IntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\\n\\n
IntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
With the desire to make book publishing more relevant for the digital age and offer innovative Open Access publishing options, we are thrilled to announce the launch of our new publishing format: IntechOpen Book Series.
\n\n
Designed to cover fast-moving research fields in rapidly expanding areas, our Book Series feature a Topic structure allowing us to present the most relevant sub-disciplines. Book Series are headed by Series Editors, and a team of Topic Editors supported by international Editorial Board members. Topics are always open for submissions, with an Annual Volume published each calendar year.
\n\n
After a robust peer-review process, accepted works are published quickly, thanks to Online First, ensuring research is made available to the scientific community without delay.
\n\n
Our innovative Book Series format brings you:
\n\n
\n\t
Topic Focused Publications - Each topic showcases high impact subject areas
\n\t
Renowned Editorial Expertise - Series Editors, Topic Editors, and a team of international Board Members that permanently support each Book Series
\n\t
Fast Publishing - quick turnaround which is unique for book publishing
\n\t
The benefit of ISSN and ISBN for increased citation and indexing possibilities
\n
\n\n\n\n
IntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\n\n
IntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
We invite you to explore our IntechOpen Book Series, find the right publishing program for you and reach your desired audience in record time.
\n\n
Note: Edited in October 2021
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\r\n\t"The evolution of human society is closely linked to scientific progress in all areas of knowledge, which occurs more efficiently if it is integrated into society as a whole. The greatest mission of science is to produce and disseminate knowledge, informing society as clearly as possible and disconnected from particular and dogmatic interests. This facilitates the promotion of one of the most cherished values of human society, which is a collaboration between peoples and nations. This search for the democratization of access to scientific knowledge finds support in the model of dissemination of information through Open Access publications, especially through the digital format through the Internet. The current accessibility by the most diverse social classes in the most diverse countries is due in large part to scientific development in semiconductors.
\r\n
\r\n\tSemiconductor research goes much further than the development of electronic devices, because it has a multidisciplinary character, with contributions from basic sciences, such as Chemistry, Physics, and Mathematics, as well as from the frontier of knowledge, such as nanotechnology. This multidisciplinarity allows that in addition to important technological advances in electronic devices, such as diodes and transistors, integrated circuits, lasers, solar and photovoltaic cells, sensors, memory devices, among many others, there are also positive impacts for the production of new scientific knowledge.
\r\n
\r\n\tIt is within this context that this book project is inserted, aiming to gather recent and unpublished discoveries, conclusive reviews and other methodological and conceptual approaches on the subject of semiconductors. The idea is to contribute so that authors can disseminate their work both to other groups of researchers in their fields of study around the world, but also to a society in general, showing how semiconductor research positively changes our lives.
\r\n
\r\n\tThe "New Advances in Semiconductors" book will cover the main themes of the area but will be limited to these, such as unpublished experimental data, description of synthesis methods and processing techniques, characterizations of traditional or new semiconductor materials, statistical and mathematical treatments, modeling and other theoretical approaches. Articles with alternative discussions will also be welcome, whether on materials, properties, or applications, or the economic and social implications in the field of semiconductors. Thus, this book project aims to gather articles with technical-scientific information and also with general views on the topic of Semiconductors."
",isbn:"978-1-80355-682-6",printIsbn:"978-1-80355-681-9",pdfIsbn:"978-1-80355-683-3",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,isSalesforceBook:!1,hash:"238b808626f765e883b9bff8b62eae18",bookSignature:"Dr. Alberto Adriano Cavalheiro",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/11158.jpg",keywords:"Doping Mechanisms, Anomalous Effects, Prediction of Properties, Crystallinity, Phase Transitions, Heterojunctions, Amorphous Materials, Applications and Performance, Composition, Multifunctional Materials, Theoretical Approaches, Early History",numberOfDownloads:114,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 5th 2021",dateEndSecondStepPublish:"November 2nd 2021",dateEndThirdStepPublish:"January 1st 2022",dateEndFourthStepPublish:"March 22nd 2022",dateEndFifthStepPublish:"May 21st 2022",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"A researcher recognized for his work in the structural elucidation of crystalline semiconductor materials, head of the Interdisciplinary Laboratory of Advanced Materials of Navirai (LIMAN), and Area Coordinator at UEMS in the PIBID program, the largest and most important Teaching Initiation Program in Brazil.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"201848",title:"Dr.",name:"Alberto Adriano",middleName:null,surname:"Cavalheiro",slug:"alberto-adriano-cavalheiro",fullName:"Alberto Adriano Cavalheiro",profilePictureURL:"https://mts.intechopen.com/storage/users/201848/images/system/201848.jpg",biography:"Alberto Adriano Cavalheiro is an associate professor at the State University of Mato Grosso do Sul (UEMS), Brazil, where he works as a permanent lecturer in the Graduate Program in Natural Resources and coordinates the LIMAN materials laboratory. 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\n\t\t\t
1. Introduction
\n\t\t\t
Acceptance sampling is a procedure used for sentencing incoming batches. Sampling plan consist of a sample size and a decision making rule. The sample size is the number of items to sample or the number of measurements to take. The decision making rule involves the acceptance threshold and a description of how to use the sample result to accept or reject the lot. Acceptance sampling plans are also practical tools for quality control applications, which involve quality contracting on product orders between the vendor and the buyer. Those sampling plans provide the vendor and the buyer rules for lot sentencing while meeting their preset requirements on product quality. Scientific sampling plans are the primary tools for quality and performance management in industry today. In an industrial plant, sampling plans are used to decide either to accept or reject a received batch of items. With attribute sampling plans, these accept/reject decisions are based on a count of the number of defective items. The sample size is assumed constant in traditional sampling plans.
\n\t\t\t
In this section, several new decision making policies for the acceptance sampling problem are introduced. The objective of these models is to find constant control thresholds for lot sentencing problem.
\n\t\t\t
The single stage acceptance sampling plan based on the control threshold policy is presented in section 2, the acceptance sampling policy based on number of successive conforming items is presented in section 3, and acceptance sampling policy using the minimum angle method is presented in sections 4. Acceptance sampling policy based on cumulative sum of conforming Items run lengths comes in section 5 and acceptance sampling policy based on Bayesian inference comes in section 6. Finally the chapter is concluded in section 7.
\n\t\t
\n\t\t
\n\t\t\t
2. Single Stage Acceptance Sampling Plan based on the Control Threshold Policy [1]
\n\t\t\t
We suppose a batch of size \n\t\t\t\t\t\n\t\t\t\t\t\tn\n\t\t\t\t\t\n\t\t\t\t is received which its proportion of the defectives items is equal to\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t. For a batch of size\n\t\t\t\t\t\n\t\t\t\t\t\tn\n\t\t\t\t\t\n\t\t\t\t, random variable \n\t\t\t\t\t\n\t\t\t\t\t\tY\n\t\t\t\t\t\n\t\t\t\t is defined as the number of inspected items and \n\t\t\t\t\t\n\t\t\t\t\t\tz\n\t\t\t\t\t\n\t\t\t\t is defined as the number of items classified as \'defective\' after inspection. The number of inspected items has an upper threshold equal to\n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t. For \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t inspected items (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) the batch will be rejected if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twhere \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t is the upper control level for batch acceptance. In the other words, when the number of defective items in the inspected items gets more than the control threshold \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t then decision making process stops and the batch is rejected.
\n\t\t\t
The probability distribution function of \n\t\t\t\t\t\n\t\t\t\t\t\tY\n\t\t\t\t\t\n\t\t\t\t is determined by the following equations,
In Eq. (1), \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tindicates that all items are inspected therefore, the number of defective items has been less than \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tor \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tdefective item has been \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t inspected item. For the case\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tdefective item has been \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t inspected item thus, the probability distribution function of \n\t\t\t\t\t\n\t\t\t\t\t\tY\n\t\t\t\t\t\n\t\t\t\tfollows a negative binomial distribution. The expected mean of the number of inspected items is determined as follows:
Since \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is a negative binomial distribution thus using the approximation method of estimating negative binomial probabilities with Poisson distribution [2], following is concluded,
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the parameter of Poisson distribution. In order to improve the accuracy of this approximation, \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tshould be sufficiently large numbers. Using the above approximation method, following is concluded,
Now, let \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t denotes the probability of rejecting the batch. The batch is rejected if the number of defective items is more than or equal to \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t thus the value of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is determined by the following equation,
In order to calculate the total cost, including the cost of rejecting the batch, the cost of inspection and the cost of defective items, assume \n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t is the cost of rejecting the batch, \n\t\t\t\t\t\n\t\t\t\t\t\tc\n\t\t\t\t\t\n\t\t\t\tis the inspection cost of one item and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the cost of one defective item, so the total cost, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, is determined by conditioning \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\ton two events of rejecting or accepting the batch, thus the objective function is written as follows:
In Eq. (7), \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the total cost of inspection and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the total cost of defective items. The optimal value of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tis determined by minimizing the value of objective function\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Using the optimization methods, it is concluded that,
Since \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t is a sufficiently large number thus the value of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΓ\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is approximately equal to zero therefore it is concluded that,
To ensure that \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t minimizes the objective function (7), it is necessary to find the value of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tthat satisfies following inequalities:
Since with increasing the value of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\t the value of binomial distribution with parameters \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t and\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t decreases thus according to the properties of binomial distribution, it is concluded that\n\t\t\t\t\t\n \n x>(\n \n m+1\n \n )p\n \n\n\t\t\t\t therefore, the optimal value of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tis determined using the following formula,
Also The objective function, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, should be minimized regarding two constraints on Type-I and Type-II errors associated with the acceptance sampling plans. Type-I error is the probability of rejecting the batch when the nonconformity proportion of the batch is acceptable. Type-II error is the probability of accepting the batch when the nonconforming proportion of the batch is not acceptable. Then, in one hand, if\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of rejecting the batch should be less than\n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t. On the other hand, in case where\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of accepting the batch should be less than\n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\t where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the AQL (Accepted Quality Level ) and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the LQL (Limiting Quality Level) and\n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t is the probability of Type-I error and \n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\t is the probability of Type-II error in making a decision, therefore, the optimal value of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tis determined using the following formula,
When\n\t\t\t\t\t\n \n mc+npc\'>R\n \n\n\n\t\t\t\t, It is concluded that Eq. (16) is positive for all values of \n\t\t\t\t\t\n\t\t\t\t\t\tx\n\t\t\t\t\t\n\t\t\t\tso\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. In this case, if one defective item is found in an inspected sample then the batch would be rejected. In this case, the rejection cost \n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t is less than the total cost of inspecting \n\t\t\t\t\t\n\t\t\t\t\t\tm\n\t\t\t\t\t\n\t\t\t\t items and the cost of defective items, hence rejecting the batch would be the optimal decision. However, in practice the rejection cost \n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t is usually big enough so that, we overlooked that case.
3. Acceptance Sampling Policy Based on Number of Successive Conforming Items [3]
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In a typical acceptance-sampling plan, when the number of conforming items between successive nonconforming items is more than an upper control threshold, the batch is accepted, and when it is less than a lower control threshold, the batch is rejected otherwise, the inspection process continues. This initiates the idea of employing a Markovian approach to model the acceptance-sampling problem. As a result, in this method, a new acceptance-sampling policy using Markovian models is proposed, in which determining the control thresholds are aimed. The notations required to model the problem at hand are given as:
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The number of items in the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The proportion of nonconforming items in the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The cost of inspecting one item
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The cost of one nonconforming item
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The cost of rejecting the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The expected total cost of the system
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The expected total cost of accepting the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The expected total cost of rejecting the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The expected total cost of inspecting the items of the batch
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The upper control threshold
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t: The lower control threshold
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Consider an incoming batch of \n\t\t\t\t\t\n\t\t\t\t\t\tN\n\t\t\t\t\t\n\t\t\t\titems with a proportion of nonconformities\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t, of which items are randomly selected for inspection and based on the number of conforming items between two successive nonconforming items, the batch is accepted, rejected, or the inspection continues. The expected total cost associated with this inspection policy can be expressed using Eq. (17).
Let \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tbe the number of conforming items between the successive \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tnonconforming items, \n\t\t\t\t\t\n\t\t\t\t\t\tU\n\t\t\t\t\t\n\t\t\t\tthe upper and \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t the lower control thresholds. Then, if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthe batch is accepted, if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthe batch is rejected. Otherwise, if \n\t\t\t\t\t\n \n L<\n Y\n i\n \n <U\n \n\n\n\t\t\t\t the process of inspecting items continues. The states involved in this process can be defined as follows.
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State 1: \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tfalls within two control thresholds L, i.e.,\n\t\t\t\t\t\n \n L<\n Y\n i\n \n <U\n \n\n\n\t\t\t\t , thus the inspection process continues.
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State 2: \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis more than or equal the upper control threshold, i.e.,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t , hence the batch is accepted.
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State 3: \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis less than or equal the lower control threshold, i.e.,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t , hence the batch is rejected.
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The transition probabilities among the states can be obtained as follows.
