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International Journal of Civil Engineering and Technology (IJCIET) Volume 10, Issue 1, January 2019, pp.509–519, Article ID: IJCIET_10_01_048 Available online at http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1 ISSN Print: 0976-6308 and ISSN Online: 0976-6316 ©IAEME Publication Scopus Indexed CONTROL CHARTS FOR MULTIPLE DEPENDENT STATE REPETITIVE SAMPLING PLAN USING FUZZY POISSON DISTRIBUTION Sreeja M Krishnan and O.S.Deepa Department of Mathematics Amrita School of Engineering, Coimbatore Amrita Vishwa Vidyapeetham, India ABSTRACT Multiple Dependent State Repetitive Sampling(MDSRS) plan is a combination of Multiple deferred state sampling plan as well as repetitive sampling plan.This paper deals with multiple dependent repetitive state sampling plan for certain attribute control chart with respect to Poisson, gamma Poisson, fuzzy Poisson andfuzzy Binomial distributions. The average run length for various distributions in MDSRS are tabulated. Graphical illustrations are also made. The average run lengths are compared for smaller shifts in the process using control charts for different parameter values. The proposed method will be much useful in industry during monitoring of manufacturing process. An example of earthquake data set from UCI respiratory is considered and the average run length is computed based on fuzzy Poisson distribution. Key words: Poisson distribution, Gamma Poisson, Fuzzy Poisson and Fuzzy Binomial distributions, Multiple Dependent Repetitive State Sampling Plan, Control charts. Cite this Article: Sreeja M Krishnan and O.S.Deepa, Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution, International Journal of Civil Engineering and Technology (IJCIET), 10 (1), 2019, pp. 509–519. http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1 1. INTRODUCTION The quality of the products in the share market plays an important role for the customers. Different statistical techniques are used to increase the quality of the product. It was Walter. A .Shewhart who invented control chart and is known as Shewhart charts. The control chart is a graph which changes depending on time. Through historical data one can find average line, upper line and lower line of the graph. Control chart is one among the seven basic tools of the http://www.iaeme.com/IJCIET/index.asp 509 [email protected] Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution quality control. This is mainly used for manufacturing purposes, laboratory purposes, industrial purposes as well as education purposes. If the number of defectives taken from a sample is less than the acceptance number c, then the lot is accepted in case of single sampling plan. But a repetitive sampling can have repeated samplings so that taking decision from the first sampling is impossible. If decisions are not taken from the first sampling, Multiple deferred state sampling (MDS) will be useful. The decision from the process state of MDS can be made when the preceding “i” subgroups can be concluded in control or not in control. There is another sampling called multiple dependent state repetitive sampling (MDSRS) which is the combination of MDS Sampling as well as repetitive sampling. If the manufactured process has shifts then by using the MDSRS, shifts can be detected. The work in this paper is carried with multiple dependent state repetitive sampling (MDSRS) for Poisson Distribution Gamma Poisson Distribution Fuzzy Poisson Distribution Fuzzy Binomial Distribution Application of Fuzzy Poisson Distribution in earth quake data set of UCI respiratory Operating Procedure The steps involved in multiple dependent state sampling plan for p charts of any distributions is based on sample size n and the number of defectives d. 1. Let LCL1,UCL1,LCL2,UCL2be the four control limits constructed during the process . If LCL1 LCL2 d UCL1 UCL2 then the process is in control and ifd UCL and d LCL then the process is out of control. 