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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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(a) Order performance tradeoff curves.

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(b) Inventory performance tradeoff curves.

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

(c) Reference of diagonals in the parameter space.

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(e) Inventory performance tradeoff curves.

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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Working Papers Beta 2009 - 2013 nr. Year Title Author(s)

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323 322 321 320 319 318 317 316 315 314 313

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in a two-echelon distribution network Inventory reduction in spare part networks by selective throughput time reduction The selective use of emergency shipments for service-contract differentiation Heuristics for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering in the Central Warehouse Preventing or escaping the suppression mechanism: intervention conditions Hospital admission planning to optimize major resources utilization under uncertainty Minimal Protocol Adaptors for Interacting Services Teaching Retail Operations in Business and Engineering Schools Design for Availability: Creating Value for Manufacturers and Customers Transforming Process Models: executable rewrite rules versus a formalized Java program Getting trapped in the suppression of exploration: A simulation model A Dynamic Programming Approach to Multi-Objective Time-Dependent Capacitated Single Vehicle Routing Problems with Time Windows

E.M. Alvarez, M.C. van der Heijden, W.H. Zijm B. Walrave, K. v. Oorschot, A.G.L. Romme Nico Dellaert, Jully Jeunet. R. Seguel, R. Eshuis, P. Grefen. Tom Van Woensel, Marshall L. Fisher, Jan C. Fransoo. Lydie P.M. Smets, Geert-Jan van Houtum, Fred Langerak. Pieter van Gorp, Rik Eshuis. Bob Walrave, Kim E. van Oorschot, A. Georges L. Romme S. Dabia, T. van Woensel, A.G. de Kok

312 2010 Tales of a So(u)rcerer: Optimal Sourcing Decisions Under Alternative Capacitated Suppliers and General Cost Structures

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311 2010 In-store replenishment procedures for perishable inventory in a retail environment with handling costs and storage constraints

R.A.C.M. Broekmeulen, C.H.M. Bakx

310 2010 The state of the art of innovation-driven business models in the financial services industry

E. Lüftenegger, S. Angelov, E. van der Linden, P. Grefen

309 2010 Design of Complex Architectures Using a Three Dimension Approach: the CrossWork Case R. Seguel, P. Grefen, R. Eshuis

308 2010 Effect of carbon emission regulations on transport mode selection in supply chains

K.M.R. Hoen, T. Tan, J.C. Fransoo, G.J. van Houtum

307 2010 Interaction between intelligent agent strategies for real-time transportation planning

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306 2010 Internal Slackening Scoring Methods Marco Slikker, Peter Borm, René van den Brink

305 2010 Vehicle Routing with Traffic Congestion and Drivers' Driving and Working Rules

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304 2010 Practical extensions to the level of repair analysis

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303 2010 Ocean Container Transport: An Underestimated and Critical Link in Global Supply Chain Performance

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302 2010 Capacity reservation and utilization for a manufacturer with uncertain capacity and demand

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300 2009 Spare parts inventory pooling games F.J.P. Karsten; M. Slikker; G.J. van Houtum

299 2009 Capacity flexibility allocation in an outsourced supply chain with reservation Y. Boulaksil, M. Grunow, J.C. Fransoo

298

2010

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297 2009 Responding to the Lehman Wave: Sales Forecasting and Supply Management during the Credit Crisis

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296 2009 An exact approach for relating recovering surgical patient workload to the master surgical schedule

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295

2009

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293 2009 Implementation of a Healthcare Process in Four Different Workflow Systems

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292 2009 Business Process Model Repositories - Framework and Survey

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291 2009 Efficient Optimization of the Dual-Index Policy Using Markov Chains

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290 2009 Hierarchical Knowledge-Gradient for Sequential Sampling

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289 2009 Analyzing combined vehicle routing and break scheduling from a distributed decision making perspective

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288 2009 Anticipation of lead time performance in Supply Michiel Jansen; Ton G. de Kok; Jan C.

Chain Operations Planning Fransoo

287 2009 Inventory Models with Lateral Transshipments: A Review

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286 2009 Efficiency evaluation for pooling resources in health care

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

285 2009 A Survey of Health Care Models that Encompass Multiple Departments

P.T. Vanberkel; R.J. Boucherie; E.W. Hans; J.L. Hurink; N. Litvak

284 2009 Supporting Process Control in Business Collaborations

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283 2009 Inventory Control with Partial Batch Ordering O. Alp; W.T. Huh; T. Tan

282 2009 Translating Safe Petri Nets to Statecharts in a Structure-Preserving Way R. Eshuis

281 2009 The link between product data model and process model J.J.C.L. Vogelaar; H.A. Reijers

280 2009 Inventory planning for spare parts networks with delivery time requirements I.C. Reijnen; T. Tan; G.J. van Houtum

279 2009 Co-Evolution of Demand and Supply under Competition B. Vermeulen; A.G. de Kok

278 277

2010 2009

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276 2009 Coordinating Supply Chains: a Bilevel Programming Approach Ton G. de Kok, Gabriella Muratore

275 2009 Inventory redistribution for fashion products under demand parameter update G.P. Kiesmuller, S. Minner

274 2009 Comparing Markov chains: Combining aggregation and precedence relations applied to sets of states

A. Busic, I.M.H. Vliegen, A. Scheller-Wolf

273 2009 Separate tools or tool kits: an exploratory study of engineers' preferences

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272

2009

An Exact Solution Procedure for Multi-Item Two-Echelon Spare Parts Inventory Control Problem with Batch Ordering

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271 2009 Distributed Decision Making in Combined Vehicle Routing and Break Scheduling

C.M. Meyer, H. Kopfer, A.L. Kok, M. Schutten

270 2009 Dynamic Programming Algorithm for the Vehicle Routing Problem with Time Windows and EC Social Legislation

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269 2009 Similarity of Business Process Models: Metics and Evaluation

Remco Dijkman, Marlon Dumas, Boudewijn van Dongen, Reina Kaarik, Jan Mendling

267 2009 Vehicle routing under time-dependent travel times: the impact of congestion avoidance A.L. Kok, E.W. Hans, J.M.J. Schutten

266 2009 Restricted dynamic programming: a flexible framework for solving realistic VRPs

J. Gromicho; J.J. van Hoorn; A.L. Kok; J.M.J. Schutten;

Working Papers published before 2009 see: http://beta.ieis.tue.nl