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Probability of inspecting more items=\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\n \n L<\n Y\n i\n \n <U\n \n\n\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
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Probability of accepting the batch=\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
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Probability of rejecting the batch=\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t13\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
\n\t\t\t
where the probabilities can be obtained based on the fact that the number of conforming items between the successive \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tnonconforming items, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, follows a geometric distribution with parameter\n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\t, i.e., \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tThen, the transition probability matrix is expressed as follows:
As it can be seen, the matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is an absorbing Markov chain with states 2 and 3 being absorbing and state 1 being transient.
\n\t\t\t
To analyze the above absorbing Markov chain, the transition probability matrix should be rearranged in the following form:
Where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the identity matrix and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t denotes the expected long-run number of times the transient state 1 is occupied before absorption occurs (i.e., accepted or rejected), given that the initial state is 1. The long-run absorption probability matrix, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, is calculated as follows [4],
The elements of the \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t matrix, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t13\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, denote the probabilities of the batch being accepted or rejected, respectively.
\n\t\t\t
The expected cost can be obtained using Eq. (17) containing the batch acceptance, rejection, and inspection costs. The expected acceptance cost is the cost of nonconforming items (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) multiplied by the probability of the batch being accepted (i.e.,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t). The expected rejection cost is the rejection cost (\n\t\t\t\t\t\n\t\t\t\t\t\tR\n\t\t\t\t\t\n\t\t\t\t) multiplied by the probability of the batch being rejected (i.e.,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t13\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t). Moreover, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the expected long-run number of times the transient state 1 is occupied before absorption occurs. Knowing that in each visit to transient state, the average number of inspections is\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (the mean of the geometric distribution), the expected inspection cost is given by
Therefore, the expected cost for acceptance-sampling policy can be expressed as a function of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t13\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tas follows:
Substituting for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t12\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the expected cost equation can be rewritten as:
\n\t\t\t\tEq. (25) can be solved numerically using search algorithms to find \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tU\n\t\t\t\t\t\n\t\t\t\tthat minimize the expected total cost. The objective function, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, should be minimized regarding two constraints on Type-I and Type-II errors associated with the acceptance sampling plans. Type-I error is the probability of rejecting the batch when the nonconformity proportion of the batch is acceptable. Type-II error is the probability of accepting the batch when the nonconforming proportion of the batch is not acceptable. Then, in one hand, if\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of rejecting the batch should be less than\n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t. On the other hand, in case where\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of accepting the batch should be less than\n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\t where \n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\tare the probabilities of Type-I and Type-II errors, hence,
The optimum values of \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tU\n\t\t\t\t\t\n\t\t\t\tamong a set of alternative values are determined solving the model given in (25), numerically, where the probabilities are obtained using the geometric distribution.
\n\t\t
\n\t\t
\n\t\t\t
4. Acceptance Sampling Policy Using the Minimum Angle Method based on Number of Successive Conforming Items [5]
\n\t\t\t
The practical performance of any sampling plan is determined through its operating characteristic curve. When producer and consumer are negotiating for designing sampling plans, it is important especially to minimize the consumer risk. In order to minimize the consumer’s risk, the ideal OC curve could be made to pass as closely through\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. One approach to minimize the consumers risks for ideal condition is proposed with minimization of angle \n\t\t\t\t\t\n\t\t\t\t\t\tϕ\n\t\t\t\t\t\n\t\t\t\t between the lines joining the points\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Therefore in this case, the value of performance criteria in minimum angle method will be [6],
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the probability of accepting the batch when the proportion of defective items in the batch is respectively\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Assume \n\t\t\t\t\t\n\t\t\t\t\t\tA\n\t\t\t\t\t\n\t\t\t\t is the point\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, \n\t\t\t\t\t\n\t\t\t\t\t\tB\n\t\t\t\t\t\n\t\t\t\tis the point \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\tC\n\t\t\t\t\t\n\t\t\t\t is the point \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthus the smaller value of\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tϕ\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the angle \n\t\t\t\t\t\n\t\t\t\t\t\tϕ\n\t\t\t\t\t\n\t\t\t\t approaching zero, and the chord \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t approaching\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tB\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the ideal condition.
\n\t\t\t
The values of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are determined as follows,
Since the values of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tare constant and\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t therefore the objective function is determined as follows,
Another performance measure of acceptance sampling plans is the expected number of inspected items. Since sampling and inspecting usually has cost, therefore designs that minimizes this measure and satisfy the first and second type error inequalities are considered to be optimal sampling plans. Since the proportion of defective items is not known in the start of process, in order to consider this property in designing the acceptance sampling plans, we try to minimize the expected number of inspected items for acceptable and not acceptable lots simultaneously. Therefore the optimal acceptance sampling plan should have three properties, first it should have a minimized value in the objective function of the minimum angle method that is resulted from the ideal OC curve and also it should minimize the expected number of inspected items either in the decisions of rejecting or accepting the lot. Therefore the second objective function is defined as the expected number of items inspected. The value of this objective function is determined based on the value of\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\twhere \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the expected number of times in the long run that the transient state 1 is occupied before absorption occurs, since in each visit to transient state, the average number of inspections is\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, consequently the expected number of items inspected is given by\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t11\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Now the objective functions \n\t\t\t\t\t\n\t\t\t\t\t\tW\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\tZ\n\t\t\t\t\t\n\t\t\t\tare defined as the expected number of items inspected respectively in the acceptable condition\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and not acceptable condition\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t.
Now one approach to optimize the objective functions simultaneously is to define control thresholds for objective functions \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tW\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and then trying to minimize the value of objective function\n\t\t\t\t\t\n\t\t\t\t\t\tV\n\t\t\t\t\t\n\t\t\t\t. For example if parameters \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tW\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are defined as the upper control thresholds for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tW\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t then the optimization problem can be defined as follows,
Optimal values of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tcan be determined by solving above nonlinear optimization problem using search procedures or other optimization tools.
\n\t\t
\n\t\t
\n\t\t\t
5. Acceptance Sampling Policy Based on Cumulative Sum of Conforming Items Run Lengths [7]
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In an acceptance-sampling plan, assume \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the number of conforming items between the successive \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tdefective items. Decision making is based on the value of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t that is defined as,
The proposed acceptance sampling policy is defined as follows,
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If \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t then the batch is accepted
If \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tthe batch is rejected
If \n\t\t\t\t\t\t\t\n \n L<\n S\n i\n \n <U\n \n\n\n\t\t\t\t\t\t the process of inspecting the items continues
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where \n\t\t\t\t\t\n\t\t\t\t\t\tU\n\t\t\t\t\t\n\t\t\t\tis the upper control threshold and \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\tis the lower control threshold.
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In each stage of the data gathering process, the index of different states of the Markov model,\n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\t , is defined as:
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\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t represents the state of rejecting the batch. In this state \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t thus the batch is rejected.
\n\t\t\t\t\t\t\n\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t where \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t2...\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t represents the state of continuing data gathering. In this state, \n\t\t\t\t\t\t\t\n \n L<\n S\n i\n \n =\n Y\n i\n \n +\n Y\n \n i−1\n \n \n <U\n \n\n\n\t\t\t\t\t\tthus the inspecting process continues.
\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t represents the state of accepting the batch. In this state \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t hence the batch is accepted.
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In other word, the acceptance-sampling plan can be expressed by a Markov model, in which the transition probability matrix among the states of the batch can be expressed as:
where, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis probability of going from state \n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\tto state \n\t\t\t\t\t\n\t\t\t\t\t\tk\n\t\t\t\t\t\n\t\t\t\tin a single step and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tdenotes the number of conforming items between the successive defective items and \n\t\t\t\t\t\n \n Pr(\n \n \n Y\n \n i+1\n \n \n =r\n \n )=\n \n (\n \n 1−p\n \n )\n \n r\n \n p\n \n \n \n \n \n \n r=0,1,2,...\n \n \n \n \n \n \n\n\n\t\t\t\t where \n\t\t\t\t\t\n\t\t\t\t\t\tp\n\t\t\t\t\t\n\t\t\t\tdenotes the proportion of defective items in the batch.
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The values of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t are determined based on the relations among the states, for example where \n\t\t\t\t\t\n \n U+2>j>1,L≥j−2,k=1\n \n\n\t\t\t\t then according to the definition of\n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\t, it is concluded that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and transition probability of going form state \n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\tto state \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis equal to the probability of rejecting the batch that is evaluated as follows,
In the other case where, \n\t\t\t\t\t\n \n U+2>j>1,U+2>k>1,U>j+k−4>L\n \n\n\t\t\t\t, based on the definition of\n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\t, we have\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthus it is concluded that
In the other case where, \n\t\t\t\t\t\n \n U+2>j>1,k=U+2\n \n\n\t\t\t\t, then according to the definition of\n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\t, we have\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthus it is concluded that,
In the other case where, \n\t\t\t\t\t\n \n U+2>j>1,U+2>k>1,j+k−4≥U\n \n\n\t\t\t\t, then according to the definition of j, we have\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthus it is concluded that,
As a result, when \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for example, the transition probability matrix among the states of the system can be expressed as:
And it can be seen the matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is an absorbing Markov chain with states 1 and 5 being absorbing and states 2, 3, and 4 being transient.
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Analyzing the above absorbing Markov chain requires to rearrange the single-step probability matrix in the following form:
where\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the identity matrix representing the probability of staying in a state that is defined as follows
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tO\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the probability matrix of escaping an absorbing state (always zero) that is defined as follows
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis a square matrix containing the transition probabilities of going from a non-absorbing state to another non-absorbing state that is defined as follows
And \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the Matrix containing all probabilities of going from a non-absorbing state to an absorbing state (i.e., accepted or rejected batch) that is defined as follows
Bowling et. al. [4] proposed an absorbing Markov chain model for determining the optimal process means. According to their method, matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthat is the fundamental matrix containing the expected number of transitions from a non-absorbing state to another non-absorbing state before absorption occurs can be obtained by the following equation,
For the above numerical example, i.e., when \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the fundamental matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tM\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t can be obtained as:
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tI\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tis the identity matrix.
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Since \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\trepresents the expected number of the times in the long-run the transient state \n\t\t\t\t\t\n\t\t\t\t\t\tj\n\t\t\t\t\t\n\t\t\t\tis occupied before absorption occurs (i.e., before accepted or rejected), and matrix \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the absorption probability matrix containing the long run probabilities of the transition from a non-absorbing state to an absorbing state. The long-run absorption probability matrix, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, can be calculated as follows:
Again when \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the elements of \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tF\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t represent the probabilities of the batch being accepted and rejected, respectively, given that the initial state is\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. In this case, the probability of accepting the batch is obtained as:
\n\t\t\t
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tProbability of accepting the batch=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tAccepting the batch\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\tthe initial state is \n\t\t\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tthe initial state is \n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t4\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tf\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t5\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tY\n\t\t\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t\t\t3\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\tE48
\n\t\t\t
Also the expected number of inspected items will be determined as follows,
\n\t\t\t
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tExpected number of inspected items =\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tthe number of inspected items in state j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\tthe number of visits to state j\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t∑\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\tE49
\n\t\t\t
This new acceptance-sampling plan should satisfy two constraints of the first and the second types of errors. The probability of Type-I error shows the probability of rejecting the batch when the defective proportion of the batch is acceptable. The probability of Type-II error is the probability of accepting the batch when the defective proportion of the batch is not acceptable. Then on the one hand if\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of rejecting the batch will be less than \n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t and on the other hand, in case where\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the probability of accepting the batch will be less than\n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\t where \n\t\t\t\t\t\n\t\t\t\t\t\tα\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tβ\n\t\t\t\t\t\n\t\t\t\tare the probabilities of Type-I and Type-II errors. Hence,
\n\t\t\t
\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\tProbability of accepting the batch\n\t\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\tα\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t\tL\n\t\t\t\t\t\t\t\t→\n\t\t\t\t\t\t\t\tProbability of accepting the batch\n\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\tE50
\n\t\t\t
From the inequalities in (50), the proper values of the thresholds \n\t\t\t\t\t\n\t\t\t\t\t\tL\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\tU\n\t\t\t\t\t\n\t\t\t\t are determined and among the feasible ones, we select one that has the least value for expected number of inspected items that is obtained using Eq. (49).
\n\t\t
\n\t\t
\n\t\t\t
6. A New Acceptance Sampling Design Using Bayesian Modelling and Backwards Induction [8]
\n\t\t\t
In this research, a new selection approach on the choices between accepting and rejecting a batch based on Bayesian modelling and backwards induction is proposed. The Bayesian modelling is utilized to model the uncertainty involved in the probability distribution of the nonconforming proportion of the items and the backwards induction method is employed to determine the sample size. Moreover, when the decision on accepting or rejecting a batch cannot be made, we assume additional observations can be gathered with a cost to update the probability distribution of the nonconforming proportion of the batch. In other words, a mathematical model is developed in this research to design optimal single sampling plans. This model finds the optimum sampling design whereas its optimality is resulted by using the decision tree approach. As a result, the main contribution of the method is to model the acceptance-sampling problem as a cost optimization model so that the optimal solution can be achieved via using the decision tree approach. In this approach, the required probabilities of decision tree are determined employing the Bayesian Inference. To do this, the probability distribution function of nonconforming proportion of items is first determined by Bayesian inference using a non-informative prior distribution. Then, the required probabilities are determined by applying Bayesian inference in the backward induction method of the decision tree approach. Since this model is completely designed based on the Bayesian inference and no approximation is needed, it can be viewed as a new tool to be used by practitioners in real case problems to design an economically optimal acceptance-sampling plan. However, the main limitation of the proposed methodology is that it can only be applied to items not requiring very low fractions of nonconformities.
