Dynamic-programming-based heuristic for multi-objective operating theater planning

Ya Liu; Zhaojin Li

doi:10.3934/jimo.2020145

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Dynamic-programming-based heuristic for multi-objective operating theater planning

Ya Liu and Zhaojin Li ^,

School of Management, Xi'an Jiaotong University, 710049, NO.28 Xianning Road, Xi'an Shaanxi, China

^* Corresponding author: Zhaojin Li

Received December 2018 Revised August 2020 Early access September 2020

This study focuses on the assignment of surgical operations to operating room (OR) blocks to minimize not only the utilization cost of the OR blocks but also the number of patterns generated. We propose a dynamic-programming-based heuristic to solve this problem. Using an iterative formulation, we separate the patterns individually. The computational results show that the proposed heuristic is efficient. Compared with "quasi-optimal" plans, the average cost gap ranges from 0.022 to 0.066 for various scenarios. A lower bound based on column generation procedure is developed. Results show that the average absolute difference of three scenarios are respectively 0.045, 0.081 and 0.115. According to evaluations based on performance indicators from the extant literature, the utilization ratio of the operating theater (OT) varies from 1% underutilization to 2% overutilization in the solutions yielded by the proposed heuristic. This study deals with a multi-objective optimization problem, the second objective of which plays a significant role in reducing the preparation cost, error probability, and staff fatigue in medical systems, where security and human issues are far more emphasized than in other sectors. To the best of our knowledge, our study is the first to investigate such a real-world multi-objective OT planning problem.

Keywords: Healthcare, dynamic programming, operating theater, heuristic, multi-objective optimization.

Mathematics Subject Classification: Primary: 90BXX; Secondary: 90B50.

Citation: Ya Liu, Zhaojin Li. Dynamic-programming-based heuristic for multi-objective operating theater planning. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020145

References:

[1]	I. Adan, J. Bekkers, N. Dellaert, J. Jeunet and J. Vissers, Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic demand and capacitated resources, European Journal of Operational Research, 213 (2011), 290-308. doi: 10.1016/j.ejor.2011.02.025. Google Scholar
[2]	J. Ansarifar, R. Tavakkoli-Moghaddam, F. Akhavizadegan and S. Hassanzadeh Amin, Multi-objective integrated planning and scheduling model for operating rooms under uncertainty, Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 232 (2018), 930-948. doi: 10.1177/0954411918794721. Google Scholar
[3]	J. Belien and E. Demeulemeester, Building cyclic master surgery schedules with leveled resulting bed occupancy, European Journal of Operational Research, 176 (2007), 1185-1204. doi: 10.1016/j.ejor.2005.06.063. Google Scholar
[4]	B. Cardoen, E. Demeulemeester and J. Belien, Optimizing a multiple objective surgical case senquencing problem, International Journal of Production Economics, 119 (2009), 354-366. Google Scholar
[5]	B. Cardoen, E. Demeulemeester and J. Belien, Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932. Google Scholar
[6]	H. F. M. Association et al., Achieving operating room efficiency through process integration., Healthcare Financial Management: Journal of the Healthcare Financial Management Association, 57 (2003), suppl 1–7. Google Scholar
[7]	C. Chu and J. Antonio, Approximation algorithms to solve real-life multicriteria cutting stock problems, Operations Research, 47 (1999), 495-508. doi: 10.1287/opre.47.4.495. Google Scholar
[8]	N. Dellaert, E. Cayiroglu and J. Jeunet, Assessing and controlling the impact of hospital capacity planning on the waiting time, International Journal of Production Research, 54 (2016), 2203-2214. doi: 10.1080/00207543.2015.1051668. Google Scholar
[9]	N. Dellaert and J. Jeunet, A variable neighborhood search algorithm for the surgery tactical planning problem, Computers & Operations Research, 84 (2017), 216-225. doi: 10.1016/j.cor.2016.05.013. Google Scholar
[10]	F. Dexter, A. Macario, R. D. Traub, M. Hopwood and D. A. Lubarsky, An operating room scheduling strategy to maximize the use of operating room block time: computer simulation of patient scheduling and survey of patients' preferences for surgical waiting time, Anesthesia Analgesia, 89 (1999), 7-20. doi: 10.1097/00000539-199907000-00003. Google Scholar
[11]	H. Fei, C. Chu and N. Meskens, Solving a tactical operating room planning problem by a column-generation-based heuristic procedure with four criteria, Annals of Operations Research, 166 (2009), 91-108. doi: 10.1007/s10479-008-0413-3. Google Scholar
[12]	H. Fei, C. Chu, N. Meskens and A. Artiba, Solving surgical cases assignment problem by a branch-and-price approach, International Journal of Production Economics, 112 (2008), 96-108. doi: 10.1016/j.ijpe.2006.08.030. Google Scholar
[13]	H. Fei, N. Meskens and C. Chu, A planning and scheduling problem for an operating theatre using an open scheduling strategy, Computers and Industrial Engineering, 58 (2010), 221-230. doi: 10.1016/j.cie.2009.02.012. Google Scholar
[14]	P. C. Gilmore and R. E. Gomory, A linear programming approach to the cutting-stock problem, Operations Research, 9 (1961), 849-859. doi: 10.1287/opre.9.6.849. Google Scholar
[15]	F. Guerriero and R. Guido, Operational research in the management of the operating theatre: a survey, Health Care Management Science, 14 (2011), 89-114. doi: 10.1007/s10729-010-9143-6. Google Scholar
[16]	R. Guido and D. Conforti, A hybrid genetic approach for solving an integrated multi-objective operating room planning and scheduling problem, Computers & Operations Research, 87 (2017), 270-282. doi: 10.1016/j.cor.2016.11.009. Google Scholar
[17]	A. Guinet and S. Chaabane, Operating theatre planning, International Journal of Production Economics, 85 (2003), 69-81. doi: 10.1016/S0925-5273(03)00087-2. Google Scholar
[18]	E. Hans, G. Wullink, M. V. Houdenhoven and G. Kazemier, Robust surgery loading, European Journal of Operational Research, 185 (2008), 1038-1050. doi: 10.1016/j.ejor.2006.08.022. Google Scholar
[19]	A. Jebali, A. H. Alouane and P. Ladet, Operating rooms scheduling, International Journal of Production Economics, 99 (2006), 52-62. doi: 10.1016/j.ijpe.2004.12.006. Google Scholar
[20]	A. Jebali and A. Diabat, A stochastic model for operating room planning under capacity constraints, International Journal of Production Research, 53 (2015), 7252-7270. doi: 10.1080/00207543.2015.1033500. Google Scholar
[21]	M. A. Kamran, B. Karimi and N. Dellaert, Uncertainty in advance scheduling problem in operating room planning, Computers & Industrial Engineering, 126 (2018), 252-268. doi: 10.1016/j.cie.2018.09.030. Google Scholar
[22]	M. A. Kamran, B. Karimi, N. Dellaert and E. Demeulemeester, Adaptive operating rooms planning and scheduling: A rolling horizon approach, Operations Research for Health Care, 22 (2019), 100200. doi: 10.1016/j.orhc.2019.100200. Google Scholar
[23]	P. Kuo, R. Schroeder, S. Mahaffey and R. R. Bollinger, Optimization of operating room allocation using linear programming technique, The American College of Surgeon, 197 (2003), 889-895. doi: 10.1016/j.jamcollsurg.2003.07.006. Google Scholar
[24]	E. Litvak and M. Long, Cost and quality under managed care: Irreconcilable differences, American Journal of Managed Care, 6 (2000), 305-312. Google Scholar
[25]	Y. Liu, C. Chu and K. Wang, A new heuristic algorithm for the operating room scheduling problem, Computers and Industrial Engineering, 61 (2011), 865-871. doi: 10.1016/j.cie.2011.05.020. Google Scholar
[26]	E. Marcon, S. Kharraja and G. Simonnet, The operating theatre planning by the follow-up of the risk of no realization, International Journal of Production Economics, 85 (2003), 83-90. doi: 10.1016/S0925-5273(03)00088-4. Google Scholar
[27]	A. Testi and E. Tanfani, A pre-assignment heuristic algorithm for the master surgical schedule problem(mssp), Annals of Operations Research, 178 (2010), 105-119. doi: 10.1007/s10479-009-0568-6. Google Scholar
[28]	M. M. Vali-Siar, S. Gholami and R. Ramezanian, Multi-period and multi-resource operating room scheduling under uncertainty: A case study, Computers & Industrial Engineering, 126 (2018), 549-568. doi: 10.1016/j.cie.2018.10.014. Google Scholar
[29]	C. Van Riet and E. Demeulemeester, Trade-offs in operating room planning for electives and emergencies: A review, Operations Research for Health Care, 7 (2015), 52-69. Google Scholar
[30]	F. Vanderbeck, Exact algorithm for minimising the number of setups in the one-dimensional cutting stock problem, Operations Research, 48 (2000), 915-926. doi: 10.1287/opre.48.6.915.12391. Google Scholar
[31]	A. Weinbroum, P. Ekstein and T. Ezri, Efficiency of the operating room suite, The American Journal of Surgery, 185 (2003), 244-250. doi: 10.1016/S0002-9610(02)01362-4. Google Scholar
[32]	W. Xiang, A multi-objective aco for operating room scheduling optimization, Natural Computing, 16 (2017), 607-617. doi: 10.1007/s11047-016-9607-9. Google Scholar
[33]	S. Zhu, W. Fan, S. Yang, J. Pei and P. M. Pardalos, Operating room planning and surgical case scheduling: A review of literature, Journal of Combinatorial Optimization, 37 (2019), 757-805. doi: 10.1007/s10878-018-0322-6. Google Scholar

show all references

References:

[1]	I. Adan, J. Bekkers, N. Dellaert, J. Jeunet and J. Vissers, Improving operational effectiveness of tactical master plans for emergency and elective patients under stochastic demand and capacitated resources, European Journal of Operational Research, 213 (2011), 290-308. doi: 10.1016/j.ejor.2011.02.025. Google Scholar
[2]	J. Ansarifar, R. Tavakkoli-Moghaddam, F. Akhavizadegan and S. Hassanzadeh Amin, Multi-objective integrated planning and scheduling model for operating rooms under uncertainty, Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 232 (2018), 930-948. doi: 10.1177/0954411918794721. Google Scholar
[3]	J. Belien and E. Demeulemeester, Building cyclic master surgery schedules with leveled resulting bed occupancy, European Journal of Operational Research, 176 (2007), 1185-1204. doi: 10.1016/j.ejor.2005.06.063. Google Scholar
[4]	B. Cardoen, E. Demeulemeester and J. Belien, Optimizing a multiple objective surgical case senquencing problem, International Journal of Production Economics, 119 (2009), 354-366. Google Scholar
[5]	B. Cardoen, E. Demeulemeester and J. Belien, Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932. Google Scholar
[6]	H. F. M. Association et al., Achieving operating room efficiency through process integration., Healthcare Financial Management: Journal of the Healthcare Financial Management Association, 57 (2003), suppl 1–7. Google Scholar
[7]	C. Chu and J. Antonio, Approximation algorithms to solve real-life multicriteria cutting stock problems, Operations Research, 47 (1999), 495-508. doi: 10.1287/opre.47.4.495. Google Scholar
[8]	N. Dellaert, E. Cayiroglu and J. Jeunet, Assessing and controlling the impact of hospital capacity planning on the waiting time, International Journal of Production Research, 54 (2016), 2203-2214. doi: 10.1080/00207543.2015.1051668. Google Scholar
[9]	N. Dellaert and J. Jeunet, A variable neighborhood search algorithm for the surgery tactical planning problem, Computers & Operations Research, 84 (2017), 216-225. doi: 10.1016/j.cor.2016.05.013. Google Scholar
[10]	F. Dexter, A. Macario, R. D. Traub, M. Hopwood and D. A. Lubarsky, An operating room scheduling strategy to maximize the use of operating room block time: computer simulation of patient scheduling and survey of patients' preferences for surgical waiting time, Anesthesia Analgesia, 89 (1999), 7-20. doi: 10.1097/00000539-199907000-00003. Google Scholar
[11]	H. Fei, C. Chu and N. Meskens, Solving a tactical operating room planning problem by a column-generation-based heuristic procedure with four criteria, Annals of Operations Research, 166 (2009), 91-108. doi: 10.1007/s10479-008-0413-3. Google Scholar
[12]	H. Fei, C. Chu, N. Meskens and A. Artiba, Solving surgical cases assignment problem by a branch-and-price approach, International Journal of Production Economics, 112 (2008), 96-108. doi: 10.1016/j.ijpe.2006.08.030. Google Scholar
[13]	H. Fei, N. Meskens and C. Chu, A planning and scheduling problem for an operating theatre using an open scheduling strategy, Computers and Industrial Engineering, 58 (2010), 221-230. doi: 10.1016/j.cie.2009.02.012. Google Scholar
[14]	P. C. Gilmore and R. E. Gomory, A linear programming approach to the cutting-stock problem, Operations Research, 9 (1961), 849-859. doi: 10.1287/opre.9.6.849. Google Scholar
[15]	F. Guerriero and R. Guido, Operational research in the management of the operating theatre: a survey, Health Care Management Science, 14 (2011), 89-114. doi: 10.1007/s10729-010-9143-6. Google Scholar
[16]	R. Guido and D. Conforti, A hybrid genetic approach for solving an integrated multi-objective operating room planning and scheduling problem, Computers & Operations Research, 87 (2017), 270-282. doi: 10.1016/j.cor.2016.11.009. Google Scholar
[17]	A. Guinet and S. Chaabane, Operating theatre planning, International Journal of Production Economics, 85 (2003), 69-81. doi: 10.1016/S0925-5273(03)00087-2. Google Scholar
[18]	E. Hans, G. Wullink, M. V. Houdenhoven and G. Kazemier, Robust surgery loading, European Journal of Operational Research, 185 (2008), 1038-1050. doi: 10.1016/j.ejor.2006.08.022. Google Scholar
[19]	A. Jebali, A. H. Alouane and P. Ladet, Operating rooms scheduling, International Journal of Production Economics, 99 (2006), 52-62. doi: 10.1016/j.ijpe.2004.12.006. Google Scholar
[20]	A. Jebali and A. Diabat, A stochastic model for operating room planning under capacity constraints, International Journal of Production Research, 53 (2015), 7252-7270. doi: 10.1080/00207543.2015.1033500. Google Scholar
[21]	M. A. Kamran, B. Karimi and N. Dellaert, Uncertainty in advance scheduling problem in operating room planning, Computers & Industrial Engineering, 126 (2018), 252-268. doi: 10.1016/j.cie.2018.09.030. Google Scholar
[22]	M. A. Kamran, B. Karimi, N. Dellaert and E. Demeulemeester, Adaptive operating rooms planning and scheduling: A rolling horizon approach, Operations Research for Health Care, 22 (2019), 100200. doi: 10.1016/j.orhc.2019.100200. Google Scholar
[23]	P. Kuo, R. Schroeder, S. Mahaffey and R. R. Bollinger, Optimization of operating room allocation using linear programming technique, The American College of Surgeon, 197 (2003), 889-895. doi: 10.1016/j.jamcollsurg.2003.07.006. Google Scholar
[24]	E. Litvak and M. Long, Cost and quality under managed care: Irreconcilable differences, American Journal of Managed Care, 6 (2000), 305-312. Google Scholar
[25]	Y. Liu, C. Chu and K. Wang, A new heuristic algorithm for the operating room scheduling problem, Computers and Industrial Engineering, 61 (2011), 865-871. doi: 10.1016/j.cie.2011.05.020. Google Scholar
[26]	E. Marcon, S. Kharraja and G. Simonnet, The operating theatre planning by the follow-up of the risk of no realization, International Journal of Production Economics, 85 (2003), 83-90. doi: 10.1016/S0925-5273(03)00088-4. Google Scholar
[27]	A. Testi and E. Tanfani, A pre-assignment heuristic algorithm for the master surgical schedule problem(mssp), Annals of Operations Research, 178 (2010), 105-119. doi: 10.1007/s10479-009-0568-6. Google Scholar
[28]	M. M. Vali-Siar, S. Gholami and R. Ramezanian, Multi-period and multi-resource operating room scheduling under uncertainty: A case study, Computers & Industrial Engineering, 126 (2018), 549-568. doi: 10.1016/j.cie.2018.10.014. Google Scholar
[29]	C. Van Riet and E. Demeulemeester, Trade-offs in operating room planning for electives and emergencies: A review, Operations Research for Health Care, 7 (2015), 52-69. Google Scholar
[30]	F. Vanderbeck, Exact algorithm for minimising the number of setups in the one-dimensional cutting stock problem, Operations Research, 48 (2000), 915-926. doi: 10.1287/opre.48.6.915.12391. Google Scholar
[31]	A. Weinbroum, P. Ekstein and T. Ezri, Efficiency of the operating room suite, The American Journal of Surgery, 185 (2003), 244-250. doi: 10.1016/S0002-9610(02)01362-4. Google Scholar
[32]	W. Xiang, A multi-objective aco for operating room scheduling optimization, Natural Computing, 16 (2017), 607-617. doi: 10.1007/s11047-016-9607-9. Google Scholar
[33]	S. Zhu, W. Fan, S. Yang, J. Pei and P. M. Pardalos, Operating room planning and surgical case scheduling: A review of literature, Journal of Combinatorial Optimization, 37 (2019), 757-805. doi: 10.1007/s10878-018-0322-6. Google Scholar

