ISSN: (p); (e) DEVELOPMENT OF FUZZY IX-MR CONTROL CHART USING FUZZY MODE AND FUZZY RULES APPROACHES

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1 DEVELOPMENT OF FUZZY IX-MR CONTROL CHART USING FUZZY MODE AND FUZZY RULES APPROACHES Azam Morad Tad, Soroush Avakh Darestan 2* Department of Industral Engneerng, Scence and Research Branch, Islamc Azad Unversty, Qazvn, Iran. 2 Faculty of Industral and Mechancal Engneerng, Qazvn Branch, Islamc Azad Unversty, Qazvn, Iran. *(Correspondng Author: Soroush Avakh Darestan: avakh@qau.ac.r ABSTRACT Shewhart s control charts are the most mportant statstcal process control tools. In statstcal control charts, for defntve data, the concept of product qualty s expressed by a bnary functon (0 and to be matched or msmatched. Snce a bnary classfcaton n many stuatons cannot be satsfactory, especally when the data are vague, fuzzy statstcal control charts are ntroduced. The purpose of ths paper s to nvestgate the fuzzy IX-MR control chart. Data were converted nto trapezodal fuzzy number and the fuzzy control lmts were trapezodal fuzzy number calculated usng fuzzy mode and Fuzzy Rules approaches. The result was grouped nto four categores (n control, out of control, rather n control and rather out of control. Fnally, a case study was presented and the method codng was done n MATLAB software. The result shown that fuzzy control chart s more senstve than classc control chart. KEYWORDS: Fuzzy control charts, Fuzzy mode method, Fuzzy Rules method, Statstcal process control, Trapezodal fuzzy numbers. INTRODUCTION In the ndustral world today, f products or servces want to meet or exceed the customer expectatons, then they should be produced by a process that s stable. Snce ts products factory produced by humans and machnes, there s a problem n the qualty of the products. And untl ths problem s present, statstcal methods for qualty control are necessary. Among the seven tools of statstcal process control, control charts, are basc and useful tools for the mplementaton of qualty mprovement programs and plays an mportant role n mprovng the qualty of processes and products. And the most common s the Shewhart control charts to montor the product or process, only s used wth a qualtatve characterstc. Shewhart provded the frst control chart n 926. In these charts, t s assumed that data and nformaton are accurate and certan. Every control chart ncludes three parameters. The central lne (CL ndcates mean values that characterze qualty control processes and the upper control lmt (UCL and lower control lmt (LCL are to ensure that the process controls condtons and follow a normal dstrbuton (Rowlands and Wang, Most cases n ndustry to control the product qualty characterstcs are used of one sample. For example:. Measure the desred characterstc s done automatcally. 2. Producton rate s low and may not be able to analyze more than one sample. 3. The samplng process s very tme consumng or expensve. 4. Tests on the prototype, runng t. 5. Changes are not vsble n the shortest tme and successve measurement values are almost the same result. In these stuatons the IX-MR control chart s used. In ths chart, the data must be normally dstrbuted. But when the data are equvocal and uncertan, t wll lose ts effectveness charts. Fuzzy logc s used n these stuatons. In statstcal control charts, for defntve data, the concept of product qualty s expressed by a bnary functon (0 and to be matched or msmatched. When the numbers of qualty levels are between normal and defectve, fuzzy statstcal control charts are ntroduced. Wang and Raz (990 and Raz and Wang (990 were the frst words of the language as very good, good, moderate, poor and bad accepted to express moderate levels of a qualtatve character. Kanagawa et al. (993 created control charts based on probablty densty functons as lngustc varables and establshed a method for montorng the process mean and varance that was gnored by Wang and Raz (990. Also (Kanagawa, 993 used lngustc varables to express the process output. Then (Wang and Chen, 995 were ntroduced a fuzzy mathematcal programmng model to desgn nnovatve economc statstcal control charts. Also (Kahraman et al., 995 used of trangular fuzzy numbers n the desgn of control charts for lngustc data. Taleb and Lmam (2002 explaned another vew about the constructon of fuzzy control charts based on lngustc data and deduced that fuzzy control charts were Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 645

