MATHEMATICAL MODELLING OF MORPHOLOGICAL ... - Core

MATHEMATICAL MODELLINGOF MORPHOLOGICAL ANALYSIS

Tomasz T. Arciszewski Department of Civil Engineering Wayne State University Detroit, Michigan 48202, U.S.A.

Abstract. A mathematical model of morphological analysis is given. A search of a morphological table is simulated by a nonhomogeneous Harkov chain. This model was used in the developed computer program, which has been successfully applied in structural engineering for the generation of a number of innovative, patentable solutions. Morphological Keywords. homogeneous Harkov chain.

analysis;

mathematical

Morphological analysis can be considered a modern heuristic method with strong ancient roots. In Greece in the 5th century B.C., a group of philosophers established a school to be known later as The best known philosophers of this the Atomists. school are Leacrippus of Miletus and Democritus of They were under the strong influence of Abdera. Parmenides and adopted his idea of the One (the They interunity and reality of the universe). preted it, however, in a different way. The Atomists thought that if the world is divided into its smallest indivisible parts called atoms (in Greek, “that which cannot be broken up”), reality will be The concept of morphological analysis, reached. was born. of unity and division, the concept

In the U.S. morphological described by Allen (1952, oped by Grant (1977) for tecture. and urban planning.

The method was initially applied by Zwicky in astronomy and later in the field of missile engines, where he discovered several new types (Zwicky, 1969). been particularly Morphological analysis has connoon in Great Britain, as described by Rickards (1980). Carson (1979) applied it to the development of new chemical products and incorporated it named SCIMITAR, approach into the innovative through which a range of new products was generMorphological models were also used by ated. Carson (1974) in a three-dimensional version for modelling business capabilities in industrial organizations. Gregory (1974) and Cross (1975) developed morphological analysis for applications in while Norris (1963) applied architectural design, Wills (1971) it to engipeering design purposes. applied the method to technological forecasting. analysis

is

the

non-

after method

analysis was first 1962) and later develapplications in archi-

In the next period the method was developed by Arciszewski (1985) into Stochastic Form Optimization. This method is a nondeterministic optimization technique, intended to optimize a structural solution in its most general formin the preliminary stage of the structural design process, when new ideas or the quick review of all feasible solutions are most desirable. The method supplements existing methods of shape optimization, which are mostly applicable in the advanced stages of the design process, that is, when the type of the solution is fixed, or simply assumed, and only local decisions regarding sizing variables are to be made. The second generation computer applications of morphological analysis in the form of Stochastic Form Optimization have also given birth to a number of patentable innovations developed by Arciszewski (1985a, 1985b). Recent applications of the method in the area of Artificial Intelligence can be considered third generation applications. The method was used to develop a Solution Generator, applicable to a large class of learning expert systems as proposed bv Arciszewski and Ziarko (forthcoming, 1986).

In the late forties Fritz Zwicky, a Swiss-born based his morphological analysis astrophysicist, although he significantly develon the same idea, oped it and adapted it to modern technical applications (see Zwicky, 1969).

morphological

solutions;

Three periods in the history of morphological analysis can be distinguished, according to the character of its applications. The first period of traditional non-computerized applications ended in the late 1970’s. These first-generation applications resulted in many discoveries and innovations, as described by Arciszewski (1976, 1977).

The same concept can be found in the papers of and Leibnitz, who were working Lullus ( Descartes on a universal method, applicable to all theoretiDescartes in particucal and practical problems. lar strongly advocated the idea of the division of problems into subproblems and formulated it as a second rule in his universal method (see Arciszewski, et al., 1977).

Germany

innovative

known and most irequently used technique, brainstormming, as an idea-generation (Geschka, 1973; Muller, 1970).

INTRODUCTION

In

modelling;

MATHEMATICAL HODEL The principles of morphological analysis are very simple. Every complex problem can be divided into a number of subproblems. These subproblems can be analyzed separately, and the combination of their solutions gives the solution to the global problem. Qualitative variables can be assigned to individually distinguished subproblems and their solutions called feasible states. A type of solution is defined by a compatible combination of feasible states, when for all variables one state only is taken at a time. Such a morphological

best52

53

HORPBOLOGICALANALYSIS

approach is referred to as “Typology by ings,” as proposed by Arciszewski (1985).

Cover-

Typology by coverings enables forsal presentation of all considered variables and their feasible states in a tabular form called a .“morphological of a morphological table. ” The general idea table is shown below. TABLE 1 Qualitative Variables

A B C D

General

Form of

a Morphological

Feasible States 2 . ..i . ..k 1

Al Bl Cl Dl

= (PlW,

In classic morphological analysis as developed by types of solution are considered by anaZwicky, lyzing different combinations of feasible states, which are taken randomly from the morphological the selected combinaUsually, however, table. tions reflect the personal preferences of the rewhich significantly reduces the prosearcher, bability of obtaining a new, innovative or “nconThis serious disadventional type of solution. can be eliminated when a mathematical vantage model of the analysis is developed and a computer used for the generation of solution types. Morphological analysis can be modelled as a nonhomogeneous Markov chain. Results of an unbiassed morphological analysis may be described by a seidentically and uniformly quence of independent, distributed random variables X X2, X , . . . . x , where r is the total &mber o t vari;bl@lx Such a sequence is a stochastic proconsidgred. cess with r trials, which represents a morphological search of a given morphological table with r rows. Markov chain, adjacent rows

