Using Difference Equation to Model
Discrete-time Behavior in System Dynamics
Modeling
Reza Hesan, Aminch Ghorbani, and Virginia Dignum
University of Technology,
Faculty of Technology, Policy and Management,
Delft, The Netherlands
Abstract. In
been used as the basic mathematical operator. Using difference equa-
tem dynamics models instead of differential equation,
modeling, di ions have
tion to build s;
behavior. In this paper we explain how we can use differen
to build system dynamics models. We illustrate the use of this
through its application to a case study in supply chain management.
1 Introduction
mulation is accepted as a powerful tool that
es where
In the field of social science
helps rese
practical experiments are not fe:
and tool we select to model a sys
the simulation may vary
Dealing with time is one of the main is
of, before selecting a tool for simulation. Some researchers s
ial equation-
archers to get more insight into the system, especially in cas
ble. However, depending on which approach
em, the quantitative and qualitative results of
ues that every modeler should think
continuous-time systems and therefore use differen
stem. In the contrary, other
screte time systems. Therefore, they select disc
System dynamics modeling (SDM) takes a continuous-time approa
models with differential equations. While flows that get in or out
of stocks can be represented continuously or in discrete points of time [14],
system dynamic modelers argue that it is an “acceptable approximation” to
consider individual items as continuous streams that can be divided infinitely
[14]. For instance, in a organization, people are individuals and are hired in
simulate a
and
constru
ste manner, but system dynamic modelers assume that the flows of people
Besides the approximation that is caused by assuming discrete flows as con-
stead
ina
tinuous streams, another source of approximation is using average delay i
of pure de i stem dynamic modelers assume that s
post office with a large number of letters, all the letter are not delivered at once
and there is a distribution around the average delivery time, it is an acceptable
to model such cases.
approximation to use average dela
System dynamics modeling is aimed to study long term behavior of systems
at the macro level. This modeling approach is commonly used to study large
organization or global phenomena. Therefore, these approximations are accept-
able. However, the concepts of stock and flow have the potential to be used for
studying systems at the micro level and to explore behavior of small organiza-
tions or phenomena (e.g, hiring system in a small organization, a supply chain
tructed by a few people). However, the problem is that the main
Gia teristi kinds of system is that their flows are dis
most of the time there are not many items in delay. Therefore, using differ
equation to construct these kinds of system can lead to inaccurate quantitative
results. In order to avoid these inappropriate approximation, we propose using
difference equations instead of differential equations as the basic mathematical
operator that determines the relation between a stock and its flows. This method
of constructing system dynamics models allows us to model discrete flows and
pure delay.
The structure of this paper is as follows. In section 2, we look into difference
equation and differential equation modeling. In section 3 we present a example
e which we will be using to illustrate difference between the quantitative
esult of both approaches. In section 4, we propose our method. In section 5,
we rebuild the working example with the help of the method. In section 6, we
compare the quantitative results of the working example. In section 7, we will
finish with some concluding remarks.
ntinuous and
ential
2 Background
2.1 Difference Equations and Differential Equations
Control theory classifies dynamical systems, whose state varies during time,
into two subdivisions: continuous-time (CT) dynamical system and discrete-
time(DT) dynamical system. In CT-systems, the state of the system changes
after every infinite short interval of time while in DT-systems the state of the
at distinct points in time.
Differential equation is the basic operator for modeling continuous time
tems. A simple population model with the growth rate r in the CT approach is
modeled by the differential equation(integral) as following:
i r x p(t)dt (1)
Difference equation is the basic operator for modeling discrete time systems.
The simple population model that we already represent with differential equation
can be modeled by a difference equation in discrete time approach as following.
Pn = (L+1) X Pra (2)
Population in time n is equal to population in previous time pp_1 plus r* Pp.
Difference equation as the main operator of discrete-time modeling has been
used recently, especially after developing digital computers(10].
