Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol66.Issue2.Jun2000:
Int. J. Production Economics 66 (2000) 185}195
Optimising product recycling chains by control theory
U. Kleineidam*, A.J.D. Lambert, J. Blansjaar, J.J. Kok, R.J.J. van Heijningen
Eindhoven University of Technology, Faculty of Technology Management, P.O. Box 513, NL-5600 MB Eindhoven, Netherlands
Received 15 March 1999; accepted 24 September 1999
Abstract
In this paper, a modelling method is described for production chains including recycling. It consists of elementary
models of standard production operations, connected by market modules. The models are analysed using methods from
control theory. These methods allow us to investigate essential properties of the chain concerning its dynamical
behaviour, particularly with respect to stability and controllability. These properties are prerequisites for e!ective chain
management, as in the evaluation of recycling policy. A case study on paper recycling was carried out to demonstrate the
applicability of this method to practice. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Recycling; Control theory; Product chain; Sustainability; Optimisation
1. Introduction
Modelling of material and energy #ows has
appeared to be a useful tool in economy, particularly in the study of the behaviour of systems
such as product life-cycle chains. A commonly applied method for chain modelling is the product}process chain [1]. This is a chain consisting of
alternate transformation processes and product
#ows between these processes. For environmental
purposes, the full material life-cycle within the technosystem, from `cradle to gravea, is frequently
modelled. In the simplest case it is linear, i.e. it
excludes cycles. These linear chains are typically
considered in gross energy requirement (GER) and
life-cycle assessment (LCA) studies [2}4]. In these
studies the environmental impact caused by a product is determined by adding the impacts of all the
* Corresponding author.
processes in the life-cycle of a product in a standard
way. In common practice, however, the situation is
growing more and more complex because of the
increasing importance of the recycling of materials.
Traditionally, the emphasis only was on the recycling of process waste, i.e. the waste that is generated
as an unintended by-product of production processes. Nowadays, there is a growing interest for
post-consumer or product waste that originates
from discarded products. This concern is closely
related to the economic bene"ts that can be
achieved by the enterprises in the chain, the concept
of product responsibility and the intention to decrease both resource consumption and waste
quantity. Such a policy is often speci"ed by catchwords such as closing the chain, although a fully
closed chain is neither obtainable nor desirable [5].
This idea, that stems from the study of the often
surprisingly e$cient functioning of the many closely interwoven processes in nature, is studied in the
interdisciplinary "eld that is named according to
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 0 - 6
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
its biological counterparts: industrial ecology,
industrial metabolism or industrial symbiosis. For
realistic modelling the inclusion of recycle loops is
indispensable.
The product}process chain model, in which physical #ows play an essential role, is controlled by
various information #ows (or signals). Each unit in
the model, typically a production enterprise, is
a partially autonomous decision unit. There are,
however, signals to the enterprise from other production units within the chain, usually based on
transactions (horizontal control). Moreover, the
authorities want to control the chain in such a way
that it acts as a whole in an optimal way (vertical
control). From the point of view of authorities,
enterprises, and consumers, a proper policy should
be implemented to obtain optimal results. Each
actor has di!erent purposes and is, moreover, able
to make the right decisions based on an optimal
adaptation to its surroundings. This implies a complicated multi-level hierarchical decision structure
and impedes a clear insight into the consequences
of decision making, and the choice of optimal
decisions.
Many studies in this "eld are restricted to descriptive chain modelling (see for example [6] for
a wide range of products from aluminium to
packaging products). In fewer cases, the possibility
is included of adaptation of internal variables of
a mathematical model to external parameters. This
holds for optimisation models in which systems
operation or technology choice are incorporated.
Such models are usually based on mathematical
programming (MP-)techniques, such as static and
dynamic linear programming, where the optimisation criterion is expressed by an objective function.
So far, little attention has been paid to methods for
model analysis that are available in systems and
control theory. These are widely applied in mechanical, physical and chemical systems, varying from
relatively simple systems like robot arms, up to
complex systems such as process industries and
power plants. Although analogies between completely di!erent kinds of systems are abundant in
modelling, this has rarely been applied in studies on
the meso, i.e. product chain, level.
A property of product chains, which can be analysed by control theory, is the controllability. It
embraces questions concerning, for instance, the
possibility for a system to reach a de"nite state
starting from some initial state or concerning the
possibility to control a system in such a way that its
state follows a de"nite trajectory over time. This is
relevant to environmental policy, because environmental policy is concerned with reaching some target, for example, a decrease of materials use by
10%, within a de"nite period of time, e.g. 5 years.
It is evident that the e!ectiveness of an environmental policy strongly depends on the possibility of
obtaining the proposed targets. In this paper, the
application of systems and control theory is elaborated on some problems that are related to product
chains. Both physical and economic aspects are
incorporated, with an emphasis on the environmental policy. The resulting model is tested on an
existing chain, viz. the Dutch paper chain. This
chain was selected because it includes an e!ective
recycling system. Moreover, it is well documented
by rather extended time series of statistical data.
2. The product chain as a system
In industrial ecology (see, e.g. [1]) the physical
world is subdivided into ecosystem and technosystem. The technosystem involves the settings that
are designed and controlled by people. There is
a mutual exchange of physical, i.e. material and
energy, #ows between these two systems. In a simpli"ed presentation of the technosystem, #ows of
matter and energy are successively extracted, transformed into products, consumed, and "nally
discharged. This is the linear chain that is depicted
in the process}product scheme of Fig. 1. In the real
world modi"cations of the linear chain are important. A fundamental extension is the inclusion of
loops that represent recycling and reuse of process
residues and discarded products. Usually, however,
the process and product wastes are only partially
recyclable due to technical and economic reasons.
There will always be some remaining #ows that are
discharged to the ecosystem. This is depicted in
Fig. 2.
Although the aggregate #ows within the total
technosystem are depicted here, in practice only
a part of the technosystem can be studied and,
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
Fig. 1. The process}product scheme of a linear chain.
187
cannot be included in the study up to the same
extent. In the paper example paper is made of pulp,
which in turn is made of wood. Clay and other
paper chemicals are added, and so on.
