ORIGINAL PAPER

A new approach of Romero’s extended lexicographic goalprogramming: fuzzy extended lexicographic goal programming

Mar Arenas-Parra • Amelia Bilbao-Terol •

Blanca Perez-Gladish • Marıa Victoria Rodrıguez-Urıa

Published online: 3 December 2009

� Springer-Verlag 2009

Abstract Goal programming (GP) is perhaps one of the

most widely used approaches in the field of multicriteria

decision making. The major advantage of the GP model is

its great flexibility which enables the decision maker to

easily incorporate numerous variations on constraints and

goals. Romero provides a general structure, extended lex-

icographic goal programming (ELGP) for GP and some

multiobjective programming approaches. In this work, we

propose the extension of this unifying framework to fuzzy

multiobjective programming. Our extension is carried out

by several methodologies developed by the authors in the

fuzzy GP approach. An interval GP model has been con-

structed where the feasible set has been defined by means

of a relationship between fuzzy numbers. We will apply

this model to our fuzzy extended lexicographic goal pro-

gramming (FELGP). The FELGP is a general primary

structure with the same advantages as Romero’s ELGP and

moreover it has the capacity of working with imprecise

information. An example is given in order to illustrate the

proposed method.

Keywords Extended lexicographic goal programming �Fuzzy goal programming � Fuzzy number �Expected interval � Interval goal programming

1 Introduction

The fuzzy programming (FP) approach for handling multi-

objective problems was first introduced by Zimmermann

(1978). Since then, numerous authors have investigated and

developed the use of the fuzzy set theory in solving problems

with multiple goals using Zimmermann’s approach as a basis

(see Ignizio 1982). Ramadan (1997) adopts this approach

when comparing goal programming (GP) with FP, and he

states that GP and FP are approaches for solving multiob-

jective programming problems that require an aspiration

level for each objective determined by the decision maker

(DM). FP needs, in the author’s opinion, in addition to the

aspiration levels or targets for the goals, admissible violation

of those expressed by means of constants indicating the

importance for the DM for each goal. Tamiz et al. (1998)

refer to FP as a ‘‘close relative’’ of MINMAX GP where the

maximum deviational is minimized.

Advances in the field of GP maintain an increase rate;

interactive GP, non-linear GP, Stochastic GP, Fuzzy GP

(FGP) and also applications of GP to other problems as

infeasibility analysis or artificial intelligence, are a few

examples of the growth of the field (Leon and Liern 2001;

Van Hop 2007; Lijun et al. 2007; Yang 2008; Lin et al.

2009; Oddoye et al. 2009).

Pioneering works on FGP are due to Narasimhan (1980,

1981), who has illustrated the application of ‘‘fuzzy sub-

sets’’ concepts to goal programming in a fuzzy environ-

ment. In contrast to a typical GP problem, Narasimhan

proposes the statement of imprecise goals when the deci-

sion environment is fuzzy.

Since then, numerous works have been published in this

field as FGP appears to be a useful tool for dealing with real

problems where goal values, technological coefficients or

achievement function weights, for example, are not known

M. Arenas-Parra � A. Bilbao-Terol � B. Perez-Gladish (&) �M. V. Rodrıguez-Urıa

Departamento de Economıa Cuantitativa,

Facultad de CC Economicas y Empresariales,

University of Oviedo, Avenida del Cristo s/n,

Oviedo, Asturias, Spain

e-mail: bperez@uniovi.es

123

Soft Comput (2010) 14:1217–1226

DOI 10.1007/s00500-009-0533-y

with certainty or are stated by the DM in an imprecise or

vague way. If we only focus on applications, a large number

of works can be found in the literature applying FGP to

different real decision problems (see Sharma et al. 2006 for

a review of a large number of applications).

In this work we shall consider a multiobjective possi-

bilistic linear programming problem in which all the

parameters are fuzzy. We will suppose that they are rep-

resented by fuzzy numbers described by their possibility

distribution estimated by the analyst from the information

supplied by the DM.

The uncertain and/or imprecise nature of the problem’s

parameters involves two main problems: feasibility and

optimality. Feasibility may be handled by comparing fuzzy

numbers. The optimality is replaced by a satisfying strategy

which attempts to meet criteria for adequacy, rather than to

identify an optimal solution. This strategy is handled

through Romero’s unifying approach: extended lexico-

graphic goal programming (see Romero 2001, 2004). We

propose the extension of this unifying model to fuzzy

multiobjective programming: fuzzy extended lexicographic

goal programming (FELGP). Our extension is a general

primary structure with the same advantages as Romero’s

ELGP and moreover it has the capacity of working with

imprecise information.

This paper is organized as follows. Section 2 is devoted

to the treatment of the feasibility problem and the satis-

fying philosophy. We present a fuzzy goal programming

approach where the fuzzy numbers have been handled

through their expected intervals.