2.2 Discrete-time modeling in Literature
tem dynamics literature rarely addr
size that instead of first order systems in continuous time modeling which can
not gencrate oscillated behavior, first order systems in discrect-time approac
late or even generate chaotic behavior. (2] points out that the equations
which construct a system dynamics model can be either differential equations or
difference equations. s that although a system dynamics model can be
continuous, discrete or hybrid, in practice, SD takes discrete systems as continu-
ous system since contimuous-discrete hybrid model can be cumbersome to build
and analyze. [3,4] point out that by replacing dt of a system dynamics model
with 1 we can have a discrete-time version of system dynami
mulation literature addresses dif-
stems with the help of differenc
some researches use Z-transform
ete-time modeling. [14] empha-
can o'
equations. Inspiring from control theory studic
to build and analyze discrete-time models. [5] analyze a discrete-time model of
a four echelon supply chain system with the help of Z-transform. [7] study the
dynamic stability of a vendor managed inventory supply chain by constructing
a discrete transfer function of the system.
Besides the Z-transform approach to study dis
searcher use mathematical representations and state space approach
ematical representations of a system is mostly used when the
structed by one equation. [6] develop a mathematical model of a
tion model called Logistic model. [1] develop a discret
model. [9] study pattern formation of the discrete-time predator prey model.
The state-space approach is a mathematical representation which is used where
constructed by a few number of difference equations. [13] applied
ate-space approach to study bullwhip effect in a simplified supply chain. [8]
rete-time
some re-
Math-
simple popula-
Discrete event simulations (DES) may also be considered as dit
methods, as they are suitable for modeling the systems in which variables change
in discrete-times [12]. DES vi rete sequence of events in time.
In other words, DES is vent-based modeling approach that is different from
the other mentioned approaches that are equation-based.
So far, we mentioned changing dt of system dynamics modeling, z-transform,
mathematical and state-space approach as the main approaches for discrete-time
modeling. We will later explain that although the first approach: changing dt = 1
in system dynamics modeling, seems a acceptable way to model a system with
iscrete time, it may result in inaccurate behavior of system. Using Z-transform
and mathematical representation of discrete time system are suitable for linear
an
stem. However, constructing system dynamics models with difference equa-
tions can be used for both linear and nonlinear system. Besides, our proposed
method takes the advantage of the diagramming aspect of the formal system dy-
modeling which can be more powerful than Z-transform or other math-
ematical approaches for part
clients.
nam:
cipatory modeling and for giving insights to the
3 Working Example
As a running example, we take a one-echelon supply chain adopted from beer
game distribution [14,15] to present the innovative aspects of our method. The
reason for selecting this example is due to the fact that in the beer game dis
tribution model we study the micro behavior of a small group of individuals.
Therefore, we can show how difference equation can benefit the result of simu-
lation. Figure 1 depicts the stock and flow diagram of this case.
Description of System The system under study is a typical cascade production-
distribution system consisting of one retailer. Customer demand is exogenous
and retailer must
If there is insufficient product in stock, the retailer will keep surplus order in
backlog until delivery can be made. All the retailer’s orders will be fully satisfied
after delay (4 weeks) and there is no limitation for the wholesaler to supply the
retailer.
supply the amount of product requested by the customer.
Order Policy The retailer tries to keep the level of inventory at the desired level
(1.5 times of order received). Every order is determined by the number of orders
that the retailer has received and two adjustments (correction) for inventory and
supply line.
Order PlacedRate = MAX (Order ReceivedRate + DesEf fectiveInventory
Correction + DesOrder PlacedCorrection, 0)
(3)
DesE f fectiveInventoryCorrection = (DesiredInventory — Ef fective
Inventory) /DelTime «)
DesiredInventory = Order ReceivedRate x InventoryCoverage
Ef fectiveInventory = Inventory — Backlog
DesOrder PlacedCorrection = WeightOnSupplyLine x (DesiredOrderPlac
ed — Order Placed) /DelTime
DesiredOrder Placed = Order ReceivedRate x DelTime
OrderPlaced
Ld
or
OrderfulfillmentTime
Fig. 1. Beer Game Example
Shipment and Demand Policy The desired shi rate is the lation
of the backlog and the order that the retailer has received. Due to limitation in
inventory, it is not always possible to satisfy desire shipment. Shipment: rate is
determined by the minimum of desired shipment rate and inventory. It means if
inventory is lower than the desires shipment rate, the retailer will support a part
of order and backlog equal to the level of inventory. Otherwise, he can satisfy all
the order and backlog.
