International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 166

ISSN 2229-5518

Optimization methodology based on neural

networks and reference point algorithm applied to fuzzy multiobjective optimization problems

a Department of Mathematics and Statistics, Faculty of sciences, Taif University, Taif, El-Haweiah, P.O. Box 888, Zip Code 21974, Saudi Arabia.

b Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Egypt

Abstract— Artificial neural networks are massively paralleled distributed computation and fast convergence and can be considered as an efficient method to solve real-time optimization problems. In this paper, we propose reference point based neural network algorithm for solving fuzzy multiobjective optimization problems MOOP. The target is to identify the Pareto-optimal region closest to the reference points. Our approach has two characteristic features. Firstly, fuzzy multiobjective optimization problem (F-MOOP) has been transformed to crisp multiobjective optimization problem (C-MOOP) by means of Alpha-cut. Secondly a neural networks based reference point algorithm is implemented to solve C-MOOP in such a way that they integrate the decision maker DM early in the optimization process instead of leaving him/her alone with the final choice of one solution among the whole Pareto optimal set. Such procedures will provide the DM with a set of solutions near her/his preference so that a better and a more reliable decision can be made. Simulation runs on engineering application problems demonstrate their usefulness in practice and show another use of a neural network methodology in allowing the DM to solve multiobjective optimization problems better and with more conﬁdence.

Index Terms— Neural network; Reference point; Fuzzy numbers.

—————————— ——————————

ulti-objective optimization is the process of simultane- ously optimizing two or more conflicting objectives subject to certain constraints. In many real world prob-

lems, there are situations where multiple objectives may be more appropriate rather than considering single objective. However, in such cases emphasis is on efficient solutions, which are optimal in a certain multiobjective sense[1-11]. The classical interactive multiobjective optimization methods de- mand the decision-makers to suggest a reference direction or reference points or other clues which result in a preferred set of solutions on the Pareto-optimal front. In these classical ap- proaches, based on such clues, a single objective optimization problem is usually formed and a single solution is found. A single solution does not provide a good idea of the properties of solutions near the desired region of the front. By providing a clue, the DM is not usually looking for a single solution, ra ther she/he is interested in knowing the properties of solu- tions which correspond to the optimum and near-optimum solutions respecting the clue[1,12]. We here argue that instead of ﬁnding a single solution near the region of interest, if a number of solutions in the region of interest are found, the

————————————————

• **A. A. Mousa ***is currently at Department of Mathematics and Statistics, Faculty of sciences, Taif University, Taif, El-Haweiah, P.O. Box 888, Zip Code 21974, Saudi Arabia*

• **B. N. AL-Matrafi ***is currently at Department of Mathematics and Statis-*

tics, Faculty of sciences, Taif University, Taif, El-Haweiah, P.O. Box 888,

Zip Code 21974, Saudi Arabia.

decision-maker will be able to make a better and more reliable decision. Moreover, if multiple such regions of interest can be found simultaneously, decision-makers can make a more ef- fective and parallel search towards ﬁnding an ultimate pre- ferred solution.

The classical reference point approaches will find a solu- tion depending on the chosen weight vector and is therefore subjective. Moreover, the single solution is specific to the cho- sen weight vector and does not provide any information about how the solution would change with a slight change in the weight vector. To find a solution for another weight vector, a new achievement scalarizing problem needs to be formed again and solved. Moreover, despite some modifications [1], the reference point approach works with only one reference point at a time. However, the decision maker may be interest- ed in exploring the preferred regions of Pareto-optimality for multiple reference points simultaneously. In the context of finding a preferred set of solutions, instead of the entire Pare- to-optimal set, quite a few studies have been made in the past. The approach by Deb [13] was motivated by the goal pro- gramming idea, and required the DM to specify a goal or an aspiration level for each objective. Based on that information, Deb modified his NSGA approach to find a set of solutions which are closest to the supplied goal point, if the goal point is an infeasible solution and find the solutions which correspond to the supplied goal objective vector, if it is a feasible one. The method did not care finding the Pareto optimal solutions cor- responding to the multiobjective optimization problem, rather attempted to find solutions satisfying the supplied goals.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 167

ISSN 2229-5518

5) strictly decreasing on [*a*3 , *a*4 ] ;

Recently, neural networks (NNs) have become widely used

tools in many fields such as decision support tool, pattern

recognition and secure communication [14-18]. Neural Net-

works is well-known as one of powerful computing systems to

6) µa (*a*) = 0 for all *a *∈[*a*4 , +∞);

Assume that a in the FM-RAP are fuzzy numbers whose

membership functions are µa (*a*) .

solve complex optimization problems. Due to the massive

numbers a is defined as the ordinary set

La (a) for which the

computing unit-neurons and parallel mechanism of neural

network, large-scale optimization problem can be solved effi-

ciently. Many neural network for solving constraint optimiza-

tion problems can be found in [19-21]. With the above princi-

ples of reference point approaches and difficulties with the

degree of their membership functions exceeds the lev- el a ∈ [0,1] :

*L*a (*a*) = {*a *| µa (*a*) ≥ a}.

