Reader, Department of Mechanical Engineering
Institute of Tachnology,B.H.U.,Varanasi- 221 005, INDIA.
Synthetic Policy Design in System Dynamics Models:
Some Observations
8.K. Sharma, M.D.Tyagi, P.K.J. Mohapatra
This paper suggest a noval approach to policy design in system
dynamics models. The approach is based on optimal control theory
to evolve synthetic policy structures and then design realistic
policies for the famous production - distribution model’ of
Forrester. New policy sets have been presented for purchase
decision rate at retail and distributor sectors and manufacturing
decision rate at factory. It is shown that the suggested policy
sets show a marked improvement of model behaviour over that
obtained by Forrester. The approach suggested here will enhance
the art of policy design.
I duction?
The poineering work of Forrester and Meadows have demonstrated
the usefulness of system dynamics as a powerful methodology.of a
large scale societal system. The application of system dynamics
encompasses complex system such as urban and world system
(Forrester 1969, 1971). This methodology for modelling the social
system has poineered new area of research in social sciences. The
greatest assets of this methodology is its qualitative foundation
and ease of modelling. The excessive dependence on intution for
model understanding and subsequent policy design is its drawback.
As compared to other system modelling techniques the policy
design phase of system dynamics way of modelling is still non-
rigorous. Much efforts needs to be directed towards the
development of better policy design procedures. Two major
problems surrounds the policy design phase, (1) The complete
dependence on the designer’s ability to intuitively generate an
exhaustive list of policy alternatives, and (2) An excessive
reliance for support on the traditional parameters sensitivity
test to gain model understanding. a
The shortcomings. of the senstivity studies and causal loop
diagrams has been well established (Coyle 1977, 237; Morecraft
1962, 22; Legasto and Maciariello 1984, 40). Fortunatelly,
alternative policy design tools are forthcoming. While some of
these tools attempt to get over come shortcomings of the
traditional design aids, the others provide entirely new approach
to policy design. The use of digraph (McLean and Shephard 1976),
eigenvalues analysis approach to locate the dominant feedback
loops (Forrester N.B. 1983), building of a senstivity model
(Vermeulen and de Jongh 1977), uses of digraph techniques to
identify the critical parameters (Starr 1981), eigenvalues
analysis for selection of policy parameters (Mohapatra and Sharma
1985) are some of the policy design aids which helps to overcome
the traditional way of policy design by trail and error and
subsequent improvements , as one gathers more insight of the
model.
Many researchers have attempted to use dynamic optimization as
policy! design tools (Burns and Malone 1974) were possibly first
1023
1024
System Dynamics '90
&
to use optimal control theory where optimization analysis are
shown to contribute model utility and policy formulation.
(Dercksen and Rademaker 1976 ) have reported the use of dynamic
optimization technique to the world model (Forrester 1971). In
these studies the original policy structures have been kept
intacked and only the contents defining the rate variable have
been optimized. Various problems in applying dynamic optimization
techniques like defining the control variables and objective
function, selecting the solution methodology and the enormous
computational time have been pointed out by these authors.
(Keloharju 1977a,1977b),has suggested a noval method in which
policy equation are developed by first defining artificial
decision parameters which link the policy variables with other
model variables, and then finding out the values of these
parameters which are optimal with respect to a predetermined
objective criterion. This helps in revealing new information
sources and the weight to be attached to this in designing new
policy.(Mohapatra 1976,1979),has used modal control theory , a
branch of control theory to design synthetic policies in multi -
stage production inventory system.(Talvage 1988), has developed a
computer program called MODEMAP which provide great assistance in
using modal control theory to policy design in system dynamics
model. (Appiah and Coyle 1983),have reported formulation and
solution of policy design problems as a model in an adoptive
control frame work. (Keloharju and Wolstenwholme 1989) have
recently used DYSMOD (Keloharju 1983) to design parameters,table
functions and structures for project model of (Richardson 1981).
