multiple objective linear programming models with interval coefficients – an illustrated overview

European Journal of Operational Research 181 (2007) 1434–1463

www.elsevier.com/locate/ejor

Multiple objective linear programming modelswith interval coefficients – an illustrated overview

Carla Oliveira a,*, Carlos Henggeler Antunes a,b

a INESC Coimbra, Rua Antero de Quental 199, Coimbra, Portugalb Dep. Engenharia Electrotecnica e de Computadores, Universidade de Coimbra, Polo II, 3030-290 Coimbra, Portugal

Received 1 January 2005; accepted 1 December 2005Available online 19 May 2006

Abstract

In most real-world situations, the coefficients of decision support models are not exactly known. In this context, it isconvenient to consider an extension of traditional mathematical programming models incorporating their intrinsic uncer-tainty, without assuming the exactness of the model coefficients. Interval programming is one of the tools to tackle uncer-tainty in mathematical programming models. Moreover, most real-world problems inherently impose the need to considermultiple, conflicting and incommensurate objective functions. This paper provides an illustrated overview of the state ofthe art of Interval Programming in the context of multiple objective linear programming models.� 2006 Elsevier B.V. All rights reserved.

Keywords: Multiple objective linear programming; Interval programming; Satisficing approach; Optimizing approach

1. Introduction

Most real-world problems are inherently characterized by multiple, conflicting and incommensurate aspectsof evaluation of the merits of alternative solutions. These axes of evaluation are generally operationalized byobjective functions to be optimized in the framework of multiple objective linear programming (MOLP)models.

Moreover, in most real-world situations, the model coefficients are not exactly known because relevant datais inexistent or scarce, difficult to obtain or estimate, the system is subject to changes, etc. Therefore, mathe-matical programming models for decision support must take explicitly into account, besides multiple and con-flicting objective functions, the treatment of the intrinsic uncertainty associated with the model coefficients.

Interval programming is one of the approaches to tackle uncertainty in mathematical programming models,which possesses some interesting characteristics because it does not require the specification or the assumptionof probabilistic distributions (as in stochastic programming) or possibilistic distributions (as in fuzzy

0377-2217/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2005.12.042

* Corresponding author.E-mail addresses: [email protected] (C. Oliveira), [email protected] (C.H. Antunes).

mailto:[email protected]

mailto:[email protected]

C. Oliveira, C.H. Antunes / European Journal of Operational Research 181 (2007) 1434–1463 1435

programming). Interval programming just assumes that information about the range of variation of some (orall) of the parameters is available, which allows to specify a model with interval coefficients.

This paper is aimed at providing an overview of the different approaches reported in the literature to dealwith uncertainty in MOLP models through interval programming, using illustrative examples.

Interval Programming has been used to tackle specific issues in MOLP. In this sense, some algorithms onlydeal with uncertainty in the objective functions, others handle uncertainty both in the objective functions andin the RHS of the constraints, and others deal with uncertainty in all the coefficients of the model.

Inuiguchi and Kume (1994) and Inuiguchi and Sakawa (1995) consider two different approaches to dealwith an interval objective function: the satisficing approach and the optimizing approach. In the satisficingapproach each interval objective function is transformed into one or several objective functions (the lowerbound, the upper bound and the central value of the intervals are usually used) in order to obtain a compro-mise solution (e.g. Inuiguchi and Kume, 1991). Although the compromise solution obtained in this way is effi-cient, it might not be the most adequate one to the interval MOLP problem. In fact, if the gradients of theobjective functions chosen are highly correlated, the scope of the search might be reduced and ultimatelythe gradient cone of each objective function becomes a ray (Antunes and Clımaco, 2000).

On the other hand, the optimizing approach extends the concept of efficiency used in traditional MOLP tothe interval objective function case (e.g. Bitran, 1980; Ida, 1999, 2000a,b, 2005; Inuiguchi and Sakawa, 1996;Steuer, 1981; Wang and Wang, 2001a,b). Bitran (1980) suggested, in this context, two kinds of efficient solu-tions to the interval MOLP: a solution is called ‘‘necessarily efficient’’ if it is efficient for any given objectivefunction coefficient vectors within their admissible range of variation (Fig. 1); a solution is said ‘‘possibly effi-cient’’ if it is efficient for at least one of the given objective function coefficient vectors within their admissiblerange of variation. The ‘‘necessarily efficient’’ solutions are the most robust and the ‘‘possibly efficient’’ solu-tions are considered to be the optimistic ones (Ida, 1999). Although the approach presented in Ida (1999)allows the enumeration of all possibly efficient solutions and/or all necessarily efficient solutions, the compu-tational burden required is considerable. Another issue is that when the decision-maker (DM) is faced with alarge set of solutions, in many cases with just slight differences among the objective function values, the deci-sion problem becomes even more complex (Antunes and Clımaco, 2000).

In the framework of an interactive approach Urli and Nadeau (1992) have considered MOLP models withinterval coefficients in the entire model. Nevertheless, the results obtained through their algorithm do notallow the DM to take into account the worst case and the best case ‘‘scenarios’’ in order to perceive the riskat stake (e.g. Chineck and Ramadan, 2000).

In the following sections we will pay a more detailed attention to the algorithms reported in the literature,categorizing them into satisficing and optimizing approaches. Examples are given aimed at illustrating themain features of each approach.

x1

z1

z2

Necessarily efficient solution

Possibly efficient

solutions

x2

Fig. 1. Necessarily and possibly efficient solutions.

1436 C. Oliveira, C.H. Antunes / European Journal of Operational Research 181 (2007) 1434–1463

2. The satisficing approach

2.1. Interval LP

The methods which might be classified in the satisficing approach are mainly reported for linear program-ming (LP) models with an interval objective function (Chanas and Kuchta, 1996; Ishibuchi and Tanaka, 1990;Rommelfanger et al., 1989).

Let us consider, without loss of generality, the following LP problem with an interval objective function:

max ZðxÞ ¼ cx

s:t: : Ax 6 b;

x P 0;

ð1Þ

where c is an interval vector, whose generic elements are cj 2 ½cLj ; c

Uj � for j ¼ 1; . . . ; n, A is an m · n matrix, b is

an m · 1 vector, x is an n · 1 vector and the superscripts L and U represent lower and upper bounds of thecoefficients, respectively.

In the framework of the satisficing approach, some authors consider the following surrogate problem (e.g.Sengupta et al., 2001):

max ZCðxÞ ¼ cCx

s:t: : Ax 6 b;

x P 0;

ð2Þ

where cC is the vector of the central values of the original interval vector c.Rommelfanger et al. (1989) proposed a method to solve problem (1), which allows reducing the infinite

number of objective functions, using only the two extreme bounds of the interval objective function. Hence,problem (1) is solved through the following bi-objective problem (see also Chineck and Ramadan, 2000):

max ZLðxÞ ¼Xn

j¼1

cLj xj

max ZUðxÞ ¼Xn

j¼1

cUj xj

s:t: : Ax 6 b;

x P 0:

ð3Þ

Ishibuchi and Tanaka (1990) transformed problem (1) considering two objective functions and assuming apessimistic procedure. In the maximizing case the central value and the lower bound of the interval objectivefunction are maximized and in the minimizing case the central value and the upper bound of the interval objec-tive function are minimized.

Chanas and Kuchta (1996) considered both the concepts of cut – u0, u1 – of an interval and parametriclinear programming. The surrogate problem obtained according to this method is

max ZðxÞ ¼ kXn

j¼1

½cLj þ u0ðcU

j � cLj Þ�xj

!þ ð1� kÞ

Xn

j¼1

½cLj þ u1ðcU

j � cLj Þ�xjÞ

!

s:t: : Ax 6 b;

0 6 k 6 1;

0 6 u0 6 u1 6 1 ðu0 and u1 are fixedÞ;x P 0:

ð4Þ

The operational approach used by Chanas and Kuchta (1996) allows enlarging the range of solutions obtainedbecause the whole range of coefficient values between cL

j and cUj can be analyzed.


However, if there is a strong correlation between the directions associated with ZL(x) and ZU(x), and con-sequently also with ZC(x), the scope of the search might be reduced and only a small number of solutions tothe interval problem would be obtained (eventually, only one) (Antunes and Clımaco, 2000).

2.2. Goal programming approach

In the context of goal programming (GP), Inuiguchi and Kume (1991) have derived four formulations for agoal programming problem with interval coefficients and target intervals, considering the concepts of neces-sary subtraction and possible subtraction between intervals (Inuiguchi and Kume, 1991; Moore, 1966), eitherminimizing the lower bound (optimistic decision procedure) or minimizing the upper bound of the regret inter-val functions obtained (pessimistic decision procedure).

Let us consider, without loss of generality, the following GP problem with interval targets and coefficients:

Xn

j¼1

ckjxj ¼ tk; k ¼ 1; . . . ; p;

s:t: : ckj 2 Ckj ðk ¼ 1; . . . p; j ¼ 1; . . . ; nÞ;tk 2 T k ðk ¼ 1; . . . ; pÞ;Ax 6 b;

x P 0;

ð5Þ

where Ckj is the closed interval ½cLkj; c

Ukj� and Tk is the closed interval ½tL

k ; tUk �. Problem (5) can assume four pos-

sible formulations, considering the concepts of possible and necessary subtractions (Inuiguchi and Kume,1991), respectively.

To obtain the two formulations of problem (5) according to the concept of possible subtraction, it is nec-essary to consider the possible deviation of the goals from their respective targets.

The operators used between real numbers can be extended to interval numbers. In this context, (�) and (+)refer to the ‘‘possibly’’ extended operators, and (�) and (+) refer to the ‘‘necessary’’ extended operators.

