2018 IJSRST Volume 4 Issue 2 Prnt ISSN: 2395-6011 Onlne ISSN: 2395-602X Themed Secton: Scence and Technology Constructon of Control Chart Based on Sx Sgma Intatves for Regresson ABSTRACT R. Radhakrshnan 1, P. Balamurugan 2 1 Advsor, Maharaja Group of Insttutons, Perundura, Tamlnadu, Inda 2 Department of Statstcs, Government Arts College (Autonomous), Salem, Tamlnadu, Inda A control chart s a statstcal devce used for the study and control of repettve process. W.A. Shewhart [21] of Bell Telephone Laboratores suggested control charts based on the 3 sgma lmts. Now the companes n developed and developng countres started applyng Sx Sgma ntatves n ther manufacturng process, whch results n lesser number of defects. The companes practcng Sx Sgma ntatves are expected to produce 3.4 or less number of defects per mllon opportuntes, a concept suggested by Motorola [22]. If the companes practcng Sx Sgma ntatves use the control lmts suggested by Shewhart, then no pont fall outsde the control lmts because of the mprovement n the qualty of the process. In ths paper an attempt s made to construct a control chart based on sx sgma ntatves for Regresson specally desgned for the companes applyng Sx Sgma ntatves n ther organzaton. Sutable Table 2 s also constructed and presented for the engneers to take quck decsons. Keywords: Control Chart, Process control, Sx Sgma, Sx Sgma Qualty Level. I. INTRODUCTION The concept of Sx Sgma was ntroduced by Motorola [22] by the engneer M.Harry who analyzed varatons n outcomes of the company s nternal procedures and realzed that by measurng varatons t wll be possble to mprove the workng of the system. The procedure was amed at takng acton to mprove the overall performance. The companes, whch are practcng Sx Sgma, are expected to produce 3.4 or less number of defects per mllon opportuntes. Radhakrshnan and Svakumaran [15-20] used the concept of Sx Sgma n the constructon of samplng plans such as sngle, double and repettve group samplng plans ndexed through Sx Sgma Qualty Levels (SSQLs) wth Posson dstrbuton as the base lne dstrbuton. Radhakrshnan [2] suggested sngle samplng plan ndexed through Sx Sgma qualty levels (SSQLs) based on Intervened Random Effect Posson Dstrbuton and Weghted Posson Dstrbuton as the base lne dstrbutons. Radhakrshnan and Balamurugan [3-14] constructed control charts based on sx sgma ntatves for defects, mean, average fracton defectves, number of defectves, X bar usng standard devaton, Exponentally Weghted Movng Average (EWMA), proporton defectves number of defectves, Fracton defectves, Standard devaton wth varable sample sze, average number of nonconformtes per multple unts and number of defects - average number of defects per unt. The control charts orgnated by W.A. Shewhart [21] was based on 3 sgma control lmts. If the same charts are used for the products of the companes whch adopt sx sgma ntatves n the process, then no pont wll fall outsde the control lmts because of the mprovement n the qualty. So a separate control chart s requred to montor the outcomes of the companes, whch adopt sx sgma ntatves. IJSRST1841264 Receved : 07 Feb 2018 Accepted : 17 Feb 2018 January-February-2018 [ (4) 2: 1142-1147] 1142
