TESTING THE DECISION RULES USED
IN STOCK MANAGEMENT MODELS
Yaman Barlas Mehmet Giinhan Ozevin
Department of Industrial Engineering IMS Software
Bogazici University Talatpasa Cad. 20/5
80815, Bebek, Istanbul, TURKEY 80640, Giiltepe, Istanbul, TURKEY
Tel: +90 (212) 263 15 40/2073 Tel:+90 (212) 270 96 50
Fax:+90 (212) 265 18 00 Fax:+90 (212) 270 96 55
ybarlas@ boun.edu.tr ozeving@ims.com.tr
Abstract
This paper evaluates the basic decision rules typically used in dynamic
decision-making modeling. The plausibility and consistency of each rule is evaluated
first. Then, the adequacy of these rules is tested empirically by comparing them against
the performances of different subjects (players) in an experimental stock management
game. For this purpose, the generic stock (inventory) management problem, one of the
most common dynamic decision problems, is chosen as the interactive gaming
environment. Experiments are designed to test the effects of three factors on decision
making behavior: two different patterns of customer demand, minimum possible
decision interval and finally representation of receiving delays. The performances of
subjects are compared with the simulated results obtained using the common linear
"Anchoring and Adjustment Rule." Next, several alternative (nonlinear) rules are
formulated and tested. Finally, some typical inventory control niles (such as (s, Q))
from the standard inventory management literature are tested. These niles, compared to
the linear stock adjustment rule, ae found to be more representative of the subjects’
decisions in certain cases. Another major finding is the fact that the well-documented
oscillatory dynamic behavior of the inventory is true not just with the linear anchor-and
adjust rule but also with the non-linear rules as well as the standard inventory
management rules.
1. Stock Management Game
The objective of the game is defined as "keeping the inventory level as low as
possible while avoiding the backorders." If there is not enough goods in the inventory at
any time, customer orders are entered as backorders to be supplied later. This situation,
is modeled very simply, by letting the inventory become negative until enough goods
are eventually received. "Order decisions” are the only means of controlling the
inventory level. The general structure of the stock management problem is illustrated in
Figure 1.1. The three empty boxes: Expectation Formation, Goal Formation and
Decision Rule are deliberately left blank, as they are unknown, since they take place in
the “minds” of the players. (Later, in the simulation version of the game, these three
boxes will have to be specified. For instance, the expectation formation will be
formulated by exponential smoothing; inventory goal will be set to inventory coverage
times expected demand and supply line goal will be order delay times expected demand.
As for the decision rule, different formulations will be tried: linear stock adjustment
rule, three different non-linear adjustment rules and finally various standard discrete
inventory control rules.)
While playing the game, subjects can monitor the system from information
displays showing their inventory, supply line levels and customer demand. As the game
progresses, they can also see the dynamics of these variables plotted on graphs. Neither
costs associated with high inventories nor costs resulting from backorders are accounted
for explicitly in the simulation games. However, the relation between keeping these
costs at a minimum and the objective of the game 5 stated in the instruction given to
subjects. Before beginning the game, each subject is given a written instruction
presenting the problem and their task. Time available to accomplish the task is not
limited. No explicit reward is used to motivate the subjects.
cal STOCK ACQUISITION SYSTEM AB TOT) EXTERNALINPULN
SUPPLY LINE sTock
INFLOW uri
Ma DECISIONPOLICY RULES 7 |
Wl Goat romain A ___ EXPECTATION FORMATIONZS
PINK NOISE
FIGURE 1.1. Stock Management Problem
2. Experimental Design
We design our experiments for testing the effects of three factors on the
decision making behavior of the subjects:
(a) minimum possible decision interval;
(b) representation of the receiving delays;
(c) two different patterns of customer demand
2.1. Decision Interval
Decision- makers were allowed to order at “each period” in the first group of
experiments (Short Game), whereas they were allowed to order “once every five
periods” in the second group of experiments (Long Game). Short Games are simulated
for 100 time periods whereas Long Games are simulated for 250 time periods (50
decision intervals).
2.2. Receiving Delay
The average receiving delay is set as four and ten in the Short and Long Games
respectively. The other factor is the nature of the delay. In different inventory
acquisition systems, continuous first-order exponential delays or discrete delay
representations may be appropriate. Since “Receiving” is the inflow to inventory, its
transient behavior may influence decision-maker ’s interpretation of the results of his/her
own order decisions. Therefore, the effects of time delay representation is tested in the
experiments. The two extremes of the exponential delay family, namely the discrete
delay and first order exponential delay are chosen as high and low levels of delay
representation factor respectively.
2.3. Patterns of Customer Demand
Until the fifth decision interval, average customer demand remains constant at
20. At the beginning of fifth decision interval (at the fifth period in the Short Game and
at the 25’th period in Long Game) an unannounced, onetime increase of 20 units
occurs in demand in the customer demand patterns used in the experiments. The
increase in demand facilitates the analysis since the subjects must react to
disequilibrium created by the disturbance. After the fifth decision interval, demand
remains constant at 40 in the first type of customer demand pattern which we call “step
up in customer demand”.
In the second type of demand pattern, ‘step up and down in customer demand,”
a second disturbance, one-time decrease in demand follows the first increase, after some
time interval, reducing the demand back to its original level of 20. The time interval
between the step up and down in customer demand is chosen as roughly half of the
natural periodicity of the model.
"Pink noise" (auto-correlated noise) is added to the pattems of average
customer demand described above to obtain more realistic demand pattems. The
standard deviation of the white noise is set to be 15 percent of average customer
demand. The delay constant of the exponential smoothing (the correlation time used to
create pink noise) is taken as two time units.