The possible deviation Dk ¼ ½dLk ; d

Uk � of ðþÞnj¼1Ckjxj ¼

Pnj¼1cL

kjxj;Pn

j¼1cUkjxj

h ifrom Tk is represented by the

following expression:

Dk ¼ jT kð�ÞðþÞnj¼1Ckjxjj ¼ tLk �

Xn

j¼1

cUkjxj; tU

k �Xn

j¼1

cLkjxj

" #��

¼

tLk �

Pnj¼1

cUkjxj; tU

k �Pnj¼1

cLkjxj

" #; if tL

k �Pnj¼1

cUkjxj P 0;

0;Pnj¼1

cUkjxj � tL

k

!_ tU

k �Pnj¼1

cLkjxj

!" #; if tL

k �Pnj¼1

cUkjxj < 0 < tU

k �Pnj¼1

cLkjxj;

Pnj¼1

cLkjxj � tU

k ;Pnj¼1

cUkjxj � tL

k

" #; if tU

k �Pnj¼1

cLkjxj 6 0:

8>>>>>>>>>><>>>>>>>>>>:

Inuiguchi and Kume (1991) consider a possible regret interval, D(x), where

DðxÞ ¼ ½dLðxÞ; dUðxÞ�

¼ kXp

k¼1

ck dL�k þ dUþ

k

� �þ ð1� kÞ_p

k¼1 dL�k þ dUþ

k

� �; kXp

k¼1

ck dLþk _ dU�

k

� �þ ð1� kÞ_p

k¼1 dLþk _ dU�

k

� �" #;

0 6 k 6 1; ck P 0 andXp

k¼1

ck ¼ 1: ð6Þ


The deviational variables, dL�k , dLþ

k , dU�k and dUþ

k , are defined in such a way that:

Xn

j¼1

cUkjxj þ dL�

k � dLþk ¼ tL

k () tLk �

Xn

j¼1

cUkjxj ¼ dL�

k � dLþk ; dL�

k � dLþk ¼ dL

k ; ð7Þ

Xn

j¼1

cLkjxj þ dU�

k � dUþk ¼ tU

k () tUk �

Xn

j¼1

cLkjxj ¼ dU�

k � dUþk ; dU�

k � dUþk ¼ dU

k ; ð8Þ

Dk ¼ j½dLk ; d

Uk �j; dL�

k � dLþk ¼ 0; dU�

k � dUþk ¼ 0: ð9Þ

If the DM wishes to follow an optimistic procedure then problem (5) is substituted by

min kXp

k¼1

ckðdL�k þ dUþ

k Þ þ ð1� kÞvL

s:t: :Xn

j¼1

cUkjxj þ dL�

k � dLþk ¼ tL

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

cLkjxj þ dU�

k � dUþk ¼ tU

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

aijxj 6 bi ði ¼ 1; . . . ;mÞ;

dL�k þ dUþ

k 6 vL ðk ¼ 1; . . . ; pÞ;0 6 k 6 1;

ck P 0 ðk ¼ 1; . . . ; pÞ;Xp

k¼1

ck ¼ 1;

xj P 0 ðj ¼ 1; . . . ; nÞ:

ð10Þ

Otherwise, if the DM wishes to minimize the upper bound of the possible regret interval, then the followingproblem is obtained:

min kXp

k¼1

ckvk þ ð1� kÞvU

s:t: :Xn

j¼1

cUkjxj þ dL�

k � dLþk ¼ tL

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

cLkjxj þ dU�

k � dUþk ¼ tU

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

aijxj 6 biði ¼ 1; . . . ;mÞ;

dLþk 6 vk ðk ¼ 1; . . . ; pÞ;

dU�k 6 vk ðk ¼ 1; . . . ; pÞ;

vk 6 vU ðk ¼ 1; . . . ; pÞ;0 6 k 6 1;

ck P 0 ðk ¼ 1; . . . ; pÞ;Xp

k¼1

ck ¼ 1;

xj P 0 ðj ¼ 1; . . . ; nÞ:

ð11Þ


In order to obtain the other two formulations of problem (5) according to the concept of necessary subtraction

it is required to consider the necessary deviation Ek ofðþÞnj¼1Ckjxj ¼Pn

j¼1cLkjxj;

Pnj¼1cU

kjxj

h ifrom Tk:(

Ek ¼jT kÞ � ððþÞnj¼1Ckjxjj; if w½T k�P w½ðþÞnj¼1Ckjxj�;jðþÞnj¼1CkjxjÞ � ðT kj; if w½ðþÞnj¼1Ckjxj�P w½T k�;

Ek ¼tLk �

Pnj¼1

cLkjxj; tU

k �Pnj¼1

cUkjxj

" #��; if tU

k � tLk P

Pnj¼1

ðcUkj � cL

kjÞxj;

Pnj¼1

cLkjxj � tL

k ;Pnj¼1

cUkjxj � tU

k

" #��; if

Pnj¼1

ðcUkj � cL

kjÞxj P tUk � tL

k ;

8>>>>><>>>>>:

Ek ¼

tLk �

Pnj¼1

cLkjxj; tU

k �Pnj¼1

cUkjxj

" #; if tU

k �Pnj¼1

cUkjxj P tL

k �Pnj¼1

cLkjxj P 0;

0; tUk �

Pnj¼1

cUkjxj

!_

Pnj¼1

cLkjxj � tL

k

!" #; if tL

k �Pnj¼1

cLkjxj < 0 < tU

k �Pnj¼1

cUkjxj;

Pnj¼1

cUkjxj � tU

k ;Pnj¼1

cLkjxj � tL

k

" #; if tL

k �Pnj¼1

cLkjxj 6 tU

k �Pnj¼1

cUkjxj 6 0;

Pnj¼1

cLkjxj � tL

k ;Pnj¼1

cUkjxj � tU

k

" #; if

Pnj¼1

cUkjxj � tU

k PPnj¼1

cLkjxj � tL

k P 0;

0;Pnj¼1

cUkjxj � tU

k

!_ tL

k �Pnj¼1

cLkjxj

!" #; if

Pnj¼1

cLkjxj � tL

k < 0 <Pnj¼1

cUkjxj � tU

k ;

tUk �

Pnj¼1

cUkjxj; tL

k �Pnj¼1

cLkjxj

" #; if

Pnj¼1

cLkjxj � tL

k 6Pnj¼1

cUkjxj � tU

k 6 0:

8>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

w applied before the intervals denotes the width of the interval.Consider E1

k and E2k such that:

ðþÞnj¼1CkjxjÞ þ ðE1kðþÞE2

k ¼ T k

E1k ¼

0; if tUk � tL

k PPnj¼1

ðcUkj � cL

kjÞxj;

tUk �

Pnj¼1

cUkjxj; tL

k �Pnj¼1

cLkjxj

" #; if

Pnj¼1

ðcUkj � cL

kjÞxj > tUk � tL

k ;

8>>><>>>:

E2k ¼

tLk �

Pnj¼1

cLkjxj; tU

k �Pnj¼1

cUkjxj

" #; if tU

k � tLk P

Pnj¼1

ðcUkj � cL

kjÞxj;

0; ifPnj¼1

ðcUkj � cL

kjÞxj > tUk � tL

k :

8>>><>>>:

Ek can also be represented by Ek ¼ E1kðþÞE2

k .Inuiguchi and Kume (1991) obtained the necessary regret interval as

EðxÞ ¼ ½eLðxÞ; eUðxÞ�

¼ kXp

k¼1

ck

�eL�

k þ eUþk

� �^ eLþ

k þ eU�k

� ��þ ð1� kÞ_p

k¼1

�eL�

k þ eUþk

� �^ eLþ

k þ eU�k

� ��;

"

kXp

k¼1

ck eL�k _ eLþ

k _ eU�k _ eUþ

k

� �þ ð1� kÞ_p

k¼1 eL�k _ eLþ

k _ eU�k _ eUþ

k

� �#: ð12Þ


The deviational variables, eL�k , eLþ

k , eU�k and eUþ

k , are defined in such a way that

Xn

j¼1

cLkjxj þ eL�

k � eLþk ¼ tL

k ;Xn

j¼1

cUkjxj þ eU�

k � eUþk ¼ tU

k ; eL�k � eLþ

k ¼ 0; eU�k � eUþ

k ¼ 0: ð13Þ

If the DM wishes to minimize the lower bound of the necessary regret interval, problem (5) is substituted bythe following non-convex problem:

min kXp

k¼1

ckððeL�k þ eUþ

k Þ ^ ðeLþk þ eU�

k ÞÞ þ ð1� kÞuL

s:t: :Xn

j¼1

cLkjxj þ eL�

k � eLþk ¼ tL

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

cUkjxj þ eU�

k � eUþk ¼ tU

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

aijxj 6 bi ði ¼ 1; . . . ;mÞ;

ðeL�k þ eUþ


k Þ 6 uL ðk ¼ 1; . . . ; pÞ;0 6 k 6 1;

ck P 0 ðk ¼ 1; . . . ; pÞ;Xp

k¼1

ck ¼ 1;

xj P 0 ðj ¼ 1; . . . ; nÞ:

ð14Þ

Problem (14) can be solved by a branch and bound algorithm, since it can be transformed into a linear mixedinteger programming problem (Inuiguchi and Kume, 1991).

If, on the other hand, the DM wishes to follow a pessimistic procedure then problem (5) becomes:

min kXp

k¼1

ckuk þ ð1� kÞuU

s:t: :Xn

j¼1

cLkjxj þ eL�

k � eLþk ¼ tL

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

cUkjxj þ eU�

k � eUþk ¼ tU

k ðk ¼ 1; . . . ; pÞ;

Xn

j¼1

aijxj 6 bi ði ¼ 1; . . . ;mÞ;

eL�k þ eLþ

k 6 uk ðk ¼ 1; . . . ; pÞ;eU�

k þ eUþk 6 uk ðk ¼ 1; . . . ; pÞ;

uk 6 uU ðk ¼ 1; . . . ; pÞ;0 6 k 6 1;

ck P 0 ðk ¼ 1; . . . ; pÞ;Xp

k¼1

ck ¼ 1;

xj P 0 ðj ¼ 1; . . . ; nÞ:

ð15Þ

Hence, it is possible to obtain four kinds of solutions depending on how the original problem is treated.


2.2.1. Illustrative example

Let us consider the following GP problem with interval targets and coefficients:

½0:5; 1:8�x1 þ ½�0:5; 0:5�y1 ¼ ½1:2; 12�;½0:3; 0:8�x2 þ ½1; 1:2�y2 ¼ ½2:333; 9:506�;x1 þ x2 6 8;

y1 þ y2 6 15;

xj P 0 ðj ¼ 1; 2Þ;yj P 0 ðj ¼ 1; 2Þ:

Let k ¼ c1 ¼ c2 ¼ 12.

Under an optimistic perspective and according to the concept of possible regret interval, the problem is for-mulated as (see (10)):

min1

4dL�

1 þ dUþ1 þ dL�

2 þ dUþ2

� �þ 1

2ðvLÞ

s:t: : 1:8x1 þ 0:5y1 þ dL�1 � dLþ

1 ¼ 1:2;

0:8x2 þ 1:2y2 þ dL�2 � dLþ

2 ¼ 2:333;

0:5x1 � 0:5y1 þ dU�1 � dUþ

1 ¼ 12;

0:3x2 þ y2 þ dU�2 � dUþ

2 ¼ 9:506;

x1 þ x2 6 8;

y1 þ y2 6 15;

dL�1 þ dUþ

1 6 vL;

dL�2 þ dUþ

2 6 vL;

xj P 0 ðj ¼ 1; 2Þ;yj P 0 ðj ¼ 1; 2Þ:

The solution obtained is (x1,x2,y1,y2)T = (0.667,0,0,1.944)T.On the other hand, under a pessimistic perspective and according to the concept of possible regret interval,

the problem is formulated as (see (11)):

min1

4ðv1 þ v2Þ þ

1

2ðvUÞ

s:t: : 1:8x1 þ 0:5y1 þ dL�1 � dLþ

1 ¼ 1:2;

0:8x2 þ 1:2y2 þ dL�2 � dLþ

2 ¼ 2:333;

0:5x1 � 0:5y1 þ dU�1 � dUþ

1 ¼ 12;

0:3x2 þ y2 þ dU�2 � dUþ

2 ¼ 9:506;

x1 þ x2 6 8;

y1 þ y2 6 15;

dLþ1 6 v1;

dU�1 6 v1;

v1 6 vU;

dLþ2 6 v2;

dU�2 6 v2;

v2 6 vU;

xj P 0 ðj ¼ 1; 2Þ;yj P 0 ðj ¼ 1; 2Þ:

The solution obtained is (x1,x2,y1,y2)T = (5.739,0,0,5.382)T.


