Date post: | 17-Feb-2018 |
Category: |
Documents |
Upload: | bambang-ariwibowo |
View: | 251 times |
Download: | 0 times |
of 51
7/23/2019 Ch10 - Simulation
1/51
1
Simulation
Chapter Ten
7/23/2019 Ch10 - Simulation
2/51
2
13.1 Overview of Simulation
When do we prefer to develop simulation modelover an analytic
model? When not all the underlying assumptions set for analytic model are valid.
When mathematical complexity makes it hard to provide useful results. When good solutions (not necessarily optimal are satisfactory.
! simulation develops a model to numerically evaluate a system
over some time period.
"y estimating characteristics of the system# the best alternative
from a set of alternatives under considerationcan $e selected.
7/23/2019 Ch10 - Simulation
3/51
3
Continuous simulation systems monitor the system each
time a change in its state takes place.
Discrete simulation systems monitor changes in a state
of a system at discrete points in time.
%imulation of most practical pro$lems re&uires the use of
a computer program.
13.1 Overview of Simulation
7/23/2019 Ch10 - Simulation
4/51
4
!pproaches to developing a simulation model'sing addins to )xcel such as *+isk or Crystal "all
'sing general purpose programming languages such as,-+T+!/# 0123# 0ascal# "asic.
'sing simulation languages such as 40%%# %56!/# %1!6.
'sing a simulator software program.
13.1 Overview of Simulation
6odeling and programming skills# as well asknowledge of statistics are re&uired when implementing
the simulation approach.
7/23/2019 Ch10 - Simulation
5/51
5
10.2 Monte Carlo Simulation
6onte Carlo simulation generates random events.
+andom events in a simulation model are needed
when the input data includes random varia$les.
To reflect the relative fre&uencies of the random
varia$les# the random number mappingmethod is
used.
7/23/2019 Ch10 - Simulation
6/51
6
7ewel 8ending Company (78C installs and
stocks vending machines.
"ill# the owner of 78C# considers the installation
of a certain product (%uper %ucker 9aw
$reaker in a vending machine located at a newsupermarket.
JEWEL VEN!N" COM#$N% &
an example for the random mapping techni&ue
7/23/2019 Ch10 - Simulation
7/517
:ata The vending machine holds ;< units of the product.
The machine should $e filled when it $ecomes half empty. :aily demand distri$ution is estimated from similar vending machine
placements. 0(:aily demand = < 9aw $reakers =
7/23/2019 Ch10 - Simulation
8/518
7/23/2019 Ch10 - Simulation
9/51
9
3.
7/23/2019 Ch10 - Simulation
10/51
10
! random demand can $e generated $y hand (for
small pro$lems from a ta$le of pseudo random
num$ers.'sing )xcel a random num$er can $e generated
$y
The +!/:( functionThe random num$er generation option (ToolsH:ata
!nalysis
Simulation of t/e JVC #ro+lem
7/23/2019 Ch10 - Simulation
11/51
11
*andom .wo 2irst .otal emand
a0 Num+er i-its emand to ate
3 C> @ @
? DDC3 DD A D
@ C3D< C3 @ 3? DA @ 3B
*andom .wo 2irst .otal emanda0 Num+er i-its emand to ate
3 C> @ @
? DDC3 DD A D
@ C3D< C3 @ 3? DA @ 3B
Simulation of t/e JVC #ro+lem
%ince we have two digit pro$a$ilities# we use the first two
digits of each random num$er.
BBAA
< 3 @ A > 3
!n illustration of generating a daily random demand.
7/23/2019 Ch10 - Simulation
12/51
12
The simulation is repeated and stops once total demand reaches
A< or more.
Simulation of t/e JVC #ro+lem
*andom .wo 2irst .otal emand
a0 Num+er i-its emand to ate
3 C> @ @
? DDC3 DD A D
@ C3D< C3 @ 3? DA @ 3B
*andom .wo 2irst .otal emanda0 Num+er i-its emand to ate
3 C> @ @
? DDC3 DD A D
@ C3D< C3 @ 3? DA @ 3B
The num$er of simulated days
re&uired for the total demand to
reach A< or more is recorded.
7/23/2019 Ch10 - Simulation
13/51
13
The purpose of performing the simulation runs is to find the
average num$er of days re&uired to sell A< 9aw $reakers.
