### Simulation - Hatem Masri

Chapter 6
Simulation
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Simulation
Modeling
Random Variables and Pseudo-Random Numbers
Time Increments
Simulation Languages
Validation and Statistical Considerations
Examples
• Risk Analysis
• Waiting Line Simulation
Slide 1
What is Simulation?
An attempt to duplicate the features, appearance, and
characteristics of a real system
1. To imitate a real-world situation mathematically
2. To study its properties and operating
characteristics
3. To draw conclusions and make action decisions
based on the results of the simulation
Slide 2
Simulation Applications
Ambulance location and
dispatching
Assembly-line balancing
Parking lot and harbor design
Distribution system design
Scheduling aircraft
Labor-hiring decisions
Personnel scheduling
Traffic-light timing
Voting pattern prediction
Bus scheduling
Design of library operations
Production facility scheduling
Plant layout
Capital investments
Production scheduling
Sales forecasting
Inventory planning and control
Slide 3
The
Process of
Simulation
Define problem
Introduce variables
Construct model
Specify values
of variables
Conduct simulation
Examine results
Select best course
Slide 4
1. Relatively straightforward and flexible
2. Can be used to analyze large and complex realworld situations that cannot be solved by
conventional models
3. Real-world complications can be included that
most OR models cannot permit
4. “Time compression” is possible
Slide 5
5. Allows “what-if” types of questions
6. Does not interfere with real-world systems
7. Can study the interactive effects of individual
components or variables in order to determine
which ones are important
Slide 6
1. Can be very expensive and may take
months to develop
2. It is a trial-and-error approach that may
produce different solutions in repeated
runs
3. Managers must generate all of the
conditions and constraints for solutions
they want to examine
4. Each simulation model is unique
Slide 7
Monte Carlo Simulation

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Select numbers randomly from a probability
distribution
Use these values to observe how a model performs
over time
Random numbers each
have an equal likelihood
of being selected
at random
Slide 8
Distribution of Demand
LAPTOPS DEMANDED
PER WEEK,
0
1
2
3
4
FREQUENCY OF
DEMAND
PROBABILITY OF
DEMAND, P(x)
CUMULATIVE
20
40
20
10
10
0.20
0.40
0.20
0.10
0.10
100
1.00
0
0.20
0.60
0.80
0.90
Slide 9
Roulette Wheel of Demand
0
90
x=4
80
x=0
x=3
20
x=2
x=1
60
Slide 10
Generating Demand from
Random Numbers
DEMAND,
x
RANGES OF RANDOM NUMBERS,
r
0
1
2
3
4
0-19
20-59
60-79
80-89
90-99
r = 39
Slide 11
Random Number Table
39
73
72
75
37
65
71
18
12
17
76
23
47
25
79
45
70
33
69
88
45
90
84
17
74
19
65
51
17
63
90
97
67
95
52
69
60
47
21
06
64
12
97
78
34
61
11
19
58
30
Slide 12
15 Weeks of Demand
WEEK
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
r
39
73
72
75
37
02
87
98
10
47
93
21
95
97
69
DEMAND (x)
REVENUE (S)
1
2
2
2
1
0
3
4
0
1
4
1
4
4
2
4,300
8,600
8,600
8,600
4,300
0
12,900
17,200
0
4,300
17,200
4,300
17,200
17,200
8,600
 = 31
\$133,300
Average demand
= 31/15
= 2.07 laptops/week
Slide 13
Computing Expected Demand
E(x) = (0.20)(0) + (0.40)(1) + (0.20)(2)
+ (0.10)(3) + (0.10)(4)
= 1.5 laptops per week
Not particularly close to simulated
result of 2.07 laptops
Difference is due to small
number of periods analyzed
Slide 14
Random Numbers in Excel
Slide 15
Simulation in Excel
Enter this formula
in G6 and copy to
G7:G20
Enter “=4300*G6”
in H6 and copy to
H7:H20
Generate random
numbers for cells
F6:F20 with the
formula “=RAND()”
in F6 and copying to
F7:F20
= AVERAGE (G6:G20)
Slide 16
Simulation in Excel
Slide 17
Example of Risk Analysis
PortaCom Project
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PortCom’s product design group has developed a
prototype for a new high-quality portable printer.
The new printer has an innovative design and the
potential to capture a significant share of the portable
printer market. Preliminary marketing and financial
analysis have provided the following information.
Selling price = \$249 per unit
PortaCom believes that the costs and the demand
range as follows:
Unit direct labor cost = \$43~\$47
Unit parts cost = \$80~\$100
First-year demand = 1500~28,500 units
Slide 18
Simulation
The advantage of simulation is that it allows us to
assess the probability of a profit and the probability
of a loss.
Procedure of simulation
1. Check parameters
2. Check controllable inputs
3. Check probabilistic inputs
* Generate random numbers

