solution of nonlinear interval vector optimization problem
TRANSCRIPT
ORI GIN AL PA PER
Solution of nonlinear interval vector optimizationproblem
Mrinal Jana • Geetanjali Panda
Received: 2 May 2013 / Revised: 8 August 2013 / Accepted: 21 August 2013
� Springer-Verlag Berlin Heidelberg 2013
Abstract In this paper a vector optimization problem is studied in uncertain
environment.The objective functions and constraints of this problem are interval
valued functions. Preferable efficient solution of the problem is defined and a
methodology is developed to derive one preferable efficient solution. The proposed
methodology is illustrated through a numerical example.
Keywords Vector optimization problem � Efficient solution � Preference
function � Interval valued function
Mathematics Subject Classification (2010) 90C29 � 90C31
1 Introduction
In a general vector optimization problem, decisions need to be taken with trade-offs
among conflicting objectives. In general the coefficients of these problems are
assumed to be real numbers. But in real life situations this assumption is not always
true due to the presence of various type of uncertainties in the domain of the
optimization model. If at least one parameter of a vector optimization problem is an
interval then this is an interval vector optimization problem, which we denote by
(IVOP). In (IVOP) at least one objective function or one constraint is an interval
valued function. Example 1 explains a real life (IVOP) model.
M. Jana
Indian Institute of Technology Kharagpur, Kharagpur 721302, West Bengal, India
e-mail: [email protected]
G. Panda (&)
Department of Mathematics, Indian Institute of Technology Kharagpur,
Kharagpur 721302, West Bengal, India
e-mail: [email protected]
123
Oper Res Int J
DOI 10.1007/s12351-013-0137-2
Since last few years existence of solution of linear interval vector optimization
problem has been discussed by many researchers like Urli and Nadeau (1992),
Oliveira and Antunes (2007), Oliveira and Antunes (2009), Han et al. (2011),
Rivaz and Yaghoobi (2012). Some of these models consider interval parameters in
the objective function only. Most of the methods find the upper and lower bounds
of every objective function in best and worst scenarios. Wu (2009) has considered
a nonlinear vector optimization model whose objective functions have interval
parameters but the constraints are free from interval uncertainty. Theory of
nonlinear vector optimization models with interval uncertainty in both objective
function and constraints has not been addressed so far. In this paper we consider a
general interval vector optimization problem which addresses both linear and
nonlinear interval valued functions in the objective function as well as in the
constraints. We propose a methodology to address the uncertainty in the model
and find an efficient solution which is acceptable by the decision maker. This
solution is called as preferable efficient solution. The uncertainties in feasible
region and objective functions are addressed separately. In this process the
original problem is transformed to a deterministic optimization problem. It is
proved that optimal solution of the transformed problem is a preferable efficient
solution of the original problem. Section 2 discusses some pre-requisites on
interval analysis and general vector optimization problem. A general interval
vector optimization model is proposed in Sect. 3 and existence of solution of this
model is discussed. In Sect. 4, a methodology is developed to derive this solution
and the proposed methodology is illustrated in a numerical example. Sect. 5
provides some concluding remarks.
Example 1 Balancing reward against risk is the base of a general mean-variance
portfolio optimization problem. Reward is measured by the portfolio expected
return and risk is measured by the portfolio variance. In the most basic form,
portfolio optimization model determines the proportion of the total investment xi
of ith asset of a portfolio x ¼ ðx1; x2; . . .; xnÞ; whereP
i=1n xi = 1. In general, rate
of expected returns of the assets of a portfolio are estimated from previous data.
Due to the presence of uncertainty in market an investor can not estimate the
exact rate of expected return. If the investor finds the lower bound (riL) and upper
bound (riR) of the return of the assets from previous data for a fixed time period
then the expected rate of return of ith asset lies in the interval [riL, ri
R]. In a
portfolio optimization problem, an investor wants to maximize the expected return
of the portfolio with minimum risk. Some more realistic factors also affect the
portfolio selection like skewness and kurtosis. In this situation, an investor needs
to maximize the expected return as well as skewness of the expected return and to
minimize the variance and kurtosis. Since variance, covariance, skewness and
kurtosis depend upon the rate of expected returns, so they are also in the form of
intervals. Suppose, the expected return, variance, skewness and kurtosis of
portfolio are denoted by RðxÞ; r2ðxÞ; SðxÞ and KðxÞ respectively, which can be
defined as follows.
M. Jana, G. Panda
123
RðxÞ ¼Xn
i¼1
½rLi ; r
Ri �xi
r2ðxÞ ¼Xn
i¼1
x2i ½r2
i
L; r2
i
R� þXn
i¼1
Xn
j¼1
½rLij; r
Rij �xixj ði 6¼ jÞ
SðxÞ ¼Xn
i¼1
x3i ½s3
i
L; s3
i
R� þ 3Xn
i¼1
Xn
j¼1
½sLiij; s
Riij�x2
i xj þXn
j¼1
½sLijj; s
Rsijj�xix
2j
!