In order to put the model in the steady-state, we set Order ReceivedRate to
4 and InventoryCoverage to 1.5. The initial amount of inventory is 6 equal to
DesiredInventory. The initial amount of Order Placed would be 16. At time 4
we increase Order ReceivedRate to 8 in order to analyze the behavior of model.
Shi tRatemin(DesiredShi tRate, (Inventory/dt) + AcquisitionRate)
4 System Dynamics Modeling with Difference Equation
Although, differential equation is traditionally used as the mathematical opera-
s and flows, difference equation
is also compatible with the concept of stock and flow [11]. Therefore, it can also
be used as the basic operator of SDM.
Using differential equation (Formula 7) in order to study the micro level
behavior of systems or the short term behavior of small organization in which
flows are not changed in every instance of time, renders the model far from
rate quantitative results of the
instance, when modeling the process of ‘making orders by retailers’, if there are
many retailers in the model, assuming that an order takes place every instance
tion that determines the relation between stoc
reality and leads to in ina imulation. For
of time is reasonable. However, when there is only one retailer in the system (or
a limited number of them), making the assumption that an order is taking place
in every instance of time is unrealistic. Therefore, for such cases, using difference
equation is a more reasonable option.
Time delays often play an important role in the dynamics
we model delay 1 in determining the behavior of models.
Using average delay (Formula 8) instead of pure delay to model a
the micro level or study short term behavior of a small system or organization
can bring some inaccuracy in quantitative result of simulation. For instance, in
very crv
em at
our working example, as we are modeling the behavior of a individual retailer,
there is no distribution of delay time. The retailer will receive his product after
a constant delay of time. Therefore, it would not be appropriate to use average
delay in these kind of cases as it is far from reality.
stock(t) = [ (inflow(t) — out flow(t)) dt + stock(0) 7)
out flow(t) = ei
In order to avoid the mentioned inappropriate approximation, we propose to
use difference equations to construct system dynamics models. In this approach,
stock(t)
D
the amount of stock is calculated by Formula 9 which calculates the amount of
stock based on the inflow and the outflow and the previous amount of stock in
every discrete point of time. In order to model D step time pure delay depicted
in Formula 10, the amount of delayed outflow is equal to amount of inflow in
time t — D. The amount of the stock that is linked to the delayed outflow is
equal to summation of all inflow which are in queue to become outflow.
stock{t] = stock|t — 1] + [inflow — out flow] (9)
out flow(t) = inflow|t — D|
e
stock{t] = }~ inflow(t)
t=t—-D
(10)
Although, in practi
culate differential equation with the help of Rungg Kutta or Euler numerical
method, it does not mean that we can change a system dynamics model con-
all simulation software use difference equation to cal-
to 1. Due to the fact that changing the sequence between events in discrete time
modeling changes the numerical result of models, setting dt = 1 may result in
model as we do not consider the
‘em dynamic
sequence between events during the steps of time. Even if we set dt to 1 in for-
mal system dynami annot model pure delay the numerical
result of our model would be different from the difference equation version of
that model.
We will discuss some Characte:
formal SDM next.
s of proposed method in comparison with
4.1 Difference Equation Approach Characteristics
Besides the discrete flows and pure delays that make our model closer to reality,
using difference equation give modelers some extra opportunity to build more
accurate and flexible models.
