1. Decision map for spatial decision making Salem Chakhar in collaboration with Vincent Mousseau, Clara Pusceddu and Bernard Roy LAMSADE University of Paris Dauphine www.lamsade.dauphine.fr 30-06-2005 CUPUM’05 – University College London, London, UK · 29 July – 01 August 2005

2. CUPUM'05 · 30-06-20052 Contents 1.Introduction 2.Decision map 3.Generation process 4.Inference model 5.Illustrative example 6.Some applications 7.Conclusion

3. CUPUM'05 · 30-06-20053 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

4. CUPUM'05 · 30-06-20054 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

5. CUPUM'05 · 30-06-20055 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

6. CUPUM'05 · 30-06-20056 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

7. CUPUM'05 · 30-06-20057 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 [g 1 (u),…,g m (u)] where: - U = {u 1,u 2,…,u n } is the study area - E : is an ordinal scale - Φ : is a multi-criteria aggregation model - g i (u): evaluation of spatial unit u according to criterion g i;

8. CUPUM'05 · 30-06-20058 Generation process Additional Infor. Multi-criteria classification Data analysis Cartographic modelingDecisional cartography Decision map Geographic map Criteria maps g(u i ) g(u i )=[g 1 (u i1 ),...,g m (u im )] u i  {C 1,…C r } u i  C j 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

9. CUPUM'05 · 30-06-20059 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 c j (u,b h ) and c j (b h,u) - Compute the global concordance indices c(u,b h ) and c(b h,u) - Compute de discordance indices d j (u,b h ) - Compute the outranking degree: S(u,b h ) = c(u,b h ), si d j (u,b h )  c(a,b h ), for all j, = c(u,b h ) *  j  J(u,b h ) [1-d j (u,b h )] / [1-c(u,b h )], otherwise. 2. Exploit the relation S in order to assign each unit to a specific category : confirm or infirm uSb h The problem: To perform ELECTRE TRI, we need several parameters from the decision makers: q j, p j, v j, k j  incorporate an inference model into the system

10. CUPUM'05 · 30-06-200510 Spatial Decision Model criterion map2 g 1 g 2 g 3 … g m-1 g m ELECTRE TRI b0b0 b1b1 b p-1 b p+1 bpbp C1C1 C p+1 CpCp INFERENCE MODEL criterion map1 criterion map m.... Decision map

11. CUPUM'05 · 30-06-200511 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 no yes Stop Start

12. CUPUM'05 · 30-06-200512 Different Inference strategies ELECTRE TRI Model Infer weights k j Infer veto thresholds v j Infer categories limits b h Partial inferenceGlobal inference Direct elicitationInfer from examples

13. CUPUM'05 · 30-06-200513 Inference model : Optimization problem Max  +   u k  U* (x k + y k ) s.t.   x k,  u k  U*   y k,  u k  U* [  j=1..m k j c j (u k,b h-1 ) /  j=1..m k j ]-x k =,  u k  U* [  j=1..m k j c j (u k,b h ) /  j=1..m k j ] + y k =,  u k  U*  [0.5, 1] g j (b h+1 )  g j (b h )+p j (b h )+p j (b h+1 ),  j  F,  h  B p j (b h )  q j (b h ),  j  F,  h  B k j  0, q j (b h )  0,  j  F,  h  B

14. CUPUM'05 · 30-06-200514 An Example didactic: the location of an incineration plant Four criteria have to be considered: CriterionDescriptionMax/Min g 1 Waste Volume  g 2 Underground water resources pollution  g 3 Air pollution  g 4 Social Acceptance  very low low average high very high c 1 < c 2 < c 3 < c 4 < c 5 We use an ordinal scale E with five categories:

15. CUPUM'05 · 30-06-200515 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:

16. CUPUM'05 · 30-06-200516 Example 1 Very low 2 Low 3 Average 4 High 5 Very high Scale : “Underground water resource pollution” criterion map: 11 22 33 44  28  25  24 66 55  27 88 77  26 99  14  20  21  22  23  11  13  15  19  17  18  10  12  16

17. CUPUM'05 · 30-06-200517 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1 22 33 44  28  25  24 66 55  27 88 77  26 99  14  20  21  22  23  11  13  15  19  17  18  10  12  16 11 22 33  14  28  27 66 55 44  15  16  29 77  17  26 99  13  25  24 88  18  21  12  20  22  10  11  19  23 11 22  16  18 44 33  15  17 55 77  23 66 99  24  21  20 88  14  19  13  25  22  26  10  11  12  27 the intermediate map Example g 1 : “Waste Volume” g 2 : “Underground water resources pollution” g 3 : “Air Pollution” g 4 : “Social Acceptance” Each unit u i is characterized by a vector of performances: [g 1 (u i ),g 2 (u i ),g 3 (u i ),g 4 (u i )] u1u1 u2u2 u 3 u4u4 u5u5 u6u6 u7u7 u8u8 u9u9 u 10 u 11 u 12 u 13 u 1 4 u 15 u 16 u 17 u 18 u 19 u 20 u 21 u 22 u 23 u 24 u 26 u 27 u 28 u 29 u 30 u 31 u 32 u 33 u 25 u 34 u 35 u 36 u 37 u 39 u 40 u 41 u 42 u 43 u 44 u 45 u 38 u 46 u 47 u 48 u 49 u 50 u 51 u 52 u 53 u 54 u 55 u 56 u 57 u 58 u 59 u 60 u 61 [3,4,4,3] [2,5,2,5]

