Detailed Modeling and
Terminating Statistical Analysis
Chapter 5
(from KSS’01)
What We’ll Do ...
•
Explore lower-level modeling constructs
•
Model 5-1: A generic call-center system
Nonstationary arrival process
Balking, three-way decisions, sets, variables, expressions, submodels, and costing
•
Debugging
•
Model 5-2: Animating the call center model
Plots, global pictures, and storages
•
Model 5-3: The model with overall performance
measures
Run conditions, model size and speed, overall performance measures
What We’ll Do ...
(cont’d.)•
Statistical analysis of simulation output
(terminating systems)
Time frame of simulations
Strategy for data collection and analysis
Confidence intervals
Comparing two alternatives
Comparing many alternatives via the Arena Process Analyzer (PAN)
Generic Call Center
(Model 5-1)
•
Single telephone number, 26 trunk lines
If all 26 lines busy, caller gets busy signal and goes away
•
Answered call gets recording asking …
Technical support? (76% of callers choose this)
Sales information? (16%)
Order-status inquiry? (8%)
•
Time for caller to choose ~ UNIF (0.1, 0.6)
Technical Support Calls
•
Get second recording asking …
Product type 1? (25% of tech support callers choose this)
Product type 2? (34%)
Product type 3? (41%)
•
Recording and choosing takes UNIF(0.1, 0.5)
•
If a qualified tech-support person is available for
chosen product, call routed for immediate service
•
If not, call placed in (electronic) queue, subjected
to annoying rock music
•
All tech support conversations ~ TRIA (3, 6, 18)
Technical Support Calls
(cont’d.)•
4% of tech support calls need further assistance
after completion of their call
Questions forwarded to another tech group that prepares a response; time to prepare this response ~ EXPO (60)
Response sent back to the same tech-support person who took the original call
This person calls the customer back and talks, which lasts TRIA (2, 4, 9)
These calls require one of the 26 trunk lines and take priority over incoming calls
If return call not completed on same day, it’s carried over to the next day
Sales Calls
•
Call automatically routed to sales staff
•
Sales staff is separate from tech-support staff
•
If a sales-staff person is available, call gets
immediate service
•
If not, call placed in (electronic) queue, treated to
soothing new-age space music
•
All sales conversations ~ TRIA (4, 15, 45)
Order-Status Calls
•
Automatically handled by phone system — no people
•
No limit on number handled at a time (but still limited
by the 26 trunk lines)
•
Time for “conversation” ~ TRIA (2, 3, 4)
•
After call, 15% take option to talk to a real person (the
rest exit the system)
These calls are routed to sales staff
Have same priority as incoming sales calls
Conversation durations ~ TRIA (3, 5, 10)
Call Arrivals
•
Calls accepted from 8 AM until 6 PM
•
Some staff available until 7 PM
Incoming calls shut out after 6 PM
But all calls that entered before 6 PM are answered
•
Call arrival rate varies substantially over the day
Staffing
•
Sales staff: 7 people with staggered schedules
3 on duty for first 90 minutes (notation: 3@90)
Then 7@90, 6@90, 7@60, 6@120, 7@120, 4@90
•
Tech support staff: 11 people variously qualified
for the three different product lines
Work eight-hour days plus 30 minutes off for lunch
Some people only qualified on one line
Some people qualified on two or maybe all three lines
New Modeling Issues
•
This is a service (not manufacturing) system
But can use same modeling capabilities
•
Nonstationary arrival process
Arrivals occur one at a time and are independent of one another
Average rate varies over time (would be constant for a stationary Poisson process)
Built into Create module (beware of popular-but-wrong methods … details in book)
•
Balking
Required because there are only 26 trunk lines
Entity arrives at queue, which is full (capacity is 0 here)
New Modeling Issues
(cont’d.)•
Three-way decisions
Entity or call can go to one of three places in model based on call type
– Similarly, tech-support calls can go to one of three places based on
product type
Capability available in Decide module
•
Sets
Groups of similar objects
Can be referenced by a common set name and index (1, 2, 3, …) into the set
– Can also be referenced by original name, independent of set
Technical-support staff requires sets
– An object can be a member of more than one set
New Modeling Issues
(cont’d.)•
Variables and Expressions
Can be referenced in model by name
Can be one- or two-dimensional arrays, indexed by one or two integers
User-defined Variables
– Store some numerical value (not a formula) – Can be initialized in Variable data module
– Can be used, reassigned during the simulation run by any entity
User-defined Expressions
– A name defined by a mathematical expression
– This name can be references anywhere in the model
