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May  2021, 17(3): 1411-1422. doi: 10.3934/jimo.2020027

Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations

Tzu-Hsin Liu , Jau-Chuan Ke , and  Ching-Chang Kuo

Department of Applied Statistics, National Taichung University of Science and Technology, Taichung, Taiwan

* Corresponding author: Jau-Chuan Ke

Received  January 2019 Revised  June 2019 Published  February 2020

The purpose of this paper is to explore the multiple-server machine interference problem with standby unsuccessful switchover and Bernoulli vacation schedule. Failure times of operating and standby machines are assumed to have exponential distributions and repair times of the failed machines and vacation times of servers are also assumed to have exponential distributions. After the completion of service, the server either goes for a vacation or may continue serving for the next machine. The vacation policy we considered is a single vacation policy. In practice, the switchover may experience a significant failure. The matrix analytical method and recursive method are employed to obtain the steady-state probability vectors, and closed-form expressions of some important system characteristics are obtained. The problem of cost optimization dealt with a number of numerical examples is provided by the Quasi-Newton method, the pattern search method, and the Nelder-Mead simplex direct search method. Expressions of various system characteristics are derived. Sensitivity analysis is performed numerically for system parameters. This paper presents the first time that machine interference problem with unsuccessful switchover for a group of repairable servers with vacations has been obtained, which is quite useful for the decision makers.

Keywords: Bernoulli vacation; Cost, Standbys unsuccessful switchover, Recursive method, Quasi-Newton method, Pattern search method, Nelder-Mead simplex direct search method.
Mathematics Subject Classification: Primary: 90B22; Secondary: 60K25.
Citation: Tzu-Hsin Liu, Jau-Chuan Ke, Ching-Chang Kuo. Machine interference problem involving unsuccessful switchover for a group of repairable servers with vacations. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1411-1422. doi: 10.3934/jimo.2020027
References:
[1]

W. L. Chen and K. H. Wang, Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy, Reliability Engineering & System Safety, 180 (2018), 476-486.  doi: 10.1016/j.ress.2018.08.011.  Google Scholar

[2]

G. Choudhury and J. C. Ke, An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule, Applied Mathematics and Computation, 230 (2014), 436-450.  doi: 10.1016/j.amc.2013.12.108.  Google Scholar

[3]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211.  doi: 10.1016/S0166-5316(96)00046-6.  Google Scholar

[4]

G. He, W. Wu and Y. Zhang, Performance analysis of machine repair system with single working vacation, Communications in Statistics – Theory and Methods, 48 (2019), 5602-5620.  doi: 10.1080/03610926.2018.1515958.  Google Scholar

[5]

Y. L. Hsu, J. C. Ke and T. H. Liu, Standby system with general repair, reboot delay, switching failure and unreliable repair facility – A statistical standpoint, Mathematics and Computers in Simulation, 81 (2011), 2400-2413.  doi: 10.1016/j.matcom.2011.03.003.  Google Scholar

[6]

Y. L. Hsu, J. C. Ke, T. H. Liu and C. H. Wu, Modeling of multi-server repair problem with switching failure and reboot delay and related profit analysis, Computers & Industrial Engineering, 69 (2014), 21-28.  doi: 10.1016/j.cie.2013.12.003.  Google Scholar

[7]

H. I. Huang, C. H. Lin and J. C. Ke, Parametric nonlinear programming approach for a repairable system with switching failure and fuzzy parameters, Applied Mathematics and Computation, 183 (2006), 508-517.  doi: 10.1016/j.amc.2006.05.119.  Google Scholar

[8]

J. B. Ke, J. W. Chen and K. H. Wang, Reliability measures of a repairable system with standby switching failures and reboot delay, Quality Technology and Quantitative Managementl, 8 (2011), 15-26.  doi: 10.1080/16843703.2011.11673243.  Google Scholar

[9]

J. B. Ke, W. C. Lee and J. C. Ke, Reliability-based measure for a system with standbys subjected to switching failures, Engineering Computations, 25 (2008), 694-706.  doi: 10.1108/02644400810899979.  Google Scholar

[10]

J. C. Ke, K. B. Huang and W. L. Pearn, A batch arrival queue under randomized multi-vacation policy with unreliable server and repair, International Journal of Systems Science, 43 (2012), 552-565.  doi: 10.1080/00207721.2010.517863.  Google Scholar

[11]