ði 6¼ jÞ
KðxÞ ¼Xn
i¼1
x4i ½k4
i
L; k4
i
R� þ 4Xn
i¼1
Xn
j¼1
½kLiiij; k
Riiij�x3
i xj þXn
j¼1
½kLijj; k
Rijj�xix
3j
!
þ 6Xn
i¼1
Xn
j¼1
½kLiijj; k
Riijj�x2
i x2j
!
ði 6¼ jÞ;
where [r2L
i ;r2R
i ] is the variance of expected returns of asset i, [rijL, rij
R] is the
covariance, [siijL , siij
R ] and [sijjL , sijj
R ] are co-skewness and [kiiijL , kiiij
R ], [kijjjL , k-
ijjjR], [kiijj
L , kiijjR ] are co-kurtosis of expected returns of assets i and j. If bRi ¼ ½rL
i ; rRi �
denotes the mean of expected return of asset i in time period t, then
½sLiij; s
Riij� ¼
1
t
Xn
i¼1
Xn
j¼1
ðRi � bRiÞ2ðRj � bRjÞ;
½sLijj; s
Rijj� ¼
1
t
Xn
i¼1
Xn
j¼1
ðRi � bRiÞðRj � bRjÞ2
½kLiiij; k
Riiij� ¼
1
t
Xn
i¼1
Xn
j¼1
ðRi � bRiÞ3ðRj � bRjÞ;
½kLijjj; k
Rijjj� ¼
1
t
Xn
i¼1
Xn
j¼1
ðRi � bRiÞðRj � bRjÞ3;
½kLiijj; k
Riijj� ¼
1
t
Xn
i¼1
Xn
j¼1
ðRi � bRiÞ2ðRj � bRjÞ2
The arithmetic operations in these expressions are sum, difference and product of
interval arithmetic operations. The above problem can be represented mathemati-
cally as:
min f�RðxÞ; r2ðxÞ;�SðxÞ; KðxÞgsubject to
Pn
i¼1
xi ¼ 1; xl� 0;
which is a non-linear interval vector optimization problem.
Solution of nonlinear interval vector optimization problem
123
2 Preliminaries
The following notations are used throughout the paper.
I(<) = The set of closed intervals on R. A 2 Ið<Þ is the set A = [aL, aR].
A is said to be a degenerate interval if aL = aR and is denoted by bA.
Ið<Þn ¼ fAv : Av ¼ ðA1;A2; . . .;AnÞT ; Aj 2 Ið<Þ; j ¼ 1; 2; . . .; ng:Ið<Þþ ¼ fA ¼ ½aL; aR� : aL� 0g:Ið<Þnþ ¼ fAv : Av ¼ ðA1A2. . .;AnÞT ; Ai 2 Ið<Þþ; i ¼ 1; 2; . . .; ng:
Ið<Þnþþ ¼ fAv : Av 2 Ið<Þnþ; Av 6¼ ðb0; b0; . . .; b0ÞTg:Kk ¼ f1; 2; . . .; kg:
In classical method, an algebraic operation ~ð� 2 fþ;�; �; =gÞ in I(<) is defined
as follows. For A = [aL, aR] and B = [bL, bR] in I(<),
A~B ¼ fa � b : a 2 A; b 2 BgFor A �B, 0 62 B. �A ¼ ½�aR;�aL�: The interval A� ð�AÞ is not equal to the
degenerate interval [0, 0] for any non zero interval A. So we use a nonstandard
difference between two intervals as defined by Markov (1979), which is denoted by
�M: For A = [aL, aR] and B = [bL, bR],
A�M B ¼ ½aL � bL; aR � bR�; if lðAÞ� lðBÞ½aR � bR; aL � bL�; if lðAÞ\lðBÞ;
�
where l(A) = aR - aL, l(B) = bR - bL are spreads of the intervals A and
B respectively. In this case A�M A ¼ ½0; 0�:
2.1 Order relations in I(<)
I(<) is not a totally ordered set. Several partial orderings in I(<) exist in literature
[see Moore et al. (2009), Hansen and Walster (2004)]. We consider the following
LR-partial order relations in I(<). For A = [aL, aR] and B ¼ ½bL; bR� 2 Ið<Þ;A ¼ B iff aL ¼ bL and aR ¼ bR;
A LR B iff aL bL and aR bR;
A†LRB iff aL bL and aR bR with A 6¼ B;
A �LR B iff aL\bL and aR\bR:
I(<)n is also not a totally ordered set. To compare the interval vectors in I(<)n we
define the following partial ordering nLR :
Definition 2.1 For Av ¼ ðA1A2. . . AnÞT and Bv ¼ ðB1B2. . . BnÞT 2 Ið<Þn;Av ¼ Bv iff Ai ¼ Bi 8i ¼ 1; 2; . . .; n;
Av nLR Bv iff Ai LR Bi 8i ¼ 1; 2; . . .; n;
Av �nLR Bv iff Ai LR Bi and Av 6¼ Bv:
M. Jana, G. Panda
123
2.2 Interval inequations
Given two intervals A = [aL, aR] and B = [bL, bR], the solution of the interval
equation Ax = B is the set fx 2 <jax ¼ b; a 2 A; b 2 Bg; provided 0 62 A: For
example, solution of the interval equation [1, 2]x = [3, 4] is the set fx 2 <jax ¼b; 1 a 2; 3 b 4g ¼ fx 2 <j 3
2 x 4g: In a similar way, solution of interval
inequation can be defined. Ax B is an interval inequation whose solution is the set
fx 2 <jax b; a 2 A; b 2 Bg: In (IVOP), the constraints are interval inequations.
