Lect19 Applicationo of Probbability

7/27/2019 Lect19 Applicationo of Probbability

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1

Probability and its applications

Engineering

Economics

Management

Agriculture

etc.

Decision Analysis under Uncertainty

Reliability

Quality Control

Queueing Systems

Production and Inventory Control

Simulation

Probability and its applications

Modeling uncertainty with Probability

Uncertainty is a critical element of many decisions

Model depends on the nature of the uncertainty

The use of probability to model uncertainty

-Uncertainty quantities (random variables)

- Chance event

Sth about which a decision maker is uncertain

More than one possible outcome

The use of decision tree

Measuring uncertainty with Probability

Bayes Theorem

)B(P)B\A(P)B(P)B\A(P

)B(P)B\A(P)A\B(P

+=

Example (Aircraft engine)

Test a part of an aircraft engine before installation

75% chance of revealing a defect if present

75% chance of passing a good part

Part may undergo an rework operation to produce a part free from

defects

1/8 of parts are initially defective

CL

Rework

test

install

A1

A2

Rewo

rk

install

Rework

install

-L

0

L

-C

-L-C

-C

CL

Decision Tree Diagram Data from Example (defect)

25.0)\P(A75.0)\P(A

25.0)\P(A75.0)\P(A

8

7)P(

8

1)P(

costrework5

L

costlossLtestofcostC

defectarevealtdoesntestAdefectarevealstestA

pre sen tisdefectnopre sen tisdefecta

1222

2111

21

21

21

==

==

==

=

==

==

==


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Posterior Distribution by Bayes Theorem

0.6875P(A2)

0.9550.656250.757/8

0.0450.031250.251/8A2:

0.3125P(A1)

0.70.218750.257/8

0.30.093750.751/8A1:

Posterior

Probability

Prior Prob. x

Likelihood

LikelihoodPrior

Probability

1

1

2

2

Decision Tree Diagram

-L-C

8

7

C5

L

Rework

test

install

A1

A2

Rewo

rk

install1

1

2

2

1

Rework

install

-L

0

5

L

-C

-L-C

-C

C5

L

8

1

0.3125

0.6875

0.3

0.7

0. 045

0. 955

[-0.3L-C]

[-0.045L-C]

[-0.045L-C]

[-0.2L-C]

[-0.093L-C]

[-0.125L]

||

||

||

2

The possible optimal decision:

1. - 0.093L C < -0.125L

C > 0.032L do not test and install

2. - 0.093L C > -0.125L

C < 0.032L

test and if test reveals a defect, reworkif test doesn t reveal a defect, install

3. - 0.093L C = -0.125L

install or test

Making Decision

Example (Oil Wildcatter problem)

Not sure about cost to drill the well

Not sure whether will find oil

Two options for conducting test

Cost = Cost of test + Cost of drilling

Revenue from selling oil

Example

Influence Diagram for the Drilling Problem

Specify

Decisions

Uncertainties

Values

Relationships


Show precise sequence of events

0.5

0.3

0.2

0

120

270

-40

-50

-70

[-50]

[70]

[220]

[40]


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Max(40,0)



Drilling_Costs

Yes [40]

No [0]

Drill

None [40]

Seismic_Structure

Core_Sample

-10

[34.3]

Exp_Seismic_Test

Exp_Seismic

-3

[37]

Test

[40]

Result from Decision Tree Analysis

Sample

size n

X1

#defective in nStop or

Continue

X2

#defective

in N-n

Loss

Y1

#additional

inspections

%defective

Influence Diagram for Deming s inspection Software Design


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Applications of Poisson process in Reliability

n21n21 t.. .tt)},t(X),...,t(X),t(X{


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Network Reliability

s:source t:terminal

1

2

3

4

5

6

7

8

9

)x,x,...,x,x,(x:arcsofstate

otherwise0

works,iarcif1x

98321

i =

Success: there is at least one working path of nodes and arcs from s to t.

Network Reliability

Coherent system:

1. is increasing coordinatewise,

2. Each component is relevant.

=otherwise0

workssystemtheif1)x,...,x,(x 921

x),(0x),(1

ifirrelevantisComponent

ii =

relevant.aresystemcoherentaincomponentsall

1)1,...,1,1( =

Given P(Xi=1)=pi

Probability that there is at least one working path at a given

instant in time between the distinguished nodes s and t:

)p,...,p,p()( n21 hhDEF

=p

1)1,...,1,1( =h

For a coherent system,

Network Reliability Network Reliability

1 2

p1p2

p1p2

p1Bp2=p1+p2-p1p2

p1

p2

1

2

M

M

M

M

M M

M M

Factoring Algorithm

Series-parallel probabilityreductions

Pivoting

Conditions(1) Arcs fail independently

(2) Nodes are perfect

Network Reliability

Pivoting operation

tlyindependenfailarcswhenvalid

),)h(0p(1),h(1p)(

)p,...,p,,1p,...,p,(pp),(1

and)p,...,p,p(

iiii

n1ii1-i21i

n21

+=

==

+

pp

p

ph

Network Reliability

1 4

2

3

5

s t

1

2

3

4

5

s t

2

3

4

5

s t

)]pp(p)[pp(1]p))pp[((pp)p,...,p,p(h)(h

5432145321

521

CCC +==p


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Probability in Quality Control

Control Chart

-p Chart: for the rate of defective items

-c Chart: for the number of defects on a group of samples

p Chart

n

)p(1p3pCL

n

npp

=

==n

npp

Probability in Quality Control

p Chart

.

.

.

.

.

.

.

.

.

.

group

p

.

.

.

Probability Application in Queueing Systems

Queue

the current demand for service exceeds the current capacity

Examples

Commercial service systems

barber shop, bank teller service, checkout points, a gas station

Business-industrial internal service systems

Maintenance systems, material handling systems,

a computer facility

Transportation service systems

Car waiting at a tollbooth or traffic light,

airplanes waiting to land or take off,

a parking lot, elevators

Social service systems

A juridical system, health-care systems

Presonal Queues

Work to do, homework assignments, etc.

Probability Application in Queueing Systems

Queueing theory

Involves the mathematical study of queues or waiting lines

Provides information for making decisions

Goal: Achieve an economic balance between

cost of service and cost associated with waiting for the service

Queueing Theory

First applications to telephone engineering, A. K. Erlang

Structure of Queueing Models

Input

sourceQueue

Service

Mechanism

Customers

Queueing system

Served

customers

Input source

Infinite

finite

Interarrival

time

Queue

Infinite

finite

Service Mechanism

First-come-first

served

Random

Priority-based

Service time

Exponential

Degenerate

Erland

General


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M/M/1 Queueing systems

Basic model of M/M/1 System

Single server

Exponential distributed, time homogeneous, state

independent interarrival times with mean 1/

Exponentially distributed service times with mean 1/

L(t) = number of customers in system at time t, t 0

In steady state,

Rate In = Rate Out


A flow balance procedure (Stochastic Balance)

n-1 n n+1

0 = 1

n+1+n-1 = n+ n

0 1


1

,

1

1p

p1p1p1,pp

1,2,...n,p

p,p

p

0n

n0

1n

n

0

1n

n0

1n

n0

0

n

n01

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Lect19 Applicationo of Probbability