American Journal of Oper ations Research, 2011, 1, 203-213
doi:10.4236/ajor.2011.14023 Published Online December 2011 (http://www.SciRP.org/journal/ajor)
203
Multiobjective Stochastic Linear Programming:
An Overview
A. Segun Adeyefa, Monga K. Luhandjula*
Department of Deci sio n Sci e n ces , Uni versi t y of So ut h Africa, Pretoria, South Africa
Received June 15, 2011; revised July 10, 2011; accept ed July 29, 2011
Abstract
Many Optimization problems in engineering and economics involve the challenging task of pondering both
conflicting goals and random data. In this paper, we give an up-to-date overview of how important ideas
from optimization, probability theory and multicriteria decision analysis are interwoven to address situations
where the presence of several objective functions and the stochastic nature of data are under one roof in a
linear optimization context. In this way users of these models are not bound to caricature their problems by
arbitrarily squeezing different objective functions into one and by blindly accepting fixed values in lieu of
imprecise ones.
Keywords: Linear Programming, Multiobjective Programming, Stochastic Programming, Expected Value
Optimality/Efficiency, Variance Optimality/Efficiency, Expected Value/Standard Deviation
Efficiency, Tammer Optimality, Minimum Risk Optimality/Efficiency, Optimality/Efficiency in
Probabilities
1. Introduction
Many concrete real life problems may be put into a Lin-
ear Programming framework (see e.g., [1-8]). For some
of these problems, the Decision maker has to ponder
conflicting objective functions. Such competing goals
cannot be arbitrarily squeezed within the narrow frame-
work of a unique objective function, without running the
risk of invalidating all implications that are supposed to
be drawn from the analysis. Simple examples (see e.g.,
[9-13]) are in line with the endorsed paradox [14] and the
Arrow’s impossibility Theorem [15], where there are no
good ways of aggregating conflicting criteria into a sin-
gle one. This has given rise to the field of Multiobjective
Programming (MOP). For discussions on Multiobjective
Programming problems, the reader may consult [16-21].
Over and above the presence of several conflicting
goals, the above mentioned problems may involve some
level of uncertainty about the values to be assigned to
various parameters. In this connection the noted phi-
losopher Nietzche was quoted as saying,
“No one is gifted with immaculate perception”.
False certainty is bad science and it could be danger-
ous if it stunts articulation of critical choices. Interested
readers are referred to [22-32] for problems where uncer-
tainty should be accommodated in an optimization setting.
Uncertainty presents unique difficulties in constrained
optimization problems, because the Decision makers are
faced with doubtful situations, requiring an analysis of
multiple outcomes in different states of nature. When the
uncertainty in question is stochastic in nature, then we
enter the field of Multiobjective Stochastic Linear Pro-
gramming (MOSLP); the subject matter of this paper.
In such a turbulent environment, the notion of “opti-
mum optimorum” no longer applies. One has, then, to
resort to the notion of satisficing solution, based upon
Simon’s bounded rationality principle [33].
Methods for singling out a compromise solution in a
MOSLP problem have been developed in the literature,
leading to three main trends, namely: the hard, the soft
and the metaheuristics. For the first trend, we refer the
reader to [34-37]. For the second one, the reader may
consult [38,39]. Examples of the third trend may be
found in [40-42].
Within each group, the original problem may be either
reduced to a single objective stochastic program (stochas-
tic approach) or converted to a deterministic multiobjec-
tive program (multiobjective approach). A third alternative
is to combine in an appropriate manner a technique of
single objective Stochastic Programming with a technique

