Parameter Estimation of Type-I and Type-II Hybrid Censored Data from the Log-Logistic Distribution

In experiments on product lifetime and reliability testing, there are many practical situations in which researchers terminate the experiment and report the results before all items of the experiment fail because of time or cost consideration. The most common and popular censoring schemes are type-I and type-II censoring. In type-I censoring scheme, the termination time is pre-fixed, but the number of observed failures is a random variable. However, if the mean lifetime of experimental units is somewhat larger than the pre-fixed termination time, then far fewer failures would be observed and this is a significant disadvantage on the efficiency of inferential procedures. In type-II censoring scheme, however, the number of observed failures is pre-fixed, but the experiment time is a random variable. In this case, at least pre-specified number of failure are obtained, but the termination time is clearly a disadvantage from the experimenter’s point of view. To overcome some of the drawbacks in those schemes, the hybrid censoring scheme, which is a mixture of the conventional type-I and type-II censoring schemes, has received much attention in recent years. In this paper, we consider the analysis of type-I and type-II hybrid censored data where the lifetimes of items follow two-parameter log-logistic distribution. We present the maximum likelihood estimators of unknown parameters and asymptotic confidence intervals, and a simulation study is conducted to evaluate the proposed methods

EDITORIAL: “Six Sigma - Its Application, Practice, and Utility”

Editorial for Special Issue on Six Sigm

Systematic Routine for Setting Confidence Levels for Mean Time to Failure (MTTF)

There are circumstances where an item is intentionally tested to destruction.  The purpose of this technique is to determine the failure rate (λ) of a tested item.  For these items, the quality attribute is defined as how long the item will last until failure.  Once the failure rate is determined from the number of survivors and total time of all items tested the mean time to failure (MTTF) which is a typical statistic for survival data analysis issues.  MTTF is calculated by dividing one by failure rate (λ).  From this one obtains the reliability function R(t) = e-λt where t is time.  This allows the cumulative density function F(t) = 1- e-λt  to be determined.  This density function, f(t) = λe-λt is a negative exponential with a standard deviation (σ) = 1/λ.  Thus setting a warranty policy for the tested item is difficult for the practitioner.  An important property of the exponential distribution is that it is memory less.  This means its conditional probability follows P(T > s + t |T > s)=P(T > t) for all s, t ≥0.  The exponential distribution can be used to describe the interval lengths between any two consecutive arrival times in a homogeneous Poisson process.  The purpose of this research paper is to present a simple technique to determine a realistic confidence level. Using the same technique the warranty level for the tested item can be predicted

Professional Perceptions of Six Sigma’s Value

At the 2008 American Society of Quality (ASQ) World Conference on Quality and Improvement in Houston, Texas, generational diversity and knowledge transfer were two areas of focus.  Quality methodology and tools have not changed much over the years. However, these practices and techniques have been repackaged and sold to upper management as the latest and greatest cure for what ails the company.  With each acceptance of a repackaged methodology aging professionals are replaced by advocates of the latest methods that promise greater productivity.  Lean Six Sigma and Six Sigma are the current corporate management approaches that have been marketed widely.  The selling point for such initiatives is the promise that a large return on investment (ROI) can be realized.  A survey was developed to compare the cost of implementing a program such as Lean Manufacturing, Six Sigma or Lean Six Sigma for various size organizations. The survey, which was divided into an initial and a follow-on survey, was administered to professionals who are employed in some field where implementations have been perceived as successful to one extent or another.  The survey’s focus was to explore whether there are situations where the promise is not kept and/or where there are situations where the impact of the cultural change does not justify the effort. Based upon the occupations of the volunteer subjects, there is a very high possibility of a positive bias in the results.  While survey results were generally quiet supportive of the methodologies and would lead to a recommendation for the adoption of one of the programs, it is necessary for there to be further study

Systematic Pedagogy to Line Balancing with EXCEL

Over the past ten years, simple and inexpensive operations research software that is user friendly to the mentor, student, and instructor has become difficult to obtain.  This is especially true since Emmons, Flowers, Khot, and Mathur’s STORM 4.0 for Windows is obsolete for current 32 and 64 bit operating systems and no longer in print.  After a diligent product and literature search, it appears there is no adequate inexpensive software that is easily available.  Assembly line balancing algorithms are heuristic methods used for balancing operations or production lines.  However, most methods employ complex calculations that are challenging to the mentor and mentee.  This paper presents a pedagogy from a systems approach using Microsoft EXCEL.  The object is to prepare a spreadsheet file with four separate worksheets that are linked to the first worksheet.  The step-by-step systematic approach allows the entry on the main worksheet of data such as an annual demand, annual time available, and process times. When the user changes these data entry points, the efficiencies of each operating or production line are automatically re-computed for all three shifts.  Worksheets use one of the several available heuristics to compute cycle times (required time between process activities) and transfers it to one, two, or three shifts (worksheets two, three, or four).  Once the spreadsheet and accompanying worksheets were completed, the results were compared to several different heuristic algorithms.  When authors were satisfied that the results were accurate and not significantly different from other examined algorithms, the final step was to develop a working pedagogy to efficiently describe the process.  This allows the user an efficient analytical tool to illustrate and explain interactions within a given process.  A local manufacturing facility used this method as a part of a monthly effort to increase line efficiency for individual workstations. The project’s results were satisfactorily tested in a production operations class.  The major advantage to the practitioner, engineer, instructor, and student is that EXCEL is readily available on all personal computers, easily understood, and is very practical. Students with very little exposure to line balancing were able to master the method within the first hour of exposure

