Acceptance sampling plans by variables are a basic quality control technique. These plans provide economical procedures to determine the acceptability of batches of product. Most of these plans are based on a single quality characteristic and are of the classical type. This work concentrates on Bayesian variable acceptance sampling plans for fraction defective. Both destructive and non-destructive sampling procedures are considered. A set of decision points are estimated and employed to make decisions about the inspected lots. Techniques to dispose of the rejected lots are provided. Components of the expected total cost relative to various decisions are estimated. The sample size required to obtain the expected optimum cost is found. An untrue assumption implicit in the measurement of the quality characteristic of items sampled is that the observed dimensions are error free. The distributions, means, and variances of a set of parameters for error free and error prone sampling is listed. Computer programs written in FORTRAN 77 are developed to compute the decision points and the costs for both destructive and nondestructive testing. Precise Bays estimate of the costs and other economic parameters involve the moments of the fraction defective p raised to the kᵗʰ power. Mathematical expressions for the conditional expectations of p|x and p|ẋ are derived and a computer program to estimate these moments is provided. Producing quality items with minimum cost requires keeping a production process under control. The quality characteristic X of each item produced is determined and the sample means are plotted on an Ẋ-control chart. A production process is assumed to start in control at time t = 0 with specific values of the mean and standard deviation. The occurrence of a single or multiple cause-failures shift the process mean outside the control limits. During the search for the causes of failure, the process is either allowed to continue in operation or shut down until the assignable cause or causes are discovered. The expected duration of time during which the process is shut down and the additional time to be taken to repair the process are considered. Computer programs are provided to estimate the optimal sample size, the interval between successive samples, the control limits, the probability of type I error, the power of the chart, and the average time the process operates in the presence of an assignable cause. The parameters estimated are employed to estimate the optimal loss-cost. The economic design of Ẋ -charts assumes one quality characteristic of interest. However a product quality in most industrial products and processes is characterized by more than one quality characteristic where the application of a Ẋ -control chart for each variable is inappropriate. In this work a Hotellings T² control chart is employed to handle cases of where products are tested relative to several quality characteristics.

Assignment (MSc)--Stellenbosch University, 2004. / ENGLISH ABSTRACT: The design of control charts refers to the selection of the parameters implied, including the sample size n, control limit width parameter k, and the sampling interval h. The design of the X -control chart that is based on economic as well as statistical considerations is presently one of the more popular subjects of research. Two assumptions are considered in the development and use of the economic or economic statistical models. These assumptions are potentially critical. It is assumed that the time between process shifts can be modelled by means of the exponential distribution. It is further assumed that there is only one assignable cause. Based on these assumptions, economic or economic statistical models are derived using a total cost function per unit time as proposed by a unified approach of the Lorenzen and Vance model (1986). In this approach the relationship between the three control chart parameters as well as the three types of costs are expressed in the total cost function. The optimal parameters are usually obtained by the minimization of the expected total cost per unit time. Nevertheless, few practitioners have tried to optimize the design of their X -control charts. One reason for this is that the cost models and their associated optimization techniques are often too complex and difficult for practitioners to understand and apply. However, a user-friendly Excel program has been developed in this paper and the numerical examples illustrated are executed on this program. The optimization procedure is easy-to-use, easy-to-understand, and easy-to-access. Moreover, the proposed procedure also obtains exact optimal design values in contrast to the approximate designs developed by Duncan (1956) and other subsequent researchers. Numerical examples are presented of both the economic and the economic statistical designs of the X -control chart in order to illustrate the working of the proposed Excel optimal procedure. Based on the Excel optimization procedure, the results of the economic statistical design are compared to those of a pure economic model. It is shown that the economic statistical