url: ijpam · objective quadratic programming problem, in which both the technological...

International Journal of Pure and Applied Mathematics

Volume 89 No. 4 2013, 511-529ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.eudoi: http://dx.doi.org/10.12732/ijpam.v89i4.6

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A COMPUTATIONAL PROCEDURE FOR SOLVING

A NON-CONVEX MULTI-OBJECTIVE QUADRATIC

PROGRAMMING UNDER FUZZY ENVIRONMENT

Shashi Aggarwal1, Uday Sharma2 §

1Department of MathematicsMiranda House, University of Delhi

Delhi, 110007, INDIA2Department of Mathematics

University of DelhiDelhi, 110007, INDIA

Abstract: The purpose of this paper is to study a non-convex fuzzy multi-objective quadratic programming problem, in which both the technological co-efficients and resources are fuzzy. A nonlinear membership function is defined.A computational procedure to find a fuzzy efficient solution of this problem isdeveloped. A numerical example is given to illustrate the procedure.

AMS Subject Classification: 90C20, 90C26, 90C29, 90C70Key Words: non-convex, fuzzy multi-objective quadratic programming, non-linear membership function, fuzzy efficient solution

1. Introduction

This paper studies the problem of maximizing a k number of quadratic objectivefunctions subject to linear and bound constraints under fuzzy environment:

Received: August 13, 2013 c© 2013 Academic Publications, Ltd.url: www.acadpubl.eu

§Correspondence author

512 S. Aggarwal, U. Sharma

(NFMOQPP) MaxZ1(x) = ct1x+1

2xtQ1x

MaxZ2(x) = ct2x+1

2xtQ2x

...

MaxZk(x) = ctkx+1

2xtQkx

subject to x ∈ X={x ∈ Rn : Ax≤ b, l≤x≤ u, x ≥ 0},

where cq, q = 1, 2, . . . , k are n-dimensional cost vectors; Qq, q = 1, 2, . . . , k are

n×n symmetric positive semi-definite matrices; A is the m×n constraint fuzzymatrix of technological coefficients; b is them-dimensional fuzzy resource vector,n-dimensional fuzzy vectors l and u are lower and upper bounds respectivelyon n-dimensional decision vector x.

Remark 1.1. The problem (NFMOQPP) is a non-convex programmingproblem because Qq, q = 1, 2, . . . , k are assumed to be n×n symmetric positivesemi-definite matrices and our problem is maximization.

In the literature section, a greater part of the study focused on convexprogramming problem; as in convex programming local optima is global op-tima but in case of non-convex programming local optima may not be globaloptima. Orden [11] was the first one who considered the maximization of anon-concave quadratic programming. He derived the necessary and sufficientconditions for the maximization of non-concave quadratic programming. Ritter[12] extended Orden’s work and gave a different approach to find the maximiza-tion of non-concave quadratic programming. Murty and Kabadi [10] discussedthat non-convex problem is an N-P Hard problem and it is very hard to get theglobal maxima. They have conferred that descent algorithms are quite practicalalgorithms for dealing with non-convex nonlinear problem. Hanson [7] used gen-eralized invexity to derive the necessary and sufficient conditions in non-convexquadratic programming for global minima. Ye [16] applied the affine scalingalgorithm to find the optimal solution for the non-convex quadratic program-ming. Burer and Vandenbussche [3] has developed a finite branch and boundalgorithm for non-convex quadratic programming via semi definite relaxation.Chen and Burer [4] extended the work of Burer and Vandenbussche to get theglobal optima of non-convex problem with linear and bound constraints.

The purpose of this paper is to develop a computational procedure for moregeneral non-convex quadratic problem similar to that of Chen and Burer [4] andOrden [11]. They have focused on the single objective non-convex quadratic

A COMPUTATIONAL PROCEDURE FOR SOLVING... 513

programming problem. But here in this paper, we have concentrated on theproblem (NFMOQPP) in which instead of single objective we have takenmultiple objectives which are fuzzified during the procedure in order to obtainthe bounds on them; the crisp linear and bound constraints have been replacedby fuzzy linear and bound constraints.

