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An analysis of the effect of response speed on the Bullwhipeffect using control theoryCitation for published version (APA):Udenio, M., Fransoo, J. C., Vatamidou, E., & Dellaert, N. P. (2013). An analysis of the effect of response speedon the Bullwhip effect using control theory. (BETA publicatie : working papers; Vol. 420). Eindhoven: TechnischeUniversiteit Eindhoven.
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An analysis of the effect of response speed on the Bullwhip effect using control theory
Maximiliano Udenio, Jan C. Fransoo, Eleni Vatamidou,
Nico Dellaert
Beta Working Paper series 420
BETA publicatie WP 420 (working paper)
ISBN ISSN NUR
804
Eindhoven May 2013
An analysis of the effect of response speed on the Bullwhip
effect using control theory
Maximiliano Udenioa, Jan C. Fransooa, Eleni Vatamidoub, and Nico Dellaerta
aSchool of Industrial Engineering, Eindhoven University of Technology, NetherlandsbEURANDOM and Department of Mathematics & Computer Science, Eindhoven
University of Technology, Netherlands
May 29, 2013
Abstract
In this paper, we use linear control theory to analyze the performance of a firm gener-
ating material orders as a function of the difference between actual inventory and pipeline
and their respective targets (APIOBPCS: Automatic Pipeline, Variable Inventory Order
Based Production Systems). We explicitly model managerial behavior by allowing frac-
tional –instead of full– adjustments to be performed in each period, thus introducing a
proxy to the firms’ reactiveness to changes. We develop a new procedure for the deter-
mination of the exact stability region for such a system, and derive an asymptotic region
of stability that gives a sufficient stability criteria independent of the lead time. We then
quantify the performance of the system by analyzing the effect of different demand signals
on order and inventory variations. We find that firms with low reactiveness perform well in
the presence of stationary demands but perform poorly when encountering finite demand
shocks. Furthermore, we find that the ratio of the reactiveness of inventories and pipeline
affects both the performance, and robustness of the system. When the inventory is more
reactive than the pipeline the system can achieve increased performance at the cost of
being sensitive to the parameters. When the pipeline is more reactive than inventories,
on the other hand, the system is robust, but at the cost of sub-optimal performance.
1
1 Introduction
The bullwhip effect is a major problem in todays’ supply chains. Lee et al. (1997b) define it as
the observed propensity for material orders to be more variable than demand signals, and for
this variability to increase the further upstream a company is in a supply chain. It is a dynamic
phenomenon that has sparked a vast body of research from a wide array of methodologies.
Empirical, experimental, and analytical studies exist of both a descriptive nature –trying to
identify and describe it– and of a normative nature –trying to overcome it–. The causes for
the bullwhip effect can be broadly separated into operational (such as order batching and
price fluctuations) and behavioral categories (such as artificially inflating orders and pipeline
under-estimation). The goal of this paper is to extend the analytical knowledge regarding the
influence of human behavior in the appearance of the bullwhip effect and the amplification of
inventory variance. We use linear control theory as a modeling methodology, and frame our
work as a descriptive work: we attempt to link existing experimental and empirical results to
insights developed through this analysis.
Regarding behavior, it has long been understood that decision-makers do not operate under
the paradigm of complete rationality. Both in real life and in experiments, humans operate
in ways that deviate from theoretical predictions. We make mistakes. We exhibit psycho-
logical biases that affect our decisions. In operations research, heuristics containing feedback
structures are commonly used to model human behavior: decisions bring about changes that
affect future decisions. The modeling of feedback structures in this context can be traced
back to Forrester (1958) and the introduction of system dynamics. Such modeling makes it
possible to understand the dynamics of the system under study and how they are affected
by non-observable parameters and their interactions. In particular, anchor and adjustment
heuristics (Tversky and Kahneman, 1974) have been proposed to model the feedback loops
introduced by decisions pertaining the generation of material orders. In these heuristics, fore-
casts act as an anchor to orders, and deviations from inventory and pipeline targets are used
to adjust orders up or down (Sterman, 1989). Human biases become apparent when studying
these adjustments: experimental work has shown that people consistently under-estimate the
pipeline when making decisions (Sterman, 1989; Croson and Donohue, 2006; Croson et al.,
2
Forthcoming). These biases have also been observed in empirical data (Udenio et al., 2012).
On the analytical front, linear control theory has long been used to model dynamic inventory
models. Herbert Simon, in 1952, studied a continuous-time system through the Laplace trans-
form method, in which inventory targets are used to derive material orders, Vassian (1955)
studied the equivalent discrete-time system using the Z-transform, and Deziel and Eilon (1967)
extended the discrete case by adding a smoothing parameter to control the variance of the
response. Towill (1982), extended and formalized these ideas with the introduction of the
Inventory and Order Based Production Control System (IOBPCS) design framework. In an
IOBPCS design, replenishment orders are generated as the sum of an exponentially smoothed
demand forecast and a fraction of the inventory discrepancy (the gap between a constant
target inventory and the actual value). His work, by representing an Inventory/Production
system in block diagram form, allowed for the straightforward application of linear control
theory methodologies to study its structural and dynamic properties.
A first extension to IOBPCS is VIOBPCS (Variable Inventory and Order Based Production
Control System), where the inventory target is no longer constant, but calculated each period
as a multiple of the demand forecast. Edghill and Towill (1990) study this system and find
that, in comparison to IOBPCS, the variable inventory targets of VIOBPCS designs intro-
duce interesting trade-offs between the “marketing” and “production” sides of a firm: increased
service levels through a better correlation of inventory and demand, at the cost of increased
variability in orders. A powerful extension, which was presented by John (1994) and adds a
second feedback loop in the form of a pipeline adjustment is APIOBPCS (Automatic Pipeline
Inventory and Order Based Production Control System). The ordering logic of this design is
a direct equivalent to the anchor and adjustment heuristic that Sterman (1989) used to model
beer-game playing behavior.
Because of this equivalence to human-decision-making heuristics, discrete APIOBPCS and
its Variable Inventory extension, APVIOBPCS, have been extensively studied through the Z-
transform method in the two decades since John’s work. These designs share a large number
of characteristics and for the sake of brevity we use the term AP(V)IOBPCS designs to refer
to both. A discrete APVIOBPCS design with full adjustments (inventory and pipeline dis-
3
crepancies are filled every period) was used by Hoberg et al. (2007a) to compare the stationary
and transient response of systems using echelon and installation stock policies. Hoberg et al.