2.3 Interval valued function
Interval valued function is defined by many authors in different ways [see Hansen
and Walster (2004), Moore et al. (2009), Wu (2008), Bhurjee and Panda (2012) etc].
In general, interval valued function is a mapping from one or more interval
arguments onto an interval number. In this paper we consider an interval valued
function f : D � <n ! Ið<Þ; as f ðxÞ ¼ ½f LðxÞ; f RðxÞ�; where f L; f R : <n ! < such
that f LðxÞ f RðxÞ 8x 2 D: For example, f : <2 ! Ið<Þ is
f ðx1; x2Þ ¼ ½�4; 8�ex1 þ ½�1; 2�x2 ¼½�4ex1 � x2; 8ex1 þ 2x2� if x2� 0
½�4ex1 þ 2x2; 8ex1 � x2� if x2\0
�
3 Interval vector optimization problem (IVOP)
A general vector optimization problem is
ðPÞ : minx2D�<n
f ðxÞ ¼ ff1ðxÞ; f2ðxÞ; . . .; fkðxÞg; where fi : <n �! <; i ¼ 1; 2; . . .; k:
ð3:1Þ
There may not exist a single optimum solution for (P), which can simultaneously
optimize all the objective functions. In this circumstance the decision maker looks
for the compromise solution. Hence in (P), the concept of optimum solution is
replaced with pareto-optimal/ compromise/ efficient solution. x� is said to be an
efficient solution of (P) if there does not exist any feasible point y 2 D such that
f(y) B f(x�) with f(y) = f(x�). This concept may be extended for vector optimization
problem with interval parameters.
Consider a general interval vector optimization problem (IVOP) with k objective
functions as
ðIVOPÞ : min f ðxÞ ¼ fbf1ðxÞ; bf2ðxÞ; . . .; bfkðxÞg (3.2)
subject to bgjðxÞ bbj ; j 2 Km; x 2 S � <n (3.3)
where fi : <n �! Ið<Þ; i 2 Kk and gj : <n �! Ið<Þ; j 2 Km are interval valued
functions as described in Sect. 2.3. fiðxÞ ¼ ½f Li ðxÞ; f R
i ðxÞ�; gjðxÞ ¼ ½gLj ðxÞ; gR
j ðxÞ�;f Li ðxÞ f R
i ðxÞ; gLj ðxÞ gR
j ðxÞ8x 2 S; bj ¼ ½bLj ; b
Rj �:
Solution of nonlinear interval vector optimization problem
123
One may observe that uncertainties are associated with (IVOP) in several forms.
x is a feasible solution of (IVOP) if x satisfies the interval inequations gjðxÞ bj:
Here, for every x; gjðxÞ is an interval [gjL(x), gj
R(x)]. So for every x, [gjL(x), gj
R(x)]
may lie before or after [bjL, bj
R] or overlap with [bjL, bj
R]. Feasibility of x depends
upon the closeness of these two intervals. A point x with certain degree of closeness
of gjðxÞ with bj can be a compromise solution of (IVOP) if it optimizes k number of
conflicting objective functions simultaneously. Since every objective function is an
interval valued mapping, so interval vectors have to be compared corresponding to
every feasible solution. All these uncertainties are addressed in the following three
major stages in next section.
(1) Uncertainty in feasible region.
(2) Uncertainty in objective function.
(3) Uncertainty in feasible region and objective function taken together.
4 Methodology for finding preferable efficient solution
4.1 Uncertainty in feasible region: Acceptable feasible solution
Feasible region of (IVOP) is determined from m number of interval inequations
gjðxÞ bj; j 2 Km: So the role of the constraints to describe it’s feasible region is
different from that in classical optimization problem, where the violation of any
single constraint by any amount renders the solution infeasible. Following the
discussion of Sect. 2.2, the feasible region becomes
S ¼\
j2Km
fx 2 D � <n : gjðxÞ bjg
¼\
j2Km
fx 2 D � <n : zj bj; where gLj ðxÞ zj gR
j ðxÞ; bLj bj bR
j g
One may observe that any x 2 <n satisfying the interval inequalities gjðxÞ LR bj for
all j, belongs to the feasible set S. But x 2 S may not satisfy the interval inequalities
gjðxÞ LR bj for all j. So the feasible region of (IVOP) can not be decided through
LR partial ordering. (Hence we use the symbol to describe an interval inequation
and LR to describe the comparison of two fixed intervals). For any x in S, the
interval [gjL(x), gj
R(x)] may lie behind or overlap or exceed [bjL, bj
R] for every
j. Accordingly the feasibility of x for (IVOP) is completely acceptable or partially
acceptable or not acceptable. Hence every point in S is associated with certain
degree of acceptability/feasibility/closeness. We will convert S to a deterministic
form to have a mathematical sense as follows.