In real world, the concept of decision-making takes place in an environmentin which the objectives and constraints are not known precisely. Such situa-tions can be tackled efficiently with the help of fuzzy set theory. Fuzzy sets werefirst introduced by Zadeh [17]. These sets were used to introduce the conceptof decision-making in a fuzzy environment by Bellman and Zadeh [2]. Theyhave defined appropriate aggregation of fuzzy sets for the fuzzy decision. Zim-mermann [18] has used fuzzy decision concept in the fuzzy linear programmingfor several objectives. Guu and Wu [5] extended the Zimmermann’s approachto two-phase approach for solving the multi-objective linear programming inthe fuzzy environment. Several authors have studied linear programming inthe fuzzy environment and applied to real world problems like transportation,production planning etc. A fuzzy decision is based on the intersection of mem-bership functions of the goals and constraints. Most of the authors have usedlinear membership functions as they are easy to tackle. Leberling [8] has de-fined the hyperbolic membership function. Li and Lee [9] have defined theexponential membership function and Yang, Ignizio and Kim [15] have definedpiecewise nonlinear membership function. In our paper, we have used trigono-metric membership function in terms of sin, for objective functions and someof the constraints and used LINGO 9.0 to find the fuzzy efficient solution of(NFMOQPP).

The paper is organized as follows: Section 2 consists of preliminaries, whichcontain a useful definition and some basic membership functions of fuzzy tech-nological coefficients and fuzzy resource vector; fuzzy efficient solution and non-linear membership functions are described in Section 3; in Section 4, we havediscussed the solution methodology for solving non-convex fuzzy multi-objectivequadratic programming problem; and illustrative numerical example to explainthe solution methodology is given in Section 5.

2. Preliminaries

In this section, we give a basic definition and membership functions correspond-ing to (NFMOQPP).

General form of Non-convex Multi-objective Quadratic Programming prob-


lem is given by

(NMOQPP) MaxZ1(x) = ct1x+1

2xtQ1x

MaxZ2(x) = ct2x+1

2xtQ2x

...

MaxZk(x) = ctkx+1

2xtQkx

subject to x ∈ X = {x ∈ Rn : Ax ≤ b, x ≥ 0},

where cq, q = 1, 2, . . . , k are n-dimensional cost vectors; Qq, q = 1, 2, . . . , kare n× n symmetric positive semi-definite matrices; A is the m× n constraintmatrix of technological coefficients; b is the m-dimensional resource vector andx is n-dimensional decision vector.

Definition 2.1. A point x∗ ∈ X is said to be Pareto optimal solution of(NMOQPP) if there does not exist any x ∈ X such that

Zq(x) ≥ Zq(x∗) ∀ q = 1, 2, . . . , k

and (1)

Zr(x) > Zr(x∗) for at least one r = 1, 2, . . . , k.

In the problem (NFMOQPP), A is the m× n constraint fuzzy matrix oftechnological coefficients; b is the m-dimensional fuzzy resource vector, l and u

are n-dimensional fuzzy vectors and x is n-dimensional decision vector. Theirmembership functions are given as below:

1. The membership function of b:

µb(y) =

1, y ≤ bi(bi + pi − y)

pi, bi ≤ y ≤ bi + pi

0, y ≥ bi + pi

(2)

where y ∈ R and pi > 0 is the tolerance level of bi for all i = 1, 2, . . . ,m.

2. The membership function of the fuzzy matrix A:

µA(y) =

1, y ≤ aijaij + dij − y

dij, aij ≤ y ≤ aij + dij

0, y ≥ aij + dij

(3)


where y ∈ R and dij > 0 is the tolerance level of aij for i = 1, 2, . . . ,mand j = 1, 2, . . . , n.

3. The membership function of u:

µu(y) =

1, y ≤ ujuj + tj − y

tj, uj ≤ y ≤ uj + tj

0, y ≥ uj + tj

(4)

where y ∈ R and tj > 0 is the tolerance level of uj for all j = 1, 2, . . . , n.

4. The membership function of l:

µl(y) =

1, lj ≤ yy − lj + rj

rj, lj − rj ≤ y ≤ lj

0, y ≤ lj − rj

(5)

where y ∈ R and rj > 0 is the tolerance level of lj for all j = 1, 2, . . . , n.

