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A General Stock Control Formulation
For Stock Management Problems Involving Delays and
Secondary Stocks
Hakan Yasarcan and Yaman Barlas
Bogazi¢i University, Dept. of Industrial Eng.
34342 Bebek, istanbul - Turkey
Tel: ++(90) 212 358 15 40/1407-1408, Fax: ++(90) 212 265 18 00
yasarcan@pboun.edu.tr, ybarlas@boun.edu.tr
It is well established that if the stock management formulation ignores the supply line
delay, the behavior of the system can be quite oscillatory. There are naturally other types
of delays in stock management problems such as information delays in decision processing
and delays caused in controlling a primary stock indirectly via a secondary stock. But
there exist no general decision rules in system dynamics that explicitly consider these
different delays in stock management structures. In this research, we first investigate the
implications of ignoring such indirect delays in the decision formulation. We show that the
behavioral consequence of ignoring information delays or ignoring the delay implicit via
secondary stock control is equivalent to ignoring the supply line delay in the standard
case: large oscillations. Next we derive a general stock control heuristic that does take
into account these more advanced types of delays and show that the result is a stabilized
dynamic behavior. Finally, we implement our decision heuristic on an example involving
all three types of delays, demonstrating the “generic” nature of the proposed formulation
structure. The combined result is a significant improvement in the stability of the system,
when compared against the standard policy that considers the supply line delays only.
Keywords: stock management, secondary stock control, supply line, information delay,
virtual supply line.
INTRODUCTION
In the standard stock management structure, there is typically a material delay (supply line)
before control flow actually reaches the stock. (Figure 1.). It is well established that, if the
stock control formulation (typically a linear anchor-and-adjust) does not take into account
the supply line delay, the behavior of this system can be quite oscillatory. (See for instance
Sterman, 1987, 1989, 2000 and Forrester 1961). It is well known that there are other type
of delays in stock management problems such as information delays in decision
processing, delays caused in trying to control a stock indirectly via a secondary stock and
combinations of these. But there are no general decision rules in system dynamics that
explicitly consider these different delays in stock management structures. In this research,
we first investigate the implications of ignoring such less common and indirect delays in
the stock control formulation. We show that the behavioral implication of ignoring
information delay in the decision stream or ignoring the delay implicit via secondary stock
control is equivalent to ignoring the supply line delay in the standard case: large
oscillations. Then we derive a general stock control heuristic that does take into account
these more advanced types of delays and show that the result is stabilized dynamic
behavior. Finally, we implement the decision heuristic on an example involving all three
types of delays, demonstrating the “generic” nature of the proposed formulation structure.
EQUIVALENCY OF THE THREE STRUCTURES IF DELAYS ARE IGNORED
There can be three types of delay structures in the stock management problem; material
supply line, information delay and indirect delay caused by secondary stock control. All
these three structures introduce a delay between the control decision and resulting control
action.
Material Supply Line Delay
Material supply line is a material delay structure that shows the actual transportation of
quantities from one stock to another. For example, supply line may be goods on order in
the inventory control problem. (For more examples see for instance Sterman, 2000).
Osl\Ordey of supply line
S\Stock
S1\Supply line 1
AF 1\AcquisitiotNlow 1 AEDAequisition fg
Tad! <r
S\Desired stock
Tsa\Stock adjustment time SA\Stock adjustment
SLS2\Supply line
2
CF\Control £5 LF\Loss flow
Figure 1. A simple stock management structure involving material supply line
Note that, abbreviated variable names and actual variable names are both shown -separated
by a ‘\’ symbol- in the models in Figure 1., Figure 2. and Figure 3. Abbreviated names are
used in mathematical equations. For abbreviations we use the following rules:
All variables are abbreviated in all-capital letters.
Desired levels and desired values are shown by appending a
All stock variables end with “S”.
All flow variables end with “F”.
Parameters are abbreviated by an initial capital letter followed by a small
letter\letters index. Parameter conventions:
* C: Coefficient
* O: Order
* T: Time
« W: Weight
cere
eoceoee
CF =LF+SA (1)
SLS, SLS,
eas (2)
Tad/Osl Tad/2
SLS SLS5
jo @)
Tad/Osl_ Tad /2
S'-S
SA= 4
Tsa ®
In the models in Figure 1., Figure 2. and Figure 3., it is assumed that there are no
perception delays in perceiving loss flows or time delays, without loss of generality.
