1

Interval Programming

Zahra Sadeghi

2

Motivation

• Motivation for developing Interval Programming

technique:

– When using mathematical programming methods

to solve practical problem, it is usually not so easy

for decision makers to determine the proper values

of model parameters;

– on the contrary, such uncertainty can be roughly

represented as an interval of confidence.

3

Bicriterion problem

• A general biobjective or bicriterion integer program (BIP)

• The set X is called the set of feasible solutions

• the space containing X is the solution space

• Generally, X is the subset of (contained in a region

defined by a combination of equality and inequality

constraints, as well as explicit bounds on individual

variables.)

2,1),( ixfi

nZ

4

interval optimization

• In the past decade, two different approaches have been proposed for interval optimization:

– Interval Analysis: interval variables and normal coefficients.

– Interval Programming: interval coefficients and normal variables.

Basic idea of Interval Programming problem:

Transform interval programming model into an

equivalent bicriteria programming model

Find the Pareto solutions of the bicriteria

programming problem using genetic algorithms

5

• Example of Interval Programming Problem:

integer :3,2,1,0

604)(

402)(

30)( t.s.

]30 ,10[]20 ,15[]17 ,15[)( max

313

3212

3211

321

jx

xxg

xxxg

xxxg

xxxz

j

x

x

x

x

6

Interval Arithmetic • An interval = an ordered pair of real numbers

A = [aL, aR] = {x | aL x aR; x R1}

)(2

1

)(2

1

LRW

LRC

aaa

aaa

aL aC

aW

aR

A

aW

A = [ aC, aW ]= {x | aC - aW x aC + aW ; x R1}

aC : center of interval A

aW :width of interval A

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definitions of interval arithmetic

0,0if),log()log()log(

0,0if,,

0,0if],,[

0if],,[

0if],,[

],[

],[

LL

LL

L

R

R

L

LLRRLL

LR

RL

LRRL

RRLL

ba

bab

a

b

a

bababa

kkaka

kkakak

baba

baba

BAAB

B

A

BA

A

BA

BA

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Interval Inequality

Definition 1:

degree for inequality Ax holding true

A : an interval

x : a real number

LR

L

aa

axxAg ,1min,0max)(

Definition 2:

degree for inequality A B holding true

A and B: an interval

LRLR

LR

abba

abBAq ,1min,0max)(

n

j=

LR

j

L

jj

R

j

j

n

j

j

qbbqxaqxqa

BxA

1

1

)1())1((

La

1

Ra

A

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Order Relation between Intervals

n

j

n

jj SxxCZ1

|)(max Rx

S is a feasible region of x

Cj is an interval coefficient which represents the uncertain unit profit from xj.

For a given x, the total profit Z(x) is an interval.

We need to make a decision based on such interval profits.

321 ]30 ,10[]20 ,15[]17 ,15[)( max xxxz x

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Order Relation between Intervals

Definition 3:

the order relation LR

A and B: two intervals RRLL

LR babaBA and if

Definition 4:

the order relation CW

A and B: two intervals

WWCC

CW babaBA and if

Definition 5:

the order relation LC

A and B: two intervals CCLL

LC babaBA and if

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solution of problem

• Definition 6:

• A vector x S is a solution of problem if and only if there is no x’ S which satisfies )()( xx ZZ LC

Theorem 2:

The solution set of problem

}|)(,)({max NCL Szz Rxxx

n

j

n

jj SxxCZ1

|)(max Rx

can be obtained as the Pareto solutions of the following

bicriteria programming problem:

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A productive problem

Theorem 3: order LC A and B : two positive intervals

n

j

n

jj SxxCZ1

|)(max Rx

)log()log( BABA LCLC

Corollary 1:

it is equivalent to the following linear interval programming problem;

n

j

n

jj SxCZ1

|)log()(max Rxx

A nonlinear interval programming problem

in which the objective takes a product form.

•By the following theorem, this kind of nonlinear interval programming problem can be transformed into an equivalent linear interval programming problem.

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1.Transforming Interval Programming

-Maximization Problem:

There are two key steps when transforming

interval programming to bicriteria linear programming:

njxxx

miBxAG

xCZ

U

jj

L

j

n

j

ijiji

n

j

jj

,,2,1 ,integer:

,,2,1 ,)( t.s.

)(max

1

1

x

x

–consider maximization interval programming problem:

],[];,[];,[ R

ij

L

ijij

R

ij

L

ijij

R

j

L

jj bbBaaAccC

•Using the definition of the degree of inequality holding true for

two intervals, transform interval constrains into equivalent crisp

constraints.

•Using the definition of the order relation between intervals,

transform interval objective into two equivalent crisp objectives.

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1.Transforming Interval Programming

njxxx

mibxag

xccz

xcz

R

jj

L

j

n

j

ijiji

n

j

j

R

j

L

j

C

n

j

j

L

j

L

,,2,1 integer,:

,,2,1,)( t.s.

)(2

1)(max

)(max

1

1

1

x

x

x

njxxx

miBxAG

xCZ

U

jj

L

j

n

j

ijiji

n

j

jj

,,2,1 ,integer:

,,2,1 ,)( t.s.

