p7 Sig Pertambangan Ahp

8/10/2019 p7 Sig Pertambangan Ahp

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SIG Pertambangan(Theory of Spatial Analysis : Metode AHP)

Oleh :

Irvani

Universitas Bangka Belitung Jurusan Teknik Pertambangan

SKS, Penilaian & Kehadiran :

Banyaknya SKS = 2 SKS (Teori)

Penilaian :

- Absensi 10%- Tugas 20%- Teori (UTS & UAS) 70%

Kehadiran minimal 75% dari 14x perkualiahan



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Referensi :


Bonham-Carter, G.F. (1994) Geographic Information System for Geoscientists: Modellingwith GIS. Delta Print ing , Ontario, 398 p.

Harris, J.R. (ed) (2006) GIS For The Earth Sciences. GAC Special Paper 44, Geological As sociati on of Can ada, 616 p.

de By, R.A. (ed) (2000) Principles of Geographic Information Systems. ITC educationalTexbook Series, Netherlands.

Huisman, O. And de By, R.A. (2009) Principles of Geographic Information Systems. ITCeducational Texbook Series, Netherlands.

Mitchel, A. (1999) The ESRI guide to GIS Analysis. Volume 1: Geographic patterns &Relationship s, ESRI Press, 186 pp.

Kennedy, H. (ed) (2001) Dictionary of GIS terminolog y. ESRI Press, Redlands, 116 p. Longley, P.A., Goodchild, M.F., Maguire, D.J. and Rhind, D.W. (2001) Geographic

Informati on Systems and Scienc e. John Wiley & Sons, 454 pp. Maguir e, D. J., Goodch ild, M. F., and Rhind, D. W. (eds) (1991) Geographical in formati on

systems: principles and applications, Longman. Zeiler, M. (1999) Modeling Our Wor ld: th e ESRI Guide to Geodatabase Design. ESRI Press,

Redlands, 198 p. ESRI Homepage ( http://esri.com /index.html ) : understand ing GIS, indu stry applicati ons,

user conference, virtual campus, ESRI Press books

Materi/Pokok BahasanI Pendahuluan (P.1)

II Overview of GIS (P.2)

III Map Projection andCoordinate System (P.3-4)

IV GIS for Geoscience (P.5)

V GIS Database (P.6)

VI Theory of Spatial Analysis (P.7-9)a. Metode AHPb. Principle Steps

in GIS Spatial

c. GIS ProcessingVII Introduction to ArcGIS or

MapInfo (P.10) (Option)

VIII Case Studies/Latihan (P.11-14)



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Teori AHP 1Teori AHP 1

Analytic Hierarchy Process Multiple-criteria decision-making

Real world decision problems multiple, diverse criteria qualitative as well as quantitative information

Comparing apples and oranges?Spend on defence or agriculture?Open the refrigerator - apple or orange?


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AHP

Information is decomposed into a hierarchy ofalternatives and criteria

Information is then synthesized to determinerelative ranking of alternatives

Both qualitative and quantitative information canbe compared using informed judgements to

derive weights and priorities

Example: Car Selection

Objective Selecting a car

Criteria

Style, Reliability, Fuel-economyCost?

Alternatives Civic Coupe, Saturn Coupe, Ford Escort,

Mazda Miata


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Hierarchi cal tree

Style Reliability Fuel Economy

Selectinga New Car

- Civic

- Saturn- Escort- Miata

- Civic


- Civic


Ranking of criteria

Weights? AHP

pair-wise relative importance

[1:Equal, 3:Moderate, 5:Strong, 7:Verystrong, 9:Extreme]

Style Reliability Fuel Economy

Style

Reliability

Fuel Economy

1/1 1/2 3/1

2/1 1/1 4/1

1/3 1/4 1/1


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Ranking of pr iorities

Eigenvector [Ax = x]Iterate

1. Take successive squared powers of matrix2. Normalize the row sums

Until difference between successive row sums

is less than a pre-specified value

1 0.5 32 1 40.333 0.25 1.0

3.0 1.75 8.05.3332 3.0 14.01.1666 0.6667 3.0

squared

Row sums

12.7522.33324.8333

39.9165

NormalizedRow sums

0.31940.55950.1211

1.0

New iteration gives normalized row sum0.31960.55840.1220

Difference is: -0.31940.55950.1211

0.31960.55840.1220

=- 0.0002

0.0011- 0.0009


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Preference Style .3196 Reliability .5584 Fuel Economy .1220

