IIVG P a p e r s
V e r ö f f e n t l i c h u n g s r e i h e d e s I n t e r n a t i o n a l e n I n s t i t u t s f ü r V e r g le i c h e n d e G e s e l l c h a f t s f o r s c h u n g
W is s e n s c h a f ts z e n tr u m B e r l i n
IIVG/dp 79-5
D e c e n t r a l i z a t i o n : A New A n a l y t i c A p p ro a c h
by
M a n fre d Kochen > a n d K a r l W. D e u ts c h U n i v e r s i t y o f M ic h ig a n H a r v a r d U n i v e r s i t y
I n t e r n a t i o n a l I n s t i t u t e f o r C o m p a ra tiv e S o c i a l R e s e a r c h S c ie n c e C e n t e r B e r l i n
P r e p a r e d f o r d e l i v e r y a t t h e Moscow IPSA C o n g r e s s o f A u g u s t 1 2 -1 8 , 1 97 9 . C o p y r i g h t @ 1979 I n t e r n a t i o n a l
P o l i t i c a l S c i e n c e A s s o c i a t i o n
P u b l i c a t i o n s e r i e s o f t h e I n t e r n a t i o n a l I n s t i t u t e f o r C o m p a ra tiv e S o c i a l R e s e a r c h
W i s s e n s c h a f t s z e n t r u m B e r l i n
D ecen tralizab io n : A New A nalytic Approach.
Manfred Kochen and Karl W. Deutsch
U n iv ersity o f Michigan Harvard U n iv ersity and WissenschaftsZentrum Berlin
1. In tro d u ctio n
Human needs a re becoming more complex. Human se rv ic e o rg a n iz a tio n s , such as goveruments, c l i n i c s , l i b r a r i e s , a re becoming more complex a ls o . I t i s u se fu l to in v e s tig a te whether o rg a n iz a tio n a l s tr u c tu r e a f f e c ts perform ance, and i f so , how. The e x te n t to which o rg a n iz a tio n a l reso u rce s a re d is tr ib u te d i s an asp ect o f s tr u c tu r e th a t i s o f i n t e r e s t to many managers and re s e a rc h e rs . (Kochen and Deutsch, 1969, 1979). Proponents o f c e n tr a liz a tio n have argued p a s s io n a te ly th a t a l l o rg a n iz a tio n a l reso u rce s and c o n tro l ought to be c o n c e n tra te d . Advocates o f d ece n tra lism favor th e w idest p o ss ib le d i s t r ib u t i o n o f c o n tro l and re s o u rc e s. Both have argued th a t tre n d s in technology and p a tte rn s o f c u ltu r e favor t h e i r p o s itio n . We explore in t h i s paper what l i g h t the a n a ly s is o f sim ple m athem atical models can shed on th ese d is c u s s io n s .
I t i s not d i f f i c u l t to reach consensus about th e g o a ls o f c e r ta in human se rv ic e o rg a n iz a tio n s when th ese a re not expressed a t a very high le v e l o f s p e c i f i c i t y . An accep tab le norm ative statem ent i s t h a t the goal should be to ensure adequate long-term n e t b e n e f its to a la rg e community a t th e same tim e t h a t no one f a l l s below acc ep tab le l im i ts in the q u a lity o f key s e rv ic e s they need. Consensus among managers and c li e n t s and s e rv ic e p ro v id ers i s le s s l ik e l y when i t comes to the meaning and in te r p r e ta tio n o f "lo n g -term ", "n e t b e n e f its " , "ad eq u ate",
" la rg e community", "ac c ep tab le lim its " and "key s e r v ic e s " . Yet, su c c e ssfu l o rg a n iz a tio n s cope w ith th e ta s k s o f providing s e r v ic e s . I t i s th e re fo re more f r u i t f u l to focus in v e s tig a tio n on th e p ro cess by which they cope in th e presence o f such lac k o f consensus and d iv e r s ity in outlook and v alu es a t th e needed le v e l o f s p e c i f i c i t y . This process i s governed by o rg a n iz a tio n a l s tr u c tu r e and m anagerial pro ced u res.
To d e riv e a r e s u l t w ith our a n a ly s is , we assume t h a t th e se rv ic e o rg a n iz a tio n s o f i n t e r e s t aim p rim a rily to respond to the needs o f t h e ir c l i e n t s . We do not claim t h a t a l l s e rv ic e o rg a n iz a tio n s should pursue t h i s aim. Nor do we a s s e r t th a t any a c tu a l o rg a n iz a tio n s do pursue i t . We b e lie v e t h a t because c l i e n t s become in c re a s in g ly informed and aware o f what q u a lity o f s e rv ic e can be provided, and because th ey r e a liz e th a t they need s e t t l e fo r no le s s than th e b e s t, th e re fo re they w ill bring p ressu re to bear on s e rv ic e o rg a n iz a tio n s to ■ meet t h e i r needs.
But we do n o t a s s e r t t h i s as a co n clu sio n . We simply ex p lo re to what e x te n t c e r ta in id e a liz e d o rg a n iz a tio n s would d e c e n tra liz e i f they pursued t h is goal in some s p e c if ic c o n te x ts .