In Sect. 3, a fuzzy extended lexicographic goal pro-

gramming approach, extending Romero’s unifying

approach (Romero 2001, 2004) to problems with fuzzy

parameters in the objective functions and the constraints, is

developed. In order to illustrate the proposed model, an

example is included in Sect. 4. Finally, Sect. 5 includes the

main conclusions and final remarks.

2 A fuzzy goal programming approach

We shall consider the following multiobjective possibilistic

linear programming problem involving fuzzy parameters:

opt ~z ¼ ~z1; ~z2; . . .; ~zkð Þ ¼ ~c1x; ~c2x; . . .; ~ckxð Þs: t:

x 2 F ~A; ~b� �

¼ ~aix� ~bi; i ¼ 1; . . .;m

x� 0

( )

ðFP�MOLPÞ

where xt ¼ x1; x2; . . .; xnð Þ is the crisp decision vector, ~ct ¼~c1; ~c2; . . .; ~ckð Þ is composed on fuzzy vectors which are the

fuzzy coefficients of the k considered objectives, ~A ¼ ~aij

� �m�n

is the fuzzy technological matrix and ~bt ¼ ~b1; ~b2; . . .; ~bm

� �is

the vector of the fuzzy right-hand sides of the constraints.

A fuzzy number is defined differently by different

authors. The most frequently used definition is that a fuzzy

number is a fuzzy set in the real line which is normal1 and

bounded convex. The fuzzy numbers represent the con-

tinuous possibility distributions for fuzzy parameters and

hence place a restriction on the possible values the variable

may assume. From the definition of a fuzzy number ~N, it is

significant to note that the set of level a, Na, can be rep-

resented by the closed interval which depends on the value

of a: Na ¼ nLa ; n

Ra

� �.

As it has been pointed out in the introduction, the

uncertain and/or imprecise nature of the technological

matrix and the resources vector which define the set of

constraints of the model leads us to compare fuzzy num-

bers. In this work we have handled fuzzy numbers through

their expected intervals2 which fulfil suitable properties

such as linearity, which are easy to apply and which give us

a good representation of fuzzy numbers.

In order to compare two fuzzy numbers ~a and ~b, rep-

resented by their expected intervals, the fuzzy relationship

defined by Jimenez (1996) has been used. This fuzzy

relationship verifies some suitable properties and it is

computationally efficient to solve linear problems because

it preserves linearity:

l ~a;~bð Þ ¼

0 if EIL ~að Þ[ EIR ~b� �

EIR ~bð Þ�EIL~að Þ

EIR~að Þ�EIL

~að ÞþEIR ~bð Þ�EIL ~bð Þ; if 0 2 EIL ~b� �� EIR ~að Þ;EIR ~b

� �� EIL ~að Þ

� �

1 if EIR ~að Þ\EIL ~b� �

8>><

>>:

ð1Þwhere l ~a;~bð Þ is the degree of preference of ~a over ~b.

If l ~a;~bð Þ � b; with b 2 0; 1½ �, we say that ‘‘~a is smaller

than ~b at least in a degree b’’ and it is denoted by ~a� b~b.

From the above definition this is equivalent to:

1� bð ÞEI ~að ÞLþbEI ~að ÞR� bEI ~b� �Lþ 1� bð ÞEI ~b

� �R ð2Þ

This leads us to the concept of b-feasibility of a decision

vector.

Definition 1 A decision vector x 2 IRn, is said to be

b-feasible for the problem FP-MOLP if x satisfies the

constraints at least in a degree b. That is ~aix � b~bi;

i ¼ 1; . . .;m.

1 A fuzzy set is normal if the supreme of its membership function is

equal to 1.2 Heilpern (1992) defines the expected interval of a fuzzy number ~N,

which will be noted EI ~N� �

. In terms of a-cuts the expected interval is:

EI ~N� �¼ EI ~N

� �L�EI ~N� �R

h i¼R 1

0nL

ada;R 1

0nR

a dah i

.

The expected interval of a fuzzy vector ~ai ¼ ~ai1; ~ai2; . . .; ~ainð Þ, as a

vector composed of the expected intervals of each fuzzy number of

the vector ~ai, that is, EI ~aið Þ ¼ EI ~ai1ð Þ;EI ~ai2ð Þ; . . .;EI ~ainð Þð Þ.

1218 M. Arenas-Parra et al.

123

This allows us to handle the problem constraints in a

flexible way as b can be considered as the degree of sat-

isfaction of them, and thus, 1 - b can be interpreted as the

level of risk of non-fulfilment of the constraints the DM

wants to allow.

The set of all b-feasible decision vectors is denoted by

F bð Þ.In accordance with the above considerations we shall

solve the FP-MOLP through a family of b-FP-MOLP

problems, where 0� b� 1:

In order to solve (b-FP-MOLP) several methods can be

applied; in this work we shall use GP. As it is well known,

GP is an analytical approach devised to address decision

making problems where targets must be assigned to all the

attributes and where the DM is interested in minimizing the

non-achievement of the corresponding targets. In other

words, the DM seeks a Simonian satisficing solution (i.e.

satisfactory and sufficient) with this strategy.

