Stable Stock Management Heuristics when the Outflow is
Proportional to the Control Stock'
Hakan Yasarcan and Yaman Barlas
Bogazici University, Dept. of Industrial Eng.
34342 Bebek, Istanbul - Turkey
Tel: ++(90) 212 359 7073, Fax: ++(90) 212 265 18 00
yasarcan@boun.edu.tr, ybarlas@boun.edu.tr
In this paper, the typical anchor (expected value of the outflow or expected loss) used in
the most popular decision rule of the stock management modeling, the “Anchoring and
Adjustment Rule” is studied for structures including a decaying stock. A new anchor
(equilibrium value of loss) is proposed and compared with the expected loss formulation.
We demonstrate that equilibrium value of loss formulation helps bringing the control stock
to its desired level more rapidly. In addition, we show that managing a decaying stock in a
stable way is difficult when the supply line is discrete. Standard stock adjustment and
supply line adjustment terms anchored around expected loss can yield highly unstable
oscillations. Counter-intuitively, for some cases, ignoring the supply line adjustment term
may completely eliminate unwanted oscillations. If equilibrium value of loss is selected as
the anchor and when the decay time (life time) is small enough, management of the stock
can even be done by ignoring all the adjustment terms.
Keywords: stock management, anchoring and adjustment rule, decaying stock, expected
value of flow, equilibrium value of flow, counter intuitive behavior, anchor, stock
adjustment term
INTRODUCTION
Anchoring and adjustment rule is a well known rule in the literature. Consider
management of a stock. People, mostly, first call a reference point that we call “anchor”
and make necessary corrections that we call “adjustment” to determine the desired order
rate (Sterman, 1987, 1989; Ozevin, 1999; Barlas and Ozevin, 2004). If there is no anchor
term in the decisions, this will yield steady state error and if there is no adjustment term,
seeking the desired level (goal) is not possible, so both terms are required to make the
stock achieve its goal. Note that, goal seeking without any adjustment term is possible, if
anchor is Equilibrium value of loss instead of Expected loss.
In stock control management, it is important to bring the stock to its desired level rapidly,
yet in a stable manner. In System Dynamics literature, the effect of adjustment terms
(Stock adjustment and Supply line adjustment) on the behavior of the control stock is well
discussed (Forrester 1961, 1973; Sterman, 1989, 2000; Yasarcan 2003). It is known that
decreasing Stock adjustment time makes the control stock more responsive, but this may
bring instability. To prevent instability, Supply line adjustment term must also be included
in the decision equation (Control flow equation). A specific and important case is when the
' Supported in part by Bogazici University Research Fund; grant no: 02R102
Supply line adjustment time is selected to be equal to the Stock adjustment time, then the
control stock and its supply line effectively reduces to a single stock, and it is a well known
fact that a single stock can not oscillate (Sterman, 2000; Yasarcan 2003). A stable and fast
response in the control stock can be obtained with the help of this fact. Note that, this
statement is not valid for a decaying stock with a discrete supply line.
Firstly, the effect of the parameters will be summarized in this paper and then we will
focus on the anchor term. The outflow of the stock can be independent from the stock or it
can be dependent to the stock like in decaying stock case. In most of the System Dynamics
models the outflow (smoothed value of the outflow; Expected loss) is included in the
control equation as the anchor for both independent and dependent outflow (i.e. decaying
stock; outflow proportional to the control stock) cases. Expected loss is a good anchor in
most of the cases but for decaying stock case, we propose using a new anchor that we call
Equilibrium value of loss. We demonstrated that for a decaying stock with a second order
supply line, the Equilibrium value of loss is more effective than Expected loss in bringing
the control stock to its desired level, when there is a minimum limit for the value of the
Stock adjustment time. Furthermore, in obtaining a good behavior one has to play with the
value of the Weight of supply line (ratio of Stock adjustment time and Supply line
adjustment time) if he is using expected loss as the anchor. This is not necessary when
equilibrium value of loss is the anchor. Note that, if Weight of supply line is equal to zero,
this means that Supply line adjustment term is ignored completely in the Control flow
equation, if it is equal to one (Supply line adjustment time is equal to Stock adjustment
time), this means that the Supply line has equal weight with the Stock in the Control flow
equation.