Geographical system boundaries also play a role.
In general, not the global system but rather the
production system of a particular country or region
is studied. This requires the de"nition of import
and export. With further detail, disaggregation of
production systems may also be needed. Paper
production, for example, involves a sequence of
di!erent production steps, which may vary according to the kind of paper product that is produced.
Di!erentiation between paper products is also
required with respect to the di!erent characteristics
these products have in the consumption and recycling phases.
Disaggregation and recycling transforms the
chains of Figs. 1 and 2 into complicated networks
that consist of many strongly interconnected
chains. Within these networks, basic modules can
be discerned. In the following, the essential features
of a basic module will be described and a model of
such a module will be presented.
3. A generalised company model
Fig. 2. The process}product scheme of a chain with recycling.
consequently, the #ows and the processes should be
disaggregated. When focusing on such a partial
#ow, for example paper and paper products, ambiguity arises in setting appropriate system boundaries. Moreover there is an exchange of physical
#ows between di!erent production chains, which
As was stated before, a process}product chain is
considered to be a network of basic units, each
represented by a generalised company model.
A characteristic of the product}process chain description is the succession of product #ows and
processes. Although the description deals with
physical #ows, these are controlled by economic
mechanisms. They are incorporated within the
model by the introduction of markets. Transactions
of products between the parts of the chain are
described by a market model * a market is the
interface between companies. Fig. 3 shows a simpli"ed #owchart of a company including two storage
facilities, accompanied by two markets that refer to
the input and the output #ows of the system. The
boundary of the company subsystem is indicated
by the dashed line. As a company's basic activity is
the transformation of material #ows; it is represented as a process (square). Storage is represented by
a triangle, see Fig. 3. Entering #ows consist of raw
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
Fig. 3. A company #owchart.
materials or semi-manufactured products and originate from a market; leaving #ows are products that
are directed towards another market.
These #owcharts are graphical representations
from which mathematical models can be derived
providing insight into the functioning of the chain.
As dynamic processes such as storage and dynamic
markets play an important role, dynamic models
are needed. Such a model is represented here as
a set of di!erential equations.
3.1. The market model
Here a market is considered to be a virtual place
in which a trade-o! between demand and supply
takes place. As a whole it is governed by some price
mechanism. Although in economic science markets
are crucial, the development of quantitative market
models has proven to be extremely di$cult [7}9].
In standard economic theory there is a demand
curve that decreases with increasing price, and
a supply curve that increases with increasing price.
The intersection of these two curves is the market
equilibrium. However, the shape of the curves is
a result of di!erent mechanisms (substitution,
change of real income, etc.) that cannot be analysed
individually. As an extra complication, the dynamics of the system plays a role. Change in supply is
often not instantaneously possible, but requires
preparations, such as investments in capital goods,
or the recruitment of an appropriately skilled
labour force. On the other hand, in the case that
demand and supply are in disequilibrium on a market, the price adjustment will not occur instantaneously but rather in a dynamic way. A normally
functioning market is in equilibrium, but the point
of equilibrium will gradually change, because the
economy as a whole is not in equilibrium.
For the purpose of a chain model we assume
a strongly simpli"ed market model, which also describes non-equilibrium situations. To this purpose
we observe that two quantities are measurable: the
trade t (in kg/yr), which represents the real product
#ow that enters or leaves the system, and the price
p (in $/kg). Each of these quantities may vary over
time according to the time scale that is chosen.
It is assumed that there are di!erent producers
(indicated by the index i) and consumers (indicated
by the index j). The total supply and demand are
denoted by s and d , respectively. The amount of
505
505
traded products then reads as follows:
t"min(s , d ).
505 505
(1)
In Fig. 4, an example of demand and supply curves
is depicted. In case t"s (d there is scarcity, in
505
505
the case t"s 'd there is excess. Competition
505
505
mechanisms between di!erent producers and consumers are not modelled so that distribution proceeds proportionally. This means that in case of
excess a producer i supplies the share
d
505 s .
s i
505
(2)
In case of scarcity a consumer j buys the share
s
505 d .
d i
505
(3)
When the production of some producer surpasses
t the remaining production is stored. This will only
i
be possible up to some extent however because
storage is associated with costs, and the storage
capacity is restricted.
As the system studied here generally is not in
equilibrium, the system dynamics have to be taken
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
189
there is only one producer and one consumer. In
this case the subscripts i, j and tot are omitted. It is
assumed that the producer produces to stock, so
that the production equals
q"c (g !g) with q*0.
(5)
1 s
Here g is the desired stock, g is the actual stock
s
and c a proportionality constant expressed in
1
yr~1. For the stock of "nished products the following equation holds, because of mass conservation:
g5 "q!t.
Fig. 4. Supply and demand curves.
into account by dealing with an adjusting mechanism that may establish an equilibrium situation.
There are two basic theories to describe such
a mechanism: the Marshallian theory of quantity
adjustment that assumes a di!erence between
hypothetical prices as a driving force for adjusting
product #ows, and the Walrasian theory of price
adjustment that assumes a di!erence between
hypothetical product #ows as a driving force for
adjusting prices [10]. As in our description hypothetical product #ows, viz. supply and demand, are
at the centre, the Walrasian description is adopted
here. Its basic dynamics are described by the
following:
p5 "c(d !s ).
(4)
505
505
Here c is a proportionality constant, measured in
$/kg2.
(6)
As a matter of fact, all storage processes are
described by di!erential equations similar to (6).
When companies and markets are combined to
a chain or network, markets are nodes for which
the entering #ows equal the leaving #ows because
markets have no storage capacity.
In the model, discontinuities in the functions or
in their derivatives occur frequently. Examples are
(1), see Fig. 4, and cases in which a splitting of
product #ows takes place. When, for example,
waste is generated, the decision for recycling or
"nal processing might be taken on the basis of the
processing costs. In standard modelling techniques
like linear programming these discontinuities do
not cause essential di$culties; in control theory,
however, they are usually smoothened.