Denote Smax ¼ fx : gLj ðxÞ bR
j j 2 Kmg and Smin ¼ fx : gRj ðxÞ bL
j j 2 Kmg. Smax
and Smin are the maximum and minimum feasible regions respectively. It is easy to
prove the following result from the definition of Smax and Smin.
M. Jana, G. Panda
123
Proposition 4.1 Smin � Smax:
From Proposition 4.1 and definition of Smax and Smin, it is true that any feasible
point of (IVOP) lies either in Smin or in SmaxnSmin but not in the complement of Smax
(which is Smaxc ), depending upon the relation between the lower and upper bound of
gjðxÞ with bjL and bj
R respectively.
(1) x 2 S is a fully acceptable feasible solution if x 2 Smin;(2) x is not at all acceptable feasible point if x goes beyond the region Smax i.e.
x 2 Scmax;
(3) x is partially acceptable feasible solution if it lies in SmaxnSmin:
In case (3), the degree of acceptability of x decreases from 100 to 0 % as it
moves closer to Smax from Smin. For this we need to define a decreasing function in
I(<) as follows. For A ¼ ½aL; aR�;B ¼ ½bL; bR� 2 Ið<Þ;A LR B holds if aR B bL. Its
converse is not necessarily true. On the basis of this logic we define increasing/
decreasing real valued function on I(<) with respect to LR below which will be
needed in sequel.
Definition 4.2 m : Ið<Þ ! < is said to be an increasing function with respect to
LR if for all A ¼ ½aL; aR�;B ¼ ½bL; bR� 2 Ið<Þ; aR bL and l(A) B l(B) implies
m(A) B m(B). Similarly m is said to be decreasing with respect to LR if for all
A ¼ ½aL; aR�;B ¼ ½bL; bR� 2 Ið<Þ; aR bL and l(A) B l(B) implies m(A) C m(B).
Example 2 m : Ið<Þ ! < defined by mðAÞ ¼ 1þaR�ðaLÞ2aR�aL ; aL� 0; for A = [aL, aR] is
decreasing with respect to LR and sðAÞ ¼ aL�1aR�aL is increasing with respect to LR :
A decreasing function mj : Ið<Þ ! ½0; 1� can be associated with every jth
constraint to explain the closeness of [gjL(x), gj
R(x)] towards [bjL, bj
R] for every x 2 S:
The closeness of [gjL(x), gj
R(x)] towards [bjL, bj
R] is denoted by sjðgjðxÞ bjÞ and
defined as follows
sjðgjðxÞ bjÞ ¼1; x 2 Smin
0; x 2 Scmax
mjðbgjðxÞÞ 2 ½0; 1� x 2 SmaxnSmin
8<
:ð4:4Þ
This concept can be explained geometrically in Fig. 1 in case of D = <. Hence it
is clear that every x 2 S is associated with certain degree of acceptability sj with
respect to jth constraint. Since S is the intersection of m number of constraints, so
every x 2 R satisfies the minimum degree of closeness/acceptability. Acceptable
degree of x, which satisfies all the m constraints is s ¼ min1 jm
fsjðgjðxÞ bjÞg:
Define a set
S0 ¼ fðx; sÞ : x 2 S; s ¼ min1 jm
fsjðgjðxÞ bjÞg:
We say x is a feasible point with acceptable degree s and S0 is the acceptable
feasible region.
Solution of nonlinear interval vector optimization problem
123
4.2 Uncertainty in objective function: acceptable objective value
Next it remains to develop a methodology to determine a preferable efficient
solution of (IVOP) over this acceptable feasible region S0. This means we need to
solve
minðx;sÞ2S0
ff1ðxÞ; f2ðxÞ; . . .; fkðxÞg ð4:5Þ
Recall that a feasible solution of (P) is an efficient solution if there is no other
feasible solution that would reduce some objective value without causing
simultaneous increase in at least one other objective value. This type situation
appears in an interval vector programming (IVOP) also. An exact efficient
solution of an interval vector optimization problem may not be found always due
to the nature of conflicting objectives. Hence the decision maker has to com-
promise with several objective values. Here each objective value is an interval
which leads to uncertainty. For this purpose partial orderings are necessary to
compare interval vectors as well as intervals in place of real vectors and real
numbers respectively. To compare interval valued objective functions in
(IVOP), we accept LR and nLR partial orderings as discussed in Sect. 2.1.
In the light of the definition of the solution of a general vector optimization
problem (P), we define efficient solution of (IVOP) with respect to kLR partial
ordering in I(<)k and call this solution as LR-efficient solution.
