Economic Design of Skip-Lot Sampling Plan of Type (SkSP 2) in Reducing Inspection for Destructive Sampling

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1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Economic Design of Skip-Lot Sampling Plan of Type (SkSP 2) in Reducing Inspection for Destructive Sampling K. Pradeepa Veerakumari 1 and M. Kokila 2 1,2 Department of Statistics, Bharathiar University, Coimbatore, Tamil Nadu. 1 sadeep 13@yahoo.co.in 2 kktejaskok@gmail.com Abstract In acceptance sampling an integral part of the quality system becomes an ingredient in the general approach to maximize the quality of the lot or product at minimum cost. This reduces the risk of producer and favours the consumer. In this paper, the inspection is carried out based on the skip-lot sampling plan of type SkSP 2 for the destructive sampling in which the cost model has been developed. The sensitivity analysis and tables are provided for the optimal inspection plan which is most economical for the producer and also delivering the quality goods to the consumer at a satisfactory cost. AMS Subject Classification: 62pxx Key Words and Phrases: Skip-lot sampling plan, Destructive sampling, Optimal inspection plan, Failure costs, Manufacturing costs. 1 Introduction Acceptance sampling procedures given by Dodge and Romig [4] helps in providing the decision about the received lots in case of testing and inspection of raw materials. In the continuous flow of production process, the items are produced in lots of equal size and testing of items may be destructive or non-destructive. It is uneconomical and timeconsuming if each and every lot is inspected. Every industry needs the feedback of their produced item within the short period after inspection. The continuous sampling plan approach extended for an individual lot received in a steady stream of production is the skip-lot sampling plan-1 introduced by Dodge [5]. In this paper in case as the products are destructive in order to reduce inspection the skip-lot sampling plan 2 by Dodge and Perry [6] is used and the cost model has been initiated to minimize the expected costs 101

2 in which the producers risk and consumers risk has been factored into skip-lot procedure using standard reference plan. Hald [10] gave a foundation for selecting single sampling plans on the basis of a prior distribution and it minimizes the average costs assuming the costs are linear with fraction defective p. Hsu [11] developed skip-lot cost model for destructive sampling which gave optimum plan that minimizes the average cost per good unit. Followed by Phelps [14] developed skip-lot model using Bayesian approach to minimize s-expected cost per good item. Deepa Menon [20] has studied the formulation of a Bayesian Special type Double Sampling Plan using acceptance probability with Gamma prior distribution for product quality and derived the standardized cost function. Sampath Kumar et al. [18] determined an optimal testing sequence for a problem with n tests on one item and obtained the expected cost to minimum. Kobilinsky and Bertheau [13] estimated a cost optimal acceptance sampling plan with least expensive on grain control with application to genetically modified organism detection. Chen and Chou [2] introduced economic design of short-run CSP 1 Plan under linear inspection cost. It has been established that the optimal sequence that minimizes the expected inspection cost is obtained by sequencing the tests. Balamurali and Jun [1] has formulated a Markov chain and derived performance measures for the skip-lot sampling plan. Certain cost models has been proposed for the SkSP 3 plan. Hsu and Hsu [12] developed the mathematical model that the product inspection, internal failure and post sale failure costs also have an effect on the choice of the economic sampling plan in a Two-Stage supply chain. Fallahnezhad and Aslam [7] proposed an economical design for the optimal decision using the Bayesian inference along with backward induction is utilized to analyze the expected cost of different decisions. Fallahnezhad et al. [8] proposed an iterative decision rule to design a optimal cost Sampling Plan for machine replacement problem which fulfils the constraint of first Type I and Type II errors. Roan s model modified by Chen et al. [3] including the Taguchi s quadratic quality loss of conforming products associated with raw material and production process sustains the optimal parameters under the minimization of the expected total relevant cost of product per unit time. In this paper an attempt has been made for SkSP 2 to model an appropriate function for the expected costs involving strength of the plan of two points having acceptance probabilities of lots with different state of quality. 2 Operating procedure of SkSP-2 The probability of acceptance under Poisson model for the single sampling plan is given by P = c e np np x x=0 x! : x = 0, 1, 2 (1) Here np is replaced by considering risks np(1 α) for good state k, δ k = 0.01 and npβ for degraded state j, δ j = Hence P k and P j are obtained. Perry [15] gave the operating procedure of SkSP-2 as, i) Start with normal inspection of each lot using the reference plan. ii) When i consecutive lots are found clear of defects to standards, then switch to skipping inspection (inspecting only f of lots) using the SSP procedure as the reference plan. iii) If any lots are found non conformance on skipping inspection then a lot is rejected, switch to normal inspection. Otherwise continue the skipping inspection using the same 102