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2. Problem F o rm u la tio n
We regard human s e rv ic e o rg a n iz a tio n s as systems th a t transform inputs in to o u tp u ts according to a " s e rv ic e production fu n c tio n ". In the follow ing ta b le , we i l l u s t r a t e what we mean by in p u ts and o u tp u ts with th re e examples.
Example o f System Government (A branch o f the
Executive arm) Medical C lin ic L ibrary or Inform ation Center
Some Outputs Executive Actions
P o lic ie s
Medical Judgments Treatm ents
Answers to Q uestions Documents
Some Inputs Funds Man-hours
Space
S p ecialized Knowledge Communications
Documents
Each o f the six in p u ts l i s t e d in th e th ir d column are required fo r each o f the th re e system s, and many more. Executive a c tio n s include d e cisio n s to accept or r e j e c t p ro p o sals subm itted to a government agency. .Another kind o f ex ecu tiv e a c tio n by such an agency i s a req u e st for more inform ation about a p ro p o sal; another i s th e d ecisio n not to take a p o sitio n on an issu e o r p ro p o sal. An im portant kind o f executive actio n i s to i n s t r u c t se le c te d persons to implement c e r ta in d e c isio n s;
another i s to form ulate a p p ro p ria te p o l ic i e s .
Medical judgments p e rta in to the d ia g n o s is , p ro g n o sis, prevention and treatm en t o f c o n d itio n s t h a t may a f f l i c t th e c l i e n t s o f a medical se rv ice o rg a n iz a tio n . To r e l a t e th ese o u tp u ts and in p u ts , i t i s necessary to sp e c ify them as v a ria b le s . Thus, fo r a c l i n i c , we may c h a ra c te riz e output le v e ls by th e number o f m edical judgments o f a c e rta in q u a lity t h a t a re made per month. For a governmental agency, i t may be th e number o f e x ecu tiv e d e c is io n s a t a c e r ta in le v e l o f responsiveness t h a t are made per month. For a l i b r a r y , output le v e ls may be s p e c ifie d by th e number o f q u e stio n s per month th a t are adequately answered, or th e expected number o f needed books th a t are d eliv ered by th e time they a re needed.
Person-hours and space a re r e a d ily s p e c ifie d in term s o f r a tio n a l numbers and square m eters, w hile funds may be c h a ra c te riz e d by th e number o f , say , d o l la r s , t h a t are budgeted and a v a ila b le fo r spending per month. S p ecialized knowledge may be measured in b i t s o f organized, r e tr ie v a b le memory. Communications a re c h a ra c te riz e d by who communicates w ith whom ( l a t e r a l l y , v e r t i c a l ly ) and how fre q u e n tly .
The fu n c tio n a l r e la tio n between ou tp u t and in p u t v a ria b le s cannot be expressed u n t i l the param eters t h a t d e sc rib e th e o rg a n iz a tio n a l s tru c tu re are e x p lic a te d . I f we a re ab le to w rite an a lg eb raic expression fo r some performance measure to be optim ized (say maximized w ithout v io la tin g c o n s t r a i n t s ) , then we can find those values o f the s tr u c tu r a l param eters th a t optim ize performance fo r given input le v e ls ,
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and to estim ate the d ir e c tio n in which th ese with changes in th e input le v e ls .
p a ra m e te r v a lu e s change
To c h a ra c te riz e a s tr u c tu r e we s t a r t w ith i t s components. Since we are d ealin g w ith human se rv ic e o rg a n iz a tio n s , we sp e cify f i r s t o f a l l the s e t o f c l i e n t s or custom ers, C, and secondly th e s e t o f persons in the se rv ic e o rg a n iz a tio n , P. I t i s in a p p ro p ria te to c a l l th e se two s e ts components, sin c e they a re the essence o f th e o rg a n iz a tio n . Tne th ir d s e t needed to s p e c ify an o rg a n iz a tio n i s n o t a component e i t h e r ; i t i s the s e t o f a l l p o in ts in a th ree-d im en sio n al space a t which persons and equipment can be lo c a te d . C all t h i s E, fo r Euclidean space. Next, we consider the s e t F o f fu n c tio n a l s p e c i a lt i e s in which th e s e rv ic e pro v id ers have e x p e r tis e . Let D be th e s e t o f document f i l e s n ecessary to record the o rg a n iz a tio n ’ s b u sin ess and p o lic ie s . F in a lly , l e t M be the s e t o f machines and item s o f equipment.
In b r i e f , we c h a ra c te riz e an o rg a n iz a tio n p a r t i a l l y by th e s e t (C,P,E,F,D,M ). We must now s p e c ify r e l a ti o n s among th ese ’’components"
and between them and input v a ria b le s . The sim p lest in p u t v a ria b le i s m : th e number o f person-hours needed to render a s e rv ic e . This i s to be d is tr ib u te d over P. The most c e n tr a liz e d d is tr ib u tio n i s to a llo c a te a l l the m hours to j u s t one person in th e s e rv ic e o rg a n iz a tio n , assuming th a t he or she can do i t a lo n e. The most d e c e n tra liz e d d i s t r ib u t i o n i s to have each o f n s e rv ic e p ro v id ers spend m/n hours on tlja t s e rv ic e (Presumably th ey spend the remainder o f t h e i r tim e on o th e r s e r v i c e s ) .