TABLE 2.1. Design of Experiments
Minimum Receiving Del: Patten of Customer
Ordering Interval ecelving Delay Demand
Runs Step-Up eed
1 5 Exponential | Discrete | Customer | 0? 0W2
Damand Customer
Demand
1 xX xX xX
2 xX xX xX
3 xX xX xX
4 xX xX xX
5 xX xX xX
6 xX xX xX
7 xX xX xX
8 xX xX xX
(X) indicates the selected level of factors for the corresponding runs.
P rece
t £000 4-
r 00
6e¢ 00 00 5000 7500
N Groph 1:pl Patan 1) Tre 1427 O7 May 1968 Per
10000
FIGURE 2.3.1. Step Up Customer Demand for Short Games
B i cemand
a)
80.0
1. 40.0
q
CII
L 0.00
0.00 25.00 50.00 75.00 100.0
ING T=? a Graph 1: p2 (Pattern 2) Time 14:29 07 May 1998 Pe|
FIGURE 2.3.2. Step Up and Down Customer Demand for Short Games
There are eight combinations of the above three factors across the two levels of
each. Each condition is played six times (random replications) yielding a total of 48
runs. Since the demand patter is discovered by the subjects once the game is played
and because they can improve their performance by practice, to obtain unbiased results,
the subjects never played two Short Games or two Long Games. However, some of the
subjects were allowed to play one Short Game and one Long Game, since transferring
experience between Short and Long Games is not easy.
BP inventory oder
L 500.00: eemaneenee
2 250.00
1 0.00 INI fe.
2 125.00
0.00: + +
eeBF >” 25.00 50.00 75.00 100.00
N Graph 1: p11 (Subject 11) Time 14:10 28 Ara 1998 Paz
FIGURE 3.1.1. Performance of a Player in Game 11 (Short Game with Orders Each
Period, Step Up and Down in Customer Demand, Exponential Delay)
-500.00
1:
2
PB r:invenory 2:order
Ts 500.00] +++eeeeeeeeee
2 500.00
2. 2 !
8eé' 0.00 6250 125.00 187.50 250.00
N Graph 1: p9 (Subject 33) Time 1518 28 Ara 1997 Paz
FIGURE 3.1.2. Performance of a Player in Game 33 (Long Game with Orders Once
Every Five Periods, Step Up and Down in Customer Demand, Exponential Delay)
2.4. Initial Conditions
Games start at equilibrium. Supply line level is initially set (80 in Short Games
and 200 in Long Games) such that no backordering occurs even when the decision
maker does not order goods during the first four decision intervals, until disturbance in
customer demand creates disequilibria. Inventory levels are initially set arbitrarily (40
and 200 respectively) so as to satisfy the average initial customer demand for the first
two decision intervals (two time periods for Short Games and ten time periods for Long
Games).
3. Analysis Of Experiments
The fundamental behavior pattem of inventory in most games is one of
oscillations. (See Figures 3.1.1., 3.1.2. and most other games illustrated in the sections
to follow). This finding is consistent with overwhelming evidence on oscillating
inventories in system dynamics literature and elsewhere (for instance, Sterman 1987,
1989 and Lee et al 1997).
3.1. Effect of Different Patterns of Customer Demand
Experiments show that the "demand pattem" (step up only or step up and
down) does not have a major effect on the ordering behavior of the subjects. There are
some numerical differences but no major qualitative differences. (See Table 3.1, below).
3.2. Effect of the Length of Decision Intervals
When subjects can order “every period,” they prefer to order intermittently.
Consequently, inventory kvels are noisy. When subjects can only order "every five
periods" (instead of each period), inventory oscillations become less “noisy”, having
longer and smoother cycles. In these games, since subjects order larger amounts for
longer periods, they order more consistently. However, fundamental behavior patterns
of the inventory do not seem to be affected by the length of decision interval.
3.3. Effect of Different Representations of Receiving Delays
The "type of receiving delay" seems to affect the difficulty level of the game.
With continuous exponential delay, subjects are able to manage the inventories in a
relatively more stable way. It seems that when the goods ordered arrive gradually over
some period of time, it prevents the players from over-ordering or under-ordering
excessively. In contrast, discrete delay representation affects their performance very
badly by causing fluctuations and noisy orders. When the receiving delay is discrete,
subjects fail to bring the oscillations under control in most cases. Subjects seem to have
difficulties in accounting for the effects of sudden receiving. They generally overreact
and their reactions are destabilizing in these games (Figures 3.3.1.and 3.3.2). The above
results are consistent with research evidence on the effect of delays on dynamic decision
making performance, in system dynamics literature (Sterman 1989), as well as in
experimental psychological research (Brehmer 1989).