Figure 1. Average number of patterns generated in different cases (n = 5, 6, 7)

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Figure 2. UROR for different cases (n = 5, 6, 7)

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Figure 3. Average accumulated overtime for different cases (n = 5, 6, 7)

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Table 1. Notations

$ Parameters: $
$ n $	number of OR block types;
$ m $	number of operation types;
$ C^{0} $	set of OR blocks available during the planning horizon;
$ B^{0} $	set of operations to be assigned;
$ T_{i} $	the regular opening time of OR block with type $ i $;
$ OT_{i} $	the maximum overtime of OR block with type $ i $;
$ c_{i}^{0} $	number of OR blocks with type $ i $;
$ \omega_{k} $	number of mandatory operations with type $ k $;
$ b_{k}^{0} $	number of operations to be assigned with type $ k $;
$ l_{k} $	the duration of operation with type $ k $;
$ p_{i} $	number of all feasible patterns using block of type $ i $;
$ \rho_{ij} $	utilization cost of pattern $ (i, j) $;
$ a_{ijk} $	number of operations of type $ k $ assigned under pattern $ (i, j) $;
$ \beta $	cost for preparing a pattern;
$ M $	an arbitrarily large number;
$ Decision \quad Variables: $
$ x_{ij} $	the number of occurrences of pattern $ (i, j) $;
$ z_{ij} $	1, if pattern $ (i, j) $ occurs at least once in the solution; 0, otherwise.

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$ Parameters: $
$ n $	number of OR block types;
$ m $	number of operation types;
$ C^{0} $	set of OR blocks available during the planning horizon;
$ B^{0} $	set of operations to be assigned;
$ T_{i} $	the regular opening time of OR block with type $ i $;
$ OT_{i} $	the maximum overtime of OR block with type $ i $;
$ c_{i}^{0} $	number of OR blocks with type $ i $;
$ \omega_{k} $	number of mandatory operations with type $ k $;
$ b_{k}^{0} $	number of operations to be assigned with type $ k $;
$ l_{k} $	the duration of operation with type $ k $;
$ p_{i} $	number of all feasible patterns using block of type $ i $;
$ \rho_{ij} $	utilization cost of pattern $ (i, j) $;
$ a_{ijk} $	number of operations of type $ k $ assigned under pattern $ (i, j) $;
$ \beta $	cost for preparing a pattern;
$ M $	an arbitrarily large number;
$ Decision \quad Variables: $
$ x_{ij} $	the number of occurrences of pattern $ (i, j) $;
$ z_{ij} $	1, if pattern $ (i, j) $ occurs at least once in the solution; 0, otherwise.

Table 2. Comparison with "quasi-optimal", $ n $ = 5

Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	16	113	27	0.047	0.116	0.069	0.50		0.116	0.200	0.084	0.52		0.186	0.281	0.095	0.56
2	20	122	24	0.049	0.065	0.016	0.56		0.122	0.138	0.016	0.56		0.195	0.210	0.015	0.55
3	15	118	24	0.048	0.074	0.026	0.43		0.121	0.146	0.025	0.47		0.194	0.218	0.024	0.45
4	19	165	28	0.043	0.085	0.042	0.70		0.106	0.202	0.096	0.70		0.170	0.323	0.153	0.71
5	16	104	26	0.049	0.051	0.002	0.47		0.125	0.127	0.002	0.53		0.200	0.202	0.002	0.51
6	19	146	28	0.045	0.053	0.008	0.65		0.112	0.120	0.008	0.66		0.179	0.187	0.008	0.67
7	17	133	26	0.044	0.089	0.045	0.53		0.111	0.183	0.072	0.56		0.177	0.246	0.068	0.54
8	19	136	26	0.046	0.070	0.024	0.58		0.115	0.152	0.037	0.58		0.183	0.234	0.051	0.61
9	18	106	26	0.047	0.078	0.032	0.50		0.118	0.158	0.041	0.53		0.188	0.228	0.040	0.54
10	18	121	24	0.052	0.108	0.056	0.47		0.131	0.204	0.073	0.51		0.209	0.280	0.071	0.50
Ave.	17.70	126.40	25.90	0.047	0.079	0.032	0.54		0.118	0.163	0.045	0.56		0.188	0.241	0.053	0.56
Case 2
1	17	114	22	0.050	0.082	0.032	0.41		0.125	0.200	0.075	0.42		0.201	0.307	0.106	0.45
2	19	174	26	0.048	0.080	0.032	0.60		0.120	0.152	0.032	0.64		0.191	0.224	0.032	0.62
3	18	147	25	0.052	0.173	0.121	0.54		0.130	0.280	0.151	0.58		0.207	0.367	0.160	0.58
4	18	122	20	0.052	0.079	0.027	0.36		0.131	0.173	0.043	0.35		0.209	0.252	0.043	0.36
5	17	121	26	0.048	0.122	0.074	0.50		0.121	0.223	0.102	0.55		0.194	0.322	0.128	0.56
6	18	164	27	0.044	0.071	0.027	0.61		0.110	0.138	0.028	0.63		0.177	0.205	0.028	0.65
7	15	114	25	0.044	0.070	0.025	0.46		0.112	0.137	0.025	0.45		0.180	0.204	0.024	0.49
8	16	143	26	0.047	0.076	0.030	0.48		0.117	0.144	0.027	0.53		0.187	0.215	0.028	0.54
9	17	147	26	0.047	0.065	0.018	0.55		0.118	0.135	0.017	0.58		0.188	0.205	0.017	0.57
10	18	123	24	0.045	0.078	0.033	0.47		0.112	0.159	0.046	0.48		0.180	0.234	0.054	0.46
Ave.	17.30	136.90	24.70	0.048	0.090	0.042	0.50		0.120	0.174	0.055	0.52		0.191	0.253	0.062	0.53
Case 3
1	17	147	24	0.047	0.092	0.045	0.50		0.117	0.162	0.044	0.49		0.188	0.231	0.043	0.52
2	17	109	24	0.045	0.129	0.084	0.38		0.112	0.206	0.095	0.41		0.179	0.265	0.086	0.45
3	18	181	27	0.048	0.057	0.009	0.47		0.120	0.129	0.009	0.60		0.192	0.201	0.009	0.61
4	17	141	24	0.049	0.126	0.077	0.47		0.122	0.216	0.094	0.49		0.195	0.316	0.121	0.48
5	17	103	22	0.060	0.107	0.048	0.39		0.149	0.195	0.046	0.43		0.239	0.282	0.043	0.45
6	20	189	26	0.046	0.058	0.012	0.59		0.115	0.136	0.020	0.67		0.185	0.206	0.021	0.68
7	17	123	27	0.043	0.077	0.034	0.52		0.110	0.142	0.033	0.53		0.175	0.207	0.031	0.54
8	16	109	23	0.057	0.133	0.075	0.42		0.144	0.235	0.092	0.44		0.230	0.338	0.108	0.43
9	17	145	24	0.046	0.072	0.026	0.48		0.115	0.169	0.054	0.48		0.184	0.275	0.091	0.50
10	17	164	28	0.045	0.063	0.018	0.61		0.112	0.130	0.018	0.63		0.179	0.197	0.018	0.63
Ave.	17.30	141.10	24.90	0.049	0.091	0.043	0.48		0.122	0.172	0.050	0.52		0.195	0.252	0.057	0.53
Case 4
1	17	157	27	0.038	0.084	0.046	0.58		0.094	0.161	0.067	0.59		0.151	0.228	0.077	0.59
2	19	110	26	0.052	0.065	0.013	0.53		0.130	0.143	0.012	0.57		0.209	0.220	0.012	0.58
3	16	100	24	0.055	0.076	0.020	0.39		0.139	0.168	0.029	0.46		0.222	0.250	0.028	0.51
4	17	136	26	0.046	0.057	0.011	0.57		0.115	0.139	0.024	0.58		0.184	0.213	0.030	0.56
5	19	125	23	0.051	0.073	0.022	0.49		0.130	0.173	0.044	0.52		0.207	0.267	0.059	0.53
6	16	111	25	0.048	0.065	0.018	0.42		0.120	0.137	0.017	0.44		0.191	0.208	0.017	0.45
7	18	117	25	0.047	0.056	0.009	0.51		0.117	0.126	0.009	0.55		0.188	0.197	0.010	0.55
8	17	107	26	0.052	0.090	0.038	0.53		0.130	0.168	0.037	0.55		0.209	0.245	0.036	0.55
9	17	159	26	0.048	0.053	0.005	0.44		0.120	0.126	0.006	0.50		0.192	0.198	0.006	0.52
10	18	105	24	0.050	0.095	0.045	0.45		0.125	0.169	0.044	0.48		0.200	0.242	0.042	0.46
Ave.	17.40	122.70	25.20	0.049	0.071	0.023	0.49		0.122	0.151	0.029	0.52		0.195	0.227	0.032	0.53
Case 5
1	17	155	27	0.039	0.059	0.020	0.61		0.098	0.130	0.031	0.62		0.157	0.200	0.043	0.65
2	17	135	27	0.038	0.104	0.066	0.49		0.095	0.163	0.068	0.54		0.152	0.222	0.069	0.54
3	18	103	24	0.050	0.070	0.020	0.31		0.125	0.146	0.021	0.45		0.200	0.223	0.023	0.46
4	16	168	28	0.040	0.064	0.023	0.61		0.101	0.137	0.036	0.62		0.162	0.210	0.048	0.63
5	18	139	27	0.044	0.051	0.008	0.61		0.110	0.117	0.007	0.61		0.175	0.183	0.007	0.65
6	19	171	24	0.051	0.055	0.004	0.51		0.128	0.132	0.004	0.53		0.205	0.209	0.004	0.55
7	18	147	26	0.045	0.049	0.005	0.58		0.113	0.117	0.005	0.60		0.180	0.184	0.005	0.59
8	20	141	25	0.049	0.093	0.044	0.55		0.122	0.215	0.092	0.62		0.195	0.318	0.122	0.61
9	17	145	26	0.042	0.054	0.013	0.58		0.105	0.190	0.085	0.59		0.168	0.279	0.111	0.63
10	16	111	25	0.045	0.066	0.021	0.45		0.112	0.152	0.040	0.47		0.180	0.234	0.054	0.47
Ave.	17.60	141.50	25.90	0.044	0.066	0.022	0.53		0.111	0.150	0.039	0.56		0.177	0.226	0.049	0.58
Case 6
1	17	125	19	0.054	0.065	0.011	0.31		0.136	0.147	0.011	0.32		0.217	0.228	0.011	0.31
2	17	164	28	0.043	0.080	0.037	0.58		0.108	0.170	0.061	0.62		0.173	0.248	0.074	0.63
3	18	168	27	0.041	0.065	0.024	0.62		0.103	0.153	0.049	0.67		0.166	0.240	0.075	0.65
4	17	138	24	0.053	0.083	0.030	0.47	0.133	0.162	0.029	0.49	0.212	0.240	0.028	0.48
5	18	139	25	0.044	0.064	0.020	0.56	0.110	0.156	0.046	0.57	0.175	0.214	0.038	0.56
6	18	130	25	0.044	0.055	0.011	0.52	0.110	0.122	0.013	0.56	0.175	0.188	0.013	0.53
7	16	130	24	0.054	0.090	0.037	0.49	0.134	0.202	0.068	0.47	0.215	0.314	0.100	0.50
8	17	149	26	0.047	0.069	0.022	0.56	0.119	0.155	0.036	0.57	0.191	0.241	0.050	0.62
9	17	120	27	0.051	0.106	0.055	0.50	0.129	0.186	0.057	0.57	0.207	0.265	0.058	0.58
10	18	124	26	0.046	0.084	0.038	0.53	0.116	0.147	0.031	0.56	0.185	0.240	0.055	0.58
Ave.	17.30	138.70	25.10	0.048	0.076	0.028	0.51	0.120	0.160	0.040	0.54	0.192	0.242	0.050	0.54