For a certain degree α, the (FM-RAP) can be represented as a nonfuzzy a -VMP as follows:

classical methods, we propose a reference point based on neu-

Min

{ f1 ( X , a), f2 ( X , a),....., fm ( X , a)}

ral networks, by which a set of Pareto-optimal solutions near a

supplied set of reference points will be found, thereby elimi-

nating the need of any weight vector and the need of applying

the method again and again. Instead of finding a single solu-

subject to g ( X , a) ≤ 0

X = ( x1 , x2 ,...xn ), a = (a1 , a2 ,...., an )

La i ≤ ai ≤ Ua i

tion corresponding to a particular weight vector, the proposed

Where constraint

La i ≤ ai

≤ Ua i gives the lower and upper

procedure will attempt to a find a set of solutions in the neighborhood of the corresponding Pareto-optimal solution,

a

bound for the parameters i

x* ∈ X is said to be

so that the DM can have a better idea of the region rather than

an α–Pareto optimal solution to the (α-VMP), if and only if

a single solution.

there does not exist another x ∈ X ,

a ∈ La (a) such that

In this paper, an attempt is made to solve Fuzzy Multi-

objective optimization with fuzzy parameters. Based on Al-

pha concept [22,23], F-MOOP can be transformed to crisp mul- tiobjective optimization problem (C-MOOP) at certain degree of α (α-cut level). Also, we combine one such preference-based strategy with a neural network methodology and demonstrate

how, instead of one solution, a preferred set of solutions near the reference points can be found parallel. Such procedures will provide the decision-maker with a set of solutions near her/his preference so that a better and a more reliable decision can be made.

Detailed A Multi-objective Optimization Problem (MOP) can

f ( x, a) ≥ f ( x* , a* ), i = 1, 2,.., k , with strictly inequality holding for at least one i, where the corresponding values of parameters a* are called α-level optimal parameters.

For papers In this section, a framework for the proposed ap- proach that involves two phases was presented. The first one transforms the fuzzy multiobjective optimization problem (F- MOOP) to the crisp multiobjective optimization (C-MOOP) by means of Alpha-cut, while the other phase employs a refer- ence point based on neural networks algorithm to solve the crisp optimization problem.

be defined as determining a vector of variables within a feasi-

ble region to minimize a vector of objective functions that

Min

{ f1 ( X , a), f2 ( X , a),....., fm

( *X *, *a*)}

usually conflict with each other. The following fuzzy vector minimization problem (FVMP) involving fuzzy parameters in

where f1 ( *X *, *a*)

subject to g ( X , a) ≤ 0

is the ith objective function; and g( *X *, *a*) is con-

the objective functions and constraints such a problem takes

straint vector, X is vector of decision variables; and

the form:

Min

{f1 ( *X *, *a*), *f*2 ( *X *, *a*),....., *f*m ( *X *, *a*)}

*a* = (*a*1 , *a*2 ,....*a*n )

problem

represented a vector of fuzzy parameters in the

where f1 ( *X *, *a*)

subject to g( X , a) ≤ 0

is the ith objective function; and g( *X *, *a*) is con-

into crisp multiobjective optimization problem using Alpha-

Level cut.

straint vector, X is vector of decision variables; and

Min

{ f1 ( X , a), f2 ( X , a),....., fm ( X , a)}

*a* = (*a*1 , *a*2 ,....*a*n ) represented a vector of fuzzy parameters in the

problem. Fuzzy parameters are assumed to be characterized as

subject to g ( X , a) ≤ 0

X = ( x , x ,...x ), a = (a , a ,...., a )

1 2 *n*

1 2 *n*

the fuzzy numbers. The real fuzzy numbers a form a convex

L ≤ a ≤ U

continuous fuzzy subset of the real line whose membership

a i i a i

function µa (*a*) is defined by:

1) a continuous mapping from

[0,1];

