Optimisation
Inmathemati aloptimisationofasingle riterion,theaimisto ndoutthebestsolution
for that riterion from a set of feasible alternatives. Using some omputational tools
the best solution is found and implemented. Therefore, unless there are alternative
optimal solutions, there is not in general a de ision pro ess involved in thesele tion of
thesolutiontoimplement. However,frequently theDM wishesto optimisenot onlyone
but multiple obje tives simultaneously. This is the ontext in whi h the multi- riteria
de ision methodologies falland the subje t ofthis work.
Inmultiple riteria de isionmaking (MCDM),there aretwo ormore obje tives toopti-
mise. Those obje tivesare on i ting, i.e., thereis not asolution whi h optimises them
allsimultaneously,andmultipleoptimal solutions( allednon-dominatedsolutions) an
befound. Therefore,anintermediatesele tion pro essisrequired beforesolution imple-
mentation -thesubje tivityoftheDMshouldbeintegratedinthisintermediatepro ess
not exist,but the preferredsolution or alternative. Theinterest for MCDM has ontin-
ually grown up in the last de ades, as proved by the several books and surveys in the
literature[3438℄.
MCDM an be divided in two sub-areas: multi-attribute de ision making (MADM)
and multi-obje tive optimisation (MOO). The dieren es between these two groups is
dis ussedinthis hapter. Fo ustoMOOisgivenforabetterunderstandingoftheMOO
methodology applied inthiswork.
4.1.1 Multi-attribute De ision Making
In MADM the aim is to order, group or sele t one alternative out of a dis rete and
nite set of non-dominated solutions whi h are mutually ex lusive alternatives. There
are several te hniques to solve multi-attribute problems. These te hniques are divided
in two main ategories: non- ompensatory and ompensatory approa hes. The former
do not permit trade-os between attributes, so an unfavorable value in one attribute
annot be oset by a favorable in some other. It in ludes methods su h as maximin,
maximax, onjun tive or disjun tive onstraint method or the lexi ographi method.
Those methods aredes ribed ingreaterdetailin[39℄.
The ompensatory methods, on the ontrary, allow trade-os between attributes. In
those methods, hanges in one attribute an be ompensated by hanges inany of the
other attributes. Thesemethods in lude s oring models su h as the analyti hierar hy
pro ess(AHP)[40℄orthesimpleadditiveweighting(SAW)[41℄and ompromisingmodels
asthe te hnique fororder preferen ebysimilarityto anideal solution(TOPSIS) [41℄or
linearprogrammingformultidimensionalanalysisofpreferen e(LINMAP)[42℄. Alsothe
on ordan emodelssu hasPROMETHEE [43℄ortheeliminationand hoi etranslation
4.1.2 Multi-obje tive Optimisation
In MOO the problem is represented as a mathemati al problem integrating the on-
straints and obje tive fun tion(s). If theproblem is of ombinatorial nature, we lay in
theframeworkofMulti-obje tiveCombinatorialOptimisation(MOCO).Inorderto nd
out a (redu ed or whole) set of trade-o solutions, an optimisation pro edure must be
used. At the end a MADMte hnique maybe applied to hoosea solution from theset
of solutions foundbythe MOOte hnique.
Asmentioned, aMOO problemis mathemati ally representedasa typi aloptimisation
problem,butwithmorethanasingleobje tive. Theobje tivesareusually on i ting. It
means that there is not one optimal solution simultaneously optimising all the riteria,
but several good solutions exist. They have been and will be referred in this text as
non-dominated or trade-o solutions. The obje tive fun tions an be either maximised
or minimised. A setof onstraints and boundsde ne the solution spa e. Without loss
of generality, in (4.1) a MOO problem is represented onsidering
n
obje tive fun tionsto minimise. minimise
F1(x), F2(x), ..., Fn(x)
subje ttogj(x) ≤ 0, ∀j ∈ 1...J
hk(x) = 0, ∀k ∈ 1...K
(4.1) 4.1.2.1 Basi Con eptsInthisse tionthebasi on eptsrelatedwithMOOne essaryforabetterunderstanding
of thework developed inthisdo ument arepresented. For a more detailedexplanation
of thetheoryrelatedwithMCDM,we referto [38,46℄.
