Table of Contents
Tampering Dynamics
Nicholas C. GEORGANTZAS Joyce N. ORSINI
Director, System Dynamics Consultancy Director, Deming Scholars MBA
Fordham University Business Schools Fordham University Business Schools
113 West 60th Street, Suite 617-D 33 West 60th Street, Suite 417
New York, NY 10023-7484, USA New York, NY 10023-7471, USA
Tel.: (212) 636-6216, Fax: (212) 765-5573 Tel.: (212) 636-6219, Fax: (212) 636-7303
E-mail: georgantzas@fordham.edu E-mail: orsini@fordham.edu
Abstract—Tampering can cause many of the tragedies that people and organizations increasingly
face. To show pervasive forms of tampering, Deming used a funnel experiment, which quality
researchers and practitioners now use to show the dysfunctional effects of not using statistical
process control (SPC) charts. Interspersed with real-world tampering examples, this paper pre-
sents a system dynamics model of the funnel experiment, useful for reproducing Sparks & Field's
(2000) SPC charts and statistical tests. The results also show multi-dimensional vistas of location
probability and Theil's (1966) inequality statistics, plus an entropy-based view of tampering.
System dynamics allows looking at the experiment causally, as opposed to merely looking at
coincidental, due to randomness, SPC charts and entropy (uncertainty) measures. Looking under
the hood, so to speak, one can see how the circular, feedback-loop relations among variables in
the funnel experiment system produce assumption-violating dynamics as multiple feedback loops
determine system behavior. Enhancing SPC with system dynamics can help detect, explain and
prevent tampering with the very processes that managers must manage.
Keywords: assumptions, Deming, entropy, feedback, funnel experiment, Shewhart charts, simu-
lation, statistical process control (SPC), system dynamics, tampering, uncertainty
Remember the Tylenol Scare? That tampering nightmare (Beck et al. 1982) was just one of the
tragedies that made Mitroff & Kilmann (1984) look at the threats governmental, industrial, edu-
cational and health-care organizations face today. Only to see that psychopathology, sabotage
and terrorism have become commonplace. Despite federal anti-tampering laws, terrorists, sabo-
teurs and psychopaths continue committing such crimes, medical researchers fiddle with genes
(August et al. 2002) and malicious troublemakers adjust, negate and remove pollution control
equipment from factories and motor vehicles. To build wealth fast, managers too engage secretly
or improperly in shady transactions and often meddle, mix up, rig and tinker with the very proc-
esses they are supposed to manage (Deming 2000a&b).
Look at Enron, for example. After a one-year old investigation, not one Enron manager has
gone to jail (Dobbs 2002). Governments always tamper with the economy to talk up markets, but
investors do not come back until they see real reforms, accountability and earnings. While ours,
according to Dobbs (2002), "fails to assure them that is serious about choking off corporate
fraud," the PricewaterhouseCoopers data show that, from 2000 to 2001, annual securities-
tampering litigation in USA increased by 240.3 percent (O'Connor 2002).
According to Merriam-Webster's Collegiate Dictionary (2001), 'tampering' entails carrying
on underhand or improper negotiations (as by bribery) and interfering so as to weaken, to make
something worse. In management, tampering entails reacting to common-cause variation or ran-
domness in a process as if it were special-cause variation. Every time a manager mistakes ran-
domness for special-cause variation, tampering makes processes, i.e., Merriam-Webster's
‘something’, worse too. Managers who understand that tampering with a process is harmful will
let the process run and study it (Latzko & Saunders 1995). But the profound knowledge required
to stop tampering (a) has no substitute and (b) is not automatic (Deming 2000a&b). It takes wis-
dom to manage the uncertainty that leads to tampering with a process. And managing uncertainty
must be learnt; must be led (Georgantzas & Acar 1995).
According to Deming (2000a&b), managers tamper with processes because they are trained
to do so. They react to each rise in performance as if things were improving and react to each dip
in performance as if things were beginning to fall apart. Typically, nothing has changed at all.
What managers react to is common randomness, which pervades all business systems and proc-
esses as "a measure of our ignorance" (Sterman 2000, p. 127).
It was Walter A. Shewhart of the Bell Telephone Laboratories who first used statistics for
quality control. Shewhart's 16 May 1924 memorandum featured the very first sketch of a modern
control chart (cf Schultz 1994, p. 5). Shewhart worked on and improved his approach until his
1931 book (Shewhart 1980) set the tone for statistical process control (SPC). To tell between
chance and assignable causes or variation, he created a method to distinguish one type of varia-
tion from the other. Following Shewhart, Deming (2000a&b) refers to 'common' and 'special’
causes of variation. Common causes are inherent in a process. Special causes are rare events that
require immediate action. When a machine starts producing defects consistently, for example, it
must be taken off line for repair. But eighty to ninety percent of product and service variation is
due to common causes, endemic to a process. The only way to reduce common variation is to
redesign the process itself.
To show subtle but pervasive forms of tampering with stable processes, Deming (2000a&b)
used a funnel experiment for which he credits Dr. Lloyd S. Nelson, Director of Statistical Meth-
ods, Nashua Corporation (Deming 2000a, p. 20). Its well-established four rules determine the
funnel and marble dynamics, i.e., behavior through time. MacGregor (1990) used the funnel ex-
periment to show the value of control versus no control for a drifting process mean, Gunter
(1993) for experimental design and Sparks & Field (2000) to check their SPC chart assumptions.
Sparks & Field see in the funnel experiment a useful tool for quality trainers to show prac-
titioners how to use SPC charts correctly. Inappropriate charts lead to excessive, unnecessary
adjustment of a process, time lost in looking for nonexistent special causes, or to a belief that the
process is out of control because of the false signaling of special causes. Well aware that deci-
sions based on 'black-box' SPC charts can be incorrect (Field & Sparks 1995), Sparks & Field
(2000) present a unidimensional analysis of the funnel experiment as a means of training workers
to become discerning SPC users. Sparks & Field also address practitioner concerns about the be-
havior that the funnel experiment SPC charts produce.
Interspersed with real-world tampering examples, the funnel experiment overview below
reiterates the practitioner concerns that Sparks & Field (2000) address because of their educa-
tional value. The model description section then presents a two-sector system dynamics
(Forrester 1958 & 1961) model of the funnel experiment. The results show the model's useful-
ness as it helps replicate Sparks & Field's (2000) SPC charts and statistical tests. Next, however,
the paper goes beyond the unidimensional analysis of SPC charts. It adds three-dimensional
graphs of location probability and Theil's (1966) inequality statistics (TIS). Also, it measures lo-
cation uncertainty using Shannon's information entropy (Shannon & Weaver 1949) and thereby
shows an entropy (uncertainty) based view of the funnel experiment.
The 3D graphs, TIS and the discerning entropy-based view of the funnel experiment are
some of the benefits from extending "the necessary mathematical framework" of Sparks & Field
(2000, p. 292) with a system dynamics model. System dynamics, the study of cause and effect
2
through time, allows looking at Deming's experiment causally, as opposed to merely looking at
coincidental, due to randomness, SPC charts and entropy measures. Looking under the hood, so
to speak, the paper shows how the circular, feedback-loop relations among variables in the fun-
nel experiment system produce assumption-violating dynamics as multiple feedback loops de-
termine system behavior. Enhancing SPC with system dynamics can help detect, explain and
prevent tampering with the very processes that managers must manage.
Deming's funnel experiment and quality practitioner concerns
An apparatus that demonstrates Deming's funnel experiment includes a mobile stand that holds a
funnel at a fixed height above a gridded paper pad (Fig. 1). The point (0, 0) on the gridded pad
designates the target. Imagine dropping the marble down the funnel. The marble rolls down in-
side and out the funnel in a random fashion. Friction, gravity and harmonics are some of the
natural forces acting on the marble to produce common-cause, i.e., random, variation (Latzko &
Saunders 1995, p. 151).
Figure | An apparatus for the Deming- Nelson funnel experiment
The apparatus:
funnel,
mobile stand,
marble,
gridded paper na 4
and rules! =e . \
eS
Source: www.knoxshops.com
With the funnel clamped to its steady stand, the marble drops on the gridded pad and rolls
until it stops somewhere near the target. A pen easily marks the spot where the marble comes to
a rest. When repeatedly dropped through the funnel, the marble does not always land nor does it
come to a rest on the same spot. Natural forces make it drop and stop randomly around the target.
Performing the funnel experiment requires dropping the marble through the funnel multiple
times, while one of the four rules on Table 1a determines the funnel's aiming point (Deming
2000a&b, Latzko & Saunders 1995, Sparks & Field 2000).
Rule 1 means no tampering (Table 1a). The funnel remains aimed at the target. Its position
is the same for each drop, "producing a stable distribution of points and a minimum variance on
any diameter drawn through the target" (Deming 2000a, p. 329).
