|
STATISTICAL METHODS FOR IMPROVING CONFIDENCE IN SYSTEM
DYNAMICS MODELS - A CASE STUDY ON BLOOD BANK INVENTORY
MANAGEMENT SYSTEMS:
* #
Shoukath Ali-K and Ramaswamy.N
ABSTRACT
The article discusses some statistical techniques
applied as confirmatory tools to the System Dynamics
modelling and analysis of Blood Bank Inventory
Management systems. Instead of using arbitrary means.
problem definition and statements are corroborated with
statistical methods of correlation and formulation of
adjacency matrices. This is extended to the estimation
of some of the. parameters of the system. Numerical
Performance Measures (NPM) used to evaluate the system
response to-various inputs are discussed. The response
of. the system is illustrated primarily as time series
plots.System trajectories or phase plane plots are
presented with statistical inferences in relation to the
model. It is concluded that for SD model refinement and
analysis statistical techniques can be used judiciously
as a confirmatory tool in. unison with judgmental
evaluation of the system.
INTRODUCTION
System Dynamics models are basically policy design tools used for
vnderstanding possible strategies to improve the system behavior.
Judgmental methods are usually adopted in building up the model and
carrying out the analysis right from problem definition to validation.
System Dynamicists believe that what is important both in explaining
the dynamics and in designing policies.is to have results whose
accuracy and validity emerge from qualitative considerations rather
than from mathematical exercises. They argue that questions related to
system boundaries,correspondence between model structure and real
system, model reproduction of the system behavior etc are not really
permeable to conventional statistical procedures, especially when
dealing with large non-linear dynamic systems found in real life.
JW.Forrester (1967) argues that statistical techniques can be
inconclusive and misleading in System Dynamics modelling. However’ the
authors feel that judgmental methods can equally be inconclusive and
misleading if the analyst has not conducted an extensive environmental
study of the system with an enormous database to supplement his rather
%* Research scholar. Department of Mechanical Engineering, Indian
Institute of Technology, Bombay - 400 076, INDIA.
# Professor, Department of Mechanical Engineering, Indian Institute
of Technology, Bombay - 400 076, INDIA.
SYSTEM DYNAMICS '93 1
philosophical abstractions on causal structures and dynamic
consequences. Though failure to pass a statistical test need not be a
sufficient condition for rejecting a model it can surely be handy as a
confirmatory tool or a reminder to search for flaws in the model.
Legasto.A.A. (1980) admits that they can be useful for parameter
estimation and statement validation and Graham A.K (1980) gives
certain guide lines for statistical procedures especially in model
formulation and validation
In the following discussion some methods are presented which can
be judiciously used by the modeler for strengthening the validity and
representativeneas of hia model. A case study is conducted on blood
bank inventory management system which haa been an elusive problem for
OR/MS practitioners for the last few decades due to its highly complex
nature of interaction of operating variables.
VALIDATION OF A MODEL.
Shoukath Ali and Ramaswamy (1991) were the first to analyze the
problem from a System Dynamics point of view. Fig (1) shows the time
series plots of simulation runs of the model developed by them using
DYNAMO. The TABLE input corresponds to the actual data on patients”
demand for blood collected from a blood bank in Bombay city. PULSE
input is in the form of a sudden moderate hypothetical hike in
patients” demand on the 10th day for an additional quantity of 20
units of blood which lasts for 5 days. STEP input corresponds to a
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Fig 1. Time series plots of simulotion results
Tor
12 SYSTEM DYNAMICS '93.
ainy Assigned IlVentory remb Rote of Cross Matching of Blood
Rate of Release of Cross Match Qly
emrq Cross Match Release Quantity rremq
dout —Deley in OUTdating rptdb Rate of PatieriTs’ Demand for Blood
dta Deloy in Filling at Assigued inventory rbcol_—_-Rote of Blood COLlection
dfpr_Geloy in Filling Physician Requisitiins — rLrfb Rate of Bonk Requisition for Blood
demb Delay in Cross Matchiny of Blood ricn Rate of TRansfusio!
dremq Deloy in Release of Cruss Match Gly —slitgb SHorTaGe of Blood :
demr Deloy in Cross Match Release trem TRallsfusion ~ Cross Match ratio
outq —OUTdating Quantity uuiny — Undysigned I!Ventory
Fig 2. Acronyms used In the inodel
continued hike in demand for 20 units of blood from the 10th day
onwards. The acronyms used in the model are shown in Fig.2.
