- Email: [email protected]
Interactive graphic floor plan layout method
Interactive g raphic floor plan layout method Robin S Liggett and William d Mitchell
A new approach to the interactive solution of optimal floor plan layout problems is presented. The method is based upon the use o f probability theory to predict the likely consequences o f activity location decisions, combined with use o f low-resolution rester graphics displays. It not only generates high quality results, but also provides the designer with a structural understanding of the space o f alternatives being explored.
Since the early 1960s numerous computer programs for the automated solution of architectural spatial allocation problems have been developed. (Surveys, from varying perspectives, are given by Francis and White I , Eastman 2 Mitchell a, and Henrion 4 .) The objectives and levels of ambition of these programs have varied widely. Some have aimed to produce solutions that are in some sense optimal, or at least very good, and hence potentially better" than those produced by skilled human designers. Others have aimed at speed rather than quality, seeking to produce reasonable results in contexts where skilled human designers are simply not available. Finally, some have been research exercises in artificial intelligence, with no immediate practical application in view. Those of the more ambitious programs that are intended for practical application must compete effectively with approaches based upon human experience and visual aids. This proves to be very difficult, for at least the following reasons: • Experienced human designers are very good at solving spatial allocation problems. • It is difficult to capture all the relevant requirements in a mathematical formulation of a spatial allocation problem. • It is often very costly and difficult to obtain reliable data upon which to base a formulation. • The problem often turns out to be a large one of combinatorial optimization, falling into a special class of mathematical problems known as NP-complete (see Garey and Johnson s, Karp 6, Lewis and Papadimitriou 7 and Sahni et ala). Computational demands thus tend to be heavy, and it is usually necessary to settle for some kind of sub-optimal solution. Consequently, few programs of this type have met with much success in practice. Those that have tend to be School of Architecture and Urban Planning, University of California, Los Angeles, CA 90024, USA
volume 13 number 5 september 1 9 8 1
restricted to rather specialized domains of application (see for example Liggett and Mitchellg). This situation has prompted many proposals for the interactive solution of spatial allocation problems. For example, Scriabin and Vergin I° suggest that 'attempts to use fully automated computer algorithms to solve the layout problem should be reexamined with a view of incorporating man's visual capability into the procedures, especially since the real layout problem involves many factors which cannot readily be incorporated into a computer program, but which a man can take into account while designing a layout'. Although such use of interactive computer graphics seems attractive in principle, it proves to be difficult to implement successfully in practice. First, there are hardware problems; cheap and widely available storage tube graphics technology does not provide an adequately responsive interface for this kind of application, while refresh tube graphics technology tends to be too expensive. Second, there are conceptual problems; it is not obvious what computer-generated information is useful to a designer attempting to solve a layout problem, or in what format it should be presented. Hence, most attempts to solve architectural spatial allocation problems (and analogous combinatorial problems) interactively have failed to yield particularly encouraging results. Certainly experimental comparisons of the performances of human designers, fully automated systems, and interactive systems have not demonstrated any clear superiority of the interactive approach (see for example Michie, et a111 and Cross 12). This paper describes a new approach to the interactive solution of floor plan layout problems that overcomes many of the difficulties. It utilizes raster graphics to realize an inexpensive but highly responsive interface, and it effectively guides the designer's decisions by providing useful information in an immediately comprehensible graphic format. Test applications to realistic problems, and comparisons with a high-performance fully automated program, have confirmed that it is a very effective practical design tool.
OVERVIEW OF THE APPROACH The spatial allocation task considered here is formulated as a quadratic assignment problem (Koopmans and Beckmann 13), in which individual activities are assigned to individual locations on a floor that is subdivided by a grid into equal-sized squares. Each activity is thought of as a set of square modules, the number determined by the activity's area requirements. The objective to be minimized is a cost
0010-4485/81/050289-I0 $02.00 © 1981 I PC Business Press
289
function that measures both the fixed costs of assigning particular activities to particular locations, and communication costs resulting from flows (of people, materials etc) between activities in a particular layout. This formulation is employed frequently in programs for automated solution of floor plan layout problems (see for example Liggett and Mitchell 9). Fixed and interactive cost data, as required in this formulation, are either entered interactively via a keyboard, read in as a file, or extracted from a database maintained by a space programming, planning and management system (for example the Computer Aided Design Group's (1980) BAPS system). The designer works interactively at a low-resolution colour raster graphics terminal, and uses a light-pen to locate activities in the plan, one at a time. Probability theory is employed to compute the expected value of the objective function for each location in the plan of any activity that the designer selects as a candidate for placement. This information is displayed in the form of a colour or grey-scale 'contour map' superimposed on the plan. Such maps can be computed and displayed almost instantaneously, They enable the designer effectively to integrate information on the cost of location with other considerations that are not incorporated in the mathematical formulation. Where fully automated algorithms for solution of quadratic assignment problems produce reasonably good solutions (which usually cannot be shown to be globally optimum) to the complete floor plan layout problem, this interactive method aids the designer by quickly providing good solutions to a sequence of individual activity optimal placement problems.
E X P E C T E D V A L U E OF T H E O B J E C T I V E This approach is closely related to the implicit enumeration method for solution of quadratic assignment problems developed by Graves and Whinston 14. The layout task is viewed as an n-stage decision process where, at each stage, an activity is selected for assignment to a particular location. Since it is impossible to tell at any point the exact cost of making a particular assignment (this will depend on the subsequent location of other activities), the expected value of the objective function that results from the assignment is calculated instead. At any step of the decision process, a partial map (solution) consists of a set of elements that have already been assigned (fixed elements) and a set of elements which are candidates for assignment (free elements). Given the sets of fixed and free elements at a specific step h in the n-stage decision process, the expected value of the objective function for the general quadratic assignment formulation given in Liggett and Mitchell 9 can be calculated very efficiently as follows. (See Graves and Whinston 14 and Liggett Is for more detailed derivations of the expected value functions.) Given that there is to be a one-to-one assignment of the set of activity modules, M = { il, i2, i 3 , . . , im }, to the set of location modules, N ={ Jl, J2, J3, • • • Jn }, then the expected value of the objective function at step k is ~"
i (fixed) 1 + in-k)j
290
lip(i)
~, ~, (free)/(free) fi/
Y,, ~ qi, i~cp(i,)p(i2) + i, (fixed) i 2 (fixed) + - -]
~"
~
~"
(qi,i~Cp(il)j2 + qi~i, Cj~p(i
( n - k ) il (fixed) i 2 (free)J2 (free) +
] (n-h)(n-h-])
~ ~ qi~i= T, i, (free) i2 (free) j, (free) J2 (freel
where fij
is the fixed cost of assigning activity i to Iocationj qi, i= is the interactive cost between activities ii and i 2 cj,j~ is'the travel time between locations j l and j2 p(i) is the location assigned to activity i under a particular map p.
The expected value is formed from the following: fixed costs associated with fixed activity-location pairs + average fixed cost of all possible mappings of free activities to free locations + interactive costs associated with pairs of fixed activities + average interactive costs between fixed and free activities over all possible mappings of free activities to free locations
+ average interactive costs between free activities over all possible mappings of free activities to free locations As with theautomated solution approach to the floor plan layout problem (Liggett and Mitchellg), the general quadratic assignmentformulation is complicated by the need to consider activity area requirements. Note that the expected value function expressed above is based on the assignmentof individual activity modules to location modules and so includes interaction penalty costs between modules of the same activity, which are introduced into the formulation to ensure activity coherence. In the fully automated approach, the expected value of the cost function is calculated at each step for each possible activity-location combination. That combination which yields the minimum value is selected for assignment at this stage. This cost information is then fixed for later stage evaluations. Execution of this algorithm traces a path through the decision tree to a very good layout, a!though the result cannot be guaranteed to be a global optimum. The expected value normally drops at each step, reflecting the effects of making 'good' location choices one-by-one. Early location decisions usually have a major effect on the expected value, while late decisions have relatively marginal effects.
D I S P L A Y S OF E X P E C T E D V A L U E S The interactive method is also viewed as an n-stage decision process, with the designer making two kinds of decisions: • Which activity should be located next? • Where should it be placed? Activities can be located in arbitrary sequence, but it is very helpful to know, at any stage, how to choose candidates for location so as to achieve large reductions in the expected value. Thus the system can be requested to generate and display a list of remaining unlocated
computer-aided design
employed. For comparison purposes, Figure 2 shows a solution generated automatically by the implicit enumeration algorithm discussed in Liggett and Mitchell ~. In this example, 29 activities of varying size have been located on an L-shaped plan. There are no fixed costs associated with this problem. The interactive costs are shown in the matrix in Figure 3. Rows and columns have been sorted in the matrix to emphasize clusters of interactive activities. Figure 4 illustrates a sequence of expected value contour maps generated during interactive solution of this problem, as follows: Step I: At the first step of the decision process, no activities have yet been placed. At this stage just two types of contour maps are generated - one in which the best activity placements are located in the centre of the plan and one in which the best activity placements are located around the perimeter. Activities that are highly interactive with other activities, or moderately interactive but large in size, are 'attracted' to the centre of the plan. Activities with low interactions and/or very small size are attracted to the perimeter. Figures 4a and 4b show these extremes for two activities in the L-shaped plan. (This observation holds true no matter what perimeter shape is used. Figure 5 shows the contour maps for these same two activities in several alternative plan shapes.) At step 1, activity 27 which is the most highly interactive activity, was selected for placement in a central location. Step 2: Once an activity has been placed, a wider variety of types of contour maps emerges at the next step. If the placement is symmetrical as on the square plan shown in Figure 6, the contour maps that are generated remain
activities, ranked according to the reductions in the expected value that their most advantageous placements in the current partially complete layout would achieve. Once a candidate has been selected for location, the expected value for placement of a module at each empty location in the plan is calculated. This information is displayed in the form of a colour or grey-scale contour map superimposed on the plan, with darker-toned areas indicating the more advantageous locations. Three different contour maps can be requested, as follows (Figure 1): A map showing activity fixed costs of assignment to locations (which does not change during the assignment process). • A map showing activity interactive costs with respect to only those activities already placed. • A map showing the expected valuesof the fixed and interactive costs combined for a single objective measure. The last map is the most valuable, since it shows the expected effects of alternative location decisions on overall plan efficiency. Generation of these maps does not require a great deal of computation. Even using a small minicomputer, such as a PDP-11/23, they can be computed and displayed almost instantaneously. EXAMPLE
The process is best illustrated by means of a worked example. A department store layout problem will be z~
................................
~ ,,~ '~"
L
+l+ =+.i z+ I +i!
iii~iii!ii!iiii!!i!ii!!ii!i!!!!iiii ::::!i!!iBS~i~"'~;;~!!iE;! :::::::::::::::::::::::::::
........................ ::I =;..:P4,'.'=~ ............
l .........
+, "! 1iiiii;!::=t?!~..=-=~i~!!~!! ........................... I ,~, :::::::::::::::::::::::::::: ,~ ........ ',Lff~:P&"~..... ] ~, ::::::::::::::::::::::::::: ,~ ::::::::::::::::::::::::::::::::::::::: Jii!!!!!iI +
,,,t, i s .s is I i+ I u I
Iii-
Ii I
i= I
,!
,!
+,! ,!
,! s! ]!
,4
~i ~!
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~--~+- j - + - L - - + - - I - - + - + - T ] 8 . ~ - ~ j - 3
j-] |-] j-I£n++;+-iF+~-;
t! r;+-;~-;t
a
b
c
Figure I. Contour maps (darker toned areas indicate more advantageous locations) (a) fixed costs (b) interactive costs with activities already placed (c) expected value of the objective function combining fixed and interactive costs 4~ 4884~444424834 ~ 7 e s 9 io is ~ I~ i~ 16 ~ 2 2 2 3 2 0 ~ 2~ Z~ 2~ 2 s ~ i 2 ,~ i~ ,3
191
beats i~,~ z,-
• .......................
, ............ ~ ~ 8 4 4 4 8 4 B
.!
"~{........i'" ..
J
iiiii:: ill
i1!
+
iz =
.___L..
.
. 2
;~
~ 4 \ 8 I
\.4 4 e\4
:il';
I-...1 r::..l z~$........ L.,
.
.
.
.
.
.
.
.
.
.
.
l,..1 J.,; IL-I
! i
.
.
-
.
-
41
i
']
I
II ', I
i
I
~s
........... ':" .......
.
";-'27
!!
.
