Transcript Liikenneteorian polku - Systeemianalyysin laboratorio: Etusivu

Helsinki University of Technology
Systems Analysis Laboratory
Incomplete Ordinal Information in
Value Tree Analysis
Antti Punkka and Ahti Salo
Systems Analysis Laboratory
Helsinki University of Technology
P.O. Box 1100, 02015 TKK, Finland
http://www.sal.tkk.fi/
[email protected]
Helsinki University of Technology
Systems Analysis Laboratory
Value tree analysis



m alternatives X={x1,…,xm}
n twig-level attributes A={a1,…,an}
Additive value
j
n
v
(
x
j
j
*
0
N
j
i
i )
V ( x )   vi ( xi )   (vi ( xi )  vi ( xi ))

w
v
(
x

i i
i )
*
0
vi ( xi )  vi ( xi ) i 1
i 1
i 1
n
n
non-normalized form

 wi
normalized form
[0,1]
Set of possible non-normalized scores
n


S0  s  [sij ]  [vi ( xij )] | 0  vi ( xij )  vi ( xi* )  0,  vi ( xi* )  1,
i 1


– vi(xi0)=0 and vi(xi*) = wi
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Helsinki University of Technology
Systems Analysis Laboratory
Preference information
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Complete information
– Point estimates for weights and scores
– Examples
» SWING (von Winterfeldt and Edwards 1986)
» SMART (Edwards 1977)
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Incomplete information
– Modeled through linear constraints on weights and scores
– Provides dominance relations and value intervals for alternatives
– Supports ex ante sensitivity analysis in view of feasible
parameters
– Examples
» PAIRS (Salo and Hämäläinen 1992)
» PRIME (Salo and Hämäläinen 2001)
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Helsinki University of Technology
Systems Analysis Laboratory
Incomplete weight information
(0,0,1)

w3
Forms of incomplete information
(Park and Kim 1997):
1.
2.
3.
4.
5.
weak ranking wi  wj
strict ranking wi – wj  a
ranking with multiples wi  awj
interval form a ≤ wi ≤ a + e
ranking of differences
wi – wj  wk – wl
w3  4w1
w3  w2
w3  2w1
Sw
 3 4 12 
 , , 
 19 19 19 
w3  3w2
3 2 6
 , , 
 11 11 11
1 4 4
 , , 
9 9 9
1 2 2
 , , 
5 5 5
w1
(1,0,0)
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A feasible region for
attribute weights Sw
(0,1,0)
w2
w1  0.25w2
w1  1.5w2
1 / 3  w2 / w3  1,
1 / 4  w1 / w3  1 / 2
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Helsinki University of Technology
Systems Analysis Laboratory
Preference Programming with ordinal information
Incomplete ordinal information about:
1. Relative importance of attributes
2. Values of alternatives w.r.t.
- A single twig-level attribute
- Several attributes (e.g. higher-level attributes)
- All attributes (holistic comparisons)
Other forms of incomplete information:
1.
2.
3.
4.
5.
Weak ranking
Strict ranking
Ranking with multiples
Interval form
Ranking of differences
Additional
information
MILP
model
Dominance relations,
Decision rules
and
Overall value intervals
no
Results sufficiently
conclusive
for the DM?
yes
Decision
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Helsinki University of Technology
Systems Analysis Laboratory
Ordinal preference information

Comparative statements between objects
– No information about how much ’better’ or more important an object is
than another
– Can be useful in evaluation w.r.t. qualitative attributes
– Complete information = a rank-ordering of all attributes or alternatives
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Uses in preference elicitation
– Rank attributes in terms of relative importance
» Obtain point estimates through, e.g., rank sum weights (Stillwell et al. 1981),
rank order centroid (Barron 1992)
– Rank alternatives with regard to one or several attributes
– Holistic comparisons: ”alternative x1 preferred to alternative x2 overall”
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Helsinki University of Technology
Systems Analysis Laboratory
Incomplete ordinal preference information

A complete rank-ordering, too, may be difficult to obtain
– Identification of best performing alternative with regard to some attribute
» which office facility has the best public transport connections?
– Comparison of attributes
» which attribute is the most important one?