designs lead to wider control limits and smaller sampling intervals than the economic designs. Furthermore, even if they are more costly than the economic design they do guarantee output of better quality, while keeping the number of false alarm searches at a minimum. It also leads to low process variability. These properties are the direct result of the requirement that the economic statistical design must assure a satisfactory statistical performance. Additionally, extensive sensitivity studies are performed on the economic and economic statistical designs to investigate the effect of the input parameters and the effects of varying the bounds on, a, 1-f3 , the average time-to-signal, ATS as well as the expected shift size t5 on the minimum expected cost loss as well as the three control chart decision variables. The analyses show that cost is relatively insensitive to improvement in the type I and type II error rates, but highly sensitive to changes in smaller bounds on ATS as well as extremely sensitive for smaller shift levels, t5 . Note: expressions like economic design, economic statistical design, loss cost and assignable cause may seen linguistically and syntactically strange, but are borrowed from and used according the known literature on the subject. / AFRIKAANSE OPSOMMING: Die ontwerp van kontrolekaarte verwys na die seleksie van die parameters geïmpliseer, insluitende die steekproefgrootte n , kontrole limiete interval parameter k , en die steekproefmterval h. Die ontwerp van die X -kontrolekaart, gebaseer op ekonomiese sowel as statistiese oorwegings, is tans een van die meer populêre onderwerpe van navorsing. Twee aannames word in ag geneem in die ontwikkeling en gebruik van die ekonomiese en ekonomies statistiese modelle. Hierdie aannames is potensieel krities. Dit word aanvaar dat die tyd tussen prosesverskuiwings deur die eksponensiaalverdeling gemodelleer kan word. Daar word ook verder aangeneem dat daar slegs een oorsaak kan wees vir 'n verskuiwing, of te wel 'n aanwysbare oorsaak (assignable cause). Gebaseer op hierdie aannames word ekonomies en ekonomies statistiese modelle afgelei deur gebruik te maak van 'n totale kostefunksie per tydseenheid soos voorgestel deur deur 'n verenigende (unified) benadering van die Lorenzen en Vance-model (1986). In hierdie benadering word die verband tussen die drie kontrole parameters sowel as die drie tipes koste in die totale kostefunksie uiteengesit. Die optimale parameters word gewoonlik gevind deur die minirnering van die verwagte totale koste per tydseenheid. Desnieteenstaande het slegs 'n minderheid van praktisyns tot nou toe probeer om die ontwerp van hulle X -kontrolekaarte te optimeer. Een rede hiervoor is dat die kosternodelle en hulle geassosieerde optimeringstegnieke té kompleks en moeilik is vir die praktisyns om te verstaan en toe te pas. 'n Gebruikersvriendelike Excelprogram is egter hier ontwikkel en die numeriese voorbeelde wat vir illustrasie doeleindes getoon word, is op hierdie program uitgevoer. Die optimeringsprosedure is maklik om te gebruik, maklik om te verstaan en die sagteware is geredelik beskikbaar. Wat meer is, is dat die voorgestelde prosedure eksakte optimale ontwerp waardes bereken in teenstelling tot die benaderde ontwerpe van Duncan (1956) en navorsers na hom. Numeriese voorbeelde word verskaf van beide die ekonomiese en ekonomies statistiese ontwerpe vir die X -kontrolekaart om die werking van die voorgestelde Excel optimale prosedure te illustreer. Die resultate van die ekonomies statistiese ontwerp word vergelyk met dié van die suiwer ekomomiese model met behulp van die Excel optimerings-prosedure. Daar word aangetoon dat die ekonomiese statistiese ontwerpe tot wyer kontrole limiete en kleiner steekproefmtervalle lei as die ekonomiese ontwerpe. Al lei die ekonomies statistiese ontwerp tot ietwat hoër koste as die ekonomiese ontwerpe se oplossings, waarborg dit beter kwaliteit terwyl dit die aantal vals seine tot 'n minimum beperk. Hierbenewens lei dit ook tot kleiner prosesvartasie. Hierdie eienskappe is die direkte resultaat van die vereiste dat die ekonomies statistiese ontwerp aan sekere statistiese vereistes moet voldoen. Verder is uitgebreide sensitiwiteitsondersoeke op die ekonomies en ekonomies statistiese ontwerpe gedoen om die effek van die inset parameters sowel as van variërende grense op a, 1- f3 , die gemiddelde tyd-tot-sein, ATS sowel as die verskuiwingsgrootte 8 op die minimum verwagte kosteverlies sowel as die drie kontrolekaart besluitnemingsveranderlikes te bepaal. Die analises toon dat die totale koste relatief onsensitief is tot verbeterings in die tipe I en die tipe II fout koerse, maar dat dit hoogs sensitief is vir wysigings in die onderste grens op ATS sowel as besonder sensitief vir klein verskuiwingsvlakke, 8. Let op: Die uitdrukkings ekonomiese ontwerp (economic design), ekonomies statistiese ontwerp (economic statistical design), verlies kostefunksie (loss cost function) en aanwysbare oorsaak (assignable cause) mag taalkundig en sintakties vreemd voordoen, maar is geleen uit, en word so gebruik in die bekende literatuur oor hierdie onderwerp.