Six Sigma SLPs; Probability and Randomness - Assignment Example

Essay, Management: Six Sigma SLPs Essay, Management: Six Sigma SLPs Six Sigma SLPs; Probability and Randomness For the session long project discuss in 2 pages how probability or randomness impacts your organization, such as scheduling, order shipments, inventory control, etc. Probability or random process is a non-deterministic system in which subsequent events are determined by probability (chance of occurrence). In operation research (OR), this technique are applied to manufacturing industries to analyse individual departments’ performance such as system throughput and bottleneck analysis, inventory control, production scheduling, capacity planning, etc. (Dennis et al., n.d). In an industry, many problems such as profit, cost and delay require finding their maximum or minimum. This requires in-depth study of the function variables and constraints to help set optimal working quantities of the parameters. These function variables are in most cases determined by both deterministic and random components. Scheduling involves establishing the type, amount and order of work to be done in a particular unit or department. This also involves determining the quantity, quality and rate of output of that particular work (John, Hildreth & Brian, 2005). The randomness of the raw materials supply may affect scheduling work in that the supply may delay thus affecting work that was to be done at a particular period. The probability that a certain number of machine will be functional and available for individual work may also impact scheduling when the machines are unexpectedly out of order. Also, it is not certain that all workers will be available for scheduled work. Inventory control is the art of maintaining the lowest possible material and product levels in the stock; this will ensure the production distribution continuity without understanding or overstocking at any particular moment. The inventory level depends on the demand and lead time among other variables. Lead time here refers to the time placing an order and the time when the order arrives. In a situation where demand and lead time are random the level of inventory will be the probability distribution of these variables. This may impact the organization since it is not certain that the ordered raw materials will arrive on time for the intended production. Customer demand in most situations is not constant. This randomness of demand affects the organization’s inventory. With a view of keeping a lean inventory, sometimes it is not possible when the demand is too high or too low at unexpected period. Focusing on PepsiCo Inc., a corporation dealing with food and beverage, keeping their inventories lean must be one of their primary concern since their products and raw materials may go bad in a short while (table1). However, this is of significant challenge when the demand and lead time are probability events. Table 1.PepsiCo Inc., Statement of Financial Position, Inventory USD $ in millions Dec 28, 2013 Dec 29, 2012 Dec 31, 2011 Dec 25, 2010 Dec 26, 2009 Raw materials 1,732 1,875 1,883 1,654 1,274 Work-in-process 168 173 207 128 165 Finished goods 1,509 1,533 1,737 1,590 1,179 Inventories 3,409 3,581 3,827 3,372 2,618 Source: Based on data from PepsiCo Inc. Annual Reports As can be deduced from the table1 above, raw materials, work in progress and finished goods all fell from 2011 to 2012 and declined further in 2013. Thus, the inventories of the corporation have been on the decline since 2011. This is done to keep a lean inventory while still satisfying the customer demand. Reference Dennis, E. Blumenfeld, Debra, A. Elkins & Jeffrey M. (n.d). Alden Mathematics and Operations Research in Industry. Retrieved from http://www.maa.org/mathemaics-and-operations-research-in-industry John, C. Hildreth, Brian, P. (2005). An Introduction to the Management Principles of Scheduling. Retrieved from http://www.virginiadot.org/business/resources/const/0504_ManagementPrinciplesofScheduling.pdf . May 2005. Munoz Virginia Tech The Gauge Verification Gauge verification is an important exercise that manufacturing factories must practice in order to maintain their customer specifications (Sloop, 2009). This process involves periodically checking the quality and quantity of their manufactured products. PepsiCo is a global food and beverage corporation with many brands that includes breakfast bars, coffee drinks, energy drinks, cereals, rice snacks, side dishes, snacks soft drinks and many more (US Brands Shopping List. PepsiCo, 2011). There is a variety of tests that can be done in a food and manufacturing company including the mineral contents tests. For this study, we sampled data from a test that was carried to verify the quantity of 1litre soft drink branded Diet Pepsi. In the manufacturing process the content of the 1litre Diet Pepsi product should be 1000mL. Due to random errors in the instruments it cannot be exactly the stated quantity. A gauge verification was done to improve the manufacturing process and maintain the customers’ specification. The data in the table1 below shows results in mL from three operators. The master content reading is 1000mL. The allowable