Definition 4.3 A feasible solution x� of (IVOP) is said to be a LR-efficient solution
of (IVOP) if there does not exists any feasible solution y of (IVOP) such that
f ðyÞ kLR f ðx�Þ with f ðyÞ 6¼ f ðx�Þ:
Fig. 1 Acceptable degree of gjðxÞ bj
M. Jana, G. Panda
123
Definition (4.3) means, for any feasible point y; f ðyÞ �M f ðx�Þ 62 �Ið<Þkþþbecause
f ðyÞ �M f ðx�Þ 62 �Ið<Þkþþ ½f LðyÞ; f RðyÞ� �M ½f Lðx�Þ; f Rðx�Þ� 62 �Ið<Þkþþ
½f LðyÞ � f Lðx�Þ; f RðyÞ � f Rðx�Þ� 62 �Ið<Þkþþ if lðf ðyÞÞ� lðf ðx�ÞÞ½f RðyÞ � f Rðx�Þ; f LðyÞ � f Lðx�Þ� 62 �Ið<Þkþþ if lðf ðyÞÞ lðf ðx�ÞÞ:
(
In other words x� is a LR-efficient solution of (IVOP) if there does not exist any
feasible point y of (IVOP) such that fiðyÞ LR fiðx�Þ 8i ¼ 1; 2; . . .; k and for some
i� 2 f1; 2; . . .; kg; fi� ðyÞ �LR fi� ðx�Þ:To solve (4.5) we will assign a target/goal to every interval valued function fjðxÞ:
These goals may be provided by the decision maker, otherwise we can assign these
goals using the following procedure.
Determination of goal to each objective function:
Consider the following single objective problems corresponding to the sets Smin
and Smax for each i 2 Kk as,
ðPLi Þ : min
x2Smin
f Li ðxÞ;ðPR
i Þ : minx2Smin
f Ri ðxÞ;
ðPL
i Þ : minx2Smax
f Li ðxÞ;ðP
R
i Þ : minx2Smax
f Ri ðxÞ:
For each i, denote the solution of individual problems ðPLi Þ; ðPR
i Þ; ðPL
i Þ and ðPR
i Þ as
xLi ; x
Ri ; x
Li and xR
i respectively and Sideal ¼ fxLi ; x
Ri ; x
Li ; x
Ri ; i 2 Kkg: Let
lLi ¼ min
x2Sideal
f Li ðxÞ; uL
i ¼ maxx2Sideal
f Li ðxÞ; lRi ¼ min
x2Sideal
f Ri ðxÞ; uR
i ¼ maxx2Sideal
f Ri ðxÞ
ð4:6Þ
liL, ui
L, liR, ui
R can be treated as their goals. One may note that the above procedure
determines target/goal for each objective function. But goals can be found by other
procedures also.
For every ðx; sÞ 2 S0; deviation of fiL(x) from the goals li
L and uiL and of
fiR(x) from the goals li
R and uiR may be more or less acceptable for the decision
maker. This implies that fiL(x) and fi
R(x) are associated with certain degree of
flexibility from their goals. For every ðx; sÞ 2 S0; the degree of flexibility of
fiL(x) is higher if deviation of fi
L(x) from liL is less and the degree of flexibility is
less if deviation of fiL(x) from ui
L is more. In other words, we may say that, the
flexible degree of fiL(x) is fully achieved if its value is less than or equal to li
L and
not acceptable if its value is greater than equal to uiL. Similar interpretation can be
made for the upper bound function fiR(x). Hence the degree of flexibility of
fiL(x) and fi
R(x) can be measured through some decreasing functions giL and gi
R
from R to [0,1] respectively. This can be explained in Fig. 2. Mathematically we
may write this function as
Solution of nonlinear interval vector optimization problem
123
gLi ðf L
i ðxÞÞ ¼1; f L
i ðxÞ lLi
2 ½0; 1� lLi f L
i ðxÞ uLi ;
0; f Li ðxÞ[ uL
i
8><
>:
gRi ðf R
i ðxÞÞ ¼1; f R
i ðxÞ lRi
2 ½0; 1� lRi f R
i ðxÞ uRi
0; f Ri ðxÞ[ uR
i
8><
>:
4.3 Uncertainty in feasible region and objective function taken together:
Preferable efficient solution
It is understood from the above discussions that every LR-efficient solution remains
feasible with certain degree of acceptability and its optimal value is flexible towards the
goal with certain flexible degree. The objective functions are characterized by their
degree of flexibility and the constraints are characterized by their degree of acceptability.
So a decision x in this uncertain environment is the selection of activities that
simultaneously satisfies all the objective functions and constraints, which is
mini;lfgL
i ðf Li ðxÞÞ; gR
l ðf Rl ðxÞÞ; ðx; sÞ 2 S0g ð4:7Þ
Every efficient solution of (IVOP) is associated with the partial orderings nLR
due to vector nature of the objective function. It is also associated with uncertainty
in each objective function which is an interval valued mapping. Due to the presence
of these uncertainties, a LR-efficient solution should provide the most preferable
compromising value for the decision maker.
