ON THE OPTIMUM STRUCTURE OF THE HIERARCHY IN AN ORGANIZATION

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Mathematical Social Sciences 14 (1987) 239-250 N o r t h - H o l l a n d 239 O N T H E O P T I M U M S T R U C T U R E O F T H E H I E R A R C H Y IN

AN

O R G A N I S A T I O N

Ming-Yueh TARNG

[#lstillue Qf Mutmy.ement Science, Chiuo-Fung University, Tuiwcm, t?.0.(. Miao-Sheng CHEN

Institute q / Manugesnent Sc,ence, Chioo-Tung University; und Deportmem ~;: Xlathemot,c~. 7~mH,ang University, 7?fiwun. RICLC.

C o m m u n i c a t e d by K.M. Kim Received 3 September !986 Revised 17 October 1986

Using the character of organizations and tl~e elementary results o[ qtneueing theory, a quantitati'~e model of the hierarchy is presented, lis aim is to minimize cost',, wifich are the sum of wage costs and costs caused by delays in decision making. With this model, the sensilivil\ analysis on key variables of the optimum structure is concretely discussed

Key words: Organization design; hierarchy structure; sensitivity analysis.

1. I n l r o d u e l i o n

In the design of organization structure, among the important factors to be considered are the quality, the speed, and the cost of the staff of job planning. For nonhierarchical organization, because there is a lack of a superior-subordinate relationship among the decision makers, the action of planning tends to bias towards the consideration of strategies (Marschak and Radnew, 1972; Nojiri, 1980). For hierarchical organizations, all the above three mentioned factors are important in the design of organization structure (Beckmann, 1960; Williamson, 196"7). However, the quality of planning jobs does not have satisfactory measurement tools, therefore, only the speed and the cost of job planning will be considered in this paper. A model is established based on these two factors which will provide the optimum organization structure by considering the trade-off between these two conflicted factors.

Eeren and Levhari (1979) try to explain both the existence of hierarchies and their structure by positing that they serve the need to reduce the planning time of the general manager. They assume that the planning time of each level in the hierarchy is linear on the level's span of control, and use the sum of level's phmning time to measure the speed of the planning of an organization. Although Keren and Levhari

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240 M.-Y. 7arng, M.-S. Chen / Hierarchy in an orgamsation

gave the topic of the hierarchy structure a general discussion, the following three factors were still ignored in their model: (1) It did not provide us with sufficient reasons that the planning time of a g r o u p leader must vary linearly with his span o f c o n t r o l . In fact, m a n y authors (Caplow, 1957; Costs and Updegraff, 1973; P u g h and H i c k s o n , 1976; Spyros and Demitris, 1982) on organization theory have suggested that c o m m u n i c a t i o n interaction, c o - o o r d i n a t i o n s and control problems increase at a faster rate than size. (2) It did not convince us with sufficient reasons that the planning time o f a group leader depends on his span o f control and in- d e p e n d s on the level where he lies. In fact, since the upper level o f group leaders must bear m o r e uncertainties for decision m a k i n g , so, with the same span of c o n t r o l , the p l a n n i n g time o f g r o u p leaders in the upper level is usually greater than that o f g r o u p leaders in the lower level (Starbuck, 1979). (3) It only uses the proceeding time o f a p l a n n i n g job to measure the speed o f the planning, and omit its waiting time. if a new j o b occurs and all group leaders have not finished their own tasks for p r i m a r y jobs, then this. new job may be p o s t p o n e d for planning in the hierarchy. A d d i t i o n a l l y , even t h o u g h the new j o b could be planning immediately in the hierarchy, the waiting time may still exist in some levels if the level's planning times are not all equal.

In this paper we shall use the results of queueing theory to f o r m u l a t e the waiting time o f a j o b , and then present a general m o d e l to discuss the sensitivity analysis on o p t i m u m structure ~ariables - the height of the hierarchy, the n u m b e r of g r o u p leaders in the hierarchy, the level's span o f c o n t r o l , and the idle time o f a g r o u p leader; with respect to the parameters - the wage rate, the organization size, the complexities o f planning fobs, and the expected interarrival times o f planning jobs, separately.

