 Welcome to session 9 of our lecture on Quality Control and Improvement Using Minitab. So, I am Professor Indrajit Mukherjee from Shailesh J. Mehta School of Management. So, last time what we are discussing about control charts which is used to differentiate between normal and abnormal scenarios with respect to time. So, what we are discussing last time we will start from there and this was the basic details that we are discussing over here if it is a normal distribution. So, in that case what is expected that 99.73 of the observation should fall within plus or minus 3 standard deviation. So, if we can draw two demarcation line one is known as upper control limit line over here which is also in short form you can write as UCL and lower control limit line as LCL. So, if we can define this line with a central line over here which is known as CL over here and that will help us to differentiate between normal and abnormal scenarios. So, this is a normal scenario this is a abnormal scenario what we discussed last time. Abnormal scenario means assignable cost due to certain assignable cost or cost that is that leads to a abrupt change in the process mean over here because central line is basically mean over here. So, overall process and this is the mean over here what you see over here. So, suddenly there is some something has gone wrong and the average has come out which is this observation over here this will be some average which is plotted at every time point and at different time points we are taking information that also we discussed last time. And we are plotting average because average tends to be much smoother and it follows normal distribution from the central limit theorem which we which we have told last time. So, if we can and this is the CTQ we are monitoring the CTQ basically this x bar is for a specific CTQ let us assume. So, if we can define the upper limit line and lower limit line and a central line. So, in that case it becomes easier for us to understand when the process has gone out of control. So, this is if the process if something goes outside the limit line we say process has gone out of control basically ok. So, then what we do is that we stop the process and take some precautionary actions and so that this does not recur this type of incidence. And if it is a mass manufacturing we may not be able to stop, but later on we can diagnose and see that which is which is the assignable cause which which we can block in next operation or next shift maybe it will not recur like that. So, it uses a signal kind of signal alarm kind of scenarios and it is a proactive actions that we generally try to take so that it does not recur basically ok. So, this is the concept of control chart. So, we plot with respect to time at different time points what we will do is that we will monitor the average values over here and for that we will define a limit lines over here based on process average overall process average and based on the information of standard deviation of the process or range of the process maybe we will try to differentiate where to use which one over here. So, mostly people prefer to use R or S like that for expressing the variability and so those things we will discuss how to implement that in Minitab. So, assuming that one CTQ we are monitoring and we want to use control chart for that and try to figure out when there is a abnormal scenario when there is a normal scenario like that ok. So, there are different types of control chart suggested by researchers like that ok. So, one is known as variable control chart what you see on the left hand side and one is known as attribute control chart over here. Variable control chart like thickness what we have mentioned so that can be considered as variable for that CTQ we can use variable control chart because variable control chart where we are interested in location and also in precision both the things are important for me because variables can be measured and in that case continuous variable mostly and in that case or ratio scale variables we can think of. So, that is what we can we can monitor and for that mean accuracy and precision both is important. So, in that case both can be measured and can be monitored like that and attribute like defects what you see over here and defectives. So, difference between defects and defectives we need to understand when to use defect control chart when to use defective control chart. So, in a product what happens is that there can be innumerable problems or defect types like that some can be rectified some may not be rectified like that. So, when we can rectify all then we have a we have revoked all the defects and we have a good products like that. But sometimes what happens is that one of the defects has gone beyond my control. So, if there is a both dimensions which has to be within certain specification over here and I have manufactured something which is having a dimension more than that one that means I have basically the both dimension has increased over here due to processing or some other mistakes like that. But this cannot be rectified you see this kind of defects cannot be rectified. So, that means this bore is of no use for the for the next operations like that ok. So, this goes as a scrap basically this goes as a scrap over here. So, that whenever it goes as a scrap and we cannot do anything about that. So, then we define that as a defective items like that then we define that as a defective item over here. So, whenever so to create so to express a defective items we we need at least one defects which cannot be rectified basically ok. So, attribute control chart talks about defects and defectives this is a primary control chart which is implemented in processes like assembly processes or any other manufacturing processes. First our identification whether defects are going abnormal defectives are going abnormal and then every so then can we link it with some CTQs and monitor that one. So, it has to be converted because defects has no meaning defects because of what some CTQs like that. So, we have to convert the defects into some CTQ and monitor the CTQ so that we take