tJrgillurillg Opl inIllQl ;QIf. 1984. Vol. 7. pp . 319-336 (t) 1984 Go rdon and Rrcach Science Publishers. Inc.
OJ05· 2ISX/84/0704-o J I9SI8.S0/0
.
Printed in G reat Britain
A COM PARISON OF THREE METHO DS
FO R GENE RATING TH E
P AR ETO OPTIMAL SET
M . BALA CHA NDRA N and J. S. GE RO
Compu ter Applications Research Unit.
Departm ent of Ar chite ctural Science, University of Sydney,
N.S. W. 2006 Australia.
(Rc( t'IlJf'd Augu.!1 20, / 9$J .. ill fi llal/o rm Ft'b,uU'f 21. 1984)
This paper describes and cc mparesmree approaches 10 solving design optimiza tion problems with muhiple
co nflicring ohjectives. The three tec hniques are de scribed in detail and then applied 10 an example which
demOnStrlllles how infor m:uion is accumulated which leads to a logical and eflicien l multicriteria op timal
design. The techniques employed (weighting. noni uferior $CI estimation and conslrai nt melhod s) are
compared 10 each other by considering their com putationa l efficiencic' and their ah ililies 10 product: an
approximat ion of the Pareto oplimal ser.
I NTRO D U CfI ON
Pareto optimization is a methodology for solving multicriteria decision prob lems.
T his methodology prov ides a systematic ap proach toward s design problems with
mul tiple co nflic ting objectives. In Pareto optima l design situations. t he designer
has mo re than one performance measure of interest. An optimal solution is genera lly
defi ned as the best solution. However, with multicriteria problems, the " best " is
o ften dependent upon a designer's preferences. The Pareto op timization meth odology
usually generates a large number of alternatives which the d esigner evaluates in order
to arrive at his best so lution often termed the bcsr co mpro mise solution.
Althou gh mult iple objective analysis was int rodu ced in the early 19505· the mo ve
toward design decision ma king based on this methodology has occurred largely
during the past decade. During th is interva l severaltechniq ues ha ve been adva nced
for this purpose.
In t his paper, Section 2 outlines the Pareto o ptimization methodology, whilst
Sectio ns 3 to 5 describe the three multiple objective solution techniq ues in detail.
Sect ion ed emonstrates the a pplicatio n of these techniques to the op timal dimensioning
of architectural floor plans for confl icting criteria. T he so lutions thus obtained ar e
co mpared with each other in orde r to show the effectiveness and usefulness of those
techniques.
.
2 PARETO O PTI M IZATION MET HODO LOG Y
2./
Formulation of Problems
Pa reto optimization prob lems de al wit h two o r mor e objective functions,The genera l
319
l~U
Pa reto o p tlm i;r;11inn problem with S de srg n \ uria bles. .\ f constra int s a nd J' objectives
IS
( II
su bjec t
\\ he re \
I'
hi g~{.\") ~
Gl k == I. 2... ,.\ 1
12J
an .\' -com pcne ru vector consisti ng o f d esign vanablcs.
!Ill"), k '" I. 2, .. . • .\1
arc .\f \.'I" 1StI aint funct ions and Z I (.\ I, Z ~ (\ ). ' , . . Z ,.(,,) ar c J' objcc uvc func tions .
T hc d C\.:ISIOII Sp;.ICC. Figu re I. IS ccmpuscd o f a ll the feasib le so lutions 10 a d esign pro btern. In genera l. for a p ro blem with S des ig n var ia bles. the d eci sion space \\.111 he a n
.v-di me nsional space.
T he cr ue na space. Figure I. rep rcsems the d esign pe rform a nces wh ic h co rres pond
to evc ry so lut io n In th e d ecisio n space. For a pro blem w ith t' o bjectives the cntena
space w 111 be a l'-dll lle nsio na l space.
Each pllin t in th e d ecisio n space is associa ted with a poi nt ill the Criteria space and
vice vcrs u. III genera l terms, a Pa re to o p tima l d esign p roblem ca n be d escr ibed quuc
simply a s the idcnufica uo n of the be st com p rom ise feavrble poi nt in the c riteria space
an d rhe co rres po nding: poin t In the d ccrsro n space .
..
..
L -_ _~ . . L -_
......._ .·.u
}.3
_
~
•.....,••·.ro
..
Paretn 0 r t intl.ll. , t,
In si ng le objective p roblems the go a l IS the ide nt ificatio n o f the o p tima l so lu tio n. tha t
is. the selec tio n (If th e fea sib le solution (or sol utions) that give s the best value of the
objecu vc Iuncuo u. This no tio n of o p tima lit) ca nno t be a ppl ied III mu lt ic riteria sn uanons.