In ths paper an attempt s made to construct a control chart based on sx sgma ntatves for Regresson. Sutable Table 3 s also constructed and presented for the engneers to take quck decsons. II. CONCEPTS AND TERMINOLOGIES Upper specfcaton lmt (USL) It s the greatest amount specfed by the producer for a process or product to have the acceptable performance. Lower specfcaton lmt (LSL) It s the smallest amount specfed by the producer for a process or product to have the acceptable performance. Tolerance level (TL) It s the dfference between USL and LSL, TL = USL- LSL Process capablty (Cp) Ths s the rato of tolerance level to sx tmes standard devaton of the process. TL USL LSL cp 6 6 Parameters for Regresson (a & b): The parameter a s an ntersecton pont of the ftted center lne wth the vertcal axs and the parameter b s the slope of the ftted center lne. Qualty Control Constant ( C6 ) The constant C6 ntroduced n ths paper to determne the control lmts based on sx sgma ntatves for Regresson chart. Subgroup sze (n & N) It s the choce of the sample sze n and the frequency of samplng and N s the total number of samples. III. CONSTRUCTION OF CONTROL CHART BASED ON SIX SIGMA INITIATIVES FOR REGRESSION Fx the tolerance level (TL) and process capablty (CP) to determne the process standard devaton ( ). Apply the value of 6 & 6 n the control lmts a C 6 6 b a C 6 6 b, to get the control lmts based on sx sgma ntatves for Regresson chart. The value of C6 s obtaned usng p(z z ) 1, 3.4 x 10 6 6 1 1 and z s a standard normal varate. For a specfed TL and CP of the process, the value of (termed as from c p 6 ) s calculated TL usng a C program and presented n 6 Table 3 for varous combnatons of TL and CP. Further the value of C6 s also obtaned usng the procedure gven above and presented n Table 3 for varous combnatons of TL and CP. The control lmts based on sx sgma ntatves for Regresson chart are UCL 6 6 6 Central Lne CL LCL 6 6 6 6 a C b ab a C b IV. CONDITIONS FOR APPLICATION Human nvolvement should be less n the manufacturng process The company adopts Sx sgma qualty ntatves n ts processes V. EXAMPLE The example provded by Amtava Mtra [1] s consdered here. The followng data are the course of machnng the dameter of steel hubs, tool wear s gradual and the sample X and the range R for 25 such samples wth subgroup sze 4: Internatonal Journal of Scentfc Research n Scence and Technology (www.jsrst.com) 1143
Sample Average Table 1. Sample mean and Range for tool wear data (n mm) Range Sample Average Range Sample Average Range X R X R X R 1 36.2 8.0 10 52.6 7.8 19 64.2 13.5 2 42.4 11.8 11 50.4 11.3 20 61.4 9.4 3 38.6 6.2 12 59.5 15.1 21 66.7 16.6 4 45.5 14.3 13 60.5 11.7 22 63.2 12.2 5 53.1 16.2 14 53.8 8.8 23 62.1 10.5 6 46.7 9.5 15 54.5 12.8 24 64.5 12.6 7 55.4 10.2 16 61.2 14.5 25 69.6 14.7 8 42.8 12.0 17 60.4 12.0 9 57.3 13.9 18 63.8 10.4 Table 2. Calculatons for determnng Trend chart for tool wear data (n mm) Sample Average X X 2 R 1 36.2 36.2 1 8.0 2 42.4 84.8 4 11.8 3 38.6 115.8 9 6.2 4 45.5 182.0 16 14.3 5 53.1 265.5 25 16.2 6 46.7 280.2 36 9.5 7 55.4 387.8 49 10.2 8 42.8 342.4 64 12.0 9 57.3 515.7 81 13.9 10 52.6 526.0 100 7.8 11 50.4 554.4 121 11.3 12 59.5 714.0 144 15.1 13 60.5 786.5 169 11.7 14 53.8 753.2 196 8.8 15 54.5 817.5 225 12.8 16 61.2 979.2 256 14.5 17 60.4 1026.8 289 12.0 18 63.8 1148.4 324 10.4 19 64.2 1219.8 361 13.5 20 61.4 1228.0 400 9.4 21 66.7 1400.7 441 16.6 22 63.2 1390.4 484 12.2 23 62.1 1428.3 529 10.5 24 64.5 1548.0 576 12.6 25 69.6 1740.0 625 14.7 Internatonal Journal of Scentfc Research n Scence and Technology (www.jsrst.com) 1144
