Abstract This paper addresses a general multi-objective fractional programmingproblem whose parameters in the objective functions and constraints are intervals.Existence of the efficient solution of this model is studied. A methodology is devel-oped to determine its efficient solutions. This methodology is illustrated through anumerical example.
A general multi-objective optimization problem optimizes several real valued func-tions simultaneously. Due to the conflicting nature of the objective functions, exactoptimum solution may not exist. Several well known methods exist in the litera-ture to derive the compromise/efficient/Pareto optimal solution of a linear/nolinearmulti-objective programming problem. Generally, the parameters in these optimiza-tion problems are considered as real numbers. However, in many real life situationsthe parameters are not fixed due to the presence of some uncertainties in the dataset. Upper and lower bound of these parameters may be estimated from the historical
A. K. Bhurjee (�)Research Scholar, Indian Institute of Technology Kharagpur, Kharagpur, WB-721302, Indiae-mail: [email protected]
G. PandaFaculty of Mathematics, Department of Mathematics, Indian Institute of Technology Kharagpur,Kharagpur, WB-721302, Indiae-mail: [email protected]
data. In other words the parameters are assumed to lie in closed intervals. Conse-quently, the real valued functions in the optimization problem become interval valuedfunctions and the said model is an interval optimization problem. Since the set ofintervals is not a totally ordered set, so justification of the existence of solution ofthese problems is a challenging research area. Depending upon various type of partialorderings in the set of intervals, multi-objective interval optimization problems areaddressed during last few years by some researchers. (see [3, 8, 9, 11, 14, 15]). Mostof these models cover linear objective functions. If at least one objective functionof a multi-objective interval optimization problem is the ratio of two interval val-ued functions then we call the corresponding optimization model as a multi-objectiveinterval fractional programming problem and denote in short as (MIFP ). Followingtransportation model explains a real life situation of (MIFP ) problem.
Example 1 Suppose, a homogeneous product is to be transported from m numberof sources to n number of destinations. The ith source can provide ai units of acertain product and the j th destination has a demand for bj units of the same prod-uct. From the historical data, it is observed that due to traffic jam, climate change,shortage of vehicles etc., the transportation cost, deterioration cost and time for trans-portation of one unit of the product from ith source to j th destination lie in thelower and upper bounds cLij and cRij ; dLij and dRij ; and, tLij and tRij , respectively. Theobjective is to minimize the total transportation expenditure per unit transportationtime to satisfy the demands and the total deterioration cost per unit transportationtime, simultaneously. If xij is the number of units transported from source i to des-tination j , then the mathematical model of the conventional transportation problembecomes
(P ) min
{∑mi=1
∑nj=1[cLij , cRij ]xij∑m
i=1∑n
j=1[tLij , tRij ]xij,
∑mi=1
∑nj=1[dLij , dRij ]xij∑m
i=1∑n
j=1[tLij , tRij ]xij
}
subject ton∑
j=1
xij = ai, i = 1, 2, . . . , m,
m∑i=1
xij = bj , j = 1, 2, . . . , n,
m∑i=1
ai =n∑
j=1
bj , xij ≥ 0, ∀ i, j.
General multi-objective fractional programming with fixed parameters are dis-cussed by several researchers in many directions (see [1, 4, 6, 7, 10, 12]). Hladik [5]considered a single objective linear fractional programming problem with intervalparameters and compute the bounds for optimal values. Nonlinear multi-objectivefractional programming problem with interval parameters (MIFP ) has not beenstudied yet. In this paper, we discuss the existence of the solution of (MIFP )
which contains both linear and nonlinear interval valued functions in the objectivefunctions as well as constraints.
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Section 2 explains some prerequisites on interval analysis. A methodology isdescribed in Section 3 in which the original problem is transformed to a single objec-tive deterministic optimization problem. Relation between the efficient solution ofthe original (MIFP ) and the transformed problem is established.
Throughout the paper, the following notations are used. Bold capital letters denoteclosed intervals.