Another case is when the Supply line is discrete (infinite order). We demonstrated that,
managing a decaying stock with a discrete supply line is not easy. Counter intuitive results
can be observed; unstable behavior can be observed even when two adjustment times
(Stock adjustment time and Supply line adjustment time) have the same value (Weight of
supply line is equal to one) or ignoring the supply line (Weight of supply line is equal to
zero) may eliminate instability. For the discrete supply line case, if the decision maker
selects Expected loss as the anchor, he has to play with two parameters, which are the
Stock adjustment time and the Weight of supply line, to obtain good results, but if he selects
the Equilibrium value of loss as the anchor, he can control the Stock with playing a single
parameter, which is the Stock adjustment time. Furthermore, if Life time (decay time) of the
control stock is small enough, Equilibrium value of loss necessitates no adjustment term in
the decision flow.
A GENERAL MODEL STRUCTURE WITH A SECOND ORDER SUPPLY LINE
AND A DECAYING STOCK
Consider the model in Figure 1. The aim is to bring the Stock to its desired level (goal) that
is the Desired stock and maintain Stock at that level. For this, orders are placed at Control
flow and after passing two stages they reach to Stock.
For completeness, first of all, the level (stock) equations will be given:
Stock(t) = Stock(t - DT)+ (Acquisition flow, — Loss flow)« DT (1)
Supply line, (0) = Supply line, (G - DT)+ (Control flow — Acquisition flow, ) °DT (2)
Supply line, (0) = Supply line, G - DT)+ (Acquisition flow, — Acquisition flow, ) “DT (3)
Expected loss(t) = Expected loss(t -DT ) + Expectation adjustment flow* DT (4)
Lif time
ad ine adjustment
Weight\pfsupply line Expected loss/averaging time
Stock adjustment time
Desired stock
Expectation adjustment flow
Figure 1. A stock management structure involving second order material supply line and a
decaying stock
The flow equations are as follows:
Control flow = Expected loss + Stock adjustment + Supply line adjustment (5)
Supply li
Acquisition flow, = MPPIY HE (6)
Acquisition delay time / Order of supply line
Ly Li
Acquisition flow, = Supply lines (7)
Acquisition delay time / Order of supply line
In the Control flow equation, the anchor is Expected loss and the adjustment term consists
of two parts; Stock adjustment and Supply line adjustment. Expected loss is obtained by
exponential smoothing:
Loss — Expected los:
Expectation adjustment flow = oss flow — Expected loss
(8)
Expected loss averaging time
Stock is decaying proportional to its own level:
Loss flow = Slack (9)
Life time
The other equations are:
Desired supply line = Acquisition delay time * Expected loss (10)
Order of supply line = 2 qd)
Stock adjustment = Desired stock — Stock (12)
Stock adjustment time
Supply line = Supply line, + Supply line, (13)
Desired supply line — Supply line
Supply line adjustment = Weight of supply line« (14)
Stock adjustment time
Supply line adjustment can also be given as
Supply line adjustment = Desired supply line — Supply line (15)
Supply line adjustment time
Equations 14 and 15 are identical since
Weishvof suppivtine= Stock adjustment time (16)
Supply line adjustment time
Weight of supply line is the weight given to Supply line with respect to Stock by the
decision maker. Here we prefer to use Equation 14 instead of Equation 15 to see the
relative importance of Supply line with respect to Stock, clearly.
All the level variables are initiated at their equilibrium levels:
. Stock(0) = Desired stock
e Expected loss(0) = Loss flow
e Supply line, (0) = Desired supply line/ Order of supply line
e Supply line, (0) = Desired supply line / Order of supply line
Desired level of Stock is set to nine and it is increased by one unit at time five arbitrarily to
perturb the system from its equilibrium:
Desired stock =9 +STEP (1,5) (17)
THE EFFECT OF THE MODEL PARAMETERS ON THE BEHAVIOR
Now we will demonstrate the effects of the model parameters; Acquisition delay time,
Stock adjustment time, Life time, Expected loss averaging time and Weight of supply line
on the model in Figure 1. We will examine the parameters one by one and set the others to
some selected values so that the effect of the parameter under examination will be seen
clearly.
Effect of Acquisition Delay Time
If the value of Acquisition delay time is low, to obtain a stable and a fast response in the
control stock is easier. If it is high it may even cause unstable oscillations. When it gets
bigger, the strength and the period of the oscillations grow. The runs for the different
values of Acquisition delay time can be seen in Figure 2. Note that if value of Weight of
supply line is one, we will not have unwanted oscillations (this true in general but as will
be present, decaying stock with a discrete supply line is an exception). In stable case, i.e.
