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July  2021, 17(4): 1973-1991. doi: 10.3934/jimo.2020054

A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure

1. 

Department of Mathematical Sciences, Kean University, New Jersey, USA

2. 

Mathematics & Natural Sciences Department, Gulf University for Science and Technology, Kuwait

3. 

Economics & Finance Department, Gulf University for Science and Technology, Kuwait

4. 

Department of Industrial and Management Systems Engineering, Kuwait University, Kuwait

* Corresponding author: Muberra Allahverdi

Received  August 2019 Revised  September 2019 Published  July 2021 Early access  March 2020

In this paper, we consider a manufacturing system with two-machine no-wait flowshop scheduling problem where setup times are uncertain. The problem with the performance measure of maximum lateness was addressed in the literature (Computational and Applied Mathematics 37, 6774-6794) where dominance relations were proposed. We establish a new dominance relation and show that the new dominance relation is, on average, about 90$ \% $ more efficient than the existing ones. Moreover, since it is highly unlikely to find optimal solutions for problems of reasonable size by utilizing dominance relations and since there exist no heuristic in the literature for the problem, we propose constructive heuristics to solve real life problems. We conduct extensive computational experiments to evaluate the proposed heuristics. Computational experiments indicate that the performance of the worst proposed heuristic is at least 20$ \% $ better than a benchmark solution. Furthermore, they also indicate that the best proposed heuristic is about 130$ \% $ better than the worst one. The average CPU time of the heuristics is significantly less than a second.

Citation: Muberra Allahverdi, Harun Aydilek, Asiye Aydilek, Ali Allahverdi. A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1973-1991. doi: 10.3934/jimo.2020054
References:
[1]

A. Allahverdi, A survey of scheduling problems with no-wait in process, European Journal of Operational Research, 255 (2016), 665-686.  doi: 10.1016/j.ejor.2016.05.036.

[2]

A. Allahverdi, The third comprehensive survey on scheduling problems with setup times/costs, European Journal of Operational Research, 246 (2015), 345-378.  doi: 10.1016/j.ejor.2015.04.004.

[3]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize makespan with bounded setup and processing times, Int. Journal of Agile Manufacturing, 8 (2005), 145-153. 

[4]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize maximum lateness with bounded setup and processing times, Kuwait Journal of Science and Engineering, 33 (2006), 233-251. 

[5]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize total completion time with bounded setup and processing times, Int. Journal of Production Economics, 103 (2006), 386-400. 

[6]

A. AllahverdiT. Aldowaisan and Y. N. Sotskov, Two-machine flowshop scheduling problem to minimize makespan or total completion time with random and bounded setup times, Int. Journal of Mathematics and Mathematical Sciences, 39 (2003), 2475-2486.  doi: 10.1155/S016117120321019X.

[7]

A. Allahverdi and M. Allahverdi, Two-machine no-wait flowshop scheduling problem with uncertain setup times to minimize maximum lateness, Computational and Applied Mathematics, 37 (2018), 6774-6794.  doi: 10.1007/s40314-018-0694-3.

[8]

A. AydilekH. Aydilek and A. Allahverdi, Increasing the profitability and competitiveness in a production environment with random and bounded setup times, Int. Journal of Production Research, 51 (2013), 106-117. 

[9]

A. AydilekH. Aydilek and A. Allahverdi, Production in a two-machine flowshop scheduling environment with uncertain processing and setup times to minimize makespan, Int. Journal of Production Research, 53 (2015), 2803-2819. 

[10]

O. BraunT.-C. LaiG. SchmidtSo tskov and N. Yu, Stability of Johnson's schedule with respect to limited machine availability, Int. Journal of Production Research, 40 (2002), 4381-4400. 

[11]

E. M. Gonzalez-NeiraD. FeroneS. Hatami and A. A. Juan, A biased-randomized simheuristic for the distributed assembly permutation flowshop problem with stochastic processing times, Simulation Modelling Practice and Theory, 79 (2017), 23-36. 

[12]

N. G. Hall and M. E. Posner, Generating experimental data for computational testing with machine scheduling applications, Operations Research, 49 (2001), 854-865.  doi: 10.1287/opre.49.6.854.10014.