3. Fuzzy Efficient Solution

Werners [13, 14] has given a definition of fuzzy efficient solution for fuzzy multi-objective linear programming problem. As in our problem (NFMOQPP),the constraint set X is described exclusively by fuzzy constraints, the varyingdegree of feasibility should be taken into account by consideration of efficientsolutions, for additional dependencies between the individual goal values andthe degree of membership to the region of feasible solutions can arise. Thatmeans we want to emphasize not only the achievement of maximum value ofobjective functions, but also the highest membership grade of fuzzy constraintsin a flexible region. In order to take care of fuzzy concept, we will extend thedefinition of Werners to (NFMOQPP).

Let the constraint sets of (NFMOQPP) be denoted by

Ci(x) = {x ∈ Rn : Aix ≤ bi}, i = 1, 2, . . . ,m

Bj(x) = {x ∈ Rn : xj ≤ uj}, j = 1, 2, . . . , n

B′j(x) = {x ∈ Rn : xj ≥ lj}, j = 1, 2, . . . , n


Define the trigonometric membership function of ith constraint Ci(x) as:

µ(Ci(x)) =

0, bi ≤n∑

j=1aijxj

sin

bi−n∑

j=1

aijxj

n∑j=1

dijxj+pi

π

2

,

n∑j=1

aijxj≤bi≤n∑

j=1(aij + dij)xj+pi

1, bi ≥n∑

j=1(aij + dij)xj + pi

(6)

where x ∈ Rn and dij > 0 and pi > 0 are respectively the tolerance level of thetechnological coefficients and resources for i = 1, 2, . . . ,m and j = 1, 2, . . . , n.This trigonometric membership function has the following properties:

1. µ(Ci(x)) is a nonlinear and monotonically decreasing function.

2. 0 ≤ µ(Ci(x)) ≤ 1 ∀ x ∈ Rn.

3. µ(Ci(x)) is a concave function on the set

{x ∈ Rn :

n∑j=1

aijxj ≤ bi

}.

We define the linear membership function of the constraint Bj(x), j = 1, 2, . . . , nas:

µ(Bj(x)) =

1, xj ≤ uj(uj + tj − xj)

tj, uj ≤ xj ≤ uj + tj

0, xj ≥ uj + tj

(7)

where x ∈ Rn and tj > 0 is the tolerance level of uj.

Define the linear membership function of the constraint B′j(x), j = 1, 2, . . . , n

as:

µ(B′j(x)) =

1, lj ≤ xj(xj − lj+rj)

rj, lj − rj ≤ xj ≤ lj

0, lj − rj ≥ xj

(8)

where x ∈ Rn and rj > 0 is the tolerance level of lj .

Definition 3.1. The point x∗ ∈ X is said to be fuzzy efficient solution for


the (NFMOQPP) if there does not exist any x ∈ X such that

µ(Zq(x)) ≥ µ(Zq(x∗)) ∀ q = 1, 2, . . . , k and

µ(Ci(x)) ≥ µ(Ci(x∗)) ∀ i = 1, 2, . . . ,m and

µ(Bj(x)) ≥ µ(Bj(x∗)) ∀ j = 1, 2, . . . , n and

µ(B′j(x)) ≥ µ(B′

j(x∗)) ∀ j = 1, 2, . . . , n

and

µ(Zq(x)) > µ(Zq(x∗)) for at least one q = 1, 2, . . . , k or

µ(Ci(x)) > µ(Ci(x∗)) for at least one i = 1, 2, . . . ,m or

µ(Bj(x)) > µ(Bj(x∗)) for at least one j = 1, 2, . . . , n or

µ(B′j(x)) > µ(B′

j(x∗)) for at least one j = 1, 2, . . . , n

As mentioned in Werners [13, 14], similarly here, this definition takes intoaccount that for a fuzzy efficient solution an improvement concerning an objec-tive function can only be reached either at the expense of an additional objectivefunction or at the expense of the membership into the constraints. In fact, wecan easily see that the fuzzy efficiency defined above comprises the classicalefficiency as special case if each of the µ(Ci(x)), µ(Bj(x)) and µ(B′

j(x)) is 1 foreach x ∈ Ci(x), x ∈ Bj(x) and x ∈ B′

j(x), respectively.