Firstlyweshouldformallyintrodu e the on eptofdominan e. AsolutionA dominates
another solution Bif A is no worse than B in all obje tives and is stri tly better than
B inat least one of the obje tives. Other dominan e-related on epts are worth being
introdu ed:
•
Dominated solution: A solution is dominatedif there existsanother solution that isbetter inat leastone riterion andnot worse intheremaining riteria.•
Non-dominated solutionorParetooptimalsolution: Asolutioninnon-dominatedif itisnotdominatedbyanyotherfeasiblesolution. Inthisway,noneoftheobje tivefun tionsofanon-dominatedsolution anbeimprovedinvaluewithoutimpairment
insome of the other obje tive values. Without any preferen e information about
the riteria,we annotsaythatonesolutionisbetter thananothersolutionifboth
arenon-dominated.
•
Non-dominated set, Pareto frontor Pareto optimal set: Representsthesetofnon- dominatedsolutions.•
Trade-o: Relationbetween values ofattributes thatmeans thene essaryamount to looseinan attributeto gain inanother attribute.•
Convex dominan e: A non-dominated solution is onvexly dominatedifthere is a onvex ombinationofother solutions thatdominates thatsolution.4.1.2.2 Approa hes and Methods for Multi-obje tive Optimisation
Thete hniquesasso iatedtoMOO an be lassi ed a ordingto theroleofthede ision
maker and the stage in the de ision making pro ess at whi h his/her preferen es are
arti ulated. These preferen es an be indi ated by asking the DM for his/her opinion
abouttheattributes(relativeimportan e,aspirationlevels,...) or aboutthealternatives
(satisfa tion level, omparisonbetween pairs, ...).
•
no arti ulation of preferen e information: in these te hniques the subje tivity of the DM is not required,and is not in luded inthe nalsele tion. The te hniquesaremostlyusedwhenthe DM annotde ne what he/sheprefers. Thisisthe ase
ofthe MinMax formulation [47℄.
•
aprioriaggregationofpreferen e information: thepreferen esareintegratedinthe formulation ofthe problem, sotheDM parti ipates onit, leading to thepreferredsolution. Those methods are usually performed by asso iating weights to ea h
obje tive and aggregating the obje tives resulting on a single obje tive fun tion.
Examplesfor thistype ofte hniquesaretheweightedsum[48℄,goalprogramming
[49℄, lexi ographi method [50℄ or the exponential weighted riterion [51℄. The
value theoryand theutility theory [45℄ are also applied insome MOO problems,
asan aggregationmethodwithpre-arti ulation of preferen es.
•
progressive or intera tive arti ulation of preferen e information: the DM a tively takespartinaniterativesolutionpro essandspe i esthepreferentialinformationgradually, until an a eptable solution is found. This group in ludes te hniques
su h astheSTEM[52℄orthe method ofSteuer [53℄.
•
a posteriori arti ulation of preferen e information: in these methods the set of non-dominated solutions is rstly generated and then presented to the DM, whoissupposedto sele tthe mostsatisfa torysolution. AMADMte hnique fromthe
se tion 4.1.1 an be usedfor that purpose. There aretwo goals to a hieve within
this approa h: 1) obtain a good approximation of thenon-dominated set; 2) the
setof solutions shouldbe maximally-spread overthePareto frontier.
MOOte hniquesfollowing thisapproa hare,forinstan e,the
ǫ
- onstraintmethod [34,35,37℄ and the normal boundary (NBI) method [54℄. There aresome multi-obje tivemeta-heuristi s basedonsimulatedannealing[46,55℄,tabusear h[56℄or
evolutionaryoptimisation[57℄su hasNSGA[58℄,NSGA-II[59℄,theni hedPareto
Some reviews on multi-obje tive optimisation te hniques an be found in theliterature
[35,36,36℄.