 If you recollect that, so far we have discussed the solution of a single objective function by using what is called different techniques with equality constraint and inequality constraint. However, in real practice many designers are willing to optimize more than two or more than two optimization problems. So, that problem is called multi objective optimization problems or multi criteria optimization problems or sometimes it is called vector optimization problems. So, if you want to minimize a objective function more than two or more than two with equality constraint and inequality constraint then we will call it is a multi objective optimization problems. So, basic definition of this problem like this problem definition in general the objective function a multi objective function optimization problem can be represented can be represented as minimize f of x, f of x is a vector of dimension k into k into 1 which we have a n k objective function f 1 of x f 2 of x and dot dot f k of x and each is a function of x design variable and x is a n dimensional design variables or one can write it in other words minimize f 1 of x f 2 of x dot dot f k of x. We have k objective function we have to minimize subject to the constraint our constraint are equality constraint and inequality constraints are there subject to h i of x is equal to 0 i is equal to 1 2 dot dot p equality constraint that inequality constraint may be linear non-linear and our objective function also can be linear non-linear. So, our inequality constraint x g j of x is less than equal to 0 for j is equal to 1 2 dot dot m where our k is the number of objective function involved in the optimization problem k is the number of objective function to be optimized. So, from this two inequality that m and p equality constraint and m equality constraint we will get the feasible region or feasible space. So, feasible set or space is defined set or space is defined as x is the feasible region or set where I will get it from x i is equal to 0 i is equal to 1 2 dot dot p and g j of x less than equal to 0 j is equal to 1 2 dot dot m. So, these are as is the feasible set. So, in general a multi objective optimization problem we cannot get a single solution which will simultaneously minimize both the functions in multi objective optimization problem. We cannot get a single optimal solution that could optimize the both the objective function simultaneously. So, because of this the multi objective optimization is not a is not to search the optimal solution, but to find out an efficient solution which will make the of both the objective function value is minimum as minimum as possible. So, multi objective you can say not multi objective optimization is not to search for optimal solution, but for efficient solution that can be best attained or that can provides the best solution or that can provide best solution. Since we know in multi objective optimization problem we cannot get a single solution that will simultaneously optimize the both the objective function. If it is two objective function it cannot simultaneously optimize the that two objective function. If it is more than two let us call n objective k objective function is that a single point solution it cannot optimize cannot optimize all the objective function simultaneously. That is why we are looking for a multi objective optimization problem is not a search for optimal solution, but efficient solution can be obtained efficient solution efficient solution can be obtained efficient best solution can be attained. So, let us see with an example what we made this statement. So, let us call we will take in one example start with a simple example two objective function minimize f 1 of x is equal to x 1 minus alpha 1 whole square plus x 2 minus alpha 2 whole square. This is one objective function another objective function f 2 of x I am writing x 1 minus beta 1 square plus x 2 minus beta 2 whole square. If you look both the objective function in its quadratic form and our subject to the constraint let us call we have just consider only for inequality constraints that g 1 of x is equal to a 1 1 x 1 plus a 1 2 x 2 plus c 1 is less than equal to 0. And g 2 of x is equal to a 2 1 x 1 plus a 2 2 x 2 plus c 2 is less than equal to 0. Look this two objective function this objective function are non-linear in R in more specifically it is in quadratic form whereas, our constraints are in linear equation form. Now, if you plot these things in a graphically look what we will see this one suppose this is the x axis x 1 axis this is the y 1 axis sorry x 2 axis this is the two equation linear equation. Let us call g 1 of x I denoted by that one agree this was our g 1 of x and its the inequality condition satisfy in this region that above this line or on the line let us say. So, this one is you can say g 1 of x is equal to 0 or it is g 1 is less than 0 in this side on the line it is g 1 is equal to 0. And we have a another line g 2 is let us call it is something like this say and this feasible region is either on the line either any point on the line when it is equal to sign is there and if it is a less than 0 it is let us call in this side g 2 x is less than 0 in this or on the line this is g 2 of x is equal to 0. So, if you see carefully the feasible region of this optimization problem for multi objective optimization problem is our this region feasible region or as feasible set is as this one. Now, let us see what is this our two objective function then one objective function the center is here let us call this center is here whose alpha 1 and alpha 2 this one another is let us call here this is call is this center is beta 1 and beta 2. Now, you see this objective function f 2 this is corresponding to f 2 objective function f 2 this is corresponding to f 2 objective function this objective function value will be minimum when you draw a circle which touches this line g 2 of x which touches the circle g 2 of x. If you draw a circle which will touches the circle g 2 of x that means the perpendicular distance from here to here it should be perpendicular distance is the let us call this is p 2 this is p 2. So, p 2 one can easily find out from the equation of this g 2 expression. So, our if you see this one I will read out this circuit read out this figure because I have to make more conclusion from this curve. So, I will just read out this one once again to make it more clear I just read out this one once again. So, let us call g 1 just going and this is our g 2 g 2 of x this