more proactive actions on the CTQs like that when that is going out of controls like that. So, because that will create defects that will create defects. So, this is a primary label of data analysis what we do by control charting, but if you want to go depth in depth like that then we go for variable control charts like that because defects are linked with some CTQs like that ok. So, if you can link defects with CTQs multiple CTQs can be and individual CTQs can be monitored or together also there are control charts to be monitored which is known as multivariate control chart ok. So, so the basic difference between defects and defective is that defectives cannot be rectified defects can be rectified, but minimum one defect is required to define a defective items like that. So, there are different types of control chart to monitor that one. Now, in variable control chart we have to understand that at a given time point t how many observations we are taking over here, how many observations we can take for calculating the average of the process like that average of the process. So, we may manufacture let us say baby foods and in that case 4 we we can take let us say 50 packets are coming out of the process at a given time point and we will take only 4 out of that. So, that 4 number which which we are taking at a given time point to calculate average weights of the packets like that. So, in that case that is known as subgroup size that is known as subgroup size in in quality control subgroup size subgroup size ok. So, this subgroup size is an important aspects. So, we need to ensure some subgroup size over here based on the process details. So, we have to define the subgroup size. So, generally subgroup size is taken as 5 or more like that 4 to 5 or more than that also can be taken depending on how much we can we can if it is destructive kind of testing in that case it becomes difficult to take more samples like that, but more of 5 is sufficient enough to draw the primary control chart like what is known as X bar R chart or X bar S chart. So, whenever the sample size some some guideline is given this is not absolute guideline what we are using over here. So, maybe if the sample size is more than 5 we can we can go for X bar and standard deviation S chart. So, this is known as X bar and S chart and this is known as X bar R chart. So, if it is less than or equals to 5 maybe we can restrict to X bar R chart like that. So, if it is a variable and more than one subgroup size is there. So, n is greater than 1 what is written over here n is known as the subgroup size and if n equals to 1 sometimes in chemical process what happens is that we do not need more than one samples because the variation will be very less sample to sample at a given time point maybe viscosity or something like that we are talking over here. In that case there is little variation even if I take more than one samples readings will be more or less same. So, in that case it is unnecessary to take more number of samples like that. So, if that is the scenario what we do is that we plot individual moving range like that we can write IMR type of chart which can be written as IMR over here or if it is more than one samples can be gathered and we know that there will be sample to sample variation. So, in that case whether we have we can take about 5 samples like that if it is within 5 samples what we will do is that we will use X bar R chart like that and in case it is more than 5 also we can we can we can afford. So, in that case more appropriate chart will be X bar S chart like that. So, we will go variable control chart we will do first and then we will see attribute control chart how to do it in MINITAB like that ok. So, variable control chart within that also we are restricting over here to for larger shift the control chart that I use to detect large shifts and in that case what she what has recommended one is X bar R chart X bar S chart and if it is n equals to 1 then how to use individual moving range chart which is known as IMR chart like that that we will discuss first then we will go to attribute chart. Within attribute charts also there are bifurcations over here what you can see is that if it is defective chart we have constant sample size the number of samples that is subgroups that is taken in whether it is constant or whether it is varying like that. So, in case it is constant in that case we can use P or NP charts like that we can also use NP charts over here or P charts. P and NP chart is more or less same only for the operators benefit of the operators what we do is that we can multiply it within and that gives the operator some sense that defectives if we write 0.00 something defectives like that we will see how calculation is done. So, NP chart is only to facilitate the operator to monitor like that they understands 2 number of defects, 3 number of defects, 2.5 defects that is understood. But if we if we write defects in decimal place or something like the defectives in decimal place that is not well understood by the operator. So, in that case what happens is that somebody prefers to use NP chart instead of P charts like that. But there is also possibility that we can vary the sample size when n is not constant at any given time point. So, n keeps on changing with respect to time like that. So, if we keep on changing then in that case we have a variable control chart P chart with variable sample sizes like that that we will also demonstrate in that if that is the scenario how the control chart needs to be used and how it is monitored like that. Similarly, in case of defectives. So, defective is like how many assemblies you are monitoring the assembly operations like that and 20 engines are manufactured and in that case you take some sample and or 100 engines are manufactured out of that you are taking let us say 5 of them and you are checking that whether it is defective or no defect 0 1 condition basically whether it is working or not working like that that is the condition defective is 0 1 condition. So, either it is working or not working type of scenario. Here we can have 1 defects, 2 defects like