To address this concept we assign a function F : S0 ! ½0; 1� to every LR-efficient
solution and consider the highest preferable LR-efficient solution and call this
solution as preferable efficient solution.
Definition 4.4 A feasible point x� of (IVOP) with acceptable degree s� of (IVOP)
is said to be a preferable efficient solution with respect to LR partial order relation if
x� is a LR-efficient solution and Fðx; sÞFðx�; s�Þ8x 2 S0; where F is preassumed
preferable function.
Best efficient solution is that solution which maximizes the minimum value
obtained in (4.7). So it is necessary to solve maxðx;sÞ2S0
Fðx; sÞ: A preference function
F : S0 �! ½0; 1� may be considered as
Fðx; sÞ ¼ min1 i k
1 l k
fgLi ðf L
i ðxÞÞ; gRl ðf R
l ðxÞÞg ð4:8Þ
For the most preferable optimal solution of (IVOP) we need to maximize F(x, s).
This is equivalent to
(IVOP)0 : max h (4.9)
subject to h gLi ðf L
i ðxÞÞ; h gRl ðf R
l ðxÞÞ (4.10)
h sjðgjðxÞ bjÞ; x 2 S; 0 h 1: (4.11)
M. Jana, G. Panda
123
This is a general nonlinear programming problem which is free from interval
uncertainty. This problem can be solved using non linear programming techniques.
Let the solution of the problem (IVOP)0 be (hopt, xopt, sopt). We prove the following
result which establishes the relation between the solution of (IVOP)0 and (IVOP).
Theorem 4.1 If (hopt, xopt, sopt) is the optimal solution of (IVOP)0, then xopt is a
preferable efficient solution with acceptable degree s� of (IVOP). In case of
alternate optimal solution of (IVOP)0, at least one of them is a preferable efficient
solution of (IVOP).
Proof Here hopt = h(xopt, sopt). Suppose xopt is not a preferable efficient solution of
(IVOP). Consequently xopt is not a LR-efficient solution. Suppose there exists a
preferable efficient solution x� 6¼ xopt with feasibility degrees�ðs� � soptÞ. Then from
Definition 4.3, this implies that f ðxoptÞ �M f ðx�Þ 62 f�Ið<Þkþþg: If s�ðs�[ soptÞ, then
s� is not the minimum of all sj. Take s� ¼ sopt. Since xopt is not a preferable efficient
solution, f ðx�Þ kLR f ðxoptÞ and f ðx�Þ 6¼ f ðxoptÞ: In other words
fiðx�Þ LR fiðxoptÞ; 8i; fi� ðx�Þ �LR fi� ðxoptÞ for some i� 2 Kk:
This is equivalent to
f Li ðx�Þ f L
i ðxoptÞ and f Ri ðx�Þ f R
i ðxoptÞ; with f Li� ðx�Þ\f L
i� ðxoptÞ and f Ri� ðx�Þ\f R
i� ðxoptÞ
Since giL and gi
R are decreasing functions so giL(fi
L(x�)) C giL(fi
L(xopt)) and
giR(fi
R(x�)) C giR(fi
R(xopt)) for all i, with gLi� ðf L
i� ðx�ÞÞ[ gLi� ðf L
i� ðxoptÞÞ and gRi� ðf R
i� ðx�ÞÞ[gR
i� ðf Ri� ðxoptÞÞ for at least one i*. Also
hopt gLi ðf L
i ðxoptÞÞ gLi ðf L
i ðx�ÞÞhopt gR
i ðf Ri ðxoptÞÞ gR
i ðf Ri ðx�ÞÞ
Since x� 2 S; x� is also a feasible point with some acceptable degree which is
s� ¼ minj
sjðgjðx�ÞÞ:
So (x�; s�) is a feasible point of (IVOP)0. Now,
hðx�Þ ¼ mini2Kk
fgLi ðf L
i ðx�ÞÞ; gRi ðf R
i ðx�ÞÞg
� mini2Kk
fgLi ðf L
i ðxoptÞÞ; gRi ðf R
i ðxoptÞÞg
¼ hðxoptÞ:) hðx�Þ� hðxoptÞ ¼ hopt
If h(x�) = h(xopt) then x� and xopt are alternate optimal solutions of (IVOP)0.Since x� is a LR-efficient solution and xopt is not a LR-efficient solution, so x� is the
preferable efficient solution of (IVOP) with feasibility degree s� (= sopt). If
h(x�) [ h(xopt) we have a contradiction to the optimality of xopt.
