24th Mternational Conference of the System Dynamics Society
July 23-27, 2006, Nijmegen, The Netherlands
Eds.: A. Grofler, E.A.J.A. Rouwette, R.S. Langer, J.I.Rowe, J.M. Yanni
Parallel Dual Problem of Optimization
Embedded in Some Model Type
System Dynamics
Elzbieta Kasperska, Damian Slota
Institute of Mathematics
Silesian University of Technology
Kaszubska 23, 44-100 Gliwice, Poland
e-mail: {e.kasperska, d.slota}@polsl.pl
Abstract
Some kind of dual problem of optimization, on the base of simulation on the model type
System Dynamics, is presented. Authors refere to question of, so called, optimization
embedded in simulation. Some new results of experiments with the comparison of the so-
lution of the minimization of cost and maximization of profit in the firm are described. The
generalization of formulation both problems in matrix form, on the example of described
model type System Dynamics is proposed.
1 Introduction
The dual problem of optimization on economic, mathematical models is widely described
in literature. Classical concept of maximization of profit and minimization of cost can be
applied to simulation models type System Dynamics [1,2,5]. First attempts were under-
taken by authors in paper [3]. Now the concept were generalized to formulate, so called,
parallel dual problem. Generally speaking, in specific matrix form the profit and cost are
conceived in one experiment. The results are compared with the individual experiments
(only the maximization or only the minimization). Authors, in their experiments, use the
simulation language Professional Dynamo 4.
2 Mathematical Model of the System
The model named DYNBALANCE(2-2
was adappted from paper [3]. Its correct
c-I) was chosen like the object of experiments. It
ed main structure is illustrated on Figure 1.
E. Kasperska, D. Slota
Parallel Dual Problem of Optimization. ..
level of raw materials
during transformation
into product P1
tprl
rprl
inventory
of product P2
desir! xg
Q Cussmel
level of inventory, \ 7
q22,q21 3
uepij E
EIQ
‘Deesir2 .
losse of
profit from
level of raw materials
during transformation
into product P2
Figure 1: Structure of model DYNBALANCE(2-2-c-I)
Specific local matrix equation formulation is:
qi 0 qi2 0 frdpi(t)
0 ql 0 q22 frdp2(t)
1 1 0 0 sourcl(t)
0 0 1 1 rmi1 sourc2(t)
ucpll- qil 0 ucp21 - q12 0 rm12 _ 05 (1)
0 ucp12 + q21 0 ucp22 + q22 rm21 66
1 0 0 0 rm22 o7
0 1 0 0 b8
0 0 1 0 9
0 0 0 1 610
E. Kasperska, D. Slota — Parallel Dual Problem of Optimization. .. 3
The main idea was to achieve the solution which locally minimize the cost of production
(the Euclidean norm of overdetermined system (1) was optimal in each step of simulation).
The results of experiments for some taken scenarios was already described in paper [3].
Now we will try to formulate the ” dual” problem for the presented model. Its mathematical
form is as follows:
qil 0 qi2 0 frdpl(t)
0 gi 0 q22 frdp2(t)
1 1 0 0 sourcl(t)
0 0 1 a: rmi1 sourc2(t)
pricel-qll 0 pricel - q12 0 rm12 b5 (2)
0 price2 - q21 0 price2 - q22 rm2. b6 °
1 0 0 0 rm22 o7
0 1 0 0 b8
0 0 1 0 9
0 0 0 1 610
The interesting idea occur. How to generalize the model, in order to achieve the simul-
taneously: minimization of cost of production and maximization of profit, in one model?
So, the proposed form of such idea is below:
qil 0 qi2 0 frdpi(t)
0 gi 0 q22 frdp2(t)
1 1 0 0 sourcl(t)
0 0 d L sourc2(t)
ucpl1-q1l 0 ucp21 - q12 0 rmil b5
0 ucp12 - q21 0 ucp22 - q22 rm12 b6 (3)
1 0 0 0 rm2. o7 ,
0 1 0 rm22 b8
0 0 1 0 b9
0 0 0 1 610
pricel - qil 0 pricel - q12 0 bli
0 price2 - q21 0 price2 « q22 b12
where 65 and 65 are small numbers, and b11 and 612 are large numbers. The results of
experiments both types (using formula (2) and formula (3) are presented in the next
section and are compared with previously results described in paper [3], for formula (1).