4. Procedure to Find A Fuzzy Efficient Solution

Our objective functions are crisp and constraints are fuzzy in nature in(NFMOQPP). To defuzzificate the problem, we will first fuzzify the objectivefunctions, which can be done by solving the following four quadratic program-ming problems:

(NQPP1q) Z1

q = MaxZq(x) = ctqx+1

2xtQqx, q = 1, 2, . . . , k

subject to

n∑

j=1

aijxj ≤ bi, i = 1, 2, . . . ,m

xj ≤ uj , j = 1, 2, . . . , n

lj ≤ xj, j = 1, 2, . . . , n


x ≥ 0.

(NQPP2q) Z2


2xtQqx, q = 1, 2, . . . , k

subject ton∑

j=1

(aij + dij)xj ≤ bi, i = 1, 2, . . . ,m

xj ≤ uj, j = 1, 2, . . . , n

lj − rj ≤ xj , j = 1, 2, . . . , n

x ≥ 0.

(NQPP3q) Z3


2xtQqx, q = 1, 2, . . . , k

subject to

n∑

j=1

(aij + dij)xj ≤ bi + pi, i = 1, 2, . . . ,m

xj ≤ uj + tj, j = 1, 2, . . . , n

lj − rj ≤ xj, j = 1, 2, . . . , n

x ≥ 0.

(NQPP4q) Z4


2xtQqx, q = 1, 2, . . . , k

subject to

n∑

j=1

aijxj ≤ bi + pi, i = 1, 2, . . . ,m

xj ≤ uj + tj, j = 1, 2, . . . , n

lj ≤ xj , j = 1, 2, . . . , n

x ≥ 0.

We have now four non-convex quadratic problems corresponding to qth (q =1, 2, . . . , k) objective function. They can be solved by using LINGO 9.0 to getthe aspiration level for the qth objective function.

Now we take the best and worst value of the optimal solutions of the fournon-convex quadratic programming problems.

Let

ZLq = Min(Z1

q , Z2q , Z

3q , Z

4q )

and q = 1, 2, . . . , k. (9)

ZUq = Max(Z1

q , Z2q , Z

3q , Z

4q ),


Now taking the interval [ZLq , Z

Uq ] as the tolerance level interval for the qth

objective (q = 1, 2, . . . , k) in (NFMOQPP), we define the trigonometric mem-bership function of the qth objective (q = 1, 2, . . . , k) as:

µ(Zq(x)) =

0, Zq(x) ≤ ZLq

sin

[(Zq(x)− ZL

q

ZUq − ZL

q

)π

2

]ZLq ≤ Zq(x) ≤ ZU

q

1, Zq(x) ≥ ZUq

(10)

This trigonometric membership function has the following properties:

1. µ(Zq(x)) is a nonlinear and monotonically increasing function.

2. 0 ≤ µ(Zq(x)) ≤ 1 in [ZLq , Z

Uq ].

3. µ(Zq(x)) is concave function in [ZLq ,∞).

Now, by using max-min fuzzy decision making approach given by Bellmanand Zadeh [2], we have

µD(x) = Minq,i,j

(µ(Zq(x)), µ(Ci(x)), µ(Bj(x)), µ(B′j(x))) (11)

Then, the optimal fuzzy decision is a solution of the problem

Maxx≥0

µD(x) = Maxx≥0

(Minq,i,j

(µ(Zq(x)), µ(Ci(x)), µ(Bj(x)), µ(B′j(x)))). (12)

The problem (12) is equivalent to the following problem as discussed byZimmermann [18]

(NPP) Max λ

subject to µ(Zq(x)) ≥ λ, q = 1, 2, . . . , k

µ(Ci(x)) ≥ λ, i = 1, 2, . . . ,m

µ(Bj(x)) ≥ λ, j = 1, 2, . . . , n

µ(B′j(x)) ≥ λ, j = 1, 2, . . . , n

0 ≤ λ ≤ 1,

x ≥ 0.

