1SPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001
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Cj, leading to the determination of a decision matrix for the
alternatives, and (d) a weighting vector (referred to as criteria
weights) representing the relative importance of the evaluation
criteria with respect to the overall objective of the problem.
Then, this research uses a FMCDM approach based on the
synthesis of the following concepts, including (a) fuzzy set
theory, (b) AHP, (c) a-cut concept, and (d) Decision maker
(District Planner), DM’s attitude towards risk. By using the fuzzy
numbers defined, a fuzzy reciprocal judgement matrix for the
decision matrix for m criteria and n alternatives is given as
X =
*11
*12-
"*1 m
*12
*22 •
”*2 m
_*„1 X n2
(1.1)
where x tj represent the linguistic assessments of the
performance rating of alternative A t (i = 1,2,...«) with respect to
criterion Cj(j = 1,2,...m) • The decision matrix is to be given by
the DM based on the term set being defined in Figure 2. 0.
z
r
a
Z
X'
2 a
X
na _
(1.5)
In this integrated shape based redistricting model, X = 1, X =
0.5, and X = 0 are used to indicate that the DM involved has an
optimistic, moderate, or pessimistic view respectively. An
optimistic DM is apt to prefer higher values of his/her fuzzy
assessments, while a pessimistic DM tends to favor lower
values.
The evaluation criteria or compactness measurement index in
this model is a relative measure and not an absolute measure.
When the district planner compare two districts, they will not ask
“How do these districts score?” but “Which district is more
compact?”. Therefore, to evaluate the similarity of relative
compactness judgments between each pair of measures,
normalization may need to be considered. Therefore, to facilitate
the vector matching process, a normalization process in regard
to each criterion is applied to (1.5) by using (1.6), resulting in a
normalized performance matrix expressed as in (1.7).
The weighing vectors for the evaluation criteria can be given
directly by the Decision maker or obtained by using pain/vise
comparison of the AHP. The weighting vectors IN of the
linguistics term for the criteria is as follow:
W = (w l ,w 2 ,...,w m )
(1.2)
A fuzzy performance matrix representing the overall
performance of all alternatives with respect to each criterion can
therefore be obtained by multiplying the weighting vector by the
decision matrix. The arithmetic operations on these fuzzy
numbers are based on interval arithmetic.
w i*n +w 2 x u +... + w n x Un
w,x ]2 + w 2 x 22 +w n x 2m
_ w .*»i +w 2 x n2 +...+ w n x nm
(1.3)
By using a a-cut
Vae[0,ll
A „ = K» a "] = [0 2 -a x )a + a,,-(a 3 -a 2 )a + a 3 ]
On the performance matrix (1.3), an interval performance matrix
can be derived as in (1.4), where 0 <a<l. The value of
a represents the DM’s degree of confidence in his/her fuzzy
assessments regarding alternative ratings and criteria weights. A
larger avalué indicates a more confident DM, meaning that the
DM’s assessments are closer to the most possible value a 2 of
the triangular fuzzy numbers (a,, a 2 , a 3 ).
z =
K.zi]
I
(1.4)
LK,r“]J
Incorporated with the DM’s attitude towards risk using an
optimism index X, an overall crisp performance matrix is
calculated as in (1.5), where
(1.6)
(1.7)
The values of N* indicate the degree of preference with respect
to the alternatives for fixed pc and X, respectively where
a e [0,1], A e [0,1] Indeed, this value is the enhanced
compactness index (ECI), which consider both of the
compactness measurements or evaluation criteria earlier.
Therefore, the larger the value, the more the preference of the
alternative.
In summarizing the Fuzzy-AHP method for getting the enhanced
compactness measurement index above, this research present
the steps required for the algorithm in Figure 4. 0.
(1) Formulate redistricting decision problem as multi
criteria problem
(2) Identify the hierarchical structure of the problem
(3) Define membership function for each criteria
(4) Apply IF-THEN decision rules to assign a linguistic
and numeric value of each criteria variable
(5) Obtain decision matrix by fuzzy number as expressed
in (1.1) using AHP method based on fuzzy number
defined and Figure 3.0
(6) Obtain weighting vector for the criteria as expressed
in (1.2) using AHP method based on fuzzy number
defined and Figure 3.0
(7) Obtain fuzzy performance matrix (1.3) by multiplying
the decision matrix obtained at step 4 by the
weighting vector determined at step 5
(8) Obtain interval performace matrix (1.4) by a-cuts on
the performance matrix determined at step 6
(9) Obtain crisp performace matix (1.5) by DM’s attitude
towards risk represented by an optimism index X
(10) Calculate normalized performace matrix (1.7) by (1.6)
(11) Get the enhanced compactness index
Figure 4. 0: Algorithm for the Fuzzy-AHP for this study