4.linear programming

8/13/2019 4.Linear Programming

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4. Linear ProgrammingConvex test: If P, Q X X then the plane X is called convex set (or) If the line segment

oining any two points of a plane entirely lies in the same plane, then the plane is called convex Set.

Ex:

are convex sets.

are not the convex sets.

The function f = ax+by, which is to be maximised or minimised is called an objective functionor Profit function.

The line which are obtained from an Objective function with the Point from Fesible region is

called Iso- Profit line.

Iso-Profit lines are parallel to each othrr.

Every point in the feasible region is called feasible solution.

Every point which makes an objective function either maximum or minimised is called optimum

solution.

If the iso-profit line moves away form the origin then the profit will be increased and it is moves

close to the origin then the profit will be decreased

2 Marks Questions

1. State the polygonal regin represented by

x 1, x 3, y 1, y 3

A : i) x 1 boundary line x=

1 which is parallel to y-axis

ii) x 3 boundary line is x=3 which is parallel to y-axis

iii) y 1 boundary line is y =1 which is parallel to x-axis

iv) y 3 boundary line is y=3 which is parallel to x-axis

PQ


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Here the four shaded region is the solution set for the given equations.

2. State the polygonal region represented by x 0, y 0, x+y 1

A : i) x 0 boundary line is x = 0 which represents y-axis

ii) y 0 boundary line is y = 0 which represents x-axis

iii) x + y 1 boundary line is x+y = 1

x + y = 1

put x = 0, y = 1 (0, 1)put y = 0, x = 1 (1, 0)

{ (0,1) (1,0)}

(0, 0):- x + y 1

0 + 0 1

0 1 (True)

Shaded region should be towards the origin


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Here triple shaded region is the solution set for the given inequation

1 Marks Questions

1. Define "Objective function?"

A : A function which is to be maximised or minimised in L.P.P is called an objective function (or)

Profit function.

Note: The solution of Linear inequation is either closed convex polygon or open convex polygon.

2. Define "Iso-profit line"

A : The system of Parallel lines obtained form the objective function are called Isoprofit lines.

Iso profit lines are Parallel to each other.

3. Define " Feasible regin"

A : The region formed by the constrains of L.P.P in the form of Closed or Open Convex set iscalled

"Feasible region".

4. State "Fundamental Theorem?

A : Maximum (or) Minimum Value of an Objective Function in L.P.P gets atleast one of the Vertices

of the Polygon.

5. Define "Convex Set" and give two examples.

A : The line Segment joining any two points in a plane entirly lies in the same plane. then the plane

is called as " Convex Set ".

(or)

X is a Plane, P X, Q X X then " X" is Called as "Convex Set".

6. Define L.P.P. (Linear programming problem)?

A. The problem which consists of an objective function (or) profit function and the constraints that

are in the form of linear inequation is called "Linear Programming Problem" (L.P.P.)

5 Marks Questions

1. Minimise f= 3x+2y subject to x+y 1, x y, 0 x 1, y 0

Sol. x+y 1, xy, 0 x 1, y 0

i.e x+y 1, xy, x0, x 1, y 0Here x 0, y 0 represent Q

1

i) x+y 1

x+y = 1

put x = 0, y = 1 (0,1)

put y=0, x=1 (1,0)

{ (0,1) (1,0)}

(0,0) 0+0 1

0 1(True)

Shaded region should be towards originii) x y x = y

put x = 0 y = 0 (0,0)put x = 0 y = 10 (10,0){(0,0) (10,10)}

PQ




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x y satisfies Q4

(+,-)

shaded region should be towards Q4

iii) x 1 boundery line x=1

which is parallel to y axis

vertices of the polygon are (1,0), (1,1), (1/2, 1/2)

objective function f = 3x+2

3(1) + 2(0) = 3+0 = 3

A+ (1,0) f = 3(1) + 2(0) = 3+0 = 3A+ (1,1,) f = 3(1) 2(1) 3+2 = 5

A+ (1/2, 1/2) f = (1/2) +2(1/2) = 3/2 +1 = 5/2 = 2.5

minimum value is 2.5 at (1/2, 1/2)


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SHI2. Minimise f = xy subject to con ditions x+yf = x+y; x + y 6; 2x+y 8; x 0 and y 0i) x 0; y 0 represents 1st Quadrant

ii) x + y 6; boundary line is x+y = 6

put x = 0 0 + y = 6 y = 6 (0, 6)

put y = 0 x + 0 = 6 x = 6 (6, 0)

= {(0, 6); (6, 0)}

(0, 0): 0 + 0 6 0 6 (F)

Shaded region is one the other side of the origin

iii) 2x + y 8, boundary line is 2x + y = 8


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2. Maximise f = x+y subject to the constraints x+y 6, 2x+y 8, x 0 and y 0

A: x 0, y 0 represents Q1

i) x+y 6 boundary line is x+y = 6

put x= 0 y= 6, (0,6)

put y=0 x=6, (6,0)

(0,0) : {(0,6), (6,0)}(0,0) : 0+0 6

0 6 False

The shaded region should be other side of origin

ii) 2x+y 8, boundary line is 2x + y = 8

put x= 0, y = 8 (0, 8)

put y = 0 2x = 8 x = 4 (4, 0)

{(0, 8); (4, 0)}

(0, 0): 2(0) + 0 80 8 False

The shaded region Should be other side of Origen.

