Car Parking & the Six-Sigma!

We can easily understand the six-sigma concept using the analogy of the car parking into the garage.

Keeping the Garage’s Width ConstantSay you have a garage of width “D” and you want to park a car of width “d1” into it. When D/d1 < 1 (i.e. car is bigger than garage) it’s not possible to enter the garage. Take another car of width “d2” but this time as D/ d1 slightly greater than 1, in this case it’s possible to enter the garage but we will end up with frequent scratches. Take third car of width “d3” but this time as D/ d1 >> 1, finally this is case where it’s possible to enter the garage even if I am drunk provided I am not too far from the center of the garage.

“We need to buy a car according to the dimension of the garage"

If the width of the garage (D) is the customer’s specification and the car’s width (d) is the product’s specification, then there can be three cases as discussed above

Case-I: Product is out of specification for the customer, hence customer won’t accept it. It means we are producing defective products.

“I am always hitting the walls of the garage”Case-II: Product specifications are just meeting the customer’s specification,

hence customer may accept it. It means that the products that we are making is at the border line of the customer’s specification. Most of the time I am ending with re-work and reprocessing.

“Most of the time I am entering the garage with scratches”Case-II: We have a robust process in hand, where product specification are

exceeding the customer’s specification, hence customer would always welcome us. It means that the products that we are making will always meet the customer’s specification.

“Almost all the time I am parking the car at the center of the garage”

Keeping the Car’s Width Constant

Some terms used in six-sigmaVoice of Customer (VOC): is analogous to the width of the garage or

simply can be described as “what customer wants”.Voice of Process (VOP): is analogous to the width of the car or is

described as “current process capability to meet customer’s specifications or VOC”.

It is always desired that VOC/VOP should be at least 1.3

You may argue that what’s the big deal if my existing car doesn’t fit into the garage, I will rebuild the garage to fit my existing car.You can always do it with your garage but you just can’t change the customer’s specifications because that is not under your control. For your existing defective product/process, customer won’t change his manufacturing process just to fit your defective products. “The only panacea for improving your business is to improve the existing process to meet the customer’s expectation or else lose your business to your competitors”

How the garage/car example and the six-sigma (6σ) process are related?Let’s measure the width of the garage and the car in terms of some arbitrary unit sigma (σ)1 instead of meters.Further assume that the width of the garage (D) is sacrosanct (like customer’s specifications) and is equal to 12σ.

In six-sigma the ratio of D/d is called as process capability index or Cp1 Sigma is the measure of standard deviation, if you are not aware of it then for the time being just assume it to be a unit of measurement for the length. Don’t waste your time on it right now, this is because the calculation of σ for Cp/Cpk is little bit tricky.

Further remember for the time beingProcess sigma level=3×C p

Also assume that the width of the car (d) can be varied. The car width is also measured in terms of σ. Based on the above two assumption, there can be three case as shown below. 2

The best case is where there is least chance of collision of the car with the garage and this scenario is represented by the case-C. In this case, car is at the center of the garage and the ratio Cp is maximum. But there is a flaw in using above calculation for Cp. This because it doesn’t consider the relative position of the car with respect to the edges of the garage. Even if we use a car of constant width 6σ, the ratio of D/d will always be equal to 2 (Case-3) irrespective whether we are missing the garage or hitting the garage’s wall or we are parking it right in the center as shown below (case A to C). It means that all three cases (A to C) represents the 6σ process (=3 x Cp) which is not correct hence, the calculations were modified taking into account the relative position of car with respect to the edges of the garage.

2 Don’t bother right now on why we have selected 15, 11 and 6 σ as the width of the car.

This problem is overcome by another developing another process capability ratio defined as C pk where we take into the account the relative position of the car with respect to the edges of the garage. Now there can be two scenarios

1. I can be close to the left wall2. I can be close to the right wall

TheC pk, tells you about the distance of the car from both the walls. In calculatingC pk, the distance of the center of the car (C2) from each of the edges of the garage is considered. Hence, there will be two values for C pk

C pk (Upper)=(D¿¿Upper−C2)/3σ=distanceof the car ¿¿wall

C pk(Lower)=(C2−DLower) /3σ=distance of thecar ¿¿

Since all processes have two C pk values hence, the least of them is taken as C pkvalue as it require more attention.

