# How to show that the lpp has optimal solution without solving it

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?

• You want to $\underline{\text{maximize}}$ $Z=20-7x$. You´re right that 20 is the upper bound. Therefore .... – callculus Nov 13 at 18:29
• 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. – HARVEER RAWAT Nov 13 at 18:54
• Then post another example. – callculus Nov 13 at 18:58
• Let us then consider maximizing Z*=2x+y-3z+4w subject to the same constraints as in original problem. – HARVEER RAWAT Nov 13 at 19:07
• If it exists a solution for the constraints, then there exists an optimal solution. – callculus Nov 13 at 19:45