 Welcome to session 39 on our course on Quality Control and Improvement with Minitab I am Professor Indrajit Mukherjee from Shailesh J. Mehta School of Management ah IIT Bombay. So, we are discussing about multiple response optimizations. So, we will just revisit the problems what we are solving and how to solve it in Minitab we will try to see ok. So, this is the example where we want to maximize the yield over here. So, one one is maximization of yield over here ah what we are ah this is to be maximized then viscosity should be on the target and molecular weight should be lower the better type of scenarios. So, one is maximization. So, this is known as ah this is known as larger the better functions like that and on target is known as nominal the best and minimization is smaller the better type of functions like that. So, all this response function we can we can define like that if I have the response surface then either one of them has to be maximized minimized or to be on the target like that any specification is like that only engineering specification we can think of ok ah. So, in this case the boundary condition is taken from Montgomery's book. So, this is given y 1 should be within 70 to 80 and the target value is around 80 then y 2 should be ah between 62 and 68 the target value is 65 over here. So, so y 1 is we want to maximize the target value is 80 and the third one is minimization problem anything below 3400 is fine, but we are giving a ah some boundary conditions over here, but any solutions less than this. So, ah and we have defined some target values over here. So, minimization smaller the better means ah any values the problem statement is 3400 anything below that is sufficient like that. So, in this case ah we have to solve this problem. So, this is a CCD design that was experimentation was done. So, we have a factorial points, we have a ah center points and we have the axial points like that. So, this is the experimental setup that we are having and we are using a composite disability to reach to the optimal solution. So, MINITAB will we will use a response optimizer of MINITAB MINITAB ah which will use a ah heuristics to solve the problems of multiple response and it will give you the final solution of setting of x 1 and x 2 ok ah and it is using disability functions that I told that for smaller the better type it will use a disability function and nominal the better there is a different disability function and this is given by Derringer. So, the for the larger the better the function is like this. So, if I can reach the target immediately the what we will get is that disability dj value will be equals to 1 and the composite score ah disability can be calculated which is nothing but the geometric mean of the of the values of dj is over here. There are different ways of doing this, but one of this is geometric mean that we are referring over here ok. So, in this case MINITAB will do it automatically for you it will generate the x 1 and x 2 conditions and it will see the disability and composite disability and based on that final solutions will be derived after the complete iteration process like that ok ah of the algorithm. So, what ah how do we how do we have MINITAB? So, this is very easy. So, what we will do is that I am taking the data set over here and this is the ah excel ah this is the worksheet file that we have the data and C 5 and C 6 is the experimentation ah design ah design matrix and this is the outcomes y 1, y 2 and y 3. So, what we have to do is that first we have to generate the response surface over here for y 1, y 2 and y 3 how do I do that design of experiments response surface and then analyze response surface over here then you mention that I want to see y 1 ah yield I want to develop a response surface for this and go to terms because we are using CCD design we can we can go for full quadratic equations and then we click ok and ah in graphs you can see the Pareto plots also and then click ok and you will get the equations of the of the final equations for this and the r square value will tell me whether the model is adequate. So, r square 97.05 is ah adjusted r square is quite significant. So, we can we can retain the model. So, this is a quadratic model that we are retaining over here and ah and in this slide you can see this is the model for y 1 this is the model that is used and MINITAB also generates this model. So, ah and in that case also the Pareto plot will show you which is important which is not. So, in this case ah we we can see that Pareto plots indicates that square a square is important a is important b square is important b is important although maybe is not so significant, but we will retain this one because in the book example it is retained like that for the response surface. So, we are retaining this one. Similarly, for the second y 2 variable viscosity we have to generate the response surface and I am generating the response surface analyze response surface over here I will take y 2 over here and terms again full quadratic model we are using and then click ok and then again you will find that r square value adjusted value is 82.8 we can retain this one and ah then in that case also Pareto over here what it shows is that ah this Pareto plot says only b square is important, but because ah then we have to retain b and we are retaining all the variables because in the in the example this monomer is example this equation was retained like that ok. We can change the equations also, but I am retaining this one full