Keeping these limitations in mind, the blood bank model has been put
to tests for validity and representativeness. The data obtained from
the simulation run of the model using the TABLE input is used for this
purpose. The input to the model is the actual rate of patients’ demand
for blood (rptd estimated from data collected over a period of 1
year. The output data from the model on rate of blood collection
(rbeol) is compared with the actual rate of blood collection in the
blood bank. Fig (3) shows the actual rate of blood collection and
simulated rate of blood collection. The hypothesis that the mean value
of the simulated data x has come from a population (actual data) whose
mean is X and whose standard deviation is to he tested. Using test
with 5% fiducial limits it is found that the observed difference in
means could arise by chance in more than 5% trials. Thus the
significance of difference is not established. The Snedecor’s F test
passes the 10 % fiducial limits thus proving that model truly
represents the actual system. The simulated output of outdating
quantity is also compared in this manner. Table (I) gives the output
results of the statistical analysis carried out. The same procedure
may be adopted for comparing other variables such as shortage, cross
match release, transfusion rate etc. provided real life data on them
are available.
= gTEAMG uum susan
4.00 =~ = OUTOATING QUANTITY ACTUAL ee
Fig 3. Outdating quantity and rate of blood collection
SYSTEM DYNAMICS '93 13
CORRELATION AND ACCUMULATION PRINCIPLE |
Statistical relations can be helpful in enggesting areas to search |
for causal relations and statistical tests can suggest possible
errors. However statistical relations should not serve as substitutes
for causal relations and statistical criteria should not be considered
sufficient for refutation of a model. Correlation of variables can be
token as an example. Increase in patients’ demand for blood may result
ACTUAL RATE OF BLOOD COLLECTION and SIMULATED RATE OF BLOOD COLLECTION
Sample 1 Sample 2 Pooled
Sample Statistics: Number of Obs. 12 12 24
Average 26.1317 26.1783 27.155
Variance 18.8803 4.48378 11.687
Std. Deviation 4.34629 2. ae
Hedi. 28. 66.
Conf. Interval For Diff. in Means: 85 Percent
Cause Vars.) Sample 1 - Sample 2 -0.941761 4.84643 22 DF.
qual Vars.) Sample 1 - Sample 2 -2.0069 4.91357 16.9 D.F.
Conf. Interval for Ratio of Variances:
_ Sample 1 v Sample 2
Hypothesis Test for HO: Diff = 1.74 Computed t statistic = 0.152856
vs Alt: NE Sig. Level = 0.079905
at Alpha = 0.05 so do not reject HO.
COMPARISON OF SIMULATED OUTPUT OF OUTDATING QUANTITY AND ACTUAL OUTDATING
SAMPLE 1 - SIMULATED OUTDATING SAMPLE a ~ ACTUAL DATA
Sample 1 Sample 2
Sample Statistics: Number of Obs. 12 12
Average 2.43747 1.66667
Variance 0.731524 1.15152
Std. Deviation 0.865292 1.07309 i
2.4 |
Conf. Interval For Diff. in Means: Percent. i
(Equal Vars.) Sample 1 ~ Sample 2 £5: 0609231 1.59252 22 D.F.
(Unequal Yars.) Sample 1 - Sample 2 -0.0533005 1.5949 21.0 D.F.
Conf. Interval for Ratio of Variances: 0 — Percent
Sample 1 v Sample 2 i
Hypothesis Test for HO: Diff
vs Al
at Alpha
Computed t statistic = 1.94562
Big. Level = 0.0645626
so do not reject HO.
Table. Statistical resuits
14 SYSTEM DYNAMICS '93
|
in decrease in inventory level. This doesn’t mean that decrease in
inventory level of blood is the cause for increase in patients” demand
for blood. Thus these two variables are correlated in only one
direction. Therefore many System Dynamics modelers focus on generic
models i.e. they attempt to provide a general theory of the behavior
of a class of systems.