(2007b) use the same system design, but instead focus on quantifying the effect of the choice
of the forecasting smoothing parameter α. They find that both echelon stock policies and
values of α close to zero contribute to a reduction of inventory and order amplification.
Dejonckheere et al. (2003) show that Order Up To policies (OUT) are equivalent to APVIOBPCS
designs with full adjustments and find that the introduction of fractional adjustments can be
used as a tool to reduce the bullwhip effect. Disney and Towill (2006) study a particular subset
of policies, christened after the work of Deziel and Eilon, in which the fractional adjustments
for inventory and pipeline are equal: DE-AP(V)IOBPCS designs. This simple constraint in
the model parameters has very attractive properties; from an optimal design perspective stable
DE-designs always produce aperiodic responses; and from a mathematical perspective, DE-
designs produce tractable, elegant expressions. General (non-DE) AP(V)IOBPCS designs, on
the other hand, exhibit none of these desirable characteristics. The complexity introduced by
fractional parameters is such that a compact, exact expression for the stability of a discrete
general AP(V)IOBPCS design has not yet been found through classical control analysis. Dis-
ney (2008) demonstrates the usage of Jury’s Inners (Jury, 1964) to derive stability conditions,
for a given lead time and derives a new simplified rule. Disney and Towill (2002) find a gen-
eral expression for stability boundaries by modeling lead times as a third order lag instead
of a pure delay. Stability boundaries have also been found through the use of an equivalent
continuous-time system (Warburton et al., 2004), and an exact stability criterion through a
continuous, time domain, differential equation approach (Disney et al., 2006). Finally, Wei
et al. (2012) find necessary and sufficient conditions for discrete APIOBPCS design through
state-space analysis.
For comprehensive literature reviews on the application of control theory to Inventory/Production
systems, we refer the reader to Ortega and Lin (2004), and Sarimveis et al. (2008). The for-
mer centers in the application of classical control, while the latter pays special attention to
advanced control methodologies.
In this paper, we use discrete-time classical control methods to analyze an APVIOBPCS de-
4
sign in light that it is the policy that most closely resembles human decision-making heuristics.
We aim to understand not only the bias shown by beer game players (Sterman (1989), and
Croson and Donohue (2006)) in under-estimating the pipeline, but also why empirical data
suggests that real-life firms operate through APVIOBPCS with low fractional adjustments
(Udenio et al., 2012). We focus on both the bullwhip and inventory variance amplification.
The rest of this paper is structured as follows: In the next section we introduce the discrete-
time model in both time, and frequency domains. We follow with a comprehensive analysis
of the stability of the system, deriving exact expressions for the general stability boundaries.
We introduce performance measures in Section 4, analyzing the dynamic, and steady state re-
sponses of both the generation of orders and the evolution of inventories. We then identify the
trade-offs inherent to these designs, and position real-life firms in this context. We conclude
in Section 5, and present all the mathematical proofs in the Appendix.
2 Model description
In this section, we analyze a discrete-time, periodic-review, single-echelon, general APVIOBPCS
design with an exponentially smoothed forecast of demand. The inventory coverage (C ∈ R+),
the delivery lead time (L ∈ N), and the forecast smoothing parameter α ∈ [0, 1] are the struc-
tural parameters of the system, while the pipeline (γP ∈ [0, 1]) and inventory (γI ∈ [0, 1])
adjustment factors are the behavioral parameters of the system. The inventory coverage rep-
resents the target inventory –measured in weeks of sales– that a firm chooses to maintain, while
the target pipeline is calculated each period as the product of the forecast and the systems
lead time. The lead time is assumed deterministic and defined as the time elapsed between
the placement and receipt of a replenishment order. The behavioral parameters specify the
fraction of the gap between target and actual values that are taken into account in the peri-
odical ordering adjustment: γI is the fraction of the inventory gap to be closed per period,
and γP is the fraction of the pipeline gap to be closed per period. For instance, a system
with γI = 1 and γP = 0 completely closes the inventory gap every period, while it ignores the
pipeline entirely.
5
Formally, the sequence of events, and the equations in the model are as follows: at the be-
ginning of each period (t) a replenishment order (ot) based on the previous period’s demand
forecast (ft−1) is placed with the supplier. Following this, the orders that were placed L
periods before are received. Next, the demand for the period (dt) is observed and served.
Excess demand is back-ordered. Then, the demand forecast is updated according to the for-
mula: ft = αdt + (1 − α)ft−1,. The forecast is used to compute the target levels of both
inventory, it = Cft, and pipeline, pt = Lft. The orders that will be placed in the fol-
lowing period (ot+1) are generated according to an anchor and adjustment-type procedure,
ot+1 = γI (it− it) + γP (slt− slt) + ft. The balance equations for inventory (i) and pipeline (p)
are: it = it−1 + ot−l − dt, and pt = pt−1 + ot − ot−l. Note that the assumptions that orders
and inventories can be negative are necessary to maintain the linearity of the model.
This sequence of events is identical to the one described in Hoberg et al. (2007a) with the
difference being that our study introduces the fractional behavioral parameters γI and γP .
Other studies of AP(V)IOBPCS designs use different order of events; e.g., Dejonckheere et al.
(2003), and Disney (2008) orders are placed at the end of each period. These changes in the
sequence of events introduce extra unit delays in the equations. However, these differences
only affect the mathematical representation of the system; the structure of the system and
the results, remain the same.
2.1 The frequency domain
The model we introduced completely describes the relationships between the parameters of
a general APVIOBPCS design. However, due to time dependencies, we cannot find a clear
relationship between the inputs and the outputs of the system. For this reason, we turn from
the time domain to the frequency domain (where these relationships become simply algebraic),
by taking the Z-transform of the system’s set of equations. The Z-transform is defined as:
Z xt = X(z) =∞∑k=0
xkz−k, (1)
where z is a complex variable and xk is the value of a time series in period k. We refer the
reader to Jury (1964), and Nise (2007) for a comprehensive background on discrete systems
6
and the Z-transform method; and to Hoberg et al. (2007a), and Dejonckheere et al. (2003) for
an introduction to their application on inventory modeling.