– Can use constants, Variables, Attributes, system state variables,
values from distribution – connected via mathematical operations
Can use Expression Builder to help define
New Modeling Issues
(cont’d.)•
Submodels
Partition simulation model into several smaller submodels – Can link them together, more manageable pieces
Just like a normal model view within a submodel
Submodels can also contain further submodels, etc. – hierarchical structure
Submodels can be externally connected to other modules or submodels
Navigate panel in Project Bar shows submodels, under Top-Level Model
•
Costing
Automatic time and cost information for entities – Wait, value-added, non-value-added, transfer, other
You must enter cost information – Entity and Resource data modules
Key Output Performance Measures
•
Count
balks
— no. of attempted incoming calls
sent away due to all 26 trunk lines being busy
Will not model reneging — customers in queue leaving the system if they get sick of waiting
•
Total time in system, by customer type
•
Time waiting for a real person, by customer type
•
Contact time, by customer type
•
Number of calls waiting, by customer type
Animation Requirements
•
No entity movement to animate here
•
Can still display queues
•
To see how well staffing matches up with load,
craft appropriate plots vs. time
Number of calls balked
Lengths of queues
Number of idle staff
•
Strategy to improve performance — alter the
staffing schedule, see if it produces a better
matchup of the plots
System or Simulation Type
•
Terminating
Known starting and stopping conditions – part of model
Time frame is known (and finite)
•
Steady-State
Initial conditions are not always well defined
No defined stopping condition (theoretically infinite)
Interested in system response over the long run
•
Call-center model
Start at 8 AM and end at 7 PM
– Some Technical support calls are held over, but not many – we’ll
ignore this aspect (sort of … fixed below)
Arena Modeling Panels
•
Basic Process panel
Highest level of modeling
•
Advanced Process panel
More detailed (and different) modeling capabilities
•
Advanced Transfer panel
Material-handling, entity-movement capabilities
•
Blocks, Elements panels
Lowest level of modeling capabilities – the underlying SIMAN simulation language itself
Other panels are created using modules from these panels
Building the Model
•
Defining the Data
•
Submodel Creation
•
Divide model in sections or submodels
Increment the Time Period
Create Arrivals and Direct to Service
Technical Support Calls
Technical Support Returned Calls
Sales Calls
Order-Status calls
Simulation Replication Data
•
Project replication parameters
Run/Setup dialog – Replication Parameters tab
10 Replications of 11 hours each
Four options for Initialization Between Replications: – Initialize system (yes), initialize statistics (yes)
10 independent and identical replications – no calls carried over Reports for each day separately
– Initialize system (yes), initialize statistics (no)
10 independent and identical replications – no calls carried over
Cumulative summary reports (day 1, days 1-2, days 1-3, …, days 1-10) – Initialize system (no), initialize statistics (yes): Selected
10 continuous days – calls carried over Reports are by replication (day)
– Initialize system (no), initialize statistics (no) 10 continuous days – calls carried over
Schedule Data
•
Schedules
Enter into Schedule data module
13 schedules required
– One for each of the 11 technical-support people – One for the sales staff overall
– The arrival process (Type = Arrival, not Capacity)
Use Graphical schedule editor (initially)
Use Edit via Dialog (or Edit via Spreadsheet) if you need trailing zeros in the capacity to fill out the cycling time
window
– We need this in this model due to not Initializing System between
Resource Data
•
Define resources
Use Resource data module
13 resources – Sales staff
– 11 technical support staff
– Trunk Line (single resource with 26 units)
Enter Schedule Name for all but Trunk Line
– For resources on a Schedule, use Ignore option for Schedule Rule
to ensure correct cross-day modeling … details in book
Sets Data
•
Use Set data module (Basic Process panel)
•
Develop three Resource sets for technical
support staff
Product 1
– Charity, Noah, Molly, Anna, Sammy
Product 2
– Tierney, Sean, Emma, Anna, Sammy
Product 3
– Shelley, Jenny, Christie, Molly, Anna, Sammy
Note that Anna and Sammy are in all three sets
Consistently listed the more versatile staff at the end of the list in each set … “save” them … discussed later
Sets Data
(cont’d.)•
Develop two Tally sets
Tech Calls
– Product 1 Call, Product 2 Call, Product 3 Call
Returned Time
– Return 1 Call, Return 2 Call, Return 3 Call
Sets used to collect statistics by product type
•
Develop a Counter set
Keep track of number of balks per half-hour period