Accumulation principle can be used in developing causal
relationships in SD models. According to this theory :-
a) All dynamic behavior arises out of explicit flow accumulation
in the form of levels or stock variables.
b) Creates a lag between inflows and outflows and between inflows
and associated level variables. The phase lag decouples the
instantaneous value of a flow from its effect on other variables in a
nae Such decoupling accounts for all dynamic behavior in SD
models.
Dynamic behavior in Econometrics models arises through distributed
lag functions. For example in an Econometrics model let,
| Y, = BS *X, + BY X, yt BS eX 5 te. t BEX
FE BRY, 1 ‘
\ where,
| Y = collection of blood in units,
i x = desired change in attitude of people towards blood donation,
B, = lag weights.
This equation represents the relationship between Y,. the
dependant variable and xX, the independent variable. The parameters By
are the lag weights which are determined by fitting the variable X and
Y. The pattern of the lag weights is called lag pattern. Distributed
i lag functions represent a general scheme for correlating current
\ values of one variable with past values of another variable. (Bell.J.P |
| and Senge.P.M , 1980 ) |
| Even if an explanation is found for establishing. a relationship
between two variables the Econometrics model is not in a position to
explain why there is the presence of delays. For example there is
always a delay present between change in the attitude of people
towards blood donation and actual collection of blood. Any model
representing these variables is forced to give an explanation for this
H delay. In other words the model is forced to give a causal explanation .
why a delay occurs between x and Y and specify the nature of the
delay.A causal theory may provide an answer why Xe affects YX but
fails to explain why there is a delay present. The accumulation
| principle provides the answer and not the distributed lag functions.
In an SD model the change in attitude of blood donation might lead
to a flow of blood into the blood inventory The model might assume a
constant average collection time after which collection will be
delayed.
This shows that the accumulation principle forces an SD modeler to
provide a causal relationship for the dynamic behavior while the
distributed lag approach overrides this aspect and considers only the
SYSTEM DYNAMICS '93 15
correlation. Correlation approach can obscure errors in a model while
causal explanation provides more points of contact with reality and
makes corroboration or refutation more possible. However correlation
helps the modeler to establish the subjectively decided relationship
between variables. Adjacency matrix method in combination with
correlation analysis makes this job easier.
ADJACENCY MATRIX METHOD
Adjacency matrix method i. based on decomposition of the text
describing the system into a sequence of inferences. All the nouns and
adjective/noun forms in each inference are identified and inserted
into a matrix to facilitate selection of variables and polarized
relationships. (Camara.S.Antonio. 1991).
The steps involved are:-
1) Break down the description into a series of inferences by
looking for inference indicator words such as because,thus,then etc or
modal words auch as must.can,can not ete. This helps in decomposing a
large text into a discrete number of units.
2) Scan for vaniables in each reference by looking to nouns.
adjective/nouns and other combinations involving nouns.
3) Develop an adjacency matrix with the entries previously
identified. These matrices are denoted by A = (a, ,] and,
ayy = 1 if values of Ry depend on Ry and polarity is positive
a5 = 0 if Ry is not related to cay
ink) =-Lif %; depends on Ry and polarity is negative.
This pair wise analysis allows one to infer from the text all the
directed relationships and their polarity. Besides this brings out the
inconsistencies if any in the verbal description.
4) Translate the adjacency matrix into a causal diagram which is
then used for developing the computer model.
This synthetic procedure for identifying variables and polarized
relationships enhance the possibility of reaching a more consensual
influence diagram and thus a better SD model.
Fig 4. ADJACENCY MATRIX FOR VARIABLES
rbcol rptdb rrcmq rtrn outq shtgb uainv
rromq 0 ° - o 1 0 1
rtrn 1 0 -1 - -1 1 “f .
outa i o al o - -t 1
shtab 1 oO o 1 “1 0
uainy oe 0 0 1 1 ss i
16 SYSTEM DYNAMICS '93.
The variables thus identified describing the blood banking system
are inserted in a matrix as given below, Fig(4). The influence diagram
for the whole system can be easily drawn based on this table.
{Shoukath Ali and Ramaswamy .N, 1993).
