Tustin systems thinking

Towill, D.R., Lambrecht, M.R., Disney, S.M. and Dejonckheere, J., (2001), "Every supply chain is a filter", in “What really
matters in Operations Research”, Proceedings of the 8th EUROMA, Vol. 1, June 3-4, Bath, UK, pp401-411, ISBN 1 85790
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Explicit Filters and Supply Chain Design
D.R. Towill1
, M. R. Lambrecht2
, S.M. Disney1
and J. Dejonckheere2
(1) Logistics Systems Dynamics Group,
Cardiff Business School,
Cardiff University, Aberconway Building,
Colum Drive,
Cardiff, CF10 3EU, UK.
DisneySM@Cardiff.ac.uk
D.R. Towill Fax: +44(0)29 2084 2292
(2) Department of Applied Economics,
Katholieke Universiteit Leuven,
Naamsestraat 69,
B – 3000 Leuven,
Belgium.
Jeroen.Dejonckheere@econ.kuleuven.ac.be
Marc.Lambrecht@econ.kuleuven.ac.be
Abstract
Due to the complexity of present day supply chains, it is important to select the simplest
supply chain scheduling Decision Support System (DSS) which will determine and place
orders satisfactorily. We propose to use a generic design framework, termed the explicit filter
methodology, to achieve this objective. In doing so we compare the explicit filter approach to
the implicit filter approach utilised in previous OR research which focused on minimising a
cost function. Although the results may well be similar with both approaches it is much
clearer to the designer, why and how, a scheduling system will reduce the Bullwhip Effect
when designed via the explicit filter approach.
Key Words: HMMS algorithm, explicit filter, aggregate planning, decision support systems
Introduction
Decision-making in manufacturing systems is a dynamic process based on accessing system
states including inventory levels and production rates. The basic principles involved in this
dynamic process are equally applicable to both the control of individual businesses, and to
complete supply chains. At the heart of the decision-making process is the desire to ensure
that the system correctly identifies and tracks genuine variations in demand (in order to
ensure high customer service levels). At the same time the decision-making process is
expected to detect and reject rogue variations in demand (in order to avoid excess costs due
to unnecessary ramping of production up/down and the associated swings between stock-outs
and excess stocks).
Supply chains are composed of a sequence of manufacturing (value added) systems extending
from the identification of customer need right through to that need being satisfied. Simulation
by Jay Forrester (1961) has long ago established that the more extended the chain, the worse
the dynamic response of that chain. Hence global operations can be particularly at risk from
poor systems design. It has since been rigorously proved that supply chains can be
mathematically described by a set of individual manufacturing systems coupled together in a
precise and particular way. As a consequence of this proof, it has been shown possible to
predict the dynamic response for an entire traditional supply chain from a knowledge of the
filter characteristics of each component system without resort to simulation (Towill & Del
Vecchio, 1994). Furthermore, this supply chain configuration has also enabled the Law of