Il....Jl .1
"
44
I I
I~
II
2 8\24
,448\ ~\8
8
~4
I 12
Figure 2. Automatically generated solution, objective = 9382
volume 13 n u m b e r 5 september 1981
Figure 3. Matrix of interactive costs
291
symmetrical. The various different types of maps shown in Figure 6 can be explained in terms of the activity's overall level of interaction with other activities as well as its interaction with the activity already placed. For example, the most desirable locations for activity 26 are in a tight ring around activity 27 (Figure 6~). This activity is directly connected to activity 27. The next map (Figure 6b) shows a wider ring of the 'best' location areas (ie a wider range of acceptable locations). These occur for activities indirectly linked to the placed activity. Contour maps with desirable locations at extreme perimeter locations still exist for small noninteractive activities (Figure 6d), and a middle range map occurs for activities not connected to the activity
placed, yet with interactions to other activities (Figure 6c). Results {or a non-symmetrical activity placement are illustrated for a courtyard plan in Figure 7. Again an immediately understandable spectrum of contour maps is generated, ranging from those for activities with direct connections to the activity already placed through those with no or very slight connections to any other activities. Returning to the assignment progression for the L-shaped plan, in Figures 4c, 40', and 4e, contour maps for the top ranking activities competing for placement at Step 2, are shown. They also vary in relation to level of connection with the placed activity. Activity 26, which is the most strongly connected to activity 27, was selected for placement.
,==,Niiiilti , ...... ,,ooo<. . . . . ~olgl=i...l
,,,
I
,4
Ji ; i;777i71711@i
,,'. . . . . . . . . . .
'~
. . . . . . . . . . . .
!iT!~77177!7~i77i77E77i~{~J
.-, ..................................
. . . . . . . . .
:::::::::llIllliUll
Itlllli~il
:::::::::lll~llllilil~ll
I I I ~ ° ~ ~°I
7i]7iii7f#:i77i777777777{i!!~il
zz!
,~, ,,] ,~, ii! ,~,
S iiiiiii iiTiii? ii i{i;ii!iil;iiiiiiiiii?ii?iiiii?iiiiiiiiiiill ,!
:::::::::~
. li~,.,,~%~ ,~,~
'
~I~::
:::: . . . . . . . . . . . . . . . . . . . . . . . . . . .
:: :: :
~o°~o °
W V ~-~ V V W - t - - l - ' ~ - I Y ~ V 4 - 4 4 " 4 - 4 * 4 ; I ; V ~ K S V 4 - $ 1 - 7 ~
a
b , ....................................
~-
=: ::i ":
iiiili iiiiii i f, iiii!iiiiiiii!iiiiiiiiiii!i!iil
11[ la I
,.! ,~' .! ,~' ,~' ,,!
...................................
l:?i:iiTiiiiiiiiii!il]iiiiiTNiN~!l
f:lllll~:~ff~Ifl:::::::::::::::::
....................................
1~.~._.._.~
~
::::::::::::::::::::::::::
ii :::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: ,, iiiiiii~iiiiiiiiii!!!iiiii!!iiiiiiiiiiiiiiii!!!i!ii~i~ii77i ~ IIIIIIIUI~'~iI'~I~<~I~I~::::::::::::
::::::III°~'I~.~ - '
d
e
t
...............................
~i
:~i
ll!
ll!
.,
;i:~i:~iiiiiii iiiiF, iiiitiiit f:i::::il ::I
,,:
'i!NF~iiiiiiiiiiii!iiiiiiii!i!::iil
I~!
ll{
ii ~
li ~ I~J
ll!
I!
I!
o! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
g
h
,,,_
=====================================
.................................
__
,o:
,: ~..... l t,t....... fll~il,~ll
........... . . .] .......... . . . ,......... I"
!"
.
!ii!iii_=~
;--lllUllltlll;I
i
Figure 4. Sample sequence o f steps in the interactive solution process (a) step I: contour maps for highly interactive activity (b) step 1: contour map for non-interactive activity (c) step 2: contour map for activity 26 which is directly connected to the planned activity (d) step 2: contour map for activity 21 which is indirectly connected to the placed activity (e) step 2: contour map for activity with no interactions (f) step 3: contour map for activity 24 which is directly connected to both placed activities (g) step 3: contour map for activity 28 which is directly connected to one o f the placed activities (h) step 6: contour map for activity 29, directly connected to activities 28, 25 and 24 (i) step 6: contour map for activity 20, connected to activities 26 and 27
292
computer-aided design
Step 3: As shown in Figure 4 f a n d 4.<3, the contour maps focus the better placement locations around the central core for activities connected to those already placed. Activities 24, 25, and 28 were placed next in this area. Step 6: A t this step, specific sides of the plan become more desirable to individual activities. For example, activity :29 is connected to 28, 25, and 24 so the contour map shifts to the right (Figure 4t7). Activity 20 is connected to activities 26 and 27, which is reflected in a concentration of desirable locations to the left of the central core of activities (Figure 4i). Activities 29, 20, and 1 are placed next. Step 9: The placement of a cluster of interacting activities is complete, and new groups are emerging for
placement. Once again, highly interactive activities are attracted to central locations in the plan. See, for example, the contour map for activity 8 (Figure 4j). Step 12: As the plan fills in, desirable locations for placement become more strictly defined. See, for example, the contour map for activity 9, which is strongly connected to activity 8 but to no other activity (Figure 4/+). Activities which have strong relations to some other activities, but not to any of those already placed, seek spots towards the centre, as seen for activity 11 (Figure 4/). Step 14: Variations in contour maps can be seen for activities 22, 23, and ] 2 (Figures 4m, 4n, and 4o). Step 21: Placement decisions become more and more limited with less area available (eg contour maps for
:_~:! _r! _~?
.'? ."_~_LLL"
iiiiiiiiiiiiii_i___i__. i!iii ...................................
~
~,*.*
....................................
"! +~
~o!
~'I',I i~ i
,~
~'i," !+.ii+N ,',"::,~! !.......... . . . . . . . . . I +_~++_.~_%-++%-- _++_+_+l ~ ~ ........ ~. ........ ,_., ............. !;7.......
'+::;:,--~...._~;.~:.::~::~
~4
I
i
"
i:!HE££:::::::: ...,. . ..........
. . . .
.........
h
I
iii
. . . . . . . . . . . . . .+'. . . . . . . . .
~ ,liez
I........ I.......... I
';.~i~i~!|
|
+iiiiiiiiii~!i~d.........: + _ i l l
t . . . . . . . . . . . . ;;"
...... ;
:::;:,~'m~::;:I;:~+~+'~i-"'++~ III ++--~Hh~.":";r
• *****+** ~oc(i+~ I ......... m+........... ~11
!!!!J+!i+!+!!!!!++++++++M+........ li+!i+!++!!!!!!!!!+!!!!i;iii
d
i
Iili!!iiil
,II,,,,U
~..HIlUUllUlIII~;
++++++++++I
1........ F. ......... !
..................................
ll-- i--!--,l--t-- ~--L--t- -1.-1-~8-~ i+.i#l-~t-~ l-~t- ~L-~ t-% L-~,~+~p-; i-;1 -~1.~#,
"!,+'
,i ,='-:i II ,H..+::,-+'.---;,
,,.llIi!......
,~
"!"!'~";;;';;;;:~°+"+'~I
I~i~iiii~iii.'+".'~!i_'!_'i~,i"
4
...-
,
...... = ..........::i~:m.-;~ ,!d II:::........................... ++~oo,I t ..................... := =:: =t -": '--+-'":::-+: -" ::: ¢+~-m-~+°l........ I .%+.% .°~-. -~- -%m--°~.+?~+<'__+°~ °°~ _
t-- r-+~-l--l--F-,t-T-l--i;%-irit-il-;t-lrlFit-il-il-iF$1-Tt-~F]t
I
k
; ~_++_~ !_LL' .t~
,+, ,,!
iil i!iii!!!!i!iiiiiiii~ ....... i :::::::::::::::::::::
i1 I
,,~ ,¢
iiiiiiiiiiiiiiiii!~+ ...... ; ,;+ . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,+,'+' I-ii~l' 4 Itli
I
I
"-] .....
I....
I
.............. ._,., . . . .
i ...........
. . . .::':"::++ .,
+~'
+,+~+==.F,,,=m;:mm
:::::::::::::::::::::::::::::::
.+' ,~, ,~,
.~=~..~iiii+ii+ I l::=.5-:lllill|llill:...,i;;
'#
~llllhllnllUlh;+;
,,!
gii+~i~++++i
,~'
t!:-~-"~++~[~; ...... i ........ L ; ......
'",,~ i+++i+...... ~l~
I. . . . . .
,:.+
F+ ++......
t+t+++++ .,+
r - UT-~--I.-TT-t
i
--I-TiI-ir~I-i$-WW$C],~-]I-;I+~
~- IIlIIIlII:l'+IihlllIlIlllh.,II.llll ++e+ + +{++Ii * llllllsll$111 iii
Ii
~-;~r;t-+l-~l
-z0:! ~L".
l l I l , , I i i I l . + . , , , ,.+o . . . . I lliili
+++IH"mm
m°
,d
I . . . . . . . .I. . . .
+,~' "!
....................................
:#P-:i"
10~ L'I"--+~
[ ....... I.
.
.
.
I
i~iiiiiiiiiiiiiiiiiiiiiiiiiiii
i
I
,,, l..'].i!W+m++l
I
l~!~+~.l-_i +-~,........ :::L.+_I ~,~, . i !l l-°-~-~{'"~--I~'l . . . . . . . . . . . . . . . . .
I........ I++I........... i ......;~o%+.coo++....:;1
--l-1-~--~-~--+~.°9.++°+-c~'.+::::::::+
,4
l
...
~ooocco J,
'o,--Mm
......
~--l-]+-++-]+-;
¢
IM+=.+iM+..=.+.+++N.+++
~-] L-++.+I.;
.,+ ..........
++
++ ++++++++ ,,~ ~iiiii~il
,
1.............. 1
+;+iii+i:+++++L_+......
-a ;+;+++H++::v' l
~+,'°
I
i
I,
I. . . . . . . "
1%+.]+-+++;
i l.. ;;..-,..
+.++_]~+;L
,~i . . . . . . . . --1.-1 c'
{ill:
::~ ................................... [ ......
" i ......
L+ .....
+........... 1 ++...... i+
I
+:........ ~...... :::t___t
ii !........ Ill!..............
'
~,,'~':ii+--'-ii::i-....... --, ~~..... t:......i ----! !ii!}!!!iiil;~ ........: : : L ! :~ l:::++~;;~::I: ....................... I........ l+=++++++,+++,+,++++++l-='+:um== !i :=:[L Lilill] ........ ~--+- }-- t-C-+-T-F-+--i-~+-]r~3~-;I-;U;£+H.~£-~I-++.;~_;~_+~_+L
p
L_l_+_,l
2OI
+;+: I
+.......................... I .........
21-I
'1:1
ii
l~l~
I
I i+,+..... +
o+
-'
!
~:i~ "-:+:i++...... +;'......... ,., I-"..P..~+=.......... ~+' +........ +........... ~
....
'd;:iilE"~
;,;;;~;;~1111,11:~o~
BLOC. P"B
.,.~'"
Ii ;~...... i ......... L+i+___ ,I
.
I
'! I+::++::+::+~+~+~::+~++++!!!++N+++I ........ l+-l--l--+-+
I;IllllI t.,,,-,
||,|:.,,*m,.~
+-i
::::::::::::::::::::::::::::::::: 8-- ~+- j--j--
I++.......... lllll~l
1
~ ......... !........... 1
O
I~
I
1.......................... r~ l+
2,-
! !!!! Ei!!i!i!i!!iiiiiit
,~
ii;iiiiii:;;'::::J
~!
=o.E .,,.
-
::::::::l
......... ;:::::::
.....................................
,~'
"!
iii~iiii;+:$i~i
1I1
+ .............................
:::::::: :i I::::::::1 ............. ; --1~° +.,
!........"='m~+~m++
i
...................
'~
n
2~
~
'
,=,it
.......
......
!..~
"!===::=+. . . .
~, I................................... l........I..................... .+_+_.+ +-v-+--rv-~.r+-+-r++-+r~r~v++-+r~+-~+-+r;r++-+r++-++-++
ITI
,.,
+,
l +"+" INigi
+ i?::::]
"
m:+J
..!
,,:
!!i!iE+iiii,,i:l........:::i....J
P+"'R"~++++++:;::':"I 1.:=,::~+%+o.++:::::::: ..................... 1 ....................................................................
~.t++
iiiii!iiiiiiiiiiii++++!!iiiT::2;:!
,~
~, iiiiilgi!!iiiiil
II!.I!HL....,LM,.P..:I+,, . i.i~. "
,+~.,,+ . . . . . . . . . . . . . . .
