Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 709 Desgnng o Combned Contnuous Lot By Lot Acceptance Samplng Plan S. Subhalakshm 1 Dr. S. Muthulakshm 2 1 Research Scholar, 2 Proessor Department o Mathematcs, Avnashlngam Insttute or Home Scence and Hgher Educaton or Women Combatore-64 Abstract Ths paper analyses the combned contnuous lot by lot acceptance samplng plan whch combnes the eatures o contnuous samplng plan and lot by lot samplng plan. The operatng procedure o the combned plan and the dervaton o plan perormance measures usng the Markov-chan approach are gven. Desgnng o the combned contnuous lot by lot acceptance samplng plan ndexed by (, AQL) and (, LQL) are presented. and are provded to the desgned plans or derent choces o the parameters. Keywords: Combned Contnuous lot by lot acceptance samplng plan, Average Outgong Qualty Lmt (), Acceptable Qualty Level (AQL), Lmtng Qualty Level (LQL), Average Fracton Inspected (). I. INTRODUCTION Contnuous samplng plans are most suted to processes where lot ormaton s dcult. However, ther applcaton s not lmted and may also be appled to contnuous producton processes nvolvng lot ormaton. Combnng the best eatures o contnuous samplng plan and lot by lot nspecton plan s desrable n modern producton processes where the racton o producng nonconormtes s n the range o parts per mllon (ppm), snce the contnuous samplng plan or low levels o racton nonconormng requres ether a large clearance nterval or large racton to be sampled. Pesotchnsky [1] proposed a scheme or low racton nonconormng whch combned the strateges o contnuous samplng plan CSP-1 o Dodge [2] and the lot by lot nspecton plan. Bebbngton and Govndaraju [3] provded exact mathematcal results and new tables o perormance measures or Pesotchnsky scheme n order to overcome the complex and dcult procedure o mplementaton n shop loor. Usng the Markovchan approach o Stephens [4], Govndaraju and Bebbngton [5] proposed a smpled scheme and provded perormance measures. In ths paper, usng Roberts [6] Markov-chan approach varous perormance measures or the combned plan s derved and the method o desgnng the plan s gven. The plans ndexed by AQL wth producer rsk o 0.95 and, LQL wth consumer rsk o 0. and are desgned. The prmary ndex and are evaluated or the desgned plan wth varous choces o parameters. The llustraton o the mplementaton o the plan s gven. The operatng procedure o combned contnuous lot by lot acceptance samplng plan has the ollowng steps Step 1: Inspect % o the unts consecutvely untl unts n successon are ound to be deect ree. Step 2: When unts n successon are ound conormng dscontnue cent percent nspecton and start ormng lots o desred sze. Apply any lot by lot samplng nspecton plan as reerence plan. Step 3: I a lot s rejected, go to Step 1; otherwse, contnue wth lot by lot nspecton. In the lot by lot samplng nspecton consder (=ASN/N) as the racton o samplng where ASN s the average sample number o the lot by lot samplng plan consdered and N s the lot sze. Ths combned contnuous lot by lot acceptance samplng plan s ndexed by the parameter and