The main idea behind our FGP approach is to minimize

the distance between the fuzzy objective vector ~z and a

fuzzy aspiration level vector ~t. The aspiration level is either

determined by the DM or equals the fuzzy ideal solution ~z�

of (b-FP-MOLP) obtained using the solving method pro-

posed by Arenas et al. (1999).

Fuzzy coefficients in the objective function and fuzzy

targets will be also handled by their expected intervals; the

expected interval of the fuzzy number which defines the rth

objective function, EI ~crxð Þ ¼ EI ~crxð ÞL;EI ~crxð ÞR� �

, should

be the nearest possible to the expected interval of the rth

fuzzy target, EI ~trð Þ ¼ EI ~trð ÞL;EI ~trð ÞRh i

, for each

r ¼ 1; 2; . . .; k, so we will have an Interval GP model.

As we are working in an imprecise environment, to

reduce the imprecision in the solution we choose the

decision that gives an objective value whose expected

interval width is less than the one of the target. Then it

should satisfy the following condition:

EI ~crxð ÞR�EI ~crxð ÞL�EI ~trð ÞR�EI ~trð ÞL ð3Þ

In order to solve an Interval GP we present the following

definitions and results (Arenas et al. 2001):

Definition 2 Let A ¼ aL; aR½ � and B ¼ bL; bR½ � be two

intervals on the real straight line, such that the width of the

first is greater than the width of the second, i.e.

bR � bL� aR � aL. We shall define the operation difference

by extremes between A and B, denoted by , as the fol-

lowing interval:

A B ¼ aL � bL; aR � bR� �

ð4Þ

Definition 3 Given A ¼ aL; aR½ � and B ¼ bL; bR½ � with

bR � bL� aR � aL, we shall define the distance between

both intervals as a new interval obtained as the absolute

value of its difference by extremes:

DðA;BÞ¼ ABj j

¼aL�bL;aR�bR½ � if aL�bL�0

0;max �ðaL�bLÞ;ðaR�bRÞð Þ½ � if aL�bL\0\aR�bR

�ðaR�bRÞ;�ðaL�bLÞ½ � if aR�bR�0

8><

>:

ð5Þ

Observe that DðA;BÞ¼0,A¼B.

By applying the definition of distance to the expected

intervals and taking into account the condition (3), we have

that

Dr ¼ DLr ;D

Rr

� �¼ EI ~trð Þ EI ~crxð Þj j 8r ¼ 1; . . .; k ð6Þ

and

EI ~crxð Þ ¼ EI ~trð Þ , Dr ¼ 0 8r ¼ 1; . . .; k ð7Þ

Then, the following fuzzy interval goal programming

model is obtained:

minPk

r¼1

xpr Dp

r

stEI ~crxð ÞR�EI ~crxð ÞL�EI ~trð ÞR�EI ~trð ÞL

x 2 F bð Þ

ðb-FIGPÞ

This leads us to the concept of b-satisfying solution.

Definition 4 A decision vector x 2 F bð Þ, is said to be a

b-satisfying solution for the problem FP-MOLP if x is a

satisfying solution for the (b-FIGP) problem.

If the aspiration level is the fuzzy ideal solution ~z� of the

(b-FP-MOLP) we have a b-compromise solution for the

FP-MOLP.

In order to work with the GP approach we are going to

introduce in the model the positive (represented by pr) and the

negative (represented by nr) deviational variables for each

opt ~z ¼ ~z1; ~z2; . . .; ~zkð Þ ¼ ~c1x; ~c2x; . . .; ~ckxð Þs: t: ðb-FP-MOLP)

1� bð ÞEI ~aið ÞLþbEI ~aið ÞR� �

x� bEI ~bi

� �Lþ 1� bð ÞEI ~bi

� �R

x� 0 i ¼ 1; . . .;m

�¼ F bð Þ

A new approach of Romero’s extended lexicographic goal programming 1219

123

goal. These variables quantify, by means of the extremes of

the intervals, how far the solution is from the aspiration levels

set by the DM. The negative deviational variables quantify

the underachievement of an objective with respect to its level

of aspiration, while the positive ones do the same with respect

to overachievement. Then we can write

EI ~crxð ÞLþnLr � pL

r ¼ EI ~trð ÞL 8r ¼ 1; . . .; k

EI ~crxð ÞRþnRr � pR

r ¼ E ~trð ÞR 8r ¼ 1; . . .; kð8Þ

and the ‘‘least imprecise’’ condition (3) can be written as

nLr � pL

r � nRr � pR

r . So substituting in (6), we have

Drðn; pÞ ¼ DLr ðn; pÞ;DR

r ðn; pÞ� �

¼ nLr � pL

r ; nRr � pR

r

� ��� ��

8r ¼ 1; . . .; k ð9Þ

Taking into account Definition 3 and (8) we have that

Drðn; pÞ ¼½nL

r ; nRr � if nL

r � pLr � 0

½0;max pLr ; n

Rr

� �� if nL

r � pLr \0\nR

r � pRr

½pRr ; p

Lr � if nR

r � pRr � 0

8<

:

ð10Þ

and then it can be stated that (Arenas et al. 2001)

Drðn; pÞ ¼ nLr þ pR

r ;max pLr ; n

Rr

� �� �ð11Þ

Once the deviational variables have been introduced, the

next step in the solving of a GP model will consist of

determining the unwanted variables for the (b-FIGP).