3 The Results of Experiments
The most interesting point of view is to compare the important (for estimation the fitting
of the balances (1), (2) and (3)) variable like norm. Its characteristics in the whole horizon
of simulation (two years) are presented on Figure 2.
E. Kasperska, D. Slota — Parallel Dual Problem of Optimization. .. 4
2500
2000 eeor—
1500
norm
1000
500be@ A RAYON
0 20 40 60 80 100
time
Figure 2: The characteristic of variable norm with the formulas of balance Ax = b given
by equation (1) (©), (2) (GJ) and (3) (A)
350 350
300 300
250 250
200 &Y 200
i ai
~ 150 ~ 150
100 100
50 50
(c) 0
ic) 20 40 60 80 100 ic) 20 40 60 80 100
time time
Figure 3: The characteristic of variables: lin] and lin2 in minimization experiment
1600 1600
1400 1400
‘2 1200 1200
g s
Hs) 3
1000 1000
800 800
6 20 40°60 ~~ 80~——CT00 () 20 40°60 ~~ 80~——C00
time time
Figure 4: The characteristic of variables: lin1 and lin2 in maximization experiment
E. Kasperska, D. Slota — Parallel Dual Problem of Optimization. .. 5
1800 1800
1600 1600
4 1400 ~ 1400
c 2
= 1200 = 1200
1000 1000
800 800
6 20 40 #60 80 100 6 20 40 «60 80 100
time time
Figure 5: The characteristic of variables: lin1 and lin2 in parallel experiment
It is also interesting to compare the dynamics of such important variables like: inventories
of product P1 and P2 (lin1 and lin2, respectively), in these three types of experiments.
The three experiments: minimization (formula (1)), maximization (formula (2)) and par-
allel (formula (3)) were undertaken under the some condition about the demand for prod-
uct Pl and P2. The systems have reacted in such a way that minimize the Euclidean
norm (sce [4]). The rates of raw materials: rm11, rm12, rm21, rm22, were optimal to fit
the balances Ax = b (given by equations (1), (2) and (3) respectively).
4 Final Remarks and Conclusions
The purpose of the paper was to present some results of experiments considered dual
problem of optimization embedded on model type System Dynamics. These results were
compared with previously results achieved in paper [3], given the base for estimation
the goodness of pseudosolution of the overdetermined systems (1), (2) and (3), in fitting
balances type Ax = b, embedded in model Forresters type. Final conclusions are as
follows:
— the generalization of formulation of the dual problem of optimization on model type
System Dynamics, seems to be interesting way to achieve in parallel form, the best
fitting of balance of raw materials (the balance models the interesting aspect of
dynamics behaviour of system);
— the opportunity for analysing the consequences of locally optimal solutions, for the
whole system (its dynamics), can’t be overestimated for complex, dynamic and
multilevel systems;
— the fruitfull Forresters idea (classical models of type System Dynamics) can be em-
bedded or linked with method of classical numerical analysis, giving new dimension
for investigation (not only in economic area).
E. Kasperska, D. Slota — Parallel Dual Problem of Optimization. .. 6
References
[1] Coyle, R.G.: System Dynamics Modelling. A Practical Approach. Chapman & Hall,
London (1996)
Coyle, R.G.: The practice of System Dynamics: milestones, lessons and ideas from
30 years experience. System Dynamics Rev. 14 (1998) 343-365
Kasperska, E., Slota, D.: Optimization embedded in simulation on models type Sys-
tem Dynamics — some case study. In: Sunderam, V.S., Albada, G.D., Sloot, P.M.A.,
Dongarra, J.J. (eds.): Computational Science, Part I. LNCS 3514, Springer-Verlag,
Berlin (2005) 837-842
Legras, J.: Methodes et Techniques De’Analyse Numerique. Dunod, Paris (1971)
Sterman, J.D.: Business dynamics — system thinking and modeling for a complex
world. Mc Graw-Hill, Boston (2000)