Using the following properties of the Z-transform:
Z a1xt + a2yt = a1X(z) + a2Y (z) (Linearity), (2)
Z xt−T = z−TX(z) (Time delay), (3)
we can write all system parameters in the frequency domain. The equation for orders is written
then as:
O(z) =
[γI(I(z)− I(z)) + γP (P (z)− P (z)) + F (z)
]z
. (4)
In control theory, the response of a system is completely characterized by its transfer function
G(z) = N(z)/C(z), that represents the change in output N(z) with regards to a change in
input C(z) in the frequency domain. In this paper we are interested in studying the properties
of the transfer function of orders (GO(z)) and the transfer function of inventories (GI(z)) The
transfer function of orders represents the change in orders O(z) in response to a change in
customer demand D(z):
GO(z) =O(z)
D(z)=
[γI(I(z)− I(z)) + γP (P (z)− P (z)) + F (z)
]1z
D(z)
=[α(γIC + γPL+ 1)(z − 1) + γI(z − 1 + α)] zL
(z − 1 + α)(zL(z − 1 + γP ) + (γI − γP )). (5)
Analogously, the transfer function of the inventory is defined as the change in inventory level
I(z) as a response to customer demand D(z):
GI(z) =I(z)
D(z)=
zz−1
[O(z)z−L −D(z)
]D(z)
=zα(γIC + γPL+ 1)(z − 1) + z(z − 1 + α)
(γP − zL(z − 1 + γP )
)(z − 1)(z − 1 + α)(zL(z − 1 + γP ) + (γI − γP ))
. (6)
Having defined the transfer functions for orders and inventories, we can now provide structural
properties of the response of the system.
3 Stability and aperiodicity of the system
We have the following definition for the stability of a system:
7
Definition 3.1 (Nise, 2007). A system is stable if every bounded input yields a bounded
output, and unstable if at least one bounded input yields an unbounded output.
In our model, the customer demand is the input, and orders and inventory are the outputs.
Thus, the system is stable if any finite demand pattern produces finite orders and finite
inventories.
From a mathematical analysis point of view, Definition 3.1 does not help us decide whether
a system is stable or not. An alternative definition-condition that connects the stability of a
system with its transfer function is:
Definition 3.2 (Jury, 1964). Suppose that G(z) = N(z)/C(z) is the transfer function of a
linear time-invariant, system and that the denominator C(z) has exactly n roots pi, namely
C(pi) = 0, i = 1, . . . , n. We call the roots pi poles of the transfer function, and we say that
a system is stable if all poles pi are within the unit circle of the complex plane (|pi| < 1),
marginally stable if at least one pole is on the unit circle (|pi| = 1), and unstable if at least
one pole resides outside the unit circle (|pi| > 1).
Consequently, judging the stability of a system is equivalent to finding the solutions to the
characteristic equation C(z) = 0.
Remark 3.1 Suppose that P with |P | ≥ 1 is a root of C(z) with multiplicity m, namely,
C(i)(P ) = 0, ∀ i = 0, . . . ,m − 1. If N (i)(P ) = 0, ∀ i = 0, . . . ,m − 1, and if all the other
roots of C(z) are inside the unit circle, then the system is called stabilizable. However this is
sometimes used alternatively as a definition for a stable system (Wunsch, 1983). This is not
the case here.
In the next section we derive conditions for the stability of the system through an analysis of
the structure of the involved characteristic polynomials, and we introduce the aperiodicity of
the system, which is a characterization of the dynamic response of a stable system. We begin
our analysis with the response of orders to changes in demand (Eq. (5)) and follow with the
analysis of the inventory response to changes in demand (Eq. (6)).
8
In the forthcoming mathematical analysis, the fractional parameters γI , and γP are not a-
priori bounded to [0, 1]. This allows us to characterize the stability of the system over a
broader range of possible values than the ones that can be found in a physical system.
3.1 Stability boundaries
By comparing equations (5) and (6) we see that the characteristic polynomials of orders and
inventories are almost equal except for the extra term (z − 1) that appears in the latter. The
pole z = 1 would render the inventory response marginally unstable, unless this is also a root
of the numerator of GI(z) (see Eq. (6) and Remark 3.1). To this effect, we use the geometric
series zL+1 − 1 = (z − 1)∑L
i=0 zi to rewrite equation (6) as:
GI(z) =zα(γIC + γSL+ 1)− zL+2 − zL+1(α+ γP − 1) + γP z + αγP
(1−
∑Li=0 z
i)
(z − 1 + α)(zL(z − 1 + γP ) + (γI − γP )). (7)
Thus, in an APVIOBPCS design, the stability of both orders and inventories is defined by the
same characteristic polynomial
C(z) = (z − 1 + α)(zL(z − 1 + γP ) + (γI − γP )), (8)
Being a polynomial in z of degree L + 2 with real coefficients, C(z) has exactly L + 2 roots.
Unfortunately, this polynomial is transcendental: it is impossible to find its roots indepen-
dently of L. Furthermore, exact solutions for C(z) = 0 can only be found for values of L ≤ 2.
Thus, we study structural properties of C(z) to derive a set of conditions that define an exact
stability boundary for the general AP(V)IOBPCS design. The proofs of all the theorems,
lemmas, and propositions are found in the Appendix.
It can be shown that APIOBPCS and APVIOBPCS policies share the same characteristic
polynomial (Disney and Towill, 2006). Thus, the insights and conclusions derived from the
analysis of C(z) hold for AP(V)IOBPCS designs.
Proposition 3.1 The stability of a general AP(V)IOBPCS system with smoothing parameter
α ∈ [0, 1] can be determined by analyzing the poles of the reduced characteristic polynomial:
C(z) = zL(z − 1 + γP ) + (γI − γP ). (9)
9
An AP(V)IOBPCS is stable if all the roots of C(z) are located inside the unit circle.
Thus, the stability of a general AP(V)IOBPCS system with the commonly used exponential
smoothing parameter range of [0, 1] is completely determined by the values of L, γI , and γP .
Theorem 3.1 For each value of L, stability is guaranteed when γI and γP satisfy the following
L+ 1 conditions:.