2. For a given i , if the preceding “i” subgroups is concluded as in-control , then the entire process is in control. Let k1 and k2 be two control coefficients and the four control limits of MDRS can be defined as: UCL1 k1 LCL1 max( 0, k1 ) UCL2 k 2 LCL2 max( 0, k 2 ) where np is the parameter for Poisson The MDSRS sampling has more advantage than the MDS sampling because in MDS sampling the declaration of the process to be in-control is done once the preceeding “i” subgroups are in –control. But the MDSRS will repeat until a proper declaration is made even if preceeding “i” subgroups do not have a conclusion. ALGORITHM FOR THE CONTROL COEFFICIENTS (K1,K2and ARL) Consider the values of ARL, r0,p0 and Iand determine the values of k1 and k2 for which ARL0 r0 . Based on k1 and k2,determine the values of ARL1. For MDS plan the probability of considering the process to be in control is given by http://www.iaeme.com/IJCIET/index.asp 510 [email protected] Sreeja M Krishnan and O.S.Deepa P 01 in,1 UCL2 e d d LCL 1 d! 2 UCL1 LCL2 e d e d d LCL 1 d! d! d UCL2 1 11 UCL2 e d d LCL 1 d! 2 where np0 For repeated sampling: UCL1 UCL2 LCL2 e d e d e d P rep 1 1 d! d UCL 1 d! d LCL 1 d! d LCL 11 2 2 For MDSRS plan the probability of considering the process to be under control if the P 0 in,1 process actually under control is given by P 01in 1 P 0 rep 01 For the process to be in control, the average run length is ARL01 1 1 P 01in ARL FOR SHIFTED PROCESS For the shifted process, p1 p0 cp0 The probability of considering in-control for MDSRS when the process shifts is given as UCL2 e d P11in,1 d LCL 1 d! 2 UCL1 LCL2 e d e d d LCL 1 d! d! d UCL2 1 11 UCL2 e d d LCL 1 d! 2 Where np1 For repeated sampling: For shifted process, the probability of repeated sampling given as UCL1 LCL2 e d e d P11rep d LCL 1 d! d! d UCL2 1 11 UCL2 d 1 e d LCL 1 d! 2 For shifted process to be in control, the MDSRS is P11in For the shifted process : ARL11 P11in,1 1 P11rep 1 1 P11in http://www.iaeme.com/IJCIET/index.asp 511 [email protected] Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution Table 1 Different values of Average Run Length values for r=30 and i=2 : p0 k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.13 0.15 0.17 0.2 0.25 0.5 0.7 0.8 1 0.1 2.8127 2.04671 90 30.7622 30.4300 19.7642 21.5419 21.4846 24.0670 26.2857 25.9144 31.2035 23.1923 27.3560 34.0999 29.9500 36.7936 28.7078 0.2 0.3 2.8244 2.9504 2.1727 2.1651 80 81 ARL 37.4334 47.4013 36.9533 58.0619 38.6264 43.9958 42.2963 53.2946 37.0645 64.7105 40.2828 47.7972 46.2045 46.4837 35.6754 53.5851 40.0402 55.9675 43.2538 52.2996 38.8027 54.2359 41.5200 59.6298 49.0899 63.9907 45.0309 66.4436 49.9821 65.4073 0.4 2.9304 2.2021 83 58.6402 65.2745 60.9321 64.0138 55.8716 59.4761 61.8043 61.6230 61.7673 67.3314 63.6404 67.8871 73.7792 74.3189 80.4005 Table 2 Different values of Average Run Length values for r=3 and i=2 : k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.13 0.15 0.5 0.7 0.8 1 p0 0.1 0.2 0.3 0.4 3.530558 3.318249 3.067156 3.034194 0.596302 0.595697 1.036403 1.975097 83 84 78 98 ARL 2.9795 3.6451 8.4067 48.930 2.9878 3.6503 8.4280 46.0873 3.0017 3.6706 8.2350 46.2865 3.0220 3.6195 8.2921 45.7233 2.2203 3.6425 9.6036 43.8623 2.2273 4.4717 8.2685 48.3511 2.9801 3.6205 8.1831 46.3517 2.9908 3.6241 9.3360 53.3124 2.9323 4.3149 11.3786 55.3986 3.7390 4.2943 10.8866 58.3943 3.7030 5.1143 10.7910 65.8548 3.6398 5.0114 11.7217 64.4092 http://www.iaeme.com/IJCIET/index.asp 512 [email protected] Sreeja M Krishnan and O.S.Deepa Example: The Average Run Length is increasing from 0.1 to 0.4 in both Table 1 and Table 2 when c is 0.01 and for various values of p0 is 0.1,0.2,0.3,0.4.Also for different values of c, the average run length increases from 0.1 to 0.4. Gamma Poisson Distribution For MDS plan the probability of considering the process to be in control is given by P 02 in,1 ( m d 1)! np d d LCL2 1 !