As an example, for the recycling percentage, the
Heaviside function is replaced by the smoothening
function, see Fig. 5:
(7)
f(p)"1 (1#tanh(c (p!p ))).
2
&1
2
Here p is the "nal processing price and f(p) is the
&1
recycling percentage, p is the price for recycling,
c is a constant of proportionality, expressed in
2
kg/$. In the model incineration is taken as the "nal
processing technology. Thus, directly in#uencing
the incineration price is a method for controlling
the recycling percentage.
3.2. The company model
3.3. Chain dynamics
The company is represented by a stock of resources, a transformation process, and a stock of
"nished products, see Fig. 3. Production is adjusted
according to the di!erence between the actual
amount of "nished products and a prede"ned
norm. Let us, for the sake of simplicity, assume that
Information #ows (signals) rather than physical
#ows are essential to control theory. The abstract
elements of a simpli"ed control system are given in
Fig. 6. It consists of a process P that is characterised by its state vector x. The process is subject to
190
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
is called the state equation, and it describes the
complete chain dynamics. Eq. (9) expresses the output as a function of state, input, and disturbances.
The set of equations is a standard way of describing
a system in control theory, be it physical, chemical,
or economic.
4. A chain model: The Dutch paper chain
In this section, a model of a chain is described
on the basis of a combination of generalised
producers. Several transformation processes are
linked, among which a consumption process. One
recycle loop is present.
Fig. 5. The recycling percentage as a function of the price p of
a product.
Fig. 6. The abstract elements of a control system.
external disturbances, which are denoted by v. The
output signal from the process is y. It is measured
by a sensor S, which produces the measured signal
y . The measured signal is compared with the
.
desired signal y by a controller C that produces
$
the control signal u. This signal is fed to the process.
The process dynamics are described by the following evolution equations:
x5 "f(x, u, v),
(8)
y"g(x, u, v).
(9)
For a generalised production system the state
vector is composed of the stocks in the di!erent
storage facilities and the market prices. Eq. (8)
4.1. The chain model
The model that is presented in Fig. 7 is based on
the present situation in the Netherlands. Actually,
used paper contributes for a considerable part to
the raw materials supply, which further consists of
primary pulp. Another secondary resource, process
waste, is not described by the model, as it is incorporated implicitly in the producer's characteristics.
For the sake of simplicity, existing control mechanisms as determined by the Dutch Paper Fibre
Convenant (including a takeback obligation for
used paper and the determination of a guaranteed
price), are omitted in the model. Fig. 7 shows the
#ows of paper, cardboard and their waste as they
currently are encountered in the Dutch paper
chain: the paper industry buys a mix of primary
pulp and processed wastepaper (secondary pulp) at
their respective markets. Subsequently, the paper
and board industry supplies the virgin paper it has
produced to the paper and board market (market
2). The paper and board market also experiences
supply from foreign suppliers. Part of the bulk
paper and board is bought by the paper and board
product industry, which converts these raw
materials into products of paper and board (printed
paper, cardboard packaging boxes, and so on).
Currently, wastes from the sector (o$ces, shops,
services and companies) are supposed to be transferred to the waste paper market. The major part
of the waste paper from households is also collected separately from other household waste by a
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
191
Fig. 7. The #owchart of the paper chain.
large and many-branched network of wastepaper
collection, largely kept up by non-pro"t associations. There is no market between paper and
board consumption and its collection. The col-
lected #ows form a supply to the wastepaper market where the resources are sold to the wastepaper
processor. Depending on the price the collector
might receive for his waste paper he will either
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
supply waste paper to the market or decide to
incinerate it.
4.2. Paper chain modelling
The dynamic variables involved in the presented
model are included in the system's state vector x.
According to Fig. 7, this vector consists of seven
elements: the prices p through p (prices of pri1
3
mary and secondary pulp, and paper) and the
stocks g through g (primary pulp, paper, waste
1
4
paper, and secondary pulp). In the model, control
takes place by economic instruments. It is e!ectuated here by in#uencing the prices: by setting the
incineration price or by imposing taxes on primary
pulp and on paper, respectively. So the input vector
u consists of three components. Disturbances v are
imposed on the system by external supply and
demand. These are present for pulp, paper, and
waste paper, so that the disturbance vector v has six
components here. When the control aims to minimise the environmental impact, the output vector
y might consist of components that represent the
material and product #ows. Combining all the
company and market models and using the notation introduced above leads to a model in the usual
state space notation (8) and (9).
5. Model evaluation
5.1. Parameter adjustment
Apart from variables, parameters are encountered in the model that is described here, for
example the proportionality constants in (4), (5)
and (7). Several standard parameter identi"cation
techniques exist in control theory, see for example
[11]. Here we adjust the parameters by minimising
the squared di!erence between the model outcomes
and available time series. The available time series,
which were collected by National Statistics, are
given by the prices of pulp, paper, and waste paper.
Unfortunately the amount of available data is
rather limited here, and the accuracy of these data
is somewhat uncertain. Therefore the result of the
adjustment must be considered as a rough estimate
of the appropriate parameter values. Fig. 8 shows
Fig. 8. Model outcomes with estimated parameters.
the course of the yearly averaged prices of the
di!erent products in the paper chain as they
were observed in the period from 1986 to 1993.
The model outcomes with a particular parameter
set are represented by lines. As can be seen from
the "gure they closely approximate the chain behaviour.
5.2. Stability analysis
An essential characteristic of dynamic systems,
which can be analysed by control theory, is its
stability. A system is said to be stable if its state
evolves into a certain constant value. Stability of
the controlled system is an absolute requirement
since its absence causes signals to grow beyond
any limit, eventually destroying and breaking
down the system. As an example, an unlimited
growth of some price will bring the economic relations in the product chain to an end, a situation
which is certainly undesirable for any of the chain
actors.
Decisive stability analysis for systems like this, in
which discontinuities and non-linearities are encountered, is a complex task, but two methods are
indicative with respect to stability. Simulation can
be carried out with the known parameters included in the system. Results of this calculation
are shown in Fig. 9, which represents the state
variables x divided by their value at the end of the
i
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
193
Fig. 10. Eigenvalues of the system matrix.