Logical Statements
system dynamics models that are constructed by differential equation, logical
‘ ch as if...then...else bring sharp discontinuities in the model and
using them is an issue of debate [14]. However, with our prope
take advantage of logical statements. Using logical statements makes our model
more flexible, especially when we want to model decision making proc:
part of the system.
ed method, we can
asa
Memory
ume dt as
ystem dynamics models constructed using differential equations
an infinite short step of time cannot represent the notion of previous time for our
models. However, in reality, the state of system in a previous time is taken into
account to reach a decision. In our working example, since the backlog presents
the accumulation value of surplus order during the simulation, we cannot deter-
mine how much surplus order has been added to backlog during every step of
time. In order to model the decision making process of a retailer who will order
whatever product he could not have supplied during the previous time in his
next order, we need to know how much surplus was added in previous step of
time. SDM constructed by difference equation gives us this opportunity to have
the state of system in every time step to make it possible for us to build more
flexible models.
Real Data
commonly conducted in discrete points of
profit, and balance sheet
of a corporation are booked every month or year. Using these data as inputs for a
Data gathering from social
time. For instance, the information about the income.
simulation or reference to examine the behavior of simulation is more compatible
by simulating a model in discrete time manner.
In the next section, we will rebuild our working example with the proposed
method.
5 Working Example Constructed by Difference Equation
In Section 3, we described our working example which has been developed with
differential equation. In this section, we will rebuild this model with the help
of difference equation and we will compare the quantitative results of both ap-
proaches. In order to apply this new approach, we developed software in Python
programming language. As in most SD tools, this software supports graphical
definition of equations.
Fig. 2. System Dynamics Supply Chain Example constructed by difference equation
The order, shipment and demand policy of this new model is the same as
the differential equation based model. The only difference is the stock and flow
relation and delay construction. Since inventory and backlog are both stocks that
are not used to model delay, we determine the mathematical relation between
these stocks and their flows using Formula 9.
To model the delay between Order PlacedRate andOrder Ful fillment Ra
Formula 9 is used to construct pure delay, depicted in Formula 11:
Order Ful aig = Order Placed Rate(t — 4]
1
Order Placed{t] = = Order PlacedRate(t] (11)
t=t-d
Besides defining mathematical operations tructing a model, another
sue that is important to determine, ce between the events in every
step’ of time. Denending-on how we define the sequence between evenis during
the time steps, the quantitative results of our model will change. For instance, in
our case we have three main events: placing new order, shipping and backlogging,
receiving previous order. How we arrange the sequence between these events will
sult in different behaviors in the system. The final issue that must be considered
is about the steady state of the state of the model must be
determined based on the arrangement between events.
6 Results Comparison
We assume that the retailer at the beginning of the week will receive their pre-
vious order in the supply line. Then, he will calculate the order that needs to be
placed and will satisfy costumer’s order by shipping and will adjust the backlog.
In order to put ate, we set the initial value of inventory to
2 based on the assumption that a retailer will ship all Order Received during the
week and what will be left in inventory would be half of Order Received which
is equal to 2.
tem in the steady
— Difference Approach
— Differential Approach
0 ten
1 4 7 10131619222528313437 4043 4649
Fig. 3. Inventory in both approaches
Figure 3 shows the difference between the level of Inventory in both ap-
proaches. As it is depicted, inventory in the first approach shows a smooth
oscillation while in the second approach there is no sign of oscillation.
Examining different W eightonSupplyLine in both approaches shows the fact
that in the differential approach, the more WeightonSupplyLine gets close to
1, the more the system presents behavior with lower oscillation. While in the
difference approach the more WeightonSupplyLine closer to 0.5, the more the
system shows lower oscillation. Figure 4 and Figure 5 present the level of inven-
tory for 3 different values of WeightonSupplyLine parameter of the differential
and difference approach models.
. Y
: V/
fa
/ —s
| Vg
odd.
