SKSP-T with Double Sampling Plan (DSP) as Reference Plan using Fuzzy Logic Optimization Techniques

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1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu SKSP-T with Double Sampling Plan (DSP) as Reference Plan using Fuzzy Logic Optimization Techniques K.Pradeepa Veerakumari 1 and S.Suganya 2 1, 2 Department of Statistics, Bharathiar University, Coimbatore-46, Tamilnadu, India 1 sadeep 13@yahoo.co.in, 2 suganstat@gmail.com Abstract This paper deals with the new system of skip-lot sampling plan of type SkSP-T with Double Sampling Plan (DSP) as reference plan using Fuzzy Logic optimization techniques. The concept of triangular fuzzy set is applied in this designing plan to plot the Operating Characteristic (OC) curve. The new designing technique comprises the Acceptable Quality level (AQL), Limiting Quality Level (LQL) and Fuzzy OC Band. The efficiency of the new proposed plan is tested on comparison with Acceptance Single Sampling Plan (SSP) fuzzy parameters. AMS Subject Classification: 62PXX Key Words and Phrases: DSP, Fuzzy Logic, OC Band, SkSP-T, Triangular Fuzzy. 1 Introduction Dodge [8] introduced the first skip-lot sampling plan of type SkSP-1. Various types of skip-lot sampling plans (SkSP-1, SkSP-2, SkSP-3, SkSP-V, SkSP-R), Continuous sampling plans and other attribute sampling plans are derived several authors namely, Dodge and Torrey [13] Lieberman and Solomon [11], Dodge and Perry [23], Cox [6], Carr [5], Schilling [29], Stephens [17], Vijayaraghavan [25,26], Fordice [10], Kandasamy and Govindaraju [16], Balamurali and Subramani [28], Balamurali et al. [2,3,27] and Aslam et al. [19,20,21]. Pradeepa Veerakumari and Suganya [24] introduced the new concept of skip-lot sampling plan of tightened plan designated as SkSP-T. It is based on the concept of CSP-T, CSP-M, MLP-T-3 and SkSP-2. In SkSP-T sampling plan the sampling frequency (f ) is minimized by every skipping inspection level. The Operating Characteristic functions for this SkSP-T plan are also derived with DSP as the reference 409 1

2 plan using Fuzzy membership functions and triangular fuzzy set. Lotfi A. Zadeh [18] introduced theory and examples of Fuzzy sets. Every element of fuzzy set has a membership. Fuzzy sets lies between [0, 1]. Fuzzy set theory is an expansion of classical set. In recent years fuzzy with Acceptance Sampling plans, statistical theory and application based problems are derived and developed by many authors namely, Kanagawa and Ohta [15], Tamaki, Kanagawa and Ohta [30], Hrniewicz [14], Chakraborty [7],, Grzegorzewski [12], Buckley [4], Bahram Sadeghpour-Gildeh et.al [1], Ezzatallah Baloui Jamkhaneh et.al [9], Palash Dutta, Hrishikesh Boruah and Tazid Ali [22] and Zdenek karpisek, petr stepanek and petr jurak., [31]. Already fuzzy probability theory and fuzzy quality control charts are developed. But this paper mainly focuses on acceptance skip-lot sampling plan using fuzzy logic. Combining Acceptance sampling with fuzzy logic reduces time to establish rules by analyzing clusters of data. 2 Operating Procedure For SkSP-T plan Operating procedure for the SkSP-T plan is stated as follows: 1. Initiate SkSP-T procedure with normal inspection using Double sampling plan as reference plan. 2. When i consecutive lots are accepted on normal inspection, discontinue the normal inspection and switch to skipping inspection. 3. On skipping inspection, inspect only a fraction f of the lots selected at random, level After i consecutive lots in succession have been found without a non-conforming at level 1, the system then switches to skipping inspection with a fraction of f/2, level After i consecutive lots in succession have been found without a non-conforming at level 2, the system then switches to skipping inspection with a fraction of f/4, level If a non-conforming lot is found on either skipping level, the system reverts to normal inspection. 7. Exchange all non-conforming lots establish with conforming once. 3 Operating Characteristics Function Of SkSP-T The Operating Characteristics function for SkSP-T plan is given as P a (p) = P i (f 2 f 3 ( 1 P i ) +f 1 f 3 P i ( 1 P i) +f 1 f 2 P 2i ) f 1 f 2 f 3 (1 P i ) +P i (f 2 f 3 (1 P i ) +f 1 f 3 P i (1 P i ) +f 1 f 2 P 2i ) (1) Where, P = Probability of Acceptance of the single sampling plan, f is the sampling frequency and i is the clearance number. 410