Review of Optimal Control Theory:
In optimal control problem the objective function is expressed in
the form of an integral over a time period, and the evolution of
the state variable is governed by a system of first-order
differential equation. A linear dynamical system is characterized
by
ig batt) + Batt wee GQ)
4(8) =Xo
where x is an n-th order state vector and uis an m-th order
control vector. A and Bare (n xn) and (m x m) ‘matrices
respectively. The quardratic performance criterion is defined, as
fob ton Leg 3
[facet Q x(t) + a(t) Ru(t) dt ] tee (2)
Q@,R,and S are real symmetric weighting matrices, and the final
time’ period “is considered here as infinity (Shultz and Melsa
1963, Luenberger 1979). In addition, Q and S are assumed to be
positive - semidefinite, whereas Ris assumed a positive -
definite. .
It is possible to evaluate the elements of an (m x n) matrix F by
solving a matrix - Riccati equation such that
u(t) = - Fx(t) vee (3)
which minimizes the quadratic cost criterion defined earlier. The
Riccati equation is actually a set kof interconnected
differential equations. whose terminal conditions are specified
and the solution of this equation requires backward integration
System Dynamics '90 : 1028
. over time. (Melsa and Jones 1973), give a computer program
necessary to carry out this integration.
General Approach to Synthetic Design Procedure :
The paper uses the matrix-Riccati equation to evaluate the closed
-loop feedback coefficient of F matrix inorder to synthetically
design the control policies in a system dynamics model. On the
basis of magnitude and direction (signs) of these coefficients,
new realistic policy structurs have been designed. It is shown
that new policy structures give results comparable to those given
by intutively designed policy set. The synthetically designed
policy are ideal in nature and cannot be implemented in the form
given by Eqn. (3). A close examination of Ean.(3) will reveal
that the magnitude of f;; indicate the strength of the impact of
state variable x; on the’control variable uj, and the sign of f;
indicate the direction and the strength of the impact. ead.
direction ,of causation must be supported by observed facts end
/or Logic ; otherwise, the design procedure and computational are
definitely at suspect. If, however, an element fey = @, that
parehoular state variable x) has no influence on control variable
u,-The presence of a non-zero f;,;implies that x, should be used
as information source for design ide policy variable uy. (Sharma
1985), discusses various ways of generating realistic policy
structure on the basis of close feedback matrix F. This method of
synthesizing policy structure does not assume a priori assumption
about the policy structure .
= Model of Forrester:
The production - distribution model of Forrester which contains
six pure level variables, three smoothed level variables and
eight third-order exponential delay is available in the
celeberated book entitled ‘Industrial Dynamics’ by Forrester
(1961). The original formulated DYNAMO policy equation for
‘purchase decision rate at retail PDR,purchase decision rate at
distributer PDD, manufacturing decision rate at factory MDF are
given below :
R PDR.KL = RRR.JK + (1/DIR) [ (IDR.K-IAR.K) + (EDR K -LAR.K)
: + (UOR.K - UNR.K)] swe @4)
R PDD.KL = RRD. JK + (1/DID) [ (IDD.K-IAD.K) + (EDD: K;LAD.K)
. ' + (UOD.K-UND.K)] es oe (C5)
R MDF.K = RREF.JK + (1/DIF) [ (IDF.K-IAF.K) + CODES K - LAF.K)
+ (UOF.K-UNF.K)] s aes (8)
where,
PDR, PDD “: purchasing rate decision at retail and distributer
sectors,respectively (units/week)
RRR,RRD,RRF =: requisition received at retail, distributer, and
factory sectors respectively (units/week)
DIR,DID,DIF : delay in inventory adjustments at retail,
distributer,and factory sectors,
respectively (weeks)
IAR,IAD,IAF : inventory actual at retail, distributer, and
<i factory sectors, respectively (units)
' LDR,LDD,LDF : pipeline orders desired in. transit at retail,
distributer,and factory sectors,
respectively (units)
1026
System Dynamics ‘90
UOR,UOD,UOF +: unfilled orders. at retail, distributer, and
factory sectors, respectively (units)
MDE : manufacturing decisions at factory
sector(units/week)
This model has been restructured and reduced as shown in figure
1. This has been done to obtained two desirable factors, the
first is that this modified model should be linear to enable one
to use the powerful techniques of linear control theory,the other
is that it should be of low order to be able to handle the
designed computation with ease. Here the basic structure of the
physical flows of orders and material are retained and the
information flow structure necessary to define the policy
variables PDR,PDD,MDF, have been eliminated. . Following
simplification have been introduced to restructure and reduce the:
model (Sharma 1986; Sharma and Mohapatra 1988):
(a)The retail and distributor are stucturally equivalent, hence
distributer sector has been eliminated for sake of simplicity.
