March  2022, 18(2): 1487-1504. doi: 10.3934/jimo.2021029

Merging short-term and long-term planning problems in home health care under continuity of care and patterns for visits

1. 

Industrial Engineering Department, Yeditepe University, Istanbul, Turkey

2. 

Department of Management, Information and Production Engineering, University of Bergamo, Dalmine (BG), Italy

* Corresponding author: Ettore Lanzarone

Received  January 2020 Revised  January 2021 Published  March 2022 Early access  February 2021

Home Health Care (HHC) human resource management is a complex process. Moreover, as patients are assisted for a long time, their demand for care evolves in terms of type and frequency of visits. Under continuity of care, this uncertain evolution must be considered even when scheduling the visits in the short-term, as the corresponding operator-to-patient assignments could generate overtimes and unbalanced workloads in the long-term, which must be fixed by reassigning some patients and deteriorating the continuity of care. On the other hand, the operator-to-patient assignment problem under continuity of care over a long time period could generate solutions that are infeasible when the scheduling constraints are considered. We analyze the trade-offs between the two problems, to analyze the conditions in which they can be sequentially solved or an integration is required. In particular, we take an assignment and scheduling model for short-term planning, an operator-to-patient assignment model over a long time horizon, and we merge them into a new combined model. Results on a set of realistic instances show that the combined model is necessary when the number of patterns is limited and the variability of patients' demands is high, whereas simpler models deserve to be applied in less critical situations.

Citation: Semih Yalçındağ, Ettore Lanzarone. Merging short-term and long-term planning problems in home health care under continuity of care and patterns for visits. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1487-1504. doi: 10.3934/jimo.2021029
References:
[1]

R. ArgientoA. GuglielmiE. Lanzarone and I. Nawajah, A Bayesian framework for describing and predicting the stochastic demand of home care patients, Flexible Services and Manufacturing Journal, 28 (2016), 254-279.  doi: 10.1007/s10696-014-9200-4.

[2]

R. ArgientoA. GuglielmiE. Lanzarone and I. Nawajah, Bayesian joint modelling of the health profile and demand of home care patients, IMA Journal of Management Mathematics, 28 (2016), 531-552.  doi: 10.1093/imaman/dpw001.

[3]

P. CappaneraM. G. ScutellàF. Nervi and L. Galli, Demand uncertainty in robust home care optimization, Omega (United Kingdom), 80 (2018), 95-110.  doi: 10.1016/j.omega.2017.08.012.

[4]

P. Cappanera and M. G. Scutellà, Joint assignment, scheduling, and routing models to home care optimization: A pattern-based approach, Transportation Science, 49 (2015), 830-852.  doi: 10.1287/trsc.2014.0548.

[5]

G. Carello and E. Lanzarone, A cardinality-constrained robust model for the assignment problem in home care services, European Journal of Operational Research, 236 (2014), 748-762.  doi: 10.1016/j.ejor.2014.01.009.

[6]

G. CarelloE. Lanzarone and S. Mattia, Trade-off between stakeholders' goals in the home care nurse-to-patient assignment problem, Operations Research for Health Care, 16 (2018), 29-40.  doi: 10.1016/j.orhc.2017.12.002.

[7]

M. CisséS. YalçındağY. KergosienE. ŞahinC. Lenté and A. Matta, Or problems related to home health care: A review of relevant routing and scheduling problems, Operations Research for Health Care, 13 (2017), 1-22. 

[8]

A. Errarhout, S. Kharraja and I. Zorkani, Caregivers' assignment problem in home health care structures, in Proceedings of 2013 International Conference on Industrial Engineering and Systems Management (IESM), IEEE, 2013, 1–8.

[9]

P. EvebornP. Flisberg and M. Rönnqvist, Laps Care–an operational system for staff planning of home care, European Journal of Operational Research, 171 (2006), 962-976.  doi: 10.1016/j.ejor.2005.01.011.

[10]

C. Fikar and P. Hirsch, Home health care routing and scheduling: A review, Computers & Operations Research, 77 (2017), 86-95.  doi: 10.1016/j.cor.2016.07.019.

[11]

F. GrenouilleauA. LegrainN. Lahrichi and L.-M. Rousseau, A set partitioning heuristic for the home health care routing and scheduling problem, European Journal of Operational Research, 275 (2019), 295-303.  doi: 10.1016/j.ejor.2018.11.025.

[12]

L. GriecoM. Utley and S. Crowe, Operational research applied to decisions in home health care: A systematic literature review, Journal of the Operational Research Society, 22 (2020), 1-32.  doi: 10.1080/01605682.2020.1750311.