J. C. Ke, S. L. Lee and C. H. Liou, Machine repair problem in production systems with spares and server vacations, RAIRO Operation Research, 43 (2009), 35-54.  doi: 10.1051/ro/2009004.  Google Scholar

[12]

J. C. Ke, T. H. Liu and D. Y. Yang, Machine repairing systems with standby switching failure, Computers & Industrial Engineering, 99 (2016), 223-228.  doi: 10.1016/j.cie.2016.07.016.  Google Scholar

[13]

J. C. Ke, T. H. Liu and D. Y. Yang, Modeling of machine interference problem with unreliable repairman and standbys imperfect switchover, Reliability Engineering & System Safety, 174 (2018), 12-18.  doi: 10.1016/j.ress.2018.01.013.  Google Scholar

[14]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894.  doi: 10.1016/j.apm.2006.02.009.  Google Scholar

[15]

J. C. Ke and C. H. Wu, Multi-server machine repair model with standbys and synchronous multiple vacation, Computers & Industrial Engineering, 62 (2012), 296-305.  doi: 10.1016/j.cie.2011.09.017.  Google Scholar

[16]

B. Keren, G. Gurevich and Y. Hadad, Machines interference problem with several operatiors and several service types that have different priorities, International Journal of Operational Research, 30 (2017), 289-320.  doi: 10.1504/IJOR.2017.087274.  Google Scholar

[17]

K. Kumar, M. Jain and C. Shekhar, Machine repair system with F-policy, two unreliable servers, and warm standbys, Journal of Testing and Evaluation, 47 (2019), 361-383.  doi: 10.1520/JTE20160595.  Google Scholar

[18]

C. C. Kuo and J. C. Ke, Comparative analysis of standby systems with unreliable server and switching failure, Reliability Engineering and System Safety, 145 (2016), 74-82.  doi: 10.1016/j.ress.2015.09.001.  Google Scholar

[19]

Y. Lee, Availability analysis of redundancy model with generally distributed repair time, imperfect switchover and interrupted repair, Electronics Letters, 52 (2016), 1851-1853.  doi: 10.1049/el.2016.2114.  Google Scholar

[20]

E. E. Lewis, Introduction to Reliability Engineering, John Wiley & Sons, Inc., New York, 1987.  Google Scholar

[21]

T. H. Liu, J. C. Ke, Y. L. Hsu and Y. L. Hsu, Bootstrapping computation of availability for a repairable system with standby subject to imperfect switching, Communications in statistics – Simulation and Computation, 40 (2011), 469-483.  doi: 10.1080/03610918.2010.546539.  Google Scholar

[22]

S. Maheshwari, R. Supriya and M. Jain, Machine repair problem with K-type warm spares, multiple vacation for repairman and reneging, International Journal of Engineering and Technology, 2 (2010), 252-258.   Google Scholar

[23]

S. J. Sadjadi and R. Soltani, Minimum-Maximum regret redundancy allocation with the choice of redundancy strategy and multiple choice of component type under uncertainty, Computers & Industrial Engineering, 79 (2015), 204-213.   Google Scholar

[24]

C. Shekhar, M. Jain, A. Raina and R. Mishra, Sensitivity analysis of repairable redundant system with switching failure and geometric reneging, Decision Science Letters, 6 (2017), 337-350.  doi: 10.5267/j.dsl.2017.2.004.  Google Scholar

[25]

R. K. Shrivastava and A. K. Mishra, Analysis of queueing model for machine repairing system with Bernoulli vacation schedule, International Journal of Mathematics Trends and Technology, 10 (2014), 85-92.  doi: 10.14445/22315373/IJMTT-V10P514.  Google Scholar

[26]

S. R. Srinivas, A multi-server synchronous vacation model with thresholds and a probabilistic decision rule, European Journal of Operational Research, 182 (2007), 305-320.  doi: 10.1016/j.ejor.2006.07.037.  Google Scholar

[27]

K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method, Journal of Computational and Applied Mathematics, 233 (2009), 449-458.  doi: 10.1016/j.cam.2009.07.043.  Google Scholar

[28]

K. H. Wang, W. L. Dong and J. B. Ke, Comparison of reliability and the availability between four systems with standby components and standby switching failure, Applied Mathematics and Computation, 183 (2006), 1310-1322.  doi: 10.1016/j.amc.2006.05.161.  Google Scholar

[29]