.,
~:~!
r' +
i
.....................................
q
[ZL!I'Ji
,
~-~--Y-$--v-!-4-~--l--r,~--,r-,~-,$--,L-,r-,L-,r-,r+r-+~-+~-,~-.|--+l
(j) step 9: contour map for activity 8 (k) step 12: contour map for activity 9 (I) step 12: contour map for activity I I (m) step 14."contour map for activity 22 (n) step 14: contour map for activity 23 (o) step 14: contour map for activity 12 (p) step 21: contour map for activity 10 (q) step 21: contour map for activity 13 (r) final plan, objective = 12 791
volume 13 number 5 september 1981
:293
ZON~ Ell~ al-
zlaol.
tam,, ....................
161 ~s .a i1_1
i1-I 13J 11 -#
Io i-
:::::::::::::::::::::::::::: ::::::::::::::::::::::::::::
~.ooo~,C.~.HIIIiiiiiiilt
,hhlllh..,... ....... , . , o o o . . . . [~ . o . ~ " ~ l l l S l l l ! l i IOlH L~"I~*~°O~'O~C~C*''°* **** °°**''*'*°'"*o***0 ococo.~ ~.~te~,~cc~o~¢cc.,°.,*.,.,.*******o***o****ooo i coc o~, o~ o. ii i ee~o~e~c~ooo~c~o,,o * o * , * • . * * , * * * * * * * , o , , , , *O0~CO~G~ , ii I,,.~,~ o~o$I:III,III c~ c .c.~. .9. ...-. o. . o. .o. ....... . .. .. .. .. ........... .. .. .. ........... .. .. .. .. .. . . . ooooo***~ o
o c
o........................... .oooo.**.~¢
~ccccc~.,,.o,o.. ccccccc~c,,.,,ooo,
................... ........................
: .......
ooo,.+oo,oo
o,o.,o,oo~
o o
o
o
~o.~cpo~ ......................................... o~ooo~co c c c ~**'**°'** ........................... **,,,,o**ou o ¢o
I~o, ,*~.......................................
~oot
.i COCCCCCOU,.o*Hoo* . . . . . . . . . . . . . . . . . . . . . . . . . . .
,!
o * * * * o o**oooooc~o
................ **o. **** oo co 1eococcccc~.o**o.oo. .................................................... ~,o ~e~e~ooooo©cooo*,* ,** ,,,*o o******* o,o,.o,ooococec ~ee*ele
d
° B ~ E~oCOUGCGCCC** * * * * * 4 * * ** *+ * * ' * * * * + + * I , ,O0000GGO~d
.! ~e
19
o o~cocuo**. ** *• *.. o.oo • . . . o. o+o....c.oooo~o~ u i ~'s~eeeeeeo~,oo~rcoocooocoooc¢.¢ oo ee ......................... | | | ! | ,]~9 ~ll~ouUdlII.K~U UO¢00~¢OO0~;r.OOC0CO000u~lle I m l ~ H ~ I I I .11|i|~. oee~eee~o~ oo~ c c c c c o o o c o o c c c c o o c c o ~ a ~ ~ m ~ l e l | l U l l l l |1 i i l | | t | e(~'en~eeeoco~o© ccoc c o o o o o c ~ c o c o o o o o o e e e ~ m ~ m l ~ l l U l l l l l l |
h,dii,,
O!
~oo~...~iill 1
o!
5--~--! "~--~--Y'~,--V-~,-'~'.6--,YW-,5"-,~,'W-,L-',Y',V;V~6--~I
b
loll
iii.
........................
:1~ ~]i]]ii]~iii?iiiiii~P~iiii5i~.~iiiiiidiiiiiiii~iiidiiiiidiimii;~i~.i~i~i~ii~-~. .................. ....................
~o,.~li~.,,..,,, o:o,o~
.......................
.................... ~ " I "
I
I
, ,
•
iiwuuiK'iiihilii[,'i ,l .............. i a
.... =.,.,,____...,,,
,,
:::::::::::::::::::::::::Ull~lllmlllllll s i
'-' :::::::::::::::......... ...... ~.........., . , ~ . ~ . . ,
~[[[~il .=.
~!
..... ~ :~.~,
.
.::::::: .......
,~
::::::::
l-T-i-t-
l--~--;&-i y~t--i~-; I-W~ l-;t-;l-i
~iiiili::iii::::?::i::::/?/]i::::i::]::::/::::i::/::iiii~iii~
...............
h]~!l.~ .............. ii,, ! ~,.,,~ .................. i~.:."-: ~ I ~ : :::':'::"
d
:ii]::ii!::i!!!i!ii~~~~iiiii!iiii::i/::: t 6 - : I~- j - i -
~}!!}!!!}!}!!}}!i!i}:~!:~i:~!~=~ ..................
- F ::::::i
lEi!i!!!!iiiiiii!!iiiiiiiiiiiiiiii!!!
~..~, . . . ~ ; . , ~ . . . . . . . , . ~ , . I J I ,.-'th.'h ~ L,..,o,,~|i~
S
t - ~ t-H t - ~ y i i - i l - ~ t - ~ l U + - i l - i l - i & 3 t - ~ - ~ U i Q V i l
¢
d
1011 I I , I
19! leJ IIIIIII
II:::'~
:
Ihlh ;,~==':" ,llllhl h III It
,q
::::::::
..............................................
:::::::':~-"~l:'lil;~ill
:::::::.:.:: ...... :::::::::::::::::::::::::::::::::::::
.........
. ,
.......... ,~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 ~,,~,~ ...................................................... S~S®~.~.,,,,,,,,, Ill--,1',1~$$~.~ ....................................................... ~ ...... lhlllh, *!
d o!
~4..........................................................................
ll-
a*-
::::::::::::::::::::::::::: ......... ~ ' ~ , , ~ . :~ • ~:t: ~:::::::::::::::::::::::::::::::::::::::::::::::::::
I.I.It.
:J!.h~:::::::::
::::::::!!:!ii! !i!
ii::::::::.~:~II
|i!l iiiiiiiiiii;:iiiii;iiiiiiiii;;iiiiii:ili;:;!il;;iiW
T*! is I .! la!
la! i1!
iiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiii!
,d 0 ¢ ,!
!!:!
6-- t " ! - ~ - T T T T T T ' I & - ; U ; ~ ' ; ~ , ' ; I ' ; I ' ; I " ~ V ; I - ~ I ' ; A ' ;
f
i l_ 7! ,!
!i
..................................
e
ITJ
. U~'W:,I
J
Figure 5. Contour maps for ~lternat/ve plan shapes (interactive and non-interactive) (a) and (b) square perimeter (c) and (d) rectangular perimeter (e) and (f) t-shaped perimeter (g) and (h) x-shaped perimeter (i) and (j) courtyard
294
computer-aided
design
activities 10 and 13 in Figures 4p and 4q). Step 29: The final configuration generated is shown in Figure 4r. This can be compared to the fully automated solution shown in Figure 1. The actual value of the objective function for a completed plan, as defined for computation of expected values, is highly inflated due to the effects of the penalty costs between modules of the same activity that are needed to prevent splitting of activities. Thus it must be adjusted to provide a more useful basis for comparison of alternative solutions. The values shown in Figures 1 and 4r were calculated using activity-activity interaction values, and straight-line distances between activity centroids, with all penalty costs disregarded. Figure 8 shows an alternative final arrangement generated by the interactive process when a different placement decision is made at the second step. Using knowledge gained from the intial interactive solution process, a considerably better plan with respect to the value of the objective function was generated. Figure 9 shows expected values for the objective function against steps in the solution process for automated solution of the problem, and for the two interactive solution processes. The general similarities between the three processes are immediately evident.
SOLUTION
EVALUATION
A single number provides an insufficient basis for c a m parison of solutions The precise details of relative location of the activities in the arrangement, and the contribution llli
fill
IGll
,'~ "7.77+iT]F+b.:ii+ii::-~ii.:+ii~ii++++++?.+:.+..:.+:~.7-777; ;" ::::::::::::::::::::::::::: :::::::::::::::::::::::::: il!
,st
....... i i ::::::* ....
l
~
l
.
l
l ~
~
ill.-
:Clt
fill
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:::::::::::::::::::::::::::::::::::::::
iiiiiiiii i ttiliitfii ,***ll***
:::::::::
~*ll**"
¢
l
,i- I..................~ < + + ~ ~
iiiiiii
t~
....~
Ist,tfmuimlIwllllll
illilllSlllilIWmill
........
.-" : : . l - l t : l : : - : : ::** ~ C=,m.O00+,,.,...., . ' + + o ~ ' + * " * * . . . . . . . ,. ,. . . ,. ,. ,. ,. . . . .
t t--t-'l-l--l-l-?-l--lit~l-~Yiii l-+t-il-;t-il-+l-it-;I
........
.......... ;";':'1
moccl¢o~lJ
uoo(¢(~o (.occc i . ~ u,ccc~c~
liWilWilillil~< lllillil~lillli~lll~
d I~°fffl~c~i~W~l I ~ ~ 1 I~ltlb~ flllmlililHtmllllHimm c~c¢cccc,o
+,o,,+++, llUllllllfllillll h ..I,I,.IMIIIMIIIIIIII
,,,,+
llmllllililWililllt I ~I llUlilllillillW~d
•
1:::::::::~:'!91BliBlll~llllllll~+~l .*+.*.*o* ~. • ****H.. I:::::::::!|=~~~~1
s+
~' Eiii!!!!i~F::::: ~1. . . . .[ .......... . . . .
o
:::::::::::::::::
. * * * . * ~ ci ! c* * * o*~ + * * ' * . *m* * *°* *°* * . . . . . . . .
¢
Jill
!
~=_=_!!!!! II ........
i+ii+iiiiiiiiiiiiiiiiiiiiiiiHiiiiiiimNii ,.',,. I!!I|I~III=,IliI~!I~ll~HHlllliillllII~llli ....
, i!ililil!~:.+: "; ,.,,| !! iii..=.: I ::....... ......... :::::::::'"++++++
'+ .'=] ;
'
i lli!! o !
:';; .~++ . . |*' | '~+* ,.|.. . ... ... . .. . ... ... . .. . .. .
' ........ : : : : : : : : e *. .
:F-";t2iiiiii]!
?iiii!!ii •
iiiiiiil
,¢
! | eeseee *e~Hee ec[o t:,*t,te*,e~eee,,e*m .III,I,I .... l,...,.. "i~,.~--~. ..... ~-'I;IIIIIIII Ill Ml~e=llV~41lmele+.cu*~ooooeilwlle~NiiwsiiImll ilell
lU
lllBiDH
~-*e~*~ei~*¢.~ooa o o . * * *.. *.. *.**** ***.. **.. * * * u u ~ u ~ o o ~ . * * N ~
,1 i.o*,,+.L~ucoc.o ........................... o~°d~.~.tl| *e*l*l***co~*~v.*C~* ** * * * • ** *** ** * ** * ** +*. ****0(+(+ C4t 15~1 I , . e . , t + * o ~ c o c c o ............................ ~m.v~l$ ~,,ee H et~¢ , *~le. .e. .. .. .*. * * . .*. .* ................... : . : [ : : " ' . . . * ** **.**** * * . ****eeeee+ eee , d b,**,,,|,+- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : * *. * * . . **..'m"*',.*1.** i~eel~ • * * . * * * o . * . . . . . . . . . . . . . . . . . . . . . . . . . . . o***.****w~li*~ 16jl
. . . . . . . . . . . . . . . . . .
,~ ....,liP,|........................................e. ¢cc£ ~coo.**.**.*. . . . . . . . . . . . . . . . . . *oo<.ouoco , ~ loot
'~
**.*** ........
:': ...... **.*****
m~.ooco
c c c c c c . c. .c . .o . *. . * . * . :. . ' -*: " * -.. . . . . . . . !~~041*t .......................... I ........ ............. lee**.*. *.+ . . . . . . . . . . I I,.,,,,I,.., . . . . . . . . . . . . . . . . l ........ I ~ee¢ + 1 * * * + * * * * * *. . . . . . . . . . . . . . . . .
I..,II............................
....... :1::::1::~...
ee
::: ......
"*****..**eeeee* i . . . . . . . . . . ****~ ~+**tee.v**,*,..** . . .*. .* .".*. .*. *.*. .+. .+.. .* . .* + * . . + .: .*';: :* * ;•; • +" 0. , , ****..imeeeeee knliii+lkl~,~OOLO~O** 0¢0C a l [.....,...¢.o+.,~o,+.o ........................... o~°~a~o,~P~,~.12 • e o e e l e ~ . v c o o o ****************************** o coo Idle* •
s_~ ,_~h,....++..coo+~o+o .......................... ~N~=+o~,|, *************************** ~occ ,
,!
