Transcript Liikenneteorian polku

Helsinki University of Technology
Systems Analysis Laboratory
RICHER – A Method for Exploiting Incomplete
Ordinal Information in Value Trees
Antti Punkka and Ahti Salo
Systems Analysis Laboratory
Helsinki University of Technology
P.O. Box 1100, 02015 HUT, Finland
[email protected]
http://www.sal.hut.fi/
Helsinki University of Technology
Systems Analysis Laboratory
Value tree analysis


m alternatives: X={x1,…,xm} , n attributes: A={a1,…,an}
Additive value function
n
V ( x j )   vi ( xij )
non-normalized form
i 1
or
n
V ( x )   wi v i ( xi )
j
N
j
i 1
n
w
i 1
i
1
normalized form
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Systems Analysis Laboratory
Preference elicitation

Complete information
– Point estimates, e.g. w1=0.5
– E.g., SMART (Edwards 1977)

Incomplete information: preference programming methods
– Weight ratio and weight intervals
» 1.5  w1 / w2  2, 0.1  w2  0.25
– Intervals for normalized scores
» 0.22  v N ( x14 )  0.34
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– PAIRS (Salo and Hämäläinen 1992), PRIME (Salo and Hämäläinen
2001), Arbel’s approach (1989)

Ordinal information
– Rank attributes in terms of relative importance
» point estimates through, e.g., rank sum weights (Stillwell et al. 1981)
» incomplete ordinal information (RICH; Salo and Punkka 2004)
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Incomplete preference information

Overall value intervals for
alternatives
 3 4 12 
 , , 
 19 19 19 
w3  w2
w3  2w1
Sw
w3  3w2
3 2 6
 , , 
 11 11 11
1 4 4
 , , 
9 9 9
1 2 2
 , , 
5 5 5
» 1/ 3  w2 / w3  1, 1/ 4  w1 / w3  1/ 2

w3
w3  4w1
Complete information hard to
acquire
– Relative importance of attributes
– Alternatives’ properties
– Incomplete information
(0,0,1)
w1
(1,0,0)
(0,1,0)
w2
w1  0.25w2
w1  1.5w2
– Smallest and largest possible value from
LP
n
min/max wi v iN ( xij )
wS w
i 1
where Sw is the feasible region for the attribute
weights
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Pairwise dominance relation

Alternative xk dominates xj in the
sense of pairwise dominance
 min[V ( x k )  V ( x j )]  0
– Two attributes,
and positive with some
feasible scores and weights
0.4  w1  0.7

Several alternatives may remain nondominated
– Additional preference statements to make
the feasible region smaller
– Decision rules assist the DM in selection
of the most preferred one
V
V ( x1 )
V ( x2 )
x1 dominates x2
w1
w2 0.4
0.6
0.7
0.3
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Incomplete ordinal preference information

Complete ordinal information is a complete rank-ordering
of attributes or alternatives
– Rankings are exactly known for each alternative
– Leads to a convex set of feasible scores and weights, when interpreted as
incomplete preference information

The RICH (Rank Inclusion in Criteria Hierarchies) method
–
–
–
–
Incomplete ordinal statements about relative importance of attributes
”Cost is the most important attribute”
”Environmental factors is among the three most important attributes”
Several rank-orderings can be compatible with the preference statements
» e.g.: either attribute a1 or a2 is the most important of the three attributes

a3 is either the second or the least important one
» may lead to non-convex feasible region of the attribute weights
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Non-convex feasible region in RICH


”Either a1 or a2 is the most
important of the three
attributes”
Calculation by dividing into
compatible rank-orderings
– Extreme points readily computed
– Lower bounds for weights wi  b  0

(0,0,1)
w3
w3  w1
w3  w2
Full support provided by RICH
Decisions ©,
http://www.decisionarium.hut.fi/
r  (1,3,2)
– Applications
» evalution of risk management tools
(Ojanen et al. 2004)
» support for setting priorities for a
(0,1,0)
research programme in wood material
science (Salo and Liesiö 2004)
w2
w1
r  (2,3,1)
r  (1,2,3)
r  (2,1,3)
(1,0,0)
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The RICHER (RICH with Extended Rankings) method

Extends incomplete ordinal information to alternatives
– ”Alternatives x1, x2 and x3 are the three most preferred with regard to
environmental factors”
– ”Alternative x1 is not among the three most preferred ones”
– ”Considering alternatives x1, x2 and x3, the least preferred with regard to
cost is x1”