23.5 7 9 111315171921232527293133 3537 3941434547
Fig. 4. Inventory with different values for WeightonSupplyLine in the
approach
- L\
\
\ =
Lo =
7 =
=
1 4 7 1013 16 19 22 25 28 31 34 37 40 43 46 49 5255
Fig. 5. Inventory with different values for WeightonSupplyLine in the difference ap-
proach
Comparing the results of both approaches indicates that with the same
WeightonSupplyLine setting, in first approach the retailer is placing order more
than what he needs to adjust the inventory and therefore it causes oscillation in
inventory behavior. However, in the second approach the oscillation in inventory
behavior is not as strong as it is presented in the differential approach which
ence of delay and the amount
y more than the retailer in the differential approach.
rve qualitative differences
For example.
means that the retailer take in to account the e
of product in del:
Besides the different quantitative results, we obs
scially when defining hypothe
ed in Figure 3, in the difference equation model with the WeightonSupplyLine
the hypothesis
e of ignoring
for simulation:
pi
equal to 0.5, inventory does not oscillate. This fact can challenges
that oscillation in the behavior of beer game distribution is b
the amount of product in supply line.
7 Conclusion
In this paper, we proposed constructing SDM models with the help of differe:
equations instead of differential equations. We illustrated this new approa
applying it to a supply chain system.
In SDM, the mathematical relationship between stocks and flows is commonly
determined by the differential equation. However,the stock and flow concept is
also compatible with difference equation and therefore, we can use difference
equation as the basic operator of SDM which leads v
titative result where we study the micro level behavior of a system or small
organization.
The proposed approach contributes to SDM in several aspec
vides the opportunity to apply the SDM concept at micro level
individual activities change the flows of systems in discrete points in time. Sec-
ondly, it can result in more accurate quantitative result for cases that are not
large enough to a
ties to use logical statements and memory as explained in Section 4, enhance
our ability to model complex s
Another contribution of thi:
to more accurate quan-
Firstly, it pro-
tems where
ume their flows as continuous streams. Thirdly, opportuni-
ems.
approach is that since we are constructing
discrete-time models with the stock and flow concepts instead of using Z-transform
or mathematical representations, it will contribute to the field of disc
and allow them to model and analyze nonlinear discrete-time models.
One final contribution of our proposed method is that since our proposed
method and ete event simulation both study the behavior of systen
iscrete points in time and use queues to model
to merge the system dynamics approach with the discrete event modeling. In
ems, it may be p
rete event
mulation, flows of entities that get through the proce:
termined by random numbers which means that we are dealing with pa
entities. However, by merging both concepts, we can develop more deterministic
to be proved as a reliable approac
In order to use the ability of thi
systems a possible option for future work can be merging this method with agent
behavior for the flows of entities so that they can be effected by the state of
system due to feedb:
‘As this method is proposed for the first time, it needs more evaluation process
for constructing s
approach for studying the micro behavior of
tem dynamics models.
based modeling.
References
1.
N
-
x
ne
©
. P. Cvitanovi
L. J. Allen. Some discrete-time epidemic models. Mathematical biosciences,
124(1):83-105, 1994.
. Y, Barlas. System dynamics: systemic feedback modeling for policy analysis. SYS-
TEM, 1:59, 2007.
Y. Barlas and B. Gunduz. Demand forecasting and sharing strategies to reduce
fluctuations and the bullwhip effect in supply chains. Journal of the Operational
Research Society, 62(3):458-473, 2011.
Y. Barlas and M. G. Ozevin, Testing the decision rules used in stock manage-
ment models. In Proceedings of the 19 th International Conference of the System
Dynamics Society 2001, 2001.
J. F. Burns and B. Sivazlian. Dynamic analysis of multi-echelon supply systems
Computers & Industrial Engineering, 2(4):181-193, 1978.
. R. Artuso, R. Mainieri, G. Tanner, G. Vattay, and N. Whelan.
Chaos: classical and quantum. ChaosBook. org (Niels Bohr Institute, Copenhagen
2005), 2005.
S. M. Disney and D. R. Towill. A discrete transfer function model to determine
the dynamic stability of a vendor managed inventory supply chain. International
Journal of Production Research, 40(1):179-204, 2002.
