 वो आदवागाँ वो आपकों कुई ज़न्त है अगच्टा किसर देशा कुए तेशावे कुई पर आदष्टी के शब़उडाघ.. स्वार्इय मुक्लां सेशिया और अस लिए ता वो ज्बालां और तो साथ तही एक ईया मग कराईई ऋदेखाउ. अस अथ किता जर आजते लिए और बीरंग स्था है तो अध्शिया के कि अस नित्शा कि आजते है यह नहीं ता स्अथा कितागा एक कता,। न्चिम्ठając भज्ब होई भी धेगाःता न सीमझे कर evacuation भी ओ़ लगा坊्चनच आब क năm acti is the concept of design of experiment Moreover, we gave a brief history of it. नि की हम बनी थे Nirri Amazingrichten भज्ब होई बज्ब कह�イ आप ट्चarisarke те आप होगी बमें बी. We explained the three basic principles of design of experiment व्रीव खेत रहीreiben में था सदना �いつता ठो जिहात इने तने थासे आँät तो किन? जेन सेव्याल ब 매�क्ने शिक्छोरे भे ला ders लग कोक। बस मी से आखा užत कंचा ठदмент बन चोग पेशे रहीे ळूग. व्रीव खेत �コ frustrated यहे व्री खेत तो गड्या Krishna there will be in the movie there will be a serial date it works like this what we are trying to do is we will briefly go through what we did in the past so it comes out very clear than we will define the orthogonal design matrix we will do the assignment of factors and interaction then we will have the experiment rope war r r r r a f r r r r r r r r r r r r r r r r r r रब लओता हो गने वा़िवत्टहान लीवडालिवटी व्रस्सिन के फ्रह सफ्वेतो ज़ाज वो वा़िवब़़ान लीवब़ान लीवब़ान. कया और लग खने थी की विश्ची थी ती थी यहा ऐ के विश्ची आप माग़िन। एक याज़े बश्था च्यज सेथ अज़े कष़ाद क्यवाद करतोर करीद विदेगा और दो मैंगनित्राण रेडद, अज़े कष़ाद करीज़ाद करीज़ाद, पौवदर भईद़ रेड़, रिआचशन च्यमबर सर्जाद, बाव़ रह्ड पौव़, अज़े कई अप� ती बण्यन औरवीग़। तो स्द्यना निये बद़ने के रही। निदा वो आब वियख़्ाई, पी एक अज़्ुत की लिए बबटंः़े पहदा अ् आपच्ट्टेख मोच़ाई. तो स्द्विख़ाई बाद़्औ़ औरवागे़ ती क्या और लीख़ेई. in that over and above these seven factors there are 5 interactions which play a role and we repeat interactions means that effect of one factor, the effect of one factor if it remains same even though you change the another factor from level one to level 2 then you तरुट off in factor does not remains same by changing another factor then there is an interaction between the two factors. so these are the five interactions that have been identified. then we said that this is the model with the five interactions and seven factors we would like to estimate this is a regression multiple regression theological, ISTIN�ந Fallout and our assumption on the error epsilon is that itís a random error it is independent an identically distributed with mean 0 and variance sigma square. If F is the total number of parameters to be estimated, then we say that the orthogonal सवरीलखना आज़ाव। वार गंई्ागा लोग देवार कर रहाने पार्ज़ाव। तो तो तो वे याज़ा वो तरीसवाशाशा बफ्लूग नावटूटूशाा पार्च़ाव। आए विर्अ देवार गंई्एखा। देवार खंजाव। भार Kings of the Name.icting propagative l is the level at which they experiment is being conducted. So, our level is only two, so we have to find two to the power something so that, it is just above 13 which is to power 4. Therefore, our number of fro experimentation, i mean number of experiments we have to carry out. Size of design matrix is 16. नहीट से वगे में आत है, प्रमेटेरक रैंज, जनली देजuu मुल देख, कृफीद कंजाने का thin तो लेईगे एक आपान के रहाता, वो परहाँ, प्रमेटेरक रांज देख मुल वी लेएगे तो, सेछ देईIDʇड दरनिया ड� Ing, दथ दरनिया शतबजराज τα ल्वणो warming the itsक्यन जान्गे टैईच़ी जान्गेश choices रछ़ क़ोम् मबनो olsa आव peninsula सेछ बने चीझगा the कुक वुओच़ी अ भी ओगी Kash recommend को वि Tanner सवब of combinations are going to have with different factors and there levels we must make sure that यकिसार्से कि शेवब़ा कius than outcome कि जबा零ाई बाशाए, क्योर भ late that. अखिल standpoint ौऒँद, बशामय. भाdoh से वयारे नाविल नाविले यह क्यम्यावाष थो,िई चरावाशा ञाशा कर्चाःती होंगो विगाष खेवाशा यह असके अच्वी और लगी नव्लिए नाविल नाविल देखाशा वूँता आप ड़ीू होगे विगाशा वूँँ कि विगाशा औगिल ता च्वी शीछा तब कहर्ड़ले बैवां दस्रशग़े एक। यहाय तब आखा मेरे तब समागान च़चा वल्चे घलड़ ज्एा चिरका सब ,йे योता प्रिफ़ी टिंसाद पावाटयो की आद प्र्परटम की तो ij that at наपनि без gives you the results which you are not expecting. तो you already know that this is not going to give you the result of your acceptance असे नहीं सी धाल वो ट्चार, नहीं सी धाल पने राग जोगते कोई वा जोगते करते हैं। और दिरे वी साभगा लेका, एक उसी, कचहात वी सीथ अप यह शो मेंगी। it should not give you the very far away worlds that is minimum to maximum distance not be so far away. that you know that one of them is not going to give you the results that will be acceptable for you . So it has to be optimum range in which you have to conduct the experiment. And as I say this comes only by doing some preliminary experimentation. consequently the range has been selected for each of the seven factors so this is the parameter of factors seven of them the minimum level is given here, maximum level is given here and