[13]

A. Hertz and N. Lahrichi, A patient assignment algorithm for home care services, Journal of the Operational Research Society, 60 (2009), 481-495.  doi: 10.1057/palgrave.jors.2602574.

[14]

E. Lanzarone and A. Matta, A cost assignment policy for home care patients, Flexible Services and Manufacturing Journal, 24 (2012), 465-495.  doi: 10.1007/s10696-011-9121-4.

[15]

E. Lanzarone and A. Matta, The nurse-to-patient assignment problem in home care services, in Advanced Decision Making Methods Applied to Health Care, Springer, 2012,121–139. doi: 10.1007/978-88-470-2321-5_8.

[16]

E. Lanzarone and A. Matta, Robust nurse-to-patient assignment in home care services to minimize overtimes under continuity of care, Operations Research for Health Care, 3 (2014), 48-58. 

[17]

E. LanzaroneA. Matta and E. Sahin, Operations management applied to home care services: the problem of assigning human resources to patients, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 42 (2012), 1346-1363.  doi: 10.1109/TSMCA.2012.2210207.

[18]

E. LanzaroneA. Matta and G. Scaccabarozzi, A patient stochastic model to support human resource planning in home care, Production Planning and Control, 21 (2010), 3-25.  doi: 10.1080/09537280903232362.

[19]

M. LinK. S. ChinX. Wang and K. L. Tsui, The therapist assignment problem in home healthcare structures, Expert Systems with Applications, 62 (2016), 44-62.  doi: 10.1016/j.eswa.2016.06.010.

[20]

S. E. MoussaviM. Mahdjoub and O. Grunder, A matheuristic approach to the integration of worker assignment and vehicle routing problems: Application to home healthcare scheduling, Expert Systems with Applications, 125 (2019), 317-332.  doi: 10.1016/j.eswa.2019.02.009.

[21]

S. NickelM. Schroderb and J. Steegb, Mid-term and short-term planning support for home health care services, European Journal of Operational Research, 219 (2012), 574-587.  doi: 10.1016/j.ejor.2011.10.042.

[22]

M. I. Restrepo, L.-M. Rousseau and J. Valle, Home healthcare integrated staffing and scheduling, Omega (United Kingdom), 95 (2020), 102057. doi: 10.1016/j.omega.2019.03.015.

[23]

J. WirnitzerI. HeckmannA. Meyer and S. Nickel, Patient-based nurse rostering in home care, Operations Research for Health Care, 8 (2016), 91-102.  doi: 10.1016/j.orhc.2015.08.005.

[24]

S. YalçındağA. MattaE. Şahin and J. G. Shanthikumar, The patient assignment problem in home health care: Using a data-driven method to estimate the travel times of care givers, Flexible Services and Manufacturing Journal, 28 (2016), 304-335.  doi: 10.1007/s10696-015-9222-6.

[25]

S. YalçındağP. CappaneraM. G. ScutellàE. Şahin and A. Matta, Pattern-based decompositions for human resource planning in home health care services, Computers & Operations Research, 73 (2016), 12-26.  doi: 10.1016/j.cor.2016.02.011.

show all references

References:
[1]

R. ArgientoA. GuglielmiE. Lanzarone and I. Nawajah, A Bayesian framework for describing and predicting the stochastic demand of home care patients, Flexible Services and Manufacturing Journal, 28 (2016), 254-279.  doi: 10.1007/s10696-014-9200-4.

[2]

R. ArgientoA. GuglielmiE. Lanzarone and I. Nawajah, Bayesian joint modelling of the health profile and demand of home care patients, IMA Journal of Management Mathematics, 28 (2016), 531-552.  doi: 10.1093/imaman/dpw001.

[3]

P. CappaneraM. G. ScutellàF. Nervi and L. Galli, Demand uncertainty in robust home care optimization, Omega (United Kingdom), 80 (2018), 95-110.  doi: 10.1016/j.omega.2017.08.012.

[4]

P. Cappanera and M. G. Scutellà, Joint assignment, scheduling, and routing models to home care optimization: A pattern-based approach, Transportation Science, 49 (2015), 830-852.  doi: 10.1287/trsc.2014.0548.

[5]

G. Carello and E. Lanzarone, A cardinality-constrained robust model for the assignment problem in home care services, European Journal of Operational Research, 236 (2014), 748-762.  doi: 10.1016/j.ejor.2014.01.009.

[6]

G. CarelloE. Lanzarone and S. Mattia, Trade-off between stakeholders' goals in the home care nurse-to-patient assignment problem, Operations Research for Health Care, 16 (2018), 29-40.  doi: 10.1016/j.orhc.2017.12.002.

[7]

M. CisséS. YalçındağY. KergosienE. ŞahinC. Lenté and A. Matta, Or problems related to home health care: A review of relevant routing and scheduling problems, Operations Research for Health Care, 13 (2017), 1-22. 

