Optimization Techniques Introduction: Linear & Nonlinear Problem Solving

in this video I'll be giving introduction to the subject optimization techniques so let's see what is

optimization it is a theory and subject related to finding the optimal points referring to maxima or minima and it

allows us to explore various methods or the algorithms that can be applied to the mathematical models that we get from

the real life situation and hence we can talk about their solutions and we can also look at the optimization of theory

from a different point of view we can see the best solutions for a given particular situation now what do I mean

by best solution in a given situation let me go first given our example with the mathematics only we all are aware

how to find a minima and Maxima of a function let us see that so considered optimization in linear and non-linear

functions both so let me first take a case in linear I want to optimize a function which takes the value small X 1

plus X 2 and the condition is that this line X 1 plus X 2 is less than or equal to 1 it is a straight line it has this

inequality it satisfy less than or equal to 1 and the decision variable are non-negative once we are saying the

decision variable are non-negative so this is non negative that means they are in the first quadrant

the points and this is a straight line which passes through the point 1 0 and it passes through the point 0 1 it is

less than so the region is the lower side but all together I get this particular region so this this triangle

is the region in which I want to maximize this value so we can see the maximum value occurs here only so this

is how you can talk about the maximum value in the linear functions and if I want to look at in the nonlinear

function we are all aware how to find the maxima or the minima involving the derivatives we can talk about the global

Maxima point or the global minimum and also we can talk about the local minima and local Maxima so this address to the

first point minimum or maximum of a function we can also look up the best solution for example in a real life

situation such as transportation problem where I want to associate the lowest cause say I have a

transportation business and I want to minimize the total transportation cost and in this situation considered this as

a transportation model so here I am showing the transportation model in the form of a network we want to supply

something from the warehouses and there is a destination where the material is going to be reached so we have a supply

and we have a destination you can see this is the supply available at each warehouse and this is the requirement at

each of the destination now the material is to be transported and there is a per unit cost associated with these

transportation so obviously the question would be how do we minimize this total transportation cost so we need to see

such real-life problem and we need to see what sort of algorithm can help us to answer these problems and in the

third case we might be interested in knowing the network and shuttling problems a little more deeper so for

example there is an infrastructure project that and in that infrastructure project we really interested in knowing

what is the time duration of the project how many years will it take to finish this big project or if we are writing a

small project maybe I'm writing a book and I divide that whole project into small activities so I need to know how

much time duration I require for finishing that project and for that I also have to invest certain cost so we

always want to minimize that cost so this means overall looking at these examples in an optimization we can have

an idea that we are trying to optimize our time or the cost or maybe if you are traveling you want to minimize the

distance also optimization technique is not limited to the examples that I mentioned here there are various

situation in which the optimization plays very important tool and so to understand optimization we also need to

understand what is mathematical formulation or the mathematical model behind that real-life situation the

mathematical formulation or the mathematical model will allow us to relate various real-life problems with

the mathematical variables tools and the solution strategies that we are going to apply while solving those problem the

one of the very important starting point in the optimization is to understand what are how they we select the

alternatives available in early life situation and the second argument that I can give here is the best

mathematical model will allow us to reach to the nearest best optimal solution let's see what do I mean by

alternatives here let me to consider an example now I have a situation one here and in this situation I'm just trying to

understand what do I mean by alternative imagine that I have a five-week business commitment between federal and then ever

I can fly out off a table on Mondays and return on Wednesday a regular round trip cost $400 but I get a discount 20% is if

the round-trip dates span a weekend a one-way ticket in either direction cost 75 percent of the regular price so these

are the condition and based on that conditions what should be my decision how should I buy the ticket for a

five-week period in the current scenario I have availability with three alternatives and let's see here in the

calendar what is given to us in the problem is we have been given five week plan so this is we need to understand

and the day that I need to cover is I have to go on Monday and I need to come back on Wednesday so say for example

this is the first week that I've selected so this is Monday to wednesday and then has choose the second week that