2. Assumptions and notations

T h e organization studied in this model is c o m p o s e d of ( f u n d a m e n t a l ) activity units which are completely dependent on the head, or general m a n a g e r , for instruc- tions. The most crucial topics for the head, is to seek a hierarchy structure which can quickly and effectively solve the problems that arise during the execution of the plan o f action. For this purpose, the head has to p r e p a r e a new set o f instructions to c o o r d i n a t e these activity units on the basis o f new observations m a d e by t h e m . To shorten the time it takes to collect the i n f o r m a t i o n and prepare the instructions, the head has to interpose additional levels o f the h i e r a r c h y between himself and ac- tivity units. The hierarchy of the o r g a n i z a t i o n is p o p u l a t e d by identical g r o u p leaders, which links head and activity units. T h e wage rate p e r g r o u p leader is d e n o t e d by w.

T h e m a i n c o n t r i b u t i o n o f the g r o u p leader is to find the relationship a m o n g his ( i m m e d i a t e ) s u b o r d i n a t e reportings, and transmit it to his (immediate) superior. To insure the sum of g r o u p leader's p l a n n i n g times that a p p e a r in a path directed f r o m

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M . - } . Tarn~', M . - S . C h e r t ./ H i e r a r c h y In a n o r g a n i s a t t o n 241

a n y activity unit to the head, are all equal; the c o m p l e x i t i e s o f tasks a n d the span o f c o n t r o l o f all g r o u p leaders in a given level, will be the same.

2.1. S t r u c t u r e variables

t~ = tile n u m b e r o f activity units; we refer to n as the size o f an organization. H = the n u m b e r o f hierarchy levels; we refer to H as the height o J t h e hierarchy. _v,,=the n u m b e r o f g r o u p leaders in level h, xl = 1 a n d XH+ L - n.

sj,=.vh+ ~/xh, is the span o f c o n t r o l o f a g r o u p leader in Level h.

M = x ~ + x 2 + ' " +XH, is the n u m b e r o f g r o u p leaders in the whole hierarchy.

In o u r m o d e l , &, a n d hence Sh are c o n s i d e r e d as c o n t i n u o u s variables; this con- s i d e r a t i o n is r e a s o n a b l e in the real w o r d , b e c a u s e the m e a s u r e m e n t o f xt, is based on t h e time which is spent in p l a n n i n g d u r i n g a clay.

2.2. The c o n t r i b u t i o n o f g r o u p leaders

a = the index o f complexilies o f an activity u n i t ' s r e p o r t i n g (with respect to a plan- ning job); we refer to a. n as the i n d e x o f the initial complexities o f a p l a n n i n g j o b .

b- a . n = t h e index o f the final c o m p l e x i t i e s o f a p l a n n i n g j o b as it is finished by g r o u p leaders, where 0 < b < l .

::.j,=the index o f complexities o f an h-level g r o u p leader's r e p o r t i n g , z~ - b- a . n a n d zH+ i = a .

c(Gh) the c o n t r i b u t i o n o f a hqevel g r o u p leader G,,, = l n p u t ( G h ) - O u t p u t ( G h ) ,

---: S h Z ' h + l -- 7~h"

.l(c(Gh)) = t h e p l a n n i n g time r e q u i r e d by G~, to c o m p l e t e his o w n task.

G r a c i u n a s (1937) presented a m a t h e m a t i c a l m o d e l to d e m o n s t r a t e h o w the c o m - plexities o f s u p e r i o r - s u b o r d i n a t e p o t e n t i a l interacts. His f o r m u l a states that as the n u m b e r o f s u b o r d i n a t e s r e p o r t i n g to a g r o u p leader increases arithmetically, the n u m b e r o f p o t e n t i a l interactions increases g e o m e t r i c a l l y . This m e a n s that f have the f o l l o w i n g p r o p e r t i e s :

f > 0 , f ' > 0 a n d f " > 0 . (2.1)

In general, the g r o u p leaders in the u p p e r levels must bear m o r e uncertainties in the p l a n n i n g d u e to the increasing c o m p l e x i t y t h a t exists in their s u b o r d i n a t e d repor- tings. H e n c e the a s s u m p t i o n we m a k e here is

Zh/Zh+l 0, 0 > 1 is a p a r a m e t e r . (2.2)

