.1016/j.ejor.2020.03.082. [12] V. Marianov, M. Ríos and M. J. Icaza, Facility location for market capture when users rank facilities by shorter travel and waiting times, European J. Oper. Res., 191 (2008), 32-44.  doi: 10.1016/j.ejor.2007.07.025. [13] M. Ndiaye and H. Alfares, Modeling health care facility location for moving population groups, Computers and Operations Research, 35 (2008), 2154-2161. [14] S. Risanger, B. Singh, D. Morton and L. Meyers, Selecting pharmacies for COVID-19 testing to ensure access, Health Care Management Science, 24 (2021), 330-338. [15] J. F. Shortle, J. M. Thompson, D. Gross ans C. M. Harris, Fundamentals of Queueing Theory, 5$^{th}$ edition, Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2018. [16] V. Verter and S. D. Lapierre, Location of preventive health care facilities, Ann. Oper. Res., 110 (2002), 123-132.  doi: 10.1023/A:1020767501233. [17] N. Vidyarthi and O. Kuzgunkaya, The impact of directed choice on the design of preventive healthcare facility network under congestion, Health Care Management Science, 18 (2015), 459-474. [18] Y. Zhang and D. Atkins, Medical facility network design: User-choice and system-optimal models, European J. Oper. Res., 273 (2019), 305-319.  doi: 10.1016/j.ejor.2018.08.008. [19] Y. Zhang, O. Berman, P. Marcotte and V. Verter, A bilevel model for preventive healthcare facility network design with congestion, IIE Transactions, 42 (2010), 865-880. [20] Y. Zhang, O. Berman and V. Verter, The impact of client choice on preventive healthcare facility network design, OR Spectrum, 34 (2012), 349-370.  doi: 10.1007/s00291-011-0280-1. [21] Y. Zhang, O. Berman and V. Verter, Incorporating congestion in preventive healthcare facility network design, European J. Oper. Res., 198 (2009), 922-935.
The Sioux Falls transport network
The evolutionary process of genetic algorithm
A sensitivity analysis with varying budget
Network characteristics for the Sioux Falls network
 Link $a$ ${{l}_{a}}$(mile) ${{t}_{a}}$(h) Link $a$ ${{l}_{a}}$(mile) ${{t}_{a}}$(h) 1, 3 3.6 0.12 33, 36 3.6 0.12 2, 5 2.4 0.08 34, 40 2.4 0.08 4, 14 3 0.1 37, 38 1.8 0.06 6, 8 2.4 0.08 39, 74 2.4 0.08 7, 35 2.4 0.08 41, 44 3 0.1 9, 11 1.2 0.04 42, 71 2.4 0.08 10, 31 3.6 0.12 45, 57 2.4 0.08 12, 15 2.4 0.08 46, 67 2.4 0.08 13, 23 3 0.1 49, 52 1.2 0.04 16, 19 1.2 0.04 50, 55 1.8 0.06 17, 20 1.8 0.06 53, 58 1.2 0.04 18, 54 1.2 0.04 56, 60 2.4 0.08 21, 24 6 0.2 59, 61 2.4 0.08 22, 47 3 0.1 62, 64 3.6 0.12 25, 26 1.8 0.06 63, 68 3 0.1 27, 32 3 0.1 65, 69 1.2 0.04 28, 43 3.6 0.12 66, 75 1.8 0.06 29, 48 3 0.1 70, 72 2.4 0.08 30, 51 4.8 0.16 73, 76 1.2 0.04
