Question

Solve the following using graphing techniques: a. Maximize 2x1 + 3x2 subject to the constraints, 2x1 + 2x2 < 8,X1 + 2x25 4, and X1 > 3, x2 > 0

Answer 1

soln consider the linear programming problem - marimize 201+382 subiect to the constraints 204+2349 58 x1+2x254 81753, 72710 Now to draw constraints treat mequalities - as eavad, we get 3917292 = 8 U 21+292 =4 - ?) X,=3 192 = 0 solving eanu , we get 2X1+292 = 8 when xazo 2x1=8 B =) *1=4 when #1=0, 2X2=8 59,=4 so for constraints (I), we get | 22 1 4 lo from when when eam (P), we get x, 20, 222=4 => N2=2. U2=0, x=4.

for constraint (3) ee 9, 73, we get a line to Parallel to y -aufs. all 3131 지 이 Similarly to X270 iwe get with the help of all let us draw these point, the we graph get este 221+22233 o co,4) + ++ uzsumudy - - +--- 11713 20+27254 W1+2912 s d (400) - N2=0 7 Y 105 i 15 $ 3's y ya's we get the extremal de Points are . A (310) ,B(410) and ((3,015). Now the value of Obiertove function at each of these extreme pornts is. At AC310) Obiective funckion value = 2(3)+300) = 6 At B(410) objectire function value =214) +3(0)=8 At c(3,05) Obiereire function value=2(3)+3(0.5) = 705 The maximum value of objectore function is 8 occurs at the externe point (4:0). Hence the required solution is ac, = 4, U2 = 0 and maximum value = 8.