(i)
|γI − γP | < 1, (10)
(ii)
(1− (γI − γP )2) |γP − 1|(n−1) Un−1(X)− |γP − 1|n Un−2
(X)> 0, n = 2, · · · , L,
(11)
where Un(X) is the Chevyshev polynomial of the second kind, defined by
Un(X) =
(X +
√X2 − 1
)n+1 −(X −
√X2 − 1
)n+1
2√X2 − 1
, (12)
with
X =1− (γI − γP )2 + (γP − 1)2
2 |γP − 1|, (13)
and,
(iii) ((1− (γI − γP )2
)2− ((γI − γP ) (γP − 1))2
)|γP − 1|L−1 UL−1
(X)−
− 2(
1− (γI − γP )2)|γP − 1|L UL−2
(X)
+ |γP − 1|L+1 UL−3(X)
+ 2 (−1)L+1 (γI − γP ) (γP − 1)L+1 > 0. (14)
Remark 3.2 It can be seen that the L conditions defined by 10 and 11 describe regions of
convergence decreasing in L. These regions are plotted in Figure 1. We observe that the
10
intersection of all the regions that are defined by the conditions in 10 is equal to the region
that is defined in 10 for n = L. Moreover, we found that the last condition can be simplified.
This allows us to pose the following conjecture.
Conjecture 3.1 For each value of L, stability is guaranteed when γI and γP satisfy the fol-
lowing conditions:
(i) |γI − γP | < 1,
(ii) (1− (γI − γP )2) |γP − 1|(L−1) UL−1(X)− |γP − 1|L UL−2
(X)> 0
(iii) If the lead time of the system, L, is odd, then the third condition simplifies to
γI > max0, 2(γP − 1), and if the lead time of the system, L, is even, then the third
condition simplifies to
0 < γI < 2.
This conjecture has been verified for L = 2, · · · , 200. Furthermore, the behavior of the condi-
tions point towards an asymptotic region of convergence, defined by
Lemma 3.1 For all values of lead times L ∈ N, stability is guaranteed in the area that is
bounded by the lines γI = 0, γI = 2, γI = 2(γP − 1), and γI = 2γP .
To build intuition for the reasoning behind Conjecture 3.1 and Lemma 3.1, we refer the reader
to Figure 1 where we plot the regions defined by the conditions from Theorem 3.1. We plot
the region defined by condition (i) in a shade of blue, the region defined by condition (iii) in
a shade of light red, and the L− 1 regions defined by condition (ii) in shades of yellow. Thus,
a system is stable where all the regions overlap, seen here as dark purple (the variation in the
color shades across plots corresponds to the amount of overlapping regions). The additional
green areas in Figs. 1a and 1b concern aperiodicity, which we define in the next sub-section.
Comparing the plots on the left column of Fig. 1 (systems with odd lead times) with the
ones on the right column (even lead times), we see that the region defined by condition (i)
11
is independent of the lead time, whereas the region defined by condition (iii) depends on the
parity of the lead time. Indeed, we observe that the latter region is the same for all even (odd)
lead times, this results in the simplified condition (iii) seen in Conjecture 3.1.
When we analyze the behavior of the L− 1 regions defined by condition (ii), we can observe
that within the overlap of the regions defined by conditions (i) and (iii) (red and blue), each
additional region includes the preceding one. In other words, the nth yellow region is included
in the n− 1st yellow region. Thus, we can ensure stability by determining the overlap of only
3 regions for any value of lead time L.
Finally, looking at the region of stability, we can see how it asymptotically converges towards
the parallelogram defined in Lemma 3.1.
12
(a) Stability and aperiodicity regions, L=1. (b) Stability and aperiodicity regions, L=2.
(c) Stability regions for lead time L=3. (d) Stability regions for lead time L=4.
(e) Stability regions for lead time L=9. (f) Stability regions for lead time L=10.
Figure 1 – Stability and aperiodicity conditions.
13
3.2 Aperiodicity
If a system has a time-domain response with a number of maxima or minima that is less than
n, the order of the system, we call such a system aperiodic (Jury, 1985). These dynamics
are also defined by the poles of the transfer function: positive real poles contribute a damp-
ing component to the response, whereas negative real poles, and poles with an imaginary
component will contribute oscillatory terms (Nise, 2007). Formally,
Definition 3.3 (Jury, 1985). Suppose that G(z) = N(z)/C(z) is the transfer function of a
stable, linear, time-invariant, system. Thus, all poles of the transfer function, pi, i = 1, . . . , n
are within the unit circle. The response of this system is aperiodic if ∀i, pi ∈ [0, 1). From
Disney (2008) we adopt the concept of a weakly aperiodic system if ∀ i, pi ∈ R and there exists
an index k ∈ 1, . . . , n such that pk < 0.
By analyzing the poles of the reduced characteristic polynomial (9) for AP(V)IOBPCS and
applying Definition 3.3 we obtain the following propositions:
Proposition 3.2 When γI = γP = γ the response of a stable system for all lead times L is:
• aperiodic when 0 < γ ≤ 1, and
• weakly-aperiodic when 1 < γ < 2.
Proposition 3.3 When L > 2 and γI 6= γP the response of a stable system is non-aperiodic.
We show the aperiodicity boundaries for the cases of γI 6= γP and L = 1, 2, in Figures 1a and
1b, the area shaded in green represents the region for which the system is aperiodic, while the
black diagonal represents the weakly-aperiodic region. The boundaries for these regions can be
found by following the same analysis as in the proof of Proposition 3.3. The analysis of stability
of an AP(V)IOBPCS design is important because a stable system guarantees bounded orders
and inventories for any possible demand, as long as it is finite. Similarly, the aperiodicity
analysis of the system is relevant because an aperiodic system avoids costly oscillations. By
14
themselves, however, stability and aperiodicity boundaries are not enough to measure the
performance of the system. The stability conditions and aperiodicity propositions, as well as
the special regions defined in the accompanying figures, must be seen, then, as a necessary
first step in the evaluation of the system. The analysis presented thus far is necessary, but it
is not enough to distinguish any performance difference between two stable systems, or two
aperiodic systems.
From an analysis of the poles of the transfer functions, we see that policies where γI <
γP will always have a dampening component, whereas when γI > γP the response can be
oscillatory. This observation suggests that the performance of the system will depend on the
ratio of the behavioral parameters; we expect an oscillatory response in systems where the
pipeline is under-estimated (γI > γP ), and an over-dampened response in systems where
the inventory is under-dampened. However, we cannot derive more general statements on the
performance of the system through pole analysis because the amount, and magnitude of the
poles (necessary to characterize the response) depend on the order of the system, and on the
behavioral parameters. This, coupled with the observation made, where humans tend to under-
estimate the pipeline, motivate the structure of the next section. We aim to characterize the
performance of a stable system as a function of the ratio of its behavioral parameters through
extensive numerical experimentation.