C. 8. Lalwani, $. M. Disney, and D. R. Towill. Controllable, o
ble state space ions of a lized order-up-to policy. International
journal of production economics, 101(1):172-184, 2006.
M. G. Neubert, M. Kot, and M. A. Lewis. Dispersal and pattern formation in
a discret predator-prey model. Theoretical F Biology, 48(1):7-43,
servable and sta-
Hall Englewood Cliffs, NJ, 1983.
itz and M. Mrotzek. The basics of system dynamics:
uous modelling of time. In Proceedings of the 26th International Conference of the
System Dynamics Society], System Dynamic Society, Wiley-Blackwell (July 2008),
2008.
jiscrete vs. contin-
. O. Ozgiin and Y. Barlas. Discrete vs. continuous simulation: When does it mat-
ter. In Proceedings of the 27th International Conference of The System Dynamics
Society, (06), pages 1-22, 2009.
. C, Papanagnou and G. Halikias. A state-space approach for analysing the bullwhip
effect in supply chains. Proceedings of ICTA, 5:79-84, 2005.
. J. Sterman. Business dynamics. Irwin-McGraw-Hill, 2000.
. J. D. Sterman. Modeling agerial behavior: Mi i of feedback in a
dynamic decision making experiment. Management science, 35(3):321-339, 1989.
Fertility Target 20 Years 50 Years 100 Years
‘Top Down Disagg. |Top Down Disagg. |Top Down Disagg.
No decrease 21.3 22.0) 41.9 47.0) 55.5 84.2)
10% (of 2001-2011) 20.9 21.5) 40.1 45.2, 50.3 78.0)
50% (of 2001-2011) 19.2 19.8} 33.2 38.0] 30.7 55.1)
100% (of 2001-2011) 17.1) 17.7] 24.9 29.4] 9.2 30.3}
Table 5 Population projection sensitivity in terms of growth rates for top down and disaggregated models
for varying fertility targets
Mortality Change 20 Years 50 Years 100 Years
‘Top Down Disagg. |Top Down Disagg. |Top Down Disagg.
5% Down 20.0 20.6 35.3 40.1 34.6 59.7]
10% Down 20.7 21.3 37.4 42.4) 38.7 64.5]
Table 6 Population projection sensitivity in terms of growth rates for top down and disaggregated models
for varying mortality targets
3.2 Sensitivity analysis
As illustrated in Figure 6, based on UN estimates, India’s fertility rate is expected to decline by 6%
in the next 10 years. Fertility in 12 states has gone below the replacement level of 2.1 (Guilmoto,
2015) but the backward states are still significant laggards. Fertility decline in the decade
preceding the 2011 census has been used as a benchmark for targeted fertility decline over the next
decade. Four differing sensitivity analysis scenarios used: 0% fertility decline compared to last
decade (2001-2011); 10% fertility decline; 50% fertility decline and a 100% fertility decline.
Observations from the fertility sensitivity charts (Figure 9 and Figure 10):
1) Atthe country level, using aggregated state level data, Figure 9 suggests that a glide path of
even a 50% of targeted fertility decline over the next decade is not enough for India’s
population curve to stabilize from an exponentially increasing function. But 50% decline in
top down estimates is enough for India’s population curve to stabilize. India’s top down
population estimates are more sensitive to fertility changes than population estimates
based on aggregated data.
2) Population trajectories are quite insensitive to mortality changes (Figure 10)
Fertility sensitivity analysis information is more revealing at the state level.
13
BR
Population trajectories of backward and most populous states are relatively insensitive to
even a 50% (of targeted) fertility decline in a decade. In comparison, high HDI states
(Dreze, 2012) are on a glide path for a secular decline for the base case sensitivity decline
of 50% (of target) over the next ten years.
Even with a 50% fertility (of targeted) decline, population growth rates of the most
populous states is significant.