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Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	16	113	27	0.047	0.116	0.069	0.50		0.116	0.200	0.084	0.52		0.186	0.281	0.095	0.56
2	20	122	24	0.049	0.065	0.016	0.56		0.122	0.138	0.016	0.56		0.195	0.210	0.015	0.55
3	15	118	24	0.048	0.074	0.026	0.43		0.121	0.146	0.025	0.47		0.194	0.218	0.024	0.45
4	19	165	28	0.043	0.085	0.042	0.70		0.106	0.202	0.096	0.70		0.170	0.323	0.153	0.71
5	16	104	26	0.049	0.051	0.002	0.47		0.125	0.127	0.002	0.53		0.200	0.202	0.002	0.51
6	19	146	28	0.045	0.053	0.008	0.65		0.112	0.120	0.008	0.66		0.179	0.187	0.008	0.67
7	17	133	26	0.044	0.089	0.045	0.53		0.111	0.183	0.072	0.56		0.177	0.246	0.068	0.54
8	19	136	26	0.046	0.070	0.024	0.58		0.115	0.152	0.037	0.58		0.183	0.234	0.051	0.61
9	18	106	26	0.047	0.078	0.032	0.50		0.118	0.158	0.041	0.53		0.188	0.228	0.040	0.54
10	18	121	24	0.052	0.108	0.056	0.47		0.131	0.204	0.073	0.51		0.209	0.280	0.071	0.50
Ave.	17.70	126.40	25.90	0.047	0.079	0.032	0.54		0.118	0.163	0.045	0.56		0.188	0.241	0.053	0.56
Case 2
1	17	114	22	0.050	0.082	0.032	0.41		0.125	0.200	0.075	0.42		0.201	0.307	0.106	0.45
2	19	174	26	0.048	0.080	0.032	0.60		0.120	0.152	0.032	0.64		0.191	0.224	0.032	0.62
3	18	147	25	0.052	0.173	0.121	0.54		0.130	0.280	0.151	0.58		0.207	0.367	0.160	0.58
4	18	122	20	0.052	0.079	0.027	0.36		0.131	0.173	0.043	0.35		0.209	0.252	0.043	0.36
5	17	121	26	0.048	0.122	0.074	0.50		0.121	0.223	0.102	0.55		0.194	0.322	0.128	0.56
6	18	164	27	0.044	0.071	0.027	0.61		0.110	0.138	0.028	0.63		0.177	0.205	0.028	0.65
7	15	114	25	0.044	0.070	0.025	0.46		0.112	0.137	0.025	0.45		0.180	0.204	0.024	0.49
8	16	143	26	0.047	0.076	0.030	0.48		0.117	0.144	0.027	0.53		0.187	0.215	0.028	0.54
9	17	147	26	0.047	0.065	0.018	0.55		0.118	0.135	0.017	0.58		0.188	0.205	0.017	0.57
10	18	123	24	0.045	0.078	0.033	0.47		0.112	0.159	0.046	0.48		0.180	0.234	0.054	0.46
Ave.	17.30	136.90	24.70	0.048	0.090	0.042	0.50		0.120	0.174	0.055	0.52		0.191	0.253	0.062	0.53
Case 3
1	17	147	24	0.047	0.092	0.045	0.50		0.117	0.162	0.044	0.49		0.188	0.231	0.043	0.52
2	17	109	24	0.045	0.129	0.084	0.38		0.112	0.206	0.095	0.41		0.179	0.265	0.086	0.45
3	18	181	27	0.048	0.057	0.009	0.47		0.120	0.129	0.009	0.60		0.192	0.201	0.009	0.61
4	17	141	24	0.049	0.126	0.077	0.47		0.122	0.216	0.094	0.49		0.195	0.316	0.121	0.48
5	17	103	22	0.060	0.107	0.048	0.39		0.149	0.195	0.046	0.43		0.239	0.282	0.043	0.45
6	20	189	26	0.046	0.058	0.012	0.59		0.115	0.136	0.020	0.67		0.185	0.206	0.021	0.68
7	17	123	27	0.043	0.077	0.034	0.52		0.110	0.142	0.033	0.53		0.175	0.207	0.031	0.54
8	16	109	23	0.057	0.133	0.075	0.42		0.144	0.235	0.092	0.44		0.230	0.338	0.108	0.43
9	17	145	24	0.046	0.072	0.026	0.48		0.115	0.169	0.054	0.48		0.184	0.275	0.091	0.50
10	17	164	28	0.045	0.063	0.018	0.61		0.112	0.130	0.018	0.63		0.179	0.197	0.018	0.63
Ave.	17.30	141.10	24.90	0.049	0.091	0.043	0.48		0.122	0.172	0.050	0.52		0.195	0.252	0.057	0.53
Case 4
1	17	157	27	0.038	0.084	0.046	0.58		0.094	0.161	0.067	0.59		0.151	0.228	0.077	0.59
2	19	110	26	0.052	0.065	0.013	0.53		0.130	0.143	0.012	0.57		0.209	0.220	0.012	0.58
3	16	100	24	0.055	0.076	0.020	0.39		0.139	0.168	0.029	0.46		0.222	0.250	0.028	0.51
4	17	136	26	0.046	0.057	0.011	0.57		0.115	0.139	0.024	0.58		0.184	0.213	0.030	0.56
5	19	125	23	0.051	0.073	0.022	0.49		0.130	0.173	0.044	0.52		0.207	0.267	0.059	0.53
6	16	111	25	0.048	0.065	0.018	0.42		0.120	0.137	0.017	0.44		0.191	0.208	0.017	0.45
7	18	117	25	0.047	0.056	0.009	0.51		0.117	0.126	0.009	0.55		0.188	0.197	0.010	0.55
8	17	107	26	0.052	0.090	0.038	0.53		0.130	0.168	0.037	0.55		0.209	0.245	0.036	0.55
9	17	159	26	0.048	0.053	0.005	0.44		0.120	0.126	0.006	0.50		0.192	0.198	0.006	0.52
10	18	105	24	0.050	0.095	0.045	0.45		0.125	0.169	0.044	0.48		0.200	0.242	0.042	0.46
Ave.	17.40	122.70	25.20	0.049	0.071	0.023	0.49		0.122	0.151	0.029	0.52		0.195	0.227	0.032	0.53
Case 5
1	17	155	27	0.039	0.059	0.020	0.61		0.098	0.130	0.031	0.62		0.157	0.200	0.043	0.65
2	17	135	27	0.038	0.104	0.066	0.49		0.095	0.163	0.068	0.54		0.152	0.222	0.069	0.54
3	18	103	24	0.050	0.070	0.020	0.31		0.125	0.146	0.021	0.45		0.200	0.223	0.023	0.46
4	16	168	28	0.040	0.064	0.023	0.61		0.101	0.137	0.036	0.62		0.162	0.210	0.048	0.63
5	18	139	27	0.044	0.051	0.008	0.61		0.110	0.117	0.007	0.61		0.175	0.183	0.007	0.65
6	19	171	24	0.051	0.055	0.004	0.51		0.128	0.132	0.004	0.53		0.205	0.209	0.004	0.55
7	18	147	26	0.045	0.049	0.005	0.58		0.113	0.117	0.005	0.60		0.180	0.184	0.005	0.59
8	20	141	25	0.049	0.093	0.044	0.55		0.122	0.215	0.092	0.62		0.195	0.318	0.122	0.61
9	17	145	26	0.042	0.054	0.013	0.58		0.105	0.190	0.085	0.59		0.168	0.279	0.111	0.63
10	16	111	25	0.045	0.066	0.021	0.45		0.112	0.152	0.040	0.47		0.180	0.234	0.054	0.47
Ave.	17.60	141.50	25.90	0.044	0.066	0.022	0.53		0.111	0.150	0.039	0.56		0.177	0.226	0.049	0.58
Case 6
1	17	125	19	0.054	0.065	0.011	0.31		0.136	0.147	0.011	0.32		0.217	0.228	0.011	0.31
2	17	164	28	0.043	0.080	0.037	0.58		0.108	0.170	0.061	0.62		0.173	0.248	0.074	0.63
3	18	168	27	0.041	0.065	0.024	0.62		0.103	0.153	0.049	0.67		0.166	0.240	0.075	0.65
4	17	138	24	0.053	0.083	0.030	0.47	0.133	0.162	0.029	0.49	0.212	0.240	0.028	0.48
5	18	139	25	0.044	0.064	0.020	0.56	0.110	0.156	0.046	0.57	0.175	0.214	0.038	0.56
6	18	130	25	0.044	0.055	0.011	0.52	0.110	0.122	0.013	0.56	0.175	0.188	0.013	0.53
7	16	130	24	0.054	0.090	0.037	0.49	0.134	0.202	0.068	0.47	0.215	0.314	0.100	0.50
8	17	149	26	0.047	0.069	0.022	0.56	0.119	0.155	0.036	0.57	0.191	0.241	0.050	0.62
9	17	120	27	0.051	0.106	0.055	0.50	0.129	0.186	0.057	0.57	0.207	0.265	0.058	0.58
10	18	124	26	0.046	0.084	0.038	0.53	0.116	0.147	0.031	0.56	0.185	0.240	0.055	0.58
Ave.	17.30	138.70	25.10	0.048	0.076	0.028	0.51	0.120	0.160	0.040	0.54	0.192	0.242	0.050	0.54

