noc19-mg15 -Lecture 44: multi-objective optimization

[Music] [Music] welcome back my dear friends a very good morning good afternoon good evening to all of you wherever you are either in India or abroad and as you know this is the da dm2 which is data analysis and decision making to course under the NPTEL MOOC series and this total course total number of contact hours over the lectures is 30 hours which is K which can be broken down into 60 lectures because each lecture is for half an hour and this course runs for for these D at least this da DM 2 runs for 12 weeks and each week we have five lectures each lecture being for half an hour as already mentioned and after each week you have assignments so we have two more lectures to go before we wrap up the ninth week and similarly will go into the 10th 11 and 12 week to wrap up this whole course so if you remember in the and and my good name is Rwandan and Sengupta from the hammi department at IIT Kanpur if you remember in the last class we were discussing about parrot optimality cons considerations and parrot opportunity considerations means that two solutions or two outcomes or two alternatives or two decisions whatever they are they are not worse off to each other with respect to your decision in the sense the overall bundle of worth which you get by taking any one of these alternatives gives you the same worth now if you decrease in one you increase another and vice versa hence the total cumulative one remains the same and I will show you examples for that as we proceed the concepts with the concepts so parrot optimality conditions are what basically of a part of multi criteria decision making which we have been discussing which is our multi attribute decision making and we consider there are many areas of work it can be say for example as I am did mention it can mean in finance finance problems we will consider when you are buying a house when you're buying a car when you're making a dish that you have to choose any one of the people who has a person has a applied for a position you want to take him or her considering there are different criterias different attributes different characteristics if an objective and subjective criteria based on which you will make a decision to select him or her so it can be in say for example in production line it can be in petroleum engineering and all these things so some of the areas where multi objective optimization czar used it can be in economics where you have to give take a decision where the total amount of resources is constrained and you have to get maximum utilization of that and you want they can be different bundle of decisions which gives you the same word so you want to make it take a decision which is best in production technology there are different products which can be produced with resource constraints capacity constraints supply constraints utilization constraints based on which you find out the best optimum solution so as that it increases your profit lowers your losses and also you try to basically make a compromise compromise in the in the sense that you want to basically make a decision which gives you the best combination of the compromise which is best worth for you in finance it can be where you want to basically invest for a portfolio portfolio consists of say for example n number of assets assets can be financial Goods can be bond stocks derivatives whatever it is and you want to minimize your loss or minimize your risk maximize your profit which is expected value so you want to make a combination which are the N number of assets you will take in order to achieve that criteria so it need not be maximization of the expected value minimization loss it can be other criterias also coming into the picture if it becomes more than two objective functions in optimal design and control when you want to basically design a very sophisticated instrument of electrical circuit a cooling tower for a thermal plant or you want to basically be assign or design the flow of heat through different pipes so there are different ways you want to be seen maximize the flow rate flow and minimize the loss also or if you want to basically also maintain the ambient temperature between the outside and the inside in such a way that your the overall process chemical process works considers a chemical process which you want to basically continue there you may basically like to minimize the total cost because insulation cost and all this thing should be minimized considering this very high cost of insulation so all this would be considered in process optimization like you're doing fractional distillation you want to get different type of products from petroleum from the ore and you want to basically will get the maximum of them utilizing minimum cost cost can be energy utilized time utilized the losses or the the overall residue which comes out from this production process process optimization process should be minimized so there are different ways you can do that but at the end of the day your main focus would be to consider more than one objective function where the objective functions are such that you want to maximize on some front minimize on some front and all do not go hand-in-hand in the sense that if you increase one of them the other would also increase that basically increasing the maximization concept the minimization value would would also be increasing in such a way that it will have a one-to-one the combined detrimental effect so if you increase one the negative sense of the other also increases if you decrease one the negative sense of the other also decreases so which one you want to choose the combination you want to choose to get the best optimum solution we'll consider a very simple example so this is the example we are considering consider that you have n number of financial assets this is from the financial engineering and our optimization problem so you have a basket of n assets and I am for the simplicity I'm considering any stocks not bonds not derivatives nothing not forward not futures we will consider basket of financial goods financial assets which is the stocks and we'll consider that for this basket the total portfolio which you formulate you want to basically maximize the return and you want to minimize the risk which is the variance so we will consider the variance is the best proxy for the loss and we are considering variance as the best proxy of the loss considering the fact the distribution on the assets you are considering as normally distributed which in the true sense