22 counters – one for each half-hour period
First defined 22 counters in Statistic data module (Advanced Process panel)
Variables and Expressions Data
•
Variables
Use Variable data module to define thee variables
– Period (the current time period)
– Busy Per Period (busy signals in current time period)
– Per Period Balk (total balks for last completed time period)
– Note – explicit use of Variables module is required only if you want
a Variable to have a non-zero initial value
•
Expressions
Use Expression data module to define three expressions
– Returned Tech Time, for duration of returned tech-support calls:
TRIA(2, 4, 9)
– Tech Time, for duration of tech-support calls : TRIA(3, 6, 18) – Available 1, Available 2, and Available 3
Sum of currently available, but idle, resources by product type, for staffing plots Use Expression Builder … details in book, model
Submodel Creation
•
Object/Submodel/Add Submodel menu option to
create a submodel … we’ll use six submodels
Define (right-click, then Properties) – Name
– Number of entry, exit points (could be 0 if there’s no flow interaction)
Move between submodels: Navigate panel, Named Views, or mouse
– Double-click on a submodel to open it
– When in a submodel, right-click in an empty place, then Close
Time Period Counter Submodel
•
Increments the time period counter
•
No entry or exit points – interacts via Variables,
not flow
•
Create Counter Entity – Create module
Time Between Arrivals – 660 minutes (constant)
•
Assign Period – Assign module
Time Period Counter Submodel
(cont’d.)•
Assign Variables – Assign module
Increment Period variable for the next half-hour period
Assign Per Period Balk to Busy Per Period variable
value (number of calls balked during previous half hour)
Set Busy Per Period variable to zero to start balk
counting during the half hour starting now
•
Check Period – Decide module
2-Way by Condition
Determine if there are still more periods in this day (i.e., if
Period < 22)
– Yes: Delay for a half hour – Delay module, then loop back – No: Dispose of entity – Dispose module
Create and Direct Arrivals Submodel
•
Creates arrivals, checks for available trunk line, and directs to appropriate type of service•
No entry points•
Three exit pointsSubmodel Logic
•
Create arriving calls
If a trunk line is available – seize one
– Assign Arrival Time attribute (for use downstream) – Delay to listen to recording
– Determine call type
– Direct call and assign entity type
Else (all trunk lines are busy) – Count balked call
– Increment Busy Per Period counter – Dispose of call
Develop Submodel
•
Create arriving calls – Create module
Time Between Arrivals – Type: Schedule
– Schedule Name: Arrival Schedule
Was defined when we defined the data for the model
•
Check for available trunk line
Queue/Seize module combination (Blocks panel)
Set queue capacity to zero
– If trunk line available, resource seized in following Seize module – If no truck line available, entity will automatically balk
Develop Submodel
(cont’d.)•
Assign arrival time – Assign module
Use Arena variable TNOW = current simulation clock
•
Delay for Recording – Delay module
Used Delay module from Blocks panel
Be careful of units – no choice here (uses Base Time Units)
•
Direct call – Decide module
Use N-way by Chance option
Enter probabilities as percents (0 – 100)
•
Assign call type – Assign module
Assign entity type to call type
Develop Submodel
(cont’d.)•
Balking entities
Count balked call – Record module – Record into counter set Busy Lines
Set index is the variable Period
Increment Busy Per Period variable – Assign module
Technical Support Calls Submodel
•
Logic for servicing technical support calls
•
One entry point
Submodel Logic
•
Delay to listen to recording – Delay module
UNIF(0.1, 0.5) minutes
•
Determine product type – Decide module
N-way by Chance
•
Seize technical support person
Seize module – Advanced Process panel
Request from appropriate set for product type
Preferred order within the set
– Save more versatile employees for other things
Save set index (particular tech-support person) in attribute
Tech Agent Index
– In case returned tech call is needed – get same tech-support person
Submodel Logic
(cont’d.)•
Save product type and call start time – Assign
module
Save type (1, 2, or 3) in attribute Product Type
Assign value from TNOW to attribute Call Start
•
Delay for call – Delay module
Use value from expression Tech Time
•
Release tech-support person and trunk line
Release module – Advanced Process panel
Use set index in attribute Tech Agent Index to release
the particular tech-support person assigned from set
Submodel Logic
(cont’d.)•
Record call and line time – Record module
Time Interval type
Tally set Tech Calls
Set index Product Type
Records only the time spent during the tech-support conversation (necessary?)
