American Institute of Mathematical Sciences

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September  2020, 16(5): 2439-2457. doi: 10.3934/jimo.2019062

Minimizing total completion time in a two-machine no-wait flowshop with uncertain and bounded setup times

Muberra Allahverdi 1,, and  Ali Allahverdi 2,

1. 

Department of Mathematical Sciences, Kean University, New Jersey, USA
2. 

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

1Corresponding Author: Muberra Allahverdi

Received  October 2018 Revised  February 2019 Published  September 2020 Early access  May 2019

We address a two-machine no-wait flowshop scheduling problem with respect to the performance measure of total completion time. Minimizing total completion time is important when inventory cost is of concern. Setup times are treated separately from processing times. Furthermore, setup times are uncertain with unknown distributions and are within some lower and upper bounds. We develop a dominance relation and propose eight algorithms to solve the problem. The proposed algorithms, which assign different weights to the processing and setup times on both machines, convert the two-machine problem into a single-machine one for which an optimal solution is known. We conduct computational experiments to evaluate the proposed algorithms. Computational experiments reveal that one of the proposed algorithms, which assigns the same weight to setup and processing times, is superior to the rest of the algorithms. The results are statistically verified by constructing confidence intervals and test of hypothesis.

Keywords: Scheduling, no-wait flowshop, total completion time, uncerain setup times.
Mathematics Subject Classification: Primary: 90; Secondary: 90B36.
Citation: Muberra Allahverdi, Ali Allahverdi. Minimizing total completion time in a two-machine no-wait flowshop with uncertain and bounded setup times. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2439-2457. doi: 10.3934/jimo.2019062
References:
[1]

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.

[2]

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.

[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 total completion time with bounded setup and processing times, Int. Journal of Production Economics, 103 (2006a), 386-400. 

[5]

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. 

[6]

A. Allahverdi, T. 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. Allahverdi and H. Aydilek, Heuristics for two-machine flowshop scheduling problem to minimize maximum lateness with bounded processing times, Computers and Mathematics with Applications, 60 (2010), 1374-1384.  doi: 10.1016/j.camwa.2010.06.019.

[9]

A. Aydilek, H. 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.  doi: 10.1080/00207543.2011.652263.

[10]

A. Aydilek, H. 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.  doi: 10.1080/00207543.2014.997403.

[11]

A. Aydilek, H. Aydilek and A. Allahverdi, Algorithms for minimizing the number of tardy jobs for reducing production cost with uncertain processing times, Applied Mathematical Modelling, 45 (2017), 982-996.  doi: 10.1016/j.apm.2017.01.039.

[12]

O. Braun, T. C. Lai, G. Schmidt and Y. N. Sotskov, Stability of Johnson's schedule with respect to limited machine availability, Int. Journal of Production Research, 40 (2002), 4381-4400. 

[13]

A. A. Cunningham and S. K. Dutta, Scheduling jobs with exponentially distributed processing times on two machines of a flow shop, Naval Research Logistics Quarterly, 20 (1973), 69-81.  doi: 10.1002/nav.3800200107.

[14]

O. Engin and A. Güçlü, A new hybrid ant colony optimization algorithm for solving the no-wait flow shop scheduling problems, Applied Soft Computing Journal, 72 (2018), 166-176.  doi: 10.1016/j.asoc.2018.08.002.

[15]

O. Engin and C. Günaydin, An adaptive learning approach for no-wait flowshop scheduling problems to minimize makespan, International Journal of Computational Intelligence Systems, 4 (2011), 521-529.  doi: 10.1080/18756891.2011.9727810.

[16]

E. M. Gonzalez-Neira, D. Ferone, S. 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.  doi: 10.1016/j.simpat.2017.09.001.

[17]

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.

[18]

P. J. Kalczynski and J. Kamburowski, A heuristic for minimizing the expected makespan in two-machine flowshops with consistent coefficients of variation, European Journal of Operational Research, 169 (2006), 742-750.  doi: 10.1016/j.ejor.2004.08.045.

[19]

I. H. Karacizmeli and S. N. Ogulata, Energy consumption management in textile finishing plants: A cost effective and sequence dependent scheduling model, Textile and Apparel, 27 (2017), 145-152. 