Pa reto optuuahr y is th e majo r co nce pt III mu lt iple c rite ria opu rniza uo n a nd it is
d efined formally as follow s :
il (nm hll' w,[ulio n 'II I I mutncriiena opl ;m;:mion pro hh'm ;5 f' aH'1U uptimal (o r '10";'"!(,',iorl (f mere t' ,\j)/l no othr It'll~ ible w it/tim ! t hlll " III .\'lela an impron>mt'nt In nnr
cr i,crlo" wi lholll 1·l1l1.~i n!lll decrease in III It'lIsl one OI Ilt'r er iE l'r ion.
In other word s, ;1 Pare to opu ma l voluuonf is a fea sible solution In th e pr o blem such
rh.u no other feasible soluuo nc exists lor which Z,(.l) ;:>: Z,(iHor so me j = 1. 2. ' , . • p
an d / Jh) ~ 7.f") for all, F- I .
I'ARl: IO Ol'lI \Pd . SU
(iE:-.: tl(!\ lln~
r (l I Jl u~ 1 rate th e pro pe r! rc~ of Pa ret o o puma l sol ut Ions. consider the cr iteria <pave
fu r a problem with two crttc ria, Ftgure 2. Accordmg to the d efimuon
Par eto li p nmalu y it is ea sil y fou nd that the set of P a reto o pt ima l perfo r mances lies a lo ng the
North-Hacrcrn bound a ry of the crite ria space. In the genera ! case o f the l' ..cntena
p ro b lem . the Pa ret o o pti ma l SCi wIll fo rm a su rface in J' -dIRlc nsio na l space. A graphica l p re-e ntatio n o f Pa reto se t for a two crite ria problem av show n in Fig u re ~ ullo.... s
ea ..y companco n o f altema tives. A vinul ar gra ph ica l p resemauo n o f higher di mensio nOi l prob lems i<; d ifficult .
or
"
-+ ..
L
Pa ret o opurm...uio n tcc hmq uc s usua lly gen era tc a lar ge nu m hcr o f alterna tives
whi c h the de cisio n mak er sho u ld In vestiga te to a r nv c a t 1m. bes t com p ronuse solu t ron .
In this pro cess the dec isio n ma ke r ha s 1(1 conside r a nother a..pcct ca lled Irad e-o lf, .
In considering t he pe rfor man ces of two Paret o so lut io ns. the amount o f one o bjec tive
tha t mu st be sa cnflccd 10 ~di n a n mcrea se In the o the r objccuv c IS ca lled a trade-off.
The ..oluuon th a i is select ed event uall y as the p referred a ltc m auvc IS called the hes l
com promise so lution.
3 Til l WEiG IITl/,;G METHOD
3.1
r ht' Gl'lIt'fuI IH'ighllllg _\f('l hod
w eight ing the o bjec tives to o ht ai n Pareto optima l sol utio ns IS pe rhaps th e o lde ..t
I' .neto opt rlT1 i/~1 1 ion tech nique." Th is met hod involves solving fo r Pa ret o solutions b)
co nve rting the m ulncruc ria pro ble m 10 a sca la r o pumizauon p ro blem. In which the
obiecuvc funcuc n bec ome s a weig hted sum o f the objec tive Iun ction -, or the multiobject ive mod el. T ha t IS.
,
vMa x '-
.' ,
s.t..x e- X
14J
(51
.... here II; ~ 0 for ..II , a nd sirictly po siuve for at Iea sl one.
Th e above proble m is a single cri terion opumiza uo n p roblem .... hich ca n be solve d
with rradruonal opurmzarion methods. The o ptimal solut ion to the we igh ted pro bIem is a Pa reto o pnmat solunon for the mult icrite ria pro blem. as Itlflg as " II o f t hc
.....eights arc no nnegati ve. F igur e l
M
Ii"LA("I IAS IH~A:"
At' ll J S (i H W
.,
",
• " ",
L
J
~
1 Jr•. U' f'ighl mq ,\ f .'l hnd ,.Ilqor i l hm
"I he .Ilgoothm fo r the weig hti ng mel hod is rather simple and strarglnforward. A
number of d ifferen t set s of wciglu s ar c used until a n adequa te representation of the
l'areto scI IS obtained . Since the soluuons which o ptimize ea ch objective sepa ra tel y
ar c th e e nd POin ts o f the Par eto set. II I!> rea so na ble to beg in by so lving th e weighted
problem P tunes with sets of weights as sho wn below.
objective n um ber
:!
J
P
o
o
I
{I
(}
(I
I
()
(I
n ()
Aftct each objecti ve is optimized individually. a systema t ic variation of the weight ..
may he follo wed For the spec ial case in wh ich one or more wetgfus arc set to zero. If
the re arc a ltcrna tivc o purn a. t hen so me o f the se o pti ma may he non- Pareto sol u tio ns.
In thj v case. (he Pareto opumahry o f the solutions mu st be checked. How ever. whe n
allof th e weight s are ..tncuy posurve. the n alt erna tive opt ima fo r the weigh ted problem ar c Pareto o p timal so lut ions.