R1 to the closed interval

2) µa (*a*) = 0 for all *a *∈ (−∞, *a*1 ];

3) strictly increasing on [*a*1 , *a*2 ] ;

4) µa (*a*) = 1 for all *a *∈[*a*2 , *a*3 ] ;

information must given to the DM, the DM suggest preferred reference point, the reference point is a feasible or infeasible point in the objective space. When decision making is empha-

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 168

ISSN 2229-5518

sized, the objective of solving a multi-objective optimization problem is referred to supporting a decision maker in finding

max >*L*( *x*, λ, µ ) , *x *∈ *R*n

*x *, λ , µ

the most preferred Pareto optimal solution according to his/her subjective preferences. The underlying assumption is that one solution to the problem must be identified to be im-

DNLP s.t

∇x *L*( *x*, λ, µ ) = 0

λ ≥ 0

T T

plemented in practice. Here, a human decision maker (DM)

Where

λ = (λ1 , λ2 ,..., λm )

, µ = (µ1 , µ2 ,..., µ p )

plays an important role. The DM is expected to be an expert in the problem domain. The reference point is used to derive achievement scalarizing functions as follows:

L( x, λ, µ ) = f ( x) −

m

∑

i =1

λi *g*i ( *x*) −

m

p

∑

j =1

µ j *h*j ( *x*) ≡ *L*( *z*),

p

Given a reference point *z *for an M-objective optimization

∇x *L*( *x*, λ, µ ) = ∇*f *( *x*) − ∑ λi∇*g*i ( *x*) − ∑ µ j ∇*h*j ( *x*)

i =1

j =1

problem of minimizing

f1 ( X ), f2 ( X ),....., fm ( X ) with X ∈ S

S, the**III- **Parameter Initialization, Let *t*=0. Arbitrary choose initial

following single-objective optimization problem is solved for

this purpose:

vector *x*(*t *) ∈ R n , λ (t) ∈ R m , µ (t) ∈ R p , ∆*t *> 0 (∆t = 0.0001)

error ε = 10−9 .

and

Minimize

m

∑ w i ( *f*i ( *x*) − *z*i )

i=1

1/ *p p *

*u*(*t *) = ∇ *E *( *z*) = λT *g *( *x*).*g *( *x*)T λ + ∇*g *( *x*)T [ *g *( *x*) − *g *( *x*)]

subject to X ∈ S

If p = 1, the sum of weighted deviations is minimized (and the

+∇2 *L*( *z*)∇ *L*( *z*) + *A*T ( *Ax *− *b*)

*v*(*t *) = ∇ *E*( *z*) = λT *g *( *x*).*g *( *x*) − ∇*g *( *x*)∇ *L*( *z*) + [λ − λ ]

problem to be solved is equal to the weighting method except

a constant). If p = 2, we have a method of least squares. The

proposed reference point approach discussed above, will find

a solution depending on the chosen weight vector and is

therefore subjective. Moreover, the single solution is specific to

λ

w(t ) = ∇µ E ( z) = − A∇ x L( z)

x(t + ∆t ) = x(t ) − ∆t.u(t ) ,

µ (*t *+ ∆*t *) = µ (*t *) − ∆*t*.*w*(*t *)

x

λ (*t *+ ∆*t *) = λ (*t *) − ∆*t*.*v*(*t *) ,

the chosen weight vector. To find a solution for another

n m

s = u 2 t r =

v2 t q =

p

w2 t

weight vector, a new achievement scalarizing problem needs

to be formed again and solved. To make the procedure inter-

active and useful in practice, Wierzbicki **[24] **suggested a pro-

∑

i =1

i ( ),

∑

j =1

j ( ),

∑

j =1

j ( ).

cedure in which the obtained solution z′ is used to create M

if *s *< ε , *r *< ε

and q < ε , then output*x*(*t *+ ∆*t *), λ (*t *+ ∆*t *), µ (*t *+ ∆*t *)

new reference points, as follows: __ __

z( j ) = z + ( z′ − z ).e( j ) ,

and draw the point

and go to step **IV**.

x1 (t + ∆t ), x2 (t + ∆t )

otherwise let t = t + ∆t

where e( *j *) is the j-th coordinate direction vector.

**Step3: **This step is a neural network phase [25] for solving convex nonlinear programming which formulated in the pre- vious step. The distinguishing features of the proposed net- work are that the primal and dual problems can be solved simultaneously. The interested reader is referred to [25] where, all necessary and sufficient optimality conditions are incorporated, and no penalty parameter is involved. Also, based on Lyapunov, LaSalle and set stability theories, Chen K. z. [25] prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the set of its equilibrium points, regardless of whether a convex nonlinear programming problem has unique or infinitely many optimal solutions.