this there can be n number of defects like that. But whenever I am saying defectives this is 0 1 and the primary distribution that it follows we are assuming is binomial. So, this is binomial distribution that underlying distribution is binomial for this and this is a Poisson distribution which is considered over here when we are talking about defects like that ok. So, this way so defects when we are talking about defects the control limit lines will be calculated and that will be based on binomial distribution and and for that different types of control chart n n p will be demonstrated and for if it is defect types also n can vary over here. So, if n is constant in that case we have a c type of control chart and in case we have a we have varied n over here. So, at a given time point there are number of samples that was inspected is different and number of defects is monitored in that case we can use huge chart. So, in software industry we will find that people are talking about lines of course. So, within that lines of course how many defects there is or error that has happened basically. So, that can be monitored because lines of course are different at given different time points like that in a projects it can vary. So, if it varies in that case u is taken as so this can be calculated at a defects by a number of observations. So, n i over here and these are the ci number of defects at a given time point t let us say and if you take the ratio of this you will get u conditions over here and this is the u variable over here and this u u will be plotted like that in a control chart instead of c charts over here ok. So, our basic assumption is that quality characteristics can be of different types, but if we can go to the highest level of precision then we have a variable control chart for that and if we are going for defects or defective attribute types of scenarios which is quality of data is not so high as compared to when we are going to variable control charts like that. So, here only defects and defective that level of differentiation is only possible. So, lowest level is defectives we can assume the data of quality is lowest level is defective then defects maybe and then variable control chart which is the highest level data we both we can gather and based on that we take a decision like that ok. So, assuming that CTQ thickness we can measure and in that case how to implement control chart that we will try to see ok. So, what do you do in statistical process control basically what we do we take some sample at a given time point that I told already that at any given time point t we will go to the process and we will try to take number of samples n equals to 4, 5 or something like that more than 5 also. So, we take a sample then inspect the samples and calculate the average of the samples maybe and range of the samples maybe of the 5, 4 or 5 samples that I have taken. And then we have a control chart first we create try to create the control chart and if it is already created then we just plot the values what we have taken over here. So, control chart we will have some limit lines upper limit lower limit that we have discussed earlier earlier and then we see whether there is abnormal or normal condition like that is it any abnormal condition is there and in that case if it is yes stop the process like that. So, this will be assignable as misprint this one so assignable cost. So, in case abnormal conditions are not there we will not stop the process and if if assignable cost is there or point is going outside the limit lines that is defined LCL like that. So, in that case what will happen is that we will say the assignable cost is there and we will try to eliminate that cost. So, we will have a 5th point diagram for this. So, this is the process CTQ which is going wrong over here and this is the cost which is basically primarily influencing and for that we are getting assignable cost how to deal with that so that it does not trigger. So, this may be assignable cost 1 so and that way we have to we have to think like that ok. So, so this is the overall process so we try to monitor whenever product is coming out of that we do not inspect everything out of that. So, we will take some 5 samples out of that and then find out the what is the average value of whatever characteristics we are monitoring over here and then plot it into control chart which is already created let us say and then figure out whether it is in control or out of control scenarios. So, if it is in control everything is fine we do not have to change the settings or adjustment is not required in case something is plotting outside figure out what is going wrong over here or adjust the process like that either I will block the cost or in presence of cost I will change the setting so that our again the process target values are satisfied. So, x bar is close to the target so that way we have to think using the control chart. So, this is a signal alarm based type of approach that is used and proactive approach that we can use over here. So, and this control limit lines will be calculated that we will show in next subsequent slides like that ok. So, variable control chart variable control chart monitors two thing one it monitors mean and also it monitors precision. So, one is accuracy it monitors one is precision it monitors accuracy means how the mean is moving. So, mean value is important because that has to be monitored with respect to target and here what we are monitoring is standard deviation or range over here. So, which talks about variability sigma that means how much variation in the process. So, maybe s type of control chart can be used or range type of control chart or r chart can be used over here. So, combination of this x bar chart with r or x bar chart with s is generally used to monitor the variable type of CTQs, variable type of CTQs whenever I have a CTQ which is variable type in that case I will use this x bar r chart to monitor that one or x bar s chart over here. General recommendation is if s bar is less than equals to 5 we can use this one. And if in case n is greater than 5 greater than 5 and in that