quadratic model we are retaining over here. So, then the third one is ah the molecular weight. So, I am going to response surface and and I am using ah what monomer has recommended the response surface that is that that equation that was taken. So, y 3 I am taking over here then the term condition what we will show we will only use the linear model over here because in the book it is linear model only that is considered when you consider only x 1 and x 2 and no interaction. So, that is the model we are using over here. So, this is around 61 percent explain variability and ah and this you can see that all the variables are important a and b is important over here. And ah the final equation is given over here. So, this is y 3. So, what we have done is that we have developed the response surface for y 1, y 2 and y 3 and now ah we can we can just optimize using response optimizer of unitab. So, what we will do is that now we will go to stat and ah then we will go to DOE response surface go to response optimizer over here. When you go there what it will ask is that it can optimize up to 25 response. So, how do you like to optimize these three variables three responses. So, y 3 is molecular weight we want to minimize this one ah y 2 is ah something y 2 is viscosity. So, it should be on the target and target value is 65 that we know and yield we have to maximize. So, we are just mentioning that one. In setup what we will do is that molecular weights ah we can take let us say the target value is around ah this is approximately 3 2 0 0 what we what we have written, but we can mention something lower than that also it does not matter like that. So, we have to minimize this one upper bound that is given as 3 4 0 0 like that. So, this is taken like that. And then target value of y 2 viscosity is 62 to ah this is taken as 68 and this is 65 that is taken as correctly and yield has to be maximized. So, target value is 80 over here and this varies from 70 to 80 like that we have taken like that. And the weights for is that means what is the importance of this variable y 1 y 2 we can keep equal weights to this weights will be equals to 1 and we can change the disability function over here. So, in this case ah we can change the shape of the disability function we are not doing that. So, importance in matrix is also taken as 111 like that we can keep it same or linear a linear disability function we are keeping over here. So, ah in this case we we click ok and then in options what we can do is that you can give restrictions to the constraint to the region. So, region is minus 1.414 that you have defined to plus 1.414 and similarly for the second one also we we are giving a constraint to the region. So, may be minus 1.414 and next one is 1.414. So, solutions can differ because it is using a heuristics to solve the problem because there is no ah single way of solving this problem. So, in this case because there are multiple response over here. So, we cannot reach global optimal solutions for all variables and it is not possible. So, in this case we have a tradeoff solution and for that we are using a ah some kind of heuristics to get the solution we need to have we will use that one internally and but disability function will be used and geometric main will be used like that. So, we will click ok and let us try to see the solutions like that. This is one example I have taken you can you can also change this option and change it to ah like this this options in in settings what you can do is that we have taken 3000 you can change this to 3200s like that that is also possible and I am just giving a wider range over here for the solution. So, in this case what what I will do is that I will click ok and when you click ok what happens let us try to see the solutions over here. So, this is the solution that we are getting over here. So, I can just copy this as and paste it in excel the solutions one of the solutions that we have got ah Minidiv has derived one of the solutions over here. So, solution is reaction time is minus 8.886 and I am assuming continuity of the variables x 1 and x 2. So, other one is minus 8 0.816 this is the coded coded information we can convert into two variables that I have told earlier also and the molecular weight that we are generating over here is approximately this solution is giving a ah predicted value of 3059 and y 2 is predicted around 65. So, it is hitting the target for y 2 and y 1 is around 78. So, we wanted 80 ah to be maximized, but it has not reached ah to that point. So, there will be a. So, all the disability values of this will not reach to 1. So, ah when you take the ah take the geometric mean over here the composite disability is coming out to be 0.84. So, it is lying between 0 to 1 also had had the better type of scenarios if composite disability reaches 1 that means, it will hit all the target values that is defined over here for y 1, y 2 and y 3. So, then only the composite disability will be equals to 1. So, it is not the case because y 1 it has not reached and also molecular weight the target value is not reached. So, in this case what composite disability that we are getting is 844 like that. So, this is ah what we wanted to explain in this ah in this session on multiple response optimization. So, you can change the equations, you can change the equations and you can you can ah just derive solutions like that because ah response surface changes and in that case ah algorithm also finds different points and in that case it will be a different solution. So, ah