[20]

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.  doi: 10.1016/S0305-0483(97)00013-3.

[21]

G. M. Kopanos, J. Miguel Lainez and L. Puigjaner, An efficient mixed-integer linear programming scheduling framework for addressing sequence-dependent setup issues in batch plants, Industrial & Engineering Chemistry Research, 48 (2009), 6346-6357.  doi: 10.1021/ie801127t.

[22]

P. S. Ku and S. C. Niu, On Johnson's two-machine flow shop with random processing times, Operations Research, 34 (1986), 130-136.  doi: 10.1287/opre.34.1.130.

[23]

T. C. Lai, Y. N. Sotskov, N. Y. Sotskova and F. Werner, Optimal makespan scheduling with given bounds of processing times, Mathematical and Computer Modelling, 26 (1997), 67-86.  doi: 10.1016/S0895-7177(97)00132-5.

[24]

X. Li, Z. Yang, R. Ruiz, T. Chen and S. Sui, An iterated greedy heuristic for no-wait flow shops with sequence dependent setup times, learning and forgetting effects, Information Sciences, 453 (2018), 408–425. doi: 10.1016/j.ins.2018.04.038.

[25]

R. Macchiaroli, S. Molè and S. Riemma, Modelling and optimization of industrial manufacturing processes subject to no-wait constraints, Int. Journal of Production Research, 37 (1999), 2585-2607.  doi: 10.1080/002075499190671.

[26]

N. M. Matsveichuk, Y. N. Sotskov, N. G. Egorova and T. C. Lai, Schedule execution for two-machine flow-shop with interval processing times, Mathematical and Computer Modelling, 49 (2009), 991-1011.  doi: 10.1016/j.mcm.2008.02.004.

[27]

N. M. Matsveichuk, Y. 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.

[28]

M. Pinedo, Stochastic scheduling with release dates and due dates, Operations Research, 31 (1983), 559-572.  doi: 10.1287/opre.31.3.559.

[29]

M. Pinedo, Scheduling Theory, Algorithms, and Systems, Prentice Hall, Englewood Cliffs, New Jersey, Page 349, 1995.

[30]

V. Portougal and D. Trietsch, Johnson's problem with stochastic processing times and optimal service level, European Journal of Operational Research, 169 (2006), 751-760.  doi: 10.1016/j.ejor.2004.09.056.

[31]

V. Riahi and M. Kazemi, A new hybrid ant colony algorithm for scheduling of no-wait flowshop, Operational Research, 18 (2018), 55-74.  doi: 10.1007/s12351-016-0253-x.

[32]

D. K. Seo, C. M. Klein and W. Jang, Single machine stochastic scheduling to minimize the expected number of tardy jobs using mathematical programming models, Computers and Industrial Engineering, 48 (2005), 153-161.  doi: 10.1016/j.cie.2005.01.002.

[33]

H. M. Soroush, Sequencing and due-date determination in the stochastic single machine problem with earliness and tardiness costs, European Journal of Operational Research, 113 (1999), 450-468. 

[34]

H. M. Soroush, Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem., European Journal of Operational Research, 181 (2007), 266-287.  doi: 10.1016/j.ejor.2006.05.036.

[35]

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.

[36]

Y. N. Sotskov, N. 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.

[37]

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.

[38]

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.  doi: 10.1080/00207543.2011.571456.

[39]

Y. Wang, X. Li, R. Ruiz and S. Sui, An iterated greedy heuristic for mixed no-wait flowshop problems, IEEE Transactions on Cybernetics, 48 (2018), 1553-1566.  doi: 10.1109/TCYB.2017.2707067.

[40]

K. C. Ying and S. W. Lin, Minimizing makespan for no-wait flowshop scheduling problems with setup times, Computers and Industrial Engineering, 121 (2018), 73-81.  doi: 10.1016/j.cie.2018.05.030.

show all references

References:
[1]

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.

[2]

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.

[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 total completion time with bounded setup and processing times, Int. Journal of Production Economics, 103 (2006a), 386-400. 

[5]

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. 

[6]

A. Allahverdi, T. 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. Allahverdi and H. Aydilek, Heuristics for two-machine flowshop scheduling problem to minimize maximum lateness with bounded processing times, Computers and Mathematics with Applications, 60 (2010), 1374-1384.  doi: 10.1016/j.camwa.2010.06.019.

