 रहाँ बाढने रहारटूी मरी अगर नेईठ गर अवजरत्या में करचान साया। Today we are going to talk about the design of experiments. उरिठा ती आच्ठारईग, विहां बने लगर गफीन इस भी ठीए। तो प्यान और गफाक तेझी एच्ठारेए ये लीके आच्ठारेग थी रहान करूें।jack and work out the analysis of its results to come up with correct set of values or the factor of levels at which you can conduct the experiments to give you the best possible results this design of experiment is a very popular application area of statistics as it is being used properly in number of cases But there are times there are certain misconceptions and therefore the use is not totally complete. So in the coming few sessions we would like to describe the method of design of experiment at a case study. So first I will give you an outline. In this session we would like to discuss what is design of experiment, and when ञाद्यात्या ओभी लिजारग ते इसा होग़ा जाद्याद। जोंगो करादर वोंगीस आश्वाया देने सोग़ों बॉश्वाश्वौई� तेतनाम संच्छना त्रीम तिष्झनटिक की च्छनादी कि के आख्वाश्वाश्वाश्वाश्वि दीजiant of experiment methodology and through this case study in the subsequent sessions ते लन functionality's will be to know the process of design of experiment gets applied to a real time situation भी च्感 लिंझ को सब ढंज है तै भी बहुं जीनागा जीक बहुंक glam enough experiment will answer is not always भी ओ� Factory whenever possible सो थू laut्जा दिनक से धो मblesब हम से भैंगा दीजouter have a system येवार भी। नहीं औो बरक्तोल करे वत्याः रहें से हहांँ। लीख करे वत्याः अपकोंध मैं से वाग्टी चीश्टी, शानचा वो सोच्ठा, हम आक आश्चला आपकोंध कि येवारोन काई वत्याघ ढूँ, यूँ अत्य nél saat kaat kaatare अत्यई Sahar input अत्यई अत्यई अतिंट यूँ ॉ अत्यई बि़श YAad ॉ yoot झोईए ठाव में ब॑ष नाए ठाऊpl शे�很有 चमर लय कताउ अतिन। यूँ आ आप क्र मsun वी आप त्रेश य Έक भों �えばon मेगazione अदर तपयकियाग़ा नहीं, नहीं समवोल्गे़ ददीं, तब वो दिने हो नहीं मुहें लिए करते नहीं तभत वैले करते तो कर्वमा वेझाग़ा की सकते रच्छेनाता, नहीं तब विन गेर आप वेशाद शब चस्फीं करते उन्च्च्कुझ़ि किते यह आप, पामत引 कर क coordinator इसंधुड कर कनी हमेंतो 설명 seeing क्स अबसु आप कक इसी मुझिं बाजा करनाय थे अबस... अबस... इस अवो दखना कनी वदycz कनायU टब वड़ाट ऎर बारा कनी मुझिं हु speeches you know conducting the experiment is also factor which have no control over in the factory set up that is in a regular production set up if you are trying to apply this technique or you try to do this kind of experiment you will find that timing of shift, the person operating the system कोरतिः श़़ के लिए �00 एक समझार क्जवी टेशके, अच्छाइनी च्छागिया वाहा, क्जवी च्छागिया घर्छिया भी एक समझार्चा। प्तदी उस हैजी बातने सकाऍित, आच्छाई Josh Shah किया बाशाट़ाता, समझार च्छाएनी, क्जवी टेश्ंवेखा, शूर, at the first experiment and which experiment will come after the second experiment, अस्भान्कब of experiments you don't even have any ideas to how many number of experiments अस्भान्कब कुँँँँँँँँँन तुछ अद ज्बचम. अन्ज़े किछ में, ज़े विया में since l'enclinique, उप्तरिजट आप और सो जच veer l'enclinique, Valli melly number of experiments we are going to conduct. This is called a ''Best guess.'' What it gives is that sum factor effect. Some factor that Αu de prenise those controllable factor, but it gives no interaction effect. I will define interaction effect soon. So, right now we I would like to emphasize that it does not give any interaction effect. Let us consider the case of one at a time. ॐ only one factor at a time ॐ ॐ ॐ the same factor at a time and conduct experiment. So, for example in this matrix, I keep B C D fixed at a level 1. for example, in this matrix, I keep bcd fixed factor as a level one. So, at some value I can keep bcd factors fixed and I vary only the factor A at two different values which I call two different levels. इक ने ज़े शिक्स living Inspector show the 2 Experiments that I find that types of आउ Morning Experiments senate of the आउron has met my requirement. मुलनक do the important आउron claim मुरितारी इक लखम feedback औ� luck लासे योग़ा करते हों। नाब उरगणे ता papier ї। लीक्चित करे करनलन तर थे से भूँस�ितियो थीसर्ईión है। तीसाव म दीसे लिकटर 나온 यहसाव नकरन तर थे शिस्रथोन कर्फ़ाहा यहसा बार्ई्या with level 2, A is fixed at 2 now and this is fixed at 1 1 and I conduct the 2 experiment and from these 2 experiment with respect to factor B, I find that this experiment has performed the best, number 2 has performed the best. So what I do is that I keep that value fixed, so now my A factor is fixed at level 2, my B factor is fixed at level 1 and then I conduct the experiment, so I have 2 1 1 experiment along with 2 1, I now I change the factor C, so I conduct the experiment with factor at 2, factor C at 2 level and I find that among the 2 with the factor 2 of level sorry, factor C at level 2 gives me the best result, so I choose that now factor C will remain at level 2 and likewise I conduct another experiment with factors A, B and C fixed at respectively level 2 1 and 2 and I conduct the experiment by varying the levels for factor D and at the end I find that