I t i s p la u s ib le th a t a s e rv ic e p ro v id er who spends a l l h is tim e on j u s t one s e rv ic e becomes q u ite competent and f a s t a t t h a t , w hile those who spend but a f r a c tio n o f t h e i r tim e do not render s e rv ic e o f as high a q u a lity in t h a t s p e c ia lty and they do i t more slow ly. Thus, i f m hours would be req u ired o f one f u ll- tim e s p e c i a l i s t , more than m hours would be req u ired i f th e ta s k were d is tr ib u te d over n s e r v e r s , th u s in creasin g the c o s t. There may be compensating b e n e f its , as shown in the next s e c tio n , and the optimum value o f n may be somewhere between 1 and an upper extrem e.
The d i s t r ib u t i o n o f th e a v a ila b le number o f p erso n -h o u rs, as an in p u t, over th e persons in the o rg a n iz a tio n , P, may be termed p lu r a l iz a ti o n , w ith n, th e number o f persons over whom th e m hours a re d is tr ib u te d , being the degree o f p l u r a l iz a ti o n .
Next, we co n sid er the mapping from M, th e s e t o f d e v ic e s , in to E, the s e t o f lo c a tio n s , or th e assignm ent o f a s i t e to each item o f equipment. This d i s t r ib u t i o n can a lso vary from one t h a t i s very concentrated w ith a l l item s on to p o f one another (s u b je c t to con
s t r a i n t s about no two bodies occupying th e same volume, o r a minimum d istan c e between item s, e t c . ) to one th a t is very d is p e rs e d . D ispersion i s a second asp ect o r dimension o f d i s t r ib u t i o n or d e c e n tr a liz a tio n . Documents and p erso n s' work s t a ti o n s can a ls o be d is tr ib u te d over space in co n cen trated or more d isp ersed ways, and i f the d is tr ib u tio n i s rep resen ted as Poisson, fo r example, then one parameter c h a ra c te riz e s i t ; i f as G aussian, then two param eters a re re q u ire d , e tc . An im portant idea to be explored in a s e p a ra te study i s based on the c la s s i f i c a t i o n o f work spaces in to public and p r iv a te , and
u
according to the fra c tio n o f tim e th a t workers spend in th ese s ta tio n s ; t h i s has im p lica tio n s fo r what tr a n s p o rta tio n and communication
are most a p p ro p ria te and most c o s t - e f f e c t iv e .
systems A th ir d dimension or a sp e c t o f d e c e n tr a liz a tio n (th e g en eric term
" d is tr ib u tio n " is p re fe ra b le to " d e c e n tr a liz a tio n " , which i s a t one end o f the scale) th a t we have considered i s th e degree o f fu n ctio n al s p e c ia liz a tio n . We may view t h a t as a mapping from P to F, th e s e t o f fu n ctio n al s p e c i a lt i e s . We should include among the s e t o f s p e c i a lt i e s an element th a t i s th e union o f s p e c i a l t i e s , so th a t i t may be more a p p ro p ria te to i n te r p r e t F as th e power s e t o f th e s e t o f elem entary s p e c i a lt i e s . Thus, i f urology, nephrology,. neurology, o to rhinolaryngology, e t c . a re elem entary m edical s p e c i a lt i e s , then the union o f sev eral o f th ese may be in te r n a l m edicine, and th e union o f a l l may be general or fam ily p r a c tic e in some sen se. I f every s e rv ic e provider in the o rg a n iz a tio n were assigned to th e same element o f F, which would presumably be g en eral p r a c tic e , then we would have a fu n c tio n a lly c e n tra liz e d s t r u c tu r e . A lte rn a tiv e ly , i f each se rv ic e provider were a s p e c i a l i s t and spent a l l h is tim e in an elem entary s p e c ia lty , we would have a c e n tra liz e d s tr u c tu r e in a d i f f e r e n t sense.
A fo u rth dimension we stu d ied was t h a t o f h ie r a r c h ic a l f la tn e s s . We regarded a f l a t s tr u c tu r e to be more d e c e n tra liz e d than a t a l l one with many le v e ls o f h ie ra rc h y . We i n te r p r e t t h i s h ere as a mapping or d is tr ib u tio n o f a t o t a l sum o f money fo r supporting the se rv ic e o rg an iz atio n over the persons in P. I f a l l o f i t i s concentrated in one person, who presumably a llo c a te s i t to su b o rd in a te s, who in tu rn a llo c a te t h e i r budgets to t h e i r su b o rd in a te s, then we have a very t a l l , c e n tra liz e d h ie ra rc h y . I f i t i s divided e q u ally among a l l the persons in P, a le s s t a l l h ie ra rc h y emerges, depending upon how much each o f th e s e rv ic e p ro v id ers chose to spend from t h e i r sh are on co o rd in atio n and m anagerial c o n tro l.
I f k b i t s o f sp e c ia liz e d knowledge a re req u ired to render adequate s e rv ic e - an input v a ria b le - , then t h a t can be concentrated in one person or P can co n tain persons each o f whom knows som ething. Even in th e extreme case in which knowledge i s d is tr ib u te d uniform ly, th e re can be v a ria tio n s from the case where everyone knows th e same th in g , which may not be enough to cope w ith th e more r a r e and e x ce p tio n a lly complex c a se s, to where everyone has unique knowledge th a t complements everyone e l s e 's . This would r e s u l t in a r e f e r r a l netw ork. I f i t i s la r g e , co o rd in a to rs are re q u ire d . G en erally , th e re are some who know more than o th e rs - a t l e a s t about s p e c if ic cases - and they d e le g a te asp ects o f such cases to o th e r s . This has led us to co n sid er d e le g a tio n as another im portant dimension o f d e c e n tr a liz a tio n .