(b)The purchase decision rate at retail PDR, the manufacturing
decision rate at factory MDF,are treated as control. variable,
Hence there is no needs to have links from auxilarly variables.
(c) Because shipment sent from retail SSR and factory SSF are
assumed to depend on order backlogs ,these are treated’ as
outflows of first-order delay. :
(d) Some of the cascaded third-order delays are combined and all
the delays are assumed to be of first-order.
(e) The variable have been defined as discrepancy from their
desired values.
It is easy to write the following vector-matrix differential
equation for the reduced model depicted in figure 1:
a(t) = A x(t) + Bult) + Ca(t) ... ... (7)
where,
* YOR
IAR . ,
- CPMPR PDR
a= MTR ae 52 5 fi Rar |
UOF MDE =
TAR
CPF...
OPE
-1/DNER a 7) ) @ @ @ )
1/DNRF ) 7) 1/DTR @ @ @
@ a -1/DCMR 2 a2 @ @
@ a @ -1/DTR @ Qo @ @
A=| @ a 1/DCMR @ -1/DNFF @ @ @ ;
r) @ @ @ -/DNEE @ @ 1/DPF
@ ) a @ 2 @® -1/DCF @
@ a @ @ @ @ 1/DCF -1/DPF
System Dynamics '90 1027
w
BSEVVAH+ag
_areeseas
°
u
Soegoaggr
The various terms used above are defined as under :
CPMPR : clerical in process and mail orders at retail (units)
MTR : material in transit to retail (units)
CPR : clerical in process manufacturing order at factory(units)
OPE : order in production at factory (units)
DNFR : delay normal in filling orders at retail (weeks)
DNFF : delay normal in filling orders at factory (weeks)
DTK : delay in transportion of goods to retail (weeks)
DCMR : delay in processing clerical and mail orders at retail(wk)
DPE : delay in production lead time at factory (weeks)
DCE : delay in clerical processing of orders at factory (weeks)
Eqn.(7) is also acceptable if the variables are defined ‘as
discrepancies from their desired values with the exception of
exogenous requisition received at retail RKK which has been
defined as the discrepancy from its initial value. Thus the stste
vector x, the control vector,. and exogenous variable z are
defined as : :
UOR - UNR
TAR -- IDR
CPMPR - CPMPDR
MTR © - MTDR PDR - RSR
x= pus ; [ere-Rey]
uor - UNF MDF - RSE
IAF ~ IDF
| CPE © - CPDE
OPF - OPDE ‘
RRI is initial value of requisition received at retail. Most of
the variable have been defined earlier. The new variables inx
are the desired values of the level variables.The value of the
constants appearing in equation (7) are the following:
DNFR = 1.4 (weeks), DTR = 1.0 (weeks), DMOR = 3.5 (weeks)
DNFF = 2.0 (weeks), DPF = 8.0 (weeks), DCF = 1.0 (weeks)
The main purpose of policy design in the Forrester’s production-
distribution model is to achieve a stable inventory behaviour at
factory sector. The performance index to be minimized for this
particular problem can be assumed to be a quadratic function of
the states and control as given by Eqn. (2). The optimal control
problem is then to designed the control vector u as a feedback of
the state vector x for a system whose dynamics is governedby the
fqn. (7) so that the performance index is minimized
Synthetic Policy
The weighting matrices Q and R are taken as unit matrices of
1028
System Dynamics '90
dimension(8 x 8) and (2 x 2) respectively. Solving the matric-
Riccati equation for evaluating the study state feedback gain F