[9]

A. Aydilek, H. 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.  doi: 10.1080/00207543.2011.652263.

[10]

A. Aydilek, H. 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.  doi: 10.1080/00207543.2014.997403.

[11]

A. Aydilek, H. Aydilek and A. Allahverdi, Algorithms for minimizing the number of tardy jobs for reducing production cost with uncertain processing times, Applied Mathematical Modelling, 45 (2017), 982-996.  doi: 10.1016/j.apm.2017.01.039.

[12]

O. Braun, T. C. Lai, G. Schmidt and Y. N. Sotskov, Stability of Johnson's schedule with respect to limited machine availability, Int. Journal of Production Research, 40 (2002), 4381-4400. 

[13]

A. A. Cunningham and S. K. Dutta, Scheduling jobs with exponentially distributed processing times on two machines of a flow shop, Naval Research Logistics Quarterly, 20 (1973), 69-81.  doi: 10.1002/nav.3800200107.

[14]

O. Engin and A. Güçlü, A new hybrid ant colony optimization algorithm for solving the no-wait flow shop scheduling problems, Applied Soft Computing Journal, 72 (2018), 166-176.  doi: 10.1016/j.asoc.2018.08.002.

[15]

O. Engin and C. Günaydin, An adaptive learning approach for no-wait flowshop scheduling problems to minimize makespan, International Journal of Computational Intelligence Systems, 4 (2011), 521-529.  doi: 10.1080/18756891.2011.9727810.

[16]

E. M. Gonzalez-Neira, D. Ferone, S. 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.  doi: 10.1016/j.simpat.2017.09.001.

[17]

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.

[18]

P. J. Kalczynski and J. Kamburowski, A heuristic for minimizing the expected makespan in two-machine flowshops with consistent coefficients of variation, European Journal of Operational Research, 169 (2006), 742-750.  doi: 10.1016/j.ejor.2004.08.045.

[19]

I. H. Karacizmeli and S. N. Ogulata, Energy consumption management in textile finishing plants: A cost effective and sequence dependent scheduling model, Textile and Apparel, 27 (2017), 145-152. 

[20]

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.  doi: 10.1016/S0305-0483(97)00013-3.

[21]

G. M. Kopanos, J. Miguel Lainez and L. Puigjaner, An efficient mixed-integer linear programming scheduling framework for addressing sequence-dependent setup issues in batch plants, Industrial & Engineering Chemistry Research, 48 (2009), 6346-6357.  doi: 10.1021/ie801127t.

[22]

P. S. Ku and S. C. Niu, On Johnson's two-machine flow shop with random processing times, Operations Research, 34 (1986), 130-136.  doi: 10.1287/opre.34.1.130.

[23]

T. C. Lai, Y. N. Sotskov, N. Y. Sotskova and F. Werner, Optimal makespan scheduling with given bounds of processing times, Mathematical and Computer Modelling, 26 (1997), 67-86.  doi: 10.1016/S0895-7177(97)00132-5.

[24]

X. Li, Z. Yang, R. Ruiz, T. Chen and S. Sui, An iterated greedy heuristic for no-wait flow shops with sequence dependent setup times, learning and forgetting effects, Information Sciences, 453 (2018), 408–425. doi: 10.1016/j.ins.2018.04.038.

[25]

R. Macchiaroli, S. Molè and S. Riemma, Modelling and optimization of industrial manufacturing processes subject to no-wait constraints, Int. Journal of Production Research, 37 (1999), 2585-2607.  doi: 10.1080/002075499190671.

[26]

N. M. Matsveichuk, Y. N. Sotskov, N. G. Egorova and T. C. Lai, Schedule execution for two-machine flow-shop with interval processing times, Mathematical and Computer Modelling, 49 (2009), 991-1011.  doi: 10.1016/j.mcm.2008.02.004.

[27]

N. M. Matsveichuk, Y. 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.

[28]

M. Pinedo, Stochastic scheduling with release dates and due dates, Operations Research, 31 (1983), 559-572.  doi: 10.1287/opre.31.3.559.