Proposition 1 If DRr ðn; pÞ ¼ 0, then:

EI ~crxð ÞL;EI ~crxð ÞR� �

¼ EI ~trð ÞL;EI ~trð ÞRh i

ð12Þ

Proof If DRr ðn; pÞ ¼ max pL

r ; nRr

� �¼ 0 then pL

r ¼ nRr ¼ 0

and substituting in the ‘‘less imprecise’’ condition nLr �

pLr � nR

r � pRr it can be observed that (12) is fulfilled. h

Therefore, the unwanted variables for the (b-FIGP) are

the ones appearing in the upper extreme of every distance

interval, so, in order to find a b-satisfying solution of FP-

MOLP, it will be sufficient to minimize the upper extreme

of each distance interval. This process can be approached in

different ways, each one giving rise to a variant of GP; the

difference between them lies on the achievement function,

remaining identical for the working constraints. In the fol-

lowing section, we present our extension of the Extended

Lexicographic Goal Programming approach developed by

Romero (2001, 2004) to fuzzy multiobjective programming.

3 A fuzzy extended lexicographic goal programming

Romero formulates a general GP structure called extended

lexicographic goal programming (ELGP). From this

approach, the connection between GP and other MCDM

models, such as conventional mathematical programming

and compromise programming, (CP) can be easily found.

This general unifying structure has been widely cited in the

academic literature. From a computer search in the SCOPUS

and ABI/Inform Global conducted to collect the relevant

studies using and/or citing Romero’s unifying framework, a

total of 28 scientific papers citing Romero’s approach can be

found. Bal et al. (2006), Yaghoobi and Tamiz (2007), Chang

(2007), Yaghoobi et al. (2008) and Akoz and Petrovic (2007)

among others, present some interesting recent theoretical

extensions of Romero’s unifying theories. Several case

studies and real applications could also be found citing Ro-

mero’s model, e.g. Elfkih et al. (2009), Leung et al. (2006),

Leung and Chan (2009), Ozcan and Toklu (2009), Ustun and

Demirtas (2008) and van Calker et al. (2008), among others.

A key element of a GP model is the achievement function

that represents a mathematical expression of the unwanted

deviational variables. The three oldest used forms of GP

achievement functions, and still most widely, are the Ar-

chimedean GP, the lexicographic GP and the MINMAX GP.

Lexicographic GP uses the concept of pre-emptive or

non-Archimedean priorities. In this structure, the DM sorts

out the objectives by grouping them in levels of importance

called priority levels, so that the objectives corresponding

to sth level have absolute priority with respect to those of

the (s ? 1) group. If several objectives share the same level

of priority, their importance will be described by means of

weighting coefficients. The extension of the ELGP to a

fuzzy framework, considering the satisfying interpretation

of FGP presented in the previous section, can be formu-

lated as follows (b-FELGP):

Lex min a¼ k1VT1þl1

X

r2P1

xrvrð Þp;...;"

ksVTsþls

X

r2Pj

xrvrð Þp;...;

klVTlþll

X

r2Pl

xrvrð Þp#

s:t:

xrvr�VTs r2Ps s2 1;...;lf gpL

r�vr;nRr �vr r2Ps s2 1;...;lf g

EI ~crxð ÞLþnLr�pL

r ¼EI ~trð ÞL r2Ps s2 1;...;lf gEI ~crx� �RþnR

r �pRr ¼EI ~tr

� �Rr2Ps s2 1;...;lf g

nLr �pL

r�nRr �pR

r r2Ps s2 1;...;lf gnL

r�pLr ¼nR

r �pRr ¼0 r2Ps s2 1;...;lf g

nLr�0;pL

r�0;nRr �0;pR

r �0 r2Ps s2 1;...;lf gx2F bð Þ

9>>>=

>>>;

ð��Þ

In this model vr ¼ max pLr ; n

Rr

� �; r ¼ 1; . . .; k; p 2 0;1½ Þ or

p ¼ 1; P1;P2; . . .;Pl l� kð Þ are the established priority

levels and Ps represents the index set of goals placed in the

1220 M. Arenas-Parra et al.

123

sth priority level; ks; ls are control parameters; ks takes

value 0, when the MINMAX (Chebyshev) option is not

considered; otherwise it takes a positive value and ls takes

value 0, when we want to consider the MINMAX option; xr

is the weight reflecting preferential and normalizing pur-

poses attached to the unwanted deviational variable of rth

goal and VTs is the maximum xrvr in the priority level Ps.