the parameters o the reerence samplng plan (N, n, c). A low dagram s gven or the proposed plan n Fg.1. www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 7 Start Inspect all the unts n the order o producton Are consecutve unts ound conormng? No Replace all the nonconormtes wth the standard ones Yes Contnue wth lot by lot nspecton usng the reerence plan No Is a lot rejected? Yes Fg.1 Flow Dagram o the Operatng procedure o combned contnuous lot by lot plan II. DERIVATION OF PERFORMANCE MEASURES Varous perormance measures o combned contnuous lot by lot acceptance samplng plan are derved usng Markov-chan approach due to Roberts (65). Let [X n ], n=1, 2, denote the dscrete parameter Markov-chan wth nte state space (S k ), k=1, 2,,(+3). The states o the process are dened as S k = A (k-1), k=1,2,3, +1 = percent nspecton s beng conducted and ncludng the latest artcle nspected and the last (k-1) consecutve artcles were ound to be conormng. S +2 = SA =Lot by lot samplng s n eect and the last lot submtted was nspected and accepted S +3 = SR = Lot by lot samplng s n eect and the last lot submtted was nspected and rejected These set o (+3) states dened above completely descrbes the mutually exclusve phases o nspecton or combned contnuous lot by lot acceptance samplng plan. The transtonal probablty matrx s presented n Table 1. The combned contnuous lot by lot acceptance samplng plan appled to an nspecton process whch s n statstcal control may be vewed as a dscrete parameter Markov-chan whch s nte, rreducble and aperodc. The vector o lmtng probabltes π can be determned usng the steady state equatons. www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 7 Table 1 Transton Probablty Matrx O Combned Contnuous Lot By Lot Plan States at (t+1) st tral States at t th tral A 0 A 1 A 2 A -1 A SA SR A 0 p q 0 0 0 0 0 A 1 p 0 0 0 0 0 0 A 2 p 0 0 0 0 0 0 A -1 p 0 0 0 q 0 0 A 0 0 0 0 0 P a 1-P a SA 0 0 0 0 0 P a 1-P a SR p q 0 0 0 0 0 Here p the process racton o nonconormng tems q 1-p, the process racton conormng P a Operatng characterstc uncton o the reerence plan The steady state probabltes π j satsy the ollowng condtons π j > 0 or j=1,2,, (+3) π j = +3 +3 k=1 πk p kj such that k=1 πk= 1 These condtons result n the ollowng equatons π 0 = p(π 0 +π 1 +π 2 + + π -1 +π SR ); π 1 = q(π 0 +π SR ); π 2 = qπ 1 ; π 3 = qπ 2 = q(qπ 1 ) = q 2 π 1 ; π = qπ -1 = q -1 π 1 ; or = 2,3,4,, π SA = P a (π +π SA ), π SR = (1 P a )(π +π SA ) On smplyng we get π 0 = p(1 q )(1 P a )/D; π k = q k p(1 P a )/D, where k = 1,2,, π SA = pq P a /D; π SR = pq (1 P a )/D, where D = (1 P a )(1 q ) + pq () () () (v) (v) The average number o unts nspected under the screenng nspecton s u = (1 q )/pq The average number o lots passed under the samplng nspecton s v = 1/(1 P a ) The OC uncton o the complex contnuous lot by lot plan s P A = pq /D I s the racton o the lot sampled,then AOQ s AOQ = 1 p 2 q /D s the maxmum o AOQ uncton. The average racton nspected () n the samplng nspecton s = ((1 P a )(1 q ) + pq )/D Tables are constructed or, and values o the combned contnuous lot by lot