[29]

M. Pinedo, Scheduling Theory, Algorithms, and Systems, Prentice Hall, Englewood Cliffs, New Jersey, Page 349, 1995.

[30]

V. Portougal and D. Trietsch, Johnson's problem with stochastic processing times and optimal service level, European Journal of Operational Research, 169 (2006), 751-760.  doi: 10.1016/j.ejor.2004.09.056.

[31]

V. Riahi and M. Kazemi, A new hybrid ant colony algorithm for scheduling of no-wait flowshop, Operational Research, 18 (2018), 55-74.  doi: 10.1007/s12351-016-0253-x.

[32]

D. K. Seo, C. M. Klein and W. Jang, Single machine stochastic scheduling to minimize the expected number of tardy jobs using mathematical programming models, Computers and Industrial Engineering, 48 (2005), 153-161.  doi: 10.1016/j.cie.2005.01.002.

[33]

H. M. Soroush, Sequencing and due-date determination in the stochastic single machine problem with earliness and tardiness costs, European Journal of Operational Research, 113 (1999), 450-468. 

[34]

H. M. Soroush, Minimizing the weighted number of early and tardy jobs in a stochastic single machine scheduling problem., European Journal of Operational Research, 181 (2007), 266-287.  doi: 10.1016/j.ejor.2006.05.036.

[35]

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.

[36]

Y. N. Sotskov, N. 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.

[37]

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.

[38]

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.  doi: 10.1080/00207543.2011.571456.

[39]

Y. Wang, X. Li, R. Ruiz and S. Sui, An iterated greedy heuristic for mixed no-wait flowshop problems, IEEE Transactions on Cybernetics, 48 (2018), 1553-1566.  doi: 10.1109/TCYB.2017.2707067.

[40]

K. C. Ying and S. W. Lin, Minimizing makespan for no-wait flowshop scheduling problems with setup times, Computers and Industrial Engineering, 121 (2018), 73-81.  doi: 10.1016/j.cie.2018.05.030.