325 X 1386.4 X 19,471.6 2 5525 R 296 Three Sgma Control lmts for Regresson chart The 3σ control lmts suggested by Shewhart (1931) are 2 & 2 a A R b a A R b From the resultng Fgure 1, t s clear that the process s n control, snce the entre subgroup numbers le nsde the control lmts. Control lmts based on sx sgma ntatves for Regresson chart For a gven TL = 10.4 (USL-LSL =16.6-6.2) & Cp = 1.5, t s found from the Table-3 that the value of 6 s 1.16. The control lmts based on sx sgma ntatves for Regresson chart for a specfed TL and C6 are & a C 6 6 b a C 6 6 b wth From the resultng Fgure 1, the sample numbers 5, 7, 9, 12 and 13 goes above the upper control lmt and the sample numbers 1, 3, 8, 23 and 24 goes below the lower control lmt. Therefore the process does not exhbt statstcal control. Fgure 1. Comparson of the process: 3 lmts and control lmts usng sx sgma ntatves VI. CONCLUSION In ths paper, a procedure s gven to construct a control chart based on sx sgma ntatves for Regresson wth an example. It s found that the process was not n control even when sx sgma ntatves are adopted. It s very clear from the comparson that when the process s centered wth reduced varaton many ponts fall outsde the control lmts than the 3 sgma control lmts, whch ndcate that the process s not n the level t was expected. So a correcton n the process s very much requred to reduce the varatons. The charts suggested n ths paper wll be very useful for the companes practcng Sx Sgma ntatves n ther process. These charts wll replace the exstng Shewhart [21] control charts n future when all the companes started mplementng sx sgma ntatves n ther organzaton. Internatonal Journal of Scentfc Research n Scence and Technology (www.jsrst.com) 1145
VII. REFERENCES [1]. Amtava Mtra, 2001, "Fundamentals of Qualty control and mprovement", Pearson Educaton, Asa. [2]. Radhakrshnan, R., 2009, "Constructon of Sx Sgma Based Samplng Plans", a D.Sc. Thess submtted to Bharathar Unversty, Combatore, Inda. [3]. Radhakrshnan, R. and Balamurugan, P. 2009a. "Sx sgma based Attrbute control charts for defects", Proceedngs of the 2009 Internatonal Marketng n Asa Pacfc: Issues and Challenges, GRD School of Commerce and Internatonal Busness, Combatore, Tamlnadu, Inda and Jontly Organzed by Unversty of central Lancashre, Nov 4-5, pp. 348-353. [4]. Radhakrshnan, R. and Balamurugan, P. 2009b. "Constructon of control charts based on sx sgma ntatves", Proceedngs of the 2009 Internatonal Conference on Trends n Informaton Technology and Busness Intellgence (ITBI 2009), Insttute of Management Technology, Nagpur, Inda. Nov 4-6, pp. 356-361. [5]. Radhakrshnan, R. and Balamurugan, P. 2009c. "Sx sgma based u chart", Proceedngs of the 2009 Internatonal Conference on Innovatve Practces n Management (IPM 2009), Paava Engneerng College, Namakkal, Tamlnadu, Inda. Dec 31st, pp. 365-367. [6]. Radhakrshnan, R. and Balamurugan, P. 2010a. "Sx Sgma based Control charts for the number of defectves", Proceedngs of the 2010 Internatonal Conference on Industral Engneerng and Operatons Management (IEOM 2010) organzed by Bangladesh Socety of Mechancal Engneers, Dhaka, Bangladesh, Jan 9-10, pp. 92. [7]. Radhakrshnan, R. and Balamurugan, P. 2010b. "Constructon of sx sgma based XBar control chart usng standard devaton", Internatonal Journal of Statstcs and System, Volume 6, Number 1, Issue date 2nd Week January, (accepted for publcaton). [8]. Radhakrshnan R. and Balamurugan, P. 2010c. "Sx Sgma based Exponentally Weghted Movng Average Control Chart", Indan journal of Scence and Technology, Issue No: 9 10, Vol. 3, pp. 1052 1055. [9]. Radhakrshnan R. and Balamurugan, P. 2010d. "Control charts based on sx sgma Intatves for proporton defectves and number of defectves", Internatonal Journal of Manufacturng Scence and Engneerng, Volume.1 No.2., (July December), pp: 49-53. [10]. Radhakrshnan, R. and Balamurugan, P. 