Statements about attribute weights incorporated as well

Comparison to the RICH method
–
–
–
–
–
Suitable also for statements about alternatives
Computationally much more efficient
Includes all features of RICH
Allows evaluation within subsets
Applicable in conjunction with other preference programming methods
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Modeling of incomplete ordinal information (1/4)

Rank-ordering –function r
– Bijection from (sub)set of alternatives X’X (or (sub)set of attributes) to set of
rankings
– E.g., r=(r(x1), r(x2), r(x3))=(1,3,2)
– The smaller the ranking, the better the alternative
r ( x j )  r ( x k )  v( x j )  v( x k )
» e.g., ”r(x4)=1  the ranking of x4 is 1, i.e. it is the most preferred”
– Several rank-orderings may be compatible with the preference information
– Incomplete ordinal statements about alternatives can be expressed with regard to
different sets of attributes A’
» single attribute, (sub)set of attributes or holistic statements considering all attributes
» e.g., one can subject statements to cost and environmental factors together
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Modeling of incomplete ordinal information (2/4)
n

Non-normalized form of value function
V ( x )   vi ( xi )
j
j
i 1

Set of feasible values V includes score vectors v=(v(x1),..., v(xm))
– v(xk) denotes the value of xk with regard to some set of attributes
» e.g., if A’={a2}, then v(xk)=v2(x2k)
» e.g., if A’=A, then v(xk)=V(xk)
– Restricted by preference statements

Feasible region associated with a rank-ordering is convex
S (r X ' )  {v  V | v( xk )  v( x j ) if r ( x k )  r ( x j ), x k , x j  X '}
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Modeling of incomplete ordinal information (3/4)

Elicitation of the preference statements is carried out through an
alternative set IX’X and a ranking set J{1,...,m’}, where m’=|X’|
– If |I||J|, the rankings in J are attained by alternatives in I
– If |I|<|J|, the alternatives in I have their rankings in J
– Sets subjected to X’ and A’ denoted by I(A’,X’) and J(A’,X’)

Examples
– x1 and x2 are among the three most preferred ones with regard to cost attribute a1.
Now A’={a1} and X’=X, I({a1}; X)={x1,x2}, J({a1}; X)={1,2,3}
– Holistically (A’=A) the two least preferred are among x4,x5, x8,x9: I(A; X)= {x4,x5, x8,x9},
J(A; X)={m-1,m}
– Holistically the most preferred of the set X’={x1, x2, x7} is x1: I(A; X’)={x1}, J(A; X’)={1}
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Modeling of incomplete ordinal information (4/4)

Sets I and J lead to compatible rank-orderings R(I,J)

The feasible region associated with many compatible rankorderings is usually non-convex
S (I , J ) 

S (r )
rR ( I , J )

Statements can be given with regard to different attribute sets
– Several rank-orderings may be compatible with each of these sets
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Mixed integer linear programming model (1/5)

Overcoming the non-convexity
– Continuous ”milestone variable” zk distinguishes between the values of
alternatives with rankings k and k+1
– If xj’s ranking is at most k, its value is at least zk and we let yk(xj)=1, else 0
– There are exactly k alternatives whose ranking is at most k
» e.g., the three rankings 1, 2 and 3 are at most 3
y2  1
z1
r 1
y3  0
z2
r2
r 3
z5
z4
z3
r4
r 5
r6
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Mixed integer linear programming model (2/5)

Formally
 z k  v( x j )  (1  yk ( x j ))M
 j
j
v( x )  z k  yk ( x ) M
k  1,...,m'1, x j  X ' , M  0
j
y
(
x
 k )  k , k  1,...,m'1
x j X '

For the sake of interpretational and computational matters, we
set
yk ( x )  yk 1 ( x ), k  2,...,m'1, x  X '
j
j
j
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Mixed integer linear programming model (3/5)

Adding preference statements into the model

Assumption |J|≤|I|
– For all rankings jJ, the respective alternative belongs to I

Because of the uniqueness of the rankings, there is exactly
one alternative in I, for which yj-1(xi)=0, and yj(xi)=1. For
other alternatives, yj-1(xi) and yj(xi) get same values
 (y ( x )  y
i
j
j 1
( x ))  1, j  J
i
xi I

E.g., I=(x1, x2, x5), J={2,4}, exactly one of the alternatives
has the ranking 2  it is the only one with different values
for y2(xi) and y1(xi)
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Mixed integer linear programming model (4/5)

Some milestone and binary variables and the respective
constraints are redundant
– Given a statement that alternatives x1 and x2 are the two most preferred,
for example variables z1, z3 and y1(xj), y3(xj) are not needed
» actually only z2 and y2(xj) are needed

