Design and Optimize Sample Plans at The Dow Chemical Company

Transcription

1 Design and Optimize Sample Plans at The Dow Chemical Company Swee-Teng Chin and Leo Chiang Analytical Technology Center The Dow Chemical Company September 12, 2011

2 Introduction The Dow Chemical Company In 2010 Dow had annual sales of $53.7 billion and employed approximately 50,000 people world wide. The Company s more than 5,000 products are manufactured at 188 sites in 35 countries across the globe. 9/12/2011 S. Chin and L. Chiang Pg. 2

3 The Manufacturing Environment Most of the sample are made in pounds not parts Quality release in the manufacturing is based on analytical inspection of grab samples against specifications and mostly one sample per lot This presentation if focus on large sample size that is applied under special circumstances 2 topics: 1. Optimize Sampling Plan to reduce number of samples 2. Design a new sample plan to understand the physical properties of catalyst 9/12/2011 S. Chin and L. Chiang Pg. 3

4 1. Optimize a New Sample Plan Background The catalyst was made by the vendor and Dow performed QC testing on them before loading it up to the plant The current sampling plan use more than half of the existing resources Based on historical data, the catalyst have been consistently well performed 9/12/2011 S. Chin and L. Chiang Pg. 4

5 1. Optimize a New Sample Plan Objective To eliminate any unnecessary catalyst QC testing for cost and labor reduction 9/12/2011 S. Chin and L. Chiang Pg. 5

6 1. Optimize a New Sample Plan Current Sampling Plan 1 million lb of catalyst in a plant unit kg (2205 lb) 1000 kg (2205 lb) 1000 kg (2205 lb) 1000 kg (2205 lb) 1000 kg (2205 lb) 1000 kg (2205 lb) Bag 1 Bag 2 Bag 18 Bag 1 Bag 2 Bag 18 1 lot Vendor QC lab Dow QC lab 1 lot Vendor QC lab Dow QC lab 1 liter 1 liter 100 mls 100 mls 9/12/2011 S. Chin and L. Chiang Pg. 6

7 1. Optimize a New Sample Plan Analysis and Results The current sampling plan is based on composite sample and no data available for each sampling stage. Use acceptance sampling plan approach to find the optimum sample size with respect to the risk involved The uncertainty of testing the catalyst is modeled using a binomial probability distribution. The approach was found in the Remund et al. 1 where it is applied on seed purity analysis 1 Remund et al. (2001), Statistical considerations in seed purity testing for transgenic traits, Seed Science Research. 9/12/2011 S. Chin and L. Chiang Pg. 7

8 1. Optimize a New Sample Plan The Binomial model 1 Let p = the proportion of defected bag of catalyst n = number of bags, N = number of segments or groups, So total number of bags = N*n Prob(no defected bag in a group of n bags) = (1 - p) n Prob(at least one defected bag in a given bag group) = P = 1 - (1 - p) n c = the maximum number of the N groups that can test defect and result in acceptance of all the groups in a reactor Probability of acceptance lot c N j N j Prob( X c P) = P ( 1 P) j= 0 j Probability of rejection Prob( X > c P) = 1 Prob( X c P) 1 Remund et al. (2001), Statistical considerations in seed purity testing for transgenic traits, Seed Science Research. 9/12/2011 S. Chin and L. Chiang Pg. 8

9 1. Optimize a New Sample Plan Assumptions The sample is random sample The analysis is solely based on the information of the sampled product and no information of the process is used (process is unchanged) The measurement data can be approximated by a normal distribution 9/12/2011 S. Chin and L. Chiang Pg. 9

10 Operating Characteristic (OC) Curve 1 1. Optimize a New Sample Plan AQL (Acceptable Quality Level) = the lowest level of defected bag in a 1 million lb catalyst that the current production practices can support LQL (Lower Quality Limit) = the lowest level of defected bag in a 1 million lb catalyst that is considered acceptable to consumer Producer s risk (or Type I error): probability of rejecting a good lot (1 million lb catalyst) or 100 Prob(AQL) Consumer s risk (or Type II error): probability of accepting a bad lot (1 million lb catalyst) or Prob(LQL) c = the maximum number of the N groups that can test defect and result in acceptance of all the groups in a reactor 1 Remund et al. (2001), Statistical considerations in seed purity testing for transgenic traits, Seed Science Research. 9/12/2011 S. Chin and L. Chiang Pg. 10

11 1. Optimize a New Sample Plan Calculating the producer and consumer risks for current sampling plan Hypothesis Testing H : µ = 590 H : µ a > Normal( , ) Producer Risk = Type I error, α = P(Ho is rejected when it is true) α = P = 1 P = ( X 591 when X ~ N( , 3.14 ) 2 ( X < 591 when X ~ N( , 3.14 ) = 1 P Z < 3.14 = or % Consumer Risk = Type II error, β = P(Not rejecting Ho when Ho is false) β = P 2 ( X < 591 when X ~ N( 597, 3.14 ) = P Z < 3.14 = or 2.801% 9/12/2011 S. Chin and L. Chiang Pg. 11

