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1. Economical Statistical Design of Combined Double Sampling and Variable Sampling Interval Joint X-bar and S Control Charts

   Student: Yi-Chun Chen Advisor: Chau-Chen Torng
2. 1 Introduction 2 Literature Review Design of control charts 3 4 Expect result Contents
3. 1 2 Background & Motivation Purposes 1.Introduction
4. Product Customers Manufacturers Background & Motivation(1/5) New Before Quantity Supply < Demand Quantity Supply>Demand Quality vs Cost Quality vs Cost Source: Business Weekly , vol. 974
5. Background & Motivation(2/5) Statistical Process Control (SPC) Control charts Shewhart (1924) development X-bar control chart Out of control μ+3σ μ μ-3σ Out of control Source: Montgomery, 2006
6. S X-bar & R X-bar & S Control chart Control chart Background & Motivation(3/5) Traditional Control chart Advantage Easy to operate, Detection large shift is sensitive Disadvantage Detection small & medium shift is not sensitive adaptive Control chart Control chart
7. Double Sampling Variable Sampling Size & Variable Sampling Intervals Variable Sampling Size Control chart Double Sampling & Variable Sampling Interval Variable Sampling Intervals Variable Parameters Background & Motivation(4/5) Costa (1997)development VSSI control chart Daudin(1992)development DS control chart Adaptive control chart Carot (2002)development DSVSI control chart better than single DS and single VSI control chart Costa(1994) development VSS control chart Reyonlds (1988)development VSI control chart Costa (1999)development VP control chart
8. Background & Motivation(5/5) On the other hand, cost is very important for Manufacturers.Duncan(1956)construct economic design of control chart, that model purpose is minimize the average unit cost. The other some researcher find that single economic design of control chart model would made higher false alarm and lower power. Saniga(1989) economical statistical design of control chart.
9. Purposes Seek appropriate economic model and statistical model used on design of control charts. By the analysis tool to find the best parametersof Economical statistical design of DSVSI X-bar & S control chart. By sensitivity analysis to find the relationship between cost parameters and statistical performance.
10. 1 2 3 Joint X-bar & S control charts Adaptive control chart Design of control chart model 2.Literature Review
11. 2.1 Joint X-bar & S control charts X-bar control chart S control chart Why do we want combination of X-bar and S control chart? Out of control σ0 σ0 μ+3σ σ1 μ μ1 LCL LCL LCL UCL UCL UCL μ0 μ0 μ0 (a) (b) (c) μ-3σ Out of control Source: Montgomery,2008
12. 2.2 Adaptive control chart Double Sampling(DS) control chart. Variable Sampling Interval(VSI) control chart. Double Sampling and Variable Sampling Interval(DSVSI) control chart.
13. DS X-bar control chart Dudin(1992) development DS X-bar control chart. Change sampling size Second stage DS X-bar control chart First stage DS X-bar control chart
14. VSI X-bar control chart Reyonlds et al. (1988) development VSI X-bar control chart. Change sampling interval VSI X-bar control chart
15. DSVSI X-bar & S control chart(1/2) Lee(2012)development DSVSI X-bar&S control chart. Design background 2002 2006 2010 2012 Carot et al. Combing DS X-bar and VSI X-bar DSVSI X-bar He et al. Combing DS X-bar and DS S DS X-bar&S Torng et al. Showed DSVSI X-bar batter than CUSUM non-normal Lee et al. Joint S and DSVSI DSVSI S
16. DSVSI X-bar & S control chart(2/2) n h h n First stage X-bar chart Second stage X-bar chart n h First stage S chart Second stage S chart
17. 2.3 Design of control chart model Economic design of control chart Statistical design of control chart
18. Ν Ν τ gn D h/(1-β) 1/λ In control Out of control Economic design of control chart Duncan(1956) Last sampling before Assignable cause First sampling after Assignable cause Assignable cause located Assignable cause removed Lack of control detected Cycle start Assignable cause
19. Statistical design of control chart Woodall(1986) find economic design of control chart will have higher false alarm and lower power. Saniga(1989)development economical statistical design of control chart.
20. 1 2 3 Symbol definition & description Model assumptions Modeling 3.Design of control charts
21. Symbol definition & description(1/2)
22. Symbol definition & description(2/2)
23. 3.2 Model assumptions The initial state of a process is in control and the mean and standard deviation are μ= μ0and σ= σ0. When an assignable cause occurs, μ1= μ0+δμ0or σ1= γσ0. Quality characteristics X is subject to the normal distribution. The process just have only one assignable cause. The relationship between assignable cause and process variation time is exponential distribution, the expected value of 1/λ. Process variation is instantaneous. The process does not have the ability to repair itself.
24. 3.3 Modeling Cycle Time Computation Cost Model Development Statistical Constraints Formulation
25. Cycle Time Computation(1/2) E(nІδ,γ) =n1+n2×P(change sampling sizeІδ,γ) P(change sampling sizeІδ,γ) A Second Warring Region
26. Cycle Time Computation(2/2) 1 2 0 j j+1 h1 j j+1 h2
27. Ν Ν ζ GE(n) D AATS 1/λ In control Out of control Cycle Time Computation Last sampling before Assignable cause First sampling after Assignable cause Assignable cause located Assignable cause removed Lack of control detected Cycle start Assignable cause E(T)=1/λ+AATS+GE(n)+D GE(n) D AATS 1/λ In control
28. 1 2 3 4 Cost Model Development defective products cost  C0 ×1/λ+C1 ×AATS False alarm times α ×s False alarm costs a3’× α ×s Search assignable cause costs a3 sampling and inspection  E(C)
29. Statistical Constraints Formulation Min s.t. X-bar control chart S control chart
30. 4.Expect result The economical statistical design of Combined DSVSI Joint X-bar and S Control Charts optimal combination of parameters obtained by solving tool. Use sensitivity analysis to find relationship between cost parameters and cost, relationship between cost and statistical performance. Lorenzen(1986) ±30%
31. Thank You !