Now, the above problem is a nonlinear convex programming problem whichcan be solved by LINGO 9.0. Let (x∗, λ∗) be the optimal solution of the problem(NPP).


Now, by two phase method proposed by Guu and Wu [5], we will solve thefollowing nonlinear problem:

(NPP1) Max

k∑

q=1

λq +

m∑

i=1

δi +

n∑

j=1

θj +

n∑

j=1

γj

subject to µ(Zq(x)) ≥ λq ≥ λ∗, q = 1, 2, . . . , k

µ(Ci(x)) ≥ δi ≥ λ∗, i = 1, 2, . . . ,m

µ(Bj(x)) ≥ θj ≥ λ∗, j = 1, 2, . . . , n

µ(B′j(x)) ≥ γj ≥ λ∗, j = 1, 2, . . . , n

λ∗ ≤ λq, δi, θi, γi ≤ 1,

x ≥ 0.

We solve the above problem by LINGO 9.0. Let (x◦, λ◦, δ◦, θ◦, γ◦) where λ◦ =(λ◦

1, λ◦2, . . . , λ

◦k), δ

◦ = (δ◦1 , δ◦2 , . . . , δ

◦m), θ◦ = (θ◦1, θ

◦2, . . . , θ

◦n) and γ◦ = (γ◦1 , γ

◦2 , . . . ,

γ◦n) be the optimal solution of (NPP1). Now we will prove that this optimalsolution (x◦, λ◦, δ◦, θ◦, γ◦) gives the fuzzy efficient solution of (NFMOQPP).

Theorem 4.1. Let (x◦, λ◦, δ◦, θ◦, γ◦) where λ◦ = (λ◦1, λ

◦2, . . . , λ

◦k), δ

◦ =(δ◦1 , δ

◦2 , . . . , δ

◦m), θ◦ = (θ◦1, θ

◦2, . . . , θ

◦n) and γ◦ = (γ◦1 , γ

◦2 , . . . , γ

◦n) be the optimal

solution of the problem (NPP1), then x◦ is the fuzzy efficient solution of the

problem (NFMOQPP).

Proof. Let if possible, x◦ be not a fuzzy efficient solution of (NFMOQPP),then there exist a y ∈ X such that

µ(Zq(y)) ≥ µ(Zq(x∗)) ∀ q = 1, 2, . . . , k and

µ(Ci(y)) ≥ µ(Ci(x∗)) ∀ i = 1, 2, . . . ,m and

µ(Bj(y)) ≥ µ(Bj(x∗)) ∀ j = 1, 2, . . . , n and

µ(B′j(y)) ≥ µ(B′

j(x∗)) ∀ j = 1, 2, . . . , n

and (13)

µ(Zq(y)) > µ(Zq(x∗)) for at least one q = 1, 2, . . . , k or

µ(Ci(y)) > µ(Ci(x∗)) for at least one i = 1, 2, . . . ,m or

µ(Bj(y)) > µ(Bj(x∗)) for at least one j = 1, 2, . . . , n or

µ(B′j(y)) > µ(B′

j(x∗)) for at least one j = 1, 2, . . . , n

As µ(Zq(.)) is the increasing function

Zq(y) ≥ Zq(x◦) ⇒ µ(Zq(y)) ≥ µ(Zq(x

◦)) for all q = 1, 2, . . . , k.


Since (x◦, λ◦, δ◦, θ◦, γ◦) is the optimal solution and the coefficients in theobjective function are positive in problem (NPP1)

So, we have

λ◦q = µ(Zq(x

◦)) q = 1, 2, . . . , k

δ◦i = µ(Ci(x◦)) i = 1, 2, . . . ,m

θ◦j = µ(Bj(x◦)) j = 1, 2, . . . , n

γ◦j = µ(B′j(x

◦)) j = 1, 2, . . . , n

Now choosing,

λyq = µ(Zq(y)) q = 1, 2, . . . , k

δyi = µ(Ci(y)) i = 1, 2, . . . ,m

θyj = µ(Bj(y)) j = 1, 2, . . . , n

γyj = µ(B′

j(y)) j = 1, 2, . . . , n

As (x◦, λ◦, δ◦, θ◦, γ◦) is the feasible solution of (NPP1) so using inequalities(13) and property of µ(Zq(x)), we get