quantity for maximum reading is 1010mLwhile the minimum reading allowed is 985mL Table 2 Appraisers results for the gauge verification Sample Inspector A Inspector B Inspector C range 1 1001 1000 1002 2 2 1003 1002 1003 1 3 1005 989 997 16 5 1004 991 998 13 6 994 993 989 5 average 1001.4 995 997.8 7.4 In this experiment, we are using three inspectors measuring 5 different randomly selected samples. From the d2 distribution, this corresponds to the value of 1.74. For a 99% confidence level, we use a study variation of value 5.15. Gauge error = ((5.15 * average range) / d2) Thus, gauge error = ((5.15*7.4) / 1.74) Gauge error = 21.9 The process tolerance is the difference between the specification limits (Richard, Connie and Douglas, 2005). For this experiment, the upper limit was 1010, and the lower limit was 985. Thus, the process tolerance is 25 We convert this error o percentage of tolerance. Thus, Gauge error = ((21.9/ 25)*100) %gauge error = 87.6% This gauge percentage is too big for a passing value. This shows that the difference between individual workers were too large. The large average operator variance, 7.4, indicates that there was a significant operator- to- the operator difference. Inexperienced operators or operator could have resulted this not following proper measurement procedures, or they are trained in a different field. Reference. Sloop, R. (2009). Understand gage R&R. Quality. Infinity QS, 48(9), 44-47. http://www.infinityqs.com/articles/understand-gage-rr-quality-magazine. Pepsi. April 2011. US Brands Shopping List. PepsiCo. Retrieved from http://www.pepsico.com/brands/Pepsi_Cola-Brands.html Richard K. Burdick, Connie M. Borror and Douglas C. Montgomery (2005). Design and Analysis of Gauge R and R Studies: Making Decisions with Confidence Intervals in Random and Mixed ANOVA Models. American Statistical Association and the Society for Industrial and Applied Mathematics. p. 2. Retrieved from http://books.google.co.ke/books?id=lsYAJWg6stcC&q=ptr&redir_esc=y#v=snippet&q=ptr&f=false Statistical concepts and six sigma principles Normal distribution curve is always symmetric and bell-shaped about the mean, thus, data corresponding to this curve mostly cluster in the middle of the range and the rest of the data scatter off symmetrically towards the right end and left end of the curve. The normal distribution is usually described using two parameters; mean (μ) and standard deviation (σ) where the mean is always the center of the distribution and the standard deviation measures how data varies from the mean. Mostly importantly for the distribution to be normal, the mean, median and mode must be equal. For many large organizations where we deal with a large quantity of data (samples), the output of most processes is generally distributed. Table 3 shows units of goods supplied to customers in one the major PepsiCo plants. By using this sample of data will test if is a normal distribution by finding it median, mode and mean. Table 3. Goods sold during 17 weeks of activities week 0 1 2 3 4 5 6 7 8 9 10 sales 13083 12392 15392 17701 12884 15157 15157 17431 16501 19413 19413 11 12 13 14 15 16 17 19 22325 19714 22723 33723 26131 29574 19413 19450 By performing basic statistics, we obtain the following results in table 4. Table 4 basic statistics Minimum: 12392 Maximum: 33723 Count: 19 Sum: 367577 Mean: 19350 Median: 19410 Mode: 19413 Standard Deviation: 5639 Variance: 31800000 Analysing the results shows mean, mode and median are slightly different. Considering that is a small sample compared to the total number of weeks that the plant is in operation we can conclude that when a significant frequency s used the mean, mode and median will approach the same number The six sigma principles are management tools that are applied to improve the quality of the process output by identifying and eliminating the causes of errors in the process (Keller, 2001) thus ensuring a lean manufacturing. These set of techniques obtained its name from the fact that the distribution have six standard deviations between the mean and either the upper limit or the lower limit; sigma being one standard deviation from the mean. In this case, the sigma process aims to attain 99.99966% of products which are statistically free from errors which in real life situation is hard to attain (Tennant, 2001). The six sigma project carried within a manufacturing organization is implemented in defined sequence of use and have predefined target are for application so as to increase the percentage of the defect- free products. PepsiCo Inc. in a drink and beverage manufacturing company. Six sigma can be involved in the process to produce a high proportion of output within the desired specifications. As we noticed in module2, one of the areas underperforming at Pepsi Co. in the department of packaging of one liter diet Pepsi. This provides us with the opportunity to apply six sigma. Being already an established manufacturing company the six sigma project to be used to improve the existing process is the DMAIC (Hahn et al., 1999). DMAIC has five phases that are applied systematically. First we define the problem; here the problem is that 1 liter Diet Pepsi brand is not conforming to the standard