In the first simple model, Supply line is ignored in the control flow equation and, LF (Loss
flow), S' (Desired stock), Tsa (Stock adjustment time), Tad (Acquisition delay time), Osl
(Order of supply line) are all assumed to be constants. The stock equations and the reduced
stock equations (after all variables and parameters are inserted) can be seen in derivative
form in Appendix 1.
A sample behavior of the model can be seen in Figure 4. Stock/ represents the Stock of the
model in Figure 1. As we mentioned before, the oscillations can be unstable as a result of
ignoring the supply line in the decision formulation.
Information Delay
Information delays are involved in the flow of information from one location to another or
in information processing. They can be part of some common stock management problems.
(i.e. Order-filling delays, ordering decision delays, mail delays, see Forrester, 1961). An
example could be the information delay in between two departments of a company.
S\Stock
@ & OH—0
CEControl flow LF\Lo4s flow
S\Desired stock
Tsa\Stock adjustment time
SA\Stock adjustment
CF'\Desired}control flow
IDS2\Infomjation delay 2
IDS \Information delay 1
TAF 2\Informatign
adjustmapt flow 2
TAF I\Int ation
adjustmeptt flow 1
Tid\Information delay time Oid\Order of information delay
Figure 2. A simple information delay structure in stock control
CF'-IDS, _ CF'-IDS,
IAF =—___. (5)
Tid / Oid Tid /2
UR, = IDS, ~ IDSy _ IDS, ~ IDS, 6
Tid | Oid Tid /2
CF = IDS, (7)
CF'= LF + SA (8)
S'-S
SA= 9
Tsa @)
In this simple model, the information delay is ignored in the decision formulation and LF
(Loss flow), S' (Desired stock), Tsa (Stock adjustment time), Tid (Information delay time),
Oid (Order of information delay) are all constants. The stock equations and the reduced
stock equations (after all variables and parameters are inserted) can be seen in derivative
form in Appendix 1.
A sample behavior of the model can be seen in Figure 4. Stock2 represents the Stock of the
model in Figure 2. As we mentioned before, the oscillations can again be unstable as a
result of ignoring the information delay in the decision formulation.
Indirect Control Via a Secondary Stock
Sometimes the stock is controlled via a secondary control. For example, production rate
may be controlled indirectly by adjusting the workforce, by hiring\firing workers. (i.e.
Customer-Producer-Employment System, Forrester, 1961, Inventory-Workforce Model, A
Generic Commodity Market Model, Sterman, 2000) The overall effect of a secondary
stock control sub-system is to introduce a delay between the desired production (control
decision) and the actual production (resulting control). This is very similar to material
supply line delay between orders (control decision) and acquisition rate (resulting control)
or, to information delay between decision (control decision) and action (resulting control).
S\Stock
Cp\Productivity
‘ondary Ste
adjustment time
Tsad\Sedo
acquisition 4
SSLS'\Desired
secondary supply line WeshWeight of
secondary supply line
Figure 3. A simple indirect stock control structure via a secondary stock
CF =Cp*SS
SCF = SLF + SSA+SSLA
_ SSLS
Tsad
SAF
Wssl * (SSLS'-SSLS )
Tssa
SSLA =
SSLS'= Tsad * SLF
ssa — 95'=S5)
Tssa
svi
Cp
CF'=LF+SA
(10)
ay
(12)
(13)
(14)
(15)
(16)
a7)
54-52
Tsa
(18)
In this simple model, the indirect delay effect of the secondary stock is ignored in the
decision formulation and LF (Loss flow), SLF (Secondary loss flow), Cp (Production
coefficient), Tsad (Secondary acquisition delay time), Wssl (Weight of secondary supply
line), Tssa (Secondary stock adjustment time), S' (Desired stock), Tsa (Stock adjustment
time) are all constants.
A sample behavior of the model can be seen in Figure 4. Stock3 represents the Stock of the
model in Figure 3. Once again, the oscillations can be unstable as a result of ignoring the
indirect delay effect via the secondary stock in the decision formulation.
Behavior of the Three Models
The above three models cause their stocks to exhibit exactly the same behaviors when
parameter values are selected appropriately; (i.e. Tssa =Tsad = a = my For example:
LF (Loss flow) = 4
SLF (Secondary loss flow) = 0.2
Cp (Production coefficient) = 12
Wssl (Weight of secondary supply line) = 1
Tsa (Stock adjustment time) = 4
Tad (Acquisition delay time) = 21
Tid (Information delay time) = Tad = 21
Tsad (Secondary acquisition delay time) = Tad/2 = 10.5
Tssa (Secondary stock adjustment time) = Tad/2 = 10.5
eoceoeoeoeoeee
All stocks are initialized to their equilibrium levels.