Considering the concept of necessary regret interval and under an optimistic perspective, the problem isformulated as (see (14)):

min1

4

X2

k¼1

ðeL�k þ eUþ


k Þ� �

þ 1

2ðuLÞ

s:t: : 0:5x1 � 0:5y1 þ eL�1 � eLþ

1 ¼ 1:2;

0:3x2 þ y2 þ eL�2 � eLþ

2 ¼ 2:333;

1:8x1 þ 0:5y1 þ eU�1 � eUþ

1 ¼ 12;

0:8x2 þ 1:2y2 þ eU�2 � eUþ

2 ¼ 9:506;

x1 þ x2 6 8;

y1 þ y2 6 15;

ðeL�1 þ eUþ

1 Þ ^ ðeLþ1 þ eU�

1 Þ 6 uL;

ðeL�2 þ eUþ

2 Þ ^ ðeLþ2 þ eU�

2 Þ 6 uL;

xj P 0 ðj ¼ 1; 2Þ;yj P 0 ðj ¼ 1; 2Þ:

The solution obtained is (x1,x2,y1,y2)T = (4,4,1.133,1.133)T.Finally, following a pessimistic perspective and according to the concept of necessary regret interval, the

problem is formulated as (see (15)):

min1

4ðu1 þ u2Þ þ

1

2ðuUÞ

s:t: : 0:5x1 � 0:5y1 þ eL�1 � eLþ

1 ¼ 1:2;

0:3x2 þ y2 þ eL�2 � eLþ

2 ¼ 2:333;

1:8x1 þ 0:5y1 þ eU�1 � eUþ

1 ¼ 12;

0:8x2 þ 1:2y2 þ eU�2 � eUþ

2 ¼ 9:506;

x1 þ x2 6 8;

y1 þ y2 6 15;

ðeL�1 þ eLþ

1 Þ 6 u1;

ðeU�1 þ eUþ

1 Þ 6 u1;

u1 6 uU;

ðeL�2 þ eLþ

2 Þ 6 u2;

ðeU�2 þ eUþ

2 Þ 6 u2;

u2 6 uU;

xj P 0 ðj ¼ 1; 2Þ;yj P 0 ðj ¼ 1; 2Þ:

The solution obtained is (x1,x2,y1,y2)T = (5.739, 2.261,3.339, 4.251)T.The possible deviations, Dk, and the possible regret intervals, D(x), are listed in Table 1. The necessary devi-

ations, Ek, and the necessary regret intervals, E(x), are displayed in Table 2.

2.3. Interactive approach

Urli and Nadeau (1992) have proposed a general methodology which enables the transformation of a non-deterministic MOLP model into a deterministic one. This latter program is then solved by an interactiveapproach derived from STEM (Benayoun et al., 1971).

Table 1Possible deviations, Dk, and possible regret intervals, D(x)

Problem D1 D2 D(x)

Optimistic perspective according to the possible regret interval [0,11.667] [0,7.562] [0,10.641]Pessimistic perspective according to the possible regret interval [0,9.13] [0,4.124] [0,7.879]Optimistic perspective according to the necessary regret interval [0,10.567] [0,7.173] [0,9.718]Pessimistic perspective according to the necessary regret interval [0,10.8] [0,4.577] [0,9.244]

Table 2Necessary deviations, Ek, and necessary regret intervals, E(x)

Problem E1 E2 E(x)

Optimistic perspective according to the possible regret interval [0.867,10.8] [0.389,7.173] [0.747,9.893]Pessimistic perspective according to the possible regret interval [0,1.67] [0,3.048] [0,2.704]Optimistic perspective according to the necessary regret interval [0,4.233] [0,4.946] [0,4.768]Pessimistic perspective according to the necessary regret interval [0,0] [0,2.596] [0,1.947]


Let us consider, without loss of generality, the following MOLP with interval coefficients and parameters:

max ZkðxÞ ¼Xn

j¼1

½cLkj; c

Ukj�xj ðk ¼ 1; . . . ; pÞ

s:t: :Xn

j¼1

½aLij; a

Uij �xj 6 bL

i bUi

� �ði ¼ 1; . . . ;mÞ;

xj P 0 ðj ¼ 1; . . . ; nÞ;

ð16Þ

where ½cLkj; cU

kj�, ½aL

ij; aUij � and bL

i bUi

� �, k = 1, . . . ,p, j = 1, . . . ,n and i = 1, . . . ,m, are closed intervals.

The surrogate problem derived by Urli and Nadeau (1992) is

max P kðdUk Þ ¼ 1� dU

k

ðtUk � tL

k Þs:t: : dU

k ¼ tUk � ZC

k ðxÞ ðk ¼ 1; . . . ; pÞ;dU

k 6 tUk � tL

k ðk ¼ 1; . . . ; pÞ;dU

k P 0 ðk ¼ 1; . . . ; pÞ;Xn

j¼1

ðaLij þ aiðaU

ij � aLijÞÞxj 6 bU

i � aiðbUi � bL

i Þ ði ¼ 1; . . . ;mÞ;

xj P 0 ðj ¼ 1; . . . ; nÞ;

ð17Þ

where tUk ¼ ZU

k ðx0kÞ; x0

k ¼ maxx2X ZUk ðxÞ; X ¼ f

Pnj¼1aL

ijxj 6 bUi ; i ¼ 1; . . . ;m; xj P 0; j ¼ 1; . . . ; ng; tL

k ¼ minr;d

ZLk ðxr

dÞ; r ¼ 0; 1; d ¼ 1; . . . ; p; dLk ¼ tL

k � ZLk ðxÞ; dU

k ¼ tUk � ZU

k ðxÞ; ZCk ðxÞ is the central value of the kth objec-

tive function and ai corresponds to the satisfaction threshold on constraint i (Urli and Nadeau, 1992). If r = 0,then problem (16) is solved considering the ‘‘most favorable’’ version of each objective function and largestfeasible region; if r = 1, then problem (16) is solved considering the ‘‘less favorable’’ version of each objectivefunction and the narrowest feasible region (Chineck and Ramadan, 2000; Urli and Nadeau, 1992).

In order to solve problem (17) it is necessary to build a double payoff table. To do this, for each objectivefunction P kðdU

k Þ of problem (17) and for each ai of the least and more constraining value of the constraints,problem (17) is solved.

The ‘‘best optimum’’ value of P kðdUk Þ in the double pay-off matrix is obtained by

P lk ¼ max

r;d1� ðt

Uk � ½ZC

k ðxrdÞ�Þ

ðtUk � tL

k Þ

; r ¼ 0; 1; d ¼ 1; . . . ; p:


In a similar way, the ‘‘worst optimum’’ value of P kðdUk Þ in the double pay-off matrix is obtained by

P ik ¼ min

r;d1� ðt

Uk � ½ZC

k ðxrdÞ�Þ

ðtUk � tL

k Þ

; r ¼ 0; 1; d ¼ 1; . . . ; p:

The terms ‘‘best optimum’’ and ‘‘worst optimum’’ refer to Chineck and Ramadan (2000).This information concerning the best and the worst possible value of the objective functions P kðdU

k Þ is con-veyed through lk and ik. These parameters are defined by

lk ¼ tLk þ P l

k ðtUk � tL

k Þ and ik ¼ tLk þ P i

kðtUk � tL

k Þ:

With this information the DM is asked to specify an aspiration level ok for each objective function Zk(x),k = 1, . . . ,p. The distance to this target vector o = (o1, . . . ,op), in the space of the objective functions, is thenminimized.

Let D0 ¼ fx 2 Rn : dUk ¼ tU

k � ZCk ðxÞ; d

Uk 6 tU

k � tLk and dU

k P 0; k ¼ 1; . . . ; pg.The first compromise solution x1 is obtained by solving the following problem:

min M � t�Xp

k¼1

ek

s:t: : pkðP 0k � P kðdU

k ÞÞ 6 t� ek ðk ¼ 1; . . . ; pÞ;x 2 D0;Xn

j¼1

ðaLij þ aiðaU



i Þ ði ¼ 1; . . . ;mÞ;

t P 0;

xj P 0 ðj ¼ 1; . . . ; nÞ;

ð18Þ

where

pk ¼/kPpk¼1/k

; /k ¼P l

k � P ik

P lk

� 1

k½cCk �k2

;

k½cCk �k2 is the Euclidean norm of the cC

k vector containing the central values of the interval vector ck, and

P 0k ¼ 1� ðt

Uk �okÞðtU

k �tLk Þ

. M is a sufficiently large number, and the terms with ek are aimed at guaranteeing the efficiency

of the solution.

The remaining compromise solutions correspond to xn, n = 2,3,4, . . . ,f.For each compromise solution the DM receives the following information:

• The value ZCk ðxnÞ. This information is supplied simultaneously with [ik,lk]. Hence, the DM obtains

½ik; ZCk ðxnÞ; lk� and determines ok.

• The DM is able to compare the different values obtained for the distinct objective functions through therelative values of ZC

k ðxnÞ; Z 0kðxnÞ. These values are, in general, comprised between 0 and 1 and translatethe achievement rate with respect to the aspiration level ok, and are given by

Z 0kðxnÞ ¼ 1� ðok � ½ZCk ðxnÞ�Þ

ðok � ikÞ:

• The DM is able to observe the evolution of the compromise solution according to different satisfactionthresholds on the constraints.

