Process Capability Index: How to Use it for Packaging Machines?

Is Your Packaging Process Stable and Reliably Repetitive?

Imagine that you are a company in the beauty product industry that supplies cleaning cream in unit doses which are then packed into carton boxes. The production management has purchased a packaging line that consists of a form-fill-seal machine (unit dose packaging machine or dose thermoforming machine) and a automatic cartoning machine to automate the process.

Or if you are not familiar with unit dosing, suppose that your factory has a packaging line that is composed of a VFFS with multi-head weigher to pack bags of cereal, which are then to be packed into cartons by a cartoning machine, and between them, there is a check weigher that collects all the data about each package produced and gets them complied in the system for later analysis.

Supposing the targetted production amount is 48,000 units which include both the qualified packages that pass the requirement and those unqualified and rejected, and the standard weight for each package is 200 gram.

$$
\begin{array}{c|lcr}
No. & \text{June 1st} & \text{June 2nd} & \text{June 3rd} \\
\hline
1 & 200.15 g & 199.8 g & 200.32 g \\
2 & 200.02 g & 199.8 g & 200.01 g \\
3 & 200.05 g & 199.8 g & 203.2 g \\
4 & 200.5 g & 199.98 g & 197.12 g \\
5 & 198.53 g & 200.5 g & 200.22 g \\
6 & 200.11 g & 201.3 g & 198.32 g \\
7 & 201.5 g & 199.87 g & 188.3 g \\
… & … & … & … \\
48000 & 200.3 g & 200.15 g & 199.95 g \\
\end{array}
$$

The above report was submitted for management’s review after a period of production and with data collected, the leadership has bumped into a headache – how the packaging process should be assessed in terms of its capability and consistency? 

That is to ask:

• To which level the packaging process can repetitively produce qualified packaging that meets the requirements and specifications?
• What is the general error in production (supposing a measurement of the weight of each package) and is it acceptable?
• How to measure the overall capability of your packaging equipment and packaging process?

Introduction – Six Sigma, PCA/PCIs, CP, CPK, PP, PPK …

Managing something that cannot be measured is surrealistic. Modern management theories have a wide range of quantitative tools that can help companies to assess, evaluate, manage, and improve their processes.

In the food and beverage industry, for example, large-scale production entails hundreds of thousands of units of packaging produced each day from filling, sealing, cartoning, and case packing. With such a significant amount of production, even a 99 percent of quality level is not acceptable by all means now that from an accumulative perspective [1] it would mean hundreds or even thousands of unqualified units: packaged food that does not meet the target weight, or bottles of nutritional shake that has volume inferior to the standard.

Fig.1 Process Capability Index (CPK) Analysis
Credit: Copar Mechanical Engineering[2]

It was in 1986 that the statistically-based method of Sig-Sigma was invented to measure the variations of processes and systems and implement the standard of “6σ” which requires a rate of 99.9997% free from defects and stands for only 3 defects out of 1 million units of production.

One of the most prominent tools provided by Six Sigma is the Process Capability Analysis (PCA) which denotes the concept of cp, cpk, pp, ppk which are indexes to assess the process’s capability – the ability of a process to meet customer expectations. All of above mentioned are concepts of Process Capability Indices (PCIs).

$$ C_\text{p} = \frac {\text{tolerance width}} {\text{process width}} = \frac{USL-LSL}{6\sigma} \qquad(1)$$
$$ C_\text{pk} = min \begin{Bmatrix} C_\text{pu} \text{，} C_\text{pl}  \end{Bmatrix} =    min\begin{Bmatrix}  \frac  {\text{USL} – \bar x}{3\sigma} \text{，} \frac{\bar x – \text{LSL}}{3\sigma}   \end{Bmatrix} \qquad(2)$$

pp and ppk are calculated by the same equitations as cp and cpk. The difference is that:

• cp and cpk are used to assess partial samples from the population, or we use these indexes for a short-term process capability assessment.
• pp and ppk engage all the measurements from the whole population that assess the whole process’s capability. That more closer the results of pp and ppk to cp and cpk, the more stable the process is.