One of the major disadvantages o f the weight ing me thod is that it always spa ns the
Pareto sol utio n space o n ly u nd er t he co nd itio n that the criteri a space i.. ..trictl y
co nvex.
4
H IE N ON IN I tKIOK SE T ESTI\IATIO N (N ISE) \ 1FI II() J)
lhe ncn infen or sci csum a uon (or f" ISE) mClh od 1 a llows fo r th e direct genera tio n
of a good ap proximation o f th e Pareto SCI. T he importan t featu re o f this method is
that the accu racy of the ap proxima tion ca n be cont rolle d th ro ugh a n a spec t called
err or c ruc ric n. T he individ ual Par et o solu tio ns ca n he fo un d in the f" ISE method
through th e use o f the weighti ng method. Un like: the weigh t ing met hod a lone wh ere
o ne ca n use a ny no nnegat ive weight s. in the NISE meth od the valu es o f the weig ht s
ar c chosen by the NISE algori thm. Alt hou gh thi s mel hod gua rantees good cov erage
of the Pareto sot uuon space, it is inca pa ble o f genera t ing non con vex Pa reto sets.
PAR ETO OPT IMA L SET GENE RAT ION
4.1
12J
T he N ISE Method for Tw o-criteria Problems
T he NISE method for two-criteria pro blems operates by fi nding a nu mber of Pareto
optimal solutio ns in the criteria space and evalua ting the pro perties of the line segments between them. Let the points A and B in Figure 4 be two Par eto optimal solutions in the criteria space. Th en the line segment A B may or may not be part o f the
Pareto optimal set. If the line segment AB is Pareto, then mo ving in a directio n out
from the line is infeasible. If the line segment A B is non -Par eto. then there are Pareto
so lutions in the o utwa rd direction. Th e weights used to generate the point A can be
used to compute the slope of a linear indifference curve corres po nd ing to the weighted
objective function . This linear ind iffe rence curve is sho wn as line A C. Line BD represents the linea r indifference curve passing thro ugh B that correspo nds to the weights
used to genera te the point B.
.,
L
FIG UR E 4
••,
Ma xima possible err or associa ted with a given line segment in the c me na space .
The lines AC and BD represent an upper bound 10 the Pareto op timal set as any
ot her Par eto solutions cannot lie above these lines. Tha t is, if there are Pareto solutions
above the line AB they sho uld lie within the triangle A EB. Th erefore. ifthe line segment
AD is taken as the approx imation oft he Pareto optimal set then the ma ximum possible
erro r is the perpendicular to A B drawn from E as shown in Figurc 4. A value for the
maximum allowable error must be selected prior to the start of the algorithm. Since
th e criteria space has axes measured in nonco mmensurable units, the d istance in
criteria space is not meaningful. Usually the value of the maximum allowable erro r is
designated as a percentage of the maximum possible erro r as sociated with the primar y
tine (see Figure S.).
.,
•
I"
-------jt-
r"
......... ". . . " ... .,
.... ...... A
L
..
/ / II
I
I
•
'--_+ "
FIGURE S Poinu P and Q rep resent the o pumel so lutions of Z, a nd Zl individ ually, respecuvely .
".
M. BALACHAN DR A N A N D J. S. GERO
In Figure 5, P and Q a re the o ptimal so lutio ns of the ind ivid ual o bjectives and R1
is t he maximum possible erro r for the line PQ. The maximum allowable er ror is set
as a percentage of RT at the beginningof the algoritbm.H t he maximum possible errol
correspond ing to an ar bitrary line segment, say A B, exceeds the maximum allowa ble
error, then t he s lope of the line segment AB is used 10 co mpute the weights (or use
in t he weighte d p rob lem. Thai is, the we ight s WI and Wz a rc chose n to sa tisfy
IV, =
- W
,
slope of A B
(6;
The new so lut ion o btained by so lving the weighted problem is then located in the
criteria space. H rhe new solution lies above AS, two new line segments arc generated
byjoining the new solution 10 A and B respectively. The case in which the new sotution
lies on AB, indicates tha t the re are no other solutions above t he line segmen t A B.
The above procedure is repeated with each line segment until the ma ximum possible
error in all parts of the Paret o optimal set is less than or equa l 10th e maxi mum allow.
able er ror.
4.1
Th e N IS£ Method Algorlthmfor Two-Cr iuria Problems
The NISE algo rithm (or the two-cr iteria problem is as fol lows.
.
"
~
" - - -A -
"
"to.
I _ 0.,
0, I ••
FIGU RE 6 Firs. IWO su.geJ or the NISE . tlorilhm (o r two.c riteri. Jlroblems.
Conoentian :-
S,
MA E
MPE Jl
N
"" Pareto optimal point having the ub hig hest value for 2 1 a t a stage.
= Maximum allowab le erro r (pre set).
= Maximum possible erro r assoc iated with the line segment SJSt .
= Nu mbe r of Pareto optimal po ints generated at a stage.