(Melsa and Jones 1973) for this case has been obtained as :
@ @ +8.9533 O +8.8161 -@.7914 -@.1124'° @
Fe
@ @® -8.1124 @ -8.4832 +8.6112 +8.9379 +8.8442)...(8)
using the Eqn. (8) the control law (eqn. 3) is obtained as :
PDR(t) = RSR(t) - @.9533 [ CPMPR(t) - CPMPDR(t) ]
- @.8161 [ UOF(t) - UNF(t) ]
+ 6.7914 { TAF(t) - IDF(t) J «e+ (9)
MDF(t) = RSF(t) + @.4832 [ UOF(t) - UNF(t) J]
~ @.6112 [ IAF(t) - IDF(t) ]
- ®.9379 [ CPF(t) - CPDF(t) ]
- 8.8442 [
OPF(t) - OPDF(t) ] vee (18)
These are the ideal policy equations.
In Ean. (9) the negative dependence of PDR on clerical-in-process
and purchase order in mail delay at retail CPMPR and unfilled
order backlog at next higher sector (i.e. factory sector in: the
praesent two sector model) UOF, have been considered by
Forrester, who treats them as pipelines order and inventories.
The positive dependence of PDR on the inventory and the pipeline
inventory at the factory sector are new facet of this ideal
policy. This implies that retail should take advantage of
comfortable inventory position at the higher sector by placing
more orders and conversely should refrain from placing high
orders at the time of low inventory. However, at a first-
glance,one will be surprise to notice the absence of the retail
inventory and backlog terms in influencing PDR, since any
practitioner will use these terms to decide his replenishment
order rates . In fact, Forrester has also considered these terms:
while designing PDR.
Eqn. (10) indicates the influence of some of the level variables
on the MDF. Most of the level variables present were introduced
by the Forrester. The only other level variables which ‘qalifies
to be present in. Eqn. (18) is CPMPR. The corresponding entry
in the matrix F is -@.1124.So the coefficient of CPMPR, when
used and included in eqn.(1@) will be +@.1124.Though the positive
sign is quite justified from causal consideration, the weightage
is considered very small, and the term has not been included.
Design of Realistic Policy
Sharma (1985), has discussed the problems and method to overcome
them while designing the realistic policies from the synthetic
policies.. In designing the realistic policies in the light of
Eqn. (9) and (10) four considerations are made:
(i) It is recalled that the original Forrester’s model has a
distributer sector which was identical to the retail sector and
was neglected while reducing the model (Fig. 1). This sector has
to be included now while designing the testing realistic
policies.
(ii) The ideal policies often need modifications to make it
meaningful to practioner. Thus the inventory and backlog terms
System Dynamics '90 1028
must be included while designing the realistic purchase decision
policies.
(iii) The influence of many of the terms which appears in
the ideal policies but do not appear in the Forrester’s original
policies, will be represented as multipliers defined through
eapie functions or defined as discrepencies from their desired
values.
(iv) The terms having small coefficient values compared to
the highest term in each row of the F-matrix are neglected as
they indicate less dominant influences.