3.4. Summary of Experiments
Some numerical characteristics of orders and inventory level obtained from the
experiments are summarized in Table 3.1. The ten characteristics are tabulated for each
of the 48 games. The averages of each characteristic for each of the six runs are also
TABLE 3.1. Selected Characteristics from Game Results
tit | Pinar | DUT®
atin, | Max. | Range Min | Max. | Rowe | patel | pil | tion Oscit
Experiment | Min. | Max.) “of” tnven- | tnven- | Of | Back- | Back | of jation
Orders tory | tory ‘ rder | Back Period
tory | Time | Time | Back
T T ] 10] 1005] |e a eT
2 o | 100 | 10 125 | 150 | 25 | 4 33 | 29
3 0 | 100 | 100175 | 150] 305 | 4 13_[_14
z 0200005 1610
5 0] 300 | 300235 | 100] 35 | 5 | 3 | 38
6 2060 [0 00) 10 Ya ET
Avg OfRunt | 33] 13 | 0 431 | © | 2 | 48 | 31 | 26
7 0 | 10 | 00 0 | i | mm | 10 | 1 | u
g o|- 100 [100-60 [250 [310] 5 13 8
9 o| | 0-0 _| 0 | 10 | 7 20_[ 13
0 0] 400 [40080180260] m1
ia 0} 150 [150100 [250350] 6 id 8
2 [70 [30-60 | 150 | _210_| 6 18 [12
Avg OfRun> | _67 | Moo | 10 70 | 1075 | 2375 | 67 | 178 | 112
ie 0] 100 | 100205 | 30 | os |S a
if o | 20220395 [295 | 50] 5 3530
1S 0150-15075 T2855
76 0-60 | 60 15 | 190[ 215 | -NA_[_NIA|_NI
17 0 [100 [1005 2503S 3833
73 0300300360105 an EI
Avg OfRun3 | 0 | 155_| 155 154 | m8 | 32 | 68 | 30 | 232
19 0 | 150 | 15025 | 10] 15 | NA | NA | NA
20 0 | 150 [150100290 [390] 6 a [15
a 0-8 [as 5 a
D 0100100 0 50300] 78 T
ra 0[ 100 [ 100 350-9050 an vl
7% 0 200200 150-2500] 5 2 [IT
Avg Ofkimd | 0 | 10 | 130 418 | 2 | wo | 08 | ma | 52
5 0] 00 | 000260) 45 | os | 20 | 30 | 10
26 oO] 400 400-350] 200 | 550] 25] 85 | 60.
oT 0 [300 300280] 20] 480] 20] 85] 5
38 0 | 400-400-300 [ 200 [500] 25] 7045
9 0560360150 [ 370] 300] 20-015
30 0 25025000 35 10
ig 0rkus [0 | as | se
3 | 3 [35200] 80) P02
3 0 [25025010020] 300] 35530
a 000 os
34 30} 250] 200-60 | 200| 260 [30 55] 25
% 50 [300 | 2500 [275 | 25 | NA | NIA | NIK
36 Tw 250_[ 150100 250_[ 025 55 30
Tyg Of Rin] 00 | 30] 3 tee
7 TH [20-1100 5008
38 20 | 500_| 480500 | 300| m0 | 25 | 90 | 65
7 30 [250 [170-40 [450 [400 [NA] _NA|_NIK
1 30] 250] 2000055 390-20 10
a 0 | tow [1000 -To00| 2000 [3000] 20 | 10080.
D 0 | 450 | 450375 | 40 | 75 | 35 | 60 | 25
Avg Of Run? | 2 | 43 | 412 369 | 083 | 102 | 26 | 66 | 40
B 1] 250 | 25015) 40] 505 | WA | NA | NIA
m1 0 | 0 | 450500] 200 | 700] 20 | 60 | 40
5 0 [35375 750} 20] 950] 20} 70] 50
46 20 [300 | 280330330 [oof 255] 30
a7 50} 250] 200-180] 300} 480} 30 45] 15
48 o | 300 | 300 375 | 350 | 75 | 30 | oo | 30
hg orkus | | 3 | aT
BP + inventory 2: order
1: 500.01
2: 250.0
1 0.09)
2 125.0
pall |
25.00 50.00 75.00 100.0
Graph 1: p2 (Subject 14) Time 12:42 23 Ara 1998 Ca
FIGURE 3.3.1. Performance of a Player in Game 14 (Short Game with Orders Each
Period, Step Up in Customer Demand, Discrete Delay)
PB + inventory 2: order
1: 500.01
2: 500.01
1
Er 1
1 0.01
2 250.0 1
|
1h — -500.0 |
2: 0.06 1 2 2
0.00 62.50 125.00 187.50 250.0
Nese Graph 1: p8 (Subject 43) Time 13:59 24 Ara 1997 Ca
FIGURE 3.3.2. Performance of a Player in Game 43 (Long Game with Orders Once
Every Five Periods, Step Up and Down in Customer Demand, Discrete Delay)
calculated. From these averages, it is possible to observe the effects of the game
parameter settings on some performance characteristics. The averages of the runs
obtained from to the low and high settings of each of the three experimental factors are
also statistically compared pair-wise (interactions ignored), using the t-test. None of the
differences between the high and low settings of a given factor are found to be
statistically significant. Thus, we can say that the high and low settings of each of the
three game factors do not have a statistically significant effect on the selected output
characteristics.
4. Linear Anchoring and Adjustment Rule
Linear Anchoring and Adjustment Rule is frequently used to model decision
making behavior in System Dynamics models (Sterman, 1987). In making certain
decisions, one can start from an initial point, called the anchor, and then make some
adjustments to come up with the final decision. In the context of inventory management,
a plausible starting point for order decisions is the expected customer demand. (When
the inventory manager can only order once every five periods, the anchor should be the
total of expected customer demand for five periods between subsequent decisions.)
When changes in either customer demand or receiving delay cause discrepancies
between desired and actual inventory levels and/or between desired and actual supply
line, adequate adjustments are made so as to bring the inventory and the supply line
back to desired levels. Therefore, the order equation based on Anchoring and
Adjustment heuristic is formulated as:
Or: =E, +?*(h-1) +?*(SLr- SL) (4.1)
When orders can be given once every five periods, then the anchor of the mule is
modified as follows, the adjustment terms being as before:
Ov= SE ct ?*(L'-1) + 2*(SLi- SLi) (4.2)
where E; represents customer demand expectations, I; represents the desired inventory,
I, the inventory, SL,’ the desired supply line and SL; the supply line. ? and ? are the
adjustment fractions.
In real life, safety stocks are determined by considering the inventory and
backordering costs. Although, an optimum inventory level minimizing these costs may
be found mathematically, more often safety stocks are set approximately. The desired
inventory I; is thus modeled as proportional to customer demand to allow adjustments
in safety stocks when changes in customer demand occur.
