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.

1. 1
Interval Programming
Zahra Sadeghi

2. 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. 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
n
Z

4. 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. 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

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


jx
xxg
xxxg
xxxg
xxxz
j
x
x
x
x

6. 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

7. 7
definitions of interval arithmetic
0,0if),log()log()log(
0,0if,,
0,0if],,[
0if],,[
0if],,[
],[
],[

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


LL
LL
L
R
R
L
LLRRLL
LR
RL
LRRL
RRLL
ba
ba
b
a
b
a
bababa
kkaka
kkaka
k
baba
baba
BAAB
B
A
BA
A
BA
BA

8. 8
Interval Inequality
Definition 1:
degree for inequality Ax holding true
A : an interval
x : a real number
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aa
ax
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Definition 2:
degree for inequality A  B holding true
A and B: an interval 
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
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

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j=
LR
j
L
jj
R
j
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n
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j
qbbqxaqxqa
BxA
1
1
)1())1((
L
a
1
R
a
A

9. 9
Order Relation between Intervals
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
 

n
j
n
jj SxxCZ
1
|)(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

10. 10
Order Relation between Intervals
Definition 3:
the order relation LR
A and B: two intervals RRLL
LR babaBA  andif
Definition 4:
the order relation CW
A and B: two intervals WWCC
CW babaBA  andif
Definition 5:
the order relation LC
A and B: two intervals
CCLL
LC babaBA  andif

11. 11
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

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



 

n
j
n
jj SxxCZ
1
|)(max Rx
can be obtained as the Pareto solutions of the following
bicriteria programming problem:

12. 12
A productive problem
Theorem 3:
order  LC
A and B : two positive intervals






 

n
j
n
jj SxxCZ
1
|)(max Rx
)log()log( BABA LCLC 
Corollary 1:
it is equivalent to the following linear interval programming problem;

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



 

n
j
n
jj SxCZ
1
|)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.

13. 13
1.Transforming Interval
Programming
-Maximization Problem:
There are two key steps when transforming
interval programming to bicriteria linear programming:
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xCZ
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,,2,1,)(t.s.
)(max
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1
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




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
x
x
–consider maximization interval programming problem:
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ij
L
ijij
R
ij
L
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R
j
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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.

14. 14
1.Transforming Interval
Programming
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xccz
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,,2,1integer,:
,,2,1,)(t.s.
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1
)(max
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1
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x
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
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)1(
)1(

15. 15
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

16. 16
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.

17. 17
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.

18. 18
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.

19. 19
Initial population
• A choromosome is defined as follows:
• K: index of choromosome
• Randomly generate within the range
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

20. 20
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.

21. 21
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 

22. 22
}|)(max{
}|)(max{
}|)(min{
}|)(min{
max
min
max
min
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Exxzz
Exxzz
Exxzz
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kkLL
kkCC
kkCC
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CC
zzw
zzw
minmax2
minmax1
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z
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z
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zmax
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zmax
1w
2w
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)())()(()( 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

23. 23
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.

24. 24
Adaptive Penalty
• Yokota
• Gen & Cheng
•Smith & Tate :

25. 25
Penalty term
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 


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 

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 
otherwise
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))((1
1)(
max
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max



26. 26