Decision map for spatial decision making Salem Chakhar in collaboration with Vincent Mousseau, Clara...

Decision map for spatial decision making

Salem Chakhar

in collaboration with Vincent Mousseau, Clara Pusceddu and Bernard Roy

LAMSADEUniversity of Paris Dauphine

www.lamsade.dauphine.fr

30-06-2005

CUPUM’05 – University College London, London, UK · 29 July – 01 August 2005

CUPUM'05 · 30-06-2005 2

Contents

1. Introduction

2. Decision map

3. Generation process

4. Inference model

5. Illustrative example

6. Some applications

7. Conclusion

CUPUM'05 · 30-06-2005 3

Introduction

• Spatial decision problems are complex: – Different participants to the decision process with conflicting objectives and

preferences

• Spatial decision problems are of multi-criteria nature:– Decision have to take account of several and conflicting territorial and urban

dimensions (e.g. social, environmental, economic)

• Traditional Spatial Models:– do not permit to represent complexity of spatial problems – neglect social, qualitative and interactive dimensions, of importance in the spatial

decision making process – do not support in their operational dimension, communicative and collaborative

decision making

CUPUM'05 · 30-06-2005 4

Research on Spatial Decision Models should:

– take into consideration the complexity and multicriteria nature of spatial problems by:

• integrating preference models to represent points of view of various actors and stakeholders involved in the spatial decision process

• providing a communicative and collaborative environment for supporting interaction and integration of expert and experiential knowledge from all the actors

CUPUM'05 · 30-06-2005 5

Objective

To propose a spatial decision model by combining GIS-based maps and

multi-criteria analysis:

– Introduction of the concept of decision map

– Introduction of an inference model

– Show how this spatial decision model can be implemented through a didactic example

CUPUM'05 · 30-06-2005 6

Decision Map: Definition

* The decision map is defined as an advanced version of conventional GIS maps which is enriched with preferential information

* It looks like a set of homogenous spatial units; each one is characterized with a global, often ordinal, evaluation that represents an aggregation of several partial evaluations relative to different criteria

* The construction of a decision map require that every criterion that represents a qualitative or quantitative territorial and urban dimension (social, demographic, geological) and which is relevant for the spatial problem under consideration, to be expressed by a criterion map

CUPUM'05 · 30-06-2005 7

Decision Map: Definition

Definition. A decision map M is defined as {(u, f(u)) : u U}, where U is a set of homogenous spatial units and f is a function defined as follows:

f : U → E

u → f(u)= Φ [g1(u),…,gm(u)]

where:

- U = {u1,u2,…,un} is the study area- E : is an ordinal scale- Φ : is a multi-criteria aggregation model

- gi(u): evaluation of spatial unit u according to criterion gi;

CUPUM'05 · 30-06-2005 8

Generation process

Additional Infor.

Multi-criteria classification

Data analysisData analysis

Cartographic modeling Decisional cartography

Decision map

Geographic map

Criteria maps

g(ui)

g(ui)=[g1(ui1),...,gm(uim)]

ui {C1,…Cr}

ui Cj

Inference of preferential parameters

Step 1. Problem definition

Step 2. Generation of an intermediate map

Step 3.

Multi -criteria Classification

Step 4. elaboration of a final map

CUPUM'05 · 30-06-2005 9

The Multi-criteria Aggregation Model: ELECTRE TRI

* ELECTRE TRI needs first to define a set of ordered categories

* ELECTRE TRI assigns spatial units to categories following two consecutive steps:

1. Construction of an outranking relation S that characterizes how units compare to the limits of the categories (Note: for two objects a and b, aSb means that “a is at least as good as b”):

- Compute the partial concordance indices cj(u,bh) and cj(bh,u)- Compute the global concordance indices c(u,bh) and c(bh,u)- Compute de discordance indices dj(u,bh)- Compute the outranking degree: S(u,bh) = c(u,bh), si dj(u,bh) c(a,bh), for all j, = c(u,bh) * j J(u,bh)[1-dj(u,bh)] / [1-c(u,bh)], otherwise.

2. Exploit the relation S in order to assign each unit to a specific category : confirm or infirm uSbh

The problem: To perform ELECTRE TRI, we need several parameters from the decision makers: qj, pj, vj, kj incorporate an inference model into the system

CUPUM'05 · 30-06-2005 10

Spatial Decision Model

criterion map2

g1 g2 g3 … gm-1 gm

ELECTRE TRI

b0

b1

bp-1

bp+1

bp

C1

Cp+1

Cp

INFERENCE MODEL

criterion map1

criterion map m

.