whichever the more accurate the response surface more accurate will be the results like that that is the that is the thing that we can say. And finally, what we can do is that this is another recommended settings that we are getting in MINITAB over here and this is giving me a composite disability of 0.94 like that and so, ah solutions can change also. So, ah finally, what you have to do is that you have to make a confirmatory trial over here. So, this is the reaction time setting condition that is MINITAB is predicting let us let us do trial after that and let us figure out what is the predictor what is the actual outputs like that. So, this is the predicted one solutions that is predicted values over here. So, actual values can only only be seen when we are doing confirmatory trials like that. So, this is a way we should do and this is the response surface optimizer plot that that you can see in MINITAB also. So, this is ah in MINITAB you can see at the end of the when you come down that this is the solutions. So, what you can see this is the final solutions current solution minus 8 ah this one is the value that is that is giving MINITAB. So, red one you can see this is the solutions like that composite disability values are individual disability values are given and individual values are also mentioned over here. So, this is the response optimizer ah that diagrammatically you can also see like that ok what is the solution although the solution is given over here this is the solution that is given over here ok. So, we will move to some different topics which is also important that we have to mention over here is known as factional factor design. So, motivation behind this is part city of effect principle that means, if you have a n number of factors many number of factors and you do not want to do full factorial experiments for all the factors because you do not know which is important which is not. So, to reduce the number of factors what we do is that we do screening screening of the factors in design of experiments before we go for optimization what we do is that we do screening of the factors there can be n number of factors there are 7 factors all may not be influencing like that there can be 10 factors 20 factors ok. So, what we have to do is that we have to reduce the number of factors based on the importance like that. So, there is a important principle that I earlier mentioned also star city of effect principle that there are lots of factors, but very few are important over here. So, system is dominated by the main effects and lower order interactions like that. So, maybe if you have factor a and b like that this main effects are important and maybe a b interactions. So, if you are let us say a b c over here. So, this is the first order interactions over here and then we can also see interactions third order interactions a b c. So, in this case what happens is that more and more you go for higher order interaction they become insignificant basically. So, that is known as part city of effect. So, individual factors are important sometimes and second order up to second order interactions can be seen like that and third order interactions this may be insignificant like that we try to avoid that one. So, this can be dropped on many of the. So, screening experimentation is and out of these factors all may not be important like that. So, we can screen maybe only one factor is important over here. So, this will help me in the final experimentation. So, sequentially what we can do take 3 or 4 factors see which is important then think another sequence another experimentation and we do and we screen the factors like that. So, we do not go all at a time we may be we do sequentially and then we try to extract the information which is required that means which other factors are important like that ok. So, how do you do that basically this is fractional factorial name is fractional factorial design and the complete experiment if you have k factors we told that it is to report k design like that. But we are not interested in running the full experimental trial let us say we want to reduce we want to see that which factor is important. So, in that case in fractional factorial we will run only. So, there are let us say 2 cube designs. So, this is 3 factors ABC. So, if you are doing full factorial design this is 2 cube over here that is 8 experimental trial but as a researcher I am not interested or as a experimenter I am not interested to run the full trial because maybe because of economic economic constraints I do not want to do that I want to figure out which factors are important or not which is. So, I want to run only half of this trial like that I want to run only half of this trial this is one half fraction this is known as one half fraction over here. So, this may be written symbolically it will be written as 3 minus 1 that means I will only run a 2 square trials whereas I should like that. So, if you have done full factorial of this it would have been all combinations would have been like this but I am only running half of the trials I do not want to run full trials over here and and based on that I will I will screen the factors based on that only I will screen the factors like that ok. So, so then you will find that a whenever a is the first trial is given as this trial experiments a is positive then b is on the lower lower level c is on the lower level like that. So, but if you multiply this symbolically ABC what will happen is that you will get a ABC over here