Communication as an in p u t v a ria b le could be measured by th e number o f hours per req u e st th a t must be spent on h o riz o n ta l communication and the numbers o f hours needed in v e r t i c a l communication, between super
v is o rs and t h e ir su b o rd in a te s, and/or c l i e n t s . I f f i s th e t o t a l number o f hours required per re q u e st in e ith e r kind o f communication, then the mapping o f f in to the union o f C and P can be in te rp re te d as the degree o f feedback. I f a l l o f f i s used up by members o f the se rv ic e o rg an izatio n communicating w ith one a n o th er, t h a t lea v es no time fo r
'S
communicating w ith c l i e n t s , and the degree o f feedback i s 0. I f i t is a l l d is tr ib u te d to c l i e n t s , then th e re i s a g r e a t deal o f feedback, but probably very l i t t l e co o rd in atio n w ith in th e o rg a n iz a tio n . Tne e x te n t o f resp o n siv e communication with c l i e n t s i s another o f our dimensions o f d e c e n tr a liz a tio n .
The l a s t dimension o f d e c e n tr a liz a tio n t h a t we have considered is th e degree o f p a rtic ip a tio n both in d e c isio n making and in o rg a n iz a tio n a l red e sig n . With regard to p a r tic ip a tio n in decision-m aking, consider d , the number o f d e c is io n s made per month, an output v a ria b le . We in te r p r e t th e e x te n t o f p a r tic ip a tio n as the d is tr ib u tio n o f d over the union o f C and P. I f t h a t d is tr ib u tio n i s concentrated in one person - se rv ic e p ro v id er o r c l i e n t then we have a c e n tra liz e d s tr u c tu r e ; i f i t i s spread over many p eople, i t i s p a r t i c i p a ti o n a l l y d e c e n tra liz e d . As in th e case o f d is tr ib u tin g m over P, e n tir e d e c is io n s may be made by many people w ith each d e c isio n made by one person, o r many people may p a r t i c i p a te to a sm all e x te n t in making each d e c is io n .
In e a r l i e r papers we rep o rted t h a t under th ese c o n d itio n s an in crease in p lu r a liz a tio n tends to be supported by c o n sid e ra tio n s o f c o st e ff e c tiv e n e s s . I f the s p a tia l d is ta n c e among c li e n t s in c re a s e s , and/or i f th e number o f req u e sts per month grows f a s te r than the speed o f tr a n s p o rta tio n or communication-, then i t pays to p lu r a liz e , to in cre ase th e number o f se rv ice p ro v id e rs . I t a lso pays to p lu r a liz e i f th e c o s ts o f lab o r r i s e f a s te r than th e c o s ts o f c a p ita l (Kochen and Deutsch, 1969-1973). A dditional r e s u l t s in regard to the f la tte n in g o f h ie ra rc h ie s and in c re a se s in d e le g a tio n o f ta s k s and reso u rces downward were rep o rted in subsequent p ap ers.
Such downward d ele g atio n and th e f la tte n in g o f h ie ra rc h ie s were found to be favored by a cheapening o f management s k i l l s and/or com putational re s o u rc e s . (Kochen and Deutsch, 1974-1977). In the p resen t p ap er, using a somewhat d i f f e r e n t m athem atical approach, we in v e s tig a te th e e f f e c t o f an in crease in th e s iz e and com plexity o f s e rv ic e and hence o f th e average amount o f tim e needed to s e rv ic e a re q u e s t, as w ell as th e d i f f i c u l t y o f s e rv ic e s (which v a rie s in v e rs e ly w ith the p ro b a b ility o f success) and a low ering o f th e c o st o f th e s e rv ic e p ro v id e rs ’ tim e. All th ese a re found to favor d e c e n tra liz a tio n as shown in what fo llo w s.
3. A nalysis o f a Simple Case.
For com pleteness, we summarize the main v a ria b le s :
m = Number o f hours required to provide a needed sp e c ia liz e d se rv ic e n = Number o f s e rv ic e p roviders assigned to provide th e needed
s e rv ic e
p = P ro b a b ility o f success in rendering needed s e rv ic e by a provider u = U t i l i t y o f s e rv ic e th a t i s s u c c e s s fu lly rendered to- a c l i e n t
($ per req u e st)
c = Cost o f m aintaining a se rv ic e p ro v id e r, ($ per h o u r).
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We now in tro d u ce se v e ra l sim p lify in g assum ptions.
(1) All n s e rv ic e p ro v id ers succeed or f a i l independently o f one a n o th er, and w ith the same p r o b a b ility . We assume th a t th e needed se rv ic e re q u ire s no c o o rd in a tio n , such as each o f n d o c to rs try in g independently to diagnose a co n d itio n or each o f n l i b r a r i a n s try in g to find a needed book in t h e i r c o lle c tio n s
I t follow s imm ediately t h a t th e p ro b a b ility w ith which th e se rv ic e o rg an iz atio n o f n such s e rv e rs succeeds in providing the s e rv ic e i s 1 - P ro b a b ility (not a l l n s e rv e rs f a i l ) or
As n grows la r g e r , t h i s in c re a s e s toward 1.