2 senstve to the degree of fuzzness, whch contradcted the concluson of Wang and Raz (990. Gullbay et al., 2004 developed fuzzy control chart usng a-cut method. Taleb and Lmam (2005 also developed polynomal fuzzy control charts. Gullbay and Kahraman (2007 provded fuzzy control charts for lngustc varables. Senturk and Ergnel (2009 desgned fuzzy control charts of Xbar-R and Xbar-S usng a-cut. In 20, Xbar-R control chart was developed usng fuzzy mode and fuzzy rules methods. Avakh Darestan et al., (204 developed fuzzy U control chart for montorng defects. As mentoned, accordng to the real-world data and nformaton are vague and mprecse expresson, ths study wth development IX-MR control chart, tryng to overcome the problem of naccurate and ambguous nformaton. Thus, accordng to the study (Kaya and Kahraman, 20 the man objectve of ths study was to calculate the fuzzy IX- MR control chart, who are more senstve than other models. MATERIALS AND METHODS After selectng a desred characterstc, f ths s not accurate, ths value can be determned by trapezodal fuzzy numbers as shown below (Kaya and Kahraman, 20. X (X a,x b,x c,x d ( X, MR and MR are calculated n Equatons 2, 3 and 4. m m m m X X X X a b c d X (,,, (X a,x b,x c,x d (2 m m m m MR (X,X,X,X (X,X,X,X (X X,X X,X X,X X a b c d a b c d a d b c c b d a MR (MR,MR,MR,MR (3 a b c d m m m m m m MR (MR, MR, MR, MR MR MR MR MR a b c d a b c d MR (,,, m m m m m m (MR a,mr b,mr c,mr d (4 Control lmts for the IX-MR chart are calculated n Equatons 5-8. MR (MR,MR,MR,MR UCL CL 3 (X,X,X,X 3 a b c d IX a b c d d2 d2 3MR 3MR 3MR 3MR (X,X,X,X (UCL, UCL, UCL, UCL (5 a b c d a b c d d2 d2 d2 d2 CL (X,X,X,X (CL,CL,CL,CL (6 IX a b c d MR (MR,MR,MR,MR LCL CL 3 (X,X,X,X 3 a b c d IX a b c d d2 d2 3MR 3MR 3MR 3MR (X,X,X,X (LCL,LCL,LCL,LCL (7 d c b a a b c d d2 d2 d2 d2 Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 646

3 UCL D4 MR D 4(MR a,mr b,mr c,mr d (UCL,UCL 2,UCL 3,UCL 4 CL MR (MR a,mr b,mr c,mr d (CL,CL 2,CL 3,CL 4 LCL D3MR D 3(MR a,mr b,mr c,mr d (LCL,LCL 2,LCL 3,LCL 4 (8 However, to determne the poston of process, we used two methods fuzzy mode and fuzzy rules. Fuzzy mode method (f The fuzzy mode of a fuzzy set mod s the value of base varable where the membershp functon equals. Ths s stated n Equaton 9 (Gulby and Kahraman, 2006, mod f f x X (x (9 Fuzzy mode for trapezodal numbers s shown n Fgure. μ F mod a b c d Fgure. Fuzzy mode for trapezodal fuzzy numbers Control lmts are calculated usng fuzzy mode for trapezodal fuzzy numbers n Equatons 0 and. UCLIXmod UCLIX2 UCLIXmod2 UCLIX3 CLIXmod CLIX2 Mode IX CLIXmod2 CLIX3 LCLIXmod LCL IX2 LCLIXmod2 LCL IX3 (0 UCLMR mod UCLMR2 UCLMR mod2 UCLMR3 CLMR mod CLMR2 Mode MR CLMR mod2 CLMR3 LCLMR mod LCL MR2 LCLMR mod2 LCL MR3 ( For each sample, the fuzzy mode s calculated accordng to the Equaton 2. Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 647