I (R)= The set of all closed intervals in R.(I (R))k= The product space I (R)× I (R)× . . .× I (R)︸ ︷︷ ︸
k times
.
Ckv= k-dimensional column whose elements are intervals.
Ckv ∈ (I (R))k , Ck
v = (C1,C2, . . . ,Ck)T , Cj = [cLj , cRj ], j ∈ �k,
�k = {1, 2, . . . , k}.The degenerate interval, η̂ is η̂ = [η, η], where η is a real number.
2 Preliminaries
For two real vectors a = (a1, a2, . . . , an)T , b = (b1, b2, . . . , bn)
T in Rn, we denote
a >v b ⇔ ai > bi, i ∈ �n and a <v b ⇔ ai < bi, i ∈ �n.
Let ∗ ∈ {+,−, ·, /} be a binary operation on the set of real numbers. The binaryoperation � between two intervals A = [aL, aR] and B = [bL, bR] in I (R), denotedby A � B is the set {a ∗ b : a ∈ A, b ∈ B}. In the case of division (A � B), itis assumed that 0 /∈ B. These interval operations can also be expressed in terms ofparameters. Any point in A may be expressed as a(t) = aL+ t (aR − aL), t ∈ [0, 1].An interval A is said to be a positive interval if a(t) is positive for every t . Thealgebraic operations of intervals in the classical form are defined in terms of eitherlower and upper bound or mean and spread of the intervals. Algebraic operations ofintervals may be explained in parametric form as follows.
A � B = {a(t1) ∗ b(t2)| t1, t2 ∈ [0, 1]}. (1)
An interval vector Ckv ∈ (I (R))k can be expressed in terms of parameters as
Ckv =
{c(t)| c(t) = (c1(t1), c2(t2), . . . , ck(tk))
T , where t = (t1, t2, . . . , tk)T ,
cj (tj ) ∈ Cj , cj (tj ) = cLj + tj (cRj − cLj ), tj ∈ [0, 1], j ∈ �k
}.
The set of intervals I (R) is not a totally order set. Consider the following partialordering in I (R) due to Bhurjee and Panda [2].For A,B ∈ I (R),
A � B ⇔ a(t) ≤ b(t), ∀ t ∈ [0, 1] and A ≺ B ⇔ a(t) < b(t), ∀ t ∈ [0, 1]. (2)
Interval valued function is defined in several ways by many authors. We considerthe interval valued function in parametric form as follows.
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Definition 1 [2] For c(t) ∈ Ckv, let fc(t) : Rn → R. For a given interval vector Ck
v,define an interval valued function FCk
v: Rn → I (R) by
FCkv(x) =
{fc(t)(x)
∣∣∣ fc(t) : Rn → R, c(t) ∈ Ckv
}.
For every fixed x, if fc(t)(x) is continuous in t then mint∈[0,1]k fc(t)(x) andmaxt∈[0,1]k fc(t)(x), exist. In that case
FCkv(x) =
[min
t∈[0,1]kfc(t)(x), max
t∈[0,1]kfc(t)(x)
].
If fc(t)(x) is linear in t then mint∈[0,1]k fc(t)(x) and maxt∈[0,1]k fc(t)(x) exist inthe set of vertices of Ck
v. If fc(t)(x) is monotonically increasing in t , then FCkv(x) =
[fc(0)(x), fc(1)(x)].
3 Methodology
Consider a general multi-objective interval fractional programming problem.
(MIFP ) min
⎧⎨⎩
F1Ck1v
(x)
H1El1v
(x),
F2Ck2v
(x)
H2El2v
(x), . . . ,
Fm
Ckmv
(x)
Hm
Elmv
(x)
⎫⎬⎭
subject to Gp
Dmpv
(x) � Bp, p ∈ �q, (3)
where Fi
Ckiv
,Hi
Eliv
,Gp
Dmpv
: Rn → I (R), i ∈ �m, partial ordering in constraints
(3) are as defined in (2), Hi
Eliv
(x) � 0, i ∈ �m and Bp ∈ I (R), p ∈ �q .