[13]

N. G. Hall and C. Sriskandarajah, A survey of machine scheduling problems with blocking and no-wait in process, Operations Research, 44 (1996), 510-525.  doi: 10.1287/opre.44.3.510.

[14]

V. N. HsuR. De Matta and C.-Y. Lee, Scheduling patients in an ambulatory surgical center, Naval Research Logistics, 50 (2003), 218-238.  doi: 10.1002/nav.10060.

[15]

Y. D. Kim, A new branch and bound algorithm for minimizing mean tardiness in two machine flowshops, Computers and Operations Research, 20 (1993), 391-401.  doi: 10.1016/0305-0548(93)90083-U.

[16]

S. C. Kim and P. M. Bobrowski, Scheduling jobs with uncertain setup times and sequence dependency, Omega, Int. Journal of Management Science, 25 (1997), 437-447. 

[17]

J. Kim, A. Kröller, J. S. B. Mitchell and G. R. Sabhnani, Scheduling aircraft to reduce controller workload, Open Access Series in Informatics, 12 (2009).

[18]

S. Q. Liu and E. Kozan, Scheduling trains with priorities: A no-wait Blocking Parallel-Machine Job-Shop Scheduling model, Transportation Science, 45 (2011), 175-198. 

[19]

C. C. LuS. W. Lin and K. C. Ying, Minimizing worst-case regret of makespan on a single machine with uncertain processing and setup times, Applied Soft Computing, 23 (2014), 144-151. 

[20]

R. MacchiaroliS. Molé and S. Riemma, Modelling and optimization of industrial manufacturing processes subject to no-wait constraints, Int. Journal of Production Research, 37 (1999), 2585-2607. 

[21]

N. M. MatsveichukY. N. Sotskov and F. Werner, Partial job order for solving the two-machine flow-shop minimum-length problem with uncertain processing times, Optimization, 60 (2011), 1493-1517.  doi: 10.1080/02331931003657691.

[22]

S. Mustu and T. Eren, The single machine scheduling problem with sequence-dependent setup times and a learning effect on processing times, Applied Soft Computing, 71 (2018), 291-306. 

[23]

Y. N. SotskovN. G. Egorova and T. C. Lai, Minimizing total weighted flow time of a set of jobs with interval processing times, Mathematical and Computer Modelling, 50 (2009), 556-573.  doi: 10.1016/j.mcm.2009.03.006.

[24]

Y. N. Sotskov and T. C. Lai, Minimizing total weighted flow under uncertainty using dominance and a stability box, Computers and Operations Research, 39 (2012), 1271-1289.  doi: 10.1016/j.cor.2011.02.001.

[25]

Y. N. Sotskov and N. M. Matsveichuk, Uncertainty measure for the Bellman-Johnson problem with interval processing times, Cybernetics and System Analysis, 48 (2012), 641-652.  doi: 10.1007/s10559-012-9445-4.

[26]

E. Vallada and R. Ruiz, Genetic algorithms with path relinking for the minimum tardiness permutation flowshop problem, OMEGA The International Journal of Management Science, 38 (2010), 57-67. 

[27]

K. Wang and S. H. Choi, A decomposition-based approach to flexible flow shop scheduling under machine breakdown, Int. Journal of Production Research, 50 (2012), 215-234. 

[28]

N. Xie and N. Chen, Flexible job shop scheduling problem with interval grey processing time, Applied Soft Computing, 70 (2018), 513-524. 

[29]

J. XuC. C. WuY. Yin and W. C. Lin, An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times, Applied Soft Computing, 52 (2017), 39-47. 

show all references

References:
[1]

A. Allahverdi, A survey of scheduling problems with no-wait in process, European Journal of Operational Research, 255 (2016), 665-686.  doi: 10.1016/j.ejor.2016.05.036.

[2]

A. Allahverdi, The third comprehensive survey on scheduling problems with setup times/costs, European Journal of Operational Research, 246 (2015), 345-378.  doi: 10.1016/j.ejor.2015.04.004.

[3]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize makespan with bounded setup and processing times, Int. Journal of Agile Manufacturing, 8 (2005), 145-153. 

[4]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize maximum lateness with bounded setup and processing times, Kuwait Journal of Science and Engineering, 33 (2006), 233-251. 