Service Quality in a Reduced payroll Environment: Applying Queuing Analysis to Customer Perception Case Study

This study was conducted in a national retail pharmacy company’s stores inWestern North Carolinato examine the impact of the reduction of store staffing, primarily pharmacists and service staff, on customers’ satisfaction with service time.    Customer arrival rates and service times for each queue were conducted to determine optimal staffing.  A random customer survey in multiple store locations provided customers’ perceptions of service quality. Analysis determined that over 30% of the customers surveyed were dissatisfied with service time. A regression analysis demonstrated a significant linear relationship (σ = 0.05) between total service time and customer satisfaction. Study results indicate that cutting staff could result in an unacceptable loss of a competitive advantage.  Payroll cost savings of less than $70,000 per year could result in lost revenue dollars in excess of$1,700,000 per year.  Thus reducing staff hours (decreasing payroll) in the short term may negatively impact long-term effectiveness and productivity

The Impact of a Material’s Inherent and Process Stress on Meeting Specification and Tolerances: A Six Sigma Case Study

This Six-Sigma Case Study was conducted in a local aerospace company that produces high quality precision-machined jet engine components.  These complex turbine components have thin walls that must meet tight tolerances. Disks, shafts, rotating seals, plates, and cases range in size from 3" to 80" in diameter.  This case focused on a 16” (diameter) rear cooling plate whose production required 18 machining processes.  The objective was to determine if it was possible to eliminate the final manual lathing process. Manual lathing was used as the last step because the material characteristics of the plate and the stress induced by the previous processes caused the final product to expand. Stress can cause unsatisfactory changes in the plate’s dimensions. Stress is not only inherent in the material’s internal properties but is also induced during machining.    It is critical that the operator’s cut is precise and does not remove too much material.  During the two most critical steps of 18, measurements were taken.  It was theorized that relaxing the first process tolerances could allow later processes to be numerically machine controlled to conform closer to the prescribed tolerance of the final product.  Plates were tested using these revised tolerances.  After the plate was shot peened (a stress redistribution process) measurements confirmed that non-conformance had been eliminated and the final machining process could be discontinued.  Cost savings for eliminating the last machining and inspection process was $268 per plate or an annual saving of approximately 11% of total cost for the item studied

A Methodology for Appropriate Testing When Data is Heterogeneous Using EXCEL

A Methodology for Appropriate Testing When Data is Heterogeneous was originally published and copy written in the mid-1990s in Turbo Pascal and a 16-bit operating system.  While working on an ergonomic dissertation (Yearout, 1987), the author determined that the perceptual lighting preference data was heterogeneous and not normal.  Drs. Milliken and Johnson, the authors of Analysis of Messy Data Volume I: Designed Experiments (1989), advised that Satterthwaite’s Approximation with Bonferroni’s Adjustment to correct for pairwise error be used to analyze the heterogeneous data. This technique of applying linear combinations with adjusted degrees of freedom allowed the use of t-Table criteria to make group comparisons without using standard nonparametric techniques.  Thus data with unequal variances and unequal sample sizes could be analyzed without losing valuable information.  Variances to the 4th power were so large that they could not be reentered into basic calculators.  The solution was to develop an original software package which was written in Turbo Pascal on a 7 ¼ inch disk 16-bit operating system.  Current operating systems of 32 and 64 bits and more efficient programming languages have made the software obsolete and unusable. Using the old system could result either in many returns being incorrect or the system terminating.  The purpose of this research was to develop a spreadsheet algorithm with multiple interactive EXCEL worksheets that will efficiently apply Satterthwaite’s Approximation with Bonferroni’s Adjustment to solve the messy data problem.  To ensure that the pedagogy is accurate, the resulting package was successfully tested in the classroom with academically diverse students.  A comparison between this technique and EXCEL’s Add-Ins Analysis ToolPak for a t-test Two-Sample Assuming Unequal Variances was conducted using several different data sets.  The results of this comparison were that the EXCEL Add-Ins returned incorrect significant differences.  Engineers, ergonomists, psychologists, and social scientists will find the developed program very useful. A major benefit is that spreadsheets will continue to be current regardless of evolving operating systems’ status

