noc19-mg15 Lecture 43: MAUT

[Music] [Music] welcome back my dear friends a very good morning good afternoon good evening to all of you wherever you are placed whether in the air abroad and as you know this is the da dm2 course which is data analysis and decision making to under the NPTEL MOOC series and this da diem - as was da diem one is basically for 12 weeks which is 60 lectures each lecture being from half for half an hour and as you already know I do repeat in in the beginning of each class each week we have five lectures each being for half an hour and after each week we have assignments so we have already completed eight weeks that is eight assignments have been solved by you and also we are in a position we are getting queries in the forum and we are considering it accordingly and today's basically the third lecture in the ninth week which as you can see from the slide is the 43rd lecture in the 93 so yes considering the concept of stochastic dominance for stochastic dominance was there was first and second the first was the case when the CDF value of for decision a 1 is greater than equal to CDF value of a 2 and in the second-order stochastic dominance was when the expected value of a 1 was greater than expected value of a 2 and I have drawn this diagram accordingly if you remember I will draw it again if required as we proceed so we will consider few of the the different forms of the different way of expressions of the Mau T which is multi attribute utility theory I'll just mention them they we would not be solving any problems apart from the very simplistic one so for the additive one the utility function would be where you add up all the so technically okay let me go back a little one step backwards so generally you have a bundle of of our decision which is bundle of criterias criteria or attributes maybe for a car I am again given the same example a color guard boot space passenger space safety style mileage price resale value per month maintenance cost all these things some are objectives some are subjective some can be expressed that equation some cannot be expressed equation they are just characteristics attributes now for the utilities of the combination of that vector X X consists of x1 to xn depending on n dimension or it can be from x1 to XK depending of K dimension of the properties if we can express that whole utility as the sum of the utility functions for each individually then we have the additive function narrative for the case if you can add up now if we basically have the concept where you add up the utility function which is good but you also give some weight edges so you will basically the weighted additive function and these lambdas based on which you you find out the weights lambda 1 lambda 2 lambda 3 lambda n on lambda 1 to lambda K they would be known as the scaling functions for the utility functions so the weighted additive functions and these are the scaling functions so and this values of small yuen-han this capital u are the same they don't make any difference I am taking a utility function as a functional form only for the multiplicative or Lord log additive one you will just consider that you will basically have the combination such that for the single units that is U 1 u am i I'm mentioning for you X 1 u 2 I am mentioning for you x2 4 u 1 u 2 u 3 T u n you will multiply with them by the scaling factor add them up 4 so this is your taking one at a time combination when you are taking to edit ROM time combination if there are n of the of them so it will be NC 2 the combinations are you one with you two you want to use u 3 till you want with you then we'll take you to with u1 u2 - with u2 u2 with you three similarly the last one you - with UN and so on and so forth so you can multiply them with the concepts of the scaling functions or scaling factors which is lambda 1 for the combinations lambda 1 lambda 2 for the combinations of inj so if you have combining u3 with you 10 it will be lambda 3 and lambda 10 so we'll combine them and similarly as you increase the combinations you take nc3 so you'll basically have lambda I lambda J lambda K for the combinations of UI UJ u K so here it is so this is I'm only marking one so I use the red color so it would be easy for me to write so this is basically you I this is you J this is U K the should be utilizing can let me use the color difference should be easy for me easy for me to explain also this is you you so the KH one goes with the Kate one the jet one goes with the jet one and the eyath one goes with the eyuth one so you have different combination if you can put the higher combination it will be given I'm using the color red only as it is able to differentiate so it will be NC for combinations so I'll write I'll do I'll write in different colors now so these are the combinations NC 4 so now if I write it it would be lambda I one I'm using the suffix I 1 I 2 so it will easy for me to differentiate not ijkl I could do could have used that but I'm not going to use that and obviously the corresponding factor so L should leave space we can write so it will be you I won when I use the second color lambda I to you I to then green lambda I three you I three so lambda I for you I for so these combinations go along this combination go along this combination go along Obama means the weightages scaling factor changing and this combination go along so a basic in