4. Formulate a model
5. Draw a flowchart
Slide 19
Simulation
1. Check parameters
Selling price = \$249 per unit
2. Check controllable inputs
Whether or not introduce the product
3. Check probabilistic inputs
Unit direct labor cost range = \$43~\$47
Unit parts cost range = \$80~\$100
First-year demand range = 1500~28,500 units
Slide 20
Simulation
4. Formulate a model
Profit=(249-c1-c2)X-1,000,000
5. Draw a flowchart
Slide 21
Probability Distribution of the
Direct Labor Cost
Direct labor cost
\$43
\$44
\$45
\$46
\$47
Probability
0.1
0.2
0.4
0.2
0.1
Slide 22
Probability Distribution of the
Parts Costs

The probability distribution for the parts cost per unit
is the uniform distribution as follows:
Slide 23
Probability Distribution of the
First-year Demand

The first-year demand is described by the normal
probability distribution with mean 15,000 units and
the standard deviation 45000 units as follows:
Slide 24
How to Generate Random
Numbers

Computer-generated random numbers
* Assign ranges of random numbers to
to corresponding values of probabilistic
inputs. The prob. of any input value is
identical to the prob. of its occurrence in the
real system.
* Placing =RAND() in a cell of an Excel
worksheet will result in a random number.
Slide 25
Generate Random Value for
Direct Labor Cost
Interval of
Direct labor cost
Probability random numbers
\$43
0.1
0.0~0.1
\$44
0.2
0.1~0.3
\$45
0.4
0.3~0.7
\$46
0.2
0.7~0.9
\$47
0.1
0.9~1.0
*Excel Statement
=Vlookup(Rand(),range, Col_index)
Slide 26
Generate Random Numbers for
Parts Cost
With a uniform probability distribution, the
following relationship between the random number
and the associated value of the parts cost is used.
Parts cost=a+r(b-a)
where r=random number
a=smallest value for parts cost
b=largest value for parts cost
Parts cost=80+r(100-80)=80+r20
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Slide 27
Generate Random Numbers for
First-year Demand
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Because first-year demand is normally distributed,
we need a procedure for generating random values
from a normal distribution.
We use the following formula of Excell
=NORMINV(RAND(),mean,standard deviation)
Slide 28
Waiting Line Simulation

HKSB Savings Bank will open several new branch
bank during the coming year. Each new branch is
designed to have one automated teller machine
(ATM). A concern is that during busy periods several
customer may have to wait to use the ATM. This
concern prompted the bank to undertake a study of
the waiting line system. The bank’s vise president
want to determine whether one ATM will be
sufficient. The bank established service guidelines for
its ATM system stating that the average customer
waiting time for an ATM should be one minute or
less
Slide 29
Waiting Line Simulation

Customer Arrival Times
Interval Time = a + r (b-a)
r = random number between o and 1
a = minimum interarrival time
b = maximum interarrival time
For the HKSB ATM System, the minimum interarrival
time is a = 0 minutes, and the maximum interarrival time
is b = 5 minutes
Interval Time = 0 + r (5 - 0)= 5r
Slide 30

Customer Arrival Times
For the HKSB ATM System, the minimum interarrival time
is a = 0 minutes, and the maximum interarrival time is b = 5
minutes
Interval Time = 0 + r (5 - 0)= 5r
Assume that the simulation run begins at time = 0,
A random number of r= 0.2804 generates an interval time of
5(0.2804) = 1.4 minutes for customer 1.
A second random number of r=0.2598 generates an
interarrival time of 5(0.2598) = 1.3 minutes, indicating that
customer 2 arrive 1.3 minutes after customer 1. Thus
customer 2 arrives 1.4 + 1.3 = 2.7 minutes after the
simulation begin.
Continuing, a third random number of r = 0.9802 indicates
that customer 3 arrives 4.9 minutes after customer 2, which
7,6 minutes after the simulation begin.
Slide 31
Waiting Line Simulation

HKSB ATM Simulation Model
Interarrival
Time
Number
of ATMs
Service
Time
Operating
Character
istic
Model
Interarrival Times (Uniform Distribution)
Smallest Value
Largest Value
Service Times (Normal Distribution)
Mean
Std Deviation
0
5
2
0.5
Slide 32
Waiting Line Simulation
• Simulation results for 10 ATM Customer
Customer
Interarrival
Time
Arrival
time
Service
Start Time
Waiting
Time
Service
Time
Completion
Time
Time in
System
1
1.4
1.4
1.4
0.0
2.3
3.7
2.3
2
1.3
2.7
3.7
1.0
1.5
5.2
2.5
3
4.9
7.6
7.6
0.0
2.2
9.8
2.2
4
3.5
11.1
11.1
0.0
2.5
13.6
2.5
5
0.7
11.8
13.6
1.8
1.8
15.4
3.6
6
2.8
14.6
15.4
0.8
2.4
17.8
3.2
7
2.1
16.7
17.8
1.1
2.1
19.9
3.2
8
0.6
17.3
19.9
2.6
1.8
21.7
4.4
9
2.5
19.8
21.7
1.9
2.0
23.7
3.9
10
1.9
21.7
23.7
2.0
2.3
26.0
4.3
Total
21.7
11.2
20.9
Averages
2.17
1.12
2.09
Slide 33