Weight of supply line is one, the only effect of Acquisition delay time is when it is
increased, the delay in the response of the control stock will also be increased. Note that
the effect of the Acquisition delay time that we summarized here is also valid for the non-
decaying stock case.
Stock: 1- 2-3-4-
1 12.009
1 10.004 ri
XuAl ar
1 8.00
0.00 50.00 100.00 150.00 200.00
IPage 1 Time
Figure 2. Runs for Acquisition delay time equal to 1 (goal seeking), 10 (stable oscillation),
21.4 (neutral oscillation) and 50 (unstable oscillation) respectively for the 1*, 2" 34 and
4" runs
The values of the parameters (except Acquisition delay time, since the different values for
this parameter are already given in the name of the figure) for the runs in Figure 2 are:
e Stock adjustment time = 3
e Life time = 40
e Expected loss averaging time =2
e Weight of supply line =0
Effect of Stock Adjustment Time
If the value of Stock adjustment time is low, this may result in unstable oscillations, and if
it is high, this may result in slow response in the control stock. When it is decreased, the
period of the oscillations also decreases but the strength of the oscillations increases. The
runs for the different values of Stock adjustment time can be seen in Figure 3.
Note that if value of Weight of supply line is one, we will not have unwanted oscillations
and decreasing the value of the Stock adjustment time will make the response of the control
stock faster. However, the response in the control stock can only be fast to a point that is
determined by the value of the Acquisition delay time. The effect of the Stock adjustment
time that we summarized here is also valid for the non-decaying stock case.
Stock: 1-2-3-4-
ag 12.009
13 10.004 Sh 4s
YA F
bet
1 8.00 T
0.00 50.00 100.00 150.00 200.00
IPage 4 Time
Figure 3. Runs for Stock adjustment time equal to 2 (unstable oscillation), 3.25 (neutral
oscillation), 5 (stable oscillation) and 40 (goal seeking ) respectively for the 1°, 2", 3 and
4 runs
The values of the parameters (except Stock adjustment time) for the runs in Figure 3 are:
e Acquisition delay time = 25
e Life time = 40
e Expected loss averaging time =2
e Weight of supply line =0
Effect of Life Time
Decreasing the value of Life time results in more stable behavior in the control stock. If the
behavior is already stable, i.e. Weight of supply line is one, then decreasing Life time may
result in over damped behavior. The runs for the different values of Life time can be seen in
Figure 4.
The values of the parameters (except Life time) for the runs in Figure 4 are:
e Acquisition delay time = 25
e Stock adjustment time = 3
e Expected loss averaging time =2
e Weight of supply line =0
Stock: 1-2-3-4-
1 124
1 8
0.00 50.00 100.00 150.00 200.00
Page 1 Time
Figure 4. Runs for Life time equal to | (goal seeking), 16 (stable oscillation), 34.7 (neutral
oscillation) and 70 (unstable oscillation) respectively for the 1, 2", 3“ and 4" runs
Effect of Expected Loss Averaging Time
The behavior is not very sensitive to Expected loss averaging time. Increasing Expected
loss averaging time just creates a bigger delay in perceiving Loss flow that slows down the
response of the control stock. The runs for the different values of Expected loss averaging
time for goal seeking (Weight of supply line is one) and oscillatory (Weight of supply line is
0.1) cases can be seen in Figure 5.
The values of the parameters (except Expected loss averaging time and Weight of supply
line) for the runs in Figure 5 are:
e Acquisition delay time = 25
e Stock adjustment time = 2
e Life time = 40
Stock: 1-2-3-4-
dy 11.004
dy 10.004
1: 9.005
0.00 50.00 100.00 150.00 200.00
IPage 4 Time
Figure 5. Runs for Expected loss averaging time equal to 1, 20, 1 and 20; Weight of supply
line equal to 0.1, 0.1, 1 and 1 respectively for the 1“, 2", 3" and 4" runs
Effect of Weight of Supply Line
Weight of supply line is a very effective control parameter in stock management decisions.
If it is equal to one, the behavior of the control stock will be stable even when Acquisition
delay time is high and Stock adjustment time is low (this true in general but as will be
present, decaying stock with a discrete supply line is an exception). Weight of supply line
equal to one means that the Supply line adjustment time is equal to Stock adjustment time
(see Equation 16). When the two adjustment times are equal to each other, the stock and its
supply line effectively reduces to a single stock, and it is a well known fact that a single
stock can not oscillate (Sterman, 2000; Yasarcan 2003). For independent outflow (i.e. non-
decaying stock) case, in general it can be said that, the behavior of the control stock can
not be oscillatory for Weight of supply line greater than or equal to one (Yasarcan 2003).