µ(Zq(y)) ≥ µ(Zq(x◦)) ≥ λq ≥ λ∗, q = 1, 2, . . . , k,

µ(Ci(y)) ≥ µ(Ci(x◦)) ≥ δi ≥ λ∗, i = 1, 2, . . . ,m,

µ(Bj(y)) ≥ µ(Bj(x◦)) ≥ θj ≥ λ∗ i = 1, 2 . . . ,m,

µ(B′j(y)) ≥ µ(B′

j(x◦)) ≥ γj ≥ λ∗ j = 1, 2, . . . , n

So, we can easily see that (y, λy, δy , θy, γy), where λy = (λy1, λ

y2, . . . , λ

yk), δ

y =(δy1 , δ

y2 , . . . , δ

ym), θy = (θy1 , θ

y2 , . . . , θ

yn) and γy = (γy1 , γ

y2 , . . . , γ

yn) is the feasible

solution of (NPP1).As (y, λy, δy , θy, γy) and (x◦, λ◦, δ◦, θ◦, λ◦) are solutions of the problem

(NPP1) and using (13), we get

k∑

q=1

λyq +

m∑

i=1

δyi +

n∑

j=1

θyj +

n∑

j=1

γyj

=

k∑

q=1

µ(Zq(y)) +

m∑

i=1

µ(Ci(y)) +

n∑

j=1

µ(Bj(y)) +

n∑

j=1

µ(B′j(y))

>

k∑

q=1

µ(Zq(x◦)) +

m∑

i=1

µ(Ci(x◦)) +

n∑

j=1

µ(Bj(x◦)) +

n∑

j=1

µ(B′j(x

◦))


=

k∑

q=1

λ◦q +

m∑

i=1

δ◦i +

n∑

j=1

θ◦j +

n∑

j=1

γ◦j

This is a contradiction as (x◦, λ◦, δ◦, θ◦, λ◦) is optimal solution of the prob-lem (NPP1).

So x◦ is the fuzzy efficient solution of (NFMOQPP).

4.1. Computational Procedure

The ideas discussed above for finding fuzzy efficient solution of (NFMOQPP)can be summarized in the form of algorithm as given below.

Step 1: Construct the four problems of the form (NQPPjq) (j = 1, 2, 3, 4)

corresponding to qth (q = 1, 2, . . . , k) objective.

Step 2: Solve them to get the aspiration level for the qth (q = 1, 2, . . . , k)objective.

Step 3: Determine the aspiration level of the qth (q = 1, 2, . . . , k) objectiveby Zu

q = Max(Z1q , Z

2q , Z

3q , Z

4q ) and Z l

q = Min(Z1q , Z

2q , Z

3q , Z

4q ) ∀ q =

1, 2, . . . , k.

Step 4: Construct the trigonometric membership functions of the form (6) and(10) for the constraints and the objectives respectively. Also linearmembership functions of the form (7) and (8) respectively for upperand lower values of x.

Step 5: Construct the problem (NPP) and solve it. Let (x∗, λ∗) be the solutionand λ∗ be the optimal value.

Step 6: Construct the problem (NPP1) and solve it. Let (x◦, λ◦, δ◦, θ◦, γ◦)where λ◦ = (λ◦

1, λ◦2, . . . , λ

◦k), δ

◦ = (δ◦1 , δ◦2 , . . . , δ

◦m), θ◦ = (θ◦1, θ

◦2, . . . , θ

◦n)

and γ◦ = (γ◦1 , γ◦2 , . . . , γ

◦n) be the optimal solution. Then x◦ is the fuzzy

efficient solution of problem (NFMOQPP) by Theorem 4.1.

5. Illustrative Example

(NFMOQPP) MaxZ1(x) = x1 + 2x2 + x21 + 2x22


MaxZ2(x) = 4x1 + 7x2 + 2x21 + 3x22

subject to 1x1 + 1x2 ≤ 10,

2x1 + 3x2 ≤ 25,

2 ≤ x1 ≤ 9,

2 ≤ x2 ≤ 8,

x1, x2 ≥ 0.