acceptance samplng plan ndexed by certan selected values o AQL wth P A =0.95, LQL wth P A =0., sample sze n and sample racton =n/n usng the search procedure. Numercal values n the tables reveal the ollowng eatures www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 7. Increase n AQL and the sample sze decrease the clearance number.. Increase n LQL and the sample sze decrease the clearance number.. Increase n clearance number decreases v. Increase n clearance number decreases () () III. SELECTION OF PLANS Suppose that one requres combned contnuous lot by lot acceptance samplng plan wth AQL o 0.0% wth probablty o acceptance 0.95, the sample racton o 5% and acceptance number c=2. From the constructed table one obtans the requred plan as(,, n, c) =(326, 1/20, 400,2). For ths plan s 0.00032 whch s the worst qualty receved by the consumer by the applcaton o the constructed plan. At ths worst qualty the amount o nspecton s 40.75%. Suppose that one requres combned contnuous lot by lot acceptance samplng plan wth LQL o 0.% wth probablty o acceptance 0., the sample racton o % and acceptance number c=1. From the constructed table one obtans the requred plan as (,, n, c) =(, 1/,,1). For ths plan s 0.0003268 whch s the worst qualty receved by the consumer by the applcaton o the constructed plan. At ths worst qualty the amount o nspecton s 58.26%. REFERENCES [1]. Pesotchnsky, L (87) Plans or low racton nonconormng, Journal o qualty technology,, pp. 1-6. [2]. Dodge, H. F. () A Samplng plan or contnuous producton, Annals o Mathematcal Statstcs,, pp, 264-279. [3]. Bebbnton, M and Govndaraju, K (98) On Pesotchnsky s Scheme or very low racton nonconormng, Journal o qualty technology, 30, pp. 2-2. [4]. Stephens, K. S. (95) How to perorm Contnuous Samplng, Vol. 2, Second edton, ASQC Basc Reerences n Qualty Control (Wnconsn, Amercan Socety or Qualty control). [5].K. Govndaraju and M. Bebbngton (0) Combned contnuous lot by lot acceptance samplng plan, Journal o appled scences, Vol. 27, No.6, pp 7-730. [6]. Roberts, S.W. (65) States o Markov chans or Evaluatng Contnuous Samplng plans, Transactons o the 17 th Annual All Day Conerence on Qualty control, Metropoltan Secton, ASQC and Rutgers Unversty, New Brunswck, N.J., pp.6-1. www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 713 Table 2 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=1 n AQL 0.0001 0.0005 0.00015 0.000175 0.05 7 3.7 0.62 263 207.1 0.50 8 2.8 0.5405 7 183.5 0.5133 263 6.2 0.00 8 201.6 0.5647 007 163.1 0.5673 263 174.4 0.6000 8 179.2 0.6130 651 246.1 0.4946 169 261.8 0.09 76 269.1 0.5478 651 233.2 0.52 169 247.9 0.56 76 4.9 0.5716 651 207.3 0.5744 169 0.4 0.6049 76 6.6 0.62 456 298.8 0.50 8 316.9 0.67 326.7 0.59 456 283.1 0.5274 8.2 0.56 309.7 0.5793 456 1.6 0.5799 8 266.8 0.6099 275.3 0.6260 337 2.1 0.5069 87 373.0 0.5423 386.6 0.5651 0.1 0.2 337 333.6 0.28 87 3.4 0.5664 366.3 0.5879 n AQL 0.0002 0.000 0.000 0.000 9 405.9 0. 67 429.4 0.5470 30 447.4 0.57 9 384.6 0.72 67 406.8 0.5709 30 423.8 0.5959 9 341.9 0.56 67 361.6 0.6186 30.7 0.6409 215 4.9 0.57 477.9 0. 