Managing Complexity by Recursion
by Bernd Schiemenz
Chair for general business management and management of the manufacturing firm
Philipps-University Marburg, Am Plan 2, 35032 Marburg, Germany
(email: schiemen@wiwi.uni-marburg.de)
Abstract:
Recursion is a well known concept within computer science and mathematics . In business management it is rarely used explicitly. Counterexamples are applications of dynamic programming and
the ´Viable System´ of Stafford Beer [Beer, 1972].
This paper demonstrates the broad applicability
of recursion to managing complexity, especially in
business. It starts by showing that recursion is – in
contrast to how it may appear in papers of e.g.
H.A.Simon [Simon, 1973] – a special case of (intra-systems-)hierarchy. It shows the two different
yet related forms of using recursion: recursive objects and recursive problem solving and gives general examples of both. It then focuses on recursive
objects and problem solving in business and finally
summarizes the advantages of recursion.
Contents
1.
Recursion as a special case of hierarchy
1.1 Hierarchy and Complexity
1.2 Recursive models
1.3 Recursive problem solving
2.
Recursive models in business
3.
Recursive problem solving in business
4.
Advantages of recursion
1. Recursion as a special case of hierarchy
1.1 Hierarchy and Complexity
In business the notion of ´hierarchy` is generally associated with a relation between a superior and her/his subordinates. This partly results from, but is not identical to
what is here meant by hierarchy. Here we follow
H.A.Simon, according to whom hierarchy is „... a system
that is composed of interrelated subsystems, each of the
latter being, in turn, hierarchic in structure until we reach
some lowest level of elementary subsystem." [Simon,
1962, p. 468]
More formally we can formulate:
A system H = (A, B, ...) of subsets A, B, ... of a set O is
called a hierarchy, when
(a)
O belongs to H;
(b)
if A, B belong to H, A and B are either disjunctive or one is contained in the other;
(c)
all subsets of one element only belong to H.
[Schneider, 1991, p. 364]
The importance of hierarchy in this sense was first
elaborated in Simon’s famous article „The Architecture
of Complexity“, showing that "Hierarchy ... is one of the
central structural schemes that the architect of complexity
uses." [Simon, 1962, p. 468] A statement that is frequently quoted in the literature.
As far as complexity is concerned, our view is a combination of the views of the German sociologist H. Willke
[Willke, 1987] and the German psychologist D. Dörner
[Dörner, 1989]. A system, problem or decision field is the
more complex, the more levels it has, the higher its connectivity is, the more important its consequences are and
the less “Superzeichen”* an actor has in order to cope
with them [Schiemenz, 1996]
1.2 Recursive Models
In a later paper Simon states that "... nature is organized
in levels because hierarchic structures - systems of Chinese boxes - provide the most viable form for any system
of even moderate complexity." [Simon, 1973, p. 27]
Similarly Grobstein speaks of "hierarchical order in Chinese boxes and subdivided triangles." [Grobstein, 1973,
p. 32] Sketches of these Chinese boxes and subdivided
triangles can be found in fig 1.
Chinese boxes and subdivided triangles are indeed of
hierarchical nature according to the definition given
above. They are, however, a specific type of hierarchy.
„An object is called recursive if it contains itself as a part
or if it is defined with the aid of itself.“ [Wirth, 1983, p.
149]. Chinese boxes contain Chinese boxes, subdivided
triangles contain subdivided triangles. They are recursive
(hierarchical) objects.
*
Complex symbols like ‘change gears’, ‘innovator’ or ‘system’.
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Fig. 1: Chinese Boxes and Subdivided Triangles as
Examples of Recursive Models
More recursive objects are surrounding us. Pieces of
music, language or pictures contain smaller, but similar
pieces of music, language or pictures respectively
[Hofstadter, 1979]. Russian dolls contain puppets. In data
management systems directories unfold into (sub-)
directories, indicated by the same symbol. (Graphtheoretical) Trees exist of (sub-) trees, independent of how
they are painted (see fig. 2) and are the usual way to bring
order out of chaos.
