taylor introms10 ppt 14
DESCRIPTION
mat 540 week 14TRANSCRIPT
-
5/27/2018 Taylor Introms10 Ppt 14
1/65
14-1Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Simulation
Chapter 14
-
5/27/2018 Taylor Introms10 Ppt 14
2/65
14-2
The Monte Carlo Process Computer Simulation with Excel Spreadsheets
Simulation of a Queuing System
Continuous Probability Distributions Statistical Analysis of Simulation Results
Crystal Ball
Verification of the Simulation Model
Areas of Simulation Application
Chapter Topics
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
3/65
14-3
Analogue simulation replaces a physical system with an analogousphysical system that is easier to manipulate.
In computer mathematical simulation a system is replaced with a
mathematical model that is analyzed with the computer.
Simulation offers a means of analyzing very complex systemsthat cannot be analyzed using the other management science
techniques in the text.
Overview
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
4/65
14-4
A large proportion of the applications of simulations are forprobabilistic models.
The Monte Carlo technique is defined as a technique for selecting
numbers randomly from a probability distribution for use in a trial(computer run) of a simulation model.
The basic principle behind the process is the same as in the
operation of gambling devices in casinos (such as those in MonteCarlo, Monaco).
Monte Carlo Process
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
5/65
14-5Table 14.1 Probability Distribution of Demand for Laptop PCs
In the Monte Carlo process, values for a random variable aregenerated by sampling from a probability distribution.
Example: ComputerWorld demand data for laptops selling for$4,300 over a period of 100 weeks.
Monte Carlo Process
Use of Random Numbers (1 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
6/65
14-6
The purpose of the Monte Carlo process is to generatethe random variable, demand, by sampling from theprobability distribution P(x).
The partitioned roulette wheel replicates the probabilitydistribution for demand if the values of demand occur ina random manner.
The segment at which the wheel stops indicates demandfor one week.
Monte Carlo Process
Use of Random Numbers (2 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
7/6514-7Figure 14.1 A Roulette Wheel for Demand
Monte Carlo Process
Use of Random Numbers (3 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
8/6514-8
Figure 14.2
Numbered Roulette Wheel
Monte Carlo Process
Use of Random Numbers (4 of 10)
When the wheel is spun, the actual demand for PCs is determined by a
number at rim of the wheel.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
9/6514-9Table 14.2 Generating Demand from Random Numbers
Monte Carlo Process
Use of Random Numbers (5 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
10/6514-10
Select number from a random number table:
Table 14.3 Delightfully Random Numbers
Monte Carlo Process
Use of Random Numbers (6 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
11/6514-11
Repeating selection of random numbers simulatesdemand for a period of time.
Estimated average demand = 31/15 = 2.07 laptop PCsper week.
Estimated average revenue = $133,300/15 = $8,886.67.
Monte Carlo Process
Use of Random Numbers (7 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
12/6514-12
Monte Carlo Process
Use of Random Numbers (8 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice HallTable 14.4
-
5/27/2018 Taylor Introms10 Ppt 14
13/6514-13
Average demand could have been calculated analytically:
per weeksPC'1.5
)4)(10(.)3)(10(.)2)(20(.)1)(40(.)0)(20(.)(
:therefore
valuesdemanddifferentofnumberthe
demandofyprobabilit)(ivaluedemand
:where
1)()(
xE
n
xP
x
n
i
xxPxE
i
i
ii
Monte Carlo Process
Use of Random Numbers (9 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
14/6514-14
The more periods simulated, the more accurate the results.
Simulation results will not equal analytical results unless enoughtrials have been conducted to reach steady state.
Often difficult tovalidateresults of simulation - that true steadystate has been reached and that simulation model truly replicatesreality.
When analytical analysis is not possible, there is no analyticalstandard of comparison thus making validation even more difficult.
Monte Carlo Process
Use of Random Numbers (10 of 10)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
15/6514-15
As simulation models get more complex they become impossibleto perform manually.
In simulation modeling, random numbers are generated by amathematical processinstead of a physical process (such as wheelspinning).
Random numbers are typically generated on the computer using anumerical technique and thus are not true random numbers but
pseudorandom numbers.
Computer Simulation with Excel Spreadsheets
Generating Random Numbers (1 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
16/6514-16
Artificially created random numbers must have the followingcharacteristics:
1. The random numbers must be uniformly
distributed.
