heuristic algorithms for automated space planning

8/13/2019 Heuristic Algorithms for Automated Space Planning

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Seseion No. 2 Applications 27

HEURISTIC ALGORITHMS FOR AUTOMATED SPACE PLANNING

Charles M. EastmanInstitute of Physical Planning

Carnegie-Mellon UniversityPittsburgh, Penna., U.S.A.

Summary

This paper reviews the major characteristicsof existing programs for automatically solvingspace planning problems. It outl ines generalfeatures of this class of problem and introducesa set of heuristic decision rules called Constraint Structured Planning, which in preliminarytests have efficiently solved a variety of problems. Developments all owing even greater e f f i ciencies are also outlined.

Introduction

I wish to focus on computer programs allowing automatic resolution of problems which humanssolve using orthograph ic drawing. Typical problems include the layout of floor plans, the arrangement of equipment in rooms, site planning,and other forms of two-dimensional design tasks.In such problems, distance, adjacency, and otherfunctions of arrangement are a principal concern.To distinguish spatial arrangement tasks fromother types of design, we call them space planningproblems.

Several programs are currently running orare under development which allow formulation

and automatic resolution of space planning problems ( 1 , 5, 6, 11 , 12). With in them, a va riet yof computer data structures have been developedwhich can represent space and objects, and operations have been implemented for manipulating andte st in g the re su lt in g arrangements. While it isstraightforward to use such systems in an interactive design mode, the benefit and challenge ofsuch programs comes from their potential forautomated space planning, for automatically combining manipulation and test operations in sucha way that acceptable arrangements are quicklygenerated. To date the complex and il l- fo rm edstructure of most space planning tasks has allowed the development of algorithms which do onlypoorly when compared with an experienced human

draftsman.

The relationship between the techniques implemented in the different programs for automaticspace planning has not been at a l l clear . Eachmust be considered an ad hoc attempt to buildfrom scratch a theory and program with capabilities originally only the province of humans.

This paper examines the structure of thespace planning task and proposes improved methodsfor dealing wit h i t . Several general issues in herent to the task are identified and the methods

used for dealing with them in the existing programs are reviewed. The space planning taskis organized into its component decision rules.One set of such rules is described which we callConstraint Struc tured Planning . Aspects ofConstraint Structured Planning have been implemented in the General Space Planner (GSP) program in operation at Carnegie-Mellon University(1 , 2) . In i t i a l t r ia ls with i t indicate thatConstraint Structured Planning has significantpotential for efficiently solving a wide varietyof space planning problems. The logic behindthe aspects of GSP not yet implemented suggestthat a program incorporating all its features

w i l l be more ef f ic ie nt than those now in operat ion.

Problem Formulation

In previously published studies of howhumans solve space planning problems (3,9) ithas been recognized that designers do not treattheir problems within the traditional frameworkof opt imi za tio n. While on the one hand thei rconcerns are usually complexly related and i n volve multiple objectives, the small amount of

information available about a problem does notpermit meaningful development of linear or nonlinear parameter weighting schemes.

In a general way, their problems arenormally self-defined in terms of constraints.The total set of constraints defines a feasibleso lu tion domain. The task is to fi nd a fea sib lesol utio n wi th in this domain. Optimization ofa sort does take place during iteration of thesearch sequence. When the i n i t i a l set of constrain ts results in a tr i v i a l search, eithernew constraints are added or old constraintsare replaced with new more restrictive ones.Conversely, when a cur rent problem seems i n tractable, the constraints are often relaxed.

Optimization takes the form of iterativelymodifying the problem definition until anappropriate balance between tractability andqu al it y of res ul ts are achieved. Problem sol ving effort thus plays an important role determining the final problem formulation.

Implicit recognition of the above procedure of working has led the designers of a l lexisting computer assisted space planning programs to rely on the following general formulati on. Some of the current programs also f a c i l i tate iterative definit ion and resolution of


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28 Session No. 2 Applications

a problem.

a Space;

a set of elements toarrange in that space;

a set of desired spatialrelations required forany acceptable arrangement of elements;

a set of operators formanipulating the locationof elements within theS pac e;

some i n i t i a l design condition (which simply maybe a)

We call this formulation the single levelspace planning task. It can depi ct many typi ca ldraft ing problems. One example is shown inFigure One. It describes a typi ca l mechanical

room layout problem. The elements to be includedin the space and the set of relations to be satisfied are shown at the bottom of the figure.

