for Spatial Multiattribute Decision Analysis Mikko Harju Juuso Liesiö Kai Virtanen Systems Analysis Laboratory Department of Mathematics and Systems Analysis Aalto University School of Science ID: 599521
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Slide1
Preference Programmingfor Spatial MultiattributeDecision Analysis
Mikko Harju
*
, Juuso Liesiö
**
, Kai Virtanen
*
* Systems Analysis Laboratory,
Department of Mathematics and Systems Analysis, Aalto University School of Science
** Department of Information and Service Economy, Aalto University School of BusinessSlide2
Spatial Decision Analysis
Consequences of alternatives are distributed across a geographical region
E.g., select the
p
osition of a rescue helicopter base,
or Alternatives imply different response times, i.e., consequences, for each locationLocations not equally important? (cf. population density)Plenty of other applicationsUrban, environmental and transportation planningWaste management, hydrology, agriculture, and forestrySee, e.g., Malczewski & Rinner 2015, Ferretti & Montibeller 2016
Long
Short
Response time
Alternative 1
Alternative 2
Slide3
Our contribution:
Axiomatic
basis for preferences that can be represented with the
spatial value function
Spatial
preference programming: Determination of dominances among alternatives based on incomplete specification of weightsSpatial Value Function
Value of decision alternative
(Simon, Kirkwood and Keller 2014):
: spatial
weight (“importance”)
of specific location
in region
S
: consequence for location when alternative
is chosen: consequence value function
Challenges:
Specifying spatial
weights 𝑎(𝑠)
for an infinite number of locations 𝑠
Only a conjecture on the underlying preference assumptions exists Slide4
Preference Assumptions
Let
be a binary relation on
the set of decision alternatives
: set of locations
: set of consequences
Assumptions
A1
is transitive and complete
A2 There exist
such that
A3 “Spatial preference
independence”
A4 “Consequence
consistency”
A5 “Spatial consistency
”
A6 “Divisibility of
subregions
”
A7 “Monotonicity”
Least preferred
Most preferred
Consequences
A3: Preference
between two alternatives does not depend on locations with equal
consequenceSlide5
Additive Spatial Value Function
Theorem.
satisfies A1-A7
iff
there exists a non-atomic measure on and a bounded function such that
w
here
Proof based
on
Savage 1954
The weighting function
Assigns a weight to each
subregion
(cf.
relative importance)Connection to Simon’s et al. weighting
:
is a cardinal value function for consequences
I.e., unique up to positive affine
scaling
E.g., additive multiattribute
Slide6
Incomplete Preference Information
”
Subregion
more
important than
”
Small
set of feasible
weighting functions can be sufficient for ranking alternatives
Avoiding the overwhelming task of specifying
the
exact weighting function
Stated preferences
between
pairs
of alternatives
Constraints on the spatial weighting function
and
the vector
of attribute weights
Multiple preference statements comparing suitable alternatives
System of
linear constraints on
where
is a partition of
Least preferred
Most preferred
Consequences
Slide7
DominanceConstraints from preference statements result in
A set
of feasible weighting
functions
A set
of feasible attribute weights
Alternative
dominates alternative
if
for all
and
for some
and
Dominance check: bi-level LP problem
where
is a partition of
Solution: Enumerate extreme points of
and solve
LP
problem in each one
and
non-dominated
Slide8
Air Defense Planning:Positioning of Air BasesSelect positions for 2 main and 3 secondary air bases to maximize air defense capability
Main bases: 3 position candidates
Secondary bases: 5 position candidates
Spatial consequences provided by
a simulation tool – input parameters:
Number of defensive flying units; fuel consumption; weapons consumption; flight speedPositions of air bases; turnaround times; refueling and rearming times; alert, taxi and scramble delaysMajor city CityPower plant
T
hreat
T
hreatSlide9
Attributes of Air Defense Capability
Positions of air bases affect…
“Engagement frontier” where hostile aircraft can first be intercepted by defensive flying units
Attribute #1: Location’s distance to south frontier
Attribute #2:
Location’s distance to west frontier“Force fulfillment”Attribute #3: Average number of defensive flying units available at the locationAttribute #4: As attribute #3 with one secondary base destroyed (cf. combat sustainability)Attribute #1Attribute #2Attribute #3Attribute #4Alternative with bases at the highlighted positions
Least preferred
Most preferred
Consequences
Slide10
Spatial preference statements (
)
Major cities > SW coastal area
Power plants > SW coastal
area
SW coastal area > NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide11
Spatial preference statements ()Major
cities
>
SW coastal area
Power plants > SW coastal
areaSW coastal area > NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide12
Spatial preference statements ()Major cities > SW coastal area
Power plants
>
SW coastal area
SW coastal area > NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide13
Spatial preference statements ()Major cities > SW coastal area
Power plants > SW coastal
area
SW coastal area
> NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide14
Spatial preference statements ()Major cities > SW coastal area
Power plants > SW coastal
area
SW coastal area > NE coastal
area
NE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide15
Spatial preference statements (
)
Major cities > SW coastal area
Power plants > SW coastal
area
SW coastal area > NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference Statements
T
hreat
T
hreatSlide16
13 non-dominated alternatives
Spatial preference statements (
)
Major cities > SW coastal area
Power plants > SW coastal
areaSW coastal area > NE coastal areaNE coastal area > Other areasAttribute preference statements ()Engagement frontier attributes >Force fulfillment attributes Preference StatementsSlide17
4
non-dominated alternatives
Additional Preference Statements
Spatial preference statements
(
)Power plants > Major citiesPower plant #1 > Power plant #2City #1 > City #2 > City #3...Attribute preference statements ()West engagement frontier > South engagement frontierForce sustainability > Initial force fulfillment
B
C
A
2
3
5
4
1
1
2
1
2
3
Major city
Power plant
Main base
Secondary
baseSlide18
Conclusion: Spatial Decision Analysis Benefits from Preference ProgrammingThe additive spatial value functionAxiomatic basisWeighting subregions rather than locationsPreference programming for spatial decision analysis
Incomplete
preference
information
& non-dominated decision alternativesBurden
of DM eased considerably by not requiring unique spatial weightingGlobal sensitivity analysis: Effect of spatial weighting on ranking of alternativesFuture developmentPractices and behavioral issues of eliciting the weighting functionSpatial decision support systems: Graphical user interface, utilization of GIS dataSlide19
ReferencesFerretti, V. and Montibeller, G., 2016. Key challenges and meta-choices in designing and applying multi-criteria spatial decision support systems.
Decision
Support Systems
,
84Malczewski, J. and Rinner, C., 2015.
Multicriteria decision analysis in geographic information science. New York: SpringerSalo, A. and Hämäläinen, R.P., 1992. Preference assessment by imprecise ratio statements. Operations Research, 40(6)Savage, L.J., 1954. The foundations of statistics. New York: John Wiley and SonsSimon, J., Kirkwood, C.W. and Keller, L.R., 2014. Decision analysis with geographically varying outcomes: Preference models and illustrative applications. Operations Research, 62(1)