[5]

A. Allahverdi, Two-machine flowshop scheduling problem to minimize total completion time with bounded setup and processing times, Int. Journal of Production Economics, 103 (2006), 386-400. 

[6]

A. AllahverdiT. Aldowaisan and Y. N. Sotskov, Two-machine flowshop scheduling problem to minimize makespan or total completion time with random and bounded setup times, Int. Journal of Mathematics and Mathematical Sciences, 39 (2003), 2475-2486.  doi: 10.1155/S016117120321019X.

[7]

A. Allahverdi and M. Allahverdi, Two-machine no-wait flowshop scheduling problem with uncertain setup times to minimize maximum lateness, Computational and Applied Mathematics, 37 (2018), 6774-6794.  doi: 10.1007/s40314-018-0694-3.

[8]

A. AydilekH. Aydilek and A. Allahverdi, Increasing the profitability and competitiveness in a production environment with random and bounded setup times, Int. Journal of Production Research, 51 (2013), 106-117. 

[9]

A. AydilekH. Aydilek and A. Allahverdi, Production in a two-machine flowshop scheduling environment with uncertain processing and setup times to minimize makespan, Int. Journal of Production Research, 53 (2015), 2803-2819. 

[10]

O. BraunT.-C. LaiG. SchmidtSo tskov and N. Yu, Stability of Johnson's schedule with respect to limited machine availability, Int. Journal of Production Research, 40 (2002), 4381-4400. 

[11]

E. M. Gonzalez-NeiraD. FeroneS. Hatami and A. A. Juan, A biased-randomized simheuristic for the distributed assembly permutation flowshop problem with stochastic processing times, Simulation Modelling Practice and Theory, 79 (2017), 23-36. 

[12]

N. G. Hall and M. E. Posner, Generating experimental data for computational testing with machine scheduling applications, Operations Research, 49 (2001), 854-865.  doi: 10.1287/opre.49.6.854.10014.

[13]

N. G. Hall and C. Sriskandarajah, A survey of machine scheduling problems with blocking and no-wait in process, Operations Research, 44 (1996), 510-525.  doi: 10.1287/opre.44.3.510.

[14]

V. N. HsuR. De Matta and C.-Y. Lee, Scheduling patients in an ambulatory surgical center, Naval Research Logistics, 50 (2003), 218-238.  doi: 10.1002/nav.10060.

[15]

Y. D. Kim, A new branch and bound algorithm for minimizing mean tardiness in two machine flowshops, Computers and Operations Research, 20 (1993), 391-401.  doi: 10.1016/0305-0548(93)90083-U.

[16]

S. C. Kim and P. M. Bobrowski, Scheduling jobs with uncertain setup times and sequence dependency, Omega, Int. Journal of Management Science, 25 (1997), 437-447. 

[17]

J. Kim, A. Kröller, J. S. B. Mitchell and G. R. Sabhnani, Scheduling aircraft to reduce controller workload, Open Access Series in Informatics, 12 (2009).

[18]

S. Q. Liu and E. Kozan, Scheduling trains with priorities: A no-wait Blocking Parallel-Machine Job-Shop Scheduling model, Transportation Science, 45 (2011), 175-198. 

[19]

C. C. LuS. W. Lin and K. C. Ying, Minimizing worst-case regret of makespan on a single machine with uncertain processing and setup times, Applied Soft Computing, 23 (2014), 144-151. 

[20]

R. MacchiaroliS. Molé and S. Riemma, Modelling and optimization of industrial manufacturing processes subject to no-wait constraints, Int. Journal of Production Research, 37 (1999), 2585-2607. 

[21]

N. M. MatsveichukY. N. Sotskov and F. Werner, Partial job order for solving the two-machine flow-shop minimum-length problem with uncertain processing times, Optimization, 60 (2011), 1493-1517.  doi: 10.1080/02331931003657691.

[22]

S. Mustu and T. Eren, The single machine scheduling problem with sequence-dependent setup times and a learning effect on processing times, Applied Soft Computing, 71 (2018), 291-306. 

[23]

Y. N. SotskovN. G. Egorova and T. C. Lai, Minimizing total weighted flow time of a set of jobs with interval processing times, Mathematical and Computer Modelling, 50 (2009), 556-573.  doi: 10.1016/j.mcm.2009.03.006.