S T EP J. Ma ximize the obj ect ives ind ividually. Lei the image in the criteria space
of the optimum for objective 2 . be P I and for objec tive 2: be Ph Figure 6:
i.e" S , '" PItS: = P 2, N = 2.
Compute MPE 12 and set the value for MAE.
ST EP 1. If MPE1• i • 1 .$ MAE fo r i ... t, 2, ... , (N - I) then STO P otherwise
proceed to S T EP 3.
STEP J. find the value of j for the maxim um of a ll M PE,.•• I ' So lve the weighted
probl em using the line segment 5/ S.-+I . If the new sol ution lies above the line
segment 5/5•• 1 designa te the new solu tion as p~ + I and proceed to STE P 4.
O the rwise, set M PE,.I• • to zero and ret urn 10 S TE P 2.
I'AR[TO O I'Tl MAL SET (i J:f\ f' k ATl OS
STEP 4, Reo rd er the poi nts P,. I = I. 2..... ( N + I)
S; = S,
/ =1 .2•...• /
5; . 1 = P" l
5;. 1 S,
I "" (i + I )...• ,....
The error terms are a lso rela be lled .
=
,\f PE;." 1 = MPE,.' . 1
' .1.2
U -I) (i > I)
.\11'£; . 1". : =- ·\11'£',"1
, = (i + I )
,\' U < tV - 11
Com pu te .\f PE,.,. I a nti ,\ fl ' £, . t. J
4
. t
N
= '"
+ I and return 10 S7F.P 2.
5 "liB: COr-;ST RAINT \ iETl IOD
5./
'1"11('
G('nalll Cumfm ;m .\ fe/hod
Another ap p roach used to identify the Pa reto o ptimal set is culled the constrai nt
approa c h. T his tec hniq ue reta ins o ne obj ect!ve 01 <; p rimary a nd lre al ~ the remaining
P - I objectives a s co nstrain IS, suc h thai
Max L j..x )
(X I
Su bject to .'( E: X
Z ,(.'() ~b ,
; = 1.1.. . .. j - 1.
j+ 1.. . .. l'
(9)
where the j th o bjec tive was arbitrarily chosen fur maxurnzauon an d b, are tbc lower
bo u nd.. o n the remaining " - I obj ective s. Figu re 7.
"
. , :> . ,
' OW
•••••
•••
. . . . . 0 ..
'----j-----. "
FIGURE 7 Conce pt of Wn\ lf.& ,nl A hound .. specified on 7.,
soluuon
~nd .7.l
,\ rnu,m'lrd 10 find II Pllir iu
T his fo r mulation is a singlc o bject ive problem s0 it ca n he: so lved by co n ven tiona l
mel hod s. Th e Par cto optimal set is the n generated by sol vmg the abo ve sm gle obj ec uve problem with parametric va riat ion o f b,.
5.l
The
C{)/l~lril;m
M e/hvc! A /flori/hm f or TKO Ob}t>CI;I t> Pmhll'1It\
S T EP l , Solve two ind ividua l ma ximiza tio n proble m.. to find the o pt ima l solunon
for each objective alo ne. FIgure 8.
1~ I.
~1
IIALAntASDRA~
A~n
J S CiU Hl
>'
.,
"
-::::-_-<> "
L.._ _--:::NO
,\T f: / ' ~ .
livnluat c the performance of the o the r object ive a t each of these optim a
solutions.
ST EP 3. Conv ert the muluobjccuvc for m o f the prob lem 10 its corres pond ing con
su amed pro blem
M a x ZI(X)
o
su bject to :
'( E X
Z l(X) 2 b 2
STf, P 4. Solve the co nstrai ned problem fur a number
Where
n2
= "2
+ [- '
(r
- ](mz-
+ 1)
" 2)
or val ues o f nl .
1 =
1. 2. . . . . r
With r "" number of va lues of bou nd b 2 that will be used in th e genera tio n 0
Pareto solutio ns. ms is th e max im um of Z 2 and " 2 is the minimum o f L 2 in tlu
Pa rc lU se t.
I\ N I L L UST RATI VE EXA M Pl.E
h
6./
Oprimal Dmlt'" \iOlli"!I of Rectunqulur Floor Plans
To cxummc and compare th e methodology using the tec hniques d iscussed p reviou sly
co nsid er a pr oblem formu lated by M itch ell. Steadman and Ligg eu ' fro m the fieh
o f o pt imizat ion in building de sig n. T he pro b le m co ncerns the optima l dimensioninj
of sma ll rect angular floor plans for w hich an optima l to po lo gy ha s bee n sepa ra tely
ide n tified. T o illustrate the formu la tion co nsider the example p resented b)' \.1 ile her
ct ul .. ~ F igu re 9 . Th e multic rite ria model provides the d esigner with a number 0
performa nces. Th ree c rite ria will be used to ide ntify Ihe opt ima l plan although onl j
two wi ll be co nsid ered a l a nyone timc . T he o bjectives ar e :
I. M inimize co ns truction cost :
2. M a xim ize to tal usable a rea ; a nd
J. Ma ximize aspect ra tio.
PARETO OPTIMAL SET GENERATION
x,
I x2
I
3
I
327
",
5
BATH
BEDt
4..IVtND
•
HALoL
2
6
7
ae02
~ITC~EN
FIGURE 9
6.2
BED3
Dimensionless representation of a small apartment unit.