This fuzzy extended goal programming model (b-FEL-

GP) encompasses at least the compromise programming

(L1 and L?) model and the following fuzzy GP approaches:

Nonlinear and linear Archimedean Fuzzy GP, Lexico-

graphic Fuzzy GP, MINMAX or Chebychev GP and

Extended GP (Romero 2001).

Table 1 displays the parameter specification for the

different approaches. In fact, by substituting the parameters

values in the model (b-FELGP) by the corresponding

specifications shown in that table, the approaches cited

below are straightforwardly obtained. The parameter l in

the fuzzy extended GP and CP models, weights the

importance attached to the minimization of the weighted

sum of unwanted variables. In the fuzzy conventional

mathematical programming and CP models the target ~z�r is

the fuzzy ideal solution of (b-FP-MOLP).

The corresponding specifications introduced in Table 1

lead to the following models:

– Fuzzy Archimedean GP:

minXk

r¼1

xrvrð Þp

s:t:

pLr � vr; n

Rr � vr r 2 1; . . .; kf g

EI ~crx� �LþnL

r � pLr ¼ EI ~trð ÞL r 2 1; . . .; kf g

EI ~crx� �RþnR

r � pRr ¼ EI ~tr

� �Rr 2 1; . . .; kf g

and ð��Þ ð13Þ

– Fuzzy Lexicographic GP:

Lex min a ¼X

r2P1

xrvrð Þ; . . .;X

r2Ps

xrvrð Þ; . . .;X

r2Pl

xrvrð Þ" #

s:t:

pLr � vs; n

Rr � vs r 2 Ps s 2 1; . . .; lf g

EI ~crxð ÞLþnLr � pL

r ¼ EI ~trð ÞL r 2 Ps s 2 1; . . .; lf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~trð ÞR r 2 Ps s 2 1; . . .; lf g

and ð��Þ ð14Þ

– Fuzzy MINMAX GP:

minVT

s:t:

xrvr �VT r 2 1; . . .; kf gpL

r � vr; nRr � vr r 2 1; . . .; kf g

EI ~crxð ÞLþnLr � pL

r ¼ EI ~trð ÞL r 2 1; . . .; kf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~trð ÞR r 2 1; . . .; kf g

and ð��Þ ð15Þ

– Fuzzy Extended GP:

min 1� lð ÞVT þ lXk

r¼1

xrvrð Þp

s.t.

xrvr �VT r ¼ 1; . . .; k

pLr � vr; n

Rr � vr r ¼ 1; . . .; k

EI ~crxð ÞLþnLr � pL

r ¼ EI ~trð ÞL r 2 1; . . .; kf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~trð ÞR r 2 1; . . .; kf g

and ð��Þ ð16Þ

For l 2 0; 1ð Þ we have intermediate solutions between the

solutions provided by the fuzzy MINMAX GP model

l ¼ 0ð Þ and fuzzy Archimedean GP model l ¼ 1ð Þ:– Fuzzy conventional mathematical programming:

min v1

s:t:

pL1 � v1; n

R1 � v1

EI ~c1xð ÞLþnL1 � pL

1 ¼ EI ~z�1� �L

EI ~c1xð ÞRþnR1 � pR

1 ¼ EI ~z�1� �R

and ð��Þ for k ¼ 1 ð17Þ

– Fuzzy CP (L1):

minXk

r¼1

xrvr

s:t:

pLr � vr; n

Rr � vr r ¼ 1; . . .; k

EI ~crxð ÞLþnLr � pL

r ¼ EI ~z�r� �L

r 2 1; . . .; kf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~z�r

� �Rr 2 1; . . .; kf g

and ð��Þ ð18Þ

Table 1 Parameter specification in the (b-FELGP) model

ks ls ~tr p l Approach

0 1 ~tr p 1 Fuzzy Archimedean GP

0 1 ~tr 1 l Fuzzy lexicographic GP

1 0 ~tr 1 1 Fuzzy MINMAX (Chebyshev)

GP

1 - l l ~tr p 1 Fuzzy extended GP l 2 0; 1½ �ð Þ0 1 ~z�r 1 1 Fuzzy conventional mathematical

programming (k ¼ 1)

0 1 ~z�r 1 1 Fuzzy CP (L1)

1 0 ~z�r 1 1 Fuzzy CP (L?)

1 - l l ~z�r 1 1 Fuzzy extended CP l 2 0; 1½ �ð Þ

A new approach of Romero’s extended lexicographic goal programming 1221

123

– Fuzzy CP (L?):

minVT

s:t:

xrvr �VTr 2 1; . . .; kf gpL

r � vr; nRr � vr r 2 1; . . .; kf g

EI ~crxð ÞLþnLr � pL

r ¼ EI ~z�r� �L

r 2 1; . . .; kf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~z�r

� �Rr 2 1; . . .; kf g

and ð��Þ ð19Þ

– Fuzzy extended CP:

min 1� lð ÞVT þ lXk

r¼1

xrvr

s:t:

pLr � vr; n

Rr � vr r ¼ 1; . . .; k

xrvr �VT r ¼ 1; . . .; k

EI ~crxð ÞLþnLr � pL

r ¼ EI ~z�r� �L

r 2 1; . . .; kf gEI ~crxð ÞRþnR

r � pRr ¼ EI ~z�r

� �Rr 2 1; . . .; kf g

and ð��Þ ð20Þ

For l 2 0; 1ð Þ we have intermediate solutions between

the solutions provided by the fuzzy CP (L?) model l ¼ 0ð Þand the fuzzy CP (L1) model l ¼ 1ð Þ:

With respect to the fuzzy conventional mathematical

programming and CP models, the following observations

should be made: since ~z�r is an ideal value then pLr ¼ 0,

pRr ¼ 0 and Dr ¼ nL

r ; nRr

� �. Therefore, from the Proposition

1, if nRr ¼ 0 the ideal expected interval EI ~z�r

� �L;EI ~z�r� �R

h i

coincides with the objective expected interval

EI ~crxð ÞL;EI ~crxð ÞR� �

and substituting in the models (17)–

(20) the fuzzy conventional mathematical programming

and the fuzzy CP models are obtained.

Although the potential generation of GP-dominated

solutions could be an important problem, there are sev-

eral methods of avoiding this problem without too many

difficulties. The following procedure based on Romero’s

test (Romero 1991) restores the efficiency of the

solution:

maxXk

r¼1

qLr þ qR

r

s:t:

EI ~crxð ÞL�pLr ¼ EI ~crx

�ð ÞL r 2 1; . . .; kf gEI ~crxð ÞR�pR

r ¼ EI ~crx�ð ÞR r 2 1; . . .; kf g

EI ~crxð ÞR�EI ~crxð ÞL�EI ~trð ÞR�EI ~trð ÞL r 2 1; . . .; kf gqL

r � 0; qRr � 0 r 2 1; . . .; kf g

x 2 F bð Þ ð21Þ

This test checks the efficiency of the optimal (b-FELGP)

solution, x*, and, if it is not efficient, then it produces an

efficient solution that dominates x*.

In the next section, an example will be provided in order

to illustrate the relationship existing between the achieve-

ment function of the (b-FELGP) model and the fuzzy

MCDM approaches presented earlier in Table 1 (models

(13)–(20)). The example has been implemented using the

Matlab and its toolbox of optimization.

4 Illustrative example

Let us consider the following multiobjective linear pro-

gram with fuzzy triangular parameters:3

max 40; 50; 80ð Þx1 þ 100x2 þ 17:5x3

max 80; 92; 120ð Þx1 þ 50; 75; 110ð Þx2 þ 50x3

max 10; 25; 70ð Þx1 þ 100x2 þ 75x3

s:t:

6; 12; 14ð Þx1 þ 17x2� 1400

3x1 þ 9x2 þ 3; 8; 10ð Þx3� 1000

10x1 þ 7; 13; 15ð Þx2 þ 15x3� 1750

4; 6; 8ð Þx1 þ 16x3� 1325

7; 12; 19ð Þx2 þ 7x3� 900

9:5x1 þ 3:5; 9:5; 11:5ð Þx2 þ 4x3� 1075

xi� 0; i ¼ 1; 2; 3

9>>>>>>>>>>>>>=

>>>>>>>>>>>>>;

F ~A; ~b� �

For each feasibility degree b fixed by the DM, a

b-satisficing solution can be obtained solving the following

model:

3 In practice, input fuzzy data are often assumed to be triangular

fuzzy numbers. A triangular fuzzy number can be denoted as~N ¼ n1; n2; n3ð Þ:

1

0n1 n2 n3

( )Ñ

xµ

For a triangular fuzzy number

~N ¼ n1; n2; n3ð Þ the expected interval is obtained as:

EI ~N� �¼ EI ~N

� �L�EI ~N� �R

h i¼ n1þn2

2; n2þn3

2

� �:

1222 M. Arenas-Parra et al.

123

Table 2 0.6-Satisfying solutions in the objective space

Approach ~z�1 ~z�2 ~z�3

FELGP

k1 ¼ k2 ¼ 0:65;ðl1 ¼ l2 ¼ 0:35Þ

(7474.77, 7995.44, 9557.44) (8558.15, 10351.02, 13444.18) (8277.93, 9058.94, 11401.95)

Fuzzy Archimedean G (6184.3, 6719.1, 8323.8) (8595.7, 10005, 12578) (7777.6, 8579.9, 10987)

Fuzzy lexicographic GP (7523.45, 8042.94, 9601.4) (8514.09, 10325.39, 13443.04) (8244.7, 9023.93, 11361.62)

Fuzzy MINMAX GP (6509.9, 6931.3, 8195.4) (8131.5, 9594.2, 12114) (8519.3, 9151.3, 11047)

Fuzzy extended GP l ¼ 0:65ð Þ (6346, 6824.5, 8260) (8365.1, 9801.2, 12348) (8146, 8863.7, 11017)

