 I am Dr. Keshav Valasey from Mechanical Engineering Department of Valchian Institute of Technology in Solarpur. In today's session, we will briefly discuss about formulation of LPP that is Linear Programming Problem. Given a particular situation, given a particular date of some industry, how can we convert that into mathematical model that is Linear Programming Problem? At the end of this session, students will be able to understand what is the basic nature of the data that we need in LPP and how exactly we can convert that data into mathematical form that is what is referred as formulation of LPP. According to the nature of data that we have, we need to really understand what is the form of the data and once we understand that then probably we will be in a position to convert this real-life situation into mathematical form. So for objective functions and for constraints, how can we get that data? Linear programming problems, they have got two main things here. One is objective function which represents something that is to be achieved and second is subjected to a certain set of constraints which represent the limitations or the restrictions of different resources. Now how do we get these two things? Now basically we need to define the variables in the problem. From the different resources available as well as from the market conditions we may use certain forecasting techniques, collect the data on all these resources and then only we will be in a position to define the variables correctly and then only we will be in a position to convert that data collector into mathematical form of objective function as well as constraints. Now the reliability of the data that we collect here will decide the reliability of success of this solution given by this model and hence this data collection has to be very, very crucial. This is a very important phase and if we go wrong in this our total problem may go wrong and the worst part of it is nobody will be in a position to tell us in advance are we really right or wrong that is the main problem. So we have to take utmost care in collecting this data for different resources data from market condition types of domains whatever the sources we have data collection has to be carried out very critically. At this point I expect you people to think for a while on how can you formulate for some real life situation you can think of from any industry be it manufacturing or service industry just think for a while and how can you define the variables how can you collect the data and how you will be in a position to then formulate then put up mathematical relationships for objective function as well as constraints. So here let us consider one example and this is a example this is a data for one manufacturing industry which is manufacturing two types of paints. So it is a paint manufacturing company you really need to understand the data given and for this if you see this table this these two rows RMA and RMB these are raw material A and raw material B and on this next row it is given as profit per ton rupees units are 5 and 4 in the table. Again if you come back to RMA and RMB very extreme right side is 24 and 6 if you read on the top of the table maximum daily availability of these raw materials in tons what does it mean is raw material A is available 24 tons per day secondly raw material B is available 6 tons per day. So this data how it is given that you really need to understand first. Next is along this row of raw material A if you see these columns exterior paint and interior paints. So this is the company which is manufacturing two types of paints exterior paint and interior paint these are the two main products companies dealing with and this raw material A and B are being used in the proportion of 6 is to 4 and 1 is to 2 for these two types of paints. So this is how we really need to understand the data given. So this data gives us the details about the profits as well as the proportions of these raw materials being used in manufacturing two types of paints. Next two lines below the table if you see they are concerned with the market survey and demand aspects which are coming out from the market situation and lastly if you see the line determine optimum product mix to maximize the daily profit. So this is what is the main question we have we are asked to determine the product mix that is which product in which what quantity we should manufacture so that our daily profit will be maximum that is the ultimate question and what are we doing is from this given data we will be trying to convert this real life data into a mathematical form and that is referred as the formulation. So here very basic thing is we need to define the variable first in any formulation the very first step we need to have is define the variables of the problem. Now if you read here x1 is turns produced daily of exterior paint now how it comes up just go back and here the last line in the question determine optimum product mix to maximize daily profit. So daily day is the time dimension this is very very important for us to understand and these are the two types of paints. So exterior paint let us say denote as x1 per day produced and interior paint x2 units x2 turns per day produced. So this is how we get the time unit also and ultimately we define the variables as x1 is turns produced daily of exterior paint and x2 as turns produced daily of interior paint. This is a very important stage defining variables once we do this then we can go ahead with objective function as you all know objective function is something that is to be achieved and here in given problem if we just go back again the last question they have said is again I repeat determine in optimum product mix so as to maximize the daily profit. So profit maximization is the main objective and here we need to go back to again these slides we will swap as we need 5x1 and 4x2 look to these two terms on the right side and on the left side we have put up maximize z. What is this z is some symbol we are using to notify the function. So z is our objective function we are using a symbol and how do we formulate this z is it is given here in the data 5 rupees actually this is could be a scaled down data 5000 rupees could be the actual data as they have maybe it seems scaled down. So as to match with the data from other resources so 5 units in rupees is the profit from exterior paint so x1 units to manufacture for exterior paint into 5 is the profit from this first type of paint similarly x2 the variable notation into 4 will be the formulation of this total profit and that we sum up and put it to z that will be again maximize z is equal to 5x1 plus 4x2 is ultimately the final objective function. 6x1 plus 4x2 this is coming out from this data 6 is to 4 is the proportion of raw material a for x1 paint and x2 paint so 6 into x1 4 into x2 that we added and you cannot consume beyond 24 tons that is mathematically put up here as 6x1 plus 4x2 less than or equal to 24 tons the daily maximum limit of consumption of raw material a similarly this is the constraint we get from second raw material 1 into x1 plus 2 into x2 restricted to 6 tons that is put up as less than equal to 6 tons. Then this line reads the table part is over after the table below this you have market survey tells us that maximum daily demand of interior paint is restricted to 2 tons it is put up as a fourth constraint here x2 demand for interior paint cannot be more than 2 tons so less than equal to 2 this is the first line after the table we get and this one line it tells us again daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton so that is mathematically put up here as daily demand for interior paint x2 suppose if we put first then cannot exceed means maybe we could have put up less than equal to then than exterior than exterior that is x1 paint it could have been on the right side with plus plus x1 plus 1 on the right side so that we have taken to the left hand side and ultimately this particular market research data gives us minus x1 plus x2 less than equal to 1 so these are when all we got four constraints two from raw materials and two from market survey and lastly if you do not manufacture anything x1 and x2 that is two types of paints at the max can take zero value but not negative it is mathematically put up as x1 x2 greater than equal to zero which is referred as non negativity constraint this is in all the formulation of LPP that is linear programming problem these are the three books we have Taha, Sherman, Gupta thank you