A. S. ADEYEFA ET AL.
204
of Multiobjective Programming (hybrid approach).
For the sake of space, this review focuses on the hard
trend. An interested reader is referred to [43-47] where
he may find details about the other two trends.
The methodological line followed in this overview
consists of discussing upstream, existing solution con-
cepts and placing extant results in a coherent and com-
putational framework. Some existing applications are
then listed downstream.
We also take a step towards comparing the approaches
mentioned. Such a comparison may help in designing a
Decision Support System for MOSLP. The above men-
tioned extension is outside the scope of the present paper,
and has therefore, been left for further research.
Despite the purely mathematical nature of many works
in the field of MOSLP as illustrated in [48-50], research
in this field has been suggested by a specific class of con-
crete, real-life problems. Such a class of problems includes
reservoir operation [51], coal mining [11], water resource
management [52] and transportation planning [14].
The paper is organized as follows. In the next section,
we give a mathematical formulation of the problem at
hand and discuss related solution concepts. Section 3
deals with some mathematical results in connection with
MOSLP. Section 4 is devoted to a discussion of meth-
odological aspects of MOSLP along with a comparison
of the above mentioned approaches. In Section 5, we
point out some existing applications. We end with a
number of concluding remarks along with suggestions
for further developments in this field.
2. Problem Formulation and Solution
Concepts
2.1. Problem Formulation
A Multiobjective Stochastic Linear Programming pro-
blem is a problem of the type:
 

1
()
,,
min K
xD cxc x



 (1)
where
 

():; 0
n
DxAxbx   
 
1,, K
cc

  are n-dimensional random vectors
defined on a probability space

, ,,


A

and

b

are respectively m × n and m × 1 random matrices
defined on the same probability space.
As an example of a concrete problem that may be put
into the form of (1), we mention the automated manufac-
turing system in a production planning situation, with
several objective functions, where the costs and time of
production are known only stochastically [53].
For other problems that may be modelled in the same
way as (1), we may mention reconfigurable manufac-
turing systems [40], distributed energy resources planning
[54], water use planning [55], manufacturing planning
[56], power systems planning [57-59] energy and reserves
markets [60] and multi-product batch plant design [61].
Owing to the presence of conflicting goals and the
randomness surrounding data, the mathematical program
described in (1) is an ill-stated problem. Therefore,
neither the notion of feasibility nor that of optimality is
clearly defined for this problem. One, then, has to resort
to the Simon’s bounded rationality principle [33] and
seek for a satisficing solution instead of an optimal one.
Before discussing some existing solution concepts for
this problems along with some related mathematical
results and methodological approaches, let us attempt to
provide some meaning to problem (1).
2.2. Transformation of the Feasible Set
One generally transform

D

to a deterministic set,
say according to the rules used in Stochastic Pro-
gramming (see e.g. [62-64]).
D
Some commonly used deterministic counterparts of


D

are listed below:
1)


  


:;
n
DxEA xEbx 
0


where
E
stands for the expected value.
2)
 




:;
n
DxPAxb x 
 0



where

is a probability level pre-defined by the Deci-
sion maker.
3)

11
,, m
mi
i
D

i
D



 
 where for each fixed


=1, ,im
  




:,
n
iiii i
DxPAxb x  0 


here i

are probability levels a-priori fixed by the
Decision maker and


i
A

,

i
b

are respectively
the row of
th
i


A

and the component of
th
i


b

.
4)




probability 1Dx:
n,Qx
 , with
iv
where


inf ;
, if
; if
qy
Qx y





 





where


q

is a penalty cost,


W

is a recourse
matrix and


 
:;
m
yWybAxy  0

 
In the next section, we discuss some existing solution

205 A. S. ADEYEFA ET AL.
concepts for MOSLP problems.
3. Solution Concepts for Multiobjective
Stochastic Linear Programming Problems
To avoid complications unrelated to our subject, we as-
sume that involved random data have known distributions
with finite expected values and variances.
3.1. Expected Value and Variance Optimalities
Consider the following deterministic mathematical pro-
grams:

 
min
xDEc x

 (2)

 
min
xD
Vc x 
 (3)
with E and V denoting the expected value and the
variance respectively.
Definition 3.1 If *
x
is an optimal solution for Pro-
gram (2), ((3)) then *
x
is called an expected value (a
variance) optimal solution for problem (1), when D is a
transformation of