Table 3. Comparison with "quasi-optimal", $ n $ = 6

Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	16	137	28	0.055	0.089	0.034	0.71		0.137	0.198	0.061	0.74		0.220	0.289	0.069	0.75
2	18	133	27	0.053	0.103	0.049	0.71		0.134	0.203	0.068	0.75		0.215	0.284	0.069	0.73
3	17	105	25	0.051	0.077	0.026	0.57		0.128	0.180	0.052	0.60		0.205	0.254	0.050	0.60
4	17	115	27	0.047	0.066	0.019	0.66		0.118	0.140	0.023	0.69		0.188	0.210	0.023	0.72
5	17	148	28	0.050	0.101	0.051	0.75		0.126	0.176	0.050	0.79		0.201	0.249	0.048	0.75
6	16	105	24	0.057	0.108	0.052	0.54		0.143	0.216	0.073	0.63		0.228	0.314	0.086	0.57
7	18	115	24	0.057	0.129	0.072	0.59		0.143	0.217	0.074	0.65		0.228	0.316	0.087	0.62
8	16	122	26	0.059	0.081	0.021	0.62		0.149	0.170	0.021	0.64		0.239	0.259	0.020	0.64
9	18	167	27	0.048	0.088	0.040	0.78		0.119	0.158	0.040	0.79		0.190	0.229	0.039	0.77
10	19	117	26	0.059	0.136	0.077	0.72		0.147	0.242	0.094	0.89		0.236	0.348	0.112	0.76
Ave.	17.20	126.40	26.20	0.054	0.098	0.044	0.67		0.134	0.190	0.056	0.72		0.215	0.275	0.060	0.69
Case 2
1	21	162	24	0.064	0.105	0.041	0.67		0.159	0.214	0.056	0.74		0.254	0.314	0.060	0.72
2	21	124	24	0.062	0.110	0.048	0.69		0.155	0.214	0.059	0.72		0.248	0.305	0.057	0.69
3	18	116	24	0.065	0.189	0.124	0.60		0.163	0.322	0.159	0.63		0.261	0.429	0.169	0.64
4	22	173	27	0.056	0.068	0.012	0.76		0.140	0.152	0.012	0.84		0.224	0.236	0.012	0.81
5	16	136	26	0.048	0.097	0.049	0.61		0.119	0.170	0.051	0.64		0.191	0.241	0.050	0.61
6	22	167	26	0.051	0.072	0.021	0.84		0.127	0.160	0.033	0.86		0.204	0.253	0.049	0.85
7	18	104	25	0.068	0.119	0.051	0.48		0.170	0.220	0.050	0.57		0.272	0.321	0.050	0.60
8	18	137	25	0.058	0.098	0.041	0.62		0.144	0.201	0.058	0.65		0.230	0.302	0.072	0.64
9	19	96	27	0.056	0.113	0.057	0.71		0.139	0.227	0.088	0.82		0.222	0.322	0.100	0.74
10	19	128	28	0.050	0.070	0.020	0.84		0.125	0.159	0.034	0.89		0.200	0.246	0.046	0.88
Ave.	19.40	134.30	25.60	0.058	0.104	0.047	0.68		0.144	0.204	0.060	0.74		0.231	0.297	0.066	0.72
Case 3
1	20	147	23	0.064	0.148	0.084	0.62		0.160	0.257	0.097	0.63		0.255	0.345	0.090	0.62
2	20	136	28	0.051	0.075	0.024	0.71		0.128	0.165	0.036	0.85		0.205	0.253	0.048	0.85
3	22	141	25	0.061	0.096	0.035	0.74		0.153	0.187	0.034	0.80		0.245	0.278	0.033	0.79
4	19	172	26	0.056	0.145	0.089	0.72		0.141	0.291	0.151	0.77		0.225	0.383	0.158	0.75
5	19	138	26	0.052	0.084	0.031	0.73		0.132	0.189	0.058	0.75		0.210	0.295	0.085	0.73
6	21	160	28	0.044	0.056	0.012	0.84		0.109	0.124	0.015	0.89		0.175	0.190	0.014	0.90
7	18	105	18	0.071	0.123	0.052	0.35		0.178	0.255	0.077	0.37		0.284	0.362	0.078	0.37
8	19	143	25	0.062	0.091	0.029	0.62		0.156	0.184	0.028	0.68		0.249	0.277	0.027	0.67
9	18	140	25	0.056	0.065	0.010	0.66		0.141	0.151	0.010	0.68		0.225	0.236	0.011	0.65
10	22	136	23	0.062	0.112	0.050	0.66		0.155	0.239	0.084	0.68		0.248	0.336	0.088	0.67
Ave.	19.80	141.80	24.70	0.058	0.099	0.042	0.67		0.145	0.204	0.059	0.71		0.232	0.295	0.063	0.70
Case 4
1	21	109	24	0.056	0.088	0.032	0.68		0.140	0.187	0.047	0.73		0.224	0.284	0.060	0.70
2	18	123	27	0.053	0.091	0.038	0.72		0.132	0.169	0.037	0.76		0.211	0.247	0.036	0.80
3	21	127	25	0.062	0.077	0.015	0.66		0.156	0.170	0.014	0.71		0.249	0.263	0.014	0.74
4	20	119	25	0.051	0.078	0.027	0.69		0.130	0.177	0.048	0.74		0.208	0.271	0.063	0.71
5	19	126	27	0.051	0.067	0.016	0.74		0.129	0.147	0.018	0.82		0.207	0.224	0.017	0.80
6	21	99	25	0.052	0.131	0.078	0.62		0.131	0.207	0.075	0.67		0.210	0.282	0.072	0.68
7	19	103	24	0.059	0.102	0.043	0.59		0.148	0.206	0.059	0.66		0.236	0.312	0.076	0.65
8	20	150	25	0.058	0.072	0.014	0.74		0.146	0.183	0.036	0.80		0.234	0.286	0.052	0.77
9	21	137	27	0.052	0.056	0.004	0.80		0.131	0.135	0.004	0.89		0.210	0.214	0.004	0.85
10	22	170	28	0.045	0.075	0.030	0.91		0.112	0.141	0.029	0.93		0.180	0.208	0.028	0.93
Ave.	20.20	126.30	25.70	0.054	0.084	0.030	0.72		0.136	0.172	0.037	0.77		0.217	0.259	0.042	0.76
Case 5
1	18	121	23	0.062	0.082	0.019	0.55		0.156	0.175	0.019	0.68		0.249	0.267	0.018	0.56
2	18	186	28	0.047	0.075	0.028	0.83		0.119	0.148	0.029	0.92		0.190	0.218	0.028	0.86
3	22	137	27	0.048	0.089	0.041	0.86		0.119	0.182	0.063	0.96		0.190	0.268	0.077	0.92
4	18	142	26	0.054	0.091	0.037	0.67		0.137	0.191	0.054	0.73		0.220	0.271	0.051	0.70
5	20	138	26	0.050	0.055	0.005	0.75		0.126	0.132	0.005	0.78		0.203	0.208	0.006	0.77
6	18	106	25	0.059	0.111	0.052	0.56		0.150	0.199	0.049	0.62		0.240	0.287	0.047	0.60
7	21	135	27	0.052	0.090	0.038	0.66		0.130	0.165	0.035	0.84		0.208	0.243	0.035	0.83
8	18	140	28	0.046	0.060	0.014	0.74		0.116	0.130	0.014	0.79		0.186	0.199	0.013	0.82
9	19	117	26	0.054	0.081	0.026	0.71		0.136	0.177	0.041	0.72		0.218	0.272	0.055	0.74
10	18	119	27	0.056	0.074	0.018	0.76		0.141	0.174	0.033	0.78		0.226	0.274	0.048	0.81
Ave.	19.00	134.10	26.30	0.053	0.081	0.028	0.71		0.133	0.167	0.034	0.78		0.213	0.251	0.038	0.76
Case 6
1	20	119	25	0.055	0.084	0.029	0.62		0.138	0.185	0.046	0.67		0.220	0.281	0.061	0.69
2	21	162	25	0.049	0.059	0.010	0.71		0.124	0.147	0.023	0.73		0.197	0.221	0.023	0.72
3	20	127	26	0.051	0.071	0.020	0.72		0.128	0.176	0.048	0.76		0.205	0.265	0.060	0.78
4	19	162	28	0.056	0.074	0.019	0.68	0.140	0.159	0.019	0.81	0.224	0.242	0.019	0.82
5	17	116	24	0.056	0.078	0.022	0.52	0.141	0.162	0.021	0.55	0.226	0.246	0.021	0.59
6	18	138	26	0.062	0.166	0.105	0.68	0.154	0.285	0.132	0.70	0.246	0.392	0.146	0.70
7	20	165	28	0.046	0.081	0.035	0.91	0.115	0.186	0.071	0.89	0.184	0.297	0.114	0.89
8	19	117	26	0.049	0.062	0.013	0.66	0.122	0.147	0.026	0.69	0.195	0.226	0.030	0.71
9	18	151	28	0.044	0.087	0.043	0.68	0.111	0.164	0.053	0.75	0.177	0.240	0.063	0.77
10	20	123	26	0.061	0.130	0.069	0.72	0.154	0.253	0.099	0.74	0.246	0.360	0.114	0.81
Ave.	19.20	138.00	26.20	0.053	0.089	0.036	0.69	0.133	0.186	0.054	0.73	0.212	0.277	0.065	0.75

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Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	16	137	28	0.055	0.089	0.034	0.71		0.137	0.198	0.061	0.74		0.220	0.289	0.069	0.75
2	18	133	27	0.053	0.103	0.049	0.71		0.134	0.203	0.068	0.75		0.215	0.284	0.069	0.73
3	17	105	25	0.051	0.077	0.026	0.57		0.128	0.180	0.052	0.60		0.205	0.254	0.050	0.60
4	17	115	27	0.047	0.066	0.019	0.66		0.118	0.140	0.023	0.69		0.188	0.210	0.023	0.72
5	17	148	28	0.050	0.101	0.051	0.75		0.126	0.176	0.050	0.79		0.201	0.249	0.048	0.75
6	16	105	24	0.057	0.108	0.052	0.54		0.143	0.216	0.073	0.63		0.228	0.314	0.086	0.57
7	18	115	24	0.057	0.129	0.072	0.59		0.143	0.217	0.074	0.65		0.228	0.316	0.087	0.62
8	16	122	26	0.059	0.081	0.021	0.62		0.149	0.170	0.021	0.64		0.239	0.259	0.020	0.64
9	18	167	27	0.048	0.088	0.040	0.78		0.119	0.158	0.040	0.79		0.190	0.229	0.039	0.77
10	19	117	26	0.059	0.136	0.077	0.72		0.147	0.242	0.094	0.89		0.236	0.348	0.112	0.76
Ave.	17.20	126.40	26.20	0.054	0.098	0.044	0.67		0.134	0.190	0.056	0.72		0.215	0.275	0.060	0.69
Case 2
1	21	162	24	0.064	0.105	0.041	0.67		0.159	0.214	0.056	0.74		0.254	0.314	0.060	0.72
2	21	124	24	0.062	0.110	0.048	0.69		0.155	0.214	0.059	0.72		0.248	0.305	0.057	0.69
3	18	116	24	0.065	0.189	0.124	0.60		0.163	0.322	0.159	0.63		0.261	0.429	0.169	0.64
4	22	173	27	0.056	0.068	0.012	0.76		0.140	0.152	0.012	0.84		0.224	0.236	0.012	0.81
5	16	136	26	0.048	0.097	0.049	0.61		0.119	0.170	0.051	0.64		0.191	0.241	0.050	0.61
6	22	167	26	0.051	0.072	0.021	0.84		0.127	0.160	0.033	0.86		0.204	0.253	0.049	0.85
7	18	104	25	0.068	0.119	0.051	0.48		0.170	0.220	0.050	0.57		0.272	0.321	0.050	0.60
8	18	137	25	0.058	0.098	0.041	0.62		0.144	0.201	0.058	0.65		0.230	0.302	0.072	0.64
9	19	96	27	0.056	0.113	0.057	0.71		0.139	0.227	0.088	0.82		0.222	0.322	0.100	0.74
10	19	128	28	0.050	0.070	0.020	0.84		0.125	0.159	0.034	0.89		0.200	0.246	0.046	0.88
Ave.	19.40	134.30	25.60	0.058	0.104	0.047	0.68		0.144	0.204	0.060	0.74		0.231	0.297	0.066	0.72
Case 3
1	20	147	23	0.064	0.148	0.084	0.62		0.160	0.257	0.097	0.63		0.255	0.345	0.090	0.62
2	20	136	28	0.051	0.075	0.024	0.71		0.128	0.165	0.036	0.85		0.205	0.253	0.048	0.85
3	22	141	25	0.061	0.096	0.035	0.74		0.153	0.187	0.034	0.80		0.245	0.278	0.033	0.79
4	19	172	26	0.056	0.145	0.089	0.72		0.141	0.291	0.151	0.77		0.225	0.383	0.158	0.75
5	19	138	26	0.052	0.084	0.031	0.73		0.132	0.189	0.058	0.75		0.210	0.295	0.085	0.73
6	21	160	28	0.044	0.056	0.012	0.84		0.109	0.124	0.015	0.89		0.175	0.190	0.014	0.90
7	18	105	18	0.071	0.123	0.052	0.35		0.178	0.255	0.077	0.37		0.284	0.362	0.078	0.37
8	19	143	25	0.062	0.091	0.029	0.62		0.156	0.184	0.028	0.68		0.249	0.277	0.027	0.67
9	18	140	25	0.056	0.065	0.010	0.66		0.141	0.151	0.010	0.68		0.225	0.236	0.011	0.65
10	22	136	23	0.062	0.112	0.050	0.66		0.155	0.239	0.084	0.68		0.248	0.336	0.088	0.67
Ave.	19.80	141.80	24.70	0.058	0.099	0.042	0.67		0.145	0.204	0.059	0.71		0.232	0.295	0.063	0.70
Case 4
1	21	109	24	0.056	0.088	0.032	0.68		0.140	0.187	0.047	0.73		0.224	0.284	0.060	0.70
2	18	123	27	0.053	0.091	0.038	0.72		0.132	0.169	0.037	0.76		0.211	0.247	0.036	0.80
3	21	127	25	0.062	0.077	0.015	0.66		0.156	0.170	0.014	0.71		0.249	0.263	0.014	0.74
4	20	119	25	0.051	0.078	0.027	0.69		0.130	0.177	0.048	0.74		0.208	0.271	0.063	0.71
5	19	126	27	0.051	0.067	0.016	0.74		0.129	0.147	0.018	0.82		0.207	0.224	0.017	0.80
6	21	99	25	0.052	0.131	0.078	0.62		0.131	0.207	0.075	0.67		0.210	0.282	0.072	0.68
7	19	103	24	0.059	0.102	0.043	0.59		0.148	0.206	0.059	0.66		0.236	0.312	0.076	0.65
8	20	150	25	0.058	0.072	0.014	0.74		0.146	0.183	0.036	0.80		0.234	0.286	0.052	0.77
9	21	137	27	0.052	0.056	0.004	0.80		0.131	0.135	0.004	0.89		0.210	0.214	0.004	0.85
10	22	170	28	0.045	0.075	0.030	0.91		0.112	0.141	0.029	0.93		0.180	0.208	0.028	0.93
Ave.	20.20	126.30	25.70	0.054	0.084	0.030	0.72		0.136	0.172	0.037	0.77		0.217	0.259	0.042	0.76
Case 5
1	18	121	23	0.062	0.082	0.019	0.55		0.156	0.175	0.019	0.68		0.249	0.267	0.018	0.56
2	18	186	28	0.047	0.075	0.028	0.83		0.119	0.148	0.029	0.92		0.190	0.218	0.028	0.86
3	22	137	27	0.048	0.089	0.041	0.86		0.119	0.182	0.063	0.96		0.190	0.268	0.077	0.92
4	18	142	26	0.054	0.091	0.037	0.67		0.137	0.191	0.054	0.73		0.220	0.271	0.051	0.70
5	20	138	26	0.050	0.055	0.005	0.75		0.126	0.132	0.005	0.78		0.203	0.208	0.006	0.77
6	18	106	25	0.059	0.111	0.052	0.56		0.150	0.199	0.049	0.62		0.240	0.287	0.047	0.60
7	21	135	27	0.052	0.090	0.038	0.66		0.130	0.165	0.035	0.84		0.208	0.243	0.035	0.83
8	18	140	28	0.046	0.060	0.014	0.74		0.116	0.130	0.014	0.79		0.186	0.199	0.013	0.82
9	19	117	26	0.054	0.081	0.026	0.71		0.136	0.177	0.041	0.72		0.218	0.272	0.055	0.74
10	18	119	27	0.056	0.074	0.018	0.76		0.141	0.174	0.033	0.78		0.226	0.274	0.048	0.81
Ave.	19.00	134.10	26.30	0.053	0.081	0.028	0.71		0.133	0.167	0.034	0.78		0.213	0.251	0.038	0.76
Case 6
1	20	119	25	0.055	0.084	0.029	0.62		0.138	0.185	0.046	0.67		0.220	0.281	0.061	0.69
2	21	162	25	0.049	0.059	0.010	0.71		0.124	0.147	0.023	0.73		0.197	0.221	0.023	0.72
3	20	127	26	0.051	0.071	0.020	0.72		0.128	0.176	0.048	0.76		0.205	0.265	0.060	0.78
4	19	162	28	0.056	0.074	0.019	0.68	0.140	0.159	0.019	0.81	0.224	0.242	0.019	0.82
5	17	116	24	0.056	0.078	0.022	0.52	0.141	0.162	0.021	0.55	0.226	0.246	0.021	0.59
6	18	138	26	0.062	0.166	0.105	0.68	0.154	0.285	0.132	0.70	0.246	0.392	0.146	0.70
7	20	165	28	0.046	0.081	0.035	0.91	0.115	0.186	0.071	0.89	0.184	0.297	0.114	0.89
8	19	117	26	0.049	0.062	0.013	0.66	0.122	0.147	0.026	0.69	0.195	0.226	0.030	0.71
9	18	151	28	0.044	0.087	0.043	0.68	0.111	0.164	0.053	0.75	0.177	0.240	0.063	0.77
10	20	123	26	0.061	0.130	0.069	0.72	0.154	0.253	0.099	0.74	0.246	0.360	0.114	0.81
Ave.	19.20	138.00	26.20	0.053	0.089	0.036	0.69	0.133	0.186	0.054	0.73	0.212	0.277	0.065	0.75