Solution of nonlinear interval vector optimization problem
123
Remark 1 In this section one may observe that the efficient solution of (IVOP) is
defined with respect to LR partial ordering. However, any partial ordering can be
used to define efficient solution. It is true that a particular partial ordering can not
compare all objective values, which are intervals vectors. To address this difficulty,
a goal has been associated with every objective function and, degree of closeness of
the objective value and the goal has been measured through a scalar valued function
in Sect. 4.2, so that all the objective values can be compared. Furthermore, it is
proved in Theorem 4.1 that a solution of (IVOP) can be found by solving (IVOP)0. A
particular partial ordering is required to prove this result. Here we have accepted
LR partial ordering because the proof depends upon the structure of (IVOP)0.However, a similar methodology in the light of the developments of this section can
be established with respect to any other type partial ordering in the set of intervals.
In that case the construction of (IVOP)0 and proof of Theorem 4.1 may be different
accordingly.
The discussion of this section may be explained through a numerical example in
the next Subsection.
4.4 Numerical example
Example 3 Consider the following (IVOP) as,
(IVOP) : min f½�2; 1�x1; ½1; 2�x1 þ ½0; 1�x22g
subject to ½1; 2�x21 þ ½�2;�1�x2 � ½0; 0� ð4:12Þ
½1; 1�x1 þ ½1; 3�x2 ½2; 4� ð4:13Þx1; x2� 0:
Fig. 2 gLi ; g
Ri : < ! ½0; 1� for different positions of the lower and upper bounds of fiðxÞ
M. Jana, G. Panda
123
Solution Here
g1ðx1; x2Þ ¼ ½gL1ðx1; x2Þ; gR
1 ðx1; x2Þ� ¼ ½x21 � 2x2; 2x2
1 � x2�g2ðx1; x2Þ ¼ ½gL
2ðx1; x2Þ; gR2 ðx1; x2Þ� ¼ ½x1 þ x2; x1 þ 3x2�
f1ðx1; x2Þ ¼ ½f L1 ðx1; x2Þ; f R
1 ðx1; x2Þ� ¼ ½�2x1; x1�f2ðx1; x2Þ ¼ ½f L
2 ðx1; x2Þ; f R2 ðx1; x2Þ� ¼ ½x1; 2x1 þ x2
2�Smin ¼ fðx1; x2Þ : 2x2
1 � x2 0; x1 þ 3x2 2; x1� 0; x2� 0gSmax ¼ fðx1; x2Þ : x2
1 � 2x2 0; x1 þ x2 4; x1� 0; x2� 0g
To determine the acceptable feasible region S0, we need to assign decreasing
functions s1; s2 : Ið<Þ ! ½0; 1� to the interval inequalities (4.12) and (4.13)
respectively. These functions have to be provided by the decision maker. For
convenience we consider the following functions s1 and s2. For x 2 SmaxnSmin;
sjðgjðx1; x2Þ bjÞ ¼bR
j � gLj ðx1; x2Þ
ðbRj � bL
j Þ þ ðgRj ðx1; x2Þ � gL
j ðx1; x2ÞÞ; j ¼ 1; 2 ð4:14Þ
So s1ðg1ðx1; x2Þ b1Þ ¼ �x21þ2x2
x21þx2
and s2ðg2ðx1; x2Þ b2Þ ¼ 4�x1�x2
2þ2x2: To find the
solution of lower and upper bound functions of each objective function corre-
sponding to the minimum feasible region (Smin) and maximum feasible region
Smax, consider the following problems.
ðPL1Þ : min
ðx1;x2Þ2Smin
�2x1; ðPL2Þ : min
ðx1;x2Þ2Smin
x1; ðPR1 Þ : min
ðx1;x2Þ2Smin
x1; ðPR2 Þ : min
ðx1;x2Þ2Smin
2x1 þ x22;
ðPL
1Þ : minðx1;x2Þ2Smax
�2x1; ðPL
2Þ : minðx1;x2Þ2Smax
x1; ðPR
1 Þ : minðx1;x2Þ2Smax
x1; ðPR
2 Þ : minðx1;x2Þ2Smax
2x1 þ x22:
The solutions of the individual problems ðPL1Þ; ðPL
2Þ; ðPR1 Þ; ðPR
2 Þ; ðPL
1Þ;ðPL
2Þ; ðPR
1 Þ; ðPR
2 Þ are provided in the following table. From (4.6), the maximum
and minimum value of the lower and upper bound functions are found as
f1L(x1, x2), f2
L(x1, x2), f1R(x1, x2), f2
R(x1, x2) are found as
lL1 ¼ �4; uL
1 ¼ 0; lL2 ¼ 0; uL
2 ¼ 2; lR1 ¼ 0; uR1 ¼ 2; lR
2 ¼ 0; uR2 ¼ 8
The degree of flexibility of the objective values from these goals, corresponding to
an acceptable feasible solution ððx1; x2Þ; sÞ 2 S0 should be provided by the decision