Fig. 9. The stable chain.
simulation horizon. The simulation indicates stability, albeit for a particular set of initial values.
The second method is based on linearisation
around the points of the system trajectory
(x (t), u (t), v (t)). Via the decomposition around
0
0
0
this reference trajectory (where *x"x!x ,
0
*u"u!u , *v"v!v ), (8)}(9) is linearised. As
0
0
a result of this transformation (8)}(9) can be written
as follows:
*x5 "A *x#B *u#E *v,
*y"C *x#D *u#F *v,
(10)
with A, B, C, D, E, F matrices of appropriate dimensions. These relations are valid in a neighbourhood
of the reference trajectory. For the linearised system
linear control systems analysis can be applied.
According to standard linear control theory a
system is stable if the eigenvalues of the matrix A
lie in the left complex half-plane. For the product chain system the eigenvalues of A are depicted
in Fig. 10 for some selected points of the same
trajectory as in Fig. 9. As the state vector has seven
components, the matrix A, at each point of the
trajectory, has seven eigenvalues, some of which
can coincide. It is shown here that the linearised
model is indeed stable because the eigenvalues
of the system matrix lie in the left complex halfplane.
5.3. Controllability analysis
In complex systems, the question of the selection
of the inputs that are used for system control is
relevant. In a typical plant, for example, many
possibilities for process control exist, but sensitiveness of the output to various inputs can be
signi"cantly di!erent. The same might be true for
production chains. The relevant question here is
whether or not the system is controllable by the
selected inputs. A system is said to be controllable if,
for a given initial situation, each value of the state
x can be reached at a chosen time ยน by an appropriate choice of the input u. In standard linear
control theory (see, e.g. [12]) controllability can be
assessed by examining the rank of the so-called
controllability matrix C"[B AB 2 An~1B] with
A and B from (10), see the Appendix for an illustrative example. For controllability the rank of C must
be of the same magnitude as the number of
elements in the state vector x. Fig. 11 shows the
rank of the product chain controllability matrix for
the given trajectory. As the rank of this matrix
varies between two and "ve, the value of the full
rank, i.e. seven, is not obtained here. The system
cannot be completely controlled by the given inputs. This means that not all the state variables can
be brought to arbitrary values by the given instruments.
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
inclusion of the non-linear structure of the models
is therefore necessary to obtain more general
results.
Appendix
As an example to illustrate the controllability
of a system, consider the simple two-dimensional
linear system de"ned by the following:
C D CD
1 0
x5 "
1 1
1
u"Ax#Bu.
0
x#
(A.1)
For this system the controllability matrix is given
by
Fig. 11. Rank of the controllability matrix.
C D
C"[B AB]"
6. Conclusions
The study of chain dynamics is relevant for
appropriate control of product chains, which is
of considerable economic, societal, and environmental interest. Control theory o!ers the tools to
analyse system models with respect to essential
properties, such as stability and controllability.
These tools have long been used to deal with this
kind of problem mainly for dynamical mechanical,
electrical and chemical systems, and appear to be
particularly suited for the analysis of product
chains, because product chains show analogues to
technical systems. Starting out with a standard
production system description, chains can be composed.
For the example of the paper chain, some results
of stability and controllability analysis are shown.
It is made clear that controllability analysis can
help us to "nd strategies to identify the ways to
in#uence product chains. On the basis of the presented model of the paper chain, it can be concluded that the chain is not controllable by the
incineration price and taxes on pulp and paper.
Additional instruments such as regulatory instruments and convenants between government and
industry should be taken into consideration.
A principle restriction of the model is that a linear approximation is applied. Further analysis with
1 1
0 1
and has full rank: two. On the other hand if the
input matrix B is changed to
CD
B"
C D
0 0
0
, then C"
with rank C"1.
1
1 1
The "rst system is controllable whereas the second
is not. This is intuitively clear because in the second
example only the second component of x5 can be
in#uenced by the input u but x , in turn, does not
2
in#uence x because the upper right element of A is
1
0. Thus x cannot be altered into any desired value.
1
References
[1] R.U. Ayres, Industrial metabolism, in: J.H. Ausubel, H.E.
Sladovich (Eds.), Technology and Environment, National
Academy Press, Washington, DC, 1989, pp. 23}49.
[2] IFIAS, Report of the International Federation of Institutes
for Advanced Study, Workshop no. 6 on Energy Analysis,
Guldsmedshyttan (S), 1973.
[3] J.B. GuineH e, H.A. Udo de Haes, G. Huppes, Quantitative
life cycle assessment of products, Part one: Goal de"nition
and inventory, Journal of Cleaner Production 1 (1) (1993)
3}13.
[4] J.B. GuineH e, H.A. Udo de Haes, G. Huppes, Quantitative
life cycle assessment of products, Part two: Classi"cation,
valuation and improvement analysis, Journal of Cleaner
Production 1 (2) (1993) 82}91.
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
[5] T.E. Graedel, B.R. Allenby, Industrial Ecology, PrenticeHall, Englewood Cli!s, NJ, 1995.
[6] R.U. Ayres, L.W. Ayres, Industrial Ecology: Towards
Closing the Materials Cycle, Edward Elgar, Cheltenham,
1996.
[7] K.J. Arrow, F.H. Hahn, General Competitive Analysis,
Holden-Day, San Francisco, CA, 1971.
[8] G. SchwoK diauer (Ed.), Equilibrium and Disequilibrium in
Economic Theory, Reidel, Dordrecht, 1978.
195
[9] C.R. Plott, G. George, Marshallian vs. Walrasian stability
in an experimental market, The Economic Journal 102
(1992) 437}460.
[10] A. Takayama, Mathematical Economics, Cambridge University Press, Cambridge, 1985.
[11] J.P. Norton, An Introduction to Identi"cation, Academic
Press, London, 1986.
[12] D.G. Luenberger, Introduction to Dynamic Systems:
Theory, Models and Applications, Wiley, New York, 1979.
