Process Reliability Plots For Solving The
Most Important Production Problems First
Using Weibull Probit Analysis
H. Paul Barringer, PE
Barringer & Associates, Inc.
Humble, Texas

The Big Picture-
In businesses, the most important problems to solve first are the big money problems with symbols such as: $, €, £, ¥, etc.  This is the reason to first build a monetized Pareto list based on the problems and cost.  The largest cost problems on the Pareto list are the first order of magnitude issues for problem solutions—this becomes the strategy.  Resolution of the problems becomes the tactics.  Solve the big money problems first!  The priority is not to work on your own personal love affairs (meaning an intense enthusiasm or liking for something).  Your love affair is not your first business priority!

In any business organization you must minimize your losses and maximize your profits to be consistently profitable.  This requires your personal discipline to only work your personal love affairs when they get to the top of the $Pareto list.  In business it is important to slay the giant lost money issues ahead of your love affairs in your specialty area.  You never get criticized by saving large monetary issues.  Keep your focus and go after the big money.  Yes, you can work on your love affair at work when it gets to the top of your $Pareto list.

Problems in production processes occur in two different buckets of problems.  Quantify the $problems as described clearly by Dr. Deming as:
              1) Special cause problems have specific names for resolving the issues.
              2) Common cause problems are not obvious with many small, unnamed issues to resolve.
The logic is simple: don’t get bogged down by nose counts of problems—think about the $’s lost to concentrate on the important metric.  If you don’t minimize your losses and maximize your profits, your competitor’s will put you out of business!  No profitable business means no jobs.  No jobs means no paycheck.  No paycheck means you’re personally in big financial problems for paying your bills. 

In short, business is a competitive battle!  This requires you to become a very involved employee to solve problems.  Help make your company very profitable, else you may soon be on the street looking for a job.  Make sure you know the strategy for attacking primary issues first by using new and better tools for your tactics.  You are a cog in the wheels of business and not just an idler.  Make every lick count!  Tempus fugit!

You must have a strategy for attacking the primary $ issues.  The strategy drives the tactics for specifically resolving issues with attacks by new tools and better techniques.  In many cases the unresolved issues have been around for a long time and you, with new tools, can permanently resolve the issues.

Remember special causes have names.  Special causes are an army of problems with names and uniforms.

Remember common cause have names yet to be discovered  Common causes are an army of problems without names and without uniforms—they are similar to terrorist!

Special causes are losses from reliability issues which form cusps on the Weibull probability plot trend lines of production.

Common causes are shown by the gaps between the demonstrated production line and the nameplate line on a Weibull probability plot of production.

Common cause problems are often identified and solved by 6-σ methods/tools which root out and give names for the problems within the common cause arena.

Special cause problems with names can be solved by use of known tools from production, maintenance, and engineering.

First And Fourth Quartile Comparisons-
First quartile companies (the winners) have minimized common cause problems compared to fourth quartile companies (the not so good companies).  First quartile companies have small gaps between the demonstrated production trend lines and few cusps to the left with the reasons for the gaps to the left of the demonstrated few in number and few in the sum of the gaps.  The first gaps between the nameplate trend lines and the demonstrated line describe the efficiencies and utilization issues.  They have few gaps to sum for few losses which tells that most common cause problems have been solved

Fourth quartile companies have huge gaps in all directions.  They are the flip side of first quartile companies.

First quartile companies have few special cause problems along with high reliability for consistent production quantities compared to fourth quartile companies.  First quartile companies have high reliability as shown by the cusps on the break away from the demonstrated production trend lines.  The gaps to the left of the main demonstrated production line are lower production outputs and describe special cause events which are easier to find the roots of the problems and easier to solve.  The small gaps between the demonstrated trend line and the nameplate trend lines describe common cause issues which are more difficult to identify and to solve but for first quartile companies the common cause problem have been identified and removed.  Fourth quartile companies have poor reliability on the demonstrated production trend lines with large gaps to the left of the demonstrated production line summing to large losses.  Fourth quartile companies also have large gaps between the name plate lines and the demonstrated production line as the common cause problems have not been identified and removed for consistent output.

The adage the rich get richer and poor get poorer is demonstrable on Weibull production plots.  To get rich you must eliminate both special cause and common causes.

First quartile companies have few lost $ issues and higher profitability.  Fourth quartile companies have many many lost $ issues and marginal profitability.

First quartile companies make their success with fewer problems and fewer excuses.  Fourth quartile companies have much misery and claim “bad luck” as the try to swim in the completive river with a mill stone around their neck without solving the easy special causes and the more difficult common cause problems.

Barringer process reliability (BPR) techniques give you tools for quantifying the misery of your problems.  BPR techniques tell you kind of problems which are eating your lunch: special causes or common causes.  It sets up the direction for where to look to prevent the loss of $’s. 