Using Definition 1, the interval valued functionFi
Using (4) and (6), (MIFP ) can be expressed in parametric form as
(MIFP ) minx∈S
{(f 1c1(t ′1)(x)
h1e1(t ′′1)(x)
,f 2c2(t ′2)(x)
h2e2(t ′′2)(x)
, . . . ,f mcm(t ′m)(x)
hmem(t ′′m)(x)
) ∣∣∣ci(t ′i ) ∈ Cki
v , ei(t ′′i ) ∈ Eliv , i ∈ �m
}.
Since (MIFP ) has several interval valued conflicting objective functions, soexact solution of (MIFP ) may not exist which minimizes all objective functionssimultaneously. Like general multi-objective problem, solution of (MIFP ) is acompromise/ Pareto optimal/ efficient solution.
For every x, the objective value of (MIFP ) is an interval vector whose compo-nents are ratio of two interval valued functions. So, for any two different points x
and y in S, the objective values of (MIFP ) can be compared componentwise likegeneral vector optimization problem as follows.
⎛⎝F1
Ck1v
(x)
H1El1v
(x),
F2Ck2v
(x)
H2El2v
(x), . . . ,
Fm
Ckmv
(x)
Hm
Elmv
(x)
⎞⎠ �v
⎛⎝F1
Ck1v
(y)
H1El1v
(y),
F2Ck2v
(y)
H2El2v
(y), . . . ,
Fm
Ckmv
(y)
Hm
Elmv
(y)
⎞⎠
⇔Fi
Ckiv
(x)
Hi
Eliv
(x)�
Fi
Ckiv
(y)
Hi
Eliv
(y)∀i ∈ �m.
One partial ordering can not compare all intervals. Due to the complexities asso-ciated with partial orderings, involved at different stages of (MIFP ), it is difficultto derive the efficient solution of (MIFP ) directly like general vector optimiza-tion problem. To avoid these complications, (MIFP ) is transformed to a generaloptimization problem in the subsequent sections. Relation between the efficientsolution of (MIFP ) and optimal solution of the transformed problem is estab-lished. We accept the partial ordering � as defined in (2) to prove the result andcall an efficient solution of (MIFP ) with respect to � as I�−Efficient solu-tion. (The results of this paper are based on the partial ordering in parametricform. However, similar theory may be developed with respect to any other partialordering.)
Like vector optimization problem, I�−Efficient solution and properly I�−Efficient solution of (MIFP ) may be defined as follows.
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Definition 2 x∗ ∈ S is called an I�−Efficient solution of (MIFP ) if there is nox ∈ S with
Fi
Ckiv
(x)
Hi
Eliv
(x)�
Fi
Ckiv
(x∗)
Hi
Eliv
(x∗)∀ i ∈ �m
andFj
Ckjv
(x)
Hj
Eljv
(x)≺
Fj
Ckjv
(x∗)
Hj
Eljv
(x∗)at least one j �= i. (7)
Definition 3 x∗ ∈ S is called a properly I�−Efficient solution of (MIFP ) withrespect to � if x∗ ∈ S is an I�−Efficient solution of (MIFP ) and there is a positive
degenerate interval η̂ such that for all i ∈ �m and every x ∈ S withFi
Ckiv
(x)
Hi
Eliv
(x)≺
Fi
Ckiv
(x∗)
Hi
Eliv
(x∗) , for at least one j �= i exists withFj
Ckjv
(x∗)
Hj
Eljv
(x∗)≺
Fj
Ckjv
(x)
Hj
Eljv
(x)and
⎡⎣Fi
Ckiv
(x∗)
Hi
Eliv
(x∗)�
Fi
Ckiv
(x)
Hi
Eliv
(x)
⎤⎦⎡⎢⎣Fj
Ckjv
(x)
Hj
Eljv
(x)�
Fj
Ckjv
(x∗)
Hj
Eljv
(x∗)
⎤⎥⎦ � η̂. (8)
3.1 Transformation of (MIFP ) into deterministic problem
As discussed earlier the parametric form of (MIFP ) is (MIFP ). For a given inter-val A = [aL, aR] with 0 /∈ A, aL > 0, 1
A = [ 1aR
, 1aL
].For x ∈ Rn, i ∈ �m,
1
Hi
Eliv
(x)= [τLi , τRi ] =
{τi ∈ R | τi∈ [τLi , τRi ], 1
hiei (t ′′i)(x)= τi
},
where τLi = mint ′′i 1hiei (t ′′i )(x)
and τRi = maxt ′′i 1hiei (t ′′i )(x)
.
