As x2 and x4 are unrestricted sign first we will replace them by nonnegtivevaria

Question

As x2 and x4 are unrestricted sign first we will replace them by nonnegtivevariables form

by taking x2=x2'-x2'' and x4=x4'-x4'' then our original problem can be written as the folowing form

Maximize z=2x1+(x2'-x2'')+3x3+(x4'-x4'')

subj to ;

x1+(x2'-x2'')+x3+(x4'-x4'') <= 5,

2x1-(x2'-x2'')+3x3 = -4,

x1-x3+(x4'-x4'') >= 1,

x1>=0, x3 >=0, x2'>=0, x2''>=0, x4'>=0, x4''>=0

Now again we make this problem as equality constraint by adding to more variable in it say

x5>=0, x6 >=0

Now for calculating duality use the following formula

Primal{ max C^T X s.t Ax <=b, X>=0} Then its dual is written in the following way

Dual{min w b such that wA >=c, w>=0}

Explanation / Answer

2.10. Consider the following linear programming problems maximize z subjectto Axb Exercises 187 and minimize z subject to Ax2 b 0) Write the duals to these problems ) If both of these problems are feasible, prove that if one of these problems has ii)If both of these problems are feasible, prove that the first objective is unhounded iv) Assume that both of these problems have finite optimal solutions Let x be a finite optimal solution then so does the other above if and only if the second objective is unbounded below feasible for the first problem and let i be feasible for the second. Prove that 115. How to do question(, (ii), and (iv)?

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