15
4 The relationhip between behavior and performance
In this section, we study the performance of an APVIOBPCS design through an analysis of its
variance amplification (stationary performance) and its response to demand shocks (transient
performance). We introduce in each case the relevant measures we need to characterize its
performance. Our objective is to gain an understanding of the influence of the behavioral
parameters on said performance measures through extensive experimentation.
4.1 Variance amplification
In supply chains, the concept of variance amplification is often studied in the context of the
bullwhip effect: the propensity of orders to be more variable than demand signals, and for
this variability to increase the further upstream a firm is in a supply chain (Lee et al., 1997a).
In general, variance amplification measures the ratio of input variance to output variance and
can be defined for any pair of input/outputs. In this section, we use concepts from control
theory to analyze the stationary performance of an APVIOBPCS design through the lens of
variance amplification for orders and inventories.
To perform our analysis, we introduce two performance measures: the amplification ratio,
which measures the ratio of the output and input standard deviations in the steady state; and
the bullwhip measure, which measures the ratio of the output variance to the input variance
when demand is stochastic and stationary.
4.1.1 Steady state performance
We study the systems’ steady state performance by evaluating its amplification ratio when
demand is a sinusoid of frequency ω. In steady state, a sinusoidal input to a linear system
produces a sinusoidal output of the same frequency but of a different magnitude and phase. For
a given linear system, the ratio between the amplitude of the input and the magnitude of the
output at a given frequency is constant and is calculated as the modulus of its transfer function
evaluated at that frequency (Dejonckheere et al., 2003). Thus, the steady state amplification
ratio of an APVIOBPCS design can be calculated directly, for any input sinusoid, from its
16
transfer functions. Furthermore, it can be shown that for sinusoidal inputs, the amplification
ratio value is exactly the same as the ratio of the standard deviations of input over output
(Jakšič and Rusjan, 2008). Formally, for our system we define AO,ω as the amplification
ratio of orders for a sinusoidal demand of frequency ω, and AI,ω as the amplification ratio of
inventory for a sinusoidal demand of frequency ω, where:
AO,ω = |GO(eiω)|, and (15)
AI,ω = |GI(eiω)|. (16)
Frequency response plots are the graphical representation of the AO,ω and AI,ω of the system
for sinusoidal demands with frequencies between 0 and π. Because any demand stream can
be decomposed into a sum of sinusoids, these plots are a very powerful tool; by showing
the response to every possible demand frequency, they provide a good intuition over the
performance of a certain behavioral policy (pairs of γI and γP values) with regards to any
possible demand pattern.
4.1.2 Stationary performance
We now introduce the bullwhip measure as a way of evaluating the performance of a system
over all possible demand frequencies. Disney and Towill (2003) define the bullwhip measure
as the ratio between input variance to output variance (BW = σ2out/σ2in) and show that if the
mean of the input is zero and its variance is unity, the bullwhip of a system can be directly
calculated through the square of the area below the frequency response plots. In particular,
this implies that if the input to our system is a stationary i.i.d. normal demand stream, then
the bullwhip of orders can be calculated through
BWO =σ2Oσ2D
=1
π
∫ π
0|GO(eiω)|2dω, (17)
and the bullwhip of inventories (BW I) is defined analogously:
BW I =σ2Iσ2D
=1
π
∫ π
0|GI(eiω)|2dω. (18)
Thus, by plotting the contours of BWO and BWI as a function of the behavioral parameters
γI and γP , we obtain a graphical representation of the stationary performance of the system
17
as a function of the behavioral policies.
4.2 Influence of behavioral policies on the amplification of variance
Since the stationary and steady state performance measures are closely related, we adopt a
two-step approach to analyze the influence of different behavioral policies on the system’s
performance: we first look at the stationary performance by plotting contours of the bullwhip
measures, and follow by analyzing the steady state performance of different policies belonging
to the same contour.
Because every frequency is equally represented in the stationary measures, the insights ob-
tained by comparing behavioral policies within a single contour, hold for every contour. The
results presented in this section correspond to a system with α = 0.3, C = 3, L = 5. Note
that we performed extensive experiments with different parameters, we chose to present only
these cases since the qualitative conclusions are similar.
Bullwhip contours Figure 2a shows the bullwhip contour plots for orders as a function of
the behavioral parameters, and Figure 2b, the equivalent plot for inventories. We see that
despite differences in the magnitude of the variance amplification (BWI>BWO for any given
behavioral policy), low values of γI and γP correlate with low values of BWI and BWO.
Apart from this tendency to increase from the lower left quadrant towards the upper right,
the variance amplification displays an asymptote in the critical stability line. These obser-
vations suggest that the behavioral policies observed in experimental and empirical research
result in good stationary performance for both inventories and orders, but do not, however,
provide any additional information about how specific policies affect the performance (i.e., we
cannot separate between different policies within a single contour line). Also note that under
stationary demand an unstable system has a finite bullwhip.
Performance within contours We use frequency response plots of different behavioral
policies within a single contour to study how these policies affect the performance of a system.
The steady state performance of the system can be grouped into three categories, according to
18
1
2
4
8
6
10
12
14
0.5
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
ΓI
ΓP
(a) Order variance amplification (BWO)
810
1214
16
18
20
2224
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
ΓI
ΓP
(b) Inventory variance amplification (BWI)
Figure 2 – Contour plots
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
ΓI
ΓP
(a) Parameter settings
0 Π
4Π
23 Π
4Π
Ω12
4
6
8
10
12
14
16AO,Ω
(b) Order amplification BWO = 2
0 Π
4Π
23 Π
4Π
Ω12
4
6
8
10
12
14
16AO,Ω
(c) Order amplification BWO = 6
Figure 3 – Bullwhip contour, and frequency plots for orders
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0
ΓI
ΓP
(a) Parameter settings
0 Π
4Π
23 Π
4Π
Ω12
4
6
8
10
12
14
16AI,Ω
(b) Inventory amplification BWI = 12
0 Π
4Π
23 Π
4Π
Ω12
4
6
8
10
12
14
16AI,Ω
(c) Inventory amplification BWI = 16
Figure 4 – Inventory amplification contour, and inventory frequency plots
whether the ratio between behavioral parameters is greater, smaller, or equal to one. Specif-
ically, γI > γP policies outperform the rest when the frequency is larger than roughly π/4
(If demand is observed daily, a frequency of π/4 represents a cyclical demand with a period
19
of 8 days), but significantly under-performs otherwise. The performance advantages observed
are a lower amplification ratio for any given frequency than all other policies, and a higher
robustness to changes in frequency (i.e., flatter response) than γI < γP policies. Conversely,
γI < γP policies offer a performance advantage with frequencies smaller than roughly π/4.
