# Questions on Linear Programming Problem (LPP) Graphical Method

Question 1: Maximize $$z = 15x_1 +10 x_2$$

Subject to,

$$4x_1\ + 6x_2 \le 360$$

$$3x_1\ +\ 0x_2 \le 180$$

$$0x_1\ + 5x_2\le 200$$

$$x_1,x_2\ge 0$$

Ans: (60,20) and 1100

Question 2: Maximize $$z=2x_1\ + x_2$$

Subject to,

$$x_1\ + 2x_2 \le 10$$

$$x_1\ -x_2 \le 2$$

$$x_1\ +x_2\le 6$$

$$x_1-2x_2 \le1$$

$$x_1,x_2\ge 0$$

Ans: (4,2) and 10

Question 3: Minimize $$z=3x_1+2x_2$$

Subject to,

$$5x_1\ + x_2 \ge 10$$

$$x_1+x_2\ge 6$$

$$x_1\ +4x_2 \ge 12$$

$$x_1,x_2\ge 0$$

Ans: (1,5) and 13

Question 4: Minimize $$z=20x_1\ + 10x_2$$

Subject to,

$$x_1 \ + 2x_2 \le 40$$

$$3x_1\ +x_2 \ge30$$

$$4x_1+3x_2 \ge 60$$

$$x_1,x_2\ge 0$$

Ans: (6,12) and 240

Question 5: Minimize $$z=6x_1+24x_2$$

Subject to,

$$x_1+2x_2 \ge 3$$

$$x+4x_2\ge 4$$

$$x_1,x_2\ge 0$$

Ans: (4,0) and (3,0.25) and 24

Question 6: Maximize $$z=45x_1 + 80x_2$$

Subject to,

$$5x_1+20x_2 \le 400$$

$$10x_1+15x_2 \le 450$$

$$x_1,x_2\ge 0$$

Question 7: Minimize $$z=3x_1 +2x_2$$

Subject to,

$$x_1-x_2\le 1$$

$$x_1+x_2 \ge 3$$

$$x_1,x_2\ge 0$$

Question 8: Minimize $$z=7x_1+8x_2$$

Subject to,

$$3x_1+x_2\ge 8$$

$$x_1 + 3x_2\ge 11$$

$$x_1,x_2\ge 0$$

Question 9: Maximize $$z=120x_1+100x_2$$

Subject to,

$$10x_1+5x_2\le 80$$

$$6x_1+6x_2\le 66$$

$$4x_1+8x_2\ge 24$$

$$x_1,x_2\ge 0$$

Question 10: Maximize $$z=5x_1+7x_2$$

Subject to,

$$x_1+x_2 \le 4$$

$$3x_1+8x_2\le 24$$

$$10x_1+7x_2\le 35$$

$$x_1,x_2\ge 0$$

Question 11: Mazimize $$z=40x_1+30x_2$$

Subject to,

$$x_1\le 16$$

$$x_2\le 8$$

$$x_1+2x_2\le 24$$

$$x_1,x_2\ge 0$$

Question 12: Maximize $$z=6x_1+4x_2$$

Subject to,

$$-2x_1+x_2\le 2$$

$$x_1-x_2\le 2$$

$$3x_1+2x_2\le9$$

$$x_1,x_2\ge 0$$

Ans:($$x_1$$=13/5 or 5/7,$$x_2$$=3/5 or2/7) and 18

Question 13: Maximize $$z=3x_1+4x_2$$

Subject to,

$$5x_1+4x_2\le 200$$

$$3x_1+5x_2\le150$$

$$5x_1+4x_2\ge 100$$

$$8x_1+4x_2\ge 80$$

$$x_1,x_2\ge 0$$

Ans: (30.8,11.5) and 138.4

Question 14: Maximize $$z=5x_1+7x_2$$

Subject to,

$$x_1+x_2\le 4$$

$$3x_1+8x_2\le24$$

$$10x_1+7x_2\le 35$$

$$x_1,x_2\ge 0$$

Ans: (1.6,2.4) and (24.8)

Question 15: Maximize $$z=5x_1+3x_2$$

Subject to,

$$3x_1+5x_2\le 15$$

$$5x_1+2x_2\le10$$

$$x_1,x_2\ge 0$$

Ans: (20/19,45/19) and 235/19

Question 16: Maximize $$z=8x_1+7x_2$$

Subject to,

$$3x_1+x_2\le 66$$

$$x_1+x_2\le 45$$

$$x_1\le 20$$

$$x_2\le 40$$

$$x_1,x_2\ge 0$$

Ans: (21/2,69/2) and 325.5

Question 17: Minimize $$z= 20x_1+10x_2$$

Subject to,

$$x_1+2x_2\le 40$$

$$3x_1+x_2\ge 30$$

$$4x_1+3x_2\ge 60$$

$$x_1,x_2\ge 0$$

Ans: (6,12) and 140

Question 18: Minimize $$z=x_1-7x_2+190$$

Subject to,

$$x_1+x_2\le 8$$

$$x_1\le5$$

$$x_2\le 5$$

$$x_1+x_2\ge4$$

$$x_1,x_2\ge 0$$

Ans: (0,5) and 155

Question 19: Minimize $$z=x_1-5x_2+20$$

Subject to,

$$x_1-x_2\ge 0$$

$$-x_1+2x_2\ge 2$$

$$x_1\ge 3$$

$$x_2\ge 4$$

$$x_1,x_2\ge 0$$

Ans: (4,4) and 4

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