Transcript The Analytical Hierarchy Process (AHP)

The Analytical Hierarchy
Process (AHP)
AHP and Risk-based Decision
Making
• A major goal of this course: “Enhance
capability of course participants to make
fully conceptualized and informed
decisions…..”
• AHP is a hybrid decision-analytic model
-- Not MAUT (but has elements of OR)
-- Not psychological model (but consistent
with some psychological theory)
AHP: a tool for decision analysis
• Helps to deal with multiple and conflicting
decision criteria
• Helps extend one’s ability to deal logically
with complex situations
• Combines decision-maker’s value
judgments with information about decision
situation
• Obtains a set of priority measures for
evaluating decision alternatives
The Aswan High Dam Project
• Near first cataract of Nile on Egypt: for control
•
•
•
•
of flood waters
First dam completed in 1902; found to be
inadequate (1946 dam almost overflowed)
Second dam planned in the 1950s, built in the
1960s
Provides electric power, mitigates risk of drought
and famine, enables fishing industry as well as
farmland and cotton fields
Entails many environmental issues (e.g. silt
deposits, erosion, increased salinity, diseases)
Numerous categories of application
• Electric utility planning
• Energy policy
• Site selection
• Portfolio analysis
• Strategic planning
• Risk assessment
• Resource allocation
• Many others
Countless instances of utilization
• Just about every federal government
agency
• Many Fortune 500 firms
• Risk managers, portfolio managers
• Dozens of doctoral dissertations, and
myriads of papers written about it, and
written using it
Conceptual Basis
• Start by identifying “decision elements”
• Decision elements include (a), alternative
feasible courses of action (choice set), (b)
criteria or attributes that will be used to
prioritize them, (c) the objective(s) of the
decision, and (d) the relations between
these elements
Categories of “elements”
• Goals
• Alternatives
• Criteria
“Structuring a hierarchy”
• The elements are used to “structure a hierarchy”
• Structure: “a persistent pattern of relations
•
•
which hold between the components of a given
system”
Hierarchy: a form of organization resembling a
pyramid, in which each level is subordinate to
the one above it
The structure is hierarchical because the
decision elements may exist at different levels of
abstraction
Hierarchies in AHP
• The simplest AHP hierarchy has three levels (one
•
choice set, one set of attributes, and one goal or
objective)
Regardless of the numbers of levels the process
is always the same: (a) identify decision
elements, structure the hierarchy, (b) weight the
attributes, (c) weight the alternatives, (d) obtain
composite weights, (e) check for consistency
Structuring using hierarchies
Decision
Criteria 1
Criteria 2
Criteria n
Alternative 1
Alternative 2
Alternative n
• Each element in a level has influence on element
in a higher level
Unidimensional vs.
multidimensional preferences
• If we have a single criteria C
• Two alternatives A1 and A2
A1 is preferred to A2 using criteria C
then chose A1
• Problem when multiple criteria are present
and when preferences conflict
Multiple alternatives, multiple
criteria: a simple example
BUY A NEW CAR
Economy
Appearance
Honda Accord
Mazda Miata
Performance
Capacity
Ford Taurus
Analytic Hierarchy Process
• Systematic method for using hierarchies
to structure a decision problem
• Elicit preferences at various levels of the
hierarchy and combines these preferences
to help make multi-criteria decisions
Systematic Analysis of Preferences
• Map qualitative scale to numeric scales
• Do so in as simple of a way as is feasible
• Do so systematically, throughout the
hierarchy
• Synthesize the judgments afterwards
• Enable estimation of consistency
Numeric Mapping
• AHP utilizes a 1-9 scale
Intensity of
importance
Definition
1
Equal importance
Two activities contribute equally to
the objective
3
Weak importance of one
Experience and judgment slightly
favor one over another activity over
another
5
Essential or strong
Experience and judgment strongly
favor one importance activity over
another
Explanation
The AHP scale
7
Very strong or demonstrated An activity is favored very strongly
importance
over another; its dominance
demonstrated in practice
9
Absolute importance
2, 4, 6, 8
Intermediate values between When compromise is needed
adjacent scale values
Reciprocals of
If activity I has one of the above A reasonable assumption
nonzero numbers assigned to it
when compared with activity J,
then J has the reciprocal value when
compared with I
The evidence favoring one activity
over another is of the highest possible
order of affirmation
An automobile selection example:
step 1
A.
Car A
Car B
Car C
Price
13100
11200
9500
MPG
18
23
29
Interior
Deluxe
Above Average
Standard
Body 4-Door Mid-size 2-Door Sport 2-door compact
Automobile selection: step 2
B.