Table 4. Comparison with "quasi-optimal", $ n $ = 7

Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	22	127	27	0.054	0.123	0.069	1.04		0.135	0.215	0.080	1.09		0.216	0.296	0.080	1.01
2	20	153	29	0.050	0.092	0.042	1.05		0.125	0.177	0.052	1.06		0.200	0.262	0.063	1.07
3	20	146	29	0.057	0.095	0.038	1.18		0.142	0.205	0.063	1.20		0.227	0.315	0.088	1.22
4	17	144	29	0.061	0.111	0.050	0.92		0.155	0.203	0.048	0.96		0.248	0.294	0.046	0.97
5	19	111	29	0.048	0.058	0.010	0.94		0.121	0.131	0.011	1.17		0.193	0.205	0.011	1.03
6	17	97	27	0.060	0.112	0.052	0.82		0.149	0.211	0.062	0.82		0.239	0.313	0.074	0.80
7	20	105	25	0.068	0.126	0.058	0.67		0.172	0.231	0.059	0.79		0.275	0.333	0.057	0.77
8	18	143	28	0.060	0.118	0.057	0.91		0.150	0.207	0.057	0.97		0.241	0.296	0.055	0.96
9	20	115	26	0.057	0.095	0.038	0.90		0.097	0.161	0.064	0.92		0.230	0.324	0.094	0.92
10	20	136	28	0.053	0.078	0.025	1.03		0.134	0.196	0.062	1.08		0.214	0.289	0.075	1.09
Ave.	19.30	127.70	27.70	0.057	0.101	0.044	0.95		0.138	0.194	0.056	1.01		0.228	0.293	0.064	0.98
Case 2
1	17	110	27	0.056	0.081	0.025	0.82		0.140	0.179	0.039	0.87		0.224	0.263	0.038	0.91
2	24	153	27	0.063	0.120	0.057	1.15		0.157	0.232	0.075	1.18		0.250	0.339	0.088	1.28
3	21	110	26	0.069	0.098	0.029	0.93		0.172	0.203	0.030	0.97		0.275	0.304	0.029	0.99
4	22	115	25	0.074	0.117	0.043	0.88		0.184	0.225	0.041	0.94		0.296	0.335	0.039	1.01
5	12	92	29	0.060	0.066	0.005	0.47		0.152	0.137	-0.015	0.47		0.243	0.201	-0.042	0.48
6	21	135	24	0.061	0.100	0.038	0.77		0.155	0.192	0.037	0.79		0.247	0.282	0.036	0.80
7	19	128	28	0.058	0.081	0.023	0.99		0.145	0.173	0.027	1.00		0.233	0.260	0.026	1.00
8	21	114	27	0.062	0.100	0.037	0.91		0.158	0.208	0.051	1.02		0.252	0.302	0.050	0.98
9	23	101	23	0.065	0.092	0.027	0.72		0.163	0.190	0.027	0.78		0.260	0.287	0.027	0.77
10	20	151	24	0.068	0.119	0.050	0.80		0.172	0.225	0.053	0.81		0.274	0.324	0.050	0.78
Ave.	20.00	120.90	26.00	0.064	0.097	0.034	0.84		0.160	0.196	0.037	0.88		0.256	0.290	0.034	0.90
Case 3
1	21	152	28	0.057	0.079	0.023	1.13		0.143	0.178	0.035	1.13		0.229	0.264	0.035	1.15
2	21	149	27	0.054	0.096	0.042	1.08		0.135	0.184	0.049	1.11		0.216	0.281	0.065	1.05
3	16	117	26	0.060	0.118	0.057	0.72		0.152	0.266	0.113	0.74		0.243	0.369	0.126	0.74
4	21	145	24	0.065	0.108	0.043	0.86		0.162	0.207	0.046	0.84		0.258	0.301	0.043	0.88
5	15	114	28	0.068	0.125	0.057	0.80		0.171	0.239	0.069	0.85		0.273	0.342	0.069	0.84
6	16	158	25	0.057	0.063	0.006	0.73		0.143	0.150	0.007	0.74		0.229	0.236	0.007	0.74
7	19	128	25	0.069	0.150	0.081	0.76		0.174	0.252	0.078	0.79		0.277	0.352	0.075	0.77
8	18	160	26	0.058	0.074	0.016	0.85		0.144	0.160	0.016	0.86		0.230	0.246	0.015	0.84
9	20	134	26	0.064	0.119	0.055	0.89		0.162	0.215	0.053	0.89		0.259	0.310	0.051	0.88
10	21	147	27	0.055	0.083	0.028	0.99		0.139	0.167	0.028	1.01		0.222	0.252	0.029	0.98
Ave.	18.80	140.40	26.20	0.061	0.101	0.041	0.88		0.152	0.202	0.049	0.90		0.244	0.295	0.051	0.89
Case 4
1	23	99	24	0.065	0.097	0.031	0.76		0.163	0.192	0.029	0.85		0.262	0.290	0.028	0.86
2	22	130	28	0.056	0.093	0.037	1.16		0.141	0.179	0.039	1.17		0.225	0.262	0.037	1.18
3	18	115	27	0.061	0.097	0.036	0.82		0.151	0.187	0.036	0.84		0.242	0.278	0.035	0.84
4	24	130	27	0.068	0.111	0.043	1.13		0.171	0.216	0.045	1.19		0.273	0.316	0.043	1.21
5	23	111	25	0.058	0.077	0.019	0.85		0.147	0.166	0.018	0.93		0.235	0.253	0.018	0.95
6	20	128	26	0.056	0.084	0.028	0.88		0.141	0.169	0.028	0.94		0.225	0.252	0.027	0.93
7	20	104	25	0.067	0.099	0.032	0.84		0.167	0.199	0.032	0.85		0.268	0.300	0.032	0.87
8	22	123	24	0.067	0.104	0.038	0.84		0.167	0.220	0.053	0.92		0.267	0.334	0.066	0.87
9	25	133	26	0.064	0.092	0.028	1.09		0.159	0.207	0.047	1.14		0.254	0.301	0.047	1.11
10	19	107	25	0.067	0.087	0.021	0.80		0.167	0.187	0.020	0.91		0.267	0.286	0.019	0.83
Ave.	21.60	118.00	25.70	0.063	0.094	0.031	0.92		0.157	0.192	0.035	0.97		0.252	0.287	0.035	0.96
Case 5
1	23	107	27	0.067	0.151	0.084	0.91		0.168	0.275	0.106	1.04		0.268	0.389	0.121	1.04
2	21	151	28	0.054	0.082	0.028	1.05		0.137	0.180	0.043	1.17		0.219	0.274	0.055	1.11
3	22	136	26	0.058	0.081	0.023	0.97		0.147	0.183	0.037	1.01		0.235	0.285	0.050	1.04
4	18	138	27	0.060	0.092	0.032	0.90		0.151	0.182	0.031	0.92		0.241	0.264	0.023	0.90
5	21	89	26	0.064	0.102	0.038	0.78		0.160	0.196	0.036	0.89		0.257	0.290	0.033	0.92
6	20	123	25	0.068	0.117	0.048	0.84		0.172	0.234	0.061	0.87		0.276	0.352	0.076	0.86
7	19	127	24	0.059	0.073	0.015	0.76		0.148	0.163	0.016	0.77		0.236	0.251	0.015	0.75
8	21	107	25	0.067	0.076	0.010	0.69		0.169	0.179	0.010	0.81		0.270	0.281	0.011	0.79
9	18	137	26	0.068	0.084	0.015	0.78		0.172	0.187	0.015	0.88		0.275	0.291	0.016	0.86
10	24	135	23	0.076	0.096	0.020	0.82		0.191	0.210	0.019	0.87		0.306	0.325	0.019	0.83
Ave.	20.70	125.00	25.70	0.064	0.095	0.031	0.85		0.161	0.199	0.038	0.92		0.258	0.300	0.042	0.91
Case 6
1	21	181	28	0.053	0.081	0.028	1.10		0.133	0.180	0.046	1.11		0.213	0.273	0.059	1.20
2	17	119	25	0.068	0.094	0.025	0.69		0.171	0.196	0.025	0.73		0.274	0.298	0.024	0.79
3	16	138	25	0.069	0.102	0.033	0.67		0.173	0.230	0.056	0.74		0.278	0.337	0.059	0.82
4	24	154	27	0.060	0.088	0.027	1.10	0.152	0.183	0.031	1.13	0.243	0.272	0.030	1.29
5	18	101	25	0.065	0.097	0.032	0.69	0.164	0.199	0.035	0.76	0.263	0.301	0.038	0.83
6	19	155	27	0.053	0.079	0.026	0.87	0.133	0.159	0.025	0.95	0.213	0.238	0.025	0.93
7	21	134	27	0.057	0.076	0.019	1.02	0.142	0.175	0.032	1.09	0.228	0.274	0.046	1.20
8	23	141	26	0.064	0.124	0.060	1.05	0.161	0.218	0.057	1.02	0.257	0.311	0.054	0.98
9	14	126	24	0.079	0.154	0.075	0.57	0.198	0.255	0.057	0.59	0.317	0.371	0.054	0.62
10	20	153	28	0.053	0.085	0.032	1.03	0.132	0.186	0.053	1.05	0.211	0.276	0.065	1.11
Ave.	19.30	140.20	26.20	0.062	0.098	0.036	0.88	0.156	0.198	0.042	0.92	0.250	0.295	0.045	0.98