At time four, S' (Desired stock) is decreased from ten to nine.
1: Desired stock 2: Stock 1 3: Stock 2 4: Stock 3
11.00%
an
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Figure 4. Output behaviors of three models without considering the different delays
In the above figure, Stock, Stock2 and Stock3 are the primary control stocks of the models
in Figure 1., Figure 2. and Figure 3. respectively. (It is a well-known fact that when supply
line is ignored in the decision rule, oscillatory behavior may be obtained. We selected
parameter values so as to demonstrate an extreme case of unstable oscillations).
It can be seen above that the three structures generate exactly the same behavior patterns.
The equivalence can also be mathematically proven (see Appendix 2).
THE VIRTUAL SUPPLY LINE CONCEPT
It is well known that to obtain stable behavior, considering the supply line in stock control
decisions is critical (Sterman, 1987, 1989, 2000 and Forrester 1961). In the same way, we
claim that considering other type of delays like information delay and indirect secondary
stock delays can also be very important. Oscillations can be dampened by incorporating
such delays in the stock control decisions.
Supply line is considered by simply including a supply line adjustment term in control
flow. It is not possible to include secondary stock and information delay in the same way,
since their role and even their units are different from the primary supply line. But there
must be a way to handle these delays since they are behaviorally and mathematically
equivalent to supply line delay when they are seen as input/output systems. To account for
these two types of delays we propose the virtual supply line concept that has unit
consistency with the primary stock.
The Supply Line Adjustment in the Model with Supply Line Delay (Figure 1.)
In the improved model, the Adjusted control flow (ACF) is used as input to supply line,
instead of Control flow (CF). Equation for Adjusted control flow can be given as
ACF = CF +SLA= LF +SA+SLA (19)
where
SLA =Wsl SES SES. (20)
Tsa
SLS'= Tad * LF (21)
SLA is Supply line adjustment, Wsl is Weight of supply line and, SLS' is Desired supply
line.
A sample behavior of the model can be seen in Figure 5. Stock/ represents the Stock of the
model in Figure 1. Note that model is modified with the above equations to consider
supply line delay in decisions and Ws/ is taken to be one. It is known that for Ws/ = 1,
supply line and stock system reduces effectively to a first order system that is perfectly
stable (Sterman, 1989 and Yasarcan 2003).
Introducing the Virtual Supply Line in the Model with Information Delay (Figure 2.)
Again, the Adjusted desired control flow (ACF’) can be used as input to information delay
instead of Desired control flow (CF). Equation for Adjusted desired control flow can be
given as
ACF'= CF'+VSLA = LF + SA+VSLA (22)
where
VStA = Wosgt® SES VSES. (23)
Tsa
VSLS'= Tid * LF (24)
VSLS = (Tid /2)* IDS, + (Tid /2)* IDS (25)
VSLA is Virtual supply line adjustment, VSLS is Virtual supply line, VSLS' is Desired
virtual supply line and Wvsl is Weight of virtual supply line. Note that the Virtual supply
line stock VSLS is a hypothetical (virtual) stock that has the same dynamic (and
mathematical) role as having a supply line delay in the stock control structure. This
concept is thus used in order to account for the information delay in the decision
formulation.
A sample behavior of the model can be seen in Figure 5. Stock2 represents the Stock of the
model in Figure 2. Note that the model is modified with the above equations to consider
the information delay in decisions and Wvs/ is taken to be one. The structure with
information delay reduces effectively to a first order system and a perfectly stable behavior
results.
Introducing the Virtual Supply Line in the Model with Secondary Stock (Figure 3.)
Once again, the Adjusted desired control flow (ACF’) can be used as input to secondary
stock sub-structure instead of Desired control flow (CF). Equation for Adjusted desired
control flow can be given as
ACF'= CF'+VSLA = LF + SA+VSLA (26)
where
VSLA = Ws) * PSESTUSES. (27)
Tsa
VSLS'= (Tssa + Tsad )* LF (28)
VSLS = Cp * ((Tssa + Tsad )* SS + Tsad * (SSLS — SSLS')) (29)
VSLA is Virtual supply line adjustment, VSLS is Virtual supply line, VSLS" is Desired
virtual supply line and Wvsl is Weight of virtual supply line. Note that the Virtual supply
line stock VSLS is a hypothetical (virtual) stock that has the same dynamic (and
mathematical) role as having a supply line delay in the stock control structure. This
concept is thus used in order to account for the implicit delay caused by the indirect
secondary stock control, in the decision formulation.