)ach simulation run ends up with (possi$ly a different num$er
of days.
! hypothesis test is conducted to test whether or not = 3.
/ull hypothesis I< , =3
!lternative hypothesis I!, 3
Simulation *esults and ,ot/esis ests
7/23/2019 Ch10 - Simulation
14/51
14
The test,:efine (the significance level.
1et n $e the num$er of replication runs."uild the tstatistic
The t-statistic can be used if the random variable observed
(number of day required for the total demand to be 40 or more) is
normally distributed, hile the standard deviation is un!non"+e9ect I
7/23/2019 Ch10 - Simulation
15/51
15
Trials = 3
6ax=JK sales = = .(
7/23/2019 Ch10 - Simulation
21/51
21
'sing the template inventory.xls for the plannedshortage model# and assuming a constant
demand of units per week (@3 per year wehave,ptimal ordering policy,
RS = A.;; (rounded to >
%S= .3> (rounded to to K +eorder when inventory isat a level of 3;.A;
$$C & /e #lanned S/orta-e Model
7/23/2019 Ch10 - Simulation
22/51
22
"ecause demand is uncertain# a simulation
models has $een developed.
! continuous review (+#R system is studiedfirst# where + = 3< and R = >.
$$C & /e Simulation Model
7/23/2019 Ch10 - Simulation
23/51
23
The random num$er mapping associated with the distri$utionsare,/um$er of !rrivals 0ro$a$ility +andom mapping
< .3< U ;AA .3> ;> U BB
:emand2customer 0ro$a$ility +andom mapping
3 .3< U AA .@> > U BB
$$C & /e Simulation Model
7/23/2019 Ch10 - Simulation
24/51
24
The simulation keeps track of the following
&uantities,
"eginning inventory for the week = )nding inventory ofthe previous week L order received.
/um$er of retailers arriving# their demand# and the total
weekly demand.
)nding inventory for the week = "eginning inventory L
order received U weekly demand.
$$C & /e Simulation Lo-i
7/23/2019 Ch10 - Simulation
25/51
25
The simulation determines whether or not an order
should $e placed as follows,
5s the ending inventory3< and is there no outstandingorder? 5f so# place an order and keep track of the lead
time.
The simulation calculates the Weekly cost,rdering cost (if applica$le L Iolding cost (if ending
inventory H
7/23/2019 Ch10 - Simulation
26/51
26
5nitial inventory = >.
Weekly cost = rder cost (if any L 3(%tock on hand L (/ew$ack orders L >(Total $ackorders
Total cost for 3< weeks = NA3> (weekly average = NA3.>.
$$C & 10 wee5 simulation results
7/23/2019 Ch10 - Simulation
27/51
27
$$C & 1000 wee5s of simulation
s,reads/eet resultsINPUTS
Q = 25 Ch = 1
R = 10 Co = 45
Cs = 5
Cb = 2
OUTPUTAverage Cost = 33.109
a! Start o" #ee$ % o" C&sto'er Tota( )*+ o" #ee$ Tota(
I*ve*tor! Arr,va(s e'a*+ I*ve*tor! Cost
1 25 3 9 1- 1-
2 1- 2 9 54
3 9 3 11 /2 144 /2 1 4 /- 3
5 19 0 0 19 19
- 19 1 4 15 15
15 1 2 13 13
7/23/2019 Ch10 - Simulation
28/51
28
10.) Simulation of a 6ueuin- Sstem
5n &ueuing systems time itself is a random varia$le.
Therefore# we use the ne$t event simulationapproach.
The simulated data are updated each time a new event
takes place (not at a fixed time periods.
Theprocess interactive approach is used in this kind of
simulation (all relevant processes related to an item as it
moves through the system# are traced and recorded.
7/23/2019 Ch10 - Simulation
29/51
29
C$#!$L 7$N8
$n e4am,le of 9ueuin- sstem simulationCapital "ank is considering opening the $ank on
%aturdays morning from B,
7/23/2019 Ch10 - Simulation
30/51
30
C$#!$L 7$N8
:ata, There are > teller positions of which only three will
$e staffed.
!nn :oss is the head teller# experienced# and fast. "ill 1ee and Carla :omingueV are associate tellers less
experienced and slower.