The Representation of Objects and Relations

The representation of spaces and elementsthat have been implemented in the various spaceplanning programs a l l def ine shape. A majorconcern has been the accuracy with which shapesare defin ed. Several are lim ite d to rec tang les. *Some implementations include predicates assignedto domains within the shape to define the type ofspace being represented. This inf ormat ion isuseful for automatically evaluating possible locations and eliminating those with overlap conf l i c t s . Usually, only overlaps of cert ain kindsof spaces are allowed . If domains of space andtheir type are included in the shape representat i o n , then the shape can be made up of more thanone type of space. This allows the area adjacentto an element and required fo r i t s use to be i n cluded in i ts ini t ial shape defini t ion, e.g., thepace that must remain free behind a desk so that

* Of the representations developed to dat e,Pfe ffe rco rn' s is the most accurate. His is li ke lyto be the model for future sophisticated spaceplanning systems (12).

the drawers can be pu ll ed out . Other implementations leave the prohibition against certain overlaps to be defined by a rel a t io n. Several of thespace planning implementations also includepoints that may be defined within a shape (e.g.the location of an electrical outlet) and refer

ence can be made to particular sides of objects.For any particular problem, one can

identify a set of relations that should be satisf ied if an arrangement is to be accepted. Forautomatic treatment, the task becomes one ofdefining a general set of tests with appropriatepredicates which are able to depict the widevariety of specific relations required in anyparticular class of space planning problems.(Throughout the res t of th is paper, we w i l l c a l lsuch tests space Relations - or S-Relations.)In the systems implemented to date, S-Relationsuseful for room arrangement (5), equipmentarrangement wi th in a room ( 1 , 12), and bu il di ngarrangement on a site (6,11), have been con

side red. The set of S-Relations tha t have beenimplemented in the GSP space planning program arepresented in Table One. The task of de fi ni ng aset of S-Relations that are able to depict thewhole set of space planning tasks, or even asingle class of such tasks, remains as a problem for further research.

For ease of description, we say that aS-Relati on which has as one of i t s predicates anelement i "belongs" to element i. MostS-Relations implemented in programs thus faract to disqualify certain arrangements of pairsof elements. Thus most S-Relat ions belong totwo elements. (A few belong to a si ng le element,

e.g., requiring that the element face southward.The IMAGE system by Johnson et al (6) incorpo rates two S-Relations which belong to threeelements.) The reason space planning is inherently diff icult is the large (potential ly infinite) number of location and orientationcombinations that are available for any singleelement. The d i f f i c u l t y involved in evaluatinglocations for a single element is severelycompounded by the interdependencies amongelements imposed by the S-Relat ions. The exhaustive enumeration of a l l combinations oflocations for each element is impractical fora l l but the most t r i v i a l arrangement problems.

Location Operations

In any space planning program, the typesof S-Relations that can be resolved and the characteristics of the location operators availableare int imat ely re lat ed. Let us consider for amoment the kinds of locations that should be generated if different S-Relations are to be satisf i ed . Almost a l l S-Relations that have been i n corporated into space planning programs accepteither a line of locations (generated along theperimeter of an element, for example) or anarea, possibly disjoint, in which any location


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30

is equall y acceptab le. For example, the areaand l ine al lowed by a SIGHT and an ADJACENT r elat ion are shown in Figure Two. The actua l areaprojected by a S-Relation is only defined if oneof the elements it belongs to is lo cated. Inthe following discussion, we limit considerationto those S-Relations for which an area can bedefin ed. (This important li mi ta ti on is considered in the final section of this paper.)

S-Relations restricting the locations foran element i w i l l define a set of line s andareas, denoted as S i,. It is easily^p roved thatthe area in te rs ec tion of any s e t, S., can be foundby examining only the points of intersection between pair s of i t s areas or li ne s. * Of course,if two S-Relations belonging to S. define li neswhich are not pa ra l l e l , then only the poin tdefined by their intersections need be considered.It is important to note that it is never necessaryto consider location points that are not at the

intersection of two boundaries of areas in S..

FIGURE TWO - The areas projected

by a sightline and an adjacentS-Relation.

In theory, one should be able to projectthose lines that define the boundaries of theareas in S, and to search th ei r point in ter sections for one sat isf yin g a l l rel ati ons . Nospace planning program has implemented such anapproach to lo ca ting elements. Two di ff er en tapproaches have thus far been relied upon.

