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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