D


obtained through technique of
stochastic optimization.
Where

c

is an aggregation of



1,, K
cc

 
based on techniques of multiattribute utility theory [65].
From now on
E

and V

stand respectively for the
set of expected value and variance optimal solutions for
problem (2).
A shortcoming of the above defined solution concepts
is that, the expected value and the variance do not
exhaust the information contained in the distributions of
involved random variables [34]. To overcome this draw-
back, other solution concepts have been proposed. We
discuss some of them in the next three subsections.
3.2. Tammer and Minimum Risk Optimalities
and Optimality in Probability
Definition 3.2 *
x
is a Tammer

-optimal solution for
Problem (1), if there is no
x
D such that




*
:1Pcxcx

   
and




*
:<Pcxcx   >0
when D is a transformation of

D

obtained through
technique of stochastic optimization. Here

is a pro-
bability level pre-defined by the Decision maker.
For details on this solution concept, we invite the
reader to consult [66].
Definition 3.3 *
x
is an

-minimum risk optimal
solution for Problem (1), if *
x
is an optimal solution
for the following program:



max
xDPc x


 (4)
when D is a transformation of

D

obtained through
technique of stochastic optimization. Where

is an
aspiration level a-priori fixed by the Decision maker.
An interested reader is referred to [67] for key facts
about the minimum risk solution concept.
Definition 3.4 *
x
is a

-optimal solution in proba-
bility for Problem (1), if there is such that
*
 


**
,x

is optimal for the program:

 
,min
subject to
=
xD
Pc

x



R

D
(5)
when D is a transformation of


obtained through
technique of stochastic optimization. Where

is a
probability level pre-defined by the Decision maker.
A reader interested to know more about this solution
concept is referred to [68].
3.3. Expected Value and Variance Efficiencies
Consider the following deterministic multiobjective pro-
grams:







1
minK
xD Ec xcx ,,E
  
(6)







1
minK
xD Vc xcx ,,V
  
(7)











 
11
,,, ,,
min
xD
K
Ec xEcxc x
cx




K

(8)
where

stands for the standard deviation.
Definition 3.5 *
x
is called an expected value, a
variance or an expected value/standard deviation effi-
cient solution for problem (1), If *
x
is efficient for
Programs (6), (7) or (8) respectively, when is a
transformation of D


D

obtained through technique of
stochastic optimization.
The sets of expected value, variance and expected
value/standard deviation efficient solutions for Program
(1) are denoted by
E

, V

and E


respectively.
The concept of expected value weak efficiency,
variance weak efficiency and expected value/standard
deviation weak efficiency and those of expected value
proper efficiency, variance proper efficiency and ex-
pected value/standard deviation proper efficiency are
obtained by replacing “efficiency” by “weak efficiency”
and by “proper efficiency” respectively.

A. S. ADEYEFA ET AL.
206
In the sequel w
E

(),
p
E

p
E

(
p
V

) and w
E


(
p
E


)
denote the sets of expected value weakly (properly)
efficient solutions, variance weakly (properly) efficient
solutions and expected value/standard deviation weakly
(properly) efficient solutions for program (1) respectively.
3.4. Minimum Risk Efficiency and Efficiency in
Probabilities
Minimum risk efficiency is defined as follows.
Definition 3.6 *
x
is an


1,,
K

 
*-minimum risk
efficient solution for problem (1), if
x
is efficient for
the multiobjective program:

 

 
1
1,,
max KK
xD Pc xPcx
 


 (9)
when D is a transformation of

D

obtained through
technique of stochastic optimization. Here

1,,
K

 
are aspiration levels a-priori fixed by the Decision maker.
Characterizations of minimum risk efficiency with
aspiration levels, may be found elsewhere [69].
As in the case of expected value efficiency, the
concepts of

1,,

K

 

-minimum risk weak efficiency
and 1

,,
K

 -minimum risk proper efficiency may be
obtained by respectively replacing “efficiency” by “weak
efficiency” or “proper efficiency” in the above definition.
In what follows