One approach responds to the interdepend-ency of S-Relations and locations by binding themtogether, one-to-one, much like the recognitionand implementation aspects of an interpreter.The IMAGE system, by Johnson et al at M.I .T . (6) ,starts with an initial (arbitrary) arrangementand upon finding a S-Relation not satisfied,evokes an operator which satisfies it by findinga new loc at io n for an element. This approachsimplifies the control structure of a space planning system and guarantees that each S-Relationhas at least one effective means for responding

that is the intersection ofS. . Each of it s edges w i l l becdefined by one

boundary of one area in F. . I ts cornersw i l l then be defined by the in ter se cti on oftwo of i t s edges.

Session No. 2 Applications

to i t . I t s disadvantage is that the operatorsused are not able to systematically explore thearea defined by any S-Relation.g Thus neither thein te rs ec ti on of the area in S. nor any othersystematic search of locations can be underta ke n. ** Grason's program si mi la rl y binds

S-Relations to location operators, but incorporates operators which generate all locationssa ti sf yi ng a S-Relat ion (5) . Each time an operator is evoked, it produces another locationwi th in the area defined by i t s S-Relation . Tofacilitate search, each S-Relation operator pairis co ntex tual ly independent. This independenceis achieved by restricting the S-Relations thatcan be represented.

Eastman (1) and Pfeffercorn (12), on theother hand, keep the binding between S-Relationsand loca tion operators loose. In th ei r programs,a limited number of location operators are provided which serve to satisfy whole sets of

S-Relations. In Pfe fferc orn 's DPS program a l lthe possible locations for an element are generated in one pass and stored on a l i s t fo reval uati on. A sing le loc ati on operator generates the list and thus it is relied on for satisfyi ng a l l S-Rela tions . Eastman's, on the otherhand, incorporates operators which sequentiallyproduce locations, one at a time, like Crason's.

A set of operators is prov id ed , each producinga different sequence of locations for an element.Thus matching is possible between a set of S-Relations belonging to an element and the mostappropriate locat ion operator for loca ting i t .In both programs, no one-to-one relation existsand thus each location produced must be evaluated.

In these programs, search of the boundaryof the areas defined by different S-Relationsis approximated by location operators that respond to particular arrangement characteristics,e.g., the projected edges of elements and theperimeters of elements. Thus they are usu all yable to generate some of the locations satisfying a set of S-Relations. (The loca tion operations are described in more de ta i l la ter .) Thelimitation imposed by the approximations usedby these programs has not been determined.

Algori thms fo r Space Planning

Two alternative but fundamental approacheshave been utilized thus far in constructing algorithms for solving space planning problems.The first takes AB its initial step the generat ion and storage of al l the poss ible loc atio nsfor each in di vi du al element. The second step isthen to eliminate all conflicting locations resulting from overlaps or S-Relations until only

**IMAGE relies on "averaging" the locations proposed by each S-Relation using a least-squares-means -fit procedure. The new loca tio n is l i ke l yto be different from any of those that are generated by the S-Relations (6).


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Session No. 2 Applications 31

S-RELATIONS:

(1) ADJACENT (A, B, SA, SB) - This S-Relation sp ec ifi es thatelement B must have a l l of i t s side SB adjacent toside SA of element A. (A may also be a space.)Parameters SA and SB are opt io nal; if one or bothare not specified, any side of the corresponding elementmay be adjacent;

(2) SIGHT (A, B, PTA, PTB) - This S-Rela tion sp ec if ie s thatPoin t PTA on element A be v is ib le from Point PTBon element B; no so lids may l i e in the space d i re nt lybetween them. Again, the points are optional and ifnot defi ned, a l l of the element facing the other elementmust be visible;

(3) DISTANCE (A, B, F, PTA, PTB) - This S-Relation definesthat Point PTA on element A be less than F un it s

away from Point PTB on element B. PTA and PTB areboth opt ion al parameters. If eit her or both are omi tte d,distance is measured from the center of the element;

(4) ACCESS (A, B, F) - This S-Relation requires th at a pathwayexi st between element A and element B. The pathwaymust be at least F un it s wide along i t s whole le ng th ;

(5) ORIENT (A, B, SA, SB) - This rela t ion def ines requiredor ie nt at io ns between elements B and A. Side SB ofelement B must face element A and side SA of element

A must face element B. Ei ther SA or SB (but notboth) are op tional parameters.

LOCATION OPERATORS:

SCAN (A, B) - Sequentially generates the complete set of locationsfor element B, id en ti fi ed by the proj ec tion of edges inthe space A. (See Figure Three.) It sequential ly considersplacement of the element in each; a l l four ori en ta ti onsof the element are considered. SCAN is the most generallocation operator available in GSP;

PERIMETER (A, B, SB) - This operator sequentially generates thesi gn if ic an t locat ions for element B along the boundaryof a specified and located element in the space, denotedby A. The side SB of element B is placed along thecommon border. One or more cal ls of th is operator cansatisfy the ADJACENT S-Relation.