Fig. 3: A television picture presented live in TV as an
example of a recursive object
1.3 Recursive Problem Solving
With regard to problem solving recursion is defined as
„... the reduction of the general task to a ‘simpler’ task of
the same class.“ [Bauer/Goos/Dosch, 1991, p. 59]
If one views the task as an object or problem to be
solved, one recognizes the relationship to recursive
objects. One can solve problems recursively if they
contain problems of the same class.
Feedback control uses this concept. The initial problem
of eliminating a larger difference between a controlled
variable and its target value YW (1) = W is stepwise
reduced to the problem of eliminating smaller differences
YW (k), k = 2, 3, ..., until YW (k) is small enough, when
(stable) discrete time feed back control systems are
employed And we all know how effective and frequently
used feed back control is. „It is difficult to conceive of
any control process not involving feedback that is
scientifically interesting and significant“ [Tou, 1964, p.
2].
target value of
the controlled
variable
W
manipulated
variable
Yw
controller
controlled
system
U
Z
YW (k) = W - R
YW (1) = W
Y
S
R
S
disturbance
variable
YW (k-1)
Fig. 4: Recursion in Feedback Control
Fig. 2: Presentation of a recursive (hierarchic) structure
by (a) nested sets, (b) nested brackets, (c) insertion, (d)
graphs
And if one feeds the picture of a man besides his television back to the television one sees an example of a recursive object too, as depicted in fig. 3.
Modern control theory goes further. Not only does it aim
at eliminating this difference but it strives to eliminate it
in an optimal way. One approach used to achieve this
purpose is dynamic programming with its underlaying
principle of optimality: „An optimal policy has the
property that whatever the initial state and initial decision
are, the remaining decisions must constitute an optimal
policy with regard to the state resulting from the first
2
decision.“ [Bellman, 1957, p. 83]. Using this approach, a
problem with n steps, e.g. planning periods, is recursively
seen as consisting of problems of the same class with (n1), (n-2), ....,1 steps and solved, beginning with the
problem with 1 step only. Tapiero developed a whole
theory of managerial planning on the basis of this
approach [Tapiero, 1977].
But also in daily life we use recursive problem solving,
quasi intuitively. One example is the insertion of cards
into a stack (see fig. 5) . An other example is climbing a
hill when it is foggy in the direction of the steepest ascent. The theoretical basis for this can be found in sort
algorithms and in books on optimum seeking [e.g. Wilde,
1964]
each other [Hall/Fagen,1956]. It is also characterized by
the fact, that the system is part of a (super-) system of a
lower resolution level, and its parts themselves are (sub-)
systems of an even higher resolution level and that supersystem and subsystems are systems in the defined sense.
(See fig. 6) [Schiemenz, 1993]
Fig. 6: A recursive view of ´System´
Fig. 5: Recursive solution of the insertion of file cards
A further utilization of recursive problem solving is in
connection with complete induction:
When the propositions A(n0), A(n0+1), A(n0+2) ...are
given it is sufficient to show that
1. the proposition A(n0) is correct and
2. that for any nonnegative integer n  n0: if A(n) then
A(n+1)
[Stowasser/Mohry, 1978, p. 19 ff.]
To always model reality into similar objects is an advantage that seems to be more and more appreciated. In
the ´Fractal Factory` [Warnecke, 1992, 1993] a factory
consists of factories. And Ìntrapreneurship` leads to enterprises within enterprises. [Oden, 1997]. Similarly, cost
centers can be resolved into cost centers, regions into
(sub-)regions etc,
3. Recursive problem solving in business
Products can also be constructed of products and projects
can be subdivided into projects. (See fig. 7 and 8)
Layer
1
2. Recursive Models in Business
For cyberneticians and systems theorists, the best known
recursive model in business is the ´Viable System´, first
introduced by Stafford Beer [Beer, 1972] and further
developed by members of the St. Gallen Business School
[e.g. Malik, 1996; Espejo/Schwaninger, 1993) and others.
According to this organizational approach a system is
viable when it has a specific informational and control
structure and when its operational subsystems are viable
themselves.