Set-1:
, Considering the above argument, the policy equation for. thrge
: sectors are given by
: R PDR.KL = { RSR.K + (1/DIR) ( (IDR.K - IAR.K) . ”
. + (DR K - LAR-K) + (UOR.K ~ UNR-K) ) 1
: + (PRMUOD.K)* (PRMID.K) we O
A PRMUOD.K = TABHL(TPRMUOD, RUOUND.K,@,2,0.5)
fig TPRMUOD = 1.8/1.0/1.6/8.85/0.5
A RUOUND.K = UOD.K/UND.K
A. PRMID.K = TABHL (TRRMID,RIADD.K,@,2,.0.5)
T TPRMID = @.5/8.75/1.8/1.8/1.8
A RIADD.K = IAD.K/IDD.K
PRMUOD : purchase decision rate multiplier at retail from
unfilled order backlog at distributer (dimensionless)
j PRMID : purchase decision rate multiplier from inventory at
distributer (dimensionless)
It may be noted here that in Eqn. (9), there is no need to
seperately consider multiplier from CPMPR as it is already
included in the original equation of PDR (Ean. 4). The above
_ multipliers are shown in figures 2 and 3. as table functions
“whose slopes are approximately equal to coefficient associated
with the variable they represent in Eqn.(9).The purchase decision
rate at distributor is similarly defined as:
R PDD.KL = { RSD.K + (1/DID) ( ( IDD.K ~ IAD.K)
+ (LDD.K - LAD.K) + (UOD.K - UND. K) d]
* (PDMUOF.K) * (PDMIF.K) wee (12)
A PDMUOF.K = TABHL (TPDMUOF, RUOUNF.K,@,2.0.5)
T TRDMUOF = 1.8/1.8/1.8/8.85/0.5
A RUOUNF.K = UOF.K/UNEF.K
A POMIF.K = TABHL ( TPDMIF,RIADF.K,0,2,0.5)
t TPDMIF = @.5/6.85/1.0/1.8/1.8
A RIADP = IAF.K/IDF.K.
where, PDMUOF and PDMIF are the multipliers and are shown as
table functions in the figures 4 and 5.It can be noticed that eqn.
11 and 12 retains the original equations with new information
sources acting as multipliers.
The MDF is modelled by retaining the original equation (6) and
modifying the time constants of this equation in the light of
Ean. (18),
MDF.KL = RRF.K + (IDF.K ~ IAF.K) /DIAF
+ (LDF.K - LAF.K)/DLAF
; : + (UOF.K - UNF.K)/DUOF +e. (13)
1030
System Dynamics ‘90
The values of above mentioned parameters are taken as DIAF = 1.75
(weeks) ,DLAF=1.0 (week), DUCF = 2.8 (weeks).
Set 3
Further refinement in the policy set I is possible by modelling
the new information sources as discrepancies from their desired
values. The new policy equation for PDR is written as :
R PDR.KL = { RSR.K + (1/DIR) ( (IDR.K - IAR.K)
+ (UOR.K-UNR.K) ) + (LDR.K - LAR.K)/DLAR
+ (UND.K - UOD.K)/DUOR
i + (IAD.K - IDD.K)/DIAR wee (14)
| The values of the parameters DLAR,DUOR,DIAR have been chosen as
1.0, 1.6, and 1.5 weeks respectively. Eqn. (14) has the usual.
terms and parameter values except the parameter value of DLAR,
the associated coefficient for this variable from Eqn. (9) is
- @.9533.This implies that the values of DLAR should be selected
as 1/8.9533 which is approximately equal to one week instead of 4
weeks in the original Eqn. (4). Apart from this it also models
the two new information sources namely the order backlog at
distributor and inventory actual at distributor as additional
features of policy equation. Similarly the equation for PDD is
expressed as :
R PDD.KL = { RSD.K + (1/DID) ( (IDD.K - IAD.K) +
(UOD.K - UND.K) )+(UOD.K ~ UND. Kw)
(LDD.K - LAD.K)/DLAD
(ONE.K - UOF.K)/DUOD
(IAF.K - IDF.K)/DIAD ] «+6 (15)
The values of DLAD, DUOD, and DIAD are 1.0,1.9, and 1.5 weeks
respectively. y .
+tet
bs: .