K. H. Wang, T. C. Yen and J. Y. Chen, Optimization analysis of retrial machine repair problem with server breakdown and threshold recovery policy, Journal of Testing and Evaluation, 46 (2018), 2630-2640.  doi: 10.1520/JTE20160149.  Google Scholar

[30]

C. H. Wu and J. C. Ke, Multi-server machine repair problems under a (V, R) synchronous single vacation policy, Applied Mathematical Modelling, 38 (2014), 2180-2189.  doi: 10.1016/j.apm.2013.10.045.  Google Scholar

[31]

D. Y. Yang and Y. D. Chang, Sensitivity analysis of the machine repair problem with general repeated attempts, International Journal of Computer Mathematics, 95 (2018), 1761-1774.  doi: 10.1080/00207160.2017.1336230.  Google Scholar

[32]

D. Yue, W. Yue and Y. Sun, Performance analysis of an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers, The Sixth International Symposium on Operations Research and Its Applications (ISORA–06), (2006), 128–143. Google Scholar

show all references

References:
[1]

W. L. Chen and K. H. Wang, Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy, Reliability Engineering & System Safety, 180 (2018), 476-486.  doi: 10.1016/j.ress.2018.08.011.  Google Scholar

[2]

G. Choudhury and J. C. Ke, An unreliable retrial queue with delaying repair and general retrial times under Bernoulli vacation schedule, Applied Mathematics and Computation, 230 (2014), 436-450.  doi: 10.1016/j.amc.2013.12.108.  Google Scholar

[3]

S. M. Gupta, Machine interference problem with warm spares, server vacations and exhaustive service, Performance Evaluation, 29 (1997), 195-211.  doi: 10.1016/S0166-5316(96)00046-6.  Google Scholar

[4]

G. He, W. Wu and Y. Zhang, Performance analysis of machine repair system with single working vacation, Communications in Statistics – Theory and Methods, 48 (2019), 5602-5620.  doi: 10.1080/03610926.2018.1515958.  Google Scholar

[5]

Y. L. Hsu, J. C. Ke and T. H. Liu, Standby system with general repair, reboot delay, switching failure and unreliable repair facility – A statistical standpoint, Mathematics and Computers in Simulation, 81 (2011), 2400-2413.  doi: 10.1016/j.matcom.2011.03.003.  Google Scholar

[6]

Y. L. Hsu, J. C. Ke, T. H. Liu and C. H. Wu, Modeling of multi-server repair problem with switching failure and reboot delay and related profit analysis, Computers & Industrial Engineering, 69 (2014), 21-28.  doi: 10.1016/j.cie.2013.12.003.  Google Scholar

[7]

H. I. Huang, C. H. Lin and J. C. Ke, Parametric nonlinear programming approach for a repairable system with switching failure and fuzzy parameters, Applied Mathematics and Computation, 183 (2006), 508-517.  doi: 10.1016/j.amc.2006.05.119.  Google Scholar

[8]

J. B. Ke, J. W. Chen and K. H. Wang, Reliability measures of a repairable system with standby switching failures and reboot delay, Quality Technology and Quantitative Managementl, 8 (2011), 15-26.  doi: 10.1080/16843703.2011.11673243.  Google Scholar

[9]

J. B. Ke, W. C. Lee and J. C. Ke, Reliability-based measure for a system with standbys subjected to switching failures, Engineering Computations, 25 (2008), 694-706.  doi: 10.1108/02644400810899979.  Google Scholar

[10]

J. C. Ke, K. B. Huang and W. L. Pearn, A batch arrival queue under randomized multi-vacation policy with unreliable server and repair, International Journal of Systems Science, 43 (2012), 552-565.  doi: 10.1080/00207721.2010.517863.  Google Scholar

[11]

J. C. Ke, S. L. Lee and C. H. Liou, Machine repair problem in production systems with spares and server vacations, RAIRO Operation Research, 43 (2009), 35-54.  doi: 10.1051/ro/2009004.  Google Scholar

[12]

J. C. Ke, T. H. Liu and D. Y. Yang, Machine repairing systems with standby switching failure, Computers & Industrial Engineering, 99 (2016), 223-228.  doi: 10.1016/j.cie.2016.07.016.  Google Scholar

[13]

J. C. Ke, T. H. Liu and D. Y. Yang, Modeling of machine interference problem with unreliable repairman and standbys imperfect switchover, Reliability Engineering & System Safety, 174 (2018), 12-18.  doi: 10.1016/j.ress.2018.01.013.  Google Scholar