To illustrate, extract collected data from such a cereal packaging line during a certain short period of time (1 hr, for example) and carry out the PCIs analysis with the above equations. With several such processes repeated we would be calculating cp and cpk.

While if I do the same thing with samples during a month, we tend to review the packaging line’s stability and capability as a whole and in the long run with pp and ppk.

Foundation to Process Capability Index

Understanding your measurement, data, and samples is the first step to assessing mathematically and quantitatively your packaging and production process.

The analysis with Process Capability Indexes requires that the data collected from the process complies with the normality or is normally distributed. Otherwise, a process that itself is not stable does not make sense to proceed with PCIs analysis.

Supposing the following data is the weight of each bag of cereal produced from the packaging line, as recorded by the checker weigher before they proceed to cartoning machine. In the following chapters, we will carry out a Process Capability Index calculation based on them to assess the stability and consistency of the packaging process and packaging line.

To specify the concepts that appear in the tablet:

• i: ref. no. of each unit of packaging
• xi: the weight of each unit of packaging
• Measuring unit: gram
• Nominal Value: 200 gram

Fig.2 Sample data for Process Capability Index calculation
Credit: ELITER

Understanding Normal Distribution

Quantitative and statistical analysis in industrial manufacturing is always carried out with the assumption that the measurement of the production, for example, sizes, weight of each packaging, etc., would always be normally distributed within the tolerance range (limit of tolerance).

Empirical facts have also proven that, for example, the more amount of units and measurements recorded, the more it appears in the shape of a curve after being converted into a graphic.

By building a histogram, the measurement will tend to gather around a “centered” value while the further from this value, the fewer measurements would appear there.

There are two components which are critical to the shape of the “bell curve”, which are:

• \( \mu  \): which stands for the mean, and determines the position of the central axis of the symmetry.

$$   \mu = \frac{\sum_{i=1}^n X_i}{n}  \qquad(3) $$

• \( \sigma  \): standard deviation and distance toward the mean by which the Curvature changes in sign. σ is the key component to determine how much the curve is centralized or decentralized.

$$  \sigma^2= \frac {\sum_{i=1}^n (X_i – \mu)^2 } {n}   \qquad(4) $$

Fig.3  Histogram of Normal Distribution – Drawn with the above data
Credit: ELITER Packaging Machinery

The normal distribution is also referred to as the Gaussian distribution, the general form of which is:

$$  f(x) = \frac {1} {\sigma\sqrt{2e}} · e^{-\frac{1}{2}(\frac{x-\mu}{\sigma}^2)}   \qquad(5)$$

the normal distribution can be converted into a standard normal distribution with the support of a conversion formula the convert the variant X:

$$  \mathit z = \frac {x-\mu}{\sigma} \qquad(6)$$

after which we will acquire the standard normal distribution as follows:

$$  \varphi(\mathit z)= \frac {e^{z^2/2}} {\sqrt{2\pi}}   \qquad(7)$$

Normality Test – Prerequisite to PCIs Analysis

“Testing for normality is often a first step in analyzing your data.” [3]

The analysis of cpk is based on the prerequisite that the tolerance distribution of the produced packaging complies with the normal distribution, for which reason all of our analysis should start with the test that our sample data collected is normal.

Q-Q Plot

Q-Q Plot is a graphical method to test the normality of a group of samples.

To carry out the test, we may convert the collected sample data (weight of each package or bag of cereal) above with the function (6), the result would be as follows:

Fig.4  Standardly converted samples
Credit: ELITER Packaging Machinery

The Q-Q plot is drawn with the results from the cumulative area function of the normal distribution, which stands for the ratio that packages with measurement within the range that takes over the total amount.

$$ \mathit e \mathit r \mathit f (\mathit x) = \frac {2}{\sqrt {\pi}} \int_o^x \mathit e \mathit x \mathit p (-t^2) \mathit d \mathit t \qquad(8)$$

while the standard form, if the sample is converted with the function (6), would be:

$$  \mathit F (x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\frac{1}{2}z^2}\mathit d \mathit z   \qquad(9)$$

The manual way of drawing a q-q plot is somehow cumbersome with a considerable amount of calculation.

