Production quantities always vary. Production personnel always optimistically expect lost production will be recovered before the "end or the week...end of the month...end of the year"and they rarely do overcome the obstacles which result in a case of sending too much gross margin dollars to corporate headquarters. Consequently, production volumes are always varying.
Production variations raise many questions such as:
1) What is
"normal" production output?
2) How do we determine
when production failures have occurred?
3) What is the
probability that we will meet or exceed the best one day record?
4) How much production
was lost during the year from cutbacks in output?
Failure definition is important. Shewhart control charts often display failure trends. However, control charts often miss information concerning failure modes. Knowing the failure mode (or where different failure modes are occurring) is important information for helping ferret out the root cause of the failurenot all failures are of equal impact.
The Problem
Look at last years daily production output described in tons per day from a
continuous processing plant where the production is "sold out", and
every ton is needed for meeting customer demand. The data is shown in a frequency
table format with tonnage * occurrence.
Daily Production Output (tons) 

0.1*5 
250*1 
500*1 
725*8 
925*10 
1125*20 
Mean = 897.48, Standard Deviation. = 254.69; Median = 925; Mode = 900 
Notice the traditional measures of central tendency (Mean = average, Mode = the
largest number of observed data points, and Median = half of the data points
lie each side) have considerable variance and give a clue that the data is not bellshaped
and symmetrical.
Questions:
1) What information does the data provide about failure modes and how
many failure modes exist for this model?
2) What is the normal range of production?
3) Where do typical cutbacks begin?
4) Where do severe cutbacks begin?
5) The data shows the best oneday production input is 1450 tons, what is
the probability the process will meet or exceed this best one day production
record?
6) How much production was lost during the year from cutbacks in output?
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Solutions:
Answer For Question #1: What information does the data provide about
failure modes and how many failures modes exist for this model?
*
Dr. Ellis Ott, in his book Process Control, says: always plot the data.
Figure 1 shows a probability plot of the production output made by WinSMITH Weibull.
Reliability experts, who are purist, will be offended by using a Weibull probability plot for anything but ageto failure data. However, a probability plot of production quantities shows departures from the straight line expected for normal production. The breaks in the curve from the desired trend rate represent "failures" to achieve the desired production rateclearly these are economic failures caused by either people, processes, procedures, or equipment (or combinations of these categories)!
The data shows three trend lines (maybe two if you're real fussy).
The steepest curve is the normal production line for about 90.78% of the data. This slope has a b = 5.5 suggest the process has an undesirable coefficient of variation. This is a troubled process because of the flat slope! It has many opportunities for improvement by reducing hidden factory losses from efficiency and utilization losses explained for the nameplate ratings.
The middle slope shows production cutbacks in output (a fairly common failure mode for about 3.6% of the data lying between 90.78% and 94.34% which covers the production range of 350 to 650 tons).
The flattest slope shows the "crash and burn" of major calamities covering about 5.66% of the data.
In short, the data shows two predominate failure modes which degrade ~10% of the production capability. So the process shows about ~90% reliability and about ~10% unreliability. Now since the data is not shown in agetofailure format, do not attempt to infer specific modes of failure in the usual manner for Weibull plots.
The breaks in the trend lines tell you to look for different reasons for the failures. By the way, the five production values at 0.1 tons/ per day were actually recorded as zero output. Since you can't plot zeros on a log scale, enter a small value which is say ~2 logs lower than the smallest real number to reflect an almost zero condition.
Answer For Question #2: What is the normal range of production?
From Figure 1, most (99.9%0.1% = 99.8%) of the expected production will lie between 284 tons/day and 1421 tons per day.
Answers For Question #3: Where do typical cutbacks begin?
Typical cutbacks occur at 650 tons and below. Until the reasons for failure are permanently removed, a watch should be established to flag reasons for the low production and root cause analysis should be initiated to eliminate the problems.
Answers For Question #4: Where do severe cutbacks begin?
Severe crash and burn problems occur at 350 tons and below. Use the same logic as described in the answer of question 3.
Answers For Question #5: The data shows the best oneday
production input is 1450 tons, what is the probability the process will meet or
exceed this best one day production record?
The normal production trend line projects the probability of achieving production at 1450 tons or more is 0.04%. That's the same as saying 99.96% chance of not making this production level unless a substantial change has been made to the process which causes a new and higher level of production to be achievedyou've got to make a change to get a change! New one day records (such as 1450 tons) are used by production management as carrots for production personnel to run faster and harderthat's a noble reason but not a realistic reason unless process changes occur to reproduce the results by cause rather than by chance.
Answers For Question #6: How much production was lost during the
year from cutbacks in output?
The shaded area between the curves shown in Figure 1 represent the losses. Had the process been in control, losses of 10,183 tons (more than 10 days of lost production) would not have occurred as shown in the table below. In most commodity businesses, profits only occur squeezing the last 10 to 15% out of the output!!
The losses are found by subtracting the predicted normal production curve from the actual data points at the specified % of occurrences are easily obtained from WinSMITH Weibull.
Notice the differences in losses between the "normal" cutback zone losses of 1,397 tons and the severe cut back zone losses of 8,786 tons. On a Pareto distribution basis, correcting the severe cutback zone is the first order of business! For this case, it emphasizes that arguing about whether cutbacks begin at 650 or 350 or 284 tons is not so important as correcting the primary problem from the severe crash & burn zone. Note that the severe cutback zone's lost of 8.8 days of production suffers from the five down daysthis emphasizes why total down days for turnarounds and other major problems are genuine killers!
Cutback zone 

Severe Cutback zone 

Normal 
Actual 
Loss 
Normal 
Actual 
Loss 

650 
650 
0 
596 
350 
246 

Cutback Zone = ~1.4 days lost, Severe Cutback Zone = ~8.8 days lost 
By the way, how much production can you afford to loose from your process because of partial or total failures? Are your cutback percentages higher or lower than this example?
Comments:
Know your actual production variations and what is possible based on demonstrated results. Identify your failures and find ways to defeat Mr. Murphy and his saboteurs. Keep plants running to generate gross margins. Plants are built to make moneynot to sit idle when failures are preventable or controllable.
· Production Reliability Example With Nameplate Ratings
· Key Performance Indicators From Weibull Production Plots
· Process Reliability Plots With Flat Line Slopes
· Process Reliability Line Segments
· Papers On Process Reliability As PDF Files For Nocharge Downloads
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