Problem Formulation
Design variables:
The components of the dimensioning vectors X and Yare the design variables:
X = (X I> X 2' X 3, X 4)
Y = (YI , Y2 , Y3 , Y4 )
(10)
Constraints:
Constraints exist on wall lengths, room areas and room proportion ratios.
Objective Functions:
1. Construction Cost
7
Minimize
L
[[2-(wj
+
lJ·(dJC 1
+ d 2C2 ) + 2(w +
j
lj),C 3.Jh
i= 1
(11)
where
(12)
4
Wi
=
L
a ik Y",
(13)
k=!
where h
Wi
t,
C I , C 2 , C 3 j , C 4 j and
c,
is the room height
are room widths
are room lengths
are unit construction costs of external walls, in ternal
walls, internal finishes, floors and ceilings respectively.
are 0-1 (binary) variables. For an external wall
d l = l,d 2 = 0. For an intemal wall d, = O,d2 = 1,
etc.
A more detailed exposition of the formulation can be found in Gero.:'
M. BALACHANDRAN AND J. S. GERQ
328
2. Total Usable Area.
7
Maximize
L
(l;wJ
(14)
(~~;)
(15)
i=1
where I; and
Wi
are as defined previously.
3. Aspect Ratio of the Building
Maximize R =
i.e.
log(R)
= log(!:X i )
-
log(!: Y;)
(16)
Since the right-hand side terms in (16) are separable, they can be Iinearised using a
piecewise linear approximation technique. In this process an additional eight variables
and four constraints are introduced in the model. Since 10g(R) is a monotonically
increasing function, maximizing 10g(R) is equivalent to maximizing R.
The above problem is formulated with two objectives at one time as a quadratic
programming optimization problem. Computer programs have been written which
carry out the analysis required by the Pareto optimization techniques discussed
previously. In this work the mathematical programming system (MPOS), developed
in Northwestern University," has been used to solve the quadratic programming
problems.
7
7.1
RESULTS OF THE THREE METHODS
Optimizing Total Usable Area and Construction Cost
TABLE 1
Summary of the results obtained by the weighting method objectives: I) Maximize
total net usable floor area; 2) Minimize construction cost
Pareto optimal solutions
Solution No.
I
2
3
4
5
Weights used
1.0
0.00
1.0
1.0
1.0
2.0
1.0
3.0
1.0
4.0
1.0
5.0
1.0
6.0
1.0
7.0
1.0
8.0
1.0
9.0
1.0
10.0
1.0
11.0
1.0
12.0
1.0
13.0
1.0
14.0
1,0
15.0
0.0
\.00
Cost ($)
12582.67
A rea (sq. ft.)
609.56
12627.93
16717.77
613.39
945.20
17309.42
17996.99
990.29
1029.68
PARETO OPTIMAL SET GENERATION
TABLE I
329
(continued)
Decision variables
Solution No.
XI
X2
I
6.00
6.00
6.00
7.74
8.95
4.42
4.47
9.30
9.04
8.16
2
3
4
5
X3
5.58
5.75
5.50
5.53
7.62
X4
9.00
9.00
11.80
11.57
12.45
YI
8.38
8.11
8.50
8.50
8.50
Y2
3.00
3.00
6.00
6.00
5.35
Y3
3.00
3.21
3.00
3.00
3.33
Y4
10.00
10.00
11.50
11.72
10.52
o
II>
..
..
..
II>
o
"'1900.
~
1800.
1700.
1600.
1500.
1400.
1300.
COST
FIGURE 10 The Pareto optimal solutions generated by the weighting method.
Table I and Figure 10 contain the results obtained using the weighting method. In
addition to optimizing each of the objectives separately the weights were incremented
by unity from 1.0: 1.0 to 1.0: 15.0. Of these 15 weights only five produced Pareto
optimal solutions. More solutions could have been produced if some a priori knowledge were available about applicable weight regimes. However, such knowledge is
not available. The solutions are largely bunched at the extremes of the Pareto optimal
solution space.
Table II and Figure 11 contain the results obtained using the noninferior set
estimation method. The maximum allowable error was set at 0.02 per cent. The NISE
method resulted in 17 Pareto optimal solutions which included all 5 of those generated
by the weighting method. The solutions are still largely bunched at the extremes of
the Pareto optimal solution space. Thus, the NISE method did not increase our
knowledge of the shape of the Pareto optimal solution space in this case.
M. BALACHANDRAN AND J. S. GERO
330
TABLE Il
Summary of the results obtained by the NISE method objectives: I) Maximize total net
usable floor area; 2) Minimize construction cost
Pareto optimal solutions
Solution No.