Optimising product recycling chains by control theory
U. Kleineidam*, A.J.D. Lambert, J. Blansjaar, J.J. Kok, R.J.J. van Heijningen
Eindhoven University of Technology, Faculty of Technology Management, P.O. Box 513, NL-5600 MB Eindhoven, Netherlands
Received 15 March 1999; accepted 24 September 1999
Abstract
In this paper, a modelling method is described for production chains including recycling. It consists of elementary
models of standard production operations, connected by market modules. The models are analysed using methods from
control theory. These methods allow us to investigate essential properties of the chain concerning its dynamical
behaviour, particularly with respect to stability and controllability. These properties are prerequisites for e!ective chain
management, as in the evaluation of recycling policy. A case study on paper recycling was carried out to demonstrate the
applicability of this method to practice. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Recycling; Control theory; Product chain; Sustainability; Optimisation
1. Introduction
Modelling of material and energy #ows has
appeared to be a useful tool in economy, particularly in the study of the behaviour of systems
such as product life-cycle chains. A commonly applied method for chain modelling is the product}process chain [1]. This is a chain consisting of
alternate transformation processes and product
#ows between these processes. For environmental
purposes, the full material life-cycle within the technosystem, from `cradle to gravea, is frequently
modelled. In the simplest case it is linear, i.e. it
excludes cycles. These linear chains are typically
considered in gross energy requirement (GER) and
life-cycle assessment (LCA) studies [2}4]. In these
studies the environmental impact caused by a product is determined by adding the impacts of all the
* Corresponding author.
processes in the life-cycle of a product in a standard
way. In common practice, however, the situation is
growing more and more complex because of the
increasing importance of the recycling of materials.
Traditionally, the emphasis only was on the recycling of process waste, i.e. the waste that is generated
as an unintended by-product of production processes. Nowadays, there is a growing interest for
post-consumer or product waste that originates
from discarded products. This concern is closely
related to the economic bene"ts that can be
achieved by the enterprises in the chain, the concept
of product responsibility and the intention to decrease both resource consumption and waste
quantity. Such a policy is often speci"ed by catchwords such as closing the chain, although a fully
closed chain is neither obtainable nor desirable [5].
This idea, that stems from the study of the often
surprisingly e$cient functioning of the many closely interwoven processes in nature, is studied in the
interdisciplinary "eld that is named according to
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 2 0 - 6
186
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
its biological counterparts: industrial ecology,
industrial metabolism or industrial symbiosis. For
realistic modelling the inclusion of recycle loops is
indispensable.
The product}process chain model, in which physical #ows play an essential role, is controlled by
various information #ows (or signals). Each unit in
the model, typically a production enterprise, is
a partially autonomous decision unit. There are,
however, signals to the enterprise from other production units within the chain, usually based on
transactions (horizontal control). Moreover, the
authorities want to control the chain in such a way
that it acts as a whole in an optimal way (vertical
control). From the point of view of authorities,
enterprises, and consumers, a proper policy should
be implemented to obtain optimal results. Each
actor has di!erent purposes and is, moreover, able
to make the right decisions based on an optimal
adaptation to its surroundings. This implies a complicated multi-level hierarchical decision structure
and impedes a clear insight into the consequences
of decision making, and the choice of optimal
decisions.
Many studies in this "eld are restricted to descriptive chain modelling (see for example [6] for
a wide range of products from aluminium to
packaging products). In fewer cases, the possibility
is included of adaptation of internal variables of
a mathematical model to external parameters. This
holds for optimisation models in which systems
operation or technology choice are incorporated.
Such models are usually based on mathematical
programming (MP-)techniques, such as static and
dynamic linear programming, where the optimisation criterion is expressed by an objective function.
So far, little attention has been paid to methods for
model analysis that are available in systems and
control theory. These are widely applied in mechanical, physical and chemical systems, varying from
relatively simple systems like robot arms, up to
complex systems such as process industries and
power plants. Although analogies between completely di!erent kinds of systems are abundant in
modelling, this has rarely been applied in studies on
the meso, i.e. product chain, level.
A property of product chains, which can be analysed by control theory, is the controllability. It
embraces questions concerning, for instance, the
possibility for a system to reach a de"nite state
starting from some initial state or concerning the
possibility to control a system in such a way that its
state follows a de"nite trajectory over time. This is
relevant to environmental policy, because environmental policy is concerned with reaching some target, for example, a decrease of materials use by
10%, within a de"nite period of time, e.g. 5 years.
It is evident that the e!ectiveness of an environmental policy strongly depends on the possibility of
obtaining the proposed targets. In this paper, the
application of systems and control theory is elaborated on some problems that are related to product
chains. Both physical and economic aspects are
incorporated, with an emphasis on the environmental policy. The resulting model is tested on an
existing chain, viz. the Dutch paper chain. This
chain was selected because it includes an e!ective
recycling system. Moreover, it is well documented
by rather extended time series of statistical data.
2. The product chain as a system
In industrial ecology (see, e.g. [1]) the physical
world is subdivided into ecosystem and technosystem. The technosystem involves the settings that
are designed and controlled by people. There is
a mutual exchange of physical, i.e. material and
energy, #ows between these two systems. In a simpli"ed presentation of the technosystem, #ows of
matter and energy are successively extracted, transformed into products, consumed, and "nally
discharged. This is the linear chain that is depicted
in the process}product scheme of Fig. 1. In the real
world modi"cations of the linear chain are important. A fundamental extension is the inclusion of
loops that represent recycling and reuse of process
residues and discarded products. Usually, however,
the process and product wastes are only partially
recyclable due to technical and economic reasons.
There will always be some remaining #ows that are
discharged to the ecosystem. This is depicted in
Fig. 2.
Although the aggregate #ows within the total
technosystem are depicted here, in practice only
a part of the technosystem can be studied and,
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
Fig. 1. The process}product scheme of a linear chain.
187
cannot be included in the study up to the same
extent. In the paper example paper is made of pulp,
which in turn is made of wood. Clay and other
paper chemicals are added, and so on.
