# DECISION THEORY Preference Programming Ratio-based Efficiency Analysis incomplete

DECISION THEORY
Preference Programming

Ratio-based Efficiency Analysis

incomplete information
in value tree analysis
preference statements

n

j

no

comparison of DMUs under incomplete information
about the output and input weights
N
value of outputs
Efficiency of DMUk Ek

value of inputs

j

V ( x ) vi ( xi )
i 1

E1
E2

yes
Recommendations
by decision rules

Overall value and
ranking intervals

u y

(0,0,1)

Display of results and
recommendations

E3 / E0 = 1.6

n nk

n 1
M

x2

v

x2

x

m mk

yes
Decision

3 4 12
, ,
19 19 19

w3 w2

interval methods:

Preference Assessment by Imprecise
Ratio Statements (PAIRS)
Interval AHP
Preference Ratios in Multiattribute Evaluation (PRIME)
Interval SMART/SWING
w2

S

0.5

DMU2

E1 / E*=0.6

DMU3

ranking 1

x1

ranking 2

DMU1

ranking 3
ranking 4

DMU4

DMU1 DMU2 DMU3 DMU4

x1

new flexibility in dynamic problems
x

1

dominance relations

attainable rankings

3 2 6
, ,
11 11 11

1 2 2
, ,
5 5 5

(0,1,0)

E0

efficiency bounds

S

w3 3w2
1 4 4
, ,
9 9 9

w3 2w1

u1/u2

x*

x*
S

E1 / E*[0.6,1.0]
E2 / E*[0.9,1.0]
...
E3 / E0[1.2,1.6]
E4 / E0[1.0,1.3]

w3
w3 4w1

Goal set (intervals): ([l1,u1],[l2,u2])

E3 / E0 = 1.2

E4
Interpretation
of results

Goal point: (g1,g2)

E*

E1 / E*=1

E3

no

extension of a goal point to a goal set

m 1

Consistent ?

Dominance
relations

Interval goal programming

w1 0.25w2

sensitivity of university rankings

w1
(1,0,0)

w1 1.5w2

(0,0,1)

- what if slightly different weigths were applied?

w3

w3 w1

Robust rankings

w3 w2

incomplete ordinal information:

w1

r (3,1, 2)
r (1,2,3)
r (2,1,3)

(1,0,0)

Different weighting would
likely yield a better ranking

t2

Alternative

Utility

20 % interval

Strategy 0

0.474

Strategy 1

0.697

Strategy 2

0.694

Strategy 3

0.748

Strategy 4

0.628

incomplete ordinal
no information

t3

tk

time

global sensitivity analysis

exact weights

30 % interval
r (1,3,2)

Rank Inclusion in Criteria Hierarchies (RICH)
RICHER = RICH with Extended Rankings

t1

Costs

Other cancers

Political cost

Soc.-Psych Negative

Soc.-Psych Positive

Thyroid cancer

(0,1,0)
w2

origins of procedural and behavioral biases
Number of attribute
levels effect in
conjoint analysis

Hierarchical
steeper weigths

Range effect

Systems

Analysis Laboratory
Updated 17.05.2010

Weighting methods
yield different
weights

Weights
derived
from
ordinal
infromation

Averages over a
group yield even
weights

Normalization

Division of
attributes changes
weights

Rank reversal in
AHP

Splitting bias

10th

442nd

web-sites and selected publications

http://www.decisionarium.hut.fi
A. Salo and A. Punkka: Ranking intervals and dominance relations for Ratio-based Efficiency Analysis, manuscript, 2010
A. Punkka and A. Salo: Preference Programming with incomplete ordinal information, manuscript, 2010
A. Salo and R. P. Hmlinen: Preference Programming - multicriteria weighting models under incomplete information,
in: Zopounidis and Pardalos (eds.): Handbook of Multicriteria Decision Analysis, Springer, New York, 2010
J. Liesi, P. Mild and A. Salo: Preference programming for robust multi-criteria portfolio modeling and project selection,
Eur. J. Oper. Res. (EJOR), 2007
J. Mustajoki, R. P. Hmlinen and M. R. K. Lindstedt: Using intervals for global sensitivity and worst case analyses in multiattribute value trees, EJOR, 2006
A. Salo and A. Punkka: Rank inclusion in criteria hierarchies, EJOR, 2005
J. Mustajoki, R. P. Hmlinen and A. Salo: Decision Support by Interval SMART/SWING - Incorporating Imprecision in the SMART and SWING Methods,
Decision Sciences, 2005
A. Salo and R. P. Hmlinen: Preference ratios in multiattribute evaluation (PRIME), IEEE Syst. Man Cybernetics, 2001
R. P. Hmlinen and J. Mntysaari: A dynamic interval goal programming approach to the regulation of a lake-river system, J. Multi-Crit. Dec. Anal., 2001
A. Salo and R. P. Hmlinen: Preference programming through approximate ratio comparisons, EJOR, 1995
A. Salo and R. P. Hmlinen: Preference assessment by imprecise ratio statements, Operations Research, 1992

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