[24]

Y. N. Sotskov and T. C. Lai, Minimizing total weighted flow under uncertainty using dominance and a stability box, Computers and Operations Research, 39 (2012), 1271-1289.  doi: 10.1016/j.cor.2011.02.001.

[25]

Y. N. Sotskov and N. M. Matsveichuk, Uncertainty measure for the Bellman-Johnson problem with interval processing times, Cybernetics and System Analysis, 48 (2012), 641-652.  doi: 10.1007/s10559-012-9445-4.

[26]

E. Vallada and R. Ruiz, Genetic algorithms with path relinking for the minimum tardiness permutation flowshop problem, OMEGA The International Journal of Management Science, 38 (2010), 57-67. 

[27]

K. Wang and S. H. Choi, A decomposition-based approach to flexible flow shop scheduling under machine breakdown, Int. Journal of Production Research, 50 (2012), 215-234. 

[28]

N. Xie and N. Chen, Flexible job shop scheduling problem with interval grey processing time, Applied Soft Computing, 70 (2018), 513-524. 

[29]

J. XuC. C. WuY. Yin and W. C. Lin, An iterated local search for the multi-objective permutation flowshop scheduling problem with sequence-dependent setup times, Applied Soft Computing, 52 (2017), 39-47. 

Figure 1.  The average errors of the heuristics with respect to n
Figure 2.  The average errors of the heuristics with respect to R
Figure 3.  The average errors of the heuristics with respect to T
Figure 4.  The average errors of the heuristics with respect to $ \Delta $
Figure 5.  The average errors of the heuristics with respect to n (Positive Exponential)
Figure 6.  The average errors of the heuristics with respect to n (Negative Exponential)
Figure 7.  The average errors of the heuristics with respect to n (Uniform)
Figure 8.  The average errors of the heuristics with respect to n (Normal)
Figure 9.  The average errors of the heuristics with respect to n (Positive Linear)
Figure 10.  The average errors of the heuristics with respect to n (Negative Linear)
Figure 11.  The average improvements of the dominance relation with respect to n
Table 1.  Evaluation of the newly established dominance relation
T=0.25 T=0.5 T=0.75
n D R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
5 122.1 139 69.1 133.2 85.5 112.1 80.8 108.7 96.3
30 10 152 156.2 133.6 153.8 93.2 131.8 125 85.3 96.7
20 137.9 96 125 172 183 174.7 135.2 75.7 110.6
5 106.6 72.1 98.6 55.5 94.6 84.2 94.4 103.4 80.7
40 10 100 66.8 90.2 116.8 112.4 91.7 68.5 86 106