•
Record tech line time – Record module
Time Interval type
Tally Tech Support Line Time (not a Tally set)
Use Arrival Time attribute set when call first arrived, so
this records the total time in the system so far
Returned Tech Calls Submodel
•
Logic for returned tech calls
Submodel Logic
•
Check for returned call – Decide module
•
If no returned call is needed
Dispose of entity
•
If a returned call is needed
Entity Type set to Returned Call – Assign module
Delay for response time – Delay module
Direct by product type
– N-way by Condition based on attribute Product Type
Seize tech-support person and trunk line
– Seize module: Seize specific member of appropriate Resource set Use Set Index Tech Agent Index attribute
– Seize Trunk Line
Note use of “==” to check for equality.
Submodel Logic
(cont’d.) Delay for call time – Delay module – Expression Returned Tech Time
Release tech person and trunk line – Release module
Record returned time – Record module
– Use beginning-time attribute Arrival Time, defined when call first
arrived, so this records total time in system
– Use Tally Set Return Time indexed by Product Type
Sales Calls Submodel
•
Logic for sales calls
•
One entry point, no exit points
•
Uses a
Shared Queue
Single queue
Shared by two or more seize activities
– In this case, the “real” incoming sales calls, as well as those
Submodel Logic
•
Seize sales person – Seize module
Shared queue declared in Queue data module
•
Delay for call – Delay module
•
Release sales person and Trunk Line – Release
module
•
Record call time – Record module
Records elapsed time from call’s original arrival until now
Order-Status Calls Submodel
•
Logic for order-status calls
•
One entry point, no exit point
Submodel Logic
•
Delay for call – Delay module
•
Decide if Sales person required – Decide module
•
If sales person is required
Seize sales person – Seize module, shared queue
Follow-up delay – Delay module
Release sales person – Release module
•
Record call – Record module
Elapsed time from call’s arrival to system up to now
•
Release trunk line – Release Module
Finding and Fixing Model Errors
•
Arena picks up “simple” errors in Check phase,
and leads you to them via Find and Edit buttons
in Errors/Warnings windows
Undefined variables, attributes, resources
Unconnected modules
Duplicate module names
Typos
•
Other kinds of errors are more complex, can’t be
detected without trying to run — options on Run
Interaction toolbar or on Run menu
•
Only mention capabilities here; see text for
details
Finding and Fixing Model Errors
(cont’d.)•
Run Controller — Command-driven window to
control, display details about model operation
and underlying SIMAN code
•
Trace — Follow active modules, selected
variables
•
Highlight active module – highlights the active
module during the simulation run
•
Layers – gives control over what you see during
the simulation run
Finding and Fixing Model Errors
(cont’d.)•
Break on Module; Break — stop run when entity
hits a selected module, at a specific time, or when
a selected entity is about to become active
•
Watch — select expressions to display in a
window as model runs
•
Look at reports when model is running or paused
Model 5-2: Animating the Model
•
No “normal” entity animation — just plots,
queues, a few other “data” animations
•
Plots (all vs. time on horizontal axis)
Queue lengths (as in earlier models)
Balks per period — reason for variable Per Period Balk
Number of tech support people available for each product type — reason for the “Available” expressions defined in Expressions module
With multiple plots, configure first one, then copy/edit for others to get consistent look and feel; snap to grid to align
Animating the Model
(cont’d.)•
Created digital clock “by hand” (details in text)
Why not ready-made animated clocks? We didn’t reset the system state between replications, so internal clock just
keeps increasing
•
Resource and queue animations
•
Just for realism — doesn’t add any analysis value
•
Resource button from Animate toolbar
Take pictures from libraries (.plb files), different states
•
Queue button from Animate toolbar
Animating the Model
(cont’d.)•
Storages for further animation
Requires using Delay modules from Blocks panel, not Advanced Process panel