1,,
M
RK

 ,

1,,
w
M
RK

 
and 1

,,
p

M
RK

   denote the sets of

1,,
K

 
,,
-
minimum risk efficient solutions,

1
K

 

1,,
-mini-
mum risk weakly efficient solutions and

K

 -
minimum risk properly efficient solutions for Program (1)
respectively.
For efficiency with given probabilities we give the
following definition.
Definition 3.7 *
x
is a

1,,

K

 -efficient solution
in probability for problem (1), if there is
1

**
=,,

*
K
 
 such that


**
,x

is efficient for the
mathematical program:




1
,
,,
min
subject to
,=1, ,
K
K
xD
kkk
Pc xkK





R


(10)
when D is a transformation of

D

obtained through
technique of stochastic optimization. Where 1,,
K


are probability levels that are a-priori fixed by the
Decision maker.
An interested reader may consult [14] for a thorough
discussion on this efficiency concept.
Concepts of

1,,
K

 
,,
-weak efficiency in pro-
bability and

1

K

 -proper efficiency in pro-
bability may also be obtained in a way similar to the one
in which minimum risk weak and proper efficiencies
were obtained.
From now on


1,,
K
TK

 ,

1,,
w
K
TK
 
 and


1,,
P
K
TK

 denote the sets of

1,,
K

 

1,,
-effi-
cient solutions in probability,
K

 
1,,
-weakly
efficient solutions in probability and

K

 -
properly efficient solutions in probability for Program (1)
respectively.
In the next section we present some theoretical results
related to problem (1).
4. Related Mathematical Results
Most stochastic constraint transformations yield noncon-
vexity on resulting deterministic feasible sets. This pre-
cludes the application of existing powerful convex opti-
mization algorithms (see e.g. [70,71]). It is therefore,
relevant to know when a deterministic counterpart of


D

is convex. The following four propositions; the
proofs of which may be found in [72], provide some
insights to this issue.
Proposition 4.1


0D

,


1D , ,

0; =1,,
i
Dim


1;1, ,
i
Dimiv
D and are convex sets.
Proposition 4.2 Consider Problem (1) and suppose
that


A

is a fixed matrix with maximal rank. Then


():(); 1,,
n
iiii i
DxFAxi   m
are convex for every probability distribution i
F
of


i
b

.
Proposition 4.3 Assume that the probability space
under consideration is discrete, that is,


1,,
L

  
and


0
ll
Pp 

, 1, ,lL

. Let


l
L
*
 max 1:1
l
pl then the set


ll
D

is
convex for any l
*
>
l

 D and


 is convex for any
*
>,


*
where

and *
l

are real numbers.
Proposition 4.4 Suppose that the probability space
under consideration is

1,,

L

   and suppose
that


=>
ll
pP  0 if and only if


1, ,lN r
N. Assume also that only one element exists such
that o
l
min
ll
olN
pp


,
then the sets


D

 and

l
D

 are convex for every
1
>1 l p


where

1\{ }
ll
lN
l
o
minpp

.
The next two results established in [73,74], bridge the
gap between solution concepts based on the first two
moments (Proposition 4.5) and establish a connection
between a minimum risk efficient solution with aspiration
levels and an efficient solution with given probabilities
(Proposition 4.6).
Proposition 4.5
1) EV E

 


207 A. S. ADEYEFA ET AL.
2) w
EV E

 

3) ww w
EV E

 

AsProposition 4.6sume that the probability distribu-
tions of the random vectors
 
1,, K
cc

  are con-
tinuous and strictly increasing

1,,
. Then for any
K
K
 ,


*
1,,
M
RK
x

  if a only if nd

*
1,,
K
TK
x

, where
k


kk
Pc

x

  ;

1, ,kK
Moreover, we have:

1
Proposition 4.7

1
,




1
1,,
,,
, ,,
KK
K
M
RK KT
B
R


K



 







with




1,,:0,1; 1,, Kk
B  

kK
Well-known characterizations of proper efficiency
have been explored to relate optimality and efficiency of
program (1). This is the subject matter of the next two
propositions.
Proposition 4.8 If *
x
is an expected value optimal
solution for problem (2 then * ),
x
is an expected value
properly efficient solution for program (6). That is,
P
E
E

 
Proposition 4.9 If is a convex set and ions, then D
,


k
c x  ; =1,kK are convex functE
*
x
is an
the expected v palueroperly efficient solution for multiobjective program (6), if and only if, *
x
is an
expected value optimal solution for the problem ). That
is, (2
=
P
E
E


An interested reader is referred to [14] for more details
on
ogical Approaches for Solving
hed in the previous sections have served
tochastic Approach
ethod described in [38] for
olving problem (1), using the stochastic approach. For
this matter.
. Methodol5Multiobjective Stochastic Linear
Programs
e ideas discussT
as guidelines in implementing efficient techniques for
solving Multiobjective Stochastic Linear Programming
problems.
In what follows we outline a method within each of
the three existing approaches namely, the stochastic
approach, the multiobjective approach and the hybrid
one.
.1. S5
In this section we present a m
s
this method the following assumptions should be met:


i
A

, =1,,im;


b

and

k
c

, =1, ,kK
are normally distributed random vectors. k

,
,
interval
=1,kK are strictly positive real numbers in the


0,1 such that =1
K

.
=1 k k
the follo notations are used: Moreovewing
1)
r,




=hxAx

,b

iii

 1,, =,im.
2)

denotes the cumulative distribution function of
the standard normal random variable.
ion of
3) 1 and 2 q are weights associated with the ex-
pected value and the standard deviat
q


c

res-
pectiv.
4)
ely


1
=,,
m

  where i

, =1, ,im pro-
bability le on m
are
bed by the Decisiaker for
constraints satisfaction
vels prescri
.
A stepwise description of the method is as follows:
Step 1. Read k

, =k1, ,;
K


k
c

, =1, ,kK;


,x  , 1, ,im
i
h

; i

, =1,,im
Step 2. Find
K
 
=1
=k
k
k
cc





D

by Step 3. Replace




  
1
(),0,1,,;0
vn
i
ii
h
hx imx

 

:,DxRE x

 
Step 4. Solve the mathematical program:







12
min
v
xD qEcqc


 (11)
Let *
x
Step 5. Stop.
be a solution of (11).
an transforms the original
prngle objective problem, that has been
pu
As cbe seen, this algorithm
oblem into a si
t in the deterministic form (11), using the expected
value model approach [63].
The solution *
x
obtained is an expected value/stan-
dard deviation efficient solution for problem (1) as de-
fin
olving problem (1) include, decomposition
m
ach
n the multiobjective
proach. For this method, we need
ed in §3.1.
Other techniques closely related to the stochastic
approach for s
ethod [75-77], chance-constrained method [4,78], si-
mulation based techniques [79-81], two stage method [61]
and multistage method [82].
5.2. Multiobjective Appro
Here we outline a method withi
ap k

, =1, ,kK;
such that >0
k

, 11
K
k
k


 as in §5.1.

A. S. ADEYEFA ET AL.
208
:
Read
The steps of the method are as follow
Step 1. k

, =1, ,kK;

k
c

,

A=1, ,kK;

;


b

.
Step 2. Replace


D

by

 

 
:0DxEA xEbx   
n
R
 ;0 (12)
Step 3. Find:




1,, K
Ec Ec

 
Step 4. Solve the mathematical program:
(13)
Let


min Kk Ecx 



=1 k
xD k


*
x
Step 5. Stop.
be a solution of (13)
is mtackle randomness, while
St objective functions. The
so
In thethod, Steps 2 and 3
ep 4 als with multiplicity of de
lution *
x
obtained is an expected value efficient solu-
tion for MOSLP problem (1) as defined in §3.3.
For a more thorough discussion of other methods for
solving MOSLP problem (1) based on the multiobjective
approach, the reader is referred to [83-89].
5.3. Hybrid Approach
In this section, we describe
for solving MOSLP proble
a hybrid method due to [90],
m (1). This method is based
on the assumptions given in §5.1. The following nota-
tions are used in the sequel.
1)
s