\n\t\t\t
\n\t\t\t\t
6.1. Notations
\n\t\t\t\t
The following notations are used throughout the paper.
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Set of decisions: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis defined the set of possible decisions where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t refer to accepting and rejecting the batch, respectively.
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State space: \n\t\t\t\t\t\t\n \n P={\n \n p\n l\n \n ;l=1,2,...;0<\n p\n l\n \n <1\n }\n \n\n\n\t\t\t\t\tis defined the state of the process where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\trepresents nonconforming proportion items of the batch in \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t state of the process. The decision maker believes the consequences of selecting decision\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t or \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t depend on \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\n\t\t\t\t\t that cannot be determined with certainty. However, the probability distribution function of the random variable \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t can be obtained using Bayesian inference.
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Set of experiments: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis the set of experiments to gather more information on \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\tand consequently to update the probability distribution of\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t. Further, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tis defined an experiment in which \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\titems of the batch are inspected.
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Sample space: \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tdenotes the outcomes of experiment \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tshows the number of nonconforming items in\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t.
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Cost function: The function \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t on \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\tA\n\t\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t denotes the cost associated with performing experiment\n\t\t\t\t\t\t\n\t\t\t\t\t\t\te\n\t\t\t\t\t\t\n\t\t\t\t\t, observing\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\n\t\t\t\t\t, making decision\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\n\t\t\t\t\t, and finding\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t.
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\n\t\t\t\t\t: The total number of items in a batch
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\n\t\t\t\t\t: The cost of rejecting a batch
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\n\t\t\t\t\t: The cost of one nonconforming item
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\n\t\t\t\t\t: The cost of inspecting one item
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\n\t\t\t\t\t: An upper bound on the number of inspected item
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6.2. Problem Definition
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Consider a batch of size \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\n\t\t\t\t\t with an unknown percentage of nonconforming \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t and assume \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\n\t\t\t\t\titems are randomly selected for inspection. Based on the outcome of the inspection process in terms of the observed number of nonconforming items, the decision-maker desires to accept the batch, reject it, or to perform more inspections by taking more samples. As Raiffa & Schlaifer [9] stated "the problem is how the decision maker chose \n\t\t\t\t\t\t\n\t\t\t\t\t\t\te\n\t\t\t\t\t\t\n\t\t\t\t\tand then, having observed\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\n\t\t\t\t\t, choose \n\t\t\t\t\t\t\n\t\t\t\t\t\t\te\n\t\t\t\t\t\t\n\t\t\t\t\tsuch that \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is minimized. Although the decision maker has full control over his choice of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\te\n\t\t\t\t\t\t\n\t\t\t\t\tand\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\n\t\t\t\t\t, he has neither control over the choices of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\n\t\t\t\t\tnor\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t. However, we can assume he is able to assign probability distribution function over these choices." They formulated this problem in the framework of the decision tree approach, the one that is partially adapted in this research as well.
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6.3. Bayesian Modelling
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For a nonconforming proportion\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t, referring to Jeffrey’s prior (Nair et al. [10]), we first take a Beta prior distribution with parameters \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0.5\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t to model the absolute uncertainty. Then, the posterior probability density function of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\t using a sample of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t inspected items is
where \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\n\t\t\t\t\tis the number of nonconforming items and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\n\t\t\t\t\t is the number of conforming items in the sample. Moreover, to allow more flexibility in representing prior uncertainty it is convenient to define a discrete distribution by discretization of the Beta density (Mazzuchi, & Soyer [11]). In other words, we define the prior distribution for \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tas
where \n \n \n p\n 1\n \n =(\n \n \n \n 2l−1\n \n 2\n \n \n \n )δ and δ=\n 1\n m\n \n for l=1,2,...,m\n \n
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Now, define \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tand\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t...\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t the experiment in which \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\t nonconforming items are found when \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\titems are inspected. Then, the sample space \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\n\t\t\t\t\t becomes\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tZ\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t:\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, resulting in the cost function representation of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t[\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t]\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t;\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tk\n\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\tthat is associated with taking a sample of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\titems, observing\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\tnonconforming and adopting \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tor \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\twhen the defective proportion is\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Using the notations defined, the cost function is determined by the following equations:
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\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1) for accepted batch \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t\tN\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t2) for rejected batch \n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tR\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tS\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tE53
\n\t\t\t\t
Moreover, the probability of finding \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\n\t\t\t\t\t nonconforming items in a sample of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\n\t\t\t\t\t inspected items, i.e., \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, can be obtained using a binomial distribution with parameters \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t as:
Hence, the probability \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t can be calculated as follows
In other words, applying the Bayesian rule, the probability \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t can be obtained by
In the next Section, a backward induction approach is taken to determine the optimal sample size.
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6.4. Backward Induction
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The analysis continues by working backwards from the terminal decisions of the decision tree to the base of the tree, instead of starting by asking which experiment \n\t\t\t\t\t\t\n\t\t\t\t\t\t\te\n\t\t\t\t\t\t\n\t\t\t\t\tthe decision maker should select when he does not know the outcomes of the random events. This method of working back from the outermost branches of the decision tree to the initial starting point is often called "backwards induction" [9]. As a result, the steps involved in the solution algorithm of the problem at hand using the backwards induction becomes
\n\t\t\t\t
1. Probabilities \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t are determined using Eq. (52) and Eq. (54), respectively.
\n\t\t\t\t
2. The conditional probability \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\tl\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is determined using Eq. (57).
\n\t\t\t\t
3. With a known history\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, since \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\n\t\t\t\t\tis a random variable, the costs of various possible terminal decisions are uncertain. Therefore the cost of any decision \n\t\t\t\t\t\t\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\n\t\t\t\t\t for the given \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is set as a random variable\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t. Applying the conditional expectation, \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tE\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t, which takes the expected value of \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t with respect to the conditional probability \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tP\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t(Eq. 57), the conditional expected value of the cost function on state variable \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is determined by the following equation.
4. Since the objective is to minimize the expected cost, the cost of having history \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t and the choice of decision (accepting or rejecting) can be determined by
5. The conditional probability \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tPr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t{\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t}\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is determined using Eq. (56).
\n\t\t\t\t
6. The costs of various possible experiments are random because the outcome \n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\n\t\t\t\t\tis a random variable. Defining a probability distribution function over the results of experiments and taking expected values, we can determine the expected cost of each experiment. The conditional expected value of function \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t on the variable \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\tj\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t is determined by the following equation.
7. Now the minimum of the values \n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tu\n\t\t\t\t\t\t\t\t*\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\te\n\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t would be the optimal decision, which leads to an optimal sample size.
Acceptance sampling plans have been widely used in industry to determine whether a specific batch of manufactured or purchased items satisfy a pre-specified quality. In this chapter, new models for determining optimal acceptance sampling plans have been presented. The relationship between the cost model and a decision theory model with probabilistic utilities has been investigated. However, the acceptance sampling plan, which are derived from the optimization of these models, may differ substantially from the plans that other economic approaches suggest but optimization of these models are simple and efficient, with negligible computational requirements. In next sections, a new methodology based on Markov chain was developed to design proper lot acceptance sampling plans. In the proposed procedure, the sum of two successive numbers of nonconforming items was monitored using two lower and upper thresholds, where the proper values of these thresholds could be determined numerically using a Markovian approach based on the two points on OC curve. In last section, based on the Bayesian modelling and the backwards induction method of the decision-tree approach, a sampling plan is developed to deal with the lot-sentencing problem; aiming to determine an optimal sample size to provide desired levels of protection for customers as well as manufacturers. A logical analysis of the choices between accepting and rejecting a batch is made when the distribution function of nonconforming proportion could be updated by taking additional observations and using Bayesian modelling.
\n\t\t
\n\t\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/37736.pdf",chapterXML:"https://mts.intechopen.com/source/xml/37736.xml",downloadPdfUrl:"/chapter/pdf-download/37736",previewPdfUrl:"/chapter/pdf-preview/37736",totalDownloads:2364,totalViews:255,totalCrossrefCites:0,totalDimensionsCites:0,totalAltmetricsMentions:0,impactScore:0,impactScorePercentile:9,impactScoreQuartile:1,hasAltmetrics:0,dateSubmitted:"March 29th 2012",dateReviewed:"June 19th 2012",datePrePublished:null,datePublished:"December 19th 2012",dateFinished:"July 6th 2012",readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/37736",risUrl:"/chapter/ris/37736",book:{id:"3074",slug:"practical-concepts-of-quality-control"},signatures:"Mohammad Saber Fallah Nezhad",authors:[{id:"150393",title:"Dr.",name:"Mohammad Saber Fallah",middleName:null,surname:"Nezhad",fullName:"Mohammad Saber Fallah Nezhad",slug:"mohammad-saber-fallah-nezhad",email:"saber9876@yahoo.com",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/150393/images/system/150393.jpg",institution:{name:"Yazd University",institutionURL:null,country:{name:"Iran"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Single Stage Acceptance Sampling Plan based on the Control Threshold Policy [1]",level:"1"},{id:"sec_3",title:"3. Acceptance Sampling Policy Based on Number of Successive Conforming Items [3]",level:"1"},{id:"sec_4",title:"4. Acceptance Sampling Policy Using the Minimum Angle Method based on Number of Successive Conforming Items [5]",level:"1"},{id:"sec_5",title:"5. Acceptance Sampling Policy Based on Cumulative Sum of Conforming Items Run Lengths [7]",level:"1"},{id:"sec_6",title:"6. A New Acceptance Sampling Design Using Bayesian Modelling and Backwards Induction [8] ",level:"1"},{id:"sec_6_2",title:"6.1. Notations",level:"2"},{id:"sec_7_2",title:"6.2. Problem Definition",level:"2"},{id:"sec_8_2",title:"6.3. Bayesian Modelling",level:"2"},{id:"sec_9_2",title:"6.4. Backward Induction",level:"2"},{id:"sec_11",title:"7. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFallahnezhad\n\t\t\t\t\t\t\tM. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHosseininasab\n\t\t\t\t\t\t\tH.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2011\n\t\t\t\t\tDesigning a Single Stage Acceptance Sampling Plan based on the control Threshold policy.\n\t\t\t\t\tInternational Journal of Industrial Engineering & Production Research\n\t\t\t\t\t22\n\t\t\t\t\t3\n\t\t\t\t\t143\n\t\t\t\t\t150\n\t\t\t\t\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHilbe\n\t\t\t\t\t\t\tJ. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tNegative Binomial Regression.\n\t\t\t\t\tCambridge, UK\n\t\t\t\t\t: Cambridge University Press\n\t\t\t\t\n\t\t\t'},{id:"B3",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFallahnezhad\n\t\t\t\t\t\t\tM. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNiaki\n\t\t\t\t\t\t\tS. T. 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S.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2012\n\t\t\t\t\tA New Approach for Acceptance Sampling Policy Based on the Number of Successive Conforming Items and Minimum Angle Method.\n\t\t\t\t\tTo Appear in Iranian Journal of Operations Research.\n3\n1\n104\n111\n\t\t\t\t\n\t\t\t'},{id:"B6",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSoundararajan\n\t\t\t\t\t\t\tV.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChristina\n\t\t\t\t\t\t\tA. L.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1997\n\t\t\t\t\tSelection of single sampling variables plans based on the minimum angle.\n\t\t\t\t\tJournal of Applied Statistics\n\t\t\t\t\t24\n\t\t\t\t\t2\n\t\t\t\t\t207\n\t\t\t\t\t218\n\t\t\t\t\n\t\t\t'},{id:"B7",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFallahnezhad\n\t\t\t\t\t\t\tM. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNiaki\n\t\t\t\t\t\t\tS. T. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAbooie\n\t\t\t\t\t\t\tM. H.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2011\n\t\t\t\t\tA New Acceptance Sampling Plan Based on Cumulative Sums of Conforming Run-Lengths.\n\t\t\t\t\tJournal of Industrial and Systems Engineering\n\t\t\t\t\t4\n\t\t\t\t\t4\n\t\t\t\t\t256\n\t\t\t\t\t264\n\t\t\t\t\n\t\t\t'},{id:"B8",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFallahnezhad\n\t\t\t\t\t\t\tM. S.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNiaki\n\t\t\t\t\t\t\tS. T. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tVahdat\n\t\t\t\t\t\t\tM. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2012\n\t\t\t\t\tA New Acceptance Sampling Design Using Bayesian Modeling and Backwards Induction.\n\t\t\t\t\tInternational Journal of Engineering, Islamic Republic of Iran\n\t\t\t\t\t25\n\t\t\t\t\t1\n\t\t\t\t\t45\n\t\t\t\t\t54\n\t\t\t\t\n\t\t\t'},{id:"B9",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRaiffa\n\t\t\t\t\t\t\tH.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2000\n\t\t\t\t\tSchlaifer R. Applied statistical decision theory.\n\t\t\t\t\tNew York\n\t\t\t\t\tWiley Classical Library\n\t\t\t\t\n\t\t\t'},{id:"B10",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNair\n\t\t\t\t\t\t\tV. N.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tTang\n\t\t\t\t\t\t\tB.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tXu\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2001\n\t\t\t\t\tBayesian inference for some mixture problems in quality and reliability.\n\t\t\t\t\tJournal of Quality Technology\n\t\t\t\t\t33\n\t\t\t\t\t16\n\t\t\t\t\n\t\t\t'},{id:"B11",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMazzuchi\n\t\t\t\t\t\t\tT. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSoyer\n\t\t\t\t\t\t\tR.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1996\n\t\t\t\t\tAdaptive Bayesian replacement strategies.\n\t\t\t\t\tProceedings of the Fifth Valencia International Meeting on Bayesian Statistics(J.M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, eds.)Elsevier\n\t\t\t\t\t667\n\t\t\t\t\t674\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Mohammad Saber Fallah Nezhad",address:"Fallahnezhad@yazduni.ac.ir",affiliation:'
Assistant Professor of Industrial Engineering, Yazd University, Iran
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1. Introduction
Oxygen is a basic need for the survival of human beings who can stand its deficiency only for a certain period of time which is shorter when compared with their other needs. Human beings receive oxygen from air through respiration. It is a right for them to have clean air around them. Pollution of air have adverse effects on the health of living things. The relation between health and air quality has been a fact acknowledged since Hippocrates. Indoor environments may pollute the air with the outdoor pollutants and indoor pollutants. Indoor air quality is of particular importance for people who spend 90% of their time in public or private indoors such as houses, schools, fitness centers, shopping malls, supermarkets, workplaces and transportation vehicles. The health risks accompanying exposure to indoor air pollution can be worse for many people when compared with those of the outdoor air pollution. Unhealthy indoor air may be harmful particularly for risk groups, such as children, the elderly or the patients suffering chronic disorders. Workplace indoor air quality is equally important for those spending at least 1/3 of their lifetime in workplace [1, 2].