Rank Inclusion in Criteria Hierarchies (RICH)
– Salo and Punkka (2005), European Journal of Operational Research
163/2, pp. 338-356
– Admits incomplete ordinal information about the importance of attributes
» ”the most important attribute is either cost or durability”
» ”environmental factors is among the three most important attributes”
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Helsinki University of Technology
Systems Analysis Laboratory
Non-convex feasible region RICH
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”Either attribute a1 or a2 is the most
important of the three attributes”
(0,0,1)
w3  w1
Four rank-orderings compatible with this
statement
w w
3

w3
2
Supported by RICH Decisions ©,
r  (1,3,2)
http://www.decisionarium.tkk.fi
http://www.rich.tkk.fi
r  (1,2,3)
r  (2,1,3)


w1
r  (3,1,2)
Selection of risk management methods
(Ojanen et al. 2005)
w
Participatory priority-setting for a
Scandinavian research program (Salo
and Liesiö 2006)
(1,0,0)
(0,1,0)
2
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Helsinki University of Technology
Systems Analysis Laboratory
RICHER - RICH with Extended Rankings

Admits incomplete ordinal information about alternatives
– ”Alternatives x1, x2 and x3 are the three most preferred ones with regard to
environmental factors”
– ”Alternative x1 is the least preferred among x1, x2 and x3 w.r.t. cost”
– ”Alternative x1 is not among the three most preferred ones overall”
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Ordinal statements w.r.t. different attribute sets
– Twig-level attributes
– Higher-level attributes A’  A
– Holistic statements w.r.t. all attributes
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Helsinki University of Technology
Systems Analysis Laboratory
Modeling of incomplete ordinal information (1/3)

The smaller the ranking, the more preferred the alternative
– r(x4)=1  the ranking of x4 is 1  it is the most preferred
r ( x j )  r ( x k )  v( x j )  v( x k )

Rank-orderings r=(r1, ..., rm’) on alternatives X’  X
– Bijections from alternatives X’  X to corresponding rankings 1,...,|X’|=m’
– Notation: ri = r(xj), s.t. j is the i-th smallest index in X’
– Convex feasible region
» A’={ai}
S (r)  {s  S0 | vi ( xij )  vi ( xik ) if r( x j )  r( xk )}
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Helsinki University of Technology
Systems Analysis Laboratory
Modeling of incomplete ordinal information (2/3)

Specified as a set of alternatives I  X ’ X and corresponding
rankings J  {1,...,m’}
– X’ = subset of alternatives under comparison and m’ = |X’| its cardinality

If |I|<|J|, alternatives in I have their rankings in J
– x4 and x5 belong to the three most preferred alternatives
– I = {x4, x5}, J = {1,2,3}

If |I||J|, rankings in J are attained by alternatives in I
– The least preferred alternative in X={x1,...,x10} is among x1, x2, x3, x4
– I = {x1, x2, x3, x4}, J = {10}
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Helsinki University of Technology
Systems Analysis Laboratory
Modeling of incomplete ordinal information (3/3)

Sets I and J lead to compatible rank-orderings R(I,J) for each
combination of X’, A’
{r | r 1 ( j )  I  j  J }, if I  J
R( I , J )  
k
k
{
r
|
r
(
x
)

J

x
 I }, if I  J

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Feasible region associated with compatible rank-orderings
S (I , J ) 

S (r )
rR ( I , J )
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Sets S(I,J) have several useful properties, for example
– S(I,J) = S(IC,JC), where IC is the complement of I in X’
– Set inclusions: I2  I1, |Ii||J| => S(I2,J)  S(I1,J)
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Helsinki University of Technology
Systems Analysis Laboratory
Linear inequality formulation for S(I,J) (1/3)
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Values of alternatives with rankings k and k+1 are separated by
milestone variable zk
– If the ranking of xj is ”worse” than k, its value is at most zk
v( x j )  zk  (1  yk ( x j ))M
 j
j
v
(
x
)

z

y
(
x
)M
k
k

M  0
– Binary variable yk(xj)=1 iff the value of xj is at least zk
– Milestone, binary and value variables subjected to A’ and X’
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Helsinki University of Technology
Systems Analysis Laboratory
Linear inequality formulation for S(I,J) (2/3)
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There are exactly k alternatives whose ranking is k or better
j
y
(
x
 k )k
x j X '