, =1,,
s
S; t

, =1, ,tT; u

, u


,
=1, ,uU denote positive,and two sided
deviations from targets
negative
s
g
, =1,,
s
S; t
g
,
=1, u ,tT;
g
, =1, ,uU respectively. S, T and
U are respectively the total number of positive, negative
and two-sided deviationsm targets fro
s
g
, t
g
anu d
g
.
2)
s

, =1,,
s
S; t

, =1, ,tT; u

,
=1, ,uU are probability levels a-priori fixed by the
Decision maker.
Herhe steps of the method.
Step 1. Read S, T, U,
e are t
s
g
,
s

, =1,,
s
S; t
g
,
t

, =1, ,tT; u
g
, u

, =1, ,uU;


k
c

,
=1, ,K; k


,hx  , =1,,im
i
Step 2. Put


D

the following form

in:

 










1
1
1
:
)0
s
cx g  

 ( ,
1,,;
1()0,
1,,;
1
0
2
vi s
sss
t
t
ttt
u
k
uu
Dx Ex
sSEcx
cxg
tTEcx
cxg



 








 



 





nc



  

1
1,,;,
,0
1,,;0,0, 0
i
ii
st
uUEhx
hx
im x

 




 




Step 3. Solve the mathematical program:
1
T

11
min
vi
US
uus t
xD ust






 





(14)
Let *
x
Step 4. Stop.
be a solution of (14).
It is clear that this method combines the goal program-
ming technique for solving a multiobjective program
with the chance-constrained method for soing a sto-
chastic optimization problem.
Other methods pertaining to the hybrid approach may
be
5.4. Comparison of Different Approach
Thinn while comparing the
abed above are as follow:
nt than
requires
lv
found in [91-93].
es
e ma lessons that can be draw
pproaches outlinove described a
1) The stochastic approach takes into account depen-
dencies between objective functions, whereas the multi-
objective approach does not (see for example [94]). This
makes the stochastic approach closer to reality. Therefore,
the stochastic approach is more effective for finding
solutions to a MOSLP problem than the multiobjective
pproach. a
2) The multiobjective approach is more efficie
he stochastic approach, in the sense that it t
fewer computations. These computations are easier to
handle than those required by the stochastic approach.
(see e.g., [49,58,95]).
3) The hybrid approach combines the strengths of the
stochastic and the multiobjective approaches. Conse-
quently, the hybrid approach could perform better than
either of the other two approaches for a given problem.
Interested readers may consult [96] for a substantiation
of this claim.
4) Methods pertaining to the hybrid approach create
more flexibility in allowing the Decision maker to
specify his preferences (see e.g., [91]).
Nevertheless, it is the nature and the structure of the
problem that determines which approach to use.
In what follows, we briefly discuss some applications
of Multiobjective Stochastic Linear Programming to
concrete real-life problems.
6. Applications
6.1. Applications of the Stochastic Approach
Production planning problems, lend themselves better to