this is the combination which gives me the best result, suppose this is called one factor at a time method, this is called one factor at a time method, it gives again partial factor effect, it is better than this because it gives you some five partial effect but again I repeat that it does not give any interaction effect, what is interaction well, interaction says that failure of one factor to produce same effect on the response at different levels of another factor, let us understand this, suppose the effect is controlled by factor A, the effect is controlled by factor A and then this effect you want to check if you vary factor B at level 1 or you keep the factor B at level 2, does it give the same effect or with the factor B, A has a different effect, it means that if both things are, if B is varied from level 1 to level 2 and factor A gives you only one effect then we say that there is no interaction effect but if you vary the factor B at level 1 and 2 and the A starts giving you two different effects then we say that this situation says that there is interaction between A and B, A and B, we will explain it once again because interaction effect is very important, so we will time and again if I explain it, so let us move on, when you follow these strategies you are not able to get any interaction effect and therefore we are suggesting that statistical design of experiment with orthogonal design matrix is the best possible strategy, historically speaking it was in 1920s when Sir R. A. Fisher did the pioneering work in design of experiment for agricultural experimentation and he developed three basic principles of design of experiment, they are called randomization, replication and blocking, we will talk about it in future, then from agriculture it moved to biological and life sciences applications and some initiation of industrial application began during 40s, so we can call this an agricultural era of application of design of experiment with GEP box the introduction to response surface methodology came and it became very popular in the industrial era, the difference why this immediacy and sequentiality came is you can imagine when it was applied by Sir Fisher in agricultural science, you do the you design the experiment and you have you saw the seeds and you have to wait until the results came, so you may have changed the watering plan, you may have changed the soil, you might have changed the seeds and you might have made some you know different experimentation using different fertilizers, but it would take at least 3 to 4 months time to realize the result and then analyze it, while applying the same technique in the industries GEP box realized that there is gear in industry the results come immediately and therefore you need not plan a very huge experiment, but you can plant a smaller experiment and sequentially keep improving yourself on making different design of experiment step by step and come to a final conclusion, so instead of having a huge design of experiment to come to a one conclusion, he suggested that in the industry we can go by what is known as response surface methodology, first you conduct a smaller experiment with lesser complicity, but more factors and select a few factors which you find is affecting the most and also you will know in which direction if this factors value are changed you will get a better result and then you start working with a fewer and fewer number of factors and come to a better conclusion after 2 or 3 sets of such experiment, so this is called immediacy and sequentiality, this technique flourished during 60s and 70s for industrial applications and other applications as well, it was in 80s that the Japanese electrical engineer Geniji Taguchi introduced a robust design concept, it would be of interest for you to know that the first lessons, the first few lessons of design of experiments Mr. Taguchi, Dr. Taguchi gained very much from India from Indian Statistical Institute and he developed this new technique which is known as Taguchi design of experiments and presently you find that this applications have proliferated in a variety of areas which includes the soft industries such as software testing, model simulation, etc. because the computing power has gone up so high that earlier smaller the design, more the information now we can afford to have a larger design and still have a better information from the experiment analysis. Now what are the 3 basic principles? Please remember in throughout this area the basic principles have not changed, they have remained the same which were laid down by Sir R.A. Fisher and that is randomization, replication and blocking. So these basic principles are the first of all is randomization. See the idea is that what you are conducting in a laboratory or on a small scale in an industry, the results and the analysis of it you would like to apply in the real life situation and as we discussed in real life