 Hello, Myself Satish Halange, Assistant Professor, Department of Civil Engineering, Vulture Institute of Technology, Swarapu. In today's session, we are going to see regarding the formulation of the problem in the linear programming problem for the minimize case. At the end of the session, the learner will be able to convert the real-world problem into the mathematical form that is the linear programming problem. Here, let's see what is the linear programming problem. It's a mathematical modeling technique in which the following terms are involved. First one is objective function. It may be of a maximizing case or it may be of a minimizing case. And next comes regarding the decision variables which are involved in the particular example or the problem which are given the restriction that is the non-negative variables we say. The objective of learning of the linear programming problem is to maximize the profit or you can say the benefits or the production or to minimize the time requirement for the manufacturing or minimize the wastage of the resource which is required for the manufacturing of the product, etc. Now coming to the point of today's session, the formulation of the problem in the linear programming problem. The linear programming problems can be solved by the following techniques that is by the graphical method, simplex method or big M method or two phase method. Now steps involved in the formulation of the problem in the linear programming problem are as follow. Step one, we have to observe, study and understand the real-world problem. We have to identify the correct problem of the case and we have to describe the objective function. After this, we have to go for the defining the decisions variables involved in the problem. The decision variables are always help us to define the values in the quantitative value. And we have to go to the next step and we have to define the objective function equation by using the decision variables. We have to write the case of the particular problem. It may be the maximize case or the minimize case. Now next come step number five. Once we have identified the objective of the particular problem, we have to define these constraints or the limitations associated with the example. The limitations or you can say the constraints set equations are in the equality form or inequality forms. And finally we have to give the restrictions to the particular variables. That is see the example, here the multiple is there which is having the case study of the water supply system. Basically they are using two methodology. One is the gravitational methodology and second one is the mechanical boosting or you can boosters states for the supplying of the water. And when we see this description, they are conducting the research and they are saying that the particular overall description that the pressure expressed by the gravitational method and the mechanical pumping station in the entire municipalities were put at the 2 kN per m2 and 8 kN per m2 respectively. And the municipality has three pumping states that is A, B and C. There are three locations pumping states A, B and C and the pumping state A is expressed to have the combined pressure of not more than 24 kN per m2. And the housing state B should have the pressure of not more than 130 kN per m2. And the house state C will have the pressure of magnitude not less than 12 kN per m2. This are the particular limitations which are defined for the housing state A, B and C regarding the pressure. Means here these are the pressures, minimum pressures or you can say limitations of pressures defined to get the proper supply of the water by the supplied apartment. Now this table is giving you overall picture of the pressure by the gravitational method and the mechanical booster states for the each housing state A, B and C. Now first step as we have seen we have identified that in this present case the municipality or you can say municipal is instead interstate to have the betterment water system or you can say increase the distribution of water properly by two methodology one is of gravitational method and by the mechanical method. As they are facing the problem by the only by the gravitational method. So we have observed the particular problem, we have studied what the exactly the example or the problem requirement is there and we have understood. Now we have to define what is our objective of the particular problem, yes in the present case the main objective of the particular municipality water system is to minimize the pressure requirement in the municipality. So we have to define the variables. Now after knowing the what the particular problem is there and what is the objective and we have to move towards the finalizing the decision variables. Now the municipality is one to supply the water by gravitational method and the by the mechanical booster states. So let us assign x1 is the variable of gravitational method and x2 is the variable of mechanical booster state. Now once we have defined the variables of the problem that is of the related to the gravitational method and the mechanical method we have to move towards the objective function and we have to write the objective function equation. The objective function equation is what to minimize the pressure. So minimization case it is there and z is equal to 2x1 plus 8x2. How it is been defined you just see what they are saying just go through the overall particular description the pressure expected by the gravitational method and the mechanical pumping station in the entire municipality where put up to the 2 kilo Newton per meter square and 8 kilo Newton per meter square we have to define the constraints or you can say set of constraints. Here after identifying the constraints are what the constraints are particular availability of the pressure with each pump stations or you can say housing stations. Here housing is state A when we move towards the example here the description is saying the municipality has 3 housing states A B and C and housing state A is expected to have combined pressure of not more than 24 kilo Newton per meter square. So this by using this table for housing state 1 A or you can say A 1x1 plus 2x2 is here what they are saying for the first housing station it should not more than means the equation will be 1x1 plus 2x2 less than or equal to 24 because 24 is the maximum it should not be more than 24. So similarly for the house is standing B when we have seen this problem the housing state B should have the pressure of not more than 130 kilo Newton per meter square. So 7x1 plus 6x2 is less than or equal to 132 similarly for the house is states C x1 plus 2x2 greater than or equal to 12 because here the when we see the description the house is state C will have the pressure of magnitude not less than they are saying means it should not be have the pressure less than 12 kilo Newton per meter square. So by using this table for housing state C 1x1 plus 2x2 it should be greater than or equal to 12 these are the three set of constraints available with this particular problem and finally we have to use the restrictions signed to the particular variables decision variables x1 and x2 is greater than or equal to 0. So this is overall formula example objective function is equal to minimize z is equal to 2x1 plus 8x2 subjected to the constraint x1 plus 2x2 less than or equal to 24 for sub or house is state A and 7x1 plus 6x2 less than or equal to 132 for house is state B and x1 plus 2x2 greater than or equal to 12 for house is state C by the restriction of the sign to that both of the variables. Just answer this particular question hope so you have selected the correct answer for the example these are the references for the today's session thank you.