)(max

1

1

x

x

L

i

R

ii

L

ij

R

ijij

qbbqb

aqqaa

)1(

)1(

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Numerical Example

integer :3,2,1,0

604)(

402)(

30)( t.s.

]30 ,10[]20 ,15[]17 ,15[)( max

313

3212

3211

321

jx

xxg

xxxg

xxxg

xxxz

j

x

x

x

x

integer :3,2,1,0

604)(

402)(

30)( t.s.

205.1716)( max

101515)( max

313

3212

3211

321

321

jx

xxg

xxxg

xxxg

xxxz

xxxz

j

C

L

x

x

x

x

x

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2.Pareto Solution for Interval Programming

Definition 7: Let F be the set of feasible solutions.

A feasible solution y F is said to be a nondominated solution if

and only if

)()()()(, yxyxFx zzzz

))(...,),(),(()( 21 xxxx qzzzz

Definition 8: The positive ideal solution (PIS)

= all the best objective values attainable; z+ = {z1+, z2+, …, zq+},

zq+ = the best value for the qth objective without considering other objectives.

Definition 9: The negative ideal solution (NIS)

= all the worst objective values attainable; z - = {z1-, z2-, …, zq-},

zq- = the worst value for the qth objective without considering other objectives.

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Pareto Solution for Interval Programming

– Three primary approaches or philosophies that form the basis for nearly all the candidate multiobjective techniques:

• Weight or utility method: This approaches that attempt to express all objectives in terms of a single measure. It is attractive from a strictly computational point of view. However, the obvious drawback is that associated with actually developing truly credible weights.

• Ranking or prioritizing methods: This methods try to circumvent the heady problem s indicated above. They assign priorities to each objective according to their perceived importance. Most decision makers can do this.

• Efficient solution or generation methods: – This avoids the problems of finding weights and satisfying the ranking.

– It generates the entire set of nondominated solutions or an approximation of this set and then allows to the decision makers to select the nondominated solution which best represents their tradeoff among the objectives.

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GA Procedure for Interval Programming

0.set population size pop_size, – mutation rate pm, crossover rate pc,and maximum number of generation max_gen.

– let t=0 and E=0

1.Initialization:Randomly generate initial population

2.crossover:uniform crossover

3.Mutation:Perform random perturbation mutation.

4.Update set E: – Compute objective function values of bicriteria for each choromosome.

– Update set E by adding new nondonimated points into E and deleting dominated points.

– Determine new special points

5.Evaluation:compute fitness values for each choromosome.

6.Selection: – Delete all duplicate choromosomes.

– Sort them in descending order.

– Select the first pop_size choromosome as new population.

7.Terminate set: if t = gen_max then stop,

otherwise let t=t+1 and go to step 2.

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Initial population

• A choromosome is defined as follows:

• K: index of choromosome

• Randomly generate within the range

],[

},...,,{ 21

U

j

L

jj

k

n

kkk

xxx

xxxx

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Crossover and Mutation

• Uniform crossover – It has been shown to be superior to

traditional crossover strategies for combinational problems.

• Mutation : – random perturbation within the permissive

range of integer variable.

• Selection: – Deterministic selection:

– delete all duplicated parents and offspring,

– sort them in descending order

– Select the first pop_size choromosome as new population.

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Evaluation

• It contains of two terms:

1- weighted sum objective function:

• Tries to give selection pressure to force genetic search

toward exploiting the set of Pareto solutions

– W1 and w2 are weights corresponding to the importance of the

objectives.

2- the penalty term:

• Tries to force genetic search to approach Pareto

solutions from both feasible and infeasible regions.

)())()(()( 21

kkCkLk xpxzwxzwxeval

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}|)(max{

}|)(max{

}|)(min{

}|)(min{

max

min

max

min

Exxzz

Exxzz

Exxzz

Exxzz

kkLC

kkLL

kkCC

kkCC

LL

CC

zzw

zzw

minmax2

minmax1

ZFF

ZFF

z

z

Czmax

Czmin

F

F

Lzmin

Lzmax

1w

2w

Cz

Lz

)())()(()( 21

kkCkLk xpxzwxzwxeval

•The feasible solution space F is correspondingly divided into two parts

•At each generation the Pareto set E is updated and the two special points may be

renewed.

•Along with the evolutionary process, the line formed with this two points will move

gradually from a negative ideal point to a positive ideal point.

•A solution in half_space has higher fitness values and a relatively larger chance to enter the

next generation

F

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Penalty • The penalty term: a measure of infeasiblity for a choromosome.

• It is used to evaluate how far the an infeasible choromosome seprates from the feasible area.

• Genetic search will approach optimum only from the feasible side.

• The proposed penalty approach can force the genetic search to approach the Pareto solutions from both feasible and infeasible regions.

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Adaptive Penalty

• Yokota

• Gen & Cheng

•Smith & Tate :

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Penalty term

otherwise

gg

sizepopkgg

otherwisexg

bxgg

bg

bgg

mxp

ki

ki

kii

k

i

i

k

i

ki

m

i ii

ikikik

1

00)(

}_,...,2,1|max{

)(

)(0

))((11)(

max

1max

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