[8]

A. Errarhout, S. Kharraja and I. Zorkani, Caregivers' assignment problem in home health care structures, in Proceedings of 2013 International Conference on Industrial Engineering and Systems Management (IESM), IEEE, 2013, 1–8.

[9]

P. EvebornP. Flisberg and M. Rönnqvist, Laps Care–an operational system for staff planning of home care, European Journal of Operational Research, 171 (2006), 962-976.  doi: 10.1016/j.ejor.2005.01.011.

[10]

C. Fikar and P. Hirsch, Home health care routing and scheduling: A review, Computers & Operations Research, 77 (2017), 86-95.  doi: 10.1016/j.cor.2016.07.019.

[11]

F. GrenouilleauA. LegrainN. Lahrichi and L.-M. Rousseau, A set partitioning heuristic for the home health care routing and scheduling problem, European Journal of Operational Research, 275 (2019), 295-303.  doi: 10.1016/j.ejor.2018.11.025.

[12]

L. GriecoM. Utley and S. Crowe, Operational research applied to decisions in home health care: A systematic literature review, Journal of the Operational Research Society, 22 (2020), 1-32.  doi: 10.1080/01605682.2020.1750311.

[13]

A. Hertz and N. Lahrichi, A patient assignment algorithm for home care services, Journal of the Operational Research Society, 60 (2009), 481-495.  doi: 10.1057/palgrave.jors.2602574.

[14]

E. Lanzarone and A. Matta, A cost assignment policy for home care patients, Flexible Services and Manufacturing Journal, 24 (2012), 465-495.  doi: 10.1007/s10696-011-9121-4.

[15]

E. Lanzarone and A. Matta, The nurse-to-patient assignment problem in home care services, in Advanced Decision Making Methods Applied to Health Care, Springer, 2012,121–139. doi: 10.1007/978-88-470-2321-5_8.

[16]

E. Lanzarone and A. Matta, Robust nurse-to-patient assignment in home care services to minimize overtimes under continuity of care, Operations Research for Health Care, 3 (2014), 48-58. 

[17]

E. LanzaroneA. Matta and E. Sahin, Operations management applied to home care services: the problem of assigning human resources to patients, IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 42 (2012), 1346-1363.  doi: 10.1109/TSMCA.2012.2210207.

[18]

E. LanzaroneA. Matta and G. Scaccabarozzi, A patient stochastic model to support human resource planning in home care, Production Planning and Control, 21 (2010), 3-25.  doi: 10.1080/09537280903232362.

[19]

M. LinK. S. ChinX. Wang and K. L. Tsui, The therapist assignment problem in home healthcare structures, Expert Systems with Applications, 62 (2016), 44-62.  doi: 10.1016/j.eswa.2016.06.010.

[20]

S. E. MoussaviM. Mahdjoub and O. Grunder, A matheuristic approach to the integration of worker assignment and vehicle routing problems: Application to home healthcare scheduling, Expert Systems with Applications, 125 (2019), 317-332.  doi: 10.1016/j.eswa.2019.02.009.

[21]

S. NickelM. Schroderb and J. Steegb, Mid-term and short-term planning support for home health care services, European Journal of Operational Research, 219 (2012), 574-587.  doi: 10.1016/j.ejor.2011.10.042.

[22]

M. I. Restrepo, L.-M. Rousseau and J. Valle, Home healthcare integrated staffing and scheduling, Omega (United Kingdom), 95 (2020), 102057. doi: 10.1016/j.omega.2019.03.015.

[23]

J. WirnitzerI. HeckmannA. Meyer and S. Nickel, Patient-based nurse rostering in home care, Operations Research for Health Care, 8 (2016), 91-102.  doi: 10.1016/j.orhc.2015.08.005.

[24]

S. YalçındağA. MattaE. Şahin and J. G. Shanthikumar, The patient assignment problem in home health care: Using a data-driven method to estimate the travel times of care givers, Flexible Services and Manufacturing Journal, 28 (2016), 304-335.  doi: 10.1007/s10696-015-9222-6.

[25]

S. YalçındağP. CappaneraM. G. ScutellàE. Şahin and A. Matta, Pattern-based decompositions for human resource planning in home health care services, Computers & Operations Research, 73 (2016), 12-26.  doi: 10.1016/j.cor.2016.02.011.