CNLP

Min f ( x) , x ∈ Rn

s.t gi ( *x*) ≥ 0, *i *= 1, 2,....., *m*

h = aT x − b , j = 1, 2,...., p( p < n)

New Pareto optimal solutions are then found by forming

new achievement scalarizing problems. If the decision-maker

is not satisfied with any of these Pareto-optimal solutions, a

new reference point is suggested and the above procedure is

repeated. It is interesting to note that the reference point may

be a feasible one or an infeasible point. If a reference point is

feasible and is not a Pareto-optimal solution, the decision-

maker may then be interested in knowing solutions which are

Pareto-optimal and close to the reference point. On the other hand, if the reference point is an infeasible one, the decision- maker would be interested in finding Pareto-optimal solutions which are close to the supplied reference point.

A case study of engineering application (The Environmen- tal/Economic Dispatch EED multiobjective problem) , was carried out to verify the feasibility and efficiency of the pro- posed approach. EED seeks to simultaneously minimize both fuel cost and the emissions produced by power plants. Envi- ronmental concerns on the effect of SO2 and NOX emissions produced by the fossil-fueled power plants led to the inclusion of minimization of emissions as an objective in the OPF formu-

where

f ( x) and gi ( x) , (i=1,2,…, m ) are convex functions.

lation. The economic emission load dispatch involves the sim-

ultaneous optimization of fuel cost and emission objec-

tives[4,5,8]. The deterministic problem is formulated as de-

scribed below.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 169

ISSN 2229-5518

n

1 ∑ *i i Gi i Gi*

The proposed approach has been applied to the standard IEEE

30-bus 6-generator test system. The single-line diagram of this

Min f ( x) =

i =1

n

(*a *+ *b P*

+ *c P*2 ) $ / *hr*

system is shown in figure **1 **and the detailed data are given in

2 ∑ *i i Gi i Gi i i Gi*

[28-30].

Min f (⋅) =

i =1

[10−2 (a + b *P*

n

+ γ *P*2 ) + x exp(λ *P *)] *ton */ *hr*

Where

s.t.

∑ PGi − PD − PLoss = 0,

i =1

PGi min ≤ PGi ≤ PGi max QGi min ≤ QGi ≤ QGi max Vi min ≤ Vi ≤ Vi max

max

i = 1,......, n i = 1,......, n i = 1,......, n

= 1,...., *n*Line ,

Cost a 10 10 20 10 20 10

b 200 150 180 100 180 150

c 100 120 40 60 40 100

a 4.091 2.543 4.258 5.426 4.258 6.131

Emission b -5.554 -6.047 -5.094 -3.550 -5.094 -5.555

γ 6.490 4.638 4.586 3.380 4.586 5.151

ζ 2.0E-4 5.0E-4 1.0E-6 2.0E-3 1.0E-6 1.0E-5

f1 ( x) is The classical economic dispatch problem of finding the

optimal combination of power generation, which minimizes

the total fuel cost while satisfying the total required demand

f2 (⋅) is The emission function can be presented as the sum of

all types of emission considered, such as NOx , SO2 , thermal emission, etc., with suitable pricing or weighting on each pol- lutant emitted.

C: total fuel cost ($/hr), Ci : is fuel cost of generator i

P : power generated (p.u)by generator i,

a i ,bi ,ci : fuel cost coefficients of generator i,

n: number of generator.

Constraints: The optimization problem is bounded by the fol- lowing constraints:

Pnower balance constraint. The total power generated must s∑upPpGliy−thPeD t−otPaLloslsoa=d0demand and the transmission losses.

i =1

λ 2.857 3.333 8.000 2.000 8.000 6.667

Where

PD : total load demand (p.u.), and

Ploss : transmission

losses (p.u.).

