The economics of organizational design

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is modal, the orSa:tizatian member ‘processes’ input xi into output yr. may be d&2rministic or stochastic. The ptobabihstic process the &grVs 0$ r&abiBtla) of symbol pao~essmtr;,which is a for determining failure rates in large message tran43ing &yond t& degree of reliability it is essential to asume that a ~~rocessing josh CLAbe more or lass di%cuh for the organizatiiort member, depending on tYrenature of the task a9 weil as on his competence for its execution, As a frox;y measure for the level of&&2&y, or, complexity of a task a5Ggned to Sh3 o~g~n~ation member we coufd use the (average) pro~ssirng trmet which he needs to form a given task, i.e., the individual processing time timer; the n,%pcetiveprobabihty with which the proassing of the symbols is tailed for. We could tiksualize an organization member, a machine, as being ccmpo~d of ar, itip~t machine and an outpuf machine -- hooked in serial conrlection. The reason for this distinction is that an organizatnon member may be called upon to do quite differen4:tasks, and may have c~ifficu:tiesof different ki:lds in performing them. One is the acquisit& aad sorting out of input symbols, thz other is the production and dispatching of (Cpucput symbols. As a weak normative requirement, we postulate th:at an orgar,iziition is required to meet its scneduie, and this is finked t3 the ucernge ~roct rsi~g time a of each member not exceeding the average time between the zrrivars . i input symbols, namely one: time unit. Therefore, 7 =

*

3) + +J) s

1

must be assured for every member of the organization, where 7”‘, 7ao) are the mean processing times of his input and output machines, respectively. If this h4,un.di$ violatell for some member. he Al be considered ‘overloaded’, in an intuitive and technical stznse. Example. Take the: sizuplest case of a singIl= decision-maker ;or decisic:l machine), Tan&d DM, hr3wcan he manage the prtx?essing task under a XX. Suppose Xjz(X~,Xj2}*

input symbols,

in su:ch a w;?? that the (mean) payoff is largest. Then SOR: IN: Q+li with prabab%ty 1. >,,s-t, 1 I

tr~r4p~swd,

HI&e, 3 a’itd’k tinge ovef the DNl’s input and output alphabets, respectively.

,

-

. -

this is wt done, i.e., if z> 1 for some number, he wifl k3e said to be ‘averload@.

Here tDM,r is the procassing time of the DM for the directive dk and Pk is the processing probability that he is doing that. must be smaller thaq they In order for this design to effective the CDM,k would be if the DM had to produce yk himself.’ Otherwise there would be no point in having a line unit. A3s0,overload must be avoided by y. That is,

zY =r,t,,P,S 1, b

whs!:e tvkis y’s intrinsic processing ticmc;for the conversion of dk to yk, This simple two-nember organization, in e%xt, divides the inpui. and output loads betwee~r the DM and &e line unit y The ar~augement* in ether words, will avoid overload on y and wiil notably reduce the DM’s output load. 4 The alternately prmh~;

he

If the designer finds that a one-member line is not sufficie,,t to ,oive the problem of overload he will expantl it to include additional members. In that case, he must decide whether to use the lint for altern&z or ~~rulkb pycxxssing. First we treat the (altenateiy processing line. Under this option the DM follows some procedural ruIe by which he issues directives to the lrarious line members once at a time, each instructing its addressee to p,*oduce a certain output. 2 The basic idea is that in this way every !ine member can be given more time to do his processing at least on Ithe average, and therefore is more likely to avoid overload than one unit, .i, actkg ky itself. Thq consider a kultipk m&nber tine, starting with y a;od S an,d end& w&11‘&i.e.:.,s==y, S, i. .,)L Let ti’le*DM, once during each time interval, issue an il~~ktkdOat ak,;k~ iilt,. . ., n ttr one (and only one) of the fine mem*ks, direk&& h4m to &&duce ihe oufptit symbol yk. The DM can act actor tig tid%tk &X, -thi: most g&e& and m&t flexible appears to be a stochastic or n&&i i%6‘ trvhic%r prey&& that the dire&iv; be issued to s with a certain @b&$Q Pai. .% ~’‘; ’ a [ 1 _ . _’

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V’

‘This relat.es to opportunity time. Sometimes you think it is better to do it yourself than to instruct sanaone who does it, in particular, if he does it wrongly. tkb wrtd bn!B nt@onal ruk, B random rub Gthout repkxmtzt task to anyomwhd,is u&xl& sufficiently.

i&w

i.httfrYget, 3”log+?CUstoniCSUjt?f~Q&QtiCM?.Qi &?Si@i

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Let -us yt :whc&m the expansiun of the !ine has actually overcome the proMem of the one-member line. Is it always possible to avoid ovedoad on the line’! It cara be shown that the answer is yes in general, provided the line b m&&ela%@2 cnou@I.