Geographical system boundaries also play a role.
In general, not the global system but rather the
production system of a particular country or region
is studied. This requires the de"nition of import
and export. With further detail, disaggregation of
production systems may also be needed. Paper
production, for example, involves a sequence of
di!erent production steps, which may vary according to the kind of paper product that is produced.
Di!erentiation between paper products is also
required with respect to the di!erent characteristics
these products have in the consumption and recycling phases.
Disaggregation and recycling transforms the
chains of Figs. 1 and 2 into complicated networks
that consist of many strongly interconnected
chains. Within these networks, basic modules can
be discerned. In the following, the essential features
of a basic module will be described and a model of
such a module will be presented.
3. A generalised company model
Fig. 2. The process}product scheme of a chain with recycling.
consequently, the #ows and the processes should be
disaggregated. When focusing on such a partial
#ow, for example paper and paper products, ambiguity arises in setting appropriate system boundaries. Moreover there is an exchange of physical
#ows between di!erent production chains, which
As was stated before, a process}product chain is
considered to be a network of basic units, each
represented by a generalised company model.
A characteristic of the product}process chain description is the succession of product #ows and
processes. Although the description deals with
physical #ows, these are controlled by economic
mechanisms. They are incorporated within the
model by the introduction of markets. Transactions
of products between the parts of the chain are
described by a market model * a market is the
interface between companies. Fig. 3 shows a simpli"ed #owchart of a company including two storage
facilities, accompanied by two markets that refer to
the input and the output #ows of the system. The
boundary of the company subsystem is indicated
by the dashed line. As a company's basic activity is
the transformation of material #ows; it is represented as a process (square). Storage is represented by
a triangle, see Fig. 3. Entering #ows consist of raw
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
Fig. 3. A company #owchart.
materials or semi-manufactured products and originate from a market; leaving #ows are products that
are directed towards another market.
These #owcharts are graphical representations
from which mathematical models can be derived
providing insight into the functioning of the chain.
As dynamic processes such as storage and dynamic
markets play an important role, dynamic models
are needed. Such a model is represented here as
a set of di!erential equations.
3.1. The market model
Here a market is considered to be a virtual place
in which a trade-o! between demand and supply
takes place. As a whole it is governed by some price
mechanism. Although in economic science markets
are crucial, the development of quantitative market
models has proven to be extremely di$cult [7}9].
In standard economic theory there is a demand
curve that decreases with increasing price, and
a supply curve that increases with increasing price.
The intersection of these two curves is the market
equilibrium. However, the shape of the curves is
a result of di!erent mechanisms (substitution,
change of real income, etc.) that cannot be analysed
individually. As an extra complication, the dynamics of the system plays a role. Change in supply is
often not instantaneously possible, but requires
preparations, such as investments in capital goods,
or the recruitment of an appropriately skilled
labour force. On the other hand, in the case that
demand and supply are in disequilibrium on a market, the price adjustment will not occur instantaneously but rather in a dynamic way. A normally
functioning market is in equilibrium, but the point
of equilibrium will gradually change, because the
economy as a whole is not in equilibrium.
For the purpose of a chain model we assume
a strongly simpli"ed market model, which also describes non-equilibrium situations. To this purpose
we observe that two quantities are measurable: the
trade t (in kg/yr), which represents the real product
#ow that enters or leaves the system, and the price
p (in $/kg). Each of these quantities may vary over
time according to the time scale that is chosen.
It is assumed that there are di!erent producers
(indicated by the index i) and consumers (indicated
by the index j). The total supply and demand are
denoted by s and d , respectively. The amount of
505
505
traded products then reads as follows:
t"min(s , d ).
505 505
(1)
In Fig. 4, an example of demand and supply curves
is depicted. In case t"s (d there is scarcity, in
505
505
the case t"s 'd there is excess. Competition
505
505
mechanisms between di!erent producers and consumers are not modelled so that distribution proceeds proportionally. This means that in case of
excess a producer i supplies the share
d
505 s .
s i
505
(2)
In case of scarcity a consumer j buys the share
s
505 d .
d i
505
(3)
When the production of some producer surpasses
t the remaining production is stored. This will only
i
be possible up to some extent however because
storage is associated with costs, and the storage
capacity is restricted.
As the system studied here generally is not in
equilibrium, the system dynamics have to be taken
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
189
there is only one producer and one consumer. In
this case the subscripts i, j and tot are omitted. It is
assumed that the producer produces to stock, so
that the production equals
q"c (g !g) with q*0.
(5)
1 s
Here g is the desired stock, g is the actual stock
s
and c a proportionality constant expressed in
1
yr~1. For the stock of "nished products the following equation holds, because of mass conservation:
g5 "q!t.
Fig. 4. Supply and demand curves.
into account by dealing with an adjusting mechanism that may establish an equilibrium situation.
There are two basic theories to describe such
a mechanism: the Marshallian theory of quantity
adjustment that assumes a di!erence between
hypothetical prices as a driving force for adjusting
product #ows, and the Walrasian theory of price
adjustment that assumes a di!erence between
hypothetical product #ows as a driving force for
adjusting prices [10]. As in our description hypothetical product #ows, viz. supply and demand, are
at the centre, the Walrasian description is adopted
here. Its basic dynamics are described by the
following:
p5 "c(d !s ).
(4)
505
505
Here c is a proportionality constant, measured in
$/kg2.
(6)
As a matter of fact, all storage processes are
described by di!erential equations similar to (6).
When companies and markets are combined to
a chain or network, markets are nodes for which
the entering #ows equal the leaving #ows because
markets have no storage capacity.
In the model, discontinuities in the functions or
in their derivatives occur frequently. Examples are
(1), see Fig. 4, and cases in which a splitting of
product #ows takes place. When, for example,
waste is generated, the decision for recycling or
"nal processing might be taken on the basis of the
processing costs. In standard modelling techniques
like linear programming these discontinuities do
not cause essential di$culties; in control theory,
however, they are usually smoothened.
As an example, for the recycling percentage, the
Heaviside function is replaced by the smoothening
function, see Fig. 5:
(7)
f(p)"1 (1#tanh(c (p!p ))).