Table 1.  Sets, parameters and decision variables for the three models. Symbol $ \checkmark $ means that the set, parameter of variable is included in the model, symbol $ \circ $ that it is included only for $ w = 1 $, and symbol $ \bullet $ that it is included only for $ w>1 $
Model I Model II Model III
Sets
$ O $ set of homogeneous operators
$ P $ set of patients
$ W $ set of weeks in the planning horizon $ \bullet $
$ D $ set of days in the first week $ w=1 $
$ \Pi $ set of patterns
Parameters
$ v_{pw} $ number of visits required by patient $ p \in P $
at week $ w \in W $ $ \circ $
$ s_p $ service time required for each visit to patient $ p \in P $
$ n_\pi $ total number of visits in pattern $ \pi \in \Pi $
$ m_{\pi d} $ 1 if pattern $ \pi \in \Pi $ includes a visit at day $ d \in D $;
0 otherwise
$ r_p $ average travel time to reach patient $ p \in P $
$ a_o $ weekly capacity of operator $ o \in O $
$ \alpha_w $ weight of week $ w \in W $ in the objective function
Decision variables
$ \lambda_{od} $ daily workload of operator $ o \in O $ at day $ d \in D $
$ \omega_{ow} $ weekly workload of operator $ o \in O $ at week $ w \in W $ $ \bullet $
$ h_w $ maximum utilization rate at week $ w \in W $ $ \circ $
$ z_{p\pi} $ 1 if pattern $ \pi \in \Pi $ is assigned to patient $ p \in P $;
0 otherwise
$ u_{op} $ 1 if operator $ o \in O $ is assigned to patient $ p \in P $;
0 otherwise
$ \mu_{op}^d $ 1 if $ o \in O $ is assigned to $ p \in P $ on day $ d \in D $;
0 otherwise
Model I Model II Model III
Sets
$ O $ set of homogeneous operators
$ P $ set of patients
$ W $ set of weeks in the planning horizon $ \bullet $
$ D $ set of days in the first week $ w=1 $
$ \Pi $ set of patterns
Parameters
$ v_{pw} $ number of visits required by patient $ p \in P $
at week $ w \in W $ $ \circ $
$ s_p $ service time required for each visit to patient $ p \in P $
$ n_\pi $ total number of visits in pattern $ \pi \in \Pi $
$ m_{\pi d} $ 1 if pattern $ \pi \in \Pi $ includes a visit at day $ d \in D $;
0 otherwise
$ r_p $ average travel time to reach patient $ p \in P $
$ a_o $ weekly capacity of operator $ o \in O $
$ \alpha_w $ weight of week $ w \in W $ in the objective function
Decision variables
$ \lambda_{od} $ daily workload of operator $ o \in O $ at day $ d \in D $
$ \omega_{ow} $ weekly workload of operator $ o \in O $ at week $ w \in W $ $ \bullet $
$ h_w $ maximum utilization rate at week $ w \in W $ $ \circ $
$ z_{p\pi} $ 1 if pattern $ \pi \in \Pi $ is assigned to patient $ p \in P $;
0 otherwise
$ u_{op} $ 1 if operator $ o \in O $ is assigned to patient $ p \in P $;
0 otherwise
$ \mu_{op}^d $ 1 if $ o \in O $ is assigned to $ p \in P $ on day $ d \in D $;
0 otherwise
Table 2.  List of patterns for the flexible alternative, with at least one pattern for each $ n_\pi $. The list for the rigid alternative is randomly extracted while keeping the condition of at least one pattern for each $ n_\pi $
Pattern $ n_{\pi d} $ (visit at day $ d $) $ n_\pi $