TABLE ONE

Elements of General Space Planner. The aboveS-Relations and Operators are those implementedin the most current version of GSP.


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one or more feasibl e sets remain. The d i f f i c u l ties with this Approach are (a) the large amountof memory consumed in st or in g a l l lo ca ti on combi na ti on s; (b) the la rge amounts of computertime required to search the combinations; and(c) the s trong assumptions that must be made

about which l oca tio ns from among a l l those possi bl e should be stor ed. Pfef ferc orn 's programrelies on this method to locate single elements.The one effort which totally relied on this approach utilized a grid for defining shapes andpotential locations (9).*

The second approach, and the one we w i l l examine in detail, views the space planning taskas one of sequentially generating locations foreach element, and adding elements or changingtheir locations, one at a time, until a singleacceptable arrangement is produced. In orderthat the perturbation of elements to locationsnot be random, consider them as i n i t i a l l y perturbed and evaluated in lexiographic order.

The task then is to al te r the i n i t i a l lex io graphic order, using available information, soas to make the search sequence as efficient aspossible.

The decisions defining the perturbation sequence and thus the efficiency of space planningin this approach are:

1. the sequence of locat ions considered fo ra particular element when it is added tothe arrangement;

2. the sequence in which the elements areadded;

3. the sequence in which S-Relations areapplied to test an arrangement or location and when in the perturbationsequence they are applied;

4. when a lo ca tion operator cannot fi nda location, or whan a S-Relation istested and fails, the rule or ruleswhich define what object/locationcombination is to be perturbed toallow further progress on the problem.

In terms of the formulation represented ineq. [1], this approach relies on operations ofthe form

ci si on rules in an ef fi ci en t manner.** In thefo ll ow in g, we present bases on which ef fi ci en tprocedures may be developed for responding tothe four dec ision co nd it io ns . In most cases,the concepts described have been incorporatedin the GSP program now in operation at Carnegie-

Mellon Un iv er si ty . Thus the fo ll owi ng can beconsidered a description of the automated designprocedures in GSP. Where the deci si on ru lesdescribed have not been implemented, we shalle x pl i c i t l y say so. Also included in the di scussion is Pfeffercorn's method for treatingthese four issues. To augment the pre senta tion,the S-Relations and location operators incorporated in GSP are described in Table One.Those in ter est ed in a fu l l e r desc ript io n of CSPshould refer to (1, 2).

The Sequence of Locations

In the previous discussion of locationoperations i n i t i a l consid eration was given to

the selection of operators and appropriatelo ca tion s. In the fo ll ow in g, we examine morefully rules for sequencing locations.

The basis for selecting a location operator and/or location for an element is the interaction between two or more of the S-Relationsbelonging to the element. If the kinds ofS-Relations that are used to specify arrangementobjectives are kept very simple, many generalizations concerning combinations of S-Relationsare poss ib le . For example, suppose adjacencyrelations require only that any two sides ofthe elements it belongs to must be in contact.If we also assume that distance relations are

always measured from the nearest perimeter ofthe two objects, then a distance relation ofzero is equivalent to an adjacency relationbetween the same two elements. In cases wherethe distance relation is not zero, the locationsavailable to the adjacency relation would be asubset of those defined by the distance relationand could be ignored.

If S-Relations are defined in such a wayas to represent the f u l l complexity of meaningful problems, then the interaction among themcannot be so simply sp ec if ie d. For example,in the S-Relations that have been incorporatedin GSP, the DISTANCE S-Relations allows dis tance

between specified points; the ADJACENT S-Relationallows specification of the side to be adjacent.If points and sides are used as references, the

* Recent contributions in the area of constraintelimination using known alternatives suggest thatth is approach may s t i l l have me ri t. See (10 ).

** The breakdown of the remaining programs areas fo ll ows: Graaon's (5) re li es on an exhaustive search; Negroponte's URBAN5 (11) operates inthe interactive design mode only (? ); andJohnson's IMAGE (6) rel ies on a least-squares -means-fit procedure for combining S-Relations.