I
2
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
Weights used
0.00
1.0
12.05
1.0
1.0
12.32
1.0
12.89
13.05
1.0
1.0
13.13
1.0
15.14
1.0
15.26
1.0
16.38
16.82
1.0
17.02
1.0
1.0
17.23
17.45
1.0
20.06
1.0
22.94
1.0
1.0
24.78
0.0
1.00
Cost ($)
12582.67
12627.93
12717.84
16717.77
16725.84
17281.41
17309.62
17406.63
17492.37
17583.85
17672.88
11761.71
17849.73
17870.88
17900.13
17948.83
17996.99
Area (sq. ft.)
609.56
613.39
620.78
945.20
945.82
988.39
990.29
996.67
1002.26
1007.73
1012.99
1018.18
1023.26
1024.47
1025.77
1027.82
1029.68
Decision variables
Solution No.
I
2
3
4
5 .
6
7
8
9
10
11
12
13
14
15
16
17
XI
6.00
6.00
6.00
6.00
6.00
7.49
7.74
7.86
7.97
8.11
8.26
8.40
8.54
8.58
8.66
8.80
895
X2
X3
4.42
4.47
4.28
9.30
9.35
9.30
9.04
8.93
8.82
8.67
8.53
8.39
8.24
8.21
8.20
8.18
8.16
5.58
5.75
5.85
5.50
5.50
5.50
5.53
5.77
5.98
6.27
6.56
6.84
7.13
7.20
7.30
7.46
7.62
X4
9.00
9.00
9.13
11.80
11.83
11.80
11.57
11.69
11.80
11.87
11.94
12.01
12.09
12.10
12.18
12.31
12.45
Yl
8.38
8.1l
7.96
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
Y2
3.00
3.00
3.00
6.00
5.97
6.00
6.00
6.00
6.00
5.93
5.85
5.78
5.71
5.69
5.62
5.49
5.35
Y3
3.00
3.21
3.62
3.00
3.03
3.00
3.00
3.00
3.00
3.07
3.15
3.22
3.29
3.31
3.31
3.32
3.33
Y4
10.00
10.00
10.00
11.50
11.44
11.50
11.72
11.60
11.50
11.35
11.21
11.07
10.92
10.89
10.80
10.66
10.52
PARETO OPTIMAL SET GENERATION
331
+
-c
w
0:::
«
"',.00.
'800.
1100.
1600.
'SOO.
1400.
'300.
COST
FIGURE 11 The Pareto optimal solutions generated by the NISE method. (cf. Figure 10).
TABLE III
Summary of the results obtained by the constraint method objectives; I) Maximize
total net usable floor area; 2) Minimize construction cost
Pareto optimal solutions
Solution No.
1
2
3
4
5
6
7
8
9
10
11
Cost ($)
12582.67
13112.37
13604.44
14114.72
14707.83
15205.58
15703.89
16236.43
16743.54
17422.00
17996.99
Area (sq. ft.)
609.56
651.62
691.07
731.98
780.24
820.46
860.68
904.66
945.32
997.67
1029.68
Decision variables
Solution No.
I
2
3
4
5
6
7
8
9
10
11
Xl
X2
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.00
6.94
7.88
8.95
4.42
4.50
4.50
4.98
6.43
6.50
7.19
7.86
8.27
8.91
8.16
X3
5.58
5.50
5.50
5.50
5.50
5.50
5.50
5.50
5.50
5.80
7.62
X4
9.00
10.33
11.92
12.77
11.32
11.25
10.56
10.55
10.77
11.71
12.45
YI
8.38
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
8.50
n
3.00
3.00
3.00
3.00
3.20
4.50
5.19
6.00
6.00
6.00
5.35
Y3
3.00
3.25
3.25
3:53
4.98
4.50
3.81
3.00
3.00
3.00
3.33
Y4
10.00
10.00
10.00
10.00
10.00
10.55
11.93
12.75
12.52
11.59
10.52
332
M. BALACHANDRAN AND J. S. GERO
•
•
•
•
•
«~
w
..
0::::
•
«0
•
•
•
o
",900.
•
1900.
1100.
liDO.
1500.
1400.
1300.
COST
FIGURE 12 The Pareto optimal solutions generated by the constraint method. (cf. Figure II).
Table III and Figure 12 contain the results obtained using the constraint method.
It was decided to set 11 constraints resulting in 11 Pareto optimal solutions only
two of which map onto those generated using the NISE method. However, there is a
good indication of the shape of the Pareto optimal solution space (which is slightly
concave).
7.2 Optimizing Total Usable Area and Aspect Ratio
Here, we have not presented the results from the weighting method since it only gave
the solutions for optimizing each objective separately even though a wide regime of
weights was tried.
Table IV and Figure 13 contain the results obtained using the NISE method with the
maximum allowable error set at 1 per cent. (In fact, the errors in the line segments
were zero.) Only four Pareto optimal solutions could be found. Again, the shape of
the Pareto optimal solution space was not well characterized by the NISE method.