$ \pi $ $ d=1 $ $ d=2 $ $ d=3 $ $ d=4 $ $ d=5 $ $ d=6 $
1 0 1 0 0 0 0 1
2 0 0 1 0 0 0 1
3 0 0 0 1 0 0 1
4 0 0 0 0 1 0 1
5 1 0 1 0 0 0 2
6 1 0 0 1 0 0 2
7 1 0 0 0 1 0 2
8 0 1 0 0 1 0 2
9 0 0 1 0 1 0 2
10 0 0 1 0 0 1 2
11 0 0 0 1 0 1 2
12 1 0 1 0 1 0 3
13 1 0 1 0 0 1 3
14 0 1 0 1 0 1 3
15 1 1 0 1 1 0 4
16 1 1 0 1 0 1 4
17 0 1 1 1 0 1 4
18 0 1 1 0 1 1 4
19 1 1 1 0 1 1 5
20 1 1 0 1 1 1 5
21 1 1 1 1 1 1 6
Pattern $ n_{\pi d} $ (visit at day $ d $) $ n_\pi $
$ \pi $ $ d=1 $ $ d=2 $ $ d=3 $ $ d=4 $ $ d=5 $ $ d=6 $
1 0 1 0 0 0 0 1
2 0 0 1 0 0 0 1
3 0 0 0 1 0 0 1
4 0 0 0 0 1 0 1
5 1 0 1 0 0 0 2
6 1 0 0 1 0 0 2
7 1 0 0 0 1 0 2
8 0 1 0 0 1 0 2
9 0 0 1 0 1 0 2
10 0 0 1 0 0 1 2
11 0 0 0 1 0 1 2
12 1 0 1 0 1 0 3
13 1 0 1 0 0 1 3
14 0 1 0 1 0 1 3
15 1 1 0 1 1 0 4
16 1 1 0 1 0 1 4
17 0 1 1 1 0 1 4
18 0 1 1 0 1 1 4
19 1 1 1 0 1 1 5
20 1 1 0 1 1 1 5
21 1 1 1 1 1 1 6
Table 3.  Combinations of hyperparameters used to generate low variability and high variability instances
Instance Percentage of patients Variability $ M_\sigma $ Trend
$ m_\kappa $ $ M_\kappa $
Low 50% -0.5 0 0.5
variability 50% 0 0.5 0.5
High 50% -1 0 6
variability 50% 0 1 6
Instance Percentage of patients Variability $ M_\sigma $ Trend
$ m_\kappa $ $ M_\kappa $
Low 50% -0.5 0 0.5
variability 50% 0 0.5 0.5
High 50% -1 0 6
variability 50% 0 1 6
Table 4.  Results for the flexible set of patterns and high variability of demands (first set of experiments). INF denotes infeasibility
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.804 0.841 0.841 0.801 INF 0.841 0.841
$ h_1 $ 0.804 0.801 0.802 0.801 0.801 0.802
$ \delta_1 $ 0.010 0.006 0.007 0.006 0.007 0.007
$ \delta_2 $ 0.018 0.016 0.016 0.016
$ \delta_3 $ 0.005 0.010 0.010 0.010
$ \delta_4 $ 0.023 0.018 0.018 0.018
Replication 2 OFV 0.850 0.805 0.806 0.853 0.879 0.805 0.806
$ h_1 $ 0.850 0.853 0.851 0.853 0.850 0.853 0.851
$ \delta_1 $ 0.011 0.022 0.016 0.022 0.011 0.022 0.016
$ \delta_2 $ 0.017 0.023 0.283 0.017 0.023
$ \delta_3 $ 0.039 0.039 0.228 0.039 0.039
$ \delta_4 $ 0.005 0.007 0.254 0.005 0.007
Replication 3 OFV 0.802 0.785 0.787 0.800 0.840 0.785 0.787
$ h_1 $ 0.802 0.800 0.800 0.800 0.802 0.800 0.800
$ \delta_1 $ 0.013 0.015 0.018 0.015 0.013 0.015 0.018
$ \delta_2 $ 0.015 0.016 0.307 0.015 0.016
$ \delta_3 $ 0.008 0.006 0.135 0.008 0.006
$ \delta_4 $ 0.015 0.025 0.135 0.015 0.025
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.804 0.841 0.841 0.801 INF 0.841 0.841
$ h_1 $ 0.804 0.801 0.802 0.801 0.801 0.802
$ \delta_1 $ 0.010 0.006 0.007 0.006 0.007 0.007
$ \delta_2 $ 0.018 0.016 0.016 0.016
$ \delta_3 $ 0.005 0.010 0.010 0.010
$ \delta_4 $ 0.023 0.018 0.018 0.018
Replication 2 OFV 0.850 0.805 0.806 0.853 0.879 0.805 0.806
$ h_1 $ 0.850 0.853 0.851 0.853 0.850 0.853 0.851
$ \delta_1 $ 0.011 0.022 0.016 0.022 0.011 0.022 0.016
$ \delta_2 $ 0.017 0.023 0.283 0.017 0.023
$ \delta_3 $ 0.039 0.039 0.228 0.039 0.039
$ \delta_4 $ 0.005 0.007 0.254 0.005 0.007
Replication 3 OFV 0.802 0.785 0.787 0.800 0.840 0.785 0.787