– We did this only for the delay listening to the first recorded message
Enter a Storage name in the Blocks Delay module
(Message 1)
Storage button from Animate Transfer toolbar
•
Animated variables
Number of available trunk lines
Number of available salespeople
Model 5-3: Model for Analysis with
Overall Performance Measure
•
Modify the call-center model for intensive study
Different run conditions – to allow valid statistical analysis
Smaller size – to continue to fit in academic version and make room for other enhancements
Faster – to allow for extensive analysis
New overall performance measures to consider both resource costs and customer-oriented performance
•
Base on Model 5-1 rather than Model 5-2 since
the latter adds only animation and we’re now
crunching numbers
Run Conditions
•
Want valid terminating statistical analysis
New replications must start independently with no model-state carry-over
•
Run/Setup/Replication Parameters
Check “Initialize System Between Replications” – Still check “Initialize Statistics Between Replications”
Get truly independent and identically distributed replications
Unreturned tech-support follow-up calls lost – unrealistic
Compromise – redefine a “replication” to be a five-day work week … Monday-Thursday returned tech-support calls
carried over, Friday-night ones lost
– Run/Setup/Replication Parameters: Replication Length = 5 Days,
specify 11 Hours Per Day
Slimming Down and Speeding Up
•
Academic version places several different kinds
of limits on model “size”
Max of 150 concurrent entities … though millions could pass through … this limit is not a problem here
Max of 150 “module instances” … includes flowchart
modules, and each entry (line) in data modules … this limit
is a problem
Also need room to add new output performance measures
•
Reduce number of module instances
Eliminate many statistical accumulators … included getting rid of lines in data modules, entire flowchart modules, and unchecking stat-collection boxes … details in book
•
This also increased speed by factor of 3 to 4
Overall Performance Measures
•
Form an overall cost measure – reduce, minimize
What controllable input parameters affect cost? How?
•
Two components to cost
Due to staffing and resources – tangible, measurable
Due to poor customer service – intangible, hard to measure
•
Staffing and resource costs
Hourly salaries: $18 for sales, $16 to $20 for tech-support depending on training (see Resource data module)
– Salaries paid whenever person is on duty, whether busy or not
Get current weekly payroll of $13,110 = Staff Cost
Generalize for More Staff
•
Try to improve service via more staff
Will certainly increase staff cost
Try to improve customer service to make it worthwhile
•
Base-model results – worst staffing shortfalls are
between 11:30AM and 3:30PM
•
Add sales and tech-support staff for that
four-hour period (half-four-hours 8 through 15)
Additional Sales Staff
•
Add variable New Sales to be the number of additionalsalespersons to add in periods 8-15
Define in Variable data module
Use in Schedule data module (under Sales Schedule) … add to
number of sales staff in base model in periods 8-15
– Must use Edit via Dialog or Spreadsheet since Graphical Schedule Editor
cannot handle Variables
•
Cost for each new salesperson: $15/hour Each will work 20 hours/week, so cost $300/week
Variable New Sales Cost set to 300
Additional cost is (New Sales) * (New Sales Cost), used in
Additional Tech-Support Staff
•
Possibly add new tech-support staff for products
1, 2, 3 (only), and for all products
Variables New Tech 1, New Tech 2, New Tech 3
– Named Larrys, Moes, and Curlys, respectively
– Each paid $14/hr * 20 hrs/week = $280/week (variable LMC cost)
Variable New Tech All
– Named Hermanns
– Each paid $17/hr * 20 hrs/week = $340/week (variable Her cost)
Resource data module – define resources Larry, Moe,
Curly, Hermann … and hourly costs for them (not used)
Set data module – add Larry, Moe, Curly, Hermann to
appropriate resource sets
Schedule data module – add Larry, Moe, Curly,
Changing the Number of Trunk Lines
•
Each trunk line costs a flat $89/week, including all
calls (even long-distance)
•
Is 26 the right number?