20 124.2 141.9 140 90.5 56.1 103.7 74.5 68.6 82.5
5 75.7 88.2 80.5 84.4 91.2 95.6 88.1 59.7 95.1
50 10 66.5 68 107.4 118.3 87.8 91.2 76.6 61.6 83.4
20 63.5 121.4 106.3 74.2 40 140 69.2 75.7 76.8
5 100 106.1 107.7 64 71.3 71.1 78 56.5 57.8
60 10 66.2 84.3 79.7 63.5 143 80.9 46.8 94.7 70.8
20 106.5 89.1 62.1 95.9 81.1 76.4 65.1 58.4 75.7
5 78.4 52.2 54.4 65.9 78.3 62.5 70.3 110.8 68.2
70 10 77.1 81.9 133.5 57.8 52.4 74.7 66.6 72.6 89.7
20 57.9 47.6 108.3 88.3 96.2 40.2 77 54.9 61.8
5 96.6 91.5 82.1 80.6 84.2 85.1 82.3 87.8 79.6
Avg 10 92.4 91.4 108.9 102 97.7 94.1 76.7 80 89.3
20 98 99.2 108.3 104.2 91.3 107 84.2 66.7 81.5
T=0.25 T=0.5 T=0.75
n D R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
5 122.1 139 69.1 133.2 85.5 112.1 80.8 108.7 96.3
30 10 152 156.2 133.6 153.8 93.2 131.8 125 85.3 96.7
20 137.9 96 125 172 183 174.7 135.2 75.7 110.6
5 106.6 72.1 98.6 55.5 94.6 84.2 94.4 103.4 80.7
40 10 100 66.8 90.2 116.8 112.4 91.7 68.5 86 106
20 124.2 141.9 140 90.5 56.1 103.7 74.5 68.6 82.5
5 75.7 88.2 80.5 84.4 91.2 95.6 88.1 59.7 95.1
50 10 66.5 68 107.4 118.3 87.8 91.2 76.6 61.6 83.4
20 63.5 121.4 106.3 74.2 40 140 69.2 75.7 76.8
5 100 106.1 107.7 64 71.3 71.1 78 56.5 57.8
60 10 66.2 84.3 79.7 63.5 143 80.9 46.8 94.7 70.8
20 106.5 89.1 62.1 95.9 81.1 76.4 65.1 58.4 75.7
5 78.4 52.2 54.4 65.9 78.3 62.5 70.3 110.8 68.2
70 10 77.1 81.9 133.5 57.8 52.4 74.7 66.6 72.6 89.7
20 57.9 47.6 108.3 88.3 96.2 40.2 77 54.9 61.8
5 96.6 91.5 82.1 80.6 84.2 85.1 82.3 87.8 79.6
Avg 10 92.4 91.4 108.9 102 97.7 94.1 76.7 80 89.3
20 98 99.2 108.3 104.2 91.3 107 84.2 66.7 81.5
Table 2.  Errors of Heuristics when $ \Delta = 5 $
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 1.21 1.81 6.67 0.4 1.09 1.47 0.32 0.55 0.58
CH2 1.2 1.54 6.58 0.32 1.26 0.82 0.28 0.64 0.37
30 CH3 0.91 2.4 3.05 0.51 1.08 1.44 0.12 0.67 0.26
CH4 0.81 1.9 2.07 0.54 0.74 1.3 0.17 0.31 0.23
CH5 0.99 2.13 1.98 0.55 0.63 1.33 0.26 0.48 0.52
CH1 0.99 5.3 13.96 0.48 0.96 2.86 0.26 0.56 0.92
CH2 0.78 4.4 13.05 0.4 1.06 1.88 0.4 0.34 0.87
40 CH3 0.76 4.55 10.53 0.5 1.27 1.53 0.26 0.46 0.37
CH4 0.59 2.72 3.95 0.33 0.85 1.03 0.27 0.39 0.34
CH5 0.97 2.9 3.7 0.4 0.86 0.96 0.32 0.63 0.47
CH1 1.19 4.49 11.86 0.33 1.76 2.28 0.29 0.69 1.2
CH2 0.92 3.04 11.18 0.39 1.91 1.69 0.24 0.47 1.3
50 CH3 0.77 4 9.17 0.31 1.36 1.07 0.27 0.52 0.94
CH4 0.61 2.76 4.17 0.3 0.61 0.66 0.23 0.46 0.33