Practically, most problematic behavior is obtained when Supply line is completely ignored;
i.e. Weight of supply line is zero. If Weight of supply line is greater than one, the result is
over damped behavior (see 5 run in Figure 6). Note that, non-oscillatory behavior is also
possible for Weight of supply line smaller than one, depending on the values of the other
parameters (see 3° run in Figure 6). The runs for the different values of Weight of supply
line can be seen in Figure 6.
The values of the parameters (except Weight of supply line) for the runs in Figure 6 are:
e Acquisition delay time = 25
e Stock adjustment time = 2
e Life time = 40
e Expected loss averaging time =2
Stock: 1- 2- 3-4-5
1 129
4 a
1 10 3m 4 | Sa
1 8 5
0.00 50.00 100.00 160.00 200.00
Page 1 Time
Figure 6. Runs for Weight of supply line equal to 0 (unstable oscillation), 0.1 (stable
oscillation), 0.6 (goal seeking), 1 (goal seeking) and 3 (over damped) respectively for the
1%, 24 3" 4" and 5" runs
DECISION AND NON-DECISION PARAMETERS
Some of the five parameters that we mentioned in this paper, can easily be changed by the
decision maker, while some of them either can not be changed or can be changed between
a range with great effort. Now we will see these parameters one by one:
Acquisition Delay Time
The value of this parameter can only be changed by great effort. i.e. if the supply line
represents delay for the goods coming from the supplier, than Acquisition delay time may
stand for the production and transportation times. This parameter is the basic parameter
that makes the control easy or hard. It must be as small as possible, but this necessitates
long term efforts. Practically it may be very hard or impossible to change the value of this
parameter. Therefore, in this paper, we do not consider this parameter as a short term
decision parameter.
Stock Adjustment Time
The value of this parameter can be chosen freely by the decision maker. It does not have
physical limitations, it can take any value greater than zero. Note that, to get meaningful
behavior in the control stock it must chosen carefully by the decision maker. In most of the
cases, it is good to choose it as small as possible (after setting Weight of supply line to
one). In practice, decisions can not be updated continuously, there is a smallest time period
that can decisions can be updated. If this parameter is smaller than the revision period of
control decisions, the behavior of the control stock most probably will be unstable
oscillations.
We consider Stock adjustment time as the primary decision parameter. We assume that the
smallest revision period of the decisions is one, so we will not assume a value for Stock
adjustment time smaller than one, starting from this point of this paper.
Life Time
In this paper, we assume that this parameter stands for the life time of goods or similar
objects and can only be changed by long term efforts, which may mean technological
improvements. If it has high value, managing the control stock is easier. If it has low value,
either it creates over damping for low order supply line (i.e. second order) or it creates
counter intuitive behavior for high order supply line (i.e. infinite order; discrete) as will be
presented in this paper.
Sometimes a decaying structure can be used to control a stock with adjusting the value of
this parameter but in those cases, another name must be chosen instead of Life time. In
practice, many decision makers tend to use both inflow and outflow control at the same
time but simultaneous inflow and outflow control is harder than controlling a single inflow
or outflow (Fey, 1974).
In this paper, we do not consider Life time as a short term decision parameter.
Expected Loss Averaging Time
This parameter stands for both perception delay and smoothing time. We assume that
perception delay can not be changed but smoothing time can take any value. Therefore,
this parameter is assumed to have a minimum level and at that minimum it only represents
the perception delay. In our models, we used single order information delay (single level
variable in the expectation formation structure) to form Expected loss. One may propose to
use second order information delay structure, but the effect of such a change will be
negligible with respect to our research aim. We assume that this parameter is partially
under control of the decision maker; i.e. Expected loss averaging time can not be smaller
than some given value (i.e. perception delay).
If outflow (loss) is deterministic, we prefer Expected loss averaging time to be small but if
it is stochastic, Expected loss averaging time must be selected big enough to smooth the
control decisions so that the behavior of the control stock becomes also smoother. In this
paper, we are working with deterministic structures, so with respect to the limits of our
research, this parameter is not a basic decision parameter.