I =*E; (4.3)
To maintain a receiving rate consistent with receiving delay and customer demand, SLt
is formulated as a function of receiving delay ? and expected customer demand Et.
SLf = ?*E; (4.4)
4.1. Comparison of Anchoring and Adjustment Rule with Experiments
The rule can mimic the subjects’ performances adequately in experiments
where they tend to order continuously (Figure 4.1.1.), which is especially the case when
orders are placed every five periods. (See Ozevin 1999 for more illustrations). However,
when they can order each period, most subjects tend to order intermittently especially
when the receiving delay representation is discrete. Their orders result in zigzagging
inventory patterns. Furthermore, some subjects tend to order suddenly very large
quantities, after some period of zero or negligible ordering (Figure 3.3.1.).The linear
“Anchoring and Adjustment Rule" can not yield such intermittert or infrequent large
orders, hence does not represent well the subjects’ performances in these cases. Such
cases will be addressed below in two separate sections: Nor linear adjustment rules and
standard inventory control rules.
5. Rules with Nonlinear Adjustment
The linear adjustment rule makes adjustments in the orders proportional to the
discrepancy between the desired and observed stock levels. Some orders are placed
regularly each period, the quantity of which depending on the discrepancy between the
desired and observed stock levels. However, some decision makers do not seem to
place such smooth orders. They cease ordering when the inventory is around the desired
level and give rather large orders as the discrepancy between the desired and actual
inventory becomes larger. In this section, three alternative decision rules, which may
represent this “nonlinear” adjustment aspect of subjects’ ordering, are tested.
5.1. Cubic Adjustment Rules
Similar to the Linear Anchoring and Adjustment rule, Cubic Adjustment rules
also start with customer demand expectations as an initial point but the adjustments are
not formulated as linear. One or both of the adjustment terms may be cubic in
discrepancies (in inventory and/or in supply line). Alternative order equations can be
mathematically expressed as:
Or =Er+?*(K' - 1)5 (5.1)
where only the discrepancy in inventory is taken into account,
Op=Ert+?*(Ie -1)°+?*(SL¢- SL) (5.2)
Or=Er+?*(hr - 1) + ?*(SL- SL)? (5.3)
where one of the adjustments is made cubic and the others are linear and finally,
O; =E, +? *(h" - 1)? + ?*(SLt'- SL)? (5.4)
where both of the adjustments are formulated as cubic. In the equations above,
represents customer demand expectations, ° and SL’ the desired inventory and supply
line levels, I;and SL, the actual inventory and supply line. ? and ? are the fraction of the
discrepancy corrected by the decision-maker at each period. The internal consistency of
the rules can be shown mathematically (See Ozevin 1999). When orders can be given
once every five periods, the adjustments are as before, however, the anchor is increased
to five times the expected customer demand.
Although choosing stable values for the adjustment fractions is not always easy
with Cubic Adjustment mules, it is possible to generate ordering patterns consisting of
intermittent orders and sudden large adjustments within quiet ordering periods, as
observed in some subjects. Dynamics of the inventory levels corresponding to these
orders are very similar with the ones observed in some of the graphs obtained in the
games (Figures 5.1.1.and 5.1.2.).
Most of the time, when orders are given once every five periods, the cubic
adjustments rules fail to generate stable orders. When simulated with exponential delay
representation, the orders are either unstable or when stable, they become very similar
with the ones obtained by simulating the Anchoring and Adjustment rule. In other
words, the performance of the nile is too sensitive to the adjustment parameter values.
Anchoring and A djustment mules are not suitable to generate certain fluctuating
inventory dynamics observed with discrete delays when orders are given once every
five periods. Although the rules can potentially yield such fluctuating behavior with
discrete delays, the range in which they can yield fluctuating yet stable dynamics is too
narrow to offer a rich repertoire of oscillations. (See Ozevin 1999).
BD «inventory 2: order
1: 500.04
2: 250.0
1 o.oo] > a a a
2 125.07 r st
i -500.0
2: 0.06 T T
0.00 25.00 50.00 75.00 100.04
6a (A) 3raph 2 (1Adjt=2, SLAdjt=50) Time 00:19 01 Oca 1994 Cu
Ss 1: inventory 2: order
1: 500.04
2: 250.0
q
|
0.
125.0
2 YW an
1: -500.0 ai
2: 0.00 +
0.00 25.00 50.00 75.00 100.00]
Nee vraph 1: pS (Subject3) Time 12:45 25 Ara 1998 Cum
FIGURE 4.1.1. Comparison of (A) the Anchoring and Adjustment Rule (? =0.5, ?=
0.02), with (B) the Performance of a Player in Game 3 (Short Game with Orders Each
Period, Step Up in Customer Demand, Exponential Delay)
5.2. Variable Adjustment Fraction Rule
Analogous to the Anchoring and Adjustment Rule and the Cubic Adjustment
Rules, Variable Adjustment Fraction Rule also starts with expectations about customer
demand. However, the adjustments are increased sharply (nonlinearly) when the
discrepancy in inventory increases. The order equation of the mle can be
mathematically expressed as
O.=E, +? (e-ld (5.5)
1
§
E 0,75 4
i 054
3 0,25 +
g
T 4 T
-1 -0,5 0 0,5 1
normalized inventory discrepancy
FIGURE 5.2.1. Graphical Adjustment Fraction Function
where the variable fraction ? is a function of the discrepancy in inventory, normalized
by the desired inventory. The shape of the function yields increased adjustments when
the discrepancy in inventory is increased. Normalized inventory discrepancy ? is
defined as
= (5.6)
The rule is not mathematically unbiased in the ideal known demand case; there
will be some small, deliberate steady state discrepancy between the inventory and its
desired level. But this may well be a "realistic" bias in order to be able to obtain a non
linear ordering behavior similar to some subjects. The rule performs quite realistically
in the "noisy" demand case, in which case the steady state bias is negligible anyway and
may be irrelevant in real life.