•
To change it, just edit the Capacity entry in the
Resource data module
•
Add variable
Line Cost
to be $89, multiply by
number of trunk lines (
MR(Trunk Line)
) to get
Total New Resource Cost
•
Define an Expression called
New Res Cost
:
New Sales * New Sales Cost
+ (New Tech 1 + New Tech 2 + New Tech 3) * LMC Cost + New Tech All * Her Cost
+ Line Cost * MR(Trunk Line)
•
Does not depend on what happens during
simulation … used only at end in Statistic module
•
Does not include cost of the base-model human
staff (sales, tech-support) … viewed as sunk, and
constant for all variants of staffing changes
Costs for Putting Customers on Hold
•
Impute a cost for making customers wait on hold
Trade off against resource costs
Use model to understand, improve, optimize this tradeoff
Such customer-dissatisfaction costs are hard to quantify
•
People have a “tolerance” for holding
Tech-support calls: 3 minutes (variable Tolerance Tech)
Sales calls: 1 min. (Tolerance Sales)
Order-status calls: 2 min. (Tolerance Order Status)
•
Beyond the tolerance point, system incurs cost of
Tech-support calls: $1.67/min. (variable TWT Cost)
Sales calls: $3.72/min. (SWT Cost)
Costs for Putting Customers on Hold
(cont’d.)
•
Accumulate “excess” waiting time (time past
tolerance) for each call type
Assign module when call is done
Use built-in Arena attribute ENTITY.WAITTIME
– Accumulates total of times in queues as entity goes along, and
other “Wait”-allocated times … but there are none upstream in this model so this attribute will have the waiting time on hold
– Requires that Costing box be checked in Run/Setup/Project
Parameters
Variable Excess Tech Wait Time accumulates via
adding in for each tech-support call
MAX( ENTITY.WAITTIME - Tolerance Tech, 0 )
At end of run, multiply Tech Wait Time by TWT Cost
Total Cost
•
Adding together all the costs, get the overall
economic (cost) performance measure
Total Cost = New Res Cost
+ Excess Sales Wait Time * SWT Cost + Excess Status Wait Time * OSWT Cost + Excess Tech Wait Time * TWT Cost
+ Staff Cost
•
This is defined in Statistic data module
Type = Output – already being computed, just report it
In Category Overview Report, get via User Specified Æ
Percent-Busy Requirement
•
Above cost performance measure ignores calls
balked away due to no trunk line … busy signal
Clearly, undesirable – very hard to put a cost on it
Instead, have a strong goal to limit this to no more than 5% of incoming calls … a model configuration not satisfying
this will be deemed unacceptable no matter how attractive (low) the cost may be
Like a constraint except it’s on an output, not an input … call it a requirement
•
Compute via two Record modules in arrival
submodel to count incoming and balked calls …
Aside – Generality of Models
•
We could have done a lot of things in very
different ways in this model
Using Arena’s costing functions more and doing fewer of our own external calculations
Reparameterize using only “primitive” parameters (e.g., hourly wage rates) and programming Arena to do the calculations
•
How much of this you do depends on model’s
intended use and users
•
Tradeoff between generality (elegance?) vs. time
spent building the model
Results
•
Base case (no more people, 26 trunk lines)
Total cost (for the week) = $34K
Percent busy signals = 11% (unacceptable)
•
Added one resource unit for each type
New Sales, New Tech 1, New Tech 2, New Tech 3,
New Tech All, and go to 27 trunk lines
Total cost (for the week) = $28K
– Added resources reduced customer waiting time by more than
enough to cover their cost
Percent busy signals = 3% (acceptable)
– Extra trunk line, plus added resources to move calls through
Statistical Analysis of Output from
Terminating Simulations
•
Random input leads to random output (RIRO)
•
Run a simulation (once) — what does it mean?
Was this run “typical” or not?
Variability from run to run (of the same model)?
