Solve the below LP using SIMPLEX method. {:[" Maximize ",-x_(1)-x_(2)+4x_(3),],[" subject to ",x_(1)+x_(2)+2x_(3)

Question

Solve the below LP using SIMPLEX method.   {:[" Maximize ",-x_(1)-x_(2)+4x_(3),],[" subject to ",x_(1)+x_(2)+2x_(3) <= 9],[,x_(1)+x_(2)-x_(3) <= 2],[,-x_(1)+x_(2)+x_(3) <= 4],[,x_(1)","x_(2)","quadx_(3) >= 0.]:}

Bytelearn · Accepted Answer

Introduce slack variables: Introduce slack variables to convert inequalities into equations. Let x_(4), x_(5), and x_(6) be the slack variables for the respective constraints.  x_(1) + x_(2) + 2x_(3) + x_(4) = 9 x_(1) + x_(2) - x_(3) + x_(5) = 2 -x_(1) + x_(2) + x_(3) + x_(6) = 4Set up initial tableau: Set up the initial simplex tableau.  | Basis | x_(1) | x_(2) | x_(3) | x_(4) | x_(5) | x_(6) | RHS | |-------|-------|-------|-------|-------|-------|-------|-----| | x_(4) | 1     | 1     | 2     | 1     | 0     | 0     | 9   | | x_(5) | 1     | 1     | -1    | 0     | 1     | 0     | 2   | | x_(6) | -1    | 1     | 1     | 0     | 0     | 1     | 4   | | Z     | -1    | -1    | 4     | 0     | 0     | 0     | 0   |Identify entering variable: Identify the entering variable. The entering variable is the one with the highest coefficient in the objective function row, which is x_(3).Identify leaving variable: Identify the leaving variable using the minimum ratio test. Ratios for x_(3): 9/2, cannot use 2/-1 (negative), 4/1. The smallest positive ratio is 4, so x_(6) will leave.Perform row operations: Perform row operations to make x_(3) the basic variable in place of x_(6).  Row x_(6) becomes: (Row x_(6) + Row x_(1)) / 1 New Row x_(6): 0x_(1) + 2x_(2) + 2x_(3) + 0x_(4) + 0x_(5) + 1x_(6) = 4

Solve the below LP using SIMPLEX method. $\newline$ $\text{Maximize } -x_{1}-x_{2}+4x_{3},$ $\newline$ $\text{ subject to } (x_{1}+x_{2}+2x_{3} \leq 9),$ $(x_{1}+x_{2}-x_{3} \leq 2),$ $(-x_{1}+x_{2}+ x_{3} \leq 4),$ ( $x_{1}, x_{2}, x_{3} \geq 0.)$

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