Weight of Supply Line
The value of Supply line adjustment time can be chosen freely like Stock adjustment time
so this means that, their ratio, which is Weight of supply line can also be chosen freely.
Although there is no limitation in selecting a value for Weight of supply line, as we
concluded before, we prefer to set it as one, to have a stable and fast behavior in the control
stock. In addition, we also concluded that it is easier to manage an effectively first order
stock system. We consider Weight of supply line as one of the basic decision parameters.
AN ALTERNATIVE ANCHOR FOR DECAYING STOCK CASE
We propose Equilibrium value of loss to be the anchor of the decision formulations for the
decaying stock case. The proposed equation for Equilibrium value of loss is:
Desired stock
Equilibrium value of loss = REE SOS (18)
Life time
If Equilibrium value of loss is the anchor, the following Equations 19 and 20 are used
instead of Equations 5 and 10.
Control flow = Equilibrium value of loss + Stock adjustment + Supply line adjustment (19)
Desired supply line = Acquisition delay time + Equilibrium value of loss (20)
COMPARING EXPECTED LOSS AND EQUILIBRIUM VALUE OF LOSS AS
ANCHORS FOR A DECAYING STOCK
Expected loss is a good anchor in many cases but we argue that Equilibrium value of loss is
a better anchor in decaying stock case. Assume that Acquisition delay time is high and Life
time is small. We will try to obtain a fast and stable behavior in the control stock towards
its goal by selecting good values for the three decision parameters (Stock adjustment time,
Weight of supply line and Expected loss averaging time). The model in Figure | will be
used. The values of the parameters for the runs in Figure 7, 8 and 9 are:
e Acquisition delay time = 25
e Life time=4
e Stock adjustment time = 1
e Expected loss averaging time = 1
Life time is small enough to create extra damping and furthermore, there is no stochasticity
in the model so it is good to set Expected loss averaging time to its minimum value that is
one. Under these conditions, instability can easily be eliminated if there is any, so to obtain
a fast response, it is good to set Stock adjustment time to its minimum value that is one.
Only one parameter remains to play with, which is Weight of supply line.
It can be said that, best run for Expected loss as the anchor can be obtained if Weight of
supply line has a value around 0.1 (see 3" run in Figure 7) and best run for Equilibrium
value of loss as the anchor can be obtained if Weight of supply line has a value around 1
6" run in Figure 8). The heuristics of selecting parameter values in obtaining the best
performance can be given as:
Stock: 1-2-3-4-5-
1: 11.004
pa
1: 10.004 oes ZT OR | ee
ame
1: 20——
0.00 75.00 100.00
IPage 4
Figure 7. Runs for Expected loss as anchor and Weight of supply line equal to 0, 0.03, 0.1
0.3 and 1 respectively for the 1°, 2", 3°, 4" and 5" runs
Stock: 1-2-3-4-5-
1: 11.004
[1
as 10.004 ie
3)
1 9.00-F =
0.00 25.00 50.00 75.00 100.00
[Page 1 Time
Figure 8. Runs for Equilibrium value of loss as anchor and Weight of supply line equal to
0, 0.03, 0.1, 0.3 and 1 respectively for the 1°, 2" 3" 4" and 5" runs
Parameter value selection heuristic when Expected loss is the anchor:
Select the minimum possible value for Stock adjustment time and play with Weight of
supply line assigning a value between zero and one, and select the value giving the best
performance.
Parameter value selection heuristic when Equilibrium value of loss is the anchor:
Set Weight of supply line to one and select the minimum possible value for Stock
adjustment time.
As it can be seen the parameter value selection heuristic for Equilibrium value of loss is
easier than the heuristic for Expected loss.
B® stock: 1-2-
fis
10.004 542
1: 9.504
1 1
1: 9.001 = 2
0.00 12.50 25.00 37.50 50.00
Page 1 Time 13:30 Sun, Sep 12, 2004
Figure 9. First run; Expected loss is the anchor and Weight of supply line is equal to 0.1,
and second run; Equilibrium value of loss is the anchor and Weight of supply line is equal
tol
The best runs of the both anchors eB" run in Figure 7 and 5" yun in Figure 8) are re-
sketched in Figure 9. It is obvious for this example that the performance obtained by
choosing Equilibrium value of loss as the anchor is better than choosing Expected loss.
This result is true in general for the decaying stock case. Either the performances of the
both anchors are close to each other (if Life time has a high value) or Equilibrium value of
loss produces a better result (if Life time has a low value).