4 IX mod X,2,...,m 2 2 b IX mod X,2,...,m c MR mod MR,2,...,m b MR mod MR,2,...,m (2 c Respectvely, ndexes (C IX and (C MR for IX - MR chart are defned by the Equatons 3 and 4 (Kaya and Kahraman, f (IX mod UCL IXmod 2 UCLIXmod 2 IX mod f (LCLIXmod IX mod UCL IXmod 2 (IX mod 2 UCL IXmod 2 IX mod 2 IX mod C IX f (LCLIXmod IX mod (IX mod 2 UCL IXmod 2 LCLIXmod IX mod f (IX mod LCLIX mod (LCLIXmod IX mod 2 UCL IXmod 2 IX mod 2 IX mod 0 f (IX mod 2 LCL IXmod (3 0 f (MR mod UCL MR mod2 UCLMR mod2 MR mod f (LCLMR mod MR mod UCL MR mod2 (MR mod 2 UCL MR mod2 MR mod 2 MR mod C MR f (LCLMR mod MR mod (MR mod 2 UCL IXmod2 LCLMR mod MR mod f (MR mod LCLMR mod (LCLMR mod MR mod 2 UCL MR mod2 MR mod 2 MR mod 0 f (MR mod 2 LCL MR mod (4 And fnally, the decson process s as follows (Kaya and Kahraman, 20. "n control" f (C IX (C MR "out of control" f (C IX 0 (C MR 0 process control (5 "rather n control" f (C IX (C MR "rather out of control" f (C IX (C MR Fuzzy Rules Method Fuzzy rules ntroduced by Kaya and Kahraman n 20. In the secton, Fuzzy rules for trapezodal fuzzy numbers are expressed. Rule- takes nto account the case where a sample mean (or a range s between control lmts. Rule-2 takes nto account the case where a sample mean (or a range s out of control lmts. Rules 3 and 4 nterpreted the case where a sample mean (or a range s partally ncluded by ether of control lmts. In ths case, some lngustc decsons such as rather n control or rather out-of control are establshed. If the percentage area of a sample mean (or a range whch stays nsde the fuzzy control lmts s equal or greater than a predefned acceptable percentage (β, then the process can be dentfed as rather n control ; f not t can be explaned as rather out of control. Rule-5 nterpreted the case where a sample mean (or a range s partally ncluded both of control lmts (Kaya and Kahraman, 20. These rules are shown n the followng fgures 2-5. Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 648

5 x d UCL Ix4 UCL Ix3 UCL Ix2 UCL Ix4 x d x c x b x a LCL Ix4 LCL Ix3 LCL Ix2 LCL Ix x c x b x a UCL Ix4 UCL Ix3 UCL Ix2 UCL Ix LCL Ix4 LCL Ix3 LCL Ix2 LCL Ix x d x c x b x a Fgure 2: Fuzzy rule- Fgure 3: Fuzzy rule-2 UCLx 4 UCLx 3 UCLx 4 UCLx 3 UCLx 2 O 4 UCLx O 3 O 2 O O 4 O 3 LCLx 4 O 2 LCLx 3 O LCLx 2 LCLx Fgure 4: Fuzzy rules 3 and 4 UCLx 2 O 4 UCLx O 3 O 2 LCLx 4 O LCLx 3 LCLx 2 LCLx Fgure 5: Fuzzy rule-5 Then for IX chart, ndex (C IX and, for MR chart, ndex (C MR are calculated by the followng Equatons 6 and 7 (Kaya and Kahraman, 20. Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 649

6 f (xd UCL IX (xa LCL IX4 0 f (xa UCL IX4 (xd LCL IX (xd UCL IX C IX f (xd UCL IX (xd x a (LCLIX x 3 a f (xa LCL IX4 (x x d a (xd UCL IX (LCL IX x 3 a Mn, f (x (xd x a (x d x a d UCL (x LCL IX a IX4 f (MRd UCL MR (MRa LCL MR 4 0 f (MRa UCL MR 4 (MRd LCL MR (MRd UCL MR C MR f (MR d UCL MR (MRd MR a (LCLMR MR 4 a f (MRa LCL MR 4 (MRd MR a (MRd UCL MR (LCL MR MR 4 a Mn, f (MRd UCL MR (MRa LCL MR 4 (MRd MR a (MR d MR a And fnally, the decson process s as follows (Kaya and Kahraman, 20. "n control" f (CIX (CMR "out of control" f (C 0 (C 0 "rather n control" f (C IX (C MR "rather out of control" f (C IX (C MR IX MR process control (8 (6 (7 RESULTS AND DISCUSSION In ths secton, n order to llustrate the proposed model, we solved a numercal example wth fuzzy mode and fuzzy rules methods. The data conssted of 25 samples whch are normally dstrbuted. Normalty test was performed by the software Mntab and s shown n Fgure 6. Accordng to Fgure 6, the data on one lne and the p-value s greater than 0.05, whch ndcates that the data are normally dstrbuted. Lngustc terms wth trapezodal fuzzy numbers are r shown n Table. Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 650

7 Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 65 Table. Trapezodal fuzzy numbers Trapezodal fuzzy numbers NO Trapezodal fuzzy numbers NO d c b a d c b a Fgure 6: normalty test usng MINITAB software