Figure 1.  Flowchart of the algorithms
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Figure 2.  The distributions used for generating $ s_{i, k} $ within $ Ls_{i, k} $ and $ Us_{i, k} $
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Figure 3.  Overall Avg. Error of Algorithms
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Figure 4.  Avg. Std. of Algorithms
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Figure 5.  Overall Avg. Error of Algorithms with respect to $H$
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Figure 6.  Overall Avg. Error of Algorithms with respect to $H$
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Figure 7.  Overall Avg. Error of Algorithm $ALG-6$ with respect to $H$
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Figure 8.  Overall Avg. Error of Algorithms with respect to distributions
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Table 1.  Error of Algorithms for Positive Linear Distribution
$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.08 9.11 9.17 9.17 9.27 9.16
$ALG-2$ 9.38 9.37 9.28 9.34 9.3 9.33
$ALG-3$ 3.05 2.98 2.95 2.96 2.97 2.98
$ALG-4$ 4.99 4.93 4.88 4.9 4.9 4.92
$ALG-5$ 4.85 4.81 4.76 4.81 4.83 4.79
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.23 4.1 4.02 4 3.95 4.06
$ALG-8$ 2.44 2.35 2.44 2.44 2.44 2.42
$ALG-1$ 30 9.17 9.33 9.42 9.4 9.4 9.34
$ALG-2$ 9.5 9.56 9.48 9.47 9.47 9.50
$ALG-3$ 3.04 3.1 3.03 3.03 3 3.04
$ALG-4$ 5.04 5.07 4.97 4.98 4.98 5.01
$ALG-5$ 4.81 4.93 4.95 4.95 4.95 4.92
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.2 4.12 4.02 4.04 3.96 4.07
$ALG-8$ 2.48 2.45 2.46 2.49 2.49 2.47
$ALG-1$ 40 9.37 9.47 9.49 9.54 9.56 9.49
$ALG-2$ 9.65 9.69 9.64 9.63 9.67 9.66
$ALG-3$ 3.01 3.07 3.06 3.05 3.08 3.05
$ALG-4$ 5.06 5.06 5.06 5.05 5.06 5.06
$ALG-5$ 4.93 5.01 4.96 5 5.05 4.99
$ALG-6$ 0.09 0.02 0 0 0 0.02
$ALG-7$ 4.24 4.17 4.03 4.1 4.12 4.13
$ALG-8$ 2.47 2.51 2.53 2.51 2.57 2.52
Avg. 4.80 4.80 4.78 4.79 4.79
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$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.08 9.11 9.17 9.17 9.27 9.16
$ALG-2$ 9.38 9.37 9.28 9.34 9.3 9.33
$ALG-3$ 3.05 2.98 2.95 2.96 2.97 2.98
$ALG-4$ 4.99 4.93 4.88 4.9 4.9 4.92
$ALG-5$ 4.85 4.81 4.76 4.81 4.83 4.79
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.23 4.1 4.02 4 3.95 4.06
$ALG-8$ 2.44 2.35 2.44 2.44 2.44 2.42
$ALG-1$ 30 9.17 9.33 9.42 9.4 9.4 9.34
$ALG-2$ 9.5 9.56 9.48 9.47 9.47 9.50
$ALG-3$ 3.04 3.1 3.03 3.03 3 3.04
$ALG-4$ 5.04 5.07 4.97 4.98 4.98 5.01
$ALG-5$ 4.81 4.93 4.95 4.95 4.95 4.92
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.2 4.12 4.02 4.04 3.96 4.07
$ALG-8$ 2.48 2.45 2.46 2.49 2.49 2.47
$ALG-1$ 40 9.37 9.47 9.49 9.54 9.56 9.49
$ALG-2$ 9.65 9.69 9.64 9.63 9.67 9.66
$ALG-3$ 3.01 3.07 3.06 3.05 3.08 3.05