$ h_1 $ 0.802 0.800 0.800 0.800 0.802 0.800 0.800
$ \delta_1 $ 0.013 0.015 0.018 0.015 0.013 0.015 0.018
$ \delta_2 $ 0.015 0.016 0.307 0.015 0.016
$ \delta_3 $ 0.008 0.006 0.135 0.008 0.006
$ \delta_4 $ 0.015 0.025 0.135 0.015 0.025
Table 5.  Results for the flexible set of patterns and low variability of demands (first set of experiments)
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.849 0.822 0.821 0.848 0.838 0.821 0.822
$ h_1 $ 0.849 0.848 0.848 0.848 0.849 0.848 0.848
$ \delta_1 $ 0.013 0.013 0.011 0.013 0.013 0.011 0.013
$ \delta_2 $ 0.013 0.011 0.037 0.011 0.013
$ \delta_3 $ 0.028 0.026 0.073 0.026 0.028
$ \delta_4 $ 0.018 0.026 0.059 0.026 0.018
Replication 2 OFV 0.776 0.764 0.764 0.776 0.793 0.764 0.764
$ h_1 $ 0.776 0.776 0.776 0.776 0.776 0.776 0.776
$ \delta_1 $ 0.010 0.010 0.011 0.010 0.010 0.010 0.011
$ \delta_2 $ 0.012 0.011 0.036 0.012 0.011
$ \delta_3 $ 0.020 0.018 0.080 0.020 0.018
$ \delta_4 $ 0.027 0.017 0.144 0.027 0.017
Replication 3 OFV 0.748 0.726 0.726 0.751 0.756 0.726 0.726
$ h_1 $ 0.748 0.751 0.748 0.751 0.748 0.751 0.748
$ \delta_1 $ 0.003 0.007 0.005 0.007 0.003 0.007 0.005
$ \delta_2 $ 0.017 0.022 0.089 0.017 0.022
$ \delta_3 $ 0.014 0.019 0.070 0.014 0.019
$ \delta_4 $ 0.006 0.004 0.108 0.006 0.004
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.849 0.822 0.821 0.848 0.838 0.821 0.822
$ h_1 $ 0.849 0.848 0.848 0.848 0.849 0.848 0.848
$ \delta_1 $ 0.013 0.013 0.011 0.013 0.013 0.011 0.013
$ \delta_2 $ 0.013 0.011 0.037 0.011 0.013
$ \delta_3 $ 0.028 0.026 0.073 0.026 0.028
$ \delta_4 $ 0.018 0.026 0.059 0.026 0.018
Replication 2 OFV 0.776 0.764 0.764 0.776 0.793 0.764 0.764
$ h_1 $ 0.776 0.776 0.776 0.776 0.776 0.776 0.776
$ \delta_1 $ 0.010 0.010 0.011 0.010 0.010 0.010 0.011
$ \delta_2 $ 0.012 0.011 0.036 0.012 0.011
$ \delta_3 $ 0.020 0.018 0.080 0.020 0.018
$ \delta_4 $ 0.027 0.017 0.144 0.027 0.017
Replication 3 OFV 0.748 0.726 0.726 0.751 0.756 0.726 0.726
$ h_1 $ 0.748 0.751 0.748 0.751 0.748 0.751 0.748
$ \delta_1 $ 0.003 0.007 0.005 0.007 0.003 0.007 0.005
$ \delta_2 $ 0.017 0.022 0.089 0.017 0.022
$ \delta_3 $ 0.014 0.019 0.070 0.014 0.019
$ \delta_4 $ 0.006 0.004 0.108 0.006 0.004
Table 6.  Results for the rigid set of patterns and high variability of demands (first set of experiments). INF denotes infeasibility
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.801 0.841 0.843 INF INF INF 0.843
$ h_1 $ 0.801 0.801 0.810 0.810
$ \delta_1 $ 0.005 0.006 0.024 0.024
$ \delta_2 $ 0.018 0.013 0.013
$ \delta_3 $ 0.005 0.013 0.013
$ \delta_4 $ 0.023 0.018 0.018
Replication 2 OFV 0.853 0.805 0.805 INF 0.902 INF 0.805
$ h_1 $ 0.853 0.853 0.854 0.853 0.854
$ \delta_1 $ 0.013 0.022 0.016 0.013 0.016
$ \delta_2 $ 0.017 0.028 0.168 0.028
$ \delta_31 $ 0.039 0.011 0.282 0.011
$ \delta_4 $ 0.005 0.012 0.256 0.012
Replication 3 OFV 0.799 0.785 0.787 INF INF INF 0.787
$ h_1 $ 0.799 0.800 0.800 0.800
$ \delta_1 $ 0.014 0.015 0.014 0.014
$ \delta_2 $ 0.015 0.012 0.012
$ \delta_3 $ 0.008 0.025 0.025