that is that is positive over here. So, similarly when combination is b only b is at high level and other a and c is a low level what will happen is that this is the combination that I am running over here and but if you multiply this a b and c what will happen is that ABC is positive over here again for c ABC will be positive like that when ABC all at high level what will happen is that ABC symbolically if you multiply the plus signs over here that is also positive like that. So, all positive signs if you accumulate that means these are the trials you have to run ABC and ABC if you run this trial this this only this block this block is known as the principal block this is known as the principal block and there is a alternate block over here. So, either I can run this one or I can run this one like that both you can run any of this you can run any of this. So, my idea is that I do not want to run the full trial I am only running the half of the trials like that half of the experimental trials like this and I will lose some information I will always lose some information because I will not I will not be able to calculate all interaction independently like that. So, in this case some information I will lose, but in this case I am I am not going for optimization over here I have my idea is to screen the factors to see in the with minimum number of trials like that. So, in this case what we do is that. So, this is known as design generator over here. So, if I have to make it into fractions which fraction we should run and which fraction we can omit like that and for that we use a design generator here ABC is used as design generator. So, when I multiply the signs of A, B and C and it will give some either plus sign or minus sign. So, one block will be plus sign one block will be minus sign like that I can take the first block which is known as principal block and only carry out these trials over here that is combination ABC and ABC like that. So, and this is the combination that we will run. So, this is basically half of the full trials over here 2 to the power k by 2, so 2 to the power k minus 1 that I mentioned. So, this is 2 to the power 3 minus 1 that we are running is symbolically we write like that ok. So, so when you run this there will be some confounding there will be some confounding in the effects that means, when I am estimating A basically I am not estimating along A because there will be some allies that means confounded because I have not run the full trial. So, this sum of the information. So, over here when I estimate the effect of A basically I will estimate A plus BC like that. So, this is A plus BC I will estimate. So, how it is coming? So, A if you multiply with ABC what happens is that this is A square and BC. So, I can I can figure out this is the main effect A with which it is confounded with that means, it is together with another interaction effects. So, when I am estimating A basically I am estimating A plus BC over here, when I am estimating B I am estimating basically B plus AC over here, when I am estimating C I am estimating C plus AB over here. So, this is known as confounding that that will happen inevitable in case of fractional factorial design that means some information independently I cannot calculate BC interaction effects. So, this will be inside when I am calculating the effects of A like that. So, this is confounded over here A and BC is confounded and B and AC is confounded like this C and AB will be confounded and I told this is the principal block that I am running all plus signs is the principal block and this is the alternate fractional that you can run also. So, either I run principal or I run the alternate fraction it does not matter like that. So, Ally structure will be shown by Minitab whenever you select a design like that a fraction of factorial design it will show like that and so, we will have this confounding effects that I told that means there will be some effects which will be when I estimate one it is not estimating that one it is estimating something along with that some other interactions like that. When I am calculating main effects I am from the 4 experimental steps I am doing. So, I am estimating A from over here it is not A it is A plus BC that I am estimating basically. So, this is also estimated when I am estimating A's effect basically. So, that is the idea. So, some all information but if you run the full trial what will happen is that I can independently estimate A I can independently estimate BC also. So, that is the idea that goes and that is known as Ally structure over here what you see and Minitab will give you automatically which is alleged with which one like that which is confounded with which other interaction effects or main effects like that will be shown ok. So, we are taking one of the examples over here where it is a 2 to the power 4 factor experimentation temperature pressure concentration and steering rate. So, 2 to the power 4 is 16 trials complete experimentation over here. So, I do not run full factorial over here. So, I will only run half of the fraction over here. So, 2 to the power 4 minus 1 half of the fraction we want to run over here and which are the which are the which are the runs that we will take. So, that is one will be taken AB will be taken. So, you can just multiply A, B, C, D over here and principal block will be whenever A, B, C, D multiplication is positive plus 1. So, that will be the principle that will come in the principal block. So, over here minus minus this will