After conveying this information to the DM, he/she is asked to reveal his/her preferences with respect to thecurrent compromise solution. If the DM considers the current solution as a satisfactory one the procedureends. Otherwise, the DM should proceed in the quest for new solutions. In this last situation, he/shechooses the objective function ZC

k ðxÞ� that he/she wishes to improve and, if possible, the maximum level of


improvement, Dk� . If the DM is not able to determine this level of improvement then this value can be fixedautomatically as Dk� ¼ ok� � ZC

k ðxnÞ�. After this step, a parametric analysis is performed to explore the conse-quences of the choices made by the DM. This procedure consists in analyzing the repercussions of the possibleimprovement level sDk� in the different objective functions ZC

k ðxÞ�, s 2 [0, 1], k = 1, . . . ,p. Consequently, this

parametric analysis is operated by means of the following problem:

min M � t�Xp

k¼1

ek

s:t: : pkðP 0k � P kðdU

k ÞÞ 6 t� ek; x 2 D0 ðk ¼ 1; . . . ; pÞ;ZC

k ðxÞ� ¼ ok� � sDk� ; s 6 1;Xn

j¼1

ðaLij þ aiðaU



i Þ ði ¼ 1; . . . ;mÞ;

t P 0;

xj P 0 ðj ¼ 1; . . . ; nÞ:

ð19Þ

By solving problem (19) a series of stability intervals of the form [si,si+1] are obtained. If the optimal solutionof this parametric problem for a certain s is xs, the value ZC

k ðxsÞ� is shown to the DM. With these values theDM may choose a certain level of improvement s*. After choosing s* the DM faces a new compromise solu-tion xs� , subsequently adding the following constraints to problem (18): pk� ¼ 0 and ZC

k ðxÞ�6 ZC

k ðxsÞ�.This interactive process proceeds in the same way and the procedure ends when the DM is satisfied with the

compromise solution obtained.This approach uses a simple mathematical formulation and allows a strong integration of the DM in the

decision phases. The following assumptions are considered for the problem transformation: the DM is lesssatisfied when the lower bound of the objective function is closer to the lower bound of the target interval(obtained from the extended pay-off table) and more satisfied when the upper bound of the objective functionis closer to the upper bound of the target interval; the DM hopes that the lower bound of the LHS of the con-straints will not be larger than the upper bound of the RHS of the constraints and his/her satisfaction level willbe even higher as much as the upper bound of the LHS will be closer to the upper bound of the RHS. In theframework of this approach, the preference structure of the DM is considered. Nevertheless, although the DMprefers the most favorable situation, the decision process should consider equally the occurrence of a pessimis-tic scenario and the occurrence of an optimistic scenario. In the context of multiobjective stochastic linear pro-gramming, Urli and Nadeau (2004) proposed a scenario approach where the probabilities of scenarios are onlyincompletely specified according to a ranking from the most probable to the less probable, with possibility ofex-aequo. This interactive method is called PROMISE/scenarios and is also derived from the STEM method.Firstly, on the basis of the scenarios, the multiobjective stochastic program (in which violation variables forthe constraints are introduced) is transformed into an equivalent multiobjective deterministic program. Then,the DM is asked to take advantage of the available information about the violation of the constraints and thepartially specified probabilities on scenarios, in order to progress interactively towards a satisfactory compro-mise solution. Nevertheless, this method is only appropriate for problems of small dimension or for problemsof relatively large dimension but with a small number of scenarios (Urli and Nadeau, 2004).


Consider the following MOLP problem with interval coefficients:

max Z1ðxÞ ¼ ½0:5; 1:8�x1 þ ½�0:5; 0:5�x2

max Z2ðxÞ ¼ ½0:3; 0:8�x1 þ ½1; 1:2�x2

s:t: : ½1:5; 2:5�x1 þ ½0:5; 1�x2 6 ½6; 10�;½0:5; 2�x1 þ ½3; 6�x2 6 ½14; 16�;xj P 0 ðj ¼ 1; 2Þ:


Step 1. Obtain the equivalent deterministic problem.In order to obtain the targets, tk, we have to solve the following problems:

TableValues

ZU1 ðxr

dZL

1 ðxrdÞ

ZU2 ðxr

dZL

2 ðxrdÞ

TableIndivid

k

12

max ZUk ðxÞ

s:t: : 1:5x1 þ 0:5x2 6 10;

0:5x1 þ 3x2 6 16;

xj P 0 ðj ¼ 1; 2Þ;for each k; k ¼ 1; 2;

max ZLk ðxÞ

s:t: : 2:5x1 þ 1x2 6 6;

2x1 þ 6x2 6 14;

xj P 0 ðj ¼ 1; 2Þ;for each k; k ¼ 1; 2:

Then, we obtain xrd , d = 1,2 and r = 0,1: x0

1 ¼ ð6:667; 0ÞT, x02 ¼ ð5:177; 4:471ÞT, x1

1 ¼ ð2:4; 0ÞT and

x12 ¼ ð0; 2:333ÞT.

This information can be organized in a table (Table 3) that contains the values of ZUk ðxr

dÞ and ZLk ðxr

dÞ, forr = 0,1 and d = 1,2. The targets tU

k and tLk are obtained from Table 4.

P l1 ¼ 0:671; P i

1 ¼ 0:089; P l2 ¼ 0:802; P i

2 ¼ 0:068, /1 = 0.755, /2 = 0.744, p1 = 0.504, p2 = 0.496, a1 =a2 = 0.8.

The equivalent deterministic problem is

min M � t�X2

k¼1

ek

s:t: : 0:191� 0:044x1 6 t� e1;

0:339� 0:031x1 � 0:062x2 6 t� e2;

2:3x1 þ 0:9x2 6 6:8;

1:7x1 þ 5:4x2 6 14:4;

dU1 ¼ 12� 1:15x1;

dU2 ¼ 9:506� 0:55x1 � 1:1x2;

dU1 6 13:167;

dU2 6 8:786;

dUk P 0 ðk ¼ 1; 2Þ;

t P 0;

xj P 0 ðj ¼ 1; 2Þ:

3of ZU

k ðxrdÞ and ZL

k ðxrdÞ

x01 x1

1 x02 x1

2

Þ 12.000 4.320 11.553 1.1673.333 1.200 0.353 �1.167

Þ 5.333 1.920 9.506 2.8002.000 0.720 6.024 2.333

4ual targets

tLk tU

k tUk � tL

k

�1.667 12.000 13.1670.720 9.506 8.786

Table 5Individual optima

x01 x1

1 x02 x1

2 lk ok ik

ZC1 ðxr

dÞ 7.667 2.760 5.953 0.000 7.667 5.000 0.000ZC

2 ðxrdÞ 3.667 1.320 7.765 2.5667 7.765 6.000 1.320

Table 6Table with the values of ZC

k ðx1Þ and Z 0kðx1Þk ZC

k ðx1Þ ok Z 0kðx1Þ1 2.509 5.000 0.5022 3.378 6.000 0.440


Step 2. Obtain the first compromise solution.The first compromise solution is obtained by solving the equivalent deterministic problem. The solution of

the equivalent deterministic problem is x1 = (x1,x2)T = (2.182, 1.98)T.For a detailed analysis of solution x1, the information of Table 6 is provided to the DM (see also Table 5).Step 3. Interactive phases.Consider, for example, that the DM is not satisfied with the compromise solution x1 and wants to obtain

another solution. The DM considers that ZC1 ðxÞ should be improved, relaxing the thresholds on the constraints

for a1 = 0.6 and a2 = 0.8, respectively. This choice is made after analyzing Table 7.Suppose that with this information the DM decides to improve ZC

1 ðxÞ and does not know the value toimpose to D1. Then, D1 ¼ o1 � ZC

1 ðx1Þ ¼ 5:000� ð3:402Þ ¼ 1:598.Hence, the parametric analysis is performed by means of the following problem:

TableEvolut

a1

0.80.80.70.6

min M � t�X2

k¼1

ek

s:t: : 0:191� 0:04x1 6 t� e1;

0:339� 0:031x1 � 0:062x2 6 t� e2;

2:1x1 þ 0:8x2 6 7:6;

1:7x1 þ 5:4x2 6 14:4;

1:15x1 ¼ 5� s1:598; s 6 1;

dU1 ¼ 12� 1:15x1;

dU2 ¼ 9:506� 0:55x1 � 1:1x2;

dU1 6 13:167;

dU2 6 8:786;

dUk P 0; k ¼ 1; 2;

t P 0;

xj P 0 ðj ¼ 1; 2Þ:

7ion of the compromise solution due to changes on the thresholds of the constraints

a2 ZC1 ðxÞ Z 01ðxÞ ZC

2 ðxÞ Z 02ðxÞ0.8 2.509 0.502 3.378 0.440.7 2.397 0.479 3.599 0.4870.8 2.936 0.587 3.453 0.4560.8 3.402 0.680 3.536 0.474

Table 8Values ZC

k ðxÞ and Z 0kðxÞ for different s

s ZC1 ðxÞ Z 01ðxÞ ZC

2 ðxÞ Z 02ðxÞ1 3.402 0.680 3.536 0.4740.85 3.641 0.728 3.049 0.3690.70 3.881 0.776 2.561 0.265


After analyzing Table 8, the DM chooses s = 0.85, obtaining compromise solution x2 = (3.166,1.188)T. Thealgorithm proceeds with the interactive phases until the DM finds a satisfactory compromise solution.

3. The optimizing approach

3.1. Possible and necessary efficiency

Let us assume that a variable # belongs to a set K. Let us consider, on the other hand, that # might belong(or not) to another set 1. If K \ 1 5 ;, then it is possible that # 2 1, knowing that # 2 K. If K � 1, then # 2 1does necessarily hold, knowing that # 2 K. Hence, the following possibility and necessity measures can bedefined, taking the value 1 if # 2 1 is possible or necessarily verified and 0 otherwise, respectively (Inuiguchiand Sakawa, 1996):

PKð1Þ ¼1; if K \ 1 6¼ ;;0 otherwise;

NKð1Þ ¼1; if K � 1;

0; otherwise:

In this case, the possibility and necessity measures are regarded as conditional measures induced by the set K.Consider, without loss of generality, the following MOLP problem with interval objective functions:

Max zðxÞ ¼ Cx

s:t: : Ax 6 b;

x P 0;

C 2 U;

ð20Þ

where U is a set of p · n matrices, with each row of the form ck, whose generic elements areckj 2 ½cL

kj; cUkj� for k ¼ 1; . . . ; p; j ¼ 1; . . . ; n, A is an m · n matrix, b is an m · 1 vector, x is an n · 1 vector

and the superscripts L and U represent lower and upper bounds of the coefficients, respectively.A solution is necessarily efficient to problem (20) if and only if it is efficient for any C 2 U. The necessarily

efficient solution set (NE) is obtained by

N E ¼\C2U

X EðCÞ;

where XE(C) is the efficient solution set for each C 2 U.On the other hand, a solution is possibly efficient to problem (20) if and only if it is efficient for at least one

C 2 U. The possibly efficient solution set (PE) is obtained by

P E ¼[C2U

X EðCÞ:

Hence, NE � PE and a necessarily efficient solution is also a possibly efficient solution.Let q(x) be a set of p · n matrices for which the solution x is efficient:

qðxÞ ¼ fC : there is no x0 2 X ¼ fx : Ax 6 b; x P 0g : Cx0 P Cx and Cx0 6¼ Cxg:
Then, x 2 NE () NUðqðxÞÞ ¼ 1() U � qðxÞ 2 X and x 2 P E () PUðqðxÞÞ ¼ 1() U \ qðxÞ 6¼ ;.