Then process capability C pkis defined as C pk=Minimum[Cpk (lower ) ,C pk (upper )]

Now consider the case-C once again in detail where we parked the car right in the center of the garage. Here half of the car (3σ) lies in both halves of the garage.

In all cases D/d = 2C p = 12/6 = 2Process Sigma level = 3xC p = 6, irrespective of whether we are on target or not!It’s not the correct measure of process capability.

Now let’s calculate C pk(lower)

C pk(lower)=Dlower−C23σ

=6σ3σ

=2

Similarly C pk(upper) is given by

C pk(upper)=C1−DUpper3σ

=6σ3σ

=2

Process sigma level = 3 xC pk = 6 for both C pk ( lower )∧Cpk (upper )

There is a 3σ gap between the edges of the car and the garage wall. This is called as margin of safety. Total gap is 6σ hence, the process is called as a six sigma process. If your driving is not influenced by external factors (like drinking, depression etc.) then there is only 3.4ppm chances of hitting the walls. In other words you need to double the size of the car if you want to hit the walls of the garage with your normal driving skills.Let’s consider case-B, where you need to leave a space in the garage for your wife’s car. Now every day you need to park your car on the left side of the garage as shown below. Now the center of the car c2 is 3σ away from the DLower (which is also half the width of the car) and C2 is 9σ away from the DUpper. Let’s calculate C pk (Upper)

C pk(lower)=DUpper−C23σ

=9σ3σ

=3

Similarly C pk(Lower) is given by

C pk(upper)=C2−DLower3σ

=3σ3σ

=1

C pk=Minimum [1 ,3 ]=1

Or process sigma level is = 3 x C pk = 3 i.e. there is a high risk of car getting scratches on the left side while parking.

There is a 3σ gap between the side of the car and the garage wall on both side. This is called as safety margin.In present case, you need to grow your car by 6σ (3σ on both side) before you can touch the garage’s wall.

You can go for a twice the size the current car to hit the walls!

There is no safety margin on left side of the garage. Damage to the car won’t be that much because speed would be ~10 Km/Hr while parking.

“Instead of garage, just imagine you are entering a tunnel at 150 Km/Hr & then re-evaluate the whole scenario discussed above to understand the gravity of the

situation”

Further development on process capabilityWe saw that how useful Cpk is where we measure the safety distance between the edge of the process control limits and the customer’s specification and we use this information to tell the risk of quality failure with respect to USL/LSL.

This is traditional way of quality control which can be called as “GOAL-POST” approach where the possible out-come was goal or no-goal. Similarly we were used to focus only on the end product quality with two possible outcomes, pass or fail.

Latter on Taguchi gave the concept of producing products with quality targeted at the center of the customer’s specifications. He stated that as we move away from the center of the specification, we incur cost either at the producer’s end or at the consumer’s end.

For example;

Buying a ready to wear suit, it is very difficult to find a suit that perfectly matches your body, hence you end up in going for alterations. This incurs cost. Whereas if you get yourself a suit stitched by a tailor meeting the specification of your body, it would not incur any extra cost.

Similarly if you are producing goods of the target i.e. you are of target either on LSL or on USL hence, chances of rejection increases which in turn increases the chances of reprocessing and rework thereby increasing the cost. Even if you manage to pass the quality on borderline then your customer has to adjust his process accordingly to accommodate your product hence increasing the set-up time. Moreover the variance in your product and the variance from the customer’s process just get adds up to given final product with more variance (remember! Variance has an additive property).

Hence question arises, why can’t we measure the deviation of the process mean with respect to the mean of the customer’s specifications? This is necessary because if I can keep the process mean and the specification mean near to each other, the chances of touching the specification limits would be less which in turn would reduce the chances of reprocessing and we can control the process in a better way.