Since hiei (t ′′i)(x) > 0 so τi > 0 and mint ′′i hiei (t ′′i )(x) ≤ hiei (t ′′i)(x) ≤maxt ′′i hiei (t ′′i )(x) ∀i. So
τi mint ′′i
hiei (t ′′i )(x) ≤ 1 and τi maxt ′′i
hiei (t ′′i )(x) ≥ 1.
Since x satisfies the above inequalities and lies in S so the feasible set can betransformed to the following set.
For any vector valued weight function w (or w(t)) = (w1(t1), w2(t2), . . . ,
wm(tm))T , wi(ti ) > 0, μi > 0, i ∈ �m, consider the following deterministic
optimization problem (MIFPμw ), which is free from the interval uncertainty.
(MIFPμw ) min
(x,τ )∈χ �(x, τ),
where �(x, τ) =∑i∈�m
μiψi(x, τi), ψi(x, τi) =∫kiwi(t ′i )τif i
ci(t ′i )(x) dt′i ,∫
ki=∫ 1
0
∫ 1
0. . .
∫ 1
0︸ ︷︷ ︸ki times
, dt ′i = dt ′1i dt ′2i . . . dt′ii , t ′i = (t ′1i , t ′2i , . . . , t ′i )T , i ∈ �m, which
can be solved using any non-linear optimization technique.
wi(ti ) may be treated as a preference weight function, which has to be providedby the decision maker. Different preference functions can be provided to estimatethe Pareto optimal value of the model. For every i, wi(ti) = 1 indicates that theinvestor’s natural attitude is to estimate the mean. If
∫kiwi(ti )dti = 1 for every i
then the investor’s inclination is to estimate in between the optimistic and pessimisticoptimal value. In the subsequent results we will see that any selection of wi withpositive value can provide an efficient solution.
Relationship between the optimal solution of (MIFPμw ) and the I�−Efficient
solution (MIFP ) is established in the following result.
Theorem 1 If (x∗, τ ∗) is an optimal solution of (MIFPμw ) for some w >v 0 and
μ >v 0, then x∗ is an I�−Efficient solution of (MIFP ).
Proof Let (x∗, τ ∗) ∈ χ be an optimal solution of (MIFPμw ) for some w >v 0
and μ >v 0. Assume that x∗ is not an I�−Efficient solution of (MIFP ) then thereis some x ∈ S satisfying (7). Using (2), the inequalities in (7) can be rewritten asfollows.
For some x in S and each t ′i ∈ [0, 1]ki , t ′′i ∈ [0, 1]li ,f ici(t ′i )(x)
hiei (ti ′′)(x)≤ f i
ci(t ′i )(x∗)
hiei (ti ′′)(x∗), i ∈ �m
andfj
cj (t ′j )(x)
hj
ej (tj ′′)(x)<
fj
cj (t ′j )(x∗)
hj
ej (tj ′′)(x∗)
at least one j �= i.
That is there exists x ∈ S so that for every ti ′ ∈ [0, 1]ki ,τif
ici (t ′i)(x) ≤ τ ∗i f i
ci (t ′i)(x∗), i ∈ �m
and τjfj
cj (t ′j )(x) < τ ∗j fj
cj (t ′j )(x∗) at least one j �= i.
