Design and Estimate of the Optimal Parameters of Adaptive Control Chart Model Using Markov Chains Technique
American Journal of Theoretical and Applied Statistics
Volume 4, Issue 2-1, March 2015, Pages: 19-26
Received: Dec. 4, 2014; Accepted: Dec. 5, 2014; Published: Mar. 11, 2015
Views 2881      Downloads 181
Authors
Vishwa Nath Maurya, Department of Pure & Applied Mathematics and Statistics, School of Science & Technology, The University of Fiji, Lautoka, Fiji Islands
Ram Bilas Misra, Division of Applied Mathematics, State University of New York, Incheon, Republic of Korea & Ex-Vice Chancellor, Dr. R. M. L. Avadh University, Faizabad, UP, India
Chandra K. Jaggi, Department of Operations Research, University of Delhi, New Delhi, India
Charanjeet Singh Arneja, Department of Agricultural Extension, Punjab Agricultural University, Ludhiana, India
Rama Shanker Sharma, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea
Avadhesh Kumar Maurya, Department of Electronics & Communication Engineering, Lucknow Institute of Technology, U.P. Technical University, Lucknow, India
Article Tools
Follow on us
Abstract
Present paper aims to plan and estimate the optimal parameters of an adaptive control chart model for monitoring the mean of a process using sample size and variable interval. Here, the X_BARRA-VSSI chart has been chosen because of its two special features- firstly being an adaptive scheme with great potential for practical application, and secondly the chart only requires knowledge of the sample size and the time between sample selections after established the optimal parameters for the chart. To estimate the optimal parameters of chosen control chart model, the Markov chains technique has been applied. Two functions written in R language are presented in order to assist the user in planning a statistical project based on the X_BARRA-VSSI adaptive scheme. Evaluating the effectiveness of the control chart of by means of Markov chains has been examined and the optimal parameters of the adaptive control chart model have been explored. In addition to this, a numerical example for application of the control chart model has also been illustrated and finally some conclusive observations with significant suggestions for its future scope are carried out.
Keywords
Sampling amplitude, hypothesis testing, statistical control of process (SCP), adaptive charts, Markov chains technique, standard normal cumulative function, statistical software, decision variable, transition matrix
To cite this article
Vishwa Nath Maurya, Ram Bilas Misra, Chandra K. Jaggi, Charanjeet Singh Arneja, Rama Shanker Sharma, Avadhesh Kumar Maurya, Design and Estimate of the Optimal Parameters of Adaptive Control Chart Model Using Markov Chains Technique, American Journal of Theoretical and Applied Statistics. Special Issue: Scope of Statistical Modeling and Optimization Techniques in Management Decision Making Process. Vol. 4, No. 2-1, 2015, pp. 19-26. doi: 10.11648/j.ajtas.s.2015040201.13
References
[1]
Arneja C.S., Maurya V.N. and Kaur Gaganpreet, Entrepreneurship development of Punjab farmers based on statistical survey, Journal of Engineering and Technology Research, Scientia Research Library, Georgia, Vol. 2(1), pp. 1-9, 2014, ISSN: 2348-0424, USA CODEN JETRB4
[2]
Bai D.S. and Lee K.T., An economic design of variable sampling interval control chart, International journal of production economics, Vol. 54, pp. 57- 64, 1998.
[3]
Celano G., On the constrained economic design of control charts: a literature review, Prod. Vol. 21(2), pp. 223-234, 2011.
[4]
Celano G., Robust design of adaptive control charts for manual manufacturing/inspection workstations, Journal of Applied Statistics, Vol. 36(2), pp. 181-203, 2009.
[5]
Costa A.F.B., charts with variable parameters, Journal of Quality Technology, Vol. 31, pp. 408-416, 1999.
[6]
Costa A.F.B., charts with variable sample size and sampling intervals, Journal of Quality Technology, Vol. 29, pp.197-204, 1997.
[7]
Costa A.F.B., charts with variable sample size, Journal of Quality Technology, Vol. 26, pp. 155-163, 1994.
[8]
Costa A.F.B., Epprecht E.K. and Carpinetti L.C.R., Controle Estatístico de Qualidade, São Paulo, Atlas, 2008.
[9]
Faraz A. and Saniga E., A unification and some corrections to Markov chain approaches to develop variable ratio sampling scheme control charts, Statistical Papers, Vol. 52(4), pp.799-811, 2011.
[10]
Leoni R.C., Costa A.F.B., O ambiente R como proposta de apoio ao ensino no monitoramento de processos, Pesquisa Operacional para o Desenvolvimento, Vol. 4(1), pp. 83-96, 2012.
[11]
Magalhães M.S., Epprecht E.K., and Costa A.F.B., Economic design of a Vp chart, International Journal of Production Economics, Vol. 74, pp.191-200, 2001.