Rule 2 implies reacting to an individual data point by correcting in an equal and opposite
direction. Adjusting the position of a gun after firing one shot at a target is one example of this
form of tampering. Another example is the tampering that takes place with the thermostat in a
conference room. Someone feels hot when coming into the room from the hallway and, on the
basis of this one data point, how the room feels upon entering it, adjusts the thermostat. That per-
3)
son is changing the system based on that one point. Someone else comes in later, when the room
is on the cool side of its cycle, and tampers the system up to a higher temperature. Over the
course of the day these tampering adjustments increase the variation of the temperature in the
room. Back to the funnel experiment, Rule 2 produces stable results, but the variance of the dis-
tribution of points along any diameter drawn through the target will, under Rule 2, be double the
expected variance under Rule 1 (Deming 2000a, Sparks & Field 2000).
Table 1 The Deming-Nelson funnel experiment (a) rules and (b) practitioner questions
(a) Rule Simplified rule description adapted from Deming (2000a, p. 328)*
1 Keep funnel aimed at the target (no tampering)
2 Move funnel from its last position to compensate for last error
3 Move funnel from the target to compensate for last error
4 Move funnel right over the previous drop location
(b) Question Quality practitioner questions adapted from Sparks & Field (2000, p. 292)
1 Under Rule 2, why does the ¥ chart hug the centerline?
2 Why does the Rule 3 ¥ chart hug the centerline even more?
3 Why does the Rule 3 ¥ chart develop a strong alternating behavior for odd sample sizes?
4 Why do later vales on the Rule 3 x chart with odd sample sizes deviate more from the cen-
terline than do earlier values? Why is this behavior not apparent for even sample sizes?
5 Why does the Rule 4 ¥ chart behave as if very out of control?
6 Given the processes under the four different rules, what are the appropriate charts?
* In addition to crediting Dr. Lloyd S. Nelson for the funnel experiment idea, Deming attributes experimental results
to Lord Rayleigh's work on vibrations and theory of sound, and to Dr. Frank $. Grubbs (Grubbs 1983) for optimum
convergence to the target (Deming 2000a, p. 329).
Although Sparks & Field (2000) do a remarkable job in answering quality practitioner questions from a statistical
viewpoint, the simple system dynamics model of the funnel experiment presented here can still offer insight in an-
swering Questions 1, 2 and 5.
Rule 3 makes one forget about the target and react only to the last event that happens. The
system explodes as the marble rest points eventually move away farther and farther in opposite
direction from the target, in a symmetrical bow-tie pattern; an accelerating swing from one di-
rection to another. Escalating warfare is an example of Rule 3 tampering. One side increases its
nuclear arms. The other sides increase theirs. The first side then reacts by increasing its arms,
and so on. Zero-based budgeting, price wars between stores, promotional competition, shouting
matches, one-upmanship and the wildcard interest rates of the late 1970s are good examples too.
Usually, escalation persists until the system explodes or outside intervention occurs or one side
quits, surrenders or goes out of business. In the case of wildcard interest rates, outside interven-
tion by a regulatory agency can bring an end to irrationally escalating rates.
Rule 4 abandons all hope of hitting the target and tries instead to be consistent with the last
outcome. The marble eventually moves farther and farther from the target as if executing a ran-
dom walk under Brownian motion. The ‘telephone game' that children play is a good example of
this tampering rule. One person whispers a story to another, who in turn whispers it to another,
who again whispers it to another, and so on, until the last person says out loud what s/he heard.
When the first person announces what was whispered initially, then everyone laughs at how the
4
story has completely changed in the game's quiet retelling. Each person whispers the story based
on the soft whisper by the person before, thereby creating a clear instance of tampering under
Rule 4 (Table 1a). The corporate version entails rumors that spread throughout a company. With
each repetition, the story changes somewhat, often wandering off farther from the truth with each
retelling. Another example is on-the-job training by employees, with each new employee training
the next. A training center avoids this type of tampering. Similarly, in real estate, a construction
firm draws blueprints before constructing a building. When it makes an addition to the building
two years later, its engineers redraw the blueprints to include the new addition. A year later,
when four offices are combined to create a new conference room, new blueprints are made from
the last set. Through time, small mistakes accumulate. Based on the last set created, new blue-
prints begin to differ substantially from the actual building and new mistakes get added with new
iterations.
Under Rule | (no tampering), each marble drop is independent of any other drop. So, the
autocorrelation, covariance and the process mean for Rule | are all zero (Sparks & Field 2000).
Under Rules 2 through 4, however, the aim for each drop depends on where the marble came to a
rest at the previous drop. Under Rules 2 and 3, the process mean remains zero, but it does change
under Rule 4, signaling a location change. Sparks & Field (2000) show exact autocorrelation and
variance results for Rules 2 through 4 and how these three rules violate their assumptions about
normality. Subsequently, they address the questions of Table 1b by comparing SPC chart limits.
Shewhart (1980) does not require that distribution characteristics be plotted on SPC charts
and shows that normality is not essential. If normality were essential, then SPC charts would not
work as well as they do. Supporting Shewhart's experimental results with the Triangular and the
Uniform distributions, Wheeler's (2000) data from 1,143 heap-, J- and U-shaped probability dis-
tributions confirm that deviations from normality have little effect on SPC chart parameters. Bol-
stered by the Central Limit Theorem, however, some statisticians deem normality desirable, if
not outright required, as Sparks & Field (2000) argue.
To answer the practitioner questions of Table 1b, Sparks & Field show a unidimensional
version of Deming's funnel experiment. They mathematically model each of its four rules (Table
la), which they use to generate experimental data using a computer. From their simulation data,
Sparks & Field (2000) then plot R and x charts, linking the behavior of their charts to the practi-
tioner questions of Table 1b.
The two-sector system dynamics model below also generates experimental data using a
computer. Based on the simulation data, the results section first replicates Sparks & Field's
(2000) unidimensional analysis and then checks their statistical assumptions. It adds three-
dimensional views of the marble's location probability for Rules | through 4 and Theil's (1966)
inequality statistics (TIS) for Rules 2 through 4 compared to Rule 1. Also, it measures entropy
(uncertainty) about the marble's location for Rules 1 through 4 using Shannon's information en-
tropy formula, thereby showing an entropy-based view of the Deming-Nelson funnel experiment.
After Boltzmann's and von Neumann's work on statistical mechanics, Shannon defined informa-
tion entropy as:
a
U = -k> p,logp;. (1)
p=
where i runs over all possible outcomes n, p; is the probability of finding the marble in state i,
and k is a positive constant (Shannon & Weaver 1949).
Model description
Extending Sparks & Field's (2000) work on the funnel experiment with system dynamics hinges
on two bases. First, Deming's (2000b) System of Profound Knowledge, which integrates systems,
statistics, knowledge theory and psychology, begins with building appreciation for a system.
Second, Deming said: "Until you draw a flow diagram, you do not understand you business" (cf.
Schultz 1994, p. 21). System dynamics does use stock and flow diagrams to depict relations
among variables in a system. A fundamental tenet of system dynamics is that the structure of re-
lations among variables in a system gives rise to its behavior (Sterman 2001, p. 16). Figure 2
shows the stock and flow diagram of the funnel (Xf, Yf) and marble (Xm, Ym) location sector,
reproduced from the simulation model built with i7hink™ Analyst 7 (Richmond et al. 2001).
There is a one-to-one association between the model diagram of Fig. 2 and its equations
(Table 2). Like the diagram of Fig. 2, the equations are also the actual output from iThink®.
Building the model entailed first diagramming the experiments structure on the glass of a com-
puter screen and then specifying simple algebraic equations and parameter values. The software
enforces consistency between the diagram and the equations, while its built-in functions help
quantify parameters and variables pertinent to the funnel experiment rules, instigated with two
switches (Fig. 2, and Eqs 2.15 and 2.16, Table 2).
In system dynamics, rectangles represent stocks or level variables that can accumulate,
such as the funnel Xf and Yf coordinates on Fig. 2. Emanating from cloud-like sources and ebb-
ing into cloud-like sinks, the double-line, pipe-and-valve-like icons that fill and drain the stocks
represent flows or rate variables that cause the stocks to change. The pulse Xm outflow (right of
Fig. 2), for example, bleeds dry the marble Xm stock after a pen marks the spot where the marble
comes to a rest. Single-line arrows represent information connectors, while circular icons depict
auxiliary converters where constants, behavioral relations or decision points convert information
into decisions. Under Rule 2, for example, the change Ym flow depends on the funnel Yf coor-
dinate stock, adjusted by the marble's random angle and random distance as it randomly rolls
down inside and out the funnel.