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relation between the two S-Relations cannot beeas ily spe ci fie d. After caref ul analysis of theinteraction of the S-Relations in GSP, theauthor has concluded that only four warrantspecial attention in its space planning algorithms.*

It is not possible in GSP to generate theboundaries of areas def ined by S-Relat ions. GSPrelies instead on selecting a location operatorthat is likely to quickly generate some of thelocati ons in the int er se ct io n. It includes anoperator that generates locations responsive tothe one l ine producing S-Relation, ADJACENT. Itgenerates locations along the perimeter of aspec ifi ed or element. Points on the perimeterare selected that are defined by projection ofthe edges in the current arrangement. SeeFigure Three. If no ADJACENT S-Relation belongsto the element to be located, a general operator ca ll ed SCAN which tr ies a l l corners definedby projections of edges within the space.

(See Figure Three.) Pfe ffe rco rn' s program re li eson one opera tor s imi la r to SCAN for making a l llocations.

FIGURE THREE - In the twenty-fourlocations shown, each element wouldbe considered in its four orientat ions. PERIMETER (in rela t ion tothe space) would generate 1-2-3-4-5-6-12-11-16-20-22-24-23-17-13-7 inthat order.

* The design of S-Relations which can depictthe complexities of the real world and whichin te ra ct in we ll s pec ifi ed ways is an important

research problem. The in ter act ions specia llyconsidered in GSP are that ORIENT defines asubcondi ti on of ADJACENT and SIGHT. Only location-orientation combinations which agree withORIENT need be considered in solving SIGHT.Two ADJACENT S-Relations interact to define oneof three condi ti ons: (1) a sing le or possiblytwo loca tion s; (2) a nul l loca tion set; (3) ifthe distance between the two elements are theexact width of the third, a line of locationsre su l ts . Last , two ORIENT S-Relations belonging to the same element must agree, else theydefine a null intersection.

The Sequence of Elements

In general, an efficient space planner isone which can find a feasible arrangement asquick ly as poss ible. An equivalent ob je cti ve ,is to show in as few tr i a l s as possible th at no

arrangement is po ss ib le .* * In the fol lowin gdiscussion we use this second objective as asubstitute of the f irst.

FIGURE FOUR - Effect of order on

the number of operations requiredto search a tree.

The sequence in which elements are addedto an arrangement can significantly effectsearch ef fi ci en cy . For inst ance, consider anarrangement involving two elements, A and B.We ignore considera tion of al l issues but thenumber of locations for each element; element

A can be located in f ive locati ons, B in two.Two search trees are pos sibl e. Both are shownin Figure Four. In order to show that noarrangement is possible, the number of locationoperations required to search the A-B tree is15, whi le the B-A tree requ ires 12. In gen eral,

the number of operations required to enumeratethe alternating location possibilities of a setof elements, with a fixed number of location

** The relationship between these two objectivesmay not be obvious. Consider each sequen tial lygenerated arrangement of elements as identifiedby It s index n, where 1 < n < N. In order toprove that no arrangement sat is fi es a l l S-Relat ions, a l l N arrangements must be generated andevaluated. On the other hand, if M arrangementsexist that satisfy the S-Relations, their locations wi th in the ordering of n can be cons idered random and uniformly distributed (assumingan exhaustive enumeration al go ri th m) . Thus theaverage number of locations that must be

evaluated is equal to


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po ss ib il it ie s is equal to W where

Because the size of an element also restrictsthe number of locat ions avai lab le to i t , wedivide the area available to locate an elementby its area, or

* In the above, we assume that the cost of generating locations for all elements is equal.See (13).

which defines the relative number of disjointlocations ava ila ble to element i, given thecurrent arrangement.

For each unlocated element, GSP computesthe above function and selects the element withthe lowest value to locate in the arrangementnext.

Pfeffercorn's program includes elementsin decreasing order of the ir size (12 ). Thus,it approximates the lefthand part of eq. [ 5 ] .

Sequencing of S-Relations Applied to a Location

The S-Relations would then be applied in increas

ing order of th ei r 0 . Pfeffe rcorn also lo cates elements in order of their severity,(though he does not state his function for determining severity).