Orders generated by the Variable Adjustment Fraction rule display very large
variability, the resulting inventory patterns being much smoother. Therefore, they may
be used to represent subjects’ behavior especially when orders are intermittent and
corresponding inventory patterns are continuous, as an alternative to Anchoring and
Adjustment nules (Figure 5.2.2.).
B® & inventory 2: order
1:
500.01
2: 250.01
t- Nt
1: 0.0q) * 4
2: 125.0
2.
i: -500.0 EE
2: 0.00 r
ou 25.00 50.00 75.00 100.0
Neer @) "Ie: p2 (SLAdjt=1, IAdjt=5Dioje 02:27 20 0ca 1994 P
1: inventory 2: order
1: 500.04
2: 250.0
1 0.0q) 4 \ eemeiomee
2: 125.0
Y
; pV
1: -500.0 fi
2: 0.00
0.00 25.00 50.00 75.00 100.09
N8& a hl: pl(Subject1) Time 12:32 04 Eyl 1986 Per
FIGURE 5.1.1. Comparison of (A) Performance of “Linear Supply Line and Cubic
Inventory Adjustment Rule” (? =1/500, ?=1) with (B) the Performance of a Player in
Game 1 (Short Game with Orders Each Period, Step Up in Customer Demand,
Exponential Delay)
B 1: inventory 2 2: order 2
1 500.04
2 250.0
+ Www YY “
o.oof wt
J rs
2 Ww
Ld j 2
as: -500.01
2: 0.004 — ,
50.00
=
0.00 25.00 75.00 100.0
Neea4 (A) 24:p4 (SLAGit=1,1adjt=75008 02:52 200ca 1994 Pd
1: inventory 2: order
1: 500.0
2: 250.0
. 1
1: 0.0q 2
2: 125.0 iN
ar)
ry 2
2
1 -500.0
2: 0.0
0.00 (B) 25.00 50.00 75.00 100.0
Né =a hl: p5 (Subject17) Time 14:14 23 Ara 1998 Car|
FIGURE 5.1.2. Comparison of (A) Performance of “Linear Supply Line and Cubic
Inventory Adjustment Rule” (? =1/7500, ?=1) with (B) Performance of a Player in
Game 17 (Short Game with Orders Each Period, Step Up in Customer Demand,
Discrete Delay)
® +: inventory 2: order
500.04
250.01
2
J]
1: :
2: 0.00
0.00 25.00 50.00 75.00 100.0
NX 8 aF* inventory and orders: p3() Time 01:25 22 Haz 1999S
(A)
Ss 1: inventory 2: order
1: 500.0
2 250.0
L1
L: 0.oq/
2: 125.0 (a ~
La
r)
\ Me wa _
1 — -500.0
2: 0.0 1
0.00 (py 25.00 50.00 75.00 100.00]
Néee h1:p3(Subject2) Time 13:44 04 Eyl 1986 Per
FIGURE 5.22. Comparison of (A) Variable Adjustment Fraction Rule with (B) the
Performance of a Player in Game 2 (Short Game with Orders Each Period, Step Up in
Customer Demand, Exponential Delay)
5.3. Nonlinear Expectation Adjustment Rule
The order equation of the expectation adjustment rule can be mathematically
expressed as
O, = ?*E, (5.7)
where the variable adjustment coefficient ? is a function of discrepancy in inventory
normalized by desired inventory and E; represents the customer demand expectations. ?
is equal to one when the inventory is at the desired level since adjustments are not
necessary when the system is in equilibrium. The shape of the ? function causes
increasing upward adjustments in orders when the inventory level is below the desired
level and it causes reductions in orders when the inventory is above its desired level.
¢
ao)
U a
£ 1
g
i 10 4
3 54
2
©
T 4 ; r r
-2 1 0 1 2 3 4
normalized inventory discrepancy
FIGURE 5.3.1. Graphical A djustment Coefficient Function
The rule is also simulated with discrete delays and similar ordering and
inventory patterns are obtained. With discrete delays, orders given previously are
received all at once. Therefore, in the discrete case, although the shape and the range of
the function is kept as before, adjustment values corresponding to positive normalized
inventory discrepancies are decreased to keep orders stable.
Similar with the Variable Adjustment rule, Nonlinear Expectations Rule can
yield intermittent orders yet continuous inventory patterns. Therefore, it may provide an
altemative to Anchoring and Adjustment rule, when subjects’ orders are intermittent
and corresponding inventory patterns are continuous. (Figures 5.3.2 and 5.3.3).
There is another important general finding in this section: the well-documented
oscillatory dynamic behavior of the inventory is true not just with the linear anchor-and-
adjust rule but also with the non-linear rules seen above. (Figures 5.1.1 through 5.3.3).
B + inventory 2: order
1: 500.09
2: 250.01
j got Ss
: | Lj-——~—__y
L: o.oo} + ¥ > ae
2 125.09
Fa i
Wo 2
1: -500.0
2: 0.00
0.00 25.00 50.00 75.00 100.0
IN| 8 Be (A) ‘ryand orders: p1() Time 00:06 010ca 1994 Cui
1: inventory 2: order
1: 500.09
2: 250.0
1,
1 0.0q) 2 \ooe qepreayesme a
2 125.0
. [Ns Og ae
1: -500.0 ai yt
2: 0.00!