In the h i e r a r c h y , the m o r e i n f o r m a t i o n that is missing m e a n s the increasing final c o m p l e x i t y o f a p l a n n i n g j o b (i.e. b increasing). T h e r e are two aspects a b o u t the missing i n f o r m a t i o n . In r.he l o n g i t u d i n a l a s p e c t , it usually d e p e n d s on the s u p e r i o r - s u b o r d i n a t e c o m m u n i c a t i o n s . In the cross section aspect, the missing in- f o r m a t i o n is usually caused by the delivery o f g r o u p leader's tasks. Given In-

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..4.. M . - Y . Tc, rl~g, A,I.-S. (.'he~l H i e r a r c h y in on or~ctms~tt~o~l

put(G/,), the smaller the value o f O implies that the m o r e detailed d a t a ' s stateme~t o n facts, c o u l d be f o u n d in Gh's r e p o r t i n g . A d d i t i o n a l l y , with the s a m e height H, lhe decrease o f 0 yields the decrease o t the final c o m p l e x i t i e s o f a p l a n n i n g j o b Icf. ~3.1)).

2.3. T h e w a i l i n g t i m e o f ce p l a n n i n g j o b

C o n s i d e r the hierarchy as a q u e u e i n g system (regarding p l a n n i n g j o b s a n d g r o u p leaders as c u s t o m e r s a n d :severs respectively). If we a s s u m e that this q u e u e i n g system has P o i s s o n input process with m e a n i n l e r a r r i v a / t i m e s e, a n d it has c o n s t a n t service times; then the expected waiting time o f a p l a n n i n g j o b (Hillier a n d L i e b e r m a n ,

1980, p. 437) is given by {2

~ . t . - , where t-- max ./( c( G j, ) ). (2.3t

2 ( e - t) I _</,_<,

2.4. T h e p r o c e e d i n g t i m e o f a p l a n n i n g j o b

In general, g r o u p leaders in the s a m e level o f a hierarchy are w o r k i n g c o n c u r r e n t - ly, a n d g r o u p leaders o f i m m e d i a t e l y s u p e r i o r (or inferior) level wait until the adja- cent level has finished. So the p r o c e e d i n g time o f a p l a n n i n g j o b is d e f i n e d by

Iq

p.t. = ~ f(c(G,,)).

1 1 - ]

2.5. C o s t s c a u s e d b v d e l a y s in d e c i s i o n m a k i n g

T h e cost o f per unit time p r o f i t s lost t h r o u g h slow p l a n n i n g , is d e f i n e d by

C ( T ) = C ( w . t . q p.[.), where C ' ( T ) > 0 a n d C " ( T ) > _ t l .

3. The m o d e l o f a hierarchy

A s s u m p t i o n (2.2) yields that

. . . 0 H - 1 7 + I - . _ o t q - / l + !

gh 0~'f1~ 1 " t 1 ! I

a n d t h e r e f o r e we have the f o l l o w i n g properties:

b . a . n = zl = Otto,

c ( G h ) = ( s h - 6)zj, ~ ~ = ( s / , - 0)0 /'h- a- I1,

p . t . = ~ /'((St,- O)O-/'b • a . n). h : I

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M . - Y . T a r n g , .~,.I+-S. C h e n /" H i e r a r c h y in a n o r g a n i s ~ t i o n

243

The objective o f this m o d e / i s to m i n i m i z e the total cost L, which are the sum o f wage costs and costs caused by delays in decision making. That ts

I m i n L = C ( w . t . + p . t . ) + w . ( H Xh)

~ s u b j e c t to: .v I 1,;\-/t+ 1 = n , .G+~=sh.G, 1 - < h _ < 14.

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{ O H = b " n , p . t . = h ~ _ , l f ( ( s I , - - O ) O h b . a . n ) ,

~ .t. 2(e- t /

where a,n, w, e, 0_> 1, 0 < b < ][, are parameters.

Model (1) is a generalization of the Keren and Levhari model. If we om~t the term w.t., restrict f r o be a linear function and set O - 1; then it is the Keren and Levhari model.