2011a. "Sx sgma based Control Chart for Fracton Defectves", Journal of Testng and Evaluaton, Volume 39, Issue 4, Issue date 11th July, (accepted for publcaton). [11]. Radhakrshnan, R. and Balamurugan, P. 2011b. "Constructon of control chart based on sx sgma ntatves for standard devaton wth varable sample sze", Proceedngs of the 2011 Natonal conference on Trends and Research n Management, Oxford Engneerng College, Trchy, Tamlnadu, Inda, 28th January, 2011, pp. 218-222. [12]. Radhakrshnan, R. and Balamurugan, P. 2011c. "Constructon of control charts based on sx sgma Intatves for average number of nonconformtes per multple unts", Internatonal Journal of Industral Engneerng & Technology, Volume 3, Number 1, pp.99-105. [13]. Radhakrshnan, R. and Balamurugan, P. 2011d. "Constructon of control chart based on sx sgma ntatves for number of defectves", Proceedngs of the 2011 Internatonal conference on Global Challenges of Emergent Inda - A Management Perspectve, Vvekanandha Insttute of Informaton & Management Studes (VIIMS), Trchencode, Internatonal Journal of Scentfc Research n Scence and Technology (www.jsrst.com) 1146
Namakkal, Tamlnadu, Inda, 14-15, February (Accepted for publcaton n Proceedngs). [14]. Radhakrshnan, R. and Balamurugan, P. 2011e. "Constructon of control charts based on sx sgma Intatves for the number of defects and average number of defects per unt", Journal of Modern Appled Statstcal Methods, (Accepted for publcaton). [15]. Radhakrshnan, R., and Svakumaran, P.K., 2008a, "Constructon and Selecton of Sx Sgma Samplng Plan Indexed through Sx Sgma Qualty Level," Internatonal Journal of Statstcs and Systems, 3(2), 53-159. [16]. Radhakrshnan, R., and Svakumaran, P.K., "Constructon of Sx Sgma Repettve Group Samplng Plan", Internatonal Journal of Mathematcs and Computaton, 1(N08), 2008b, 75-83. [17]. Radhakrshnan, R., and Svakumaran, P.K., 2008c, "Constructon and Selecton of Condtonal Double Samplng Plan ndexed through Sx Sgma Qualty Levels," Sr Lankan Journal of Appled Statstcs, 9, 74-83. [18]. Radhakrshnan, R. and Svakumaran, P.K., 2009a, "Constructon of Sx Sgma Double Samplng Plans", Proc. of Natonal Semnar IT and Busness Intellgence (ITBI-08), Nagpur, Inda. [19]. Radhakrshnan, R., and Svakumaran, P.K., 2009b, "Constructon of Double Samplng Plans through Sx Sgma Qualty Levels," Proc. of the IEEE Second Internatonal Jont Conference on Computatonal Scences and Optmzaton (IEEECSO2009), Sanya, Hanan, Chna, 1027-1030., Aprl 24-26. [20]. Radhakrshnan, R. and Svakumaran, P.K., 2010, "Constructon and Selecton of Tghtened- Normal-Tghtened Schemes of Type TNT- (n1,n2; c) ndexed through Sx Sgma Qualty Levels", Proceedngs of the 2010 Internatonal Conference on Industral Engneerng and Operatons Management (IEOM 2010), Dhaka, Bangladesh, Jan 9-10, 2010. [21]. Shewhart, W.A., 1931, "Economc Control of Qualty of Manufactured Product", New York: Van Nostrand. [22]. Motorola: http://www.motorola.com TL C p Table 3. 6 Values for a specfed Cp and TL 10.1 10.2 10.3 10.4 10.5 1.0 1.68 1.7 1.72 1.73 1.75 1.1 1.53 1.55 1.56 1.58 1.59 1.2 1.40 1.42 1.43 1.44 1.46 1.3 1.29 1.31 1.32 1.33 1.35 1.4 1.20 1.21 1.23 1.24 1.25 1.5 1.12 1.13 1.14 1.16 1.17 1.6 1.05 1.06 1.07 1.08 1.09 1.7 0.99 1.00 1.01 1.02 1.03 1.8 0.94 0.94 0.95 0.96 0.97 1.9 0.89 0.89 0.90 0.91 0.92 2.0 0.84 0.85 0.86 0.87 0.88 2.1 0.80 0.81 0.82 0.83 0.83 2.2 0.77 0.77 0.78 0.79 0.80 2.3 0.73 0.74 0.75 0.75 0.76 2.4 0.70 0.71 0.72 0.72 0.73 2.5 0.67 0.68 0.69 0.69 0.70 Internatonal Journal of Scentfc Research n Scence and Technology (www.jsrst.com) 1147