case we can use x bar chart more or less the concept or the formulation over here is more or less same. So, in that case we do not have to some variable some constant will be changed otherwise the process remains same process remains same ok. So, then what we will do is that we will try to see. So, what we do is that we try to first decide on the subgroup size that is n. So, what we do is that we decide on the subgroup size n over here. And based on the subgroup size n then what we do is that we record the observation what is the x bar and what is the range at a given time point t let us say at a given time point t and with the all information. So, here also we calculate x bar and r like that. So, here record observation maybe I have 5 observation over here y 1 2 y 5 like that. And then after recording the observation I calculate the mean of the observation and range of the observation at a given time point t at a given time point t ok. So, then we calculate so this is the same thing what we are seeing over here. So, more or less then we calculate the average and range and that will be plotted basically that will be plotted like that ok. So, what we can see out of this is that whenever the mean let us say this is the mean over here and mean can shift over here mean can shift that means, location can shift over here and mean may be same and then standard deviation may change over here. So, sigma z is the standard deviation variation can also change like that. So, both will impact the control chart whenever there is a variation problem it will be indicated whenever is a we have mean shifting problem there also we will have situation like that. So, that is also we will come across when we are using the control chart like that ok. So, so here we will take a specific example to illustrate this one and this is taken from Montgomery's example statistical quality control book and in this case it is it is saying that heartbreak process is used in conjugation with the photo lithography in semiconductor manufacturing and we want to establish control chart statistical control and try to see and for that some data was gathered like that and flow width is the parameter which was monitored over here. So, this is the CTQ which was monitored and we want to use x bar r chart for this process like that. So, we will see at a given time point T1 let us say first sample observation was taken and given number of subgroups over here n equals to 5. So, n equals to 5 this observation. So, 5 observation was measured at a given time point T1 over here and at this at this instance what we can do is that we can calculate the average over here which is x over here. So, what is the average of this? So, all this 5 observation that we can calculate average and we can also calculate range maximum minus minimum maximum of this observation and minus minimum of this observation. So, this can be easily done when we are using excels like that ok. So, let me take this example and show you to excel what it does. So, then we have 25 number of averages we will get we will get 25 average over here and we will have 25 ranges over here. So, then what we can do is that x double bar can be the average of all x bar over here and divided by 25. So, this will give me a overall average or grand average over here. Also range average we can calculate. So, we can calculate whatever r values we can also calculate range average. So, we will have 25 range and average of that range can also be calculated like that ok. And then we will use the control chart to monitor this one. So, let us go to the what we will do is that we will see that this data is already available with us and we will just demonstrate that one ok. So, flow with x bar r chart. So, this is the data set that we are having what I mentioned is that with at different time points what is done is that this is the data set the same data set that we have shown from the examples over here 25 data sets and then x bar was calculated which can be calculated as we can we can just write x bar is the average observation of this. So, we can write what average. So, I can highlight this one 5 averages over here and then range can also be calculated as maximum what you can see on the top maximum minus minimum of this observation. So, maximum of this observation. So, B 3 to F 3. So, this is B 3 to F 3 and minus minimum of this. So, range can also be calculated for a specific row over here. So, similarly we can calculate all average and all range and grand average can also be calculated which is the average of all this observation that you see over here ok. Similarly, range average can also be calculated. So, these two values are required to develop the control chart limit things like that. So, x double bar is 1.51 over here and r bar is 0.33 over here. So, this will help me. So, I have 25 observations with subgroup size of 5 over here and then I can calculate individual values. So, these are the individual x bar r values. This will be plotted in the control chart and I have a grand average and average of the range and which will help me to define the control limit lines that is which is normal which will differentiate between what is normal and what is abnormal basically ok. So, this will be required when we. So, let us take this data set into a mini tab. So, we will just see and it is already there in the mini tab. So, this is the observation wafer 1 to wafer 5 I think. So, this is the same observation wafer 1 to 5. So, 1.3 to 3 5 and this is 1.3 to 3 5. So, that you can see over here ok. So, and this is a new sheet which is named over here x bar s chart. So, you can also create from excel and you can just type in the sample observation over here. So, whenever you have typed in the data then we can we can create the control charts and what is a formula that is used to create the control chart over here. So, what you see formulation over here is that there is a mini tab will use a calculation method to get the upper control limit line over here and this is x double bar 1.50 and this is same as what we have calculated 1.2 grand average what you what we have calculated over here. So, maybe we