You can download all of the process reliability papers on this web site concerning use of the Weibull technique for analysis and quantification of the losses.

The Beginning Of The Barringer Process Reliability Technique-
The first use of BPR began in 1995 at a refinery.  The data supplied for 365 consecutive days was daily charge to the crude tower in the refinery.  This measuring point was chosen on purpose as the oil going into the crude tower is easy data to obtain and easy to use.  However output streams from the crude tower are many and depending on the crude source the output lines have large changes in the many output streams. 

The charge to the first crude tower showed a good straight line with a steep slope meaning high consistency of the charge with few efficiency and utilization losses.  The slope of the demonstrated production line showed a beta, slope, of 125 so the common cause gap between the demonstrated line and the nameplate line was very small telling the common cause problems were very small.  The beta of 125 said this was 1st quartile capability.  The major problem was some reliability issues which displayed special causes issues to be solved for reducing losses.

Cusps on the actual charge data points showed opportunities for improvement with elimination of special causes—the easier problem to solve.  Immediately we knew where to attack the issues—it certainly was not in the difficult area of common cause issues such as efficiency and utilization.  Rather the Weibull plot said focus in the area of special causes.  It’s important to know where to attack the problems beginning with reliability problems which ae special cause problems!  If the charge to the crude tower is smooth with little variability and high reliability the losses are small and you’re going to have a happy business situation.  Knowing where to focus corrective action is important.  Of course with high production variability of production plus high losses from reliability issues it will make you pull out your hair because of fourth quartile performance!

Making A Barringer Process Reliability Plot-
A “cook book” recipe for first making a BPR Weibull probability plot follows for a typical analysis.  Then the Weibull plot is converted to probit data for giving the plot different colors to the data points for “selling” the details to your audience for motivating improvements.  

1.      Gather 365 days of prime production from your process.  Production by the minute or hour is better for showing the process problems.  Weekly or monthly data is too course for clearly identifying problems.  Only count the prime product.  Do not count off grade or scrap production.  We only get real paid for prime product.  Off grade product or scrap produced is not what we’re paid to produce as it represents a failure of defective product.

2.      On days of trivial poduction, represent the ~0 data with a small positive number as the X-axis of the logarithmic scale does not allow a zero or a negative number.

3.      Plot the data on a SuperSMITH Weibull software tool.  You may be tempted to use the DEMO version of the software for your input of raw data.  NOTE: the no-cost DEMO version will randomize your input data.  However, if you send one Excel file by Email to Paul Barringer, I will send you back by Email an authentic file.  You can import the authentic SuperSMITH Weibull file into the DEMO software and avoid randomization.  DON’T resave the file from the DEMO version as that will destroy the integrity of the authentic file.  You can convert the DEMO version to an authentic software version with use of an electronic key and we can handle the financial transaction by credit card with the key appearing on the Invoice.  You can read the EULA (license agreement) with software prices posted.

4.      Using the SuperSMITH Weibull process reliability icon (bottom row of icons, 4th from the right) zoom in on the upper regions of the data and fit a trend line to the high output regions—use good engineering judgment in selecting the region as speed runs and catalyst burn-in can screw-up the analysis.

5.      Select the cusps via multiple mouse clicks or by manual input of the cusps.  This will give you the losses between the cusps and the demonstrate production line.

6.      Add the nameplate line with the nameplate line crossing the demonstrated production line at the altitude of the highest point so the trend line is steeper than the demonstrated production.  Suggested starting nameplate betas are 25, 50, 75, and 100 for 4th quartile, 3rd quartile, 2nd quartile to the 1st quartile results.  The name plate line should always be steeper than the demonstrated production line.  This will produce the metrics you need for the analysis.

7.      Remember the nameplate line must always be at a steeper slope than the production line.  Remember if also if you have a 4th quartile demonstrated production line with a 4th quartile beta =15 and you choose a 1st quartile nameplate line of beta = 100, the production team will be demoralized and they won’t even try to achieve better performance as they will see the problem as impossible so it means you must set the nameplate line a little steeper to “seduce” the production team into trying harder.  Don’t set a production line of beta = 15 to a 1st quartile beta = 100 as that will more closely represent rape rather than seduction!  Use the old fable of the carrot and the stick to keep the horse moving faster.