[14]

J. C. Ke and K. H. Wang, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 31 (2007), 880-894.  doi: 10.1016/j.apm.2006.02.009.  Google Scholar

[15]

J. C. Ke and C. H. Wu, Multi-server machine repair model with standbys and synchronous multiple vacation, Computers & Industrial Engineering, 62 (2012), 296-305.  doi: 10.1016/j.cie.2011.09.017.  Google Scholar

[16]

B. Keren, G. Gurevich and Y. Hadad, Machines interference problem with several operatiors and several service types that have different priorities, International Journal of Operational Research, 30 (2017), 289-320.  doi: 10.1504/IJOR.2017.087274.  Google Scholar

[17]

K. Kumar, M. Jain and C. Shekhar, Machine repair system with F-policy, two unreliable servers, and warm standbys, Journal of Testing and Evaluation, 47 (2019), 361-383.  doi: 10.1520/JTE20160595.  Google Scholar

[18]

C. C. Kuo and J. C. Ke, Comparative analysis of standby systems with unreliable server and switching failure, Reliability Engineering and System Safety, 145 (2016), 74-82.  doi: 10.1016/j.ress.2015.09.001.  Google Scholar

[19]

Y. Lee, Availability analysis of redundancy model with generally distributed repair time, imperfect switchover and interrupted repair, Electronics Letters, 52 (2016), 1851-1853.  doi: 10.1049/el.2016.2114.  Google Scholar

[20]

E. E. Lewis, Introduction to Reliability Engineering, John Wiley & Sons, Inc., New York, 1987.  Google Scholar

[21]

T. H. Liu, J. C. Ke, Y. L. Hsu and Y. L. Hsu, Bootstrapping computation of availability for a repairable system with standby subject to imperfect switching, Communications in statistics – Simulation and Computation, 40 (2011), 469-483.  doi: 10.1080/03610918.2010.546539.  Google Scholar

[22]

S. Maheshwari, R. Supriya and M. Jain, Machine repair problem with K-type warm spares, multiple vacation for repairman and reneging, International Journal of Engineering and Technology, 2 (2010), 252-258.   Google Scholar

[23]

S. J. Sadjadi and R. Soltani, Minimum-Maximum regret redundancy allocation with the choice of redundancy strategy and multiple choice of component type under uncertainty, Computers & Industrial Engineering, 79 (2015), 204-213.   Google Scholar

[24]

C. Shekhar, M. Jain, A. Raina and R. Mishra, Sensitivity analysis of repairable redundant system with switching failure and geometric reneging, Decision Science Letters, 6 (2017), 337-350.  doi: 10.5267/j.dsl.2017.2.004.  Google Scholar

[25]

R. K. Shrivastava and A. K. Mishra, Analysis of queueing model for machine repairing system with Bernoulli vacation schedule, International Journal of Mathematics Trends and Technology, 10 (2014), 85-92.  doi: 10.14445/22315373/IJMTT-V10P514.  Google Scholar

[26]

S. R. Srinivas, A multi-server synchronous vacation model with thresholds and a probabilistic decision rule, European Journal of Operational Research, 182 (2007), 305-320.  doi: 10.1016/j.ejor.2006.07.037.  Google Scholar

[27]

K. H. Wang, W. L. Chen and D. Y. Yang, Optimal management of the machine repair problem with working vacation: Newton's method, Journal of Computational and Applied Mathematics, 233 (2009), 449-458.  doi: 10.1016/j.cam.2009.07.043.  Google Scholar

[28]

K. H. Wang, W. L. Dong and J. B. Ke, Comparison of reliability and the availability between four systems with standby components and standby switching failure, Applied Mathematics and Computation, 183 (2006), 1310-1322.  doi: 10.1016/j.amc.2006.05.161.  Google Scholar

[29]

K. H. Wang, T. C. Yen and J. Y. Chen, Optimization analysis of retrial machine repair problem with server breakdown and threshold recovery policy, Journal of Testing and Evaluation, 46 (2018), 2630-2640.  doi: 10.1520/JTE20160149.  Google Scholar

[30]

C. H. Wu and J. C. Ke, Multi-server machine repair problems under a (V, R) synchronous single vacation policy, Applied Mathematical Modelling, 38 (2014), 2180-2189.  doi: 10.1016/j.apm.2013.10.045.  Google Scholar

