Integer Programming, Goal Programming, and Nonlinear Programming

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Transcript of Integer Programming, Goal Programming, and Nonlinear Programming

  • Chapter 11

    Integer Programming,

    Goal Programming,

    and Nonlinear

    Programming. Prepared By: Salah A. Skaik

  • In this Chapter:

    Integer Programming.

    Goal Programming.

    Nonlinear Programming.

  • Integer Programming

    IP is the extension of LP that solves problems requiring Integer Solutions.

    Ex. (Airline)

    There are two ways to solve IP Problems:

    1- Graphically.

    2- The Branch & Bound Method.

  • Goal Programming

    GP is the extension of LP that permits

    Multiple Objectives to be stated.

    Ex. (Max profit & Max market share).

  • Nonlinear Programming

    NLP is the case in which Objectives

    or Constraints are Nonlinear.

    Ex. (Maximizing profit =

    25X1 - 0.4X1 + 30X2 0.5X2 ).

    Ex. (12X1 - 0.6X1 3,500 ).

  • Integer Programming

    There are three types of Integer

    Programs:

    1- Pure Integer Programs.

    2- Mixed-Integer Programs.

    3- Zero-One Integer Programming.

    (special cases).

  • Example of Integer Programming

    Harrison Electric Company

  • Harrison Electric Company

    Chandeliers

    2 hrs to wire

    6 hrs to assembly

    Ceiling Fans

    3 hrs to wire

    5 hrs to assembly

  • 12 hrs of wiring

    30 hrs of assembly Production Capability

    7$ each chandelier

    6$ each fan Net Profit

    Harrison Electric Company

  • Harrison Electric Company

    Let:

    X1 = Number of chandeliers produced.

    X2 = Number of ceiling fans produced.

  • Harrison Electric Company

    Objective Function:

    Maximize Profit = $7 X1 + $6 X2

    Subject to:

    2X1 + 3X2 12 (wiring hours)

    6X1 + 5X2 30 (assembly hours)

  • 6

    5

    4

    3

    2

    1

    0 | | | | | | |

    1 2 3 4 5 6 X1

    X2

    +

    + +

    + + + +

    +

    6X1 + 5X2 30

    2X1 + 3X2 12

    + = Possible Integer Solution

    Optimal LP Solution

    (X1 =3.75, X2 = 1.5,

    Profit = $35.25)

    Harrison Electric Company

  • In case the company doesnt produce a

    fraction of the product:

    1- Rounding Off.

    2- Enumeration Method.

    Harrison Electric Company

  • Rounding Off has two problems:

    1- Integer solution may not be in feasible

    region. (X1=4, X2=2). (unpractical solution)

    2- May not be the optimal feasible Integer

    solution. (X1=4, X2=1).

    Harrison Electric Company

  • Enumeration Method

    Harrison Electric Company

  • Optimal solution to integer programming problem

    Solution if rounding is used

    CHANDELIERS (X1) CEILING FANS (X2) PROFIT ($7X1 + $6X2)

    0 0 $0

    1 0 7

    2 0 14

    3 0 21

    4 0 28

    5 0 35

    0 1 6

    1 1 13

    2 1 20

    3 1 27

    4 1 34

    0 2 12

    1 2 19

    2 2 26

    3 2 33

    0 3 18

    1 3 25

    0 4 24

    Harrison Electric Company

  • An integer solution can never be better

    than the LP solution and is usually a lesser

    solution.

    Harrison Electric Company

  • Six Steps in Solving IP

    Maximization Problems by

    Branch and Bound

    Branch-and-Bound Method

  • Step(1):

    Solve the original problem using LP. If the

    answer satisfies the integer constraints, we

    are done. If not, this value provides an

    initial upper bound.

    Branch-and-Bound Method

  • Step(2):

    Find any feasible solution that meets the

    integer constraints for use as a lower

    bound. Usually, rounding down each

    variable will accomplish this.

    Branch-and-Bound Method

  • Step(3):

    Branch on one variable from step 1 that does

    not have an integer value. Split the

    problem into two sub-problems based on

    integer values that are immediately above

    or below the non-Integer value.

    Branch-and-Bound Method

  • Step(4):

    Create nodes at the top of these new

    branches by solving the new problem.