CH5 1.01 2.68 4.27 0.53 0.65 0.8 0.36 0.59 0.34
CH1 1.23 4.26 6.22 0.71 1.37 2.1 0.38 0.69 1.07
CH2 1.26 3.57 4.93 0.59 1.11 1.76 0.34 0.71 0.82
60 CH3 0.98 3.85 4.68 0.59 0.82 1.05 0.4 0.72 0.91
CH4 0.69 2.54 4.17 0.35 0.79 0.61 0.21 0.4 0.7
CH5 0.72 2.64 4.8 0.47 0.96 0.64 0.23 0.54 0.71
CH1 1.28 3.95 9.17 0.48 0.76 1.49 0.33 0.56 0.77
CH2 1.08 3.7 4.04 0.53 0.78 1.56 0.3 0.5 0.57
70 CH3 1.18 2.62 3.2 0.36 0.79 1.32 0.3 0.38 0.57
CH4 0.87 1.9 3.11 0.3 0.61 0.64 0.28 0.34 0.38
CH5 0.95 2.75 2.58 0.37 0.66 0.67 0.26 0.67 0.62
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 1.21 1.81 6.67 0.4 1.09 1.47 0.32 0.55 0.58
CH2 1.2 1.54 6.58 0.32 1.26 0.82 0.28 0.64 0.37
30 CH3 0.91 2.4 3.05 0.51 1.08 1.44 0.12 0.67 0.26
CH4 0.81 1.9 2.07 0.54 0.74 1.3 0.17 0.31 0.23
CH5 0.99 2.13 1.98 0.55 0.63 1.33 0.26 0.48 0.52
CH1 0.99 5.3 13.96 0.48 0.96 2.86 0.26 0.56 0.92
CH2 0.78 4.4 13.05 0.4 1.06 1.88 0.4 0.34 0.87
40 CH3 0.76 4.55 10.53 0.5 1.27 1.53 0.26 0.46 0.37
CH4 0.59 2.72 3.95 0.33 0.85 1.03 0.27 0.39 0.34
CH5 0.97 2.9 3.7 0.4 0.86 0.96 0.32 0.63 0.47
CH1 1.19 4.49 11.86 0.33 1.76 2.28 0.29 0.69 1.2
CH2 0.92 3.04 11.18 0.39 1.91 1.69 0.24 0.47 1.3
50 CH3 0.77 4 9.17 0.31 1.36 1.07 0.27 0.52 0.94
CH4 0.61 2.76 4.17 0.3 0.61 0.66 0.23 0.46 0.33
CH5 1.01 2.68 4.27 0.53 0.65 0.8 0.36 0.59 0.34
CH1 1.23 4.26 6.22 0.71 1.37 2.1 0.38 0.69 1.07
CH2 1.26 3.57 4.93 0.59 1.11 1.76 0.34 0.71 0.82
60 CH3 0.98 3.85 4.68 0.59 0.82 1.05 0.4 0.72 0.91
CH4 0.69 2.54 4.17 0.35 0.79 0.61 0.21 0.4 0.7
CH5 0.72 2.64 4.8 0.47 0.96 0.64 0.23 0.54 0.71
CH1 1.28 3.95 9.17 0.48 0.76 1.49 0.33 0.56 0.77
CH2 1.08 3.7 4.04 0.53 0.78 1.56 0.3 0.5 0.57
70 CH3 1.18 2.62 3.2 0.36 0.79 1.32 0.3 0.38 0.57
CH4 0.87 1.9 3.11 0.3 0.61 0.64 0.28 0.34 0.38
CH5 0.95 2.75 2.58 0.37 0.66 0.67 0.26 0.67 0.62
Table 3.  Errors of Heuristics when $ \Delta = 10 $
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 1.48 4.64 15.59 0.79 1.46 2.05 0.43 1.02 2.82
CH2 1.36 4.14 10.67 0.77 0.92 1.52 0.33 0.97 2.73
30 CH3 1.39 3.33 8.59 0.54 0.72 1.47 0.37 0.69 2.23
CH4 0.8 2.64 4.42 0.56 0.51 1.1 0.35 0.83 0.81
CH5 1.64 2.53 3.96 0.7 0.73 0.99 0.42 0.91 0.82
CH1 1.44 3.81 23.96 0.59 1.91 2.04 0.51 1.12 1.41
CH2 1.58 2.74 17.86 0.62 1.4 2.22 0.4 0.77 1.27
40 CH3 1.06 2.16 13.1 0.51 1.56 1.97 0.18 0.83 1.01
CH4 0.85 2.31 1.92 0.56 0.65 1.4 0.28 0.57 0.44
CH5 1.42 3.14 3.68 0.69 0.87 1.7 0.36 0.53 0.55