3 reference plan and accept the lot. iv)the rejected lots are replaced by the good ones and revert to 100% inspection under step (i).the average number of lots inspection in screening stage for SkSP-2 during one period of inspection is, U = 1 P i P i (1 P ) (2) Hence, V is the average number of lots passed under sampling fraction on one inspection cycle. V = fq Q 2 f 2 = 1 f(1 P ) (3) The measure of SkSP-2 for the probability of acceptance is P a (p) = (1 f)p i + fp (1 f)p i + f (4) The strength of the sampling plan is obtained by satisfying the following conditions P a (p 1 ) &P a (p 2 ) β (5) which helps the producer and the consumer economically and also quality of the product is up to the satisfactory level. 3 Notation and Glossary C m unit production cost. C i unit inspection cost. C p penalty costs associated with replacing a defective unit shipped to customer. S salvage value per unit of scrapped item. N number of items in the lot. n number of items taken for inspection. 1 α confidence level for being a good state k. β satisfactory level for being a degraded state j, j > 0. δ j lot being proportion defective in a degrade state j. δ k lot being proportion defective in a good state k, k = 0. λ k mean rate of change per lot from good state k to bad state j. λ j mean rate of change per lot from degrade state j to good state k. c acceptance number. E(C) average cost of a good unit. G good units received after inspection to the customer. i clearing interval of lots under the skip-lot sampling plan. P a k probability of accepting the lot when the fraction defective is in state k under the skipping inspection of type-2. P r k probability of rejecting the lot when the fraction defective being in state k. P a j probability of accepting the lot when the fraction defective 103

4 P r j P is in state j under the skipping inspection of type-2. probability of rejecting the lot when the fraction defective being in state j. probability of acceptance of good or degraded lot under the reference plan. 4 Economic Design for the Inspection of Destructive Lot The quality cost system determined is a function which satisfies the quality of the product assuring costs involved economically. In the inspection of destructive items, rejected lots are not sorted but scrapped or reprocessed. The various costs associated with defects discovered before and after the product has been shipped to the customers. The costs are of two types internal and external costs regarding failure include waste, rework costs, scrapping cost, returns of rejected lots plans for quality, reliability, operations, production, and inspection in case of non-destructive product. Manufacturing costs includes the cost of the direct material, factory overhead charges and wages of labour become a part of the finished product. Appraisal costs incurred includes verification, quality audits and supplier rating to find the degree of conformance to the satisfactory level evaluated by the dealer and customer of purchased materials, products, and services to ensure that they conform to the standard. These quality cost allows an organization to prevent from poor quality to the extent and to determine the savings by implementing process improvement. Here follows Hsu [11] procedure as the long run minimum average unit cost subject to n, i, f and c is given by E(C) = MC + AC + T F C G Skip-lot sampling plan helps in reduced inspections which automatically signify the minimum average unit cost favours the producer and also be safeguard to the customer from accepting the unsatisfactory lot. There may be two states k and j in the production process where in inspection lot one is good state k having very least proportion defective and the other proportion defective are considered as the degraded state j. (6) 5 Numerical procedure The Average cost behaviour is calculated for each n, c, i and f. The algorithm is as Hsu [11] follows and this paper constructs a model which differs in the following steps. 1. Given the values of δ k, δ j, λ k and λ j. 2. Initially, set i = 1, f = 1/2. 3. Set c = 0. Gradually increase n by a fixed quantity and hold c, i constant, the lowest average cost is obtained. Repeat the process for various values of f. 4. Increase c by one and again search for a minimum by increasing n and keeping i constant repeating the steps 2 and 3. When the minimum cost for a given c indicates that the minimum point for n and c together has been reached, the best plan is obtained for i. 5. Increase i by 1 and set f, c constant and varying n. Repeat steps 2, 3 and 4 to find the best i and n for SkSP-2 and a good estimate of the plan in the new interval. 6. Determine the optimal inspection plan of SkSP-2 characterized by n, c, i and f by repeating step 5 until the minimum E(C) has been reached. The various costs are included and their relationship is carried out to show the effectiveness of the skip-lot sampling plan- 2 is given: 104