(2) The number o f hours t h a t each s e rv ic e provider spends i s = m/n + m^
We in te r p r e t nu as th e minimal time a se rv er must spend per req u e st no m atter now many o th e rs he sh ares th e load w ith . I f m^ = 0, then each o f th e n persons works on n/m d i f f e r e n t s p e c ia liz e d s e rv ic e s to s a t is f y t h a t many sim ultaneous re q u e sts to remain f u lly occupied. The load o f re q u e s ts per u n it time t h a t i s j u s t equal to the c a p a c ity o f persons i s n/m re q u e sts per hour. With nonzero m_ only the f ra c tio n 1/(1+m_n/m) o f th e n h o u rs' worth o f work t h a t th e o rg an iz atio n was capable o f w ith a l l persons f u l ly occupied w ill now g e t done. Hence n(1+mnn/m) persons a re now needed to cope w ith th e same load as b e fo re , c o stin g nc( 1+rrun/m) d o lla r s /h o u r . Consequently, th e n et u t i l i t y can be expressed a s:
U = u (1 -(1 -p )n)n/m-nc(1+mon/m) =
u (1 -e - Pn )n/m-nc(1+mgn/m)($/hour) Eq. (1)
Eq. (1) i s sim ple to analyze i f th e n-person ta s k fo rce has to s e rv ic e j u s t one req u e st per hour, re q u irin g m person-hours to s e r v ic e . Then the load f a c to r , n/m, in th e f i r s t term on th e rig h t-h an d s id e ^ is replaced by 1.
Assume t h a t U as a fu n ctio n o f n i s continuous and tw ice d i f f e r e n t i a b l e . D if f e r e n tia te with re s p e c t to n and s e t th e d e riv a tiv e equal to 0 to o b tain th e value o f n t h a t maximizes U. I f we s e t mQ=0 as a f i r s t approxim ation, we can solve the r e s u ltin g equation e x p li c i t ly fo r n to o b tain
n = (1 /p )ln (p u /c ) I f m^ = 0 , then we o b tain
upe_Pn = c(1+2mgn/ra)
Eq. .(2)
Eq. (3 )
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I t i s easy to see t h a t t h i s reduces to Eq. (2) when mg=0. While we cannot solve t h i s tra n s c e n d e n ta l equation e x p l i c i t l y fo r n, we observe th a t the l e f t hand sid e d ecreases as a n eg ativ e ex ponential with n. The r ig h t hand s id e in c re a s e s l in e a r l y with n, w ith a slope th a t d ecreases toward the h o riz o n ta l as m in c re a s e s . The optimum value o f n is th a t p o in t where th e two curves i n t e r s e c t . Geometric c o n sid e ra tio n s perm it us to conclude t h a t t h i s p o in t o f in te r s e c tio n s h i f t s to the r ig h t as m in c re a s e s , because th e s t r a ig h t l in e becomes more h o riz o n ta l, w hile the negative ex ponential sta y s fix e d .
I f we i n te r p r e t n as th e degree o f p lu r a liz a tio n (our f i r s t dimension o f d e c e n tr a liz a tio n ) i t follow s t h a t in cre asin g m, th e number o f hours needed to s e rv ic e a re q u e s t, fav o rs d e c e n tr a liz a tio n . I f th e re i s a trend toward more complex s e rv ic e re q u e s ts , i . e . needs t h a t take more time to s a t i s f y , then o rg a n iz a tio n s th a t aim to be responsive in the sense o f maximizing n et u t i l i t y should employ more se rv ic e p ro v id ers. I f th e u t i l i t y o f th e s e rv ic e in c re a s e s , t h a t a lso favors increased d e c e n tr a liz a tio n . Increased c o s ts o f th e s e rv ic e p ro v id ers.
I f the u t i l i t y o f th e s e rv ic e in c re a s e s , t h a t a lso favors increased d e c e n tr a liz a tio n . Increased c o s ts o f th e s e rv ic e p ro v id ers and an in creasin g success p r o b a b ility , on th e o th e r hand, favor c e n tr a liz a tio n , in the sense o f causing th e optimum n to d e c re a se .
In our e a r l i e r p ap ers, a key v a ria b le (fo r which we had also used the l e t t e r c) was th e o p e ra tin g c o st o f d e liv e rin g the s e rv ic e , which included th e o p p o rtu n ity c o s t o f th e c l i e n t ’s tim e w hile he was w aiting id ly , such as p a tie n ts in a d o c to r’s w aiting room. We had found th a t the most c o s t- e f f e c tiv e degree o f p lu r a liz a tio n d ecreases as th a t c o st in c re a s e s . I f th e wages o f a s e rv ic e provider a re h ig h , i t pays to c e n tr a liz e . I f th e re i s a scarce s e rv ic e based on a r a r e s k i l l , such as h ig h ly sp e c ia liz e d su rg ery , i t may pay to d e c e n tr a liz e . Such r a r e se rv ic e s k i l l s a re analogous to fixed c a p i t a l . I f th e g en eral wages o f a l e t t e r c a r r i e r in c re a s e , on th e o th e r hand, i t may pay to c e n tr a liz e .
