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