Workplace indoor factors are among the factor which affect the health of workers most in all sectors. Another important one of these factors is the air quality of the workplace. One of the focal points of occupational health is environmental effect on the workers working in the indoors of workplaces, when compared to those working in industrial workplaces, such as construction, mining and agricultural workplaces which are very dangerous or dangerous. Furthermore, workers working indoors are less prepared or experienced against environment risks in comparison to the ones working in industrial workplaces. Overview of relevant legislation shows control mechanisms used in such workplaces are not sufficient [3]. Workplace indoor air quality is an optimal indoor requirement which ensures health, comfort and wellbeing of workers and includes minimum air pollutants. Indoor air quality varies according to air temperature, relative humidity, air speed and chemicals at workplace [4]. In today’s workplaces, materials, equipment, various cleaning products and chemical and particle emissions determine the indoor air quality. Indoor air quality affect workers’ health, incidence of occupational accidents, nonattendance and productivity. For this reason, ensuring that the workplace indoor air quality conforms to the norms is of particular importance. Indoor air quality affects workers’ health, emergence of occupational accidents, in attendance and productivity [5]. This article aims at shedding light on the adverse effects of workplace indoor air quality on the health of workers and also the measures which need to be taken for preventing such effects. For this reason, this part covers information on factors deteriorating indoor air pollution, health problems which might develop in workers as a result of the indoor air quality, workplace indoor air quality assessment criteria stemming from international and national source data, the measures to be taken to protect workers from indoor air pollution, indoor air quality and Covid-19.
2. Factors affecting workplace indoor air quality
Indoor air pollutants originating from environment/outdoor environment: These are biological factors, industrial pollutants, fire products, ammonia, ozone, traffic pollutants such as nitrogen dioxide and particles, radon, methane and humidity.
Indoor air pollutants originating from workplace: They were grouped as asbestos, biological pollutants, carbon monoxide (CO), carbon dioxide (CO2), formaldehyde/pressed wooden products, lead (Pb), nitrogen dioxide, (NO2), Radon (Rn), indoor particle substances, environmental tobacco smoke, volatile organic compounds (VOCs), humidity, odor and wood smoke [6, 7, 8]. Workers working in the workplace indoor environment may be exposed to various air pollutants (both in gas form and particle form), including organic, inorganic and biological ones. The workplace indoor air pollutants are among the primary indoor air pollutants with serious effects on health and the potential to deteriorate the workers’ health. Indoor air quality is affected by the tobacco and nicotine products, chemicals used for cleaning purposes, heating, construction materials and humidity. The pollutants in the indoor air can be present in the form of gas or particle and they may cause various diseases in the respiration system. The most important indoor air pollutant is the smoke of cigarette. The smoke produced by smokers includes many hazardous substances, just like the smoldering cigarette. Most of them are carcinogenic. The people who are exposed to this smoke may develop serious diseases, although they do not consume these products. All workers are affected by the smoke of cigarette consumed in the environment. Limiting the act of smoking to particular rooms would not protect other workers. The particles of tobacco products hang in the air for five hours. Hookah, warmed up tobacco products and electronic cigarette consumption have the same effects on environment and people [6, 7, 8]. The construction materials, equipment and chemicals present in the workplace deteriorates the indoor air quality. There are sources of pollutants which deteriorates the indoor air quality and health of workers can be protected only if their effects are eliminated. There are three main groups of workplace indoor pollutants: biological ones, chemical ones and particles (non-biological):
Biological factors: They may stem from excessive bacteria, virus, fungus, dust mite, animal hair, pollen concentrations, insufficient maintenance and cleaning, water split, insufficient humidity control, condensation or leakages at the building envelope or water leakage caused by flood.
Chemical Factors: Chemical pollutants (gas and vapor), emissions stemming from the products used in the building (like floor or walls covering, office equipment, furniture, insecticide, cleaning products), accidental spillage of chemicals and products used for construction purposes, adhesives, paints and combustion products such as carbon monoxide, formaldehyde and nitrogen dioxide are included in this group.
Particle-associated factors (non-biological): Particles are substances which are light enough to hang in the air, in solid or liquid form, and they are non-biological. The building may extract dust, dirt or other substances. Particles may be produced by activities such as construction, wooden punching, drywall, printing, duplication, copying and operation equipment [9].
3. Health problems associated with workplace indoor air pollution
Healthy indoor air quality (IAQ) is defined as the indoor air which does not contain hazardous substances and does not create sense of discomfort in at least 80% of the workers in the workplace [8, 9, 10]. Poor indoor may result in a variety of health problems, from headache, dizziness and nausea, to asthma, cardiovascular diseases, cancer and death. The typical effect of some common indoor air pollutants on the health and wellbeing of the residents can be seen in Table 1. Poor air quality may also have adverse impact on workplace performance, learning at the education/training institutions and improvement of health services, in addition to being hazardous in terms of health and comfort. The impact of indoor air pollutants on the health of the people depends on the concentration of the concerned pollutant, exposure duration, age and gender of the people exposed to it. As for the industrial workplaces, most of the primary risks are evaluated in terms of use of personal protection equipment (PPE), exposure risk and local air conditioning [7, 8, 9, 10].
Pollutant
Impact on health
Carbon monoxide (CO)
Carbon monoxide can cause headaches, dizziness, nausea and at very high levels, death. Elderly people, pregnant women, young children and people with heart disease and lung disease are more sensitive to the adverse effects of carbon monoxide.
Formaldehyde
Formaldehyde can cause eye, nose and throat irritation and is considered a potential human carcinogen.
Nitrogen dioxide (NO2)
Exposure to nitrogen dioxide can cause inflammation of the airways, respiratory illnesses and possibly increases the risk of lung infections. Young children and people with asthma are the most sensitive to NO2. It plays a major role in the development of chronic obstructive pulmonary disease in adults which will affect more people than heart disease by 2020 (Environmentalist 2012). Long-term exposure may also affect lung function and can enhance responses to allergens in sensitized individuals.
Odor
Odorous discharges are subjective and cause nausea and irritation for some people.
Ozone (O3)
Ozone exposure can cause asthma, irritation and damage to the eyes, nose and airways. Prolonged exposure to high levels may result in damage to the lungs and airway linings.
Particulate matter
Inhalable particles have been linked with a number of respiratory illnesses, including asthma and chronic bronchitis. Long-term exposure to fine particles can cause premature death from heart disease and lung disease including cancer. Short-term exposure to higher levels of fine particle concentrations have also been linked with cardio-vascular problems and increased death rates. Exposure to fine particles has also been linked to prevalent anxiety and hypertensive disorders.
Volatile organic Compounds (VOCs)
Key symptoms associated with exposure to VOCs include eye irritation, nose and throat discomfort, headache and allergic skin reaction.
Table 1.
Typical health impacts of some common pollutants found indoors.
EU-OFFICER research Project showed the association between the indoor chemicals in the office and sick building syndrome-SBS symptoms. The most expressed complaints are ocular irritation (dry eyes, watering eyes or itching, burning or irritation), headache, lethargy, extraordinary tiredness. The researchers also reported that xylene, ethylbenzene, α-pinene, d-limonene, styrene, formaldehyde, acrolein, propionaldehyde, hexane and ozone might increase in the incidence of the symptoms. They concluded by underlining the need for further research in order to better depict the complicated relationship between IAQ and health interaction symptoms [11]. A research conducted on the effects of indoor air quality on the health of workers in Middle East showed that the first most affected part of workers’ bodies is their respiratory system; the second most affected one is their cardiovascular system, and the third most affected one is their visual system [12].
4. Indoor air pollution for health professionals
People living in developed European and American countries are reported to be more exposed to airborne substances deriving from indoors where they spend most of their time (>90%) causing environmentally associated symptoms that should be evaluated by health professionals. However, this percentage is expected to be higher for infants and the elderly, chronically ill people and in urban settings [13]. Many pollutants present with higher concentrations indoor than outdoor, especially in case of longer and non-intermittent exposure like in the home, workplace and school. It should be taken into account that some of the signs and symptoms presented in the text may occur only in the case of significant exposures. However, lower or shorter exposures with milder or indeterminate symptoms, or atypical presentation (noted in the text) in younger aged children render the diagnosis more difficult. The cooperation of the individual and the health care professional is essential for the correct diagnosis noting clues suggestive of indoor air pollution, like time patterns or location of occurrence by the help of a log or diary of symptoms. In the absence of this cooperation, the following questions in addition to the medical history may be useful.
Start, duration and periodicity (diurnal, daily, weekly, seasonally) of symptom or complaints.
Their relation with location under consideration (cessation when away or reoccurrence when returning)
The work type, work place, any change including moving or decoration.
Exposure to environmental tobacco smoke at work, school, home, etc.
Place of residence including internal change or moving
Exposure to a new hobby etc., a new pet.
Similar problem in anybody in close contact at home or work (Table 2).
With marked hypersensitivity reactions and Legionnaires’ Disease.
4.1 Health problems related to environmental tobacco smoke (ETS)
It is the most observed indoor air contaminants. It easily disperses and it is hard not be inhaled by workers in the workplace.
Key Signs/Symptoms in Adults; conjunctival irritation, headache, persistent cough, wheezing, rhinitis/pharyngitis, nasal congestion, exacerbation of chronic respiratory conditions.
Key Signs/Symptoms in Infants and Children; asthma onset, snoring, bronchitis, repeated pneumonia, persistent middle-ear effusion, frequent upper respiratory infections and/or episodes of otitis media, increased severity of, or difficulty in controlling, asthma.
ETS is also defined as Group A human carcinogen by the U.S. Environmental Protection Agency (EPA) and related to three thousand pulmonary cancers per year among people who do not smoke in the U.S [14, 15, 16]. Among very young children, the incidence of pneumonia, bronchitis, and bronchiolitis is reported to increase two-fold and the effects to be proportional with the frequency of smoking and smokers at the home [17].
The odor of ETS can be eliminated by ventilation, but not meaning that health risks are also removed as it is not possible to totally remove tobacco smoke [18]. The most effective remedy is strict smoking prohibition in the work-place or adapting special smoking rooms with separate ventilation to the outside [19].
4.2 Other combustible products causing health problems
Carbon monoxide is an odorless and colorless asphyxiant due to carboxyhemoglobin (COHb) resulting from CO binding to Hb, impeding oxygen transport.
Nitrogen dioxide (NO) and sulfur dioxide (SO2) particularly irritate ocular, nasal, pharyngeal and respiratory tract mucosa. Acute bronchoconstriction by sulfur dioxide can be observed in asthma cases or as a hypersensitivity reaction. Continued exposure to elevated levels of nitrogen dioxide may result in acute or chronic bronchitis [20].