$ \delta_4 $ 0.015 0.018 0.018
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.801 0.841 0.843 INF INF INF 0.843
$ h_1 $ 0.801 0.801 0.810 0.810
$ \delta_1 $ 0.005 0.006 0.024 0.024
$ \delta_2 $ 0.018 0.013 0.013
$ \delta_3 $ 0.005 0.013 0.013
$ \delta_4 $ 0.023 0.018 0.018
Replication 2 OFV 0.853 0.805 0.805 INF 0.902 INF 0.805
$ h_1 $ 0.853 0.853 0.854 0.853 0.854
$ \delta_1 $ 0.013 0.022 0.016 0.013 0.016
$ \delta_2 $ 0.017 0.028 0.168 0.028
$ \delta_31 $ 0.039 0.011 0.282 0.011
$ \delta_4 $ 0.005 0.012 0.256 0.012
Replication 3 OFV 0.799 0.785 0.787 INF INF INF 0.787
$ h_1 $ 0.799 0.800 0.800 0.800
$ \delta_1 $ 0.014 0.015 0.014 0.014
$ \delta_2 $ 0.015 0.012 0.012
$ \delta_3 $ 0.008 0.025 0.025
$ \delta_4 $ 0.015 0.018 0.018
Table 7.  Results for the rigid set of patterns and low variability of demands (first set of experiments). INF denotes infeasibility
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.848 0.822 0.821 INF 0.866 INF 0.821
$ h_1 $ 0.848 0.848 0.847 0.848 0.847
$ \delta_1 $ 0.012 0.013 0.010 0.012 0.010
$ \delta_2 $ 0.013 0.010 0.101 0.010
$ \delta_3 $ 0.028 0.018 0.173 0.018
$ \delta_4 $ 0.018 0.019 0.173 0.019
Replication 2 OFV 0.777 0.764 0.764 INF 0.799 INF 0.764
$ h_1 $ 0.777 0.776 0.781 0.777 0.781
$ \delta_1 $ 0.017 0.010 0.018 0.017 0.018
$ \delta_2 $ 0.012 0.010 0.072 0.010
$ \delta_3 $ 0.020 0.010 0.089 0.010
$ \delta_4 $ 0.027 0.010 0.196 0.010
Replication 3 OFV 0.749 0.726 0.726 INF 0.746 INF 0.726
$ h_1 $ 0.749 0.751 0.750 0.749 0.750
$ \delta_1 $ 0.003 0.007 0.008 0.003 0.008
$ \delta_2 $ 0.017 0.016 0.033 0.016
$ \delta_3 $ 0.014 0.016 0.105 0.016
$ \delta_4 $ 0.006 0.006 0.054 0.006
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
Replication 1 OFV 0.848 0.822 0.821 INF 0.866 INF 0.821
$ h_1 $ 0.848 0.848 0.847 0.848 0.847
$ \delta_1 $ 0.012 0.013 0.010 0.012 0.010
$ \delta_2 $ 0.013 0.010 0.101 0.010
$ \delta_3 $ 0.028 0.018 0.173 0.018
$ \delta_4 $ 0.018 0.019 0.173 0.019
Replication 2 OFV 0.777 0.764 0.764 INF 0.799 INF 0.764
$ h_1 $ 0.777 0.776 0.781 0.777 0.781
$ \delta_1 $ 0.017 0.010 0.018 0.017 0.018
$ \delta_2 $ 0.012 0.010 0.072 0.010
$ \delta_3 $ 0.020 0.010 0.089 0.010
$ \delta_4 $ 0.027 0.010 0.196 0.010
Replication 3 OFV 0.749 0.726 0.726 INF 0.746 INF 0.726
$ h_1 $ 0.749 0.751 0.750 0.749 0.750
$ \delta_1 $ 0.003 0.007 0.008 0.003 0.008
$ \delta_2 $ 0.017 0.016 0.033 0.016
$ \delta_3 $ 0.014 0.016 0.105 0.016
$ \delta_4 $ 0.006 0.006 0.054 0.006
Table 8.  Results for the larger instances (second set of experiments). INF denotes infeasibility
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
$ |P|=150 $; $ |O|=10 $ OFV 0.887 0.888 0.889 0.890 INF 0.888 0.889
$ h_1 $ 0.887 0.890 0.889 0.890 0.890 0.889
$ \delta_1 $ 0.019 0.036 0.039 0.036 0.036 0.039
$ \delta_2 $ 0.035 0.038 0.035 0.038
$ \delta_3 $ 0.027 0.025 0.027 0.025
$ \delta_4 $ 0.015 0.021 0.015 0.021
$ |W|=8 $ OFV 0.776 0.744 0.744 0.780 0.813 0.744 0.744
$ h_1 $ 0.776 0.780 0.779 0.780 0.776 0.780 0.779