3 Definition 1 (α - cut). An α-level set of a fuzzy set C of Z is a non-fuzzy set denoted by [C] α and is defined by [C] α = m Z / C (m) = α if α > 0 cl (spc) if α = 0 Where cl (spc) denotes the closure of the sport of C. Definition 2 (triangular fuzzy number). A fuzzy set C is called triangular fuzzy number with peak (or center) c, left width α > 0 and right width β > 0 if its membership function has the following form 1 c k if c α = k = c α C (t) = 1 c a if c = k = c + β β 0 otherwise And we use the notation C = (c, α, β). It can easily be verified that [B] γ = [c (1 γ) α, c + (1 γ) β], γ [0, 1]. The sport of B is (c α, d + β). The triangular fuzzy number with center a may be seen as a fuzzy quantity. 4 Skip Lot Sampling Plan of Type SkSP-T With Double Sampling Plan as Reference Plan Using Fuzzy Parameters The Operating Characteristics function for SkSP-T plan is given as Pa (p) = P i (f 2 f 3 ( 1 P i ) +f 1 f 3 P i ( 1 P i) +f 1 f 2 P 2i ) f 1 f 2 f 3 (1 P i ) +P i (f 2 f 3 (1 P i ) +f 1 f 3 P i (1 P i ) +f 1 f 2 P 2i ) Where i is the clearance number, f is sampling fraction and P- double sampling plan as reference plan using fuzzy parameters. Double sampling plan (DSP) that the second chance is given for the lot either to accept or reject the whole lot. In DSP necessary parameters are N, n 1 (First sample ), n 2 (Second Sample), c 1 (Acceptance number for first sample), c 2 (Acceptance number for second sample), d 1 (number of defective found in the first sample)and (number of defective found in the second sample). Take a first sample n 1 from the lot size N. If d 1 c 1, accept the lot. If d 1 >c 1 reject the whole lot and suggest the lot for skipping inspection. For c 1 +1 d 1 c 2 take a second sample n 2 from the remaining lot and c 2 accept the whole lot. If >c 2 reject the whole lot and suggest the normal inspection. For DSP using λ 1 = n 1 p and λ 2 = n 2 p. Since λ= λ 1 + λ 2. Probability of acceptance of Double Sampling Plan as follows P a (p) = c 1 d 1 =0 + c 2 d 1 =c 1 +1 c 2 d1 e n2p (n 2 p)! (2) Where, P a (p) is including two parts for first sample and second sample. 411

4 P a (p) = P I a(p) + P II a(p) Therefore, P I a (p) = c 1 d=0 (3) P II a (p) = c 2 d 1 =c 1 +1 c 2 d1 e n2p (n 2 p)! (4) In these situations fuzzy membership parameter with fuzzy number P is introduced. If P as follows P = (a 1, a 2, a 3 ) Poisson parameters are replaced into fuzzy parameters. The Poisson parameters λ 1 and λ 2 is modified as λ 1 and λ 2. Such that λ 1 = n 1 p and λ 2 = n 2 p. Fuzzy probability with fuzzy number is using in DSP then the function is modified and as follows. P a (p) = P a I II (p) + P a (p) P II a P (d) [α] = [ P L [α] P U [α] ] c1 P a I (p) = P L e n1p (n 1 p) d 1 [α] = min / λ 1 ɛ λ 1 [α] d 1 =0 c1 P a I (p) = P U e n1p (n 1 p) d 1 [α] = max / λ 1 ɛ λ 1 [α] (p) = P L [α] = min P a II (p) = P U [α] = max c2 d 1 =c 1 +1 c2 d 1 =c 1 +1 d 1 =0 c 2 d1 e n2p (n 2 p)! c 2 d1 e n2p (n 2 p)! / λ 2 ɛ λ 2 [α] / λ 2 ɛ λ 2 [α] Also the probability of acceptance P a (p) is defined by fuzzy probability of acceptance P a (p) and it is given as follows c1 P a I e n1p (n 1 p) d 1 (p) = d 1 =0 / λ 1 ɛ λ 1 = n 1 p[α] P II a (p) = c2 d 1 =c 1 +1 c 2 d1 e n2p (n 2 p)! / λ 2 ɛ λ 2 = n 2 p[α] 412

5 5 Numerical Illustration First sample of size n 1 =50, c 1 =0, p = [0.001, 0.005, 0.01]. λ1 = n 1 p. Then the fraction defective p take three values. Hence λ 1 = [50 p, 50 p, 50 p] = [0.05, 0.25, 0.5]. Second sample of size n 2 =75, c 2 =1. And λ 2 = n 2 p. λ2 = [75 p, 75 p, 75 p]=[0.075, 0.375, 0.75]. After using α cut and it becomes, λ [α] = [(a 2 a 1 ) α + a 1, a 3 (a 3 a 2 ) α] Hence λ 1 [α] = [0.20α , α] λ 2 [α] = [0.3α , α] Acceptance number for first sample c 1 = 0 we get the probability of acceptance is P a I (p) = e λ 1 /λ 1 ɛ λ 1 = n 1 p[α] And probability of acceptance P I a (p) is defined by per and lower bound values. So P I a (p) is calculating by below format. P I a (p) = λ 1 [α] = [ e ((a 3 (a 3 a 2 )α)), e (a 3 (a 3 a 2 )α) ] e ( α), e (0.20α+0.05) /λ 1 ɛ λ 1 = n 1 p[α] Acceptance number for second sample c 2 = 1 we get the probability of acceptance is P II a (p) = ( α) e ( α), (0.20α ) e (0.3α+0.075), λ 2 ɛ λ 2 = n 2 p[α] Put different values of α between [0, 1]. First we put α = 0, and we get P a I (p) = II [0.6066, ]. And P a (p) = [0.2361, ] After take the combined sample size n=n 1 +n 2 wed get the probability of Acceptance is P a (p) = P a I II (p) + P a (p) = e ( α) +( α) e ( α), e (0.20α+0.05) +(0.20α ) e (0.3α+0.075) Therefore P a (p) = [0.9901, 1]. (a) (b) Figure 1: (a). Fuzzy Probability of Acceptance for First and Second Sample (b). Fuzzy Probability of Acceptance for Combined Sample 413