Let Q(x) be the set of n-row vectors for which x is a solution maximizing a linear objective function cy,subject to y 2 X:

QðxÞ ¼ c : cx ¼ maxy2X

cy; x 2 X

:

Let

RNðUÞ ¼ fc : for all C 2 U; there is v P 0 such that c ¼ vCg ¼\C2U

RðCÞ

and

RPðUÞ ¼ fc : there are v P 0 and C 2 Usuch that c ¼ vCg ¼[C2U

RðCÞ;

where R(C) = {c : there is v P 0 such that c = vC}.If RN(U) \ Q(x) 5 ;, then x 2 NE. On the other hand, x 2 PE if and only if RP(U) \ Q(x) 5 ; (Inuiguchi

and Sakawa, 1996). Furthermore, if RN(U) is not empty, then x 2 NE if and only if RN(U) \ Q(x) 5 ; (Inu-iguchi and Sakawa, 1996).

3.1.1. Illustrative examples

The concepts of necessarily and possibly efficient solutions can be illustrated using the followings examples:

(a) Consider the following interval MOLP problem (Inuiguchi and Sakawa, 1996):

max ½2; 3�x1 þ ½1:5; 2:5�x2

max ½3; 4�x1 þ ½0:5; 0:8�x2

s:t: : 3x1 þ 4x2 6 42;

3x1 þ x2 6 24;

x1 P 0; 0 6 x2 6 9:

In this case, U = {C = (cij) : 2 6 c11 6 3, 1.5 6 c12 6 2.5, 3 6 c21 6 4, 0.5 6 c22 6 0.8} and RN(U) ={c : c = k1(3,1.5) + k2(3, 0.8),k1 > 0, k2 > 0}.Consider the solution (x1,x2)T = (6, 6)T. The set for which (6,6)T is the solution that maximizes the linearobjective function cy, with y 2 {x : 3x1 + 4x2 6 42, 3x1 + x2 6 24, x1 P 0, 0 6 x2 6 9} can be repre-sented by

Qðð6; 6ÞTÞ ¼ fc : c ¼ k1ð3; 1Þ þ k2ð3; 4Þ; k1 > 0; k2 > 0g:
As shown in Fig. 2, RN(U) \ Q(x) 5 ;. Thus, the solution (6,6)T is efficient for any C 2 U, being a nec-essarily efficient solution.
(b) Consider the following interval MOLP problem:

max ½1; 2:5�x1 þ ½3; 4�x2;

max ½2; 3�x1 þ ½1:5; 2:5�x2;

s:t: : 3x1 þ 4x2 6 42;

3x1 þ x2 6 24;

0 6 x2 6 9;

x1 P 0:

In this case, U = {C = (cij) : 1 6 c11 6 2.5, 3 6 c12 6 4, 2 6 c21 6 3, 1.5 6 c22 6 2} and RP(U) = {c : c =k1(1, 4) + k2(3,1.5), k1 > 0, k2 > 0}.

Consider the solution (x1,x2)T = (2, 9)T. The set according to which (2,9)T is the solution that maximizesthe linear objective function cy, with y 2 {x : 3x1 + 4x2 6 42, 3x1 + x2 6 24, x1 P 0, 0 6 x2 6 9} can be rep-resented by

6

9

0.8

0.5

2.5

1.5

4

RN( )

c1

862 x1

c2

Q((6, 6)T)

32

x2

Φ

Fig. 2. A necessarily efficient solution.

Q((2, 9)T)

9

6

3

2.5

2.5

1

1.5

RP( )

c13

862 x1

c2

Φ

Fig. 3. A possibly efficient solution.


Qðð2; 9ÞTÞ ¼ fc : c ¼ k1ð3; 4Þ þ k2ð0; 1Þ; k1 > 0; k2 > 0g:
Through the observation of Fig. 3 we conclude that RP(U) \ Q((2,9)T) 5 ;. Hence, the solution (2,9)T is effi-cient for at least one C 2 U, being a possibly efficient solution.


3.2. Necessarily efficiency solutions: Methods and extensions

The optimizing approach usually considers the exhaustive computation of the entire set of possibly and/ornecessarily efficient solutions to (20).

Bitran (1980) proposed an implicit enumeration algorithm that uses a subproblem to test the necessary effi-ciency of a given basic feasible solution and a branch and bound algorithm to solve the subproblem.

Let V be the subset of matrices of U having all elements of each column at the upper bound or at the lowerbound. Then, if C 2 V, for j = 1, . . . ,n, either C :j ¼ CU

:j or C :j ¼ CL:j, where CU

:j and CL:j are the column vectors

of CU (matrix with the elements cUkj; k ¼ 1; . . . ; p; j ¼ 1; . . . ; n) and CL (matrix with the elements

cLkj; k ¼ 1; . . . ; p; j ¼ 1; . . . ; n), respectively. The maximum number of elements in V is 2n. Bitran (1980) has also

proved that (20) is equivalent to

max zðxÞ ¼ Cx

s:t: : Ax 6 b;

x P 0;

C 2 V :

ð21Þ

The necessary and sufficient conditions for the efficiency of a feasible solution in vector-maximum problemscan be obtained by Tucker’s and Motzkin’s theorems of the alternative (Mangasarian, 1969). In this context,let x* be a feasible solution to problem (21) and E be an n · n diagonal matrix with all the elements

ejj ¼1; if x�j ¼ 0;0; otherwise:

Then, x* is an efficient solution to problem (21) if and only if the system Cl = 0, Cl 5 0, El = 0, Al = 0

has no solution l 2 Rn (Theorem 9.4 in Steuer (1986)). Therefore, if A, C and l are partitioned into basic andnon-basic parts, x* is an efficient solution if and only if the system

CBlB þ CNlN = 0;

CBlB þ CNlN 6¼ 0;

El = 0;

BlB þ NlN ¼ 0

is inconsistent, where the subscripts N and B designate the non-basic and the basic parts associated to C and l,respectively, and B and N denote the sub-matrices of A associated with the basic and the non-basic variables,respectively.

Let lB =� B�1NlN. If x* is a non-degenerate extreme point, then El = 0. Thus, x* is an efficient solution ifand only if the system

CNlN � CBB�1lN = 0;

CBlB þ CNlN 6¼ 0;

lN P 0;

is inconsistent.Bitran’s test subproblem follows from the fact that a non-degenerate extreme point of the feasible region of

(20) is necessarily efficient if and only if for any C = (CN,CB) 2 V, the following system is inconsistent:

ðCN � CBB�1NÞl P 0; l P 0; A ¼ ðN ;BÞ: ð22Þ
Let R = (CBB�1N � CN) denote the reduced cost matrix. Hence, (22) can be written in an operational form as:a non-degenerate extreme point of the feasible region of (20) is necessarily efficient if and only if, for anyC = (CN,CB) 2 V, the optimal value to problem
P ðx0Þ : maxfz ¼ iv : Rlþ Iv ¼ 0;l P 0; v P 0;C 2 V g
is zero, where i is a vector of ones and I is the identity matrix, both with convenient dimensions.


If there is an optimal solution to P(x0), there is always an optimal solution with CN ¼ CUN, where

CU ¼ ðCUN;C

UB Þ is the sub-matrix of C with the upper bounds of the intervals. Therefore, the problem P(x0)

is nonlinear since besides l and v the matrix CB is also unknown. A naive method to solve P(x0) consistsin solving this problem for all matrices CB with each column having all elements at the upper or lower bound.Since there can be 2m of such matrices this procedure would require a considerable computational burden.Instead, Bitran (1980) uses an implicit enumeration algorithm.

Let P(x0,g1 = 1,x), x = 1 (x = 0) be the problem where the column, in CB, corresponding to g1 = 1 has allits elements at the upper (lower) bound. This algorithm starts by solving the problem P(x0,g1), with g1 = 0(notice that for g1 = 0 (g1 = m) the first (second) summation vanishes and P(x0,g1) = P(x0)), where

R:j ¼Xg1

i¼1

CB:iðB�1i: N:jÞ þ

Xm

h¼g1þ1

CBðjÞ:hðB�1h: N:jÞ � CU

N:j;

CBðjÞ:h ¼CL

B:h; if B�1h: N:j P 0; h ¼ 1; . . . ;m;

CUB:h; if B�1

h: N:j < 0; h ¼ 1; . . . ;m;

(

CLB:i 6 CB:i 6 CU

B:i, CL ¼ ðCLB;C

LNÞ, CU ¼ ðCU

B ;CUNÞ, R.j is the jth column of R, CL

B:i is the ith column of CLB, CU

B:i

is the ith column of CUB , B�1

h: is the hth row of B�1, B�1i: is the ith row of B�1, N.j is the jth column of N, K

denotes the set of indices corresponding to the non-basic components of x0 and CB(j) is the ideal matrix cor-responding to index j (this matrix when multiplied by any B�1N.j gives the p vector with the lowest componentvalues) (Bitran, 1980).

If z = 0 then the algorithm stops and x0 is necessarily efficient to (20). If z > 0, g1 is set equal to one and thefollowing two problems are generated: P(x0,g1 = 1,1) and P(x0,g1 = 1,0). The method proceeds branching onproblems with positive optimal values and fathoming those with zero optimal values until it is possible to con-clude that x0 is either necessarily efficient to (20) or not (Fig. 4). This algorithm can then be extended to com-pute the entire set of necessarily efficient solutions, considering a subproblem to test whether an adjacent edgeto a necessarily efficient point is also necessarily efficient or not. This subproblem is also solved through theimplicit enumeration algorithm. To obtain the entire set of necessarily efficient solutions to a MOLP modelwith interval objective functions, it is only necessary to choose a matrix C 2 V, apply any of the existing algo-rithms to solve the MOLP and at each of its efficient extreme points x0 solve P(x0) to determine whether it isnecessarily efficient. This procedure will generate the entire set of necessarily efficient solutions since this iscontained in the set of solutions to the problem with the chosen matrix C. Nevertheless, this algorithmmay result in a high computational burden if the solution being analyzed is not necessarily efficient.

Ida (1999) suggested an extension of the implicit enumeration algorithm in order to manage this shortcom-ing. The proposed method uses two efficiency tests based on the extreme ray generation method (Chernikova,1965). One of the tests checks necessary efficiency and the other checks non-necessary efficiency.

P(x0, g1 = 3, 0, 1, 0); z = 0

P(x0, g1 = 1, 1); z > 0 P(x0, g1 = 1, 0); z > 0

P(x0, g1 = 2, 0, 1); z > 0 P(x0, g1 = 2, 1, 0);

z = 0

P(x0, g1 = 2, 1, 1);

z = 0

P(x0, g1 = 3, 0, 1, 1); z = 0

P(x0, g1 = 0); z> 0

P(x0, g1 = 2, 0, 0); z = 0

Fig. 4. Example of a tree generated by the implicit enumeration algorithm proposed by Bitran.