Independent of this, German economists [Zschocke,
1974; Eichhorn, 1993] suggested the concept of `Production System`. It is productive in so far as it takes goods
(or also bads) up from its environment, combines these
with others from within to new goods (or bads) of higher
value for its own use or especially that of the environment. And it is a system in so far as it – recursively
– consists of production systems on the next resolution
level. This view fits very well into a (specific, recursive,
multi-level) view of `System´. According to this view a
system is not only characterized by the fact that it
consists of elements with attributes and relations with
2
3
4
Fig. 7: Recursive Bill Explosion
Therefore one can compute the demand for the products
on level n and the time for their production recursively,
starting with the final product on level 1. The level of a
product here is determined according to the path with the
maximum number of sectors between the product concerned and the final product.
If one, correspondingly, wants to compute a project
with multiple levels, e.g. its critical path, one first has to
compute the critical paths between all incoming and
outgoing relations of each (sub-) project of level n before
3
one can compute the projects on level (n-1). The same
holds true if one wants to reduce the duration of projects
with minimum additional cost.
[Bauer, Goos and Dosch, 1991] F.L. Bauer, G. Goos and
W. Dosch. Informatik - Eine einführende Übersicht.
Springer Verlag, Berlin u.a., 1991.
[Beer, 1972] St. Beer. Brain of the firm – The managerial
cybernetics of organization. The Penguin Press,
London, Herder and Herder, USA, 1972.
[Bellmann, 1957] R. Bellman. Dynamic Programming.
1957
[Bohr, 1993] K. Bohr. Effizienz und Effektivität. in: W.
Wittmann et al. (Hrsg.). Handwörterbuch der Betriebswirtschaft. 5. Aufl., Sp. 855 – 869. Schäffer-Poeschel
Verlag, Stuttgart, 1993.
[Dörner, 1989] D. Dörner. Die Logik des Mißlingens Strategisches Denken in komplexen Situationen. Verlag
Rowohlt, Reinbek 1989.
[Eichhorn, 1993] W. Eichhorn. Produktionskorrespondenzen. in: W. Wittmann et al. (Hrsg.).
Handwörterbuch der Betriebswirtschaft, 5. Aufl. Bd. 2,
Sp. 3443 – 3450. Schäffer-Poeschel Verlag, Stuttgart,
1993.
Fig. 8: Recursive Computation of Project Networks
Finally, recursive thinking can help to clarify the relation
between the conceptions of efficiency and effectiveness
that are so important in business management.
According to Barnard efficiency means to do the things
right (extrinsic aspects), effectiveness means to do the
right things (intrinsic aspects). [Barnard, 1975, p. 19 ff.;
Bohr, 1993]
However, different levels of „right things“ exist, e.g.
social welfare  maximizing profit  cost leadership 
cost minimization. Therefore „doing the right things“ is
„doing the things right“ in reference to the higher level.
4. Advantages of Recursion
[Espejo and Schwaninger, 1993] R. Espejo, M. Schwaninger (Ed.). Organisational Fitness – Corporate
Effectiveness through Management Cybernetics.
Campus Verlag, Frankfurt et al., 1993
[Grobstein, 1973] C. Grobstein. Hierarchical Order and
Neogenesis. in: H.H. Pattee (ed.). Hierarchy Theory The Challenge of Complex Systems. New York,
Braziller, 1973.
[Hall and Fagen, 1956] A.D. Hall, R.E. Fagen. Definition
of System. in: General Systems 1956, p. 18 – 28.
[Hofstadter, 1979] D.R. Hofstadter, Gödel, Escher, Bach:
an Eternal Golden Braid. Basic Books, New York,
1979.
[Malik, 1996] F. Malik. Strategie des Managements
komplexer Systeme. 5. Aufl., Haupt Verlag, Bern et al.
1996.
Recursion is, as we have seen, a special case of hierarchy.
It therefore has all the advantages of hierarchy, pointed
out by Simon [Simon, 1962]. However, it has additional
advantages.
The recursive system structure makes it possible to use
the same way of thinking on each system level. It
contains principles to structure details, methods,
techniques, programs etc. This accounts to a remarkable
improvement in creative variety and leads to considerable
rationalization effects. [Malik, 1996, p. 102]
[Oden, 1997] H.W. Oden. Managing corporate culture,
innovation, and intrapreneurship. Quorum Books,
Westport et al., 1997.
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