Graphical method to solve LPPGraphical method to solve LPP

Prof. Keyur P Hirparay pAssistant Professor

keyur.hirpara@rku.ac.in9879519312

1

C t d N tConvex set and Non-convex set

Convex setif any two points of the polygon are selected arbitrarily thena straight line segment joining these two points liescompletely within the polygoncompletely within the polygon.Extreme points of convex set are the basic solution to theLPP

2

Convex Non Convex

x2 Max. Z = 3 x1 + 5 x2

s/t, 3x1 + 2x2 ≤ 18

10 x1 ≤ 4

x2 ≤ 6

8 x1 ≥ 0

x2 ≥ 0

6 (2, 6)

4Z = 36

22

02 4 6 8 10 x1

0Z 3

Max. Z = 10 x1 + 6 x2

s/t, x1 + x2 ≤ 10x2

11 x1 + 8 x2 ≤ 88

x1 ≥ 012

x2 ≥ 0

8

10

(8/3 22/3)

6

8 (8/3, 22/3)

Z = 76.67

4

6

Z

Z = 80

2(8, 0)

‐12 4 6 8 10‐1‐2 12

x1

‐24

Max. Z = 6 x1 + 5 x2

s/t, 2x1 ‐ 3x2 ≤ 5

x1 + 3x2 ≤ 11

4 x1 + x2 ≤ 15

x2

x1 ≥ 0

x2 ≥ 0

4

5

3

4

(3.09, 2.64)

2

3

Z = 31.73

1

Z

‐11 2 3 4 5‐1‐2 x1

‐25

x2 Min. Z = 3 x1 + 5 x2

s/t, 3x1 + 2x2 ≥ 18

10 x1 ≤ 4

x2 ≤ 6

Z

8 x1 ≥ 0

x2 ≥ 0

6

4Z = 27

2(4, 3)

2

02 4 6 8 10 x1

06

Diff t f l ti f LPP Different cases of solution of LPP

Unique finite solution Unique, finite solution The example demonstratedhere is an example of LPPhere is an example of LPPhaving a unique, finitesolution. In such cases,optimum value occurs at anextreme point or vertex ofth f ibl ithe feasible region.

7

Different cases of solution of LPP, cont cont..

Unbounded solution If the feasible region is notbounded it is possible thatbounded, it is possible thatthe value of the objectivefunction goes on increasing

ith t l i th f iblwithout leaving the feasibleregion.

8

Different cases of solution of LPP, cont cont..

Multiple (infinite) solutions Multiple (infinite) solutions If the Z line is parallel to anyside of the feasible region allgthe points lying on that sideconstitute optimal solutions.

9

Different cases of solution of LPP, cont cont..

Infeasible solution Infeasible solution Sometimes, the set ofconstraints does not form af i ifeasible region at all due toinconsistency in theconstraints.

10

Different cases of solution of LPP, cont cont..

Unique feasible point Unique feasible point This situation arises whenfeasible region consist of ai isingle point.

This situation may occur onlywhen number of constraints iswhen number of constraints isat least equal to the numberof decision variables.

11