A sample behavior of the model can be seen in Figure 5. Stock3 represents the Stock of the
model in Figure 3. Note that the model is modified with the above equations to consider
secondary stock sub-structure in decisions and Wvs/ is taken to be one. The secondary
stock and primary stock system reduces effectively to a first order system and a perfectly
stable behavior results.
Behavior of the Three Models (Figure 1., Figure 2. and Figure 3.) with Supply Line
Adjustment and/or with Virtual Supply Line Adjustment
® +: desired stock 2: Stock 1 3: Stock 2 4: Stock 3
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Bens
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Figure 5. Output behaviors of three structures after considering the different delays
In the above figure, Stockl, Stock2 and Stock3 are the primary stocks of the models in
Figure |., Figure 2. and Figure 3. respectively. Note that the models are modified to take
delays into consideration and all supply line weights are set to one. As it can be seen
above, the primary stocks in the three structures generate exactly the same behaviors. This
proves that the Virtual supply line concept is structurally equivalent to Supply line. The
equivalence can also be mathematically proven (see Appendix 3).
Note that parameter values are the same as for the run in Figure 4. It can be seen that
output behaviors in Figure 5. are stable. Behavior is stabilized by incorporating the effects
of delays in control decisions.
EXAMPLE MODEL WITH ALL THREE DELAY STRUCTURES
In this example, we incorporate supply line, information delay and secondary stock sub-
structures simultaneously. Information delay is in between inventory control department
and human resources department. Production start rate is controlled by changing the
numbers of workers (labor). There is a supply line delay representing the manufacturing
process. For simplicity, we assume that there are no delays in perceiving the quit rates or
delay times and, no layoffs are allowed, without loss of generality.
Work in process inventory Inv entory
ProdGictifn start rate Production rate
Minimum shjpment time
” tomer orders
lanuf acturing ey cle time
Inventory adjui
WIPI Adjustment
Weight of WIPI
Information adjust
Perceived DPSR
wéight of VSL for 1D
ct desifefvsi for 88
VSL sau)ue) Weight of VSL for SS
oo
Tapér adjustmem
Later aan meen .)
ie ual spply line for SS
Labor Trainees
Cinees adjustment Weight of trainees,
Desired trainees
Figure 6. Example model using all three delay structures
Equations of The Example Model
Desired_pr oduction_s tart_rate (30)
Adjusted_D PSR =
~ +VSL_adjus tment_for_SS+VSL_adj ustment_fo r_ID
Average_duration_of_employement = 40
Average _time _to_ join _to_labor =9
Customer _ orders = 4000 + STEP (500,4)
Desired _ inventory = 60000
Perceived _DPSR
Desired_labor =
~ Productivity
Desired _ production _ start _rate =
Shipment_r ate+Invent ory_adjust ment
+WIPI_adju stment
Desired_trainees = Average_time_to_join _to _labor*Quit_rate
Desired_VSL _ for _ ID = Information _ delay _time*Shipment_rate
Labor_adjustment_time
Desired_VSL _ for _SS =
ne +Average_time_to_join_to_labor
}pstomen t_rate
Desired _WIPI = Manufactur ing_cycle_time*Shipm ent_rate
Hiring _ rate = Quit _ rate + Labor _ adjustment + Trainees _ adjustment
. Desired_inventory-In ventory
Inventory_adjustment = ——@
Inventory_adjustment_time
Adjusted _ DPSR — Perceived _DPSR
Informatio n _ delay _ time
Informatio n _ adjustment _ flow =
Inventory_adjustment_time =7
sf Trainees
Joining _rate=
Average _time _to _ join _to _labor
GB)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
Desired_Labor-Labor
Labor_adju stment = —¥§\
Labor_adjustment_time
Labor_adjustment _ time =16
Manufactoring _cycle_time =9
Minimum _ shipment _ time =1.6
ss Work _in_ process _ inventory
Production _rate = >=
Manufacturing _cycle_time
Production _ start _rate = Productivity * Labor
Productivity =10
Labor
Average _duartion _of _employement
Quit _rate=
Shipment _ rate = MIN(Customer_orders, Inventory/Minimum_shipment_time)
‘ Desired_trainees-Tra inees
Trainees_adjustment = Weight_of_trainees*
Labor _adju stment_time
VSL_adjustment _ for _ID=
Desired_VSL _ for _ID-Virtual_supply_line _ for _ID
Weight_of VSL _ for _ID*
~~ ~~ Inventory_adjustment_time
VSL_adjustment _ for _SS =
Desired_VSL _ for _SS-Virtual_supply_line_ for _ SS
Weight_of VSL_ for _SS*
~~ ~— Inventory_adjustment_time
Virtual_su pply_line _ for _ID = Information _ delay _ time * Perceived _DPSR
Virtual_supply_line _ for _SS =
Productivity *| . _ ; .