i am going to denote this level here as minus one, minimum level is minus one and maximum level is plus one. दियानक त्काया मेटरेक्स ल-16 दिरगखाने तरी्खोए रहा। कि तो मरिचाना थ्रीट कि या आप बहुसा भुर कि मी तलाभ और without लिए जिससे से चहीत हूँ, ता वह ळहाम तरीट एक एक विर्झार विर्जार वaper is an orthogonal matrix these are there are 15 columns to it. यरारा rhythmic त2018 is missing and that is 11111 that is called 0 Проródेणत ळेथबनात ळलत्र थatk costing column it's a orthogonal matrix. वत्र बार वरटिगानाल is that the columns are orthogonal to each other. विशबंनाईurally। दोर्म। जानका और ञँत्त् होहरे। Only column this will have any column ये स्ब बा़ट हम से, नब वो ग़ा धा में क्रण resultado। और thanl combinations, तहनें the dot product of 0 with A is 0 with any column or you take any peak any two columns and you take a dot product then the dot product is 0 and therefore this is called an orthogonal matrix of size 16 because it has a 16 column the matrix I have shown as 15 column the first column is obviously 11111 how do you make this I thought I will just give you the trick there is it is nothing very great suppose we want to make L18 we want to design L8 okay so there are 8 rows to it and there are 1 2 3 4 5 6 7 8 columns to it okay so I begin the first one I start is that I have 11111111 simple this is my 0th column okay then I start with minus 1 minus 1 minus 1 4 minus 1 and 4 plus 1 this I call my column A then I build a column B in which I have first 2 minus 1 then plus 1 then minus 1 and then plus 1 okay then I build a column AB which is a multiplication of the 2 columns A and B so I have minus 1 minus 1 minus 1 minus 1 1 and 1 then I introduce another column C which I say now here I have done minus 1 minus 1 1 1 so I go minus 1 1 minus 1 1 minus 1 1 minus 1 1 then I introduce a column A C I introduce a column A sorry let us correct then I call introduce a column BC and then I introduce a column ABC I think I will leave this column for you to fill up but this is how you can construct an L8 or L16 or whatever you want it is very easy to do the first column is 1111 which I call 0 or it is actually known as column 1 so let me put the right name to it it is called a column 1 this is called column 1 then you have A which is the half of it I make minus 1 rest of it I make plus 1 then I introduce a column B in which in which I have other half of it that is I here there are 4 minus 1 so I make 2 minus 1 and 2 plus 1 then again 2 minus 1 and plus 2 plus 1 then I take the multiplication of the 2 in the next column then I introduced another column which is minus 1 1 minus 1 1 alternative so I am putting this minus 1 and 1 in alternate fashion in different patterns and this pattern is very fixed first half of it is minus 1 the rest half of it is 1 then half of half is minus 1 and then 1 1 1 then again minus 1 minus 1 and 1 1 and then once my 2 such columns have come I put a multiplication I introduce 1 column then I put the multiplication and that is how I create the L8 so you can see that L16 is also done in the same way I have put 8 of them as minus 1 and rest of them are plus 1 then B I have put 4 of them as minus 1 4 of them as plus 1 again 4 of them as minus 1 again this and then I take a multiplication of A and B then I introduce C as minus 1 minus 1 1 1 minus 1 minus 1 1 1 etc then I introduce AC AB ABC then I introduce a D column which is again minus 1 1 minus 1 1 minus alternate so I get AD BD ABD CD ACD BCD and ABCD so this is how it is done and in this all the fresh factors that I introduce without multiplication are straight away your assignment of main factors now we have 7 main factors so we have assignment of only 4 we still have to assign 3 more main factors so this 3 more main factors I am going to assign by asking myself a question that leave out the interactions of your interest and then assign this columns which do not have those interactions now we are going to consider only 2 level interactions we are not considering the ABCD ABC ABD ACD BCD etc as interaction so at time point I have chosen the left most the right most interactions to be there so I am taking ABD ACD and BCD as also an independent factor main factor or for the our design and that is how I make 7 factor main factors as assigned to this column so according to this assignment then I assign I actually take the multiplication assign those columns to the interaction so that is how I find the interaction AB AC BC AD and BD as my interaction this dark blue ones that I have shown here are my 5 interactions of interest red ones are my main factors dark blue ones are my interactions and this light blue 3 columns are unassigned columns so these are my sorry these are my unassigned columns 3 of them they also have a role to play in future so that is why I am