unclassified the surface-weighted options ranking technique peter williams, peta erbacher and fred...
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The Surface-Weighted Options Ranking Technique
Peter Williams, Peta Erbacher and Fred DJ Bowden
Land Operations DivisionAs presented at the 21st MCDM Conference,
Finland, June 13 – 17, 2011.
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The Evolution of SWORT
Began with some work on TOPSISFilar, J.A., Gaertner, P.S. and Lu, W.M, “Aggregation of
Tactical Performance Measures: An Operations Research Perspective”
TOPSIS is a Data Envelop Technique. Key aspect include: Defining the frontier solution boundary using LP Determining each option’s relationship to this boundary Ranking the options based on this relationship
The issues with this were: Lack of flexibility in how criteria are combined Amount of processing power required to generate frontier
boundarySWORT solves these issues and provides non-linear weightings
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SWORT at its Basic
Data Envelop Analysis Technique
Allows for:1. Non-linear weights on criteria
Able to weight regions2. Computationally inexpensive3. Provides degree of separation between the Options4. Sensitivity analysis on weightings
Can be explained intuitively to decision makerEnables them to describe how weighting regions should look
Will use 2 criteria to illustrate the technique
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Once the frontier surface, S, has been chosen appropriate for the problem, the SWORT ‘value’ for an option P is calculated as:
Distance to PDistance to S through P
SP
V =
SWORT Description
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SWORT Description
Once the frontier surface, S, has been chosen appropriate for the problem, the SWORT ‘value’ for an option P is calculated as:
SP1
P3
P4
P2Option SWORT
Ranking
1 2 (0.67)
2 3 (0.61)
3 1(0.90)
4 4 (0.40)
Distance to PDistance to S through P
V =
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SWORT Description
It can be shown using a parametric representation that this value V is given by:
V = 1/t
where,
S(Pt) = 0
Hence, the calculations for most surfaces are very simple and can be solved analytically.
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Example Surface –Plane
Features:•Equivalent to the SAW method•A constant weight for each attribute•A weighted sum is calculated
Preferred Options:•Score well in most of the criteria
0
1
1
S
P
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Example Surface –Plane
Features:•Equivalent to the SAW method•A constant weight for each attribute•A weighted sum is calculated
Preferred Options:•Score well in most of the criteria
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
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Example Surface –Plane
Features:•Equivalent to the SAW method•A constant weight for each attribute•A weighted sum is calculated
Preferred Options:•Score well in most of the criteria
ExamplePurchasing a car which needs to have good ratings for fuel efficiency, cost, size, colour, safety, engine capacity and age.
b
papV
bpapt
btaptptS
baxxS
21
21
12
12
0:)(
:)(
P
x
0
1
1
S
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Example Surface – Ellipse
Features:•Non-linear behaviour •A greater emphasis placed on "individual" attributes.
Preferred Options:•Option which ranks very highly in one or more areas.
ExampleRecruiting an athlete for a sports team. They need to have speed, strength, intelligence, skill and height. However, if one person is particularly good in one attribute, they may be perfect for a specific position on the field.
ab
apbpV
baapbpt
b
tp
a
tptS
b
x
a
xS
222
221
22222
221
2
2
222
2
221
2
22
2
21
1:)(
1:)(
P
x
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Example Surface – Inverse Function
Features:•Extreme emphasis on scoring well in all attributes•Any poor score will have a large and negative impact•Through the application of asymptotes it is possible to enforce minimum 'cut-off' values for attributes
Preferred Options:•Contributions from all attributes are better than individual brilliance
ExampleEvaluating a military system which must reach minimum levels of armour, firepower, speed and deployability.If it fails in any one category it is not acceptable. Ideally, looking for a system that has it all.
a
ppV
tp
atptS
x
axS
21
12
12
.
:)(
:)(
P
x
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Example Surface - Parabolas
Features:•Two upper and right parabolas•Effectiveness decreases if you have even representation in both attributes
Preferred Options:•Very good in one Attribute and average in the other. •Effectiveness decreases if both Attributes are prominent, or one is absent
ExampleInterior design for a new office building. The Attributes could be the architecture and the design. Highly detailed and intricate construction is good. Lavish and expensive furniture and decorations are good. Having both overwhelms the senses and creates an eyesore.
21
222121
21
21
21
222121
221
221
212
212
422
2
2
422
02
:)(
:)(
bpapappap
pV
p
bpapappapt
batpaptp
batptptS
baxxS
P
x
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Extension to n Criteria
Most surfaces are readily extensible to a higher number of criteria.
E.g. n-Dimensional Ellipsoids:
n
i i
i
n
i i
i
n
i i
i
n
i i
i
a
pV
ap
t
a
tpPtS
a
xS
12
2
12
2
12
2
12
2
1
1 :)(
1 :)(x
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Working Example: Background
Considered the “value” of Seven different Body Armours (C1, C2,…, C7).
Problem had 12 Attributes by which to rank the Options.
Data was collected from participants for each of the Attributes.
The associated weightings for each of the Attributes were established during field experimentation and trials.
The central tendency of each of the Attributes, for each Body Armour, were used to get the attribute values.
The SWORT value for each Option was then calculated.
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Working Example: SWORT Tool - Ellipse
C1 C2 C3 C4 C5 C6 C7
Rank1
(1.00)5
(0.43)6
(0.18)4
(0.87)2
(0.94)3
(0.92)7
(0.00)
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Working Example: Results
The Body Armours ranked according to the surface used.
C1 C2 C3 C4 C5 C6 C7
Plane1
(1.00)5
(0.50)6
(0.26)4
(0.84)3
(0.91)2
(0.93)7
(0.00)
Ellipse1
(1.00)5
(0.43)6
(0.18)4
(0.87)2
(0.94)3
(0.92)7
(0.00)
Inverse3
(0.93)7
(0.00)5
(0.05)6
(0.40)2
(0.94)1
(1.00)4
(0.70)
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Working Example: Results
The Body Armours ranked according to the surface used.
C1 C2 C3 C4 C5 C6 C7
Plane1
(1.00)5
(0.50)6
(0.26)4
(0.84)3
(0.91)2
(0.93)7
(0.00)
Ellipse1
(1.00)5
(0.43)6
(0.18)4
(0.87)2
(0.94)3
(0.92)7
(0.00)
Inverse3
(0.93)7
(0.00)5
(0.40)6
(0.05)2
(0.94)1
(1.00)4
(0.70)
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Summary
Presented a new MCDM method – SWORT.
This method:1. Non-linear weights on Criteria.
Able to weight regions.2. Computationally inexpensive.3. Provides degree of separation between the Options.4. Sensitivity analysis on weightings.
Is Easily extendable to n Criteria.
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Questions