Let Rðg;x1;...;xgÞ ¼ Cðg;x1;...;xgÞB B�1N � CU

N, where the columns of the interval matrix, Cðg;x1;...;xgÞB , are defined as

Cðg;x1;...;xgÞB:j ¼

CUB:j; if j 6 g and xj ¼ U ;

CLB:j; if j 6 g and xj ¼ L;

½CLB:j;C

UB:j�; if g < j 6 m:

8>><>>:

In this context, g is the tree level (Fig. 5), m is the number of basic variables with interval coefficients, xj is L

(U) if the column elements of the matrix within the tree level j are in the lower bounds CLB:j (upper bounds CU

B:j)of the intervals and j = 1, . . . ,g.

Let RLðg;x1 ;...;xg Þand RUðg;x1 ;...;xg Þ

be composed of the lower and the upper bounds of each element belonging tothe interval matrix Rðg;x1;...;xgÞ, respectively. The operator ‘‘sc’’ is defined as

scðRUðg;x1 ;...;xgÞ Þ ¼ fRUðgþ1;x1 ;...;xg ;LÞ;RUðgþ1;x1 ;...;xg ;UÞg

and

scðRLðg;x1 ;...;xg Þ Þ ¼ fRLðgþ1;x1 ;...;xg ;LÞ;RLðgþ1;x1 ;...;xg ;UÞg:

The tree obtained by these definitions is shown in Fig. 5.Since Rðgþ1;x1;...;xg ;xgþ1

Þ � Rðg;x1;...;xgÞ, if the efficiency condition (see (22)) holds for the entire set of matricescontained in RLðg;x1 ;...;xgÞ

, then RLðg;x1 ;...;xg Þis efficient and Rðg;x1;...;xgÞ is necessarily efficient. Namely, if RL(0) is effi-

cient, then the matrices obtained through the branching process (sc(RL(0))) are necessarily efficient. Therefore,considering the operator ‘‘sc’’ sequentially as shown in Fig. 5, the entire set of matrices obtained after branch-ing is efficient, and, hence, R is necessarily efficient (Ida, 1999). Analogously, if RUðg;x1 ;...;xg Þ

is not efficient thenRðgþ1;x1;...;xg ;LÞ and Rðgþ1;x1;...;xg ;UÞ are not necessarily efficient (Ida, 1999).

The necessary efficiency test based on Bitran’s algorithm is obtained in the following way:

Step 1. Let SL ¼ fRLð0Þg.Step 2. Select one element RLðg;x1 ;...;xgÞ

from SL and check whether it is efficient.(a) If it is efficient then remove the element from SL.(b) Otherwise, add scðRLðg;x1 ;...;xgÞ Þ to SL. If Rðg;x1;...;xgÞ ¼ Rðm;x1;...;xmÞ, then R is not necessarily efficient.

Step 3. If the set SL is empty, then R is necessarily efficient.Step 4. Return to step 2.

This algorithm is based on the following theorem (Ida, 1999): If all the elements of SL are efficient, then R isnecessarily efficient.

If the solution being analyzed is not necessarily efficient, the implicit enumeration algorithm may be out ofacceptable computational limits due to the branching required. In this context, Ida (1999) proposed anotheralgorithm to test the necessary efficiency of a non-degenerate basic feasible solution.

sc ( U),1(R ) sc ( L)(1,R )

sc (R(0))

R(0)

L)(1,RU),1(R

L)L,(2,R U)L,(2,R L)U,(2,RU)U,(2,R

Fig. 5. Illustration of the branching process with the operator ‘‘sc’’.


The necessary efficiency test developed by Ida is carried out in the following way:

Step 1. Let SU ¼ fRUð0Þg.Step 2. Select one element RUðg;x1 ;...;xg Þ

from SU and check whether it is efficient.(a) If it is not efficient, then R is not necessarily efficient.(b) Otherwise, add scðRUðg;x1 ;...;xgÞ Þ to SU. If Rðg;x1;...;xgÞ ¼ Rðm;x1;...;xmÞ, do not add anything to SU.

Step 3. If the set SU is empty, then R is necessarily efficient.Step 4. Return to step 2.

This algorithm is based on the following theorem (Ida, 1999): If there is an element of SU that it is not effi-cient, then R is not efficient.

The efficiency check is based on Chernikova’s algorithm (Chernikova, 1965) and proceeds as follows:

Step 1. Compute R.Step 2. Analyze columns and rows of R and proceed as follows:

(a) If there are any columns in R such that R.j P 0, then eliminate these columns.(b) If there are any rows in R such that Ri. 6 0, then eliminate these rows.

Step 3. Analyze columns and rows of R and proceed as follows:(a) If there is a column in R such that R.j 6 0, then R is not efficient.(b) If there is a row in R such that Ri. > 0, then R is efficient.(c) If there is a row in R such that Ri. P 0 and a row i0 5 i such that Ri0j > 0 (Rij = 0), then R is efficient.

Step 4. Calculate the summation of the columns (R.R) and rows (R.R) of R.(a) If R.R 6 0, then R is not efficient.(b) If RR. > 0, then R is efficient.

Step 5. Let D =�R and process the rows by using the extreme ray generation method (Chernikova, 1965).Let, simultaneously, D = RT and process the rows in parallel by using the extreme ray generationmethod (D is defined in Chernikova (1965) and Ida (2000b)).

Step 6. Return to step 2.

The two efficiency tests can operate simultaneously, obtaining the result in a more efficient manner.Although the structure of this algorithm is similar to the previous one because it uses a branch and boundmethod, the efficiency tests do not require solving linear programming problems.

Ida (2000a,b, 2005) also suggested a solution generation procedure based on the extreme ray generationmethod that sequentially generates efficient points and rays by adding inequality constraints to the polyhedralfeasible region.

The proposed method is a non-pivoting method, which allows to generate the entire solution set. It is analternative method to the conventional pivoting algorithms, where linear programming problems are solvedto test whether an extreme point (or ray) is efficient or not. This method uses the properties of efficiency inthe objective space to conclude about the efficiency of the extreme points or rays obtained through the raygeneration method (see also Ida, 2003). Nevertheless, this method holds the disadvantages inherent to theextreme ray generation method, which is very sensitive to the ordering of rows of the input matrix. In fact,the use of this method in its primitive form has several drawbacks (Fukuda and Prodon, 1996): it can leadto prohibitive growth of intermediate generator matrices; it may generate an enormous amount of redundantgenerators, easily becoming beyond any tractable computational limit; if the input data is perturbed to solvedegeneracy, the size of perturbed output can grow exponentially.


Consider the following MOLP with interval coefficients in the objective function (Ida, 1999):

max Cx

s:t: : Ax 6 b;

x P 0;


C ¼

½1; 2� ½2; 3� ½�2;�1� ½3; 4� ½2; 3� ½0; 1� ½1; 2�

½�1; 0� ½1; 2� ½1; 2� ½2; 3� ½3; 4� ½1; 2� ½0; 1�

½3; 4� ½0; 1� ½1; 2� ½1; 2� ½0; 1� ½�2;�1� ½�2;�1�

2664

3775;

A ¼

1 2 1 1 2 1 2

�2 �1 0 1 2 0 1

�1 0 1 0 2 0 �2

0 1 2 �1 1 �2 �1

2666664

3777775; b ¼

16

16

16

16

2666664

3777775:

If we want to test the necessary efficiency of the extreme point ð0; 0; 323; 16

3; 0; 0; 0; 0; 32

3; 16

3; 0ÞT, then we obtain

B�1 ¼

13

0 0 13

23

0 0 � 13

� 23

1 0 13

� 13

0 1 � 13

2666664

3777775; N ¼

1 2 2 1 2 1 0

�2 �1 2 0 1 0 0

�1 0 2 0 �2 0 0

0 1 1 �2 �1 0 1

2666664

3777775 and

CUN ¼

2 3 3 1 2 0 0

0 2 4 2 1 0 0

4 1 1 �1 �1 0 0

2664

3775:

Thus,

Rð0Þ ¼ Cð0ÞB B�1N � CUN ¼

½�2;�1� ½3; 4� 0 0

½1; 2� ½2; 3� 0 0

½1; 2� ½1; 2� 0 0

2664

3775B�1N � CU

N

¼

� 23; 1

3

� �½�2; 0� ½�2; 0� 10

3; 5

� �73; 13

3

� �43; 7

3

� ��2;� 4

3

� �53; 8

3

� �½1; 3� ½�1; 1� ½0; 5

3� 8

3; 14

3

� �53; 8

3

� �� 2

3; 0

� �½�3;�2� ½1; 3� ½1; 3� 5

3; 10

3

� �½3; 5� ½1; 2� � 1

3; 1

3

� �2664

3775:

Hence,

RUð0Þ ¼

13

0 0 5 133

73� 4

383

3 1 53

143

83

0

�2 3 3 103

5 2 13

264

375 and RLð0Þ ¼

� 23�2 �2 10

373

43�2

53

1 �1 0 83

53� 2

3

�3 1 1 53

3 1 � 13

264

375:

Since RUð0Þ

2: P 0 and there is an element RUð0Þ

37 > 0 ðRUð0Þ

27 ¼ 0Þ, then RUð0Þ is efficient. On the other hand, sinceRLð0Þ

:7 6 0, then RLð0Þ is not efficient. Therefore, by the two efficiency tests, it may be necessary to examinethe efficiency of both R(1,U) and R(1,L).

Thus,

Rð1;UÞ ¼ Cð1;UÞB B�1N � CUN ¼

½�1;�1� ½3; 4� 0 0

½2; 2� ½2; 3� 0 0

½2; 2� ½1; 2� 0 0

2664

3775B�1N � CU

N

¼

� 13; 1

3

� �½�1; 0� ½�1; 0� 10

3; 14

3

� �83; 13

3

� �53; 7

3

� �� 5

3;� 4

3

� �2; 8

3

� �½2; 3� ½0; 1� 0; 4

3

� �3; 14

3

� �2; 8

3

� �� 1

3; 0

� �� 8

3;�2

� �½2; 3� ½2; 3� 5

3; 3

� �103; 5

� �43; 2

� �0; 1

3

� �2664

3775;


Rð1;LÞ ¼ Cð1;LÞB B�1N � CUN ¼

½�2;�2� ½3; 4� 0 0

½1; 1� ½2; 3� 0 0

½1; 1� ½1; 2� 0 0

264

375B�1N � CU

N

¼� 2

3; 0

� �½�2;�1� ½�2;�1� 11

3; 5

� �73; 4

� �43; 2

� ��2;� 5

3

� �53; 7

3

� �½1; 2� ½�1; 0� 1

3; 5

3

� �83; 13

3

� �53; 7

3

� �� 2

3;� 1

3

� ��3;� 7

3

� �½1; 2� ½1; 2� 2; 10

3

� �3; 14

3

� �1; 5

3

� �� 1

3; 0

� �264

375:

Hence,

RUð1;UÞ ¼

13

0 0 143

133

73� 4

3

83

3 1 43

143

83

0

�2 3 3 3 5 2 13

264

375; RLð1;UÞ ¼

� 13�1 �1 10

383

53� 5

3

2 2 0 0 3 2 � 13

� 83

2 2 53

103

43

0

264

375;

RUð1;LÞ ¼� 2

3�2 �2 11

373

43�2

53

1 �1 13

83

53� 2

3

�3 1 1 2 3 1 � 13

264

375 and RLð1;LÞ ¼

0 �1 �1 5 4 2 � 53

73

2 0 53

133

73� 1

3

� 73

2 2 103

143

53

0

264

375:

Since RUð1;UÞ

2: P 0 and there is an element RUð1;UÞ

37 > 0 (RUð1;UÞ

27 ¼ 0), then RUð1;UÞ is efficient. However, RLð1;UÞ is notefficient since RLð1;UÞ

:7 6 0. On the other hand, RLð1;LÞ is not efficient since RLð1;LÞ

:7 6 0. Finally, R(2,L,L) and R(2,L,U)

are not efficient because RUð1;LÞ

:7 6 0. Therefore, by the two efficiency tests we conclude that R is not necessarilyefficient (there is an element in SU, RUð1;LÞ , that it is not efficient) and the solution ð0; 0; 32

3; 16

3; 0; 0; 0; 0; 32

3; 16

3; 0ÞT

is not necessarily efficient.