Acceptable Noise Levels for Typical Outdoor Leisure Activities

Throughout the Appalachian Mountain chain, entrepreneurs have capitalized on the outdoor sporting experience.  Both summer and winter sports such as fly fishing, hiking, skiing and snowboarding are very popular. One popular emerging activity is skeet, trap, and sporting clays shooting.  Major tourist enterprises are developing this opportunity for both tourists and the local residents.  To meet the increasing demand, ranges are being constructed throughout the region from Alabama to Maine.  Within North Carolina that are approximately 12 ranges, with an additional 10 in east Tennessee. Although these activities have had few opponents, sporting clays establishments have received an insignificant number of complaints that the noise level is above the noxious level.  A research experiment was designed to test noise levels at one sporting clays course situated near a vacation resort. The first phase of this experiment determined that the noise levels at the source from 12 and 20 gauge shotguns were insignificant. The allowable rounds used are the standard reduced load target load. At the source there was a significant difference between trap (85.85 dBA) and skeet (74.95 dBA). At the closest guest location (second phase) 333 samples were taken between August 2015 and March 2016.  The noise level averaged 53.7 dBA (σ = 7.43dBA). During the third phase (January to March 2016) 44 samples were obtained at the main resort lodging site.  At Site 2 the noise level averaged 31.9 dBA (σ = 5.45).  The fourth and final phase of this study was to determine the noise levels at the closest guest location for each of the 12 sporting clays stations.  Ninety-six samples (8 per station) were collected in October 2016. The average noise level was 53.0 dBA (σ 4.19 dBA). An analysis at the statistical confidence level of 97.5% was conducted.  First all stations were analyzed in a pairwise comparison analysis.  Stations that were not significantly different from one another were combined.  Thus the 12 stations were reduced to three combined stations.  The analysis determined that the noise level from stations 1 through 5 (48.9dBA, σ =1.50 dBA) was significantly lower than from the other stations.  Next loudest was the combinations of stations 6, 7, 10, and 11 (54.1 dBA, σ = 1.69). The last loudest combination was stations 8, 9, and 12 (58.6 dBA, σ = 1.50 dDA).  Thus there was a difference of 9.7 dBA between the lowest and the loudest station.  Since every 5 dBA difference equates to 2 to 3 times louder, depending on frequency, the loudest station is approximately 4 to 6 times louder that the lowest noise level.  However, all stations are below the OSHA Standard.  With an annoyance level (60.0 dBA) for the guest activity, the gunshots are less than the nuisance level and significantly less than the permissible level. The few complaints about the noise levels are not supported by the noise level data. A logical conclusion is that the objection is the act of shooting and not the noise level of the shotgun’s sharp or impact report. Potential solutions might be to examine stations 8, 9, and 12 and considering possible relocation, orientation, or emplacement of a thick tree barrier to absorb the reports. Regression analysis was used to develop a highly significant model to predict noise levels with the variables collected. However, this will not solve the complaints from those who object to reports under any circumstance

Impacts of Leisure Activity Noise Levels A Revised Case Study

An article originally published in the International Journal of Industrial Ergonomics in 1991 discussed the effects of damaging noise levels exposed during the leisure time of industrial workers. With a noise level change preferences, this study was designed to work in collaboration with a 2013 study (Claire Drummond, 2013) which began the groundwork of measuring and examining the consistency of the data collected in 1990. Attributed to an increase in popularity of electronic music and improved digital sound equipment, indoor leisure environment sound levels have increased since the original study. The purpose of this study is to expand the collection of data on exposure levels in current leisure activities reported in the 2012 study to enhance the statistical significance that levels are indeed higher than the noise level preferences and tolerances specified by the Occupational Health and Safety Administration (OSHA) and International Standards Organization (ISO) guidelines. With strenuous legal standards, industries have taken substantial steps to follow the OSHA and ISO recommended guidelines; however, places of leisure such as bars, movie theaters, churches, and sporting events are not required to adhere to the same standards. The noise data was collected using a Quest SPL (2800) calibrated dosimeter. This device is a great improvement from the GenRad Sound Level Meter originally used in the 1990 study for the fact that the Quest SPL (2800) gives a complete printout at the end of a data run that takes in account the duration and average dose of noise exposure run as well as average and peak noise levels. The GenRad device was limited to measuring low and high noise level readings. Average leisure noise levels by category were as follows: peak levels in a sample of thirty-nine leisure activities: churches and concerts (118.06 dB (σ = 2.62 dB)), bars (117.48 dB (σ = 9.32 dB)), and sporting events (122.99 dB (σ = 10.97 dB)). Following the OSHA and ISO standards, any exposure noise level over 115.00dB exceeds the recommended allowance. Adult industrial workers exposing themselves to these levels after an eight hour workday in an OSHA-controlled environment are exceeding the allowable exposure. Industrial business are spending large sums of money to make sure they are in compliance with OSHA standards yet their employees are potentially permanently damaging hearing due to their choice of leisure activity. These individuals are thus increasing their risk of a permanent threshold shift. As a side observation to the adult exposures, many parents had their children under the age of 4 in attendance at the high level peak level leisure activities. Hearing is fully developed at birth, but extended exposure times to high peak noise levels can cause an earlier permanent threshold shift in children or permanent hearing loss at a younger age