2. The numerical technique for generating the numbersmust be efficient.
3. The sequence of random numbers should reflect nopattern.
Computer Simulation with Excel Spreadsheets
Generating Random Numbers (2 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
17/6514-17Exhibit 14.1
Simulation with Excel Spreadsheets (1 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
18/6514-18Exhibit 14.2
Simulation with Excel Spreadsheets (2 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
19/6514-19Exhibit 14.3
Simulation with Excel Spreadsheets (3 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
20/6514-20
Revised ComputerWorld example; order size of one laptop each week.
Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (1 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.4
C S S d
-
5/27/2018 Taylor Introms10 Ppt 14
21/6514-21
Order size of two laptops each week.
Computer Simulation with Excel Spreadsheets
Decision Making with Simulation (2 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.5
Si l i f Q i S
-
5/27/2018 Taylor Introms10 Ppt 14
22/6514-22
Table 14.5 Distribution of Arrival Intervals
Table 14.6 Distribution of Service Times
Simulation of a Queuing System
Burlingham Mills Example (1 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Si l i f Q i S
-
5/27/2018 Taylor Introms10 Ppt 14
23/6514-23
Average waiting time = 12.5days/10 batches= 1.25 days per batch
Average time in the system = 24.5 days/10 batches
= 2.45 days per batch
Simulation of a Queuing System
Burlingham Mills Example (2 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Si l i f Q i S
-
5/27/2018 Taylor Introms10 Ppt 14
24/6514-24
Simulation of a Queuing System
Burlingham Mills Example (3 of 3)
Caveats: Results may be viewed with skepticism.
Ten trials do not ensure steady-state results.
Starting conditions can affect simulation results.
If no batches are in the system at start, simulationmust run until it replicates normal operating system.
If system starts with items already in the system,simulation must begin with items in the system.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
C Si l i i h E l
-
5/27/2018 Taylor Introms10 Ppt 14
25/6514-25
Exhibit 14.6
Computer Simulation with Excel
Burlingham Mills Example
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
C i P b bili Di ib i
-
5/27/2018 Taylor Introms10 Ppt 14
26/6514-26
minutes2.254x.25,rif:Example
.determinedistime""forxvaluear,number,randomageneratingBy
r4x16x2r
rnumberrandomtheF(x)Let16
2xF(x)
x
02x21
x
0 81dxx81dx
x
08xF(x)
:xofyprobabilitCumulative
(minutes)timexwhere4x0,8xf(x)
:Example
ons.distributicontinuousforusedbemustfunctioncontinuousA
Continuous Probability Distributions
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M hi B kd d M i S
-
5/27/2018 Taylor Introms10 Ppt 14
27/6514-27
Machine Breakdown and Maintenance System
Simulation (1 of 6)
Bigelow Manufacturing Company must decide if it shouldimplement a machine maintenance program at a cost of $20,000 peryear that would reduce the frequency of breakdowns and thus timefor repair which is $2,000 per day in lost production.
A continuous probability distribution of the time between machinebreakdowns:
f(x) = x/8, 0 x 4 weeks, where x = weeks between
machine breakdownsx = 4*sqrt(ri), value of x for a given value of ri.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M hi B kd d M i S
-
5/27/2018 Taylor Introms10 Ppt 14
28/6514-28
Table 14.8
Probability Distribution of Machine Repair Time
Machine Breakdown and Maintenance System
Simulation (2 of 6)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M hi B kd d M i t S t
-
5/27/2018 Taylor Introms10 Ppt 14
29/65
14-29
Table 14.9
Machine Breakdown and Maintenance System
Simulation (3 of 6)Revised probability of time between machine breakdowns:
f(x) = x/18, 0 x6 weeks where x = weeks between machinebreakdowns
x = 6*sqrt(ri)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M hi B kd d M i t S t
-
5/27/2018 Taylor Introms10 Ppt 14
30/65
14-30Table 14.10
Machine Breakdown and Maintenance System
Simulation (4 of 6)
Simulation of system without maintenance program
(total annual repair cost of $84,000):
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M hin Br kd n nd M int n n S t
-
5/27/2018 Taylor Introms10 Ppt 14
31/65
14-31Table 14.11
Machine Breakdown and Maintenance System
Simulation (5 of 6)Simulation of system with maintenance program (total annualrepair cost of $42,000):
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
M chine Bre kdo n nd M inten nce S stem
-
5/27/2018 Taylor Introms10 Ppt 14
32/65
14-32
Machine Breakdown and Maintenance System
Simulation (6 of 6)
Results and caveats: Implement maintenance program since cost savings appear to be
$42,000 per year and maintenance program will cost $20,000 peryear.