3.3. Possibly efficient solutions: Methods and extensions

According to the theorems of the alternative, a basic feasible solution x is efficient to problem (20), withoutthe non-negativity constraints, if and only if there are g and b, such that (Inuiguchi and Sakawa, 1996):

Ax 6 b;

gA ¼ ð1þ bÞC;gðAx� bÞP 0;

g P 0; b P 0;

ð23Þ

where 1 is a vector of ones with convenient dimensions, g is an 1 · m vector and b is an 1 · p vector. If thesolution is feasible, then Ax 6 b holds and A and b can be represented by

A ¼ A0

A�

" #; b ¼ b0

b�

" #;

where A0 is a sub-matrix composed by the rows of A, such that Ai.x � bi = 0, Ai. is the ith row of A andA�x � b� < 0.

Let g = (g0,g�). Thus,

g0A0 þ g�A� ¼ ð1þ bÞC;g�ðA�x� b�ÞP 0;

g0 P 0; g� P 0; b P 0:

ð24Þ

In order to satisfy the system, g� should be zero. Then, a solution x is efficient if and only if there is (g0,b),such that:

g0A0 ¼ ð1þ bÞC;g0 P 0; b P 0:

ð25Þ


In this context, Inuiguchi and Sakawa (1996) have proposed a possibly efficiency test for a given feasible solu-tion. The possible efficiency of a given feasible solution, x, can be confirmed through the existence of (g0,b,C)such that:

g0A0 ¼ ð1þ bÞC; g0 P 0; b P 0; C 2 U: ð26Þ
Let the set U also be represented by
U ¼ fC : CL6 C 6 CUg: ð27Þ

Since b P 0, the system is consistent if and only if there is a solution (g0,b) such that

ð1þ bÞCL6 g0A0

6 ð1þ bÞCU; g0 P 0; b P 0: ð28Þ
Therefore, the possible efficiency of a given feasible solution can be confirmed by the consistency of a system ofinequalities.
Steuer (1981, 1986) developed three algorithms for solving LP problems in which some or all the objectivefunction coefficients are specified as intervals. The algorithms suitability depends upon the number of intervalobjective function coefficients, the number of nonzero objective function coefficients and whether the feasibleregion is bounded or not. These algorithms are designated by F-cone, E-cone and All Emanating Edges,respectively. The algorithms output the entire set of basic feasible solutions and the unbounded edges whichare possibly optimal (optimal for at least one of the coefficient vectors within their admissible range of vari-ation). In the case of problem (20) the F-cone algorithm can be adapted to output the entire set of possiblyefficient solutions.

Consider a closed and convex hyper-rectangle Uk with 2n extreme points (where n is the number of intervalcoefficients and, without loss of generality, all the coefficients are defined as intervals) such asff1

k ; f2k ; . . . ; f2n

k g ¼ ffk 2 Rn : fkj 2 fcLkj; c

Ukjg; j ¼ 1; . . . ; ng. Therefore, each ck 2 Uk can be expressed as a convex

combination of fqk and each linear convex combination of f

qk is an element of Uk. Hence, for each ck 2 Uk there

is a k 2 R2n, k P 0,

P2n

q¼1kq ¼ 1, such that ck ¼P2n

q¼1kqfqk . Let F be a matrix of p · 2n objective functions, since

(20) has p objective interval functions. A solution x* is weakly efficient with respect to F if and only if thereexists k 2 Rp�2n

, k P 0,Pp�2n

q¼1 kq ¼ 1, such that there is an x* that optimizes the following LP problem:

max zðxÞ ¼ kF x

s:t: : x 2 X ¼ fx 2 Rn : Ax 6 b; b 2 Rmg;

k 2 K ¼ k 2 Rp�2n: kq 2 ½0; 1�;

Xp�2n

q¼1

kq ¼ 1

( );

x P 0:

ð29Þ

In this way, problem (20) is equivalent to the following vector maximum problem:

max zðxÞ ¼ F x;

s:t: : x 2 X ¼ fx 2 Rn : Ax 6 b; b 2 Rmg;x P 0:

ð30Þ

The F-cone algorithm uses the convex cone generated by the rows of F, but because of the exponential growthof the number of objective functions in the subproblem used for testing the weak efficiency of each non-basicvariable it becomes easily beyond acceptable computational limits.

Wang and Wang (2001a,b) have used Telgen’s method (Telgen, 1982) for reducing the number of objectivefunctions in matrix F and, thus, obtain the irreducible generator of the criterion cone. Firstly, the semi-positivepolar cone of the criterion cone is determined through the concept of cone convexity (Yu, 1974). Furthermore,since the criterion cone of problem (29) corresponds to the closed convex cone generated by p · 2n gradients itis a polyhedron with at least p · 2n rays. The semi-positive polar cone of the criterion cone, XP, is obtainedthrough the solution of the following inequalities: f

qky P 0, q = 1, . . . , 2n, k = 1, . . . ,p. Let XP ¼

fy 2 Rn : �F y 6 0; �F y 6¼ 0; where �F ¼ ð�f1k ;�f2

k ; . . . ;�f2nk Þ

T; k ¼ 1; . . . ; pg [ f0g, and T means transpose.

Since the convex polyhedral set of XP is determined through a system of linear inequalities, reducing the


number of inequalities by means of Telgen’s method corresponds to reducing the number of objective func-tions. In this sense, the inequality �f

qky 6 0 is redundant if and only if the system �F v ¼ 0; vj P 0, "j 5 i,

vi =� 1, is consistent. If this system is consistent, then the corresponding objective function can be deleted.After the redundant relation is obtained, the size of the subsequent system decreases. Finally, the remaininginequalities, fq

ky P 0, for some q, are the irreducible representation of XP. Furthermore, since these fqk ’s are

irreducible, if there are no implicit equalities, a minimal representation of XP is obtained. Let {f1, f2, . . . , fs}represent an irreducible generator. Then problem (29) can be rewritten as the following reduced problem:

max zðxÞ ¼ ðf1x; f2x; . . . ; fsxÞT

s:t: : x 2 X ¼ fx 2 Rn : Ax 6 b; b 2 Rmg;x P 0:

ð31Þ

Wang and Wang (2001a,b) use the ‘‘e-constraint’’ method and the intra-parametric analysis method to obtainthe entire solution set of possibly efficient solutions for problem (20).

Consider that the first objective function of problem (31) is chosen to be optimized. Let hi correspond to anachievement rate for the optimum value reached for the remaining s � 1 objective functions, respectively.Therefore, problem (31) can be transformed in the following problem:

max z1ðxÞ ¼ f1x

s:t: :f ix� ziðxÞ

�ziðxÞ � ziðxÞP hi; i ¼ 2; . . . ; s;

hi 2 ½0; 1�; i ¼ 2; . . . ; s;

Ax 6 b;

x P 0;

ð32Þ

where �ziðxÞ ¼ maxx2X f ix; ziðxÞ ¼ minx2Ki fix; Ki ¼ fu : f iu ¼ maxx2X fjx; 8j 6¼ ig; �ziðxÞ and zi(x) are the

bounds for objective i in the possibly efficient set. When hi = 0, the lower bound of the ith objective functionzi(x) is selected as the minimal value. When hi = 1, the lower bound of the ith objective function is selected asthe maximal value �ziðxÞ. Consequently, the value of objective i is constrained by the maximum value in thepossibly feasible solution set. When problem (32) is solved through intra-parametric analysis to find the entirefeasible set, all possibly optimal bases are detected. When a basis does not satisfy the optimality condition, itcannot be an optimal basis and should be deleted. The largest tolerance regions of all possible bases can befound and the distribution of all solutions and corresponding regions can be obtained, yielding all possiblesolution-mixes within different regions (Wang and Wang, 2001b).

All possibly efficient solutions can be found by continuous changes of hi. For each optimal solution derivedfrom any hi 2 [0,1], i = 2, . . . , s, three cases might occur:

(1) All the objective constraints are binding at all solutions derived from the same h. In this case, the solu-tion satisfies the binding condition and is efficient. Let W be the region where the binding condition issatisfied. If a solution is obtained within the region h 2 W, then it is efficient.

(2) The solution does not satisfy the binding condition and there are alternative optima. Let H be the regionwhere h does not satisfy the binding condition and there are alternative optima. Therefore, two subcasesare considered:
(2a) The solution is efficient, then h 2 H;(2b) The solution is non-efficient.
(3) The solution is non-binding and unique. Let N be the region where h does not satisfy the binding con-dition and the solution is unique. If the solution is efficient then h 2 N.

As a result, every h corresponding to an efficient solution must belong to W [ H [ N = C. If a h 0 existswhich does not belong to this set, then all solutions derived from h 0 cannot be efficient. Apparently, generatingthe efficient set through the largest tolerance region of h is rather time-consuming. Wang and Wang (2001b)have demonstrated that an efficient solution generated by h 2 N can be obtained through another h 0 belonging


to H or W. In fact, the generation of the entire set of possibly efficient solutions can be reduced to continuouschanges of hi within W [ H, for i = 2, . . . ,q.

Furthermore, the possibly efficient set can also be obtained using W [ H and {h 0}. The h’s which are dupli-cated in H can be obtained and deleted from H. After this procedure let the remaining region of H be denotedby H1. Secondly, if more than one efficient solution is available in H1 all pairs of them are considered and all hduplicates are deleted. Let H2 be the remaining set of H1 and let P = W [ H2. Two cases can be considered: (i)h,h 0 2 W; (ii) h 2 W, h 0 2 H1 or h, h 0 2 H1 (Wang and Wang, 2001b).

The first case is not possible because if h, h 0 2 W, with h 5 h 0, then x*(h) 5 x*(h 0) and, hence, there are noduplicates (Wang and Wang, 2001b). On the other hand, the second case is also impossible, since all pairs ofefficient solutions obtained from W and H with duplicate h corresponding to the same efficient solution aredeleted. Therefore, P is the smallest region of h corresponding to the possibly efficient set.

