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Computer-Aided Comparative Analysis of Structural Properties of Large-Scale Organization Systems
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COMPUTER-AIDED COMPARATIVE ANALYSIS OF STRUCTURAL PROPERTIES OF LARGE-SCALE ORGANIZATION SYSTEMS P. Droidiyk, E. Michalewski,
J.
Ostrowski and M. Stankiewicz
Abstract . iiecently a package of canpute r programs, calle d DIANA- 6 , has been '.
INTRODUCTION
work is concerned wi th some proble.ll s of t he analysis of structural properties of manage:nent systens.
In general a practical purpose of studies of :nanage:nent systems is the dia,;nostic analysi sand d.esign of i:nproven ents of real sys te:ns. Sl
!)ES:~~:~
•
J ·t?RO'YEME!ITS •
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The course 0 f st udies carried on with the use of the DIliliA- 6 package.
:ET",.JORK OF ElTi':ROPERATION nJFORMA. TIO;'l CON]ECTIO,IS
ia~a a2~~isi:.i=~ : :.:i2hal e~sk i, 1975) as well as :':-.e a:;p~i2a:.icr. 2:-" a cc;:).p',.;.ter :. . OY tneir ;::-::eess::.s si:;r:::-":'::ar.:.l.y red....:.ees the tL-:le ne -
The ".1 ethodolc:gy ~sed to constrc;.ct the DIAl,A6 packag e re li es on the aSS-l'nption that a ".1o:i"el of a ".1anage".1ent syste::l under in vesti "ation can be presenteci in 1:he for:n of so called network of interoperation i nfor:nation cocnec~ions ( IIC). ]ocies of this network represent rela1:ively si".1ple operations acoc:::plis hed at partic;'lar '.
eje:' :'8 28.Y:-:'; 0·...;.: :~e5e 3:·-.:~ie3 an:i ::1B-:.r..es ::-.e ·.,·::r~ 2:-' e.~.a~::S1:s ~.cre e f:""'icier::... l~.I.e '.lse :: :-. 2. .::::~.~·..::er So'::"'::";:·,,·5 :"'::r eas:,: :::.:;:ie .::...::.. i:. s of :"'''":".;::-2·:~.e:-_::: ;r.:;::se:" . :: e:-.2e :..:--.e:'r e:"':"'ec:s .: s.:-. ': :,.~ :;: ar:.:' a.'::"'::":.. -:"2:-:' :"': e:i ':: e :. . cre a:: :i..::-~ p..:.e ::"."2:-.: 9.:.:'8:': .
.:. . :.::;: :' :a=.. :;.!~:::e:'...::e :.:... a;; _:,::.n5 ::-le ==.~::.:.... - 6 :;:a2 ~~e :"'::r s:'..::i:es ei::-.e:i s.:. :.::'e 6..:'ag:1csis a:-.:' :...-::. pr~ '.e::-. e:-.: -: :.. == :,'S :e::-. s :. s S:'--!0·~; r:. :':-. ?i~ , ,::,,. ?rc·:~e-::.s &"SS0.::"8o:e': .".:' :':-. ::'--.e 8o:-.a2.j"s:s an:: i. '"":'!. prc:~,pe~ e:-.: 0:"' :::r;;8o:-.:" :s.:, :' ::-,e.: pr2cess ::iY:1Er."l ics
.:u-. :::1: netwo r k is stored in the ::lass ::le::lory
(::--~e
iia;r.csis a:-.:" i~~rc':e::e "";: of ~:-.e syste."J f·.;.nc-:icning ; 6.:-e '::"s2 -..:. ssei e s e·,.,·:""~ere .. ::atko ani others , 1?7E ; :,::".::~a'::" e~sk , 19~2) . This
o~'
a CO".1p ·..i ce r in the fnn of an appropriately organized iata base . There are two types of data base records. The firs t one contains ,
507
P. Dro~dzyk et al .
508
among others , the f ollowing data concerning nodes of an IIC netw o rk : the code of a given ex eC'J.t i ve node and i ts coded para'llcters . The second one contains dat a concerning arcs of an IIC network: the code of an executive node , the code o f a delivery node ( i . e . tha t stipp1yin:s infor:!lation to an executive node) and appropriately coded panL'llete rs of a n arc j oinin:s the::: . Denote the set of nodes of an I:C network by V={ i 1i =l , .. ,;::1} an::: -: :-:e se-: ::::" \ :::cien-:ed) ar::: s of it by S ,S ~ { ( i , j : li E \' , j E ': J , For the p..:rpose 0:' -:"e s-:r'..;c-:'..;re a na::'ysis the followi'1';; para.-:le-:e r s co!;-:air.ej i:-: t~e dat a base are '..lsed:
( i i ) A function P descr i bing the st r eIl?; th of connections 8.'llong ele:nents of the struc t ure Z is de fined as follows : l: l: p ( i , j ) , when i EA(k) j 6 A( 1) l;
!
( 2)
In this formula p ( i , j) denotes the so called strEn~th of conne c tion between the pair of nodes ( i , j) , i,j V, where p ( i , j ) =a ( i , j ) ~ ( i , j ) ;
( i ) t,( i ) - t:-:e ~:';::l::er je: er:"i~,i:-:; ::-:02 one year labo'.:r ccr:s'..:..:: p: :'C!1 2'o!"re:3~2::i :r.~ -:0 the operation represe:-:-:ej '~ y -:::e r.~je i . \' ; ( i i) f ( i ) - t~'1e rn.n ':::er injica'.: in; ::~." ::lany ti1les the operati~n represer.:ed by t~e node i E V 1 S acccnp li s t.ed j''';!'in; cne - year perio d ; ( iii ) b( i ) - the code o f an actlvlty area (delivery, planning , personel, sales et c) to which the op erat i on represented by the node i ~ V belongs ; (iv) r ( i , j) - the nunber characterizing the functional relati onsh ip between the nodes i f! V and j E V joined by the arc ( i , j )" S. It is assumed that - I , when i is subordinate to j , r(i , j) = 0 , when i is equivalent to j , { 1 , when j 1S subbordinate to i,
(3 )
I , when b ( i ) #b( j); ( 4)
a ( i , j )=
{2 , when b(i) =b \ j) ;
'f ( i , j) =
t
'
when ( i , j)
.Sand
\ j , i) CS ;
in [ f(i) , f ( j)],when (i , j) ES or ( j , i) (: S . (5)
( iii) A function R descri birg the functional relationship among ele.n er.ts of the struct ure Z is de fined in the following way :
[ r ( i , j) +r ( j , i) ] } , whe r e l ;
0 , when l =k ; k ,l=l, ... , no
(6) STRUCTURE OF A MANAGEMENT SYSTEM
It is aS~'lled that r ( i , j) =O, when ( i, j) . S.