[31]

D. Y. Yang and Y. D. Chang, Sensitivity analysis of the machine repair problem with general repeated attempts, International Journal of Computer Mathematics, 95 (2018), 1761-1774.  doi: 10.1080/00207160.2017.1336230.  Google Scholar

[32]

D. Yue, W. Yue and Y. Sun, Performance analysis of an M/M/c/N queueing system with balking, reneging and synchronous vacations of partial servers, The Sixth International Symposium on Operations Research and Its Applications (ISORA–06), (2006), 128–143. Google Scholar

Figure 1.  Plot of the average cost function versus the mean service rate and mean vacation rate
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Table 1.  The average cost function for given $ (W, S) $ with $ M = 6 $, $ \lambda = 0.5 $, $ \alpha = 0.2\lambda $, $ \theta = 0.02 $, $ \beta = 0.02 $
$ (S, W) $ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (1, 7) (1, 8)
$ TAC $ 99.72 98.43 99.20 100.74 102.52 104.32 106.04 107.65
$ (S, W) $ (1, 9) (1, 10) (1, 11) (1, 12) (2, 1) (2, 2) (2, 3) (2, 4)
$ TAC $ 109.13 110.49 111.74 112.88 92.32 $ \mathbf{90.90} $ 92.45 94.40
$ (S, W) $ (2, 5) (2, 6) (2, 7) (2, 8) (2, 9) (2, 10) (2, 11) (2, 12)
$ TAC $ 96.70 98.92 100.97 102.83 104.51 106.02 107.39 108.62
$ (S, W) $ (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) (3, 8)
$ TAC $ 97.43 94.99 96.04 97.96 100.03 102.00 103.83 105.47
$ (S, W) $ (3, 9) (3, 10) (3, 11) (3, 12) (4, 1) (4, 2) (4, 3) (4, 4)
$ TAC $ 106.95 108.28 109.46 110.53 98.44 96.39 97.75 99.94
$ (S, W) $ (4, 5) (4, 6) (4, 7) (4, 8) (4, 9) (4, 10) (4, 11) (4, 12)
$ TAC $ 102.17 104.22 106.04 107.65 109.07 110.33 111.45 112.44
$ (S, W) $ (5, 9) (5, 10) (5, 11) (5, 12) (6, 1) (6, 2) (6, 3) (6, 4)
$ TAC $ 109.33 110.60 111.72 112.72 95.48 94.17 96.15 98.83
$ (S, W) $ (6, 5) (6, 6) (6, 7) (6, 8) (6, 9) (6, 10) (6, 11) (6, 12)
$ TAC $ 101.39 103.66 105.63 107.35 108.84 110.16 111.33 112.37
$ (S, W) $ (7, 2) (7, 3) (7, 4) (7, 5) (7, 6) (7, 7) (7, 8) (7, 9)
$ TAC $ 91.92 94.27 97.24 100.01 102.43 104.53 106.35 107.93
$ (S, W) $ (7, 10) (7, 11) (7, 12) (8, 3) (8, 4) (8, 5) (8, 6) (8, 7)
$ TAC $ 109.32 110.54 111.64 91.97 95.24 98.25 100.85 103.10
$ (S, W) $ (8, 8) (8, 9) (8, 10) (8, 11) (8, 12) (9, 4) (9, 5) (9, 6)
$ TAC $ 105.03 106.71 108.19 109.49 110.65 92.94 96.20 99.01
$ (S, W) $ (9, 7) (9, 8) (9, 9) (9, 10) (9, 11) (9, 12) (10, 5) (10, 6)
$ TAC $ 101.41 103.48 105.28 106.85 108.24 109.48 93.93 96.95
$ (S, W) $ (10, 7) (10, 8) (10, 9) (10, 10) (10, 11) (10, 12) (11, 6) (11, 7)
$ TAC $ 99.53 101.75 103.67 105.36 106.84 108.16 94.73 97.49
$ (S, W) $ (11, 8) (11, 9) (11, 10) (11, 11) (11, 12) (12, 7) (12, 8) (12, 9)
$ TAC $ 99.87 101.92 103.72 105.31 106.71 95.32 97.86 100.06
$ (S, W) $ (12, 10) (12, 11) (12, 12)
$ TAC $ 101.98 103.67 105.17
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$ (S, W) $ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (1, 7) (1, 8)