. The o p l i m n m solution of the hierarchy m o d e l

I.et

t h = f ( ( s h - 0)O-hb- a- n), 1 _<t7_<H.

d(w.t.) d t l 2 ) t ( 2 e - t ) d ~ = d t 2 ( e - O , = 2 ( e - t ) : > 0

and therefore w.t. increases with t. This implies that, for fixed p . t . - V tj,, the necessary condition of the minimization of w.t. is

1 = 1 1 = l ~

. . .

l t t .

Add this condition in Model (I), and it leads to the following properties:

l) = ( S 1 -- O ) O - I b ° Lg ° n == ( s 2 - O ) O - 2 b . a . n . . . . (4.1) : ( s , - 0)0 rib. a. n, v- M = u(x 1 +x~ + --- + xH),

=(S I - 0 ) 0 - 1 b

" a " n " X I + ( S

2 - 0 ) 0 - 2 b

" a " n ' .v~ + . - + (SH-- 0)0- ttb. a- n. xH, - b . a . n [ ( x : O I + X 3 0 - : + . . . + X H t I O H)--(XI+.WO I + . . . + X H 0 H+l)]; by (3.1), = (1 - b)a. n, (4.2) It is valid that

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2 4 4 M . - Y . TartTg, A//.-S. C h e n / t t i e r a m ' l ( v in an ork, ant.sultotz

p.t. =

Hf(u):;

by (3.1),

In b + In

n

- f ( o ) .

l n 0

l-herefore Model (I) can be written as the following form

I

m i n L

C ( T ) + w . M , /

(f(o))_-

(1') ~ s u b j e c t lo: T =

, 2 ( e

-f(v))

L a n d v. M=-:(I -

b)a. t7. In b

+

In n +

.1(o)

l n 0 (4.3) (4.4)

The necessary condition of this optimality is

d L / d M = - C ' ( T ) T ' ( u ) ( 1 - b ) a , n . M 2 + w . O.

(4.5)

S. T h e s e n s i t i v i t y a n a l y s i s o f t h e o p t i m u m solution

The total differentiation of (4.5) yields

d w C " ( T ) T"(o) db - - - , : i T + - - d u - - - w C ' ( T ) T'(v) 1 - b

da

dn

2dM

4 - - - + . . .

u

n

~1

'Fhe total differentiation of (4.3) yields

a T OT OT OT OT

d T = - - do+ --- d b + - - d n + - - - + - - de.

Oo Ob On O0 Oe

The total differentiation of (4.4) yields

M . d o = ( 1 - b ) ( a - d n + n .

da) - a . n . db -

(1

- b ) . a . n

M

Substituting (5.2) and (5.3) in (5.1), leads to

d w _ [ u ( 1 - b ) . a . n

+--21] d M +

C " ( T ) OT w M:: M C ' ( T ) O0

dO

+ [ u ' ° ' ( 1 - b ) M + C " ( T )

0 T c , ( T )

On + ~ ] dn l u . c , . n C " ( T ) OT 1 ] M C ' ( T ) Ob 1 - b db u . n 1 I C " ( T ) OT + da + de, M a C ' ( T ) Oe

where

dM.

(5.1t

(5.2t

(5.3)

(5.4)

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M . - Y . Tarng, M.-,5. Chen .," t h e r a r c h v in an organLsa/lotl 245 I I - - C " ( T ) T'(v)-+- ---- T ' ( v ) m e a n s ~ . C ' ( T ) T ' ( v )

'

, F r o m (4.3), it can be s h o w n that

f(u)f'(u){2e-f(u))

In b + In n T ' ( u ) " 2 ( e - f l u ) ) : + In 0

T"(v)

=

f(la)f'(u)(2e-f(u))

[ f ' ( u ) f " ( i , )

2 (e -f l u ))-

f ( u ) f ' ( t , ) In b + l n n + . / " I v ) > O. In 0 T h i s implies u > 0 . # ' ( u ) > 0,

5 .1 The effect of changing the wage rate w

T h e o r e m .