0.00 (B) 25.00 50.00 75.00 100.00
Nee hl: p5 (Subject3) Time 12:45 25 Ara 1998 Cum
FIGURE 5.3.2. Comparison of (A) Nonlinear Expectation Rule with (B) the
Performance of a Player in Game 3 (Short Game with Orders Each Period, Step Up in
Customer Demand, Exponential Delay).
2
inventory 2 2: order 2
500.04
250.0
oh
|_/
1 -500.0
2 0.00
0.00 25.00 50.00 75.00 100.0
6eF nventory and orders () Time 00:40 01 Oca 1994 Cu
atinventor: A) 2: order
1: 500.09
2 250.0
1,
1: 0.0)" \
2: 125.0
2
a
L — -500.0
2 0.00 1
0.00 25.00 50.00 75.00 100.00]
N al 1: p11 (Subject 11) Time 14:10 28 Ara 1998 Paz
FIGURE 5.3.3. Comparison of (A) Nonlinear Expectation Rule with (B) the
Performance of a Player in Game 11 (Short Game with Orders Each Period, Step Up
and Down in Customer Demand, Exponential Delay).
6. Standard Inventory Control Rules
Intermittent ordering of subjects and the resulting zigzagging inventory
patterns suggest that the discrete inventory control rules used in inventory management
may be suitable. The following four policies are most frequently used in inventory
management literature:
Order Point, Order Quantity (s, Q) Rule;
Order Point, Order Up to Level (s, S) Rule;
Review Period, Order Up to Level (R, S) Rule;
(R, s, S) Rule.
Two fundamental questions to be answered by any inventory control system
are “how many” and “when” (or “how often”) to order. “Order-Point” systems
determine how many to order in contrast to “Periodic-Review” systems, which
determine how often to order (Silver and Peterson, 1985), (Tersine, 1994). When
subjects can order every period, they are free to order at any time they desire. Therefore,
order point systems, rather than periodic review systems, are more appropriate to
represent the ordering behavior in these situations. In contrast, when subjects can order
only once every five periods, periodic -review systems with five as review period may
be more appropriate as decision rules. These inventory control rules implicitly assume
that time is discrete. Therefore, these rules should be tested only with discrete delays.
6.1. Order PointOrder Quantity (s, Q) Rule:
Order Point Order Quantity (s, Q) rule can be mathematically expressed as
O.=Q, ifEh?s
0, — otherwise (6.1)
where EI, represents the effective inventory and s the “order point”. Effective inventory
and order point are calculated as follows:
Eh=h+SLt (6.2)
s =DAVGSL: + DMINI (6.3)
I, and SL; represent goods in inventory and in supply line respectively. DAV GSLirefers
to the desired average supply line and DMINI; to desired minimum inventory. Desired
average supply line and desired minimum inventory are calculated as
DAVGSL = ?*E; (6.4)
DMINEL=E: + SS (6.5)
B i inventory 2: DMINI 3: order
200.04
2
2.
3 250.0
A
3 125.0
4 3
E
y 3
2 -200.01 i]
3 0.00
.00
25.00 50.00 75.00 100.0
Nee Graph 2: p2 (SQ rule2) Time 03:17 20 Oca 1994 Pe
FIGURE 6.1.1. Performance of Order Point-Order Quantity Rule (Q=4*D 1p, DAVGSL;
=4*Dt, DMINI =0) in Short Game with Orders Each Period, Known
Step Up in Customer Demand, Discrete Delay.
in terms of receiving delay ?, expected demand E, and safety stocks SS. Order quantity
Q and safety stock SS are fixed arbitrarily as constants in this research. Desired
minimum inventory DMINI, is defined as the sum of a constant safety stock and
demand expectation. With such a definition, desired minimum inventory can be adapted
to variations in customer demand. The performance of this rule with deterministic
demand is seen in Figure 6.1.1. Observe that the Order Point-Order Quantity (s, Q) rule
can not prevent the inventory falling below the desired minimum inventory when
demand is not constant, even when no noise is added to the demand. This particular rule
is therefore not suitable for our purpose (i.e. comparative evaluation against the
continuous stock adjustment rules).
6.2. Order PointOrder Up to Level (s, S) Rule
Order Point Order Up to Level (s, S) rule can be mathematically expressed as
O.=S-s, ifEk?s
0, otherwise (6.6)
where EI, represents the effective inventory, s the order point and S the upper level of
inventory. Effective inventory, order point and upper level S of inventory are calculated
as follows
Ehk=I+SLt (6.7)
s =DAVGSL, + DMINI, (6.8)
S=s+Q (6.9)
I, and SL; represent goods in inventory and in supply line respectively. DAVGSL; refers
to the desired average supply line and DMINI, to desired minimum inventory. Desired
average supply line and desired minimum inventory are calculated as
DAVGSLt = Et (6.10)
DMIN]=E,+SS (6.11)
in terms of receiving delay ?, expected demand — and safety stocks SS. Order size Q
and safety stock SS are initially set as constants. But note that the actual order quantity
O; (eq. 6.6.) is a variable in this rule. Desired minimum inventory DMINI; is defined as
the sum of expectations and safety stocks. As such, desired minimum inventory DMINk
may be adapted to variations in customer demand. This rule can be shown to be
unbiased and mathematically consistent in the deterministic case. (See Ozevin 1999).
The inventory reaches equilibrium at the desired minimum inventory level, but in the
“noisy” case it may fall below the desired minimum due to discrepancies between
"expected" and "actual" customer demand (Figures 6.2.1. and 6.2.2.).