situation there are many factors which are uncontrollable, they are kind of a random factors. Some factors you can account for in what we will now discuss as blocking but there will be certain human error there will be certain machine error built into your experimental result. These are called random errors and if you want to control the random error it is suggested that instead of conducting the experiment in a one particular sequential order you try to conduct the experiment in a randomized order. You do it in a random way so that the random effect also gets mixed up. It does not remain a systematic effect that from experiment 1 to experiment 16 systematically the machine for example if you use the machine often and often the machines certain working parameters get derailed they slowly get away from their average way of working. Therefore in order to avoid this kind of a systematic error if you randomize the experiment then you take into account this change in a random fashion which may happen in real life and therefore this is called to make sure that the observations are independently distributed random variables to make sure that what you are getting does not have any systematic error but it has a random error in it you go through what is called randomization. Then comes replication this is not something very new we are quite habituated that you do a simple physics experiment you take 3 observations and you take an average of it. This is called observations replication you repeat the observations and then you the idea is again the same that there will be some random error in it and you want to average it out. Well when you consider when you do the experiment the whole experiment itself need to be repeated the whole experiment I am repeating whole experiment need to be repeated each experiment need to be repeated so that you conducted the experiment or I conducted the experiment in a certain circumstances the effect of those circumstances which is also random can be neutral out or it can be averaged out if you do the replication. So replication is says that to estimation of experimental error and precise estimate of a sample mean as an effect. So what you are estimating that itself has an error in it your experimentation itself has an error in it that whole experiment if you repeat as many times as you can afford it is called replication. It is very common to mix up the repetition with the replication observations are repeated in order to get get away from the random errors in the observation which is you average out the random error in the observation by replication we mean that the whole experiments are repeated so that the experimental error gets averaged out then comes blocking you remember we talked about certain factors which are uncontrollable some we have taken care in randomization and replication but for example if you are conducting an experiment in a laboratory and there are two operators for the machine each one have their own style of working and they cause their kind of an error but you cannot really control it particularly when you conduct the experiment in the industrial environment the shift matters what time of the day the person started working on your experiment matters if it is his first experiment of the day it is one kind if it is just before he wants to go home and his shift is getting over his mental set setup will be likely to be different and so his experiment way of conduct you know handling the equipment would be different. So there are many such situations which are uncontrollable so for that it says that you block them it means that you systematically assign certain experiments to different shifts or so that there are different persons in it and you reduce or at least if not eliminate you reduce the variability caused by uncontrollable factors which are known nuisance factors. So here is a general guideline I have picked it up all this picked it up from Montgomery's book you are all welcome to go through it it gives a extremely good treatment of design of experiments. So coming to general guideline it says that first recognize the stay and make the statement of the problem. So you must understand what is the problem that we are trying to solve you give you should recognize it and should make a clear statement of it. You should know what are your factors what levels you would like to vary them at which range you can vary them and then you know what is your response variable. These three are pre-experimental planning before you plan anything you must have this basic idea see number of times this seems very simple and very straight forward but in the haste of doing experiment number of times if these things are not clear the design itself is not having any clarity and the analysis becomes a problem and this is what has been observed several times. So it is very important to go through this first three stages before you design the experiment. Afterwards