( m 1 p )! m np UCL2 d m m m np d m UCL1 LCL2 ( m d 1)! np d m m (m d 1)! np m d LCL1 1 d !(m 1 p )! m np m np d UCL2 1 d !( m 1 p )! m np m np ( m d 1)! np * d d LCL2 1 !( m 1 p )! m np UCL2 d m m np at m For repeated sampling: P 02 rep d m UCL1 LCL2 m d 1! np d m m m d 1! np m d LCL1 1 d ! ( m 1 p )! m np m np d UCL2 1 d ! ( m 1 p )! m np m np d m UCL2 ( m d 1 )! np m 1 d LCL2 1 d!(m 1 p)! m np m np at For MDSRS plan the probability of considering the process to be under control if the process actually under control is given by P 02 in P 02 in,1 1 P 02 rep Hence ARL01 1 1 P 01in For shifted process, the process to be in control is P12 in P12 in,1 ( m d 1)! np d LCL2 1 d !( m 1 p )! m np UCL2 d P12 in,1 where 1 P12 rep m m m np d m UCL1 LCL2 ( m d 1)! np d m m (m d 1)! np m d d !(m 1 p )! m np m np d UCL2 1 d !( m 1 p )! m np m np LCL1 1 ( m d 1)! np d LCL2 1 d !( m 1 p )! m np UCL2 * d m m np m http://www.iaeme.com/IJCIET/index.asp at 513 [email protected] Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution For repeated sampling: P 12 rep d m UCL1 LCL2 m d 1! np d m m m d 1! np m d d!(m 1 p)! m np m np d UCL2 1 d ! ( m 1 p )! m np m np LCL1 1 d m UCL2 (m d 1)! np m 1 d LCL2 1 d!(m 1 p)! m np m np at Table 3 Different values of Average run length for Gamma Poisson when m=1 p0 k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.13 0.15 0.17 0.2 0.1 2.812753 2.046714 90 2.3922 2.6831 2.5993 2.5233 2.4541 2.3611 2.4103 2.3545 2.3029 2.2324 0.2 0.3 2.824495 2.950435 2.172771 2.165179 80 81 ARL 1.6607 1.5845 1.8239 1.5757 1.7972 1.559 1.7721 1.5432 1.7486 1.5883 1.7595 1.5462 1.7274 1.5406 1.7075 1.5263 1.6886 1.5365 1.6996 1.5165 Figure 1 0.4 2.930443 2.20219 83 1.4692 1.4626 1.45 1.4555 1.4436 1.427 1.4115 1.4171 1.4075 1.3938 Figure2 The graph for table 3 with m=1 and p=0.3 The graph for table 3 with m=2 and p=0.4 http://www.iaeme.com/IJCIET/index.asp 514 [email protected] Sreeja M Krishnan and O.S.Deepa Table 4 Different values of Average run length for Gamma Poisson whenm=2: p0 k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.13 0.15 0.17 0.2 0.25 0.3 0.4 0.5 0.7 0.8 1 0.1 2.812753 2.046714 90 3.6859 4.2413 4.0628 3.9015 3.7551 3.5594 3.745 3.6209 3.5068 3.3518 3.1295 3.1831 2.8427 2.7557 2.4859 1.6753 2.3829 0.2 0.3 2.824495 2.950435 2.172771 2.165179 80 81 ARL 2.1995 2.0632 2.5354 2.0411 2.4762 2.0078 2.4211 1.9739 2.3697 2.1032 2.4023 1.9883 2.3319 1.9939 2.2886 1.9619 2.2478 1.9857 2.2775 1.941 2.1859 1.8746 2.1059 1.8367 2.1656 1.7961 2.0902 1.7441 1.9241 1.6716 2.0194 1.6392 1.7828 1.5871 0.4 2.930443 2.20219 83 1.8377 1.8236 1.7966 1.8084 1.7833 1.7484 1.7163 1.7279 1.708 1.6801 1.6654 1.6263 1.6056 1.5914 1.5265 1.5056 1.446 Table 5 Different values of Average run length for Gamma Poisson whenm=1: p0 k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.1 3.530558 0.596302 83 1.5414 1.5238 1.5121 1.4774 1.5562 1.5148 http://www.iaeme.com/IJCIET/index.asp 0.2 0.3 3.318249 3.067156 0.595697 1.036403 84 78 ARL 1.2356 1.3041 1.23 1.299 1.2196 1.289 1.2697 1.2949 1.2581 1.2852 1.2423 1.2718 515 0.4 3.034194 1.975097 98 1.3657 1.3607 1.3656 1.356 1.3468 