Here, A = (aij) =

(1 12 3

), (dij) =

(1 11 2

), (aij + dij) =

(2 23 5

),

b =

(1025

), p =

(510

), (b+ p) =

(1535

), u =

(98

), t =

(32

),

u+ t =

(1210

), l =

(22

), r =

(11

), l − r =

(11

)

5.1. Computational Procedure

Step 1: Construction of the four problems of the form (NQPPjq) (j = 1, 2, 3, 4)

corresponding to qth (q = 1, 2) objective as in Table 1.

Step 2: Now by solving the above quadratic programming problems usingLINGO9.0, we get

Z1 = (Z11 , Z

21 , Z

31 , Z

41 ) = (118, 42, 96.72 and 228.75)

Z2 = (Z12 , Z

22 , Z

32 , Z

42 ) = (212, 82, 173.68 and 392.5)

Step 3: Now using (9) for q = 1, 2, we get the interval [ZLq , Z

Uq ] as the aspira-

tion level interval for the qth objective (q = 1, 2) in (NFMOQPP)as

[ZL1 , Z

U1 ] = [42, 228.75] and [ZL

2 , ZU2 ] = [82, 392.5]

Step 4: Construction of the membership functions of the form (6) to (8) and(10) for the constraints and objectives, as follows:

µ(C1(x)) =

0, 10 ≤ x1 + x2

sin

[(10 − x1 − x2

x1 + x2 + 5

)π

2

], x1 + x2 ≤ 10 ≤ 2x1 + 2x2 + 5

1, 10 ≥ 2x1 + 2x2 + 5


(NQPP11)

Z11 = Max x1+2x2+x21+2x22

subject to x1 + x2 ≤ 102x1 + 3x2 ≤ 252 ≤ x1 ≤ 92 ≤ x2 ≤ 8x1, x2 ≥ 0

(NQPP12)

Z12 = Max 4x1+7x2+2x21+3x22


(NQPP21)

Z21 = Max x1+2x2+x21+2x22

subject to 2x1 + 2x2 ≤ 103x1 + 5x2 ≤ 251 ≤ x1 ≤ 91 ≤ x2 ≤ 8x1, x2 ≥ 0

(NQPP22)

Z22 = Max 4x1+7x2+2x21+3x22


(NQPP31)

Z31 = Max x1+2x2+x21+2x22


(NQPP32)

Z32 = Max 4x1+7x2+2x21+3x22


(NQPP41)

Z41 = Max x1+2x2+x21+2x22


(NQPP42)

Z42 = Max 4x1+7x2+2x21+3x22


Table 1

µ(C2(x)) =

0, 25 ≤ 2x1 + 3x2

sin

[(25 − 2x1 − 3x2

x1 + 2x2 + 10

)π

2

]2x1 + 3x2 ≤ 25 ≤ 3x1 + 5x2 + 10

1, 25 ≥ 3x1 + 5x2 + 10


µ(B1(x)) =

1, x1 ≤ 912− x1

39 ≤ x1 ≤ 12

0 x1 ≥ 12

µ(B2(x)) =

1, x2 ≤ 810− x2

28 ≤ x2 ≤ 10

0 x2 ≥ 10

µ(B′1(x)) =

0, x1 ≤ 1(x1 − 1)

11 ≤ x1 ≤ 2

1, x1 ≥ 2

µ(B′2(x)) =

0, x2 ≤ 1(x2 − 1)

11 ≤ x2 ≤ 2

1, x2 ≥ 2

µ(Z1(x)) =

0, Z1(x) ≤ 42

sin

[(Z1(x)− 42

186.75

)π

2

]42 ≤ Z1(x) ≤ 228.75

1, Z1(x) ≥ 228.75

µ(Z2(x)) =

0, Z2(x) ≤ 82

sin

[(Z2(x)− 82

310.5

)π

2

]82 ≤ Z2(x) ≤ 392.5

1, Z2(x) ≥ 392.5

From the graph drawn in Figure 1 corresponding to the membershipfunction µ(C1(x)) we can observe that

(i) µ(C1(x)) is a nonlinear and monotonically decreasing function.

