Certain Results and Tables Relating to QSS-1 with Multiple RGS Plan as Reference Plan

This paper enlarges the concept of Quick Switching System and Multiple Repetitive Group Sampling Plan in to a new procedure for construction and selection of Quick Switching System with Multiple Repetitive Group Sampling Plan as reference plan indexed through Acceptable Quality Level (AQL), Limiting Quality Level (LQL), Indifference Quality Level (IQL) and its operating ratio. Poisson unity values have been tabulated to facilitate the operation and construction of the plan. Illustrations are also provided for selection of plan parameters.


Introduction
Acceptance sampling plan is used to either accept or reject a lot based on the sampling inspection.The primary objective of sampling inspection is to reduce the cost of inspection while at the same time assuring the customer to satisfy an adequate level of quality on items being inspected.Inspection of raw materials, semi finished products, or a finished product is an important part of quality assurance.When inspection is done for the purpose of acceptance or rejection of a product, and it is based on adherence to a standard the type of inspection procedure employed, such a procedure is usually called acceptance sampling.Sampling is widely used in government sector and industry for controlling the quality of shipment of components, supplies and final products.

Quick Switching System
Dodge [1969] has proposed a new sampling inspection-involving normal and tightened inspection plans which are usually referred to as two-plan system.Extensively, Romboski [1969] presents a system of immediate switching to tightened inspection when the rejection comes under normal inspection.Due to instantaneous switching between normal and tightened plans, this system is referred to as 'Quick Switching System (QSS)'.Romboski [1969] has studied the merits and demerits of switching rules of QSS when it is compared with two-plan system (m, d).The rule of QSS is retained at m = 1 where as tightened rule is made when d > 1.Further, Romboski [1969] introduces Quick Switching System Plan of type QSS-1 (n, C N , C T ) with single sampling plan as a reference plan (n, C N ) and (n, C T ) are respectively the normal and tightened single sampling plan with C T < C N .

Multiple Repetitive Group Sampling Plan
The concept of Repetitive Group Sampling (RGS) plan was introduced by Sherman [1965] in which acceptance or rejection of a lot is based on repeated sample results in the same lot.Soundararajan and Ramaswamy [1984] have derived the operating characteristics curve and various designing procedure for the selection of plan parameters.The disadvantage of RGS plan is that it has only a fixed group sample size but no fixed sample size, the second or subsequent samples have to be taken when the lot is neither good nor bad.Gaurishankar and Mohapatra [1993] have developed a new repetitive group sampling plan named as Conditional RGS plan.Gaurishankar and Joseph [1994] proposed another new RGS plan which is an extension of Conditional RGS plan designated as Multiple Repetitive Group Sampling plan in which disposal of a lot on the basis of repeated sample results is dependent on the outcome of the inspection of the immediate preceding i lots.Further they derived the formulae for OC and ASN function.Later Gauri Shankar and Joseph [1994] have studied the MRGS plan using the inflection point for the OC curve.
Devraj Arumainayagam [1991] has studied QSS-1 with RGS plan as reference plan, Suresh K. K. [1993] has studied Acceptance Sampling using Acceptable and Limiting Quality Levels.Suresh.K.K and Kaviyarasu.V [2008] have studied QSS-1 with Conditional RGS Plan as Reference plan indexed with Acceptable Quality Level (AQL), Limiting Quality Level (LQL), Indifference Quality Level (IQL) and its Operating Ratio.This paper provides a new procedure for selection of QSS-1 with Multiple RGS plan indexed through certain basic quality levels, which are tailor made for industrial shop floor applications.

The Quick Switching System with Multiple Repetitive Group Sampling Plan as reference plan
As proposed by Romboski [1969] and also by Devaraj Arumainayagam [1991], a Quick Switching system with Multiple Repetitive Group Sampling plan of type QSMRGSP-1 plan is carried out through the following steps: Step 1: Draw a random sample of size n from a large lot of size N and test each unit for conformity towards specified requirements.
Step 2: Under normal inspection, inspect the plan under Multiple RGS with the parameters n, u 1 and u 2 .If a lot is accepted, continue step 2, otherwise go to step 3.
Step 3:Under tightened inspection, inspect the plan under Multiple RGS with the parameters n, v 1 and v 2 .If a lot is accepted go to step 2 for the next lot, otherwise continue with step 3.
Thus a Quick Switching Multiple Repetitive Group Sampling Plan of type QSMRGSP-1 plans is characterized with six parameters namely, n, u 1 , u 2 , v 1 , v 2 and acceptance criteria i.When i = 0 one find that QSMRGSP-1 plan is reduced to QSRGSP-1.Here it may also be noted that QSMRGSP-1 plan is applicable only to a stream of lots but not for isolated lots, where as the QSRGSP-1 plan is applicable for both.The OC function of the Quick Switching Multiple RGS plan is based on Romboski [1969] andDevaraj Arumainayagam [1991].The expression for OC function of Quick Switching MRGS plan(QSMRGSP)-1 plan is derived from the following equations, where, and (3)