I choose the third week for three and the fifth week okay and each week is independent so I can purchase the ticket

independently and one roundtrip so for example the first alternative available with me is Fatima in every Fatima so

it's a round-trip departure on Monday come back on Fitness day and it's the same week so this will cost me a regular

price and the regular price is $400 and five weeks so this will cost me $2,000 that's first alternative available with

me but I have also another alternative available with me as buy one fatty will deliver ticket for round-trip then ever

federal deliver that span weekend and one again then ever federal ticket so this is one way journey this this is

also one way journey and we got four round-trip so in this one we journey it is going to cost us seventy-five percent

of the regular price that is same five percent of the $400 and forth it round-trip that span weekend there is a

discount of 20% so let's see says suppose if I started the departure from here that is the Monday and we reached

today never so this is the one-way booking from Fatima to Denver and then we reach to deliver and on fifth from

Denny where I'm going to departure again and I'm going to arrive on the federal and I'm going to now look at at this

trip here it is departure and at this stage this would be arrival and I'm going to follow this procedure now for

the next four trips and we can see that in these trips there is a weekend that is going to come up there are weekends

that are going to come up and because of these weekend we get a discount of 20% and the last journey months we have

reached from this last journey that is a day never we can come back to federal so the cost that is going to come up here

is the we have one-way trip so this is 75% of the regular price of the round-trip plus there are four for

round-trip journey that span weekend because it is span weekend so 20% discount will give me actually the cost

as 80 percent of the regular price and plus the last one VIP trip this will again cost me 75 into 400 and this total

will cost me 1880 in third alternative I'm going to buy one federal day never set ever it's a round trip again and

here I'm going to choose Monday first weekend witnessed a of last week so for example here I want to select this

Monday and the last Wednesday of the filing so this is my departure and on this date I am going to have arrival and

the remaining duration so once I have started departure here I have reached her den ever and on fifth I'll start my

departure found anywhere and I will come back on 10th and the remaining will be again round-trips so this is what we are

going to follow up so in this case we got five round trips but the round trips are little different you see here fit

able to federal and this second down trip is deliver to deliver covering Fayetteville and covering

deliver so this is five weeks and in each case this is going to give me a discount of 20% as it overlay over the

weekend this will cost me 1600 so by comparing now the cost of the other routes $2000 1880 and 1600 we see

clearly that this one is the optimal we will choose this alternative now in the second situation I want to understand

models so this will lead me to more mathematics so that is how we will ensure that we are going to construct a

mathematical model in the optimization technique and we are going to solve the problems the consider forming a maximum

area rectangle out of a piece of a wire l inch what should be the best width and height so there is a wire of length L

this is to be bent this is to be bent so I am bending this length this total length is L and this bending is to be

done in a way that I get a rectangle so here this is say width and this one is the height so the question is what

should be the maximum area in the rectangle so we can see this what are the restriction in this situation width

of the tangle plus height of rectangle is half of the length or I may say twice the width plus half is L and I want to

maximize the area's length into breadth or width into height and we know that width and height can never be negative

you know width and height of the rectangle will be positive if there is no bit if there is no width and there is

no height then the area is zero so obviously we are going to introduce now here the non and negative restrictions

these are what we call non negative restrictions and the one that we have been written here this is what we

understand it has a constraint and the first one is understanding as the objective function so this is what we

are trying to see that from the practical situation there are type of the problem that gives us alternative or

there are type of the problem that give us a constraint based on that we want to formulate the mathematical model and

then we want to see how does this model look like does this model look like linear quadratic

in nonlinear and we try to search their solution strategies so I can now see the journal modeling and optimization

techniques involve an objective function that may be maximization or minimization it depends on the problem and there are

certain constraint so we if I associate my objective function as FX and constraint as GI X so I can say that if

FX and all the constraint I X for every i if all are linear if they all are linear then we can talk about that this

is a linear programming problem this is LPP linear programming problem if any one of them if FX or GI X or both

or at least any one of them so I have put it all here if any one of them are nonlinear so because we do not know at

this moment what is the structure of a real-life problem whenever we are going to construct we will see that from the

function and this what come up from the construction so if any one of them is nonlinear we will put this into the

category of non linear and then we will learn these strategies or the solution techniques that is available in non

linear and similarly we can look up what sort of model do we have and we can start applying the programming problem

techniques