is is not right but we'll still continue doing that because it does it gives us good results so you want to maximize R star some value for the returns over which you want the port will return to be and you want to minimize Sigma square star which is some value of of standard variance below which you want the variance of the portfolio to be so our studies are such such a value that you want the portfolio returns to be more and Sigma square star to be such a value you want the variance of the portfolio to be less because loss obviously you will try to reduce deduce profit you also want to increase profit and loss I'm using in a very generic sense profits is something positive loss is something negative now the constraints have a very simple meaning so I'll go one by one so we will consider the weights which you have that the total amount of money which you have say for example 100 rupees 100 years hundred dollars whatever it is you are able to proportion in some proportions divided into those N number of assets say for example if you have ten assets and you decide that you will spend any 100 rupees and you spend one tenth of proportion of your total money in each of these assets so one tenth of 100 would be ten rupees would be in to invest in an asset 1 similarly ten rupees in asset to ten rupees in asset three and so on and for till the last asset which is the tenth one you invest ten rupees so D here the W 1 2 W n where n is 10 would be equal to 1 by 10 such that that the sum of all the weights actually should be 1 which is true so if you consider so I am NOT going sequentially poor for each and every car I'm just discussing from the simple one and then make it much more logic so it is much more logical to all of you so the first one is that weight of all the asset weights which have invested which I just gave an example w1 to WN they sum up to one so if the first part which we decide that the weights of the sum of the weights of all the assets is equal to investment being done is one the second one is which I will consider and mark a different color second one is this which is also very simple to understand what does it mean that the weight for each and every asset which you are going to invest out of that hundred rupees which we had would always be between 0 and 1 so 0 is here that means the weights cannot be negative which is right and it cannot be more than 1 which is also ology very logical so we are considering each of these W 1 to W 10 are bounded between 0 and 1 now you to make it a lot a little bit more realistic you what you can have is the weights for each would be bounded by the minimum value and the maximum value so if say for example W 1 will be bounded below by W 1 minimum and bounded above by a W 1 maximum that means there are some values corresponding to W 1 between which W 1 value has to be say for example it can be that for the first asset which you are going to invest consider that we start a steel stock or Tata Motors stocks or say for example Hindustan Unilever stock in in I am talking from the Indian context and the weights which you have to invest in this stock has to be greater than 20% but they cannot exist 80 percent in that case W 1 for these three cases or examples which I gave would be greater than equal to 0.2 and less than equal to 0.8 other case can be say for example you consider the State Bank of Indians stock in the stock market and you are under the obligation or under the condition when you are doing trying to invest is that the weight of State Bank in India of India stocks cannot exceed more than 50 percent in that case the value of W 2 which is the state bank when India would be greater than equal to zero but less than equal to 0.5 that means the values would always be between zero to less than equal to 0.5 so similarly you can have some constant nor w3 some constant and w-4 some constraint w5 and so on and say for example for the w tenth one whatever the asset is you will say that I have to invest or you put the condition that you have to invest more than 70% of your money in that W 10 stock in that case W 10 would be greater than equal to 0.7 and less than equal to 1 so based on that you can all these ideas which I gave you can have different bounds for the w i's such that the sum of the weights which you have already discussed the first priority condition which is marked in yellow would always be followed now we'll come into the subsequent third fourth and fifth constraint in the sequence or the consequence that how easy it is for us to understand the first one which was e low yellow second one was red was very easy for us to understand now consider the third one and I'll mark the color as orange so what does it mean consider the returns so now here the our eyes are the returns for each and every stock which we are investing so the turns are the if you are investing for example 100 rupees on day 1 and if I get hundred and 10 rupees in day two I will consider the returns and we found it in two ways one would be 110 minus 100 divided by 100 is the rate of return which will be denoted by small R and if I want to find out the value of capital R then the rate of the rate of the return would be generated by 110 divided by 100 so we are considering here the small R which is the rate of return which is given by 110 minus 100 divided by R so which is actually equal to the investment value which have done which will come in in the denominator and the difference in the net outcome which will get after one day minus the initial investment would give would come into the numerator so the numerator value can be both positive and negative now if I consider the returns as RI for each and every stock consider these are the 10 stocks which I am just mentioned n is 10 then if I want to find out the overall return on the portfolio which is being formulated by this 10 stock would be the sum of the corresponding returns multiplied by the weights which I am doing for stock 1 similarly first talk to stop 3 till the tenth stop which means I will multiply R 1 into W 1 added up with R 2 into W 2 then I can add it up with our three play into W 3 and so on and so till I have all the values of the our eyes and and employers multiplied together hence the left hand side would mean that overall portfolio return will be the sum of the multiplicative values of the returns and the weights now why it is greater than R star because consider I have told to myself before I invest that I want the return of the portfolio where I am going to invest has to be greater than equal to say for