is g 1 of x and our feasible region is that we have assumed this feasible region is that one for feasible region of this one is above this line just we have a point here the circle of this one is beta 1 and beta 2 centre of this one and we have a another is here you can say alpha 1 and alpha 2. Now, if you look at this one this is our feasible region and feasible set if you draw a circle which touches this line g 2 of x then it is that one and let us call this distance from here to here is p 2. So, p 2 we can easily find out and that is the if you see the p 2 is the minimum value of the function f 2 this is the minimum value of the function f 2 that one and it this point is b let us call and what is the minimum value of the function from the centre of this objective function is that one agree. Let us call this point is something else let us call it is a this, but when this is the minimum no doubt this function value is minimum, but that point that point which is lying on the g 1 x is it satisfy the first inequality condition, but it does not satisfy the second inequality conditions g 2. So, this cannot be a optimal solution of this one agree. So, on the other hand if you see this one point b for function f 2 is the optimum value of the function and it is a feasible point because b 2 lies b 2 satisfy the what is called g 2 condition as well as the g 1 inequality conditions b 2 satisfy the equality condition of g 2 and the inequality condition of g 1. So, this is the feasible point, but that is not the feasible point. So, what we can do it here that we draw a circle of a radius this one if you draw a circle of a radius circle like this way of radius let us call this point is a this is a circle. So, this if you draw a circle with this one then this point a is the minimum value of the function f 1 and which is a lies in the feasible region because it satisfy both equality condition of g 1 and g 2. Similarly, that this one I told is satisfy the point b satisfy the equality condition of g 2, but inequality condition of g 1. Now, the objective function of this f 2 is minimum no doubt, but objective function of that one is we cannot say the minimize both are that at this point is simultaneously it minimizes that one. So, we can point out now the point a is the minimum value for f 1 function f 1 objective function if you just draw another circle which radius is greater than this one then this is the feasible in this region in this region that in this region, but that is not the minimum value of the functions where whereas, this value this value is the minimum point, but they are not considering with the same point. So, therefore, next is I can tell you this one the point b individually if you think point b is the is the minimum point value for f 2 objective function. Now, because this at this point this is the minimum value of the function whereas, the function f 2 as a minimum value of the function is b 2, but at a single point both the function are not simultaneously minimized. So, we cannot expect that there is a single point for which that there is a single point for which both the function will be both the objective function will be simultaneously optimized. So, we are looking for not for we are not searching for a optimal solution for a single point optimal solution we are looking for best efficient solution that will satisfy that will give you the best optimal solution. So, let us see so we can make it conclusion now, but which look at this point as I mentioned here this point is minimum for this f 1, but this point is minimum for this point of this one. Now, which point will give you the simultaneously both the function if you just add it will give you the minimum value of function. So, we have a so many points are there from there we have to picked up which one is the minimum of objective function value minimum sum of the two objective functions. So, but which design that minimizes both f 1 and f 2 simultaneously, but this is not clear even for a simple two objective functions. You see two objective functions we have considered, but it is not clear which at which point we will get the f 1 and f 2 simultaneously minimum. So, therefore, what we are looking for therefore, if one wishes if one wishes to minimize f 1 and f 2 simultaneously, then pin point a point is not possible. Then particular point or pin pointing a single point single point is single optimum point you can write single optimum point is not possible. So, we have to do some compromise. So, in fact, there are infinite points many possible solution possible solution points called Pareto the Pareto. Look at this point suppose I am telling you at this point a point this function is f 1 function is minimum, but it satisfy the wall constraints at this point this function is minimum. So, if you increase this one this function is increasing function value is increasing agree, but this function if you make it same, but this function is value. So, function value is now increasing. So, you have to find out some point which directly not minimizing the both the function simultaneously, but some of the function value will be minimum. So, now let us see this one what is called we plot it here criteria space that is f 1 of x in this direction, then f 2 of x in the directions 0 the criteria this figure represent the graphic graphical representation of two objective function in criteria space. The graphical representation of two objective function because we have taken the example of two objective function two objective function functions in criteria space and criteria space is generated by z. What is mean by criteria space that we have a two inequality constraint will move along this line g 1 and also g 2 and we will plot the value of function corresponding to g 1 if you move along the g 1 along the g 1. We can easily find out what is the function value of f 1 and f 2 let us call at this point we know x 1 is what x 2 is what. So, we can find out the value of f 1 and f 2 plot it f 1 versus f 2 then next another point you take it you know x 1 x 2 find out the value of f 1 of x f 2 of x plot it in that space. So, that space what will get it f 1 versus f 2 that is called criteria space. So, if you plot it this one let us call we are getting this curve another curve this is for let us call it is a q 2. So, q 2 represents the g 2 boundary g 2 of x boundary what does it mean this is the g 2 of x. So, any point on this one let us call if you take this point I know x 1 x 2 then correspondingly f 1 f 2 you plot