Rule 1 requires that both switches equal zero to keep the funnel's position the same for
each marble drop. Without tampering, the funnel stays aimed at the fixed (0, 0) target. As long as
Rule | plays, the funnel's Xf and Yf coordinates (Eqs 2.1 and 2.2) retain their initial (INIT) zero
value (Eqs 2.1.1 and 2.2.1). Only the Xm and Ym stocks (Eqs 2.3 and 2.4) and their associated
flows and converters on Fig. 2 are active under Rule 1. Each marble drop entails generating a
random angle between zero and 2 radians (Eq. 2.13) and a random distance between zero and
one inch (Eq. 2.14). Together, the product of the random angle cosine (COS) and random dis-
tance then determines the change Xm inflow (Eq. 2.9), which feeds the marble Xm coordinate
(Eq. 2.3). Similarly, the product of the random angle sine (SIN) and random distance determines
the change Ym inflow (Eq. 2.11), which feeds the marble Ym coordinate (Eq. 2.4). "No tamper-
ing" (Deming 2000a, p. 328) means, however, independent marble drops (Sparks & Field 2000,
p. 293). That is why the pulse Xm and pulse Ym outflows (Eqs. 2.10 and 2.12) are quick to com-
pletely drain the marble Xm and Ym stocks (Eqs 2.3 and 2.4), respectively, after a pen marks the
spot where the marble comes to a rest.
Rule 2 calls for setting switch 1 =—1 to enable the funnel to move, erroneously correcting
in an equal but opposite (—) direction in response to an individual (Xm, Ym) data point. But
switch 2 again equals zero (Eq. 2.16) to let the funnel Xf and Yf coordinate stocks (Eqs 2.1 and
2.2) accumulate. Deming calls the accumulation of the funnel Xf and Yf coordinate stocks
"Memory 1" in his description of Rule 2 (Deming 2000a, p. 328). With the aim for each drop
now dependent on where the marble came to a rest at the previous drop, the cumulative Xf and
Yf coordinates affect the change Xm and change Ym inflows (Eqs 2.9 and 2.11). Feeding these
flows with the previous drop location's cumulative data means adding to and actually doubling
the variance of the marble's Xm and Ym coordinates (Sparks & Field 2000, p. 293).
Figure 2 The funnel (Xf, Yf) and marble (Xm, Ym) location sector
Rule 1: Keep funnel aimed at the target
Rule 2: Move funnel from its last position to compensate for last error
Rule 3: Move funnel from the target to compensate for last error
Rule 4: Move funnel right over the previous drop location
Rule Switch 1 Switch 2
1 i) i)
3 1 i)
- -1 1
4 3 1
change Xf fel change Xm pulse Xm
oe OHA om fe
pulse Xf
switch 1 fi
random angle
pulse Yf
random distance
ox Oo
change YF change Ym
switch 2 , oO
+0>
Xf = Funnel X coordinate O
Yf = Funnel ¥ coordinate
Xm = Marble X coordinate i
Ym = Marble Y coordinate 2)
ue)
pulse Ym
= Stock or level variable
= Flow or rate variable
= Auxiliary converter
= Connector or wire arrow
= Sink or source
Table 2 The funnel (Xf, Yf) and marble (Xm, Ym) location sector equations
Stocks: level or state variables Eq. #
X{(t) = Xf(t — dt) + (change Xf— pulse Xf) * dt (2.1)
INIT Xf=0 {funnel X coordinate; unit = inch} (2.1.1)
Yf(t) = Yf(t — dt) + (change Yf— pulse Yf) * dt (2.2)
INIT Yf=0 {funnel Y coordinate; unit = inch} (2.2.1)
Xm(t) = Xm(t— dt) + (change Xm — pulse Xm) * dt (2.3)
INIT Xm = 0 {marble X coordinate; unit = inch} (2.3.1)
Ym(t) = Ym(t — dt) + (change Ym ~ pulse Ym) * dt (2.4)
INIT Ym = 0 { marble X coordinate; unit = inch} (2.4.1)
Flows: rate variables
change Xf = switch 1 * Xm {unit = inch / drop; t = TIME = 200 drops or trials} (2.5)
pulse Xf= PULSE(switch 2 * Xf, 1, 1) {unit = inch / drop} (2.6)
change Yf = switch 1 * Ym {unit = inch / drop} (2.7)
pulse Yf= PULSE(switch 2 * Yf, 1, 1) {unit = inch / drop} (2.8)
change Xm = Xf + COS(random angle) * random distance {unit = inch / drop} (2.9)
pulse Xm = PULSE(Xm, 1, 1) {unit = inch / drop} (2.10)
change Ym = Yf + SIN(random angle) * random distance {unit = inch / drop} (2.11)
pulse Ym = PULSE(Ym, 1, 1) {unit = inch / drop} (2.12)
Converters: auxiliary variables and constants
random angle = RANDOM(0, 2 * PI) {unit = radian / drop} (2.13)
random distance = RANDOM(@, 1) {unit = inch / drop} (2.14)
switch 1 = 0 {unit = dimensionless} (2.15)
switch 2 = 0 {unit = dimensionless} (2.16)
Rule 3 again makes the aim for each drop depend on where the marble came to a rest at the
previous drop. But there is no funnel stock accumulation now; "No memory" (Deming 2000a, p.
328). Setting switch 2 = 1 activates the pulse Xf and pulse Yf outflows (Eqs 2.6 and 2.8) and
thereby eliminates the funnel Xf and Yf coordinate accumulation. Once activated, the pulse Xf
and pulse Yf outflows render the funnel Xf and Yf coordinates memoryless. The lack of memory
makes one forget about the (0, 0) target and react only to the last event. Like an escalating war-
fare, the marble consecutive rest points eventually move away farther and farther in opposite di-
rection from the target, in an accelerating swing from one direction to another. The marble Xm
and Ym coordinate variance rises with each marble drop (Sparks & Field 2000, p. 293).
Rule 4 not only wants the funnel Xf and Yf coordinates memoryless too, but completely
abandons all hope of hitting the target and tries instead to be consistent with the last outcome.
Like the muddling through of the telephone game and the Jogical incrementalism some strategy
theorists prescribe (e.g., Mintzberg 1994 and Quinn 1980), the marble eventually moves farther
and farther from the target as if executing a random walk. Its Xm and Ym coordinate variance
rises proportionally with each marble drop as the aim for each drop again depends on where the
marble came to a rest at the prior drop (Sparks & Field 2000, p. 293).
Marble location probability (Pml) and entropy/uncertainty (Uml) sector
Computing the entropy (uncertainty) about the marble location with Eq. | is equivalent to
assessing its state after each drop. To make it so requires gridding or tiling a paper pad (Fig. 1)
and then computing location probabilities by counting how frequently the marble hits each tile,
i.e., comes to a rest there. The marble location probability (Pml) and uncertainty (Uml) sector
computes the uncertainty about marble location after each drop (Fig. 3 and Table 3).
One can choose a tiling scheme empirically after observing running simulations. A 15[]15
tiling would, for example, require a total of 225 array elements, but Table 3 gets away with only
49, thanks to its 7[]7 tiling scheme. Extreme cases, i.e., a I[]l grid, notwithstanding, grid resolu-
tion has no effect on the shape of the entropy/uncertainty increase, but simply reduces time to
saturation, i.e., Uml reaching its upper bound of one (Georgantzas 2002). It is possible to com-
pute location probabilities by averaging over multiple simulation runs, but the model sector of
Fig. 3 computes marble location probabilities and uncertainty (entropy) dynamically.
Initialized at zero (Eq. 3.1.1), the one-dimensional arrayed Marble Drops[-] stock (Eq. 3.1)
counts how many times the marble hits each tile of the superimposed 7[]7 tiling scheme. Each
time the marble hits the first tile, for example, the change Marble Drops[1] inflow array (3.3)
adds a hit to the Marble Drops[1] array (3.1). Subsequently, the Pml[1] converter array (3.102)
computes the probability of the marble coming to a rest on the first tile as the dimensionless ratio
of the Marble Drops[1] stock array value over the total drops array sum (3.151). Adding a one to
this array sum converter prevents its dividing by zero because of empty tiles.
While the arrayed Pml[-] converters (Eqs 3.102 through 3.150) compute marble location
probabilities, the log Pml[-] converters (Eqs 3.53-3.101) calculate their respective logarithms.
Together, the probabilities, their logarithms and the total marble location probability (total Pml)
array sum (3.152) determine the change Uml (3.52) inflow, which feeds the marble location en-
tropy/uncertainty Uml stock (3.2), using Shannon's entropy formula (Eq. 1). Shannon's positive
constant k gets smaller as Uml increases and its difference from the total Pml auxiliary converter
(3.152) decreases. Dividing the reported marble location uncertainty by logio(10) in the change
Uml flow makes the logarithm base irrelevant (Eq. 3.52).