Table V and Figure 14 contain the results obtained using the constraint method again
with II constraint values resulting in 11 Pareto optimal solutions only two of which
map onto those generated by the NJSE method. A uniform distribution of solutions
is obtained which gives a good indication of the shape of the Pareto optimal solution
space (which has concave regions).
8
DISCUSSION
Three different methods have been employed in the generation of two-dimensional
Pareto optimal performances and the corresponding design solutions: the weighting
method, the noninferior set estimation (NISE) method and the constraint method.
PARETO OPTIMAL SET GENERATION
333
TABLE IV
Summary of the results obtained by the NISE method objectives: 1) Maximize total net
usable floor area; 2) Maximize aspect ratio
Pareto optimal solutions
Area (sq. ft.)
1029.67
101395
981.95
830.72
Weights used
1.0
0.00
604.18
1.0
808.78
1.0
1.00
1.0
Solution No.
I
2
3
4
Aspect ratio
1.342
1.387
1.452
1.716
Decision variables
X2
XI
8.95
6.00
8.95
9.46
Solution No.
I
2
3
4
8.16
5.30
8.82
6.50
X3
7.62
6.42
6.96
8.50
X4
YI
n
n
Y4
12.45
9.72
12.78
13.30
8.50
7.27
7.52
6.00
5.35
3.00
6.00
3.00
3.33
10.52
10.00
10.52
10.00
3.00
3.00
3.00
'"":
•"!
+
0
"!
I-
U
to
W "l
0U)
·-
+
4: "":
+
'"~
+
•
..+---,.-r---r-r-....,....--,.____-.---,.-~....,....__,._____r_
-800.00
840.00
880.00
820.00
960.00
1000.00 1040.00
AREA
FIGURE 13 The Pareto optimal solutions generated by the NISE method.
M. BALACHANDRAN AND J. S. GERO
334
TABLE V
Summary of the results obtained by the constraint method objectives: I) Maximize
total net usable floor area; 2) Maximize aspect ratio
Pareto optimal solutions
Solution No.
Cost ($)
17996.99
17783.27
17567.63
17404.76
17085.16
16803.68
16713.48
16316.52
16136.41
15906.26
15734.23
I
2
3
4
5
6
7
8
9
10
II
Area (sq. ft.)
Aspect rati 0
1029.67
1016.38
997.48
982.92
956.99
931.97
908.23
885:66
869.46
848.76
830.72
1.342
1.380
1.420
1.450
1.490
1.530
1.570
1.610
1.640
1.680
1.716
Decision variables
XI
8.95
8.95
9.21
9.45
9.46
9.46
9.46
9.46
9.46
9.46
9.46
Solution No.
I
2
3
4
5
6
7
8
9
10
II
..
..
X2
X4
12.45
12.73
13.04
13.28
13.30
13.30
13.30
13.30
13.30
13.30
13.30
X3
7.26
7.20
7.20
7.20
5.52
5.50
8.50
8.50
8.50
8.50
8.50
8.16
8.58
8.18
7.83
9.48
9.50
6.50
6.50
6.50
6.50
6.50
YI
8.50
7.62
7.25
7.02
8.48
8.50
8.05
5.50
5.50
5.50
6.00
Y2
5.35
5.95
6.00
6.00
3.87
3.18
3.00
4.95
4.53
3.98
3.00
Y3
3.33
3.05
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
Y4'
10.52
10.52
10.25
10.02
10.00
10.00
10.00
10.00
10.00
10.00
10.00
~
~
+
+
+
0
+
~
I-
U
W
(L
(/)
c:x::
+
'"
"!
+
..
+
+
.
+
+
"?
•
..
~
-800.00
840.00
8'0.00
820.00
1160.00
1000.00 1040.00
AREA
FIGURE 14 The Pareto optimal solutions generated by the constraint method. (cf. Figure 13).
PARETO OPTIMAL SET GENERATION
335
All of these generating techniques have one element in common, that they all involve conversion of the multiobjective problem into a single objective form in order
to generate Pareto solutions.
Among these techniques the weighting and constraint methods involve fairly
straightforward computations. In the weighting method, the weighted problem is
set up and several Pareto solutions are generated by changing the objective function
coefficients. The constraint method is used to generate Pareto solutions by changing
the right hand sides of the constraints on the objectives. These two methods have a
computational burden that is directly related to the number of objectives (P) and the
number of weights or bounds (r) on the objectives. The number of different combinations of weights or bounds, which is also the number of mathematical programming
problems that to be solved is r P - 1 . This indicates that their computational burden
increases exponentially with the number of objectives.
The NISE method requires computation in addition to that which is required for
the solution of the weighted problem. At every step of the algorithm, error terms and
new weights must be computed. The NISE method, however, generally requires
fewer solutions than the number of solutions required by the weighting and constraint
methods because of its exploitation of the shape of the Pareto set.