$ TAC $ 99.72 98.43 99.20 100.74 102.52 104.32 106.04 107.65
$ (S, W) $ (1, 9) (1, 10) (1, 11) (1, 12) (2, 1) (2, 2) (2, 3) (2, 4)
$ TAC $ 109.13 110.49 111.74 112.88 92.32 $ \mathbf{90.90} $ 92.45 94.40
$ (S, W) $ (2, 5) (2, 6) (2, 7) (2, 8) (2, 9) (2, 10) (2, 11) (2, 12)
$ TAC $ 96.70 98.92 100.97 102.83 104.51 106.02 107.39 108.62
$ (S, W) $ (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6) (3, 7) (3, 8)
$ TAC $ 97.43 94.99 96.04 97.96 100.03 102.00 103.83 105.47
$ (S, W) $ (3, 9) (3, 10) (3, 11) (3, 12) (4, 1) (4, 2) (4, 3) (4, 4)
$ TAC $ 106.95 108.28 109.46 110.53 98.44 96.39 97.75 99.94
$ (S, W) $ (4, 5) (4, 6) (4, 7) (4, 8) (4, 9) (4, 10) (4, 11) (4, 12)
$ TAC $ 102.17 104.22 106.04 107.65 109.07 110.33 111.45 112.44
$ (S, W) $ (5, 9) (5, 10) (5, 11) (5, 12) (6, 1) (6, 2) (6, 3) (6, 4)
$ TAC $ 109.33 110.60 111.72 112.72 95.48 94.17 96.15 98.83
$ (S, W) $ (6, 5) (6, 6) (6, 7) (6, 8) (6, 9) (6, 10) (6, 11) (6, 12)
$ TAC $ 101.39 103.66 105.63 107.35 108.84 110.16 111.33 112.37
$ (S, W) $ (7, 2) (7, 3) (7, 4) (7, 5) (7, 6) (7, 7) (7, 8) (7, 9)
$ TAC $ 91.92 94.27 97.24 100.01 102.43 104.53 106.35 107.93
$ (S, W) $ (7, 10) (7, 11) (7, 12) (8, 3) (8, 4) (8, 5) (8, 6) (8, 7)
$ TAC $ 109.32 110.54 111.64 91.97 95.24 98.25 100.85 103.10
$ (S, W) $ (8, 8) (8, 9) (8, 10) (8, 11) (8, 12) (9, 4) (9, 5) (9, 6)
$ TAC $ 105.03 106.71 108.19 109.49 110.65 92.94 96.20 99.01
$ (S, W) $ (9, 7) (9, 8) (9, 9) (9, 10) (9, 11) (9, 12) (10, 5) (10, 6)
$ TAC $ 101.41 103.48 105.28 106.85 108.24 109.48 93.93 96.95
$ (S, W) $ (10, 7) (10, 8) (10, 9) (10, 10) (10, 11) (10, 12) (11, 6) (11, 7)
$ TAC $ 99.53 101.75 103.67 105.36 106.84 108.16 94.73 97.49
$ (S, W) $ (11, 8) (11, 9) (11, 10) (11, 11) (11, 12) (12, 7) (12, 8) (12, 9)
$ TAC $ 99.87 101.92 103.72 105.31 106.71 95.32 97.86 100.06
$ (S, W) $ (12, 10) (12, 11) (12, 12)
$ TAC $ 101.98 103.67 105.17
Table 2.  The minimum average cost function for varying values of $ \lambda $ with $ M = 6 $, $ \alpha = 0.2\lambda $, $ \theta = 0.02 $, $ \beta = 0.02 $
QN method
$ \lambda $ 0.2 0.4 0.6
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 1204 1427 1583
CPU Time 6.26 5.62 5.64
MN method
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 6741 6074 6176
CPU Time 6.39 5.77 5.62
PS method
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 11679 13179 13294
CPU Time 22.44 23.57 25.80
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QN method
$ \lambda $ 0.2 0.4 0.6
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 1204 1427 1583
CPU Time 6.26 5.62 5.64
MN method
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 6741 6074 6176
CPU Time 6.39 5.77 5.62
PS method
$ TAC $ 68.34 85.47 95.95
$ (W^*, S^*) $ (1, 1) (2, 2) (2, 2)
$ (\mu^*, \delta^*) $ (2.37, 5.43) (2.38, 2.14) (3.09, 4.09)
Iterations 11679 13179 13294
CPU Time 22.44 23.57 25.80
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