1.334 [email protected] Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution Table 6 Different values of Average run length for Gamma Poisson whenm=2: p0 k1 k2 n c 0 0.01 0.03 0.05 0.07 0.1 0.1 3.530558 0.596302 83 2.0589 2.0006 1.9619 1.8697 2.1659 2.022 0.2 0.3 3.318249 3.067156 0.595697 1.036403 84 78 ARL 1.5035 1.5916 1.4881 1.5788 1.4597 1.5544 1.5876 1.5848 1.5541 1.56 1.5101 1.5264 0.4 3.034194 1.975097 98 1.6457 1.635 1.6449 1.6246 1.6055 1.5789 Fuzzy Poisson Distribution: The fuzzy Poisson distribution is given by ~ ~ e d e d U PK [ ] min | [ ] and PK [ ] max | [ ] A d! A d! L ~ Where [ ] [1 [ ], 2 [ ]] [nK na2 , nK na3 n(a3 a 2 ) K ~ ~ p a P( A)[ ] [ PKL [ ], PKU [ ]] ARL for Poisson process,(in-control): UCL2 e d P 03in,1 d LCL 1 d! 2 UCL1 LCL2 e d e d d LCL 1 d! d! d UCL2 1 11 UCL2 e d d LCL 1 d! 2 Where np0 For repeated sampling: UCL1 UCL2 LCL2 e d e d e d P 03 rep 1 d LCL 1 d! d! d LCL2 1 d! d UCL2 1 11 The UCL and LCL values are MDSRS when the process is in-control is : P 03in P 03in,1 1 P 03 rep For the process is in-control then: 1 1 P 03 in Then to find ARL1i.e., the shifted process when the process is in control is ARL03 http://www.iaeme.com/IJCIET/index.asp 516 [email protected] Sreeja M Krishnan and O.S.Deepa UCL2 e d P in,1 d LCL 1 d! 2 13 UCL1 LCL2 e d e d d LCL 1 d! d! d UCL2 1 11 UCL2 e d d LCL 1 d! 2 Where np1 For repeated sampling: UCL1 UCL2 LCL2 e d e d e d P13 rep 1 d LCL 1 d! d! d LCL2 1 d! d UCL2 1 11 For the process to be in-control in case of shifted processis: 13 P in P13in,1 1 P13 rep Hence ARL13 1 1 P13in Table 7 Different values of Average run length for Fuzzy Poisson Distribution p0 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.4 c 0.1 0.11 0.111 0.23 0.24 0.243 0.38 0.6 n 90 90 90 80 80 80 81 83 k1 2.812753 2.812753 2.812753 2.824495 2.824495 2.824495 2.950435 2.930443 k2 2.046714 2.046714 2.046714 2.172771 2.172771 2.172771 2.165179 2.20219 ARL,L 20.70716 26.21117 26.32208 31.93199 37.65789 38.59999 45.20291 48.38394 ARL,U 21.27237 26.31821 26.34777 32.54151 37.98218 38.78242 46.02123 48.98715 Example: Consider the value of p0= 0.1 , c = 0.1 and n = 90 then the ARL lower limit value is 20.70 and the ARL upper limit value is 21.27and is represented as (20.70,21.27). Fuzzy Binomial Distribution: For MDS plan the probability of considering the process to be in control is given by UCL1 UCL2 n d UCL2 n d LCL2 n d n d nd nd nd nd P 04 in,1 p0 1 p0 p0 1 p0 p0 1 p0 p0 1 p0 d LCL 1 d d LCL 1 d d UCL2 1 d 2 11 d LCL2 1 d or repeated sampling: P 04 rep UCL1 UCL2 LCL2 n d n d n d nd nd nd p0 1 p0 p0 1 p0 1 p0 1 p0 d LCL 1 d d LCL 1 d d UCL2 1 d 11 2 For the MDSRS, the process to be in control is P 04 in http://www.iaeme.com/IJCIET/index.asp 517 P 04 in,1 1 P 04 rep [email protected] Control Charts For Multiple Dependent State Repetitive Sampling Plan Using fuzzy Poisson Distribution The average run length for the process to be in control is ARL04 1 1 P 04 in For the shifted process, p1 p 0 cp 0 Then to find ARL1 UCL1 UCL2 n d UCL2 n d LCL2 n d n d nd nd nd nd P14 in,1 p1 1 p1 p1 1 p1 p1 1 p1 p1 1 p1 d LCL 1 d d LCL 1 d d LCL 1 d d UCL2 1 d 2 2 11 For repeated sampling: 14 P rep UCL1 UCL2 LCL2 n d n d n d nd nd n d p1 1 p1 p1 1 p1 1 p1 1 p1 d LCL 1 d d LCL 1 d d UCL2 1 d 11 2 For shifted process, the process to be in-controlis P14 in Hence P14 in,1 1 P14 rep ARL14 1 1 P14 in Table 6 The Average Run Length for the Fuzzy Binomial: p0 0.1 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.2 c 0.1 0.12 0.13 0.14 0.15 0.02 0.03 0.04 0.05 0.06 n 90 90 90 90 90 32 32 32 32 32 k1 2.812753 2.812753 2.812753 2.812753 2.812753 2.824495 2.824495 2.824495 2.824495 2.824495 k2 