5 1. Manufacturing Costs is given as MC = NC m i (7) 2. Appraisal Costs: AC = nc i (8) 3. Total Failure Costs T F C = (Ni n){c P {P k δ k P α + P j δ j P β } {P k P r k + P j P r j }} (9) where P a k (1 α) = P α and P a j β = P β 5. Total cost in a screening and sampling inspection under SkSP-2 is T C = MC + AC + T F C (10) The Average cost per unit is calculated by equation 6 evaluated for obtaining minimum average cost and are given in Tables with numerical illustration. 6 Sensitivity Analysis and Numerical Illustration Consider the sampled items which are taken for inspection as a destructive one. For example, cutting operations carried out by the cutting tool or in factory holding screws and nuts of the equipment to find loosening ratio of chopped item. The lot are submitted for inspection in the order of their production process. There may be two states k and j where there are changes in the process fraction defective is continuous as the lots are produced in each inspection interval from the past records. There may be transition from one state to another per lot. If the sample size n and the acceptance number c exceeded as the SSP reference plan, the measurement is stopped and the lot is rejected. The acceptance and rejection is based on as per the skip-lot sampling plan-2 procedure. Let N = 1000,C m = $1, C i = $0.10, S = $0.20,C p = $50, α = 0.05,β = 0.10, δ k = 0.01,δ j = 0.10,λ k = 0.01 and λ j = From Table 1 Let i = 4 and c = 1 are fixed and the sampling fraction f varies, then E(C) is increased up to certain values of n and decreases at particular sample size. The minimum average cost exist for the optimum plan is i = 4, c = 1, f = 1/2, n = 120, G = 1276, E(C) = $2.77. From Table 2 Let i = 4 and c = 3 are fixed and the sampling fraction f varies, then E(C) is increased up to certain values of n and decreases at particular sample size. The minimum average cost exist for the optimum plan is i = 4, c = 3, f = 1/3, n = 102, G = 3203, E(C) = $2.35. From Table 1 and 2 it is revealed that if we relax the acceptance number and f, the sample size n gets decreases and cost decreases by $0.42. In Table 3 Let i, f fixed and the acceptance number c varies, then E(C) is increased up to certain values of n and decreases at particular sample size but not decreased in the plan as c varies. The minimum average cost exist for the optimum plan is i = 4, c = 1, f = 1/3, n = 103, E(C) = $ For n fixed, i = 4, f = 1/3 and c varies and increases, thene(c) is a decreasing function of c. From Table 4 Let f and c are fixed and the clearance interval i varies, the E(C) is an increased up to certain values of n and decreases at particular sample size. The minimum average cost exist for the optimum plan is i = 1, c = 1, f = 1/3 and n = 134. E(C) = $ Let n and f are fixed, c = 1 and i varies then E(C) is an increasing function of i. From Table 5 Let c is fixed and i and sampling fraction f varies, then E(C) is increased up to certain values of n and decreases 105

6 at particular sample size but stable and later increased for the plan i = 2, i = 3 and i = 4 with respective f variation. Hence the minimum average cost exist for the optimum plan is i = 4, c = 1, f = 1/2, n = 105, E(C) = $3.625 gives more protection to the consumer from accepting unsatisfactory level. For n is fixed, c = 3 and i and f varies, then E(C) is an decreasing function of i. From the study the minimum mean cost obtained for the optimum plan is i = 4, c = 3, f = 1/3, n = 102, G = 3203, E(C) = $2.35. Even though the cost and sample size n in some situation of inspection changes to obtain minimum mean cost it protects the consumer as well as the producer with the consideration of required plan with strength (δ k, 1 α) and (δ j, β). Table 1: Determination of Optimal Cost for the SkSP-2 Plan i = 4, c = 1 n f = 1 f = 1 f = 1 f = 1 n f = 1 f = 1 f = 1 f = Figure 1: The optimum cost for the plan keeping i, c fixed and f varies 7 Conclusion This paper attempted to model a appropriate function for the expected costs involving strength of the plan (δ k, 1 α) and (δ j, β) points with acceptance probabilities of lots with different state of quality. In this paper, the inspection is carried out based on the skip-lot sampling plan 2 for the destructive sampling in which the cost model has been 106

7 Table 2: Determination of Optimal Cost for the SkSP-2 Plan i = 4, c = 3 n f = 1 f = 1 f = 1 f = 1 n f = 1 f = 1 f = 1 f = Figure 2: The optimum cost for the plan keeping i, c fixed and f varies Table 3: Determination of Optimal Cost for the SkSP-2 Plan i = 4, f = 1 3 n i = 1 i = 2 i = 4 n i = 1 i = 2 i = developed. The sensitivity analysis are performed for the cost parameters of proposed model and tables are provided for the optimal inspection plan which is most economical 107