•
Need statistical analysis of output data
From a single model configuration
Compare two or more different configurations
Search for an optimal configuration
•
Statistical analysis of output is often ignored
This is a big mistake – no idea of precision of results
Not hard or time-consuming to do this – it just takes a little planning and thought, then some (cheap) computer time
Time Frame of Simulations
•
Terminating
: Specific starting, stopping
conditions
Run length will be well-defined (and finite)
•
Steady-state
: Long-run (technically forever)
Theoretically, initial conditions don’t matter (but practically they usually do)
Not clear how to terminate a simulation run
•
This is really a question of intent of the study
•
Has major impact on how output analysis is done
•
Sometimes it’s not clear which is appropriate
Strategy for Data Collection and
Analysis
•
For terminating case, make IID replications
Run/Setup/Replication Parameters: Number of Replications field
Check both boxes for Initialize Between Replications
•
Separate results for each replication – Category
by Replication report
Strategy for Data Collection and
Analysis
(cont’d.)•
Category Overview report will have some
statistical-analysis results of the output across
the replications
•
How many replications?
Trial and error (now)
Approximate number for acceptable precision (below)
Sequential sampling (Chapter 11)
•
Turn off animation altogether for max speed
Confidence Intervals for
Terminating Systems
•
Using formulas in Chapter 2, viewing the
cross-replication summary outputs as the basic data:
•
Possibly most useful part – 95% confidence
interval on expected values
•
This information (except standard deviation) is in
Category Overview report
If > 1 replication specified, Arena uses cross-replication data as above
Half Width and Number of Replications
•
Prefer smaller confidence intervals —
precision
•
Notation:
•
Confidence interval:
•
Half-width =
•
Can’t control t or s
•
Must increase n — how much?
Want this to be “small,” say < h where h is prespecified
Half Width and Number of Replications
(cont’d.)
•
Set half-width = h, solve for
•
Not really solved for n (t, s depend on n)
•
Approximation:
Replace t by z, corresponding normal critical value
Pretend that current s will hold for larger samples
Get
•
Easier but different approximation:
s = sample standard
deviation from “initial” number n0 of replications
h0= half width from “initial” number n0of replications n grows quadratically as h decreases.
Interpretation of Confidence Intervals
•
Interval with random (data-dependent) endpoints
that’s supposed to have stated probability of
containing, or covering, the expected valued
“Target” expected value is a fixed, but unknown, number
Expected value = average of infinite number of replications
•
Not an interval that contains, say, 95% of the data
That’s a prediction interval … useful too, but different
•
Usual formulas assume normally-distributed data
Never true in simulation
Might be approximately true if output is an average, rather than an extreme
Central limit theorem
Comparing Two Alternatives
•
Usually, want to compare alternative system
configurations, layouts, scenarios, sensitivity
analysis … here just two alternatives
•
Base case of Model 5-3, vs. adding one resource
unit for each type
New Sales, New Tech 1, New Tech 2, New Tech 3,
New Tech All, and go to 27 trunk lines
Earlier, one run of each suggested big differences … real?
•
Reasonable but not-quite-right idea: Make
confidence intervals on expected outputs from
each alternative, see if they overlap
Compare Means via the Output Analyzer
•
Output Analyzer is a separate application that
operates on .dat files produced by Arena
Not installed by default from book CD – need custom install
Launch separately from Windows, not from Arena
•
To save output values (Expressions) of entries in
Statistic data module (Type = Output) – enter
filename.dat in Output File column
Just did for Total Cost, not Percent Busy
Will overwrite this file name next time … either change the name here or out in Windows before the next run
Compare Means via the Output Analyzer
(cont’d.)
•
Start Output Analyzer, open a new data group
Basically, a list of .dat files of current interest
Can save data group for later use – .dgr file extension
Add button to select (Open) .dat files for the data group
•
Analyze/Compare Means menu option
Add data files … “A” and “B” for the two alternative
Select “Lumped” for Replications field
Compare Means via the Output Analyzer
(cont’d.)