495.9 0.5816 215 4.3 0.5402 452.8 0.5760 469.8 0.6036 215 378.0 0.5913 402.5 0.6231 417.6 0.6476 167 5.5 0.56 546.9 0.89 5.7 0.5965 167 7.5 0.5449 518.1 0.58 0.1 0.6177 167 3.3 0.5954 460.5 0.6286 9.0 0.6602 86 737.1 0.26 796.8 0.5852 861.6 0.6734 86 698.3 0.72 754.9 0.6071 816.2 0.6906 337 296.5 0.5847 87 3.1 0.66 3.6 0.6338 86 620.7 0.6064 670.9 0.6507 7.5 0.7249 Table 3 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=2 n AQL 0.0001 0.0005 0.00015 0.000175 0.05 0.1 0.2 7866 9.5 0.30 7866 3.2 0.3379 7866.6 0.45 4566 1.9 0.3158 4566 5.8 0.18 4566 9.6 0.42 2868 1.9 0.3302 2868 178.9 0.36 2868 159.1 0.59 4.1 0.3389 2.3 0.3736 1.7 0.42 400 40 6.2 0.3324 2329 130.2 0.38 40 9.5 0.3675 2329 3.4 0.3831 40 6.3 0.78 2329 9.7 0.4516 68 161.4 0. 52 165.6 0.82 68 152.9 0.3785 52 156.9 0. 68 1.9 0.4475 52 1.4 0.4595 1 6.8 0.27 7 201.1 0.3657 1 186.4 0.3867 7 0.5 0.91 1 165.7 0.4549 7 169.3 0.4659 896 232.3 0.3602 2 236.8 0.37 896 0.0 0. 2 4.3 0.4050 n LQL 0.0002 0.000 0.000 0.000 1334 9.5 0.3462 1334 245.9 0.37 1334 218.6 0.4495 28 287.9 0.13 28 272.8 0.38 28 242.5 0. 7 330.8 0.77 7 313.4 0.15 7 278.5 0.4591 2 474.9 0.3726 2 449.9 0.4056 400 615 267.9 0.3651 329 272.6 0.1 615 2.8 0.85 329 8.3 0.4089 615 5.6 0.4654 329 9.6 0.4746 469 296.7 0.36 0 301.7 0.3793 469 281.0 0.4020 0 285.8 0.49 469 249.8 0.4685 0 4.0 0.4773 326 3.8 0.3742 345.5 0.3841 326 321.9 0.4072 327.3 0.4165 326 286.1 0.4730 290.9 0.13 4 6.8 0.3869 66 494.6 0.82 4 461.2 0.42 66 468.6 0.4299 896 5.6 0.46 2 9.4 0.47 2 9.9 0.4717 4 409.9 0.37 66 416.5 0.4933 www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 7 Table 4 Values o (,, ) or Combned plan ndexed by AQL [P a (p)=0.95], and c=3 n AQL 0.0001 0.0005 0.00015 0.000175 0.05 0.1 0.2 9246 1.1 0.69 9246 170.6 0.2676 9246 151.7 0. 6589 209.2 0.22 6589 8.2 0.2792 6589 176.2 0.93 51 232.9 0.2464 51 0.6 0.2861 51 6.1 0.3654 49 268.8 0. 49 4.6 0.2947 49 6.3 0.3730 750 0 3444 1.9 0.2676 1370 8.2 0.2897 3444 181.8 0.3061 1370 187.8 0.3271 3444 161.6 0.3832 1370 166.9 0.4018 31 2.2 0.2761 849 8.5 0.2946 31 2.5 0.32 849 216.5 0.3318 31 187.1 0.04 849 2.5 0.4060 1618 246.6 0.28 600 2.9 0.2974 1618 233.6 0.30 600 2.6 0.3344 1618 207.6 0.47 600 213.0 0.4083 283.1 0.28 289.5 0.3024 268.2 0.3262 274.3 0.31 n LQL 0.0002 0.000 0.000 0.000 37 0.3 0.2786 37 369.8 0.3165 37 328.7 0. 5 513.1 0.2923 5 6.1 0.3295 5 2.1 0.4040 0 760.9 0.32 0 720.8 0.3465 0 640.7 0.41 16.0 0.3245 962.5 0.3600 750 0 3 405.5 0.3032 9 413.6 0.3164 3 384.2 0.39 9 1.8 0.24 3 341.5 0.4132 9 3.3 0.42 3 0.6 0.34 544.0 0.3309 3 502.7 0.05 515.4 0.3661 3 446.9 0.47 458.1 0.65 32 792.1 0.33 8.9 0.3663 32 750.4 0.3705 791.9 0.96 32 667.0 0.4404 703.9 0.4663 81.6 0.36 4.6 0.4427 24.7 0.58 4 16.8 0.47 238.4 0.40 2.8 0.45 8.6 0. 9.8 0.4629 4.4 0.07 Table 5 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=1 n LQL 0.0015 0.00 0.004 0.005 0.05 266.7 0.4977 207 2.1 0.5279 9 1.9 0.5405 245 2.7 4.6 418.3 0.5242 0.5770 0.54 207 2.7 0.28 9 2.2 0.5647 207 7.9 0.60 9 186.9 0.6131 370.7 0.5417 344.9 0. 