Duality 5 Dual of an LPP with Equality constraint

hello friends welcome now we are going to take a new case of duality and the speciality of this case is this is a linear programming problem having different types of constraints or say mixed constraints the sign of the first constraint is less than or equal to in case of second it is equality and in case of 30 is greater than or equal to but let us take what is the object you it is minimization and now we are well aware that in case of minimization the sine of all the constraints should be greater than or equal to now in this case these two constraints are with different sign so we have to change the sign of these two constraints and they must be greater than or equal to that will be very easy for the first constraint it is less than or equal to we just need to River the sine 2h will be minus 2 X 1 minus 4 X 2 greater than or equal to minus 160 but the unique case is the second constraint that is an equality what should happen equality is X 1 minus X 2 equals to 30 that can be segregated into 2 inequalities with different signs 1 will be X 1 minus X 2 greater than or equal to 30 another will be X 1 minus X 2 less than or equal to 30 now what should we write first greater than or equal to or less than or equal that depends upon the objective if objetive is minimization then write the inequality of greater than or equal to first and if the objective is maximize the son then write the inequality with less than or equal sign first so it will be X 1 minus X 2 greater than or equal to 30 and X 1 minus X 2 less than or equal to 30 but our objective is minimization so the sign of this inequality should also be greater than or equal to and what will happen it will be minus X 1 plus X 2 less than or equal to minus 30 so now our revised primal see that rival original linear programming problem is always known as rival but we have to rewrite the primal in its revised de form what will happen minimize Z equal to X 1 plus 2 X 2 subject to minus 2 X 1 minus 4 X 2 greater than or equal to minus 160 second will be now X 1 minus X 2 greater than or equal to 30 third will be minus X 1 plus X 2 greater than or equal to minus 40 and the last will be as it is because it's sign matches with the objective X 1 greater than or equal to 10 and X 1 X 2 both are non-negative now sign of our signs of all the constraints are greater than or equal to directly matching with the objective we can write you all of this revised prime what will we do first of all number of variables in the duel will be exactly equal to number of primals so the number of constraints in the primal so there will be 4 y1 y2 y3 and y4 similarly now the right hand side will be 4 Z X Z Y or Z star let us first write the doer of this reverse friend against the objective of minimization the objective of viewer will be maximizes and maximize Z star or zy whatever you write yes now this right-hand side will become the coefficients of the respective variables in the Z functional objective function so it will be fast minus 160 y1 plus 30 Y 2 minus 30 y 3 + 10 y 4 ok subject to the constraints and the constraints will be only 2 because there are only two variables in the private number of constraints in that ul will be exactly equal to the number of variables or decision variables in the crime from X 1 we can form the first constraint minus 2 into I 1 minus 2 I 1 plus 1 y 2 minus 1 by 3 plus 1 y 4 less than or equal to opposite sign or sign matching with the objective of maximization less than or equal to nothing is 1 and the second constraint will be from X 2 minus 4 XY - sorry and they show it - 4 y 1 minus 1 by 2 plus 1 y 3 no I - here so less than or equal to 2 and all for y1 y2 y3 and y4 all are non-negative now the two very principles are followed number of constraints in the primal will be number of variables in the dual four constraints for variables number of variables in the primal will be exactly equal to the number of constraints in that your two against two but there is an equality at number two in the original primal what the rule says if there is an equality in the primal there must be a variable unrestricted in signing the dual but there is no variable unrestricted in sign in the viewer so this cannot be treated as final viewer this is not the final you are not final this is not final now what how can we say write unrestricted variable what is the sign of unrestricted variable if there is any unrestricted variable in the dual we have to split it into two variables which are always non-negative say in this case observed y2 and y3 see the value of the coefficients are equally all the cases but the signs are opposite to each other this is the sign that y2 and y3 are the variables say substituted in place of a variable which is unrestricted in sign so take or substitute y dash equals 2 y 2 minus y3 where why this is unrestricted in sign now the dual will be maximize Z star equals to minus 160 y1 sine of pi/2 will be there plus 30 y dash plus 10y for now subject to minus 2 I 1 plus y dash y plus if we have maintain the world of sine here properly matching with the objective then the sine of this unrestricted variable will be the sine of the first variable which is taken as the representative variable one of the two representative variables here Y 2 is positive so why guess will be positive 2 minus 2 y 1 plus y dash plus y 4 less than or equal to 1 minus 4 y 1 minus y dash because sine of Y 2 is minus no wife 1 less than or equal to 2 yes it is - now y 1 and y 3 are non-negative and y dash unrestricted now anyways what is our original primal it has two variables the final view LS two constraints original primal s three constraints now the final viewer has three variables original driver has an equality at number two the viewer has unrestricted variable at a number to see order is y 1 y dash and wife so exactly at number two there is unrestricted variable and all other rules minimizes and maximization lasts forever right hand side becomes the coefficient of objective function so this is the final view we can say that this is final rule of the original right so this is the dual of this primal these two are parts of working or we can read them as working on that's it thank you

Channel: PUAAR Academy

Views: 77,831

Keywords: OR, Operarions Research, Linear Programming, Duality, Dual, MBA, MCA, CA, CS, CWA, CPA, CFA, CMA, BBA, BCom, MCom, CAIIB, FIII

Id: GVk46HOneAI

Length: 11min 48sec (708 seconds)

Published: Sun May 29 2016