The t riplet Z=( W, P,R) is called the ~ nagement system structure , W={A ( k)lk=l ,." n }, where n<:n , denotes the set o f struct ure el~ ments , P is a f unct i on describing the strerg th o f connectio:1s 8.'llong structure elen ents , P : W~W ~ R+, where R+ denotes the set of non ne~ati ve real nun b ers , R is a func t ion des cribing the fmctional relationshi p among structure ele:nents , R:Wx W~{ -1 , 0 ,I } , where -I R[A ( k ) , A( l )] = {
when A(k) 1S subordinate , to A( l ) , 0 , when A( k ) is equivalent to A( l) o r P!A ( k ) , A(l) 1=0 , 1, wh en A(l) is subordinate to A( k) ,
In t he ::: et ho dology used to construct the DIANA- 6 packcge the following assu:npt i ons are adopted: ( i ) ElEments A(k) . W of the structure Z are identi fied with non- overlapping sub s ets of nodes 0 f an IIC network. The following re l a ti o ns hold: A( k ) C. V ; k =l , .. . , n ; A(k) " A(l) = ~ ; k , l =l , ... , n ; k# l ;
~ A( K) =V.
( 11
COMPARATIVE ANALYSIS OF STRUCTURAL PROPERTIES The DIANA- 6 packcge contains , among others , the following four prcgrams: ( i) The S- Ol program. For a given W={A(k) 1 k =l , . . . , n} i t com~utes the strrogth if connec tion P [A( k) , A( l) J for every pair of struc ture elements , i . e . for k , l =l , ... , n . This plugr8.'ll makes use of the para.'lleters f ( i) and b ( i) char acterizing nodes of an IIC network , whose values are stored in the data b a se . Using them , the approp r iate calculations a re perfonned on the basis of the formula ( 2). ( ii) The S- 02 prcgram . For a given W={A ( k) 1 k =l , . . . , n} it evaluates the function al relationship R [A( k) , A(l )] between every pair o f st r ucture e lements, i. e . fo r k , l =l , ... , n o This prcg r a'll makes use of the param e te rs f ( i) characterizing nodes and paI13mete r s r ( i , j) describing arcs of an IIC networ k , stored in the da ta base . Using tlEm , the appropriate calculations are performed on the bas i s of the formul a (6) .
509
Computer Analysis of Large Organizations ( iii) The S-03 progra-n. I t is used to determine "the ideal" (i. e . opti:nal with respect to an adopted perfonr.ance index) basic structure ele'nents . Usually these ele.llents are constructed of work stations. The~ro gra'Il deternines the optLllal set WX={A ( k) k=l, ... ,n~ ( one- person o r :nulti-person work stations) .
I
( i v) The 8- 04 progra:n. It is used to de ter:nine " the ideal" ( i.e. optLnal with repect to an adopted peri'ormance index) aggregation tree for the basic structure ele:nents i.e. it is ai:ned at co:nbining elenents of a given set WO={ AO ( k) l k=l, ... ,n O ~ to f~ larger , co:nplex structure ele:nents (organlzational units , sections , departnents, divi sions etc ) . The S- 03 and S- 04 are nore co:npl icated . They are discussed in :nore detail in further sections . The application of the above nen tioned prograns results in data making it possible to carry on the co:nparative analysis of structural properties. The basic aspects of such an analysis are presented in Table 1 . It follows fro!!'! Table 1 that four basic types of structures can be distinguished and compared. They are usually referred to as: formal, real, :nixed and ideal . The process of compar ison is divided into two stage s . Firstly, the properties of these structures constructed of basic elements are compared. Secondly , the properties of these structures composed of more complex ele:nents, obtained as a result of aggregation of basic elenents, are co:n pared. At each stag e all three structure aspects, i. e . all the term s of the tri plet Z=< W,P,R> are analyzed. TABLE 1
Aspects of
a; H
....,::\ ~ ....,H en
.....u Ul
1l
Ul ...., c
a;
E
~
a;
"-< 0
x
a; ....,
...... C 0. a;
....,
u E E o~
::\
H ...., D
U
The ideal structure (denoted by the symbol I) can be completely defined by the use of suitable programs. In th is approach basic structure ele:nents are represented by some subs ets of nodes of an lIe network and are determined with the use of the S-03 program. COm plex ele:nents, i. e. those obtained as a result of basic elements aggregation, are fc:und out vi th the use of the s-04 prog ram . The strength of connection and functional relationship among ele:nents are co:nputed by the programs S-Ol and S-02 respectively.
Real
R(F) docunents
Mixed
R( F) documents
Ideal
I S-03
Basic ele:nent s
F doctment s
Strengt h of connection
F docunents
R S- Ol
R S-Ol
I S- Ol
Functional relationship
F docu'llents
R S- 02
R 8'-02
I 8 - 02
Co:nplex s t ructure ele'n ent s
F docunents
I 8- 04
I 8- 04
Strensth of connection
F docu:nent s
R 8- 01
8- 01
I 8-01
?'..:nc t ional relations h ip
F docunents
R 8- 02
I 8 - 02
S-02
R( F) documents
'0
a; H ::\
Elenents of the real structure (denoted by the ~llbol R) are the same as those of the formal one. They are identi fled with elements of the appropriate subsets of nodes of an lIe net work. Owing to it, the programs S-Ol and S- 02 can be used to determine the real strength of connection and functional relationship among structure ele:nents. For this purpose the knowledge of pammeters characterizing nodes and arcs of an lIe network , whose values are stored in the data base, is to be utilized.