and along this line when g 1 is 0 along this line you plot if you plot it that is q 2. Similarly, if you move along the g 1 axis g 1 will straight line on the line you will get q 1 q 1 represents the boundary of g 1. So, let us call q 1 is like this way. So, this is q 1 that is represents the boundary represents the boundary of the boundary of represents the g 1 of x boundary. So, this is the boundary of this one. So, inside this g 1 is this region and inside the g 2 is this region. So, what is the common space of that one if you see. So, our criteria space is that one only. So, that is the criteria space that is the criteria space. Now, look at this point that is what I am telling it here if I make a circle here if I make a circle here with a radius b this function value f 1 function value this I will call this is the function value of f 1 at b and this function is f 1 at a. So, this function value will be what this is the common point for both the common point for both the objective function of this you can say this is the one of the solution. But whether solution is the Pareto optimal or not we do not know and one another thing we can draw it here by centering this one with this point of this one. So, this is the f 2 of a and this point is f 1 of b point. So, now you may say that this point whether it is a sum of the two objective function value is less than the sum of the objective function of. So, what we can say the f 1 a sorry we can write it now based on this one that f 1 a and f 2 a this point some of this function objective function whether it is less than the f 1 b plus f 2 b this is f 1 b is this is the f 2 b sorry this is the f 2 b and this is the f 2 b this is f 1 b which one is less this we do not know, but we have to search for this one for which we will get the there are infinite number of points that for which the function value you see this function value is increased, but this function value is decreased. Whereas, if you see in this case at this point the function value of what is called f 1 a is minimum whereas, the f 2 a value f 2 a value is f 2 value a is increased. So, we do not have a clear cut answer at what point whether a point b point or other point we will get some of the two objective function is minimum. So, if you take in another point let us call this point is that point if you can take it we draw a circle of this one this point, which crosses this one and that is if you see this circle if you draw a circle over there our feasible region is from here to here and this portion is infeasible region. So, let us call this circle I am considering this one now. So, we have a so now let us see this one if you want to represent this thing in here. So, let us call our d point is here or let us call our b point is here a point is here we draw a circle which is passing through this one another point is that one it cuts here let us call this is the cuts is here is d point is here which cuts that circle which passes to the a point of f 1 of a d point then we have a it cuts that point here that is our we can draw a circle. Then say this point is our c point agree this is the c point and this is the another point which cuts is here let us call this point is this cuts here here that is the e let us call this is f point. Now, see this one and I told as I mentioned it here that this region is the our feasible region this region is our feasible region and this is this point e point and f point is satisfy the g 1 it satisfy the g 1 condition, but it does not satisfy the f 1 f does not satisfy the g 2 condition. Whereas, e satisfy the g 2 conditions agree and if you see the d point and as well as the c point d point satisfy the what is called g 1 condition, but c point does not satisfy the g 2 conditions. So, in this way we will see that there are some points are there which will give you the sum of the two objective function value is minimum. So, if you plot it this one from there we can find out the region where is the our what is called criteria space this is the criteria space of that one and this is we got it from the design space by finding out the objective function value along the constraint that g 1 along the constraint g 1 of x 0 and g 2 of x 0 this is corresponding to g 2 of x 0 and this is corresponding to g 1 of x 0. So, you will just give some definition of that one now some of the important definition of that in context to the what is called multi objective optimization problems. In general a curve in the design space in the form of g j x is equal to 0 is translated into a curve q j in the criteria space simply by evaluating the value simply by computing the values of of the objective functions at different points along the g 1 on the surface of g 1 or g 2 functions at different points on the constraint curve on the constraint curve on the constraint curve in the design space that is what we have done it this q 1 q 2. So, there are some of the terminology generally used in objective what is called multi objective optimization problems that is called solution concept. So, first is Pareto optimality what is this suppose a point x star is a point x star in the feasible or design space in the feasible region or in the design space x is the feasible or design space s is the Pareto optimal Pareto optimal is the Pareto optimal Pareto optimal if and only if necessary and sufficient condition if and only if there exist there if and only if there does not exist does not exist. Another point on the in the feasible region another point x in the design space x star set s s is the feasible design space such that f of x is less than equal to f star of x this there are no such x is there other than x star that which function value will be less than this we have a how many functions are there k functions are there any function of that f 1 f 2 should not be less than this if it is less than this should not be less than this one agree with at least one f 1 of x is less than equal to f 1 rather that is called in the other more general f i of x star i is equal to 1 2 dot dot f 1 k in any one of this is violates this one should not be less than this one without increasing another one. So, a point x star is said to be optimal what is the Pareto optimal if there does not exist any feasible point inside a feasible region such that this condition does not satisfy this condition does not satisfy without increasing any other function values. Then we will call it is a Pareto optimal solution another this one is weak Pareto optimality this definition is