example 12% so in that case R star is the value which have basically fixed for myself for my investment is 12% says that the sum of RI into WI would be greater than 12 star but the question would obviously come up in your mind is that why are we trying to maximize R star also it means that what I am doing is that I am trying to basically push consider the real line from my side so if this is the value of 0 if I go on to the right hand side it increases if I go onto the left hand side it decreases consider that 12% value which is there so I consider the 12% value which would be 0.12 and I am trying to basically increase this value of 0.2 0.1 to more and more on to the right that it becomes point 1 3 1 4 1 5 point one five point 1 6 1 and forth so as that the values of the different type of investment which I do and the net return on the portfolio which I get for each in every cases which is the sum of our eyes inaudible eyes would be always greater than the art star value which I am trying to fix for this problem so as I keep increasing R star the overall combination of the weights for those before that portfolio will be such that the summation of which into the returns would always be greater than equal to that R star value which I have now in the as as I'm dimension the third constraint in the similar way let us consider the fourth concern this will now make things are much more simpler for you to understand consider that if your returns are to be increased then in the same case since the risk of the loss has to be decreased so can again consider the real line now consider what is risk risk I mentioned is the variance so consider Sigma 2 star star is the basically fixed value which I have considered is say for example point to zero in positive sense so what I won't want for my portfolio it overall variance would be the value of that variance for the portfolio will be always be less than equal to point 2 so it will be on to the left so what I will try to do is that I'll formulate my portfolio in such a way that the variance value would be pushed more on to the left that will keep decreasing and it as it decreases the total combination the weights which I have for the portfolio when multiplied by the corresponding variance covariance matrix or by the standard deviation of each and every portfolios taken together two together is multiplied along with the correlation coefficient would be such that the value of that overall portfolios risk would always be less than that Sigma square star value which I have now the value which I have which is double summation and I am NOT talking about the summation well is I 1 from 1 to N and I 2 from 1 to n it is basically W 1 I and W 2 W I 1 WI 2 into Sigma I 1 and I 2 is basically the multiplicative alyou of the weights multiplied by the covariance variance matrix which I have for the whole set of combination of the assets which are being taken into consideration to formulate the portfolio where I am going to invest so obviously that value would always be on to the left hand side less than equal to Sigma square star the final values of the constraint is this one that we use the blue color is this one which is also very logical in the sense consider that initially when I start investing my total amount of returns which I have would be such that the final return after one time period to time period whatever I have would always be greater than equal to the total investment which have already done in the initial time so say for example if I have invested say for example 100 rupees I would ensure that the combination of the total portfolio would be such that the total returns would be hundred plus some positive value such that it be makes sense to invest in some combination in those in that portfolio such that I am able to meet all the free criterias point 1 the overall return on the portfolio is always greater than equal to R star value I am trying to maximize R star value point number two the overall variance covariance or over a risk of the portfolio is always less than equal to Sigma star square star value which I am trying to minimize the Sigma square star value and number three is that the overall return which I am getting from the from the overall from the investment would be always be greater than the total amount which a mall we are based on which I am already investing from from point one or point zero which means that if I as I mentioned if I have 100 rupees the total return which I will get in the value sense from the total investment which is being done in that portfolio would be definitely greater than equal to hundred rupees now here in the last line it mentions our iwi Sigma I 1 Sigma 2 Sigma I 1 I 2 R star and Sigma star have these usual meaning where R I am again I am mentioning is the return of each and every stock WI is the weight which you are investing in each and every stock where I is equal to 1 to n Sigma I 1 I 2 is basically the covariance values for the combination of the I 1 and I 2 stop taken to get 2 together so if there are capital n number of such number of stocks the total number of combinations would be NC 2 so for any values where I 1 and I 2 are unequal then we basically have the covariance values if they are equal we'll have the variance values of each ticket with itself so you will basically go through the principal diagram in the covariance variance matrix when I 1 is equal to I 2 and if they are not obviously will go to the of the diagonal element which is the covariance values now based on that what we do is that we analyze the data and so you have to optimize it this problem it utilizes the data for the Dow Jones Industrial Average we take the data from the Wikipedia it is easily available and we take the time frame from 1st January 2012 to 31st January 2015 and we basically format at the port for 4th folio based on the fact that you want to maximum I asked our star minimize Sigma squared star subject to the constraints as mentioned I am again mentioning 1 will be the initial investment which I am doing should always be less than equal to the final investment value which I get number two is that the overall portfolio returns is always greater than R star such that we are trying to push R star more on to the right that is increasing it the overall variance or the portfolio will always be less than equal to Sigma square star says that we are pushing Sigma square star more on to the left and finally the weights or the portfolio should add up to 1 along with the fact that the weights are always bounded between wi max for the