When a given morphological field r x m is consida maximum nwnber of entries, including posered, sible intermediate empty entries, is equal to “m” for the most numerous single row of this field. It can then be formally assumed that each trial has m possible outcomes, called states. For some in the field states may include feasible NWS states as well as certain outcomes, representing which are states that will never empty entries, occur. It is convenient, however, to make the size of all rows in the field uniform and to assume that it is m. The analysis is a sequence of r-l random transitions, performed successively between adjacent rclws . It starts from the first row, when for the variable X its absolute probabilities at time 1 random11y generated and their complete and are ordered set, called the absolute probability vector at time 1, is to he determined so that:

(1)

- the absolute probability time 1, the probability the event Xl = jl

at of

. . . . Pj(‘),

where pj(I)

= p(X,

= jl)

- the tor

P(l)

absolute

m - the

. ..m

In the rows of the morphological table successive variables and their feasible states are specified. Such a table allows a simple and formal type identification and is very useful in computer-aided applications. When a given problem is considered, a morphological table can be used to present qualitative variables and their feasible states. Subsequent qualitative variables are considered independently and all intervariable relations are temporarily suspended.

. . . . P,(l)1

P2(1),

Table

A2 . . . Ai . . . Ak . . . Am B2 . . . Bi . . . Bk C2...Ci...Ck...Cm D2 . . . Di

This process is a non-homogeneous when subsequent transitions between are independent.

P(I)

total

probability

number

of

vec-

states

This first trial is also called the “initial trial,” and its results are initial probabilities that define the initial probability vector. The next step of the analysis is the transition from the first to the second row, and the absolute probability vector at time 2, for the random variable X2, is calculated as follows: ~(2)

= p(l)

T;l)

(2)

where

:

p(2)

- the

p(l)

- the absolute or the initial

TW 1

- the one-step matrix for trial

absolute

probability

vector

at

time

2

probability probability

vector vector

at

time

1

randomly the first

All subsequent transitions same manner and may be p(n)

= p(n-1)

where

:

are performed in the described in general:

Tr!!i

(3)

p(n) =

P (n),..., p.(n), (p,(n), ahsol” t e probabillity

p&l)

-

the n-l

absolute

T(l)

_ the

one-step matrix

n-l

tion trial n-

generated transition second transition,

the

. . . . p (n)) - the vector mat time n

probability

vector

at

randomly generated transifor the n-l transition, nth

number of

a trial,

n = 1, 2,

r - the number of parameters of rows field, the total number of trials The

one-step

m x m matrix transition PTJI(l) =

where

transition with entries probabilities

= P(X,

T(l) n-l

= jnlXn_l

3,..

. ,r

in

the

(1) mat+ Tn_l is a square pn_ (l), called one-step aid defined as follows:

=

in) (4)

(PijW

:

P(X, = jnlXn_l = in_l)

- the conditional ity of given

state

i,

time

j = 1, 2,

10-

the

the X ,,i*-l

event is

probabilX = j l”n tsC

. .., m size

of

the

square

transition

matrix.

Since in an ideal morphological analysis there are no relations between states of adjacent rows, transitions between them should have a purely random character. In the general case, individual rows may have different numbers of feasible states because certain entries, intermediate or last, may

5th xQ4M

54

was developed later type shown in Fig. Arciszewski (1985),

be empty. For this reason not all transitions are and this must be reflected in the form possible, It may have zero-rows and of a transition matrix. zero-columns, corresponding to empty entries in the pre-transition row and in the post-trans_ition Such a matrix should be called row respectively. stochastic matrix,” since for zeroa “modified rows sums of entries are not equal to unity. For individual transition analysis all the whole matrices should be different as well as indepedently and randomly generated. When this kL:?;s~‘,‘,)

stochastic ,obt,;;pe(dn),

process

is

completed

a

CONCIJSIONS The developed approach to formal identification of the morphological complex engineering problems, table and typology by coverings, are very practical and can be used for conceptual representation purposes in the area of expert systems.

set

The prepared mathematical model of morphological analysis has been successfully used in the area of steel structures and has proved to be useful for Its application practical engineering purposes. to other areas of engineering is feasible and This model can be used should also be effective. randomly generating solution generator, as a solutions to a given problem from the prepared Such a solution generator morphological table. can be used in learning expert systems.

. . . , p(r)1 of probability

and thus discrete distribution functions for ail random variables X (n = 1, Z..r) are known. Then a final selection”of feasible states can be made using a number of different decision criteria as proposed by Arciszewski. APPLICATIONS A computer program has been developed, based on the proposed mathematical model of morphological analysis. It has been used successfully in the area of structural engineering for the development of new types of structures and connections. This program has also been used as a “Solution Generator” in a learning expert system for preliminary structural design being developed by Arciszewski and Ziarko (1986).

REFERENCES Allen, Prentice

1.