However, there are potentialproblemscaused by simulatingboth systems onlyonce.
Simulation results could exhibit significant variation since timebetween breakdowns and repair times are probabilistic.
To be sure of accuracy of results, simulations of each systemmust be run many times and average results computed.
Efficient computer simulation required to do this.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Machine Breakdown and Maintenance System
-
5/27/2018 Taylor Introms10 Ppt 14
33/65
14-33
Exhibit 14.7
Machine Breakdown and Maintenance System
Simulation with Excel (1 of 2)
Original machine breakdown example:
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Machine Breakdown and Maintenance System
-
5/27/2018 Taylor Introms10 Ppt 14
34/65
14-34Exhibit 14.8
Machine Breakdown and Maintenance System
Simulation with Excel (2 of 2)
Simulation with maintenance program.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistical Analysis of Simulation Results (1 of 2)
-
5/27/2018 Taylor Introms10 Ppt 14
35/65
14-35
Outcomes of simulation modeling are statisticalmeasuressuch as averages.
Statistical results are typically subjected to additionalstatistical analysisto determine their degree of accuracy.
Confidence limitsare developed for the analysis of thestatistical validity of simulation results.
Statistical Analysis of Simulation Results (1 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Statistical Analysis of Simulation Results (2 of 2)
-
5/27/2018 Taylor Introms10 Ppt 14
36/65
14-36
Formulas for 95% confidence limits:upper confidence limit
lower confidence limit
where is the mean and s the standard deviation from a
sample of size n from any population.
We can be 95% confident that the true population mean will be
between the upper confidence limit and lower confidence limit.
)/)(.( nsx 961
)/)(.( nsx 961
x
Statistical Analysis of Simulation Results (2 of 2)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Simulation Results
-
5/27/2018 Taylor Introms10 Ppt 14
37/65
14-37
Simulation Results
Statistical Analysis with Excel (1 of 3)
Simulation with maintenance program.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.9
Simulation Results
-
5/27/2018 Taylor Introms10 Ppt 14
38/65
14-38
Simulation Results
Statistical Analysis with Excel (2 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.10
Simulation Results
-
5/27/2018 Taylor Introms10 Ppt 14
39/65
14-39Exhibit 14.11
Simulation Results
Statistical Analysis with Excel (3 of 3)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
40/65
14-40
Crystal Ball
Overview
Many realistic simulation problems contain more complexprobability distributionsthan those used in the examples.
However there are several simulation add-insfor Excel that
provide a capability to perform simulation analysis with avariety of probability distributions in a spreadsheet format.
Crystal Ball, published by Decisioneering, is one of these.
Crystal Ball is a risk analysis and forecasting program that
uses Monte Carlo simulation to provide a statistical range of
results.Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
41/65
14-41
Recap of Western Clothing Company break-even and profitanalysis:
Price (p) for jeans is $23
variable cost (cv) is $8
Fixed cost (cf) is $10,000
Profit Z = vp - cfvc
break-even volume v = cf/(p - cv)
= 10,000/(23-8)
= 666.7 pairs.
Crystal Ball
Simulation of Profit Analysis Model (1 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
42/65
14-42
Modifications to demonstrate Crystal Ball Assume volume is now volume demandedand is defined by a
normal probability distributionwith mean of 1,050 andstandard deviation of 410 pairs of jeans.
Price is uncertain and defined by a uniform probabilitydistributionfrom $20 to $26.
Variable cost is not constant but defined by a triangularprobability distribution.
Will determineaverageprofit and profitability with givenprobabilistic variables.