If set J is ”sequential”, i.e., it constitutes of consecutive positive
integers, the number of variables and constraints can be
substantially decreased
– For example sets {3,4,5} and {1} are sequential, set {1,3} not

For the compatible rank-orderings associated with sets I and J,
k
k
|I||J|, it holds
R( I , J )   R( I , J i ), J   J i
i 1
i 1
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Mixed integer linear programming model (5/5)

Partitioning of J into a minimal number of sequential sets Ji
– For example, J={1,2,6,7} is partitioned into J1={1,2} and J2={6,7}
– At most 2 milestone and 2m’ binary variables needed to represent the
statement associated to a sequential set Ji

Representation of the feasible region S(I,J) as the intersection
of the feasible regions S(I,Ji)
– Constraints for all pairs I, Ji are set in the same model

If contradictionary to the assumption it holds |J|>|I|, the feasible
region is constructed with the help of complement sets IC=X’\I
and JC={1,...,m’}\J
– Now |JC|≤|IC|
– S(I,J)=S(IC,JC)

All linear inequalities can be included in the model
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An illustrative example (1/6)

A company is about to choose the facility for a new office
– Eight attributes relevant: a1: size of the office, a2: rental costs, a3: renovation
need, a4: car park opportunities, a5: means of communication, a6: distance to
city center, a7: other facilities in the neighborhood, a8: habitability
– 12 alternatives
alt.
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
Area
Rent / €
180 m2 2000
240 m2 3000
210 m2 2800
214 m2 2000
300 m2 3200
170 m2 1800
250 m2 2600
260 m2 2650
262 m2 2400
241 m2 2500
198 m2 2200
201 m2 2000
Renovation need
considerable
none
intermediate
very small
considerable
considerable
small / considerable
intermediate
big
small
considerable
intermediate
Car park
13
13
2
13
13
5
13
10
13
11
13
7
Garage
no
no
no
yes
yes
no
yes
no
yes
yes
no
no
Public
transport
quite bad
good
great
bad
good
good
intermediate
good
good
intermediate
bad
bad
Distance
to center
12 km
15 km
0 km
25 km
4 km
0 km
7 km
10 km
10 km
7 km
17 km
22 km
Other facilities
intermediate
good
great
intermediate
good or great
great
good
intermediate
intermediate
good
good
quite bad
Habitability
great
bad
good
good
very good
good
intermediate
intermediate
very good
good
good
intermediate
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An illustrative example (2/6)

Attributes size, rental costs, car park opportunities and distance to city
center are assessed through [0,1]-normalized scores or score
intervals
alt.
x1
x2
x3
x4
x5
x6
x7
x8
x9
x10
x11
x12
v1
0.13
0.66
0.43
0.47
1.00
0.00
0.73
0.79
0.80
0.67
0.32
0.35
v2
0.86
0.14
0.29
0.86
0.00
1.00
0.43
0.39
0.57
0.50
0.71
0.86
v4
[0.90,0.95]
[0.90,0.95]
[0.20,0.30]
[1.00,1.00]
[1.00,1.00]
[0.40,0.50]
[1.00,1.00]
[0.80,0.88]
[1.00,1.00]
[0.95,0.98]
[0.90,0.95]
[0.55,0.65]
v6
[0.27,0.40]
[0.17,0.30]
[1.00,1.00]
[0.00,0.00]
[0.70,0.80]
[1.00,1.00]
[0.50,0.65]
[0.30,0.45]
[0.30,0.45]
[0.50,0.65]
[0.10,0.20]
[0.01,0.05]
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An illustrative example (3/6)

Other information is turned into incomplete ordinal statements
– E.g., alternative x2 is the only one with no renovation need (a3), hence the ranking 1
– E.g., alternative x2 is the least preferred w.r.t. habitability (a8)
attr. I
a3 {x2}
{x4}
{x10}
{x7}
{x3,x8,x12}
{x9}
{x1,x5,x6,x11}
a5 {x3}
{x2,x5,x6,x8,x9}
{x7,x10}
{x1}
{x4,x11,x12}
J
attr. I
{1}
a7 {x3,x6}
{2}
{x5}
{3,4}
{x2,x7,x10,x11}
{3,4,8,9,10,11,12}
{x1,x4,x8,x9}
{4,5,6,7}
{x12}
{7,8}
{8,9,10,11,12}
{1}
a8 {x1}
{2,3,4,5,6}
{x5,x9}
{7,8}
{x3,x4,x6,x10,x11}
{9}
{x7,x8,x12}
{10,11,12}
{x2}
J
{1,2,3}
{1,2,3,4,5,6,7}
{3,4,5,6,7}
{8,9,10,11}
{12}
{1}
{2,3}
{4,5,6,7,8}
{9,10,11}
{12}
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An illustrative example (4/6)