A MODEL WITH A DISCRETE SUPPLY LINE AND A DECAYING STOCK
Consider the model in Figure 1. The aim is to bring the Stock to its desired level that is
Desired stock and maintain Stock at that level. For this, the orders are placed at Control
flow and after passing two stages they reach to Stock. Assume the number of the stages are
infinite with the same delay type. This kind of supply line is said to have infinite order or
simply it is called discrete supply line. The supply line of the model in Figure 10 is a
discrete supply line instead of a second order supply line, which is the only difference
between the two models.
Even though most of the equations of the model in Figure 10 are the same with the
equations of the model in Figure 1, we will give all equations of the model in Figure 10,
for sake of completeness. The level (stock) equations are as follows:
Stock(t) = Stock(t -DT ) + (Acquisition flow — Loss flow)« DT
Supply line(t) = Supply line(t - DT)+ (Control flow — Acquisition flow)+ DT
Expected loss(t) = Expected loss(t - DT) + Expectation adjustment flow* DT
The flow equations are as follows:
Control flow = Expected loss + Stock adjustment + Supply line adjustment
Acquisition flow\t) = Control flow(t — Acquisition delay time)
Expectation adjustment flow = Loss flow = Expected loss
Expected loss averaging time
Stock
Loss flow =——
fi Lifetime
Q))
(22)
(23)
(24)
(25)
(26)
(27)
The most important difference between the first model in Figure | and the second model in
Figure 10 is the equations of Acquisition flow. The rest of the equations are similar.
Supply line Stock
O—O O xO
Contr
Acquisition flow Lfss flow
Lif time
Acquisition}delay time
Supply ling adjustment
Expected loss
Desired supply line
Weight of supply line
Expectation adjustihent flow
Stock adjustment
Stock adjustment time
Desired stock
Expected loss averaging time
Figure 10. A stock management structure involving infinite order material supply line
(discrete supply line) and a decaying stock
The other equations are:
Desired supply line = Acquisition delay time + Expected loss
(28)
Stock adjustment = Desired stock — Stock (29)
Stock adjustment time
Desired supply line — Supply line
Supply line adjustment = Weight of supply line (30)
Stock adjustment time
All the level variables are initiated at their equilibrium levels:
. Stock(0) = Desired stock
e Expected loss(0) = Loss flow
e Supply line(0) = Desired supply line
Desired level of Stock is set to nine and it is increased by one unit at time five arbitrarily to
perturb the system from its equilibrium:
Desired stock =9 +STEP (1,5) G1)
SURPRISE BEHAVIOR WITH EXPECTED LOSS AS THE ANCHOR FOR A
DECAYING STOCK AND A DISCRETE SUPPLY LINE
The only difference between the first model in Figure 1 and the second model in Figure 10
is that, the order of the supply line is two in the first and infinite (discrete) in the latter. One
may not expect big differences, but counter intuitively the difference in the behavior of the
control stock is great.
Instability When Weight of Supply Line is One
For example lets assume the following parameter values:
e Acquisition delay time = 16
e Life time=7
e Stock adjustment time = 5
e Expected loss averaging time = 1
e Weight of supply line =1
Weight of supply line is one, so we expect a stable behavior. Moreover, Life time is small
so we expect over damping in the behavior of the control stock. The first run (second order
supply line) in Figure 11 produces an over damped behavior as one should expect.
However, the second run (discrete supply line) in Figure 11 produces an unstable behavior
that is contradictory to what we expect.
Stock 1 2: Stock 2
4] 124
;
2J 8 r
0.00 50.00 100.00 150.00 200.00
IPage 4 Time
Figure 11. Runs for Expected loss as anchor in models with second order supply line (first
run) and with discrete supply line (second run)
Stock: 1- 2-3-4-
a 12.004
a 10.004
ft Demme Seed
1: 8.00 T
0.00 20.00 40.00 60.00 80.00
[Page 1 Time
Figure 12. Runs for discrete supply line; Expected loss is the anchor and Weight of supply
line is equal to 0, 0.2, 0.5 and | respectively for the 1“, 2", 3 and 4" runs
Stability in the Behavior When the Supply Line is Ignored Completely
For example lets assume the following parameter values:
e Acquisition delay time = 12
e Life time =3
e Stock adjustment time = 4
e Expected loss averaging time = 1
In Figure 12, there are four runs; in the first run Weight of supply line is zero and counter
intuitively the behavior is stable; in the fourth run Weight of supply line is one and again
counter intuitively the behavior is unstable. The second and the third runs demonstrate the
transition from stability to instability as Weight of supply line increases. These results are
contradictory with the previous general results. All the system dynamists would argue that
supply line must not be ignored to have a stable behavior in the control stock but the
results in Figure 12 is just the opposite. Here we must make a warning that Weight of
supply line equal to zero does not necessarily give stable results for all parameter values
(see first run in Figure 13).