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Industrial Dynamics originally established by Burbidge (1984) on an empirical basis, to be
put on a rigorous mathematical foundation.
The relevance of the explicit filter approach to supply chain design is emphasised by the
current move towards expanding global operations, the pressure to enable agile supply, and
the realisation that minimising a cost function at an individual echelon within the chain is not
the way to optimise response (or profitability) of the entire system. These pressures combine
to force the supply chain champion to adopt, market, and implement a holistic solution
(McHugh et. al., 1995). A consequence is the realisation that it is the total supply chain that
competes for business (Christopher, 1997). So the focus is changing from local cost
minimisation to enabling the complete chain to operate smoothly and thereby effectively. DSS
have a major role to play in achieving this objective (Towill & McCullen, 1999). Therefore
selecting the appropriate DSS to suit the needs of the supply chain becomes a critical part of
the overall design process (Dejonckeere et al, 2000). It is the purpose of this paper to show
how choice of such a DSS is greatly simplified via the “explicit filter” approach.
OR and DSS Selection
The study of DSS has a long history particularly amongst the Operations Research
community. This dates back to the classic HMMS algorithm (Holt et al, 1960) and its many
variants and updates typified by Jones (1967), Bertrand (1986) and Lambrecht et al, (1982).
Corresponding mathematical analysis based on what has become known as the
servomechanism approach started with Tustin (1952). Early approaches concentrated on
attempting to control the dynamic response via placement of the system poles (i.e.. roots of
the characteristic equation) using Laplace Transforms (Simon, 1953) and a short while later z
Transforms (Vassian, 1954). The problem was subsequently re-cast in the form of
minimising a cost function composed of storage and production adaptation costs (Adelson,
1966 and Deziel & Eilon, 1967). Despite the considerable difficulty in establishing a realistic
cost model, this HMMS style approach became extremely popular. We term it the implicit
filter design method because it concentrates on minimising the cost function and hoping that
the consequential filter performance is acceptable. Table I compares the salient features of
the OR and Filter approaches. Note that both could result in similar optimal designs for a
manufacturing system, but the routes to accomplishment are very different.
Our methodology is based on the same mathematical principles as the OR approach. However
we reverse the procedure since we are concerned upfront with explicit filter design in which
good dynamic performance is a prerequisite. The argument here is that cost control follows
from good dynamic design, to ensures high customer service level currently with small
swings in capacity requirements. At the heart of the explicit filter design approach is the
complete understanding of how the system transfer function governs dynamic response. For
example, the dominant roots of the characteristic equation are important factors in
determining system bullwhip, but it is only one factor. Thus a conservative placement of the
dominant poles of the feedback loop can fail to adequately dampen the bullwhip in the
presence of exponential smoothing of feed-forward demand (Towill, 1982). In system
engineering terms, pole-placement as a controller of dynamic response breaks down in the
presence of predictive elements within the system. In contrast, transfer function techniques
are all embracing. Hence given the transfer function of a manufacturing system the filter
characteristics are uniquely defined for all inputs. So the core design problem becomes
“given the expected inputs and desired outputs, what transfer function will deliver the
necessary performance?”

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The Concept of the ‘Ideal’ Filter
Dynamic systems must be designed to follow command signals (the ‘signal’), and yet at the
same time reject the unwanted disturbances (the ‘noise’). To achieve this objective, systems
may be designed using time domain concepts, or via frequency domain concepts. In either
case, the required performance assessment can be obtained via mathematical analysis (at least
in theory), or by simulation. An advantage of the filter concept is that it forces the “client”
and the system designer to discuss and think carefully about their definitions of “command”
and “noise” signals as appropriate to this specific application. For example, do we or do we
not consciously change workforce levels for weekly changes in demand? Should the answer
be “no” then the designer needs to ensure that weekly variations are adequately filtered out by
the ordering algorithms. If the answer is “yes” then some volatility is permitted.
The concept of the ideal filter may thus be represented as an envelope of amplitude ratio
values that are either one or zero. This is depicted in Fig. 1, which is for the particular case of
the Low Pass Filter. For instance, in supply chains this might correspond to the desired
relationship between marketplace consumption and consequential orders placed on the
factory. So slow stable changes in demand pattern would be considered important, and hence
“followed”. In contrast rapid fluctuations would be seen as random “noise” and be filtered
out.
System Trade-Off Between Capacity Requirements and Stock Fluctuations
Of course, the frequency bands of signal and noise are rarely known with certainty. Indeed,
their ranges may overlap. Nevertheless the ‘ideal filter’ (and especially the departure of a
‘sup-optimal filter’ from this target) is a useful concept in system design and has already been
successfully included in a manufacturing system CAD design suite (Edghill and Towill,
1989). Here the departure of the candidate filter from the ideal filter is used, for a suitable
range of frequencies, as input data to a dynamic performance index to be minimised in the
search for the ‘best’ compromise.
Fig. 2 shows how the filter concept may be applied. The graphs are equally applicable to
individual manufacturing systems, individual non-manufacturing but value-added activities,
and to complete supply chains. It is also possible to deliberately engineer the supply chain to
enable substantially different dynamic responses at different echelons within the chain.
Typical of such an approach is the leagile supply chain which has a carefully selected material
flow decoupling point, usually based on product configuration considerations (Naylor, et al,
1999). Upstream of the decoupling point, orders conform to the Level Scheduling mode.
Downstream of the decoupling point (i.e. nearer the marketplace), orders conform directly to
end customer requirements.
In Fig. 2 we compare the waveforms to be expected at discrete frequencies selected either side
of the cut-off frequency where the “ideal” filter would switch from “transmit” to “reject”.
Here the “order-up-to” model (Lee et al, 1997) amplifies the order pattern across the
frequency range, and has volatile stock behaviour also. In contrast Level Scheduling (Suzaki,
1987) just takes a scythe to all order variations by holding the production order rate constant,