+Average_time_to_join_to _labor*(Trainees-Desired_trainees)
(46)
(47)
(48)
(49)
(50)
G6)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(Labor_adjustment_tim e+Average_time_to_join_to_labor mer) (59)
WIPL. adjustment =Weight of WIPT* Desired_WIPI-Work_in_process_inventory (60)
Inventory_adjustment_time
All stocks are at their equilibrium levels initially. At time four Customer_orders is
increased from 4000 to 4500 to disturb the system from its equilibrium level.
Runs of The Model
First run:
e Weight_of_trainees =0
¢ Weight_of WIPI=0
¢ Weight_of VSL_for_ID=0
e Weight_of VSL_for_SS=0
Second run:
e Weight_of_trainees = 1
¢ Weight_of WIPI=0
e Weight_of VSL_for_ID=0
¢ Weight_of VSL_for_SS=0
Third run:
e Weight_of_trainees = 1
© Weight_of WIPI=1
¢ Weight_of VSL_for_ ID=0
¢ Weight_of VSL_for_SS=0
Fourth run:
e Weight_of_trainees = 1
¢ Weight_of WIPI=1
¢ Weight_of VSL_for ID=1
e Weight_of VSL_for_SS=0
Fifth run:
e Weight_of trainees = 1
¢ Weight_of WIPI=1
¢ Weight_of VSL_for ID=1
¢ Weight_of_ VSL_for_SS=1
BP ioventory: 1-2-3-4-5-
1: 800004
x 600004
+ 40000:
T
0.00 50.00 100.00 150.00 200.00 250.00
Time 22:00 28 Mar 2003 Cum
a = Pa ? Effect of weights
Figure 7. Output behaviors of the example model with different supply line and virtual
supply line weight values
A typical system dynamics decision formulation run is the third run, consisting of
oscillations. In this run, supply line of the primary stock and supply line of the secondary
stock are both considered in the decisions, but the information delay and indirect secondary
stock delay effects are ignored. When the information delay and indirect secondary stock
delays are also considered in decisions using our formulations, the behavior is improved
significantly (fifth run). When all delays are considered optimally, oscillations are
completely eliminated.
IMPLEMENTATION ISSUES IN VIRTUAL SUPPLY LINE ADJUSTMENT
Virtual Supply Line adjustment necessitates delay durations, orders and the stock values in
the delay structure to be known or estimated.
In the secondary stock control structure, the stocks of the structure is already monitored
and known, so the only unknown may be the delay durations. In this case we propose to
use the estimates of the delay durations for calculation of Virtual Supply Line. Ignoring
delays are a far bigger mistake than using estimates. Furthermore, the Virtual Supply Line
adjustment is quite robust, so it works quite well with estimates of delay durations. (See
Yasarcan 2003).
Information delay structure may be a harder case. For some cases decision maker can only
perceive the last stock of the delay structure. In this case, not only the delay durations but
also the stock values in the delay structure must be estimated, which may not be a simple
task. Furthermore, in some cases even delay order may not be available to the decision
maker. For these more complicated cases, if we know the initial value of the delay stock,
we can continuously update the value of the stock by using a “stock type virtual supply
line” assuming that we have access to the outflow of the delay. (This structure is skipped in
this article; see Yasarcan, 2003).
CONCLUSION
It is well known that, if the stock management formulation ignores the supply line delay,
the behavior of the system can be quite oscillatory. There are naturally other types of
delays in stock management problems such as information delays in decision processing
and delays caused in controlling a primary stock indirectly via a secondary stock. But there
are no general decision rules in system dynamics that explicitly consider these different
delays in stock management structures. In this research, we first investigate the
implications of ignoring such indirect delays in the stock control formulation. We show
that the behavioral consequence of ignoring information delay in the decision stream or
ignoring the delay implicit via secondary stock control is equivalent to ignoring the supply
line delay in the standard case: large oscillations. Next we define the ‘virtual supply line’
concept and derive a general stock control heuristic that does take into account these more
advanced delays and show that the result is stabilization of the dynamic behavior. We
prove again that the improvement obtained by incorporating these less common delays is
equivalent to the improvement obtained by incorporating the supply line delay in the
standard case. Finally, we implement our decision heuristic on an example involving all
three types of delays, demonstrating the “generic” nature of the proposed formulation
structure. The combined effect is a significant improvement in the stability of the system,
when compared against the standard policy that considers the supply line delays only.