12 1. Optimize a New Sample Plan Possible sample sizes for the QC test c 1 AQL 0.08 LQL 1.23 N n Producer Consumer (number of segments) (number of bags) Risk (%) Risk (%) /12/2011 S. Chin and L. Chiang Pg. 12

13 1. Optimize a New Sample Plan Summary Results In the possible ranges of sample sizes, the producer and consumer risks do not seem to have a big change in values for both tests Optimized sample plan has 60% reduction in sample size From 25 samples to 9 samples! 9/12/2011 S. Chin and L. Chiang Pg. 13

14 2. Design a New Sample Plan Background Dow buys a catalyst from a vendor. Catalyst is shipped in drums. The current sampling plan is not sufficient to distinguish between good and bad catalyst based on the specification Sometimes batches pass the spec but they performed poorly in the plant. This costs $$! The current plan calls for pulling a small sample which is a composite across multiple drums 9/12/2011 S. Chin and L. Chiang Pg. 14

15 2. Design a New Sample Plan Objective Develop a sampling plan for the fresh catalyst that we received from the vendor to understand the underlying distribution of its physical properties. 9/12/2011 S. Chin and L. Chiang Pg. 15

16 2. Design a New Sample Plan Analysis The quality test on the catalyst are usually done by the vendor Need to determine an appropriate sample size: Use information from the ASTM method to estimate the measurement variation. Sample size calculation is based Power analysis on one sample mean Standard deviation Difference to Detect Sample Size 5 10 Type I error (probability of rejecting a good lot), α = 0.05 Type II error (probability of accepting a bad lot), β = 0.2 9/12/2011 S. Chin and L. Chiang Pg. 16

17 2. Design a New Sample Plan Analysis But the estimated sample size is for one location in a drum rather than the whole lot This is to ensure that we fully understand the variance structure of the product. There might be a significant drum to drum variation or within drum variation that we might not be aware Total number of pellets per lot 1 location sampling Number of drums 91 Number of pellets per drum 10 Total pellets location sampling Number of drums 5 Number of pellets per location within a drum 10 Number of pellets per drum 30 Total pellets 150 Grand total pellets location (top, middle and bottom of a drum) samples to estimate within drum variation. 9/12/2011 S. Chin and L. Chiang Pg. 17

18 2. Design a New Sample Plan Results Results on the data analysis shows that 1. There is no significant drum to drum variation and within drum variation. 2. New specification was defined based on the results 9/12/2011 S. Chin and L. Chiang Pg. 18

19 2. Design a New Sample Plan Specification analysis Quantiles Moments 100.0% maximum 46.8Mean % Std Dev % Std Err Mean % 24.3Upper 95% Mean % quartile Lower 95% Mean % median N % quartile % % % % minimum 5.2 Old specification is based on a lower specification on average value of 15 on one of the critical physical test Data is skewed to the left, hence a spec based on average is not sufficient especially it is based on small composite sampling plan New specification is based on Median not mean Some small fraction of lower right end of the same physical property 9/12/2011 S. Chin and L. Chiang Pg. 19

20 2. Design a New Sample Plan Results New batch is made and has been tested by the vendor as good batch based on the proposed sampling plan Dow need to validate this batch using a reduce sampling plan with the same level of protection but with less number of sample Variable sampling plan Double sampling plan 9/12/2011 S. Chin and L. Chiang Pg. 20

21 2. Design a New Sample Plan Variables sampling plan 2 Requires that the measurement to be normally distributed Assuming the standard deviation is unknown START Take a representative sample of n units, measure each unit and calculate average (X) and standard deviation (S) If USL and X USL k S No REJECT 2 Wayne Taylor, Successful Acceptance Sampling, Taylor Enterprise, Inc., www. variation.com If LSL and X LSL + k S Yes ACCEPT 9/12/2011 S. Chin and L. Chiang Pg. 21 No REJECT

22 2. Design a New Sample Plan Double sampling plan Characterized by five parameters: n1 = first sample size a1 = first accept number r1 = first reject number n2 = second sample size a2 = second accept number 9/12/2011 S. Chin and L. Chiang Pg. 22

23 2. Design a New Sample Plan START Double sampling plan 2 Take a representative sample of n1 units and count number of defectives Count a1 ACCEPT Compare count Count r1 REJECT Take second representative sample of n2 units and count the total number of defectives in both samples Continue To Second Stage Count a2 Compare count Count > a2 2 Wayne Taylor, Successful Acceptance Sampling, Taylor Enterprise, Inc. www. variation.com ACCEPT REJECT 9/12/2011 S. Chin and L. Chiang Pg. 23

24 2. Design a New Sample Plan Comparison between the Sampling Plans Both of the variable and double sampling plan requires less sample to test but the execution procedure is more completed. 9/12/2011 S. Chin and L. Chiang Pg. 24

25 2. Design a New Sample Plan Conclusion The original spec based on the sample plan from the vendor did not work Performance didn t always match the predicted performance based on quality analysis. Vendor will be using a new sample plan and Dow will be using a new reduced sample plan to validate the results. 9/12/2011 S. Chin and L. Chiang Pg. 25

26 General Conclusions At The Dow Chemical Company, we use statistical methods for designing and optimizing sampling plans. Need to take into account the risk and costs. 9/12/2011 S. Chin and L. Chiang Pg. 26