Question

Consider the linear Programme (LP)

Max 4x + 6y
Subject to
$3x+2y\le 6$
$2x+3y\le 6$
$x,y\ge 0$

The dual for the LP in question is

• A. Max 6u + 6v
  subject to
  $3u+2v\ge 4$
  $2u+3v\ge 6$
  $u,v\ge 0$
• B. Min 6u + 6v
  subject to
  $3u+2v\ge 4$
  $2u+3v\ge 6$
  $u,v\ge 0$
• C. Min 4u + 6v
  subject to
  $3u+2v\le 4$
  $2u+3v\le 6$
  $u,v\ge 0$
• D. Max 6u + 6v
  subject to
  $3u+2v\le 4$
  $2u+3v\le 6$
  $u,v\ge 0$

Answer 1

The correct option is B Min 6u + 6v

subject to
$3u+2v\ge 4$
$2u+3v\ge 6$
$u,v\ge 0$
$\text{Primal is}$

$MaxZ=\left(46\right)\left(\frac{x}{y}\right)i.e.MaxZ=CX$

$\text{Subject to}$

$\left(\begin{array}{cc}3& 2\\ 2& 3\end{array}\right)\left(xy\right)\le \left(\begin{array}{c}6\\ 6\end{array}\right)i.e.\mathrm{AX}\le B$

$\text{Dual is}$

$MinZ=B^{T}Y$

$\mathrm{Min}Z=\left(66\right)\left(\begin{array}{c}u\\ v\end{array}\right)\Rightarrow \mathrm{Min}Z=6u+6v$

${\text{A}}^{T}Y\ge C^T$

$\left(\begin{array}{cc}3& 2\\ 2& 3\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right)\ge \left(\begin{array}{c}4\\ 6\end{array}\right)\Rightarrow \begin{array}{c}3u+2v\ge 4\\ 2u+3v\ge 6\end{array}$

$\text{and}u,v\ge 0$

Note:

$\left(\begin{array}{c}\mathrm{If}\mathrm{Primal}\mathrm{is}\mathrm{given}\mathrm{by}\mathrm{Min}Z=\mathrm{CX}\\ \mathrm{AX}\ge B\\ \mathrm{the}\mathrm{corresponding}\mathrm{dual}\mathrm{for}\mathrm{LP}\mathrm{is}\mathrm{given}\mathrm{by}\\ \begin{array}{c}\mathrm{Max}Z=B^{T}Y\\ A^{T}Y\le \mathrm{CT}\end{array}\end{array}\right]$

$\left(\begin{array}{c}\mathrm{And}\mathrm{if}\mathrm{the}\mathrm{Primal}\mathrm{is}\mathrm{given}\mathrm{by}\mathrm{Max}Z=\mathrm{CX}\\ \mathrm{AX}\le B\\ \mathrm{Then}\mathrm{corresponding}\mathrm{dual}\mathrm{is}\mathrm{given}\mathrm{by}\\ \begin{array}{c}\mathrm{Min}Z=B^{T}Y\\ A^{T}Y\ge \mathrm{CT}\end{array}\end{array}\right]$