S ystemsAnalysis LaboratoryHelsinki University of Technology

Practical dominance and process support in the Even

Swaps method

Jyri MustajokiRaimo P. Hämäläinen

Systems Analysis LaboratoryHelsinki University of Technology

www.sal.hut.fi

S ystemsAnalysis LaboratoryHelsinki University of Technology

Presentation outline

• Introduction to the Even Swaps method• Hammond, Keeney and Raiffa (1998, 1999)

• Two new techniques to support the method• New concept based on the PAIRS method

• Salo and Hämäläinen (1992)• Aim to provide support for tasks needing

mechanical scanning• Smart-Swaps software

• The first software for supporting the method

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Even Swaps method

• Multicriteria method to find the best alternative

• Based on even swaps• Value trade-off, where the value change in

one attribute is compensated in some other attribute

• The alternative with these changed values is equally preferred to the initial one

It can be used instead

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Elimination process• Aim to carry out even swaps that make

• Alternatives dominated• Some other alternative is equal or better than this

one in every attribute, and better at least in one attribute

• Attributes irrelevant• Every alternative has the same value on this

attribute These can be eliminated

• Process continues until one alternative (i.e. the best one) remains

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Practical dominance

• If alternative x is better than alternative y in several attributes, but slightly worse in one attribute x practically dominates y y can be eliminated

• Aim to reduce the size of the problem in obvious cases• No need to carry out an even swap task

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Example

• Office selection problem (Hammond et al. 1999)

78

25

Practicallydominated

byMontana

Dominatedby

Lombard

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Two new techniques

• Modeling of the practical dominance• Support for looking for efficient even swaps

• New concept based on the PAIRS method• Aim to provide support for tasks needing

mechanical scanning• Computer support to help in these tasks

• For supporting the process – not for automating it

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PAIRS Method• Additive value function• Imprecise statements by intervals on

• Attribute weight ratios (e.g. 1 w1 / w2 5) Feasible region of the weights• Ratings of the alternatives (e.g. 0.6 v1(x1) 0.8)

Intervals for overall values• Lower bound for the overall value of x:

• Upper bound correspondingly

n

iiii xvwxv

1

)(min)(

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Pairwise dominance

• x dominates y in a pairwise sense if

i.e. if the overall value of x is greater than the one of y with any feasible weights of attributes and ratings of alternatives

0])()([min1

n

iiiiiiw

yvxvw

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Modeling practical dominance

• General constraints for the weight ratios and value functions• These should cover all the plausible weights

and values• If x dominates y in a pairwise sense with

these general constraints y can be seen as practically dominated

S ystemsAnalysis LaboratoryHelsinki University of Technology

General constraints

• On weight ratios

• On value functions• E.g. exponential value

function constraints• Any value function within

the constraints allowed• Additional constraints, e.g.

for the slope

jirww

j

i ,,

1

0 xi

vi(xi)

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Use of even swaps information

• With each even swap the user reveals information about his/her preferences

• This information can be utilized in the process

Tighter weight ratio constraints elicited from the given even swaps

Better estimates for practical dominances

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Support for looking for efficient even swaps

• Aim to carry out as few swaps as possible to eliminate alternatives or attributes

Scanning through the consequences table

• There may also be other objectives• E.g. easiness of the swaps

Different types of suggestions of even swaps for the decision maker

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Irrelevant attributes

• Look for an attribute in which the most alternatives have the same value

Carry out such even swaps that make the values of all the alternatives the same in this attribute

• Compensation in attribute with which new dominances could also be obtained• Possible reduction also in the number of the

alternatives

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Dominated alternatives

• Look for such pair of alternatives, where dominance between these could be obtained with fewest swaps• E.g., if x outranks y only in one attribute, carry

out an even swap that makes the values of these alternatives the same in this attribute

• However, the ranking of the alternatives can change in compensating attribute

We cannot be sure that the other alternative is dominated after the swap

S ystemsAnalysis LaboratoryHelsinki University of Technology

Dominated alternatives

• An estimate for each swap, how far we relatively are from dominance• The ratio between

• The allowed value change in compensating attribute, and

• The maximum estimated value change in this• Estimated from general constraints

• d(y, x) = 'likelihood' of y dominating x after this even swap

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Example

36 different options to carry out an even swap which may lead to dominanceE.g. change in Monthly Costs of Montana from 1900 to 1500:Compensation in Client Access:

d(Mon, Bar) = ((85-78)/(85-50)) / ((1900-1500)/(1900-1500)) = 0.20d(Mon, Lom) = ((85-80)/(85-50)) / ((1900-1500)/(1900-1500)) = 0.14

Compensation in Office Size:d(Mon, Bar) = ((950-500)/(950-500)) / ((1900-1500)/(1900-1500)) = 1.00d(Mon, Lom) = ((950-700)/(950-500)) / ((1900-1500)/(1900-1500)) = 0.56(Assumptions: linear estimates for value functions; weight ratios = 1)

Initial Range:

85 - 50

A - C950 - 500

1500 -1900

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Use in practice

• The proposed techniques assume an additive value function• Not explicitly assumed in the Even Swaps

method• Can still be used approximatively Suggestions should be confirmed by the

decision maker

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Smart-Swaps softwarewww.smart-swaps.hut.fi

• Support for the proposed approaches• Identification of practical dominances• Suggestions for even swaps

• Additional support• Information about what may happen with

each swap• Notification of dominances• Rank colors• Process history

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Smart-Swaps software

S ystemsAnalysis LaboratoryHelsinki University of Technology

www.Decisionarium.hut.fi

Software for different types of problems:• Smart-Swaps (www.smart-swaps.hut.fi)• Opinions-Online (www.opinions.hut.fi)

• Global participation, voting, surveys & group decisions

• Web-HIPRE (www.hipre.hut.fi)• Value tree based decision analysis and support

• Joint Gains (www.jointgains.hut.fi)• Multi-party negotiation support

• RICH Decisions (www.rich.hut.fi)• Rank inclusion in criteria hierarchies

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Conclusions• Techniques to support the even swaps

process presented• Modeling the practical dominance• Support for looking for efficient even swaps• New concept based on the PAIRS method

• Support for tasks needing mechanical scanning• Especially useful in large problems

• Computer support needed in practice• Smart-Swaps software introduced

S ystemsAnalysis LaboratoryHelsinki University of Technology

ReferencesHammond, J.S., Keeney, R.L., Raiffa, H., 1998. Even swaps: A rational

method for making trade-offs, Harvard Business Review, 76(2), 137-149.Hammond, J.S., Keeney, R.L., Raiffa, H., 1999. Smart choices. A practical

guide to making better decisions, Harvard Business School Press, Boston, MA.

Mustajoki, J., Hämäläinen, R.P., 2003. Practical dominance and process support in the Even Swaps method. Manuscript. Downloadable soon at www.sal.hut.fi/Publications/

Salo, A., Hämäläinen, R.P., 1992. Preference assessment by imprecise ratio statements, Operations Research, 40(6), 1053-1061.

Applications of Even Swaps:Gregory, R., Wellman, K., 2001. Bringing stakeholder values into

environmental policy choices: a community-based estuary case study, Ecological Economics, 39, 37-52.

Kajanus, M., Ahola, J., Kurttila, M., Pesonen, M., 2001. Application of even swaps for strategy selection in a rural enterprise, Management Decision, 39(5), 394-402.

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S ystemsAnalysis LaboratoryHelsinki University of Technology

xi x'i

v(xi)-

v(xi)-v(x'i)-

v(x'i)-

mini maxi0

1

Value function constraints

• Exponential value function constraint

where a (0, 1)xN = (xi – mini) / (maxi – mini)vi(maxi)=0, vi(maxi)=1

(here a=0.15)

11)(

a

axvNx

ii

Appendix

S ystemsAnalysis LaboratoryHelsinki University of Technology

xi x'i

min(v(x'i)-v(xi))

max(v(x'i)-v(xi))

xD

mini maxi0

1

Value function constraints• Slope

constraints

wheres (0, 1)Dx = (x'i – xi) / (maxi – mini)vi(maxi)=0, vi(maxi)=1

(here s=0.5)

sxxvxvs iiii 1)()'(

D

Appendix

S ystemsAnalysis LaboratoryHelsinki University of Technology

New constraints from the given trade-offs• E.g. change xi x'i is compensated with

the change xj x'j• Assume an additive value function:

wi v(xi) + wj v(xj) = wi v(x'i) + wj v(x'j)• General constraints for value functions New weight ratio constraint:

))()'()'()(

max(iiii

jjjj

j

i

xvxvxvxv

ww

Appendix

S ystemsAnalysis LaboratoryHelsinki University of Technology

The use of practical dominance in practice• Suggestions - not automatization

• The user should confirm the dominances• Strict gereral constraints

Smaller feasible region Alternatives may become incorrectly identified

as dominated ones• Loose general costraints

Larger feasible region Not as many dominances, but all these should

be real ones

Appendix

S ystemsAnalysis LaboratoryHelsinki University of Technology

Estimate how far we are from dominance• Assume, e.g. that

• The change xi yi (vi(xi) > vi(yi)) is compensated with the change xj x'j (vj(x'j) > vj(xj))

• x'j should remain under yj to make y dominate x • The allowed value change in j:

The maximum plausible value change in j:

• Derived from general constraints in PAIRS

)()( jjjj xvyv

)))()((max())()'(max( iiiij

ijjjj yvxv

wwxvxv

Appendix

S ystemsAnalysis LaboratoryHelsinki University of Technology

Estimate how far we are from dominance

• An estimate how close we are relatively to make y dominate x• The ratio between the allowed compensation

and the maximum plausible value change

• The bigger the ratio is, the better the dominance would be obtained

• Strict constraints can also be used instead of intervals

)))()(())()((

min(),(iii

jjj

yvxvwxvyvw

xyd

Appendix