8.      Here is the process Weibull plot for the dataset before conversion to probit data.


Barringer Process Reliability


Total Production (TP) = 201,774.7 [tons/yr] ß55.33% of capability (TP/NP)

Production Line (PL) = Eta(998.4187)  Beta(106.7372) ß World class beta  > 100

Nameplate Line (NL) = Eta(1000.929)  Beta(125) ßName plate is slightly larger (NP >PL)

Process Reliability (BPR%) = 50.0 %ßPoor reliability performance with deficient output



     Reliability (RL) Loss = 161,712 [tons/yr] ß#1 problem!! = 162 days lost production

          50.0-60.00%: Loss = 21,039 [tons/yr] ßCutback transition = 21 days lost production

          60.00-80.0%: Loss = 69,608 [tons/yr] ßCompressor failure = 69.7 days lost production

          80.0-90.0%: Loss =   36,237 [tons/yr] ßLack of orders = 36.3 days lost production

          90.0-100.0%: Loss = 34,828 [tons/yr] ßTurnaround = 35.9 days lost production

          Efficiency + Utilization (EUL) Loss = 1,193 [tons/yr] ßMinor problem = 1.2 days lost

     Total (TL) Loss = 161,712 + 1,193 = 162,906 [tons/yr] ßTotal losses = 163.2 days lost


     Nameplate Production (NP) = 364,680 [tons/yr]

     Effectiveness = (TP / NP) = 55.33%  ßLow effectiveness mainly due to reliability issues

Figure 1: Process Reliability Plot Before Conversion to Probit Data

9.      Convert the dataset to probit data for obvious named problems as follows:
a.  Stacks of data with the same X-value will not plot because the line slope β is infinite.
b.  Change the data stack by adding 0.0001 in sequence to unstack the data.
c.  Click on Methods Icon (Analysis section, top row one the left).
d.  Choose Option I and select probit , and then option 3 followed by auto data transform.
e.  The probit plot will appear with numbers in lieu of the desired points which we want.
f.  Click on the magnifying icon, choose Option P to change the number to point symbols.
g  Follow by click on Option P.
h. Delete the trend lines by going to the Analysis section click on the marker line.
i.  Choose line 1 brining up a menu delete lines with small beta values.
j.  Put the groups of similar data into different columns to produce Figure 2 with colors

10.  Click on the labels icon and add the problem reasons with losses for the year—See Figure 2
with different colors and reasons for the discrepancies—this is the sales graphic to make your point!

Figure 2: Process Reliability Plot With Probit For Different Colors

The old saying is: the opera is rarely over until the fat lady sings.  Business issues are not over until we quantify the money lost.

Cost out improvement proposals with alternatives (note the word is plural!) for solving the lost money.  The first alternative is do nothing and cost it out with $ signs.  The production process in Figure 1 is a 1st quartile beta = 106.7.  Unfortunately it only operates at consistently high output 50% of the time.  Even with a 1st quartile beta > 100 survival is tenues because of cutbacks and other issues.  In short, low reliability has $consequences!

Economic Facts Of Life-
Some economic facts of life and their terms:
Net Sales
= Sales-Returns
Cost of Goods Produced = Cost at the plant level for Net Sales
Gross Profit = (Net Sales) – (Cost Of Goods Produced) = A measure of a company’s efficiency in use of labor and supplies which is generally thought of as a variable cost but in reality contains some fixed cost plus a major component of variable cost.
Gross Profit Margins = (Gross Profit)/(Net Sales) = (Net Sales – Cost Of Goods Sold)/(Net Sales) shown as a percentage value.
              Commodities businesses have small percentage values, say 10% to 15%.
              High margin businesses have larger percentage values, say 40% to 60%.
              Gross margins are the $ left overs to pay for sales, development, research and net profits.
In Figure 2 we two defect things to connect such as 69,600 tons/yr for compressor failures + 21,039 tons/yr of cut back transitions which are the in and out of failures for a total of 90,639 tons/yr all associated with compressor failures.  Suppose the do nothing gross profits are $15/ton.  Then the do nothing loss is (-$15/ton)(90,630 tons/yr) =   $1,359,450/yr.  We  leave this big loss on the table by doing nothing to eliminate the problems.  Of course this assumes we can sell the product if we could make it.

How much capital could we spend to correct this $1,359,450/yr problem for a 20 year project.  Since capital is tight, Accounting says we need the extra restriction that imposed the requirement of achieving a 20% internal rate of return because capital is tight (in good times the requirement was only a discount rate of 12%).

Download the life cycle cost Excel spreadsheet and unZIP to help answer the question of what capital can we spend to get back the cost restriction and save all funds lost of $1,39,450 per year.  Of course this will wipe out our losses if the expenditure goes to preventing the cutbacks and failure for the compressor.  The Excel worksheet has tabs for two worked out examples plus a clean worksheet for our life cycle cost problems.

In the downloaded worksheet we will use an iterative processor call Goal Seek.  So we must setup iteration details before we commence the calculations.  Open the Excel worksheet.  Click on File.  Click on Options.  Click on Formulas.  Click on the box to enable iterative calculations.  Set up maximum iterations = 10,000 (we’ll have an answer before we reach this large value).  Set up maximum change = 0.00001 and then click on OK.  Save the worksheet on your computer.  Go to the Excel tab for LCC Worksheet.