On the other hand, with a base case of just 50% fertility decline over ten years, population
trajectories of the high HDI states are peaking out (in case of Tamil Nadu, the state has
already peaked out) in the next 50 years. High HDI states like Tamil Nadu could have a
negative population growth over the longer term. The state’s fertility rate has been on a
decline for some time now. (Savitri, 1996).
Itis to be noted that total fertility rates for Bihar, Rajasthan and Madhya Pradesh declined
by 0.6% points in the period 2006 to 2011 (Kawadia et al., Aug 2014). While the states
have been economically progressive, the ask rate for a more than 50% (of targeted) decline
in fertility levels cannot be glossed over.
am)
gS
&.
Top Down Fertility Decline Target
(Sensitivity)
Population (
be ee
Bib Bs
Bees
8sss
HAOFOHHOAKDHTOKNHAHHHYTD HAD
SASARBRFLSHRRSESRRBRES
Years
—0% —10% 50% —100%
Figure 9 Population projection sensitivity in terms of growth rates for top down model for varying fertility
target decline relative to the decline in last decade
14
Disaggregated Fertility Decline Target
(Sensitivity)
_. 2600
€
Population (M
—0% —10% 50% ——100%
Figure 10 Population projection sensitivity in terms of growth rates for disaggregated model for varying
fertility target decline relative to the decline in last decade
REFERENCES
Christophe Z Guilmoto, S Irudaya Rajan, Economic & Political Weekly. Fertility at the district
level in India, Lessons from the 2011 census. June 2013.
Debraj Ray, Development Economics, Pages 60-61. 1998.
Estimates of Mortality Indicators, Census of India 2011, Registrar General of India, 2012.
Government of India, Press Information Bureau, Ministry of Statistics & Programme
Implementation, http://pib.nic.in/newsite/PrintRelease.aspx?relid=123563, July, 2015.
Jean Dréze and Reetika Khera, Economic & Political Weekly. Regional patterns of human and
child deprivation in India. September 2012.
John D Sterman, Systems thinking and modelling for a complex world, Business Dynamics, page
108. 2000.
Jorgen Randers, Elements of the System Dynamics Method, MIT Press, Page 35. 1980.
Juan Esteban Montero, Implication of the Jensen's Inequality for System Dynamic Simulations:
Application to a Plug & Play Refinery Supply Chain Model, MS Thesis, MIT. February 2014.
415:
Kanhya L. Gupta, Intemational Economic Review, Vol 12/2. Aggregation bias in linear economic
models. June 1971.
Kjersti-Gro Lindquist, Statistics Norway, The importance of disaggregation in Economic
Modelling. June 1999.
Methods of population projection by sex and age, Manual III, UN, 1956
Michael Lipka, PEW Research Center, Muslims & Islam: Key findings in the US and around the
world. Feb 2017.
R Savitri, Economic & Political Weekly, Fertility Rate Decline in TamilNadu. July 1994.
Report of the Technical Group on Population Projections constituted by the National Commission
on Population, May 2006
Robert Cassen, Leela Visaria, Page 1.Twenty First Century India, Tim Dyson, Page 1. 2004.
Sheena Sara Philips, Economic & Political Weekly. Shedding the BIMA RU tag, Ganesh Kawadia.
August 2014.
Shreenivas Kunte, Om Damani. Population projection for India— A System Dynamics approach.
System Dynamics Conference. 2015.
Stacey Tevlin, Karl Whelan, Explaining the Investment Boom of the 1990s, Journal of Money,
Credit and Banking. Feb 2003.
Tim Dyson, Robert Cassen, Leela Visaria. Twenty First Century India, Page 75. 2004.
United Nations, Department of Economic and Social Affairs, Population Division. 2015.
World Population Prospects: The 2015 Revision, Methodology of the United Nations
Population. 2015.
Vinita Sharma, Economic & Political Weekly. Are BIMARU states still Bimaru? May 2015.
Weisstein, Eric W. "Jensen's Inequality." From MathW orld--A Wolfram Web Resource.
http://mathworld.wolfram.com/] ensensInequality html
16
17
