Ch10 - Simulation

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Simulation

Chapter Ten


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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.


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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.



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!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.


6odeling and programming skills# as well asknowledge of statistics are re&uired when implementing

the simulation approach.


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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.


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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

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: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 =


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3.


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! 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

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*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


%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.

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The simulation is repeated and stops once total demand reaches

A< or more.


*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.

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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


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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


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Trials = 3

6ax=JK sales = = .(


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'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

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"ecause demand is uncertain# a simulation

models has $een developed.

! continuous review (+#R system is studiedfirst# where + = 3< and R = >.

$$C & /e Simulation Model

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The random num$er mapping associated with the distri$utionsare,/um$er of !rrivals 0ro$a$ility +andom mapping

< .3 ;> U BB

:emand2customer 0ro$a$ility +andom mapping

3 .3 > U BB

$$C & /e Simulation Model


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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

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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

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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


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$$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


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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.


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C$#!$L 7$N8

$n e4am,le of 9ueuin- sstem simulationCapital "ank is considering opening the $ank on

%aturdays morning from B,


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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.


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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 .


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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


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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

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C$#!$L & Simulation emonstration

6apping 5nterarrival time

; minutes

6apping !nns %ervice time

@> U A minutes

3.:!nn"ill

1.:1.:

1.: 1.: 1.:1.:

1.:

1.:1.:1.:

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6apping 5nterarrival time

; minutes

6apping "ills %ervice time

A minutes

!nn"ill 3 :.:

3.:1.:


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3.:3Waiting time


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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


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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.;


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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

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Ch10 - Simulation