Style.3196

Reliability.5584

Fuel Economy.1220

Selectinga New Car

1.0

Ranking alternatives

Style

Civic

Saturn

Escort

1/1 1/4 4/1 1/6

4/1 1/1 4/1 1/4

1/4 1/4 1/1 1/5

Miata 6/1 4/1 5/1 1/1

Civic Saturn Escort Miata

Miata

Reliability

Civic

Saturn

Escort

1/1 2/1 5/1 1/1

1/2 1/1 3/1 2/1

1/5 1/3 1/1 1/4

Miata 1/1 1/2 4/1 1/1

Civic Saturn Escort Miata

.1160

.2470

.0600

.5770

Eigenvector

.3790

.2900

.0740

.2570


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Fuel Economy(quantitative

information)

Civic

Saturn

Escort

MiataMiata

34

27

24

28113

Miles/gallon Normalized

.3010

.2390

.2120

.24801.0

Style.3196 Reliability.5584 Fuel Economy.1220

Selectinga New Car

1.0

- Civic .1160- Saturn .2470- Escort .0600- Miata .5770




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Ranking of alternatives

Style Reliability FuelEconomy

Civic

Escort

MiataMiata

Saturn

.1160 .3790 .3010

.2470 .2900 .2390

.0600 .0740 .2120

.5770 .2570 .2480

*.3196

.5584

.1220

=

.3060

.2720

.0940

.3280

Handling Costs

Dangers of including Cost as another criterion political, emotional responses?

Separate Benefits and Costs hierarchical trees Costs vs. Benefits evaluation

Alternative with best benefits/costs ratio


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Cost vs. Benefits

MIATA $18K .333.9840

CIVIC $12K .2221.3771

SATURN $15K .2778.9791

ESCORT $9K .1667 .5639

CostNormalized

CostCost/Benefits

Ratio

Complex decisions

Many levels of criteria and sub-criteria


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Application areas strategic planning resource allocation source selection, program selection business policy etc., etc., etc..

AHP software (ExpertChoice) computations sensitivity analysis graphs, tables

Group AHP

Teori AHP 2


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Analytical Hierarchy Process (AHP)- by Saaty

Another way to structure decision problem Used to prioritize alternatives Used to build an additive value function Attempts to mirror human decision process Easy to use

Well accepted by decision makers Used often - familiarity Intuitive

Can be used for multiple decision makers Very controversial!

What do we want to accomplish?

Learn how to conduct an AHP analysis Understand the how it works Deal with controversy

Rank reversal Arbitrary ratings

Show what can be done to make it useable

Bottom Line: AHP can be a useful tool. . . but itcant be used indiscriminately!


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AHP Procedure Build t he Hierarchy

Very similar to hierarchical value structure Goal on top (Fundamental Objective) Decompose into sub-goals (Means objectives) Further decomposition as necessary Identify criteria (attributes) to measureachievement of goals (attributes and objectives)

Alternatives added to bottom Different from decision tree Alternatives show up in decision nodes Alternatives affected by uncertain events Alternatives connected to all criteria

Building the Hierarchy

SecondaryCriteria

Ford Taurus

Goal

Lexus Saab 9000

General C riteria

Alternatives

Braking Dis t Turning Radius

Handling

Purchase Cost Maint Cost Gas Mileage

Economy

Time 0-60

Power

Buy the bestCar

Note: Hierarchy corresponds to decision maker values No right answer Must be negotiated for group decisions

Example: Buying a car

Affinity

Diagram


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AHP Procedure Judgments andComparisons

Numerical Representation Relationship between two elements that share a common

parent in the hierarchy Comparisons ask 2 questions:

Which is more important with respect to the criterion? How strongly?