I f p = .9 r u=1000 and c=10, then n, computed according to Eq. ( 2 ) , i s approxim ately 4. This v e r i f i e s our i n tu i t io n th a t th e re i s an optimum degree o f p lu r a liz a tio n ly in g between th e extreme o f having one person handle th e e n ti r e load o f th e one re q u e st req u irin g m hours (assuming th a t he could w hile th e o th e r n-1 persons s i t by id ly ) and the o th er extreme o f having very many persons w ith each one spending s l i g h tl y more than m^ hours on the re q u e s t.
I f we d i f f e r e n t i a t e U in Eq. ( 1 ) , w ith n/m not replaced by 1, w ith re s p e c t to n and s e t the d e riv a tiv e equal to 0, we o b ta in :
u(1+pne_pn) = c(nn-2mQn)+ue-pn Eq. (4)
The le ft-h a n d sid e may be in te r p r e te d as th e m arginal in crease in the expected u t i l i t y o f s e rv ic e s rendered due to adding 'th e l a s t s e rv ic e provider (except fo r th e term ue- Pn , th e expected d i s u t i l i t y when a l l n persons f a i l , which i s transposed to the rig h t-h an d s id e .) The right-hand sid e may be in te rp re te d as th e m arginal in crease in expected
3
COST ue-p n
!ue to adding the l a s t s e r v ic e p r o v i d e r , e x c e p t f o r ?rrn- F ig . 1 shows a p lo t o f the le ft-h a n d s id e in dashed l i n e s ( ---- ) and the rig h t-h an d sid e in a so lid l i n e .
A
The s o lid and th e dashed l in e w ill n o t i n t e r s e c t a t a l l i f th e low est p o in t o f th e s o lid curve i s above th e peak o f th e dashed cu rv e, i . e . when:
cm + 2cmg/p + cmg/pln(up/2cmg) >U + 1 /e . Eq. (5)
The term cm i s th e c o st o f providing th e s e rv ic e to s a t is f y a re q u e s t.
I f we take m to be 50 and c=10 and u=1000, as above, then t h a t c o s t i s
$500 w hile th e s e rv ic e i s valued a t $1000. I f m0 , th e l e a s t time t h a t a s e rv ic e provider must spend on th e re q u e st i s 1 hour, and p=.9 as above, th en :
cm + 2cmg/p + cmQ/pln(up/2cmg) = 564,
s t i l l le s s than u + 1 /e , which i s about 1000.4.
There may be j u s t one s o lu tio n fo r n when the in e q u a lity in t Eq.
(5) i s rep laced by e q u a lity . This o c c u rs , fo r example, when m is increased from 50 in the above example to near 94 hours per re q u e s t.
The minimum o f th e s o lid curve occurs a t n = (1 /p )ln (u p /2 cm g ).
The maximum o f th e dashed curves occurs a t n = 1/p.
For the above numbers 1/p = 1 .1 and ln(up/2cm Q) = 3 .8 .
4 q
Thus, when the bottom o f th e s o lid curve i s a t th e same le v e l as th e top o f the dashed cu rv e, th e two curves are no longer ta n g e n t. That w ill occur a t a value below m - 94. Below t h a t , th e re are two valu es o f n th a t maximize u t i l i t y . The sm aller is due to the a r t i f a c t o f tr e a tin g n as continuous. We are in te r e s te d , th e r e f o re , in th e la r g e r one. 'That value o f n, n^ in F ig . 1 corresponding to p o in t A, w ill in cre ase - p o in t A w ill s h i f t to the r ig h t -
as m d e crea se s, because the s o lid curves s h i f t s down;
as c d e crea se s, fo r th e same reason
as m^ d e crea se s, because th e s o lid curve i s ro ta te d to the r i g h t , being asym ptotic to a l in e with slope 2cmn .
as u in c re a s e s , because th e dashed curve s h i f t s up f a s t e r than the so lid curve.
In o th er words, p lu r a liz a tio n or d e c e n tr a liz a tio n i s favored by le s s complex, more ro u tin e b u t more v alu ab le re q u e sts tak in g le s s tim e, by se rv ic e p ro v id ers w ith lower wages, who need to spend l e s s minimum time on the s e rv ic e . For th e num erical values assumed above, namely, u=1000, m=50, e=10, p = .9 , nig = 1, Eq. (1) becomes
Ü = 10n - 20ne“ *9n - .2n2 . The d e riv a tiv e i s
U’ = 10-20e“ ,9n + 18ne- ’9n- . 4n
This i s near 0 when n i s about 25 sin c e the ex p o n en tial term s a re le s s than 10"°.- Thus, 25 s e rv ic e p ro v id ers should be used. As long as these approxim ations h o ld , n = (u-mc)/2cnig.
(3) The next assumption m o d ifies assumption (1) toward a l i t t l e - more re a lis m . In stead o f assuming th a t p i s th e same no m atter how many se rv ic e p ro v id ers th e re a r e , we now re p la c e p by p^/n + pG. We
in te r p r e t Pg as th e p ro b a b ility th a t a G e n e ra l-p ra c tic e se rv ic e p ro v id er, d iv id in g h is tim e among n/m jo b s , succeeds in rendering adequate s e rv ic e . A s p e c i a l i s t who did not know what a g e n e r a lis t knows would succeed w ith p ro b a b ility p^. The kind o f s p e c i a l i s t s we consider are assumed to know what th e g e n e r a lis t knows as w ell as t h e ir s p e c ia lty . Such a s p e c i a l i s t ’ s .p ro b a b ility o f ren d erin g adequate se rv ic e i s p^ + Pg. Note th a t i f n=1, then Pg + Pg
re p la c e s p and i s in te r p r e te d as th e p r o b a b ility t h a t a s p e c i a li s t working alone succeeds. I f n i s very la r g e , then p i s rep laced by j u s t PG.