Building with the samples we have about the weight of each cereal bags produced from the packaging line a sequence as folows:

$$ \mathit X = (x_1, x_2, x_3, … x_n)， n\gt0  \qquad(10)$$

where,

$$ x_1  \lt x_2 \lt x_3\lt …\lt x_n  \qquad(11)$$

Calculate the \( \mu \) and \( \sigma \) with functions (3) and (4), then proceed with:

$$\mathit Q _i = \frac {x_i – \bar x} {\sigma} \qquad(12) $$

$$ t_i =\frac {i – 0.5}{N} \qquad(13) $$

Locate the corresponding result on the normal distribution table

Fig.5  Normal Distribution Table

Draw the result on a coordination system and compare their shape with the y=x. In the case that the trajectory is similar to the latter we can confirm that the sample is normally distributed and is available for process capability analysis for the packaging line.

Alternatively, the Q-Q Plot can be generated by a lot of tools with your computer, the result of which will be as follows:

Fig.6  q-q plot based on the sample data of cereal bag weight
Credit: ELITER Packaging Machinery

Once upon confirming that each sample plot on the graphic does align with the line on the plot would we be able to confirm that the sample data is not different statistically from a normal distribution – leading to a conclusion that our samples are available for cp, cpk and various PCIs analysis.

Anderson-Darling Test

Invented by T. W. Anderson and D. A. Darling in 1954. AD test is similar to QQ-plot in the sense that it proceeds to a probability plot to test the normality.

$$ z =  \int_{-\infty}^{+\infty}  [F_n(x) – F(x)]^2 w(x)f(x)dx  \qquad(14) $$

where the f(x) is the Cumulative Distribution Function (CDF) of the standard normal distribution while w(x) is a weighting function whose formula is as follows:
$$ w(x) =[F(x)(1-F(x))]^{-1}  \qquad(15) $$

Equation (14) is simplified by calculation into the following form:

$$ A^2 = -n-\sum_{i} \frac{2i-1}{n} [\ln F(Y_i)+\ln(1-F(Y_{n+1-i}))]  \qquad(16) $$

in which \( Y_i \) is:

$$  \mathit Y_i = \frac {X_i – \hat\mu}{\hat\sigma} \qquad(17)$$

The first step with the AD test is starting with sorting the sample date (the weight of bag of cereal produced from the packaging line) just like how it is for q-q plot:

$$ \mathit X = (x_1, x_2, x_3, … x_n)， n\gt0  \qquad(18)$$

$$ x_1  \lt x_2 \lt x_3\lt …\lt x_n  \qquad(19)$$

In the case that we are partially taking samples from the population, the above equations can be written into:

$$ A^{*2} = A^2(1 + \frac{4}{n} – \frac{24}{n^2} ) \qquad(20)$$

Our raw measurements about the production from the packaging line are now able to be processed into the following tablet, where the vector (x) is weight of each bag of cereal, and Pr is calculated from AD*

Fig.7   Result from AD test
Credit: ELITER Packaging Machinery

By plotting all the data of X and their corresponding p-value on a coordination system. The graphic is assessed in the same way as a q-q plot to test the normality. Note that the p-value calculated from AD* must be greater than 0.05 otherwise it the hypothesis of normality would be rejected.

Fig.8  Probability Plot with Anderson-Darling Method
Credit: ELITER Packaging Machinery

Analyze the Process Capability of your Packaging Equipment

Process Capability Index was not invented at the beginning for packaging but for the automotive industry and various indsutrial manufacturing sectors to measure and provide estimations on how a production process is capable of consistent outputs and remaining stable within the limits of the permitted specifications, after all, all management would like to have their process “stably within the control”.

Fig.9  Stability and Capability
Credit: ELITER Packaging Machinery

Both stability and capability are essential factors in a manufacturing process. Capability refers to and measures the ability of a process to meet a specification, and stability measures the consistency that such a process is repetitive and stays reliable around some key characteristics and features such as the average measurement (size, weight, etc.).

