24th Mternational Conference of the System Dynamics Society
July 23-27, 2006, Nijmegen, The Netherlands
Ed: . GroBler, E.A.J.A. Rouwette, R.S. Langer, J.1. 2, JM. Yanni
Extended Sensitivity Analysis of Parameters
and Structure in System Dynamics
Models — Some Case Study
Elzbieta Kasperska, Elwira Mateja-Losa
Institute of Mathematics
Silesian University of Technology
Kaszubska 23, 44-100 Gliwice, Poland
e-mail: {e.kasperska, e.mateja}@polsl.pl
Abstract
The problem of sensitivity analysis of parameters and structures in System Dynamics
models is rather new for field modelers. The possibilities of packet COSMIC and COSMOS
allows to apply extended sensitivity analysis not only of parameters of the simulation
models but structures of these models too.
1 Introduction
Analysis of the changing of the values of the parameters in simulation model type System
Dynamics [1-6, 14] is one of the fundamental stage of building such a model. Detection of
set of, so called, sensitive parameters of the model is not easy in case of complex, nonlinear,
dynamical and multilevel systems. First prof. Coyle has undertaken this problem and
proposed the tool for analysis (see [1-4]). Authors have applied the idea of prof. Coyle
and extended the evolutionary aspect of such analysis, specially in the context of process
of learning (sce {13]). In paper some new results with using COSMIC and COSMOS are
presented and some conclusions about subject of consideration are drown.
2 Extended Sensitivity Analysis of Parameters and Structure —
Case Study
Three different kinds of analysis were applying to the model DYNBALANCE (3-1-IH).
First analysis was, so called, Direct Optimization, that allows to choose the optimal value
E. Kasperska, E. Mateja-Losa — Extended Sensitivity Analysis. . . 2
PRODUCTION Sources of raw material
ehn 4] Techn 2] Tchn 3 Source] Sourc?2 | Soure3
STORADGE *)
Invcost
SELLING
OBJECTIVE FUNCTION
MIN(cost) | MAX(profit)
to TOT
PRICES
Figure 1: Simplified idea of the structure of model DYNBALANCE (3-1-III)
of searching parameters, which optimized the objective function. Second analysis was:
Base Vector Analysis that allow to test the sensitivity of parameters in model in context
to improving the value of objective function. Third analysis was, so called, Simplification,
which try to find simpler model structure.
The model DYNBALANCE (3-1-III) was created by Kasperska and is one from the family
of models type System Dynamics, associated with problem of balancing of raw materials
and production (see [9-13]. The general structure of model, in illustrative form the reader
can see on Figure 1.
The model contains some levels, which accumulate the production and store and such
information like: losse of profit, cost of raw materials, cost of production. The objective
function consists of the total cost of production and ” penalty” factor (accumulation of
the cost of the inventoring of product, accumulation of lose of profit). The ” penalty” has
weights factors, which modelled the preferences about contributing of items of the penalty
to final value. It is also possibly to apply different objective function: profit from sell and
the aim will be searching to maximize it. In next section the results of some experiments
will be presented in undertaken context of consideration.
3 Some Results of Sensivity Analysis on Model
DYNBALANCE (3-1-III)
Below authors present the main results of experiments, mentioned three kinds: Direct
Optimization, Base Vector, Simplification.
E. Kasperska, E. Mateja-Losa — Extended Sensitivity Analysis. . . 3
Table 1: The main results of experiment 1
parameters | final value | original value | lower limit } upper limit
ucrl 50.000 100.000 50.000 100.000
ucr2 50.000 50.000 50.000 100.000
ucr3, 10.000 10.000 10.000 100.000
tchnl 15.280 20.000 0.000 40.000
tchn2 39.866 10.000 0.000 40.000
tchn3 39.999 20.000 0.000 40.000
ucprl 200.000 500.000 200.000 500.000
ucpr2 200.000 500.000 200.000 500.000
ucpr3 100.000 100.000 100.000 700.000
Initial value of ” fob”: 0.29128E + 08
Final value of ” fob”: 0.2098E + 08
Final value of inventory: 260.60
Final value of price: 1118.1
Final value of penalty: 17.5640E + 06
Final value of demand: 253.50
Experiment 1
The set of searching parameters was rather large. In optimization dialog we assume:
e number of iteration: 100
e value of step multiplier: 0,3
e length of simulation: 100 weeks.
After 100 iteration we have obtained the results presented in Table 1.
It will be interesting to compare the results of this Direct Optimization with next exper-
iment, so called, Base Vector Analysis.
Experiment 2
In this experiment, like the ” active parameters” was chosen: tchn1, tchn2, tchn3 and after
100 iteration we have obtained the results presented in Table 2.
Comparing the final value of objective function in both experiment (1) and (2) we can see
that this value in Base Vector is worse to value in Direct Optimization. Probably there
are others sensitive parameters in model (not only: tehn1, tchn2, tchn3).
E. Kasperska, E. Mateja-Losa — Extended Sensitivity Analysis. .. 4
Table 2: The main results of experiment 2
parameters | final value | original value | lower limit } upper limit
ucrl 55 100 50 100
ucr2 50 50 50 100
ucr3 10 10 10 100
tchnl 32 20 0 40
tchn2 22 10 0 40
tchn3 32 20 0 40
ucprl 500 500 200 500
ucpr2 500 500 200 500
ucpr3 100 100 100 700
Initial value of ” fob”: 0.29128E + 08
Final value of ” fob”: 0.25939E + 08
Final value of inventory: 244
Final value of price: 1118.10
Final value of penalty: 20.8607E + 06
Final value of demand: 253.50
Experiment 3
The assumptions and main results of experiments type ”simplification” are presented in
Table 3. The accepted values of parameters in simplification type Y (see [1]) gives better
results then in type X.
4 Final remarks and conclusions
The purpose of the paper was to present some results of experiments considered the
problem of extended sensitivity analysis of parameters and structures in some model type
System Dynamics. Final remarks are as follows:
e the Direct Optimization allows to choose the optimal value of searching parameters,
which optimized the objective function,
e the Base Vector Analysis allows to test the sensitivity of parameters in model in
context to improving the value of objective function,
e the Simplification try to find simpler model structure,
e the evolutionary aspect of sensitivity analysis (both: parameters and structure) can’t
be overestimated; in building, simulation and testing models of complex, dynamical,
nonlinear systems this aspect seem to be of great importance (the scope of paper
not allow to develop this problem). We plan to undertake it in near future.
E. Kasperska, E. Mateja-Losa — Extended Sensitivity Analysis. . . 5
Table 3: The main results of ”SIMPLIFICATION” for maximization
type PARAMETERS
of Simplified parameter Ordinary parameters
Simplification | initial value | final value | initial value final value
a= 0.3 a=1 tchn1 = 0.40 | tehn1 = 38.800
type X B=0.3 B=1 tchn2 = 0.40 | tehn2 = 39.986
y= 0.3 y=1 tchn3 = 0.40 | tehn3 = 39.997
value of objective function: 0.184 + 08 (initial: 0.513 + 06)
a=0.5 a=0 tchn1 = 23.200
type Y B=0.5 B=1 tchn2 = 39.849
7 = 0.5 7=0 tchn3 = 39.849
value of objective function: 0.191 + 08 (initial: 0.614 + 07)
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