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and absorbs all marketplace fluctuations from stock. Conversely, passing-on-orders (used as
an MIT Beer Game benchmark by Sterman, 1989) tries to maintain stock levels constant by
ordering the factory to respond directly to the marketplace. But the “ideal filter” switches so
that it behaves as level scheduling at high frequencies but passes-on-orders at low
frequencies.
In theory the “ideal” filter experiences the best of both worlds i.e. respond only to real
changes in demand, but buffer out and take from stock the random changes that do not
influence actual needs. In practice the “ideal” filter cannot be physically realised, some
rounding of the corners is inevitable although a good approximation is achievable and we
thereby design an ordering algorithm that gives a good compromise. Such a practical filter
response is also shown in Fig. 2. Hence although there are production order rate swings both
above and below the idealised cut-off frequency, they approach the target behaviour in both
cases. Thus low frequency marketplace signals are reasonably tracked, whilst high frequency
“noise” is reasonably rejected. However, such a practical filter cannot be arbitrarily selected;
it has to be matched to customer and system needs, otherwise the “Forrester” effects or
demand amplification (now known as “bullwhip”) and rogue seasonality results (Berry and
Towill, 1995).
Synergies with Aggregate Planning Decision Making
Aggregate planning increases the range of alternatives for capacity usage that must be
considered by management and our review follows the description by Buffa (1969). Despite
the passage of time, his statement of the problems faced by management remain largely
unchanged. The term “aggregate planning” includes scheduling in the sense of a programme;
the terms “aggregate planning” and “aggregate planning and scheduling” are used almost
interchangeably. The economic significance of these ideas is by no means minor. The
concepts raise such broad basic questions as: To what extent should inventory be used to
absorb the fluctuation in demand? Why not absorb all these fluctuations by simply varying
the size of the work force? Why not maintain a fairly stable work force size and absorb
fluctuations by changing production rates through varying work hours? Why not maintain a
fairly stable work force and production rate and let subcontractors wrestle with the problem of
fluctuating order rates? Should the business purposely not meet all demands or seek to
smooth them via price changes (Hay, 1970). In most instances it is probably true that any one
of these extreme strategies would not be as effective as a balance among them. Each strategy
has associated costs and, therefore, we seek an astute combination of the alternatives.
If inventories are used to absorb seasonal changes in demand, capital and obsolescence costs
as well as the costs associated with storage, insurance, and handling tend to increase. Beyond
the question of seasonal factors, the use of inventories to absorb short-term fluctuations incurs
increases in these same costs compared to some ideal or minimum inventory level necessary
to maintain the production process. When inventories fall below this ideal or minimum level,
stock-out costs and all costs associated with short runs increase. Changes in the size of the
work force affect the total costs of labour turnover. When new workers are hired, costs arise
from selection, training, and lower production effectiveness. Learning curve factors may then
become dominant. The firing of workers may involve unemployment compensation or other
separation costs, as well as an intangible effect on public relations and public image. Large
changes in the size of the work force may mean adding or eliminating an entire shift; the
incremental costs involved are shift premiums as well as incremental supervision costs and
other overheads.