Now we can Talk about Six-SigmaBut before that, let’s understand the word “process”. It is a set of procedures used to accomplish any work, it may be any business process or manufacturing process. The main point is that even if we repeat a well-established process again and again, we will not get all the products of same specification. Variation will always be there and we need to learn to live with it. This happens because you can’t control everything involved in any process. There are some uncontrollable factors known as “common causes” in six-sigma.For example

1. Receptionist might be trained in dealing with visitors politely, but on some occasions she couldn’t do it, may be because of some personal issues.

2. You are producing some part to be used in automobiles, there will be a variation in product specification as there will be wear and tear of machines, change of operators etc.

If we plot a histogram of the product specification from a stabilized process, it would look like

We can see that maximum products would be clustered around the mean and as we move away from the mean, number of products decreases. Point to be noted here is that we can’t have all products exactly at the mean!

Why? Because there will always be some variation as discussed above. From now onwards above histogram would be represented by a curve. Width of the customer’s specification is analogous to garage’s width and the process variation is analogous to car’s width.If you don’t have proper control on your process (driving) you are going to crash your process (car) against the customer’s specifications (garage walls).

Now I feel that everyone agrees that variation is a part of life and we need to learn to cope with it. The only thing we can do is to minimize it by using some proven methodology so that whatever we are producing (product or services) should always meets customer’s specifications or should have enough safety margin. This proven methodology of reducing variability is called as six-sigma.

Till now whatever we have learned, we can describe six-sigma as Six-Sigma is a methodology by which we reduce the process variation so that the output of the process always remains well within the customer’s specifications. So six-sigma is a clamping tool that compresses the variance in the process till we have enough margin of safety.

Any process whether it is a business or manufacturing, when it is scaled-up for the first time would face lot of variability in the output as every process takes its own time to settle down. The R&D environment in which product was made is different from the manufacturing atmosphere. Practically it is not always possible to study everything at the R&D stage. But team at manufacturing makes every efforts to bring the process on track, ultimately with everyone’s effort process get stabilized. Both stages are shown below. Most important point is that even if the process is stabilized, there will be

variations due to wear and tear of machines, operator etc. if this variation is within ±1.5σ, it is acceptable.3

What do we mean by garage’s width = 12σ and car’s width = 6σ?

Right now we are not in a position of going into the details of the standard normal distribution hence, for the time being let’s assume that my manufacturing process is stabilized, which is represented by a symmetrical curve shown below

3 For the time being accept this statement, it will be dealt separately when we are discussing the control charts.

The main characteristic of this curve is that the 99.7% of the product would be between LCL & UCL or within ±3σ distance from the mean (μ). Only 0.3% or 3000ppm products would be beyond ±3σ or defective products. So width of the car is equivalent to the width of the process = UCL-LCL = voice of the process = VOP = 6σ = ±3σ.

Second point is that the curves never touches the x-axis means that there will always be some probability of failure even if you move to infinity from the mean (probability can be negligible but will be there).

Now let’s overlap the above process curve with the customer’s specifications (=12σ = ±6σ) or the garage’s specifications.

We can see that there is a safety margin of 3σ on both side of the process control limits (LCL & UCL). In layman words, in order to produce a defective product, my process has to deviate by another 3σ, which has very remote possibility. Statistically ±6σ (position of LSL & USL) from the mean would account for only ~3.4 ppm failure (don’t bother about the calculation right now, just understand the concept). For this has to happen, someone has to disturb the process deliberately. Compare this failure of 3.4 ppm at ±6σ level with 3000ppm at ±3σ level!

Even if the mean of the process deviate by ±1.5σ, there is enough margin of safety and it will not impact the quality and in regular production, this deviation of ±1.5σ is quite common.

But How Six-Sigma Tools Compresses the Variation?In order to understand this, let’s take the following equation

y=5 x1+3 x2+2

Now if I ask you the value of ‘y’ for x1 = 3 and x2 = 7. The value of ‘y’ would be given by

y=5×3+3×7+2=38

Point to be noted here is that “you were in a position to calculate the value of ‘y’ because you have a mathematical equation describing the relationship between ‘y’ and x1 & x2.