245 245 1 6.3 2.3 632.5 0.81 0.5894 0.5269 1.7 0.5659 326.8 0.5820 3.2 0.60 290.5 0.6285 546.9 0.89 495.9 0.5816 1 1 599.2 2.7 766.7 0.18 0.6016 0.42 518.1 0.58 469.8 0.6036 460.5 0.6286 417.6 0.6476 31 6.9 0.5696 5.7 0.5965 0.1 0.2 726.3 0.87 31 621.3 0.5923 0.1 0.6177 n LQL 0.0065 0.008 0.01 0.013 960.4 0.5444 769.8 0.5852 696.5 0.6207 909.9 0.5684 754.9 0.6071 659.8 0.6406 8.8 0.6163 670.9 0.6507 586.5 0.66 38.2 0.53 17 926.1 0.60 766.1 0.63 78.3 0.5768 17 877.3 0.60 7.7 0.6578 958.5 0.6238 17 779.8 0.6640 645.1 0.6959 29 1344.1 0.56 40.4 0.6169 9.6 0.6524 29 73.4 0.5869 985.6 0.6371 766.9 0.6707 29 31.9 0.6328 876.1 0.6774 681.7 0.7073 20 1663.6 0.5816 09.0 0.6469 861.6 0.6734 20 1575.9 0.6036 15.4 0.66 816.2 0.6906 645.6 0.6078 31 2.3 0.6 9.0 0.6602 20 00.9 0.6477 18.1 0.7027 7.5 0.7249 www.jsret.org
Internatonal Journal o Scentc Research Engneerng & Technology (IJSRET), ISSN 78 02 715 Table 6 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=2 n LQL 0.0015 0.00 0.004 0.005 0.05 0.1 0.2 16 517.5 0.2901 16 490.3 0.3275 16 5.8 0.40 972 771.5 0.3166 972 730.9 0.26 972 649.7 0.4245 427.6 0.39 427.1 0.3747 427 936.1 0.4441 282 1323.9 0.01 282 54.2 0.38 282.9 0.4527 38 4.3 0.3404 7 370.8 0.3629 38 2.5 0.3751 7 1.3 0.64 38 3.8 0.4451 7 3.3 0.46 623.5 0.36 176 565.6 0.3798 590.7 0. 176 5.8 0.45 5.1 0.4620 176 476.3 0.4777 0 913.1 0.37 65 823.6 0.62 0 865.0 0.45 65 7.3 0.4279 0 768.9 0.4769 65 693.6 0.4915 90.7 0.3866 42 973.7 0.40 33.3 0.41 42 9.5 0.61 n LQL 0.0065 0.008 0.01 0.013 171 16.2 0.3617 171.6 0. 171 1368.6 0.46 1.9 0.3707 1 18.3 0.40 1 16.9 0.4701 74 72.2 0.34 74 2152.6 0.49 74 13.4 0.4782 45 2769.1 0.23 45 2623.4 0.42 52 13.5 0.71 26 74.7 0.4170 52 65.2 0.42 26.9 0.4477 52 24.6 0.4923 26 989.2 0.5091 18.3 0.4064 1332.1 0.4268 76.2 0.76 61.9 0.4569 13.2 0.1 21.8 0.5 24 19.7 0.4168 15.4 0.89 24 17.5 0.4475 30.9 0.4685 24 1523.9 0.5089 71.9 0.5275 16 2134.2 0.09 1672.8 0.4516 16 2021.9 0.4608 1584.7 0.04 918.5 0.34 42 8.9 0.49 45 2331.9 0.83 16 1797.2 0.5207 08.6 0.82 Table 7 Values o (,, ) or Combned plan ndexed by LQL [P a (p)=0.], and c=3 n LQL 0.0015 0.00 0.004 0.005 0.05 85 7.5 0.18 2629 6.1 0.92 1671 549.1 0.79 85 709.1 0.84 2629 592.2 0.2698 1671 520.2 0.2969 85 630.3 0.31 2629 526.4 0.09 1671 462.4 0.3751 2051 38.1 0.2079 49 920.9 0.67 7.1 0.2826 2051 78.2 0.2497 49 872.5 0.2958 764.6 0.3204 2051 958.4 0.3330 49 775.6 0.3740 679.6 0.59 962 1644.9 0.2328 9 1308.3 0.23 181.4 0.3023 962 18.3 0.2732 9.5 0.3182 181 94.6 0.3383 962 1385.2 0. 9 01.8 0. 181 972.9 0.44 6.1 0.2449 242 1545.6 0.2907 4 1368.3 0.37 0.1 0.2 6 1845.6 0.2847 242 64.3 0.32 4 96.3 0.3469 n LQL 0.0065 0.008 0.01 0.013 406 2369.4 0.93 131 11.2 0.30 1661.1 0.3215 406 44.7 0.2983 131 1782.2 0.3389 1573.7 0.72 406 95.3 0.3 131 1584.1 0.44 18.8 0.4286 272 2761.9 0.2705.3 0.32 08.8 0.3302 272 2616.6 0.3089 2079.7 0.3474 18.4 0.36 272 23.8 0.3857 18.6 0.49 1607.4 0.59 32.7 0.2821 73.9 0.3209 08.0 0.3409 3082.5 0.38 28.4 0.66 2091.8 0.3756 27.9 0.54 2167.5 0.4281 1859.4 0.4449 29.9 0.29 28 30.7 0.3326.3 0.41 3723.1 0.3324 28 28.3 0.3677 2420.8 0.31 6 1640.5 0.3641 242 1301.6 0.4027 4 52.3 0.45 3309.4 0.4066 28 67.3 0.79 2151.8 0.4561 www.jsret.org