In the hierarchy model, if we consider w as a variable and keep other

parameters ,fixed," then

d M

[ ( u ( 1 - b ) . a . n

2 ) 1 - ' - w ~ + - - < 0 , (1) d w , M -~ M du ( 1 - b ) - a . n d M (2) d w - M 2 d w > 0 , dsh Oh(l - b) d M (3)

dw -

b. M 2

d w > 0 ' d I f ' ( u ) ( l - b)" a . n d M - < 0 , ( 4 ) d w M 2 d w

where I= e f(u) is the idle time o f a group leader.

P r o o f . (I) By setting dn = d e : : d a = d b = d 0 = 0 in (5.4).

(2) B y d i f f e r e n t i a t i n g (4.2~: v = (1 - b ) - a . n - M -~ with respect t o w.

(3) By d i f f e r e n t i a t i n g (4.1): ~ = ( s / , - O ) O hb. a . n with respect to

w,

a n d using t h e r e s u h o f d v / d w .

(4) By d i f f e r e n t i a t i n g : l = e - f ( u ) with respect to w, a n d using the result o f d u / d w .

5.2. The effect o f changing the organization size n

[ h e o r e m .

In the hierarchy model, if we consider n as a variable and keep other

parameters fixed," then

a M

[ u ( 1 - b ) . a

C ' ( T ) J ( u ) ~

[ u ( l - b ) . a . n

2 ] - '

= + - - - - - + ~ + - > 0 ,

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246 M . - Y . 7arng, M . - S . C h e n / Hierarchy in an organisation

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ds~ - (l - b)O h d M - - < 0 , dn b. M ~" dn d H 1 (3) - - - > 0 , dn n In 0 dI f ' ( v ) ( 1 - b ) . a ( d M ) (4) - - = M ~ M - n - - - . dn dn

Proof. (1) By setting d w = d e = d a = d b = d 0 = 0 in (5.4), and using the differentiation of (4.3) with respect to n.

(2) Together with the differentiation of (4.1):

d s h v - h b . d s h (1 - b ) . a d V - o - h b . a . n . - - + - = O a n . - - +

dn dn n dn M

and the differentiation of (4.2):

dv ( l - b ) . a ( d M t

dn - A/~ M - n dnn " (5.5)

(3) By differentiating (3.11) with respect to n.

(4) By differentiatiang idle time: I = e - f ( v ) with respect to n, and using (5.5).

Remark. A simple c o m p u t a t i o n yield s that dI/dn>_O if and only if C"(T)f(u)>_ C ' ( T ) In 0. In particular if C is a linear function then dI/dn<O.

5.3. The effect o f changing the mean interarrival time e

T h e o r e m . In the hierarchy model, i f we consider e as a variable and keep other parameters fixed," then

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(2)

d M d e d s h de C " ( T ) f2(t)) 2M 2 < 0 , C ' ( T ) ( e - f ( v ) ) 2 u(1 - b ) . a. n + 2 M - (1 - b ) . a . n . 0 e d M - - > 0 , a . n- M 2- b de d I f'(t))(l - b ) - a - n d M - - = 1 - " de M 2 de

Proof. (1) By setting d w = d a - - d n = d b = d O = O in (5.4), and using the differentia- tion of (4.3) with respect to e.

(2) Together with the differentiation of (4.1) and the differentiation of (4.2) (with respect to e).

(3) T o g e t h e r with the differentiation of the equation: I = e - f ( v ) and the differen- tiation of (4.2).

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,~4.-}'. TattT~, ,~,I.-S. C/ten / H i e t a r c h v it? oft organi.satl(.7 247

R e m a r k . A simple c o m p u t a t i o n yields that i f C is a linear function then d M / d e = O

and d l / d e = 1.

5.4. The e f f e c t o f changing ,,he initial c o m p l e x i t i e s a

T h e o r e m . M the hierarchy m o d e l , i f we consider a as a variable and k e e p o t h e r p a r a m e t e r s f i x e d ; then (1) d a + + > 0, (2) ds h 1 - b 0 / , d M - - - - - < 0 , da b. M 2 da dI (3) da f ' ( v ) ( l - b ) . n - ( u - b - a , n - M ) M(u(l - b ) - a - n + 2 M ) d l ; a n d - - > O i f f u . b . a . n > M . da l ' r o o f . (1) By setting d w = d e = d n = d b = d O = O in (5.4). (2) Together with the differentiation of (4.1):

d&, v = 0 - / ' b . ds h (1 - b). t7 dt~ _ O_/,b. a- n - - - + - a. n - - +

da da a da m

and the differentiation of (4.:2): dr) (1 - b n d a - M 7 M - a daa ' (5.6) d l du 13) da f ' ( v ) d a ' by (5.6) f ' ( o ) ( l - b). n ( d M M ) M ~ ,\ da ,

and then by using the result of d M / d a .

