DECISION MODELING WITH
MICROSOFT EXCEL
Chapter 10
DISCRETE EVENT
SIMULATION
Part 2
Copyright 2001
Prentice Hall
EXAMPLE 4 – HERVIS CAR
RENTAL MODEL REVISITED
Based on the previous results, Hervis came up
with an idea of pulling some of the car service
people off the car lot to staff more rental stations
whenever the queue of customers became large.
In this way, the staffing of the rental stations
would increase to three or even four clerks
whenever a peak in customer demand occurred.
To model this dynamic change in behavior in
Extend, we would need a way to alter the value in
the Maximum Number in Activity field of the
Activity, Multiple block dialog during the
simulation.
To do this, double-click the Queue, FIFO block to
display its dialog window. For explanations, click
on the
Help button.
The L value
output
connector
reads out the
queue length
throughout the
simulation run.
The C connector can be used to input a capacity
value that overrides the number recorded in the
Maximum Number in Activity field.
What is needed is a way to convert the L value
from the Queue, FIFO block into the desired
number of rental stations and feed that dynamic
number into the C connector of the Activity,
Multiple block during the simulation.
Extend’s Conversion Table block is similar to
Excel’s VLOOKUP in that it will map the input L
values to the output C values.
To open the Conversion Table block, click on
GENERIC.lix - Math - Conversion Table.
Connect the L value
output connector of the
Queue, FIFO block to
the Conversion Table’s
input connector.
Next, connect the
Conversion Table’s
output connector to
the C value of the
Activity, Multiple block.
The text box shows Hervis’ proposed decision rule
for increasing the number of rental station clerks.
Double-click on the Conversion Table block to
open its dialog window.
Choose Stepped
under the Table
is: option.
These values
(0 - 30 for x in
and 1 - 4 for y
out) represent
the proposed
decision rule.
A value of 4 for y out is assigned to values of x in
greater than 6. This places a cap on the
maximum number of rental clerks.
Click on the Run Simulation
icon.
Here is the resulting graph.
Remember, results will differ due to the random
effects.
Now, compare the following results for the
Queue, FIFO block:
Up to 4 Clerks during
2 Clerks
peak periods
Compare the following results for the Activity,
Multiple block:
Up to 4 Clerks during
2 Clerks
peak periods
To verify these results, we could run the
simulation many more times. However, it would
be difficult to compare each set of results.
Extend’s Queue Stats block, will collect queuing
statistics from multiple runs of a simulation
model.
Using this, we can systematically assess the
influence of the model’s inherent randomness on
a performance measure.
Open this block by
going to the Library –
Open Library menu
and choosing
Stats.lrx from the
resulting dialog.
Now, open the Queue Stats block and place it
anywhere in the Extend model window.
Double-click on the Queue Stats block to open its
dialog window:
To tabulate the results of multiple runs, select the
Append new updates and Update at end of
simulation options.
Now, go to Run – Simulation Setup dialog and
change the Number of runs to say, 20.
This instructs Extend to make 20 independent
runs of the model, tabulating the queuing
statistics from each run into the Queue Stats
block.
Clicking on the Run Simulation icon will now
cause the model to automatically run 20 times.
After starting the simulation, a status bar and
small graph will appear on the bottom of the
screen, from which you can monitor the progress
of the simulations.
After the runs are finished, double-click on the
Queue Stats block to open the dialog and review
the results.
These results confirm that increasing the number
of clerks will not help the model.
Clicking on the Confidence Interval button in this
dialog will further summarize the results giving
95% confidence intervals.
EXAMPLE 5 – FINAL HERVIS
CAR RENTAL MODEL
Now, return to the original 2 clerk rental station
policy.
Instead of increasing the number of clerks at the
rental station, let’s consider serving only two
types of customers:
Those who desire smaller compact cars
(Economy cars)
Those who desire larger, more powerful cars
(Deluxe cars)
Assume that 70% of customers will opt for
economy cars and the balance for deluxe cars.
The challenge is to decide how many of each type
of car to have available at the beginning of the
week to avoid car shortages during the week and
ultimately customer dissatisfaction.
Past data on car rentals shows:
Economy car customers rent their car for
between one and two days.
Deluxe car customers rent their cars typically
for two days plus or minus a half day.
Since the time interval for economy car rentals is
fairly evenly distributed, we can approximate the
distribution as being Uniform.
For deluxe car renters, a Normal distribution for
the rental times is a good fit to the data, with a
standard deviation of ½ a day.
We will use Extend’s Hierarchical blocks to
minimize visual complexity. This will allow us to
create models made up of submodels (logical
groupings of Extend blocks).
To do this, select all of the blocks except the
Executive and Exit blocks and choose the Make
Selection Hierarchical item on the Model menu.
A dialog will appear
in which you can
specify a name for
the new super-block.
Enter the name and click on Make H-Block to
create the super-block.
You can double-click on the super-block to
expand it into a new window and add a label.
Now, begin building the expanded model in place
of the original Exit block, so that customers
flowing out of the Rental Office submodel will
enter the rental car lot of the simulated model.
Delete the Exit block in the Extend model and add
two new blocks:
Input Random Number block
This block uses an Empirical Table to assign
a 0 value (to signify an economy car rental)
with probability .7 and a 1 value (deluxe car
rental) with probability .3.