8 Figure 3: The optimum cost for the plan keeping i, f fixed and c varies Table 4: Determination of Optimal Cost for the SkSP-2 Plan c = 1f = 1 3 n i = 1 i = 2 i = 4 n i = 1 i = 2 i = Figure 4: The optimum cost for the plan keeping c, f fixed and i varies 108

9 Table 5: Determination of Optimal Cost for the SkSP-2 Plan i = 1 i = 2 i = 3 i = 4 i = 1 i = 2 i = 3 i = 4 n f = 1 f = 1 f = 1 f = 1 n f = 1 f = 1 f = 1 f = Figure 5: The optimum cost for the plan keeping i, f vary and c fixed for the producer and also delivering the quality goods to the consumer at a satisfactory cost. For product characteristics that involve costly or destructive testing by attributes, SkSP-2 with SSP having smaller acceptance numbers, such as c = 1 and c = 3, is desirable. It is very economical and time consumer for the shop-floor industrialist. References [1] S. Balamurali, C.H. Jun, A new system of skip-lot sampling plans having a provision for reducing normal inspection, Applied Stochastic Models in Business and Industry, 27 (2010), [2] C.H. Chen, C.Y. Chou. Economic design of continuous sampling plan under linear inspection cost, Journal of Applied Statistics, 29 (2006), [3] C.H. Chen, C.Y. Chou, C.C. Kan, Modified economic production and raw material model with quality loss for conforming product,journal of Industrial and Production Engineering, 32 (2015), [4] H.F. Dodge, H.G. Romig, A method of sampling inspection. Bell system Technical Journal, 8 (1929),

10 [5] H. F. Dodge, Skip-lot sampling plan, Industrial Quality Control, 11 (1955), 3-5. [6] H.F. Dodge, R.L. Perry, A system of skip-lot plans for lot by lot inspection, ASQC Technical Conf. Trans.(1971), [7] M.S. Fallahnezhad, M. Aslam, A new economical design of acceptance sampling models using Bayesian inference, Accred Qual Assur., 18 (2013), [8] M.S. Fallahnezhad, M.S. Sajjadieh, P. Abdollahi, An iterative decision rule to minimize cost of acceptance sampling plan in machine replacement problem, International Journal of Engineering (IJE) Transactions A: Basics, 27 (2014), [9] A. Golub, Designing Single Sampling Inspection Plans when the sample size is fixed, Journal of American Statistical Association, 48 (1953), [10] A. Hald, On the Theory of Single Sampling Inspection by Attributes Based on Two Quality Level, Review of the International Statistical Institute, 35 (1967), [11] J.I.S. Hsu, A cost model for skip-lot destructive sampling, IEEE Transactions on Reliability, 26 (1977), [12] L.F. Hsu, J.T. Hsu, Economic Design of Acceptance Sampling Plans in a Two-Stage Supply Chain, Advances in Decision Sciences, Article ID , (2012), [13] A. Kobilinsky, Y. Bertheau, Minimum cost acceptance sampling plans for grain control, with application to GMO detection, Chemometrics and Intelligent Laboratory System, 75 (2005), [14] R.L. Phelps, Skip-lot Destructive Sampling with Bayesian Inferences, IEEE Transactions on Reliability, 31 (1982), [15] R.L. Perry, Skip-lot sampling plans, Journal of Quality Technology, 5 (1973), [16] K. Pradeepa Veerakumari, R. Resmi, Designing optimum plan parameters for continuous sampling plan of type (CSP-2) through GERT analysis, Journal of Statistics and Management Systems, 19 (2016), [17] S.M. Ross, Stochastic processes, 2nd edition. NewYork: Wiley (1996). [18] V.S. Sampath Kumar, K.V.S. Sarma, K. Sekhar. The Least Cost Testing Sequence Problem, Economic Quality Control, 19 (2004), [19] E.G. Schilling, D.V. Neubauer, Acceptance Sampling in Quality Control. BocaRaton: CRC Press. Marcel Dekker, NewYork (2008). [20] O.S. Deepa Menon, Studies on certain classical and Bayesian Sampling Plans. Ph.D.Thesis, Bharathiar university, Tamil Nadu (2002). [21] R. Vijayaraghavan, Construction and selection of skip-lot sampling inspection plans of type SkSP-2 indexed by IQL and MAPD, Journal of Applied Statistics, 21 (1994),

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