The transmission losses are given by[27]:

n n

PLoss = ∑∑[Aij (Pi Pj + Qi Q j ) + B ij (Qi Pj − Pi Q j ]

i =1 i =1

Where Pi = PGi − PDi , Qi = QGi − QDi ,

Fig. 1: Single line diagram of IEEE 30-bus 6-generator test system The values of fuel cost and emission coefficients are given in Table 1. Naturally, these data (cost and emission) involve many controlled parameters whose possible values are vague and uncertain. Consequently each numerical value in the do-

Aij =

Rij

ViV j

cos(δi − δ j ), Bij =

Rij

ViV j

sin(δi − δ j )

main can be assigned a specific "grade of membership" where

0 represents the smallest possible grade of membership, and 1

is the largest possible grade of membership. Thus fuzzy pa-

n : number of buses

R ij : series resistance connecting buses *i *and *j*

V i : voltage magnitude at bus i

δi : voltage angle at bus i

Pi : real power injection at bus i

Qi : reactive power injec- tion at bus i

rameters can be represented by its membership grade ranging between 0 and 1.

The fuzzy numbers shown in figure 2 have been obtained from interviewing DMs or from observing the instabilities in the global market and rate of prices fluctuations. The idea is to transform a problem with these fuzzy parameters to a crisp

version using a -cut level. This membership function can re-

power generated

PGi

by each generator is constrained be-

write as follows:

tween its minimum and maximum limits, i.e.,

PGi min ≤ PGi ≤ PGi max ,

QGi min ≤ QGi ≤ QGi max ,

1,

a = aij

Vi min ≤ Vi ≤ Vi max ,

i = 1,......, n

20*a *− 19 0.95*a*

jk ≤ *a *≤ *a*ij

where

PGi min : minimum power generated, and

PGi max : maxi-

*a*ij

mum power generated.

µ (*a*ij ) =

20*a*

− ≤ ≤

• **Security Constraints. **For secure operation, the transmis- sion line loading S l is restricted by its upper limit as

21

*a*ij

aij

a 1.05aij

S ≤ S , = 1,...., n

Where n is the number of transmission line.

For comparison purposes with the reported results, the system is considered as losses and the security constraint is released.

0

a < 0.95aij or a > 1.05aij

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 170

ISSN 2229-5518

a − Level →

↓ ↓

would be able to make a better and more reliable decision than with a single solution. An attempt is made to solve Fuzzy Multiobjective optimization with fuzzy parameters. Based on Alpha concept, F-MOOP can be transformed to crisp multi- objective optimization problem (C-MOOP) at certain degree of α (α-cut level). Also, we combine one such preference-based strategy with a neural network methodology and demonstrate

aij ij

aij

ij 1.05aij

a (Parameter values)

how, instead of one solution, a preferred set of solutions near

the reference points can be found parallel. Such procedures

will provide the decision-maker with a set of solutions near

So, every fuzzy parameter

aij can be represented using the

her/his preference so that a better and a more reliable decision can be made.

membership function. By using a -cut level, these fuzzy pa-

rameters can be transformed to a crisp one having upper and

[*a *L , *a*U ]

The main features of the proposed algorithm could be summa- rized as follows:

(a) The main crux of this paper is exploitation of reference

lower bounds

ij ij

, which declared in figure 2. Conse-

point base neural network procedure in finding more than one

quently, each a -cut level can be represented by the two end

points of the alpha level.

Here, the problem is how to determine the optimal power flow for considering the minimum cost and the minimum emission objectives simultaneously. In order to efficiently and effectively obtain the solution, the search for the optimal solu- tion is carried out in two steps. Firstly transforming the fuzzy multiobjective optimization problem (F-MOOP) to the crisp multiobjective optimization (C-MOOP) by means of Alpha- cut, In order to study the influence of fuzzy parameters on the obtained Pareto optimal solutions, all the range of the parame- ter fluctuation were scanned, two bounds of Alpha value have

been considered a = 0,1 , and also we take some values be-

tween these bounds a = 0.2, 0.4, 0.6, 0.8 . While the other phase

employs a neural networks based reference point algorithm to

solve C-MOOP, where the decision maker (DM) plays an im-

portant role. The DM is expected to be an expert in the prob-

lem domain and provide us with different preferred reference

point for each case as in figures(3-8) . A partial set of nondom-

inated solutions is obtained by exploring the optimal Pareto frontier using different a cut level and certain preferred refer-

ence point. Graphical presentations of the experimental results are presented in figures (3-8) for six cases with different three preferred reference point. It is obvious from figures (3-8) that

the results maintain the diversity and convergence for all a

cut level. On the basis of the application, we can conclude that

the proposed method can provide a sound optimal power

flow by simultaneously considering multiobjective problem.

On the basis of the application, we can conclude that the pro-

posed method can provide a sound optimal power flow by

simultaneously considering multiobjective problem.