‘The assign&nt probabilities can c.lc;rtainly be made equal, a>, - l/p. Overload will then be avoided if

l&murk.

which certainly is possible if y is made large enongh. Load dependence would imply here, for a general definition see Gotringer (1980), that the Yne unit’s mean processing time is a function of the size n of his input alphabei and of the probabilities with which he receives his input symbols. These probabilities may bc interpretetl in two ways ?xre, namely as P,P, or a’ Q(dkIs). The second m effect assurrles that his nnput machiac adapts to tb: relative frequency with yvhich dk occurs amon-s his assigrmr:Jents,white the first makes the machine adaptive to the relative frequency with which dk i::, processed by him axiong all assignments. Cor&lJa~y I.

&et, Qx iqut &chine

of the sth line member be kud-de~enrknt:

(c;). if this depend ence i:s on the probabilities @ik 1s) Theorem 1 remnirzs t&id, (b) if it is on the prtibabilities P&P,. 7’heorem I remainzg t;alil if a&o lim,,, P&(P)=0 us P, goes to zero.

with the und~standing that Q is to be issued to ;j, G2 to 6, etc.’ The rule ~3 furthermore prescri’be which among ail (m!) possible permutations the 13M may UN, and finahy with what assignment probability Ptrk satisfying ~a)P&Qr;lPj@l (k=1,2 ,..., n) it is to be used. The rule would oonstituto tL$e.SOR for the lEb’,kl. A line member, on receipt of his instruction dg ~ouid turn out tk subq&~~l J$ a~ld transmit it to the Post Office. There, the subsymbois ~ruk!l be ~cl@ed a&to the complete symbol y, and dispatched to the ~rcq~r destination. The SOR for this arrangfmenl is ta *n,tlatedas follows: /’

DM: xk+~J&. SOR s: 6I;-+y;, I

with probability P,,k. to P.0. with probability 1,

. ., d;f, d;: to s

fi

a==1,2,..., m,

k=1,2 ,..., n.

where u is the number of a&wed permutafion~. What can he said about the effectiveness of such an SOR? So far as the DM is c+ncerned, it wit1 be successful if it reduces his mean processing time as far as pssible, with tn,,,,$now interpreted as the time needed by him to generate the m-triple! of instructions &, and to .address them to the proper line mkm’bers.&th the prescribed probability. As far as the line members :WZ c&eke& the, SOR v& be successful if it eliminates overload among theci. The conditiork for thalt &II be obtained as follcws. Among the permutaiions ‘@rich the D?& may use there ~$1 be a certain subset which assigns the subsy&o~.d~ of gk to unit s, for example. The probability of this happening $ .‘ , / ,.

with the sum extend& over all u that belong to that subset. In analo,;ous manner one defines kjt ,. ., Pd. Then if

(*cl t*=

cT=4,2,..., m. k-l,2,...,n

i j , ‘I’.? 8. ‘I1.’ .::.x , ‘sThs, it is askvnmedthat the task aI&ation’ permutes acc;ording to tb: number OFrtxeivcrs, at?d&is ia ttm’tdapondaon to which extent the outptA symbols, and hen-, the dire-civa arc de&mp&abk

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ati assun~ for the moment that ii is non-empty. Select a point in that subs& 0~1 which the ~comnion. value assumes its minimum r*. It is claimed that ihis is a point al: which the tar st of the z,,s=y, ii, i: is as small as possible. For suppose not, one couid find another point of CP at Which the hwgest 2, is small[er than 2%.Wowever, it would then also be larger than any o&e], T*at that point for if they w sre ali equal, z” would not have b&n the sm;alleSt.Tt follows th.at T* minimizes the largest rs_ [Z) It r~ow remains to be shown that there exist equii,tzing SORs. i.e., that the subset on which o,,= td-’ --2, is non-empty. Here we c:an use a line of reasoning by NatA as exposited by Lute and g.aifla (1957, p. 391). Sqpose that some mixed SOR ha:; been chosen by the designer but that q, T,, and z, are not equal. In that case he can cboc~se a new SOR which is the same except for th:: assignment probabilities of two symbols, Ji dnd & say. Consider the change in qobabiiities

The transformation from the PUi and Pstk to the P$ and P& is a continuous transfcrmation, and it carries CP into itselL By Brouwer’s fix-point theorem it has a fixed point, i.e., a set r~f assignment prob&Wies for which Eli and &, as well as P, and Ylykare the same. EGdently for this parti?qJlar set,

it follows that there exist equalizing SORs. vote_ that the prouf is v,alia regydiesr of whether the input machines of the line, mer@crs,a~, 5 er ar@not, load-&pend,znt. If :.&eyare not, the problem of deter&&g &e MinimFx SOR ii of a, well-kr own type which is called an ‘a&&neat problem’, and moN particularly, of the variety of ‘%ottleneck assiqmer$ probl,eq$. Algori&qs exisi by .&ch pure SORs can be $$&n$nt?&~~t J-t 19 @ti$ple [e.gepFord acd ?Ukerson (1981, p. 57)J. . 9s t3Q.b. ~OW~Y~J’ may, produce, overlai;l6ion I .e line in cases in which mix& ..Qn%$ilX avoid jt.. ,73&zdeter@natioa of a b:;t mixed SOR, can bc &&n&ted ti the f&&&g LP problem: determine Gdues for the assignment ~~b&litics, .subject to, the probaWty canstraints, and +o zy =zd =zl, such that 21,,say, is minimized.