Future work will involve testing our “generalized” stock adjustment formulation in actual
models that involve further loops and non-linearities.
APPENDIX 1. EQUATIONS OF SIMPLE DELAY STRUCTURES
Stock Equations of Model with Supply Line Delay (Figure 1.)
S = AF) - LF (61)
SLS, = CF - AF, (62)
SLSy = AF, — AF) (63)
Reduced Stock Equations of Model with Supply Line Delay (Figure 1.)
$2 SS
Tad /2
(64)
S'-S SLS,
SLS, = LF + -
Tsa Tad /2
1s, = SESi__SUS2
~ Tad/2— Tad /2
Stock Equations of Model with Information Delay (Figure 2.)
S=CF-LF
IDS, = IAF,
IDS = IAF,
Reduced Stock Equations of Model with Information Delay (Figure 2.)
S = IDS, - LF
. LF +5=5_ ips,
IDS, = ——sa__
Tid /2
1Ds, = 1PSi= DS.
Tid /2
Stock Equations of Model with Secondary Stock (Figure 3.)
S=CF-LF
SS = SAF - SLF
SSLS = SCF ~ SAF
Reduced Stock Equations of Model with Secondary Stock (Figure 3.)
S=Cp*SS—LF
(65)
(66)
(67)
(68)
(69)
(70)
(7)
(72)
(73)
(74)
(75)
(76)
- SLF (77)
LF +(S'-S)/Tsa _ ss
Cp , Wssl* (Tsad * SLF -SSLS) _ SSLS
Tssa Tssa Tsad
SSLS = SLF + (
(78)
APPENDIX 2. MATHEMATICAL EQUIVALENCY OF THE THREE
STRUCTURES WHEN DELAYS ARE IGNORED
It can be shown mathematically that the three structures can be made equivalent. We will
consider material supply line, information delay and secondary stock structures as separate
input\output systems and ignore primary stocks for simplicity.
For material supply line structure input is CF and output is AF 2 and, for secondary stock
and information delay structures input is CF’ and output is CF.
We will re-write reduced stock equations ignoring primary stock.
Reduced Supply Line Equations (Figure 1.)
SLS, = CF -— SLS\ (79)
Tad |2
Ls, = SLS, __ SLS> (80)
Tad/2 Tad /2
From Equation (80) the following two equations are obtained:
SLS, = (Tad / 2)* SLS,+ SLS5 (81)
SLS1 = (Tad /2)* SLS,+ SLS, (82)
Equation (81) and Equation (82) can be inserted in Equation (79) to obtain the following
equation:
(Tad /2)* SIS, +SLSy
(Tad /2)* SLS+ SLS, = CF —
Tad |2
(83)
Equation (83) can be simplified to the following:
(Tad 12) * SLS)+ Tad * SLS)+ SLSy = (Tad /2)* CF (84)
Equation (84) can be re-written for AF2 with using the relationship given for SLS2 and
AF? in Equation (3)
(Tad /2)° * AF, + Tad * AF, + AF) = CF (85)
Reduced Information Delay Equations (Figure 2.)
ws, = EP (86)
Tid /|2
IDS, = P= DS. (87)
Tid /2
From Equation (87) the following two equations are obtained:
IDS, = (Tid /2)* IDS, + IDS, (88)
IDS, = (Tid /2)* IDS, + IDS, (89)
Equation (88) and Equation (89) can be inserted in Equation (86) to obtain the following
equation:
. . "_(Tid /2)*
(Tid /2)* IDS)+ IDS2. = a a TB (90)
if
Equation (90) can be simplified to the following:
(Tid /2)° * IDS + Tid * IDS,+ IDS = CF' (91)
Equation (84) can be re-written for CF with using the relationship given for JDS2 and CF
in Equation (7).