.

Decision map

CUPUM'05 · 30-06-2005 11

Inference process

Choose U*

Assign spatial units from U* to the categories

Optimize to obtain a model

Fix value or interval of variation for one or several parameters

Additional information on some model parameters?

Model accepted?Revise assignment examples

nono

no

yes

yes

Stop

Start

CUPUM'05 · 30-06-2005 12

Different Inference strategies

ELECTRE TRI Model

Infer weights kj Infer veto thresholds vj Infer categories limits bh

Partial inference Global inference

Direct elicitationInfer from examples

CUPUM'05 · 30-06-2005 13

Inference model : Optimization problem

Max + ukU* (xk + yk)

s.t. xk, uk U*

yk, uk U*

[j=1..m kjcj(uk,bh-1) / j=1..m kj]-xk= , uk U*

[j=1..m kjcj(uk,bh) / j=1..m kj] + yk = , uk U*

[0.5, 1]

gj(bh+1) gj(bh)+pj(bh)+pj(bh+1), j F, h B

pj(bh) qj(bh), j F, h B

kj 0, qj(bh) 0, j F, h B

CUPUM'05 · 30-06-2005 14

An Example didactic: the location of an incineration plant

• Four criteria have to be considered:

Criterion Description Max/Min

g1 Waste Volume

g2 Underground water resources pollution

g3 Air pollution

g4 Social Acceptance

very low low average high very high

c1 < c2 < c3 < c4 < c5

We use an ordinal scale E with five categories:

CUPUM'05 · 30-06-2005 15

Example

Slope Lithology

Overflowing

“Underground water resource pollution”Criterion map

Spatial operations (e.g. overlay)

Wetland

Landslide

Digital Elevation Model

Geology

Other sources (e.g. satellite imagery)

• Flowchart of “Underground water resource pollution” criterion map:

CUPUM'05 · 30-06-2005 16

Example

1 Very low

2 Low

3 Average

4 High

5 Very high

Scale :

• “Underground water resource pollution” criterion map:

1 2 3 4 28 25 24

6 5 27

8 7 26

9 14 20 21 22 23

11 13 15 19

17 18

10 12 16

CUPUM'05 · 30-06-2005 17

1 2 3 21

20 23 24

7 4 22

8 6 5 19 25

10 16 26 27

9 13 14 17 28

11 12 15 18 29

1 2 3 4 28 25 24

6 5 27

8 7 26

9 14 20 21 22 23

11 13 15 19

17 18

10 12 16

1 2 3 14 28 27

6 5 4 15 16 29

7 17 26

9 13 25 24

8 18 21

12 20 22

10 11 19 23

1 2 16 18

4 3 15 17

5 7 23

6 9 24 21 20

8 14 19

13 25 22 26 10 11 12 27

the intermediate map

Example

g1: “Waste Volume”

g2: “Underground water resources pollution”

g3 : “Air Pollution”

g4 : “Social Acceptance”

Each unit ui is characterized by a vector of performances: [g1(ui),g2(ui),g3(ui),g4(ui)]

u1 u2 u3 u4 u5 u6 u7 u8 u9 u10

u11 u12 u13 u1 4 u15 u16 u17 u18 u19 u20

u21 u22 u23 u24 u26 u27 u28 u29 u30

u31 u32 u33 u25 u34 u35 u36 u37

u39 u40 u41 u42 u43 u44 u45

u38 u46 u47 u48 u49 u50 u51 u52

u53 u54 u55 u56 u57 u58 u59 u60 u61

[3,4,4,3]

[2,5,2,5]

CUPUM'05 · 30-06-2005 18

Example

[2,4,4,2]

u1

[3,4,4,3]

u2

[3,3,4,1]

u3

[5,1,1,1]

u4

[1,5,5,1]

u5

[1,5,1,4]

u6

[3,1,3,4]

u7

[3,1,3,5]

u8

[3,4,2,3]

u9

[5,3,2,1]

u10

[2,4,4,5]

u11

[5,5,4,5]

u12

[3,3,4,3]

u13

[5,1,1,1]

u1 4

[3,3,5,3]

u15

[1,4,11]

u16

[3,3,5,3]

u17

[3,1,2,5]

u18

[3,1,2,3]

u19

[5,3,1,1]