Table 3 Expected intervals of the 0.6-satisfying solutions

Approach EI ~z�1� �

EI ~z�2� �

EI ~z�3� �

FELGP

k1 ¼ k2 ¼ 0:65;ðl1 ¼ l2 ¼ 0:35Þ

[7735.10, 8776.44] [9454.58, 11897.60] [8668.44, 10230.44]

Fuzzy Archimedean GP [6451.7, 7521.5] [9300.5, 11292] [8178.7, 9783.4]

Fuzzy lexicographic GP [7783.19, 8822.17] [9419.74, 11884.21] [8634.32, 10192.77]

Fuzzy MINMAX GP [6720.6, 7563.4] [8862.8, 10854] [8835.3, 10099]

Fuzzy extended GP l ¼ 0:65ð Þ [6585.3, 7542.3] [9083.1, 11074] [8504.8, 9940.3]

Table 4 0.6-Satisfying

solutions in the decision spaceApproach x�1 x�2 x�3

FELGP 52.07 46.72 41.13

k1 ¼ k2 ¼ 0:65;ðl1 ¼ l2 ¼ 0:35Þ

Fuzzy Archimedean GP 53.49 30.71 55.62

Fuzzy lexicographic GP 51.95 47.52 39.65

Fuzzy MINMAX GP 42.14 38.28 56.93

Fuzzy extended GP l ¼ 0:65ð Þ 47.85 34.47 56.27

Table 5 0.6-Compromise solutions in the objective space

Approach ~z�1 ~z�2 ~z�3

Fuzzy CP (L1) (6051.1, 6514.3, 7903.6) (8267.9, 9612, 12012) (8095.9, 8790.5, 10875)

Fuzzy CP (L?) (6173.1, 6593.7, 7855.6) (8094.1, 9458.1, 11839) (8373.5, 9004.4, 10897)

Table 6 Expected intervals of

the 0.6-compromise solutionsApproach EI ~z�1

� �EI ~z�2� �

EI ~z�3� �

Fuzzy CP (L1) [6282.7, 7208.9] [8939.9, 10812] [8443.2, 9832.6]

Fuzzy CP (L?) [6383.4, 7224.6] [8776.1, 10648] [8689, 9950.8]

Table 7 0.6-Compromise

solutions in the decision spaceApproach x�1 x�2 x�3

Fuzzy CP (L1) 46.31 31.54 59.72

Fuzzy CP (L?) 42.06 34.37 60.24

A new approach of Romero’s extended lexicographic goal programming 1223

123

45x1 þ 100x2 þ 17:5x3; 65x1 þ 100x2 þ 17:5x3½ �� EI ~t1ð Þ ¼ EI t1

1; t12; t

13

� �

86x1 þ 62:5x2 þ 50x3; 106x1 þ 92:5x2 þ 50x3½ �� EI ~t2ð Þ ¼ EI t2

1; t22; t

23

� �

17:5x1 þ 100x2 þ 75x3; 47:5x1 þ 100x2 þ 75x3½ �� EI ~t3ð Þ ¼ EI t3

1; t32; t

33

� �

s:t:

9 1� bð Þ þ 13bð Þx1 þ 17x2� 1400

3x1 þ 9x2 þ 5:5 1� bð Þ þ 9bð Þx3� 1000

10x1 þ 10 1� bð Þ þ 14bð Þx2 þ 15x3� 1750

5 1� bð Þ þ 7bð Þx1 þ 16x3� 1325

9:5 1� bð Þ þ 9bð Þx2 þ 7x3� 900

9:5x1 þ 6:5 1� bð Þ þ 10:5bð Þx2 þ 4x3� 1075

xi� 0; i ¼ 1; 2; 3

9>>>>>>>>>>>=

>>>>>>>>>>>;

F bð Þ

By introducing the deviational variables in the problem

and assuming that the DM considers two priority levels,

P1 ¼ 1; 3f g and P2 ¼ 2f gð Þ, the following (b-FELGP)

formulation is obtained:

Lex min a

¼ k1VT1 þ l1

X

r2P1

xrvrð Þp; k2VT2 þ l2

X

r2P2

xrvrð Þp" #

s:t:

xrvr �VTs r 2 Ps s 2 1; 2f gpL

r � vr; nRr � vr r 2 Ps s 2 1; 2f g

45x1 þ 100x2 þ 17:5x3 þ nL1 � pL

1 ¼ EI ~t1ð ÞL

65x1 þ 100x2 þ 17:5x3 þ nR1 � pR

1 ¼ EI ~t1ð ÞR

92x1 þ 72:5x2 þ 50x3 þ nL2 � pL

2 ¼ EI ~t2ð ÞL

92x1 þ 82:5x2 þ 50x3 þ nR2 � pR

2 ¼ EI ~t2ð ÞR

17:5x1 þ 100x2 þ 75x3 þ nL3 � pL

3 ¼ EI ~t3ð ÞL

47:5x1 þ 100x2 þ 75x3 þ nR3 � pR

3 ¼ EI ~t3ð ÞR

nLr � pL

r � nRr � pR

r r 2 Ps s 2 1; 2f gnL

r � pLr ¼ nR

r � pRr ¼ 0 r 2 Ps s 2 1; 2f g

nLr � 0; pL

r � 0; nRr � 0; pR

r � 0 r 2 Ps s 2 1; 2f gx 2 F bð Þ

9>>>>=

>>>>;