We see this in Figure 3 (Figure 4), where we plot the frequency responses of the same system
under 7 different behavioral policies, grouped into Figures 3b and 3c (4b and 4c) according to
their BWO (BWI).
We see that the influence of behavioral parameters on the system’s performance depends on
the demand pattern. In particular, the behavioral policies cause systems to react very differ-
ently towards high- and low-frequency demands, with a very clear trade-off in performance.
The observed empirical behavior of under-estimating the pipeline, for example, is consistent
with a desire to buffer short term fluctuations in demand. In the next section, we study a
special case of demand: a one-time shock.
4.3 Influence of behavioral policies on the response to demand shocks
We perform this study in the time-domain, by keeping track of the actual changes in orders
and inventories after a step demand increase. We quantify the system’s transient performance
through the integral time weighted absolute error (ITAE ), a measure of the dynamic per-
formance of the system in terms of time-weighted deviations from the ideal response (Hoberg
et al., 2007a). We use the ITAE to quantify the transient behavior of the system after a step
change in demand. The ITAE is defined as:
ITAE =∞∑t=0
t|ε(t)|, (19)
where ε(t) represents the absolute error at time t. This measure penalizes deviations from the
new steady state, and introduces a linear penalty for longer lasting deviations. Thus, both
the amplification and the settling time of the system play a role in its determination. Due to
the transcendental nature of the transfer functions of the system, it is not possible to derive a
general, closed form expression for its ITAE. An analytical expression for fixed values of L can
be calculated through a double transform of the transfer functions (Hoberg et al., 2007a), but
20
(a) log of ITAE for orders. (b) log of ITAE for inventories.
3
6
9
12
15
(c) Log scale
Figure 5 – ITAE as a response to a step increase in demand.
the complexity of the general transforms makes this procedure unsuitable for anything but
trivial values of L. Hence, we continue to build upon our results thus far through numerical
experimentation.
Thus far, the influence of behavioral parameters on the orders and inventories have been
comparable. When responding to shocks, on the other hand, we find that γP = γI policies
perform best when looking at ITAEO, while the performance of ITAEI is best for a policy
of the type γP < γI . Furthermore, and in direct contrast with the findings of the previous
section, for a given γI/γP ratio, the system’s performance increases towards the top-right
quadrant of the behavioral parameter space. This means that the behavioral policies that
were found to work best for stationary demands, are at a disadvantage when shocks occur.
Figure 5 shows the transient performance of a APVIOBPCS design as measured by the ITAE
of orders (ITAEO) and inventory (ITAEI) of a design with α = 0.3, C = 3, and L = 5 for
different behavioral parameters (γI and γP ). The ITAE is calculated for the first 50 periods.
Due to the extreme variation in values, we plot the logarithm of ITAE and clip the values of
exploding series.
21
4.4 Insights
The majority of systems encountered in real life seem to operate in what we describe as the
lower left quadrant of the behavioral parameter space, with γI > γP (Sterman, 1989; Udenio
et al., 2012). These systems offer performance advantages under stationary demands, and
demands with high frequency components but at the same time offer performance disadvan-
tages when there is a demand shock. This is to be expected: the slow behavioral response
that buffers high frequency demand fluctuations causes the system to be too slow to respond
to a shock, while the rapid behavioral response that allows the system to quickly adapt to a
demand shock causes the system to overreact when demand is stationary or cyclic. To under-
stand and analyze this performance trade-off we introduce performance trade-off curves for
orders and inventories.
4.5 Stationary and transient performance tradeoff
To quantify the trade-off we group behavioral policies through the γI/γP ratio (diagonals in
the behavioral parameter space) and plot BWO against IATEO, and BWI against IATEI .
As expected, DE-policies (γP = γI) offer the best trade-off between stationary and transient
performance. However, under-estimating the pipeline (γP < γI) can give a performance edge
if the objective is to minimize the stationary error. This slight performance edge comes
at the cost of an increased sensitivity: a deviation from the optimal policy brings about
significant performance penalties. Over-estimating the pipeline, on the other hand, decreases
the performance of the system, but displays increased robustness.
Figures 6a and 6b show the performance trade-off curves of order and inventories, Figure 6c
gives the reference for the position of the curves within the behavioral parameters, and Figures
6d and 6e show the trade-off curves limited to the [0, 0.4]2 area in the behavioral space.
22
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ΓP
(c) Reference of diagonals in the parameter space.
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Figure 6 – Performance tradeoff curves between stationary and transient responses.
23
5 Conclusions
We have used classical control theory to model general AP(V)IOBPCS systems and analyze
the impact of fractional behavioral parameters γI and γP on their performance. Behavioral
parameters in the context of these systems are what in the control theory world are known
as feedback controllers, and represent the incomplete closure of inventory and pipeline gaps
at the moment of generating replenishment orders. The study of the behavioral influence is
motivated by the existence of a large body of experimental, and empirical, research that iden-
tifies human biases in ordering decisions. The quantification of these biases suggests that both
individual decision makers and firms operate in a clearly defined zone within the parameter
space. We attempt, throughout this paper, to understand what –if any– advantages this zone
brings to the performance of the system.
To achieve this, we first modeled a discrete time, general APVIOBPCS design with inde-
pendent behavioral parameters. We then derived its order and inventory transfer functions,
analyzed the stability of such designs and developed a new procedure for the determination
of the exact stability region of a system with a given lead time. This test has the advantage
of avoiding the direct calculation of determinants, or matrix-based procedures that charac-
terize previous exact solutions of the problem (Jury, 1964; Disney, 2008). Additionally, this
procedure allowed us to find an asymptotic region of stability, providing us with a sufficient
condition for stability that is independent of L. From the work of Disney (2008), we adopted
a characterization of the response of the system based on the position of the poles of the
transfer function in the complex plane with which we completely identified the regions for
which a system complies with the aperiodicity and weak aperiodicity conditions.