8 Table. Trapezodal fuzzy numbers NO a Trapezodal fuzzy numbers b c d NO Trapezodal fuzzy numbers a b c d In ths secton, trapezodal fuzzy control lmts are calculated for IX-MR control chart. Modelng of the control chart was coded by MATLAB software. Control lmts are calculated as shown n Table 2. Table 2. Fuzzy IX-MR control lmt IX control chart UCL ( , , , CL (60.007, , 60.00, 60.0 LCL ( , , , MR control chart UCL (4.257, , , CL (.2904,.2924,.2964,.2984 LCL (0,0,0,0 Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 652

9 After calculated the trapezodal fuzzy control lmts, we must evaluate the process. For ths purpose, we used fuzzy mode and fuzzy rules method. The result was grouped nto four categores (n control, out of control, rather n control, rather out of control. Then, the results respectvely were gven n Tables 3 and 4. Accordng to Table 3, by usng Fuzzy mode method, data 92 was "out of control". Also, accordng to Table 4, by usng Fuzzy rules method, data 92 was "out of control" and data 93 was "rather out of control". Therefore, these data should be deleted and re-calculate fuzzy control lmts for these data. New control lmts are shown n Table 5. Table 3. Results of the process control by usng Fuzzy mode method NO IX MR NO IX MR NO IX MR n control 43 n control n control 85 n control n control 2 n control n control 44 n control n control 86 n control n control 3 n control n control 45 n control n control 87 n control n control 4 n control n control 46 n control n control 88 n control n control 5 n control n control 47 n control n control 89 n control n control 6 n control n control 48 n control n control 90 n control n control 7 n control n control 49 n control n control 9 n control n control 8 n control n control 50 n control n control 92 n control Out of control 9 n control n control 5 n control n control 93 n control n control 0 n control n control 52 n control n control 94 n control n control n control n control 53 n control n control 95 n control n control 2 n control n control 54 n control n control 96 n control n control 3 n control n control 55 n control n control 97 n control n control 4 n control n control 56 n control n control 98 n control n control 5 n control n control 57 n control n control 99 n control n control 6 n control n control 58 n control n control 00 n control n control 4 n control n control 59 n control n control 0 n control n control 8 n control n control 60 n control n control 02 n control n control 9 n control n control 6 n control n control 03 n control n control 20 n control n control 62 n control n control 04 n control n control 2 n control n control 63 n control n control 05 n control n control 22 n control n control 64 n control n control 06 n control n control 23 n control n control 65 n control n control 07 n control n control 24 n control n control 66 n control n control 08 n control n control 25 n control n control 67 n control n control 09 n control n control 26 n control n control 68 n control n control 0 n control n control 27 n control n control 69 n control n control n control n control 28 n control n control 70 n control n control 2 n control n control 29 n control n control 7 n control n control 3 n control n control 30 n control n control 72 n control n control 4 n control n control 3 n control n control 73 n control n control 5 n control n control 32 n control n control 74 n control n control 6 n control n control 33 n control n control 75 n control n control 7 n control n control 34 n control n control 76 n control n control 8 n control n control 35 n control n control 77 n control n control 9 n control n control 36 n control n control 78 n control n control 20 n control n control 37 n control n control 79 n control n control 2 n control n control 38 n control n control 80 n control n control 22 n control n control 39 n control n control 8 n control n control 23 n control n control 40 n control n control 82 n control n control 24 n control n control 4 n control n control 83 n control n control 25 n control n control 42 n control n control 84 n control n control Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 653