(Tid 2)? * CF + Tid * CF+ CF = CF' (92)
Letting Tid = Tad the following is obtained:
(Tad 12) * CF+ Tad * CF+ CF = CF" (93)
If Equation (93) is compared with Equation (85) it can be seen that the two differential
equations are the same except for the variable names. This proves that as input-output
systems, supply line and information delay structures can be identical with appropriate
selection of parameter values.
Reduced Secondary Stock Equations (Figure 3.)
ss = SSES _ gy p (94)
Tsad
. 1 * * —
SRS (Ss'-SS) ssl (Tsad * SLF —SSLS) _ SSLS (95)
Tssa Tssa Tsad
From Equation (94) the following two equations are obtained:
SSLS = Tsad * SS+Tsad * SLF (96)
SSLS = Tsad * SS (97)
Equation (96) and Equation (97) can be inserted in Equation (95) to obtain the following
equation:
Wssl* [rsa * SLF —Tsad * SS. + Tsad * SLF }
Toad * $5 = sLF + S783) ,
Tssa Tssa (98)
_ Tad * $8+ Tad * SLF
Tsad
Equation (98) can be simplified to the following:
Tsad * Tssa * $8+ Tssa* SS+Wssl* Tsad * SS+ SS = SS' (99)
Letting Wss/ = 1 and Tsad = Tssa = ue the following is obtained:
(Tad /2) * 88+ Tad * SS+ SS = SS" (100)
Equation (100) can be re-written for CF with using the relationship given for SS and CF in
Equation (10).
(Tad 12) * CF + Tad * CF + CF = CF" (101)
If Equation (101) is compared with Equation (85) and Equation (93) it can be seen that it is
same with these equations. This proves that as input-output systems, supply line,
information delay and secondary stock structures can be identical with appropriate
selection of parameter values.
APPENDIX 3. MATHEMATICAL EQUIVALENCY OF THE THREE
STRUCTURES WHEN DELAYS ARE CONSIDERED
It can be shown mathematically that the three structures can made equivalent. We will
consider material supply line, information delay and secondary stock structures as separate
input\output systems and ignore primary stocks for simplicity.
For material supply line structure input is CF and output is AF2 and, for secondary stock
and information delay structures input is CF’ and output is CF.
Reduced Supply Line Equations when Supply Line is Considered (Figure 1.)
. * * pen
SiS, =CF + Wsl* (Tad * LF —SLS) _SLS, (102)
Tsa Tad /2
sis, = SLS, __SLS> (103)
Tad/2 Tad /2
From Equation (103) the following two equations are obtained:
SLS, = (Tad /2)* SLS,+ SLS> (104)
SLS, = (Tad /2)* SLSy+ SLS5 (105)
Equation (104) and Equation (105) can be inserted in Equation (102) to obtain the
following equation:
(Tad /2)* SLS+ SLSy =
. (106)
Wsl*| Tad * LF —(Tad /2)* SLS,—SLS .
Cr+ _ (Tad /2)* SLS, + SLSy
Tsa Tad /2
Equation (106) can be simplified to the following:
(Tad /2)° *Tsa* SLS> + (rad *Tsa +Wsl* (Tad /2)° )s SLSy+ (Tsa+Wsl*Tad)*SLSy (107)
= Tsa* (Tad /2)* CF +Wsl *(Tad? 2) LF
Equation (84) can be re-written for AF2 with using the relationship given for SLS2 and
AF? in Equation (3)
3
(Tad /2)° *Tsa* AFy+ (Tad *Tsa + Wsl * (Tad /2) )s AF) +(Tsa+Wsl*Tad)* AF, (108)
=Tsa* CF +Wsl* Tad * LF
Reduced Information Delay Equations when Virtual Supply Line is Considered
(Figure 2.)
Tid * LF —(Tid /2)* IDS, —(Tid /2)* IDS,
. CF'+Wysl* —IDS;
IDS, = _ (109)
Tid |2
*. _ IDS, — IDS,
IDS, = 110
2 Tid /2 (rm)
From Equation (110) the following two equations are obtained:
IDS, = (Tid /2)* IDS, + IDS, (1)
IDS, = (Tid /2)* IDS) + IDS} (112)
Equation (111) and Equation (112) can be inserted in Equation (109) to obtain the
following equation:
CF
4 Wl a| Tid * LF —(Tid /2) * IDS,
Tsa_ | _(Tid /2)* IDS) —(Tid /2)* IDS,
_(Tid|2)* IDS,
- IDS,
(Tid /2)* IDS)+ IDS, = (113)
Tid /2
Equation (113) can be simplified to the following:
(Tid /2)° *Tsa* IDS)+ (ria *Tsa +Wvsl* (Tid /2)° )s IDS»+(Tsa+Wvsl*Tid)* IDS, (114)
= Tsa* CF'+Wvsl* Tid * LF
Letting Tid = Tad and using the relationship given for JDS2 and CF in Equation (7),
Equation (84) can be re-written for CF:
(Tad (2)? * Tsa* CF + (Tad * Tsa + Wosl * (Tad /2)*)* CF+ (Tsa+Wvsl *Tad)*CF (115)
= Tsa * CF'+Wvsl * Tad * LF
If Equation (115) is compared with Equation (108) it can be seen that the two differential
equations are the same except for the variable names. This proves that, with appropriate
selection of parameter values, Virtual supply line for information delay is mathematically
equivalent to Supply line.