6.3. Review Period, Order Up to Level (R,S) Rule
Order Point Order Up to Level rule can be mathematically expressed as:
0; =S-El; ift=R*k
0 otherwise (6.12)
where EI, represents the effective inventory, t the time, S the upper level of inventory. k
is an integer and R is the review period. Effective inventory, order point and the upper
level S of inventory are calculated as follows
Ek =h+SLe (6.13)
S =DAVGSLt+DMINI + R*Et (6.14)
I, and SL; represent goods in inventory and in supply line respectively. Review period R
Ss 1: inventory 2: DMINI 3: order
y 200.04
2
3 250.0
ra
0.00 25.00 50.00 75.00 100.01
Nee Graph 2: p6 (SS rule4NE) Time 04:26 20 Oca 1994 Pe|
FIGURE 6.2.1. Performance of Order Point-Order Up to Level (s, S) Rule (DAVGSL;
=4*E,, DMINI =E,) in Short Game with Orders Each Period, Step Up in Customer
Demand, Discrete Delay.
B & inventory 2: DMINI 3: order
1 200.09
2
3 250.0
1
Lf ;
ave
y came 2
j 0.0
3 125.01
B
y 3
2 -200.0
3: 0.064 3
0.00 25.00 50.00 75.00 100.0
Neeé Graph 2: p7 (SS ruleSNE) Time 04:25 20 0ca 1994 Pe|
FIGURE 6.2.2. Performance of Order Point-Order Up to Level Rule (DAVGSL; =
4*E,, DMINI = E)) in Short Game with Orders Each Period, Step Up and Down in
Customer Demand, Discrete Delay.
is five. DAVGSL; refers to the desired average supply line and DMINI; to desired
minimum inventory. Desired average supply line is calculated as
DAVGSL, = ?*E; (6.15)
B® » inventory
2: DMINI 3: order
5] 500.04
2
2 1000.0
z 1
AbtuAtAt WAN
y iN il
0.00)
2 Wyyret ia i ey
3 500.01
3
q]
2 -500.01
3 0.00 3 3h
0.00 62.50 125.00 187.50 250.01
Nee Graph 5: p5 (RS ruleS5NE) Time 00:52 010ca1994Cu
FIGURE 6.3.1. Performance of Review Period-Order Up to Level Rule (DAVGSL; =
10*E;, DMINI = 0) in Long Game with Orders Every Five Periods, Step Up in
Customer Demand, Discrete Delay.
B 2: inventory
2: DMINI 3: order
1 500.04
2
3 1000.0
1 \
1
t
3 0.004 7) —— yi a meer
3 500.0 |
3
: JULIA
2 -500.0
3 0.06 3 3 3
0.00 62.50 125.00 187.50 250.0
Neeaé Graph 5: p6 (RS rule6NE) Time 00:51 01 0ca 1994 Cu
FIGURE 6.3.2. Performance of Review Period-Order Up to Level Rule (DAVGSL; =
10*E,, DMINI =0) in Long Game with Orders Every Five Periods, Step Up and Down
in Customer Demand, Discrete Delay.
interms of receiving delay ?, expected demand Et Desired minimum inventory DMINk
corresponds to the safety stock. DMINItis arbitrarily fixed as constant. This rule can be
shown to be unbiased and mathematically consistent in the deterministic case. (See
Ozevin 1999). The inventory reaches equilibrium at the desired minimum inventory
level, but in the “noisy” case it may fall below the desired minimum due to noise
effects. (Figures 6.3.1. and 6.3.2.).
6.4. (R, 5s, S) Rule:
(R, s, S) rule can be mathematically expressed as
O,=S-EL ift=k*R and EI? s
0 otherwise (6.16)
where S represents the upper level of inventory, El the effective inventory, R the
review period, t the time and s the order point. Review period R is five. Effective
inventory, order point, upper level S of inventory and safety stock SS are calculated as
follows
Ek =Ik+SLt (6.17)
SS=R*E,+D MIN|, (6.18)
s=DAVGSL, + SS; (6.19)
S =s +R*E; (6.20)
I, and SL; represent goods in inventory and in supply line respectively. DAVGSLyrefers
to the desired average supply line, DMINk to desired minimum inventory, SS to safety
stock, Et to expected demand. Desired average supply line is calculated as:
DAVGSL, = #E; (6.21)
in terms of receiving delay ?, expected demand E;. DMINI, is determined as a constant.
This mule can be shown to be unbiased and mathematically consistent in the deterministic
case (See Ozevin 1999). Due to the difference between the expected and actual customer
B & inventory 2: DMINI 3: order
y 500.04
2
3: 1000.01
1
W h 1
1 coql—.J
3: 500.0
| | |
y
i Ky: r
:
0.00 62.50 125.00 187.50 250.00
NeeF Graph 7: p4 (RSS rule4NE) Time 01:46 01 0ca 1994 Cu
FIGURE 6.4.1 Performance of (R, s, S) Rule (DAVGSL; =10*E;, DMINI =0) in Long
Game with Orders Once Every Five Periods, Step Up in Customer Demand, Discrete
Delay.
demand in the noisy case, (R, s, S) rule may not result in ordering each time the
effective inventory falls to or below the order point; orders are sometimes delayed until
the following period. But the inventory never drops below the desired minimum level as
can be seen in Figures 6.4.1 and 6.4.2.
B v inertor 2: DMINI 3: order
y 500.004-+
3 1000.00
q 0.00 |.
3 500.00
y
| 500.00 H i i Ll H
3 0.00 —_
|, 0.00 62.50 125.00 18750 250.00
N =a Graph 7: p5 (RSS rule5NE) Time 01:45 01 Oca 1994 Cum
FIGURE 6.4.2. Performance of (R,s, S) nile (Divest = 10*E,, DMINI =0) in Long
Game with Orders Once Every Five Periods, Step Up and Down in Customer Demand,
Discrete Delay.