Correlation does not represent the causal relationships between
variables but the extent to which two variables are interrelated.
However the causal relations attributed subjectively to the variables
in the adjacency matrix may be checked with the correlation matrix
(Fig 5). For example from the adjacency matrix it is seen that an
increase in rate of blood collection (rbcol causes an increase in
unassigned inventory, (uainy. The coefficient of correlation between
these two variables is 0.3982. Similarly relations between other
variables listed can be checked.
Fig 5. CORRELATION MATRIX - HODEL OUTPUT
RBCOL RETUB RRCHQ RTRN ouTe SITGB UAINY
RBCOL 1.0000 0.8994 0.0746 0.6994 0.7137 0.9778 0.3928
RPTDR 0.8994 1.0000 -0.4122 1.0000 -0.7783 0.9140 0.4953
RRCHQ)-0.0746 -0.4122 1.0000 -0.4122 0.6241 0.0202 0.8069
RTRN 0.8994 1.0000 -0.4122 1.0000 -0.7783 0.9140 -0.4953
ouTa 0.7137 -0.7783 0.6241 0.7783 1.6000 -9.6181 0.8511
SHTGB 0.9778 0.9140 -0.0202 0.9140 -0.6181 1.0000 0.2442
AINV. -0.3928 -0.4953 0.8069 -0.4953 0.6511 0.2442 1.0000
PARAMETER ESTIMATION
When parameters of a model are too aggregated to be set reliably
from available unaggregate data statistical techniques are used. i.e
restructure the model so that its parameters corresponds to observable
changing characteristics of the system.(Graham.A.K,1980). A few
plausible methods are presented here for arriving at the parameters of
the system and evaluating them.
Delay in outdating (dout) is one of the most important parameters
in the model. This represents the time elapsed between the collection
of blood from the donor’s body and the discarding of blood after it
decays. As per the standards specified by the American Association of
Blood Banks (AABB) the age of whole blood is 35 days under standard
preservation conditions. These standards are internationally accepted
and followed in India too. Thus a value of 35 is assigned to dout
When blood components are considered the value of dout will assume
SYSTEM DYNAMICS '93 17
values corresponding to their age.
Another parameter of importance is delay in cross match release of
blood (dremq which is the period elapsed between. cross matching of
blood for a particular patient for transfusion and returning of this
quantity without transfusion to the unassigned inventory or free
inventory. When this delay is more chances of outdating is more. dremq
is dependent on trnemsince,
Cross match release quantity
tenem: Cross match quantity
Thus it becomes necessary to determine both dremqand trnenfhe
data on average cross match release quantities and cross match
quantities were taken and trncmwas computed. Delphi method suggested
by Graham.A.K (1980) was used to determine the value of dremq as 2
days. The effect of varying drcmq on outdating of blood was also
studied by the authors. (Shoukath Ali and Ramaswamy. 1993).
Other important parameters of the system are dfpr, the delay in
filling physician requisitions and dfa, delay in filling requisitions
from assigned inventory. As no blood bank. keeps records of these
information, expert opinion was sought on the values of these
parameters and average values fixed as 0.125 days. Thus collection of
real system data and estimating the mean yields model parameters
reliably. Statistical tests like t test may also be conducted to
check if the values have really come from the real system.
SYSTEM TRAJECTORIES
System Trajectories can be drawn for studying the response of the
system. They can be of two types namely, time series plots and phase
plane plots.Plots of level variables against each other with time as
parameter are called system trajectories or phase plane plots.
(Aburdene, 1982). The Fig (6) shows phase plane plots obtained by
plotting rate of blood collection, x(t) against level variables.
x(t) i.e. unassigned inventory level (uainy, and outdating quantity
(outy . All the above variables are time response outputs from the
model developed for blood bank inventory system. This process is
repeated for all possible initial conditions of x, (0) and x,(0).