Information on attributes’ relative importance
– Complete rank-ordering of the attributes is r(a1,a2,...,a8)=(1,2,...,8)
– A weight of 0.50 is assigned to the most important attribute, size of the office
– Weights are lower bounded by wi  1/3n
1
0.50  w1  w2  ...  w8 
24

A holistic preference for x4 over x1 over x3
V ( x 4 )  V ( x1 )  V ( x3 )
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An illustrative example (5/6)

PB(x1,x)
PB(x2,x)
PB(x3,x)
PB(x4,x)
PB(x5,x)
PB(x6,x)
PB(x7,x)
PB(x8,x)
PB(x9,x)
PB(x10,x)
PB(x11,x)
PB(x12,x)
Pairwise bounds (minima of overall value differences) indicate that
there are 5 non-dominated alternatives x5, x7, x8, x9 and x10
x1
0.033
-0.040
0.000
0.143
-0.238
0.076
0.119
0.172
0.143
-0.152
-0.115
x2
-0.263
-0.263
-0.244
0.049
-0.390
-0.043
-0.089
-0.042
-0.070
-0.298
-0.349
x3
0.000
0.033
0.000
0.143
-0.238
0.076
0.119
0.172
0.143
-0.152
-0.115
x4
-0.282
-0.123
-0.282
0.029
-0.397
-0.079
-0.115
-0.028
-0.099
-0.310
-0.361
x5
-0.428
-0.312
-0.428
-0.376
-0.655
-0.309
-0.215
-0.173
-0.239
-0.569
-0.480
alternative x
x6
x7
-0.058
-0.350
-0.025
-0.171
-0.058
-0.350
0.007
-0.229
0.137
-0.051
-0.464
0.058
0.103
-0.177
0.203
-0.129
0.085
-0.158
-0.150
-0.382
-0.079
-0.433
x8
-0.354
-0.178
-0.354
-0.236
-0.061
-0.481
-0.137
-0.135
-0.067
-0.388
-0.439
x9
-0.323
-0.274
-0.323
-0.314
-0.155
-0.561
-0.217
-0.188
-0.164
-0.465
-0.418
x10
-0.368
-0.199
-0.368
-0.264
-0.038
-0.482
-0.150
-0.192
-0.106
-0.399
-0.452
x11
-0.089
-0.056
-0.089
-0.071
0.105
-0.308
0.016
0.067
0.167
0.054
-0.177
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x12
-0.153
-0.048
-0.153
0.004
0.102
-0.263
0.035
0.039
0.087
0.102
-0.183
-
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An illustrative example (6/6)

To discriminate between non-dominated alternatives, decision rules
are applied
maximax
maximin
central values
minimax regret
x5
0.892
0.571
0.732
0.155
x7
0.758
0.505
0.632
0.309
x8
0.762
0.511
0.637
0.215
x9
0.834
0.556
0.695
0.173
x10
0.768
0.485
0.627
0.239

Each rule recommends alternative x5

XPress-MP was used in solving the example

Calculation of the example (pairwise bounds, overall value intervals for
each 12 alternatives; 156 MILPs) took 14 seconds on a Pentium III at
800 MHz with 256 MB RAM
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Conclusion

The DM can give incomplete ordinal information about the alternatives with
regard to a single attribute, a set of attributes or holistically

Statements about relative importance of attributes are allowed, as well

Based on a linear model and hence it can be used in conjunction with other
preference programming methods

Computationally far more efficient than RICH, and more flexible as it contains
all features of RICH

Software implementation of RICHER Decisions © ongoing

Future research directions
–
–
–
Modeling of classification procedures with RICHER methodology (cf. the example in this
presentation)
Application of RICHER methodology to voting or other group decision processes
Application of incompelete ordinal information in Robust Portfolio Modeling (RPM)
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Related references
Arbel, A., “Approximate Articulation of Preference and Priority DerivationApproximate Articulation of Preference and
Priority Derivation”, European Journal of Operations Research 43 (1989) 317-326.
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 (to appear).
Salo, A. ja R. P. Hämäläinen, "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 Programme”, submitted
manuscript.
Salo, A. and Punkka, A., “Rank Inclusion in Criteria Hierarchies”, European Journal of Operations Research (to appear).
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.
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