Figure 3 The marble location probability (Pml) and entropy/uncertainty (Uml) sector
Pmt
total drops total Pml
a
5 Marble iat
Drops
change ed change
} Marble Uml
| Drops log Pmt
Table 3 The marble location probability (Pml) and uncertainty (Uml) sector equations
Stocks: level or state variables Eq. #
Marble Drops[-]() = Marble Drops[-(t — dt) + (change Marble Drops[-]) * dt ED)
INIT Marble Drops{-] = 0 {unit = hit} G.1.1)
Umil(t) = Umi(t - dt) + (change Uml) * dt (3.2)
INIT Uml = 0 {unit = dimensionless} (3.2.1)
Flows: rate variables
change Marble Drops[1] = IF (7 <= Xm AND (Xm <-5) AND (-7 <= Ym) AND (Ym <-5)) 3)
THEN (1) Else (0) {unit = hit / drop}
change Marble Drops[2] = IF (-5 <= Xm AND (Xm <-3) AND (-7 <= Ym) AND (Ym <-5)) (3.4)
THEN (1) Else (0) {unit = hit / drop}
change Marble Drops[49] = IF (5 <= Xm AND (Xm <7) AND (5 <= Ym) AND (Ym <7)) (3.51)
THEN (1) Else (0) {unit = hit / drop}
change Uml = (total Pml — Uml) * (— ((Pml[1] * log Pml[1] + Pml[2] * log Pml[2] +... + (3.52)
Pmil[49] * log Pml[49]) / LOG10(10))) {unit = dimensionless / drop}
Converters: auxiliary variables and constants
log Pmil[I] = IF (Pmi[1] = 0) THEN (0) ELSE (LOG10(Pmil[1)) {unit = dimensionless} G53)
log Pml[2] = IF (Pml[2] = 0) THEN (0) ELSE (LOG10(Pmi[2])) {unit = dimensionless} (3.54)
log Pml[49] = IF (Pml[49] = 0) THEN (0) ELSE (LOG10(Pml[49])) {unit = dimensionless} (3.101)
Pmi[1] = Marble Drops[1] / total drops {unit = dimensionless (3.102)
Pml[2] = Marble Drops[2] / total drops {unit = dimensionless} (3.103)
Pml[49] = Marble Drops[49] / total drops {unit = dimensionless} (3.150)
total drops = ARRAYSUM(Marble Drops[*]) + 1 {unit = hit} (3.151)
total Pml = ARRAYSUM(Pml[*]) {unit = dimensionless (3.152)
Simulation results
The results here entail setting the iThink® run specs as follows: Euler's integration method with
simulation length TIME [] [1, 200] (drops) and computation interval dt = 1. These settings match
Sparks & Field's (2000) specifications. The phase plots of Fig. 4 show how marble (Xm, Ym)
location patterns form under Rules | through 4. Although Sparks & Field do not show any phase
plots, the ones on Fig. 4 are similar to those of Deming (2000a, p. 328) and Latzko & Saunders
(1995, pp. 151-154).
Rule 1 (no tampering) keeps the funnel stationary, aimed at the (0, 0) target, as the marble
drops. The marble then rolls until it stops somewhere near the target. As the marble comes to a
rest near the target for 200 successive drops, its marked rest points form a circular gradient pat-
tern (top left, Fig. 4). This circular pattern is the result of a process in control with common-
cause or random variation.
Figure 4 Marble location (Xm, Ym) phase plots for Rules 1 through 4 (200 drops per rule)
Rule 1: 3 Rule 2: 3 s
eo, .
error ® 3 ane
no tampering 3 0 3 tampe
tamy
Xm
Rule 2 erroneously compensates for the last error in an equal but opposite (—) direction in
response to the marble (Xm, Ym) location data. The switch 1 =—1 value enables the funnel to
move now, but switch 2 still equals zero, letting the funnel Xf and Yf stocks accumulate. The top
right of Fig. 4 shows that Rule 2 produces stable results, but the variance of the distribution of
points along any diameter drawn through the target is double the expected variance under Rule |
(Deming 2000a and Sparks & Field 2000).
Rule 3 makes the aim for each drop depend on where the marble came to a rest at the pre-
vious drop, but without any stock accumulation; no memory now. The lack of memory due to
switch 2 = | causes the marble rest points to move away farther and farther in opposite direction
from the target. Like those irrational, escalating price wars between stores, the funnel's and mar-
ble's accelerating swing from one direction to another form the symmetrical bowtie-like pattern
on the lower left of Fig. 4.
Rule 4 causes the marble rest points to move farther and farther from the target as if exe-
cuting a random walk under Brownian motion. The phase plot on the lower right of Fig. 4 shows
a typical instance of tampering under Rule 4. Not only the funnel Xf and Yf coordinates are
memoryless now but, much like muddling through and logical incrementalism proponents and
managers, Rule 4 abandons all hope of hitting the target and tries instead to be consistent with
the last outcome. The Xm and Ym coordinate variance rises proportionally with each marble
drop as the aim for each drop again depends on where the marble came to a rest at the previous
drop (Sparks & Field 2000, p. 293).
Statistical behavior and test reproduction
Sparks & Field's (2000) unidimensional results entail 200 simulated consecutive marble drops
through the funnel under each rule. They then take 50 successive samples of size n = 4 from their
200 marble drops and plot R (range) and x (x-bar) charts. Applied to the data of Fig. 4, Sparks &
Field's sampling procedure produces the R and x charts of Fig. 5, Fig. 6 and Fig. 7.
The normality assumptions Sparks & Field (2000, p. 295) choose to test for are:
R chart: sample values are both independent and approximately normally dis-
tributed with constant variance.
x chart: sample values are independent and their means are approximately
normally distributed with constant variance.
Figure 5 R (range) charts for Rules | through 4 (sample size n = 4)
217 Rxm (inches) 21) Rxm (inches)
144 Rulei 144 Rule 2
7 7?
1 8 15 22 29 36 43 50 1 8 15 22 29 36 43 50
Sample # # (range) charts Sample #
(a= 4)
217 Rym (inches) 21 Rym (inches)
144 Rule3 144 Ruled
? 7
Or ao eater WA ne
1 8 15 2 29 3% 4 50 1 8 15 2 29 36 43 50
Sample # Sample #
Figure 6 x (x-bar) charts for Rules | through 4 (sample size n = 4)
5/Xm (inches) 51 Xm (inches)
ON eae? Oem AAA AA
5 5
1 Rule 1 1) Rule 2
-15 15
1 8 15 22 29 3% 43 50 1 8 15 22 29 3% 43 50
Sample # Sample #
® charts (7 = 4)
54Xm (inches) 54Xm (inches)
Ob prrenneerrrntyetonennnn 0
5 5
10) Rule 3 7°) Rule 4
15 15
1 8 15 22 29 36 43 50 1 8 15 22 29 3% 43 50
Sample # Sample #
Figure 7 R (range) and x (x-bar) charts for Rule 3 (sample size n = 5)
2, xm (inches) Rule 3 Xm (inches)
5
% OANA AAA,
5
7
10
X chart (77 = 5)
SEER
s] 50 1 8 15 22 29 36 43 50
Sample # Sample #
The between values correlation coefficient (r), which measures the degree to which data
co-vary, gives the lag j sample autocorrelation coefficient (rj) between lagged samples. A zero
autocorrelation means uncorrelated values. If uncorrelated values are also normally distributed,
then they are independent. To check for autocorrelation and independence, Sparks & Field
(2000, p. 296) compute the autocorrelations between lagged samples that the funnel experiment
tules generate. Using this approach, Fig. 8 shows the Xm lag / autocorrelations 7; that Rules |
through 4 generate (j = 0, 1, ..., 9). After lag zero, all Rule 1 autocorrelations are nonsignificant
(top left, Fig. 8). But the rest of Fig. 8 shows significant Xm autocorrelations after lag zero for
Rules 2, 3 and 4, indicating that the adjacent Xm values these rules generate are dependent.
Figure 8 The lag 7 autocorrelations rj for Xm, Rules | through 4 (j= 0, 1, ..., 9)
1)m 77, Rule 1 1)_ 77, Rule 2
0.5 0.5
0 O,-fe64°9°- C) 1 .
1 1
012345 6 7Lag9 012345 6 7Lag9
1 "I 1
’ . DUNNO 000
t) 0
O78 ' ih os Ty Rule 4
a
012345 6 7Lagd 012345 6 7Lago
The way Sparks & Field (2000, p. 296) check for the constant variance assumption in indi-
vidual sample values entails comparing parallel boxplots for the first half of the data in a sample
with the other half. Using their approach, Fig. 9 shows the parallel boxplots of the marble Xm
coordinate data for Rules | through 4. It is worth noting that while Rules 3 and 4 both show in-
creasing variance (lower panel, Fig. 9), Rules 1 and 2 do not (top panel, Fig. 9). The results of
Fig. 9 confirm those of Sparks & Field. While Rules 1 and 2 produce data with homogeneous
variance, the Rule 3 and 4 data are likely to show variance heterogeneity.