Formal
Structure types characteristics
0
So called f=al structure ( denoted by the symbol F) constitutes the reference point for the analysis to be performed. It is a structure of a :nanage.-nent system under investig ation as it is defined by valid formal documents determining the system partition into structure ele.-nents as well as their mutual connections (relationship ) , e .g. legal reg ulations, instructions, organi zational diagrams, bills of duties etc.
comparative analysis
I~ "-<
In Table 1 the character of each term of the triplet W,P ,R is denoted by the symbol F,R or I. Under this symbol a source of informat ion on a given unit is shown.
CJ
:t is aSs.l:ned t h at basic ele::l ents of the ::Ji xed str;;.ct'Jre are the sa:ne as in the real one . To aggregate basic ele::J.ents into co:n ple.x struct.: re ones , the progra.':J. 8- 0 4 is u sed .
I
I
It should be enphasized that all the prog-
ra.llS discussed can be used not only for the analysis of the whole network , but also for examination of any segnent of it.
P. Droz dzyk et al .
510
ALGORITHM FOR THE SYNTHESIS OF BASIC STRUCTURE ELEMENTS If in the fonnula (2) , defining the strength of connections a'll mg ele:nents of the struc ture Z , it is assumed that n =l (i . e . one- ele:nent structure is cons i dered) , then the ex press ion de ter.nining the so called total strength of connections In an IIC network is obtained : P(V , V) =
~ L i ('V
L p ( i , j). j 6V
( 7)
If structu res Z with more than one ele'llent (n>l) are taken into consideration , then P(V , V) is d i vided into two co:nponents (Fig . 2). One of then is called the stren;>;th of iotraelement connections and is denoted by pl(W). The second one is called the stren;>;th of interele.'llent connections
A(2.)
.···F'l-------t':~"'" ;,~t-::'--:~t) --) A(4) \
.....
\
-..
,/ .... . .... /'
--.. ...A(~) ....\~;;\ ".. . .....
_pl(wl ---- pt(W)
w= tA(1)/Al2),Al~n
n p i (W) = l: P [A(k) , A(k)], k =l n n pe(w) = L L P [A(k) , A(l) ] k =l l ) k
(8) (9)
and pi(W) +pe(W) =p(V , V) ,
( 10)
'1e nce fo r an I I C networ k the following pro~le.'ll can be fonnulated . Given the function P and the number n * of st r ucture ele:nents d2te:rmine such a set W~ of structure elements that p i (W* ) i s maximized (or equivalently , pe(w* ) is mi nim i zed) , i. e. the follo wing relations a r e to be satisfied: or
pe (WIf ) =ru inpe ( W)
(ll)
W
In other words this proble:n consists in de - . ter.nining " the ideal " basic s t ructu r e elene nts . An algo r i th:n p r esent ed , constituting the ba-
s i s of the S- 03 prcg r am , makes it poss i ble to find a subopti'lla l so l ut i on to this probJe:n •
where :n T= l: t(i) i= l
*
( 13)
denotes the total one- year labour consu.'llpt i on assoc i ated with al l the operations performed in a 'llanage:nent syste:n analyzed , rep r esented by the nodes i (: V of an lIC network . ( iii) The composition of so called seed
The algori th.'ll assigns all " free " nodes (i . e . those which do not belong to seeds) of an IIC network to particular basic structure eleme nts. Three steps of this algorithm are to be distinguished. At each step the constraints imposed on T( k) , k =l , .. " n have to be satisfied . Step 1 resu l ts in the construction of a set of basic st r ucture ele:nents . At Step 2 and 3 this set is iteratively improved . The algorithm is as follows: Step 1 . Every free node is assi gned to that seed fo r which the strength of connection between them is :naximal. Nodes , for which the strengths of connections with al l the seeds a r e equal zero , are distributed at random . Step 2 . Every free node is assigned to that basic structure ele:nent for wh i ch the strength of connection between tre:n is 'llaximal . Step 3 . Step 2 is re~eated unt il it results In the increase in pl(W) or decrease in pe ( W). It can be seen that the algo r i th:n presented makes use of the sa.'ne principle as sequ ential procedures of the cluster analysis (Duran and Odell , 1974) . Hence it can be applied , as procedu r es ment i oned , in cases when :n is large ( several thousands or mor e) .
ALGORITHM FOR THE SYNTHES I S OF COMPLEX STRUCTURE ELEMENTS The S- 04 prcg r B..'ll is based on an a1gor ith.'ll for the synthes i s of plex structure elements . It :nakes poss i ble to determ i ne " the ideal" ( i . e . optimal wi th r espects to an adopted per fomance index and agg r eg at ion p rinc iple) I'g gr eg at ion of simple structur es ele:nent s into comp l ex~ ones . As a result of agg r egation
=
To app ly this algori th.'ll , the following so ur ce data a r e to b e known : (i) The number n ments.
I T( k»T , k =l
where T ( k) i s the one-year labour consunp tion as~ociated with a given seed.
and is denoted by pe(W) . The following relations hold :
w
n it
(14)
Fig . 2 . The intraelement and interele:nent strengths of connections .
pi(W It) =ruax pi(W)
( ii) The :nax i :nal one- year labour consu.'Ilption T(k) , k =l, . . . , n ~; associated with each basic elEment . The following condition is to be satisfied :
of bas i c structure ele-
Comput e r Ana l y sis o f Large Or ga ni za ti on s an H-level binary tree is obtained ( Fig. 3) , whose levels correspond to consecutive ste p s of the algorit~~ . Its vertices rep r esent si:nple and co:nplex str'J.cture ele:::tents . At the h- th step of the algori t~'!l a pair of vertices of the ( h- l) - th level , for which ohe followi.n;\ ~easure p[ h- 1, ) h - l ( )] ~ h- l 1 h- 1 , _ - A \k , A ,1 ,![A ( K) ,A \ l)J - h-l h- l ' ( 15) T ( k) +T ( 1) where: l~ h~n ; k , l =l , .. . , Kh- l; k¥l assU::tes the -:laxi:nal value is joined .