Hierarchy of Decisions
Select the best car
Price
M PG
C om fort
Style
C ar A
C ar B
C ar C
C ar A
C ar B
C ar C
C ar A
C ar B
C ar C
C ar A
C ar B
C ar C
Automobile selection: step 3
C.
Establish Priorities
1. The priorities of the four criteria in terms of the overall goal.
2. The priorities of the three cars in terms of the purchase-price criterion.
3. The priorities of the three cars in terms of the MPG criterion.
4. The priorities of the three cars in terms of the comfort criterion.
5. The priorities of the three cars in terms of the style criterion.
Automobile selection: step 4
D. Pairwise Comparison Scale
Verbal Judgement of Preference
Extremely Preferred
Very strong to extremely
Very strongly preferred
Strongly to very strongly
Strongly preferred
Moderately to strongly
Moderately preferred
Equally to moderately
Equally preferred
Numerical Rating
9
8
7
6
5
4
3
2
1
Automobile selection: step 5
E. Pairwise comparison matrix showing preferences for the three cars in terms of comfort
Comfort
Car A
Car B
Car C
Car A
1
2
8
Car B
0.5
1
6
Car C
0.125
0.166666667
1
Automobile selection: step 6
F. Synthesizing Judgments
Comfort
Car A
Car B
Car C
Car A
1
0.5
0.125
1.625
Car B
2
1
0.166666667
3.166666667
Car C
8
6
1
15
Sum the columns in the pairwise comparison matrix.
Automobile selection: step 7
Comfort
Car A
Car B
Car C
Car A
0.615384615
0.307692308
0.076923077
1
Car B
Car C
0.631578947 0.533333
0.315789474
0.4
Divide elements by the column totals.
0.052631579 0.066667
1
1
Automobile selection: step 8
Priority Vector for Comfort
Comfort
Car A
Car A
0.615384615
Car B
0.307692308
Car C
0.076923077
Car A
Car B
Car C
0.593
0.341
0.065
Car B
0.631578947
0.315789474
0.052631579
Car C
0.533333 0.593432
0.4
0.341161
0.066667 0.065407
The priority vector for the cars with respect to comfort.
Sum the rows.
Gather other pairwise comparison
judgments
G. Other Pairwise Comparisons
Price
Car A
Car A
1
Car B
3
Car C
4
Car B
0.33333333
1
2
Car C
0.25
0.5
1
Priority Vectors
0.123
0.32
0.557
MPG
Car A
Car B
Car C
Car A
1
4
6
Car B
0.25
1
3
Car C
0.166667
0.333333
1
0.087
0.274
0.639
Style
Car A
Car B
Car C
Car A
1
3
0.25
Car B
0.33333333
1
0.142857143
Car C
4
7
1
0.265
0.655
0.08
Automobile selection: step 9
H. Pairwise comparison matrix for the four criteria in the car selection problem.
Criterion
Price
MPG
Comfort
Style
Price
1
3
2
2
MPG
0.333333333
1
0.25
0.25
Comfort
0.5
4
1
0.5
Style
0.5
4
2
1
I. Priorities for the overall goal.
Price
0.398
MPG
0.085
Comfort
0.218
Style
0.299
Automobile selection: step 10
J. Developing an overall priority ranking.
Criterion
Car A
Alternative
Car B
Car C
Price
0.123
0.32
0.557
MPG
0.087
0.274
0.639
Comfort
0.593
0.341
0.065
Style
0.265
0.655
0.08
Overall car A priority = .398(0.123)+0.085(0.087)+).218(0.593)+0.299(0.265)=.265
Overall car B priority = .398(0.320)+0.085(0.274)+).218(0.341)+0.299(0.655)=.421
Overall car C priority = .398(0.557)+0.085(0.639)+).218(0.066)+0.299(0.080)=.314
Automobile selection: step 11
K. Final AHP Ranking of Alternatives
Car B
Car C
Car A
0.421
0.314
0.265
1
Final choice: a mid-life crisis Maserati
A hazardous facility siting example
• Assume 5 alternative feasible sites (j =
1,2,3,4,5) (S1,S2,S3,S4,S5)
• Assume 4 decision criteria (i=1,2,3,4)
• Decision criteria:
– (1) number of families within 1000 yards of a
site,
– (2) environmental degradation at a site,
– (3) aesthetic degradation,
– (4) cost of locating a facility.