 Link $a$ ${{l}_{a}}$(mile) ${{t}_{a}}$(h) Link $a$ ${{l}_{a}}$(mile) ${{t}_{a}}$(h) 1, 3 3.6 0.12 33, 36 3.6 0.12 2, 5 2.4 0.08 34, 40 2.4 0.08 4, 14 3 0.1 37, 38 1.8 0.06 6, 8 2.4 0.08 39, 74 2.4 0.08 7, 35 2.4 0.08 41, 44 3 0.1 9, 11 1.2 0.04 42, 71 2.4 0.08 10, 31 3.6 0.12 45, 57 2.4 0.08 12, 15 2.4 0.08 46, 67 2.4 0.08 13, 23 3 0.1 49, 52 1.2 0.04 16, 19 1.2 0.04 50, 55 1.8 0.06 17, 20 1.8 0.06 53, 58 1.2 0.04 18, 54 1.2 0.04 56, 60 2.4 0.08 21, 24 6 0.2 59, 61 2.4 0.08 22, 47 3 0.1 62, 64 3.6 0.12 25, 26 1.8 0.06 63, 68 3 0.1 27, 32 3 0.1 65, 69 1.2 0.04 28, 43 3.6 0.12 66, 75 1.8 0.06 29, 48 3 0.1 70, 72 2.4 0.08 30, 51 4.8 0.16 73, 76 1.2 0.04
Healthcare demand data for the Sioux Falls network
 Population node ($i$) Demand${{h}_{i}}$(clients/hr) 1 37 2 30 4 21 5 26 13 37 14 32 15 39 20 24
 Population node ($i$) Demand${{h}_{i}}$(clients/hr) 1 37 2 30 4 21 5 26 13 37 14 32 15 39 20 24
Optimal network design with client flows at equilibrium state
 Selected facility location (associated capacity) Population node 3 (20) 7 (5) 21 (13) 23 (12) 1 36.01 0.33 0.33 0.33 2 19.55 9.91 0.27 0.27 4 20.44 0.19 0.19 0.19 5 25.3 0.23 0.23 0.23 13 1.65 0.33 6.94 28.08 14 0.29 0.29 0.29 31.14 15 0.35 0.35 37.96 0.35 20 0.21 7.07 16.5 0.21
 Selected facility location (associated capacity) Population node 3 (20) 7 (5) 21 (13) 23 (12) 1 36.01 0.33 0.33 0.33 2 19.55 9.91 0.27 0.27 4 20.44 0.19 0.19 0.19 5 25.3 0.23 0.23 0.23 13 1.65 0.33 6.94 28.08 14 0.29 0.29 0.29 31.14 15 0.35 0.35 37.96 0.35 20 0.21 7.07 16.5 0.21
Utility matrix between population node and selected facility location
 Selected facility location Population node 3 7 21 23 1 -0.272 -0.515 -0.550 -0.547 2 -0.392 -0.395 -0.630 -0.667 4 -0.272 -0.415 -0.550 -0.487 5 -0.312 -0.375 -0.590 -0.527 13 -0.332 -0.575 -0.330 -0.327 14 -0.472 -0.555 -0.370 -0.287 15 -0.572 -0.455 -0.310 -0.367 20 -0.592 -0.315 -0.310 -0.387
 Selected facility location Population node 3 7 21 23 1 -0.272 -0.515 -0.550 -0.547 2 -0.392 -0.395 -0.630 -0.667 4 -0.272 -0.415 -0.550 -0.487 5 -0.312 -0.375 -0.590 -0.527 13 -0.332 -0.575 -0.330 -0.327 14 -0.472 -0.555 -0.370 -0.287 15 -0.572 -0.455 -0.310 -0.367 20 -0.592 -0.315 -0.310 -0.387
The estimated parameters for the polynomial regression
 Coefficients Estimate Standard Error $t$ value Pr(${> \left| t \right|)}$) ${{\alpha }_{0}}$ -308.431 51.678 -5.968 0.004 ${{\alpha }_{1}}$ 7.253 1.757 4.129 0.015 ${{\alpha }_{{\rm{2}}}}$ -0.054 0.015 -3.718 0.021
 Coefficients Estimate Standard Error $t$ value Pr(${> \left| t \right|)}$) ${{\alpha }_{0}}$ -308.431 51.678 -5.968 0.004 ${{\alpha }_{1}}$ 7.253 1.757 4.129 0.015 ${{\alpha }_{{\rm{2}}}}$ -0.054 0.015 -3.718 0.021
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