$ALG-4$ 5.06 5.06 5.06 5.05 5.06 5.06
$ALG-5$ 4.93 5.01 4.96 5 5.05 4.99
$ALG-6$ 0.09 0.02 0 0 0 0.02
$ALG-7$ 4.24 4.17 4.03 4.1 4.12 4.13
$ALG-8$ 2.47 2.51 2.53 2.51 2.57 2.52
Avg. 4.80 4.80 4.78 4.79 4.79
Table 2.  Error of Algorithms for Negative Linear Distribution
$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.08 9.13 9.22 9.26 9.27 9.19
$ALG-2$ 9.38 9.26 9.33 9.35 9.27 9.32
$ALG-3$ 3.05 2.97 3.04 3.02 2.99 3.01
$ALG-4$ 4.99 4.85 4.92 4.9 4.87 4.91
$ALG-5$ 4.75 4.73 4.89 4.92 4.86 4.83
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.23 4.1 4.06 4.07 3.95 4.08
$ALG-8$ 2.44 2.41 2.44 2.5 2.43 2.44
$ALG-1$ 30 9.17 9.37 9.37 9.37 9.41 9.34
$ALG-2$ 9.5 9.51 9.53 9.5 9.51 9.51
$ALG-3$ 3.04 3.05 3.04 3.04 3.05 3.04
$ALG-4$ 5.04 5 5.03 5 4.99 5.01
$ALG-5$ 4.81 4.92 4.9 4.94 4.93 4.90
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.2 4.15 4.1 4.05 4.01 4.10
$ALG-8$ 2.48 2.54 2.46 2.55 2.53 2.51
$ALG-1$ 40 9.37 9.37 9.5 9.51 9.55 9.46
$ALG-2$ 9.65 9.54 9.59 9.64 9.62 9.61
$ALG-3$ 3.01 2.96 3.04 3.09 3.07 3.03
$ALG-4$ 5.06 4.97 5.01 5.06 5.05 5.03
$ALG-5$ 4.93 4.85 4.95 4.97 5.01 4.94
$ALG-6$ 0.09 0.01 0 0 0 0.02
$ALG-7$ 4.24 4.12 4.08 4.07 4.08 4.12
$ALG-8$ 2.47 2.44 2.58 2.54 2.57 2.52
Avg. 4.80 4.76 4.80 4.81 4.79
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$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.08 9.13 9.22 9.26 9.27 9.19
$ALG-2$ 9.38 9.26 9.33 9.35 9.27 9.32
$ALG-3$ 3.05 2.97 3.04 3.02 2.99 3.01
$ALG-4$ 4.99 4.85 4.92 4.9 4.87 4.91
$ALG-5$ 4.75 4.73 4.89 4.92 4.86 4.83
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.23 4.1 4.06 4.07 3.95 4.08
$ALG-8$ 2.44 2.41 2.44 2.5 2.43 2.44
$ALG-1$ 30 9.17 9.37 9.37 9.37 9.41 9.34
$ALG-2$ 9.5 9.51 9.53 9.5 9.51 9.51
$ALG-3$ 3.04 3.05 3.04 3.04 3.05 3.04
$ALG-4$ 5.04 5 5.03 5 4.99 5.01
$ALG-5$ 4.81 4.92 4.9 4.94 4.93 4.90
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.2 4.15 4.1 4.05 4.01 4.10
$ALG-8$ 2.48 2.54 2.46 2.55 2.53 2.51
$ALG-1$ 40 9.37 9.37 9.5 9.51 9.55 9.46
$ALG-2$ 9.65 9.54 9.59 9.64 9.62 9.61
$ALG-3$ 3.01 2.96 3.04 3.09 3.07 3.03
$ALG-4$ 5.06 4.97 5.01 5.06 5.05 5.03
$ALG-5$ 4.93 4.85 4.95 4.97 5.01 4.94
$ALG-6$ 0.09 0.01 0 0 0 0.02
$ALG-7$ 4.24 4.12 4.08 4.07 4.08 4.12
$ALG-8$ 2.47 2.44 2.58 2.54 2.57 2.52
Avg. 4.80 4.76 4.80 4.81 4.79
Table 3.  Error of Algorithm for Uniform Distribution
$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.05 9.32 9.34 9.37 9.41 9.30
$ALG-2$ 9.39 9.45 9.43 9.49 9.43 9.44
$ALG-3$ 3.09 3.02 2.97 3.04 3.04 3.03
$ALG-4$ 4.98 4.96 4.92 4.98 4.96 4.96
$ALG-5$ 4.86 4.88 4.87 4.94 4.95 4.90
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.24 4.08 4.09 4.09 4.05 4.11
$ALG-8$ 2.42 4.08 4.09 4.09 4.05 4.11
$ALG-1$ 30 9.09 9.47 9.52 9.64 9.56 9.46
$ALG-2$ 9.59 9.61 9.63 9.67 9.59 9.63
$ALG-3$ 3.1 3.04 3.04 3.09 3.03 3.06