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Since wi(t′i ) > 0, the above relations imply that
∫ki
wi(t ′i )τif ici(t ′i )(x)dt ′i ≤
∫ki
wi(t ′i )τ ∗i f ici(t ′i )(x
∗)dt ′i ,i ∈ �m and for at least one j �= i∫
kj
wj (t ′j )τj f j
cj (t ′j )(x)dt ′j <
∫kj
wj (t ′j )τ ∗j f j
cj (t ′j )(x∗)dt ′j .
For μi > 0, i ∈ �m, there exists some x ∈ S such that
∑i∈�m
μiψi(x, τi) <∑i∈�m
μiψi(x∗, τ ∗i ) ⇒ �(x, τ) < �(x∗, τ ∗).
This contradicts the assumption that (x∗, τ ∗) is an optimal solution of (MIFPμw ).
Hence x∗ is an I�−Efficient solution of (MIFP ).
Theorem 2 If (x∗, τ ∗) ∈ χ is an optimal solution of (MIFPμw ), w >v 0, wi are
continuous functions satisfying∫kiwi(ti ) dti = 1, ∀ i ∈ �m, μ >v 0 then x∗ is a
properly I�−Efficient solution of (MIFP ).
Proof Suppose that x∗ is not properly I�−Efficient solution of (MIFP ). So eitherx∗ is not an I�−Efficient solution of (MIFP ) or x∗ is an I�−Efficient solution butnot satisfy the condition for properly I�−Efficient solution (8).(I) Suppose x∗ is not an I�−Efficient solution of (MIFP ). Then by Theorem 1,(x∗, τ ∗) ∈ χ is not an optimal solution of (MIFP
μw ) for any w >v 0 and μ >v 0,
which contradicts that (x∗, τ ∗) ∈ χ be an optimal solution of (MIFPμw ).
(II) Let x∗ be an I�−Efficient solution of (MIFP ) but does not satisfy the conditionfor properly I�−Efficient solution (8). One can choose
η = (m− 1)maxi,j
maxt ′i ,t̄ ′i ,t ′j ,t̄ ′j
{μjwj (t ′j )wj (t̄ ′j )μiwi(t ′i )wi(t̄ ′i )
},
for m ≥ 2, i �= j, t ′i , t̄ ′i ∈ [0, 1]ki . Since wi is continuous so η exists. Then from
Definition 3, for some i ∈ �m and some x ∈ S withFi
Ckiv
(x)
Hi
Eliv
(x)≺
Fi
Ckiv
(x∗)
Hi
Eliv
(x∗) ,
⎡⎣Fi
Ckiv
(x∗)
Hi
Eliv
(x∗)�
Fi
Ckiv
(x)
Hi
Eliv
(x)
⎤⎦⎡⎢⎣Fj
Ckjv
(x)
Hj
Eljv
(x)�
Fj
Ckjv
(x∗)
Hj
Eljv
(x∗)
⎤⎥⎦ � η̂, ∀j ∈ �m, i �= j, (9)
withFj
Ckjv
(x∗)
Hj
Eljv
(x∗)≺
Fj
Ckjv
(x)
Hj
Eljv
(x)holds for η̂ = [η, η].
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Expression (9) means, for every t ′i , t ′′i ,[f ici(t ′i)(x
∗)hiei (t ′′i )(x
∗)−
f ici(t̄ ′i )(x)
hiei ( ¯t ′′i )(x)
]/⎡⎣ fj
cj (t ′j )(x)
hj
ej (t ′′j )(x)−
fj
cj (t̄ ′j )(x∗)
hj
ej ( ¯t ′′j )(x∗)
⎤⎦ > η
≥ (m− 1)
{μjwj (t ′j )wj (t̄ ′j )μiwi(t ′i )wi(t̄ ′i)
}∀ j ∈ �m/{i},
which is ∀ j ∈ �m/{i},
μiwi(t ′i )wi(t̄ ′i )[f ici(t ′i )(x
∗)hiei (t ′′i )(x
∗)−
f ici(t̄ ′i )(x)
hiei ( ¯t ′′i )(x)
]
≥ (m− 1)μjwj (t ′j )wj (t̄ ′j )⎡⎣ f
j
cj (t ′j )(x)
hj
ej (t ′′j )(x)−
fj
cj (t̄ ′j )(x∗)
hj
ej ( ¯t ′′j )(x∗)
⎤⎦ .
