system from the visual point of view. The contrast
between figure 5 and figure 6 shows that the range of
decision parameter α
WIP
in ninth-order delay system
is smaller than that in PTD system with the same
delay time and range of decision parameter α
i
. That
is, in high order delay system, the decision maker
can consider more about inventory and consider less
about WIP. Beer game has shown that most
decision-makers think that the actual inventory is
more important than WIP. Therefore, even if the
decision maker has bias against the above conclusion,
the high order delay system may still be stable. In
other words, the decision strategies that lead PTD
system to be unstable may make high order delay
system stable. So, high order delay system is more
stable than PTD system.
Through simulations, we can find the critical
stable state of n-th order delay system with a given
DT and obtain the critical stable points (α
,
, α
,
).
Therefore, the critical stable condition of inventory
control system under n-th order delay is defined as
following:
Definition: Suppose the n-th order delay
inventory control system is stable at the initial time.
With a given DT, when imposing a small step
disturbance on demand, if there exists the decision
parameters (α
,
, α
,
) that can keep the inventory
curve oscillate with equi-amplitude, then the state is
called critical stable state, (α
,
, α
,
) is the critical
stable point of the system under the given DT.
As high order delay system considers the order of
delay, its critical stable points are determined by four
dimensional vectors, (DT, α
i
, α
WIP
, n). Through the
traversal simulation of α
i
and α
WIP
under certain DT,
several critical stable points are found. After
connecting these points in the plane that takes α
i
as
horizontal axis and α
WIP
as vertical axis, we obtain
the stability boundary of the tenth-order delay
system named s curve. Figure 7 shows some s curves
under different DT and the index of s represents the
value of DT. For comparison, the corresponding
stability boundaries of PTD system are also given.
Figure 7: Comparison between the stability boundaries of
tenth-order delay system and PTD system.
Furthermore, s curve of high order delay system is
approximate to linear property and the lower right of
s curve is the unstable region. For comparison, figure
8 shows the stability boundaries of five high order
delay systems with different delay orders
(n=2,4,6,10,25) under certain DT (DT=9).
Figure 8: The effect of delay order on system stability
(DT=9).
After running simulations for the high order
delay inventory control model under different DT
and comparing with PTD system, we can draw the
following conclusions:
First, the stable region of high order delay
system is larger than that of PTD system. With the
same delay time, the larger order of delay, the closer
the stability behavior of high order delay system to
PTD system, and the oblique line (α
WIP
=α
i
/2) is the
upper bound when s curve moves up to top left.
Second, as the oblique line (α
WIP
=α
i
/2) is the
upper bound when s curve of n-th order delay
inventory control system moves up to top left with
certain DT, it can be concluded that the upper left
area of oblique line (α
WIP
=α
i
/2) is the stable region
which is independent of delay (IoD), and the oblique
line (α
WIP
=α
i
/2) can be defined as IoD stability
boundary which is only determined by systemic
structure.
Third, under the same delay condition, the
smaller delay time, the larger the stable region of
high order delay time; and under the same delay time,
the smaller the order of delay, the larger the stable
region. Therefore, the stability of high order delay
system with short delay time and small order of
delay is closer to first-order system.
Further analysis on figure 8 reveals that when
the order of delay is small, the s curves tend to be
relative dispersive, and when the order of delay is
large, the s curves are comparatively concentrated.
This shows that the sensibility of decision
parameters to the change of system order is
nonlinear.
Based on figure 8, when the decision parameter
α
i
takes a certain value, figure 9 shows the relation
THE RESEARCH ON STABILITY OF SUPPLY CHAIN UNDER HIGH ORDER DELAY
365