quantity and the quantity in each bottle is inconsistent. The consumers would like to have products of standard measurement. The goal of six sigma in this situation will be to ensure that the content of the bottle is of standard litres without much variance. The second phase of DMAIC is to measure the aspects of the current process obtaining data that that can be used to assess the situation. These data will help to analyze the condition and possibly identify the cause of defects that is the third step in our project. For PepsiCo defects, the major culprits were the automation device that was being used to fill the content and the personnel operating the machine. Since the machine is automated, the specific areas to check on the machine includes assessing the valve of the filler tank if it is opening and closing on time. The valve opening too late or closing too early may be a course of the underweight content. The second last step involves improving and optimizing the current process. Since most of the bottles had less content, one of the principal remedies is to increase the machine changeover time. Lastly, control measures have be laid down to ensure that the future of the process will not show any deviations from the target as corrected. Such measures include continual monitoring of the bottle content to. References Keller & Paul A. (2001). Six Sigma Deployment: A Guide for Implementing Six Sigma in Your Organization. Tucson, AZ: Quality Publishing Tennant, G. (2001). SIX SIGMA: SPC and TQM in Manufacturing and services. Gower Publishing Ltd. p. 6. Retrieved from http://books.google.co.ke/books?id=O6276jidG3IC&printsec=frontcover&hl=en#v=onepage&q&f=false Hahn, G. J., Hill, W. J., Hoerl, R. W. & Zinkgraf, S. A. (1999) The Impact of Six Sigma Improvement-A Glimpse into the Future of Statistics, The American Statistician, 53 (3), pp. 208–215. PepsiCo Inc. Hypothesis testing A statistical hypothesis test is a probability inference method that be used to predict an occurrence. One the data available for the PepsiCo Inc. is a data about a case study “coke vs. Pepsi challenge.” A study was conducted to see if the Diet coke drinkers also prefer the taste of the Diet Pepsi. This challenge was a blind test in which the drinkers did not know the identity of the flavour they were testing. This was done by using unmarked cups Pepsi and coke, and the drinkers were asked to identify which drink they prefer. Pepsi representatives reported that more than half of the Diet coke drinkers surveyed said that they preferred his test of Diet Pepsi (Woolfolk, Castellan & Brooks1983). In the survey, 100 Diet coke drinkers participated in the Pepsi challenge and 56 indicated they preferred the taste of Diet Pepsi. In this taste, we identify the null hypothesis that there was no preference between Diet coke and Diet Pepsi among the Diet coke drinkers. The alternate hypothesis is that the more than half the Diet coke drinkers prefer the Diet Pepsi over Diet coke. We can express this hypothesis mathematically as; Ho: p =0.5 Ha: p >0.5 To perform the hypothesis test, we use the binomial distribution since the distributions are for independent events and for each event there were only two possible outcomes i.e. preferring either the taste of Diet coke or Diet Pepsi. We can define preference over Diet Pepsi as a success; therefore, we have 56 successes in 100 trials. To obtain our P- value, we sum all the probabilities from zero success outcome to 56 success outcomes and compare this value to the alpha value given (Fisher, 1925). The value obtained will be coupled to a significance level of 5% i.e. we will apply a confidence level of 95% to our distribution (Lehmann, 1993). This will form the basis of the decision on whether to reject or not to reject the null hypothesis. By use of a binomial calculator, we obtain the results in the table 5 below; Table 5 results of binomial calculation Probability of success on a single trial =0.5 Number of trials = 100 Number of successes (x) =56 Binomial Probability: P(X = 56) = 0.039 Cumulative Probability: P(X < 56) = 0.864 Cumulative Probability: P(X < 56) = 0.903 Cumulative Probability: P(X > 56) =0.097 Cumulative Probability: P(X > 56)= 0.136 The main objective of performing the survey was to find if more than half the Diet coke drinkers also prefer Diet Pepsi hence our main area of concern is the right tail. At 56 success, the cumulative probability above for the number of drinkers above 56 is 0.097. This value is to the right of the 0.05 of the right tail. Hence we fail to reject the null hypothesis. Thus, we cannot relay claim that the Diet coke Drinkers prefer Diet Pepsi. References Woolfolk, Castellan, W & Brooks, CI (1983). Pepsi versus Coke: Labels, not tastes, prevail. Psychological Reports 52: 185–186. Retrieved from http://wayback.archive.org/web/20060914114727/http://www.psy.jhu.edu/~lapd/dl/coke2.pdf Fisher, R. A. (1925).Statistical Methods for Research Workers, Edinburgh: Oliver and Boyd, 1925, p.57. Retrieved from http://psychclassics.yorku.ca/Fisher/Methods/chap5.htm Lehmann, E. L. (1993). "The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?” Journal of the American Statistical Association 88 (424): 1242–1249 Read More