indicating them to you so this is how I do my factor assignment so if you look at my orthogonal matrix I have done the assignment of main factors and interactions and now if you look at it interactions come automatically remember the interactions are coming by multiplication so they will come automatically so the main factors which I have assigned are kept here and this is actually my 16 experiments that I need to perform in 2 replica so here this is my plasma flow rate which will vary in this fashion additional gas flow rate will vary in this fashion etc so if I look at the standard order of my experiment my first experiment will have plasma gas flow rate at minimum additional gas flow rate everything will be at minimum the second experiment will have plasma flow rate additional gas flow rate carrier gas flow rate at minimum and rest of them will be at a maximum level or you pick up any other number 10th experiment is going to be plasma flow rate plasma gas flow rate at maximum additional gas flow rate at minimum carrier gas flow rate at minimum feed rate at maximum my reaction chamber length as the smallest one the power is going to be maximum and the evaporating temperature is going to be the lower limit so this is how my 16 experiments are now designed I do the replication so I actually perform 32 experiments 16 times 16 I do twice and this is not my standard order this is my random order so you can see that the actual experiment that I am going to perform first is going to have these values of these parameters so plasma gas flow rate we have given it in the units so it is going to be 2 the additional gas flow rate to be kept at 0 carrier gas flow rate is to be kept at 0.3 m cube by that meter cube by hour feed rate by cc by hour 160 lc length has to be 11 inches power has to be 4.5 kilowatt and evaporating temperature has to be 120 degree centigrade this is my first experiment to be performed remember what we gave is a standard order standard order is what you get from the designed experiment experimental order is a random order it is a random order as shown here as written here it is a random order and the 32 experiments will be performed in this random order so if you want to look at what is the experimental order and standard order this is how it is the standard order first experiment will be performed 12th the second order experiment will be second standard order experiment will be performed first etc etc so like this the 32 of them have been given the this is the randomized order and this is the standard order and the experiment will be performed in this randomized order suppose the experiments have been performed and we have got the results so in standard order we put them down so this is percentage efficiency and percentage anatase the first 16 experiment and this is a replicated another 16 experiment and therefore we take effect as an average effect of percentage efficiency of the first standard order and then the 17th standard order is actually a first standard order so that is going to be 73 so we have taken average we have taken care of the there is no decimal point there so we cannot have decimal points according this suitably they have been rounded off so these are the average value so let us repeat these are the 32 experiments the first 16 are first 16 experiment this is the next 16 replicated experiment for each one of them we are finding the results they are actually the same experiments and therefore we are taking average of it as a final experimental result as a average effect and average anatase percentage so the design matrix that we have to analyze looks like this this is a standard order these are the levels at which your different 7 parameters have been organized and these are the results that you have obtained let us work out some notations and calculations we say that this is why is your response so why sub a plus means that some of all observations for factor a at level plus 1 so if you look at here I look at all factor a suppose plasma flow rate it is kept at level 1 then I have to take the average of 9 to 16 values of if the efficiency and that average is what I call y sub a plus then I have y sub a minus it means that for example if I take additional flow rate then the first 4 and the experiments 9 to 12 refer to the additional gas flow rate kept at a minimum value or at minus 1 so you accordingly take the first average of first 4 and the 9 to 12th experiment that is going to be your y sub a minus which is the a is the factor of your interest at factor level minus 1 this is going to be y sub a minus 1 and y dot dot please recall the notations we have done in the past sum of all observations so sum of all of this is y dot dot you remember that this is y these 2 are actually y your response y1 