2
&1
2
Here p is the "nal processing price and f(p) is the
&1
recycling percentage, p is the price for recycling,
c is a constant of proportionality, expressed in
2
kg/$. In the model incineration is taken as the "nal
processing technology. Thus, directly in#uencing
the incineration price is a method for controlling
the recycling percentage.
3.2. The company model
3.3. Chain dynamics
The company is represented by a stock of resources, a transformation process, and a stock of
"nished products, see Fig. 3. Production is adjusted
according to the di!erence between the actual
amount of "nished products and a prede"ned
norm. Let us, for the sake of simplicity, assume that
Information #ows (signals) rather than physical
#ows are essential to control theory. The abstract
elements of a simpli"ed control system are given in
Fig. 6. It consists of a process P that is characterised by its state vector x. The process is subject to
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
is called the state equation, and it describes the
complete chain dynamics. Eq. (9) expresses the output as a function of state, input, and disturbances.
The set of equations is a standard way of describing
a system in control theory, be it physical, chemical,
or economic.
4. A chain model: The Dutch paper chain
In this section, a model of a chain is described
on the basis of a combination of generalised
producers. Several transformation processes are
linked, among which a consumption process. One
recycle loop is present.
Fig. 5. The recycling percentage as a function of the price p of
a product.
Fig. 6. The abstract elements of a control system.
external disturbances, which are denoted by v. The
output signal from the process is y. It is measured
by a sensor S, which produces the measured signal
y . The measured signal is compared with the
.
desired signal y by a controller C that produces
$
the control signal u. This signal is fed to the process.
The process dynamics are described by the following evolution equations:
x5 "f(x, u, v),
(8)
y"g(x, u, v).
(9)
For a generalised production system the state
vector is composed of the stocks in the di!erent
storage facilities and the market prices. Eq. (8)
4.1. The chain model
The model that is presented in Fig. 7 is based on
the present situation in the Netherlands. Actually,
used paper contributes for a considerable part to
the raw materials supply, which further consists of
primary pulp. Another secondary resource, process
waste, is not described by the model, as it is incorporated implicitly in the producer's characteristics.
For the sake of simplicity, existing control mechanisms as determined by the Dutch Paper Fibre
Convenant (including a takeback obligation for
used paper and the determination of a guaranteed
price), are omitted in the model. Fig. 7 shows the
#ows of paper, cardboard and their waste as they
currently are encountered in the Dutch paper
chain: the paper industry buys a mix of primary
pulp and processed wastepaper (secondary pulp) at
their respective markets. Subsequently, the paper
and board industry supplies the virgin paper it has
produced to the paper and board market (market
2). The paper and board market also experiences
supply from foreign suppliers. Part of the bulk
paper and board is bought by the paper and board
product industry, which converts these raw
materials into products of paper and board (printed
paper, cardboard packaging boxes, and so on).
Currently, wastes from the sector (o$ces, shops,
services and companies) are supposed to be transferred to the waste paper market. The major part
of the waste paper from households is also collected separately from other household waste by a
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
191
Fig. 7. The #owchart of the paper chain.
large and many-branched network of wastepaper
collection, largely kept up by non-pro"t associations. There is no market between paper and
board consumption and its collection. The col-
lected #ows form a supply to the wastepaper market where the resources are sold to the wastepaper
processor. Depending on the price the collector
might receive for his waste paper he will either
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
supply waste paper to the market or decide to
incinerate it.
4.2. Paper chain modelling
The dynamic variables involved in the presented
model are included in the system's state vector x.
According to Fig. 7, this vector consists of seven
elements: the prices p through p (prices of pri1
3
mary and secondary pulp, and paper) and the
stocks g through g (primary pulp, paper, waste
1
4
paper, and secondary pulp). In the model, control
takes place by economic instruments. It is e!ectuated here by in#uencing the prices: by setting the
incineration price or by imposing taxes on primary
pulp and on paper, respectively. So the input vector
u consists of three components. Disturbances v are
imposed on the system by external supply and
demand. These are present for pulp, paper, and
waste paper, so that the disturbance vector v has six
components here. When the control aims to minimise the environmental impact, the output vector
y might consist of components that represent the
material and product #ows. Combining all the
company and market models and using the notation introduced above leads to a model in the usual
state space notation (8) and (9).
5. Model evaluation
5.1. Parameter adjustment
Apart from variables, parameters are encountered in the model that is described here, for
example the proportionality constants in (4), (5)
and (7). Several standard parameter identi"cation
techniques exist in control theory, see for example
[11]. Here we adjust the parameters by minimising
the squared di!erence between the model outcomes
and available time series. The available time series,
which were collected by National Statistics, are
given by the prices of pulp, paper, and waste paper.
Unfortunately the amount of available data is
rather limited here, and the accuracy of these data
is somewhat uncertain. Therefore the result of the
adjustment must be considered as a rough estimate
of the appropriate parameter values. Fig. 8 shows
Fig. 8. Model outcomes with estimated parameters.
the course of the yearly averaged prices of the
di!erent products in the paper chain as they
were observed in the period from 1986 to 1993.
The model outcomes with a particular parameter
set are represented by lines. As can be seen from
the "gure they closely approximate the chain behaviour.
5.2. Stability analysis
An essential characteristic of dynamic systems,
which can be analysed by control theory, is its
stability. A system is said to be stable if its state
evolves into a certain constant value. Stability of
the controlled system is an absolute requirement
since its absence causes signals to grow beyond
any limit, eventually destroying and breaking
down the system. As an example, an unlimited
growth of some price will bring the economic relations in the product chain to an end, a situation
which is certainly undesirable for any of the chain
actors.
Decisive stability analysis for systems like this, in
which discontinuities and non-linearities are encountered, is a complex task, but two methods are
indicative with respect to stability. Simulation can
be carried out with the known parameters included in the system. Results of this calculation
are shown in Fig. 9, which represents the state
variables x divided by their value at the end of the
i
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
193
Fig. 10. Eigenvalues of the system matrix.