JOINT

M.S. Hall.

Allen, M.S. The Miracle of Cliffs, Prentice

The results of the application of Stochastic Form Optimization in the area of steel skeleton and space structures are particularly interesting. A morphological table for wind bracings in steel skeleton structures had been prepared and used for the preparation of new types of wind bracings. A number of previously-unknown types was found, some of them are quite interesting, as described by Arciszewski (1985). Also, analysis of the problem of structural shaping of joints in skeleton structures and computer generation of solutions brought many new solutions. One of them has already been patented (see Arciszewski, 1976), while others are being processed now (Fig. 1).

FIG.

into the even more innovative described by This joint, 3. is being patented now.

(1952).

Morphological

(1962). Morphological Your Hidden Brainpower, Hall.

Creativity,

Creativity: Englewood

Arciszewski, T. and Kisielnicka, J. (1974). MOP phological Method and Quasi-Algorithms. Proceedings Of 1st Conference on Heuristic Methods, Warsaw, Poland. New T. and Pancewicz, Z. (1976). Arciszewski, Skeleton Beam-to-Column Connection in TYPO of Structures. Bulletin of the Patent Office, No. 1, (55)) Warsaw, Poland. Systems Approach to the Arciszewski, T. (1977). Proceedings Analysis of Steel Space Structures. Warsaw, of 3rd Conference on Heuristic Methods, Poland.

IN SKELETON STRUCTURES

Steel space structures are considered among the most complex of sLrucLura1 systems, and the development of new types of joints is particularly difficult. However, the application of Stochastic Form Optimization has enabled the preparation of a formal typology of joints (Arciszewski, 1984) and the development of many interesting joints. One of them, shown in Fig. 2, generated by a computer,

Artiszewski, T., Bojarski, T., Dorosinski, W., Coralski, A., Mapierska, S., Mizinski, J.,,Ez;F;;(1978). and Tarpowska, M. ski, If. t Technics of Creative Thinkink Method, Solution. Part II PUN-Pbl. House, Warsaw, Poland. _-_--’

~RPHOLOGIG~

FIG.

2.

FIG.

Arciszcwski, T. Space Structures. ence Publishers,

SPHERICAL JOINT IN SPACE

3.

STRUCTURES

-

INITIAL

VERSION

SPHERICAL JOXNT IN SPACE STRUCTURES- FINAL VERSION

(1984). Design of Joints in Steel Space Structures, Applied SciGreat Britain.

Arciszewski, T. (1984). Application of Stochastic Processes in Innovative Structural Design. Proceedings of the 5th Engineering Mechanics Division Specialty Conference, ASCE, Wyoming. Arciszewski, T. (1985). Decision-Making Parameters and Their Computer-Aided Morphological Analysis for Wind Bracings in Steel Skeleton Structures. Advances in Tall Buildings, Hutchinson Ross Publish‘ing Company. Arciszewski, Structures. versitv.

55

ARALYSIS

T. (1985). Spherical Joint in Space Patent Application, Wayne State Uni-

Arciszewski, T. (1985). Skeleton Structures. State University.

Prefabricated Joint in Patent Application, Wayne

Arciszewski, T. and Ziatko, W. (forthcoming, 1986). Learning Expert System for Preliminary Design of Wind Bracings in Skeleton Structures. Second Century of the Skyscraper, Van Nostrand Reinhold. Arciszewski, T. (forthcoming). Optimization. Paper submitted Structural Division, ASCE. Carson, tation

Stochastic Form to Journal of

J.W. (1974). Three-Dimensional Represenof Company Business and Investigation Ac-

56

5th ICm

tivities. Vo. 5, No.

Resaerch 1, U.K.

and Development

Carson, J.W. and Rickards, New Products Development. ough, U.K.

Cross, Manual, Geschka, Battelle,

T. (1979). Industrial Gower Press, Farnbor-

N. and Roy, R. (1975). The Open University, U.K. H. Ideenfindung (1973). Frankfurt, GDR.

Grant, D.P. (1977). ical Box. Journal Vol. 11. No. 3.

of

Management,

How to Construct Design Methods

Design

Methods

Forecasting.

a Morphologand Theories,

Gregory, S.A. (1974). The Shape of Ends and Means: ProceedSome Aspects of Morphological Analysis. ~. ings of ASLIB Conference, Oxford, U.K.

Huller, J. (1970). Grundlagen Heuristic, Dietz Verlag, CDR.

der

Systematischen

The Morphological Approach Norris, K.W. (1963). in Proceedings of the Conto Engineering Design, ference on Design Methods, Pergamon, U.K. Designing for Creativity: Rickards, T. (1980). State of the Art Review. Design Studies, Vol. No. 5, U.K. Wills, Pelican

Technological

G. (1971). Books, U.K.

Zwicky, F. (1957). Morphological lin, Springer Verlag. (1962). &icky, F. Power. Monograph of Research, Pasadena. &icky, rhrough fork.

the

A 11,

Forecasting,

Astronomy,

Ber-

Morphology of Propulsive Society for Morphological

F. (1969). Discovery, Invention, Research the Morphological Analysis, MacMillan, iir*