(ii) 0 ≤ µ(C1(x)) ≤ 1 ∀ x ∈ Rn.

(iii) µ(C1(x)) is a concave function on the set {x ∈ Rn : x1+x2 ≤ 10}.

Also from the graph drawn in Figure 2 corresponding to the member-ship function µ(Z1(x)) we can observe that

(i) µ(Z1(x)) is a nonlinear and monotonic increasing function.

(ii) 0 ≤ µ(Z1(x)) ≤ 1 ∀ x ∈ Rn.


(iii) µ(Z1(x)) is concave function on the interval [42,∞).

Similar properties can be observed for all other membership functionswith the help of the graphs.

Step 5: Now the problem (NPP) becomes

(NPP) Max λ

subject to sin

[((x1 + 2x2 + x21 + 2x22 − 42)

186.75

)π

2

]≥ λ,

sin

[((4x1 + 7x2 + 2x21 + 3x22 − 82

)

310.5

)π

2

]≥ λ

sin

[(10− x1 − x2

x1 + x2 + 5

)π

2

]≥ λ,

sin

[(25− 2x1 − 3x2x1 + 2x2 + 10

)π

2

]≥ λ

(12− x1

3

)≥ λ,

(10− x2

2

)≥ λ,

(x1 − 1) ≥ λ

(x2 − 1) ≥ λ

0 ≤ λ ≤ 1,

x1, x2 ≥ 0.

We solve the above nonlinear problem by LINGO 9.0. We get theoptimal solution as (λ∗ = 0.3310024, x∗1 = 1.331002, x∗2 = 5.804374).

Step 6: Now construct the problem (NPP1) as follows:

(NPP1) Max λ1 + λ2 + δ1 + δ2 + θ1 + θ2 + γ1 + γ2

subject to sin

[((x1 + 2x2 + x21 + 2x22 − 42)

186.75

)π

2

]≥ λ1 ≥ 0.3310024,

sin

[((4x1 + 7x2 + 2x21 + 3x22 − 82)

310.5

)π

2

]≥ λ2 ≥ 0.3310024,

sin

[(10− x1 − x2

x1 + x2 + 5

)π

2

]≥ δ1 ≥ 0.3310024,


sin

[(25− 2x1 − 3x2x1 + 2x2 + 10

)π

2

]≥ δ2 ≥ 0.3310024,

(12− x1

3

)≥ θ1 ≥ 0.3310024,

(10− x2

2

)≥ θ2 ≥ 0.3310024,

(x1 − 1) ≥ γ1 ≥ 0.3310024,

(x2 − 1) ≥ γ2 ≥ 0.3310024,

0.3310024 ≤ λ1, λ2, δ1, δ2, θ1, θ2, γ1, γ2 ≤ 1,

x1, x2 ≥ 0.

We solve the above problem (NPP1) by LINGO 9.0 and get the op-timal values as (λ◦

1 = 0.3310024, λ◦2 = 0.3401065, δ◦1 = 0.362496,

δ◦2 = 0.3310024, θ◦1 = 1, θ◦2 = 1, γ◦1 = 0.3310024, γ◦2 = 1, x◦1 =1.331002, x◦2 = 5.804373). Hence the fuzzy efficient solution of theproblem (NFMOQPP) is x◦1 = 1.3310024, and x◦2 = 5.804373. Thecorresponding values of the objectives are Z1(x

◦) = 82.092806 andZ2(x

◦) = 150.569993.

Acknowledgments

We are indebted for very helpful discussion to Prof. Davinder Bhatia (Rtd.),Department of Operational Research, Faculty of Mathematical Sciences, Uni-versity of Delhi, Delhi 110007, India. The second author is thankful to theCouncil of Scientific and Industrial Research (CSIR), New Delhi, India, for thefinancial support (09/045(1153)/2012-EMR-I).

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531

Figure 1: Graph of µ(C1(x))

Figure 2: Graph of µ(Z1(x))

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