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Ex.	$ m $	#Operation	#OR blocks	Scenario 1					Scenario 2					Scenario 3
Ex.	$ m $	#Operation	#OR blocks	$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time		$ rc_1 $	$ rc_2 $	Gap	Time
Case 1
1	22	127	27	0.054	0.123	0.069	1.04		0.135	0.215	0.080	1.09		0.216	0.296	0.080	1.01
2	20	153	29	0.050	0.092	0.042	1.05		0.125	0.177	0.052	1.06		0.200	0.262	0.063	1.07
3	20	146	29	0.057	0.095	0.038	1.18		0.142	0.205	0.063	1.20		0.227	0.315	0.088	1.22
4	17	144	29	0.061	0.111	0.050	0.92		0.155	0.203	0.048	0.96		0.248	0.294	0.046	0.97
5	19	111	29	0.048	0.058	0.010	0.94		0.121	0.131	0.011	1.17		0.193	0.205	0.011	1.03
6	17	97	27	0.060	0.112	0.052	0.82		0.149	0.211	0.062	0.82		0.239	0.313	0.074	0.80
7	20	105	25	0.068	0.126	0.058	0.67		0.172	0.231	0.059	0.79		0.275	0.333	0.057	0.77
8	18	143	28	0.060	0.118	0.057	0.91		0.150	0.207	0.057	0.97		0.241	0.296	0.055	0.96
9	20	115	26	0.057	0.095	0.038	0.90		0.097	0.161	0.064	0.92		0.230	0.324	0.094	0.92
10	20	136	28	0.053	0.078	0.025	1.03		0.134	0.196	0.062	1.08		0.214	0.289	0.075	1.09
Ave.	19.30	127.70	27.70	0.057	0.101	0.044	0.95		0.138	0.194	0.056	1.01		0.228	0.293	0.064	0.98
Case 2
1	17	110	27	0.056	0.081	0.025	0.82		0.140	0.179	0.039	0.87		0.224	0.263	0.038	0.91
2	24	153	27	0.063	0.120	0.057	1.15		0.157	0.232	0.075	1.18		0.250	0.339	0.088	1.28
3	21	110	26	0.069	0.098	0.029	0.93		0.172	0.203	0.030	0.97		0.275	0.304	0.029	0.99
4	22	115	25	0.074	0.117	0.043	0.88		0.184	0.225	0.041	0.94		0.296	0.335	0.039	1.01
5	12	92	29	0.060	0.066	0.005	0.47		0.152	0.137	-0.015	0.47		0.243	0.201	-0.042	0.48
6	21	135	24	0.061	0.100	0.038	0.77		0.155	0.192	0.037	0.79		0.247	0.282	0.036	0.80
7	19	128	28	0.058	0.081	0.023	0.99		0.145	0.173	0.027	1.00		0.233	0.260	0.026	1.00
8	21	114	27	0.062	0.100	0.037	0.91		0.158	0.208	0.051	1.02		0.252	0.302	0.050	0.98
9	23	101	23	0.065	0.092	0.027	0.72		0.163	0.190	0.027	0.78		0.260	0.287	0.027	0.77
10	20	151	24	0.068	0.119	0.050	0.80		0.172	0.225	0.053	0.81		0.274	0.324	0.050	0.78
Ave.	20.00	120.90	26.00	0.064	0.097	0.034	0.84		0.160	0.196	0.037	0.88		0.256	0.290	0.034	0.90
Case 3
1	21	152	28	0.057	0.079	0.023	1.13		0.143	0.178	0.035	1.13		0.229	0.264	0.035	1.15
2	21	149	27	0.054	0.096	0.042	1.08		0.135	0.184	0.049	1.11		0.216	0.281	0.065	1.05
3	16	117	26	0.060	0.118	0.057	0.72		0.152	0.266	0.113	0.74		0.243	0.369	0.126	0.74
4	21	145	24	0.065	0.108	0.043	0.86		0.162	0.207	0.046	0.84		0.258	0.301	0.043	0.88
5	15	114	28	0.068	0.125	0.057	0.80		0.171	0.239	0.069	0.85		0.273	0.342	0.069	0.84
6	16	158	25	0.057	0.063	0.006	0.73		0.143	0.150	0.007	0.74		0.229	0.236	0.007	0.74
7	19	128	25	0.069	0.150	0.081	0.76		0.174	0.252	0.078	0.79		0.277	0.352	0.075	0.77
8	18	160	26	0.058	0.074	0.016	0.85		0.144	0.160	0.016	0.86		0.230	0.246	0.015	0.84
9	20	134	26	0.064	0.119	0.055	0.89		0.162	0.215	0.053	0.89		0.259	0.310	0.051	0.88
10	21	147	27	0.055	0.083	0.028	0.99		0.139	0.167	0.028	1.01		0.222	0.252	0.029	0.98
Ave.	18.80	140.40	26.20	0.061	0.101	0.041	0.88		0.152	0.202	0.049	0.90		0.244	0.295	0.051	0.89
Case 4
1	23	99	24	0.065	0.097	0.031	0.76		0.163	0.192	0.029	0.85		0.262	0.290	0.028	0.86
2	22	130	28	0.056	0.093	0.037	1.16		0.141	0.179	0.039	1.17		0.225	0.262	0.037	1.18
3	18	115	27	0.061	0.097	0.036	0.82		0.151	0.187	0.036	0.84		0.242	0.278	0.035	0.84
4	24	130	27	0.068	0.111	0.043	1.13		0.171	0.216	0.045	1.19		0.273	0.316	0.043	1.21
5	23	111	25	0.058	0.077	0.019	0.85		0.147	0.166	0.018	0.93		0.235	0.253	0.018	0.95
6	20	128	26	0.056	0.084	0.028	0.88		0.141	0.169	0.028	0.94		0.225	0.252	0.027	0.93
7	20	104	25	0.067	0.099	0.032	0.84		0.167	0.199	0.032	0.85		0.268	0.300	0.032	0.87
8	22	123	24	0.067	0.104	0.038	0.84		0.167	0.220	0.053	0.92		0.267	0.334	0.066	0.87
9	25	133	26	0.064	0.092	0.028	1.09		0.159	0.207	0.047	1.14		0.254	0.301	0.047	1.11
10	19	107	25	0.067	0.087	0.021	0.80		0.167	0.187	0.020	0.91		0.267	0.286	0.019	0.83
Ave.	21.60	118.00	25.70	0.063	0.094	0.031	0.92		0.157	0.192	0.035	0.97		0.252	0.287	0.035	0.96
Case 5
1	23	107	27	0.067	0.151	0.084	0.91		0.168	0.275	0.106	1.04		0.268	0.389	0.121	1.04
2	21	151	28	0.054	0.082	0.028	1.05		0.137	0.180	0.043	1.17		0.219	0.274	0.055	1.11
3	22	136	26	0.058	0.081	0.023	0.97		0.147	0.183	0.037	1.01		0.235	0.285	0.050	1.04
4	18	138	27	0.060	0.092	0.032	0.90		0.151	0.182	0.031	0.92		0.241	0.264	0.023	0.90
5	21	89	26	0.064	0.102	0.038	0.78		0.160	0.196	0.036	0.89		0.257	0.290	0.033	0.92
6	20	123	25	0.068	0.117	0.048	0.84		0.172	0.234	0.061	0.87		0.276	0.352	0.076	0.86
7	19	127	24	0.059	0.073	0.015	0.76		0.148	0.163	0.016	0.77		0.236	0.251	0.015	0.75
8	21	107	25	0.067	0.076	0.010	0.69		0.169	0.179	0.010	0.81		0.270	0.281	0.011	0.79
9	18	137	26	0.068	0.084	0.015	0.78		0.172	0.187	0.015	0.88		0.275	0.291	0.016	0.86
10	24	135	23	0.076	0.096	0.020	0.82		0.191	0.210	0.019	0.87		0.306	0.325	0.019	0.83
Ave.	20.70	125.00	25.70	0.064	0.095	0.031	0.85		0.161	0.199	0.038	0.92		0.258	0.300	0.042	0.91
Case 6
1	21	181	28	0.053	0.081	0.028	1.10		0.133	0.180	0.046	1.11		0.213	0.273	0.059	1.20
2	17	119	25	0.068	0.094	0.025	0.69		0.171	0.196	0.025	0.73		0.274	0.298	0.024	0.79
3	16	138	25	0.069	0.102	0.033	0.67		0.173	0.230	0.056	0.74		0.278	0.337	0.059	0.82
4	24	154	27	0.060	0.088	0.027	1.10	0.152	0.183	0.031	1.13	0.243	0.272	0.030	1.29
5	18	101	25	0.065	0.097	0.032	0.69	0.164	0.199	0.035	0.76	0.263	0.301	0.038	0.83
6	19	155	27	0.053	0.079	0.026	0.87	0.133	0.159	0.025	0.95	0.213	0.238	0.025	0.93
7	21	134	27	0.057	0.076	0.019	1.02	0.142	0.175	0.032	1.09	0.228	0.274	0.046	1.20
8	23	141	26	0.064	0.124	0.060	1.05	0.161	0.218	0.057	1.02	0.257	0.311	0.054	0.98
9	14	126	24	0.079	0.154	0.075	0.57	0.198	0.255	0.057	0.59	0.317	0.371	0.054	0.62
10	20	153	28	0.053	0.085	0.032	1.03	0.132	0.186	0.053	1.05	0.211	0.276	0.065	1.11
Ave.	19.30	140.20	26.20	0.062	0.098	0.036	0.88	0.156	0.198	0.042	0.92	0.250	0.295	0.045	0.98

Table 5. Comparison with lower bound

Ex.		Scenario 1				Scenario 2				Scenario 3
Ex.		$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $	$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $	$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $
n=5	case1	0.079	0.039	0.040	145.800	0.163	0.096	0.067	132.000	0.241	0.128	0.113	152.610
	case2	0.090	0.042	0.048	116.200	0.174	0.102	0.072	116.300	0.253	0.153	0.100	132.560
	case3	0.091	0.040	0.051	118.200	0.172	0.097	0.075	120.200	0.252	0.155	0.097	118.800
	case4	0.071	0.031	0.040	110.200	0.151	0.072	0.079	116.400	0.227	0.107	0.120	119.600
	case5	0.066	0.038	0.028	94.000	0.150	0.083	0.067	103.800	0.226	0.118	0.108	111.100
	case6	0.076	0.037	0.039	110.600	0.160	0.091	0.069	97.000	0.242	0.130	0.112	119.000
n=6	case1	0.098	0.040	0.058	120.400	0.190	0.089	0.101	132.500	0.275	0.150	0.125	126.800
	case2	0.104	0.054	0.050	127.400	0.204	0.123	0.081	124.100	0.297	0.194	0.103	124.800
	case3	0.099	0.057	0.042	118.500	0.204	0.131	0.073	127.300	0.295	0.195	0.100	120.800
	case4	0.084	0.046	0.038	106.900	0.172	0.089	0.083	119.300	0.259	0.150	0.109	115.700
	case5	0.081	0.036	0.045	120.900	0.167	0.084	0.083	128.000	0.251	0.126	0.125	132.700
	case6	0.089	0.035	0.054	140.100	0.186	0.070	0.116	155.900	0.277	0.105	0.172	157.700
n=7	case1	0.101	0.051	0.050	126.700	0.194	0.111	0.083	145.100	0.293	0.176	0.117	135.900
	case2	0.097	0.051	0.046	109.000	0.196	0.116	0.080	98.400	0.290	0.174	0.116	114.300
	case3	0.101	0.058	0.043	98.000	0.202	0.145	0.057	101.800	0.295	0.218	0.077	103.700
	case4	0.094	0.044	0.050	133.900	0.192	0.096	0.096	138.200	0.287	0.138	0.149	143.200
	case5	0.095	0.056	0.039	100.800	0.199	0.124	0.075	120.800	0.300	0.199	0.101	109.700
	case6	0.098	0.044	0.054	111.500	0.198	0.093	0.105	112.800	0.295	0.162	0.133	117.900
	Average	0.090	0.044	0.045	117.172	0.182	0.101	0.081	121.661	0.270	0.154	0.115	125.382

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Ex.		Scenario 1				Scenario 2				Scenario 3
Ex.		$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $	$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $	$ rc_2 $	$ rc_3 $	$ Gap_1 $	$ Time_1 $
n=5	case1	0.079	0.039	0.040	145.800	0.163	0.096	0.067	132.000	0.241	0.128	0.113	152.610
	case2	0.090	0.042	0.048	116.200	0.174	0.102	0.072	116.300	0.253	0.153	0.100	132.560
	case3	0.091	0.040	0.051	118.200	0.172	0.097	0.075	120.200	0.252	0.155	0.097	118.800
	case4	0.071	0.031	0.040	110.200	0.151	0.072	0.079	116.400	0.227	0.107	0.120	119.600
	case5	0.066	0.038	0.028	94.000	0.150	0.083	0.067	103.800	0.226	0.118	0.108	111.100
	case6	0.076	0.037	0.039	110.600	0.160	0.091	0.069	97.000	0.242	0.130	0.112	119.000
n=6	case1	0.098	0.040	0.058	120.400	0.190	0.089	0.101	132.500	0.275	0.150	0.125	126.800
	case2	0.104	0.054	0.050	127.400	0.204	0.123	0.081	124.100	0.297	0.194	0.103	124.800
	case3	0.099	0.057	0.042	118.500	0.204	0.131	0.073	127.300	0.295	0.195	0.100	120.800
	case4	0.084	0.046	0.038	106.900	0.172	0.089	0.083	119.300	0.259	0.150	0.109	115.700
	case5	0.081	0.036	0.045	120.900	0.167	0.084	0.083	128.000	0.251	0.126	0.125	132.700
	case6	0.089	0.035	0.054	140.100	0.186	0.070	0.116	155.900	0.277	0.105	0.172	157.700
n=7	case1	0.101	0.051	0.050	126.700	0.194	0.111	0.083	145.100	0.293	0.176	0.117	135.900
	case2	0.097	0.051	0.046	109.000	0.196	0.116	0.080	98.400	0.290	0.174	0.116	114.300
	case3	0.101	0.058	0.043	98.000	0.202	0.145	0.057	101.800	0.295	0.218	0.077	103.700
	case4	0.094	0.044	0.050	133.900	0.192	0.096	0.096	138.200	0.287	0.138	0.149	143.200
	case5	0.095	0.056	0.039	100.800	0.199	0.124	0.075	120.800	0.300	0.199	0.101	109.700
	case6	0.098	0.044	0.054	111.500	0.198	0.093	0.105	112.800	0.295	0.162	0.133	117.900
	Average	0.090	0.044	0.045	117.172	0.182	0.101	0.081	121.661	0.270	0.154	0.115	125.382