All the generating techniques treat the solution procedure as one in which the articulation of preferences is postponed until the range of choice is identified. When an
approximation of the Pareto set is sufficient for decision making purposes any of
these techniques can be implemented. However, it is found that each technique has
some weaknesses as well as some advantages.
The weighting method can give poor coverage of the Pareto set by "getting stuck"
at an extreme point or in a small range of the Pareto set and by skipping over large
portions of the Pareto set. On the other hand if the weights themselves are considered
important results, then some degree of control over their values is a significant
attribute of the solution method. For instance, it may be worthwhile to communicate
to decision makers that this solution implies that objective Z 1 has twice the weight
of objective Z 2' In similar situations the weighting method is an advantageous approach since the weights can be completely controlled. However, the major disadvantages of the weighting method is that it always spans the Pareto solution space only
under the condition that the Pareto set is strictly convex and it does not guarantee to
produce any new solutions for any particular set of weights. Without an a priori
knowledge of the applicable weights the method can be very expensive computationally.
.
The NISE method is a powerful approach to generate a good approximation of
the convex portion of the Pareto set in a manner that allows the accuracy of the approximation to be controlled. More importantly, every Pareto solution that it identifies is chosen so as to reduce the error in the approximation as much as possible.
The NISE method guarantees to identify the exact shape of the convex portions of
the Pareto set. However, like the weighting method, the NISE method is also incapable
of generating information on nonconvex portions of the Pareto set. The strength of
the NISE method is most dramatic when t he Pareto optimal set has sharp" elbows"
in it.
The constraint method shows a specific and interesting feature that it will always
generate the shape of the whole Pareto set. By contrast, the weighting and NISE
method will span only the convex portion of the Pareto set. Furthermore, the constraint method provides complete control of the spacing and coverage of the Pareto
set. Its major weakness, however, is the rather high occurrence of infeasible formulations for higher dimensional problems. Furthermore, there may be difficulties in
336
M. BALACHANDRAN AND 1. S. GERO
finding a suitable formulation if the design variables do not appear explicitly in the
objectives.
The multiobjective approach for the floor layout dimensioning problem presents
a range of possible solutions, none of which is unambiguously better than another.
The techniques demonstrated here provide information on decision space, criteria
space and trade-offs to the decision maker. Among the techniques illustrated the
constraint method seems to be sufficiently powerful to provide information on the
whole Pareto optimal set while the NISE method is capable of generating information
on a convex Pareto optimal set more quickly and more efficiently than the other
techniques.
The example used as a vehicle for the comparisons was set up as a two objective
problem. For problems with more than two objectives the weighting method performs,
in general, no better and no worse than it does for two objective problems. The NISE
method was originally developed for two objective problems, it has been extended to
three objectives. Balachandran and Gero, 1 but the authors are not aware of published
extensions beyond this. For higher dimensioned problems the constraint method
becomes increasingly less effective as a means of providing a good characterization
of the Pareto hypersurface.
One possible way to improve the performance of all three methods is to provide
better controls based on the available knowledge of the shape of the Pareto optimal
set. Such an approach moves away from an algorithmic view of control to a declarative,
symbolic (i.e. non-numeric) representation which makes use of knowledge engineering
principles."
For the general P objective multicriteria problem multicriteria dynamic programming provides a good technique."
ACKNOWLEDGEMENT
The work described here is directly supported by UNESCO and the Australian Research Grants Scheme
(ARGS Project No. F80jI5542).
10 REFERENCES
I. M. Balachandran and J. S. Gero, .. The Noninferior Set Estimation (NISE) Method for Three Objective
Problems" (in preparation).
2. J. L. Cohon, Multiobjectioe Programming and Planning, Academic Press, New York {I 978).
3. J. S. Gero, Note on 'The Optimization of Small Rectangular Floor Plans' ofMitchell, Steadman, and
Liggett, Environment and Planning B, 4, 81-87 {I 977).
4. H. W. Kuhn and A. W. Tucker, "Nonlinear Programming," hoc. Second Berkeley Symposium on
Mathematical Statistics and Probability, University of California Press, Berkeley, California, 481-491
(1951).
5. W. J. Mitchell, J. P. Steadman, and R. S. Liggett, "Synthesis and Optimization of Small Rectangular
Floor Plans." Environment and Planning B, 3, 37-70 (1976).
6. A. D. Radford, J. S. Gero, M. A. Rosenman, and M. Balachandran, .. Pareto Optimization as a Computer-Aided Design Tool," Optimization in Computer-Aided Design, J. S. Gero (ed), North-Holland,
Amsterdam (to appear).
7. M. A. Rosenman and J. S. Gero, "Pareto Optimal Serial Dynamic Programming," Engineering
Optimization, 6, (4), 177-183 (1983).
8. J. Stein and C. Cohen, Multi-Purpose Optimization System User's Guide, Version 3, Northwestern
University, Evanston, Illinois, (1976).