• wee* e~¢ooooooc
i
!ImlHMIMHIIIIIIHIN!
+, ..+.++!++!., ................................................
!..
•
•
eel c c o o o o o o o
*eel*el
•
,: I.ILIll . . . . . . . ! ~ ........ ~ i M . . gi * .1-0 lal l .l g~i l iJ* Iiom lll || ,
Illll,,lil..: . . . . . . . . . . . . . . . . . .
*l~141k *tl4•4111k IltiO OCO000CO0
II.. . . . . . . . . .
li|fil21111
+-- ~-- j-- + - ~+-~--i--}+- }--l-~+-;l+~|-+]-+l-++-.+i-+ #-~,~.q+-++-~l
d
e
Figure 6. 5election o f contour maps after an activity has been placed in a symmetrical plan (a) contour map for activity directly connected to the placed activity (b) contour map for activity indirectly connected to the placed activity (c) contour map for activity indirectly connected to the placed activity (d) contour map for activity with other interactions (e) contour map for very weakly connected or small activities
volume 13 number 5 september 1981
I
+-- ++- j - 5 - i - U - l - - + - - i - - i-+~,---; U.~I-;,,-+~i - ; £ ~ y ; y.+l-; i-~+-i1
b ZCll
I
................. I
~ o * * * * * * * * *
',~ =~lf=~dlitlllllgl~lll"
~!!!!!!!!~m.~+.==iii~!~iiiiii~!!!!!!! 7:::]]:::!!:2!5!!!!!...... -: ................ "5"5!!:2]]!!]i!
::::
* * * * ** * **uo~oouc¢o+,¢
mmmm
+IWillill
d d
.........
": ! ````:```;~=m~
.l~+,i~ ' ' ' I l l '
/
,,2++++++++ '
Jill
. I
. . . . . . . . . . . . . ."~ . *'*~ l * * ; ; l i i i i ; ~ ........
=
each activity makes to the total interactive and fixed costs should be examined. To aid in the detailed evaluation of solution quality, and to provide guidance for possible further modification and refinement, additional visual displays are available. First, a network diagram illustrating the basic flow structure of the problem can be generated. Figure 10 shows the network diagram overlaid on the plans produced by the automated and interactive solution processes. The most important communication links between activities are displayed. Note that the second interactively generated plan (Figure 10c) is clearly superior with respect to both the cost function and the details of relative location of the activities in the plan. A second visual aid (also shown in Figure 10), which can be used in conjunction with the network display to evaluate the solution generated, is a colour or grey-scale map indicating each activity's contribution to the total cost function. This map complements information in the network diagram by highlighting those activity placements which are contributing most significantly to the overall cost of the arrangement. Larger contributions will generally be made by the larger, highly interactive activities even in an optimum placement (see for example the cost levels associated with the centrally placed activities in Figure 1Of), and thus in this case the information presented is not of much value for the generation of alternative arrangements. However, in Figure lOe the high costs displayed for activities 16 and 17 reflect interaction requirements not adequately dealt with in the solution, and so suggest a potential improvement.
295
DISCUSSION The distinctive feature of this method is that it employs an optimization technique to provide the designer with a structural understanding of the decision-space being explored. It makes graphically evident the important branch-points in the exploration process, reveals symmetries that suggest the existence of distinct solutions with approximately the same cost, develops an excellent understanding of sensitivity to variations in placement and shape of activities, and shows how distributions of expected values across a plan tend to fall into a few recognizable and frequently-recurring types of patterns. All this yields much more insight than a single automatically-generated plan. Such insight is vitally important in practice, since the cost function captures only one aspect of a designer's concerns in any realistic context (and that only imperfectly), and since quadratic assignment problems of this type notoriously have many local optima. The task is not to find one least-costly plan, but to integrate an understanding of the cost consequences of location decisions, in a well-structured way, into a search for a solution that responds to a broad spectrum of complex and often ill-defined criteria.
terms of values achieved for the objective function, it is competitive with the best available fully automated technique. In terms of guidance of the design process, it provides valuable information, of a kind that is not available in a manual process, in striking and instantly comprehensible graphic format. It appears to have excellent potential for practical application to problems of office, department store, hospital, warehouse and factory floor layout, where communication efficiency is a central design concern, but not the only one.
REFERENCES 1 Francis, R L and White, J A Facility layout and location: an analytical approach Prentice-Hall, Englewood Cliffs, NJ, USA (1974) 2 Eastman, C M Spatial synthesis in computer-aided building design John Wiley and Sons, New York, NY,
USA (1975) 3 Mitchell, W J Computer~ided architectural design Petrocelli Charter, New York, NY, USA (1977)
CONCLUSIONS
4 Henrion, M 'Automatic space-planning: a postmortem' in Latombe, J C (ed) Artificial intelligence and pattern recognition in computer-aided design North-Holland Publishing Company, Amsterdam, The Netherlands (1978)
This technique is computationally efficient, can be implemented on inexpensive hardware, and allows very fluid interactive exploration of layout alternatives. In
5 Garey, M R and Iohnson, D S 'Strong NP-completeness results: motivation, examples and implications' ].'Assoc. Comput. Mach. Vol 25 No 3 (July 1978) pp 499-508
:i ...................................................................................................................................................
::[ .........................................................................
u!
,~1
s!
s!
b
21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~0-
,,, :~:. ~:, ~:.:. : : : : : : : : : : : : : : : *: : : : : :~: : ~g: .:. .:. . . . . . . . . . . . . . ,~! [
,,I
F
I
"-"-'-'-'-¢-~-~'~
;:::;;:::::::::::
:::~]:q~::::::::
,~,,,: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ,~
I,IIIIIt~,!~°¢~1
,,' m!!l=~::::::::
1
,,, l~i~jll~lil
I]]]]]]]]][i]i]i]]
~I||11
.............................. :::::::::!]!]!!ii ,: : : : : : : : @ . . . : . : . . : . : : : : : : : : ,: : : . %~:~'~!l..::::::::F:::::::::::::::
I:::::::::::::::::
d
. . . . . . ~[
1:::.....
:::::::::::::::::
......................................................................
e d Figure 7. Selection of contour maps after an activity has been placed in a non-symmetrical plan (a) contour map for activity directly connected to the placed activity (b) contour map for activity indirectly connected to the placed activity (c) contour map for activity indirectly connected to the placed activity (d) contour map for activity with other interactions (e) contour map for very weakly connected or small activities
296
computer-aided design
~',,,
lil]i]iiiiiiii!]iii]iiiii!iTiiiiil]]
i?i
i"~......... FY]%;mll
t ~o!
~ |
,~,
l~l~mll~::::::{iii::::::
"~
1
., 1~ ,,I
l l l ! ! l l .[ .I. . .l. . . .l. . .l. . . l. . . .l l l l ,=.=.,.
...................................
,,' I:::::::::~+,~III~=I~,UlIII"
I
I '+
........... :=-mr,m,i
~'
:,~
i
m,,,m_
,:::::::::++,=mlHlt ........ l
l ...........
t
'4
I
IlUlIIIIW:,~I
:+1
! .J ................. I ............ 0,
~
~
........ +t ..... '~ .............. +~ I TM I--! ........ I'+ '~
I I
!_.# ........ i U
.J
1721........ l
~.
II
"
t::::::i::]?i]iii!iii!!ii!!iiii~i~N~N~:ii!i!ii ;it ........ ! ................. ......................................................................
I-----I"
/
I
t ......... I I
I
' -+! I..............................
,!
:::::::::::::::::::::::::::::::::::::::::
I
.............. I++..... i ~'....... ~........ I
"
t
::: .................. ot~P~llo,I t111~I,~o ............. ~ o o ~ oo~.. m . . ~ . , , ~ , . I. . . . . . . iiI__~,,~ooo~o . . . . ...+.+ ......... .. +........ tltl ::,::::: .......... ~ J .+o,o ~ ! l l l l l l l , n , l m : m . . mIIII~II ~+ .+>o
,.
I~'
l
:ii"~~+'
IP!ltr. m ,~m!
,, l : : : : : : : : . . ~ < - - m m
I
-
'~!
~3o~o °'
.~
.-sl-----I ' 1--.
"
I
~........ !"
! ............. ! ........... ! ..............
I
mi
&-- i- 1-+|-4~- ~--I--t--V-~-1631-~-]~-11-;~-i1-it-ii-14-iFi1-5~-5~-5~
b
8
Figure 8. A Iternative plan resulting from a different placement decision at the second step (a) alternative placement o f activity 26 (b) final plan, objective = 9108
~ .......
x,
/'.,,,~.~
1st interactive
~x \~",~
2rid mterocttve s~ut,on
2
4
6
8
10
12
~.
,.
"~
14 16 Step
18
20
22
24
26 28
Figure 9. Graph of expected values for the objective function against steps in
the solution process
-: ::!
~-~
L.
@iiiiii
[m~,,L +i! ......lm::LTl I "-'i.......... i~+;. . . . . . . . . .
';i~ ........ l..+~ ~ ' . "
~-t', i:--~
! _o_u! !?~_,
I
~-"
:i i+-*/i
"~--
........ [ ................ c
a
ZONE I(I~
ZOI~ I l l 7
........
o .......
ii_I
.
a,~ao!
,#
:::::::::::ff!lllIh'l,
::~
::::::::::::::::::::::::::::::::::
T~
:::::::::::::::::::::::::::::::::::
i,I isI
.
.
.
iii{i: ......... iid+~+~~ 3 a '
iii.:
lllIi,~l~
iliiiiiiii!iiii?tliiii!iiiiiiiiiii ..t
!i!iiiiiii!!iiii!!iiiiiiiiiii!iiiii o~coooooo+.ooo*i****~., o. ooooo..., •
::::::::::::::::::::::::::::::::::: ::::::::::::::::::;~;:::::::::::::: ..................................... ::::::::::::::::::::::::::::::::::: ,,! :;:::::::;:::::::::::::::::::::::::::::::::::::::::X:::::::~:::::::::: ,~ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: i*! ,~
~!
~!
~~,6~,~,&~.~,~~,~~~;~
~.~..t,~,~l.:
:::: ..::::: ::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
'<~ iiiiiiii{}iiiiii
s!
...............
iiiiiiiiiiiii
iiiiiiiiiiiiii
:::::::::::::::::::::::::::::::::::::::: ............. ::::::::::::::::
i=ii"'i'iiilllii
]! ,!
.~!!t!,,,1~:::....::=:
d
e
...... : ...............
::::.....::::.:::,
:} ram.::: :
I
::
im.,~a,~i ..........................
f
Figure 10. Visual aids for solution evaluation (a) network diagram overlaid on generated solution (b) network diagram overlaid on 1st interactive solution (c) network diagram overlaid on 2nd interactive solution (d) activity contribution to total cost~enerated solution {e) activity contribution to total cost - 1st interactive solution (f) activity contribution to total cost - 2nd interactive solution volume
2 3 number
S september
1981
297
6 Karp, R M 'Reducibility among combinatorial problems' in Miller, R B and Thatcher, J W (eds) Complexity of computer computations Plenum Press, New York, NY, USA (1972) pp 85-104 7 Lewis, H R and Papadimitriou, C H 'The efficiency of algorithms' ScL Am. (January 1978) pp 96-109 8 Sahni, S and Gonzales, T 'P-complete approximation problems'J. Assoc. Comput. Mac& Vol 23 No 3 (July 1976) pp 555-565 9 Liggett, S and Mitchell, W J 'Optimal space planning in practice' Comput. Aided Design Vol 13 No 5 (September 1981 ) 10 Scriabin, M and Vergin, R C 'Comparison of computer algorithms and visual based methods for plant layout' Manage. ScL Vol 22 No 2 (October 1975) pp 172-181 11 Michie, D, Fleming, J G and Oldfield, J V 'A comparison of heuristic, interactive, and unaided methods of solving a shortest route problem' in Michie, D (ed) Machine Intelligence 3 Edinburgh University Press, Edinburgh, UK (1968)
298
12 Cross, N Human and machine roles in computer~ided design 1he Open University, Milton Keynes, UK (1974) 13 Koopmans, J C and Beckmann, M J 'Assignment problems and the location of economic activities' Econometrica Vol 25 (1957) pp 53-76 14 Graves, G W and Whinston, A 'An algorithm for the quadratic assignment problem' Manage. Sci. Vol 17 No 7 (March 1970) pp 453-471 15 Liggett, R S An exploration of approximate solution strategies for combinatorial optimization problems PhD dissertation, University of California, Los Angeles, CA, USA (December 1978)
BIBLIOGRAPHY Basic architectural programming system (BAPS) user manual Computer Aided Design Group, Santa Monica, CA, USA (I 980)
computer-aided design
A new approach to the interactive solution of optimal floor plan layout problems is presented. The method is based upon the use o f probability theory to predict the likely consequences o f activity location decisions, combined with use o f low-resolution rester graphics displays. It not only generates high quality results, but also provides the designer with a structural understanding of the space o f alternatives being explored.