Figure 9 Parallel boxplots of Xm for Rules | through 4 (200 drops per rule)
4
7 Rule 1 __ Rule 2
Xm Xm
a es wa —
0 —— (yi | | } i
| —— bee
First 100 Last 100 “4° First 100 Last 100
Drops Drops Drops Drops
8 4
Rule 3 +— Rule 4
— 0
Xm =
_— = Xm
alb a| — _
| i , =—— —|
First 100 Last 100 16" First 100 Last 100
Drops Drops Drops Drops
To test for the variance homogeneity and normality assumptions, Sparks & Field use Wilk
& Gnanadesikan's (1968) Q-Q plot, which computes the ith quintile from standard normal tables.
When both the variance homogeneity and the normality assumptions hold, then the Q-Q plot is
linear. Deviations from linearity mean outlying data, variance heterogeneity or non-normality.
Figure 10 shows the Q-Q plots for the individual sample data of the marble Xm coordinate under
Rules | through 4. The Q-Q plots for Rules 1 and 2 (top panel, Fig. 10) show normally distrib-
uted data with homogeneous variances. But the Rule 3 and Rule 4 Q-Q plots (lower panel, Fig.
10) show sample data either not normally distributed or with heterogeneous variance.
Figure 10 Q-Q plots of marble X coordinate (Xm) for Rules | through 4 (200 drops per rule)
15) Xm (inches) 15) Xm (inches)
| Rule 1 Rule 2
3-2-1 0 1 2 523109 4 8 3
Standard Normal Quantiles Standard Normal Quantiles
Xm (inches) 15) Xm (inches)
met ee Rule 4
= 4
3 1 6 a 4 1 7 3
Standard Normal Quantiles Standard Normal Quantiles
Answers to quality practitioner questions with statistics
Statistical process control charts like the ones Fig. 5, Fig. 6 and Fig. 7 show make quality practi-
tioners ask the questions of Table 1b. To answer Question 1: «Under Rule 2, why does the x¥
chart hug the centerline?» Sparks & Field explain that the large variance of tampering Rule 2 in-
flates the control limits of both the R (top right, Fig. 5) and the x (top right, Fig. 6) charts. But
what causes the large variance under Rule 2? How does the structure of relations among vari-
ables in the funnel experiment system gives rise to dynamics that violates Sparks & Field's as-
sumptions of independence and homogeneous variance?
In response to Question 2: «Why does the Rule 3 x chart hug the centerline even more?»
Sparks & Field explain that the large variance of tampering Rule 3 inflates the control limits of
both the R (lower left, Fig. 5) and ¥ (lower left, Fig. 6) charts, causing them to hug the center-
line. Again, why the large variance under tampering Rule 3? What causes it?
Sparks & Field (2000) answer all questions of Table 1b, at least from their perspective. The
system dynamics model of the funnel experiment presented here can still offer insight in an-
swering Questions 1, 2 and 5 (marked with the filled diamond symbol ' ' on Table 1b). In re-
sponse to Question 3: «Why does the Rule 3 ¥ chart develop a strong alternating behavior for
odd sample sizes?» for example, there is little insight this simple model can add because, to keep
it simple, SPC sampling and plotting procedures are external and posterior to the model.
To answer Question 3 (Table 1b), Sparks & Field explain that because the sample means!
autocorrelation is close to —1 for odd lags and +1 for even lags, the x values alternate above and
below the centerline. Differences in the alternating behavior of the sample means are seen in
comparing the sample size four (n = 4) ¥ chart (lower left, Fig. 6) to its sample size five (n = 5)
counterpart (right panel, Fig. 7). Their response to Question 4 (Table 1b) is that, although R
charts generally behave similarly for even and odd sample sizes (lower left of Fig. 5 and left
13
panel of Fig. 7), the sample size four (n = 4) means (lower left, Fig. 6) have a much smaller and
constant variance than the variance of sample means with n = 5 (right panel of Fig. 7) for large i
(number of marble drops) values (Sparks & Field 2000, p. 298).
In response to Question 5: «Why does the Rule 4 x chart behave as if very out of con-
trol?» Sparks & Field explain that the sample mean variance under Rule 4 increases with each
marble drop. The larger the number of marble drops, the more the x chart centerline or process
mean will differ from its zero target. The "increasing uncertainty" about + values as the marble
drops increase, "together with the increased autocorrelation between these values explains why
the [x] values are likely to wander away from zero as i [the number of marble drops] increases"
(Sparks & Field 2000, p. 298). But what causes the large variance of tampering Rule 4? How
does the structure of relations among variables in the funnel experiment system give rise to dy-
namics that violates the "approach of using within-group variation to estimate the process vari-
ance [and] makes matters a great deal worse" (Sparks & Field 2000, p. 298)?
Sparks & Field conclude by answering Question 6. They recommend using "CUSUM
charts of residuals for identifying special cause behavior particularly for processes that wander"
(2000, p. 298).
Beyond unidimensional analysis: marble location probability (Pml)
The three-dimensional surface plots of the marble location probability (Pml) confirm the roughly
normally distributed data with homogeneous variance under Rules | and 2 (top panel, Fig. 11).
Naturally, since the marble Xm and Ym coordinates are two different variables, one must no
longer talk of a normal distribution but, rather, of a bivariate normal distribution. And the tails of
the bivariate normal distribution that Rule 2 produced (top right, Fig. 11) are fat. Indeed, they are
much fatter than the tails of the bivariate normal distribution that Rule | produced (top left, Fig.
11). This corroborates Deming's (2000a) and Sparks & Field's (2000) argument that, under Rule
2, the variance of the distribution of points along any diameter drawn through the target is double
the expected variance under Rule 1.
Figure 11 Marble location probability (Pml) for Rules | through 4 (2,000 drops per rule)
Rule 3
0 7 = ~ °
Ym °xm Ym pa Xm
The three-dimensional surface plots of the marble location probability (Pml) for Rules 3
and 4 (lower panel, Fig. 11) confirm the possibility of outlying observations, heterogeneity and
non-normality that the Q-Q plots for Rules 3 and 4 show (lower panel, Fig. 10). Again, the dis-
tribution one must talk of is the bivariate normal because Xm and Ym are two different vari-
ables. But the results Rules 3 and 4 produce do not look like anything close to a bivariate normal
distribution (lower panel, Fig. 11).
As Rule 3 makes the marble rest points move away farther and farther in opposite direction
from the (0, 0) target, Pml gets low near the target and fat at the tails of its would-have-been bi-
variate normal distribution. Indeed, the Rule 3 Pml shows much more dispersion than one would
expect to see in a bivariate normal distribution (lower left, Fig. 11). This reconfirms the variance
heterogeneity that the parallel boxplot on the lower left of Fig. 9 shows.
Similarly, Rule 4 moves the marble rest points farther and farther from the target as if exe-
cuting a random walk. Consequently, Pml shifts toward a right skew, showing a long, big, fat tail
to the right (lower right, Fig. 9). This corroborates Sparks & Field's (2000, p. 293) prediction that
Pml will rise proportionally with each marble drop as the aim for each drop depends on where
the marble came to a rest at the previous drop.
Theil's inequality statistics (TIS)
Sparks & Field's (2000, p. 298) recommendation to use residual (error) plots to detect processes
that wander prompts Theil's (1966) inequality statistics (TIS). TIS use the mean square error
(MSE), which measures the average error between competing data series in the same units as the
variable itself and weights large errors much more heavily than small ones. TIS provide an ele-
gant decomposition of the MSE into three components: bias (U™), unequal variation (US) and
unequal co-variation (U‘), so that UM + US + US = 1.
Bias arises when competing data have different means. Unequal variation implies that the
variances of two time series differ. Unequal covariation means imperfectly correlated data that
differ point by point. Dividing each component by the MSE gives the MSE fraction due to bias
(U™), due to unequal variation (US) and due to unequal covariation (US).
A large U™ reveals a potentially serious systematic error. US errors can be systematic too.
When unequal variation dominates the MSE, the data match on average and are highly correlated
but the variation in two time series around their common mean differs. One variable is a
stretched out version of the other. US may be large either because of trend differences, or because
the data have the same phasing but different amplitude fluctuations (Sterman 2000, p. 876).
If most of the error is concentrated in unequal covariation, then the data means and trends
match but individual data points differ point by point. When US is large, then most of the error is
unsystematic and, according to Sterman: «a model should not be faulted for failing to match the
random component of the data» (2000, p. 877).