Key Signs/Symptoms; nausea/emesis, dizziness, headache, fatigue, ocular and upper respiratory tract irritation, tachycardia, chronic cough, confusion wheezing, hypercarboxyhemoglobinemia, increased frequency of angina in cardiovascular patients.
Diagnostic Leads
Types of heating, cooking or similar equipment and used combustion material (especially charcoal).
Similar findings/symptoms among households in heating season.
Odor felt during heating or any damage in the equipment, if they undergo periodic professional inspection.
Remedial Action.
All equipment should be periodically checked by specialized services, especially before each cold season. The ventilation of equipment (including kitchens) is required to be connected to the outdoor environment.
Health Problems Caused By Volatile Organic Compounds (VOCs).
Even at room temperature certain solids or liquids may emit VOCs like formaldehyde, benzene, perchloroethylene for different length of time. They have been observed indoors than outdoors up to 10 times in six locations of the United States as reported by the EPA, even where there were petrochemical plants in use [21].
Presence and quantity of pressed wood products at the resident.
Exposure to VOCs at work, home, school.
Exposure to pesticides, paints, or solvents.
Formaldehyde.
Formaldehyde is a possible human carcinogen (EPA). It may irritate ocular (burning or tingling sensations) or respiratory mucosa (dyspnea or wheezing). Formaldehyde vapor may result in hypersensitivity reactions including asthmatics [22].
Pesticides.
They are used in daily life as pesticides and harmful when inhaled or exposed to their vapors or contaminated dusts. Cephalgia, dizziness, muscular weakness, and nausea are the main symptoms. Some of them are considered possible human carcinogens [23].
Remedial Action.
A forced ventilation is required when such products are used. Avoid storage of opened containers of unused paints etc. at home or workplace and similar materials within home or office.
Health Problems Caused By Heavy Metals: Airborne Lead And Mercury Vapor.
In Young Children; abdominal pain, irritability, seizures/loss of consciousness, ataxia, hyperactivity, reduced attention span, (chronic) learning deficits.
Key Signs/Symptoms of Mercury Poisoning; headache, tachycardia, muscle cramps or tremors, acrodynia, intermittent fever, neurological dysfunction, personality change.
Diagnostic Leads
Housing or working in old or restored buildings or nearby busy highway or industrial area.
Working with lead material (automobile radiators, solder etc.)
Lead poisoning among people in close contact.
Exposure to mercury in latex paints or in religious or cultural activities
Remedial Action.
The possible lead dust should be cleaned by wet-mopping. Professional intervention should be sought when handling paints containing lead and adequate protective gear and good-ventilation provided in work areas.
Health Problems Caused By SICK BUILDING SYNDROME.
Key Signs/Symptoms; headache, dizziness, nausea, sensitivity to odors, lethargy or fatigue, mucosal irritation.
Diagnostic Leads
Temporal ceasing or aggravation of problems in relation to exposure frequency to suspected building, or seasonality
Similar complaints in co-workers or peers.
Remedial Action.
The building, HVAC systems or possible conditions should be investigated and examined appropriately.
4.3 Health problems caused by two long-term risks: asbestos and radon
Asbestos and radon are among the most publicized indoor air pollutants. Both are known as carcinogens. Their carcinogenic effects are not immediate after prolonged exposure.
Asbestos.
Materials containing asbestos can lose its integrity with time releasing microscopic fibers into the environment. If they remain present in the lungs for many decades as in the case of heavy occupational exposure, they may lead to asbestos-caused pulmonary fibrosis, pulmonary, pleural or peritoneal (including gastrointestinal) carcinoma, or mesothelioma [24].
Radon.
Radon is a naturally occurring radioactive gas resulting from the decay of radium, itself a decay product of uranium, follows smoking for causing pulmonary malignancies due to the emitted alpha-particles during the decay. It has no odor, color, and taste. Tobacco smoke has a synergistic effect to radon exposure putting smokers and ex-smokers in increased risk.
4.4 Health Problems Caused By ANIMAL DANDER, MOLDS, DUST MITES, OTHER BIOLOGICALS
Every home, school, and workplace are subject to biological air pollutants. Some reside outdoor or in human (viruses and bacteria), some in animals or insects (allergens), and some indoor and in water reservoirs (fungi and bacteria), such as humidifiers. High relative humidity is the most important factor contributing to the growth and dissemination of biological agents like house-dust mite populations or fungal growth on damp surfaces. They may cause infections by invading human tissues; hypersensitivity by activating the immune system; and toxicosis by direct effects of toxins [25].
Mounting evidence regarding the workplace, home, etc. as a source place (although very difficult) like presence of a reservoir or disseminator of biologicals
Evidence of mold growth (visible growth or odors)?
Hypersensitivity disease:
Relative humidity consistently above 50%.
Presence of humidifiers or other water-spray systems, proper maintenance.
History of flooding or leaks or other sources of surface wetting.
Pets, cockroaches or rodents in the place.
Toxicosis and/or irritation:
Appropriate ventilation with fresh air.
Relative humidity consistently above 50% or below 30%.
Presence of humidifiers or other water-spray systems.
Evidence of mold growth (visible growth or odors)?
Presence of bacterial odors (fishy or locker-room smells)?
Remedial Action
Adequate outdoor air ventilation.
Cleaning of water reservoirs and chlorination of potable water systems
Repairing of leaks and seepage.
Keeping relative humidity below 50%
Controlling exposure to pets.
Regular vacuuming of carpets and furniture.
Covering of mattresses. Washing in hot water (>54.4°C to kill dust mites in soft materials)
Distinguishing whether indoor air pollution originates from the home or workplace.
Some information may help to determine the presence of an indoor air quality problem at workplace:
Symptoms observed to occur at workplace and to disappear when leaving the workplace, their temporal or locality pattern (day, season or location at work)
There are some international guiding principles set for indoor air quality. The recommended guidelines define indoor air quality issues with legal standards. These guidelines are prepared and updated by professionals. There is limited information in the World, particularly on concentration guidelines and standards proposed for indoor air pollutants. Only in the United Kingdom and USA, there are concentration guidelines and standards proposed for indoor air pollutants.
World Health Organization: WHO issued various guidelines aiming at protecting the public health from risks arising from some indoor pollutants such as benzene, carbon monoxide, formaldehyde, nitrogen dioxide, polycyclic aromatic hydrocarbon (PAH), benzo [α] pyrene, radon, trichloroethylene, tetrachloroethylene. Reference values offer basic information allowing assessor to decide whether lifelong exposure to these pollutants or exposure to them for a certain approximate period of time impose a significant risk for the health and wellbeing of people [7].
Committee on the Medical Effects of Air Pollutants (COMEAP) issued “Report on the Impact of Air Pollution on Health for public institutions and agencies. It determined allowed amounts of indoor air pollutants (COMEAP-2004);, formaldehyde, benzene, PAHs (as the equivalent of benzo[α]piren), NO2 ve CO for indoors. Air Quality Strategy for England, Wales and Northern Ireland (DETR, 2000; Defra, 2007) sets out policies for the management of indoor air quality. These include air quality targets for ten basic air pollutants for protecting the health of people and the environment, without bearing unacceptable “economic and social costs. These are Particles (PM10 and PM2.5), NO2, O3, Sulfur dioxide (SO2), PAHs, benzene, 1,3 butadiene, carbon monoxide (CO) and lead. Health and Safety Executive (HSE) supports the regulatory framework for the workplace health and safety in England, Wales and Scotland, in line with the Occupational Health and Safety Law (HSE, 1974).
Regulation on Control of Substances Hazardous on Health 2002 (HSE, 2002) set out Official Workplace Exposure Limits (WELs) for 500 substances which are listed in the EH40 document (HSE, 2011 and the following revisions), as an action against specific pollutants. These limits include maximum concentrations for short term (15 minutes) and long term (8 hours) exposure in any period of 24 hours. Although it is mostly related about indoor emissions, exposure limits determine the indoor values which should not be exceeded, no matter what the source is. HSE does not set limits for continuous (24 hours) exposure. For this reason, WELs are not considered as safe concentrations for periods longer than those specified [7].
A research conducted in Europe in 2005 collected formaldehyde, CO, NO2, benzene and naphthalene under “Group 1: High Risk Chemicals”, as they can form in high concentration and impose a significant risk for the health of residents of the building (INDEKS, 2005). “Group 2” included acetaldehyde, toluene, xylene, styrene as the chemicals of second highest risk. These compounds may occur in high concentrations in indoors, but they require less urgent action under risk management practice [7]. Leading institutions regulating the national official rules are American Conference of Governmental Industrial Hygienists (ACGIH) and American Society of Heating and Air-Conditioning Engineers (ASHRAE) [8, 9, 10].
The focus point of EPA air quality is to protect the human health against outdoor air. The objective of this Standard is to control emissions of six pollutants during the release of large amounts of vehicle exhaust gas and industrial waste. These standards can be used for the indoor air quality researches as outdoor air quality offer potential contribution to the indoor exposure (Table 3) [8, 9, 10].
Indoor air pollutants
Permissible concentrations
Carbon monoxide (CO)
< 9 ppm
Carbon dioxide (CO2)
< 800 ppm
Mold
Indoor and outdoor values should be the same
Formaldehyde (CH2O)
< 20 μg/m3*
Total volatile organic compounds (VOC)
< 200 μg/m3*
4-Phenyl Cyclohexane (4-PC)
< 3 μg/m3
Total particles (PM)
< 20 μg/m3
Regular pollutants
< National indoor standard
Other pollutants
< 5% of the limit value
Table 3.
EPA maximum indoor air standards.
Above outdoor air concentrations.
OSHA claims that it has jurisdiction in all workplace environments. These standards are concerned about indoor air quality at office buildings, industrial and construction workplaces. However, OSHA standards have limitations in terms of knowledge of pollutants and limited exposure limits, as OSHA’s standards are based on old limits issued by ACGIH in 1968. Original OSA exposure limits were developed out of ACGIH recommendations dated 1968. Up to now, only limits for some chemical pollutants (for example, asbestos and benzene) have been updated. For this reason, general tendency of industrial hygienists to prefer ACGIH Instructions to OSHA limits. Although backed up by federal laws, OSHA limits are rarely exceeded in office environments where one or more pollutant substances are correctly defined. The complicated nature of the indoor air quality is not supported by the OSHA limits [8, 9, 10].
ACGIH is a professional institution which revises and recommends user manuals used for evaluation of Professional workplace exposure by industrial hygienists every year. There are approximately 400 chemicals that are listed with exposure limits of 15 minutes and 8 hours. These directives were prepared to treat the workplace exposure. Professional exposure is generally limited with a period of 8 hour exposure for healthy individuals aged between 18 and 65. For this reason, ACGIH exposure rules do not apply for house exposure for which exposure parameters are different [7, 8, 9, 10].
ASHRAE issued a revised mechanical ventilation standard namely “Ventilation for Acceptable Indoor Air Quality Standard” in 1981. ASHRAE developed consensus principles for the indoor air quality in public buildings. The Standard aims at “stating minimum ventilation rates and indoor air quality”. Health effects and acceptable exposure limits are based on specific authorized people and their recommendations. For this reason, ASRAE Standard “Ventilation for Acceptable Indoor Air Quality” has become the guideline which is most widely used for the evaluation of indoor air quality in commercial facilities and enterprises. ASHRAE previously issued Standard 62 which is a ventilation standard. This Standard was revised a few times in the following years. The amount of fresh air was specified for smokers and non-smokers separately in 1981. This value was 2,5 L/s foreseen for non-smokers and 10 L/s for smokers, which is four times higher than the one for nonsmokers. Cigarette monopolies prevented the recognition of this Standard by American National Standards Institute (ANSI) and its integration to the building regulations, by conducting intensive propagandas. Application of this Standard will increase the ventilation cost by four times in the buildings where smokers work. In 1989, the acceptable fresh air was accepted to be the amount for which 80% of the people did not express dissatisfaction. The dissatisfaction rate was increased to 20% of the people in the place.