ayats stock and double and it will be bounded lower by the case of w i mean which is the height stocks lower values of the weights so once we have this when once we plot it so what we do is that we in the simulation problem we generate about say for example hundred values for the simulation case and as we keep changing the values of R star and Sigma square star values which are pretty but determined from my end or from their investors end I have a very simple parrot optimal solution which means that the expected return if I consider that we use the red color for 4.8 the overall return value comes or this can look to be about minus six point two five if I consider a return expected return a four point five the overall value of risk so this remember this is 10 to the power minus 4 this is the expected value there is no and it is 10 to the power minus 5 so the values which I have here based on the fact I want to maximize our star and minimize Sigma square star would be such that under those constant consideration I'm equally disposed both for E as well as for B so they can be different combinations along the Barrett optimal solutions where I am plotting expected returns and risk to get the best combination of the stocks which will give me this optimization value which is true to my concentration now we will consider a different idea which is basically known as the reliability based optimization I will just give the background of that so whenever you are solving a problem they would be uncertainties both in the parameter values as well as in the external variables based on which you are trying to formulate the problem so what can this can be say for example I am trying to optimize the design of a car and the parameter values can be say for example which are internal to me can be the strength of the steel which I am going to use can be say for example the type of quality of door or dough handle which I am going to use I'm giving an example or it can be say for example how thick or how strong the glass is the windscreen and and or means or it can also be what is the overall speed at which the crowd would be traveling on an average and average as it continues either covers 100 kilometers or 200 kilometers at one stage and the external variables of the video can we say for example what is the road condition what is the wind speed or what is the say for example filtration of the road or what is the actual say for example the effect of humidity what is the effect of rain what is the effect of temperature so these are variables which are not under a control when you are designing it but you consider these variables are even though they are external they are not fixed they will change and some of the parameter values you which you want to design for yourself are are in general under your control in the sense that you know the distribution based on which you are going to model model them hence uncertainties in parameter values which are internal to the system and external variables are inevitable and you have to take them into consideration for optimal decision the uncertainties of parameters both the internal one and the variables as well as the sensitivity Ness of these uncertainties uncertainties for the parameters and the variables must be taken into account logically to arrive at some logical conclusion the uncertain parameters and variables used in the observation problem or in the optimization means month.the multi-criteria decision problem which I am going to solve or which we will discuss they may be estimated through analytical means that means you have say for example some statistical tools you have the mathematical formulation you either differentiated use the concept of maximum likelihood estimation use the concept of general methods of movements whatever it is you are able to find out and integral solutions for those parameters from the samples which are based on which you are trying to work if not say for example you have good computing facilities you you have some very nice ideas of bootstrapping yeah and you use the sample in such a way that based on the sample size based on the way you want to bootstrap you are able to estimate these parameters from the the simulation runs and the worst case scenario would be where you are you do not have any mathematical formulation to solve them you do not have any simulation result to solve them you cannot assume any values they are just kept uncertain and based on that you have to basically sort of proceed with the problem in order to consider those uncertain values in your optimization problem what are the consequences the consequences of these three things if there is the parameters and the variables both are unknown they can be found out eg as using some parametric parametric estimation method or using some computing method or at the worst case they are unknown so in that case the optimum solutions may violate some of the constraints based on which you are trying to solve the problem so what are the constants if you remember that when you are trying to solve the problem it can be that the total number of trucks which you can use cannot exceed for I am just given an example say for example you want to transport goods or see for example total number of inventory which you can have in factory one cannot exceed say for example hundred units whatever the type of goods are so if those constraints are there it may be possible that if you do not consider the concept of uncertainty in the parameters in the variables and they are kept unknown then you may violate some of the constraints based on which you are trying to solve the problem and find out the optimum value even if you do not violate the objective function could be such that the actual value based on which you will try to deduce the answer would not be true because it would give you suboptimal results so obviously the answer would be how would you solve it so here the overall comment n emphasis would be to find out a way how you can basically formulate the uncertainty in the parameters uncertainty into the variable says that you do not violate the constraints as well as on the other hand you find out the best possible solution from the objective function based on this conditions so with this I will end the forty-fourth lecture and continue in the discussion the forty-fifth one and give us some and give you some idea about reliability based optimization which is a tool under the ambit of multi criteria decision making have a nice day and thank you very much for your age [Music] [Music] [Music]