$ \delta_1 $ 0.010 0.036 0.018 0.036 0.010 0.036 0.018
$ \delta_2 $ 0.001 0.003 0.036 0.001 0.003
$ \delta_3 $ 0.018 0.018 0.080 0.018 0.018
$ \delta_4 $ 0.018 0.018 0.144 0.018 0.018
$ \delta_5 $ 0.018 0.018 0.179 0.018 0.018
$ \delta_6 $ 0.017 0.017 0.167 0.017 0.017
$ \delta_7 $ 0.002 0.002 0.167 0.002 0.002
$ \delta_8 $ 0.001 0.003 0.185 0.001 0.003
$ |P|=150 $; $ |O|=10 $; $ |W|=8 $ OFV 0.887 0.891 0.890 0.887 INF 0.890 0.891
$ h_1 $ 0.887 0.887 0.889 0.887 0.889 0.887
$ \delta_1 $ 0.019 0.017 0.021 0.017 0.017 0.021
$ \delta_2 $ 0.036 0.019 0.036 0.019
$ \delta_3 $ 0.037 0.034 0.037 0.034
$ \delta_4 $ 0.010 0.019 0.010 0.019
$ \delta_5 $ 0.011 0.020 0.011 0.020
$ \delta_6 $ 0.034 0.039 0.034 0.039
$ \delta_7 $ 0.034 0.040 0.034 0.040
$ \delta_8 $ 0.013 0.022 0.013 0.022
$ |P|=150 $; $ |O|=10 $; $ |W|=8 $; V OFV 0.911 0.913 0.913 0.908 INF 0.913 0.913
$ h_1 $ 0.911 0.908 0.908 0.908 0.908 0.908
$ \delta_1 $ 0.023 0.022 0.017 0.022 0.022 0.017
$ \delta_2 $ 0.032 0.029 0.032 0.029
$ \delta_3 $ 0.022 0.024 0.022 0.024
$ \delta_4 $ 0.012 0.035 0.012 0.035
$ \delta_5 $ 0.021 0.033 0.021 0.033
$ \delta_6 $ 0.060 0.054 0.060 0.054
$ \delta_7 $ 0.044 0.032 0.044 0.032
$ \delta_8 $ 0.030 0.054 0.030 0.054
Model solutions Executions
Model I Model II Model III Model II on I Model I on II Model II on III Model III on II
$ |P|=150 $; $ |O|=10 $ OFV 0.887 0.888 0.889 0.890 INF 0.888 0.889
$ h_1 $ 0.887 0.890 0.889 0.890 0.890 0.889
$ \delta_1 $ 0.019 0.036 0.039 0.036 0.036 0.039
$ \delta_2 $ 0.035 0.038 0.035 0.038
$ \delta_3 $ 0.027 0.025 0.027 0.025
$ \delta_4 $ 0.015 0.021 0.015 0.021
$ |W|=8 $ OFV 0.776 0.744 0.744 0.780 0.813 0.744 0.744
$ h_1 $ 0.776 0.780 0.779 0.780 0.776 0.780 0.779
$ \delta_1 $ 0.010 0.036 0.018 0.036 0.010 0.036 0.018
$ \delta_2 $ 0.001 0.003 0.036 0.001 0.003
$ \delta_3 $ 0.018 0.018 0.080 0.018 0.018
$ \delta_4 $ 0.018 0.018 0.144 0.018 0.018
$ \delta_5 $ 0.018 0.018 0.179 0.018 0.018
$ \delta_6 $ 0.017 0.017 0.167 0.017 0.017
$ \delta_7 $ 0.002 0.002 0.167 0.002 0.002
$ \delta_8 $ 0.001 0.003 0.185 0.001 0.003
$ |P|=150 $; $ |O|=10 $; $ |W|=8 $ OFV 0.887 0.891 0.890 0.887 INF 0.890 0.891
$ h_1 $ 0.887 0.887 0.889 0.887 0.889 0.887
$ \delta_1 $ 0.019 0.017 0.021 0.017 0.017 0.021
$ \delta_2 $ 0.036 0.019 0.036 0.019
$ \delta_3 $ 0.037 0.034 0.037 0.034
$ \delta_4 $ 0.010 0.019 0.010 0.019
$ \delta_5 $ 0.011 0.020 0.011 0.020
$ \delta_6 $ 0.034 0.039 0.034 0.039
$ \delta_7 $ 0.034 0.040 0.034 0.040
$ \delta_8 $ 0.013 0.022 0.013 0.022
$ |P|=150 $; $ |O|=10 $; $ |W|=8 $; V OFV 0.911 0.913 0.913 0.908 INF 0.913 0.913
$ h_1 $ 0.911 0.908 0.908 0.908 0.908 0.908
$ \delta_1 $ 0.023 0.022 0.017 0.022 0.022 0.017
$ \delta_2 $ 0.032 0.029 0.032 0.029
$ \delta_3 $ 0.022 0.024 0.022 0.024
$ \delta_4 $ 0.012 0.035 0.012 0.035
$ \delta_5 $ 0.021 0.033 0.021 0.033
$ \delta_6 $ 0.060 0.054 0.060 0.054
$ \delta_7 $ 0.044 0.032 0.044 0.032
$ \delta_8 $ 0.030 0.054 0.030 0.054
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