Figure 12 shows Theil's inequality statistics for Rules 2 through 4 compared to Rule 1,
Rule | being a process in control with common-cause or random variation. The Xm time series
data used for the three-dimensional TIS column chart of Fig. 12 are the same with those used for
the Fig. 4 phase plots and the R and x charts of Fig. 5 and Fig. 6. Overall, moving from the Rule
2 vs. Rule 1 to Rule 3 vs. Rule 1 to Rule 4 vs. Rule 1 comparisons, TIS discount the randomness
in the funnel experiment and ascribe the MSE fraction to increasingly rising systematic error.
Specifically, most of the Rule 2 vs. Rule 1 MSE fraction soars above unequal variation US,
with a still large remaining MSE fraction over unequal covariation US. The MSE fraction over
US shows that the Rule 1 and Rule 2 grand means match, but the individual data points these
rules produce differ point by point, showing a possibility of unsystematic, i.e., random, error. But
the larger MSE fraction over unequal variation US shows systematic error: the Rule | and Rule 2
sample means are the same, but much more dispersion characterizes Rule 2 than Rule 1.
Figure 12 TIS for Rules 2 through 4 compared to Rule 1 (UM + US + US = 1)
08
0.6 MSE
fraction
04
@ 0.2
0.0
uc Rule 4 rs, Rule 1
unequal
covariation ys
unequal
variation
Rule 3 vs. Rule 1
UM Rule 2 vs, Rule 1
bias
The Rule 2 vs. Rule | comparison of Fig. 12 supports Sparks & Field's answer to quality
practitioner Question 1 (Table 1b). The SPC charts hug their centerline because tampering Rule
2 causes the large unequal variation US that TIS pick up and which translates into the large vari-
ance that inflates the R and x chart control limits (top right, Fig. 5 and Fig. 6). Still, what causes
the large variance under tampering Rule 2? How does the structure of relations among variables
in the funnel experiment system give rise to its assumption-violating dynamics?
Similar to the Rule 2 vs. Rule 1 comparison, most of the Rule 3 vs. Rule 1 MSE fraction
again ascends above unequal variation U®, but the remaining MSE fraction over unequal co-
variation US is now smaller than before. The small MSE fraction over unequal covariation US
shows that the Rule 3 and Rule | grand means match, yet the sample means these rules produce
differ point by point, again showing possible unsystematic error. The large remaining MSE frac-
tion over unequal variation US shows systematic error: the Rule 1 and Rule 3 sample means
match, but Rule 3 creates more dispersion than Rule 3 and much, much more than Rule 1.
These results further support Sparks & Field's answer to Question 2 (Table 1b). The Rule 3
X chart hugs the centerline even more because the variance of tampering Rule 3, which inflates
the SPC chart control limits, is so large compared to the stable Rule | variance that TIS pick it up
as serious systematic error. Yet, how does the large variance under tampering Rule 3 happen?
What causes it? How do the relations among the funnel system variables cause behaviors that
violate Sparks & Field's normality assumptions?
Moving on to the Rule 4 vs. Rule 1 comparison of Fig. 12, the 3D TIS column chart tells an
entirely different, serious systematic error story. The high U™ MSE fraction shows that the data
Rules | and 4 generate differ far beyond unequal variation. The competing Rule 4 vs. Rule 1
process data have different means. Once more, TIS corroborate Sparks & Field's answer to
Question 5: the larger the number of marble drops under Rule 4, the more the ¥ chart centerline
or process mean will differ from the Rule | zero target. But what is it that causes such a large
variance under Rule 4 that the process mean shifts?
An entropy-based view of tampering
In response to Sparks & Field's (2000, p. 298) concern about the increasing x value uncertainty
as marble drops increase, the model computes marble location uncertainty (Uml) after each drop
dynamically, using Shannon's entropy (uncertainty) formula (Eq. 1). Typically, as marble drops
increase, average Uml increases until it saturates at its maximum value of one. The more times
the marble drops through the funnel, the more its final rest points disperse throughout the 7[]7
tilling grid, the higher average Uml grows (Fig. 13).
Initially, average Uml is zero and it starts rising as the marble drops increase, coming to a
rest randomly around the (0, 0) target. It takes a few marble drops until the tampering Rules 2
through 4 take their toll. Until then, as if uncoupled from Deming's tampering rules, the average
Uml behavior is identical for all four rules. After the first few drops, however, tampering begins
to show. Uncertainty about the marble's location increases rapidly, its final landing points now
guided by Deming's tampering rules. Average Uml continues to rise, but does so at a lower rate
for Rule | than for Rules 2 through 4 (Fig. 13).
Figure 13 Average marble location uncertainty (Uml) for Rules 1 through 4 (500 drops per rule)
1 So
1
3
Average ——
no ti
Uml 2
4 r
tampering
(n=10)
‘ampering
22 9 36
awrele
” Drops” .
Even if sequestered, there is some uncertainty about the marble's location when Rule 1
plays. When Rules 2 through 4 play, however, average Uml rises rapidly and concomitantly,
quickly reaching for its maximum value of one. Under Rules 2 through 4, average Um is already
very close to its saturation point before marble drop 30. But, without tampering, despite its ran-
domness, process Rule | continues to sequester the entropy/uncertainty about the marble's
whereabouts long past marble drop 30 (Fig. 13).
Feedback loop structure
How structure causes behavior
Figures 14 though 17 revisit the model's funnel (Xf, Yf) and marble (Xm, Ym) location sector
(Fig. 2) to show how the feedback-loop structure of relations among variables in the funnel ex-
periment system causes its dynamics. Phase plots highlight the causal relations among the spe-
cific variables embedded in feedback loops and confirm loop polarity. Both the funnel and the
marble X and Y location dimensions are in perfectly symmetrical feedback loops, so Fig. 14
through 17 show relations (arrow links) that involve the Xf (funnel X coordinate) and Xm (mar-
ble X coordinate) dimensions only.
Rule 1 (no tampering) entails one single feedback loop (Fig. 14). Its structure depicts the
process of first marking the Xm coordinate where the marble comes to a rest and then purging its
value since consecutive marble drops are independent. The Xm stock and the pulse Xm outflow
are the two variables embedded in this compensating (negative) feedback loop, its negative pulse
Xm-Xm arrow link so marked on Fig. 14 because the link emanates from an outflow. System
dynamics accepts this causal loop or influence diagramming (ID) convention because link po-
larities depict system structure, not behavior (Richardson 1995; Sterman 2000, p. 139).
Parenthetically, positive (+) links left unmarked is another ID convention. But the loop's
behavior depends on conditions set outside the Xm-pulse Xm loop. Namely on change Xm, an
inflow that in turn depends on the product of random angle by random distance. Their product
adds bipolar (+/-) random input to the change Xm inflow, which in turn feeds the Xm stock. The
three phase plots on the top right of Fig. 14 confirm the bipolar random behavior of the three
positive causal links outside the Xm-pulse Xm loop. They turn the negative Rule | loop positive
as the two phase plots on the lower right of Fig. 14 show.
Figure 14 The marble X coordinate (Xm) feedback loop with phase plots for Rule 1 (200 drops)
change Xm, change Xm
< Xm
= 0.08 3| r= 0.0453
3.16 6.28 se 3 5
B witch 1-0 random random ie
switch 1 = angle distance r= -0.00
ig itch 2-0 NS ben 8.0031
change Xm 4 3
=e change Xm
°
é
Xm ** pulse Xm
e Xm
pulse Xm J
48 = Compensating (n 0 ye [re
= Reinforce pc e) loo ro 3 0 3
© = Reinforcing (positive) loop 3| eee
© «Time lag or delay a
° Xm
The Xm-pulse Xm feedback loop is negative because it compensates for the random input
it receives. And does so both consistently and successfully, not only for Rule 1, but also for
Rules 2 through 4. As the marble drops increase, however, its random input turns the negative
Xm-pulse Xm loop positive: the more the loop compensates for the bipolar random input it re-
ceives, the more it ends up reinforcing its component variables, Xm and pulse Xm, to move in
the same direction.
This is an interesting first-order feedback loop that the no tampering Rule | of the Deming-
Nelson funnel experiment harbors. It makes neither the formal nor the intuitive definitions of
loop polarity break off but, rather, an exciting case of shifting loop polarity (Richardson 1995).
The phase plots on the lower right of Fig. 14 confirm the behavior of the Xm-pulse Xm loop as
the sign of its stock-versus-flow slope, and vice versa.
Rule 2 adds a second feedback loop to the structure of relations among variables in the
funnel experiment system. The funnel can now move from its last position to compensate for the
last error. The change Xf inflow and Xf stock are the new variables embedded in this second
compensating (negative) feedback loop. Its Xm-change Xf negative link is so marked on Fig 15
because of the switch 1 = —1 auxiliary converter, which activates the new loop. Switch 2 still
equals zero to let the funnel Xf coordinate stock accumulate (Deming's "Memory 1", 2000, p.