Since the early 1960s numerous computer programs for the automated solution of architectural spatial allocation problems have been developed. (Surveys, from varying perspectives, are given by Francis and White I , Eastman 2 Mitchell a, and Henrion 4 .) The objectives and levels of ambition of these programs have varied widely. Some have aimed to produce solutions that are in some sense optimal, or at least very good, and hence potentially better" than those produced by skilled human designers. Others have aimed at speed rather than quality, seeking to produce reasonable results in contexts where skilled human designers are simply not available. Finally, some have been research exercises in artificial intelligence, with no immediate practical application in view. Those of the more ambitious programs that are intended for practical application must compete effectively with approaches based upon human experience and visual aids. This proves to be very difficult, for at least the following reasons: • Experienced human designers are very good at solving spatial allocation problems. • It is difficult to capture all the relevant requirements in a mathematical formulation of a spatial allocation problem. • It is often very costly and difficult to obtain reliable data upon which to base a formulation. • The problem often turns out to be a large one of combinatorial optimization, falling into a special class of mathematical problems known as NP-complete (see Garey and Johnson s, Karp 6, Lewis and Papadimitriou 7 and Sahni et ala). Computational demands thus tend to be heavy, and it is usually necessary to settle for some kind of sub-optimal solution. Consequently, few programs of this type have met with much success in practice. Those that have tend to be School of Architecture and Urban Planning, University of California, Los Angeles, CA 90024, USA
volume 13 number 5 september 1 9 8 1
restricted to rather specialized domains of application (see for example Liggett and Mitchellg). This situation has prompted many proposals for the interactive solution of spatial allocation problems. For example, Scriabin and Vergin I° suggest that 'attempts to use fully automated computer algorithms to solve the layout problem should be reexamined with a view of incorporating man's visual capability into the procedures, especially since the real layout problem involves many factors which cannot readily be incorporated into a computer program, but which a man can take into account while designing a layout'. Although such use of interactive computer graphics seems attractive in principle, it proves to be difficult to implement successfully in practice. First, there are hardware problems; cheap and widely available storage tube graphics technology does not provide an adequately responsive interface for this kind of application, while refresh tube graphics technology tends to be too expensive. Second, there are conceptual problems; it is not obvious what computer-generated information is useful to a designer attempting to solve a layout problem, or in what format it should be presented. Hence, most attempts to solve architectural spatial allocation problems (and analogous combinatorial problems) interactively have failed to yield particularly encouraging results. Certainly experimental comparisons of the performances of human designers, fully automated systems, and interactive systems have not demonstrated any clear superiority of the interactive approach (see for example Michie, et a111 and Cross 12). This paper describes a new approach to the interactive solution of floor plan layout problems that overcomes many of the difficulties. It utilizes raster graphics to realize an inexpensive but highly responsive interface, and it effectively guides the designer's decisions by providing useful information in an immediately comprehensible graphic format. Test applications to realistic problems, and comparisons with a high-performance fully automated program, have confirmed that it is a very effective practical design tool.
OVERVIEW OF THE APPROACH The spatial allocation task considered here is formulated as a quadratic assignment problem (Koopmans and Beckmann 13), in which individual activities are assigned to individual locations on a floor that is subdivided by a grid into equal-sized squares. Each activity is thought of as a set of square modules, the number determined by the activity's area requirements. The objective to be minimized is a cost
0010-4485/81/050289-I0 $02.00 © 1981 I PC Business Press
289
function that measures both the fixed costs of assigning particular activities to particular locations, and communication costs resulting from flows (of people, materials etc) between activities in a particular layout. This formulation is employed frequently in programs for automated solution of floor plan layout problems (see for example Liggett and Mitchell 9). Fixed and interactive cost data, as required in this formulation, are either entered interactively via a keyboard, read in as a file, or extracted from a database maintained by a space programming, planning and management system (for example the Computer Aided Design Group's (1980) BAPS system). The designer works interactively at a low-resolution colour raster graphics terminal, and uses a light-pen to locate activities in the plan, one at a time. Probability theory is employed to compute the expected value of the objective function for each location in the plan of any activity that the designer selects as a candidate for placement. This information is displayed in the form of a colour or grey-scale 'contour map' superimposed on the plan. Such maps can be computed and displayed almost instantaneously, They enable the designer effectively to integrate information on the cost of location with other considerations that are not incorporated in the mathematical formulation. Where fully automated algorithms for solution of quadratic assignment problems produce reasonably good solutions (which usually cannot be shown to be globally optimum) to the complete floor plan layout problem, this interactive method aids the designer by quickly providing good solutions to a sequence of individual activity optimal placement problems.
E X P E C T E D V A L U E OF T H E O B J E C T I V E This approach is closely related to the implicit enumeration method for solution of quadratic assignment problems developed by Graves and Whinston 14. The layout task is viewed as an n-stage decision process where, at each stage, an activity is selected for assignment to a particular location. Since it is impossible to tell at any point the exact cost of making a particular assignment (this will depend on the subsequent location of other activities), the expected value of the objective function that results from the assignment is calculated instead. At any step of the decision process, a partial map (solution) consists of a set of elements that have already been assigned (fixed elements) and a set of elements which are candidates for assignment (free elements). Given the sets of fixed and free elements at a specific step h in the n-stage decision process, the expected value of the objective function for the general quadratic assignment formulation given in Liggett and Mitchell 9 can be calculated very efficiently as follows. (See Graves and Whinston 14 and Liggett Is for more detailed derivations of the expected value functions.) Given that there is to be a one-to-one assignment of the set of activity modules, M = { il, i2, i 3 , . . , im }, to the set of location modules, N ={ Jl, J2, J3, • • • Jn }, then the expected value of the objective function at step k is ~"
i (fixed) 1 + in-k)j
290
lip(i)
~, ~, (free)/(free) fi/
Y,, ~ qi, i~cp(i,)p(i2) + i, (fixed) i 2 (fixed) + - -]
~"
~
~"
(qi,i~Cp(il)j2 + qi~i, Cj~p(i
( n - k ) il (fixed) i 2 (free)J2 (free) +
] (n-h)(n-h-])
~ ~ qi~i= T, i, (free) i2 (free) j, (free) J2 (freel
where fij
is the fixed cost of assigning activity i to Iocationj qi, i= is the interactive cost between activities ii and i 2 cj,j~ is'the travel time between locations j l and j2 p(i) is the location assigned to activity i under a particular map p.
The expected value is formed from the following: fixed costs associated with fixed activity-location pairs + average fixed cost of all possible mappings of free activities to free locations + interactive costs associated with pairs of fixed activities + average interactive costs between fixed and free activities over all possible mappings of free activities to free locations
+ average interactive costs between free activities over all possible mappings of free activities to free locations As with theautomated solution approach to the floor plan layout problem (Liggett and Mitchellg), the general quadratic assignmentformulation is complicated by the need to consider activity area requirements. Note that the expected value function expressed above is based on the assignmentof individual activity modules to location modules and so includes interaction penalty costs between modules of the same activity, which are introduced into the formulation to ensure activity coherence. In the fully automated approach, the expected value of the cost function is calculated at each step for each possible activity-location combination. That combination which yields the minimum value is selected for assignment at this stage. This cost information is then fixed for later stage evaluations. Execution of this algorithm traces a path through the decision tree to a very good layout, a!though the result cannot be guaranteed to be a global optimum. The expected value normally drops at each step, reflecting the effects of making 'good' location choices one-by-one. Early location decisions usually have a major effect on the expected value, while late decisions have relatively marginal effects.
D I S P L A Y S OF E X P E C T E D V A L U E S The interactive method is also viewed as an n-stage decision process, with the designer making two kinds of decisions: • Which activity should be located next? • Where should it be placed? Activities can be located in arbitrary sequence, but it is very helpful to know, at any stage, how to choose candidates for location so as to achieve large reductions in the expected value. Thus the system can be requested to generate and display a list of remaining unlocated
computer-aided design
employed. For comparison purposes, Figure 2 shows a solution generated automatically by the implicit enumeration algorithm discussed in Liggett and Mitchell ~. In this example, 29 activities of varying size have been located on an L-shaped plan. There are no fixed costs associated with this problem. The interactive costs are shown in the matrix in Figure 3. Rows and columns have been sorted in the matrix to emphasize clusters of interactive activities. Figure 4 illustrates a sequence of expected value contour maps generated during interactive solution of this problem, as follows: Step I: At the first step of the decision process, no activities have yet been placed. At this stage just two types of contour maps are generated - one in which the best activity placements are located in the centre of the plan and one in which the best activity placements are located around the perimeter. Activities that are highly interactive with other activities, or moderately interactive but large in size, are 'attracted' to the centre of the plan. Activities with low interactions and/or very small size are attracted to the perimeter. Figures 4a and 4b show these extremes for two activities in the L-shaped plan. (This observation holds true no matter what perimeter shape is used. Figure 5 shows the contour maps for these same two activities in several alternative plan shapes.) At step 1, activity 27 which is the most highly interactive activity, was selected for placement in a central location. Step 2: Once an activity has been placed, a wider variety of types of contour maps emerges at the next step. If the placement is symmetrical as on the square plan shown in Figure 6, the contour maps that are generated remain
activities, ranked according to the reductions in the expected value that their most advantageous placements in the current partially complete layout would achieve. Once a candidate has been selected for location, the expected value for placement of a module at each empty location in the plan is calculated. This information is displayed in the form of a colour or grey-scale contour map superimposed on the plan, with darker-toned areas indicating the more advantageous locations. Three different contour maps can be requested, as follows (Figure 1): A map showing activity fixed costs of assignment to locations (which does not change during the assignment process). • A map showing activity interactive costs with respect to only those activities already placed. • A map showing the expected valuesof the fixed and interactive costs combined for a single objective measure. The last map is the most valuable, since it shows the expected effects of alternative location decisions on overall plan efficiency. Generation of these maps does not require a great deal of computation. Even using a small minicomputer, such as a PDP-11/23, they can be computed and displayed almost instantaneously. EXAMPLE
The process is best illustrated by means of a worked example. A department store layout problem will be z~
................................
~ ,,~ '~"
L
+l+ =+.i z+ I +i!
iii~iii!ii!iiii!!i!ii!!ii!i!!!!iiii ::::!i!!iBS~i~"'~;;~!!iE;! :::::::::::::::::::::::::::
........................ ::I =;..:P4,'.'=~ ............
l .........
+, "! 1iiiii;!::=t?!~..=-=~i~!!~!! ........................... I ,~, :::::::::::::::::::::::::::: ,~ ........ ',Lff~:P&"~..... ] ~, ::::::::::::::::::::::::::: ,~ ::::::::::::::::::::::::::::::::::::::: Jii!!!!!iI +
,,,t, i s .s is I i+ I u I
Iii-
Ii I
i= I
,!
,!
+,! ,!
,! s! ]!
,4
~i ~!
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ~--~+- j - + - L - - + - - I - - + - + - T ] 8 . ~ - ~ j - 3
j-] |-] j-I£n++;+-iF+~-;
t! r;+-;~-;t
a
b
c
Figure I. Contour maps (darker toned areas indicate more advantageous locations) (a) fixed costs (b) interactive costs with activities already placed (c) expected value of the objective function combining fixed and interactive costs 4~ 4884~444424834 ~ 7 e s 9 io is ~ I~ i~ 16 ~ 2 2 2 3 2 0 ~ 2~ Z~ 2~ 2 s ~ i 2 ,~ i~ ,3
191
beats i~,~ z,-
• .......................
, ............ ~ ~ 8 4 4 4 8 4 B
.!
"~{........i'" ..
J
iiiii:: ill
i1!
+
iz =
.___L..
.
. 2
;~
~ 4 \ 8 I
\.4 4 e\4
:il';
I-...1 r::..l z~$........ L.,
.
.
.
.
.
.
.
.
.
.
.
l,..1 J.,; IL-I
! i
.
.
-
.
-
41
i
']
I
II ', I
i
I
~s
........... ':" .......
.
";-'27
!!
.
