(TOPSIS) TOPSIS Method in MADM - faez.irfaez.ir/CourseFile/TOPSIS.pdf · (TOPSIS) TOPSIS Method in...
23
Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) 1 TOPSIS Method in MADM
Transcript of (TOPSIS) TOPSIS Method in MADM - faez.irfaez.ir/CourseFile/TOPSIS.pdf · (TOPSIS) TOPSIS Method in...
Technique for Order Preference by Similarity to Ideal Solution
(TOPSIS)
1TOPSIS Method in MADM
Origin and History1980: development by Kwangsun Yoon and Hwang Ching-Lai
– Yoon, K., “System Selection by MultipleAttribute Decision Making,” Ph. D.Dissertation, Kansas State University,Manhattan, Kansas, 1980.
– Yoon, K. and C. L. Hwang, “TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)- A Multiple Attribute Decision Making,” a paper to be published, 1980.
2TOPSIS Method in MADM
Basic Concept
l The chosen Alternative should have the shortest distance from the ideal solution and the farthest from the negative-ideal solution.
3TOPSIS Method in MADM
Basic Concept …
It is very difficult to justify the selection of A1 or A2
A+
A-
A1 A2 .
.
.
. .
. .
. .
Attribute X
2 (increasing preference)
Attribute X1 (increasing preference)
4TOPSIS Method in MADM
Decision Matrixlm Alternative, n Attributes (or criteria)
=
xxxx
xxxxxxxxxxxx
A
AAA
xxxx
mnmmm
n
n
n
m
n
D
...............
...
...
...
.
.
.
...
321
3333231
2232221
1131211
3
2
1
321
5TOPSIS Method in MADM
Hypothesis-1
l Each Attribute in the Decision Matrixtakes either monotonically increasing ormonotonically decreasing utility
6TOPSIS Method in MADM
Hypothesis-2
l A Set of Weights for the Attributes isrequired
7TOPSIS Method in MADM
Hypothesis-3
l Any Outcome which is expressed in anon-numerical way, should be quantifiedthrough the appropriate scalingtechnique
8TOPSIS Method in MADM
Steps-1
l Construct the Normalized Decision Matrix– To transform the various attribute dimensions
into non-dimensional attributes, which allowscomparison across the attributes
∑=
=m
iij
ijij
xxr1
2
9TOPSIS Method in MADM
Steps-2
l Construct the Weighted Normalized DecisionMatrix
=
=
rwrwrwrw
rwrwrwrw
rwrwrwrw
vvvv
vvvv
vvvv
mnnmjjmm
innijjii
nnijj
mnmjmm
inijii
nj
V
..................
....
....
..........
..................
....
....
..........
2211
2211
1122111
21
21
111211
10TOPSIS Method in MADM
Steps-3l Determine Ideal and Negative-Ideal Solutions
==
==
=
=∈∈=
=
=∈∈=
−−−−
−
++++
+
criteriatwithassociatedjnj
criteriabenefitwithassociatedjnjJwhere
mijJj
mijJj
J
vvvvJvvA
vvvvJvvA
nj
ijiiji
nj
ijiiji
cos,...,2,1
,...,2,1
},...,,...,,{
},...2,1)max(),min{(
},...,,...,,{
},...2,1)min(),max{(
'
21
'
21
'
11TOPSIS Method in MADM
Steps-4
l Calculate the Separation Measure:– Ideal Separation
– Negative-Ideal Separation
min
jjiji vvs ,...,2,1)(
1
2 =−= ∑=
++
min
jjiji vvs ,...,2,1)(
1
2 =−= ∑=
−−
12TOPSIS Method in MADM
Steps-5
l Calculate the Relative Closeness to the Ideal Solution
AAcAAc
csssc
ii
ii
iii
ii
if
if
mi
−
+
+
−+
−
==
==
=<<+
=
0
1
,...,2,1,10,)(
*
*
*
13TOPSIS Method in MADM
Steps-6
l Rank the preference order
– A set of alternatives can now be preference
ranked according to the descending order of ci
*
14TOPSIS Method in MADM
An example of using TOPSIS Method
Weight 0.1 0.4 0.3 0.2
Civic
Saturn
Ford
Mazda
7 9 9 8
8 7 8 7
9 6 8 9
6 7 8 6
Style Reliability Fuel Eco. Cost
15TOPSIS Method in MADM
Steps of TOPSISl Step 1: Calculate (Σx2
ij )1/2 for each column and divide each column by that to get rij
Civic
SaturnFordMazda
0.46 0.61 0.54 0.530.53 0.48 0.48 0.46
0.59 0.41 0.48 0.59
0.40 0.48 0.48 0.40
Style Rel. Fuel Cost
16TOPSIS Method in MADM
Steps of TOPSIS
l Step 2 : Multiply each Column by wj to get Vij.
Saturn
Ford
Civic
Mazda
Style Rel. Fuel Cost0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
0.040 0.192 0.144 0.080
17TOPSIS Method in MADM
Steps of TOPSIS
Step 3 (a): Determine Ideal Solution A*. A* = {0.059, 0.244, 0.162, 0.080}
Style Rel. Fuel
Saturn
Ford
Civic
Mazda
Cost
0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
0.040 0.192 0.144 0.08018TOPSIS Method in MADM
Steps of TOPSIS
Step 3 (b): Find Negative Ideal Solution A-. A- = {0.040, 0.164, 0.144, 0.118}
Style Rel. Fuel
Saturn
Ford
0.046 0.244 0.162 0.106
0.053 0.192 0.144 0.092
0.059 0.164 0.144 0.118
Civic
Mazda 0.040 0.192 0.144 0.080
Cost
19TOPSIS Method in MADM
Steps of TOPSIS
Step 4 (a): Determine Separation From Ideal Solution A* = {0.059, 0.244, 0.162, 0.080}
Si* = [ Σ (vj
*– vij)2 ] ½ for each row j
Style Rel. Fuel
Saturn
Ford
(.046-.059)2 (.244-.244)2 (0)2 (.026)2Civic
Mazda
Cost
(.053-.059)2 (.192-.244)2 (-.018)2 (.012)2
(.053-.059)2 (.164-.244)2 (-.018)2 (.038)2
(.053-.059)2 (.192-.244)2 (-.018)2 (.0)2
20TOPSIS Method in MADM
Steps of TOPSIS
Step 4 (a): Determine Separation From Ideal Solution Si*
Saturn
Ford
0.000845 0.029
0.003208 0.057
0.008186 0.090
Civic
Mazda 0.003389 0.058
Σ(vj*–vij)2 Si* = [ Σ (vj
*– vij)2 ] ½
21TOPSIS Method in MADM
Steps of TOPSIS
Step 4: Determine Separation From Negative Ideal Solution Si
-
Σ(vj-–vij)2 Si- = [ Σ (vj
-– vij)2 ]½
Saturn
Ford
0.006904 0.083
0.001629 0.040
0.000361 0.019
Civic
Mazda 0.002228 0.04722TOPSIS Method in MADM
Steps of TOPSIS
Step 5: Calculate the relative closeness to the ideal solution Ci* = S-
i / (Si* +S-
i )S-
i /(Si*+ S-
i) Ci*
Saturn
Ford
0.083/0.112 0.74 ← BEST
0.040/0.097 0.41
0.019/0.109 0.17 ← WORSE
Civic
Mazda 0.047/0.105 0.4523TOPSIS Method in MADM