This method can be extended when the RHSs of constraints are defined as intervals (for more details seeWang and Wang (2001b)). Although the methodology herein described allows obtaining the irreducible crite-rion cone, the starting step has the same drawbacks of the F-cone algorithm, because the number of objectivefunctions may be easily out of acceptable computational limits.

3.3.1. Illustrative examples

Consider the interval multiobjective problem given in the previous example. Suppose we want to test thepossible efficiency of the extreme point (16, 0, 0, 0, 0, 0, 0, 0, 48, 32, 16)T. Inuiguchi and Kume (1989) havealso proposed a simpler test for basic solutions.

Note that, since the MOLP problem considered by Inuiguchi and Sakawa (1996) does not include non-neg-ativity constraints, these must then be taken into account in matrix A. Hence,

A ¼

1 2 1 1 2 1 2

�2 �1 0 1 2 0 1

�1 0 1 0 2 0 �2

0 1 2 �1 1 �2 �1

�1 0 0 0 0 0 0

0 �1 0 0 0 0 0

0 0 �1 0 0 0 0

0 0 0 �1 0 0 0

0 0 0 0 �1 0 0

0 0 0 0 0 �1 0

0 0 0 0 0 0 �1

26666666666666666666666664

37777777777777777777777775

:

Therefore,

A0 ¼

1 2 1 1 2 1 2

0 �1 0 0 0 0 0

0 0 �1 0 0 0 0

0 0 0 �1 0 0 0

0 0 0 0 �1 0 0

0 0 0 0 0 �1 0

0 0 0 0 0 0 �1

266666666666664

377777777777775:


Then, the solution is possibly efficient if and only if the following system is consistent:

g1 � 2b1 � 4b3 6 6;

g1 � b1 þ b2 � 3b3 P 3;

2g1 � g2 � 3b1 � 2b2 � b3 6 6;

2g1 � g2 � 2b1 � b2 P 3;

g1 � g3 þ b1 � 2b2 � 2b3 6 3;

g1 � g3 þ 2b1 � b2 � b3 P 0;

g1 � g4 � 4b1 � 3b2 � 2b3 6 9;

g1 � g4 � 3b1 � 2b2 � b3 P 6;

2g1 � g5 � 3b1 � 4b2 � b3 6 8;

2g1 � g5 � 2b1 � 3b2 P 5;

g1 � g6 � b1 � 2b2 þ b3 6 2;

g1 � g6 � b2 þ 2b3 P �1;

2g1 � g7 � 2b1 � b2 þ b3 6 2;

2g1 � g7 � b1 þ 2b3 P �1;

g1; g2; g3; g4; g5; g6; g7; b1; b2; b3 P 0:

Since the system has the solution g1 = 7.5, g2 = 12, g3 = 6, g4 = 0, g5 = 10, g6 = 7, g7 = 14.5, b1 = 0, b2 = 0and b3 = 1.5, we conclude that this solution is possibly efficient.

Consider also the following example (Wang and Wang, 2001b):

max z1ðxÞ ¼ ½1; 2�x1 þ ½2; 3�x2

max z2ðxÞ ¼ ½�3;�2�x1 þ ½0; 1�x3

s:t: : 2x1 þ x3 6 6;

� x1 þ 3x2 6 6;

� 4x1 þ 6x2 þ 3x3 6 18;

x1; x2; x3 P 0:

In order to compute all possible solution-mixes within different regions, the irreducible gradient cone must beobtained. In this problem there are two objective functions, with two interval coefficients each. Hence, it isnecessary to consider the following (2 · 22) 8 objective functions: z1 = x1 + 2x2; z2 = x1 + 3x2; z3 =2x1 + 2x2; z4 = 2x1 + 3x2; z5 =� 3x1; z6 =� 3x1 + x3; z7 =� 2x1; z8 =� 2x1 + x3. The gradients of z7 andz5 have the same direction and can be omitted. Therefore, the �F matrix that defines the semi-positive polarcone is

�F ¼�1 �1 �2 �2 3 3 2

�2 �3 �2 �3 0 0 0

0 0 0 0 0 �1 �1

264

375:

In order to verify the redundancy of z1 the following system is solved:

�1 �1 �2 �2 3 3 2

�2 �3 �2 �3 0 0 0

0 0 0 0 0 �1 �1

264

375�

�1

v2

v3

v4

v5

v6

v7

2666666666664

3777777777775¼

0

0

0

264375; vi P 0; i ¼ 2; . . . ; 7:


This system is consistent and one of its solutions is v2 = 0.5, v3 = 0.25, v4 = 0, v5 = 0, v6 = 0, v7 = 0. In thiscase, z1 is redundant and can be omitted. In a similar way, we conclude that z4 and z6 are redundant. Hence,the irreducible gradient is composed of z1(x) = 2x1 + 2x2, z2(x) =� 3x1 and z3(x) =� 2x1 + x3.

If the ‘‘e-constraint’’ method is used, the following problem is solved:

max z1ðxÞ ¼ 2x1 þ 2x2

s:t: : � 3x1 P 9h2 � 9;

� 2x1 þ x3 P 12h3 � 6;

2x1 þ x3 6 6;

� x1 þ 3x2 6 6;

� 4x1 þ 6x2 þ 3x3 6 18;

x1; x2; x3 P 0:

If h2 = 0 and h3 = 0, then

B�11 ¼

13

0 0 0 0

� 23

1 0 0 0

� 23

0 1 0 019

0 0 13

023

0 0 �2 1

26666664

37777775:

With b1 = 9 � 9h2 and b2 = 6 � 12h3, b1 and b2 corresponding to the two first elements of the RHS vector,x1(h) = (3 � 3h2,3 � h2,0) if and only if h2, h3 2 G1 = {(h2,h3) : h2 � 2h3 P 0, h2,h3 2 [0,1]}. Similarly, the en-tire set of possibly efficient solutions and their corresponding domains are obtained through continuouschanges of h2 and h3 (Fig. 6). Solutions x1(h) and x6(h) are two alternative optimal solutions within the regionG1 \ G6, solutions x1(h) and x4(h) are two alternative optimal solutions within the region G1 \ G4, and soforth.

θ2

θ3

G5 ∩ G6:

x5(θ),

x6(θ)

G7 :x7(θ)

G1 ∩ G6:x1(θ), x6(θ)

G1 ∩ G4 :x1(θ), x4(θ)

G4 ∩ G5:

x4(θ), x5(θ)

G3: x3(θ)

(1, 1)

0.5

0.5(0, 0)

G2: x2(θ)

Fig. 6. Efficient solutions with respect to different regions.


If we analyze each one of the solutions obtained with respect to the objective constraints we verify that forsolution x1(h): 3(3 � 3h2) = 9h2 � 9 and 2(3 � 3h2) � 0 6 6 � 12h3 if h2 = 2h3. After repeating this procedurefor the entire set of solutions, we conclude that when h is selected from the line h2 = h3 or from the region G3

the binding condition is satisfied. Thus, W = {(h2,h3) : h2 = h3, h2,h3 2 [0, 1]} [ G3. For those alternative solu-tions which do not satisfy the binding condition, for example, x1(h) and x6(h), when we choose h from the lineh2 = 2h3, x6(h) does not satisfy the binding condition and x1(h) satisfies the binding condition. Then, in thiscase, solution x1(h) is dominated by solution x6(h). In the same way, we conclude that x1(h) is dominatedby x4(h), x5(h) is dominated by x4(h), and solution x5(h) is dominated by x6(h). Thus, H = G6 [ G4. For theunique optimal solutions which do not satisfy the binding condition x2(h) and x7(h), since it is possible toobtain h 0 2 H, these solutions are possibly efficient.

Thus, N = G2 [ G7. The largest tolerance region is C = G6 [ G4 [ G3 [ G2 [ G7 and the corresponding solu-tions are

x6ðhÞ ¼ ð3� 3h2; 3� h2; 6h2Þ; if ðh2; h3Þ 2 G6;

x4ðhÞ ¼ ð3� 3h2; 3� h2; 4� 2h2Þ; if ðh2; h3Þ 2 G4;

x3ðhÞ ¼ ð3� 3h2; 5þ h2 � 6h3;�6h2 þ 12h3Þ; if ðh2; h3Þ 2 G3;

x2ðhÞ ¼ ð3� 3h2; 5� 5h3; 6h3Þ; if ðh2; h3Þ 2 G2;

x7ðhÞ ¼ ð3� 3h3; 3� h3; 6h3Þ; if ðh2; h3Þ 2 G7:

8>>>>>><>>>>>>:

The smallest tolerance region, P, is

x6ðhÞ ¼ ð3� 3h2; 3� h2; 6h2Þ; if h2 2 ½0; 0:5�; h2 ¼ h3;

x3ðhÞ ¼ ð3� 3h2; 5þ h2 � 6h3;�6h2 þ 12h3Þ; if ðh2; h3Þ 2 G3:

If (h2,h3) = (h2,0.25) for any h2 2 [0,0.25] in G7, the solution x7 = (2.25,2.75, 1.5) is obtained. This solutioncan also be obtained with (h2,h3) = (0.25,0.25) 2 G6. Hence, x7(h) and its corresponding domain can be omit-ted. If (h2,h3) = (0.8,h3), for any h3 2 [0,0.6] in G4 the solution x4 = (0.6,2.2,2.4) is obtained. This solution canalso be obtained with (h2,h3) = (0.8,0.6) 2 G3. Therefore, x4(h) and its corresponding domain can be omitted.Finally, if (h2,h3) = (h2,0.6) for any h2 2 [0, 0.6] in G2, the solution x2 = (0.6,2,3.6) is obtained. This solutioncan also be obtained with (h2,h3) = (0.6,0.6) 2 G3. Consequently, x2(h) and its corresponding domain can beomitted. Thus, the complete possibly efficient set can be obtained from region P (Fig. 6).

4. Conclusions

Interval programming is a relevant tool to address the intrinsic uncertainty in models of real-world prob-lems, mainly because it does not require the specification or the assumption of possibilistic or probabilisticdistributions for the coefficients.

This paper provides an illustrated overview of the different approaches to deal with uncertainty in MOLPmodels using interval programming. The satisficing and optimizing approaches have been reviewed and illus-trated by means of small numerical examples. The future course of our work will consist in the study of pro-cedures aimed at making the most of the distinct methodological approaches herein reviewed in order toprovide effective decision support to DMs, paying also attention to the minimization of the computer effort.

Acknowledgements

This research has been partially supported by FCT and FEDER under Project Grant POSI/SRI/37346/2001 and by FCT under PhD Grant SFRH/BD/17540/2004. The authors are indebted to Prof. M. Inuiguchi(Osaka University) for his insightful help in the example in Section 3.3.1. The authors are also grateful to twoanonymous reviewers for their constructive comments on an earlier version of this paper.


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