I have got to maximize $$Z=x+2y-3z+4w$$

subject to constraints $$x+y+2z+3w=12$$ $$y+2z+w=8$$

$x,y,z,w\geq 0$ .

The question asks to show without actually solving the lpp that it has an optimal solution. For that I added the two constraints to get $x+2y+4z+4w=20$ and from this I put the value of $x+2y+4w$ in objective function which became $Z=20-7z$.

Thereafter I concluded that Z could not be greater than $20$ i.e. Z is bounded above. Corresponding to this maximum value of Z we have z=0 and using this I can find solutions such as $(0,6,0,2)$ and $(4,8,0,0)$ which are feasible. So the optimal solutions exist. But I want to know what are the other methods of showing it. Can we use duality theory?

  • $\begingroup$ You want to $\underline{\text{maximize}}$ $Z=20-7x$. You´re right that 20 is the upper bound. Therefore .... $\endgroup$ – callculus Nov 13 '19 at 18:29
  • $\begingroup$ This is a case where I was fortunate enough to write the objective function in terms of only one variable using the given constraints. But all cases won't be as simple as this. I am therefore looking for a general method of showing existence of optimal solution. $\endgroup$ – HARVEER RAWAT Nov 13 '19 at 18:54
  • $\begingroup$ Then post another example. $\endgroup$ – callculus Nov 13 '19 at 18:58
  • $\begingroup$ Let us then consider maximizing Z*=2x+y-3z+4w subject to the same constraints as in original problem. $\endgroup$ – HARVEER RAWAT Nov 13 '19 at 19:07
  • 2
    $\begingroup$ If it exists a solution for the constraints, then there exists an optimal solution. $\endgroup$ – callculus Nov 13 '19 at 19:45

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