Fig. 9. The stable chain.
simulation horizon. The simulation indicates stability, albeit for a particular set of initial values.
The second method is based on linearisation
around the points of the system trajectory
(x (t), u (t), v (t)). Via the decomposition around
0
0
0
this reference trajectory (where *x"x!x ,
0
*u"u!u , *v"v!v ), (8)}(9) is linearised. As
0
0
a result of this transformation (8)}(9) can be written
as follows:
*x5 "A *x#B *u#E *v,
*y"C *x#D *u#F *v,
(10)
with A, B, C, D, E, F matrices of appropriate dimensions. These relations are valid in a neighbourhood
of the reference trajectory. For the linearised system
linear control systems analysis can be applied.
According to standard linear control theory a
system is stable if the eigenvalues of the matrix A
lie in the left complex half-plane. For the product chain system the eigenvalues of A are depicted
in Fig. 10 for some selected points of the same
trajectory as in Fig. 9. As the state vector has seven
components, the matrix A, at each point of the
trajectory, has seven eigenvalues, some of which
can coincide. It is shown here that the linearised
model is indeed stable because the eigenvalues
of the system matrix lie in the left complex halfplane.
5.3. Controllability analysis
In complex systems, the question of the selection
of the inputs that are used for system control is
relevant. In a typical plant, for example, many
possibilities for process control exist, but sensitiveness of the output to various inputs can be
signi"cantly di!erent. The same might be true for
production chains. The relevant question here is
whether or not the system is controllable by the
selected inputs. A system is said to be controllable if,
for a given initial situation, each value of the state
x can be reached at a chosen time ยน by an appropriate choice of the input u. In standard linear
control theory (see, e.g. [12]) controllability can be
assessed by examining the rank of the so-called
controllability matrix C"[B AB 2 An~1B] with
A and B from (10), see the Appendix for an illustrative example. For controllability the rank of C must
be of the same magnitude as the number of
elements in the state vector x. Fig. 11 shows the
rank of the product chain controllability matrix for
the given trajectory. As the rank of this matrix
varies between two and "ve, the value of the full
rank, i.e. seven, is not obtained here. The system
cannot be completely controlled by the given inputs. This means that not all the state variables can
be brought to arbitrary values by the given instruments.
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U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
inclusion of the non-linear structure of the models
is therefore necessary to obtain more general
results.
Appendix
As an example to illustrate the controllability
of a system, consider the simple two-dimensional
linear system de"ned by the following:
C D CD
1 0
x5 "
1 1
1
u"Ax#Bu.
0
x#
(A.1)
For this system the controllability matrix is given
by
Fig. 11. Rank of the controllability matrix.
C D
C"[B AB]"
6. Conclusions
The study of chain dynamics is relevant for
appropriate control of product chains, which is
of considerable economic, societal, and environmental interest. Control theory o!ers the tools to
analyse system models with respect to essential
properties, such as stability and controllability.
These tools have long been used to deal with this
kind of problem mainly for dynamical mechanical,
electrical and chemical systems, and appear to be
particularly suited for the analysis of product
chains, because product chains show analogues to
technical systems. Starting out with a standard
production system description, chains can be composed.
For the example of the paper chain, some results
of stability and controllability analysis are shown.
It is made clear that controllability analysis can
help us to "nd strategies to identify the ways to
in#uence product chains. On the basis of the presented model of the paper chain, it can be concluded that the chain is not controllable by the
incineration price and taxes on pulp and paper.
Additional instruments such as regulatory instruments and convenants between government and
industry should be taken into consideration.
A principle restriction of the model is that a linear approximation is applied. Further analysis with
1 1
0 1
and has full rank: two. On the other hand if the
input matrix B is changed to
CD
B"
C D
0 0
0
, then C"
with rank C"1.
1
1 1
The "rst system is controllable whereas the second
is not. This is intuitively clear because in the second
example only the second component of x5 can be
in#uenced by the input u but x , in turn, does not
2
in#uence x because the upper right element of A is
1
0. Thus x cannot be altered into any desired value.
1
References
[1] R.U. Ayres, Industrial metabolism, in: J.H. Ausubel, H.E.
Sladovich (Eds.), Technology and Environment, National
Academy Press, Washington, DC, 1989, pp. 23}49.
[2] IFIAS, Report of the International Federation of Institutes
for Advanced Study, Workshop no. 6 on Energy Analysis,
Guldsmedshyttan (S), 1973.
[3] J.B. GuineH e, H.A. Udo de Haes, G. Huppes, Quantitative
life cycle assessment of products, Part one: Goal de"nition
and inventory, Journal of Cleaner Production 1 (1) (1993)
3}13.
[4] J.B. GuineH e, H.A. Udo de Haes, G. Huppes, Quantitative
life cycle assessment of products, Part two: Classi"cation,
valuation and improvement analysis, Journal of Cleaner
Production 1 (2) (1993) 82}91.
U. Kleineidam et al. / Int. J. Production Economics 66 (2000) 185}195
[5] T.E. Graedel, B.R. Allenby, Industrial Ecology, PrenticeHall, Englewood Cli!s, NJ, 1995.
[6] R.U. Ayres, L.W. Ayres, Industrial Ecology: Towards
Closing the Materials Cycle, Edward Elgar, Cheltenham,
1996.
[7] K.J. Arrow, F.H. Hahn, General Competitive Analysis,
Holden-Day, San Francisco, CA, 1971.
[8] G. SchwoK diauer (Ed.), Equilibrium and Disequilibrium in
Economic Theory, Reidel, Dordrecht, 1978.
195
[9] C.R. Plott, G. George, Marshallian vs. Walrasian stability
in an experimental market, The Economic Journal 102
(1992) 437}460.
[10] A. Takayama, Mathematical Economics, Cambridge University Press, Cambridge, 1985.
[11] J.P. Norton, An Introduction to Identi"cation, Academic
Press, London, 1986.
[12] D.G. Luenberger, Introduction to Dynamic Systems:
Theory, Models and Applications, Wiley, New York, 1979.