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Statistical Analysis of Real Manufacturing Process Data Statistické Zpracování Dat Z Reálného Výrobního Procesu

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Statistical Analysis of Real Manufacturing Process Data Statistické Zpracování Dat Z Reálného Výrobního Procesu View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Digital library of Brno University of Technology BRNO UNIVERSITY OF TECHNOLOGY VYSOKÉ U ČENÍ TECHNICKÉ V BRN Ě FACULTY OF MECHANICAL ENGINEERING DEPARTMENT OF MATHEMATICS FAKULTA STROJNÍHO INŽENÝRSTVÍ ÚSTAV MATEMATIKY STATISTICAL ANALYSIS OF REAL MANUFACTURING PROCESS DATA STATISTICKÉ ZPRACOVÁNÍ DAT Z REÁLNÉHO VÝROBNÍHO PROCESU MASTER’S THESIS DIPLOMOVÁ PRÁCE AUTHOR Bc. BARBORA KU ČEROVÁ AUTOR PRÁCE SUPERVISOR Ing. JOSEF BEDNÁ Ř, Ph.D VEDOUCÍ PRÁCE BRNO 2012 Vysoké učení technické v Brně, Fakulta strojního inženýrství Ústav matematiky Akademický rok: 2011/2012 ZADÁNÍ DIPLOMOVÉ PRÁCE student(ka): Bc. Barbora Kučerová který/která studuje v magisterském navazujícím studijním programu obor: Matematické inženýrství (3901T021) Ředitel ústavu Vám v souladu se zákonem č.111/1998 o vysokých školách a se Studijním a zkušebním řádem VUT v Brně určuje následující téma diplomové práce: Statistické zpracování dat z reálného výrobního procesu v anglickém jazyce: Statistical analysis of real manufacturing process data Stručná charakteristika problematiky úkolu: Statistické zpracování dat z konkrétního výrobního procesu, předpokládá se využití testování hypotéz, metody ANOVA a GLM, analýzy způsobilosti procesu. Cíle diplomové práce: 1. Popis metodologie určování způsobilosti procesu. 2. Stručná formulace konkrétního problému z technické praxe (určení dominantních faktorů ovlivňujících způsobilost procesu). 3. Popis statistických nástrojů vhodných k řešení problému. 4. Řešení popsaného problému s využitím statistických nástrojů. Seznam odborné literatury: 1. Meloun, M., Militký, J.: Kompendium statistického zpracování dat. Academica, Praha, 2002. 2. Kupka, K.: Statistické řízení jakosti. TriloByte, 1998. 3. Montgomery, D.,C.: Introduction to Statistical Quality Control. Chapman Vedoucí diplomové práce: Ing. Josef Bednář, Ph.D. Termín odevzdání diplomové práce je stanoven časovým plánem akademického roku 2011/2012. V Brně, dne 24.11.2011 L.S. _______________________________ _______________________________ prof. RNDr. Josef Šlapal, CSc. prof. RNDr. Miroslav Doupovec, CSc., dr. h. c. Ředitel ústavu Děkan fakulty ABSTRACT This diploma thesis deals with statistical process control. Its goal is to analyze data from real manufacturing process of the revolver molding machine. The analysis is accomplished using the statistical hypothesis testing, the analysis of variance, the general linear model and the analysis of the process capability. The analysis of the data is done in statistical software Minitab 16. KEYWORDS Analysis of variance, capability of the process, statistical hypothesis testing ABSTRAKT Tématem této diplomové práce je statistická regulace výrobního procesu. Cílem bylo analyzovat data z reálného technologického procesu revolverového vstřikovacího lisu. Analýza byla provedena za užití statistického testování hypotéz, analýzy rozptylu, obec- ného lineárního modelu a analýzy způsobilosti procesu. Analýza dat byla provedena ve statistickém softwaru Minitab 16. KLÍČOVÁ SLOVA Analýza rozptylu, způsobilost procesu, statistické testování hypotéz KUČEROVÁ, Barbora Statistical analysis of real manufacturing process data: master’s thesis. Brno: Brno University of Technology, Faculty of Mechanical Engineering, Depart- ment of Mathematics, 2012. 49 p. Supervised by Ing. Josef Bednář, Ph.D. I hereby declare that I have elaborated my master’s thesis on my own under a guidance of Ing. Josef Bednář, Ph.D., and using the sources listed in the bibliography. Barbora Kučerová I would like to express a word of gratitude to my supervisor Ing. Josef Bednář PhD. for all his contributive suggestions and valuable comments which led to the completion of my thesis. Barbora Kučerová CONTENTS Introduction 7 1 Basic definitions 8 1.1 Probability space, random variable . .8 1.2 Random sample . 11 1.3 Important types of continuous probability distribution . 11 1.4 Statistical hypothesis testing . 13 1.5 t-test . 15 1.6 F-test . 15 2 Analysis of variance 16 2.1 Two-way ANOVA with interactions . 16 2.2 General linear model . 19 2.3 General linear model with fixed factor effects . 20 3 Capability of the process 23 3.1 The process capability indices . 23 4 Technical data processing 27 4.1 Analysis of variance . 27 4.2 Capability of the process . 