$ALG-4$ 5.13 5.03 5.02 5.06 5 5.05
$ALG-5$ 4.86 4.94 4.96 5.05 5.02 4.97
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.17 4.17 4.08 4.13 4.06 4.12
$ALG-8$ 2.44 2.49 2.53 2.59 2.56 2.52
$ALG-1$ 40 9.32 9.68 9.68 9.7 9.79 9.63
$ALG-2$ 9.67 9.89 9.85 9.87 9.89 9.83
$ALG-3$ 3.05 3.06 3.06 3.08 3.1 3.07
$ALG-4$ 5.07 5.14 5.13 5.13 5.13 5.12
$ALG-5$ 4.86 5.09 5.04 5.07 5.09 5.03
$ALG-6$ 0.09 0.01 0 0 0 0.02
$ALG-7$ 4.31 4.28 4.23 4.19 4.21 4.24
$ALG-8$ 2.43 2.61 2.64 2.68 2.68 2.61
Avg. 4.81 4.86 4.86 4.89 4.88
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$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.05 9.32 9.34 9.37 9.41 9.30
$ALG-2$ 9.39 9.45 9.43 9.49 9.43 9.44
$ALG-3$ 3.09 3.02 2.97 3.04 3.04 3.03
$ALG-4$ 4.98 4.96 4.92 4.98 4.96 4.96
$ALG-5$ 4.86 4.88 4.87 4.94 4.95 4.90
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.24 4.08 4.09 4.09 4.05 4.11
$ALG-8$ 2.42 4.08 4.09 4.09 4.05 4.11
$ALG-1$ 30 9.09 9.47 9.52 9.64 9.56 9.46
$ALG-2$ 9.59 9.61 9.63 9.67 9.59 9.63
$ALG-3$ 3.1 3.04 3.04 3.09 3.03 3.06
$ALG-4$ 5.13 5.03 5.02 5.06 5 5.05
$ALG-5$ 4.86 4.94 4.96 5.05 5.02 4.97
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.17 4.17 4.08 4.13 4.06 4.12
$ALG-8$ 2.44 2.49 2.53 2.59 2.56 2.52
$ALG-1$ 40 9.32 9.68 9.68 9.7 9.79 9.63
$ALG-2$ 9.67 9.89 9.85 9.87 9.89 9.83
$ALG-3$ 3.05 3.06 3.06 3.08 3.1 3.07
$ALG-4$ 5.07 5.14 5.13 5.13 5.13 5.12
$ALG-5$ 4.86 5.09 5.04 5.07 5.09 5.03
$ALG-6$ 0.09 0.01 0 0 0 0.02
$ALG-7$ 4.31 4.28 4.23 4.19 4.21 4.24
$ALG-8$ 2.43 2.61 2.64 2.68 2.68 2.61
Avg. 4.81 4.86 4.86 4.89 4.88
Table 4.  Error of Algorithm for Normal Distribution
$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.04 9.3 9.32 9.35 9.38 9.28
$ALG-2$ 9.5 9.46 9.45 9.46 9.5 9.47
$ALG-3$ 3.03 3 3 3.03 3.06 3.02
$ALG-4$ 5.01 4.97 4.92 4.96 4.95 4.90
$ALG-5$ 4.81 4.88 4.9 4.94 4.95 4.90
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.27 4.09 4.13 4.08 4.06 4.13
$ALG-8$ 2.47 2.46 2.45 2.5 2.55 2.49
$ALG-1$ 30 9.43 9.43 9.54 9.57 9.71 9.54
$ALG-2$ 9.8 9.71 9.66 9.7 9.72 9.72
$ALG-3$ 3.11 3 3.09 3.08 3.12 3.08
$ALG-4$ 5.16 5.06 5.06 5.05 5.11 5.09
$ALG-5$ 4.99 4.92 5 4.99 5.08 5.00
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.38 4.15 4.17 4.17 4.17 4.21
$ALG-8$ 2.52 2.52 2.59 2.57 2.62 2.56
$ALG-1$ 40 9.57 9.7 9.77 9.78 9.86 9.74
$ALG-2$ 10.04 9.96 9.9 9.86 9.86 9.92
$ALG-3$ 3.12 3.03 3.05 3.07 3.05 3.06
$ALG-4$ 5.25 5.17 5.14 5.11 5.12 5.16
$ALG-5$ 5.01 5.03 5.05 5.11 5.1 5.06
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.47 4.2 4.21 4.24 4.19 4.26
$ALG-8$ 2.69 2.6 2.73 2.71 2.7 2.69
Avg. 4.91 4.86 4.88 4.89 4.91
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$n$
Algorithm $H$ 100 200 300 400 500 Avg.
$ALG-1$ 20 9.04 9.3 9.32 9.35 9.38 9.28
$ALG-2$ 9.5 9.46 9.45 9.46 9.5 9.47
$ALG-3$ 3.03 3 3 3.03 3.06 3.02