multiply them and have that value so here n is the number of utilize for lambda lambda x' are the scaling values which I have for each function so you what I am doing I am taking and see one combinations and see two combinations and see three combinations till then last one which is n CN combinations it's basically trying to find out the joint probability situation first case is the marginal case then e of the Joint Distribution bivariate case drive idiot case so on and so forth in the decomposable form the utility function UX 1 to u xn is given by a functional form where it is now a function of the utilities taken individually so it is u 1 u 2 u 3 till the last one which is taken as u n so I have a functional form which basically gives me in an output is the decomposable form of the mau t for the additive non-transitive decomposition if we have the functional form where X is bread greater than y in the sense that it dominates y X is a bundle y is a bundle of Decision bundle means those is a vector so you will basically have for any combination of you I for X and u j4x you will basically have the functional form all was dominating and and greater than zero similarly or the multiplicative util function where you multiply the utilities individually considering that u 1 u 2 u 3 u 4 and till the UN 8th one and you have been able to break them into individual part in the multi linear case you'll basically have a combination of some of the UK's where you multiply and then multiply those factors by a lambda J corresponding to the fact that you have been able to differentiate the whole utility function in to say for example three components of I has been divided into n and sorry n has been divided into three components which is N 1 n 2 n 3 such that and union of n N 1 n 2 n 3 is the universal set N and the intersection I am considering for the time being as N 1 intersection n 2 or n 2 intersection n 3 or n 1 intersection n 3 are our null set then for N 1 n 2 n 3 I have the multiplicative factors as PI J's depending of J which is equal to I 1 J is equal to I 2 in J is equal to I 3 so if I have that then then the beautiful function which I get is the multi linear utility function the utility functions which I'm just mentioning is the quasi additive bilateral hybrid quasi pyramid semi cube inter dependent variable and multi mean which have already considered now we will consider the properties for the multi attribute utility theory so there will be two properties I will come to that the pair of attributes of x and y so x and y are basically the bundle of decision which are vectors so pair of attributes X or Y which are elements of capital X then we will say that X is preferentially independent of attribute Z if the all the trade-offs between X and Z is not affected by a given level of offset so if I have the decision Z and it is not going to affect the ranking system on X 1 Y y at stage will be secrecy they are preferentially independent based on the fact and x and y are not affected by the decision or level of said while will say that utilities are independent how x will be independent of of the attribute y when the conditional preferences of the lotteries of x bundle of decision x or bundle of decision y where x and y element of the subs the overall universal set x or y whichever you know so this x and y are a subset of x and y is X small X and small Y a subset of either capital X or capital whichever you do you do not so capital X and capital y are the universal set and the elements are given by the small values and they are obviously vectors so when conditional repre friends of the lotteries in on X given Y do not do not depend on the particular level of Y so they would be independent in the sense that you can express them as as multiplicative factors so two attributes are additive independent when you can basically mention so the properties of the of the usual function so of you x1 to xn can be expressed as the sum of you until u n so that will be a property of additive independence where you to be function of any two bundles x1 and x2 can be expressed as the combination of u1 and u2 and it can go to n n different combinations so n RK number of combinations depending on how you denote the vector so if dimensions are N and you are going to combine them n number of times depending on different type of realized values so obviously it will be an additive independent to the factor of K now we will consider background of multi objective optimization theory and Here I am mentioning again I'm just giving you the picture no solutions would be done because that will be considered in the in the area of of MC well dat M 3 which is decision analysis and data analysis and decision making 3 where you will consider operation research as the main focus so let us consider a multi objective optimization problem and where the form of the optimization is that you have K number of functions and each is based on the fact that for some of the XS you basically take the functional form from F 1 to F K so consider I have the decision to find out the total cost total distance moved and consider the total say for example number of cars a lot is used so what is the background of the problem background the problem I am transporting goods from any one of the cities cities considered that's the address if for example Calcutta but now boom banish word Delhi Kanpur Lucknow Chennai