In applying the S-Relations to locations,it is useful to distinguish between two types,those that are not affected by the arrangementof elements which are not its predicates, andthose that are. An adjacency re la ti on betweentwo elements, no matter how qualified by arguments, w i l l not be affected by the placement ofother elements. On the other hand a sight l in erelation, defining that an open line of visionmust exist between two points or areas, may be

affected by any change in the total arrangementof elements. We ca ll the f i r s t type of S-Relatio n lo ca l, and the others gl ob al . A loc alS-Relation can be evaluated immediately afterplacement of the elements which are i t s predica tes . If it is sa ti sf ie d, no lat er evaluation s are requ ired . Global re lat io ns can beaffected by any perturbation of the arrangementand thus must be tested after any alterationof the arrangement. In GSP, ADJACENT, ORIENTATION, and DISTANCE are local S-Relations.

where p[1 ], P[2]'...p[m ] is a Par tic ula r ordering of the location possibilities associatedwi th each of the m elements. It is ea si lyproved that W is minimized when elements areadded to an arrangement in ascending order oftheir p. Thus if we can determine the numberof potential locations for each element, we canderive the sequence for searching their combinatorial locations most eff iciently.*

In reality, no fixed number of locations canbe specified for an element; the number changesaccording to the arrangement of the other ele

ments. We may approximate the ascending orderof p by dynamically evaluating the re la ti venumber of locations available after each changein the ex is ti ng arrangement. Af te r each change,we add to the arrangement next that element withthe relatively fewest number of locations available to i t .

The locations available to an element arethose within the intersection of areas and linesdefined by each S-Relation belonging to thatelement. These areas can be defined for anyS-Relation which has one of its element predicates already located; for those without onethe resulting acceptable area cannot be defined.

We can compute the approximate area for thoseS-Relations which def ine them. We can also combine these areas by assuming that the points ineach area are distributed independently of thepoints in other areas. (Independence is assumedfor all but the exceptions listed in the firstfootno te on the previous page.) We c a l l theto ta l area ava ilabl e to an element i, Si*.For an unlocated element i there is a setoof S-Relations c ., that belong to i t . c0denotes the S-Relations from among the cwhich have their other predicate*element alreadylocated.c The set of areas S id en ti fi edby c w i l l define an int erse cti on which isequal to


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Backup Procedures.

When a location operator fails to generatean acceptable location for an element, it is due

to one of two causes. Either the operator fa il edto generate a loc ation wit hin an exis ting i nt ersection of the S-Relation areas, or the intersection of the relevant S-Relations was null.In both GSP and Pf eff er corn 's DPS program,the f i r s t source of fai lu re is assumed to benegated with careful selection of the locationoperator . (DPS has only one operator ; thus nomistake is possible). Thus any failure is assumed to be caused by a null intersection.

The l ik el ih oo d that a S-Relation w i l l causefailure of a location operator is inverselyproportional to the area it defines; the S-Relation with the smallest area is thus mostc r i t i c a l . One way to resolve fa il ur e, then, is

to alter the area defined by the most criticalS-Relati on. If that S-Relation happens to beof the local type, then the only way to changeits area is to alter the location of its previou sly located element. If the S-Relation isglobal, then moving any one of several elementsmay alter the S-Relation's acceptable area.GSP, in both cases, selects to relocate firstthe most restrictive element related to thefa il ed one by a local S-Relat ion. If none isfound, then the global S-Relation projecting thesmallest acceptable area is considered and itsother element selected for re -l oc at io n. Thisbackup procedure can be applied recursively.

Pf ef fe rc orn' s DPS program backtracks byremoving both elements of the S-Relation assumedto be most re s t r i c t i ve . It ingeniously composesan arrangement of the two elements that satisfiestheir S-Relation and attempts to relocate a composi te pa ir back in to the arrangement (12 ).

Implementation of Constraint StructuredPlanning in GSP

GSP incorp ora tes no means for d i rect ly comput ing the area pro jected by a S-Rela tion. Thusit cannot take advantage of the location generation method outlined in the description of loca tion operati ons . When conside ring which ele ment to add next to the arrangements GSP also

can only approximate the areas projected by thedifferent S-Relations.

For each S-Relat ion, GSP has a heu ri st icfun ct ion which estimates it s allowed area. Theseestimates make assumptions about shapes (thatall are rectangular) and the distribution ofelements. They are s ta t i c , in that they rel yon the particular problem definition and not onany pa rt ic u la r arrangement. The cur rent formof these estimating functions are presented inTable Two. We expect to improve these methodsfo r estima ting areas as our experience wi th GSPgrows.

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We have found that incorporation of the previously described decision rules is facilitatedif the major information they utilize is represented in the form of a directed graph, hence

the name Cons tra int Structured Planning. Eachelement or space defined as part of the problemis represented as a node in the graph. EachS-Relation is represented as one or two directededges. If a S-Relation belongs to two elements,it defines two edges, one in each directionbetween them. If a S-Relation is unary or between an element and the space, then only oneedge is def in ed . Each edge has a value co rres ponding to the area allowed for i t s successorelement when i t s predecessor element is alreadyloca ted. Each node then takes a 0 - 1 valuedenoting whether it has been loca ted. Forexample, the problem shown in Figure One canbe represented by the graph shown In FigureFive.