The original production-distribution model was simulated with the
suggested policy sets. These results are then compared with the
behaviour of the model with original policies. The exogenous
input. RRR is subjected to a 10% increase over the steady state
value of 1800 units per week in the fourth week. The variations
ot SRF,IAF, are shown in figures 6 and 7. It is evident from the
figure 6 that the requisition received at different sectors are
@rowing progressively but have far less pronounced peaks as
compared to those obtained by Forrester. The policy set II gives
better results than policy set I.The peak values for factory
output SRF for policy set I and II are 27% at 21st week and 15%
at 12th week as compared 45% at 21st week in the original policy
set. As with the ordering policy and production rates the
inventory fluctuations are also greatly attenuated as shown in
Fig.8- . The inventory at factory sector for policy set I has a
peak of 19% at 53rd week and for policy set II a peak of 16.5% in
178th week above the initial value without any overshoot as
compared to 32% at 32nd weein the original policy set. From
figures 8 and 9, similar observation can be made for SRF and
inventory at factory for periodic variation in retail sales which
rise and fall gradually over a one year interval.
Fig. 18. and 11 shows the comparision of SRF and IAF for
Suggested policy sets with initial policy set respectively, for
random fluctuations in retail sales. .
4
System Dynamics '90 1031
This paper presents the usefulness of optimal control theory to
assists in the policy design phase in system dynamics modelling.
The time behaviour obtained with various policy sets have given
improved results. The time behaviour of inventory at three
sectors in general is slugish but free from fluctuations. In this
study of production-distribution system Forrester has identified
the delays in the inventory adjustments as the most critical
Parameters and for larger values of delays the system shows
‘improvement. It is observed clearly that the proposed policies
yield more atteniuted response with low values of inventory
adjustment delays. However, it is very difficult to say which
policy give better result when overall results for three sectors
are taken. The fact that most of the causal loops developed |
syntheticaly were considered by Forrester points out to the
robustness of Forrester’s policy.
The proposed policy sets, however, suffer from the difficulty
that it is possible to implement these policies whend complete
intersector information is available. Also the dapproach lined
out is restricted to linear quadratic problem and at present
suited for problems of low order.The paper demonstrates the use
uf optimal control theory and stresses on the Inter-disiplenary
transfer of ideas to enrich the paradigm of system dynamics.
References:
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Analysis of System Dynamics Models of Social Systems, Journal of
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Coyle,R.G.1977,Management System Dynamics,New York:John Wiley &
Sons.
Dercksén, J.W. 1976, Project Group Global Dynamics, Progress
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Loops, The 1983 International System Dynamics Conference Plenary
Session Papers, July 27-38, 1983, : 178-202.
Forrester, J.W. 1961, Industrial Dynamics, U.S.A: MIT Press.
Forrester, J.W. 1969, Urban Dynamics,Cambridg : MIT Press.
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Press.
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ics '90
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Keloharju R. and Wolstonwholm 1989, A case study in System
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System Dynamics '90
1033
Hor)
MANUFACTURING
OROER AT FACTORY
OPF ORDERS IM PRODUCTION OCs!
IN FACTORY (UNITS) MOE {23
a
OPF+6 CLERICAL IN-PROCESS
© ot SRF MANUFACTURING onpens | CPF
Oo AT FACTORY (UNITS)
9 MANUFACTURING
ry SRE] SHIPMENT RECEIVED DECISION at FACTORY
AT THE FACTORY
>
a Inventory actuat | OELAY NORMAL
° aT FacToRY — o<“X IN FILLING ORDERS
ad "ee AT FACTORY
rs) TAF Ps OWFFS2
5 = “ UNFILLED ORDERS
SHIeMENrsci@ AT FACTORY (UNITS }
FROM FACTORY :
DKF" NT =e" UoF ;
REQUISITION RECEIVED
MATERIALS IN TRANSIT -
To RETAIL AACTORY
MIR PowR?s) eRe | Ot
DELAY IN CLERICAL .