Selection of QSMRGSP-1
For construction and evaluation of the Quick Switching Multiple Repetitive Group Sampling plan, the np values presented in tables were derived under the procedure stated by Duncan [1965].Tables are used to derive individual plan to meet specified values of fraction defectives and probability of acceptance.It requires the specifications of AQL (p 1 ), LTPD (p 2 ), Producers risk (α), Consumers risk (β) and acceptance criteria i.The steps to be followed are, (1) Specify p 1 -Acceptable Quality Level (AQL), p 2 -Lot Tolerance Proportion Defective (LTPD), producer risk (α) and consumer risk (β).
(2) The operating ratio is OR = p 2 / p 1 and m = np.
(3) Choose the plan parameters having u 1 , u 2 , v 1 , v 2 and i associated with an operating ratio which is nearest in the corresponding table.
(5) The OC Curve may be drawn by dividing the values of np shown for the plan by sample size n to obtain p associated with 0.95 for Pa(p).
(6) Thus, the plan consists of six parameters namely: n, u 1 , u 2 , v 1 , v 2 and i may chosen from the given tables.
(3) The operating ratio is OR = 4 in the table 4 which is nearest to the desired ratio is 4.4294 for which the plan parameters are u 1 = 1, u 2 = 5, v 1 = 1,v 2 =4 when i = 2 and np 1 = 0.9510.

Designing the Systems with Given Sample Size and a Point on the OC Curve
It can be used to design a system of type QSMRGSP-1 when the sample size is fixed at a point on the OC curve [p, Pa(p)] is specified.To design a system let n=90 p=0.01 and Pa(p) = 0.95, scan the column headed with Pa(p) = 0.95, when i = 1 to find the np value which is nearer to the desired value 90*0.01 = 0.9.The value approximately equals to 1.0427 which corresponds to the parameters

Glossary of Symbols Used
p = incoming quality of the submitted lot P a (p) = Probability of Acceptance P N = probability of acceptance under normal inspection P T = probability of acceptance under tightened inspection u 1 and u 2 = acceptance numbers under normal inspection v 1 and v 2 = acceptance numbers under tightened inspection i = preceding lots x = number of non-conforming units found in sample p 1 = Acceptable Quality Level (AQL) at which P a (p) ≈ 0.95 p 2 = Limiting Quality Level (LQL) at which P a (p) ≈ 0.10 p 0 = Indifference Quality Level (IQL) at which P a (p) ≈ 0.50

Construction of Tables
The expression for probability of acceptance under the assumption with Poisson model, the composite OC function of QSMRGSP-1 is given through equation ( 1) with For various values of AQL and LTPD with their associated risk, one can solve the equation ( 4) and ( 5), by substituting in equation (1) using iterative technique for selected combinations of u 1 , u 2 , v 1 , v 2 and i. Accordingly the operating values are listed in Table 4, 5 and 6.Using computer programming with C++, the iterative solutions for m = np are obtained and provided respectively in table 1, 2 and 3.

Conclusion
Traditionally, Quick Switching System and Multiple RGS Plan have wide potential applications in industries to ensure a higher standard of quality attainment and increased customer satisfaction.Here, an attempt is made to apply the concept of Quick Switching System with Multiple RGS plan as a reference plan to propose a new plan designated as Quick Switching Multiple Repetitive Group Sampling Plan of type QSMRGSP-1 in which disposal of a lot on the basis of normal and tightened plan.Repetitive Group Sample result is dependent on the immediate proceeding stream of lots.
Poisson unity values have been tabulated for a wider range of plan parameters.The present development would be a valuable addition to the literature and a useful device to the quality practitioners.The concept of this article may be used for assistance to quality control engineers and plan designers in the development of further plans, which were useful and tailor made for industrial shop-floor situations.
5) The OC curve is obtained by dividing the values of np is given below