Select DE Output block
This block splits the stream of customers to
one of two output connectors based upon
the value present at its Select connector.
To insert the Input Random Number block, go to
Library – GENERIC.lix – Inputs/Outputs – Input
Random Number.
Double-click on this icon to open the respective
dialog.
Choose Empirical
Table as the
Distribution.
Give the block a
label.
Specify the values which signify whether an
economy or deluxe car is rented and the
respective probabilities.
To insert the Select DE Output block, go to Library
– DE.lix – Routing – Select DE Output.
Double-click on this icon to open the respective
dialog.
The incoming customer is directed to the top (“a”
connector) output if the random number at its
Select connector is 0 and is directed to the
bottom (“b” connector) output if the random
number at its Select connector is 1.
Connect the blocks accordingly:
Now, introduce a Resource Pool block from the
DE.lix library,
A Resource Pool is a variable that
indicates how many of a named
constrained resource (e.g., Deluxe cars) are
available.
Double click on the Resource Pool block to open
its dialog window.
Initialize the resource pool of Deluxe Cars to 100,
based on the weekly maximum demand for
deluxe cars.
Insert a special form of a queuing block, a Queue,
Resource Pool block (from DE.lix – Queues).
The Queue, Resource Pool block will hold items
(deluxe car customers) until the specified number
of Resource Pool units from the Deluxe Cars pool
are available (one in this case).
The item is then released and the count in the
pool is decremented, thus in effect, a deluxe car
is assigned to the customer.
Note that the Resource Pool block is a globally
accessible variable, tracking the number of
deluxe cars on hand in this case, and has no
drawn connections to other bocks.
Now that a deluxe customer has received a
deluxe car, we must model the rental interval
until it is returned.
We will do this using the Activity, Multiple block
and a new Input Random Number block.
Specify a
normal
distribution
with a mean of
2 days (2880
minutes) and
standard dev.
of .5 days (720
minutes).
The capacity of the Activity, Multiple block
defaults to a large number, 1000, to assure that
no items would be stopped from entering this
block.
After the rental time random delay, the car is
returned to the deluxe car pool to be available for
re-renting by the Release Resource Pool block
(from the DE.lix – Resources menu).
Now, model the economy car customers similar to
the deluxe car customers.
Add another Resource Pool block and set the
initial capacity to 140.
Add another Queue, Resource Pool block to
assign economy cars from the pool, if available.
Next, add an Activity, Multiple block, Input
Random Number block and a Release Resource
Pool block to return the economy car to its pool.
Finally, insert an Exit (4) block (from DE.lix –
Routing menu) to allow both streams of customer
types to exit the model.
To preserve customer goodwill, let’s implement a
policy of upgrading an economy car to receive a
deluxe car, if possible, in the event no economy
cars are available.
Open the Queue, Resource Pool block’s dialog
window for the economy car.
Specify Car Type under pool name: by clicking on
the edit field and entering it in the resulting
dialog.
Extend will start at the top of the Pool name list
and attempt to assign a car from the Economy
Car pool.
If that pool is empty, it will go to the Deluxe Car
pool and assign a deluxe car if available.
Otherwise, it will hold the customer until one of
either car type becomes available.
During the simulation run, economy car
customers assigned a car from one of the two
pools will have the name of that pool carried with
them as a Car Type attribute for later reference.
In the Release
Resource Pool block,
the returned rental car
is credited to whatever
pool name was used
earlier and carried by
the attribute Car Type.
The Input Random
Number block shows
the random rental time
(from 1440 to 2880
minutes) from a
Uniform distribution,
for an economy car
customer.
Now, to complete the model, add Queue Stats
(from STATS.lrx), Help (from GENERIC.lix Inputs/Outputs) and Plotter, Discrete Event
(from PLOTTER.lix) blocks.
An Activity Stats block
(from STATS.lrx) has
been added to record
statistics for the
Activity delay blocks
just as the Queue Stats
block does for the
Queue blocks.
After running the simulation (with initial resource
pools of 140 economy cars and 100 deluxe cars),
double click on the Plotter, Discrete Event block
to view the graph below:
This line
Thisrepresents
is the
the number of Economy
cars number
being rented
of
during the week.
Deluxe Car
customers
arriving on the
This is the
lot to find it
number of
empty of
Economy Car
deluxe cars.
customers
arriving on the
lot to find it
This line represents the number of Deluxeempty of any
cars being rented during the week.
car to rent.
The graph indicates (by the thick line traces) that
the initial car pool sizes are grossly inadequate
for customer demands.
The Queue Stats and Activity Stats windows
confirm this conclusion.
For example, after 20 simulation runs, the 95%
confidence interval for the average wait by a
deluxe car customer was 140 + 49 minutes.
In order to improve on the wait time in this
model, we need to increase the pool size.
Let’s experimentally increase each pool’s initial
size by about 15%, increasing the Deluxe Cars
pool to 115 and the Economy cars pool to 160 and
the re-run the simulation.
This increase in pool size shows much
improvement as illustrated by the average and
maximum wait time.
In order to reduce the maximum wait time,
change the initial resource pools to 170 Economy
Cars and 125 Deluxe Cars and re-run the
simulation.
These results are even better. It seems though,
that waiting times may be sensitive to small
underestimates in the distributions’ parameter
assumptions and that much more simulation
analysis is needed.