A. S. ADEYEFA ET AL. 209
matter of fact,
th
ns and
la
wer system security problem
preventive maintenance scheduling
ing [99], hydro-thermal electricity
res-
ant
esign [61].
problems [86],
ar
rtation network design problem [85] and
.
6.3.s within the Hybrid Approach
s, it is
be
ch, to
e
thods described in this field are valuable
re
caricature the underlying problem by
itional (deterministic)
Paraphrasing Howard [110], the scientific approach to
es, along with the meaning of
in
of MOSLP. To cater best for a
br
logical aspects and applications)
ha
ration Research techniques ignore managerial
ne
been
us
nagement needs have evolved and are more
co
uided
an
cy
ob
velopments in this field we
m
to
ltiobjective Fuzzy Linear Pro-
gr
der uncertainty.
the use of the stochastic approach. As a
e structure of these problems dictates that one starts
dealing with the multiplicity of objective functio
ter tackles the randomness in data [97].
Some other applications of the stochastic approach to
MOSLP problems include po
[98], power plant
[75], capacity plann
generation [100], deployment of roadway incident
ponse vehicles [101] and multi-product batch pl
d
6.2. Applications along the Multiobjective
Approach
Water resource planning and management
e most appropriately dealt with using the multiobjective
approach. Random parameters are first transformed into
appropriate fixed data, before the conflicting goals are
sorted out. The literature is rich in models using the
multiobjective approach. We list a few of them:
Water use planning [55], workforce scheduling model
[102], transpo
nuclear generation of electricity problem [57,103]
Application
To significantly bridge the dangerous gap between the
problems of designing reliable portfolio assets and the
mathematical programming models used to solve them,
the Decision maker should be able to consider different
objective functions and incorporate imprecision into the
model. Owing to the complexity of such problem
st to couple different techniques in an appropriate way
to solve them.
There are several good papers using this approa
which the reader may refer. The papers [96,104-109] ar
some of them.
7. Concluding Remarks
Multiobjective Stochastic Linear Programming is a
worthwhile topic. It provides a glimpse into what it
means to jostle with the complicated issue (which is
nevertheless useful for applications) of combining ran-
domness and multiplicity of objectives into an optimiza-
tion setting. Me
sources for those facing optimization problems in-
volving conflicting goals and random parameters and
wishing not to
blindly replacing it with a trad
optimization problem.
decision making and problem solving has demonstrated
that, it can provide efficient tools to those few who have
the resources and the will to use it. The new challenge is
to provide this help at an affordable price to all who
could benefit from it.
There is a rich array of methods that can be used to
deal with both Multiobjective Programming and Stochas-
tic Programming problems. This paper has somewhat
demonstrated that, the Howards view applies to Multi-
objective Stochastic Programing. Nevertheless, theoretical
and computational issu
troduced solution concepts, play a crucial role in such a
turbulent environment.
In this paper we have presented the main principle of
MOSLP. We have also indicated that there are concrete
realizations in this field. We have also discussed oppor-
tunities and limitations
oad readership, the paper has the following distinctive
features:
1) It is organized towards the technique-oriented for-
mat in contrast to the theoretically speculative one.
2) Practical aims take precedence over mathematical
niceties.
3) The basic ideas (solution concepts, related mathe-
matical results, methodo
ve been presented in an understandable manner.
4) The paper is filled with references for those whose
appetite have been sufficiently wetted.
Kirby [111] has argued that the main objections
against Operation Research techniques are as follows:
1) Ope
eds (perversion criticism).
2) Operation Research methods have already
ed wherever they were needed (obsolescence criticism).
3) Ma
mplex than those which Operation Research caters for
(inadequacy criticism).
4) Operation Research’s practice has been misg
d has undermined the confidence managers had in it
(counter-performance criticism).
This paper makes some contributions towards reme-
dying the above mentioned perversion and inadequa
jections.
Among lines for further de
ay mention:
1) Extension of the theory and methods outlined here
the nonlinear cases.
2) Comparison of Multiobjective Stochastic Linear
Programming with Mu
amming [28,50,112].
3) Design of a user-friendly Decision Support System
for Multiobjective Programming un
4) Incorporation of both randomness and fuzziness

A. S. ADEYEFA ET AL.
210
pment of Intelligent Hybrid Algorithms for
ta
sful developments in the above
m
en the language used in Multiob-
je
8.
[1]
er, New
raw Hill,
into a multiobjective optimization context [113].
5) Develo
ckling these complex optimization problems.
Let us hope that succes
entioned directions will proceed in the near future, thus
reducing the gaps betwe
ctive Stochastic Programming techniques and the lan-
guage used by potential users of these techniques.
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