Crespo-Marquez, Adolfo with Rafael Ruiz-Usano, Gloria Elena Pena-Zapata and Javier Aracil, "System Dynamics Production Models. A Qualitative Analysis", 1997 August 19-1997 August 22

System Dynamics Production Models.
A Qualitative Analysis,

Authors:

Adolfo CRESPO-MARQUEZ
Rafael RUIZ-USANO
Gloria Elena PENA-ZAPATA
Javier ARACIL

Escuela Superior de Ingenieros de Sevilla.
Avda. Reina Mercedes, s/n.
41012 Sevilla. Spain.
adolfo@ pluto.us.es_; usano@ obelix.cica.es ; gpena@pluto.us.es ; aracil@esi.us.es ;

Abstract

The result of the formalization of a production system SD model is a set of first order
differential equations, which can be numerically solved using the computer. As a result of that,
the time evolution of the model’s variables can be determined. Sensitivity analysis can be then
utilized to evaluate the impact that the variations in an element’s numerical value have in
system behavior.

The analysis of the structural stability of the production system offers a wider
perspective of its dynamic behavior, allows a new approach to the model sensitivity analysis.
This new approach is extremely helpful in order to study global aspects of the system’s
behavior, while sensitivity analysis is more appropriated to deal with local issues.

In this paper, we try to investigate the conditions under which important qualitative
changes in production models’ behavior can be expected.

Introduction

In this paper, the analysis of the structural stability of the system is applied to a very
simple example. This example is a production system containing only two production blocks,
one represents a manufacturing process that can be performed by one single machine (WIP),
the second models the stock of end products (INV).

We assume that the lead-time (LT) of the manufacturing process depends on the
amount of work-in-process (WIP) in the first block. The higher the WIP, the longer an item
takes to be processed. LT can be represented as a function of WIP showing different non-
linear behaviors.

We will analyze the phase state WIP-INV, at the same time that the sensitivity analysis,
in order to determine the stability points of the system. Moreover, we will try to identify the
system behavior topologies! leading, under certain conditions, to system stability (Aracil &
Toro, 93)

Qualitative analysis will also allow us to know when a topology will lead to unstable
equilibrium points and bifurcation, and therefore, the possibilities for a catastrophe in the
production system (Hale & Kocak, 91; Kutnetsov, 95).

' Topology: representation of the equilibrium points as a function of the system’s parameters.

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Causal Diagram

The causal diagram of the model (Figure 1) shows the existence of four feedback loops.
The first two negative ones have to do with the control of the output flow of the level
variables. The third negative one controls the production flow in order to reach a desired
inventory. The last feedback loop and positive one is very important for our study once
conditions the level of WIP according to the lead time. .

ae tad
Aiscreps desired inventory time table_9
" ae? Sane A
s
A demand forecast
ior a)
time to forecast

Figure 1. Causal diagram of the production SD model.

Stock & Flow diagram
In this diagram, a more detailed representation of the nature of the variables is shown.
(Figure 2).

Wah tae

Te mw
end product flow demand ow

3

oS
production fow

Figure 2. Stock & flow diagram.

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Model differential equations and system stability

The equations of the state variables of the system are:

ae ) [end _product_ flow] - [demand _flow] a)
ate) = [production_flow|-[end_product_flow] (2)

Setting both of these equations to zero, we obtain the values for the variables under
system stability:

d(inv) _| wip we
an EE | ~(D*valve(inv)) = 0 @B)
d(wip) = p+[ Pea wip _|_o a
ae TAI LT (wip) |

If the lead-time can be obtained, as a function of WIP, as follows:
bop

LT (wip) = eel (5)
a

Then, we can plot (Figure 3) the values of WIP as a function of the market demand and
for the system in stability conditions, just introducing (5) in (3).

WIP*(1+b*e™"(A*WIP)/a=D

dg TH

750

250

Figure 3. WIP stability points.

Figure 3 shows how, for different values of the parameters in (5) (considering a=6,
4=12.3, constants), we obtain different representations of WIP meeting the stability conditions
(3). The topology of these curves may allow in some cases the existence of several equilibrium

541

points for WIP (E.g. three points for the interval 72<D<85 aprox.). For instance, in case that
the value of the demand is equal to 80 units/day, Figure 4 shows how WIP will reach two
different equilibrium points depending on its initial condition. This gives an extra information
about the unstability of one of the equilibrium points of WIP shown in Figure 3.

700

400

0 25.00 50.00 75.00 100.00}
‘Time (Month)

Figure 4. WIP stability points. Sensitivity analysis.

Conclusions
The evolution of a production system, where the lead-time is a function of the work-in-
process, has been deeply analyzed. The qualitative analysis is a very adequate approach to

identify possible behaviors of the system under stability, especially in cases where the shape of
the function LT(wip), and the value of the market demand, change. From the analysis of the

References

Abrahan, R. and Shaw, Ch.D. “Dynamics: The Geometry of Behavior”. “Second Edition,
Addison-Wesley. 1992.

Aracil, J., y Toro, M,. “Métodos Cualitativos en Dinamica de Sistemas”, Serie Ingenieria n°.8.
Secretaria de Publicaciones. Universidad de Sevilla. 1993,

Aracil, J., “Structural Stability of low-orden System Dynamicas Models. Int. J. System
Science” 12:423-441. 1981.

Hale, J.K., Kocak, H., “Dynamics and Bifurcations”. Springer-Verlag. 1991.

Kuznetsov, Y.A. “Elements of. Applied Bifurcation Theory”. Springer-Verlag. 1995.
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