Diet IN PROCESS OROERS
D1 SRR PLUS PURCHASE CPMPR
ORDER IN Mail
[4 a PURCHASE DECISION
roy HIPMENT RECEIVED eae
ra 1
LF AT RETAIL ceuay w
ed FILUNG OROERS
oO mwvenory acTua Var aerait
AT RETAIL LN NERO TS
’
2 1AR ,
ca 4
E / UNFILLED ORDERS
Ww AT RETAIL
ae SHIPMENT SENT
FROM RETAIL
UOR
REQUSITION RECEIVED
AL RETAIL
(vw) RRR
oe) neat
Input
Fig.1: The
reduced production distribution model
1034 System Dynamics '90
20 7 2.0
1.6 1.6
x 4 4
8 1.2 q i2
2 = 4
Fd Fa
& 08 & 08 gy
0.4 0.4
+ 0.9 Lar ae a TT 0.0 ne T
0.0 04 0.8 1.2 1.6 2.0 0.0 04 08 1.2 1.6 2.0
RUOUND.K=UOD.K/UND.K RIADD.K=1AD.K/IDD.K
Fig. 2 Fig.
Purchase decision rate multiplier
yultly Purchase decision rate multiplier
from order backlog at distributor
from inventery at distributer
1
°o
2.0
1.6 1.6
¥ a 4
§ 1.2 at
8 “12
= 2 1
2 08 & 0.8
0.4 04
0.0 4a 4 0.0 4S
0.0 0.4 08 1.2 1.6 2.0 0.0 04 08 12 1.6 2.0
RUOUNF.K=U0F.K/UNF.K RIADF.K =1AF.K/IDF.K
Pi Pig, 5
Purchase decision rate multiplier Aas
Purchase decisi Itiplier
from order backlog at factory pee Wahu tay
System Dynamics '90 1038
1500
a
q
q
1000 4 ¢
ao q
= q
Zz q
= = |
= 1
uo 4 —— a1 Forrester's policy
a q —4— Bt Policy set I
5005 —@— ¢ 1 Policy set It
4
q
CO ee 5 oo ~
100 150 200
TIME (WEEKS)
Fig. 6: comparision of shipment received at factories SRF for varicus
policies for step input.
fe} 50
6000
o
E
2
2 A4 Forrester's policy
ire q —+— Bs Policy set I
SS —e— ¢ 4 Policy set 11
20004
j
{
Opp
0 50 100 4150 200
TIME (WEEKS)
Fig. 7: comparision of inventories at factory IAF for various
policies with step input.
1036
System Dynamics ‘90
2000
SRF (UNITS)
r-7
8
;
Aa Forrester's policy
—a— B, Policy set I
—¢— ¢ i Policy set IT
OTT TTT TT
° . 100 150 200
TIME (WEEKS)
Fig. 8:comparision of Shipment received at factories SRF for
various policies for sinusoidal input.
8000
6000 +
a yy
4 8B
eo 4
bad
3 004
a
u q
< q
2000 4 aes At Porrester's policy
4 —s— B 8 Policy set I
—— ¢ 4 Policy set Ir
toe TT RRA RAAAAS CR
50 150 200
Fige
100
TIME (WEEKS)
B88 :
‘omParision of inventories at factory IFA for various
01
Policies with sinusoidal input.
System Dynamics '90 ins,
1500
—— A+ Forrester's policy
—#— B1 Policy set I
—@— ¢ 1 Policy set IT
SRF (UNI
° TTT
0 $0 100 150 200
TIME (WEEKS)
Pig. 1Ocomparision of Shi,pment received at factories SRF for
various policies for Fandam fluctuation,
—— a+ Forrester's policy
7 —«— B14 Policy set I
—@— c 4 Policy set It
lar (UNITS)
3
3000 St
00 150
aT
TIME (WEEKS)
Fig. if 4 $ Comparision of inventories at factory IAF for various
policies with random fluctuation.
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