Graphical Method of LP

7/29/2019 Graphical Method of LP

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LINEAR PROGRAMMING

Boni Sena,S.T.,M.Eng.


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WHATISLINEARPROGRAMMING ?

How much fuelthat we need ?

How many tirewhich can be carried on ?


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LINEAR PROGRAMMING (STUDY CASE)

Suppose a consumer has income $60 per week. Hewants to buy food A costs $2 per kilogram and food Bcosts $3 per kilogram. Arrange linear programming forthis problem !

First step : Make table which describe the problem.You have to determine parameter or decision variable.Cost become parameter in this problem. See the table

below :

Parameter A B MaximumCapacity

Cost $2 $3 $60


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Second Step : State the table in the form ofmathematical expressions. Assume the consumerwants to use all of his budget.

2 x + 3 y = 60 (where x 0 and y 0)

Third step : Make graph from the math expressions.(Cartesian coordinates)


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LINEAR INEQUALITY

Definition

A linear inequality in the variables x and y is an inequality that canbe written in one of the form

ax + by + c < 0 ,

ax + by + c > 0,

ax + by + c 0 ,

ax + by + c 0

a, b and c areconstant anda and b 0


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See the graph below :

y = mx + b

y < mx + b

y > mx + b

Below the line,all points (x,y)

satisfyy < mx+b

Above the line,

all points (x,y)satisfy

y > mx+b

the line itself,all points (x,y)

satisfyy = mx+b


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x > ax < a

x = a

See the graph below :


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SYSTEMOF INEQUALITIES

The solution of system of inequalities consists of allpoints whose coordinates

Solve the system

Draw the equation on the graph

2 x + 3 y > 3

x y

2 y 1 > 0

y > -2x + 3

y x

y > 1/2


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y x

Solution satisfy

y > -2x +3

y > 1/2

THEANSWERIS..

2 x + 3 y > 3

x y

2 y 1 > 0


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LinearProgramming

Constraints2 x + 3 y > 32 y 1 > 0

Objective functionP (x) = ax + by

Feasible points

Solution

Maximize / minimize

Not only find the feasible region but alsoFind the way to maximize/minimize the

Objective function


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STUDY CASEA company produces two types of can openers : Manual

and electric. Each require in its manufacture the use ofthree machines : A, B and C. Table 7.1 gives data relatingto the manufacturer of these can openers.

Manual Electric HoursAvailable

A 2 hr 1 hr 180B 1 hr 2 hr 160C 1 hr 1 hr 100Profit/Unit $4 $6


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40

80

120

160

40 80 160120

2x + y = 180

FeasibleRegion

x + y = 100

x + 2y = 160

Find the feasible region

AB

C

D


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40

80

120

160

40 80 160120

FeasibleRegion

Objective function (P) : 4 x + 6yy = - 2/3 x + P/6y = - 2/3 x + 100

Try with P = 600Has no point in feasible region

y = - 2/3 x + 50Try with P = 300

AB

Has infinitely many such points


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40

80

120

160

40 80 160120

FeasibleRegion

Objective function (P) : 4 x + 6yy = - 2/3 x + P/6y = - 2/3 x + 100

Try with P = 600Has no point in feasible region

y = - 2/3 x + 50Try with P = 300

AB

Has infinitely many such points

C

DIsoprofit lines

Maximum profit line

x + 2y = 160x + y = 100

The line whose y intercept is farthest from the origin and thathas at least one point in common with feasible region

x = 40 and y = 60


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40

80

120

160

40 80 160120

FeasibleRegion

Objective function (P) : 4 x + 6y

AB

C

D(40,60)

(90,0)(80,20)

(0,80)

P =4 x + 6y = 4(90)+6(0) = 360P =4 x + 6y = 4(80)+6(20) = 440P =4 x + 6y = 4(40)+6(60) = 520P =4 x + 6y = 4(0)+6(80) = 480

Thus, P has maximum value at (40,60)

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Graphical Method of LP