2.046714 2.046714 2.046714 2.046714 2.046714 2.172771 2.172771 2.172771 2.172771 2.172771 ARL,L 5.81629 5.816294 3.837679 3.837679 3.837679 2.539819 2.539819 2.539819 2.539819 2.539819 ARL,U 36.30648 36.30806 30.93731 30.93731 30.93731 5.887213 5.887213 5.887213 5.887213 5.887213 Example: Consider the second row when the value of p = 0.1 and c= 0.12 and n = 90 then the ARL lower limit value is 5.816 and the ARL upper limit value is 36.308 and can be represented as (5.816,36.308). EARTHQUAKE DATASET Procedure 100 data was selected from UCI respiratory dataset. A target was fixed for each criteria (genergy, gpuls, gdenergy, gdpuls). After fixing the target, the data was again classified into 10 sub data of the 100 data. By checking the values against the target, the defects were estimated from each sub data. Fuzzy Poisson is then applied to the data with the n as 100 and the p value as the total defects by n. By this procedure the average run length of lower limit and upper limit of the data set can be obtained. http://www.iaeme.com/IJCIET/index.asp 518 [email protected] Sreeja M Krishnan and O.S.Deepa Table 7 The average run length for the fuzzy Poisson distribution p0 0.27 0.27 0.27 c k1 k2 ARL,l ARL,u 0.37 2.930443 2.20219 61.62233 61.32813 0.378 2.930443 2.20219 62.7359 62.07409 0.38 2.988402 2.319525 69.66456 68.89455 CONCLUSION The average run length increases for various c values in case of Poisson, Gamma Poisson and Fuzzy Poisson distributions and decreases for Fuzzy Binomial distributions. The efficiency of various distributions are compared. It is seen that the average run length increases very slowly for Gamma Poisson distribution and drastically for Poisson distribution and Fuzzy Poisson distribution. Compared to the proposed distributions and existing distribution for multiple deferred state repetitive sampling plan, the average run length using Poisson distribution and fuzzy Poisson distribution are found to be better. This method can be applied to industry for better monitoring of the process. REFERENCES [1] Ajorlou, S., Ajorlou, A. (2009),” A fuzzy based design procedure for a single-stage sampling plan,. FUZZ-IEEE, Korea, August 20-24. [2] Aslam, M., Jun, C.H. (2010), “A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters”,Journal of Applied Statistics, 37(3),405–414. [3] B.P.M. Duate, P.M. Saraiva,2008,“An optimization based approach for designing attribute acceptance sampling plans, ” Int. journal of quality & reliability management.vol. 25 no. 8. [4] Baloui Jamkhaneh, E., B. Sadeghpour Gildeh and Gh. Yari, 2011. Inspection Error and its Effects on Single Sampling Plans with Fuzzy Parameters. 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Deepa , 2015: Optimal production policy for the design of green supply chain model, International Journal of Applied Engineering Research,10(2 Special Issue), pp. 1600-1601. [11] Shruthi. G and O.S.Deepa 2018: Average run length for exponentiated distribution under truncated life test, International Journal of Mechanical Engineering and Technology (IJMET), Volume 9, Issue 6, pp.1180-1188. [12] Anand. Ayyagari, Srinivasa Rao.Kraleti and Lakshminarayana. Jayanti, Determination of Optimal Design Parameters For Control Chart with Truncated Weibull In-Control Times, International Journal of Production Technology and Management (IJPTM), 7(1), 2016, pp. 1–17. [13] Jose K Jacob and Dr. Shouri P.V, Application Of Control Chart Based Reliability Analysis In Process Industries, Volume 3, Issue 1, January- April (2012), pp. 01-13, International Journal of Mechanical Engineering and Technology (IJMET) http://www.iaeme.com/IJCIET/index.asp 519 [email protected]