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Time Varying and
Multi-Phase Throughput Analysis Reliability Block Diagram (RBD)
models are generally used to describe the reliability-wise configuration of
a system but they can also be used to represent the system's production/processing
sequence. In such cases, we can take system analysis through RBDs a step
further and utilize them to study the productivity or yield of the system
via throughput analysis. This is applicable when the system can process
(or make) something, such as manufacturing systems, chemical processes and
oil refineries. Throughput analysis has been discussed before in this
e-magazine; in this article we discuss the modeling and analysis of phased
and time varying throughput with the help of
BlockSim 7.
This article combines concepts of
throughput analysis and Reliability Phase Diagrams (RPDs). For more
information about throughput analysis, click
here.
To find out more about RPDs, click
here. You can also read an
article
about RPD analysis in the latest issue of the Reliability Edge.
Applications for This Type of Analysis
Time varying and multi-phase throughput can
be found in many industry applications. For example, consider a textile
factory that receives different quantities of raw materials during different
seasons. These seasons could be treated as different phases.
Another example is the start up period in a
processing plant. When the plant stops operating, due to planned or
unplanned maintenance, the equipment requires a warm up period before
reaching its maximum production capacity. The methodology used for
variable throughput is discussed next.
Variable (Time Varying) Throughput
In BlockSim 7, time varying throughput
can be specified at the phase level through the Variable Throughput
property of an operational phase. Variable throughput permits modeling of
scenarios where the throughput changes over time.
Three general models for variable
throughput are available in BlockSim. Each of these models
has two parameters, which are specified by the user. These models are
discussed below:
1) Linear model:
y = ax
+ b
This model describes the
change in throughput y as a
linear function of time x.
The total throughput processed between any two points of time
x1 and
x2
is obtained by integration of the linear function as:
2) Exponential model: y = beax
This model
describes the change in throughput y as an exponential function of
time x. The total throughput processed in a period of time between
any two points
x1 and
x2 is obtained as:
3) Power model: y = bxa
This model
describes the change in throughput y as a power function of time x.
The total throughput processed between two points of time
x1 and
x2 is obtained as:
All of the
above models also have a user-defined
maximum throughput capacity value. If
this maximum throughput capacity value is reached, the throughput per unit
time becomes constant at that value. In this situation,
the variable throughput model then acts
as a constant throughput model. The above models may at first
glance seem limited, but they do in fact provide ample modeling
flexibility. This flexibility is achieved by using these functions as
building blocks for more complex functions.
For example, a step model can
be easily created by using multiple phases, each with a constant throughput.
A ramp model would use phases with linearly increasing functions in
conjunction with constant phases, and so forth. In addition, decreasing
throughput can be modeled by manipulation of the parameters.
Example:
We will
use a simple example to illustrate the use of BlockSim 7 to model
phase and time varying
throughput.
A production
machine follows a Weibull distribution with β = 1.55
and η =
250.4 days. When an outage happens, it takes about 12 days to restore the
machine.
The above
figure represents the Reliability Block Diagram (RBD) of the machine.
The
failure properties are shown next.
Every time
the machine fails, the machine is repaired then the throughput of the
machine goes through a warm up period of 5 days until it reaches normal
production capacity again. The throughput
in units per day can be described
as follows:
y(x) =20x (within warm up,
t <= 5)
y(x) = 100 (during normal operation, t > 5) |
The normal
operation throughput is specified in the machine's properties window, as
shown next (no backlog is specified for this example).
To model the
two-phase operation (warm up and normal operation) of this machine, a
Reliability Phase Diagram (RPD) is used,
as shown next.
Both the Warm
Up Period and Normal Operation phases in the above RPD are linked to the
same Diagram Sheet, which contains the machine's RBD. The following figures
show the phase properties of these first two phases.
In addition to the linked diagram,
the phase properties also
specify the duration of the
phase and what action is to be taken if a system failure occurs.
In this case, the action upon system
failure is to go to a maintenance
phase, which is specified in the next field. Once the maintenance phase has
been executed, the system will then go through warm up
again.
The warm up
phase has a variable throughput, specified as follows.
The
normal operation phase has a constant throughput,
as shown next.
The
maintenance phase is defined in BlockSim 7 as a template that
contains the blocks to be fixed in the RBD and their associated repair
properties (such as repair times, resources, etc.) Note that while repairs
may be performed during an operational phase, a maintenance phase is
exclusively dedicated to the execution of maintenance tasks.
The repair properties
for the block in the maintenance template are shown next.
The system is
simulated for 1092 days (3 years) with
156 increments for weekly results. On
the Throughput Settings tab of the Maintainability/Availability
Simulation window, Report Throughput Point Results
is selected, as
shown next.
A System Overview summary of the simulation
is shown next (note that your results may
vary slightly since they were obtained via simulation).
The mean
availability of the system for the next three years is estimated to be
95.41% and the expected throughput of the plant at the end of three years is
expected to be 102,719.31 units.
For an
overview of the plant’s availability as a function of time, you can generate
a point availability plot like the one shown next.
The point
results of the throughput as a function of time can also be obtained from
the System Point Results report. The following is a plot of the
cumulative throughput over time.
The following
is a plot of the weekly throughput.
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