Table 6. By performance indicators, $ n = 5 $

Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	65	132	1.01		16	250	1.03		16	250	1.03
2	0	71	1.01		0	71	1.01		0	71	1.01
3	0	121	1.01		0	121	1.01		0	121	1.01
4	62	107	1.00		165	216	1.00		71	273	1.02
5	15	0	1.00		15	0	1.00		15	0	1.00
6	60	12	1.00		60	12	1.00		60	12	1.00
7	89	71	1.00		89	71	1.00		207	181	1.00
8	9	64	1.01		9	64	1.01		9	64	1.01
9	67	40	1.00		49	118	1.01		49	118	1.01
10	16	147	1.02		96	279	1.02		96	279	1.02
Ave.	38.30	76.50	1.00		49.90	120.20	1.01		52.30	136.90	1.01
Case 2
1	44	21	1.00		124	18	0.99		124	18	0.99
2	112	108	1.00		112	108	1.00		112	108	1.00
3	25	422	1.04		0	645	1.07		0	645	1.07
4	107	8	0.99		108	1	0.99		89	130	1.01
5	42	182	1.02		43	185	1.02		42	182	1.02
6	147	58	0.99		147	58	0.99		147	58	0.99
7	25	93	1.01		25	93	1.01		25	93	1.01
8	24	110	1.01		18	105	1.01		24	110	1.01
9	15	84	1.01		15	84	1.01		15	84	1.01
10	106	50	0.99		148	21	0.99		194	132	0.99
Ave.	64.70	113.60	1.01		74.00	131.80	1.01		77.20	156.00	1.01
Case 3
1	33	60	1.00		26	196	1.02		26	196	1.02
2	0	259	1.04		0	261	1.04		0	320	1.05
3	13	37	1.00		13	37	1.00		13	37	1.00
4	190	56	0.98		14	203	1.02		14	203	1.02
5	0	191	1.03		0	191	1.03		0	191	1.03
6	26	0	1.00		104	55	1.00		104	55	1.00
7	0	169	1.02		0	169	1.02		0	169	1.02
8	89	164	1.01		89	164	1.01		89	164	1.01
9	64	0	0.99		64	0	0.99		49	55	1.00
10	52	77	1.00		52	77	1.00		52	77	1.00
Ave.	46.70	101.30	1.01		36.20	135.30	1.01		34.70	146.70	1.01
Case 4
1	180	48	0.99		194	120	0.99		413	120	0.97
2	3	51	1.01		3	51	1.01		3	51	1.01
3	35	16	1.00		46	89	1.01		46	89	1.01
4	7	4	1.00		7	4	1.00		42	138	1.01
5	38	27	1.00		53	43	1.00		53	43	1.00
6	12	56	1.01		12	56	1.01		12	56	1.01
7	71	0	0.99		71	0	0.99		71	0	0.99
8	26	146	1.02		26	146	1.02		26	146	1.02
9	52	0	1.00		52	0	1.00		52	0	1.00
10	2	144	1.02		2	184	1.02		2	184	1.02
Ave.	42.60	49.20	1.00		46.60	69.30	1.00		72.00	82.70	1.00
Case 5
1	32	48	1.00		32	48	1.00		32	48	1.00
2	379	117	0.97		379	117	0.97		379	117	0.97
3	115	1	0.98		115	1	0.98		115	1	0.98
4	63	42	1.00		63	42	1.00		63	42	1.00
5	17	30	1.00		17	30	1.00		17	30	1.00
6	31	0	1.00		31	0	1.00		31	0	1.00
7	9	17	1.00		9	17	1.00		9	17	1.00
8	77	65	1.00		156	97	0.99		156	97	0.99
9	2	19	1.00		186	95	0.99		186	95	0.99
10	63	14	0.99		117	1	0.99		117	1	0.99
Ave.	78.80	35.30	0.99		110.50	44.80	0.99		110.50	44.80	0.99
Case 6
1	34	18	1.00		34	18	1.00		34	18	1.00
2	73	48	1.00		116	14	0.99		95	132	1.00
3	59	6	0.99		59	6	0.99		59	6	0.99
4	29	137	1.01	29	137	1.01	29	137	1.01
5	22	0	1.00	19	0	1.00	27	0	1.00
6	20	0	1.00	56	28	1.00	56	28	1.00
7	4	69	1.01	4	69	1.01	4	69	1.01
8	28	41	1.00	28	41	1.00	28	41	1.00
9	234	77	0.98	234	77	0.98	234	77	0.98
10	48	104	1.01	45	120	1.01	3	275	1.03
Ave.	55.10	50.00	1.00	62.40	51.00	1.00	56.90	78.30	1.00

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Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	65	132	1.01		16	250	1.03		16	250	1.03
2	0	71	1.01		0	71	1.01		0	71	1.01
3	0	121	1.01		0	121	1.01		0	121	1.01
4	62	107	1.00		165	216	1.00		71	273	1.02
5	15	0	1.00		15	0	1.00		15	0	1.00
6	60	12	1.00		60	12	1.00		60	12	1.00
7	89	71	1.00		89	71	1.00		207	181	1.00
8	9	64	1.01		9	64	1.01		9	64	1.01
9	67	40	1.00		49	118	1.01		49	118	1.01
10	16	147	1.02		96	279	1.02		96	279	1.02
Ave.	38.30	76.50	1.00		49.90	120.20	1.01		52.30	136.90	1.01
Case 2
1	44	21	1.00		124	18	0.99		124	18	0.99
2	112	108	1.00		112	108	1.00		112	108	1.00
3	25	422	1.04		0	645	1.07		0	645	1.07
4	107	8	0.99		108	1	0.99		89	130	1.01
5	42	182	1.02		43	185	1.02		42	182	1.02
6	147	58	0.99		147	58	0.99		147	58	0.99
7	25	93	1.01		25	93	1.01		25	93	1.01
8	24	110	1.01		18	105	1.01		24	110	1.01
9	15	84	1.01		15	84	1.01		15	84	1.01
10	106	50	0.99		148	21	0.99		194	132	0.99
Ave.	64.70	113.60	1.01		74.00	131.80	1.01		77.20	156.00	1.01
Case 3
1	33	60	1.00		26	196	1.02		26	196	1.02
2	0	259	1.04		0	261	1.04		0	320	1.05
3	13	37	1.00		13	37	1.00		13	37	1.00
4	190	56	0.98		14	203	1.02		14	203	1.02
5	0	191	1.03		0	191	1.03		0	191	1.03
6	26	0	1.00		104	55	1.00		104	55	1.00
7	0	169	1.02		0	169	1.02		0	169	1.02
8	89	164	1.01		89	164	1.01		89	164	1.01
9	64	0	0.99		64	0	0.99		49	55	1.00
10	52	77	1.00		52	77	1.00		52	77	1.00
Ave.	46.70	101.30	1.01		36.20	135.30	1.01		34.70	146.70	1.01
Case 4
1	180	48	0.99		194	120	0.99		413	120	0.97
2	3	51	1.01		3	51	1.01		3	51	1.01
3	35	16	1.00		46	89	1.01		46	89	1.01
4	7	4	1.00		7	4	1.00		42	138	1.01
5	38	27	1.00		53	43	1.00		53	43	1.00
6	12	56	1.01		12	56	1.01		12	56	1.01
7	71	0	0.99		71	0	0.99		71	0	0.99
8	26	146	1.02		26	146	1.02		26	146	1.02
9	52	0	1.00		52	0	1.00		52	0	1.00
10	2	144	1.02		2	184	1.02		2	184	1.02
Ave.	42.60	49.20	1.00		46.60	69.30	1.00		72.00	82.70	1.00
Case 5
1	32	48	1.00		32	48	1.00		32	48	1.00
2	379	117	0.97		379	117	0.97		379	117	0.97
3	115	1	0.98		115	1	0.98		115	1	0.98
4	63	42	1.00		63	42	1.00		63	42	1.00
5	17	30	1.00		17	30	1.00		17	30	1.00
6	31	0	1.00		31	0	1.00		31	0	1.00
7	9	17	1.00		9	17	1.00		9	17	1.00
8	77	65	1.00		156	97	0.99		156	97	0.99
9	2	19	1.00		186	95	0.99		186	95	0.99
10	63	14	0.99		117	1	0.99		117	1	0.99
Ave.	78.80	35.30	0.99		110.50	44.80	0.99		110.50	44.80	0.99
Case 6
1	34	18	1.00		34	18	1.00		34	18	1.00
2	73	48	1.00		116	14	0.99		95	132	1.00
3	59	6	0.99		59	6	0.99		59	6	0.99
4	29	137	1.01	29	137	1.01	29	137	1.01
5	22	0	1.00	19	0	1.00	27	0	1.00
6	20	0	1.00	56	28	1.00	56	28	1.00
7	4	69	1.01	4	69	1.01	4	69	1.01
8	28	41	1.00	28	41	1.00	28	41	1.00
9	234	77	0.98	234	77	0.98	234	77	0.98
10	48	104	1.01	45	120	1.01	3	275	1.03
Ave.	55.10	50.00	1.00	62.40	51.00	1.00	56.90	78.30	1.00

Table 7. By performance indicators, $ n = 6 $

Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	39	66	1.00		39	66	1.00		30	186	1.01
2	66	120	1.01		295	163	0.99		295	163	0.99
3	15	62	1.01		3	214	1.03		3	214	1.03
4	55	12	0.99		22	75	1.01		40	64	1.00
5	27	261	1.02		27	261	1.02		27	261	1.02
6	71	100	1.00		90	163	1.01		90	163	1.01
7	154	129	1.00		69	174	1.01		69	174	1.01
8	24	72	1.01		24	72	1.01		24	72	1.01
9	79	164	1.01		79	164	1.01		79	164	1.01
10	332	123	0.98		332	123	0.98		332	123	0.98
Ave.	86.20	110.90	1.00		98.00	147.50	1.01		98.90	158.40	1.01
Case 2
1	20	145	1.01		20	145	1.01		0	343	1.04
2	31	100	1.01		70	220	1.02		70	220	1.02
3	49	238	1.03		86	482	1.06		86	482	1.06
4	10	65	1.01		10	65	1.01		10	65	1.01
5	108	108	1.00		81	165	1.01		81	165	1.01
6	23	52	1.00		23	52	1.00		35	64	1.00
7	80	125	1.01		80	125	1.01		80	125	1.01
8	52	74	1.00		96	108	1.00		96	108	1.00
9	103	95	1.00		56	239	1.02		56	239	1.02
10	31	0	1.00		18	56	1.00		18	56	1.00
Ave.	50.70	100.20	1.01		54.00	165.70	1.01		53.20	186.70	1.02
Case 3
1	30	219	1.02		0	343	1.04		0	459	1.05
2	26	48	1.00		26	48	1.00		57	23	1.00
3	61	145	1.01		61	145	1.01		61	145	1.01
4	5	298	1.03		11	676	1.07		11	676	1.07
5	91	13	0.99		73	21	0.99		73	21	0.99
6	30	7	1.00		28	53	1.00		28	53	1.00
7	78	64	1.00		208	172	1.00		208	172	1.00
8	31	120	1.01		31	120	1.01		31	120	1.01
9	85	0	0.99		85	0	0.99		85	0	0.99
10	94	81	1.00		51	127	1.01		91	173	1.01
Ave.	53.10	99.50	1.01		57.40	170.50	1.01		64.50	184.20	1.01
Case 4
1	23	46	1.00		10	101	1.01		10	101	1.01
2	47	144	1.01		47	144	1.01		47	144	1.01
3	20	55	1.00		20	55	1.00		20	55	1.00
4	60	10	0.99		11	114	1.01		213	154	0.99
5	33	15	1.00		13	78	1.01		13	78	1.01
6	6	286	1.04		6	286	1.04		6	286	1.04
7	136	10	0.98		206	0	0.97		206	0	0.97
8	32	2	1.00		75	16	0.99		73	23	1.00
9	27	2	1.00		27	2	1.00		27	2	1.00
10	19	148	1.01		19	148	1.01		19	148	1.01
Ave.	40.30	71.80	1.00		43.40	94.40	1.01		63.40	99.10	1.00
Case 5
1	4	78	1.01		4	78	1.01		4	78	1.01
2	35	100	1.01		19	171	1.01		19	171	1.01
3	142	35	0.99		92	53	1.00		120	145	1.00
4	76	0	0.99		6	146	1.02		6	250	1.03
5	42	0	1.00		42	0	1.00		42	0	1.00
6	6	174	1.03		6	174	1.03		6	174	1.03
7	234	0	0.97		91	107	1.00		91	107	1.00
8	3	63	1.01		3	63	1.01		3	63	1.01
9	36	57	1.00		73	38	1.00		73	38	1.00
10	65	0	0.99		65	0	0.99		65	0	0.99
Ave.	64.30	50.70	1.00		40.10	83.00	1.01		42.90	102.6	1.01
Case 6
1	67	0	0.99		91	32	0.99		91	32	0.99
2	37	29	0.99		81	68	1.00		81	68	1.00
3	14	34	1.00		5	16	1.00		8	104	1.01
4	28	34	1.00	26	90	1.01	26	90	1.01
5	17	71	1.01	17	71	1.01	17	71	1.01
6	70	386	1.03	70	386	1.03	21	595	1.06
7	91	22	0.99	91	22	0.99	131	42	0.99
8	27	6	1.00	27	6	1.00	26	118	1.01
9	34	135	1.01	34	135	1.01	34	135	1.01
10	47	160	1.02	162	192	1.00	162	192	1.00
Ave.	47.40	87.70	1.00	60.40	101.80	1.00	59.70	144.70	1.01