ð��Þ

If the DM fixes b ¼ 0:6 and the following fuzzy targets

~t1 ¼ 6037; 7984; 12325ð Þ;EI ~t1ð Þ ¼ 6917:5; 9891½ �~t2 ¼ 9784; 10057; 13984ð Þ;EI ~t2ð Þ ¼ 9889:8; 11881½ �~t3 ¼ 7164; 9356; 13880ð Þ;EI ~t3ð Þ ¼ 8180:3; 11345½ �

The 0.6-satisfying solutions are given in Tables 2, 3 and 4.

If the fuzzy targets are the fuzzy ideal solution of the

(0.6-FP-MOLP)

EI ~z�1� �¼ 7824:4; 8822:2½ �

EI ~z�2� �¼ 10374; 12246½ �

EI ~z�3� �¼ 8933; 10322½ �

we obtain the 0.6-compromise solutions (Tables 5, 6 and 7).

Once the obtained results are shown to the DM he/she

may want to change the feasibility degree establishing a

higher one. If the DM fixes b ¼ 0:8 and the same fuzzy

targets that for b ¼ 0:6 the satisfying and compromise

solutions are shown in Tables 8, 9 and 10.

The fuzzy ideal solutions of the (0.8-FP-MOLP) are

EI ~z�1� �¼ 7530; 8518:7½ �

EI ~z�2� �¼ 10335; 12231½ �

EI ~z�3� �¼ 8673:3; 9998:1½ �

To analyse the obtained solutions, we are going to use

the information in the Tables 11 and 12.

Taking into account the information in Table 11 and

that our objective is to minimize the achievement function,

we can observe the 0.8-optimal solutions are worse than

that corresponding to b ¼ 0:6 in the three approaches.

On the other hand, observing Table 12, it can be seen

that the fuzzy extended GP solution, for the degree of

feasibility b ¼ 0:6, provides a good compromise between

Table 8 0.8-Solutions in the objective space

Approach ~z�1 ~z�2 ~z�3

FELGP (7084.84, 7606.21, 9170.33) (8368.30, 10063.73, 13021.27) (7887.12, 8669.18, 11015.36)

k1 ¼ k2 ¼ 0:65;ðl1 ¼ l2 ¼ 0:35Þ

Fuzzy Archimedean GP (6218.9, 6731.2, 8268.2) (8417.5, 9837.8, 12400) (7796, 8564.6, 10870)

Fuzzy lexicographic GP (6943.86, 7469.15, 9045.04) (8526.84, 10166.01, 13049.19) (8020.81, 8808.75, 11172.59)

Fuzzy MINMAX GP (6386.2, 6838.9, 8197.1) (8163.2, 9611.2, 12146) (8169.6, 8848.7, 10886)

Fuzzy CP (L1 and L?) (6149.1, 6590.7, 7915.5) (8096.7, 9470.7, 11889) (8131.5, 8793.9, 10781)

1224 M. Arenas-Parra et al.

123

the two opposite views: the Archimedean solution that

provides the maximum aggregate achievement (maximum

efficiency) and the MINMAX solution that provides the

most balanced solution between achievements of different

goals (maximum equity).

Due to the existing conflict between feasibility and

satisfying/optimality the DM will have to choose his/her

preferred solution.

5 Conclusions

In this paper we have extended the general structure ELGP

developed by Romero (2001, 2004) including in the model

uncertain and/or imprecise data represented by fuzzy

numbers that gives rise to a fuzzy multiobjective pro-

gramming problem that has been handled with the meth-

odology developed by the authors in previous works. This

leads to a more general structure than ELGP, called

(b-FELGP).

The (b-FELGP) presents some attractive properties.

First, it encompasses the Fuzzy Archimedean and Fuzzy

Chebychev GP variants in a unified format (parameter limplements trade-offs between efficiency and equity

through the achievement function). Second, due to its

lexicographic character (b-FELGP) allows the modelling

of problems where preferences among some of the attri-

butes under consideration are non-continuous, i.e. trade-

offs among attributes are not finite. Also the integration

into a unifying approach of some fuzzy GP methodologies

and other MCDM programs, such as the fuzzy conven-

tional mathematical programming and the fuzzy compro-

mise programming contributes to clarify the close

relationship between them.

Acknowledgments We would like to thank two anonymous refer-

ees for their valuable suggestions and comments which have con-

tributed to the improvement of the paper. A previous version of this

research was presented and published in the proceedings of the

Second International Congress in Soft Methods in Probability and

Statistics, Oviedo 2004. The authors wish to gratefully acknowledge

financial support from the Spanish Ministry of Education, project

MTM2007-67634.

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