36 Conclusion 47 Bibliography 48 List of symbols and abbreviations 49 INTRODUCTION Manufacturing process control and quality assurance are elements of a quality ma- nagement system, which is the set of policies, procedures, and processes used to ensure the quality of a product or a service. It is widely acknowledged that qua- lity management systems improve the quality of the products and services produ- ced, thereby improving market share, sales growth, sales margins, and competitive advantage, and helping to avoid litigation. Quality control methods in industrial production were first developed by statistician W. Edwards Deming. The International Organization for Standardization has outlined quality prin- ciples and the procedures for implementing a quality management system in ISO 9000:2000, ISO 9001:2000 and other documents. These documents have become the gold standards of best practices for ensuring quality and, in many fields, serve as the basis for regulation. An important element of quality assurance is the collection and analysis of data that measure the quality of the raw materials, components, products, and assembly processes. Statistical Process Control (SPC) is an effective method of monitoring a pro- duction process through the use of control charts. By collecting in-process data or random samples of the output at various stages of the production process, one can detect variations or trends in the quality of the materials or processes that may affect the quality of the end product. Because data are gathered during the production process, problems can be detected and prevented much earlier than methods that only look at the quality of the end product. Early detection of problems through SPC can reduce wasted time and resources and may detect defects that other me- thods would not. Additionally, production processes can be streamlined through the identification of bottlenecks, wait times, and other sources of delay by use ofSPC. In this diploma thesis we will process the data from concrete technological process to determine whether our data are influenced by some of the observed factors. The mathematical theory used to analyze the process is described in first chapters. For the data processing is used statistical hypothesis testing. Big part of the theory is focused on the analysis of variance, the general linear model and the capability of the process, as the most important parts of the statistical process control analysis, [1] and [2]. The practical part of the diploma thesis is application of the theory, which is introduced in the previous chapters. The analysis is done in statistical software Minitab 16. In Minitab we will be using the Quality Tools and the ANOVA. 8 1 BASIC DEFINITIONS 1.1 Probability space, random variable The nonempty set Ω of all the possible outcomes ! of an experiment is called the sample space of the experiment and its subsets are denoted as events. A 휎-algebra is a collection U of subsets of Ω with these properties: (i) ;, Ω 2 U. (ii) If A 2 U, then AC 2 U. (iii) If A1, A2, ::: 2 U, then 1 1 \ [ Ak; Ak 2 U: k=1 k=1 Where AC = Ω − A is the complement of A. Let U be a 휎-algebra of subset of Ω. We call P : U! [0; 1] a probability measure provided: (i) P (;) = 0;P (Ω) = 1. (ii) If A1, A2, ::: 2 U, then 1 ! 1 [ X P Ak ≤ P (Ak) : k=1 k=1 (iii) If A1, A2, ::: are disjoint sets in U, then 1 ! 1 [ X P Ak = P (Ak) : k=1 k=1 It holds that if A,B 2 U, then A ⊆ B implies P (A) ≤ P (B). A triple (Ω; U;P ) is called a probability space provided Ω is any set, U is a 휎 algebra of subsets of Ω, and P is a probability measure on U. A set A 2 U is called an event; points ! 2 Ω are sample points. P (A) is a probability of the event A. A property which is true except for an event of probability zero is said to hold almost surely (usually abbreviated “a.s.“). Let (Ω; U;P ) be a probability space. A mapping n X :Ω ! R is called an n-dimensional random variable if for each B 2 B, where B denotes the collection of Borel subsets of Rn, which is the smallest 휎-algebra of subsets of Rn containing all open sets, we have X−1(B) 2 U: 9 We equivalently say that X is U-measurable. Distribution function of random variable X is real function F (x) = P fX < xg = P fX 2 (−∞; x)g ; 8x 2 (−∞; 1) with following properties: (i) F is non-decreasing, (ii) F is right-continuous, (iii) limx!1 F (x) = 1, limx→−∞ F (x) = 0, (iv) P (a < X ≤ b) = F (b) − F (a); a < b, (v) P (X = x0) = F (x0) − lim − F (x), x!x0 (vi) F has finite number of discontinuities on (−∞; 1), (vii) 8x 2 (R) : 0 ≤ F (x) ≤ 1. Random variable X is uniquely determined with its distribution function and its probability distribution is given. Random variable X defined on probability space (Ω; A;P ) is discrete, if there P exists countable set M ⊂ R such that x2M P (X = x) = 1. Set M is then called the range of discrete random variable X and function p(x) = P (X = x), x 2 M and p(x) = 0, x 2 R − M is called probability function and we denote X ∼ (M; p). Probability function has following properties: P (i) p(x) ≥ 0; x 2 R, x2M p(x) = 1, P (ii) P (X 2 B) = x2B\M p(x);B 2 B, P (iii) F (x) = t2(−∞;x)\M p(t). Random variable X is continuous, if there exists non-negative function f such R 1 R x that −∞ f(x) dx = 1 and it holds for distribution function F (x) = −∞ f(t) dt.
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