ASHRAE’s Standard numbered 1989–2062 introduced the limit of 1000 ppm carbon dioxide for office workers. OHSA in USA defined a limit of 5000 ppm, on the condition that it does not exceed 40 hours a week. ASHRAE’s Standard 62 recognized that carbon dioxide is not a pollutant by itself, but it is one of the indicators of air polluted by people. This CO2 amount was 280 ppm before the industrial revolution and it has been continuously increasing due to the combustion of fossil fuel. As a result, global warming caused by greenhouse effect has become a very important public problem. The rate of carbon dioxide in today’s air is around 390 ppm and it increases by 2 ppm every year [8, 9, 10, 11, 12]. Workplace Exposure Limits (WELs) apply to healthy people who are at working age and directly exposed to pollutants at their workplaces. Generally lower exposure limits are imposed for people who are not healthy or of working age or those who are older than the working age. These lower limits apply even if a person is exposed to a pollutant for a period which is significantly longer than 8 hours or even if the work activities do not directly include pollutant [7, 8, 9, 10]. Attention should be paid to the selection of air quality standards applicable to a particular workplace during the selection of the most appropriate air quality standard and guideline. In schools and hospitals which are open to public Access, imposition of the targets set by the World Health Organization’s guidelines and the targets introduced by the United Kingdom Air Quality Strategy are more appropriate. As for the industrial environments, Health and Safety Executive (HSE) and Workplace Exposure Limits (WELs) are most appropriate (HSE, 2011) [8, 27]. HSE WELS defines official exposure limits for physically healthy people who are exposed for a nominal period of eight hours a day, five day a week in industrial workplaces. For this reason, the elderly, the young and the disabled who are sensitive to some pollutants should be excluded, when the limits are determined. The people who work in the office environment including the ones who are physically less in form and talented, represent a wider proportion of society, when compared to the workers in the industrial environments. Furthermore, workers may not be aware of the fact that they are exposed to a pollutant, if they are not in contact with it as a direct part of the work they perform, and thus they may not take any measure to protect themselves. Her Majesty’s Inspectorate of Pollution (MHIP) recommended that a part of the guidelines given in HSE WELs for the exposure of the general society (1993). In the light of this, it is more appropriate to apply lower outdoor air quality guidelines set by the World Health Organization and United Kingdom Air Quality Strategy (Defra 2007) for exposure of the general population also for the indoor environments. The limits determined by HMIP (1993) may be used for the pollutants which do not fall under this scope [8, 9, 10, 11, 12, 28].
6. Protection from indoor air pollution at workplaces
An employer should use a systematic approach is needed when treating the air quality at the workplace. The systematic approach to indoor air quality (IAQ) comprise commitment of the management, training, participation of employees, hazard definition, control and program inspection. A management coordinator needs to be assigned for IAQ and a management plan needs to be developed.
7. IAQ control methods
There are three main control methods used to decrease the concentration of the indoor air pollutants:
Source Management
It includes eliminating the pollutant or replacement of pollutant with a less hazardous one. It is the most effective control method in practice. For example, an employer may install temporary barriers in order to prevent pollutants during construction activity or impose negative pressure on the field in Ref. to the adjacent fields.
Engineering Controls
Local exhaust: Use of local exhaust, such as shading and fume hood are effective in eliminating pollutants which are very concentrated.
General ventilation: When designed, operated, maintained properly, general ventilation is a measure which control air pollutants of normal amount. A well designed and operating HVAC system ensures comfort, by controlling temperature and relative humidity levels, distributes the amount of air sufficient to meet the needs of ventilation for the building habitants and alleviate and eliminates odors and other pollutants.
Air cleaning: Firstly, it requires elimination of particles in the air when they pass through HVAC equipment. Generally, HVAC system filtering is used to keep the dirt away from adjustable surfaces during the process of ensuring heat transfer effectiveness.
Management Controls
Working Chart: Managers may significantly decrease the amount of exposure to pollutants in their respective buildings, by using charts. For example, they can take the following actions:
Eliminate or decrease the duration in which one worker is exposed to a pollutant (in other words, programming the maintenance or cleaning work in the absence of inhabitants)
Decrease the amount of chemical substances used by workers or used near to workers (limit the amount of chemicals used by workers for maintenance or cleaning activities during the activity).
Control the place where the chemicals are used (conduct maintenance on moving equipment in a maintenance workshop or place equipment- printers, copy machines to a separate room).
Training: It is important to give IAQ training to workers. Workers need to informed about the sources and effects of pollutants under their control and smooth operation of ventilation system. Employers can make warnings and/or take measures to decrease personal exposure.
Cleaning: Cleaning practices should contain preventing entry of dirt to the environment (using walking doormats), cleaning it when it enters the workplace, discharging the litter, storing the food properly and using minimum amount of cleaning products [7, 8, 9, 29].
8. Indoor air and coronavirus (COVID-19)
COVID-19 spreads through particles and droplets in the air. Individuals infected by COVID may release particles and droplets of inspiration liquids containing SARS CoV-2 virus to the air (by breathing, talking, singing, exercising, coughing and sneezing). Droplets- particles may continue to disseminate and accumulate indoors of workplaces. Infection may happen in case of inhaling the COVID-19 virus from air in a distance shorter than six feet. The particles from an infected person may move along all room or closed area. The particles may hang in the air for hours even after the person leaves the room. A worker may be exposed to it, if respiration liquids directly jump to the mucosa membrane and if it is sprayed on him or her. The following cases may increase the infection risk:
Spending time indoors where the amount of outdoor air and ventilation is poor
Performing activities which increases emission of respiration liquids, such as talking loudly, singing, exercising
Long term exposure (longer than a few minutes)
Spending time in crowded areas (especially without proper mask protection).
Measures to decrease the infection potential of COVID-19:
Layout, design of a building, occupancy state, heating, ventilation and acclimatization (HVAC) system may affect the spread potential of virus through air. Although improvements made on ventilation and air cleaning do not alone eliminate of risk of spread for the SARS-CoV-2 virus, EPA recommends that physical distance should be maintained and ventilation should be improved by using outdoor air and air filtering, as the important components of a strategy which includes hygiene and clothing.
Cloth masks, face guards or masks should be used. Attention should be paid to surface cleaning, hand washing, disinfection, personal and environment hygiene [8, 30].
9. Conclusion
WHO/ILO Joint Estimates of the Work-related Burden of Disease and Injury (2000–2016: Global Monitoring Report) revealed that approximately 450.000 workers’ deaths were associated with air pollution (particle substances, gas, smoke, etc.). This association was reported to be the second most important factor which comes after the factor of working for long hours among the risk factors causing death of workers. Workplace indoor air pollution can significantly increase the health risks of workers, including asthma, allergenic reactions, lung cancer and death as a result of occupational accident [1, 29].
1989 EPA Report showed that improved indoor air quality may result in higher productivity and less working day loss. EPA stated that the poor indoor air quality may bring a cost of tens of million dollars to the respective country, employer and the enterprise every year, due to the loss of productivity and medical care cost [8].
Further research is needed to detect new indoor pollutants which are increasing in number and control their effects. Lifelong awareness, elimination of potential indoor hazards, increased awareness of health service providers and professionals are reported to be important to encourage long term lung health and wellbeing [1].
Indoor air quality can be defined as an optimal indoor requirement specifying the possible minimum amount of air pollutants to ensure the health, comfort and wellbeing of majority of the workers in any closed workplace, at any given time. Temperature at the workplace depends on relative humidity and flow of air in industry. In addition, indoor air at industrial facilities is associated with the technological processes conducted and contents of the chemicals used. Workplace risk assessment is a means which helps creation of a safer environment and it is a process allowing determination of potential adverse effects imposed on the health of workers. The obligation to determine risk assessments which are both correct and simple led to the development of approaches to asses and control risks, including COSHH (Control of Substances Hazardous to Health) and “Chemical Control Kit” designed to assess chemical risks. Enterprises developed Process Route Healthiness Index (PRHI) to analyze new processes which are not yet in implementation [3].
It was seen that some people had health symptoms although concentrations of indoor air pollutants are below the indoor air quality guidelines. For this reason, further research is needed to better understand and explain the complicated relationships between IAQ and health symptoms [11].
A multidisciplinary team comprising experts of occupational medicine, IAQ, building physics and toxicology is recommended for evaluation and management of IAQ problems [5].
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They were grouped as asbestos, biological pollutants, carbon monoxide (CO), carbon dioxide (CO2), formaldehyde/pressed wooden products, lead (Pb), nitrogen dioxide, (NO2), Radon (Rn), indoor particle substances, environmental tobacco smoke, volatile organic compounds (VOCs), humidity, odor and wood smoke. The workplace indoor air pollutants are among the primary indoor air pollutants with serious effects on health and the potential to deteriorate the workers’ health. Healthy indoor air quality is defined as the indoor air which does not contain hazardous substances and does not create sense of discomfort in at least 80% of the workers in the workplace. 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Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Seguel JM, Merrill R, Seguel D, Campagna AC. Indoor air quality. American Journal of Lifestyle Medicine. 2016;11(4):284-295. DOI: 10.1177/1559827616653343'},{id:"B2",body:'Cincinelli A, Martellini T. Indoor air quality and health. International Journal of Environmental Research and Public Health. 2017;14(11):1286. DOI: 10.3390/ijerph14111286'},{id:"B3",body:'Cullen MR, Kreiss K. In: Levy BS, Wegman DH, Baron SL, Sokas RK, editors. Indoor Air Quality. Occupational and Environmental Health. Sixth ed. UK: Oxford University Press; 2011. pp. 141-153'},{id:"B4",body:'Reinhold K, Tint P, Munter R. Indoor air quality in industrial premises. RTU scientific articles Material Science and Applied Chemistry. 2009;20:48-57'},{id:"B5",body:'Carrer P, Wolkoff P. Assessment of indoor air quality problems in office-like environments: Role of occupational health services. International Journal of Environmental Research and Public Health. 2018;15:741. DOI: 10.3390/ijerph15040741'},{id:"B6",body:'Türk Toraks Derneği. İç ortam hava kalitesi. [Internet]. Available from: https://www.toraks.org.tr/site/news/10355 [Accessed: 02.12.2021]'},{id:"B7",body:'Kukaida V, Upton S. Ensuring Good Indoor Air Quality in Buildings. England: BRE Trust; March 2019'},{id:"B8",body:'EPA. Indoor Air Quality (IAQ). [Internet]. Available from: https://www.epa.gov/indoor-air-quality-iaq/office-building-occupants-guide-indoor-air-quality [Accessed: 01.12.2021]'},{id:"B9",body:'OSHA. Indoor Air Quality in Commercial and Institutional Buildings. [Internet]. Available from:https://www.osha.gov/sites/default/files/publications/3430indoor-air-quality-sm.pdf [Access: 05.12.2021]'},{id:"B10",body:'Bulgurcu H. İç Ortam Hava Kirliliği. Havalandırma ve İç Ortam Hava Kalitesi Notları. [Internet]. Available from: file:///C:/Users/Lenovo/Downloads/ Mevzuat%20ve%20standartlar %20(2).pdf [Access: 03.12.2021]'},{id:"B11",body:'Sakellaris I, Saraga D, Mandin C, de Kluizenaar Y, Fossati S, Spinazzè A, et al. Association of subjective health symptoms with indoor air quality in European office buildings: The OFFICAIR project. Indoor Air. 2021;2:426-439. DOI: 10.1111/ina.12749'},{id:"B12",body:'Al-Zboon KK, Forton OT. Indoor air quality in steel rolling industries and possible health effects. Environment and Natural Resources Journal. 2019;17(4):20-29. DOI: 10.32526/ennrj.17.4.2019.28'},{id:"B13",body:'U.S. Environmental Protection Agency, Office of Air and Radiation. Report to Congress on Indoor Air Quality, Volume II: Assessment and Control of Indoor Air Pollution, pp. I, 4-EPA 400-1-89-001C. 1989'},{id:"B14",body:'U.S. Department of Health and Human Services. The Health Consequences of Involuntary Smoking, A Report of the Surgeon General. DHHS Publication No. (PHS) 87-8398. 1986'},{id:"B15",body:'National Research Council. Environmental Tobacco Smoke: Measuring Exposures and Assessing Health Effects. USA: National Academy Press; 1986'},{id:"B16",body:'National Institute for Occupational Safety and Health. Environmental Tobacco Smoke in the Workplace: Lung Cancer and Other Health Effects. U.S. Department of Health and Human Services, Current Intelligence Bulletin 54. 