U20

[3,5,3,5]

u21

[1,5,3,1]

u22

[3,4,4,3]

u23

[5,4,3,2] [4,2,5,4][4,4,2,4]

u26

[4,3,2,4]

u27

[3,2,3,5]

u28

[4,3,3,2]

u29

[3,4,4,4]

[2,5,4,5]

u31

[1,5,1,1]

u32

[3,2,2,4]u24

u25

[1,3,5,1]

u34

[4,2,2,2]

u35

[2,4,4,2]

u36

[4,2,4,3]

u37

u30

[2,5,2,5][5,5,1,5]

u39

u33 [3,3,3,2] u40

[1,5,4,1]

u41

[4,2,2,4]

u42

[1,5,5,2][4,2,1,3]

u44

[3,4,3,4]

u38

[3,1,1,2]

u46

[3,1,3,5]

u47

[2,3,4,5]

u48

[4,2,2,5]

u49

[1,1,5,1]

u50

[2,4,5,1]

u51

u43

[5,5,2,5]

u52

u45

[1,2,2,5]

u53

[3,1,3,2]

u54

[3,1,1,2]

u55

[4,2,1,4]

u56

[2,2,2,4]

u57

[2,2,4,2]

u58

[2,2,4,1] u59

[1,5,2,3]

u60

[1,5,1,3]

u61

?

Final map

i.e. for each unit ui in the intermediate map, we associate a global evaluation g(ui) = Φ[gj(ui)]jF

Intermediate map

Multi-criteria sorting model: Φ : Em E [g1(u),g2(u),…,gm(u)] g(u)

CUPUM'05 · 30-06-2005 19

Example

g1 g2 g3 g4

g(b4) 4.5 1 1 4.5

q4 0.2 0.2 0.2 0.2

p4 0.3 0.3 0.3 0.3

g(b3) 3.5 2 2 3.5

q3 0.2 0.2 0.2 0.2

p3 0.3 0.3 0.3 0.3

g(b2) 2.5 3.5 3.5 2.5

q2 0.2 0.2 0.2 0.2

p2 0.3 0.3 0.3 0.3

g(b1) 0.25 4 4 0.25

q1 0.2 0.2 0.2 0.2

p1 0.3 0.3 0.3 0.3

• Φ = ELECTRE TRI

• Inference of the weight kj only.

g1 g2 g3 … gm-1 gm

bp

bp+1

Cp-1

bp+pp

bp+qp

bp-pp

bp-qp

CUPUM'05 · 30-06-2005 20

Example

• Result without additional information:(c2)

u1

(c2,c3)

u2

(c2,c3)

u3

(c2,c5)

u4

(c1,c2)

u5

(c1,c2, ,c3)

u6

(c3,c4 ,c5)

u7

(c3,c4)

u8

(c2, c3)

u9

(c2,c3, c4)

u10

(c2)

u11

(c1,c2,c3)

u12

(c2,c3)

u13

(c2,c5)

u1 4

(c1,c3)

u15

(c2)

u16

(c1,c3)

u17

(c3,c4,c5)

u18

(c3,c4)

u19

(c2,c3,c4)

u20

(c1,c3)

u21

(c1,c2)

u22

(c2,c3)

u23

(c2,c3) (c1,c4)(c2,c4)

u26

(c3,c4)

u27

(c3,c4)

u28

(c2,c3)

u29

(c2,c3)

(c1,c2)

u31

(c1,c2)

u32

(c3,c4) u24 u25

(c1,c2)

u34

(c2,c4)

u35

(c2) u36

(c2,c3,c4)

u37

u30

(c1,c2,c3) (c1,c5)

u39

u33(c2,c3) u40

(c1,c2)

u41

(c4)

u42

(c1,c2)(c3,c4)

u44

(c2,c3)

u38

(c2,c3,c4)

u46

(c3,c4)

u47

(c2,c3)

u48

(c4)

u49

(c1,c2)

u50

(c1)

u51

u43

(c1,c4,c5)

u52

u45

(c2,c4)

u53

(c2,c3)

u54

(c2,c3,c4)

u55

(c4)

u56

(c2,c4)

u57

(c2)

u58

(c2) u59

(c1,c2, ,c3)

u60

(c1,c2, ,c3) u61

Very low Low Average High Very high

• Additional information: u33c4 ; u40 c1-c3 ; u61 c1-c2

CUPUM'05 · 30-06-2005 21

Example

• Result with additional information:

(c2)

u1

(c2)

u2

(c2)

u3

(c5)

u4

(c1)

u5

(c2)

u6

(c4)

u7

(c3)

u8

(c3)

u9

(c3)

u10

(c2)

u11

(c2)

u12

(c3)

u13

(c5)

u1 4

(c3)

u15

(c2)

u16

(c3)

u17

(c4)

u18

(c3)

u19

(c4)

u20

(c3)

u21

(c1)

u22

(c2)

u23

(c2) (c4)(c4)

u26

(c4)

u27

(c3)

u28

(c4)

u29

(c2)

(c2)

u31

(c2)

u32

(c4) u24 u25

(c2)

u34

(c4)

u35

(c2) u36

(c3)

u37

u30

(c2) (c5)

u39

u33(c3) u40

(c1)

u41

(c4)

u42

(c1)(c4)

u44

(c4)

u38

(c3)

u46

(c3)

u47

(c2)

u48

(c4)

u49

(c2)

u50

(c2)

u51

u43

(c4)

u52

u45

(c4)

u53

(c3)

u54

(c3)

u55

(c4)

u56

(c4)

u57

(c2)

u58

(c2) u59

(c1)

u60

(c2) u61


CUPUM'05 · 30-06-2005 22

Example

• Result after grouping:

u4

u5 u13 u14

u29u30

u6 u12 u28 u31 u35

u3

u7 u11

u15u27 u32

u2 u19 u26 u33

u34

u8

u10u16 u20 u25

u9 u17

u21 u24

u1 u18 u22 u23


CUPUM'05 · 30-06-2005 23

Applications: Generation of alternatives in Multicriteria Analysis

→ The problem: find a corridor for a tramway between an origin o and a destination d:

?o d

* Phase 1. Elaborate a decision map

* Phase 3. Apply a classical algorithm to identify the corridors (s).

* Phase 2. Construct a connectivity graph G=(X,U) :

X = {elementary spatial units}.

U = {(x,y) : x,y X and x and y have a common frontier}.

CUPUM'05 · 30-06-2005 24

Applications: Collaborative and communicative planning

→The spatial decision model may be used as a support for communicative and collaborative planning since it permits:

* to include, by construction, the preferences of the entire participants.

* to visually and spatially represent the preferences of all the participants.

* to perform an effective what-if analysis.

* to implement a constructive spatial decision making approach.

→Two approaches may used for communicative and collaborative planning:

* Approach 1. aggregation at the input level.

* Approach 2. aggregation at the output level.

CUPUM'05 · 30-06-2005 25

Approach 1: Aggregation at the input level

Multi-criteria classification for groups method

Preference parameters inferences

…Criterion Map for criterion gi and group 1

Composite Criterion Map for criterion gi

Composite Intermediate Map

Composite Decision Map

Composite Intermediate Map

Composite Criterion Map for criterion g1

Composite Criterion Map for criterion gm

Criterion Map for criterion gi and group K

…Criterion Map for criterion g1

and group 1

Criterion Map for criterion g1 and group K

…Criterion Map for criterion gm

and group 1

Criterion Map for criterion gm

and group K

Additional information

CUPUM'05 · 30-06-2005 26

Approach 2: Aggregation at the output level



and group k

Composite Decision Map

Intermediate Map for group 1


and group k


and group 1


and group 1

…Criterion Map for criterion g1 and group K


and group K

Intermediate Map for group 1

Decision Map for group 1


Intermediate Map for group k

Intermediate Map for group k

Decision Map for group k


Intermediate Map for group K

Intermediate Map for group K

Decision Map for group K







CUPUM'05 · 30-06-2005 27

Conclusion

→ We have introduced the concept of decision map :

Cartographic modelling Decision map

an automatic process largely controlled by the decision maker(s)

presentation-oriented “visual” decision-aid-oriented

preferences are often reduced to a tabular representation

preferences are explicitly and spatially represented

aggregation is performed in early steps

aggregation is performed in latter steps

weighted sum-like aggregation outranking relations-based aggregation

preference parameters are directly provided

Preference parameters are indirectly provided

→ Perspective: Extend the inference model for groups

Decision map for spatial decision making Salem Chakhar in collaboration with Vincent Mousseau, Clara...

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