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azizahkamar2007

LINEAR PROGRAMMING

THE CONCEPT OF LINEAR PROGRAMMING

The problems related to linear programming can be solved by the following steps:

(i) Write linear inequalities and equations that describe a situation.

(ii) Draw the graphs and shade the region where the points in the region

are feasible solutions.

(iii) Determine and draw the object function.

(iv) Determine graphically the optimum value of the objective function.

SKILL 2Identifying and shading the region in which every point satisfies a linear inequality.

EXAMPLES1) x ≤ 4 x ≥ 4

2) y ≤ 7 y ≥ 7

3) y ≤ x + 3 y ≥ x + 3

3) 2x +3 y < 6 2x +3 y > 6

+y < below +y > above

4 4

7 7

3

3 3

3

2 3

SKILL 1

Drawing a graph of

a straight line.

EXAMPLES1) x = 4

2) y = 7

3) y = x + 3

x 0 1y 3 4 (0,3) (1,4)

3) 2x +3 y = 6

x 0 3y 2 0 (0,2) (3,0)

2x +3 y > 6

x

y

y

x

x

X (1,4)

y

y

x

SKILL 3Writing linear inequality or equation that describe a situation.

EXAMPLESDescription Mathematical

representation1) y is more or equal to x. y ≥ x2) y is at least x. y ≥ x3) y is at most two times x y ≤ 2 x4) y is at not more than x. y ≤ x5) x + y has a maximum value of 10 x + y ≤ 106) y is at least 20 more than x. y- x ≥ 20 7) The minimum value of the total of x and y is 40.

x + y ≥ 40

4

7

2

3

1 2 3 4 5 6

R

Maximum value of x = 6Maximum value of y = 8Minimum value of x + y =

Maximum value of 2x + 3y = 34Minimum value of y when x=2 :Ans y = 2 Maximum value of 2x + y when y = 4 : Ans: 14

Given (x,y) € R ,x and y are integers:

1

2

3

5

6

7

8

SKILL 4y

x

0

0

0

0

0

3