Following the stability and aperiodicity results we performed an extensive set of numerical
experiments to help us understand the influence of behavioral parameters in the performance
of the system. Due to the large amount of parameters of the system, we needed to specify
values for non-behavioral parameters in our experiments. We then performed extensive tests
that suggest that the insights developed through the paper hold in general.
The performance of a system was measured first in the frequency domain, and then in the
time domain. Through the frequency domain analysis we found insights related to the steady
24
state response of the system to cyclical inputs (i.e., sinusoidal demands of varying frequency),
and the stationary response to white noise (i.e., i.i.d normally distributed demands). Through
these analyses, we found that the heavy smoothing (low behavioral parameters) and pipeline
underestimation (γP < γI) found in practice favors the reduction of both the bullwhip for
orders and the amplification of inventory variance, as well as increasing the robustness of the
system to short term demand fluctuations.
In contrast, the time domain analysis identifies serious performance issues of the system when
confronted to a demand shock. The smoothing that we found beneficial in lowering the vari-
ance amplification of the system, lowers the reaction time of the system thus decreasing its
reactiveness in reaching a new steady state after a shock. Similarly, the pipeline underestima-
tion that contributes to the robustness of the system to short term demand variations, causes
performance-decreasing oscillations following a demand shock.
Finally, we showed the tradeoff between the stationary and transient performances and found
that both inventories and orders achieve good performance around the same values of behav-
ioral parameters. The best combined performance occurs in the lower left quadrant of the
behavioral parameter space, which suggests that the smoothing observed in real life data is
indeed beneficial. With regards to the pipeline underestimation, also predominant in the data,
further research needs to be undertaken to further quantify the tradeoffs, taking into account
realistic demand streams. Since demand observed in real life is neither purely stationary nor
composed entirely of shocks, how should firms approach the tradeoffs? What is the weight
that should be placed upon different demand types? Are the benefits of increased high fre-
quency robustness achieved through underestimating the pipeline offset by the decrease in
performance following shocks? We believe our modeling framework provides a good basis to
address these and similar highly relevant research questions.
25
Appendix: Proofs
Proof of proposition 3.1:
We denote each root of the characteristic polynomial (pole of the transfer function) by pLi ,
where i = 1, 2, ..., L+ 2.
It follows from (8) that pL1 = (1− α) is a root of the characteristic polynomial that does not
depend on L.
When α ∈ (0, 1], pL1 is inside the unit circle and the remaining L+1 roots of the characteristic
polynomial C(z) will be equal to the L + 1 roots of the reduced characteristic polynomial
C(z). Thus the condition for stability when α ∈ (0, 1] reduces to checking that all roots of
C(z) be inside the unit circle.
When α = 0, then∣∣pL1 ∣∣ = 1, which means that the system will be marginally stable unless∣∣pL1 ∣∣ = 1 is also a root of the numerator of the transfer function. The transfer function for
orders when α = 0 can be rewritten as:
GO(z) =γI(z − 1)zL
(z − 1) ˆC(z)=γIz
L
ˆC(z). (20)
The transfer function for inventories when α = 0 can be rewritten as:
GI(z) =γI − z(zL+1 + zL(γP − 1)− γP
(z − 1) ˆC(z)=γI − z(zL + γP z
L−1 + . . .+ γP z + γP )
ˆC(z). (21)
Thus, since z = 1 is a root of the denominator of both GO(z) and GI(z), ∀L ∈ N, the
conditions for stability when α = 0 reduce to checking that all roots of C(z) be inside the unit
circle.
Proof of proposition 3.2:
When γI = γP = γ we can rewrite the reduced characteristic polynomial:
C(z) = zL(z − 1 + γ). (22)
Its L+ 1 zeroes are:
pL2 = (1− γ), (23)
pL3 = pL4 = ... = pLL+2 = 0. (24)
26
When γ ∈ (0, 2),∣∣pL2 ∣∣ < 1, therefore the system is stable. More precisely, for γ ∈ (0, 1], pL2 ≥ 0,
and for γ ∈ (1, 2), pL2 < 0. Thus, the system is respectively aperiodic, and weakly aperiodic.
Proof of proposition 3.3:
We know that all the roots of the reduced characteristic polynomial C(z) of a stable system
lie inside the unit circle of the complex plane. To judge on the aperiodicity of such a system,
we need to know whether the roots of C(z) lie on the negative or positive half plane.
For this reason, we apply Descartes’ rule of signs to the polynomial C(z). Descartes’ rule of
signs states that:
When the terms of a single variable polynomial with real coefficients are ordered
by descending variable exponent, then the number of positive roots of the polyno-
mial is either equal to the number of sign differences between consecutive nonzero
coefficients, or is less than it by a multiple of 2. As a corollary, the number of
negative roots is the number of sign changes after multiplying the coefficients of
odd powers by (-1), or less than it by a multiple of 2. Finally, for a polynomial
of degree n, the minimum number of complex roots equals n− (p+ q) where p is
the maximum number of positive roots, and q the maximum number of negative
roots. (Struik, 1969)
To apply Descartes’ rule of signs, we write the reduced characteristic polynomial as
C(z) = zL+1 − zL(1− γP ) + γI − γP . (25)
We assume that the system is stable and distinguish between two cases: γI < γP , and γI > γP .
The case of γI < γP For all values of L, γP this polynomial will have one sign change.
Therefore we will always have one positive and real root.
To find the negative and real roots of C(z), we separate between odd and even lead times L:
For L odd, the polynomial C(−z) = zL+1 + zL(1 − γP ) + γI − γP has 1 sign change for all
values of γP . Therefore, there exists a real and negative root of C(z) and the remaining L−1
roots come in pairs of complex conjugates. Only for L = 1 does C(z) have no complex roots.
27
For L even, the polynomial C(−z) = −zL+1 − zL(1 − γP ) + γI − γP does not have any sign
change when γP ∈ [0, 1] and thus no negative and real roots. This means that it has at least
1 pair of conjugate complex roots. When γP ∈ (1, 2), it has 2 sign changes and thus 0 or
2 negative and real roots. If it has 2 negative roots and L = 2, then the system is weakly
aperiodic. In any other case, there exists at least 1 pair of complex roots and the system is
thus unstable.