maker depending upon his/her choice. Here for the sake of convenience we assign
decreasing functions giL, gi
R, i = 1, 2 as
gtiðf t
1ðx1; x2ÞÞ ¼ut
1 � f t1ðx1; x2Þ
ut1 � lt
1
; t 2 fL;Rg ð4:15Þ
Then for �4 f L1 ðx1; x2Þ 0; gL
1ðf L1 ðx1; x2ÞÞ ¼ x1
2; for 0 f L
2 ðx1; x2Þ 2;
gRi ðf R
i ðx1; x2ÞÞ ¼ 1� x1
2; for 0 f R
1 ðx1; x2Þ 2; gR1 ðf R
1 ðx1; x2ÞÞ ¼ 1� x1
2; and for
0 f R2 ðx1; x2Þ 8; gR
2 ðf R1 ðx1; x2ÞÞ ¼ 1� x1
4� x2
2
8:
Solution of nonlinear interval vector optimization problem
123
Here
S0 ¼ fððx1; x2Þ; sÞ : ðx1; x2Þ 2 S; s ¼ minfs1ðg1ðx1; x2ÞÞ; s2ðg2ðx1; x2ÞÞg
and from (4.8),
Fððx1; x2Þ; sÞ ¼ minfgL1ðf L
1 ðx1; x2ÞÞ; gL2ðf L
2 ðx1; x2ÞÞ; gR1 ðf R
1 ðx1; x2ÞÞ; gR2 ðf R
2 ðx1; x2ÞÞg
So (IVOP)0 becomes maxððx1;x2Þ;sÞ2S0 Fððx1; x2Þ; sÞ; which is equivalent to
(IVOP)’: max hsubject to �x1 þ 2h 0; x1 þ 2h 2;
2x1 þ x22 þ 8h 8; ðhþ 1Þx2
1 þ ðh� 2Þx2 0;x1 þ ð2hþ 1Þx2 � 2hþ 4; x1; x2� 0; 0 h 1:
It is easy to verify that this is a deterministic convex nonlinear programming
problem for which KKT optimality conditions are sufficient for the existence of the
solution. Consider the Lagrange function L(x, k, l) as
Lðx; k; lÞ ¼ �hþ k1ð�x1 þ 2hÞ þ k2ðx1 þ 2h� 2Þ þ k3ð2x1 þ x22 þ 8h� 8Þ
þ k4ððhþ 1Þx21 þ ðh� 2Þx2Þ þ k5ðx1 þ ð2hþ 1Þx2 þ 2h� 4Þ
� l1x1 � l2x2 � l3h
where k, l0s are dual variables. The KKT optimality conditions are
� k1 þ k2 þ 2k3 þ 2k4x1ðhþ 1Þ þ k5 � l1 ¼ 0
2x2k3 þ k4ðh� 2Þ þ k5ð2hþ 1Þ � l2 ¼ 0
� 1þ 2k1 þ 2k2 þ 8k3 þ k4x21 þ k4x2 þ 2k5x2 þ 2k5 � l3 ¼ 0
k1ð�x1 þ 2hÞ ¼ 0; k2ðx1 þ 2h� 2Þ ¼ 0
k3ð2x1 þ x22 þ 8h� 8Þ ¼ 0; k4ððhþ 1Þx2
1 þ ðh� 2Þx2Þ ¼ 0
k5ðx1 þ ð2hþ 1Þx2 þ 2h� 4Þ ¼ 0; l1x1 ¼ 0; l2x2 ¼ 0; l3h ¼ 0
kt� 0; t ¼ 1; 2; 3; 4; 5:
This system is solved in Lingo 13.0. The optimal solution of this problem is found
as hopt = 0.5, sopt = 0.5, x1opt = 1 and x2
opt = 1. (1, 1) is a preferable efficient
solution of the given interval vector optimization problem.
Table 1 Solutions of the
individual problemsProblem Solution Problem Solution
P1L x1
L = (0.5,0.5) PL
1xL
1 ¼ ð2; 2ÞP2
L x2L = (0,0) P
L
2xL
2 ¼ ð0; 1:235ÞP1
R x1R = (0,0) P
R
1xL
1 ¼ ð0; 1:235ÞP2
R x2R = (0,0) P
R
2xL
1 ¼ ð0; 0Þ
M. Jana, G. Panda
123
5 Conclusion
This paper provides a method to solve a vector optimization problem when the
parameters in the objective functions and constraints are intervals. The original
problem is transformed to a general optimization problem and it is theoretically
(Theorem 4.1) proved that the optimal solution of the transformed problem is a
preferable efficient solution of the original interval optimization problem. Proof of
Theorem 4.1 uses LR partial ordering. However, in the light of this methodology
similar deterministic equivalent problems can be constructed and similar result in
the light of Theorem 4.1 can be established using any other partial ordering. The
methodology of the present work is applicable to both linear or non-linear interval
vector optimization problems. This method provides one solution of the problem,
which is feasible and efficient up to certain acceptable degree. Here, the decision
maker has to provide suitable functions m, giL and gi
R to construct the deterministic
problem (IVOP)0. Like vector optimization problem, preferable efficient solution of
(IVOP) is not necessarily unique. For different choices of m, giL and gi
R, (IVOP)0
can be simplified accordingly and a preferable efficient solution can be found.
Acknowledgments The authors are greatly indebted to the anonymous referee for valuable comments
and remarks.
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