Crystal Ball
Simulation of Profit Analysis Model (2 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
43/65
14-43
Crystal Ball
Simulation of Profit Analysis Model (3 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
44/65
14-44
Crystal Ball
Simulation of Profit Analysis Model (4 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.12
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
45/65
14-45
Crystal Ball
Simulation of Profit Analysis Model (5 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.13
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
46/65
14-46
Crystal Ball
Simulation of Profit Analysis Model (6 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.14
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
47/65
14-47
Crystal Ball
Simulation of Profit Analysis Model (7 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.15
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
48/65
14-48
Crystal Ball
Simulation of Profit Analysis Model (8 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.16
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
49/65
14-49
Crystal Ball
Simulation of Profit Analysis Model (9 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.17
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
50/65
14-50
Crystal Ball
Simulation of Profit Analysis Model (10 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.18
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
51/65
14-51
C ystal Ball
Simulation of Profit Analysis Model (11 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.19
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
52/65
14-52
y
Simulation of Profit Analysis Model (12 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.20
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
53/65
14-53Exhibit 14.21
y
Simulation of Profit Analysis Model (13 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
54/65
14-54
y
Simulation of Profit Analysis Model (14 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall Exhibit 14.22
Crystal Ball
-
5/27/2018 Taylor Introms10 Ppt 14
55/65
14-55
y
Simulation of Profit Analysis Model (15 of 15)
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Exhibit 14.23
Verification of the Simulation Model (1 of 2)
-
5/27/2018 Taylor Introms10 Ppt 14
56/65
14-56
Analyst wants to be certain that model is internally correct and
that all operations are logical and mathematically correct.
Testing procedures for validity:
Run a small number of trials of the model and compare
with manually derived solutions.
Divide the model into parts and run parts separately to
reduce complexity of checking.
Simplify mathematical relationships (if possible) for
easier testing. Compare resultswith actual real-world data.
( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Verification of the Simulation Model (2 of 2)
-
5/27/2018 Taylor Introms10 Ppt 14
57/65
14-57
Analyst must determine if model starting conditions are correct
(system empty, etc).
Must determine how long model should run to insure steady-stateconditions.
A standard, fool-proof procedure for validation is not available.
Validity of the model rests ultimately on the expertise andexperience of the model developer.
( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Some Areas of Simulation Application
-
5/27/2018 Taylor Introms10 Ppt 14
58/65
14-58
Queuing
Inventory Control
Production and Manufacturing
Finance Marketing
Public Service Operations
Environmental and Resource Analysis
pp
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example Problem Solution (1 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
59/65
14-59
Willow Creek Emergency Rescue Squad
Minor emergency requires two-person crew
Regular emergency requires a three-person crewMajor emergency requires a five-person crew
p ( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example Problem Solution (2 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
60/65
14-60
Distribution of number of calls per night and emergency type:
Calls Probability
0123456
.05
.12
.15
.25
.22
.15
.061.00
Emergency Type Probability
MinorRegularMajor
.30
.56
.141.00
p ( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
1. Manually simulate 10 nights of calls
2. Determine average number of calls
each night
3. Determine maximum number ofcrew members that might be needed
on any given night.
Example Problem Solution (3 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
61/65
14-61
Calls ProbabilityCumulativeProbability
Random NumberRange, r1
01
23456
.05
.12
.15.25
.22
.15
.061.00
.05
.17
.32.57
.79
.941.00
1 56 17
18
3233 5758 7980 94
95 99, 00
EmergencyType
ProbabilityCumulativeProbability
Random NumberRange, r1
MinorRegularMajor
.30
.56
.141.00
.30
.861.00
1 3031 86
87 99, 00
Step 1: Develop random number ranges for the probability distributions.
p ( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example Problem Solution (4 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
62/65
14-62
Step 2: Set Up a Tabular Simulation (use second column of random
numbers in Table 14.3).
p ( )
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example Problem Solution (5 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
63/65
14-63
Step 2 continued:
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
Example Problem Solution (6 of 6)
-
5/27/2018 Taylor Introms10 Ppt 14
64/65
14-64
Step 3:Compute Results:
average number of minor emergency calls per night = 10/10 =1.0
average number of regular emergency calls per night =14/10 = 1.4
average number of major emergency calls per night = 3/10 = 0.30
If calls of all types occurred on same night, maximum number of
squad members required would be 14.
Copyright 2010 Pearson Education, Inc. Publishing as Prentice Hall
-
5/27/2018 Taylor Introms10 Ppt 14
65/65