come in the principal block here also if you multiply that will be positive A, B, C, D will be positive. So, this will come this will come like this you can see which are the trials that will come over here. So, out of 16 trials only this 8 trials if you run in that case we will be having a having an interpretation that we will have some some information that which is which is basically prominent out of A, B, C and D. And that specificity of effect principle we are using over here that if you consider all interaction it is you have to run the full experimental trials like that factorial design like that. We do not want to do that we have run half of the fraction and based on that we will just screen the variables over here. So, how do you run the trials? So, we will only run this one and minute I will do it automatically for you you do not have to multiply anything like that. So, minute I will do it. So, this is the experimental run where I want to see that which are the factors affecting the filter rate which is Y basically over here and we want to analyze that one. So, we will see in the experimental design so, I have to create a fraction of factorial design. So, what I will do is that stat design of experiment factorial design create factorial design. So, I will go to create factorial design two level factors number of factor is 4 because I have four variables over here that is A, B, C, D and then what I have to do is that four factor then design I will go to design. So, it will say full factorial or fractional factorial. So, I will say half of the fraction 8 trials I will run over here and it will give you some resolution resolution 4 over here. So, this is the resolution 4 design basically in fractional factorial there is a term which is known as resolution of the design. So, higher the resolution better is the fractional factorial design. So, in this case what I am doing is that number of replicates corner point is nothing over here. So, I will click ok and then factors we can add over here. So, this is factors we can write then in options what we do not pull randomize we do not randomize use principle block. So, either you can use fractions over here use principle block I am using the principle block over here which is all positive that is that is the principle block. So, in options so then you click ok the design will be created. So, if you see that this is the design and it will say which is allied with which one like that this is the allied structure that is given. So, E is confounded with B, C, D, B is with A, C, D like that and A, B is confounded with C, D. So, when you estimate A, B, A, B interaction it is actually estimating A, B plus C, D interaction basically. When you estimate A, it is estimating A plus B, C, D basically ok and we told square C, D of effects B, C, D is maybe ignored. So, effectively you are making a good judgment and only the estimation will be correct and close to effect of A basically. So, because higher ground interaction has little impact in the system. So, in that case we can consider that one as ignoring we can ignore that one basically ok. So, when you run this so 8 trials are given over here. So, there is no blocking over here. So, in this case 8 trials and we have got the design. So, I have to respond C, D give you right over here and then run the trial and then run the analyze factorial design like that. So, this is already created. So, if I go to the examples factorial design it is already created. So, in this case filter rate experimentation. So, this is the principle block is given over here. So, this is the trial. So, how do I analyze this one? I go to stat DOE factorial design, analyze factorial design. So, you have to write what is the response then you go to terms and in this case what we are interested in that we we can go to the second order terms all all terms we can see over here. And in graphs we can see the Pareto plots over here and then click ok over here what you see that estimation is not done because we do not have degree of freedom here over here. But which is prominent we can see A C D A C and D is prominent over here and A C and A D is prominent over here. So, B is not prominent AB interaction is not prominent. So, what we can do is that this is the this is the plot what we are seeing. So, initially what we are seeing is that magnitude of A D A A C and D and C is quite prominent. So, we will keep only these variables in our next analysis. So, what we will do is that go to stat design a factorial design analyze factorial design over here and in terms what we will do is that we will go to first order interactions and first order factor effects and then A C we will put A D we will put A C A D we will put and based on that we will see A C and A D. So, we will click ok and we will click ok and then we can see the estimations over here and what we see is that B is not coming out to be prominent. So, we can eliminate B like that. So, B is very very insignificant. So, what is important over here from this analysis? So, what we are seeing is that A A D A C D C and only B we can eliminate. So, next what we can do is that we can eliminate this one factorial design and then we can analyze and then in terms. So, we have to remove let us say B I will remove from here only A C and D is important and their interactions. So, in this case the model explains R square value is around 99.2 C that is very good that means other terms even if we ignore all other terms like second order, third order all other interactions higher order