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Solutions of the given linear programming problem

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,047
Hello!! :eek:
Given the following linear programming problem
$$\min(2x + 3y + 6z + 4w)$$
$$x+2y+3z+w \geq 5$$
$$x+y+2z+3w \geq 3$$
$$x,y,z,w \geq 0$$
I am asked to find all the solutions using the simplex method.

To solve this problem we use the Two-Phase method, don't we?
Then I found that there are infinite many solutions, $(x,y,z,w)=(1-m, 2+0.4m, 0, 0.2m), 0≤m≤1$ with $min=8$.
Could you tell me if this is right?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,715
Hello!! :eek:
Given the following linear programming problem
$$\min(2x + 3y + 6z + 4w)$$
$$x+2y+3z+w \geq 5$$
$$x+y+2z+3w \geq 3$$
$$x,y,z,w \geq 0$$
I am asked to find all the solutions using the simplex method.

To solve this problem we use the Two-Phase method, don't we?
Then I found that there are infinite many solutions, $(x,y,z,w)=(1-m, 2+0.4m, 0, 0.2m), 0≤m≤1$ with $min=8$.
Could you tell me if this is right?
Your answer is certainly correct, because if you add the two inequalities you get $2x + 3y + 5z + 4w \geqslant 5+3=8$. Therefore $2x + 3y + 6z + 4w \geqslant 8+z$, which is minimised by taking $z=0$. Your solutions, with $z=0$, clearly satisfy all the given conditions, so they must be right.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,793
Hello!! :eek:
Given the following linear programming problem
$$\min(2x + 3y + 6z + 4w)$$
$$x+2y+3z+w \geq 5$$
$$x+y+2z+3w \geq 3$$
$$x,y,z,w \geq 0$$
I am asked to find all the solutions using the simplex method.

To solve this problem we use the Two-Phase method, don't we?
Then I found that there are infinite many solutions, $(x,y,z,w)=(1-m, 2+0.4m, 0, 0.2m), 0≤m≤1$ with $min=8$.
Could you tell me if this is right?
Looks good! ;)

EDIT: Aargh, overtaken by Opalg.
 

mathmari

Well-known member
MHB Site Helper
Apr 14, 2013
4,047
Your answer is certainly correct, because if you add the two inequalities you get $2x + 3y + 5z + 4w \geqslant 5+3=8$. Therefore $2x + 3y + 6z + 4w \geqslant 8+z$, which is minimised by taking $z=0$. Your solutions, with $z=0$, clearly satisfy all the given conditions, so they must be right.
Great!!! Thank you for your answer!!! :eek:

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Looks good! ;)

EDIT: Aargh, overtaken by Opalg.
Nice!!! Thank you!!! :eek: