Transcript Spatial Compromise Programming

Spatial Compromise
Programming
RESM 575
Spring 2011
Lecture 4
Last time
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Weighting techniques
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Point allocation
Ranking methods
Pairwise comparison
Class exercise
Lab: Criteria weighting for suitability models
and spatial sensitivity
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Review of question 3
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Why nonparametric?
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Nonparametric tests:
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Chi-square
U-test
Wilcoxon
Sign test
Kruskal-Wallis
Friedman
Mann-Whitney
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Building suitability models, steps
1.
2.
3.
4.
5.
6.
Define the problem or goal
Decide on evaluation criteria
Normalize and create utility scales
Define weights for criteria
Calculate a ranking model result
Evaluate result
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What is multiple objective decision
making?
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A form of decision analysis that seeks to
analyze complex decision problems by
dividing the problem into smaller
understandable parts
Then, we integrate the parts in a logical
manner to produce a meaningful solution
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Generally speaking….
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Individuals, groups, and organizations, in
their decision making efforts,
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pursue multiple objectives
set multiple goals
evaluate their options according to multiple criteria
and as a consequence experience conflict.
Decision making under these such conditions
is characterized by incessant attempts at
conflict resolution and the simultaneous
attainment of goals.
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Goal Programming (GP)
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A form of linear programming that allows for
consideration of multiple goals
GP can be used to determine the optimal
solution to a multi-objective decision problem
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Illustration (single objective decision
making)
Skilled labor
Technical
know how
Raw materials
Build an auto of
maximum
horsepower
Energy
“a purely technical problem”
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Illustration (multiple objective
decision making)
Build the “best”
auto
Safety
Price
Depreciation
rate
Reliability
Weight
Size
No single end or no single criterion, a multiple criteria decision problem or
an economic problem according to Freidman’s definition
Human value judgments, trade off evaluations, and assessments of criteria 14
now become integral to the problem
Compromise programming (CP)
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Similar to goal programming in that it uses
the concept of minimum distance
A distance based technique that depends on
the point of reference or “ideal” point
Attempts to minimize the “distance” from the
ideal solution for a satisficing solution
The closest one to the ideal across all criteria
is the compromise solution or compromise
set
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CP notes
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The concept of non dominance is used in
distance-based techniques to select the best
solution or choice of alternative.
A solution is said to be non dominated if there
exists no other feasible solution that will
cause an improvement in a value of the
objective or criterion functions without making
a value of any other objective function worse
(Tecle and Yitayew, 1990).
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CP notes
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The non dominance solution concept,
originating with Pareto in 1906, has been one
of the cornerstones of traditional economic
theory. It is usually stated as the Pareto
principle:
“A state of the world A is preferable to a state of the world B if at
least one person is better off in A and nobody is worse off. A
state is said to be Pareto optimal or Pareto efficient when there is
no other state in which one individual can obtain higher
satisfaction without at the same time lowering the satisfaction of
at least one other individual” (Just et al., 1982).
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CP model
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CP model
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CP model notes
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In (3), the parameter p can have values from
zero to infinity and represents the concern of
the decision maker over the maximum
deviation (Tecle and Yitayew, 1990)
(Duckstein and Opricovic, 1980).
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CP model notes
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The larger the value of p, the greater the
concern becomes.
For p = one, all weighted deviations are assumed to
compensate each other perfectly.
For p = two, each weighted deviation is accounted
for in direct proportion to its size.
As p approaches the limit of infinity, the alternative
with the largest deviation completely dominates
the distance measure (Zeleny, 1982).
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Using the CP model
1.
Assemble data for all evaluation criteria, this
becomes the evaluation matrix
Alternatives Criteria 1
Criteria 2
Criteria 3
A
B
C
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Evaluation Matrix
More of Criteria 1 is preferred
Lower costs for Criteria 2 is preferred
Low Percentages for Criteria 3 are preferred
Alternatives Criteria 1
Criteria 2
Criteria 3
A
1000
$65
35%
B
800
$25
15%
C
500
$90
28%
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Using the CP model
2.
Normalize the matrix based on rules, this
becomes the payoff matrix
More of Criteria 1 is preferred
1 - (65/90)
Lower costs for Criteria 2 is preferred
Low Percentages for Criteria 3 are preferred
Alternatives Criteria 1
Criteria 2
Criteria 3
A
1.000
.277
0
B
.800
.722
.58
C
.500
0
.2
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Using the CP model
3.
Find the best and worst for each alternative
across the criteria from the payoff matrix which
has been already normalized
Alternatives Criteria 1
Criteria 2
Criteria 3
f*= best
1.000
.722
.58
f**=worst
.500
0
0
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Using the CP model
4.
Integrate the criteria weights, f* and f** and
values for the alternative into the CP model
for a parameter value of p (1, 2, oo)
Using criteria weights C1= .4, C2=.5, C3=.1
For Alternative A and p = 1
(.4) [(1.00-1.00)/(1.00-.500)] + .(.5) [(.722 – .277)/(.722-0)] + (.1)[(.58-0)/(.58 – 0)]
Which is
0 + .3081 + .1
or
.4081
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For Alternative B and p = 2
(((.4) [(1.00-.800)/(1.00-.500)])2 + ((.5) [(.722 – .722)/(.722-0)])2 + ((.1)[(.58-.58)/(.58 – 0))2])1/2
Which is
.1600 + 0 + 0
or
.1600
For Alternative C and p = 1
(.4) [(1.00-.500)/(1.00-.500)] + .(.5) [(.722 – 0)/(.722-0)] + (.1)[(.58-.200)/(.58 – 0)]
Which is
.4 + .5 + .655
or
1.555
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Results
Alternatives
A
B
C
CP metric
.4081
.1600
1.555
Therefore since B is the lowest value
(closest to the ideal values across all the
criteria, it would be the preferred alternative
for the weights and when p = 1
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Solving the CP model
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The preferred alternative has the minimum Lp
distance value for each p and weight set that
may be used.
Thus, the alternative with the lowest value for
the Lp metric will be the best compromise
solution because it is the nearest solution
with respect to the ideal point.
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CP advantages
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Simple conceptual structure
Simplicity makes it particularly useful for
spatial decision problems in which decision
makers tend to rely on their intuition and
insight
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CP limitation
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Except at the two extremes where p = 0 and
p = oo there is no clear interpretation of the
various values of the parameter p.
Therefore, use different weights or values for
p to test overall robustness of results
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ArcGIS CP Extension
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