10 Table 4. Results of the process control by usng Fuzzy rules method NO IX MR NO IX MR NO IX MR n control 43 n control n control 85 n control n control 2 n control n control 44 n control n control 86 n control n control 3 n control n control 45 n control n control 87 n control n control 4 n control n control 46 n control n control 88 n control n control 5 n control n control 47 n control n control 89 n control n control 6 n control n control 48 n control n control 90 n control n control 7 n control n control 49 n control n control 9 n control n control 8 n control n control 50 n control n control 92 n control Out of control 9 n control n control 5 n control n control 93 n control Rather out of control 0 n control n control 52 n control n control 94 n control n control n control n control 53 n control n control 95 n control n control 2 n control n control 54 n control n control 96 n control n control 3 n control n control 55 n control n control 97 n control n control 4 n control n control 56 n control n control 98 n control n control 5 n control n control 57 n control n control 99 n control n control 6 n control n control 58 n control n control 00 n control n control 4 n control n control 59 n control n control 0 n control n control 8 n control n control 60 n control n control 02 n control n control 9 n control n control 6 n control n control 03 n control n control 20 n control n control 62 n control n control 04 n control n control 2 n control n control 63 n control n control 05 n control n control 22 n control n control 64 n control n control 06 n control n control 23 n control n control 65 n control n control 07 n control n control 24 n control n control 66 n control n control 08 n control n control 25 n control n control 67 n control n control 09 n control n control 26 n control n control 68 n control n control 0 n control n control 27 n control n control 69 n control n control n control n control 28 n control n control 70 n control n control 2 n control n control 29 n control n control 7 n control n control 3 n control n control 30 n control n control 72 n control n control 4 n control n control 3 n control n control 73 n control n control 5 n control n control 32 n control n control 74 n control n control 6 n control n control 33 n control n control 75 n control n control 7 n control n control 34 n control n control 76 n control n control 8 n control n control 35 n control n control 77 n control n control 9 n control n control 36 n control n control 78 n control n control 20 n control n control 37 n control n control 79 n control n control 2 n control n control 38 n control n control 80 n control n control 22 n control n control 39 n control n control 8 n control n control 23 n control n control 40 n control n control 82 n control n control 24 n control n control 4 n control n control 83 n control n control 25 n control n control 42 n control n control 84 n control n control Table 5. New fuzzy IX-MR control lmt IX control chart UCL (63.353, , , CL ( , , , LCL ( , , , MR control chart UCL (4.0925, 4.099, 4.2, 4.87 CL (.2527,.2547,.2587,.2607 LCL (0,0,0,0 Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 654

11 RESULTS AND DISCUSSION In ths research, we developed fuzzy IX-MR control chart, and we decson-makng about process by two approaches. The proposed model was coded n MATLAB. Classcal control charts were defned n two groups of n- control and out-of-control. The results show that the proposed model was more senstve than Classcal control charts and the process nto four categores, n control, out of control, rather n control and rather out of control. After analyzng the data, t was concluded that fuzzy rules method was more senstve than fuzzy mode n number 93 and the rest of the results obtaned by both methods were dentcal. For future research, capablty ndces for IX-MR chart can be recommended n the fuzzy envronment. REFERENCES Avakh Darestan S., Morad Tad A., Taher S. and Raeszadeh M. (204. Development of fuzzy U control chart for montorng defects. Internatonal J. Qualty Relablty Management. 3(7:8-82. Gulbay M., Kahraman C. and Ruan D. (2004. A-cut fuzzy control charts for lngustc data. Int. J. Intellgent Systems. 9: Gulbay M. and Kahraman C. (2006. Development of fuzzy process control charts and fuzzy unnatural pattern analyses. Computatonal Statstcs Data Analyss. 5(: Gullbay M. and Kahraman C. (2007. An alternatve approach to fuzzy control charts: Drect fuzzy approach. Informaton Scences. 77(6: Kanagawa A., Tamak F. and Ohta H. (993. Control charts for process average and varablty based on lngustc data. Int. J. Producton Res. 3(4: Kahraman C., Tolga E. and Ulukan Z. (995. Usng trangular fuzzy numbers n the tests of control charts for unnatural patterns, n Proceedngs of INRIA / IEEE Conference on Emergng Technologes and Factory Automaton. October, 0-3. Pars-France. 3: Kaya I. and Kahraman C. (20. Process capablty analyses based on fuzzy measurements and fuzzy control charts. Expert System Applcatons. 38(4: Raz T. and Wang J.H. (990. Probablstc and membershp approaches n the constructon of control charts for lngustc data. Producton Plannng Control. : Rowlands H. and Wang L.R. (2000. An approach of fuzzy logc evaluaton and control n SPC. Qualty and Relablty Engneerng Int. 6: Shewhart W.A. (926. Qualty Control Charts. Bell Systems Techncal J. 5(4: Senturk S. and Ergnel N. (2009. Development of-and-control charts usng a-cuts. Info. Sc. 79(0: Taleb H. and Lmam M. (2002. On fuzzy and probablstc control charts. Int. J. Producton Res. 40(2: Taleb H. and Lmam M. (2005. Fuzzy multnomal control charts. In Bandn and Manzon. S. (Eds.. AI*IA, LNAI, Sprnger-Verlag, Berln Hedelberg. 3673: Wang J.H. and Raz T. (990.On the constructon of control charts usng lngustc varables. Int. J. Producton Res. 28(3: Wang R.C. And Chen C.H. (995. Economc statstcal np-control chart desgns based on fuzzy optmzaton. Int. Qualty Relablty Management. 2(: Volume-3 (Specal Issue DAMA Internatonal. All rghts reserved. 655