Il....Jl .1
"
44
I I
I~
II
2 8\24
,448\ ~\8
8
~4
I 12
Figure 2. Automatically generated solution, objective = 9382
volume 13 n u m b e r 5 september 1981
Figure 3. Matrix of interactive costs
291
symmetrical. The various different types of maps shown in Figure 6 can be explained in terms of the activity's overall level of interaction with other activities as well as its interaction with the activity already placed. For example, the most desirable locations for activity 26 are in a tight ring around activity 27 (Figure 6~). This activity is directly connected to activity 27. The next map (Figure 6b) shows a wider ring of the 'best' location areas (ie a wider range of acceptable locations). These occur for activities indirectly linked to the placed activity. Contour maps with desirable locations at extreme perimeter locations still exist for small noninteractive activities (Figure 6d), and a middle range map occurs for activities not connected to the activity
placed, yet with interactions to other activities (Figure 6c). Results {or a non-symmetrical activity placement are illustrated for a courtyard plan in Figure 7. Again an immediately understandable spectrum of contour maps is generated, ranging from those for activities with direct connections to the activity already placed through those with no or very slight connections to any other activities. Returning to the assignment progression for the L-shaped plan, in Figures 4c, 40', and 4e, contour maps for the top ranking activities competing for placement at Step 2, are shown. They also vary in relation to level of connection with the placed activity. Activity 26, which is the most strongly connected to activity 27, was selected for placement.
,==,Niiiilti , ...... ,,ooo<. . . . . ~olgl=i...l
,,,
I
,4
Ji ; i;777i71711@i
,,'. . . . . . . . . . .
'~
. . . . . . . . . . . .
!iT!~77177!7~i77i77E77i~{~J
.-, ..................................
. . . . . . . . .
:::::::::llIllliUll
Itlllli~il
:::::::::lll~llllilil~ll
I I I ~ ° ~ ~°I
7i]7iii7f#:i77i777777777{i!!~il
zz!
,~, ,,] ,~, ii! ,~,
S iiiiiii iiTiii? ii i{i;ii!iil;iiiiiiiiii?ii?iiiii?iiiiiiiiiiill ,!
:::::::::~
. li~,.,,~%~ ,~,~
'
~I~::
:::: . . . . . . . . . . . . . . . . . . . . . . . . . . .
:: :: :
~o°~o °
W V ~-~ V V W - t - - l - ' ~ - I Y ~ V 4 - 4 4 " 4 - 4 * 4 ; I ; V ~ K S V 4 - $ 1 - 7 ~
a
b , ....................................
~-
=: ::i ":
iiiili iiiiii i f, iiii!iiiiiiii!iiiiiiiiiii!i!iil
11[ la I
,.! ,~' .! ,~' ,~' ,,!
...................................
l:?i:iiTiiiiiiiiii!il]iiiiiTNiN~!l
f:lllll~:~ff~Ifl:::::::::::::::::
....................................
1~.~._.._.~
~
::::::::::::::::::::::::::
ii :::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::: ,, iiiiiii~iiiiiiiiii!!!iiiii!!iiiiiiiiiiiiiiii!!!i!ii~i~ii77i ~ IIIIIIIUI~'~iI'~I~<~I~I~::::::::::::
::::::III°~'I~.~ - '
d
e
t
...............................
~i
:~i
ll!
ll!
.,
;i:~i:~iiiiiii iiiiF, iiiitiiit f:i::::il ::I
,,:
'i!NF~iiiiiiiiiiii!iiiiiiii!i!::iil
I~!
ll{
ii ~
li ~ I~J
ll!
I!
I!
o! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
o! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
g
h
,,,_
=====================================
.................................
__
,o:
,: ~..... l t,t....... fll~il,~ll
........... . . .] .......... . . . ,......... I"
!"
.
!ii!iii_=~
;--lllUllltlll;I
i
Figure 4. Sample sequence o f steps in the interactive solution process (a) step I: contour maps for highly interactive activity (b) step 1: contour map for non-interactive activity (c) step 2: contour map for activity 26 which is directly connected to the planned activity (d) step 2: contour map for activity 21 which is indirectly connected to the placed activity (e) step 2: contour map for activity with no interactions (f) step 3: contour map for activity 24 which is directly connected to both placed activities (g) step 3: contour map for activity 28 which is directly connected to one o f the placed activities (h) step 6: contour map for activity 29, directly connected to activities 28, 25 and 24 (i) step 6: contour map for activity 20, connected to activities 26 and 27
292
computer-aided design
Step 3: As shown in Figure 4 f a n d 4.<3, the contour maps focus the better placement locations around the central core for activities connected to those already placed. Activities 24, 25, and 28 were placed next in this area. Step 6: A t this step, specific sides of the plan become more desirable to individual activities. For example, activity :29 is connected to 28, 25, and 24 so the contour map shifts to the right (Figure 4t7). Activity 20 is connected to activities 26 and 27, which is reflected in a concentration of desirable locations to the left of the central core of activities (Figure 4i). Activities 29, 20, and 1 are placed next. Step 9: The placement of a cluster of interacting activities is complete, and new groups are emerging for
placement. Once again, highly interactive activities are attracted to central locations in the plan. See, for example, the contour map for activity 8 (Figure 4j). Step 12: As the plan fills in, desirable locations for placement become more strictly defined. See, for example, the contour map for activity 9, which is strongly connected to activity 8 but to no other activity (Figure 4/+). Activities which have strong relations to some other activities, but not to any of those already placed, seek spots towards the centre, as seen for activity 11 (Figure 4/). Step 14: Variations in contour maps can be seen for activities 22, 23, and ] 2 (Figures 4m, 4n, and 4o). Step 21: Placement decisions become more and more limited with less area available (eg contour maps for
:_~:! _r! _~?
.'? ."_~_LLL"
iiiiiiiiiiiiii_i___i__. i!iii ...................................
~
~,*.*
....................................
"! +~
~o!
~'I',I i~ i
,~
~'i," !+.ii+N ,',"::,~! !.......... . . . . . . . . . I +_~++_.~_%-++%-- _++_+_+l ~ ~ ........ ~. ........ ,_., ............. !;7.......
'+::;:,--~...._~;.~:.::~::~
~4
I
i
"
i:!HE££:::::::: ...,. . ..........
. . . .
.........
h
I
iii
. . . . . . . . . . . . . .+'. . . . . . . . .
~ ,liez
I........ I.......... I
';.~i~i~!|
|
+iiiiiiiiii~!i~d.........: + _ i l l
t . . . . . . . . . . . . ;;"
...... ;
:::;:,~'m~::;:I;:~+~+'~i-"'++~ III ++--~Hh~.":";r
• *****+** ~oc(i+~ I ......... m+........... ~11
!!!!J+!i+!+!!!!!++++++++M+........ li+!i+!++!!!!!!!!!+!!!!i;iii
d
i
Iili!!iiil
,II,,,,U
~..HIlUUllUlIII~;
++++++++++I
1........ F. ......... !
..................................
ll-- i--!--,l--t-- ~--L--t- -1.-1-~8-~ i+.i#l-~t-~ l-~t- ~L-~ t-% L-~,~+~p-; i-;1 -~1.~#,
"!,+'
,i ,='-:i II ,H..+::,-+'.---;,
,,.llIi!......
,~
"!"!'~";;;';;;;:~°+"+'~I
I~i~iiii~iii.'+".'~!i_'!_'i~,i"
4
...-
,
...... = ..........::i~:m.-;~ ,!d II:::........................... ++~oo,I t ..................... := =:: =t -": '--+-'":::-+: -" ::: ¢+~-m-~+°l........ I .%+.% .°~-. -~- -%m--°~.+?~+<'__+°~ °°~ _
t-- r-+~-l--l--F-,t-T-l--i;%-irit-il-;t-lrlFit-il-il-iF$1-Tt-~F]t
I
k
; ~_++_~ !_LL' .t~
,+, ,,!
iil i!iii!!!!i!iiiiiiii~ ....... i :::::::::::::::::::::
i1 I
,,~ ,¢
iiiiiiiiiiiiiiiii!~+ ...... ; ,;+ . . . . . . . ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,+,'+' I-ii~l' 4 Itli
I
I
"-] .....
I....
I
.............. ._,., . . . .
i ...........
. . . .::':"::++ .,
+~'
+,+~+==.F,,,=m;:mm
:::::::::::::::::::::::::::::::
.+' ,~, ,~,
.~=~..~iiii+ii+ I l::=.5-:lllill|llill:...,i;;
'#
~llllhllnllUlh;+;
,,!
gii+~i~++++i
,~'
t!:-~-"~++~[~; ...... i ........ L ; ......
'",,~ i+++i+...... ~l~
I. . . . . .
,:.+
F+ ++......
t+t+++++ .,+
r - UT-~--I.-TT-t
i
--I-TiI-ir~I-i$-WW$C],~-]I-;I+~
~- IIlIIIlII:l'+IihlllIlIlllh.,II.llll ++e+ + +{++Ii * llllllsll$111 iii
Ii
~-;~r;t-+l-~l
-z0:! ~L".
l l I l , , I i i I l . + . , , , ,.+o . . . . I lliili
+++IH"mm
m°
,d
I . . . . . . . .I. . . .
+,~' "!
....................................
:#P-:i"
10~ L'I"--+~
[ ....... I.
.
.
.
I
i~iiiiiiiiiiiiiiiiiiiiiiiiiiii
i
I
,,, l..'].i!W+m++l
I
l~!~+~.l-_i +-~,........ :::L.+_I ~,~, . i !l l-°-~-~{'"~--I~'l . . . . . . . . . . . . . . . . .
I........ I++I........... i ......;~o%+.coo++....:;1
--l-1-~--~-~--+~.°9.++°+-c~'.+::::::::+
,4
l
...
~ooocco J,
'o,--Mm
......
~--l-]+-++-]+-;
¢
IM+=.+iM+..=.+.+++N.+++
~-] L-++.+I.;
.,+ ..........
++
++ ++++++++ ,,~ ~iiiii~il
,
1.............. 1
+;+iii+i:+++++L_+......
-a ;+;+++H++::v' l
~+,'°
I
i
I,
I. . . . . . . "
1%+.]+-+++;
i l.. ;;..-,..
+.++_]~+;L
,~i . . . . . . . . --1.-1 c'
{ill:
::~ ................................... [ ......
" i ......
L+ .....
+........... 1 ++...... i+
I
+:........ ~...... :::t___t
ii !........ Ill!..............
'
~,,'~':ii+--'-ii::i-....... --, ~~..... t:......i ----! !ii!}!!!iiil;~ ........: : : L ! :~ l:::++~;;~::I: ....................... I........ l+=++++++,+++,+,++++++l-='+:um== !i :=:[L Lilill] ........ ~--+- }-- t-C-+-T-F-+--i-~+-]r~3~-;I-;U;£+H.~£-~I-++.;~_;~_+~_+L
p
L_l_+_,l
2OI
+;+: I
+.......................... I .........
21-I
'1:1
ii
l~l~
I
I i+,+..... +
o+
-'
!
~:i~ "-:+:i++...... +;'......... ,., I-"..P..~+=.......... ~+' +........ +........... ~
....
'd;:iilE"~
;,;;;~;;~1111,11:~o~
BLOC. P"B
.,.~'"
Ii ;~...... i ......... L+i+___ ,I
.
I
'! I+::++::+::+~+~+~::+~++++!!!++N+++I ........ l+-l--l--+-+
I;IllllI t.,,,-,
||,|:.,,*m,.~
+-i
::::::::::::::::::::::::::::::::: 8-- ~+- j--j--
I++.......... lllll~l
1
~ ......... !........... 1
O
I~
I
1.......................... r~ l+
2,-
! !!!! Ei!!i!i!i!!iiiiiit
,~
ii;iiiiii:;;'::::J
~!
=o.E .,,.
-
::::::::l
......... ;:::::::
.....................................
,~'
"!
iii~iiii;+:$i~i
1I1
+ .............................
:::::::: :i I::::::::1 ............. ; --1~° +.,
!........"='m~+~m++
i
...................
'~
n
2~
~
'
,=,it
.......
......
!..~
"!===::=+. . . .
~, I................................... l........I..................... .+_+_.+ +-v-+--rv-~.r+-+-r++-+r~r~v++-+r~+-~+-+r;r++-+r++-++-++
ITI
,.,
+,
l +"+" INigi
+ i?::::]
"
m:+J
..!
,,:
!!i!iE+iiii,,i:l........:::i....J
P+"'R"~++++++:;::':"I 1.:=,::~+%+o.++:::::::: ..................... 1 ....................................................................
~.t++
iiiii!iiiiiiiiiiii++++!!iiiT::2;:!
,~
~, iiiiilgi!!iiiiil
II!.I!HL....,LM,.P..:I+,, . i.i~. "
,+~.,,+ . . . . . . . . . . . . . . .