Similarly in six-sigma we find out the variables (x) that impact the variation (y) and then we find a quantitative relationship between them. In six-sigma language we describe it as “y is a function of x1, x2,….”

y=f (x1 , x2 ,……)

For exampleTime taken to reach office (y) is a function of following variables

1. When he slept last night? (x1)2. Did he had drinks last night? (x2)3. When he woke-up? (x2)4. When he started from the home? (x4)5. How was the traffic in the morning? (x5)6. How fast he was driving? (x6)7. Which route he took? (x7)Let’s assume for the time being that x2, x4, x5 and x7 were found to be important during investigation using six-sigma4 and the relationship between time taken to reach office and all 4 factors can be described arbitrary for the time being as

Time taken=β0+β1x1+β2 x2+ β3 x3+ β4 x4

Using the above equation, the response (time taken to reach office) could be optimized.

4 This investigation is usually done using a famous methodology called as DIMAC. I hope everyone is acquainted with it.

Choosing a University!

As a parent you must have had an argument with your child that he should work hard in order to ensure a seat in top 3 universities. Your child is also good at studies and he is aware of his current capability that he can easily get admission in universities with ranking from 4 to 6 but has to work hard to compete for top 3 universities.If we consider parent as a customer, then what customer is demanding is the admission in top 3 universities. This is called as “voice of customer” (VOC).

If you consider your child as a supplier, then with his current efforts (current process) he can guarantee admission in the universities with ranking 4-6. This is called as "voice of process" (VOP).

There is a gap between what customer wants (VOC) and what your current process (VOP) can deliver. Both are independent processes but it is desired that VOP should match VOC.Your child understood this and made a plan to fill the gap by making more efforts.Above methodology of bridging the gap between customer's specification (VOC) and the current process capability (VOP) is called as Six-Sigma methodology.Let’s look at this issue from a different angle. Now consider student as the customer and parent as supplier and student (or customer) is trying to convince his father by arguing “look dad, all universities with ranking 1 to 10 are same, it is just a statistical rating that is

done for attracting students, it changes every year and the universities that you are talking about are best in science, but I want to study law for which a particular university with 6th rank is the best”. As an understanding father (supplier or vendor), he finds the argument too strong to be opposed any further and agrees to his son’s specification i.e. he modifies his current process (expectation). Here Student is setting the specifications (VOC) and father is accepting it (VOP). The process of convincing the father is six-sigma which bridges the gap between them.

Let’s be little philosophical

All of us had a dream during college days that I want to be this, I want to be that. What we did is to provide ourselves with a specification about our future life or VOC. We were also aware about our current capability (VOP) but we never took pain of performing a gap analysis and as a result we couldn’t take appropriate steps to reduce the gap between our desire and our capability, ultimately landing somewhere else in our life. Our desires are still our desires only.

Can we apply six-sigma to build our career? Or at least help our children in doing so?

Getting Late for the Office

Our office timings is 8:30 AM to 5:00PM. Company requires us to reach the office between 8:00 and 8:305 otherwise it will lead to a pay loss of 1 hour if late for two consecutive days.

Following are the arrival time of my new colleague to the office for last 35 days.

Figure-1: Arrival time at the office with same starting time of 7:30 AM every day

What is “Voice of Customer” (VOC) Or the Customer’s Specifications? Here customer is the company to whom we are providing our services and in return we are getting the salary. Now customer’s requirement is that we should be in the office between 8:00 to 8:30 AM. This is called as voice of customer or customer’s specifications. It can be represented as

5 For simplicity, assume that you can’t enter the office before 8:30AM.

USL Upper specification limits

LSL Lower specification limits

Customers have right to demand anything in this world, but it is most important to recognize the current process capability?

Figure-2: Voice of the customer

What is Voice of Process (VOP) or Process Control Limits?Now we need to understand that the time my colleague took to reach the office is independent of the customer’s requirement. It is a different process by itself but it is desired that the output of this process (arrival time at office) or VOP should comply with the customer’s specifications (VOC).