Using the transformation as discussed in Subsection 3.1, the above relationbecomes,
μiwi(t ′i )wi(t̄ ′i)(τ ∗i f ici(t ′i )(x
∗)− τifici(t̄ ′i )(x)) >
(m− 1)μjwj (t ′j )wj (t̄ ′j )(τjf j
cj (t ′j )(x)− τ ∗j fj
cj (t̄ ′j )(x∗)).
Integrating with respect to ti , t̄i , tj , t̄j on both sides and using∫kiwi(ti ) dti = 1,
That is, �i(x∗, τ ∗) > �i(x, τ ). This contradicts the assumption that x∗ is an
optimal solution of (MIFPμw ). Hence x∗ is a properly I�−Efficient solution of
(MIFP ).
Note If all the parameters of (MIFP ) are considered as degenerate intervals then(MIFP ) becomes a conventional multi-objective fractional programming problemand the entire methodology, discussed in this section will reduce to conventionalfractional programming technique due to Schaible [13] and general multi-objectiveprogramming technique. In that case I�−Efficient(properly I�−Efficient) solutionbecomes efficient(properly efficient) solution.
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3.2 Numerical example
The methodology of Section 3 is explained in the following example.
Example 2 Consider the following optimization problem whose objective functionsare nonlinear interval valued functions.
Consider the weight functions w1(t′11 , t ′21 , t ′31 ) = 1 + t ′11 t ′21 t ′31 , w2(t
′′11 , t ′′21 ) = 1 +
t ′′11 t ′′21 and μ1 = 0.25, μ2 = 0.75. Then
ψ1(x) =∫
2w1(t
′11 , t ′21 )((4 + 6t ′11 )x2
1 + (−1 + 2t ′21 )x1x2 + (10 + 10t ′31 )x22)dt
′11 dt ′21
= 8x21 + 1
24x1x2 + 205
12x2
2
ψ2(x) =∫
2w2(t
′′11 , t ′′21 )((−1 + 3t ′12 )x1 + x2
2 ) =3
4x1 + 5
4x2
2 .
The deterministic problem corresponding to (MIFP ) becomes
(MIFPμw ) min τ1x
21 + 1
96τ1x1x2 +
(205
48τ1 + 15
16τ2
)x2
2 + τ2x1
subject to τ1(−5x1 + x2) ≤ 1, τ1(−3x1 + 2x2) ≥ 1,
τ2(2x1 + 1.5x2) ≤ 1, τ2(3x1 + 2.5x2) ≥ 1,
x1, x2 ∈ S, τ1 > 0, τ2 > 0.
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Using Lingo the optimal solution of the above problem is found as x1 = 0, x2 =3.333, τ1 = 0.15, τ2 = 0.12. From Theorem 1 it follows that (0, 3.333) is anI�−Efficient solution of (MIFP ).
4 Conclusion
In this paper, a multi-objective interval fractional programming problem is trans-formed to a general optimization problem in which the parameters have fixed realvalues. Using a specific partial ordering, connection between the solution of originalproblem and the transformed problem is established under some assumptions. Sim-ilar methodologies may be developed in the light of present work using any otherpartial ordering. However, in case of a different partial ordering, the formulation ofthe transformed problem and the proof of the theorems may be different. Duality the-ory plays an important role for the existence of solution of a general multi-objectiveinterval fractional programming problem. In the light of this methodology dualitytheory for a general multi-objective interval fractional programming problem can beestablished which is the future scope of the presents work.
Acknowledgments The authors thank the anonymous referees whose justified critical remarks on theoriginal version led to an essential improvement of the paper.
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