Bengaluru Mumbai and all these things I am basically transporting goods from each location to the other so there are some distributors they are basically transporting goods to the retailer retailers so and I am the main decision maker I am the owner of the company so what is my main task is to find out what is the minimum number of lorries which I utilized transport goods what are the goods consider there I am trying to transport some cards say for example or parts of the cars or it can be say for example I am trying to transport different type of motors motors being of different ratings now when I do that I need to find out what is the minimum number of Lord is used because I want to I want to have to hire so if the hiring cost is there I want to minimize that Oh point one point number two is that if I am going to transfer goods from any one of the cities which I mentioned to the other retailers point from the disturber to the retailers or from the main storehouses of the factory to the other retailer point then basically I have to utilize the lorries in a such a way that I am able to visit maximum the hubs with the least number of lorries hubs means the places where I want to deliver and I want to basically travel in such a way that the minimum amount of some of the distance travelled by all the Lord is taken together is minimum that's why number two and consider the total cost is also minimum total cost food may may depend on say for example the roads which I follow the toll tax the charges per hour kilometers traveled and all these things so these are cost which I want to minimize on the other hand maximization can be what is the total profit what is the total sales revenues they can be many things so consider for some decisions where consider the decisions are to manufacture some quantum of the motors remote discount the ratings are say for example there are 20 my 50 horsepower 100 horsepower 150 horsepower so X is a bundle of decision where X 1 X 2 X 3 are the corresponding numbers which I produced for 50 hundred and 150 horsepower motors so for different values of X these functional forms of F 1 2 F 2 K are the total cost of travel total number of GML orders which I use total total con distance travelled it can be total revenues like a generator and so on and so when I want to optimize I want to optimize the combination of them in such a way that for the cost I want to minimize them for the revenues of the profits I want to maximize them so the optimization word I am using is in a very general sense here n X which I mentioned x1 x2 x3 which is 50 hundred and 150 HP Moto's they are basically belong to the real line or in the n dimension n where n is basically the dimension if there are 10 products obviously they will be 10 such items the individual objective functions are given as F 1 to F K I am not mentioning F 1 X still F KX because as you put the take out the values of X you get the functional form of F 1 to F K in the numeric value and what you do is that for someone bundled off of X's which I have I get the functional form of F I eyes got 2 1 2 K into the real line and get one value in dupeys or dollars or hence whatever it is so there may be many instances when the objective function are at least partially in conflict to each other because if you increase the cost obviously to mean the increase the it will also have an effect on the revenues or let me put it the other way if I want to increase the revenues I have to transport more Goods more lorries would be utilized Travel has to be more so obviously revenues will also incur and higher cost so that's why we do not have any decision vector X which will optimize in the best possible manner the objective function in all the respect that means it is able to maximize the the positive values and as well as minimize the negative values all at the same time but I will try to find out the best optimum solution so we are interested to solve the optimization problem in order to find out some of area or some of the values of X such that we have a multi criteria multi objective problem based on the fact I want to maximize or minimize or optimize some of the objective function based on an outcome which is X so some X's are given which are elements of capital X and I want to choose to some of them X such that I am able to optimize the fact of the optimization is that they are to be optimized individually but collectively the overall fact would be that the best solutions we shall have may not be same in the quantum front on on the vector sense that means if I have to bundle of outputs that means I have X star 1 and X star 2 and then the individual values of x star 1 and X star 2 may not be equal so what I'll try to show is this so I have let me mention so that you mention it in so high of X is an element of capital X and this X is in the real line and now the functional forms are okay let me mention the chain the color so it's easy so all negative costs are red consider if one f3 f4 and the positive are f2 that's why it's green it's poor f5 f6 f7 now what we will do is that if I find out say for example some I so what we'll have is that they would be a positive sense positive values would accrue from f2 f5 f6 and f7 negative values with a crew from so I'm trying to draw it f1 f3 so increasing one would also have a decreasing effort on the other if it is true then obviously will try to