Now he want to understand the efficiency of his current process (arrival time at office). Simply he wants to know what his routine or the current process can offer if he makes 100 trips to the office?

The statistical calculation shows that on 95 occasions out of 100, he would land in the office between 7:41 AM and 8:54 AM.6 This is called as lower control limit (LCL) and upper control limit (UCL) of the process respectively. This is also known as voice of the process (VOP).

Overlap of the process efficiency (VOP) and the customer’s specification (VOC) is shown below in figure-3. It is clear from the chart below that the current process is incapable to meet the customer’s specification.

6 It is based on the normal distribution that mean±3σ contains ~95% of the population. Right now don’t bother about the calculations.

Figure-3: VOC Vs. VOPNow what we need to do is to analyze why his process is incapable of meeting customer’s requirement.

Where is the GAP?It is evident from the figure-3 that the control limits of his process is way beyond the customer’s specifications. In other words there is a huge gap between customer’s expectation and his current process efficiency, it’s the time that he needs to improve his process by optimizing the following variables that can influence his arrival time to the office.

8. When he slept last night?9. Did he had drinks last night?10. When he woke-up?11. When he started from the home?12. How was the traffic in the morning?13. How fast he was driving?14. Which route he took?

How to Reduce the GAP?Above mentioned variables were studied and optimized using DMAIC process,7 this enables him to improve his process so that on 95% of the occasions he would be landing in the office between 8:02 and 8:26 AM, hence he would be complying with the customer’s specifications as shown in 7 This might be a small statement made here! But this is the whole crux of the six-sigma – it helps us in optimizing the process.

figure-4. It is also evident that the control limits of the process is inside the customer’s specifications. But still on 5% of the occasions he would be outside the specification limits. Hence process needs further improvements. If the process is improved to such an extent that there is only 3.4ppm failures then it is called as six sigma process which means that if he make one million trip to office, then I will be late only on 3.4 of the occasions.8

Figure-4: Customer’s specification Vs process control limits after process improvement.Summary

8 Let’s not bother about this statement at this moment.

http://6sigma-concepts.com/

Soap Manufacturing Company

A company is in the business of making soaps with a specification of 50-55 Gms/cake. Anything less than 50 Gms may invite litigation from consumer forum and anything beyond 55 Gms would hit their bottom line. They started the manufacturing and found huge variation in the mean weight of the cakes week after week (see figure-1, January-February period). They were taking one batch per week and producing 250000 soap cakes per batch. From each batch they draw a random samples of 100 soaps for weight analysis. Average weight of 100 samples drawn per batch for the month of Jan-Feb is given below in figure-1.

Figure-1: Average weight of 100 sample/batch for Jan-Feb period.In order to evaluate the performance of the process, a control chart is plotted with VOP & VOC (Figure-2). Presently it represents the case-I scenario, Figure-6 where VOP is beyond VOC.

Figure-2: Control chart for the month of Jan-Feb9

They started continuous improvement program to reduce the variability in the process using DMAIC process.10 They were able to reduce the variability to some extent but still majority of the soap cakes were out of specifications (March-April period, Figure-3). They continued their endeavor and reduced the variability further and for the first time the control limits of the process was within the specification limits (May-June period, Figure-3). At this point their failure rates were reduced as 95% of the soaps would be meeting the specifications. We can further reduce the variability to reach the 6 sigma level where the failure rates would be 3.4ppm. But now, we need do a cost benefit analysis as improvement beyond a limit would involve investment. If 5% failure rate is acceptable to the management then we would stop here.

9 USL & LSL is customer’s requirement. Don’t bother about the calculation of LCL & UCL at this moment.10 Once again, it uses DMAIC process which can’t be discussed right now. Right now assume that they were successful in continuous improvement.

Figure-3: Continuous Process Improvement.Comments:

It is not always desirable to achieve 6 sigma level, a 3 sigma process is good enough. But there are cases where human life is involved like passenger aircraft, automobile brakes and airbags, medical devices etc. and in these cases it worth going to 6 sigma and beyond to ensure the safety of the human life.