6.5. Comparison of the Standard Inventory Rules with Experiments
Order point, Order Quantity (s, Q) rule is not a plausible decision nile
formulation when demand is not constant, as seen above. Order Point, Order Up to
Level (s, S) rule can provide an adequate representation of subjects’ performance in
some cases where the continuous Anchoring and Adjustment Rule is inadequate. One
such case is depicted in Figures 6.5.1. and 6.5.2.
The orders patterns generated by the Review Period, Order Up to Level (R, S)
nue are in general very similar to the ones generated by the Anchoring and Adjustment
rule. Therefore, (R, S) rule does not provide any novel behavior patterns which Linear
Anchoring and Adjustment rule fails to represent. On the other hand, when orders are
given every five periods, (R, s, S) nile can represent subjects' decisions in some cases
where Anchoring and Adjustment rule fails. One such case is depicted in Figures 6.5.3.
and 6.5.4. Finally, it is important to observe that the well known oscillatory dynamic
behavior of the inventory is true not just with the linear anchor-and- adjust rule, but also
with the standard inventory management rules. (Figures 6.2.1 through 6.5.3).
Ss 1: inventory 2: DMINI 3: order
y 200.04
2.
3 250.0
50.00 75.00 100.00
Graph 2: p6 (SS rule4NE) Time 04:26 20 Oca 1994 Pe}
FIGURE 6.5.1. Performance of Order PointOrder Up to Level (s, S) Rule (DAVGSL,
= 4*E;, DMINI = &) in Short Game with Orders Each Period, Step Up in Customer
Demand, Discrete Delay.
Fy 1: inventory 2:order
1 500.00) +++ ++ +4 preset egee etter eee
2 250.00
1 0.00}-2)
2 125.00
M
1 -500.00| 2 F H H
2 0.00: + :
ef 0.00 25.00 50.00 75.00 100.00
Ng “ Graph 1: pé (Subject 18) Time 14:36 23 Ara 1998 Car
FIGURE 6.5.2. Performance of a Player in Game 24 (Short Game with Orders Each
Period, Step Up in Customer Demand, Discrete Delay).
» 1: inventory 2: DMINI
¥y 500.00;
3 1000.00
y 0.00} 2H
500.00
-500.00
0.00
w
wre
oy el os
0.00 62.50 125.00 187.50 250.00
Né BF Graph 7: p5 (RSS ruleSNE) Time 01:45 01 0ca 1994 Cum
FIGURE 6.5.3. Performance of (R, s, S) nile (DAVGSLt=10Et, DMINI = 0) in Long
Game with Orders Once Every Five Periods, Step Up and Down in Customer Demand,
Discrete Delay.
BP +: inventory 2: order
1: 800.0
2: 500.0
lag
2
Se
250 N
2
1: -800.01 ill
2: 0.0 2 2.
0.00 62.50 125.00 187.50 250.0
Née BF Graph 1: p13 (Subject 45) Time 12:57 09 Eyl 1986 Sal
FIGURE 6.5.4. Performance of a Player in Game 14 (Long Game with Orders Once
Every Five Periods, Step Up and Down in Customer Demand, Discrete
Delay).
7. Conclusion
In this research, performances of subjects in an experimental stock
management game are compared with simulated results obtained using some typical
stock management rules. The research shows that the common linear anchoring and
adjustment rule is not always adequate in representing decision-making behavior of
subjects in a dynamic stock control environment. The nile can mimic the subjects
performances’ when they tend to order continuously. However, most people tend to
order intermittently when they are allowed to order each period and/or the receiving
delays are discrete in nature, causing sharp inventory oscillations. To address this
particular behavior, three different types of nonlinear adjustment formulations have
been designed and tested. Some "nonlinear" adjustment rules have been found to be
more representative of subjects’ decisions in many cases. Finally, some standard
inventory management rules have been tested and in some cases, the classical rules such
as (s, S) and R, s, S) are found to represent better the players’ ordering behavior.
Another major finding is the fact that the well-documented oscillatory dynamic
behavior of the inventory is true not just with the linear anchor-and- adjust rule, but also
with the nonlinear mules, as well as the standard inventory management niles. More
research is needed to formulate and test other nonlinear formulations. It would also be
interesting to test these rules in more complex and realistic game environments (such as
multi player supply chains).
References
Barlas, Y. and A. Aksogan, “Product Diversification and Quick Response Order
Strategies in Supply Chain Management,” Proceedings of the International System
Dynamics Conference, 1997
Brehmer, B., “Feedback Delays and Control in Complex Dynamic Systems,” in Milling,
P. and E. Zahn (eds.), Computer Based Management of Complex Systems, Berlin:
Springer-Verlag, pp.189-196, 1989.
Lee, H.L., Padmanabhan V. and Wang S., “The Bullwhip Effect in Supply Chains”,
Sloan Management Review 1997.
Ozevin, M. G.,”Testing The Decision Rules Frequently Used In System Dynamics
Models”, M.S. Thesis, Bogazici University, 1999.
Sterman, J. D., “Testing Behavioral Simulation Models by Direct Experiment,”
Management Science, Vol. 33, No.12, pp. 1572-1592, December 1987.
Sterman, J. D., “Modeling Managerial Behavior: Misperceptions of Feedback in a
Dynamic Decision Making Experiment,” Management Science, Vol. 35, No. 3, pp. 321-
339, March 1989
Tersine, R. J., Principles of Inventory and Materials Management, Prentice-Hall,
Englewood Cliffs, New Jersey, 1994.
Silver, E. A. and R. Peterson, Decision Systems for Inventory Management and
Production Planning, John Wiley and Sons, New Y ork, 1985.