(4) We now modify (2) toward a l i t t l e more re a lis m . Like p, m^
should also depend on n . In p a r t i c u l a r , rrig should be 0 when n=1.
We th e re fo re assume th a t m^ can be replaced by -k(n-1)/nw
mg u -e )
where k i s a c o n s ta n t, mg i s now th e la r g e s t e x tra tim e i t tak es a n o n -s p e c ia lis t to provide a s e rv ic e fo r req u e sts t h a t he sees le s s o ften
10
than does a s p e c i a l i s t , and k i s th e r a te a t which th a t maximum t in e is reached as n in c re a s e s .
I f the formulas from assum ptions 3 and 4 are s u b s titu te d fo r p and m^ re s p e c tiv e ly in Eq. ( 1 ) , we o b tain
U = u [1 -(1 -(p s /n + pG) ) n]n/m
r nc[1 + (mGn/m) ( l - e ^ ^ e ^ 01^ ) ] Eq. (6)
Using our previous approxim ations, and s e ttin g the d e riv a tiv e with re s p e c t to n equal to zero , we o b tain a very com plicated tra n s c e n d e n ta l equation th a t we cannot even analyze g e o m e tric a lly .
Using the same num erical valu es as above, w ith PG = .8 and k = 1, Eq. (6) becomes
U = 10n - 20ne- ’ 1e~ ’8n
with th e d e riv a tiv e
U’ = 10 - 20e- , 1 e~ ,8n + 16 n e " ’ 1e " ' 8n
— . 4n + . 4ne e — . 02n e e
n=25 i s s t i l l a good approxim ation to . _ making U’ = 0, w ith th e sum o f a l l the exponential term s being le s s than .22 in ab so lu te v a lu e .
Even t h i s sim ple example i l l u s t r a t e s how q u ick ly th e a n a ly s is becomes r a th e r complex, and i t i s n e ce ssa ry to r e s o r t to the h elp o f computers. Considering th a t t h i s i s b u t one o f many p o ss ib le s p e c ia l problems th a t can be analyzed by t h i s g en eral method, we w ill not pursue the te c h n ic a l d e t a i l s f u r th e r . S u ffic e i t to s t a t e th a t many such cases have been analyzed and reported (Kochen and Deutsch, 1979)-
4. Conclusions
We envisage a network o f la b o r a to r ie s where s tu d ie s o f th e kind i l l u s t r a t e d above w ill be made and where hypotheses to be e m p iric a lly te s te d w ill be form ulated. In te re s te d p a r t i e s , which in clu d e th e p lan n ers, managers, s e rv ic e p ro v id ers and c li e n t s o f human s e rv ic e o rg a n iz a tio n s, would c o n su lt th e la b o ra to ry c lo s e s t to them fo r advice and a s s is ta n c e in determ ining optimum s tr u c tu r e . In doing so , th ey would provide the la b s with needed in p u t l e v e ls . H opefully, th ey would also perm it the la b o r a to r ie s to observe t h e i r performance fo r a number o f y e ars, thus enabling them to c o l l e c t v alu es o f output and performance v a ria b le s and to c o r r e la te them w ith th e s tr u c tu r a l param eters chosen by the s u b je c t- c lie n t o rg a n iz a tio n s . In t h i s way, some o f the hypotheses could be t e s t e d .
11
These a n a ly tic resea rch la b o r a to r ie s would o f course keep a l l p riv a te d ata about c l i e n t o rg a n iz a tio n s very c o n f id e n tia l. Tney would take pains to remove a l l id e n tify in g f e a tu r e s , sin c e th a t i s not needed to make v a lid g e n e r a liz a tio n s . Any q u a lifie d and concerned p a rty w ill be ab le to supply in p u ts to t h i s network o f la b o r a to r ie s , which r e f l e c t s d iv e rs e p o in ts o f view and valu es on th e meaning o f "lo n g -term ", "net b e n e f its " , "adequate", "community", "ac cep tab le lim its " and "key s e r v ic e s ." Equation ( 1 ) , fo r example, c o n ta in s th e im p lic it assumption th a t . th e only c o s ts t h a t m atter are those o f m aintaining th e se rv ic e p ro v id ers in the o rg a n iz a tio n . In our previous models we have alread y taken in to account the c o s ts to the c l i e n t s , in clu d in g t h e i r o p p o rtu n ity c o s ts o f w aiting fo r s e r v ic e . We have a lso included the fixed c o s ts o f m aintaining the o rg a n iz a tio n , as w ell as th e expected c o s ts to s o c ie ty when c l i e n t s are not provided adequate s e r v ic e .