Matrix shows results of all such comparisons Typically uses a 1-9 scale Requires n(n-1)/2 judgments Inconsistency may arise

1 -9 ScaleIntensity of Importance Definition

1 Equal Importance

3 Moderate Importance

5 Strong Importance

7 Very Strong Importance

9 Extreme Importance

2, 4, 6, 8 For compromises between the above

Reciprocals of above In comparing elements i and j- if i is 3 compared to j- then j is 1/3 compared to i

Rationals Force consistencyMeasured values available


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Example - Pairwise Comparisons

Consider following criteriaPurchase Cost Maintenance Cost Gas Mileage

Want to find weights on these criteria AHP compares everything two at a time

(1) Compare Purchase Cost to Maintenance Cost

Which is more important?Say purchase cost

By how much? Say moderately 3

Example - Pairwise Comparisons

(2) Compare Purchase Cost to

Which is more important?Say purchase cost

By how much? Say more important 5

Gas Mileage

(3) Compare to

Which is more important?Say maintenance cost

By how much? Say more important 3

Gas MileageMaintenance Cost


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Consistency And Weights So consistent matrix for the car example

would look like:

P

M

G

P M G

1

1

1

3 5

1/3

1/5

5/3

3/5

Note that matrixhas Rank = 1 That means thatall rows are multiplesof each other

Weights are easy to compute for this matrix Use fact that rows are multiples of each other Compute weights by normalizing any column

We getw P 1523 0.65 , w M 523 0.22 , w G 323 0.13

Weights for Inconsistent Matrices

More difficult - no multiples of rows Must use some averaging technique Method 1 - Eigenvalue/Eigenvector Method

Eigenvalues are important tools in several math,

science and engineering applications- Changing coordinate systems- Solving differential equations- Statistical applications

Defined as follows: for square matrix A and vector x, Eigenvalue of A when Ax = x, x nonzerox is then the eigenvector associated with

Compute by solving the characteristic equation:det( I A) = | I A | = 0


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Weights for Inconsistent Matrices Properties: - The number of nonzero Eigenvalues for a matrix is

equal to its rank (a consistent matrix has rank 1) - The sum of the Eigenvalues equals the sum of the diagonal elements of the matrix (all 1s for consistent matrix) Therefore: An nx n consistent matrix has one

Eigenvalue with value n

Knowing this will provide a basis of determiningconsistency Inconsistent matrices typically have more than 1 eigen value - We will use the largest, , for the computationmax

Weights for Inconsistent Matrices

Compute the Eigenvalues for the inconsistentmatrix

P

M

G

P M G

1

1

1

3 5

1/3

1/5

3

1/3

= A

w = vector of weights Must solve: Aw = w by solving det( I A) = 0 We get:

10.0,26.0,64.0 G M P www Different than before!

max = 3.039find the Eigen vector for 3.039 and normalize


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Measuring Consistency

Recall that for consistent 3x3 comparisonmatrix, = 3

Compare with from inconsistent matrix Use test statistic:

max

C.I. n

n 1 Consistency Index

max

From Car Example:C.I. = (3.0393)/(3-1) = 0.0195

Another measure compares C.I. with randomly generatedonesC.R. = C.I./R.I. where R.I. is the random index

n 1 2 3 4 5 6 7 8R.I. 0 0 .52 .89 1.11 1.25 1.35 1.4

Measuring Consistency For Car Example:

C.I. = 0.0195n = 3

R.I. = 0.52 (from table)So, C.R. = C.I./R.I. = 0.0195/0.52 = 0.037

Rule of Thumb: C.R. 0.1 indicates sufficientconsistency

Care must be taken in analyzing consistency Show decision maker the weights and ask forfeedback


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Weights for Inconsistent Matrices(continued)

Method 2: Geometric Mean Definition of the geometric mean:

Given values x 1, x 2, , x n

xg xii 1

n

n geometric mean

Procedure:

(1) Normalize each column(2) Compute geometric mean of each row

Limitation: lacks measure of consistency

Weights for Inconsistent Matrices(continued)

Car example with geometric means

P

M

G

P M G

1

11

3 5

1/3

1/5

3

1/3

Normalized P

M

G

P M G

.65

.23.11

.69 .56

.22

.13

.33

.08

w

w

w

p

M

G

= [(.65)(.69)(.56)]1/3

= [(.22)(.23)(.33)]1/3

= [(.13)(.08)(.11)]1/3

= 0.63

= 0.26

= 0.05

Normalized

0.67

0.28

0.05


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p7 Sig Pertambangan Ahp

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