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Maximize $w=2x+3y+6z$ subject to

$2x+y+z \le 5$

$3y+2z \le 6$

$x,y,z \ge 0$

Is the optimal solution unique? Justify your answer.

I tried to solve it by Simplex method. In the second iteration, I got the optimal solution condition and the optimum solution is $z=3,\; x=1, y=0$. . Correct me if I have done anything wrong.

Also how to justify that there exists or doesn't an alternate optimum?

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    $\begingroup$ i have got $$30,x=0,y=0,z=5$$ $\endgroup$ – Dr. Sonnhard Graubner Sep 18 '17 at 12:55
  • $\begingroup$ That's right, $(0,0,5)$ is the optimal and unique. $\endgroup$ – A.Γ. Sep 18 '17 at 13:04
  • $\begingroup$ Can you please provide the solution? Also how to justify that the optimal solution is unique? $\endgroup$ – Balaji Sep 18 '17 at 13:08
  • $\begingroup$ There was some mistake in the second constraint. I have corrected it. Please check it. $\endgroup$ – Balaji Sep 18 '17 at 13:17
  • $\begingroup$ @Balaji OK, the corrected problem has indeed the solution after two iterations as $(1,0,3)$. No non-basic column has zero in the top row of the simplex tableau, therefore, the solution is unique. $\endgroup$ – A.Γ. Sep 18 '17 at 13:30
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/* Objective function */ max: 2x + 3y + 6z; 2x+y+z <= 5; 3y+2z <= 6;

/* Variable bounds */ x >= 0; y >= 0; z >= 0;

x = 1, y = 0, z = 3

Optimal solution 20 after 2 iter. Relative numeric accuracy ||*|| = 0

MEMO: lp_solve version 5.5.2.5 for 32 bit OS, with 64 bit REAL variables. In the total iteration count 2, 0 (0.0%) were bound flips. There were 0 refactorizations, 0 triggered by time and 0 by density. ... on average 2.0 major pivots per refactorization. The largest [LUSOL v2.2.1.0] fact(B) had 3 NZ entries, 1.0x largest basis. The constraint matrix inf-norm is 3, with a dynamic range of 3. Time to load data was 0.003 seconds, presolve used 0.008 seconds, ... 0.006 seconds in simplex solver, in total 0.017 seconds.

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