6 Fig a and Fig b represents fuzzy probability of acceptance values of skip-lot sampling plan of type SkSP-T with DSP as reference plan using fuzzy parameters. From fig a Probability of Acceptance for first sample is P a I (p) = [0.6066, ] II and Fuzzy Probability of acceptance for second sample is P a (p) = [0.2361, ]. From Fig b Fuzzy Probability of Acceptance for the Combined sample is P a (p) = [0.9901, 1]. It is concluded that skip-lot sampling plan of type SkSP-T with DSP as reference plan using fuzzy parameters gives the better result. (i.e.) it is expected that for every 100 lots in such a process, 99 to 100 lots will be accepted. Next we draw the operating characteristic curve for SkSP-T with DSP as reference plan using fuzzy parameter. Consider fuzzy number p and the fraction defective p it follows P a = (m, a 2 + m, a 3 + m ) We consider λ = n p Where, λ= λ 1 + λ 2, λ 1 = n 1 p, λ 2 = n 2 p and n=n 1 +n 2. n p = ( nm, na 2 + nm, na 3 + nm) Where, m is the domain of [0, 1-a 3 ]. For OC band is calculating as follows p [α] = [p 1 [α], p 2 [α]] [p 1 [α], p 2 [α]] = [m + a 2 α, a 3 + m (a 3 a 2 ) α] The above fuzzy probability condition is extended to λ [α] and P a [α]. For the above example define c 1 =0, c 2 =1, a 2 = 0.001, a 3 = and α = 0. Then the values are substituted in the fuzzy poisson per and lower band equation after we get the values. Figure 2: Operating Characteristic curve for SkSP-T with DSP as reference plan using fuzzy parameters 414

7 Table 1: SkSP-T with DSP as reference plan using fuzzy parameters is comparison with SkSP-T with SSP as reference plan [fuzzy probability of acceptance table] m p Pa Pa [SkSP-T with SSP as reference [SkSP-T with DSP as reference plan using Fuzzy parameters] plan using Fuzzy parameters] 0 [0,0.01] [ ,1] [0.9967,1] 0.01 [0.01,0.02] [ , ] [0.9857,0.9967] 0.02 [0.02,0.03] [ , ] [0.9621,0.9857] 0.03 [0.03,0.04] [ , ] [0.9186,0.9621] 0.04 [0.04,0.05] [ , ] [0.8492,0.9186] 0.05 [0.05,0.06] [ , ] [0.7536,0.8492] 6 Conclusion In this proposed skip lot sampling plan of type SkSP-T with Double Sampling Plan as Reference plan using Fuzzy Parameters (SkFDSP-T) and triangular fuzzy using the production process along with the eventual objective of producing the serior quality of products. SkFDSP-T with Fuzzy logic techniques is compared with SkFSSP-T. It concludes that SkFDSP-T is more efficient than SkFSSP-T. That is, the new scheming plan has high probability of acceptance and maintains the product quality. Also OC curve and OC band of SkSP-T with DSP using Fuzzy Parameters were constructed which shows efficiency of the proposed plan. References [1] Bahram Sadeghpour-Gideh, Gholamhossein Yari and Ezzatallah Baloui Jamkhaneh, Acceptance Double Sampling plan with fuzzy parameter, Proceedings of the 11 th joint conference in information sciences (2008). [2] Balamurali, S, Modified Tightened Three level Continuous sampling plan, Economic Quality Control, (2002) Vol. 17, p [3] Balamurali, S., and Chi-Hyuck Jun, Modified CSP-T sampling procedures for continuous production process, Quality Technology and Quantitative Management, (2004), Vol. 1, No. 2, pp [4] Buckley J. J. Fuzzy probability: New approach and application, physica-velage, Heidelberg, (2003), Germany. [5] Carr W.E, Sampling plan adjustment for inspection error and skip-lot plan, Journal of Quality Technology, (1982), 14(1), [6] Cox D.C, Skip-lot sampling plan, Journal of Quality Technology, (1980), 21(8), [7] Chakraborty T.K., A class of single sampling plan based on fuzzy Optimization, Opsearch, 29(1), (1992),

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