B B Tripathy SIOM 1

Linear Programming by Graphical Method

Algorithm of finding solution by Graphical Method:

I : Formulate the appropriate LPP.II : Draw the graph of LPP.III : Obtain a feasible region ( a region which is common to all the constraints of LPP) .IV: Obtain the solution points( the corner points and intersections points of feasible region)V : Calculate the values of objective function at the solution points.VI : For maximization problem, the optimum solution is the solution point which gives the maximum value of the objective function and for minimization problem solution is the solution point that gives the minimum value of objective function.

B B Tripathy SIOM 2

Maximize Z = 3x1 + 2 x2

subject to x1 ≤ 4 x1 + 3x2 ≤ 152x1 + x2 ≤ 10

andx1 ≥ 0, x2 ≥ 0.

Solving LPP by Graphical Method

B B Tripathy SIOM 3

B B Tripathy SIOM 4

The exact value of (x1, x2) for each of these nine corner-point solutions (A, B, ..., I)

Corner-point feasible solutions

(x1, x2) Objective Value Z

A (0, 5) 3*0+2*5 = 10

C (3, 4) 3*3+2*4 = 17

E (4, 2) 3*4+2*2 = 16

F (4, 0) 3*4+2*0 = 12

I (0, 0) 3*0+0*0 = 0

B B Tripathy SIOM 5

Since point C has the largest value of Z, (x1, x2) = (3, 4) must be an optimal solution.The objective function value is 17.

B B Tripathy SIOM 6

Problem – I (Solve by Graphical Method)

Maximize : Z = 250x1 + 200x2

Subject to constraints

6x1 + 4x2 ≤ 2402x1 + 5x2 ≤ 1504x1 + 3x2 ≤ 120 x2 ≤ 20 x1 , x2 ≥ 0

B B Tripathy SIOM 7

Minimize : Z = 3x1 + 2x2

Subject to constraints

5x1 + x2 ≥ 10 x1 + x2 ≥ 6x1+ 4x2 ≥ 12 x1 , x2 ≥ 0

Problem – II (Solve by Graphical Method)By Using TORA Software

B B Tripathy SIOM 8

Maximize : Z = 10x1 + 3x2

Subject to constraints

2x1 + 3x2 ≤ 186x1 + 5x2 ≥ 60

x1 , x2 ≥ 0

Special Case in LPP Graphical Method

B B Tripathy SIOM 9

Maximize : Z = 12x1 + 25x2

Subject to constraints

12x1 + 3x2 ≥ 3615x1 - 5x2 ≤ 30

x1 , x2 ≥ 0

Special Case in LPP Graphical Method

B B Tripathy SIOM 10

Special Case in LPP Graphical Method

Maximize : Z = 20x1 + 10x2

Subject to constraints

10x1 + 5x2 ≤ 506x1 + 10x2 ≤ 604x1 + 12x2 ≤ 48

x1 , x2 ≥ 0

B B Tripathy SIOM 11

Maximize : Z = 5x1 + 7x2

Subject to constraints

x1 + x2 ≤ 43x1 + 8x2 ≤ 2410x1 + 7x2 ≤ 35 x1 , x2 ≥ 0

Problem – II (Solve by Graphical Method)By Using QM for Window Software

B B Tripathy SIOM 12

Maximize : Z = 3x1 +4x2

Subject to constraints

5x1 + 4x2 ≤ 2003x1 + 5x2 ≤ 1505x1 + 4x1 ≥ 1008x1 + 4x2 ≥ 80 x1 , x2 ≥ 0

Home Assignment (GLPP) - IBy using graph paper

B B Tripathy SIOM 13

Minimize : Z = 20x1 + 10x2

Subject to constraints

x1 +2x2 ≤ 40 3x1 + x2 ≥ 304x1 + 3x2 ≥ 60 x1 , x2 ≥ 0

Home Assignment (GLPP) - IIBy Using TORA Software

B B Tripathy SIOM 14

Maximize : Z = 1170x1 +1110x2

Subject to constraints

9x1 + 5x2 ≥ 450

7x1 + 9x2 ≥ 315

5x1 + 3x1 ≤ 1500

7x1 + 9x2 ≤ 1890 2x1 + 4x2 ≤ 1000

x1 , x2 ≥ 0

Home Assignment (GLPP) – IIIBy Using QM for Window software