Fig.10  Sample Chronological Plot Drawn from Data in Fig.2
Credit: ELITER Packaging Machinery

Assess Process’s Stability with cp and pp

To measure how stable your packaging machines are in terms of production, for example, that the multi-head weight can always proportionate as most precisely as possible the cereal to be packed into plastic bags by the vertical flow wrapper.

cp and pp denote the capability of a process or that the packaging machines can produce outputs within a given range of limits and specifications:

$$ \text{cp or pp} = \frac{\text{tolerange width}}{\text{process width}} = \frac {USL-LSL}{6\sigma}  \qquad(21) $$

where,

• USL is the upper limit of specification
• LSL is the lower limit of specification

To explain the concept with our previous case of packaging line that consists of multihead weigher, VFFS, cartoner machine and a check weigher, our standard or design weight of cereal bag of this process is 200 gram, and the limits for the weight to be varaible is from 196 gram to 204 gram, beyond which the outputs should be considered as unqualified and rejected from the packaging process.

For both cp and pp we would have for sure the same USL and LSL, while the difference between the two indexes is how the \( \sigma \) is defined – by sampling from the population to approximate the standard deviation or to calculate exactly the process’s \( \sigma \) with the whole set of data (the population).

For example,

with the 200 results we extracted from the whole production which is \( X_i=\begin{Bmatrix}x_1, x_2, x_3, …, x_n  | N=200 \end{Bmatrix} \), we will be calculating the cp, the process of which is:

$$
\begin{align}
\text{cp} & = \frac {USL-LSL}{6\sigma}  \qquad(22)\\
& =  \frac {204-196} {6 \sqrt{  \frac { \sum_i^{200}x_i – \bar x}  {N}}} \\
& = \frac {8}{5.58} \\
& =1.4337
\end{align}
$$

The same equation will work as well for pp if proceeded with the whole population – data of all the 48,000 pcs of packages produced.

Assess Process Capability with cpk and ppk

cpk and ppk contain a further component of the sample’s mean to measure the process capability in terms of the ability that it can produce output according to the design measurement. With our case of the packaging line, it refers to the capability that the equipment can accordingly produce cereal bags around the weight of 200 grams.

$$ \text{cpk or ppk} =    min\begin{Bmatrix}  \frac  {\text{USL} – \bar x}{3\sigma} \text{，} \frac{\bar x – \text{LSL}}{3\sigma}   \end{Bmatrix} \qquad(23)$$

For example,

with the 200 results we extracted from the whole production which is \( X_i=\begin{Bmatrix}x_1, x_2, x_3, …, x_n | N=200 \end{Bmatrix}  \), we will be calculating the cpk, the process of which is:

$$
\begin{align}
\text{cpk} & =    min\begin{Bmatrix}  \frac  {\text{USL} – \bar x}{3\sigma} \text{，} \frac{\bar x – \text{LSL}}{3\sigma}  \end{Bmatrix}  \qquad(24)\\
& = \frac {3.84}{2.97} \\
& =1.2929
\end{align}
$$

The same equation will work as well for ppk if proceeded with the whole population – data of all the 48,000 pcs of packages produced.

The nature that cpk and ppk take into consideration the ability that the process can stay efficient with output around the standard and design measurement can be spotted by comparing the normal distribution is 2 sets of samples with the same cp and pp but with different cpk and ppk.

That is to say, the higher the cpk or ppk is, the better our packaging process is able to produce packages close to the standard weight of 200 grams.

Fig.11  Difference between cp and cpk, or pp and ppk
Credit: Dr. Ron Lasky – Dartmouth Engineering Thayer School[4]

How are cp, cpk, pp, ppk Related to Each Other?

The relation between cp, pp, and cpk, ppk is all about:

• How capable is the process in terms of producing output with design measurement – how capable is our packaging line of producing cereal bags of around 200g
• How stable is the process in terms of producing output around the design measurement – the portion that the outputs of our line is strictly within the limited specification?