5.5. The e f f e c t o f changing the ratio o f c o m p l e x i t i e s 0

T h e o r e m . In the hierarchy m o d e l , i f we c o n s i d e r 0 as a variable a n d k e e p o t h e r p a r a m e t e r s .fixed, then d M M2j(tJ) C ' ( T ) In b + In n ( 1 ) - < _ 0 , u ( 1 - b ) . a . n + 2 M C ' ( T ) 0(lnO): (2) (3) dO d5/~ dO d H dO - 1 + ( b - a . n ) l l h . l n b + l n n - < 0, 0(In 0) 2 0/, I ( l - b ) . a . n . 0 h d M D. - - M ~ d O > 0 ,

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248 M.-Y. Tarng, M.-S. Chet~ " ttierarcht' iH o~7 or,~aHi.sat~ot;

d I f'(v)(1 - b ) . a. n d M

- - = _<0.

(4) dO M e dO

P r o o f . (1) By setting d w - d a = d n = d b = d e = O in (5,4), and using lhe differentia- Iion o f (4.3) with respect to 0.

(2) T o g e t h e r with the d i f f e r e n t i a t i o n o f (4.1) a n d the d i f f e r e n t a t i o n of (4.2) (with respect to 0).

(3i By d i f f e r e n t i a t i n g (3.1) with respect to 0.

(4) T o g e t h e r with the d i f f e r e n t i a t i o n o f the e q u a t i o n : 1 = ~ , - / ( ~ ) a n d the differen- tiation o f (4.2) (with respect to 0).

5 6. The e f f e c t o J changing the f i n a l c o m p l e x i t i e s b

l h e o r e m . In the hierarchy m o d e l , ~f we consider b as a variable a n d k e e p o t h e r

p a r a m e t e r s fixed," then ~1) db M C ' ( T ) b i n 0 + - - 1 - b M - ~ + - - M ds/, O h d M (2) d b - ( b i n ) 2 ( 1 - b ) - b db- d H 1 ~3) - >.0, d b b In 0

+M l

d l - f ' l v ) ---a n + (1 - b)-__ a- n (4) d b /I,4 M 2 "

P r o o f . (1) By setting d w : : d a = d n = d 0 = d e - 0 in (5.4), a n d using the d i f f e r e n t i a t i o n o f (4.3) with respect to t7.

(2) T o g e t h e r with the d i f f e r e n t i a t i o n o f (4.1) a n d the d i f f e r e n t i a t i o n o f (4.2) (with respect to b).

(3) Bv d i f f e r e n t i a t i n g (3.1) with respect to b.

(4) T o g e t h e r with the d i f f e r e n t i a t i o n o f tile e q u a t i o n : 1 = e -/(t~! and the differen- liation o f (4.2) (with respect to b).

R e m a r k . (I) If C is a linear f u n c t i o n , then d , k / / d b < 0 , ill) If d M / d b > 0 , then d s h / d b < 0 a n d d l / d b > O .

6. C o n c l u s i o n s

T h e m o d e l p r e s e n t e d in this p a p e r discusses the u n i t i n g o f the f u n d a m e n t a l activi- Iv units in an o r g a n i a t i o n , to o b t a i n c o o r d i n a t i o n b e t w e e n the speed o f the p l a n n i n g

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,¢I.-}. 7o#lzg, .~J .S' Chet~ " H i e r a r c l u ' itJ an orA'oni.sd~',,,*,u 249

.iobs a n d the cost o f staffing. U n d e r these a s s u m p t i o n s , a q u a n t i t a t i v e model o f the h i e r a r c h y s t r u c t u r e could be f o r m e d .