FIGURE FIVE - Constraint Graphfor the problem described inFigure One.

The Constraint Graph depicting any particular problem is treated not as an alternativerepresentation of the problem, but as a planningrepresentation that allows analysis of import

ant ch ar ac te ri st ic s. Thus we use the Const raintGraph in a manner similar to Gelernter's planning model for geometry theorem proving (4).

The matrix form of the Constraint Graph isutilized and consists of an edge by node matrix,denoted l((n + 3, m + 1) . The extra rows andcolumns are used as fol lo ws : R(n + 1 , i) =the estimated area of element i; R(n + 2, i)denotes whether the element has been locatedand R ( j , m + 1) denotes the type of S-Relationinvolved. See Figure Six .

On se lect in g a new element to lo ca te , GSP


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Current functions for estimating the area allowedby an S-Relation.The area is multiplied by the number of orientationsallowed.

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computes for each element the inter sec tion of known areas belonging to i t , definedin terms of the estimated disjoint locationsallowed, e.g.

FIGURE SIX - Matrix form of the Constraint Graph for theproblem in Figure One.

[7] In practice, we apply this algorithm to theconstraint graph twice, first subject to theaddi tiona l constraint that R( j, m 4+ 1) designates a loca l S-Relation, then again witho ut i t .

If an element i cannot be located acceptably, the backup procedure is implemented byrel oca ting that element i * , selected fromamong those where

As above, two i terat ions may be requi red, f i r s tfor local then for global S-Relations.

An Example

Constraint Structured Planning seems to have

largest area of a S-Relation(belonging to element i) testedduring this iteration.

and se lec t j which

Eq. [7] corresponds to eq. [5].

Af ter an element has been selected and a loca tion proposed, we wish to sequentially evaluateit according to Slagle's cri te ri on , e.g. eq. [ 6] ,Cu rr en tl y, we assume a l l te st s to have the samecost. Thus we make


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(f)

FIGURE SEVEN - Example problem and thesequence of alternative states generatedby GSP to solve i t .

powerful capabilities for dealing with a generalclass of space planning problems. Formal ev al uation and empirical testing of its capabilitiesare cu rren tl y underway. In order for the readerto better understand the characteristics of its

operation, we offer a detailed description of itsoperation on one t r i a l problem. We use the problem presented in Figure One and whose ConstraintGraph is shown in Figures Five and Si x. We assign to GSP the problem of arranging the machinery in Figure One to the space shown in FigureSeven (a), subject to the S-Relations defined inFigure One.

Row n+3 in Figure Six shows the computedintersection areas for the S-Relations uponi n i t i a t i o n of the program. Element One is se lect ed. Locations for it are generated by SCAN, whichrotates it and locates it as shown in Figure

Seven (6 ). The in te rse ct ions of allowed areas(eq . [ 7] ) are recomputed fo r each element. Ele ment Six is selected next, which is located withSCAN as shown in Figure Seven ( c) . Of the sixremaining elements, Seven has the smallest allowed area. Because it must be adjacent to elementSix, PERIMETER attempts to locate it but fails todo so for lack of room. The backup algo ri thm,eq. [9], is initiated and it identifies elementSix as the best candidate for reloc at io n. It ismoved to the location shown in Figure Seven (d),the only other acceptable one. Element Sevenis then located, followed sequentially by Eight,Five and Four. The resul t is shown in FigureSeven (e) . The last two elements are added,first Three then Eight, after Eight is rotatedto sat is fy the ORIENT S-Rela tion. The acceptedarrangement is shown in Figure Seven ( f ) , asprinted on our pl ot te r. In preliminary te sts ,specif ication, solution, and all plott ing forthis task took about three and a half minutes

of CPU time on Carnegie-Mellon Un iv er si ty 'sIBM 360/67 computer, using TSS FORTRAN.

Its general performance indicates that Constraint Structured Planning works most efficiently when the task is highl y struc tu red. Searchis s ign i f i cant l y speeded up if many ADJACENTS-Relations are inv olv ed. If no S-Relationsproject areas for unlocated elements, they arechosen according to their approximate size,biggest f i r s t . GSP is known to fa i l on someproblems, due to poor handling of global S-Rela tions. Debugging and re-cod ing are expectedto significantly lower current running times.