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If fluctuations are absorbed through changes in the production rate, overtime premium costs
for increases and probably idle labour costs (higher average labour cost per unit) for decreases
also will be absorbed. Usually, however, managers try to maintain the same average labour
costs by reducing hours worked to somewhat below normal levels. When under-time
schedules persist, labour turnover and the attendant costs are likely to increase. Many costs
affected by aggregate planning and scheduling decisions are difficult to measure and are not
segregated in accounting records. Some, such as interest costs on inventory investment, are
alternatives costs of opportunity. Other costs such as those associated with public relations
and public image (the role of the good employer) are now measurable directly. However, all
of the costs are real and bear on aggregate planning decisions.
In order to achieve a good balance between the conflicting demands of production and
inventory, the OR community has developed a raft of solutions to this problem. These
include those by Holt, et al, (1960) subsequently known as the HMMS algorithm and others
by Jones, (1967), Orr (1962), Elmaleh and Eilon (1974), Silver (1974) and Lambrecht et al
(1982). Whilst all of these propositions undoubtedly helped to smooth production (sometimes
under very restricted conditions of demand volatility), it was in all cases achieved indirectly.
That is, a cost function (again sometimes dubious in validity) was optimised and it was later
found that some smoothing actually resulted from using the resulting algorithm. We shall
now take a fresh look at one of these classic algorithms from a frequency domain perspective.
HMMS Revisited
In Fig. 3 we have reconstructed the famous HMMS application using their Paint Factory Data.
Time Series representing the consumption order rate inventory level and workforce level are
also shown. The Coefficient of Variation (Fransoo and Wouters, 2000) has been calculated as
a metric summarising the smoothing capability of the algorithm. The HMMS is clearly
effective. Yet Holt et al (1960) content themselves with statements as “With a perfect
forecast (sic) the decision rule avoids, almost completely, sharp month-to-month variations in
productions, but responds to fluctuations in orders that have a duration of several months”.
Their statement implies that there is a break point frequency corresponding to a period of
(say) 3 months. As we shall now see, obtaining the frequency response characteristics is
much more insightful.
Figure 4 summarises the frequency response characteristics implicit in the HMMS algorithm
applied to the Paint Factory scenario. We have obtained this estimate via Fourier analysis of
the time series. Hence for a range of integer frequencies we can calculate the amplitude
associated with a particular series. Then for a particular spot frequency the amplitude ratio is
simply the relationship between the CONSUMPTION (the system driver) and (say) ORDER
RATE (the resultant dependent variable. To simplify presentation, Fig 4(a) relates the Fourier
Coefficients for ORATE and CONS, whilst Fig. 4(b) similarly relates LABOUR and CONS.
We have also superimposed, for comparison, the guidelines for perfect rejection and perfect
tracking. Also shown are the somewhat arbitrary “best fit” ideal filters for both cases.
From Fig. 4 and related frequency response plots we are able to “explicitly” describe the
filtering responses of the HMMS algorithm. For example, for ORATE, to a first
approximation we may regard the transition as occurring at a period of 8 months, i.e. shorter
periods are “rejected” and longer periods “tracked” i.e. HMMS is “better” then implied by
Holt et al, (1960). Of course the switch is not instantaneous, but the concept gives us a real

