# 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

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