Fig . 3 . fggregation of structure ele:nents .
r h-l ( k) , Ah- 1 (1) J denotes In fOJ:llula (15) PLA the strength of connection between the pair of ele:nents Ah- l ( k) and Ah- l ( l) , defined according to the relation (2) and h-l , ) ") h- l T \ k = l: t\l ; k=l , ... , K i EAh- l ( k) denotes the one- year labour con~'!lption as sociated with the ele:nent Ah-l ( k) . Fr cm the consideration presented it follows that the nunber of binary tree levels is equal to ( 17) and the mmber of vertices of the h - th level is gi ven ':Jy
~h = KO- h ; O ~ h~ n , '",r_ere KO denot es the rl"..;.:nber of ve rti ces of the ':Jotto::: ::'e'rel \ Fig .3 ) , i. e. the n un':Jer of leaves of one ':Jinary t ree . ~Q
apply -:his algoritn.-:l , the fo l lo'",ing daoa
are :'0 be knc·,.;n :
\i )
r h e se-:
Ging
:'0
0""
s-:r'.;c:'.;re ele-::en:s ::orrespor.-
511
so called hie rar chical procedures of the chr ster analysis ( Juran and Odell , 1974) .
CO:;CLU i:l II.::; RLNARKS The paper is concerned with so:ne aspects of a cO::Jputer - aided diagnostic analys i s and d: sign of i:nprovenents of co:nplex syste:ns , based on the use of the DIANA- 6 package. A :nethod of the co:nparati ve analysis of stroc ture properties constitutes the :nain topic of the paper. Using this :::tethod a tean of analysts can acco:nplish a :nultiaspect c o:::tparison of different stroctures . Three types of structures, m..'!led in the theory of :nanage:nent and erg a nization as fcr::Jal , real and ideal ( standard, optimal) are considered . The :nethod takes into account a regularity , resulting from ex perience , that errors in structure construc tion very often ste:n from inadequate organi zatio n of its basic ele:nents , i . e . work stations, and that i:nprovenent of the whole structure reduces to improve:nent of work stations and connections among them. The method :nakes it possible to identify every work station ( or :nore co:nplex structure ele:nents) with a detailed list of relatively simple operations perfc:nnf'd at this work station . It allows for satis tYing imv)rtant re quirenents of the theo r y of management and organization that functions and tasks to be performed by a given structure have to be pre cisely defined and adequately assigned. However the theory of manage:nent and erg ani zation does not offer explicit guiding ru les , which can form a basis for the structure ele:nent synthesis and const ruction of a mea sure of their :nutual connections . Hence alg ori t~llS and :neasures proposed in the par ' !' are to be considered as exa:nples , wrose practical usefulness has been verified only partially . Ft;rther efforts aimed at their verification can result in sane :nod i fications o r application of other algori tn.'TIS and :neasures . For ~ h ese reasons and restric tions on the paper voJl;.::Je a detailed j usti fication of -:leas-.:res a!1d. algorit.n.-ns proposed is omitted . :1owe're~ it see:::tS -:hat t h eir nature can be easily ;·J.esseG .
lea":es o.f :::e "8i::a!"y :ree:
,.,o= po:kl !k=:' , . .. , ,:o~ ; "
( ll
,
assc -
,
cia:e:' -,.;i::-. :::'ese
( iii )
e2e ::-~e~:s:
T~e stre:;t~
c:~
conr:ec:io!1
"::e:-..:e e :-.
every pair of different st.r'.lc:-..:re ele:7.e;.:s correspond.ir~ :'0 i.eaves 0 : :'~'i.e bi!1a !"·y :!"ee: O,, ) ~ K
P [..A
,0 t' 1 )J., "K , ..... -- 1 ,
,.'"1..
,.0 ,, .. . , ..
lKr-....:.... -J. ·
Algori th:::ts of s uc'1 type are ccu'!only \,;Sed 1"
::~::',a2.e'.-s:"i , ::: .
; ::'::1 75) . SO::Je aspects of co:np.;:er die.@;!1ostic analysis of the :nanage::Je:--,: sys-:e::Js . Control I< Cybern. , 3 - 4 ,
~a:ko, ~. ,
:::, ~ichalewski, J. Ost rowski , E . ?a:'.;szy:-;s ka , ',;. ::iosa , :-1. Stankiewicz , and ,: ,. ::::~aszak ( =-9 7:) . A ::Jet hod of co:nputeraiiei a:--,a::'ysis ani design of large scale :~,a~,age:::te~, o ir':onation syste:!ls . In A . ~ :--asza:" , :::i . ) , Syste:n Analysis Applicaor, : 0 CO::Jplex ?rogra-:J.:nes , North Holla r: ?-.;', . (o . , imsterdall . pp. 281- 287 .
51 2
F. Dr o zdz yk et al.
Michalevski , E. (198 0 ) . Co:np uter - aided optimization of large - scale infor:nation ne tvork. In S . G. T z afestas ( Ed . ) , I~~CS Meeting on SLll ulat i on of DPS- LSS , No rt h Holl a nd Pub . Co ., Allsterda'll. pp . 241248 .
Duran , B . S ., and P. 1. Odell ( 1914) . The cluster problell and preli:ninar y ideas. In 1-1 . Be dclann and H. P . llinz i ( Eds ) , Cl u ster Analysi s , A Survey , Springer- Ve rlag , Be rlin - He i delb erg - Nev York. pp. 1 9 - 32.
( .op\lighl IF .\< 1..1Igt" .~( . d t· S\ ... lnn ... \'·. 11·.... 1\\. Pol. lI HI I q ;·u
COMPUTER-AIDED COMPARATIVE ANALYSIS OF STRUCTURAL PROPERTIES OF LARGE-SCALE ORGANIZATION SYSTEMS P. Droidiyk, E. Michalewski,
J.