[12]
Maurya A.K. and Maurya V.N., Linear regression and coverage rate analysis for optimization of received signal strength in antenna beam tilt cellular mobile environment, International Journal of Electronics Communication and Electrical Engineering, Algeria, Vol. 3(7), pp. 1-14, 2013, ISSN: 2277-7040
[13]
Maurya V.N., Arora D.K. and Maurya A.K., A survey report of parameter and structure learning in Bayesian network inference, International Journal of Information Technology & Operations Management, Academic and Scientific Publishing, New York, USA, Vol. 1(2), pp. 11-28, 2013, ISSN: 2328-8531
[14]
Maurya V.N., Arora D.K. and Maurya A.K., Significant role and special aspects of inference mechanism in Bayesian network, International Open Journal of Operations Research, Academic and Scientific Publishing, New York, USA, Vol. 1 (2), pp. 16-39, 2013, ISSN: 2328-8582
[15]
Maurya V.N., Arora D.K., Maurya A.K. and Gautam R.A., Exact modeling of annual maximum rainfall with Gumbel and Frechet distributions using parameter estimation techniques, World of Sciences Journal, Engineers Press Publishing, Vienna, Austria, Vol. 1(2), pp.11-26, 2013, ISSN: 2307-3071
[16]
Maurya V.N., Covariance analysis of a non-homogeneous Mx(t)/G(t)/∞:(∞;FCFS) queueing system with bulk arrival and arbitrary service time distribution, Acta Ciencia Indica Mathematics, Vol. 32(3), pp. 1183-1188, 2006 (Citation No. 015878, Indian Science Abstract, Vol. 43, No. 16, 2007), ISSN: 0970-0455
[17]
Maurya V.N., Determination of expected busy periods in faster and slower arrival rates of an interdependent M/M/1:(∞; GD) queueing model with controllable arrival rates, International Journal of Engineering Research and Technology, Engineering Science & Research Support Academy (ESRSA) Publication, Vadodara, India, Vol. 1(5), pp. 1-5, 2012, ISSN: 2278-0181
[18]
Maurya V.N., Further results on queue’s characteristics in power supply problems, Acta Ciencia Indica Mathematics, Vol. 32(2), pp. 805-814, 2006, ISSN: 0970-0455
[19]
Maurya V.N., Inferences on operating characteristics of the queue in power supply problems, International Research Journal Acta Ciencia Indica Mathematics, Vol. 32(4), pp. 1391-11395, 2006, ISSN: 0970-0455
[20]
Maurya V.N., Inferential analysis for some discrete distributions of the M/M/1: (∞; FCFS) queueing system in equilibrium state, Acta Ciencia Indica Mathematics, Vol. 32(4), pp. 1433-1442, 2006, ISSN: 0970-0455
[21]
Maurya V.N., Maurya A.K. and Arora D.K., Elements of Advanced Probability Theory and Statistical Techniques, Scholar’s Press Publishing Co., Saarbrucken, Germany, 2014, ISBN 978-3-639-51849-8
[22]
Maurya V.N., Maurya A.K. and Kaur D., A survey report on nonparametric hypothesis testing including Kruskal-Wallis ANOVA and Kolmogorov–Smirnov goodness-fit-test, International Journal of Information Technology & Operations Management, Academic and Scientific Publishing, New York, USA, Vol. 1(2), pp. 29-40, 2013, ISSN: 2328-8531
[23]
Maurya V.N., On the form of some distributions for non-homogeneous M(t)/M(t)/∞ and MX(t)/G(t)/∞ queueing systems, International Journal of Management & Systems, New Delhi, India, Vol. 19 (3), pp. 265-278, 2003
[24]
Maurya V.N., Jaggi Chandra K., Singh Bijay, Arneja C.S., Maurya A.K. and Arora D.K., Empirical analysis of work life balance policies and its impact on employee’s job satisfaction and performance: Descriptive statistical approach, Special Issue: Scope of Statistical Modeling and Optimization Techniques in Management and Decision Making Process, American Journal of Theoretical and Applied Statistics, USA, 2014.
[25]
Maurya V.N., Singh Bijay, Reddy N., Singh V.V., Maurya A.K., and Arora D.K., Cost-effective perspective and scenario development on economic optimization for multiple-use dry-season water resource management, American Open Journal of Agricultural Research, Academic & Scientific Publishing, New York, USA, Vol. 2(1), pp. 1-21, 2014, ISSN:2333-2131
[26]
Park C. and Reynolds M.R. Jr, Economic design of a variable sample size X chart, Communications in statistics – simulation and computation, Vol. 23, pp. 467- 483, 1994.
[27]
Park C. and Reynolds M.R. Jr., Economic design of a variable sampling rate chart, Journal of Quality Technology, Vol. 31, pp. 427-443, 1999.
[28]
Prabhu S.S., Montgomery D.C., and Runger G.C., A combined adaptive sample size and sampling interval control scheme, Journal of Quality Technology, Vol. 26, pp.164-176, 1994.
[29]
Prabhu S.S., Montgomery D.C., and Runger G.C., Economic-statistical design of an adaptive chart, International Journal of Production Economics, Vol. 49, pp. 1-15, 1997.
[30]
R Development Core Team: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria, 2011, ISBN 3-900051-07-0, URL http://www.R-project.org/.
[31]
Reynolds M.R. Jr., Arnold J.C., and Nachlas J.A., charts with variable sampling intervals, Technometrics, Vol. 30, pp. 181-192, 1988.
[32]
Shewhart W.A., Economic control of quality of manufactured product, D. Van Nostrand Company, 1st Edition, New York, USA, 1931.
[33]
Zimmer L.S., Montgomery D.C., and Runger G.C., Guidelines for the application of adaptive control charting schemes, International Journal of Production Research, Vol. 38(9), pp. 1977-1992, 2000.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186