The Vertices of convex polygon are (6,0); (2, 4); (0,8)

f= x+y

at (6, 0); f = 6+ 0 = 6

at (2, 4); f = 2 + 4 = 6

at (0,8); f = 0+8 = 8

The Minium value for f is 6 and it occurs at (6,0) and (2, 4).


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3. Certain manufacturer has 75 kg of cashew and 120kg of groundnuts. These are to be mixed

is 1kg packages as follows: A low grade mixture 250 gms of cashew and 500 grms of

peanuts. If the profit on the low grade mixture is Rs. per package and that of high grade

mixture is Rs.3 per package and that of high grade mixture is Rs. 3 per package.

Howmany package of each mixture be made for a maximum profit?

A.

Let the No. of low grade packets be x i.e. x 0

high grade packets be y i.e. y 0

cashew = 75 kgs, ground nut =120 kgs

1/4 x+1/2 y 75

75

x+2 y 300 -------- (I)

x+ 2y= 300

put x=0, 2y= 300

y= 150 (0,150)put y=0, x= 300 (300, 0)

{(0,150) (300,0)}

(0,0) : 0+2(0) 300

0 300 (T)


profit on lowgrade packet "x" Rs, High grade profit "y" Rs,

Objective Function f= 2x+3y

Vertices of the polygon are(0,0)(100,0)(0,0)(0,150)

at (0,0) ; f = 2 (0) + 3 (0) = 0at (100,0) ; f = 2 (100) + 3 (0) = 200

at (90, 105) ; f = 2 (90) + 3 (105) = 495 Rs.

3/4 x+ 1/2 y 120

120

3x+2 y 480 ------- (II)

3x+2y = 480

put x = 0 2y = 480

y = 240 (0,240)

put y = 0 3x+ 2(0) = 480

3x 2y

4

+

x 2y

4

+

Product Low grade (x) High grade (y)

Cashew 250gm = 1/4 kg 500 gm =1/2 kg

Ground nut 750gm = 3/4 kg 500 gm = 1/2kg


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3x = 480

x = 160 (160,0)

{(0,240) (160,0)}

(0, 0): 3(0) + 2(0) 480

0 480 (True)


packets and 105 high grade packets.

At (0,150), f = 2(0) + 3(150) = 450

Maximum Profit is 495.

by Selling making go lowgrade Packets and 105 high grade Packets.


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5. Maximise f = 5x +7y, subject to condition 2x+3y 12, 3x+y 12, x 0 and y 0.

Sol: x 0, y 0 represents Q1

Vertices of the polygon are (0,0) (4,0), (0,4) (24/7, 12/7) objective function f=5x+7y

At (0,0), f = 5(0)+ 7(0) = 0

At (4,0) f = 5(4) + 7(0) = 20+0 = 20

At (0,4) f = 5(0) + 7(4) = 0+28 = 28

At f = + = + =

Hence maximum value is at24 12

,7 7

204

7

204

7

84

7

120

7

247

7

245

7

24 12,

7 7

(i) 2x+3y 12 (ii) 3x+y 12

2x+3y= 12 3x+y= 12

put x = 0, 3y= 12 put x= 0y= 4, (0,4) y= 12 ; (0,12)

put y = 0, 2x = 12 put y=0

x= 6 (6,0) 3x = 12

{(0,4), (6,0)} x = 4 ; (4,0)

(0,0): 2x+3y 12 {(0,12), (4,0)}

2 (0) + 3(0) 12 (0,0) : 3x+y 12

0 12 (True) 3 (0) + 0 12

Shaded region showed be towards 0 12 (True)

the rigion Shaded region showed be towards the rigion


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Multiple Choice Questions

1. The point which satisfies 3x4y+12 > 0 is ( )

a) (4,1) b) (0,4) c) (1,4) d) (0,5)

2. The point belongs to 2x3y>0 is ( )

a) (3,2) b) (2,3) c) 3,2) d) (3,2)

3. If f= 20x+ 25y. then minimising point of f is ( )

a) (100, 0) b) (0,100) c) (10, 20) d) (50,60)

4. The point which satisfies x+2y 4 is ( )

a) (2,1) b) (3,1) c) (3,2) d) (3,4)

5. F= 2x+3y is maximum at ( )

a) (5,6) b) (6,5) c) (2,6) d) (2,3)

6. If x>0 , y


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12. Iso-profit lines are ----------- to eath other

a) perpendicular b) parallel c) interesting d) None

13. The function which is to be minimised (or) maximised is called ----------- ( )

a) Objective function b) On to function

c) Constant function d) Real function

14. x = k is parallel to ----------- axis ( )

a) x- axis b) y- axis

c) neither x, nor y d) None

15. If x>0, y 0, y> 0 and (x,y) lies in ................. quadrant

2. The line y= mn passes through ..................

3. At (2,3) p = 4x+7y value is ...............

4. Iso- profit lines are ................... to each other

5. Everey point in the feasible region is called ..................

6. If iso-profit line coinsides with any one of the boundary line of convex polygon then the L.P.P.

has .............. solution

7. The point (7,6) lies on ............... axis

8. An optimum solution makes the objective formation either ................... or ..........................

9. If x=0, y0 (x,y) is a point on .............. axis

Answers

1) II 2) origin 3) 29 4) parabala 5) feasible solution

6) infinite 7) x- 8) maximise or minimise 9) y


4.linear programming

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