CH1 1.47 5.64 15.22 0.68 1.97 2.68 0.42 0.89 2.04
CH2 1.32 5.29 11.23 0.4 1.41 2.37 0.32 0.59 1.64
50 CH3 0.85 4.06 7.54 0.42 1.27 1.85 0.27 0.74 1.3
CH4 0.74 3.81 4.92 0.43 0.89 1.26 0.27 0.37 0.92
CH5 1.23 4.04 9.89 0.63 0.84 1.29 0.52 0.45 0.91
CH1 1.4 4.32 11.36 0.56 2.63 2.47 0.43 0.84 1.95
CH2 1.24 5.41 10.53 0.57 1.87 2.23 0.34 0.62 1.41
60 CH3 1.21 2.73 7.78 0.38 1.34 0.97 0.29 0.49 0.96
CH4 1.27 2.63 2.8 0.37 1.17 0.78 0.32 0.33 0.61
CH5 1.37 2.43 5.1 0.51 1.16 0.9 0.4 0.65 0.9
CH1 1.23 5.9 11.68 0.77 1.81 2.68 0.31 1.22 1.2
CH2 1.05 3.84 8.2 0.59 1.33 2.24 0.25 0.78 0.98
70 CH3 1.08 3.58 5.97 0.44 0.95 1.46 0.24 0.57 0.74
CH4 0.9 3.1 4.29 0.37 0.69 0.9 0.28 0.54 0.54
CH5 1.78 3.13 3.71 0.55 1.45 1.58 0.4 0.65 0.69
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 1.48 4.64 15.59 0.79 1.46 2.05 0.43 1.02 2.82
CH2 1.36 4.14 10.67 0.77 0.92 1.52 0.33 0.97 2.73
30 CH3 1.39 3.33 8.59 0.54 0.72 1.47 0.37 0.69 2.23
CH4 0.8 2.64 4.42 0.56 0.51 1.1 0.35 0.83 0.81
CH5 1.64 2.53 3.96 0.7 0.73 0.99 0.42 0.91 0.82
CH1 1.44 3.81 23.96 0.59 1.91 2.04 0.51 1.12 1.41
CH2 1.58 2.74 17.86 0.62 1.4 2.22 0.4 0.77 1.27
40 CH3 1.06 2.16 13.1 0.51 1.56 1.97 0.18 0.83 1.01
CH4 0.85 2.31 1.92 0.56 0.65 1.4 0.28 0.57 0.44
CH5 1.42 3.14 3.68 0.69 0.87 1.7 0.36 0.53 0.55
CH1 1.47 5.64 15.22 0.68 1.97 2.68 0.42 0.89 2.04
CH2 1.32 5.29 11.23 0.4 1.41 2.37 0.32 0.59 1.64
50 CH3 0.85 4.06 7.54 0.42 1.27 1.85 0.27 0.74 1.3
CH4 0.74 3.81 4.92 0.43 0.89 1.26 0.27 0.37 0.92
CH5 1.23 4.04 9.89 0.63 0.84 1.29 0.52 0.45 0.91
CH1 1.4 4.32 11.36 0.56 2.63 2.47 0.43 0.84 1.95
CH2 1.24 5.41 10.53 0.57 1.87 2.23 0.34 0.62 1.41
60 CH3 1.21 2.73 7.78 0.38 1.34 0.97 0.29 0.49 0.96
CH4 1.27 2.63 2.8 0.37 1.17 0.78 0.32 0.33 0.61
CH5 1.37 2.43 5.1 0.51 1.16 0.9 0.4 0.65 0.9
CH1 1.23 5.9 11.68 0.77 1.81 2.68 0.31 1.22 1.2
CH2 1.05 3.84 8.2 0.59 1.33 2.24 0.25 0.78 0.98
70 CH3 1.08 3.58 5.97 0.44 0.95 1.46 0.24 0.57 0.74
CH4 0.9 3.1 4.29 0.37 0.69 0.9 0.28 0.54 0.54
CH5 1.78 3.13 3.71 0.55 1.45 1.58 0.4 0.65 0.69
Table 4.  Errors of Heuristics when $ \Delta = 20 $
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 2.47 6.96 24.94 0.87 3.03 4.50 0.78 1.22 2.51
CH2 1.38 3.93 8.76 0.54 2.46 3.79 0.66 0.88 1.80
30 CH3 1.22 3.42 12.21 0.56 1.47 3.14 0.47 0.49 1.52
CH4 0.91 4.07 10.30 0.69 1.51 1.66 0.46 0.69 1.28
CH5 1.73 4.18 14.58 1.19 1.42 1.33 0.65 0.92 1.23
CH1 1.71 7.57 19.56 0.94 1.86 4.27 0.64 1.43 2.98