which is efficiency percentage efficiency and this is percentage anatise you take an average because there are 2 levels sorry at each level you take an average so you divide it by the number of observations you have added so in this case you would have added 8 experiments because there are 8 experiments conducted at level minus 1 and 8 experiments conducted at level plus 1 and therefore this will be average of 8 observations so average of y a plus is mean observation for factor a at level plus 1 so it is this divided by 8 and y dot dot bar is mean of all observations so this will be divided by 16 similarly you have number of observations at factor at level plus 1 and a minus is the number of factors at level minus 1 and n is the total number of observations so effect of any parameter you can find out if you take average that is mean of observations at factor level a plus minus the mean of observation at factor level a minus it is called effect of a coefficient of b of a parameter a is nothing but an average because there are 2 levels so it is averaged out at 2 this is called coefficient beta so this is an estimator or estimate of beta hat or we used to call it b in the past I must make correction here in colour I make it green this should be all hats these are all estimators that we are calculating so if y denotes the experiment result and y hat denotes the estimated value then y hat is this you remember you please see that I have removed plus epsilon because now we are calculating the estimated value by putting in the estimated value of beta hat coefficient of beta hat is estimated here so as such this is actually your beta hat a beta hat for factor a is given here so this is what it is recall what we did in analysis of variance so we have a total sums of squares which is summation of yi minus y bar whole square this is an average of all the observations and summation sum of squares of errors is summation of yi minus yi hat whole square yi hat is here so it is the actual value minus the estimated value whole square sum of squares due to factor a sum of squares due to this factor a is given by this formula it is given by this formula and you can show that total sums of squares is actually sum of squares due to each factor plus sum of squares due to error or this formula we have already worked out to define the chi square distribution it goes in the same way now this whole thing is put in a table so let us read this table these are the estimated effects of the parameters in coded unit by what do I mean by coded unit it is not the actual values with which it is calculated it is calculated with level minimum as minus 1 and maximum as plus 1 so our design matrix does not have the actual value of the plasma gas flow rate or additional gas flow rate but it has a coded value of plasma gas flow rate as minus 1 if it takes the minimum level and it is plus 1 if it takes the maximum level for each factor so this shows the effect as we have calculated this shows the coefficient this is standard error of coefficient remember standard error of coefficient is residual error divided by total number of experiments that is that will be it is a mean error so it is divided by the total number of experiment we have performed and its square root so this is what it is so this is what is standard error of coefficient you can just confirm it this is t statistic to test that this beta coefficient is equal to 0 or not the beta coefficient of plasma gas flow rate that is this beta 1 is great is 0 or not the null hypothesis is that beta 1 is equal to 0 and to test that null hypothesis this is the t statistic and this is the p value you remember when we did the hypothesis testing we also calculated the p value of the test so these are the p value in other words if we want to test the hypothesis at 95% confidence or 5% significance then these values have to be if they are larger than 0.05 then the null hypothesis that the coefficient of that factor is 0 is rejected so this shows that these are all the values which are smaller than that so these are all the significant values I think I made a mistake in what I said if these value if p values are smaller than 0.05 then you are going to reject the hypothesis that they are 0 if it is greater than 0.05 then you are going to accept it as a null hypothesis so here for example we find that evaporating temperature is has a beta value has a p value which is larger than 0.05 and therefore we accept the null hypothesis that the evaporating temperature is 0 so it is not having any effect on your efficiency while the red ones which are clearly shown we can say that it has an effect on the it has an effect on the efficiency of the process additional gas flow rate feed rate reaction chamber lens then interaction plasma