7/23/2019 Ch10 - Simulation
31/51
7/23/2019 Ch10 - Simulation
32/51
32
C$#!$L 7$N8
:ata, Customer interarrival time distri$ution
interarrival time 0ro$a$ility
.> 6inutes .>
3 .3>
3.> .3>
.
%ervice priority rule is first come first served
! simulation model is re&uired to analyVe the service .
7/23/2019 Ch10 - Simulation
33/51
33
Calculating expected values, )(interarrival time = .>(.>L3(.3>L3.>(.3>L(. = .; customers arrive per hour on the average#
(Q
)(service time for !nn = .3(.L3(.3(. =
minutes P!nn can serve
7/23/2019 Ch10 - Simulation
34/51
7/23/2019 Ch10 - Simulation
35/51
35
5f no customer waits in line# an arriving customer seeks service $ya free teller in the following order, !nn# "ill# Carla.
5f all the tellers are $usy the customer waits in line and takes then
the next availa$le teller.The waiting time is the time a customer spends in line# and iscalculated $y
&Time service beginsQ minus&'rrival Time
C$#!$L 7$N8 & /e Simulation lo-i
7/23/2019 Ch10 - Simulation
36/51
36
C$#!$L & Simulation emonstration
6apping 5nterarrival time
;< U BA 3.> minutes
6apping !nns %ervice time
@> U A minutes
3.:!nn"ill
1.:1.:
1.: 1.: 1.:1.:
1.:
1.:1.:1.:
7/23/2019 Ch10 - Simulation
37/51
37
C$#!$L & Simulation emonstration
6apping 5nterarrival time
;< U BA 3.> minutes
6apping "ills %ervice time
A< U B .> minutes
!nn"ill 3 :.:
3.:1.:
7/23/2019 Ch10 - Simulation
38/51
38
C$#!$L & Simulation emonstration
3.:3Waiting time
7/23/2019 Ch10 - Simulation
39/51
39
Average Waiting Time in Line = 1.-0
Average Waiting Time in System = 3.993
Waiting Waiting
Random Arrival Random Time Time
Customer Number Time Number Start Finish Start Finish Start Finish Line System
1 0. 1.5 0.9- 1.5 5 0 3.5
2 0.1 2.0 0.- 2 5 0 3.0
3 0.49 2.5 0. 2.5 5.5 0 3.0
4 0.- 4.0 0.49 5 1 3.0
5 0.54 4.5 0.5 5 .5 0.5 4.0
- 0.-1 5.0 0.55 5.5 0.5 3.0
0.91 -.5 0.90 10 0.5 3.5 0.-4 .0 0.-2 10.5 1 3.5
Ann Bill Carla
C$#!$L & 1000 Customer Simulation
7/23/2019 Ch10 - Simulation
40/51
40
Average Waiting Time in Line = 1.-0
Average Waiting Time in System = 3.993
Waiting Waiting
Random Arrival Random Time Time
Customer Number Time Number Start Finish Start Finish Start Finish Line System
1 0. 1.5 0.9- 1.5 5 0 3.5
2 0.1 2.0 0.- 2 5 0 3.0
3 0.49 2.5 0. 2.5 5.5 0 3.0
4 0.- 4.0 0.49 5 1 3.0
5 0.54 4.5 0.5 5 .5 0.5 4.0
- 0.-1 5.0 0.55 5.5 0.5 3.0
0.91 -.5 0.90 10 0.5 3.5 0.-4 .0 0.-2 10.5 1 3.5
Ann Bill Carla
C$#!$L & 1000 Customer Simulation
This simulation estimates two performance measures, !verage waiting time in line (W& = 3.D minutes
!verage waiting time in the system W = @.BB@ minutes
This simulation estimates two performance measures, !verage waiting time in line (W& = 3.D minutes
!verage waiting time in the system W = @.BB@ minutes
To determine the other performance measures# we can use1ittles formulas, !verage num$er of customers in line 1& =(32.;
7/23/2019 Ch10 - Simulation
41/51
41
Ma,,in- for Continuous *andom Varia+les
)xampleThe )xplicit inverse distri$ution method can $e used to
generate a random num$er E from the exponentialdistri$ution with = (i.e. service time is exponentiallydistri$uted# with an average of customers perminute.
+andomly select a num$er from the uniform distri$ution $etween