\Ve have discussed tt~e sensitivity analysis o f the o p t i m u m s t r u c t u r e , and o b t a i n e d m a n \ ' interesting properties: i l l If the wage rate increases t h e n so does lhe span o f c o n t r o l o f a given level, while b o t h the idle time o f a g r o u p leader and the n u m b e r o f ~_'roup leaders (in the whole hierarchy) decreases. (cf. Section 5.l). (2) If the o r g a n i z a t i o n size increases then so does the n u m b e r o f g r o u p leaders and the height o f the h i e r a r c h y , while the span o f c o n t r o l o f a given level decreases: a n d the idle iime o f a g r o u p leader decreases if the indirect cost f u n c t i o n is a liriear f u n c t i o n . tc.f. Section 5.2). (3) If the m e a n interarrival time o f p l a n n i n g jobs increases, then so does the s p a n o f c o n t r o l o f a given level, while the n u m b e r o f g r o u p leaders decreases: a n d the idle time o f a g r o u p leader increases if the indirect cost ftmction is a linear f u n c t i o n . (cf. Section 5.3). (4) If the initial c o m p l e x i t i e s of planning j o b s increase then so does the n u m b e r o f g r o u p leaders, while the span o ! c o n t r o l o f a given level decreases. (cf. Section 5.4). (5) If the ratio between the complexities o f iwo i m m e d i a t e l y s u p e r i o r - s u b o r d i n a t e r e p o r t i n g s increases, then so does the span o f c o n t r o l o f a given level, while the n u m b e r o f g r o u p leaders, the height o f the hierarchy a n d the idle time o f a g r o u p leader all decrease. (cf. Section 5.5). (6) If lhe final c o m p l e x i t i e s o f a p i a m l i n g j o b as it is finished by g r o u p leaders increase, t h e n so does the height o f the hierarchy, while the n u m b e r o f g r o u p leaders decreases if the indirect cost l u n c t i o n is a linear f u n c t i o n , tcf. Section 5.61.

Finally we are obliged to r e m a r k that this m o d e l is still i n c o m p l e t e iri ihe following rcspccts: It ignores the q u e s l i o n o f h o w the o r g a n i z a t i o n s h o u l d be classified, and the d i f f e r e n c e in the wage rales in d i f f e r e n t levels o f the hierarchv. These considera- lions arc i m p o r t a n t for the i m p l e m e n t a t i o n o f the theoretical restihs to the real situa- tion, which are valuable q u e s t i o n s to be s t u d i e d f u r t h e r .

R e f e r e n c e s

XI.]. Beckmaim, Some aspects of returns to scalc iil btlSiiless administration, Quart. I. E~.'(Jno|'ll. 74 (196(I) 464-471.

1-. ~,aplow, Organizational size, Admin. Science Quart. 1 (1957) 484-505.

R. CTosls and D. Updegraff, The relationship between organization size and the admmi';trati,,e compo- i~ent of banks, J. Business 46 (1973) 576-588.

V.,\. Graicunas, Relationship in organization, in: L. Oulick and L. Urwick, eds.. Papers on the Science u l .Administration (Institute of Public Administration, New York) pp. I,~1-1,~7.

I..S l t i l l i e r a n d G . J . l . i e b e r m a n , l n t r o d u c i i o n l o O p e r a t i o n s R e s e a r c h ( H o l d e r - 1 9 a ~ , ll~c..SanFlancisco,

1980).

M. Keren and D. Levhari, The O p t i m u m span of control in a pure hierarchy, Management Science 25, !1 (1979) 1162-1172.

.1 Marschak and R. Radnew, Economic Theory of Teams (Yale Univ. Press, New Haven and London, 1972).

Ii. Nojili, ()n the fuzzy team decision in a changing e n v i r o n m e n t , F u z z \ Seis and Sysienl~ 3 (1980)

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250 M . Y. T a r n e, M . - S . C/Tetl :~ t t i e r a r c h ~ , m a n o r g a m s a [ i o t ~

D Pugh and D. Hickson, Organization Structure in its Context (Saxon House, England, 1976). K. Spyros and A. Demitris, Size and administrative intensity in organizational divisions, Management

Science 28, 8 {1982) 854-868.

\¥.H. Starbuck, Organizational growth and development, in: W.H. Starbuck, ed., Organization Growth a~ld Development (Penguin Books, Harmondsworth, t979).

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