you design you make a choice of experimental design experimental design is that how many levels you want to vary your factors number two in that case how many minimum number of experiments you must conduct number three how many replications you are going to make number four how you are going to randomize it all these aspects you have to cover in choice of experimental design then comes performing the experiments then statistical analysis of the data you will come up at the end you remember with what is called predictive interval so you will say that if i do everything in this manner my result should fall in this interval so you are actually giving a interval estimation of your result then you must run the validation runs we must make sure that what i predicted by doing the first six steps what i did is indeed true so i must have some validation run and at the end i can make conclusion and recommendation so now we are ready to start with the case of microwave plasma synthesis of nano titania this experiment design of experiment was conducted at arci in hyderabad and it has been published in this paper in the materials and manufacturing processes here the idea is to process we have to optimize the process to produce nano titania with several inputs parameters and we have two responses one is a production efficiency and the second one is percentage of anatase in nano titania powder produced based on the result we have to give a commercial viability of the process to the experimenter this is the schematic of the microwave plasma synthesis apparatus there are seven parameters or seven factors that need that can be controlled first one is called power of magnetron so this is a magnetron power then there is a plasma forming gas which goes from here and enters into the chamber then there is a carrier gas which carries the the feed the feed powder or the whether if it is in a powdered form or ticl4 which is in the either powder form or liquid form this is a carrier gas if it is liquid it has there is an evaporator in it so there is a carrier gas there is a precursor feed rate so this is a precursor ticl4 is the precursor its feed rate is another factor which can be controlled reaction tube length this is a reaction tube and this is whole thing is a reaction chamber so there is a reaction tube length which you can change and depending on that the powder quality may differ there is evaporating temperature in case for liquid precursor only so there is an evaporating temperature here and eventually what is collected here is nanotitania powder and we would like to look into the percentage of anatase in the powder and the yield in terms of percentage efficiency of the process so the objective is to optimize nanotitania production using microwave plasma synthesis synthesis these are the seven factors which is plasma gas flow rate i have given you here everyone as a short form so there is a p actually it is a pfr plasma gas flow rate additional gas flow rate now additional gas flow rate is comes straight to the reaction chamber and it increases the time to for the powder to go through the reaction go through the reaction chamber then there is a carrier gas flow rate there is a powder feed rate reaction chamber length the power of magnetron and evaporating temperature and there are two responses one is a yield as a percentage of efficiency and the second one is percentage of anatase in the powder interaction i repeat failure of one factor to produce the same effect on the response at the different levels of another factor is called an interaction so here we found that the plasma flow rate with additional gas flow rate interaction interaction of plasma flow rate with carrier gas flow rate plasma flow rate with feed rate additional gas flow rate with carrier gas flow rate and additional gas flow rate with the feed rate these were the five interactions found important in this case next so these are the interactions which are shown in the schematic बadox we have to select the levels. रही grenade. रहीद ब femmes low to high. பेराज़ी बब इंत.- कहठा रही? रहीद बther- बब इंकृट�ग.ी सुबध्ठिстिकाओ audio- dulh- बरज़्म का प्रि मेना लोगा बरажу Wherever you इंऋ tonight no? अगड़ा ड़ कौछ़剅च यूवे बínhा Brockיב pleased ॐो यूबु, for This requires you to follow this approach and find that the response graph is something like this- which is a steadily increasing function or it may be a steadily decreasing function. If that is the case, then there is no point taking more than two levels of experiment because it is a steadily increasing function, so optimum value is going to occur either यह at 100000ации ची पrendre अध ह!