•
Results:
•
Confidence interval on difference misses 0, so
conclude that there is a (statistically) significant
difference
Evaluating Many Alternatives with the
Process Analyzer (PAN)
•
With (many) more than two alternatives to
compare, two problems are
Simple mechanics of making the possibly many parameter changes, making the runs, keeping track of the many
output files
Statistical methods for drawing reliable and useful conclusions
•
Process Analyzer (PAN) addresses these
•
PAN operates on program (.p) files – produced
when .doe file is run (or just checked)
•
Start PAN from Arena (Tools/Process Analyzer) or
via Windows
PAN Scenarios
•
A
scenario
in PAN is a combination of:
A program (.p) file
Set of input controls that you choose
– Chosen from Variables and Resource capacities – think ahead – You fill in specific numerical values
Set of output responses that you choose
– Chosen from automatic Arena outputs or your own Variables – Values initially empty … to be filled in after run(s)
To create a new scenario in PAN, double-click where indicated, get Scenario Properties dialog
– Specify Name, Tool Tip Text, .p file, controls, responses
– Values of controls initially as in the model, but you can change them
in PAN – this is the real utility of PAN
– Can duplicate (right-click, Duplicate) scenarios, then edit for a new
one
PAN Projects and Runs
•
A
project
in PAN is a collection of scenarios
Program files can be the same .p file, or .p files from different model .doe files
Controls, responses can be the same or differ across scenarios in a project – usually will be mostly the same
Think of a project as a collection of scenario rows – a table
Can save as a PAN (.pan extension) file
•
Select scenarios in project to run (maybe all)
•
PAN runs selected models with specified controls
•
PAN fills in output-response values in table
Equivalent to setting up, running them all “by hand” but much easier, faster, less error-prone
Running Model 5-3 with PAN
•
Scenarios
Base case (no additional resources)
Imagine $1200/week to spend on each additional resource type, one at a time (no mixed enhancements)
7 scenarios in all (details in book)
Select all to run (click on left of row, Ctrl-Click or Shift-Click for more)
Statistical Comparisons with PAN
•
Model 5-3 alternatives were made with 10
replications each
Better than one replication, but what about statistical validity of comparisons, selection of “the best”?
•
Select Total Cost column, Insert/Chart (or or
right-click on column, then Insert Chart)
Chart Type: Box and Whisker
Next, Total Cost; Next defaults
Next, Identify Best Scenarios
– Smaller is Better, Error Tolerance = 0 (not the default) – Show Best Scenarios; Finish
Statistical Comparisons with PAN
(cont’d.)•
Vertical boxes: 95% confidence intervals•
Red scenarios statistically significantly better than blues More precisely, red scenarios are 95% sure to contain the best one
Narrow down red set – more replications, or Error Tolerance > 0
•
The scenarios just considered with PAN are just 7
of many, many possibilities
•
Try to find input-control values that minimize
Total Cost while keeping Percent Busy < 5%
•
Formulate as an optimization problem:
Minimize Total Cost
Subject to 26 ≤ Trunk Lines ≤ 50
New Sales, New Tech 1, New Tech 2, New Tech 3, New Tech All ≥ 0
New Sales + New Tech 1 + New Tech 2 + New Tech 3 + New Tech All ≤ 15 Percent Busy < 5%
Reasonable starting place – best acceptable scenario so far: Add 3 New Tech All
Where to go from here? Explore all of feasible six-dimensional space exhaustively? No.
Searching for an Optimal Alternative
with OptQuest
An output requirement, not an input constraint
Objective function is the simulation model
Contractual obligation, space limitation
Space limitation Nobody’s fired
OptQuest
•
OptQuest searches intelligently for an optimum
Like PAN, OptQuest
– Runs as a separate application … can be launched from Arena – “Takes over” the running of your model
– Asks that you identify the input controls and the output (just one)
response objective
Unlike PAN, OptQuest
– Asks that you specify constraints on the input controls – Asks that you specify requirements on outputs
– Decides itself what input-control-value combinations to try
– Uses internal heuristic algorithms to decide how to change the input
controls to move toward an optimum configuration
Using OptQuest
•
Tools/OptQuest for Arena
•
New session (File/New or Ctrl+N or )
Make sure the desired model window is active
•
Select controls – Variables, Resource levels
Trunk Line, New Tech 1, 2, 3, and New Tech All
Bounds: 26 ≤ Trunk Line ≤ 50, others between 0 and 15
Type is Discrete for all, Input Step Size 1
•
Constraints – enter
New Sales + New Tech 1 + New Tech 2 + New Tech 3 + New Tech All <= 15
•
Objective and Requirement
Total Cost Response – Select Minimize Objective
Percent Busy Response – Select Requirement, enter 5 for Upper Bound
Using OptQuest
(cont’d.)•
Options window – computational limits,
procedures
Time tab – accept Run for 10 minutes default
Precision tab – vary number of replications from 3 to 10
Preferences tab – various settings (accept defaults)