$ALG-4$ 5.01 4.97 4.92 4.96 4.95 4.90
$ALG-5$ 4.81 4.88 4.9 4.94 4.95 4.90
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.27 4.09 4.13 4.08 4.06 4.13
$ALG-8$ 2.47 2.46 2.45 2.5 2.55 2.49
$ALG-1$ 30 9.43 9.43 9.54 9.57 9.71 9.54
$ALG-2$ 9.8 9.71 9.66 9.7 9.72 9.72
$ALG-3$ 3.11 3 3.09 3.08 3.12 3.08
$ALG-4$ 5.16 5.06 5.06 5.05 5.11 5.09
$ALG-5$ 4.99 4.92 5 4.99 5.08 5.00
$ALG-6$ 0.1 0.02 0 0 0 0.02
$ALG-7$ 4.38 4.15 4.17 4.17 4.17 4.21
$ALG-8$ 2.52 2.52 2.59 2.57 2.62 2.56
$ALG-1$ 40 9.57 9.7 9.77 9.78 9.86 9.74
$ALG-2$ 10.04 9.96 9.9 9.86 9.86 9.92
$ALG-3$ 3.12 3.03 3.05 3.07 3.05 3.06
$ALG-4$ 5.25 5.17 5.14 5.11 5.12 5.16
$ALG-5$ 5.01 5.03 5.05 5.11 5.1 5.06
$ALG-6$ 0.08 0.01 0 0 0 0.02
$ALG-7$ 4.47 4.2 4.21 4.24 4.19 4.26
$ALG-8$ 2.69 2.6 2.73 2.71 2.7 2.69
Avg. 4.91 4.86 4.88 4.89 4.91
Table 5.  Error of Algoithms with respect to $ n $
n
Algorithm 100 200 300 400 500 Avg.
$ALG-1$ 9.27 9.39 9.45 9.47 9.51 9.42
$ALG-2$ 9.64 9.58 9.56 9.58 9.57 9.59
$ALG-3$ 3.07 3.02 3.03 3.05 3.05 3.04
$ALG-4$ 5.09 5.02 5.01 5.02 5.01 5.03
$ALG-5$ 4.88 4.92 4.94 4.97 4.99 4.94
$ALG-6$ 0.09 0.02 0.00 0.00 0.00 0.02
$ALG-7$ 4.28 4.14 4.10 4.10 4.07 4.14
$ALG-8$ 2.49 2.49 2.53 2.55 2.56 2.52
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n
Algorithm 100 200 300 400 500 Avg.
$ALG-1$ 9.27 9.39 9.45 9.47 9.51 9.42
$ALG-2$ 9.64 9.58 9.56 9.58 9.57 9.59
$ALG-3$ 3.07 3.02 3.03 3.05 3.05 3.04
$ALG-4$ 5.09 5.02 5.01 5.02 5.01 5.03
$ALG-5$ 4.88 4.92 4.94 4.97 4.99 4.94
$ALG-6$ 0.09 0.02 0.00 0.00 0.00 0.02
$ALG-7$ 4.28 4.14 4.10 4.10 4.07 4.14
$ALG-8$ 2.49 2.49 2.53 2.55 2.56 2.52
Table 6.  Error of Algorithms with respect to $ H $
$H$
Algorithm 20 30 40 Avg.
$ALG-1$ 9.24 9.42 9.59 9.42
$ALG-2$ 9.40 9.60 9.77 9.59
$ALG-3$ 3.01 3.06 3.06 3.04
$ALG-4$ 4.94 5.04 5.10 5.03
$ALG-5$ 4.86 4.94 5.01 4.94
$ALG-6$ 0.02 0.02 0.02 0.02
$ALG-7$ 4.10 4.13 4.19 4.14
$ALG-8$ 2.46 2.52 2.59 2.52
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$H$
Algorithm 20 30 40 Avg.
$ALG-1$ 9.24 9.42 9.59 9.42
$ALG-2$ 9.40 9.60 9.77 9.59
$ALG-3$ 3.01 3.06 3.06 3.04
$ALG-4$ 4.94 5.04 5.10 5.03
$ALG-5$ 4.86 4.94 5.01 4.94
$ALG-6$ 0.02 0.02 0.02 0.02
$ALG-7$ 4.10 4.13 4.19 4.14
$ALG-8$ 2.46 2.52 2.59 2.52
Table 7.  Confidence Intervals for Average Errors
Algorithm 95$\%$ Confidence Interval on the Avg. Error
$ALG-1$ (09.33-9.51)
$ALG-2$ (9.50-9.68)
$ALG-3$ (2.98-3.11)
$ALG-4$ (4.95-5.10)
$ALG-5$ (4.86-5.01)
$ALG-6$ (0.02-0.03)
$ALG-7$ (4.07-4.21)
$ALG-8$ (2.45-2.59)
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Algorithm 95$\%$ Confidence Interval on the Avg. Error
$ALG-1$ (09.33-9.51)
$ALG-2$ (9.50-9.68)
$ALG-3$ (2.98-3.11)
$ALG-4$ (4.95-5.10)
$ALG-5$ (4.86-5.01)
$ALG-6$ (0.02-0.03)
$ALG-7$ (4.07-4.21)
$ALG-8$ (2.45-2.59)
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