CH2 1.16 4.79 14.49 0.66 1.06 4.13 0.47 1.00 2.43
40 CH3 1.13 5.53 11.18 0.62 0.74 3.01 0.36 0.88 1.35
CH4 1.07 4.63 7.06 0.63 0.82 1.75 0.37 0.55 0.65
CH5 1.67 5.90 7.55 0.84 1.67 1.77 0.57 0.89 0.85
CH1 2.39 4.18 14.61 1.07 2.78 4.73 0.82 1.63 2.65
CH2 1.51 4.17 12.04 0.73 2.86 3.57 0.50 1.17 2.11
50 CH3 1.32 2.37 7.56 0.71 1.55 2.27 0.31 0.76 1.16
CH4 1.30 3.05 4.35 0.67 0.83 1.52 0.41 0.65 1.00
CH5 2.24 4.15 8.13 0.81 1.43 1.65 0.77 0.95 1.17
CH1 1.97 7.43 20.36 1.11 2.23 4.04 0.56 1.64 2.99
CH2 1.38 5.07 15.71 0.62 1.66 2.71 0.32 1.16 2.34
60 CH3 1.25 5.21 6.99 0.47 1.20 2.20 0.39 0.67 1.75
CH4 1.13 2.82 6.56 0.48 0.82 1.33 0.33 0.76 0.68
CH5 1.77 4.61 13.20 1.00 1.61 1.81 0.71 1.06 0.85
CH1 2.06 6.87 17.54 0.87 2.87 4.90 0.69 1.32 2.97
CH2 1.32 4.18 12.66 0.38 1.90 2.90 0.40 0.83 1.85
70 CH3 1.11 2.76 7.95 0.42 1.61 2.17 0.37 0.69 1.20
CH4 1.01 2.74 2.49 0.45 1.31 1.37 0.41 0.78 0.79
CH5 2.21 4.74 6.85 0.94 1.75 2.02 0.58 0.86 0.98
T=0.25 T=0.5 T=0.75
n Heuristic R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75 R=0.25 R=0.5 R=0.75
CH1 2.47 6.96 24.94 0.87 3.03 4.50 0.78 1.22 2.51
CH2 1.38 3.93 8.76 0.54 2.46 3.79 0.66 0.88 1.80
30 CH3 1.22 3.42 12.21 0.56 1.47 3.14 0.47 0.49 1.52
CH4 0.91 4.07 10.30 0.69 1.51 1.66 0.46 0.69 1.28
CH5 1.73 4.18 14.58 1.19 1.42 1.33 0.65 0.92 1.23
CH1 1.71 7.57 19.56 0.94 1.86 4.27 0.64 1.43 2.98
CH2 1.16 4.79 14.49 0.66 1.06 4.13 0.47 1.00 2.43
40 CH3 1.13 5.53 11.18 0.62 0.74 3.01 0.36 0.88 1.35
CH4 1.07 4.63 7.06 0.63 0.82 1.75 0.37 0.55 0.65
CH5 1.67 5.90 7.55 0.84 1.67 1.77 0.57 0.89 0.85
CH1 2.39 4.18 14.61 1.07 2.78 4.73 0.82 1.63 2.65
CH2 1.51 4.17 12.04 0.73 2.86 3.57 0.50 1.17 2.11
50 CH3 1.32 2.37 7.56 0.71 1.55 2.27 0.31 0.76 1.16
CH4 1.30 3.05 4.35 0.67 0.83 1.52 0.41 0.65 1.00
CH5 2.24 4.15 8.13 0.81 1.43 1.65 0.77 0.95 1.17
CH1 1.97 7.43 20.36 1.11 2.23 4.04 0.56 1.64 2.99
CH2 1.38 5.07 15.71 0.62 1.66 2.71 0.32 1.16 2.34
60 CH3 1.25 5.21 6.99 0.47 1.20 2.20 0.39 0.67 1.75
CH4 1.13 2.82 6.56 0.48 0.82 1.33 0.33 0.76 0.68
CH5 1.77 4.61 13.20 1.00 1.61 1.81 0.71 1.06 0.85
CH1 2.06 6.87 17.54 0.87 2.87 4.90 0.69 1.32 2.97
CH2 1.32 4.18 12.66 0.38 1.90 2.90 0.40 0.83 1.85
70 CH3 1.11 2.76 7.95 0.42 1.61 2.17 0.37 0.69 1.20
CH4 1.01 2.74 2.49 0.45 1.31 1.37 0.41 0.78 0.79
CH5 2.21 4.74 6.85 0.94 1.75 2.02 0.58 0.86 0.98
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