Ostrowski and M. Stankiewicz
Abstract . iiecently a package of canpute r programs, calle d DIANA- 6 , has been '.
INTRODUCTION
work is concerned wi th some proble.ll s of t he analysis of structural properties of manage:nent systens.
In general a practical purpose of studies of :nanage:nent systems is the dia,;nostic analysi sand d.esign of i:nproven ents of real sys te:ns. Sl
!)ES:~~:~
•
J ·t?RO'YEME!ITS •
F~ . 1.
::'--.e pa2~e J::: ..l.::A-6 i.s in:.er.d.ed. to red:lce ii :':. . i 2-..:1:.i es ji s:, ....:.r.:::.n; t.:--le ''''erk 0 f t:-:is "tea:::.. ~:-:e ·..:.~e 2: e. speci fie ::letrio.:i of so"'.Zce
The course 0 f st udies carried on with the use of the DIliliA- 6 package.
:ET",.JORK OF ElTi':ROPERATION nJFORMA. TIO;'l CON]ECTIO,IS
ia~a a2~~isi:.i=~ : :.:i2hal e~sk i, 1975) as well as :':-.e a:;p~i2a:.icr. 2:-" a cc;:).p',.;.ter :. . OY tneir ;::-::eess::.s si:;r:::-":'::ar.:.l.y red....:.ees the tL-:le ne -
The ".1 ethodolc:gy ~sed to constrc;.ct the DIAl,A6 packag e re li es on the aSS-l'nption that a ".1o:i"el of a ".1anage".1ent syste::l under in vesti "ation can be presenteci in 1:he for:n of so called network of interoperation i nfor:nation cocnec~ions ( IIC). ]ocies of this network represent rela1:ively si".1ple operations acoc:::plis hed at partic;'lar '.
eje:' :'8 28.Y:-:'; 0·...;.: :~e5e 3:·-.:~ie3 an:i ::1B-:.r..es ::-.e ·.,·::r~ 2:-' e.~.a~::S1:s ~.cre e f:""'icier::... l~.I.e '.lse :: :-. 2. .::::~.~·..::er So'::"'::";:·,,·5 :"'::r eas:,: :::.:;:ie .::...::.. i:. s of :"'''":".;::-2·:~.e:-_::: ;r.:;::se:" . :: e:-.2e :..:--.e:'r e:"':"'ec:s .: s.:-. ': :,.~ :;: ar:.:' a.'::"'::":.. -:"2:-:' :"': e:i ':: e :. . cre a:: :i..::-~ p..:.e ::"."2:-.: 9.:.:'8:': .
.:. . :.::;: :' :a=.. :;.!~:::e:'...::e :.:... a;; _:,::.n5 ::-le ==.~::.:.... - 6 :;:a2 ~~e :"'::r s:'..::i:es ei::-.e:i s.:. :.::'e 6..:'ag:1csis a:-.:' :...-::. pr~ '.e::-. e:-.: -: :.. == :,'S :e::-. s :. s S:'--!0·~; r:. :':-. ?i~ , ,::,,. ?rc·:~e-::.s &"SS0.::"8o:e': .".:' :':-. ::'--.e 8o:-.a2.j"s:s an:: i. '"":'!. prc:~,pe~ e:-.: 0:"' :::r;;8o:-.:" :s.:, :' ::-,e.: pr2cess ::iY:1Er."l ics
.:u-. :::1: netwo r k is stored in the ::lass ::le::lory
(::--~e
iia;r.csis a:-.:" i~~rc':e::e "";: of ~:-.e syste."J f·.;.nc-:icning ; 6.:-e '::"s2 -..:. ssei e s e·,.,·:""~ere .. ::atko ani others , 1?7E ; :,::".::~a'::" e~sk , 19~2) . This
o~'
a CO".1p ·..i ce r in the fnn of an appropriately organized iata base . There are two types of data base records. The firs t one contains ,
507
P. Dro~dzyk et al .
508
among others , the f ollowing data concerning nodes of an IIC netw o rk : the code of a given ex eC'J.t i ve node and i ts coded para'llcters . The second one contains dat a concerning arcs of an IIC network: the code of an executive node , the code o f a delivery node ( i . e . tha t stipp1yin:s infor:!lation to an executive node) and appropriately coded panL'llete rs of a n arc j oinin:s the::: . Denote the set of nodes of an I:C network by V={ i 1i =l , .. ,;::1} an::: -: :-:e se-: ::::" \ :::cien-:ed) ar::: s of it by S ,S ~ { ( i , j : li E \' , j E ': J , For the p..:rpose 0:' -:"e s-:r'..;c-:'..;re a na::'ysis the followi'1';; para.-:le-:e r s co!;-:air.ej i:-: t~e dat a base are '..lsed:
( i i ) A function P descr i bing the st r eIl?; th of connections 8.'llong ele:nents of the struc t ure Z is de fined as follows : l: l: p ( i , j ) , when i EA(k) j 6 A( 1) l;
!