interactions like that I am getting a model which is having a predictivity around 99.26 like that. So, you see the factors which are important over here what you can see is that only main effect A D and C and A D interaction and A C is important others are ignored over here. So, you could have done up to four interaction effects A multiplied by B and C and D like that if I have done full trials. So, I do not need full trials over here out of these four factors what I am seeing is that only A D and C is important. So, I can eliminate B from here. So, that is the objective of fractional factorial design that is objective of fractional factorial design what we can do is that we can we can eliminate this one and what we can do is that when we when we have done this then also we can see the factor effects factorial plots over here and if you want to get the settings like that. So, what we can see is that main effect plots interaction is prominent. So, we will see interaction plots over here and based on that we can get the settings like that. So, in this case filter rate if you have to maximize. So, in this case what what is required is that C is minus 1 and also we can see that A is at plus 1 like that A and C at A at plus 1, C at minus 1 that is a combination that we have seen over here. And in this case for case of D, D is at plus 1 over here, D is at plus 1 over here. So, A was at plus 1 that is the condition A plus 1, D plus 1 and C at minus 1 that is a combination that we are getting over here and B is not significant. So, in that case it can be at any level it can be at any level and which which we can see. So, if you want to see the B's effect. So, what we can do is that I know analysis you go to interact sorry you go to main effect plots that will give you the idea. So, in this case what what we are doing is that main effect plots over here and we can consider filter rates over here and the factors are considered from A to D like that you click ok what you will see is that this is the main effect plots that you are seeing. Do not go by settings. So, what I wanted to show is that I wanted to show B's effect over here that it is quite flat over here quite flat and slope is quite flat over here. So, in this case B's effect is not significant that is also reflected after the fraction of factorial design that is also reflected after the fraction of factorial design. So, this experimentation is done for screening factor fraction of factorial design is used for screening. So, out of four factors we can see A, C and D is important and we can screen that one. So, this is the fraction that we have run. So, this is the experimental trial that is done and all the all the combinations. So, this is principal plot. So, all will be plus 1 over here. So, if you multiply A, B, C and D it will be plus 1 like that ok. So, this is the interaction plot that I showed to you and based on that this is the equation filter rate equation and this is about 99 percent explained variability that we have seen when we have ignored B and higher order terms over here. So, that is the sparsity of effects. So, then there is design resolution over here it can be resolution 3 and resolution 4. We have used 2 to the power 4 minus 1 is a resolution 4 design where two factor interactions are allied with each other and the main effects are allied with three factor interaction that we have seen that means allied structure if you see. So, A is allied with B, C, D that is third order interactions and A, C is allied with B, D like that two second order interactions are allied with each other like that ok. Two factor interactions are allied with each other like that. So, this is resolution 4. So, MINITAB will generate automatically. So, higher the resolution better is the interpretation you can you can have better interpretation better better screening like that. So, resolution 4 and resolution 5 designs are also possible. So, this is the resolution of design. So, resolution this is a resolution 3 design where you can see that there are 7 factors A, B, C, D, E, F, G like that and if you run the full factor delta what happens is that this is a 2 to the power 7 experimental trial. So, 128 trials has to be run to see all effects under interaction. But if I use the theory of that specificity of effects principle. So, in that case I can use fraction of this. So, in this example what was used is that 2 to the power 7 minus 4. So, only 8 trials was considered over here to find out which factor is important and which is not. So, in this case what was figured out is that only AB and D is significant over here only main effects are significant no other interaction nothing was significant like that. So, we have arrived at a conclusion that only AB, D is important and that can be screened like that. So, that is also possible. So, this is fractional factorial design more you study you can see resolution 3, resolution 5 designs like that and how to define resolutions like that. But I told that if you go to the higher resolution it is better always because then main effects are not aligned with lower order interactions like that. So, that is the principle we will use so that we can safely assume that this effect is significant like that this effect or this interaction lower order interaction is significant like that we go for higher resolutions like that. So, we will continue discussion on a different topics that is known as Taguchi's experimentation and that will be the final topic that we will cover. So, thank you for listening.