A second way is by iso-cline method which is based on slope of the
trajectories in the phase plane. The ratio ( DX, 73%, ) represents the
slope of the trajectories in Ky o% plane. Aided by equation for
(dx/dx,) the regions of (Xy-%y) planes that have the same values for
the trajectory slope are determined. With constant values attached to
(dx,/dx, ) the slope becomes an algebraic equation of the form ,
a= f( x: X,) or the equation gives loci for states of equal
slope. :
Let dx,/ dt = 2 and dx, / dt = 0.25 * (x4-X_)
18 SYSTEM DYNAMICS '93
I
i
pron
Equilibrium conditions are X4=%q=0. Thus -trajectories should
approach the origin of the (Xy> Xp) plane. This can be shown by
forming the following equations:-
aig/at ay %
dx. /dt - dx -0.25 x -x
Letting,
s = dx, / ax, and solving for Xp results in
2
_ 0.25x,
2° e+
where,
8 = slope of the trajectory.
This type of sketches can be used to quantitatively compare the
results either with analytic solutions or with solutions obtained by
semen ahs OATES
simulation. “
—— step eu --= PULse mpuT
440.00 9 ums oF sLooD Ta BSE inbur $9 7 unns oF L000 ca Ae oY
140.00 2.00
a
gen
g 2000
Fa
2
1900
3
5
ooo erremnrrarsierininiounberiuatninenittian imine
tots
8.00" "30.00.3300" e000. 63.00 7000. 75.00
RATE OF BLOOD COLLECTION
Fig.6 Phase plane plots
NUMERIC PERFORMANCE MFASURES
Computation of certain numerical values are useful to know how
well the system has performed during the simulation run. Single
performance measures (PM) or multiple performance measures (MPM) may
be calculated from the model output and examined. Coyle.R.G (1980)
discusses various such measures like cumulative values, point values
etc. The values may be obtained either by introducing macros into the
SYSTEM DYNAMICS '93 19
seuesetethommrenmeneensenrnn SDS Saeco
model or by external computation.
The cumulative value of transfused quantity is thus obtained as
9502.9 from the model run using a macro in the following form:
1 ocrtrnk& crtrm j dt * Crtrn. jk)
nm crtrn= rptdb
a maxui.k max Cuainyv.k, omaxui.k)
1 omaxui.k omaxuk. j dt * Cmaxui. j omaxui. jk dt?
n omaxui= 0
where, crtrn= cumulative rate of transfusion
rtrn = rate of transfusion
rptdb= rate of patients” demand for blood and
uainv= unassigned inventory.
From the actual data collected the value of total quantity of
blood transfused is found to be 9502 units. Thus the method provides a
validation scheme as well.
RESULTS AND CONCLUSION
The methods described are mostly statistical in nature, though,
some are outside the conventional definition. The use of these
techniques need not necessarily bring out positive results always. But
an attempt at changing the parameter values and averaging constants is
worth trying. Even small changes in the model structure may bring out
satisfactory results. The analysis can be performed using any
available statistical packages such as STATGRAPHICS, SPSS etc.
REFERENCES
ABURDENE.M.F (1988), Coputer Simulation of Dynamic Systems ,
Wm.C.Brown Publishers, Iowa.
BELL.J.A and SENGE.P.M (1980), Methods for enhancing refutability
in System Dynamics Modeling, TIMS Studies in Management
Sciences, Vol.14,61-73.
CAMARA.S.ANTONIO (1991), Formulation of System Dynamics models,
Proceedings of the 1991 Int.S.D. Conferenece.
Conference82-87.
COYLE.R.G (1978), Management System Dynamics,
John Wiley.
FORRESTER.J.W (1967) Industrial. Dynamics,
M.I.T.Press, Cambridge, Massachusetts.
GRAHAM.A.K (1980) Parameter Estimation in System Dynamics
modeling, TIMS Studies in Management Sciences,
Vol.14, 125-142.
LEGASTO.A.A and MACIARIELLO.J.A (1980), System Dynamies : A Critical
Reciew.TIMS Studies in Management Sciences,
Vol.14, 23-44.
SHOUKATH ALI.K and RAMASWAMY.N (1991), Setting Inventory levels
for a centrally located blood bank of a metropolis - A
Simulation Approach, Proceedings of 1991 Int S D Conference
SHOUKATH ALI.K and RAMASWAMY.N (1993), Simulation of Blood bank
Inventory Management Systems : System Dynamics Approach,
Unpublished paver.
20 SYSTEM DYNAMICS '93