Strange Behaviors for Different Values of Weight of Supply Line
For example lets assume the following parameter values:
e Acquisition delay time = 25
e Life time=4
e Stock adjustment time = 1
e Expected loss averaging time = 3
The runs in Figure 13 demonstrate that, for some parameter values, whatever the value of
Weight of supply line is, the resulting behavior is always oscillatory. The behavior of a
decaying stock is very reach, when the supply line is discrete and the anchor is Expected
loss.
Stock: 1-2-3-4-
Ay 20.004
=
As 10.004
[1 2—3—-4. Ay
A: 0.00
T
0.00 40.00 80.00 120.00 160.00
IPage 1 Time
Figure 13. Runs for discrete supply line; Expected loss is the anchor and Weight of supply
line is equal to 0, 0.05, 0.15 and 1 respectively for the 1°, 2"4, 3 and 4" runs
The Possible Reason for the Surprise Behaviors Observed
The causal loop diagram of a decaying stock and Expected loss as the anchor can be seen
in Figure 14. There are two positive loops that might be the cause of the unexpected
behavior. We believe that discrete supply line is increasing the effect of these positive
loops. Note that when the outflow (Loss flow) is independent from the stock (i.e. constant
outflow), there is no positive loop. For a decaying stock we propose using Equilibrium
value of loss as the anchor, which terminates the positive loops.
Loss flow Stock
{ adjustment
Fe
- 4) Expectation
4-) Supply line
adjustment flow
adjustment Desired CA 2
&— supply line +
¥ “<<
EL.
+ Expected loss
Control flow —_—_t —
Figure 14. Causal loop diagram of a structure with a decaying stock and Expected loss as
the anchor
EQUILIBRIUM VALUE OF LOSS AS THE ANCHOR FOR A DECAYING STOCK
AND A DISCRETE SUPPLY LINE
If Equilibrium value of loss is the anchor, Equations 19 and 20 are used instead of
Equations 24 and 28. The behavior can also be problematic, but not like the behavior for
Expected loss. If supply line is ignored (Weight of supply line is zero) the behavior is
mostly oscillatory and can be unstable (see first run of Figure 15). One is a good value for
Weight of supply line, but still stable oscillations can be observed (see third run of Figure
15; see first and second runs of Figure 16). Note that, if Weight of supply line is one,
unstable oscillation is not possible, whatever the values of the other parameters. If value of
Stock adjustment time is low, oscillations (stable) can be observed (see first and second
runs of Figure 16). For low values of Life time, it is not necessary to include any
adjustment term in the Control flow equation, just using the anchor is enough to obtain a
good behavior (see fourth run of Figure 16). For the fourth run of Figure 16, the following
equation is used instead of Equation 19.
Control flow = Equilibrium value of loss (32)
Parameter values for the runs in Figure 15:
e Acquisition delay time = 16
e Life time=7
e Stock adjustment time = 3
Stock: 1- 2-3-
1 18.00 5
s} 10.00 4 fA
f1—2—3
1 2.00
0.00 30.00 60.00 90.00 120.00
Page 1 Time
Figure 15. Runs for discrete supply line; Equilibrium value of loss is the anchor and
Weight of supply line is equal to 0, 0.2 and 1 respectively for the 1°, 2"! and 3“ runs
Stock: 1- 2-3-4-
1 11.00 4
C1
1 10.004 Fak eal
1 9.00 1 2 3
0.00 75.00 100.00
[Page 1
Figure 16. Runs for discrete supply line; Equilibrium value of loss is the anchor and Stock
adjustment time is equal to 3, 10, 100 and © respectively for the 1°, 2", 3 and 4" runs
Parameter values for the runs in Figure 16:
e Acquisition delay time = 16
e Life time=7
e Weight of supply line =1
COMPARING EXPECTED LOSS AND EQUILIBRIUM VALUE OF LOSS AS
ANCHORS FOR A DECAYING STOCK WITH A DISCRETE SUPPLY LINE
Our experiments showed that, if the following heuristics are used, it is possible to obtain
good results with both anchors.