*- अधी यह वेर यह तलेवीन को नध वेर वेर � repeat अध यह डर की उज़ा related 1 rabbits & profits despite Policy को ढने दर के एक हyn तसससन longer तो जग अन त�gun lal 不是 उच़ थते है ढ़क Gaza उण वेर ई़ोह conjugate जी abode 1 lakh णचि ऱनकूज Ge शना चढ़ च cinq Peters . . . . , . . . . . . . . . . . . . . . . . . − − . साई्रा उनलेख लग़ाना रहाँ च़्या है जो वो अगर रही वो वो जो ग़ा सब मैंखात। वो वो ता थे भी आप कर रहात ग़ाना वो वी वॉक ईस वह माऒ्ट्वनी होगो के तो साएब रही वे अगर गार च़्फ। और एक जिजट मर्वयर प्लास्माश अंतिस्स सिस्டम monetary plasma synthesis system, निस्तिर्प्स, निस्ट्दिश स्ञिजट मैंनोखी, अच्काई थिस्टिष्ट्टाइ आप ग़ि मिस्तिर्प्सुऊँइवाओ�! that this cannot happen unless he has done some experiments before and let me emphasize that designed experiment is never your first experiment designed experiment is your experiment which comes after several of those preliminary experimentation to understand the system itself to understand the process then only you would know what are the interactions which are आपそれग़ क्यछितेसे हो desen के चारवार, ऐधशन्že अच्ठेरवान向तेउ ग़े अच्ठे á मेंसा नहीं और ओ़ वरे ज्र ग़ात गयतेते runway फ्रेक्गे रड़िस़ार चाना कर इ kilogram अच्ठेरवां खढि़ सेबile भ NGO ती आप देऊछा रचा बा digestive पन्टबन के नेदवक लिएठ देश्यात का नेदवक के वो आपटेंएक योंगे सास्पड्रग सास वो आपटेंटा भी बन कोई भी देश्पिर्ठष्द भी आप देश्औब खुएचाझाँ उस्टॉआपिरिमें देश्ध प्र्स्नाझाई के रहे हैद। ौ ismatic ौ  Partners ौ  Wave on ौ ौ ौ ौ ौ ौ ौ ौ ौ ौ ौ ौ ौ A ौ ौ ौ ौ ौ ौ ौ ौ ौ ौ It is it cannot be the first experiment and as we saw that at times if we are conducting the response surface kind of analysis then it is actually a sequential set of experimentation so any designed experiment may not be the last one either but this is worth remembering that design of experiment is never the first experiment one has to have an understanding of that. टु़टाम के लीक को वोगित हो संट। भी वो चम्या लना कर्एगावांत् ہیں. तूंले दहा काई जी�假 क् après था और आरसाइवय हो गलीत escaगा और आर आए प्लं गबनो coordinates घब तु़टी। संथ. बीटळेशकी लव�矆त मैग्बर प्रल tears the monó 144 44 49 50 6 60 50 6 7 0 9 7 6 3 4 6 8 4 3 4 7 6 7 6 8 x3, etc. And for example, x1, x4 it means that I am talking about interaction between powder feed rate, it is a plasma flow rate, I am sorry, it is a plasma flow rate with the feed rate of the powder. So it is the interaction between these two. So likewise and then at the end we put an epsilon and I assume that epsilon is a random error, it is IID that is if I conduct n experiments here then I will have n1, n2, n3, y1, y2, y3, yn results and therefore I will have the random error epsilon1, epsilon2, epsilon3, epsilonn. All of them are independently identically distributed with mean 0 and a variance sigma square where I do not know what is sigma square, sigma square is unknown. I call f as a total number of parameters to be estimated then the size of orthogonal matrix, this is my model. How many experiments do I need to conduct? So I calculate it in this way that if f is a total number of parameters that I have to estimate then the size of orthogonal matrix should be say l to the power n where l is the number of levels that you have chosen and l is the size of the experiment then you have to choose your l in this case is 2 because we have taken 2 there and you have to choose n so that your number of parameters that you are going to estimate is smaller than l to the power n. So here we have 13 parameters so f is equal to 13 your levels are 2 so 13 is less than 16 which is 2 to the power 4 and therefore you need to conduct 16 experiments we need to conduct 16 experiment so you have to choose ln large enough so that 2 to the l to the power n becomes smaller than f then l to the power n are the number of experiments you have to conduct and therefore in this case we have to conduct 16 experiments. There are other methods of calculating the size of design matrix suppose number of factors are p number of levels are l then factor degrees of freedom is l minus 1 second order interaction we call it degrees of freedom of interaction then the degree of freedom f is p multiplied by p is the number of factor multiplied by l minus 1 plus q which is total number of interactions multiplied by degrees of freedom of interaction and then the size of orthogonal matrix is l to the power n where n is large enough so that f is smaller than l by n so accordingly we have 7 factors at 2 levels so degrees of freedom is 7 we have 5 second level interaction so degrees of freedom is 5 1 degree of freedom for the constant of the equation you remember we have a constant here beta 0 so for that we have to add 1 into it so total degrees of freedom is 13 and therefore you have to choose n large enough so that l to the power n is just larger than 13 and therefore the size of your orthogonal array is 16 so let us summarize we introduce the concept of design of experiment we gave a brief history of it the three basic principles of design of experiment randomization randomization replication and blocking we explained we explained the general guideline and we introduce the case of optimization of titanium powder through micro-plasma synthesis factor and level selection we discussed and selection of design of size of design matrix we discussed thank you