In the worksheet in cell C2 set a typical discount rate of 12%.  In cell F2 use a project life of 20 years.  In cell I2 use a typical tax provision of 38%.  Next comes our savings of $1,359,450 $/yr for input into cells E19 to X19 input an annual savings of $1,359,450.  In cell D5 put a small number to commence the Goad Seek iteration, say 10,000.  Now we’re setup and ready for letting Goad Seek solve our problem.

Go to the upper tab for data.  Click on “What If Analysis” and select the “Goal Seek” tool.  Set Cell Input  as F3.  Set To Value input as 20.000%.  Set Changing Cell = $d$5.  Clicking on the What If Analysis icon, again under the Goal Seek Status choose Set Cell = f3, Set To Value = 20.000%, set Changing Cell = $d$5 and watch the iterations occur for about 3 to 7 seconds.  Excel will show cell D5 = $4,522,823.  (Good practice says verify cell f3=20.00%--which it does).  If we perfectly solved the compressor problem and the transition problem perfectly then we could afford to spend $4,522,823 for capital to fix the compressor problems.

Others on our tem believe that we may be able to save ½ of the compressor and transition losses for an annual savings of $679,725 for cells E19 to X19 and cell D5 set to $10,000.  Again clicking on the What If Analysis and inputting Set Cell = f3, Set To Value =20.000, set Changing Cell = $d$5.  The answer for the amount of capital we can afford to spend is $2,261,326 if we can only save $679,725/yr.

The next “yes, but question” involves probabilities to the actual expected savings.  Remember the full savings used previously was $1,359,450 for the best case of solving all the problems or the ½ savings = $679,725.  We gather some of our knowledgeable experts for use of the Delphi method to arrive at the chances for making the full savings or the ½ savings.  The expert team knowing our previous experiences came to this conclusion after multiple iterations.  They said we’ve got a 75% chance for the case of perfect savings and 97% chance for getting ½ of the savings.  Nate Silver in this book The Signal and The Noise says if use 0% or 100% you usually have made a biased decision.  He says most likely the true answer for real life decisions usually lay between the two extremes.  So here are the “real” savings we should use for our Excel worksheet.   Instead of annual savings of $1,359, 450 the number becomes (0.75*$1,359,450) = $1,019,588 savings per year and the ½ savings becomes (0,97*($1,359,450/2)) = $659,333/yr.

Here’s what the Excel Life Cycle Cost says:
For $1,019,588 savings/yr with an internal rate of return at 20%: Capital acquisition cost = $3,392,063, Net Present Value = $1,811,105.
For $659,333 savings/yr with an internal rate of return at 20%: Capital acquisition cost = $1,171,200, Net Present Value = $1,171,200
The net present value winner is the 75% chance of achieving the big savings.

The next largest potential saving is from lack of orders with a loss of 36,237 tons/yr.  At ($15/ton*36237 tons) = $543.555/yr discrepancy.  Production can’t solve this problem alone, we need the entire business team (Sales, Marketing, and Production along with the appropriate managers) to discuss the issues for filling this 1st quartile capable plant while Production solves their problem.  This solution requires a team effort, with intent of taking business away from our 4th quartile competitor.  Don’t be surprised to learn that some members of the team didn’t know the size of the problem and the money we didn’t make in one year!  Marketing and Sales departments must solve this problem.

The final problem is the loss of 34,828 tons/yr for annual turnaround costs.  Based on the data costs for failure, it is reasonable to infer the root of the turnaround problem lies with the compressor issues.  We can assume the common compressor problem starts with inferior knockout drums.  Compressors need excellent knockout drums to removed fluids prior to the compression cycle—see knockout drum requirements as fluids don’t compress well at all which in turn adds very high loads to compressors.  Based on a similar case of compressor problems I asked the Maintenance Manger to describe their knockout drums.  He said we don’t have traditional knockout drums they are more like knockout aneurisms!  So immediately we got a good clue for the root of compressor problems was the lack of adequate knockout drums—but no one was taking corrective action to solve the root of compressor issues.

In today’s business environment you cannot survive with annual shutdowns for repairs.  Most business have abandoned the 1950’s annual turnaround to use a 4-8 year interval.  Some plants are on 10 year intervals.  Change from spending ($15/ton)*(34,828 tons/yr) = $522,420/yr + other associated maintenance costs to starting with a turnaround every 4 years (or longer) to cut the cost for a four year interval (or longer) it becomes $522,420/4 = $130,605/yr + other associated maintenance costs.  The requirement is to attack and solve the root cause which now requires annual turnarounds.

Download a copy of this problem as a PDF file.

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Last revised April 30, 2016
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