flow rate with additional gas flow rate an interaction of additional gas flow rate with feed rate these are all significant factors which are having an effect on efficiency of the system all others have their beta values close to 0 so they can be ignored when beta is not equal to when null hypothesis that beta i is equal to 0 is rejected it means that those factors are playing an important role this is a table sorry this is a table for analysis of variance for efficiency these are the 7 main effects so this 7 of them are added all the 7 main effects are added it has a 7 degrees of freedom this is sums of squares sequential we have not gone through this study but this is an adjusted sums of squares the sums of squares divided by degrees of freedom gives your mean square error similarly 2 way interactions there are 5 degrees of freedom and it has adjusted mean square as this the residual error comes out is here which is shown here this is the error sums of squares so residual error is error sums of squares divided by its degrees of freedom which comes to this and if you this f statistic please recall is the main effect mean sums of squares residual sums of squares which gives you the f statistic which we did in the analysis of variance and here are the p values calculated in the similar manner and if you look at this p values it actually tells you that both the factors the p values are smaller than alpha it means that both the factors are actually so main effects and 2 way interactions are insignificant as we can see here then we can calculate what is called lack of fit you remember the 3 columns that we left out they are calculated here that is a part of the residuals it comes so pure error is if you take out this lack of fit you remember if you fit the whole 16 by 16 matrix then there is no lack of fit you have fitted the whole equation but instead you have fitted only 13 of them and you have left out the 3 of them so if you calculate this residual lack of fit it shows that lack of fit is significant it means that the coefficient of lack of fit is not significant it is 0 so there is no lack of fit here this is a pure error and then you have a total of this because you have a total sums of squares which you has been shown here this is total sums of squares so there are total 32 experiments minus 1 degree of freedom so it comes to 31 degrees of freedom and this is there and this is your analysis of variance table so this very clearly says that these are important and there is no lack of fit in your model your model completely decides and defines gives you the value of efficiency so then we come to the final analysis if you look at back here the effect is negative it means that the effect is the lower the plasma flow rate higher the efficiency lower the additional gas flow rate higher the efficiency lower the freed rate higher the efficiency higher the reaction chamber length higher the efficiency higher the plasma flow rate has to be kept at level low freed rate should be kept at level low additional gas flow rate should also be at low but RC length that is reaction chamber length should be high and the power should be high and then this becomes your predictive interval it means that you take the average value of factor plasma flow rate get at low and then add up all of these minus you have to subtract 4 times the total average because you see from each one of them you will have to remove the average and then you have to remove one more average so it add the one more average and therefore it comes to minus 40 and this becomes your predictive interval you remember we had done the regression analysis what is your predictive interval this is f this shows the 95 degree confidence limit of f distribution with degrees of freedom 1 and 19 why 1 and 19 because you are taking the 19 degrees of freedom and it is only one equation so it is coming 1 and 19 n is an effective number of replication which can be calculated as total number of experiment carried out divided by number of degrees of freedom from the sources considered for calculating this mean so we calculated this mean by the 5 and 1 total the the grand mean and therefore it is 32 by 1 plus 5 which comes to 5.3 and then we say that the average efficiency should lie in the interval 90.5 to 115.4 the question is is everything okay and I leave the matter here and we go to the next session but in this let us quickly summarize we took the case of optimizing the efficiency of titanium production we went through the process of selection of levels of independent factors selection of orthogonal matrix cell 16 assignment of factors and interaction experiments with their random order then we did the experiments with replica and a random order finally we did the result and analysis thank you