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Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	39	66	1.00		39	66	1.00		30	186	1.01
2	66	120	1.01		295	163	0.99		295	163	0.99
3	15	62	1.01		3	214	1.03		3	214	1.03
4	55	12	0.99		22	75	1.01		40	64	1.00
5	27	261	1.02		27	261	1.02		27	261	1.02
6	71	100	1.00		90	163	1.01		90	163	1.01
7	154	129	1.00		69	174	1.01		69	174	1.01
8	24	72	1.01		24	72	1.01		24	72	1.01
9	79	164	1.01		79	164	1.01		79	164	1.01
10	332	123	0.98		332	123	0.98		332	123	0.98
Ave.	86.20	110.90	1.00		98.00	147.50	1.01		98.90	158.40	1.01
Case 2
1	20	145	1.01		20	145	1.01		0	343	1.04
2	31	100	1.01		70	220	1.02		70	220	1.02
3	49	238	1.03		86	482	1.06		86	482	1.06
4	10	65	1.01		10	65	1.01		10	65	1.01
5	108	108	1.00		81	165	1.01		81	165	1.01
6	23	52	1.00		23	52	1.00		35	64	1.00
7	80	125	1.01		80	125	1.01		80	125	1.01
8	52	74	1.00		96	108	1.00		96	108	1.00
9	103	95	1.00		56	239	1.02		56	239	1.02
10	31	0	1.00		18	56	1.00		18	56	1.00
Ave.	50.70	100.20	1.01		54.00	165.70	1.01		53.20	186.70	1.02
Case 3
1	30	219	1.02		0	343	1.04		0	459	1.05
2	26	48	1.00		26	48	1.00		57	23	1.00
3	61	145	1.01		61	145	1.01		61	145	1.01
4	5	298	1.03		11	676	1.07		11	676	1.07
5	91	13	0.99		73	21	0.99		73	21	0.99
6	30	7	1.00		28	53	1.00		28	53	1.00
7	78	64	1.00		208	172	1.00		208	172	1.00
8	31	120	1.01		31	120	1.01		31	120	1.01
9	85	0	0.99		85	0	0.99		85	0	0.99
10	94	81	1.00		51	127	1.01		91	173	1.01
Ave.	53.10	99.50	1.01		57.40	170.50	1.01		64.50	184.20	1.01
Case 4
1	23	46	1.00		10	101	1.01		10	101	1.01
2	47	144	1.01		47	144	1.01		47	144	1.01
3	20	55	1.00		20	55	1.00		20	55	1.00
4	60	10	0.99		11	114	1.01		213	154	0.99
5	33	15	1.00		13	78	1.01		13	78	1.01
6	6	286	1.04		6	286	1.04		6	286	1.04
7	136	10	0.98		206	0	0.97		206	0	0.97
8	32	2	1.00		75	16	0.99		73	23	1.00
9	27	2	1.00		27	2	1.00		27	2	1.00
10	19	148	1.01		19	148	1.01		19	148	1.01
Ave.	40.30	71.80	1.00		43.40	94.40	1.01		63.40	99.10	1.00
Case 5
1	4	78	1.01		4	78	1.01		4	78	1.01
2	35	100	1.01		19	171	1.01		19	171	1.01
3	142	35	0.99		92	53	1.00		120	145	1.00
4	76	0	0.99		6	146	1.02		6	250	1.03
5	42	0	1.00		42	0	1.00		42	0	1.00
6	6	174	1.03		6	174	1.03		6	174	1.03
7	234	0	0.97		91	107	1.00		91	107	1.00
8	3	63	1.01		3	63	1.01		3	63	1.01
9	36	57	1.00		73	38	1.00		73	38	1.00
10	65	0	0.99		65	0	0.99		65	0	0.99
Ave.	64.30	50.70	1.00		40.10	83.00	1.01		42.90	102.6	1.01
Case 6
1	67	0	0.99		91	32	0.99		91	32	0.99
2	37	29	0.99		81	68	1.00		81	68	1.00
3	14	34	1.00		5	16	1.00		8	104	1.01
4	28	34	1.00	26	90	1.01	26	90	1.01
5	17	71	1.01	17	71	1.01	17	71	1.01
6	70	386	1.03	70	386	1.03	21	595	1.06
7	91	22	0.99	91	22	0.99	131	42	0.99
8	27	6	1.00	27	6	1.00	26	118	1.01
9	34	135	1.01	34	135	1.01	34	135	1.01
10	47	160	1.02	162	192	1.00	162	192	1.00
Ave.	47.40	87.70	1.00	60.40	101.80	1.00	59.70	144.70	1.01

Table 8. By performance indicators, $n = 7$

Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	232	171	0.99		247	252	1.00		247	252	1.00
2	81	140	1.01		81	140	1.01		81	140	1.01
3	111	51	0.99		11	49	0.99		111	51	0.99
4	34	243	1.02		34	243	1.02		34	243	1.02
5	81	0	0.99		81	0	0.99		81	0	0.99
6	96	90	1.00		74	130	1.01		74	130	1.01
7	91	116	1.00		87	160	1.01		87	160	1.01
8	50	196	1.02		50	243	1.02		50	243	1.02
9	66	20	0.99		151	66	0.99		53	259	1.02
10	118	24	0.99		223	91	0.99		320	19	0.97
Ave.	96.00	105.10	1.00		103.90	137.60	1.00		113.80	149.70	1.00
Case 2
1	53	7	0.99		43	139	1.01		43	139	1.01
2	28	116	1.01		137	148	1.00		137	148	1.00
3	40	58	1.00		28	117	1.01		28	117	1.01
4	16	171	1.02		16	171	1.02		16	171	1.02
5	6	53	1.01		0	119	1.02		0	119	1.02
6	36	158	1.01		36	158	1.01		36	158	1.01
7	17	64	1.00		9	140	1.01		9	140	1.01
8	37	53	1.00		74	155	1.01		74	155	1.01
9	77	62	1.00		77	62	1.00		77	62	1.00
10	68	163	1.01		6	278	1.03		6	278	1.03
Ave.	37.80	90.50	1.01		42.60	148.70	1.01		42.60	148.70	1.01
Case 3
1	74	38	1.00		90	144	1.01		90	144	1.01
2	59	28	1.00		89	119	1.00		63	164	1.01
3	65	118	1.01		195	245	1.01		195	245	1.01
4	53	108	1.01		23	184	1.02		20	181	1.02
5	6	192	1.02		6	192	1.02		36	294	1.03
6	52	3	0.99		52	3	0.99		52	3	0.99
7	55	243	1.03		55	243	1.03		55	243	1.03
8	15	31	1.00		6	79	1.01		6	79	1.01
9	31	211	1.02		31	211	1.02		31	211	1.02
10	84	91	1.00		84	91	1.00		39	131	1.01
Ave.	49.40	106.30	1.01		63.10	151.10	1.01		58.70	169.50	1.01
Case 4
1	57	88	1.00		22	104	1.01		22	104	1.01
2	21	129	1.01		2	198	1.02		2	198	1.02
3	53	81	1.00		91	92	1.00		91	92	1.00
4	7	216	1.02		20	229	1.02		20	229	1.02
5	17	58	1.01		17	58	1.01		17	58	1.01
6	31	83	1.01		15	138	1.01		15	138	1.01
7	69	82	1.00		69	82	1.00		69	82	1.00
8	76	42	1.00		45	112	1.01		45	112	1.01
9	72	12	0.99		26	120	1.01		27	254	1.02
10	20	71	1.01		20	71	1.01		20	71	1.01
Ave.	42.30	86.20	1.01		32.70	120.40	1.01		32.80	133.80	1.01
Case 5
1	334	44	0.96		376	66	0.96		384	38	0.95
2	38	37	1.00		72	73	1.00		72	73	1.00
3	113	5	0.99		113	5	0.99		113	5	0.99
4	47	139	1.01		47	139	1.01		20	113	1.01
5	67	34	1.00		18	129	1.02		37	110	1.01
6	22	157	1.02		22	157	1.02		5	182	1.02
7	50	2	0.99		26	63	1.00		26	63	1.00
8	48	6	0.99		48	6	0.99		48	6	0.99
9	54	41	1.00		54	41	1.00		54	41	1.00
10	20	74	1.01		20	74	1.01		20	74	1.01
Ave.	79.30	53.90	1.00		79.60	75.30	1.00		77.90	70.50	1.00
Case 6
1	48	0	1.00		194	27	0.98		194	27	0.98
2	19	82	1.01		21	84	1.01		21	84	1.01
3	49	28	1.00		107	71	1.00		6	283	1.03
4	6	90	1.01	0	166	1.02	0	166	1.02
5	202	0	0.97	202	0	0.97	202	0	0.97
6	41	109	1.01	41	109	1.01	41	109	1.01
7	79	0	0.99	79	0	0.99	188	71	0.99
8	9	291	1.03	9	291	1.03	9	291	1.03
9	123	203	1.01	26	207	1.03	26	207	1.03
10	111	77	1.00	118	130	1.00	118	130	1.00
Ave.	68.70	88.00	1.00	79.70	108.50	1.00	80.50	136.80	1.01

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Ex.	Scenario 1				Scenario 2				Scenario 3
Ex.	IT	Overtime	UROR		IT	Overtime	UROR		IT	Overtime	UROR
Case 1
1	232	171	0.99		247	252	1.00		247	252	1.00
2	81	140	1.01		81	140	1.01		81	140	1.01
3	111	51	0.99		11	49	0.99		111	51	0.99
4	34	243	1.02		34	243	1.02		34	243	1.02
5	81	0	0.99		81	0	0.99		81	0	0.99
6	96	90	1.00		74	130	1.01		74	130	1.01
7	91	116	1.00		87	160	1.01		87	160	1.01
8	50	196	1.02		50	243	1.02		50	243	1.02
9	66	20	0.99		151	66	0.99		53	259	1.02
10	118	24	0.99		223	91	0.99		320	19	0.97
Ave.	96.00	105.10	1.00		103.90	137.60	1.00		113.80	149.70	1.00
Case 2
1	53	7	0.99		43	139	1.01		43	139	1.01
2	28	116	1.01		137	148	1.00		137	148	1.00
3	40	58	1.00		28	117	1.01		28	117	1.01
4	16	171	1.02		16	171	1.02		16	171	1.02
5	6	53	1.01		0	119	1.02		0	119	1.02
6	36	158	1.01		36	158	1.01		36	158	1.01
7	17	64	1.00		9	140	1.01		9	140	1.01
8	37	53	1.00		74	155	1.01		74	155	1.01
9	77	62	1.00		77	62	1.00		77	62	1.00
10	68	163	1.01		6	278	1.03		6	278	1.03
Ave.	37.80	90.50	1.01		42.60	148.70	1.01		42.60	148.70	1.01
Case 3
1	74	38	1.00		90	144	1.01		90	144	1.01
2	59	28	1.00		89	119	1.00		63	164	1.01
3	65	118	1.01		195	245	1.01		195	245	1.01
4	53	108	1.01		23	184	1.02		20	181	1.02
5	6	192	1.02		6	192	1.02		36	294	1.03
6	52	3	0.99		52	3	0.99		52	3	0.99
7	55	243	1.03		55	243	1.03		55	243	1.03
8	15	31	1.00		6	79	1.01		6	79	1.01
9	31	211	1.02		31	211	1.02		31	211	1.02
10	84	91	1.00		84	91	1.00		39	131	1.01
Ave.	49.40	106.30	1.01		63.10	151.10	1.01		58.70	169.50	1.01
Case 4
1	57	88	1.00		22	104	1.01		22	104	1.01
2	21	129	1.01		2	198	1.02		2	198	1.02
3	53	81	1.00		91	92	1.00		91	92	1.00
4	7	216	1.02		20	229	1.02		20	229	1.02
5	17	58	1.01		17	58	1.01		17	58	1.01
6	31	83	1.01		15	138	1.01		15	138	1.01
7	69	82	1.00		69	82	1.00		69	82	1.00
8	76	42	1.00		45	112	1.01		45	112	1.01
9	72	12	0.99		26	120	1.01		27	254	1.02
10	20	71	1.01		20	71	1.01		20	71	1.01
Ave.	42.30	86.20	1.01		32.70	120.40	1.01		32.80	133.80	1.01
Case 5
1	334	44	0.96		376	66	0.96		384	38	0.95
2	38	37	1.00		72	73	1.00		72	73	1.00
3	113	5	0.99		113	5	0.99		113	5	0.99
4	47	139	1.01		47	139	1.01		20	113	1.01
5	67	34	1.00		18	129	1.02		37	110	1.01
6	22	157	1.02		22	157	1.02		5	182	1.02
7	50	2	0.99		26	63	1.00		26	63	1.00
8	48	6	0.99		48	6	0.99		48	6	0.99
9	54	41	1.00		54	41	1.00		54	41	1.00
10	20	74	1.01		20	74	1.01		20	74	1.01
Ave.	79.30	53.90	1.00		79.60	75.30	1.00		77.90	70.50	1.00
Case 6
1	48	0	1.00		194	27	0.98		194	27	0.98
2	19	82	1.01		21	84	1.01		21	84	1.01
3	49	28	1.00		107	71	1.00		6	283	1.03
4	6	90	1.01	0	166	1.02	0	166	1.02
5	202	0	0.97	202	0	0.97	202	0	0.97
6	41	109	1.01	41	109	1.01	41	109	1.01
7	79	0	0.99	79	0	0.99	188	71	0.99
8	9	291	1.03	9	291	1.03	9	291	1.03
9	123	203	1.01	26	207	1.03	26	207	1.03
10	111	77	1.00	118	130	1.00	118	130	1.00
Ave.	68.70	88.00	1.00	79.70	108.50	1.00	80.50	136.80	1.01

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American Institute of Mathematical Sciences

Dynamic-programming-based heuristic for multi-objective operating theater planning

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Dynamic-programming-based heuristic for multi-objective operating theater planning

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