The case of γI > γP For all values of L, and for γP ∈ [1, 2) the reduced characteristic
polynomial C(z) will have no sign changes and consequently it does not have any positive and
real roots. To find the negative and real roots of C(z) with γP ∈ [1, 2), we separate between
odd and even lead times L: For L odd, the polynomial C(−z) has 2 sign changes and therefore
either 2 or 0 real and negative roots. When L = 1 and it has 2 negative real roots, the system
is weakly aperiodic. In all other cases, there exist at least 1 pair of complex roots and the
system will consequently be non-aperiodic. For L even, the polynomial C(−z) has 1 sign
change and therefore 1 real and negative root. Thus in this case the polynomial will always
have at least 1 pair of complex roots and the system will consequently be non-aperiodic.
For all values of L, and for γP ∈ [0, 1) the reduced characteristic polynomial C(z) will have 2
sign changes and consequently it has either 2 or 0 positive and real roots. To find the negative
and real roots of C(z) with γP ∈ [0, 1), we separate once more between odd and even lead
times L: For L odd, the polynomial C(−z) has 0 sign changes and therefore either 0 real and
negative roots. When L = 1 and it has 2 positive real roots, the system is aperiodic. In all
other cases there exist at least 1 pair of complex roots and the system will consequently be
non-aperiodic. For L even, the polynomial C(−z) has 1 sign change and therefore 1 real and
negative root. Only the combination L = 2 and 2 positive real roots gives a weakly aperiodic
response. In all other cases, there exists at least 1 pair of complex roots and thus the system
is non-aperiodic.
Proof of Theorem 3.1:
According to Theorem 43.1 of Marden (1969), the number of roots of our reduced characteristic
polynomial C(z) (Eq. 9) inside the unit circle is equal to the number of negative signs in
28
the sequence
∆1,∆2
∆1, . . . ,
∆L+1
∆L, (26)
where
∆n := det
An A∗Tn
A∗n ATn
, (27)
and
An :=
a 0 0 · · · 0
0 a 0 · · · 0
0 0 a · · · 0
......
......
...
0 0 0 · · · a
, n = 1, . . . , L, AL+1 :=
a 0 0 · · · 0
0 a 0 · · · 0
0 0 a · · · 0
......
......
...
b 0 0 · · · a
,
A∗n :=
1 0 0 · · · 0
b 1 0 · · · 0
0 b 1 · · · 0
......
......
...
0 0 0 · · · 1
, n = 1, . . . , L+ 1,
with a = γI − γP , and b = γP − 1. Here ATn denotes the transpose of An where the dimension
of these matrices is n× n.
To guarantee stability, we need all the roots of Eq. (9) to be inside the unit circle. Thus, we
need to have L+ 1 negative signs in the sequence (26). Consequently, we need to have:
(−1)n∆n > 0, ∀n = 1, . . . , L+ 1. (28)
Since the matrices An and A∗n commute, according to Silvester (2000), we have that for
29
n = 1, . . . , L+ 1, (−1)n∆n = det(AnA
Tn −A∗nA∗Tn
). Thus, for n = 1, . . . , L,
(−1)n∆n = det
1− a2 b 0 · · · · · · −ab
b 1− a2 + b2 b 0 · · · 0
0 b 1− a2 + b2 b · · · 0
.... . . . . . . . . . . .
...
0 · · · 0 b 1− a2 + b2 b
−ab 0 · · · 0 b 1− a2 + b2
, (29)
and also ,
(−1)L+1∆L+1 = det
1− a2 b 0 · · · · · · −ab
b 1− a2 + b2 b 0 · · · 0
0 b 1− a2 + b2 b · · · 0
.... . . . . . . . . . . .
...
0 · · · 0 b 1− a2 + b2 b
−ab 0 · · · 0 b 1− a2
. (30)
If we denote Mn as the square n× n matrix with diagonal elements equal to 1− a2 + b2, and
all the elements on the upper and lower diagonal are equal to b, then we can find recursively
that,
(−1)n∆n = (1− a2)det Mn−1 − b2det Mn−2, n = 2, . . . , L. (31)
The determinant det Mn can be calculated through formula (3) of Marr and Vineyard (1988)
as
det Mn = Dn(1− a2 + b2, b, b) = |b|Un
(1− a2 + b2
2 |b|
), (32)
where Un is the nth degree Chebyshev polynomial of the second kind, defined by
Un(Z) =
(Z +√Z2 − 1
)n+1 −(Z −√Z2 − 1
)n+1
2√Z2 − 1
. (33)
If we set X = (1− a2 + b2)/(2 |b|), then
(−1)n∆n = (1− a2) |b|(n−1) Un−1(X)− |b|n Un−2
(X). (34)
30
Similarly, for the (L+ 1)st determinant, it holds that
(−1)L+1∆L+1 =(1− a2)2det ML−1 − 2(1− a2)b2det ML−2 + b4 detML−3 + 2(−1)L+1abL+1 − (ab)2 detML−1
=((
1− (γI − γP )2)2 − ((γI − γP ) (γP − 1))2
)|γP − 1|L−1 UL−1
(X)−
− 2(1− (γI − γP )2
)|γP − 1|L UL−2
(X)
+ |γP − 1|L+1 UL−3
(X)
+ 2 (−1)L+1 (γI − γP ) (γP − 1)L+1 . (35)
Finally, observe that −∆1 = 1− a2, which completes the proof.
Proof of Lemma 3.1:
In a compact form, the region defined by Lemma 3.1 can be written as |b| ≤ 1 − |a| with
a = γI − γP , and b = γP − 1. We observe that condition (iii) of Conjecture 3.1 always defines
two boundary lines in this region. Therefore, inside this region, condition (iii) is always going
to be satisfied. In order to show that this is indeed the asymptotic region defined by Lemma
3.1 when L goes to infinity, it is sufficient to show that when we set |b| = 1− |a|,
limL→+∞
(−1)L∆L = 0. (36)
Knowing that X = 1 (see proof of Theorem 3.1) and, using that UL(1) = L+ 1 (Abramovich
and Stegun, 1965, Table 22.3.7), we can rewrite (34) as:
(−1)L∆L = (1− |a|)L(1 + L |a|), (37)
which goes to 0 as L goes to ∞ since |a| < 1 and the proof is complete.
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307 2010 Interaction between intelligent agent strategies for real-time transportation planning
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305 2010 Vehicle Routing with Traffic Congestion and Drivers' Driving and Working Rules
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Working Papers published before 2009 see: http://beta.ieis.tue.nl