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    Graphical Method

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    Goutham Reddy
    1. The document describes three graphical optimization problems involving maximizing or minimizing objective functions subject to constraints. 2. The first problem involves maximizing Z=6x1+8x2 within the feasible region defined by two constraints, finding the optimal solution is x1=8, x2=2 for Zmax=64. 3. The second problem involves minimizing Z=2x-y within its feasible region, finding the optimal solution is x=0, y=5 for Zmin=-5. 4. The third problem is shown to have no feasible region and therefore an unbounded/infeasible solution.

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    Graphical Method

    Uploaded by

    Goutham Reddy
    16 views8 pages
    1. The document describes three graphical optimization problems involving maximizing or minimizing objective functions subject to constraints. 2. The first problem involves maximizing Z=6x1+8x2 within the feasible region defined by two constraints, finding the optimal solution is x1=8, x2=2 for Zmax=64. 3. The second problem involves minimizing Z=2x-y within its feasible region, finding the optimal solution is x=0, y=5 for Zmin=-5. 4. The third problem is shown to have no feasible region and therefore an unbounded/infeasible solution.

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    Graphical_Method

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    1. The document describes three graphical optimization problems involving maximizing or minimizing objective functions subject to constraints. 2. The first problem involves maximizing Z=6x1+8x2 within the feasible region defined by two constraints, finding the optimal solution is x1=8, x2=2 for Zmax=64. 3. The second problem involves minimizing Z=2x-y within its feasible region, finding the optimal solution is x=0, y=5 for Zmin=-5. 4. The third problem is shown to have no feasible region and therefore an unbounded/infeasible solution.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    16 views8 pages

    Graphical Method

    Uploaded by

    Goutham Reddy
    1. The document describes three graphical optimization problems involving maximizing or minimizing objective functions subject to constraints. 2. The first problem involves maximizing Z=6x1+8x2 within the feasible region defined by two constraints, finding the optimal solution is x1=8, x2=2 for Zmax=64. 3. The second problem involves minimizing Z=2x-y within its feasible region, finding the optimal solution is x=0, y=5 for Zmin=-5. 4. The third problem is shown to have no feasible region and therefore an unbounded/infeasible solution.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 8

    Graphical Method

    Graphical Method
    ① Max Z = 6x1 + 8x2
    Sub . to
    5x1 + 10x2 ≤ 60
    4x1 + 4x2 ≤ 40
    x1, x2 ≥ 0
    Find out the boundary values of x1 & x2

    I. 5x1 + 10x2 = 60
    x1 = 0 ; x 2 = 6
    x1 = 12 ; x2 = 0

    II. 4x1 + 4x2 = 40


    x1 = 0 ; x2 = 10
    x1 = 10 ; x2 = 0
    x2 Scale
    15 1 cm = 5 units
    II

    10 O - (0,0)
    (0,6) C
    I Solution Region A - (10,0)
    B - (8, 2)
    5 B (8,2) C - (0, 6)

    A
    0 5 15 x1
    (10,0)
    (0,0)
    Max Z = 6x1 + 8x2

    Substitute O, A, B & C in obj. function,


    Z (0) = 0
    Z (A) = 60
    Z (B) = 64
    Z (C) = 48

    • Optimal solution, Z max = 64


    x1 = 8 & x2 = 2
    Graphical Method
    2) Find the min. of the function, z = 2x – y subject to the constraints :

    x+y≤5
    x + 2y ≥ 8
    x, y ≥ 0

    I. x + y =5
    x=0;y=5
    x=5;y=0

    II. x + 2y = 8
    x= 0; y=4
    x=8; y= 0
    A (0, 4) ; Z = - 4
    y
    B (2,3) ; Z = 1
    C
    x+y=5 C (0,5) ; Z = - 5
    5
    • Opt. soln. Z min = - 5

    4 x=0
    A
    B
    3 x + 2y = 8 y=5

    1 2 3 4 5 6 7 8 9 x
    0
    Graphical Method
    3) Max. z = x1 + x2
    Sub. to
    x1 + x2 ≤ 1
    -3x1+ x2 ≥ 3
    x1, x2 ≥ 0

    I. x1 + x2=1
    x1 = 0 ; x2 = 1
    x1 = 1 ; x2 = 0

    II. -3x1 + x2 = 3
    x1 = 0 ; x2 = 3
    x1 = -1 ; x2 = 0
    There is no point (x1,x2) that satisfies both the
    x2 constraints simultaneously.

    • So, it is having an unbounded or infeasible


    II solution.
    4

    0 1 2 3 x1
    -1
    I

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