.,
~:~!
r' +
i
.....................................
q
[ZL!I'Ji
,
~-~--Y-$--v-!-4-~--l--r,~--,r-,~-,$--,L-,r-,L-,r-,r+r-+~-+~-,~-.|--+l
(j) step 9: contour map for activity 8 (k) step 12: contour map for activity 9 (I) step 12: contour map for activity I I (m) step 14."contour map for activity 22 (n) step 14: contour map for activity 23 (o) step 14: contour map for activity 12 (p) step 21: contour map for activity 10 (q) step 21: contour map for activity 13 (r) final plan, objective = 12 791
volume 13 number 5 september 1981
:293
ZON~ Ell~ al-
zlaol.
tam,, ....................
161 ~s .a i1_1
i1-I 13J 11 -#
Io i-
:::::::::::::::::::::::::::: ::::::::::::::::::::::::::::
~.ooo~,C.~.HIIIiiiiiiilt
,hhlllh..,... ....... , . , o o o . . . . [~ . o . ~ " ~ l l l S l l l ! l i IOlH L~"I~*~°O~'O~C~C*''°* **** °°**''*'*°'"*o***0 ococo.~ ~.~te~,~cc~o~¢cc.,°.,*.,.,.*******o***o****ooo i coc o~, o~ o. ii i ee~o~e~c~ooo~c~o,,o * o * , * • . * * , * * * * * * * , o , , , , *O0~CO~G~ , ii I,,.~,~ o~o$I:III,III c~ c .c.~. .9. ...-. o. . o. .o. ....... . .. .. .. .. ........... .. .. .. ........... .. .. .. .. .. . . . ooooo***~ o
o c
o........................... .oooo.**.~¢
~ccccc~.,,.o,o.. ccccccc~c,,.,,ooo,
................... ........................
: .......
ooo,.+oo,oo
o,o.,o,oo~
o o
o
o
~o.~cpo~ ......................................... o~ooo~co c c c ~**'**°'** ........................... **,,,,o**ou o ¢o
I~o, ,*~.......................................
~oot
.i COCCCCCOU,.o*Hoo* . . . . . . . . . . . . . . . . . . . . . . . . . . .
,!
o * * * * o o**oooooc~o
................ **o. **** oo co 1eococcccc~.o**o.oo. .................................................... ~,o ~e~e~ooooo©cooo*,* ,** ,,,*o o******* o,o,.o,ooococec ~ee*ele
d
° B ~ E~oCOUGCGCCC** * * * * * 4 * * ** *+ * * ' * * * * + + * I , ,O0000GGO~d
.! ~e
19
o o~cocuo**. ** *• *.. o.oo • . . . o. o+o....c.oooo~o~ u i ~'s~eeeeeeo~,oo~rcoocooocoooc¢.¢ oo ee ......................... | | | ! | ,]~9 ~ll~ouUdlII.K~U UO¢00~¢OO0~;r.OOC0CO000u~lle I m l ~ H ~ I I I .11|i|~. oee~eee~o~ oo~ c c c c c o o o c o o c c c c o o c c o ~ a ~ ~ m ~ l e l | l U l l l l |1 i i l | | t | e(~'en~eeeoco~o© ccoc c o o o o o c ~ c o c o o o o o o e e e ~ m ~ m l ~ l l U l l l l l l |
h,dii,,
O!
~oo~...~iill 1
o!
5--~--! "~--~--Y'~,--V-~,-'~'.6--,YW-,5"-,~,'W-,L-',Y',V;V~6--~I
b
loll
iii.
........................
:1~ ~]i]]ii]~iii?iiiiii~P~iiii5i~.~iiiiiidiiiiiiii~iiidiiiiidiimii;~i~.i~i~i~ii~-~. .................. ....................
~o,.~li~.,,..,,, o:o,o~
.......................
.................... ~ " I "
I
I
, ,
•
iiwuuiK'iiihilii[,'i ,l .............. i a
.... =.,.,,____...,,,
,,
:::::::::::::::::::::::::Ull~lllmlllllll s i
'-' :::::::::::::::......... ...... ~.........., . , ~ . ~ . . ,
~[[[~il .=.
~!
..... ~ :~.~,
.
.::::::: .......
,~
::::::::
l-T-i-t-
l--~--;&-i y~t--i~-; I-W~ l-;t-;l-i
~iiiili::iii::::?::i::::/?/]i::::i::]::::/::::i::/::iiii~iii~
...............
h]~!l.~ .............. ii,, ! ~,.,,~ .................. i~.:."-: ~ I ~ : :::':'::"
d
:ii]::ii!::i!!!i!ii~~~~iiiii!iiii::i/::: t 6 - : I~- j - i -
~}!!}!!!}!}!!}}!i!i}:~!:~i:~!~=~ ..................
- F ::::::i
lEi!i!!!!iiiiiii!!iiiiiiiiiiiiiiii!!!
~..~, . . . ~ ; . , ~ . . . . . . . , . ~ , . I J I ,.-'th.'h ~ L,..,o,,~|i~
S
t - ~ t-H t - ~ y i i - i l - ~ t - ~ l U + - i l - i l - i & 3 t - ~ - ~ U i Q V i l
¢
d
1011 I I , I
19! leJ IIIIIII
II:::'~
:
Ihlh ;,~==':" ,llllhl h III It
,q
::::::::
..............................................
:::::::':~-"~l:'lil;~ill
:::::::.:.:: ...... :::::::::::::::::::::::::::::::::::::
.........
. ,
.......... ,~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 ~,,~,~ ...................................................... S~S®~.~.,,,,,,,,, Ill--,1',1~$$~.~ ....................................................... ~ ...... lhlllh, *!
d o!
~4..........................................................................
ll-
a*-
::::::::::::::::::::::::::: ......... ~ ' ~ , , ~ . :~ • ~:t: ~:::::::::::::::::::::::::::::::::::::::::::::::::::
I.I.It.
:J!.h~:::::::::
::::::::!!:!ii! !i!
ii::::::::.~:~II
|i!l iiiiiiiiiii;:iiiii;iiiiiiiii;;iiiiii:ili;:;!il;;iiW
T*! is I .! la!
la! i1!
iiiii!iiiiiiiiiiiiiiiiiiiiiiiiiiii!
,d 0 ¢ ,!
!!:!
6-- t " ! - ~ - T T T T T T ' I & - ; U ; ~ ' ; ~ , ' ; I ' ; I ' ; I " ~ V ; I - ~ I ' ; A ' ;
f
i l_ 7! ,!
!i
..................................
e
ITJ
. U~'W:,I
J
Figure 5. Contour maps for ~lternat/ve plan shapes (interactive and non-interactive) (a) and (b) square perimeter (c) and (d) rectangular perimeter (e) and (f) t-shaped perimeter (g) and (h) x-shaped perimeter (i) and (j) courtyard
294
computer-aided
design
activities 10 and 13 in Figures 4p and 4q). Step 29: The final configuration generated is shown in Figure 4r. This can be compared to the fully automated solution shown in Figure 1. The actual value of the objective function for a completed plan, as defined for computation of expected values, is highly inflated due to the effects of the penalty costs between modules of the same activity that are needed to prevent splitting of activities. Thus it must be adjusted to provide a more useful basis for comparison of alternative solutions. The values shown in Figures 1 and 4r were calculated using activity-activity interaction values, and straight-line distances between activity centroids, with all penalty costs disregarded. Figure 8 shows an alternative final arrangement generated by the interactive process when a different placement decision is made at the second step. Using knowledge gained from the intial interactive solution process, a considerably better plan with respect to the value of the objective function was generated. Figure 9 shows expected values for the objective function against steps in the solution process for automated solution of the problem, and for the two interactive solution processes. The general similarities between the three processes are immediately evident.
SOLUTION
EVALUATION
A single number provides an insufficient basis for c a m parison of solutions The precise details of relative location of the activities in the arrangement, and the contribution llli
fill
IGll
,'~ "7.77+iT]F+b.:ii+ii::-~ii.:+ii~ii++++++?.+:.+..:.+:~.7-777; ;" ::::::::::::::::::::::::::: :::::::::::::::::::::::::: il!
,st
....... i i ::::::* ....
l
~
l
.
l
l ~
~
ill.-
:Clt
fill
.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
:::::::::::::::::::::::::::::::::::::::
iiiiiiiii i ttiliitfii ,***ll***
:::::::::
~*ll**"
¢
l
,i- I..................~ < + + ~ ~
iiiiiii
t~
....~
Ist,tfmuimlIwllllll
illilllSlllilIWmill
........
.-" : : . l - l t : l : : - : : ::** ~ C=,m.O00+,,.,...., . ' + + o ~ ' + * " * * . . . . . . . ,. ,. . . ,. ,. ,. ,. . . . .
t t--t-'l-l--l-l-?-l--lit~l-~Yiii l-+t-il-;t-il-+l-it-;I
........
.......... ;";':'1
moccl¢o~lJ
uoo(¢(~o (.occc i . ~ u,ccc~c~
liWilWilillil~< lllillil~lillli~lll~
d I~°fffl~c~i~W~l I ~ ~ 1 I~ltlb~ flllmlililHtmllllHimm c~c¢cccc,o
+,o,,+++, llUllllllfllillll h ..I,I,.IMIIIMIIIIIIII
,,,,+
llmllllililWililllt I ~I llUlilllillillW~d
•
1:::::::::~:'!91BliBlll~llllllll~+~l .*+.*.*o* ~. • ****H.. I:::::::::!|=~~~~1
s+
~' Eiii!!!!i~F::::: ~1. . . . .[ .......... . . . .
o
:::::::::::::::::
. * * * . * ~ ci ! c* * * o*~ + * * ' * . *m* * *°* *°* * . . . . . . . .
¢
Jill
!
~=_=_!!!!! II ........
i+ii+iiiiiiiiiiiiiiiiiiiiiiiHiiiiiiimNii ,.',,. I!!I|I~III=,IliI~!I~ll~HHlllliillllII~llli ....
, i!ililil!~:.+: "; ,.,,| !! iii..=.: I ::....... ......... :::::::::'"++++++
'+ .'=] ;
'
i lli!! o !
:';; .~++ . . |*' | '~+* ,.|.. . ... ... . .. . ... ... . .. . .. .
' ........ : : : : : : : : e *. .
:F-";t2iiiiii]!
?iiii!!ii •
iiiiiiil
,¢
! | eeseee *e~Hee ec[o t:,*t,te*,e~eee,,e*m .III,I,I .... l,...,.. "i~,.~--~. ..... ~-'I;IIIIIIII Ill Ml~e=llV~41lmele+.cu*~ooooeilwlle~NiiwsiiImll ilell
lU
lllBiDH
~-*e~*~ei~*¢.~ooa o o . * * *.. *.. *.**** ***.. **.. * * * u u ~ u ~ o o ~ . * * N ~
,1 i.o*,,+.L~ucoc.o ........................... o~°d~.~.tl| *e*l*l***co~*~v.*C~* ** * * * • ** *** ** * ** * ** +*. ****0(+(+ C4t 15~1 I , . e . , t + * o ~ c o c c o ............................ ~m.v~l$ ~,,ee H et~¢ , *~le. .e. .. .. .*. * * . .*. .* ................... : . : [ : : " ' . . . * ** **.**** * * . ****eeeee+ eee , d b,**,,,|,+- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : : : * *. * * . . **..'m"*',.*1.** i~eel~ • * * . * * * o . * . . . . . . . . . . . . . . . . . . . . . . . . . . . o***.****w~li*~ 16jl
. . . . . . . . . . . . . . . . . .
,~ ....,liP,|........................................e. ¢cc£ ~coo.**.**.*. . . . . . . . . . . . . . . . . . *oo<.ouoco , ~ loot
'~
**.*** ........
:': ...... **.*****
m~.ooco
c c c c c c . c. .c . .o . *. . * . * . :. . ' -*: " * -.. . . . . . . . !~~041*t .......................... I ........ ............. lee**.*. *.+ . . . . . . . . . . I I,.,,,,I,.., . . . . . . . . . . . . . . . . l ........ I ~ee¢ + 1 * * * + * * * * * *. . . . . . . . . . . . . . . . .
I..,II............................
....... :1::::1::~...
ee
::: ......
"*****..**eeeee* i . . . . . . . . . . ****~ ~+**tee.v**,*,..** . . .*. .* .".*. .*. *.*. .+. .+.. .* . .* + * . . + .: .*';: :* * ;•; • +" 0. , , ****..imeeeeee knliii+lkl~,~OOLO~O** 0¢0C a l [.....,...¢.o+.,~o,+.o ........................... o~°~a~o,~P~,~.12 • e o e e l e ~ . v c o o o ****************************** o coo Idle* •
s_~ ,_~h,....++..coo+~o+o .......................... ~N~=+o~,|, *************************** ~occ ,
,!