increase the green one and an automatically decrease the red one green and red being the positive and negative one but it's not that increasing one also would have an increasing out on another and vice versa because as I said increasing the revenue would also mean increasing the cost because you have to sell more of the profit products you have to basically utilize the transportation mode and so on and so now what I want to do is that consider there are some excise and consider this exercise are [Music] they do X let me mention them as X 1 star this this is X 2 star this X 1 and 2 have nothing to do with the values of the evening and consider X 3 star so K they give me the best solution so what is best it may be possible that the bundle of output which I have so they may be equal so what is that I'm mentioning consider this is X 1 star X 2 star X 3 star and I have these values which I have F 2 F 5 f6 f7 f 2 f 5 f6 [Music] and the negative values you have I'm drawing the using F 1 F 3 F 4 F 1 F 3 F 4 F 1 the combinations of the output which I get the bundle networth if they are equal but the individual factor of f1 f2 f3 f4 f5 and f6 and f7 individual they may not be equal that means the total benefit which I get for f7 either for x1 star or x2 star or x3 star you are definitely different but collectively if I if all of them are same I'll say that I'm going using the very simple concept the net worth if I get the same but then increase or decrease on individual front then I will say that they are para para to optimal says that I'm equally disposed by taking the decision x1 star or x2 star or x3 star because individually I get the network I'm combine I get the net for the individual may be different that's a different question so now here is the diagonal Ana and it's only possible for me to draw in the two-dimensional case I'll try to draw it in the 3 dimension because let me see how I am able to do that so consider the objectives you mentioned one is along the x axis object to and along objective one is along the y axis and if you see there are in this overall space of optimization there are four colors I have utilized so colors if you can see a B is red in color C D is green in color please note it down it will be for your own good you will understand so if you draw the diagram and if you are seeing this diagram again I mentioning mark a B is red then area or the line CD as green because the color scheme will have important fact while the line EF is blue and the line gh is violet so again I am mentioning a B is red C D is green EF is blue and gh is violet while the other portions of the line for this whole area B C is black so it would not have any meaning because you could not be possible for to us to differentiate so black color I am using for the fact that I am NOT going to draw any conclusions from here BC is black again I am mentioning de is black FG is black and H a is black so if you have drawn this now we will proceed based on which I will mention the concept so let us pay more attention on the colored line which is a B red C D green EF blue and gh as violet so in this hypothetical example which you have drawn in a two dimension it can be extend of the three dimension also as I mentioned there are two objectives it will illustrate the Pareto optimality plots for four different combinations of the objective functions where the objective functions are OB o 1 and O 2 which is objective function 1 and objective function to the boundary areas depict the hypothetical feasible set and the boundary is considered to be the optimal set of feasible points depending on whether it is a maximization or a minimization problem based on the fact that you're considering the object function 1 or 2 so the color schemes will now become important 0.1 the red line which you considered which was I for the first time I'll switch between this graph so let me make it just next to each other so this is the grind so here it is the red one so a B if I consider is for the case when we consider minimization objective 1 along with the minimization objective to let us see so I am trying to minimize both of them so considering this i am i would have this area if i similarly consider the green light blue or the blue and the violet one they are the instances which are like this maximization of objective one with minimization of the two two which is for the green one for the light blue one it is it is maximization of objective one and minimize a maximization objective two and finally violet one will be minimization of objective one and my maximization of objective two so violet one or minimization of objective one and maximization objective two so for the light blue it is a maximization of both from green one it is basically minimization of of objective two and maximization objective one while the red one is minimization of both so in the third three dimension how it looks I will consider that in the subsequent into class so this will be more of understanding the concept of multi objective problem and in the last two class for the ninth week I will try to wrap it up with some interesting examples as required to make the things much easier for you to understand from the very simplistic point of view obviously will solve such problems in t-80 m3 which I had already mentioned with this earlier in this class and have a nice day thank you very much [Music] [Music] [Music]