If the ranking of xj is better than k-1, it is also better than k
yk ( x )  yk 1 ( x )
j
j
y2  1
z1
r 1
y3  0
z2
r2
r 3
z5
z4
z3
r4
r 5
decreasing value
r6
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Helsinki University of Technology
Systems Analysis Laboratory
Linear inequality formulation for S(I,J) (3/3)
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

Feasible region S(I,J) characterized by linear constraints
on binary variables
By using milestone and binary variables for each set pair
(A’, X’) used in elicitation, all constraints are in the same
linear model
Characteristics of incomplete ordinal information used to
enhance computational properties
– E.g., only the relevant milestone and binary variables are
introduced
» given a statement that alternatives x1 and x2 are the two most
preferred, only z2 is needed
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Helsinki University of Technology
Systems Analysis Laboratory
Pairwise dominance

Value intervals may overlap, but
Alternative xk dominates xj
 min[V ( x k )  V ( x j )]  0
and strictly positive with some
feasible scores and weights

V
Example with two attributes
– Interval statement on weights
3
V ( x1 ) V ( x )
0.4  w1  0.7
– Point estimates for scores
– x1 dominates x2
– x3 is also non-dominated

Non-dominated alternatives
– Calculation through LP
V ( x2 )
Value
intervals
w1
w2 0.4
0.6
x1 dominates x2
0.7
0.3
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Helsinki University of Technology
Systems Analysis Laboratory
Decision rules

V
Maximize max overall value
(’maximax’) => x1
3
V
(
x
)
V (x )
maximax
1

Maximize min overall value
(’maximin’) => x3
central values
minimax regret

Maximize avg of max and min
values (’central values’) => x1

Minimize greatest possible loss
relative to another alternative
(’minimax regret’) => x1
maximin
w1
0.4
w2
0.6
0.7
0.3
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Helsinki University of Technology
Systems Analysis Laboratory
RICHER
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Key features
– Extends preference elicitation techniques by admitting incomplete ordinal
information about attributes and alternatives
– Converts preference statements into a linear inequality formulation
» can thus be combined with any other Preference Programming methods
– Offers recommendations through pairwise dominance and decision rules

Decision support tools
– Experiments suggest that MILP model is reasonably efficient
– Software implementation of RICHER Decisions© ongoing

Future research directions
– Sorting / classification procedures in score elicitation
– Analyses of voting behavior (e.g., acceptance voting)

Submitted manuscript downloadable at
http://www.sal.hut.fi/Publications/pdf-files/mpun04.pdf
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Systems Analysis Laboratory
References
Barron, F. H., “Selecting a Best Multiattribute Alternative with Partial Information about Attribute Weights”, Acta
Psychologica 80 (1992) 91-103
Edwards, W., “How to Use Multiattribute Utility Measurement for Social Decision Making”, IEEE Transactions on Systems,
Man, and Cybernetics 7 (1977) 326-340.
Ojanen, O., Makkonen, S. and Salo, A., “A Multi-Criteria Framework for the Selection of Risk Analysis Methods at Energy
Utilities”, International Journal of Risk Assessment and Management 5 (2005) 16-35.
Park, K. S. and Kim, S. H., “Tools dor Interactive Decision Making with Incompletely Identified Information”, European
Journal of Operational Research 98 (1997) 111-123.
Salo, A. and Hämäläinen, R. P., "Preference Assessment by Imprecise Ratio Statements”, Operations Research 40 (1992)
1053-1061.
Salo, A. and Hämäläinen, R. P., “Preference Ratios in Multiattribute Evaluation (PRIME) - Elicitation and Decision
Procedures under Incomplete Information”, IEEE Transactions on Systems, Man, and Cybernetics 31 (2001) 533545.
Salo, A. and Liesiö, J., “A Case Study in Participatory Priority-Setting for a Scandinavian Research Program”, International
Journal of Information Technology and Decision Making (to appear).
Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research 163 (2005)
338-356.
Stillwell, W. G., Seaver, D. A. and Edwards, W., “A Comparison of Weight Approximation Techniques in Multiattribute Utility
Decision Making”, Organizational Behavior and Human Performance 28 (1981) 62-77.
von Winterfeldt, D., Edwards, W., ”Decision Analysis and Behavioral Research”, Cambridge University Press (1986).
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