Conclusion

While Constraint Structured Planning, asimplemented in GSP, begins to effectively dealwith several of the issues inherent in spaceplanning, many remain for more detailed explora t ion. Wi th in the problem domain consideredhere, issues s t i l l remain in each of the fourdec ision processes. In the element se lect ionprocess, only S-Relations which have one oftheir predicate elements located can currentlybe brought into the analysis, yet the otherS-Relations act to delimit allowed areas also.No means has yet allowed us to estimate the restrictions imposed by all S-Relations.

GSP was implemented before the author appreciated or understood how S-Relations and locati on operators in te ra ct . Thus no means is bu i l tin to it for pro jec tin g areas defined by S-Relations, nor for looking at the boundaryin te rsec tions between such areas. We an ti ci pa tethat a system allowing these capabilities wouldgreatly increase the efficiency of search, aswell as to allow more varied shapes of elements.Pfeffercorn's method of representation wouldeasily handle these requirements.

Effective backup procedures have been developed for treating local S-Relations, but there


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has been l i t t l e progress in char acte rizi ng ortreating global S-Relations; thev seem to be thecause of most search difficulties.

Future ef fo rt s at expanding GSP w i l l i nclude

the introduction of variable shape elements andthe appropriate operators and tests to deal withthem. We also propose to expand GSP so as tohandle hierarchical problem domains, e.g."arrange the equipment in the rooms, which arearranged and shaped within a building, which isarranged and shaped on a si t e " (1 ). Such formulations clearly open up new problem domainsand exciting challenges for future research.

Ap pl ica ti on s fo r space planning programsshould be obvious. As greater ca pa bi li ti esdevelop for processing spatial arrangement tasks,better analyzed, better structured, and evenmore beautiful physical environments should

resul t .

References

(1) Eastman, C M . - "Prelimin ary Report on aSystem for General Space Planning",Communications of the Association forComputing Machinary (i n press) (1971).

(2) Eastman, C. M. - "GSP: A System forComputer Assisted Space Planning",Proceedings Design Automation Workshop,June 20-30, 1970, Atlantic City, N. J.

(3) Eastman, C . M . - "Cogn itiv e Processes andIl l -D efi ned Problems: a Case Study fromDesign", Proceedings of the Joint Internat ional Conference on Art i f ic ia l Intel l i gence", D. Walker and L. Norton eds.Washington, D. C, May 7-9, 1969.

(4) Gele rnter, H. - "Real izati on of a Geometry-Theorem Proving Machine", Computers andThought, E. Feigenbaum and J. Feldman (eds.)McGraw-Hill, N. Y. 1963, pp 134-152.

(5) Grason, John - "An Approach to ComputerizedSpace Planning Using Graph Theory", Proceedings Design Automation Workshop, June

28-30, 1971, Atlantic City, N.J.

(6) Johnson, T. , G. Weinzapfel, et a l . -"IMAGE: An Interactive Graphics-BasedComputer System for Multi-ConstrainedSpa tial Synt hes is". Report from theDept. of Architecture, M.I.T., Sept. 1970.

(7) Krauss, R. and J. Myer, et a l . - "Design:A Case Hi story " in Emerging Methods ofEnvironmental Design and Planning.G. Moore, (ed.) The M.I.T. Press, Cambridge,Mass., 1970.

(8) K re jc i r i k , M. - "Computer-aided PlantLayout ", Computer Aided Design 2 : 1 , 1969,pp 7-19.

(9) M i l l e r , W. R. - "Computer-Aided SpacePlanning", Proceedings of the SeventhAnnual Design Automation Workshop, SanFrancisco, Calif., 1970.

(10) Montanar i, V. - "Networks and Constraints:Fundamental Properties and Applicationsto Picture Processing", Dept. of ComputerScience Working Paper, Carnegie-MellonUniversity, January 1971.

(11) Negroponte, N., and L. Grossier - "URBAN5:A Machine th at Discusses Urban Design",in Emerging Methods of EnvironmentalDesign and Planning, G. Moore (ed.),The M. I. T. Press, Cambridge, Mass. 1970.

(12) Pf ef fe rc or n, Charles - "Computer Design ofEquipment Layouts Using the Design ProblemSolver", unpublished ph. d. dissertation,Dept. of Computer Science, Carnegie-MellonUniversity, May 1971.

(13) Sla gle , James - "An Ef f ic ie nt Algo rithmfor Finding Certain Minimum Cost Procedures for Making Binary Relations",Journal of the Association for ComputingMachinary, 1964, pp 253-264.

(14) Zehna, Peter - Probab il it y Di st ri bu ti on sand Statistics, Allyn and Bacon Inc.,

Boston 1970.

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heuristic algorithms for automated space planning

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