At a word:

• The more stable is our packaging line, the closer are pp and ppk to cp and cpk
• The more capable is our packaging line, the closer are ppk and cpk, to pp  and cp.

Fig.12  Relationship between cp cpk pp ppk
Credit: ELITER Packaging Machinery

How May I Use cp, cpk, pp, ppk for my Packaging Process and Equipment?

Process Capability Indices are in nature some yardsticks that only serve as references against which you can acquire an idea of their capability and stability either in the long term or in the short term.

Apart from being used as metrics to assess the packaging line’s performance, process capability indices can be used indirectly as a tool to spot potential problems of the packaging equipment and to facilitate preventive maintenance. [5]

Metrics to assess your packaging line’s performance

Interpreting PCIs can reveal the performance of your packaging line and automation process.

• capability index > 2.0, the process is excellent at “6 sigma” level
• capability index > 1.66, the process is capable
• 1.66> capability index >1.33, the process is limitedly capable
• 1.33 > capability index, the process is not capable

cp and cpk of the packaging machine can spot the potential deterioration

The below tablet shows PCI analysis with cp and cpk acquired with samples sporadically collected during a given period of time of the cereal bag’s weight produced from the packaging line.

A potential problem is perceived by the downward trend of cp and cpk, which stands for a probable issue that the packaging machine’s capability and stability are deteriorating.

$$
\begin{array}{c|lcr}
\text{Test Ref.} & \bar x_i  &  \sigma & cp & cpk \\
\hline
1 & 199.96 g & 0.93 & 1.47  & {1.38}\\
2 & 199.76 g & 0.91  & \color{#000}{1.43} & \color{#005}{1.37} \\
3 & 199.75 g & 0.95  & \color{#500}{1.40} & \color{#505}{1.31} \\
4 & 199.77 g & 0.96  & \color{#A00}{1.39} & \color{#A05}{1.31} \\
5 & 198.53 g & 0.95 & \color{#F00}{1.37} & \color{#F05}{1.29} \\
… & … & … & … & …\\
\end{array}
$$

Against such a discovery it is suggested that the maintenance team to find out the system’s hidden cause that is gradually dragging the packaging line’s performance. The benefits from frequent cp and cpk analysis lay down a foundation for positive preventive maintenance and lead to constant improvement.

Fig.13  Shifts in Process Capability
Credit: ELITER Packaging Machinery

Submit Report to Management: Is Our Packaging Line Capable?

The final question is, how capable the previously mentioned cereal packaging and cartoning line is?

Follow all of the processes introduced in this guide and do it yourself with the data shown in Fig. 2, we can the following results:

$$
\begin{array}{c|lcr}
\text{Indices} & \text{Value}  \\
\hline
\bar x & 199.84 g \\
Cpk_{l} &  1.37    \\
Cpk_{u} & 1.49  \\
Cp & 1.43  \\
Cpk & 1.37 \\
\end{array}
$$

Now the last step us to elaborate your own report and give this topic a conclusion based on your discovery.

Bibliography

• [1] Process Capability–A Managers Tool for 6 Sigma Quality Advantage, Dr. Vinod N Sambrani, Karnatak University, ISSN: 2249-4588 & Print ISSN: 0975-5853, 2016
• [2] Process Capability Index (CPK), Copar Mold & Mechanical Engineering, Doriane Copar Posted November 20, 2018, http://www.doriane-copar.com/process-capability-index-cpk/
• [3] What is the Anderson-Darling Normality Test (AD test)? , November 7, 2018, Ken Feldman, iSixSigma, https://www.isixsigma.com/dictionary/anderson-darling-normality-test/
• [4] Ronald C. Lasky, Professor of Engineering, Senior Technologist, Indium Corporation, Clinton, NY, https://engineering.dartmouth.edu/community/faculty/ronald-lasky
• [5] Process capability improvement of an eight-head liquid filling machine by the reduction in volume variation, C K Adarsh, Sudheer D Kulkarni, https://www.researchgate.net/publication/348605635_Process_capability_improvement_of_an_eight-head_liquid_filling_machine_by_the_reduction_in_volume_variation