1991'},{id:"B17",body:'U.S. Environmental Protection Agency. Respiratory Health Effects of Passive Smoking: Lung Cancer and Other Disorders'},{id:"B18",body:'Repace JL, Lowrey AH. An indoor air quality standard for ambient tobacco smoke based on carcinogenic risk. New York State Journal of Medicine. 1985;85(7):381-383'},{id:"B19",body:'American Society of Heating, Refrigeration and Air-conditioning Engineers. Ventilation for Acceptable Air Quality; ASHRAE Standard 62-1989'},{id:"B20",body:'Samet JM, Marbury MC, Spengler JD. Health effects and sources of indoor air pollution. Part I. Am Rev Respir Dis. 1987;136(6):1486-1508. DOI: 10.1164/ajrccm/136.6.1486'},{id:"B21",body:'U.S. Environmental Protection Agency, Office of Acid Deposition, Environmental Monitoring and Quality Assurance. Project Summary: The Total Exposure Assessment Methodology (TEAM) Study. EPA-600-S6-87-002. 1987'},{id:"B22",body:'U.S. Environmental Protection Agency, U.S. Public Health Service, and National Environmental Health Association. Introduction to Indoor Air Quality: A Reference Manual, p. 87. EPA-400-3-91-003. 1991'},{id:"B23",body:'U.S. Environmental Protection Agency Office of Research and Development. Final Report: Nonoccupational Pesticide Exposure Study (NOPES), p. 60. EPA-600-3-90-003. 1990'},{id:"B24",body:'Rom WN, Travis WD, Brody AR. Cellular and molecular basis of the asbestos-related diseases. The American Review of Respiratory Disease. 1991;143(2):408-422. DOI: 10.1164/ajrccm/143.2.408'},{id:"B25",body:'Burge HA, Feely JC. Indoor air pollution and infectious diseases. In: Samet JM, Spengler JD, editors. Indoor Air Pollution, a Health Perspective. Baltimore MD: Johns Hopkins University Press; 1991. pp. 273-284'},{id:"B26",body:'EPA. Indoor Air Pollution: An Introduction for Health Professionals [Internet]. Available from: https://www. osha.gov/indoor-air-quality/faqs [Accessed: 05.12.2021]'},{id:"B27",body:'HSE. Workplace exposure limits. [Internet]. Available from: https://www.hse.gov.uk/coshh/basics/exposurelimits.htm [Accessed: 04.12.2021]'},{id:"B28",body:'Kosa KH. Indoor Air Quality the Latest Sampling and Analytical Methods. London, New York, USA: CRC press, Taylor & Francis Group; 2011'},{id:"B29",body:'WHO/ILO joint estimates of the work-related burden of disease and injury, 2000-2016: global monitoring report. [Internet]. Available from: https://www.who.int/publications/i/item/9789240034945 [Accessed: 01.12.2021]'},{id:"B30",body:'EPA. Indoor Air and Coronavirus (COVID-19). [Internet]. Available from: https://www.epa.gov/ coronavirus/indoor-air-and-coronavirus-covid-19 [Accessed: 05.12.2021]'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Ferdi Tanir",address:"ftanir@gmail.com",affiliation:'
Department of Public Health, Çukurova University, Adana, Turkey
Department of Public Health, Çukurova University, Adana, Turkey
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She obtained a BSc from the University of Derby, England, a master’s degree from Technische Universität München, Germany, and a Ph.D. from the University of Nottingham. She undertook a post-doctoral research fellowship in the School of Medicine before accepting tenure in Veterinary Medicine and Science. Dr. Rutland also obtained an MMedSci (Medical Education) and a Postgraduate Certificate in Higher Education (PGCHE). She is the author of more than sixty peer-reviewed journal articles, twelve books/book chapters, and more than 100 research abstracts in cardiovascular biology and oncology. She is a board member of the European Association of Veterinary Anatomists, Fellow of the Anatomical Society, and Senior Fellow of the Higher Education Academy. 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He studied \r\nchemistry at the Universidad Nacional de La Plata, Argentina, where received aPh.D. degree in chemistry (Biological Branch) in 1965. From\r\n1964 to 1974, he worked as Assistant in Biochemistry at the School of MedicineUniversidad Nacional de La Plata, Argentina. From 1974 to 1976, he was a Fellowof the National Institutes of Health (NIH) at the University of Connecticut, Health Center, USA. From 1985 to 2004, he served as a Full Professor oBiochemistry at the Universidad Nacional de La Plata, Argentina. He is Member ofthe National Research Council (CONICET), Argentina, and Argentine Society foBiochemistry and Molecular Biology (SAIB). His laboratory has been interested for manyears in the lipid peroxidation of biological membranes from various tissues and different species. Professor Catalá has directed twelve doctoral theses, publishedover 100 papers in peer reviewed journals, several chapters in books andtwelve edited books. Angel Catalá received awards at the 40th InternationaConference Biochemistry of Lipids 1999: Dijon (France). W inner of the Bimbo PanAmerican Nutrition, Food Science and Technology Award 2006 and 2012, South AmericaHuman Nutrition, Professional Category. 2006 award in pharmacology, Bernardo\r\nHoussay, in recognition of his meritorious works of research. Angel Catalá belongto the Editorial Board of Journal of lipids, International Review of Biophysical ChemistryFrontiers in Membrane Physiology and Biophysics, World Journal oExperimental Medicine and Biochemistry Research International, W orld Journal oBiological Chemistry, Oxidative Medicine and Cellular Longevity, Diabetes and thePancreas, International Journal of Chronic Diseases & Therapy, International Journal oNutrition, Co-Editor of The Open Biology Journal.",institutionString:null,institution:{name:"National University of La Plata",institutionURL:null,country:{name:"Argentina"}}},editorTwo:null,editorThree:null,editorialBoard:[{id:"186048",title:"Prof.",name:"Ines",middleName:null,surname:"Drenjančević",slug:"ines-drenjancevic",fullName:"Ines Drenjančević",profilePictureURL:"https://mts.intechopen.com/storage/users/186048/images/5818_n.jpg",institutionString:null,institution:{name:"University of Osijek",institutionURL:null,country:{name:"Croatia"}}},{id:"187859",title:"Prof.",name:"Kusal",middleName:"K.",surname:"Das",slug:"kusal-das",fullName:"Kusal Das",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSBDeQAO/Profile_Picture_1623411145568",institutionString:"BLDE (Deemed to be University), India",institution:null},{id:"79615",title:"Dr.",name:"Robson",middleName:null,surname:"Faria",slug:"robson-faria",fullName:"Robson Faria",profilePictureURL:"https://mts.intechopen.com/storage/users/79615/images/system/79615.png",institutionString:null,institution:{name:"Oswaldo Cruz Foundation",institutionURL:null,country:{name:"Brazil"}}},{id:"84459",title:"Prof.",name:"Valerie",middleName:null,surname:"Chappe",slug:"valerie-chappe",fullName:"Valerie Chappe",profilePictureURL:"https://mts.intechopen.com/storage/users/84459/images/system/84459.jpg",institutionString:null,institution:{name:"Dalhousie University",institutionURL:null,country:{name:"Canada"}}}]},{id:"12",title:"Human Physiology",coverUrl:"https://cdn.intechopen.com/series_topics/covers/12.jpg",editor:{id:"195829",title:"Prof.",name:"Kunihiro",middleName:null,surname:"Sakuma",slug:"kunihiro-sakuma",fullName:"Kunihiro Sakuma",profilePictureURL:"https://mts.intechopen.com/storage/users/195829/images/system/195829.jpg",biography:"Professor Kunihiro Sakuma, Ph.D., currently works in the Institute for Liberal Arts at the Tokyo Institute of Technology. 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His research interest focuses on computational chemistry and molecular modeling of diverse systems of pharmacological, food, and alternative energy interests by resorting to DFT and Conceptual DFT. He has authored a coauthored more than 255 peer-reviewed papers, 32 book chapters, and 2 edited books. He has delivered speeches at many international and domestic conferences. He serves as a reviewer for more than eighty international journals, books, and research proposals as well as an editor for special issues of renowned scientific journals.",institutionString:"Centro de Investigación en Materiales Avanzados",institution:{name:"Centro de Investigación en Materiales Avanzados",country:{name:"Mexico"}}},{id:"76477",title:"Prof.",name:"Mirza",middleName:null,surname:"Hasanuzzaman",slug:"mirza-hasanuzzaman",fullName:"Mirza Hasanuzzaman",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/76477/images/system/76477.png",biography:"Dr. Mirza Hasanuzzaman is a Professor of Agronomy at Sher-e-Bangla Agricultural University, Bangladesh. He received his Ph.D. in Plant Stress Physiology and Antioxidant Metabolism from Ehime University, Japan, with a scholarship from the Japanese Government (MEXT). Later, he completed his postdoctoral research at the Center of Molecular Biosciences, University of the Ryukyus, Japan, as a recipient of the Japan Society for the Promotion of Science (JSPS) postdoctoral fellowship. He was also the recipient of the Australian Government Endeavour Research Fellowship for postdoctoral research as an adjunct senior researcher at the University of Tasmania, Australia. Dr. Hasanuzzaman’s current work is focused on the physiological and molecular mechanisms of environmental stress tolerance. Dr. Hasanuzzaman has published more than 150 articles in peer-reviewed journals. He has edited ten books and written more than forty book chapters on important aspects of plant physiology, plant stress tolerance, and crop production. According to Scopus, Dr. Hasanuzzaman’s publications have received more than 10,500 citations with an h-index of 53. He has been named a Highly Cited Researcher by Clarivate. He is an editor and reviewer for more than fifty peer-reviewed international journals and was a recipient of the “Publons Peer Review Award” in 2017, 2018, and 2019. He has been honored by different authorities for his outstanding performance in various fields like research and education, and he has received the World Academy of Science Young Scientist Award (2014) and the University Grants Commission (UGC) Award 2018. He is a fellow of the Bangladesh Academy of Sciences (BAS) and the Royal Society of Biology.",institutionString:"Sher-e-Bangla Agricultural University",institution:{name:"Sher-e-Bangla Agricultural University",country:{name:"Bangladesh"}}},{id:"187859",title:"Prof.",name:"Kusal",middleName:"K.",surname:"Das",slug:"kusal-das",fullName:"Kusal Das",position:null,profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0030O00002bSBDeQAO/Profile_Picture_1623411145568",biography:"Kusal K. Das is a Distinguished Chair Professor of Physiology, Shri B. M. Patil Medical College and Director, Centre for Advanced Medical Research (CAMR), BLDE (Deemed to be University), Vijayapur, Karnataka, India. Dr. Das did his M.S. and Ph.D. in Human Physiology from the University of Calcutta, Kolkata. His area of research is focused on understanding of molecular mechanisms of heavy metal activated low oxygen sensing pathways in vascular pathophysiology. He has invented a new method of estimation of serum vitamin E. His expertise in critical experimental protocols on vascular functions in experimental animals was well documented by his quality of publications. He was a Visiting Professor of Medicine at University of Leeds, United Kingdom (2014-2016) and Tulane University, New Orleans, USA (2017). For his immense contribution in medical research Ministry of Science and Technology, Government of India conferred him 'G.P. Chatterjee Memorial Research Prize-2019” and he is also the recipient of 'Dr.Raja Ramanna State Scientist Award 2015” by Government of Karnataka. He is a Fellow of the Royal Society of Biology (FRSB), London and Honorary Fellow of Karnataka Science and Technology Academy, Department of Science and Technology, Government of Karnataka.",institutionString:"BLDE (Deemed to be University), India",institution:null},{id:"243660",title:"Dr.",name:"Mallanagouda Shivanagouda",middleName:null,surname:"Biradar",slug:"mallanagouda-shivanagouda-biradar",fullName:"Mallanagouda Shivanagouda Biradar",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/243660/images/system/243660.jpeg",biography:"M. S. Biradar is Vice Chancellor and Professor of Medicine of\nBLDE (Deemed to be University), Vijayapura, Karnataka, India.\nHe obtained his MD with a gold medal in General Medicine and\nhas devoted himself to medical teaching, research, and administrations. He has also immensely contributed to medical research\non vascular medicine, which is reflected by his numerous publications including books and book chapters. Professor Biradar was\nalso Visiting Professor at Tulane University School of Medicine, New Orleans, USA.",institutionString:"BLDE (Deemed to be University)",institution:{name:"BLDE University",country:{name:"India"}}},{id:"289796",title:"Dr.",name:"Swastika",middleName:null,surname:"Das",slug:"swastika-das",fullName:"Swastika Das",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/289796/images/system/289796.jpeg",biography:"Swastika N. Das is Professor of Chemistry at the V. P. Dr. P. G.