Parameter value selection heuristic when Expected loss is the anchor:
Set Stock adjustment time to be equal to Life time and Weight of supply line equal to zero
(completely ignore the Supply line). In most of the cases (i.e. Life time value not high), this
setting will give a good starting point and the optimum value will be close. Decrease the
value of Stock adjustment time and seek a good Weight of supply line value by increasing
it. Continue decreasing the value of Stock adjustment time and seeking better Weight of
supply line value, until the behavior is satisfactory.
Parameter value selection heuristic when Equilibrium value of loss is the anchor:
Set Weight of supply line to one and set the value of Stock adjustment time to infinity
(completely ignore the adjustment term). In most of the cases (i.e. Life time value not
high), this setting will give a good starting point and the optimum value will be close.
Decrease the value of the Stock adjustment time, until the behavior is satisfactory.
As it can be seen, like in the second order supply line heuristics, the parameter value
selection heuristic for Equilibrium value of loss is again easier than the heuristic for
Expected loss.
The starting points of the two heuristics will give very similar results (see first runs of
Figures 17 and 18). The final near optimum behavior is also very similar (see second runs
of Figures 17 and 18).
Parameter values for the runs in Figures 17 and 18:
¢ Acquisition delay time = 20
e Life time = 10
e Expected loss averaging time = 1
Note that, both the starting point and the final near optimum behaviors may show strange
shocks, if the value of Expected loss averaging time is high and when Expected loss is the
anchor (see the starting point run comparison of the two anchors in Figure 19).
Stock: 1-2-3-
iB 11.004
: 4 , = — 2 3
iB 10.00. rn SS | eS
1
a 9.00 <1 2am T
0.00 25.00 50.00 75.00 100.00
[Page 1
Figure 17. Runs for Stock adjustment time equal to 10, 8 and 6; Weight of supply line
equal to 0, 0.05 and 0.12 respectively for the 1“, 2"! and 3 runs; Expected loss is the
anchor
Stock: 1-2-3-
£ 11.004
-——|
—
1: 10.004 3 ae 1—2—3
2,
1
1: 700-- = r
0.00 25.00 50.00 75.00 100.00
Page 1 Time
Figure 18. Runs for Stock adjustment time equal to 2, 140 and 40 respectively for the 1",
2" and 3" runs; Weight of supply line equal to 1; Equilibrium value of loss is the anchor
Parameter values for the runs in Figure 19:
e Acquisition delay time = 20
e Life time =2
e Expected loss averaging time = 10
Stock: 1- 2-
i 10.334
1
1 Ny
ae 9.675
1
fiz 900-- i=
0.00 50.00 100.00 150.00 200.00
[Page 1 Time
Figure 19. First run; Expected loss is the anchor, Stock adjustment time is equal to Life
time (2) and Weight of supply line is equal to 0; second run; Equilibrium value of loss is the
anchor, Stock adjustment time is equal to «© and Weight of supply line is equal to 1
CONCLUSIONS
In this paper, we showed that decaying stock necessitates special care. The usual anchor,
the Expected loss can give unexpected results, especially when the Supply line is discrete.
For discrete supply line case, there is a counter intuitive finding; completely ignoring the
supply line (Weight of supply line equal to zero) eliminates oscillations, provided that Stock
adjustment time is equal to Life time. Unexpectedly, setting Supply line adjustment time
equal to Stock adjustment time (Weight of supply line equal to one) may produce unstable
oscillations, depending on the values of the other parameters. We argue that the reason for
the surprise behavior is a result of the two positive loops (formed by using Expected loss as
the anchor) acting together with the discrete supply line.
We developed another anchor that we call Equilibrium value of loss that gives better
results than Expected loss. This anchor is much more robust in terms of parameter value
selections. If Life time is small enough, control is even possible only with the anchor and
without adjustment terms.
For both discrete and non-discrete supply line cases and for both type of anchors, we
developed parameter value selection heuristics that help to make the control stock to be
responsive and non-oscillatory.
We did not focus on what may happen if Life time is an unknown parameter to the decision
maker. We believe that, with a good estimation procedure, Equilibrium value of loss can be
used without losing much in the resulting behavior of the control stock. This will be
investigated as future research. Another further research topic is potential destabilizing
effects of stochastic elements in the outflow.
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