can just cross check in excel. So, grand average is 1.51 and range average is 0.33. So, in this case also you will find that the mini tab calculates 1.5056 and that is close and also range is 0.3 to 3 3 or that is also close. So, mini tab does the same calculation. So, from the same data set. So, what we have done in excel? So, x double bar and r bar is calculated in same way only we have to understand how this UCL and LCL is calculated over here and UCL has a specific formula. So, UCL formulation over here you see this is x double bar information. So, that is the grand average is taken over here and there is this is multiplied plus sign is over here and this is a constant factor that is multiplied with the r bar or r average range average what we have calculated already over here which is given over here. So, this is used in the formulation to calculate UCL and this is also used to calculate LCL over here. So, one is plus a factor and this can be written as this is the I can replace this one with A2. So, I can write this as A2 and this is also A2 like that. So, formulation changes like this x double bar plus or minus A2 r bar like that. So, we can just write that one replacing this one. So, what is this A2 over here? There is a constant which you see as D2 over here observation D2 and there is a n is the subgroup size like that n equals to 5 for our case over here this example flow width because 5 observation was taken over here. So, n equals to 5 and this D2 is a basically function of. So, D2 is a function of n over here. So, depending on the subgroup size this D2 value will change like that 3 is a constant. So, the 3 does not change over here, but these two has a relationship with n and D2 are related with n. So, this constant will change. So, this is for x bar chart over here is for x bar chart. So, I will get a upper control limit line using this formula that is given over here I will get a lower control limit line using the formula over here because every information is with me n is with me D2 value is with me which is we can get from table and then r bar is there x double bar is there. So, I have defined UCL as 1.69 and LCL comes out to be 1.31. Similarly, for range chart over here and central line will be x double bar in this case. So, this is x double bar what you see. Similarly, range average will be what we have calculated will be the central line and the upper control limit line is calculated based on certain formulas that is D4 r bar and lower control limit line is D3 r bar like that. Now D3 and D4 is again a constant which depends on what subgroup size I have taken over here which is equals to 5. So, this will define the D3 D4 value which is a function of basically n subgroup size like that. So, that formulation is also we can we can see. So, this is central line will be r bar over here and one side upper control limit line will be D4 r bar and D3 r bar. So, this is monitoring sample range what you see and this is monitoring mean over here. So, one is monitoring accuracy, one is monitoring precision over here. So, and this is now this calculation can be done by hand also because all the charts are available and this is the this is the chart where you will get the values of D2 for a given n over here. So, the n is the subgroup size and for a given value of n equals to 5 and I can get what is the value of D2 I can also value of A2 and I can get what is the value of D3 and she what has recommended this value. So, what is the value of D4? So, using this values to corresponding n changing n over here I will go to that particular specific row and I can get all the values. So, I want A2 values, I want D3 and D4 values. So, that will define the control limits for X accuracy and also to monitor precision like that. So, Minitab does it automatically for you. So, in that case it is easy for you to get that. So, how does Minitab does it? So, this data set is taken into Minitab and this is very easy. So, when we have the data in Minitab, so this is the same data set and what you do is that stat, control chart, variable chart, X bar, R chart, this is the option that you have to go because I am monitoring fine less than equals to 5. So, I am using X bar R chart over here. So, I will use X bar R chart then it will ask is it in same column, then I mentioned no it is in different columns like that. So, in that case I will mention that wafer 1 to wafer 5 is the data set you take and I select this one and X bar any point going outside the limit line we will and this is by default which is given over here as test. So, any point going outside. So, I have clicked only one options over here we will discuss about this in next sessions. So, this is if it is going beyond 3 sigma limit we will take corrective action. So, that will be a will be an assignable cause. So, I have taken one single condition that if point goes outside the 3 sigma limit line this is abnormal scenario. So, you you monitor that one only. So, I click ok over here and I click ok what you will see is that you have this control chart like that ok. So, whenever you do that so X bar R control chart the same information. So, you see all the points in mean is falling within the control limit line. So, X double bar is same R bar I have the control limit line upper and lower control limit line lower control limit line is 0 because d 3 is 0. So, in that case this is a 0 line or range can be equals to 0 it cannot be negative, but it can be 0. So, there is no variation. So, that is the lower control limit line. So, and this is upper control limit line. So, all the points are within the control limit lines. So, in this case there is no red red points that is going outside this limit time what you see. So, everything is going fine over here. So, there is no out of control or abnormal scenarios over here. So, we will stop over here and we will continue for a discussion on this topic from here. Again, we will take some more examples to illustrate and how it is happening. So, abnormal scenarios also we will try to see. So, let us stop over here and we will continue in the next session. Thank you for listening ok. See you in next session.