    Branch-and-Bound Method

  • Step(5-a):

    If a branch yields a solution to the LP

    problem that is not feasible, terminate

    the branch.

    Branch-and-Bound Method

  • Step(5-b):

    If a branch yields a solution to the LP

    problem that is feasible, but not an integer

    solution, go to step 6.

    Branch-and-Bound Method

  • Step(5-c):

    If the branch yields a feasible integer solution, examine the value of the objective function. If this value equals the upper bound, an optimal solution has been reached. If it not equal to the upper bound, but exceeds the lower bound, set it as the new lower bound and go to step 6. finally, if its less than the lower bound terminate this branch.

    Branch-and-Bound Method

  • Step(6):

    Examine both branches again and set the

    upper bound equal to the maximum value

    of the objective function at all final nodes. If

    the upper bound equals the lower bound,

    stop. If not, go back to step 3.

    Branch-and-Bound Method

  • NOTE:

    Minimization problems involved reversing the

    roles of the upper and lower bounds.

    Branch-and-Bound Method

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Recall that the Harrison Electric Companys integer

    programming formulation is

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    where

    X1 = number of chandeliers produced

    X2 = number of ceiling fans produced

    And the optimal non-integer solution is

    X1 = 3.75 chandeliers, X2 = 1.5 ceiling fans

    profit = $35.25

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Since X1 and X2 are not integers, this solution is not

    valid.

    The profit value of $35.25 will provide the initial upper

    bound.

    We can round down to X1 = 3, X2 = 1, profit = $27,

    which provides a feasible lower bound.

    The problem is now divided into two sub-problems.

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 4

    Subproblem A

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 3

    Subproblem B

  • Branch-and-Bound Method Harrison Electric Company Revisited

    If you solve both sub-problems graphically

    [X1 = 4, X2 = 1.2, profit = $35.20] Sub-problem As optimal solution:

    Sub-problem Bs optimal solution: [X1 = 3, X2 = 2, profit = $33.00]

    We have completed steps 1 to 4 of the branch-and-bound method.

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Harrison Electrics first branching:

    subproblems A and B

    Subproblem A

    Next Branch (C)

    Next Branch (D)

    Upper Bound = $35.25 Lower Bound = $27.00 (From

    Rounding Down)

    X1 = 4

    X2 = 1.2

    P = 35.20

    X1 = 3

    X2 = 2

    P = 33.00

    Stop This Branch Solution Is Integer, Feasible Provides New Lower Bound of $33.00

    X1 = 3.75

    X2 = 1.5

    P = 35.25

    Infeasible (Noninteger) Solution Upper Bound = $35.20 Lower Bound = $33.00

    Subproblem B

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 4

    X2 2

    Subproblem C

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 4

    X2 1

    Subproblem D

    Subproblem A has branched into two new subproblems, C

    and D.

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Subproblem C has no feasible solution because the all the

    constraints can not be satisfied

    We terminate this branch and do not consider this

    solution

    Subproblem Ds optimal solution is X1 = 4.17, X2 = 1,

    profit = $35.16

    This noninteger solution yields a new upper bound of

    $35.16

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 4

    X1 4

    X2 1

    Subproblem E

    Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12

    6X1 + 5X2 30

    X1 4

    X1 5

    X2 1

    Subproblem D

    Finally we create subproblems E and F

    Optimal solution to E:

    X1 = 4, X2 = 1, profit = $34

    Optimal solution to F:

    X1 = 5, X2 = 0, profit = $35

  • Branch-and-Bound Method Harrison Electric Company Revisited

    Subproblem F

    X1 = 5

    X2 = 0

    P = 35.00

    Subproblem E

    X1 = 4

    X2 = 1

    P = 34.00

    Harrison Electrics full branch and bound solution

    Feasible, Integer Solution

    Optimal Solution

    Upper Bound = $35.25

    Lower Bound = $27.00

    Subproblem C

    No Feasible Solution Region

    Subproblem D

    X1 = 4.17

    X2 = 1

    P = 35.16

    Subproblem A

    X1 = 4

    X2 = 1.2

    P = 35.20

    Subproblem B

    X1 = 3

    X2 = 2

    P = 33.00

    X1 = 3.75

    X2 = 1.5

    P = 35.25