In this paper, we have addressed an important task of combin-

ing neural network methodologies with a classical reference

point approach to not find a single optimal solution, but to

find a set of solutions near the desired region of decision-

maker’s interest. With a number of trade-off solutions in the

region of interests we have argued that the decision-maker

solutions not on the entire Pareto-optimal frontier, but in the regions of Pareto-optimality which are of interest to the DM.

(b) With a number of trade-off solutions in the region of inter- ests we have argued that the decision-maker would be able to make a better and more reliable decision than with a single solution

(c) Such methodology allows the DM to first make a higher- level search of monitoring a region of interest on the Pareto- optimal front, rather than using a single solution to focus on a particular solution.

(d) Since there is instabilities in the global market, implications of global financial crisis and the rapid fluctuations of prices, for this reasons a fuzzy representation of economic emission load dispatch problem has been defined.

(e)Eliminating the need of any weight vector and the need of applying the method again and again.

(f)The trade-off solutions in the obtained Pareto-optimal set are well distributed and have satisfactory diversity character- istics. This is useful in giving a reasonable freedom in choos- ing operating point from the available finite alternative.

(g) If a reference point is feasible and is not a Pareto-optimal

solution, the decision-maker may then be interested in know-

ing solutions which are Pareto-optimal and close to the refer-

ence point. On the other hand, if the reference point is an in-

feasible one, the decision-maker would be interested in find-

ing Pareto-optimal solutions which are close to the supplied

reference point.

(h) On the basis of the application, we can conclude that the

proposed method can provide a sound optimal power flow by

simultaneously considering multiobjective problem.

For future work, we intend to test the algorithm on more complex real-world applications. Also, conduct research on the parallel mechanism of multi-reference point algorithms and multi-criteria decision group problems so that it improves the efficiency of such approaches which are very relevant for real- world scenarios.

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 171

ISSN 2229-5518

0.2

0.2

0.2

Reference Point

0.195

0.195

0.195

Reference Point

0.19

0.19

0.19

0.185

Reference Point

0.185

0.185

0.18

0.18

0.18

0.175

580 600 620 640

Cost($/h)

0.175

580 600 620 640

Cost($/h)

0.175

570 580 590 600 610 620 630 640

Cost($/h)

**Fig. 3**. Pareto optimal set for a cut level =0

0.23

0.23

0.23

0.225

0.225

0.225

0.22

0.22

0.22

0.215

0.215

0.215

0.21

Reference Point

0.21

0.21

0.205

0.205

0.205

0.2

0.2

Reference Point

0.2

0.195

0.195

0.195

0.19

0.19

0.19

Reference Point

0.185

0.185

0.185

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

**Fig. 4**. Pareto optimal set for a cut level =0.2

0.22

0.22

0.22

0.215

0.215

0.215

0.21

Reference Point

0.21

0.21

0.205

0.205

0.205

0.2 __ __

0.2

0.2

0.195

0.195

Reference Point

0.195

0.19

0.19

0.19

Reference Point

0.185

0.185

0.185

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

**Fig. 5**. Pareto optimal set for a cut level =0.4

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International Journal of Scientific & Engineering Research, Volume 4, Issue 11, November-2013 172

ISSN 2229-5518

0.215

0.215

0.215

0.21

Reference Point

0.21

0.21

0.205

0.205

0.205

0.2

0.2

Reference Point

0.2

0.195

0.195

0.195

Reference Point

0.19

0.19

0.19

0.185

0.185

0.185

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

0.18

580 590 600 610 620 630 640

Cost($/h)

**Fig. 6**. Pareto optimal set for a cut level =0.6

0.22

0.23

0.22

0.215

Reference Point

0.225

0.22

0.215

0.21

0.215

0.21

0.205

0.21

0.205

0.2

0.205

Reference Point

0.2

Reference Point

0.2

0.195

0.195

0.195

0.19

580 590 600 610 620 630 640

0.19

580 590 600 610 620 630 640

0.19

580 590 600 610 620 630 640

**Fig. 7**. Pareto optimal set for a cut level =0.8

0.215 __ __

0.215

0.215

0.21

Reference Point

0.21

0.21

0.205 __ __

0.205

0.205

Reference Point

0.2

0.2

0.2

Reference Point

0.195

0.195

0.195

0.19

580 590 600 610 620 630 640

Cost($/h)

0.19

580 590 600 610 620 630 640

Cost($/h)

0.19

580 590 600 610 620 630 640

Cost($/h)

**Fig. 8**. Pareto optimal set for a cut level =1.0

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