3.2.4. Speciālgadījumi

$\displaystyle L=x_{1}+x_{2}\longrightarrow min,$
$\displaystyle x_{1}+2x_{2}\geq 10,$
$\displaystyle 2x_{1}+x_{2}\leq 4,$
$\displaystyle x_{1}\geq 0,\;\;x_{2}\geq 0.$

$\displaystyle G_1=$	$\displaystyle \;\{(x_{1};x_{2}):\;x_{1}+2x_{2}\geq 10 \},$
$\displaystyle G_2=$	$\displaystyle \;\{(x_{1};x_{2}):\;2x_{1}+x_{2}\leq 4\},$
$\displaystyle G_3=$	$\displaystyle \;\{(x_{1};x_{2}):\;x_{1}\geq 0,\;-\infty <x_{2}<+\infty\},$
$\displaystyle G_4=$	$\displaystyle \;\{(x_{1};x_{2}):\;x_{2}\geq 0,\;-\infty<x_{1}<+\infty\}.$

$\displaystyle L=x_{1}+2x_{2}\longrightarrow max,$
$\displaystyle x_{1}+3x_{2}\geq 6,$
$\displaystyle 3x_{1}+2x_{2}\geq 12,$
$\displaystyle x_{1}\geq 0,\;\;x_{2}\geq 0.$

Dotajai problēmai nav atrisinājuma, jo mērķa funkcija nav ierobežota pieļaujamajā apgabalā, kurš ir attēlots 3.5. zīmējumā. Atzīmēsim, ka minimizācijas problēmai

$\includegraphics[width=8cm]{C:/TEXfiles/felikss/3n5zcol.eps}$

3.5. zīm.

$\includegraphics[width=12cm]{C:/TEXfiles/felikss/3n6zcol.eps}$

3.6. zīm.

$\displaystyle L=10x_{1}+30x_{2}\longrightarrow min,$
$\displaystyle x_{1}+10x_{2}\geq 10,$
$\displaystyle x_{1}+3x_{2}\geq 6,$
$\displaystyle 5x_{1}+x_{2}\geq 4,$
$\displaystyle x_{1}\geq 0,\;\;x_{2}\geq 0.$

Dotās problēmas pieļaujamo apgabalu (skat. 3.6. zīm.) nosaka koordinātu asis un taisnes:

$\displaystyle CD:$	$\displaystyle \;\;x_{2}=-\frac{1}{10}x_{1}+1,$
$\displaystyle BC:$	$\displaystyle \;\;x_{2}=-\frac{1}{3}x_{1}+2,$
$\displaystyle AB:$	$\displaystyle \;\;x_{2}=-5x_{1}+4.$