exactly same as this one only inequality sign only it is a strictly inequality sign will be replaced. So, a point x star in the feasible space in the feasible design space the feasible design space s is weakly Pareto optimal if only if and only if necessary sufficient there does not exist there does not exist any x another point that does not exist another point x in the design space s such that f of x is less than equal to f star of x that is there is no point in other words what does it mean there is no point exist in the design space for which the objective function value improves means less than that one this implies that there is no point exist no point that improves means the value of the function value is reduced further that improves all the objective function all the objective function simultaneously. The difference between this one is here that if you further if you further decrease of design when the other cases without increasing the another one it may be same, but not increasing this one, but here it does not improve all the function values simultaneously this is the weakly Pareto optimal and basically there are two more definitions are there which are called efficiency efficiency and dominance efficiency is the another primary concept in multi objective optimization problems this is the another you can say another primary concept in multi objective optimization problems. So, let us call what is this efficiency indicates this one definition a point this basically efficiency and Pareto optimal definitions are exactly same a point x star is the in the feasible design space as is efficient if one only if there does not exist there does not exist another point x in the set x is such that f of x is less than f star of x with at least f i of x f i star of x that does not exist with at least otherwise x star is inefficient. So, this next is dominance, so I just mentioned is that there are two concept is a generally used in efficient and dominance. So, another common concept is the non dominance and dominant points in a multi objective optimization problems. So, non dominant non dominated and dominated points, so this are defined in based on the what is called criteria space a vector because our objective function is a vector f 1 f 2 dot dot f n a vector of the objective function f star of x in the feasible criteria space z star z a vector objective function f star in the feasible criteria z is non dominating netted mind it when we are talking about the feasible criteria space. That means, we are going along the that area if you see this one this is the criteria space this boundary is indicates the along the g 2 constraint and this boundary constraint the g 1 constraints and along this side is the criteria less than this indicates the value of the function value of the objective function inside this region indicates the value of the objective function which is which is on a inside the feasible region. So, is dominated non dominated if and only if there does not exist another vector another f f in the solution set in the set z means criteria space such that f of x is less than equal to f star of x. But it in the criteria space with at least one f i of x is less than is less than f i of x star otherwise it is dominant otherwise f star is dominant and last point associated with the multi objective point contested with this u 2 u 2 p a point. Now, you see this one u 2 p a point is if you look at this expression now let us call this one agree. If you see this function f 1 when it is passing through this a 1 the function value is minimum when it is function f 2 when it is passing through the point b it is a optimal. But that two points are not the solution of this one is not the common point that solution of this one and that point you can easily see it must be the outside the what is called criteria space. How that see this one this point this point is on the surface of the on the boundary of q 2 and this is on the boundary of if you see this one this is on the boundary of what is called q 1. One is on the boundary of q 1 another is on the boundary of this is on the boundary of q 2 and this is on the boundary of q 1. So, where is the u 2 p a point will be definition of this one a point f 0 in the criteria space the end space is called the u 2 p a point if f i f i we have a how many function of j 2 and k functions are there f i is minimum of f i x i this for this indicates for all I am talking about f i for all x in the design space s for i is equal to 1 to k. So, if you see this curve u 2 p a point of this one let us call this is our z 2 and this is our q 1 and this is our q 2 and this is our q 2 point this one and our u 2 p a u 2 p a point is outside the what is called our feasible region this is the outside the our feasible region this is in this direction f 1 of dot this is a f 2 of dot. See this one if you see this one that is and this point it is a lies on the q 2 and this point lies on the what is called q 1 point. So, q 1 point and q 2 point in other words what you say this one this is the minimum value of the function and this is the minimum value of the function if you plot it here you will be here minimum value of the function and this is the q 2 u 2 p a point of this one by definition it is nothing but a point in the criteria space is called u 2 a point when f i is minimum f i is minimum for all x in the design space because this is the minimum for all feasible region this is the minimum point and this is the minimum point for all but that point is outside the what is called our z space which is called criteria space. So, we will stop it here and this is correspondingly this point you can say this point f 1 of a and this is f 2 of b point this u 2 a point. So, from next class we will start with a what is called the dynamic optimization problems. So, far we have discussed last few lectures is a static optimization problems starting from single variable case non-linear how to solve the optimization problem using the what is the numerical techniques then linear programming non-linear programming all these things we have discussed and lastly we have given the some basic idea of multi objective optimization problems and related some definitions. We have seen that there is not possible at all to find a single optimum point single point for which both the objective function or all the objective function multiple objective function will be simultaneously minimized. So, there must be a what is called Pareto optimal point. So, here we will stop here.