• wee* e~¢ooooooc
i
!ImlHMIMHIIIIIIHIN!
+, ..+.++!++!., ................................................
!..
•
•
eel c c o o o o o o o
*eel*el
•
,: I.ILIll . . . . . . . ! ~ ........ ~ i M . . gi * .1-0 lal l .l g~i l iJ* Iiom lll || ,
Illll,,lil..: . . . . . . . . . . . . . . . . . .
*l~141k *tl4•4111k IltiO OCO000CO0
II.. . . . . . . . . .
li|fil21111
+-- ~-- j-- + - ~+-~--i--}+- }--l-~+-;l+~|-+]-+l-++-.+i-+ #-~,~.q+-++-~l
d
e
Figure 6. 5election o f contour maps after an activity has been placed in a symmetrical plan (a) contour map for activity directly connected to the placed activity (b) contour map for activity indirectly connected to the placed activity (c) contour map for activity indirectly connected to the placed activity (d) contour map for activity with other interactions (e) contour map for very weakly connected or small activities
volume 13 number 5 september 1981
I
+-- ++- j - 5 - i - U - l - - + - - i - - i-+~,---; U.~I-;,,-+~i - ; £ ~ y ; y.+l-; i-~+-i1
b ZCll
I
................. I
~ o * * * * * * * * *
',~ =~lf=~dlitlllllgl~lll"
~!!!!!!!!~m.~+.==iii~!~iiiiii~!!!!!!! 7:::]]:::!!:2!5!!!!!...... -: ................ "5"5!!:2]]!!]i!
::::
* * * * ** * **uo~oouc¢o+,¢
mmmm
+IWillill
d d
.........
": ! ````:```;~=m~
.l~+,i~ ' ' ' I l l '
/
,,2++++++++ '
Jill
. I
. . . . . . . . . . . . . ."~ . *'*~ l * * ; ; l i i i i ; ~ ........
=
each activity makes to the total interactive and fixed costs should be examined. To aid in the detailed evaluation of solution quality, and to provide guidance for possible further modification and refinement, additional visual displays are available. First, a network diagram illustrating the basic flow structure of the problem can be generated. Figure 10 shows the network diagram overlaid on the plans produced by the automated and interactive solution processes. The most important communication links between activities are displayed. Note that the second interactively generated plan (Figure 10c) is clearly superior with respect to both the cost function and the details of relative location of the activities in the plan. A second visual aid (also shown in Figure 10), which can be used in conjunction with the network display to evaluate the solution generated, is a colour or grey-scale map indicating each activity's contribution to the total cost function. This map complements information in the network diagram by highlighting those activity placements which are contributing most significantly to the overall cost of the arrangement. Larger contributions will generally be made by the larger, highly interactive activities even in an optimum placement (see for example the cost levels associated with the centrally placed activities in Figure 1Of), and thus in this case the information presented is not of much value for the generation of alternative arrangements. However, in Figure lOe the high costs displayed for activities 16 and 17 reflect interaction requirements not adequately dealt with in the solution, and so suggest a potential improvement.
295
DISCUSSION The distinctive feature of this method is that it employs an optimization technique to provide the designer with a structural understanding of the decision-space being explored. It makes graphically evident the important branch-points in the exploration process, reveals symmetries that suggest the existence of distinct solutions with approximately the same cost, develops an excellent understanding of sensitivity to variations in placement and shape of activities, and shows how distributions of expected values across a plan tend to fall into a few recognizable and frequently-recurring types of patterns. All this yields much more insight than a single automatically-generated plan. Such insight is vitally important in practice, since the cost function captures only one aspect of a designer's concerns in any realistic context (and that only imperfectly), and since quadratic assignment problems of this type notoriously have many local optima. The task is not to find one least-costly plan, but to integrate an understanding of the cost consequences of location decisions, in a well-structured way, into a search for a solution that responds to a broad spectrum of complex and often ill-defined criteria.
terms of values achieved for the objective function, it is competitive with the best available fully automated technique. In terms of guidance of the design process, it provides valuable information, of a kind that is not available in a manual process, in striking and instantly comprehensible graphic format. It appears to have excellent potential for practical application to problems of office, department store, hospital, warehouse and factory floor layout, where communication efficiency is a central design concern, but not the only one.
REFERENCES 1 Francis, R L and White, J A Facility layout and location: an analytical approach Prentice-Hall, Englewood Cliffs, NJ, USA (1974) 2 Eastman, C M Spatial synthesis in computer-aided building design John Wiley and Sons, New York, NY,
USA (1975) 3 Mitchell, W J Computer~ided architectural design Petrocelli Charter, New York, NY, USA (1977)
CONCLUSIONS
4 Henrion, M 'Automatic space-planning: a postmortem' in Latombe, J C (ed) Artificial intelligence and pattern recognition in computer-aided design North-Holland Publishing Company, Amsterdam, The Netherlands (1978)
This technique is computationally efficient, can be implemented on inexpensive hardware, and allows very fluid interactive exploration of layout alternatives. In
5 Garey, M R and Iohnson, D S 'Strong NP-completeness results: motivation, examples and implications' ].'Assoc. Comput. Mach. Vol 25 No 3 (July 1978) pp 499-508
:i ...................................................................................................................................................
::[ .........................................................................
u!
,~1
s!
s!
b
21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~0-
,,, :~:. ~:, ~:.:. : : : : : : : : : : : : : : : *: : : : : :~: : ~g: .:. .:. . . . . . . . . . . . . . ,~! [
,,I
F
I
"-"-'-'-'-¢-~-~'~
;:::;;:::::::::::
:::~]:q~::::::::
,~,,,: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ,~
I,IIIIIt~,!~°¢~1
,,' m!!l=~::::::::
1
,,, l~i~jll~lil
I]]]]]]]]][i]i]i]]
~I||11
.............................. :::::::::!]!]!!ii ,: : : : : : : : @ . . . : . : . . : . : : : : : : : : ,: : : . %~:~'~!l..::::::::F:::::::::::::::
I:::::::::::::::::
d
. . . . . . ~[
1:::.....
:::::::::::::::::
......................................................................
e d Figure 7. Selection of contour maps after an activity has been placed in a non-symmetrical plan (a) contour map for activity directly connected to the placed activity (b) contour map for activity indirectly connected to the placed activity (c) contour map for activity indirectly connected to the placed activity (d) contour map for activity with other interactions (e) contour map for very weakly connected or small activities
296
computer-aided design
~',,,
lil]i]iiiiiiii!]iii]iiiii!iTiiiiil]]
i?i
i"~......... FY]%;mll
t ~o!
~ |
,~,
l~l~mll~::::::{iii::::::
"~
1
., 1~ ,,I
l l l ! ! l l .[ .I. . .l. . . .l. . .l. . . l. . . .l l l l ,=.=.,.
...................................
,,' I:::::::::~+,~III~=I~,UlIII"
I
I '+
........... :=-mr,m,i
~'
:,~
i
m,,,m_
,:::::::::++,=mlHlt ........ l
l ...........
t
'4
I
IlUlIIIIW:,~I
:+1
! .J ................. I ............ 0,
~
~
........ +t ..... '~ .............. +~ I TM I--! ........ I'+ '~
I I
!_.# ........ i U
.J
1721........ l
~.
II
"
t::::::i::]?i]iii!iii!!ii!!iiii~i~N~N~:ii!i!ii ;it ........ ! ................. ......................................................................
I-----I"
/
I
t ......... I I
I
' -+! I..............................
,!
:::::::::::::::::::::::::::::::::::::::::
I
.............. I++..... i ~'....... ~........ I
"
t
::: .................. ot~P~llo,I t111~I,~o ............. ~ o o ~ oo~.. m . . ~ . , , ~ , . I. . . . . . . iiI__~,,~ooo~o . . . . ...+.+ ......... .. +........ tltl ::,::::: .......... ~ J .+o,o ~ ! l l l l l l l , n , l m : m . . mIIII~II ~+ .+>o
,.
I~'
l
:ii"~~+'
IP!ltr. m ,~m!
,, l : : : : : : : : . . ~ < - - m m
I
-
'~!
~3o~o °'
.~
.-sl-----I ' 1--.
"
I
~........ !"
! ............. ! ........... ! ..............
I
mi
&-- i- 1-+|-4~- ~--I--t--V-~-1631-~-]~-11-;~-i1-it-ii-14-iFi1-5~-5~-5~
b
8
Figure 8. A Iternative plan resulting from a different placement decision at the second step (a) alternative placement o f activity 26 (b) final plan, objective = 9108
~ .......
x,
/'.,,,~.~
1st interactive
~x \~",~
2rid mterocttve s~ut,on
2
4
6
8
10
12
~.
,.
"~
14 16 Step
18
20
22
24
26 28
Figure 9. Graph of expected values for the objective function against steps in
the solution process
-: ::!
~-~
L.
@iiiiii
[m~,,L +i! ......lm::LTl I "-'i.......... i~+;. . . . . . . . . .
';i~ ........ l..+~ ~ ' . "
~-t', i:--~
! _o_u! !?~_,
I
~-"
:i i+-*/i
"~--
........ [ ................ c
a
ZONE I(I~
ZOI~ I l l 7
........
o .......
ii_I
.
a,~ao!
,#
:::::::::::ff!lllIh'l,
::~
::::::::::::::::::::::::::::::::::
T~
:::::::::::::::::::::::::::::::::::
i,I isI
.
.
.
iii{i: ......... iid+~+~~ 3 a '
iii.:
lllIi,~l~
iliiiiiiii!iiii?tliiii!iiiiiiiiiii ..t
!i!iiiiiii!!iiii!!iiiiiiiiiii!iiiii o~coooooo+.ooo*i****~., o. ooooo..., •
::::::::::::::::::::::::::::::::::: ::::::::::::::::::;~;:::::::::::::: ..................................... ::::::::::::::::::::::::::::::::::: ,,! :;:::::::;:::::::::::::::::::::::::::::::::::::::::X:::::::~:::::::::: ,~ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: i*! ,~
~!
~!
~~,6~,~,&~.~,~~,~~~;~
~.~..t,~,~l.:
:::: ..::::: ::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
'<~ iiiiiiii{}iiiiii
s!
...............
iiiiiiiiiiiii
iiiiiiiiiiiiii
:::::::::::::::::::::::::::::::::::::::: ............. ::::::::::::::::
i=ii"'i'iiilllii
]! ,!
.~!!t!,,,1~:::....::=:
d
e
...... : ...............
::::.....::::.:::,
:} ram.::: :
I
::
im.,~a,~i ..........................
f
Figure 10. Visual aids for solution evaluation (a) network diagram overlaid on generated solution (b) network diagram overlaid on 1st interactive solution (c) network diagram overlaid on 2nd interactive solution (d) activity contribution to total cost~enerated solution {e) activity contribution to total cost - 1st interactive solution (f) activity contribution to total cost - 2nd interactive solution volume
2 3 number
S september
1981
297
6 Karp, R M 'Reducibility among combinatorial problems' in Miller, R B and Thatcher, J W (eds) Complexity of computer computations Plenum Press, New York, NY, USA (1972) pp 85-104 7 Lewis, H R and Papadimitriou, C H 'The efficiency of algorithms' ScL Am. (January 1978) pp 96-109 8 Sahni, S and Gonzales, T 'P-complete approximation problems'J. Assoc. Comput. Mac& Vol 23 No 3 (July 1976) pp 555-565 9 Liggett, S and Mitchell, W J 'Optimal space planning in practice' Comput. Aided Design Vol 13 No 5 (September 1981 ) 10 Scriabin, M and Vergin, R C 'Comparison of computer algorithms and visual based methods for plant layout' Manage. ScL Vol 22 No 2 (October 1975) pp 172-181 11 Michie, D, Fleming, J G and Oldfield, J V 'A comparison of heuristic, interactive, and unaided methods of solving a shortest route problem' in Michie, D (ed) Machine Intelligence 3 Edinburgh University Press, Edinburgh, UK (1968)
298
12 Cross, N Human and machine roles in computer~ided design 1he Open University, Milton Keynes, UK (1974) 13 Koopmans, J C and Beckmann, M J 'Assignment problems and the location of economic activities' Econometrica Vol 25 (1957) pp 53-76 14 Graves, G W and Whinston, A 'An algorithm for the quadratic assignment problem' Manage. Sci. Vol 17 No 7 (March 1970) pp 453-471 15 Liggett, R S An exploration of approximate solution strategies for combinatorial optimization problems PhD dissertation, University of California, Los Angeles, CA, USA (December 1978)
BIBLIOGRAPHY Basic architectural programming system (BAPS) user manual Computer Aided Design Group, Santa Monica, CA, USA (I 980)
computer-aided design