Reduced Secondary Stock Equations when Virtual Supply Line is Considered (Figure
3.)
*, SSLS
SS = —SLF (116)
Tsad
SLF +
(Tssa + Tsad)* LF
Wvsl
CF'+ * (Tssa + Tsad)* SS
Tsa | -—Cp*
+Tsad * (SSLS —Tsad * SLF )
. CP 117
SSLS = am
Tssa
_, Wssl* (Tsad * SLF —SSLS) SSLS
Tssa ” Tsad
From Equation (116) the following two equations are obtained:
SSLS = Tsad * SS+ Tsad * SLF (118)
SSLS = Tsad * SS (119)
Equation (118) and Equation (119) can be inserted in Equation (117) to obtain the
following equation:
SLF +
(Tssa + Tsad )* LF
cry Wile (Tssa + Tsad )* SS
Tsa |-Cp* :
+Tsad *| Tsad * SS+ Tsad * SLF —Tsad * SLF
-SS
Cp (120)
Tsad * SS =
Tssa
Wssl *{ Tsad * SLF —Tsad * SS—Tsad * SLF .
_ Tsad * $8+Tsad * SLF
Tssa Tsad
db
Equation (120) can be simplified to the following:
Cp *Tsad * Tssa* Tsa* SS+ Cp* (Wsst *Tsad * Tsa + Tssa* Tsa + Wvsl * Tsad* ) SS (121)
+Cp* (Tsa +Wvsl* (Tssa +Tsad )* SS =Tsa* CF'+Wvsl * (Tssa +Tsad )* LF
Letting Wss/ = 1 and Tsad = Tssa = x the following is obtained:
Cp* (Tad /2)° * Tsa* 85+ Cp * (Tad * Tsa + Wvsl * (Tad /2)°)* SS (12)
+Cp*(Tsa + Wvsl* Tad )* SS = Tsa* CF'+Wvsl* Tad * LF
Equation (122) can be re-written for CF with using the relationship given for SS and CF in
Equation (10).
(Tad /2)° *Tsa* CF + (Tad * Tsa + Wosl * (Tad /2)*)* CF+ (Tsa + Wvsl *Tad)*CF (123)
= Tsa* CF'+Wvsl * Tad * LF
If Equation (123) is compared with Equation (108) and Equation (115) it can be seen that it
is same with these equations. This proves that, with appropriate selection of parameter
values, Virtual supply line for secondary stock is mathematically equivalent to Supply line
and Virtual supply line for information delay.
REFERENCES
Barlas, Y. and M. G. Ozevin, 2001, Testing the Decision Rules Used in Stock
Management Models, Proceedings of the 19th International Conference of the System
Dynamics Society, 23 — 27 July, Atlanta, Georgia.
Forrester, J. W., 1961, Industrial Dynamics, MIT Press, Massachusetts.
Forrester, J. W., 1973, Principles of Systems, Wright-Allen Press, Massachusetts.
Ozevin, M. G., 1999, Testing the Decision Rules Frequently Used in System Dynamics
Models, M.S. Thesis, Bogazi¢i University.
Sterman, J. D., 1987, Testing Behavioral Simulation Models By Direct Experiment,
Management Science, Vol. 33, No.12, pp. 1572-1592, December.
Sterman, J. D., 1989, Modeling Managerial Behavior: Misperceptions of Feedback in a
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339, March.
Sterman, J. D., 2000, Business Dynamics: Systems Thinking and Modeling for a
Complex World, McGraw-Hill.
Yasarcan, H., 2003, Feedback, Delays and Non-linearities in Decision Structures, PhD
Thesis, Bogazi¢i University, (to be printed).
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