088X.
feel for the filter characteristics of the HMMS algorithm. It is the up front consideration of
the dynamic properties of scheduling algorithm which distinguishes the filter and OR
approaches. In contrast the “switch” occurs at a period of 16 months for LABOUR. Shorter
periods are rejected. This is as expected, and means that the Paint Company is basically
planning the workforce over an annual period. Shorter fluctuations are then met by a
combination of inventory usage and overtime/under time working.
Relevance of the “Old” Aggregate Planning Approach to the “New” Supply Chain
Dynamics
We saw how Aggregate Planning Decision making was posed by the OR community as a
balancing problem between the competing needs of production control and inventory
management. Later the filter characteristics of one esteemed (HMMS) algorithm were
determined a posterioi from time series generated from the classic paint factory case study.
Why is this relevant to modern supply chain dynamics? The answer is contained in Fig. 2,
since there is no basic difference (except time scale) between aggregate planning decision
making and ordering algorithms relevant to modern supply chains such as those one discussed
in Dejonckheere, Disney, Lambrecht and Towill, (2000 and 2002) which has exploited the
explicit filter approach. Thus Fig. 2, in one simple sketch, has covered the whole spectrum of
possibilities from which the supply chain designer can select the appropriate decision rules.
For example we showed that a Fixed Production Rate corresponds to Lean Production; Pass
on Orders corresponds to Agile Production; and Leagile Production is the appropriate mix of
Lean and Agile modes. Note that here we have used the description “Production” to describe
a value-added process within a supply chain. Our arguments clearly apply to such other
value-added processes as Acquisition and Delivery and in new configurations such as VMI.
These value-added processes are then combined (and ideally integrated, Stevens, 1989) to
deliver in accordance with end-customer expectations. Hence in seeking modern supply chain
solutions we need not entirely abandon lessons learned from the aggregate planning research
era. In particular, our practical explicit filter dynamically relates to the HMMS algorithm.
So far we have been discussing what is common between aggregate planning and modern
supply chain operations. But since the early breakthrough in supply chain dynamics was
made by Jay Forrester in 1958, it is reasonable to also highlight the differences. Christopher
and Towill (2000) have argued that the last two decades have seen major changes driven by
marketing and customer pressures. The result is a move away from “traditional” supply
chains (we know what the customer wants) through “product driven” supply chains (we
supply what we think the customer wants) to market driven supply chains (we supply what is
selling well) to “customised” supply chains (we supply what the individual customer wants).
Quite apart from the general driving out of waste using Lean Thinking Principles (Womack
and Jones, 1996) there has been tremendous pressure to increase the speed of response of the
delivery process (Lowson et al, 2000). It also needs to be matched to what the customer
actually wants, and hence avoid obsolescence via excess stock holding in the chain. So it is
back to the same balancing problem already met in aggregate planning.
Conclusion
This paper has compared a modern supply chain design procedure (termed the explicit filter
approach) to a traditional cost minimisation approach that has been applied to production
scheduling (termed the implicit filter approach). Although the results of both design

088X.
strategies were broadly similar in both cases, the route to the solution was different. The
implicit filter approach, developed to minimise a pseudo cost function has the helpful side
effect of reducing Bullwhip. The explicit filter approach was developed with the specific aim
of investigating and avoiding the Forrester Effect or Bullwhip Effect and as a helpful side
effect that it also reduces a pseudo cost function. The real advantage, however, with the
explicit filter approach is that it is possible to determine where (in the frequency range) and
how much (the amplitude) Bullwhip is generated by the scheduling algorithm, whereas, with
the implicit filter approach based on cost minimisation this is not so transparent. Thus
shaping the response to meet supply chain requirements is easier since it becomes central to
our design methodology.
References
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orders to the needs and economics of your supply chain.” Proc. EUROMA 2000 Conf,
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Lowson, R., King, R., and Hunter, A., 1999. Quick response managing the supply chain to
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CHARACTERISTICS OR APPROACH FILTER APPROACH
System Model Integral/difference
equations
Transfer functions
Typical Assumed
Stimuli
Random excitation Sinusoidal excitation
Methods of Analysis s/ z transforms
Probability theory
s/ z transforms
Fourier transforms
Performance Criteria Production/inventory
variances
Production/inventory power
spectra
Optimisation
Procedure
Minimise quadratic cost
function
Minimise deviation from
“ideal” filter
Design Emphasis Implicitly smooth
production/inventory
swings
Explicitly smooth
production/inventory swings
Bullwhip
Consequences
Somewhat arbitrary Reduce by design
Financial Implications Precise according to
cost function
Somewhat arbitrary
Table I Comparison of the OR and Filter Approaches to DSS Selection in Supply Chain
Design
2
1
0
“Message” frequency
range
Cut -off frequency wc
“Noise” frequency
range
0 p
Frequency
AmplitudeRatio
Fig 1 The “Ideal” Low Pass Filter

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Ordering Strategy
Production
order rate
Inventory
swings
Demand
pattern
Or
fixed production rate
(Level Scheduling)
Or
fixed inventory
level
(Pass on orders)
Or
practical
filter
Either
“Order-up-to“
Low frequency demand
Or
ideal
filter
Production
order rate
Inventory
swings
Demand
pattern
High frequency demand
Fig. 2. Application of Filter Concept to Supply Chains

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