We expect the re s e a rc h e rs in a lo c a l la b o ra to ry to be s e n s itiv e to the p r i o r i t i e s and p ercep tio n s o f a l l persons in th e region to be served by t h a t c e n te r , and to r e f l e c t t h i s s e n s i t i v i t y in th e assum ptions used to form ulate th e a n a ly tic model. A reg io n i s not n e c e s s a r ily a geographic or ju r is d ic t io n a l a re a ; i t may in clu d e a community o f s p e c i a l i s t s s c a tte re d over th e e n ti r e w orld. Tnat i s why we propose a network o f such c e n te r s . We expect t h a t , in tim e, as th e c e n te rs exchange t h e i r models, d a ta and r e s u l t s , th e re w ill be convergence upon a u se fu l corpus o f r e l i a b l e th e o r e tic a l and em p irical item s upon which to base both advice to o rg a n iz a tio n a l c l i e n t s and fu rth e r re s e a rc h . We do n o t expect consensus upon a s in g le , u n ify in g s e t o f p r in c ip le s , but we could expect some consensus about th e method o f re s e a rc h and a p p lic a tio n .
We b e lie v e th a t th e main t h r u s t o f re s e a rc h in such an in v e s tig a tiv e community should be guided in th e d ir e c tio n th a t i n te r p r e t s b e n e f its according to hum anistic v alu es t h a t allow fo r in d iv id u a l p referen ces as w e ll. There should be an open and unhampered search fo r c o n d itio n s about the e x te n t to which o rg a n iz a tio n -s tru c tu r a l reso u rce s should be d is tr ib u te d to provide b e n e f its to the community as a whole fo r th e lo n g est p o s s ib le f u tu r e . We a n tic ip a te t h a t hypotheses o f th e follow ing kind w ill be te s te d and probably v e r i f i e d .
1. When work lo a d , funds, c o n tro l over sp ace, and knowledge i s co ncentrated in one or very few p eo p le, th e r e s u ltin g performance o f the s e rv ic e o rg an iz atio n is not as resp o n siv e and b e n e f ic ia l to i t s c li e n t s as needed and as d e sired by most concerned p a r t i e s .
2. When work lo ad , funds, c o n tro l over space and knowledge - to name but some o f the in p u ts - a re d is tr ib u te d as w idely as p o ssib le (in clu d in g the c l i e n t s ) , then the system i s too c o s tly r e l a t i v e to the q u a lity and responsiveness o f th e r e s u ltin g s e r v ic e .
3. Optimal degrees o f p l u r a l i z a t i o n , d is p e r s io n , fu n c tio n a l s p e c ia liz a tio n , feedback, h ie r a r c h ic a l f l a t n e s s , ■ d e le g a tio n , and p a r tic ip a tio n in decision-m aking and o rg a n iz a tio n a l red esig n a re l ik e l y to f a l l between the extremes o f complete c e n tr a liz a tio n and d e c e n tr a liz a tio n in each o f th e s e e ig h t a sp ec ts o f th e d i s t r ib u t i o n .
Approximation»to optim al so lu tio n s can be computed in many c ase s.
At l e a s t d ir e c tio n s toward improvements can be shown. I f s u f f ic ie n t data and tren d s a re known, some reasonable e x p ec ta tio n s and' p ro v isio n s
fo r the fu tu re can be in d ic a te d .
M. "G enerally, optima tend toward a g re a te r degree o f d is trib u te d n e s s (d e c e n tra liz a tio n ) w ith th e main se c u la r tre n d s , p rim a rily with the trend toward in c re a sin g req u est lo a d s , toward more complex re q u e sts for se rv ic e s th a t tak e longer and more s p e c ia liz e d knowledge to pro v id e, and toward demands fo r more responsive se rv ic e o f a q u a lity t h a t i s no le s s than the b e st a v a ila b le .
13
R eferences
Kochen, M. and Deutsch, K.W.: Toward a R ational Theory o f D e c e n tra liz a tio n : Some Im p lic a tio n s o f a Mathematical Approach. Am. P o l. S e i. Rev. , 63:734—749, 1969.
Kochen, M., and Deutsch, K.W.: D e c e n tra liz a tio n and Uneven Service Loads. J . Regional S e i . , 10:153-173, 1970.
Kochen, M. and Deutsch, K.W.: P lu r a liz a tio n : A Mathematical Model. Oper. Res. , 20:276-292, 1972.
Kochen, M., and Deutsch, K.W.: D e c e n tra liz a tio n by Function and L ocation. Manag. S e i . , 19:841-856, 1973.
Kochen, M.K., and Deutsch, K.W.: A Note on H ierarchy and Co
o rd in a tio n : An Aspect o f D e c e n tra liz a tio n . Manag. S e i.
21:106-114, 1974.
Kochen, M., and Deutsch, K.W.: D elegation and Control in O rganizations With Varying Degrees o f C e n tra liz a tio n . B ehavioral S e i. , 22:258-269, 1977.
Kochen, M., and Deutsch, K.W.: D e c e n tra liz a tio n : Toward a R ational Theory (Under Review by Harvard U n iv ersity P re ss, 1978-79).
Other S elected R eferences
Lawrence, P.R. and Lorsch, J.W ., Developing O rganizations:
D iagnosis and A ction, Addison Wesley, Reading, Mass., 1 9 ^
M orris, W.T., D e c e n tra liz a tio n in Management System s, Chio S ta te U n iv e rsity , Columbus, Ohio, 1968.