( 2)
In this formula p ( i , j) denotes the so called strEn~th of conne c tion between the pair of nodes ( i , j) , i,j V, where p ( i , j ) =a ( i , j ) ~ ( i , j ) ;
( i ) t,( i ) - t:-:e ~:';::l::er je: er:"i~,i:-:; ::-:02 one year labo'.:r ccr:s'..:..:: p: :'C!1 2'o!"re:3~2::i :r.~ -:0 the operation represe:-:-:ej '~ y -:::e r.~je i . \' ; ( i i) f ( i ) - t~'1e rn.n ':::er injica'.: in; ::~." ::lany ti1les the operati~n represer.:ed by t~e node i E V 1 S acccnp li s t.ed j''';!'in; cne - year perio d ; ( iii ) b( i ) - the code o f an actlvlty area (delivery, planning , personel, sales et c) to which the op erat i on represented by the node i ~ V belongs ; (iv) r ( i , j) - the nunber characterizing the functional relati onsh ip between the nodes i f! V and j E V joined by the arc ( i , j )" S. It is assumed that - I , when i is subordinate to j , r(i , j) = 0 , when i is equivalent to j , { 1 , when j 1S subbordinate to i,
(3 )
I , when b ( i ) #b( j); ( 4)
a ( i , j )=
{2 , when b(i) =b \ j) ;
'f ( i , j) =
t
'
when ( i , j)
.Sand
\ j , i) CS ;
in [ f(i) , f ( j)],when (i , j) ES or ( j , i) (: S . (5)
( iii) A function R descri birg the functional relationship among ele.n er.ts of the struct ure Z is de fined in the following way :
[ r ( i , j) +r ( j , i) ] } , whe r e l ;
0 , when l =k ; k ,l=l, ... , no
(6) STRUCTURE OF A MANAGEMENT SYSTEM
It is aS~'lled that r ( i , j) =O, when ( i, j) . S.
The t riplet Z=( W, P,R) is called the ~ nagement system structure , W={A ( k)lk=l ,." n }, where n<:n , denotes the set o f struct ure el~ ments , P is a f unct i on describing the strerg th o f connectio:1s 8.'llong structure elen ents , P : W~W ~ R+, where R+ denotes the set of non ne~ati ve real nun b ers , R is a func t ion des cribing the fmctional relationshi p among structure ele:nents , R:Wx W~{ -1 , 0 ,I } , where -I R[A ( k ) , A( l )] = {
when A(k) 1S subordinate , to A( l ) , 0 , when A( k ) is equivalent to A( l) o r P!A ( k ) , A(l) 1=0 , 1, wh en A(l) is subordinate to A( k) ,
In t he ::: et ho dology used to construct the DIANA- 6 packcge the following assu:npt i ons are adopted: ( i ) ElEments A(k) . W of the structure Z are identi fied with non- overlapping sub s ets of nodes 0 f an IIC network. The following re l a ti o ns hold: A( k ) C. V ; k =l , .. . , n ; A(k) " A(l) = ~ ; k , l =l , ... , n ; k# l ;
~ A( K) =V.
( 11
COMPARATIVE ANALYSIS OF STRUCTURAL PROPERTIES The DIANA- 6 packcge contains , among others , the following four prcgrams: ( i) The S- Ol program. For a given W={A(k) 1 k =l , . . . , n} i t com~utes the strrogth if connec tion P [A( k) , A( l) J for every pair of struc ture elements , i . e . for k , l =l , ... , n . This plugr8.'ll makes use of the para.'lleters f ( i) and b ( i) char acterizing nodes of an IIC network , whose values are stored in the data b a se . Using them , the approp r iate calculations a re perfonned on the basis of the formula ( 2). ( ii) The S- 02 prcgram . For a given W={A ( k) 1 k =l , . . . , n} it evaluates the function al relationship R [A( k) , A(l )] between every pair o f st r ucture e lements, i. e . fo r k , l =l , ... , n o This prcg r a'll makes use of the param e te rs f ( i) characterizing nodes and paI13mete r s r ( i , j) describing arcs of an IIC networ k , stored in the da ta base . Using tlEm , the appropriate calculations are performed on the bas i s of the formul a (6) .
509
Computer Analysis of Large Organizations ( iii) The S-03 progra-n. I t is used to determine "the ideal" (i. e . opti:nal with respect to an adopted perfonr.ance index) basic structure ele'nents . Usually these ele.llents are constructed of work stations. The~ro gra'Il deternines the optLllal set WX={A ( k) k=l, ... ,n~ ( one- person o r :nulti-person work stations) .
I
( i v) The 8- 04 progra:n. It is used to de ter:nine " the ideal" ( i.e. optLnal with repect to an adopted peri'ormance index) aggregation tree for the basic structure ele:nents i.e. it is ai:ned at co:nbining elenents of a given set WO={ AO ( k) l k=l, ... ,n O ~ to f~ larger , co:nplex structure ele:nents (organlzational units , sections , departnents, divi sions etc ) . The S- 03 and S- 04 are nore co:npl icated . They are discussed in :nore detail in further sections . The application of the above nen tioned prograns results in data making it possible to carry on the co:nparative analysis of structural properties. The basic aspects of such an analysis are presented in Table 1 . It follows fro!!'! Table 1 that four basic types of structures can be distinguished and compared. They are usually referred to as: formal, real, :nixed and ideal . The process of compar ison is divided into two stage s . Firstly, the properties of these structures constructed of basic elements are compared. Secondly , the properties of these structures composed of more complex ele:nents, obtained as a result of aggregation of basic elenents, are co:n pared. At each stag e all three structure aspects, i. e . all the term s of the tri plet Z=< W,P,R> are analyzed. TABLE 1
Aspects of
a; H
....,::\ ~ ....,H en
.....u Ul
1l
Ul ...., c
a;
E
~
a;
"-< 0
x
a; ....,
...... C 0. a;
....,
u E E o~
::\
H ...., D
U
The ideal structure (denoted by the symbol I) can be completely defined by the use of suitable programs. In th is approach basic structure ele:nents are represented by some subs ets of nodes of an lIe network and are determined with the use of the S-03 program. COm plex ele:nents, i. e. those obtained as a result of basic elements aggregation, are fc:und out vi th the use of the s-04 prog ram . The strength of connection and functional relationship among ele:nents are co:nputed by the programs S-Ol and S-02 respectively.