328). Formal and intuitive loop polarity definitions notwithstanding, the very purpose of this new
18
loop, i.e., to compensate, gives away its intended dominant polarity, which the phase plots sur-
rounding the influence diagram (ID) of Fig. 15 confirm.
The phase plot on the top left of Fig. 15 shows that the Xf-change Xm link is positive, ac-
cording to the stock-versus-flow slope definition of link polarity. The higher the funnel Xf coor-
dinate stock is, the more its accumulation amplifies the effect of the random angle and random
distance product on the change Xm inflow to Xm. The corresponding phase plots on the top
middle of Fig. 15 look very different from those on the top middle of Fig. 14. Although still bi-
polar, neither the cosine pattern of random angle nor the triangular pattern of random distance
effects on change Xm of Fig. 14 are visible on Fig. 15. This is how destructive the tampering of
Rule 2 is on the common-cause variation of the funnel experiment.
Figure 15 Funnel X; and marble Xm feedback loops with phase plots for Rule 2 (200 drops)
change Xm change Xm change Xm
o|
0.0131 3| 7
¢ 0 3.14 6.28 3 r 3
Xf random andom 2
angle NX to distance
= 0.4736
change Xm a 8
change Xm
id \
‘\7N
Xf aa Xm & pulse Xm
@ switch 1 = -1 ? P . se
fa switch 2= 0 ee
‘gl ‘id change Xf ° Xm 5
af : change Xf ,) Pulse Xm oad
x J
zt oe / . 3° 0 3
3] 0.468 ob REEBE:. 3h ; = pulse Xm
3 ; 3 3 > 0 3
change Xf Xm Xm
Worth noting on Fig. 15 is how the flow-versus-stock phase plots of the change Xm-Xm
and change Xf-Xf links are almost symmetrical. Their respective flow-versus-stock correlation
coefficient (r) values confirm this almost perfect syzygy (top right and lower left, Fig. 15). But
the negative Xm-pulse Xm feedback loop continues to compensate for the amplified bipolar ran-
dom input it receives as consecutive marble drops stay close to their zero mean. Again, as the
marble drops increase, the amplified randomness outside the negative Xm-pulse Xm loop turns it
positive (lower right, Fig. 16). The more the Xm-pulse Xm loop compensates for the now tam-
pered with bipolar random input it receives, the more its component variables Xm and pulse Xm
move in the same direction, causing the loop's polarity to shift from negative to positive, with a
much wider dispersion than Rule | did (Fig. 14).
Rule 3 adds a third compensating feedback loop to the structure of relations among the
variables in the funnel experiment system (Fig. 16). The purpose of this new negative feedback
loop is to bring the funnel back to the (0, 0) target before compensating for the last error (Latzko
& Saunders 1995, p. 153). It sounds like zero-based budgeting; does it not? While switch 1 still
equals —1, it is switch 2 = 1 that activates this third feedback loop and prevents the Xf stock from
accumulating (Deming's "No memory", 2000, p. 328).
The Xf stock and the pulse Xf outflow are the variables embedded in this compensating
(negative) feedback loop, its negative pulse Xf-Xf causal link so marked on Fig. 16 because it
emanates from the pulse Xf outflow. The loop's behavior again depends on conditions set out-
side. Namely on the change Xf inflow that depends on the marble Xm coordinate stock, which in
turn depends not only on its pulse Xm outflow, but also on its change Xm inflow (Eq. 2.3, Table
2). Change Xm is itself the sum of the funnel Xf coordinate stock plus the product of random
angle by random distance, which adds bipolar (+/—) random input to change Xm. The three phase
plots on the top right of Fig. 16 confirm the bipolar random input generated outside the three
negative feedback loops of Rule 3. Propagated through the network of positive and negative
links of Fig. 16, this bipolar random input turns the negative Rule 3 loop positive as the two
phase plots on the left middle show. The dynamic complexity that the feedback-loop structure
and behavior of the funnel experiment blend can account for the intricate phase plots of the
change Xm-Xm and change Xf-Xf causal links (top right and lower left, Fig. 16).
Figure 16 Funnel X; and marble Xm feedback loops with phase plots for Rule 3 (200 drops)
change Xm » change Xm change Xm Xm
i ee |
} | #®
3 0 3.16 628 9 3 > 0 9
Xf random random change Xm
Xf angle ." i distance Xm
9 y 9 F
, 4 change Xm , ,
/ ° y,
7 , change X¢
Xf change Xf
witch 1=-1
switch 2= 1
9 | r= 0.0046
ye change X¢ Xm
The Xf-pulse Xf feedback loop is negative nonetheless. It compensates for the random in-
put it receives, causing consecutive funnel moves to spread around the (0, 0) target. And does so
successfully for Rule 3, but not for Rule 4 (Fig. 17). As the marble drops increase, however, the
bipolar random input under Rule 3 turns the negative Xf-pulse Xf loop positive: the more the
loop compensates for the bipolar random input it receives, the more it ends up reinforcing Xf and
pulse Xf to move in the same direction. This is another interesting first-order feedback loop that
Rule 3 of the funnel experiment hides, one more exciting case of shifting loop polarity
(Richardson 1995). The phase plots on the left middle of Fig. 16 confirm the behavior of the Xf-
pulse Xf loop as the sign of its stock-versus-flow slope, and vice versa.
20
Rule 4 does not add a new loop to the structure of relations among variables in the funnel
experiment system but turns the compensating feedback loop that Rule 2 added into a reinforcing
one (Fig. 17). The purpose? Trying perhaps to cluster marble drops as close as possible, even if
away from the (0, 0) target or, as Deming says: "off to the Milky Way" (cf. Latzko & Saunders
1995, p. 154). While switch 2 = 1 still prevents the Xf stock from accumulating (no memory),
setting switch 1 = 1 changes the polarity of the middle loop from negative (Fig. 16) to positive
(Fig. 17). The phase plot on the lower right of Fig. 17 confirms that the Xm-change Xf link turns
positive because switch | = 1.
Figure 17 Funnel X; and marble Xm feedback loops with phase plots for Rule 4 (200 drops)
change Xm ,, change Xm change Xm )
# OD a | OR
at i ‘ & "rt
Los ore | rso.fise 9 [r= = 0.8166
9 0 9 0 3.16 6.28 9 0 9 9 0 9
Xf random random change Xm
Xf angle \ i distance Xm
se ° change Xm o|
/ P /
7 re a ° | ts
oY P) ¥
9 oo 9
pulse X; x é m
- P f Se Af (+) Xm 83 pulse X
‘A
if ) Pd
ia change X¢ :
X¢ change X¢ +
9 9 Cy 9 9
Xf {
Rule 4: (> wera 1) 7
. | ms
e--l 9 | r=-0,8102 9 |e rot
me 7 0 9 9 > FP
ti " ering change X¢ Xm
The Xf-change Xm link is again positive under Rule 4 as the phase plot on the top left of
Fig. 17 shows. The higher the Xf bias from a previous marble drop, the more it amplifies ran-
domness on the next marble drop. The phase plots on the top middle of Fig. 17 show this off-to-
the-Milky-Way bias of Rule 4, which is negative in this instance of the funnel experiment and
consistent with the marble location (Xm, Ym) phase plot for Rule 4 (lower right, Fig. 4).
The destructive bias of Rule 4 also shows on the flow-versus-stock, symmetrical phase
plots of the change Xm-Xm and change Xf-Xf links. Almost in perfect syzygy, their respective
negative r values confirm the negative, in this instance, bias (top right and lower left, Fig. 17).
The negative Xm-pulse Xm loop continues to compensate for the amplified bipolar random input
it receives, even though consecutive marble drops no longer stay close to the (0, 0) target. Nei-
ther can the compensating Xf-pulse Xf loop cause consecutive funnel moves to stay around the
target. As marble drops increase, the now reinforcing middle loop (Fig. 17) amplifies the bipolar
random input it receives, making consecutive marble drops and funnel moves behave as if exe-
cuting a random walk under Brownian motion. The amplification biases the negative Xm-pulse
Xm and Xf-pulse Xf loops to compensate away from instead of toward the (0, 0) target.
21
Answers to quality practitioner questions with system dynamics
The preceding section shows how the structure of relations among variables in the Deming-
Nelson funnel experiment system gives rise to its statistical-assumption violating dynamics. It is
now possible to add system dynamics insight to answering Questions 1, 2 and 5, marked with the
filled diamond symbol' 'on Table 1b.