Weighting the attributes
• First, generate priority (importance) weights for
•
•
•
•
each attribute (wi, where i = 1,2,3,4)
Utilize standard AHP “pairwise comparison”
judgments of importance
Compare all possible pairs of attributes (there
are always n * (n-1) / 2 pairs)
Pairwise comparisons articulated in terms of the
standard AHP measurement scale
Create a k x k (in this case 4 x 4) attribute
weight matrix
A hypothetical attribute weight
matrix
Population
Environment
Aesthetics
Economics
Population
1
1/5
6
3
Environment
5
1
9
7
Aesthetics
1/6
1/9
1
1/4
Economics
1/3
1/7
4
1
The “rule” is that row dominates column, so for example the “9” in cell a23
indicates that the decision maker considers environmental degradation to be
“extremely” more important than aesthetic degradation.
Computing attribute weights
• Weights may be calculated using
normalized geometric row means
• Population (1*.2*6*3).25 = 1.38
• Environment (5*1*9*7).25 = 4.21
• Aesthetics = (.167*.111*1*.25).25 = .26
• Economics = (.333*.143*4*1).25 = .66
• Population (1.38/6.51) = .21, Environment
(4.21/6.51) = .65, Aesthetics (.26/6.51) =
.04, Economics (.66/6.51) = .10
Weighting the alternatives
• Determination of the priority of each site
with respect to each criterion
• How “important” is alternative j in terms
of attribute i?
• Construct one weight matrix per
alternative for each criterion (5 matrices)
Computing the overall priority of
each site
The overall priority of each site may be computed as:
4
j = Σ wi (qij), where wi is the weight for attribute j
i=1
and qij is the weight for site i with
respect to attribute j
Thus:
Population Environment Aesthetics Economics
Site 1
.21*.24 + .65 * .14 + .04*.07 + .1 * .07 = .16
Site 2
.21*.06 + .65 * .58 + .04*.11 + .1 * .04 = .38
Site 3
.21*.03 + .65 * .06 + .04*.57 + .1 * .54 = .14
Site 4
.21*.56 + .65 * .04 + .04*.18 + .1 * .24 = .17
Site 5
.21*.11 + .65 * .18 + .04*.06 + .1 * .11 = .16
Consistency
• Measurement consistency: a potential
problem with pairwise preferences
• Transitivity e.g..
a1 > a2 and a2 > a3 implies a1 > a3
Measurement consistency
a1 = 4a2 and a1 = 8a3 implies 4a2 = 8a3
Since achieving consistency is hard, AHP
introduces the notion of a deviation from
consistency
Consistency Measure
• Consistency index
A measure of the deviation from consistency
An empirically derived measure. If it is less
than .1, the preferences are deemed to be
consistent.
If the preferences are inconsistent then the
decision analyst must go back to the decision
maker and attempt to obtain more consistent
judgments
Consistency index (a rough way to
measure it for each matrix)
• The first step is to matrix multiply the
preference matrix by its priority vector
• Eg, for the site attribute judgments
|1.0
| 5.0
|.17
|.33
.2
1.0
.11
.14
6.0 3.0
9.0 7.0
1.0 .25
4.0 1.0
|
| 0.21|
| 0.88 |
| X | 0.65| = | 2.76|
|
|0.04 |
|0.17 |
|
|0.10 |
|0.42 |
Consistency index II
The second step is to compute an “L-value”
using the following equation
n
L = 1/n Σ Awi / wi
i=1
where A is the pairwise comparison matrix and
wi is the corresponding weight vector
In this case, L = ¼ (.88/.21 +2.76/.65+.17*.04
+.42/.1) = 4.22
Consistency index III
• The consistency index is then computed
using the formula: CI = (L-n)/(n-1)
• Using the siting example: CI = (4.22 –
4)/3 = 0.07
• This value of CI is then compared to a
random index, and the ratio used to
evaluate the level of consistency. If the
ratio is less than .1 then the matrix is
transitive enough to be deemed reliable.
A couple of extensions
• Hierarchies may have many levels of
attributes and sub-attributes (e.g. Saaty
1987 on the cost hierarchy for siting a
nuclear power plant)
• AHP may be used in group settings by
either eliciting group judgments or by
eliciting individual judgments and
averaging judgments across members
My work with AHP
• Three research papers
• One demonstrating it is mathematically
identical with a form of math
programming
• One formulating it formally in terms of
Thurstone’s Theory of Comparative
Judgment, and comparing it to a more
holistic approach to decision analysis
• One developing a probabilistic form of
AHP
Software
• Trial version available at:
http://www.expertchoice.com
Evaluation of AHP
• Although widely acclaimed, also controversial
• Some critique use of any mathematical model at
•
all – humans are political, emotional, spiritual,
and occasionally moral, and as such the human
condition may not submit itself to such an
approach
Some question whether this is the correct
model: one of the big objections has to do with
the fact that when an additional alternative is
added to the choice set, rank reversals may
occur