\nHalakatti College of Engineering and Technology, BLDE (Deemed\nto be University), Vijayapura, Karnataka, India. She obtained an\nMSc, MPhil, and PhD in Chemistry from Sambalpur University,\nOdisha, India. Her areas of research interest are medicinal chemistry, chemical kinetics, and free radical chemistry. She is a member\nof the investigators who invented a new modified method of estimation of serum vitamin E. She has authored numerous publications including book\nchapters and is a mentor of doctoral curriculum at her university.",institutionString:"BLDEA’s V.P.Dr.P.G.Halakatti College of Engineering & Technology",institution:{name:"BLDE University",country:{name:"India"}}},{id:"248459",title:"Dr.",name:"Akikazu",middleName:null,surname:"Takada",slug:"akikazu-takada",fullName:"Akikazu Takada",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/248459/images/system/248459.png",biography:"Akikazu Takada was born in Japan, 1935. After graduation from\nKeio University School of Medicine and finishing his post-graduate studies, he worked at Roswell Park Memorial Institute NY,\nUSA. He then took a professorship at Hamamatsu University\nSchool of Medicine. In thrombosis studies, he found the SK\npotentiator that enhances plasminogen activation by streptokinase. He is very much interested in simultaneous measurements\nof fatty acids, amino acids, and tryptophan degradation products. By using fatty\nacid analyses, he indicated that plasma levels of trans-fatty acids of old men were\nfar higher in the US than Japanese men. . He also showed that eicosapentaenoic acid\n(EPA) and docosahexaenoic acid (DHA) levels are higher, and arachidonic acid\nlevels are lower in Japanese than US people. By using simultaneous LC/MS analyses\nof plasma levels of tryptophan metabolites, he recently found that plasma levels of\nserotonin, kynurenine, or 5-HIAA were higher in patients of mono- and bipolar\ndepression, which are significantly different from observations reported before. In\nview of recent reports that plasma tryptophan metabolites are mainly produced by\nmicrobiota. He is now working on the relationships between microbiota and depression or autism.",institutionString:"Hamamatsu University School of Medicine",institution:{name:"Hamamatsu University School of Medicine",country:{name:"Japan"}}},{id:"137240",title:"Prof.",name:"Mohammed",middleName:null,surname:"Khalid",slug:"mohammed-khalid",fullName:"Mohammed Khalid",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/137240/images/system/137240.png",biography:"Mohammed Khalid received his B.S. degree in chemistry in 2000 and Ph.D. degree in physical chemistry in 2007 from the University of Khartoum, Sudan. He moved to School of Chemistry, Faculty of Science, University of Sydney, Australia in 2009 and joined Dr. Ron Clarke as a postdoctoral fellow where he worked on the interaction of ATP with the phosphoenzyme of the Na+/K+-ATPase and dual mechanisms of allosteric acceleration of the Na+/K+-ATPase by ATP; then he went back to Department of Chemistry, University of Khartoum as an assistant professor, and in 2014 he was promoted as an associate professor. In 2011, he joined the staff of Department of Chemistry at Taif University, Saudi Arabia, where he is currently an assistant professor. His research interests include the following: P-Type ATPase enzyme kinetics and mechanisms, kinetics and mechanisms of redox reactions, autocatalytic reactions, computational enzyme kinetics, allosteric acceleration of P-type ATPases by ATP, exploring of allosteric sites of ATPases, and interaction of ATP with ATPases located in cell membranes.",institutionString:"Taif University",institution:{name:"Taif University",country:{name:"Saudi Arabia"}}},{id:"63810",title:"Prof.",name:"Jorge",middleName:null,surname:"Morales-Montor",slug:"jorge-morales-montor",fullName:"Jorge Morales-Montor",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/63810/images/system/63810.png",biography:"Dr. Jorge Morales-Montor was recognized with the Lola and Igo Flisser PUIS Award for best graduate thesis at the national level in the field of parasitology. He received a fellowship from the Fogarty Foundation to perform postdoctoral research stay at the University of Georgia. He has 153 journal articles to his credit. He has also edited several books and published more than fifty-five book chapters. He is a member of the Mexican Academy of Sciences, Latin American Academy of Sciences, and the National Academy of Medicine. He has received more than thirty-five awards and has supervised numerous bachelor’s, master’s, and Ph.D. students. Dr. Morales-Montor is the past president of the Mexican Society of Parasitology.",institutionString:"National Autonomous University of Mexico",institution:{name:"National Autonomous University of Mexico",country:{name:"Mexico"}}},{id:"217215",title:"Dr.",name:"Palash",middleName:null,surname:"Mandal",slug:"palash-mandal",fullName:"Palash Mandal",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/217215/images/system/217215.jpeg",biography:null,institutionString:"Charusat University",institution:null},{id:"49739",title:"Dr.",name:"Leszek",middleName:null,surname:"Szablewski",slug:"leszek-szablewski",fullName:"Leszek Szablewski",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/49739/images/system/49739.jpg",biography:"Leszek Szablewski is a professor of medical sciences. He received his M.S. in the Faculty of Biology from the University of Warsaw and his PhD degree from the Institute of Experimental Biology Polish Academy of Sciences. He habilitated in the Medical University of Warsaw, and he obtained his degree of Professor from the President of Poland. Professor Szablewski is the Head of Chair and Department of General Biology and Parasitology, Medical University of Warsaw. Professor Szablewski has published over 80 peer-reviewed papers in journals such as Journal of Alzheimer’s Disease, Biochim. Biophys. Acta Reviews of Cancer, Biol. Chem., J. Biomed. Sci., and Diabetes/Metabol. Res. Rev, Endocrine. He is the author of two books and four book chapters. He has edited four books, written 15 scripts for students, is the ad hoc reviewer of over 30 peer-reviewed journals, and editorial member of peer-reviewed journals. Prof. Szablewski’s research focuses on cell physiology, genetics, and pathophysiology. He works on the damage caused by lack of glucose homeostasis and changes in the expression and/or function of glucose transporters due to various diseases. He has given lectures, seminars, and exercises for students at the Medical University.",institutionString:"Medical University of Warsaw",institution:{name:"Medical University of Warsaw",country:{name:"Poland"}}},{id:"173123",title:"Dr.",name:"Maitham",middleName:null,surname:"Khajah",slug:"maitham-khajah",fullName:"Maitham Khajah",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/173123/images/system/173123.jpeg",biography:"Dr. Maitham A. Khajah received his degree in Pharmacy from Faculty of Pharmacy, Kuwait University, in 2003 and obtained his PhD degree in December 2009 from the University of Calgary, Canada (Gastrointestinal Science and Immunology). Since January 2010 he has been assistant professor in Kuwait University, Faculty of Pharmacy, Department of Pharmacology and Therapeutics. His research interest are molecular targets for the treatment of inflammatory bowel disease (IBD) and the mechanisms responsible for immune cell chemotaxis. He cosupervised many students for the MSc Molecular Biology Program, College of Graduate Studies, Kuwait University. Ever since joining Kuwait University in 2010, he got various grants as PI and Co-I. He was awarded the Best Young Researcher Award by Kuwait University, Research Sector, for the Year 2013–2014. He was a member in the organizing committee for three conferences organized by Kuwait University, Faculty of Pharmacy, as cochair and a member in the scientific committee (the 3rd, 4th, and 5th Kuwait International Pharmacy Conference).",institutionString:"Kuwait University",institution:{name:"Kuwait University",country:{name:"Kuwait"}}},{id:"195136",title:"Dr.",name:"Aya",middleName:null,surname:"Adel",slug:"aya-adel",fullName:"Aya Adel",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/195136/images/system/195136.jpg",biography:"Dr. Adel works as an Assistant Lecturer in the unit of Phoniatrics, Department of Otolaryngology, Ain Shams University in Cairo, Egypt. Dr. Adel is especially interested in joint attention and its impairment in autism spectrum disorder",institutionString:"Ain Shams University",institution:{name:"Ain Shams University",country:{name:"Egypt"}}},{id:"94911",title:"Dr.",name:"Boulenouar",middleName:null,surname:"Mesraoua",slug:"boulenouar-mesraoua",fullName:"Boulenouar Mesraoua",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/94911/images/system/94911.png",biography:"Dr Boulenouar Mesraoua is the Associate Professor of Clinical Neurology at Weill Cornell Medical College-Qatar and a Consultant Neurologist at Hamad Medical Corporation at the Neuroscience Department; He graduated as a Medical Doctor from the University of Oran, Algeria; he then moved to Belgium, the City of Liege, for a Residency in Internal Medicine and Neurology at Liege University; after getting the Belgian Board of Neurology (with high marks), he went to the National Hospital for Nervous Diseases, Queen Square, London, United Kingdom for a fellowship in Clinical Neurophysiology, under Pr Willison ; Dr Mesraoua had also further training in Epilepsy and Continuous EEG Monitoring for two years (from 2001-2003) in the Neurophysiology department of Zurich University, Switzerland, under late Pr Hans Gregor Wieser ,an internationally known epileptologist expert. \n\nDr B. Mesraoua is the Director of the Neurology Fellowship Program at the Neurology Section and an active member of the newly created Comprehensive Epilepsy Program at Hamad General Hospital, Doha, Qatar; he is also Assistant Director of the Residency Program at the Qatar Medical School. \nDr B. Mesraoua's main interests are Epilepsy, Multiple Sclerosis, and Clinical Neurology; He is the Chairman and the Organizer of the well known Qatar Epilepsy Symposium, he is running yearly for the past 14 years and which is considered a landmark in the Gulf region; He has also started last year , together with other epileptologists from Qatar, the region and elsewhere, a yearly International Epilepsy School Course, which was attended by many neurologists from the Area.\n\nInternationally, Dr Mesraoua is an active and elected member of the Commission on Eastern Mediterranean Region (EMR ) , a regional branch of the International League Against Epilepsy (ILAE), where he represents the Middle East and North Africa(MENA ) and where he holds the position of chief of the Epilepsy Epidemiology Section; Dr Mesraoua is a member of the American Academy of Neurology, the Europeen Academy of Neurology and the American Epilepsy Society.\n\nDr Mesraoua's main objectives are to encourage frequent gathering of the epileptologists/neurologists from the MENA region and the rest of the world, promote Epilepsy Teaching in the MENA Region, and encourage multicenter studies involving neurologists and epileptologists in the MENA region, particularly epilepsy epidemiological studies. \n\nDr. Mesraoua is the recipient of two research Grants, as the Lead Principal Investigator (750.000 USD and 250.000 USD) from the Qatar National Research Fund (QNRF) and the Hamad Hospital Internal Research Grant (IRGC), on the following topics : “Continuous EEG Monitoring in the ICU “ and on “Alpha-lactoalbumin , proof of concept in the treatment of epilepsy” .Dr Mesraoua is a reviewer for the journal \"seizures\" (Europeen Epilepsy Journal ) as well as dove journals ; Dr Mesraoua is the author and co-author of many peer reviewed publications and four book chapters in the field of Epilepsy and Clinical Neurology",institutionString:"Weill Cornell Medical College in Qatar",institution:{name:"Weill Cornell Medical College in Qatar",country:{name:"Qatar"}}},{id:"282429",title:"Prof.",name:"Covanis",middleName:null,surname:"Athanasios",slug:"covanis-athanasios",fullName:"Covanis Athanasios",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/282429/images/system/282429.jpg",biography:null,institutionString:"Neurology-Neurophysiology Department of the Children Hospital Agia Sophia",institution:null},{id:"190980",title:"Prof.",name:"Marwa",middleName:null,surname:"Mahmoud Saleh",slug:"marwa-mahmoud-saleh",fullName:"Marwa Mahmoud Saleh",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/190980/images/system/190980.jpg",biography:"Professor Marwa Mahmoud Saleh is a doctor of medicine and currently works in the unit of Phoniatrics, Department of Otolaryngology, Ain Shams University in Cairo, Egypt. She got her doctoral degree in 1991 and her doctoral thesis was accomplished in the University of Iowa, United States. Her publications covered a multitude of topics as videokymography, cochlear implants, stuttering, and dysphagia. She has lectured Egyptian phonology for many years. Her recent research interest is joint attention in autism.",institutionString:"Ain Shams University",institution:{name:"Ain Shams University",country:{name:"Egypt"}}},{id:"259190",title:"Dr.",name:"Syed Ali Raza",middleName:null,surname:"Naqvi",slug:"syed-ali-raza-naqvi",fullName:"Syed Ali Raza Naqvi",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/259190/images/system/259190.png",biography:"Dr. Naqvi is a radioanalytical chemist and is working as an associate professor of analytical chemistry in the Department of Chemistry, Government College University, Faisalabad, Pakistan. Advance separation techniques, nuclear analytical techniques and radiopharmaceutical analysis are the main courses that he is teaching to graduate and post-graduate students. In the research area, he is focusing on the development of organic- and biomolecule-based radiopharmaceuticals for diagnosis and therapy of infectious and cancerous diseases. Under the supervision of Dr. Naqvi, three students have completed their Ph.D. degrees and 41 students have completed their MS degrees. He has completed three research projects and is currently working on 2 projects entitled “Radiolabeling of fluoroquinolone derivatives for the diagnosis of deep-seated bacterial infections” and “Radiolabeled minigastrin peptides for diagnosis and therapy of NETs”. He has published about 100 research articles in international reputed journals and 7 book chapters. Pakistan Institute of Nuclear Science & Technology (PINSTECH) Islamabad, Punjab Institute of Nuclear Medicine (PINM), Faisalabad and Institute of Nuclear Medicine and Radiology (INOR) Abbottabad are the main collaborating institutes.",institutionString:"Government College University",institution:{name:"Government College University, Faisalabad",country:{name:"Pakistan"}}},{id:"58390",title:"Dr.",name:"Gyula",middleName:null,surname:"Mozsik",slug:"gyula-mozsik",fullName:"Gyula Mozsik",position:null,profilePictureURL:"https://mts.intechopen.com/storage/users/58390/images/system/58390.png",biography:"Gyula Mózsik MD, Ph.D., ScD (med), is an emeritus professor of Medicine at the First Department of Medicine, Univesity of Pécs, Hungary. He was head of this department from 1993 to 2003. His specializations are medicine, gastroenterology, clinical pharmacology, clinical nutrition, and dietetics. His research fields are biochemical pharmacological examinations in the human gastrointestinal (GI) mucosa, mechanisms of retinoids, drugs, capsaicin-sensitive afferent nerves, and innovative pharmacological, pharmaceutical, and nutritional (dietary) research in humans. He has published about 360 peer-reviewed papers, 197 book chapters, 692 abstracts, 19 monographs, and has edited 37 books. He has given about 1120 regular and review lectures. He has organized thirty-eight national and international congresses and symposia. He is the founder of the International Conference on Ulcer Research (ICUR); International Union of Pharmacology, Gastrointestinal Section (IUPHAR-GI); Brain-Gut Society symposiums, and gastrointestinal cytoprotective symposiums. He received the Andre Robert Award from IUPHAR-GI in 2014. Fifteen of his students have been appointed a