Real
R(F) docunents
Mixed
R( F) documents
Ideal
I S-03
Basic ele:nent s
F doctment s
Strengt h of connection
F docunents
R S- Ol
R S-Ol
I S- Ol
Functional relationship
F docu'llents
R S- 02
R 8'-02
I 8 - 02
Co:nplex s t ructure ele'n ent s
F docunents
I 8- 04
I 8- 04
Strensth of connection
F docu:nent s
R 8- 01
8- 01
I 8-01
?'..:nc t ional relations h ip
F docunents
R 8- 02
I 8 - 02
S-02
R( F) documents
'0
a; H ::\
Elenents of the real structure (denoted by the ~llbol R) are the same as those of the formal one. They are identi fled with elements of the appropriate subsets of nodes of an lIe net work. Owing to it, the programs S-Ol and S- 02 can be used to determine the real strength of connection and functional relationship among structure ele:nents. For this purpose the knowledge of pammeters characterizing nodes and arcs of an lIe network , whose values are stored in the data base, is to be utilized.
Formal
Structure types characteristics
0
So called f=al structure ( denoted by the symbol F) constitutes the reference point for the analysis to be performed. It is a structure of a :nanage.-nent system under investig ation as it is defined by valid formal documents determining the system partition into structure ele.-nents as well as their mutual connections (relationship ) , e .g. legal reg ulations, instructions, organi zational diagrams, bills of duties etc.
comparative analysis
I~ "-<
In Table 1 the character of each term of the triplet W,P ,R is denoted by the symbol F,R or I. Under this symbol a source of informat ion on a given unit is shown.
CJ
:t is aSs.l:ned t h at basic ele::l ents of the ::Ji xed str;;.ct'Jre are the sa:ne as in the real one . To aggregate basic ele::J.ents into co:n ple.x struct.: re ones , the progra.':J. 8- 0 4 is u sed .
I
I
It should be enphasized that all the prog-
ra.llS discussed can be used not only for the analysis of the whole network , but also for examination of any segnent of it.
P. Droz dzyk et al .
510
ALGORITHM FOR THE SYNTHESIS OF BASIC STRUCTURE ELEMENTS If in the fonnula (2) , defining the strength of connections a'll mg ele:nents of the struc ture Z , it is assumed that n =l (i . e . one- ele:nent structure is cons i dered) , then the ex press ion de ter.nining the so called total strength of connections In an IIC network is obtained : P(V , V) =
~ L i ('V
L p ( i , j). j 6V
( 7)
If structu res Z with more than one ele'llent (n>l) are taken into consideration , then P(V , V) is d i vided into two co:nponents (Fig . 2). One of then is called the stren;>;th of iotraelement connections and is denoted by pl(W). The second one is called the stren;>;th of interele.'llent connections
A(2.)
.···F'l-------t':~"'" ;,~t-::'--:~t) --) A(4) \
.....
\
-..
,/ .... . .... /'
--.. ...A(~) ....\~;;\ ".. . .....
_pl(wl ---- pt(W)
w= tA(1)/Al2),Al~n
n p i (W) = l: P [A(k) , A(k)], k =l n n pe(w) = L L P [A(k) , A(l) ] k =l l ) k
(8) (9)
and pi(W) +pe(W) =p(V , V) ,
( 10)
'1e nce fo r an I I C networ k the following pro~le.'ll can be fonnulated . Given the function P and the number n * of st r ucture ele:nents d2te:rmine such a set W~ of structure elements that p i (W* ) i s maximized (or equivalently , pe(w* ) is mi nim i zed) , i. e. the follo wing relations a r e to be satisfied: or
pe (WIf ) =ru inpe ( W)
(ll)
W
In other words this proble:n consists in de - . ter.nining " the ideal " basic s t ructu r e elene nts . An algo r i th:n p r esent ed , constituting the ba-
s i s of the S- 03 prcg r am , makes it poss i ble to find a subopti'lla l so l ut i on to this probJe:n •
where :n T= l: t(i) i= l
*
( 13)
denotes the total one- year labour consu.'llpt i on assoc i ated with al l the operations performed in a 'llanage:nent syste:n analyzed , rep r esented by the nodes i (: V of an lIC network . ( iii) The composition of so called seed
The algori th.'ll assigns all " free " nodes (i . e . those which do not belong to seeds) of an IIC network to particular basic structure eleme nts. Three steps of this algorithm are to be distinguished. At each step the constraints imposed on T( k) , k =l , .. " n have to be satisfied . Step 1 resu l ts in the construction of a set of basic st r ucture ele:nents . At Step 2 and 3 this set is iteratively improved . The algorithm is as follows: Step 1 . Every free node is assi gned to that seed fo r which the strength of connection between them is :naximal. Nodes , for which the strengths of connections with al l the seeds a r e equal zero , are distributed at random . Step 2 . Every free node is assigned to that basic structure ele:nent for wh i ch the strength of connection between tre:n is 'llaximal . Step 3 . Step 2 is re~eated unt il it results In the increase in pl(W) or decrease in pe ( W). It can be seen that the algo r i th:n presented makes use of the sa.'ne principle as sequ ential procedures of the cluster analysis (Duran and Odell , 1974) . Hence it can be applied , as procedu r es ment i oned , in cases when :n is large ( several thousands or mor e) .
ALGORITHM FOR THE SYNTHES I S OF COMPLEX STRUCTURE ELEMENTS The S- 04 prcg r B..'ll is based on an a1gor ith.'ll for the synthes i s of plex structure elements . It :nakes poss i ble to determ i ne " the ideal" ( i . e . optimal wi th r espects to an adopted per fomance index and agg r eg at ion p rinc iple) I'g gr eg at ion of simple structur es ele:nent s into comp l ex~ ones . As a result of agg r egation
=
To app ly this algori th.'ll , the following so ur ce data a r e to b e known : (i) The number n ments.
I T( k»T , k =l
where T ( k) i s the one-year labour consunp tion as~ociated with a given seed.
and is denoted by pe(W) . The following relations hold :
w
n it
(14)
Fig . 2 . The intraelement and interele:nent strengths of connections .
pi(W It) =ruax pi(W)
( ii) The :nax i :nal one- year labour consu.'Ilption T(k) , k =l, . . . , n ~; associated with each basic elEment . The following condition is to be satisfied :
of bas i c structure ele-
Comput e r Ana l y sis o f Large Or ga ni za ti on s an H-level binary tree is obtained ( Fig. 3) , whose levels correspond to consecutive ste p s of the algorit~~ . Its vertices rep r esent si:nple and co:nplex str'J.cture ele:::tents . At the h- th step of the algori t~'!l a pair of vertices of the ( h- l) - th level , for which ohe followi.n;\ ~easure p[ h- 1, ) h - l ( )] ~ h- l 1 h- 1 , _ - A \k , A ,1 ,![A ( K) ,A \ l)J - h-l h- l ' ( 15) T ( k) +T ( 1) where: l~ h~n ; k , l =l , .. . , Kh- l; k¥l assU::tes the -:laxi:nal value is joined .