273

needed far ‘personal attention pr

iurthe mean

however, will thWi in &ect merelj

proces&g

what follows.

Exampte. In order to explain the connection bet-~xn la2&-dependence anci line structure, suppose that a line has been set up for al~rnatc processing Suppose further that it has been possible to solve the sequeucz ol prog&xrrming problems described previously fsection 5) and that a four-man line identified by y, Q, $ 4, has been found adequate. This ~~ohld mean PM an SOR has been derived which minimizes the mean outsbut processing time rDMl and which at the same time, avoids over&d anx::~~ the ifvur lane mizlmbers. Suppose, fInally, that it is intended to reducrk So i’arther, for whatever reason.. In that case, an echelon of new line men&r-x, ca\l them ‘liaison managers‘, can be interposed between the DM ara the e,iisting four, such as i and j. LM links the DM to ‘yor 6, LbIj does it for E +r 4. A table of the SORs governing these operations looks as follows:

SOR

.I 1.

&--+& dl. to y or S with probabilitie: Pyk,P,,,

*. J-

dk-+dk,dk

to 8 or 46with probabilities P,:, P+,

y,6, G,#xd,--+y,,,y, to P-0. with probabilities 1, k = 1, ?, .

e

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n.

‘12tis-SOR comp&ates the above ment’oned programming probicms irx the fobllowmgway: The largest possible redx%ion of the DM’s a&put load can be +chievtxi by solving a cxxrespondirq, newly structured pr oblem: min rDu with r&&on to assignment probablities Pu and Pir( subject to the cqnstraints:

(+*)

e~~ogpt/2-t 1.

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maximum numkr of r fcxthe indicat,Lx of order to bring ~r~~d~ on in the DIM’Soutput load. it a

liaison managers, coilectively will dso )

s he might have to ach&5v;abkreduction 8re the organization the line. ~~hatel~era introduced through a proper design of the taken up in a subsequent paper. The discussion here has been limited to the alternately processing line. ihh s~t~~~l~changes, it applies to parallel processing as well, a~&houghthe ar~tunentatisn &comes more i nvdwd.

f@w the de

many si t&

I%e basic objective of the economics of organizational &sign as pertaining to line organizatiom is the reduction of the output io& on the IBM(s), without generating oylerload among the Einemembers. As Arrow (IF4r9,p. 37) has remarked, ‘the scarcity of information-handling ability is an essential ftia&urefor the understanding of both individua! and organizational behavior and :; is to be explic? ly de& with by the orgaGzational designer. The limit results CYII avciding overload, or loss to the o;!,ganization appear to :?.gree,in principle, w~:h other limit results oln tasge te&ms obtained ,4rrow and Radner 1(19V), particularly in view of our Theorem 3 in :he prc:tious sectiosl, where it r,$ ‘(*41ows a that the performance cf a centralized lint: ia equivalent to a dp:entralized line, structured ia e ~xhelons, with ‘2 Miciently large nu m’&er of fiaison managers assuming partial d&siotP ,x&ing functions. Tlx designer has im; fundamental options of how to use the line. He caq dc it3 proc;Lssing altlt:r.natee\yor in parallel. He can also uw mixtures of the two If alternate proces,,Q~gis chosen, overload can always !XZavoided arrrong the line by making the line la.rge enough. Moreover, if the output (machine) of the I% is not load-dependent, all line menbel~ can receive their instructions from him dir&y without adverse &ct on his output load. On the a ther ha nd, if his output is Soad-dependent sddftional line whelons must be :introduced as the size of the Sine is increase4 to keep the Dhrl’s output l’nad u.r&r C~Orrttol. Similar remarks apply, in principle to a line that does its processing in par a.K& Overload among the,,line members however, may pot be avoi:!able unless ihe ~~~oduCti&$ of thlt 0igs&zationJ output can be re~~‘cvedinto sufficiently mcny subtasiks. IWe SOlRs have &,a estabtishcd that reqtire the optimal solution of progfamminig problents to avoid overload. l[n possibly more rexlisti;: situ&xrs, b,y setting u 3 SOR!; for large organizations we may face difftc3ultir.s