Answer to Question 1: Rule 2 adds a second feedback loop to the structure of relations
among variables in the funnel experiment system. In hopes of hitting the target more, the purpose
of this negative feedback loop is to compensate for the last error by letting the funnel move from
its last position. The addition of the new negative feedback loop causes common-cause variance
amplification. The more the funnel moves from its last position, the more its Xf stock accumu-
lates, the more its accumulation causes dependence among consecutive marble drops and thereby
amplifies the randomness associated with each marble drop. The amplified randomness in turn
causes the large dispersion that inflates the x chart control limits, making it hug the centerline
(top right, Fig. 6).
Answer to Question 2: Rule 3 adds yet another feedback loop to the structure of relations
among variables in the funnel experiment system. Again hoping to hit the target more, this third
negative feedback loop brings the funnel back to the (0, 0) target before compensating for the
last error. So, it prevents Xf from accumulating. Discarding the funnel's last position before
compensating for the last error should take care of the dependence among consecutive marble
drops owed to Xf accumulation. Right?
Wrong. The stock and flow structure of the funnel experiment system says otherwise. In
trying to decouple dependence due to stock accumulation, the structure of relations among vari-
ables under Rule 3 makes consecutive marble drops even more dependent on each other, thereby
causing not only a larger dispersion than that of Rule 1 but also variance heterogeneity. The
more tightly coupled consecutive marble drops become, the more their dependence amplifies the
common-cause variance associated with each marble drop, causing again large and now hetero-
geneous dispersion that inflates the x chart control limits even more than under Rule 2. So it
hugs the centerline even more (lower left, Fig. 6).
Answer to Question 5: Rule 4 does not add any new loops to the structure of relations
among variables in the funnel experiment system but turns the compensating feedback loop that
Rule 2 added (Fig. 15) into a reinforcing one (Fig. 17). Hoping not to hit the target more, but
perhaps to cluster marble drops close together, the Rule 4 structure makes the funnel aim at the
previous marble rest point, while still preventing the funnel Xf stock from accumulating. Again,
having discarded the dependence of consecutive funnel moves owed to Xf accumulation, Rule 4
makes consecutive marble drops even more dependent on one another, thereby causing both a
larger dispersion than that of Rule | and variance heterogeneity.
Worse yet, purposefully reinforcing the randomness associated with each marble drop
causes the marble rest points to gradually build a bias, moving in a single direction away from
the target as if under Brownian motion. The random component of the funnel experiment stays
consistently exogenous to its deterministic causal loop structure but, apparently, the resolute shift
in loop polarity now changes the funnel experiment system goal (Richardson 1995).
Conclusion
Enhancing SPC with system dynamics? Clearly, combining these two fields of scientific inquiry
can help detect, explain and prevent tampering with the processes managers must manage. When
22
managers react to variation due to common causes as if it were due to a special cause, two very
regrettable outcomes become certain. First, their tampering degrades the process and the results
are more variable than if the process had been left alone. Second, tampering makes it impossible
to identify the common causes of variation, leaving no information about how the process can
improve. Sometimes variation can be reduced, by understanding what is causing it in a particular
situation. Vibration can be damped or prevented, for example, tools can be sharpened more of-
ten, worn parts can be adjusted or replaced, and so on. In other cases, however, such as in cus-
tomer requirements or student interests, variation can be accommodated because is welcome. But
to deal appropriately with variation, one must first understand it.
Most managers use, for example, the words ‘accurate! and 'precise' as if they were the same.
But quality practitioners distinguish between these two very different quality aspects. Imagine an
archery student and a target set up at a distance with a clearly marked aim point. Once released,
an arrow flies to strike the target. One can measure the distance from the arrow to the aim point
to estimate the shot's quality, or the skill of the archer (or the archer's teacher). With only one
shot, there is no basis for distinguishing between precision and accuracy. After several shots,
however, say 10, the target's cover has a cluster of 10 holes. Precision measures the dispersion of
the results (arrow holes). Low variability equals high precision. Accuracy is the deviation of the
center of the group (sample) of holes from the point of aim.
Marks-person trainers also distinguish between these two quality aspects. First, they aim a
rifle at a target and clamp it in position. Then, by firing several times, they can measure the aver-
age deviation of the rounds from the target and adjust the sights so that the next rounds cluster
centers on the target. This is called 'sighting in' a rifle. Marks-persons use the sights consistently
to produce as small a cluster as possible. Once they see how rounds clusters deviate on average
from the target, they adjust the aim to bring the center of the rounds pattern in line with the tar-
get. To improve accuracy, one must not tamper. Improving accuracy is only possible by holding
the same aim for a group of rounds and, using the center of their resulting cluster, to estimate the
deviation of the entire process of aiming, firing and the rifle and round functioning with the tar-
get. Adjusting the aim after each round makes things worse because of the variation endemic to
the process.
Back to the funnel experiment. Compared to Rule 1 (no tampering, Fig. 14), the feedback
loop structure of tampering Rule 2 (Fig. 15) degrades the precision of the funnel experiment re-
sults. The deterministic feedback loop that Rule 2 adds to compensate for the last error from the
(0, 0) target amplifies the randomness this negative feedback loop receives exogenously. The
more the funnel moves from its last position, the more its Xf and Yf stocks accumulate, the more
their accumulation causes dependence among consecutive marble drops (top right, Fig. 8) and
thereby amplifies the common-cause variance associated with each marble drop. Accuracy is still
high under tampering Rule 2 because the marble rest points' center does not deviate from the
point of aim (top right, Fig. 4, 6, 9, 10, and 11).
Compared to Rule 2 (Fig. 15), the feedback loop structure of tampering Rule 3 (Fig. 16)
degrades precision even more than Rule 2 does. Hoping to hit the target more, Rule 3 adds a
third deterministic feedback loop to the structure of the experiment to bring the funnel back to
the target before compensating for the last error and thereby to prevent the funnel Xf and Yf
stocks from accumulating. The rationale being that discarding the funnel's last position before
compensating for the last error should eliminate dependence among consecutive marble drops
owed to Xf and Yf accumulation. Erroneously, however, trying to reduce dependence due to
funnel stock accumulation, the structure of tampering Rule 3 makes consecutive marble drops
23
depend on each other even more than before (lower left, Fig. 8) and thereby degrades precision.
The degradation is owed not only to a larger dispersion than that of Rule | but also to variance
heterogeneity (lower left, Fig. 9). Accuracy again is high under Rule 3 because the marble rest
points' center does not deviate from the point of aim (lower left, Fig. 4, 6, 9, 10, and 11).
Compared to Rule 3 (Fig. 16), the feedback loop structure of tampering Rule 4 (Fig. 17)
degrades not only precision but also accuracy. Hoping not to hit the target, but to cluster marble
drops close together, Rule 4 turns the deterministic, negative feedback loop that Rule 2 added
(Fig. 15) into a reinforcing one (Fig. 17). With the increased dependence of consecutive funnel
moves, owed to preventing Xf and Yf from accumulating, on the one hand, Rule 4 makes con-
secutive marble drops depend on each other even more than before (lower right, Fig. 8). So, it
reduces precision by causing both dispersion larger than that of Rule 1 and variance heterogene-
ity (lower right, Fig. 9). On the other hand, with the funnel Xf and Yf coordinates memoryless,
reinforcing randomness causes the marble rest points' center to build a bias away from the point
of aim, implicitly changing the goal state of the funnel experiment system (lower right, Fig. 4, 6,
9, 10 and 11).
The tampering dynamics of Rule 4 resembles the situation that managers face when they
muddle through. Led by logical incrementalism theorists, they too believe that strategic changes
envisioned in the complexity theory literature are unrealistic. So, they simply sidestep the sys-
temic leverage analysis and synthesis necessary in strategy design. Anchored in system dynam-
ics, however, systemic leverage analysis and synthesis can help managers align multiple, system
goal aiming tactics that mix action with communication in corporate-, business- and functional-
level strategy (Georgantzas & Ritchie-Dunham 2002).
Complexity theory and the exponential increase in computational power make simulation a
critical fifth tool in addition to the four tools used in science: observation, logical/mathematical
analysis, hypothesis testing and experiment (Turner 1997). Simulation modeling with system dy-
namics permits organization researchers and practitioners to examine the aggregate, dynamic and
emergent implications of the multiple, nonlinear, generative mechanisms embedded in the proc-
esses capabilities and resources of every modern organization (e.g., Oliva & Sterman 2001 and
Repenning & Sterman 2002). Together, SPC and system dynamics can help detect, explain and
prevent tampering and its negative effects, which include precision and accuracy degradation in
organizational processes. Complementing statistics with system dynamics can help managers
sequester the entropy (uncertainty) of the processes they must manage.
Acknowledgement: We are grateful to Dr. Gipsie B. Ranney for her invaluable suggestions that
have helped improve this paper immensely.
24
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