Fig . 3 . fggregation of structure ele:nents .
r h-l ( k) , Ah- 1 (1) J denotes In fOJ:llula (15) PLA the strength of connection between the pair of ele:nents Ah- l ( k) and Ah- l ( l) , defined according to the relation (2) and h-l , ) ") h- l T \ k = l: t\l ; k=l , ... , K i EAh- l ( k) denotes the one- year labour con~'!lption as sociated with the ele:nent Ah-l ( k) . Fr cm the consideration presented it follows that the nunber of binary tree levels is equal to ( 17) and the mmber of vertices of the h - th level is gi ven ':Jy
~h = KO- h ; O ~ h~ n , '",r_ere KO denot es the rl"..;.:nber of ve rti ces of the ':Jotto::: ::'e'rel \ Fig .3 ) , i. e. the n un':Jer of leaves of one ':Jinary t ree . ~Q
apply -:his algoritn.-:l , the fo l lo'",ing daoa
are :'0 be knc·,.;n :
\i )
r h e se-:
Ging
:'0
0""
s-:r'.;c:'.;re ele-::en:s ::orrespor.-
511
so called hie rar chical procedures of the chr ster analysis ( Juran and Odell , 1974) .
CO:;CLU i:l II.::; RLNARKS The paper is concerned with so:ne aspects of a cO::Jputer - aided diagnostic analys i s and d: sign of i:nprovenents of co:nplex syste:ns , based on the use of the DIANA- 6 package. A :nethod of the co:nparati ve analysis of stroc ture properties constitutes the :nain topic of the paper. Using this :::tethod a tean of analysts can acco:nplish a :nultiaspect c o:::tparison of different stroctures . Three types of structures, m..'!led in the theory of :nanage:nent and erg a nization as fcr::Jal , real and ideal ( standard, optimal) are considered . The :nethod takes into account a regularity , resulting from ex perience , that errors in structure construc tion very often ste:n from inadequate organi zatio n of its basic ele:nents , i . e . work stations, and that i:nprovenent of the whole structure reduces to improve:nent of work stations and connections among them. The method :nakes it possible to identify every work station ( or :nore co:nplex structure ele:nents) with a detailed list of relatively simple operations perfc:nnf'd at this work station . It allows for satis tYing imv)rtant re quirenents of the theo r y of management and organization that functions and tasks to be performed by a given structure have to be pre cisely defined and adequately assigned. However the theory of manage:nent and erg ani zation does not offer explicit guiding ru les , which can form a basis for the structure ele:nent synthesis and const ruction of a mea sure of their :nutual connections . Hence alg ori t~llS and :neasures proposed in the par ' !' are to be considered as exa:nples , wrose practical usefulness has been verified only partially . Ft;rther efforts aimed at their verification can result in sane :nod i fications o r application of other algori tn.'TIS and :neasures . For ~ h ese reasons and restric tions on the paper voJl;.::Je a detailed j usti fication of -:leas-.:res a!1d. algorit.n.-ns proposed is omitted . :1owe're~ it see:::tS -:hat t h eir nature can be easily ;·J.esseG .
lea":es o.f :::e "8i::a!"y :ree:
,.,o= po:kl !k=:' , . .. , ,:o~ ; "
( ll
,
assc -
,
cia:e:' -,.;i::-. :::'ese
( iii )
e2e ::-~e~:s:
T~e stre:;t~
c:~
conr:ec:io!1
"::e:-..:e e :-.
every pair of different st.r'.lc:-..:re ele:7.e;.:s correspond.ir~ :'0 i.eaves 0 : :'~'i.e bi!1a !"·y :!"ee: O,, ) ~ K
P [..A
,0 t' 1 )J., "K , ..... -- 1 ,
,.'"1..
,.0 ,, .. . , ..
lKr-....:.... -J. ·
Algori th:::ts of s uc'1 type are ccu'!only \,;Sed 1"
::~::',a2.e'.-s:"i , ::: .
; ::'::1 75) . SO::Je aspects of co:np.;:er die.@;!1ostic analysis of the :nanage::Je:--,: sys-:e::Js . Control I< Cybern. , 3 - 4 ,
~a:ko, ~. ,
:::, ~ichalewski, J. Ost rowski , E . ?a:'.;szy:-;s ka , ',;. ::iosa , :-1. Stankiewicz , and ,: ,. ::::~aszak ( =-9 7:) . A ::Jet hod of co:nputeraiiei a:--,a::'ysis ani design of large scale :~,a~,age:::te~, o ir':onation syste:!ls . In A . ~ :--asza:" , :::i . ) , Syste:n Analysis Applicaor, : 0 CO::Jplex ?rogra-:J.:nes , North Holla r: ?-.;', . (o . , imsterdall . pp. 281- 287 .
51 2
F. Dr o zdz yk et al.
Michalevski , E. (198 0 ) . Co:np uter - aided optimization of large - scale infor:nation ne tvork. In S . G. T z afestas ( Ed . ) , I~~CS Meeting on SLll ulat i on of DPS- LSS , No rt h Holl a nd Pub . Co ., Allsterda'll. pp . 241248 .
Duran , B . S ., and P. 1. Odell ( 1914) . The cluster problell and preli:ninar y ideas. In 1-1 . Be dclann and H. P . llinz i ( Eds ) , Cl u ster Analysi s , A Survey , Springer- Ve rlag , Be rlin - He i delb erg - Nev York. pp. 1 9 - 32.