18. CUPUM'05 · 30-06-200518 Example [2,4,4,2] u 1 [3,4,4,3] u 2 [3,3,4,1] u 3 [5,1,1,1] u 4 [1,5,5,1] u 5 [1,5,1,4] u 6 [3,1,3,4] u 7 [3,1,3,5] u 8 [3,4,2,3] u 9 [5,3,2,1] u 10 [2,4,4,5] u 11 [5,5,4,5] u 12 [3,3,4,3] u 13 [5,1,1,1] u 1 4 [3,3,5,3] u 15 [1,4,11] u 16 [3,3,5,3] u 17 [3,1,2,5] u 18 [3,1,2,3] u 19 [5,3,1,1] U 20 [3,5,3,5] u 21 [1,5,3,1] u 22 [3,4,4,3] u 23 [5,4,3,2][4,2,5,4] [4,4,2,4] u 26 [4,3,2,4] u 27 [3,2,3,5] u 28 [4,3,3,2] u 29 [3,4,4,4] [2,5,4,5] u 31 [1,5,1,1] u 32 [3,2,2,4] u 24 u 25 [1,3,5,1] u 34 [4,2,2,2] u 35 [2,4,4,2] u 36 [4,2,4,3] u 37 u 30 [2,5,2,5] [5,5,1,5] u 39 u 33 [3,3,3,2]u 40 [1,5,4,1] u 41 [4,2,2,4] u 42 [1,5,5,2] [4,2,1,3] u 44 [3,4,3,4] u 38 [3,1,1,2] u 46 [3,1,3,5] u 47 [2,3,4,5] u 48 [4,2,2,5] u 49 [1,1,5,1] u 50 [2,4,5,1] u 51 u 43 [5,5,2,5] u 52 u 45 [1,2,2,5] u 53 [3,1,3,2] u 54 [3,1,1,2] u 55 [4,2,1,4] u 56 [2,2,2,4] u 57 [2,2,4,2] u 58 [2,2,4,1] u 59 [1,5,2,3] u 60 [1,5,1,3] u 61 ? Final map i.e. for each unit u i in the intermediate map, we associate a global evaluation g(u i ) = Φ [g j (u i )] j  F Intermediate map Multi-criteria sorting model: Φ : E m  E [g 1 (u),g 2 (u),…,g m (u)]  g(u)

19. CUPUM'05 · 30-06-200519 Example g1g1 g2g2 g3g3 g4g4 g(b 4 )4.511 q 4 0.2 p 4 0.3 g(b 3 )3.522 q 3 0.2 p 3 0.3 g(b 2 )2.53.5 2.5 q 2 0.2 p 2 0.3 g(b 1 )0.2544 q 1 0.2 p 1 0.3 Φ = ELECTRE TRI Inference of the weight k j only. g 1 g 2 g 3 … g m-1 g m bpbp b p+1 C p-1 b p +p p b p +q p b p -p p b p -q p

20. CUPUM'05 · 30-06-200520 Example Result without additional information: (c 2 ) u 1 (c 2,c 3 ) u 2 (c 2,c 3 ) u 3 (c 2,c 5 ) u 4 (c 1,c 2 ) u 5 (c 1,c 2,,c 3 ) u 6 (c 3,c 4,c 5 ) u 7 (c 3,c 4 ) u 8 (c 2, c 3 ) u 9 (c 2,c 3, c 4 ) u 10 (c 2 ) u 11 (c 1,c 2,c 3 ) u 12 (c 2,c 3 ) u 13 (c 2,c 5 ) u 1 4 (c 1,c 3 ) u 15 (c 2 ) u 16 (c 1,c 3 ) u 17 (c 3,c 4,c 5 ) u 18 (c 3,c 4 ) u 19 (c 2,c 3,c 4 ) u 20 (c 1,c 3 ) u 21 (c 1,c 2 ) u 22 (c 2,c 3 ) u 23 (c 2,c 3 )(c 1,c 4 ) (c 2,c 4 ) u 26 (c 3,c 4 ) u 27 (c 3,c 4 ) u 28 (c 2,c 3 ) u 29 (c 2,c 3 ) (c 1,c 2 ) u 31 (c 1,c 2 ) u 32 (c 3,c 4 )u 24 u 25 (c 1,c 2 ) u 34 (c 2,c 4 ) u 35 (c 2 ) u 36 (c 2,c 3,c 4 ) u 37 u 30 (c 1,c 2,c 3 )(c 1,c 5 ) u 39 u 33 (c 2,c 3 )u 40 (c 1,c 2 ) u 41 (c 4 ) u 42 (c 1,c 2 ) (c 3,c 4 ) u 44 (c 2,c 3 ) u 38 (c 2,c 3,c 4 ) u 46 (c 3,c 4 ) u 47 (c 2,c 3 ) u 48 (c 4 ) u 49 (c 1,c 2 ) u 50 (c 1 ) u 51 u 43 (c 1,c 4,c 5 ) u 52 u 45 (c 2,c 4 ) u 53 (c 2,c 3 ) u 54 (c 2,c 3,c 4 ) u 55 (c 4 ) u 56 (c 2,c 4 ) u 57 (c 2 ) u 58 (c 2 )u 59 (c 1,c 2,,c 3 ) u 60 (c 1,c 2,,c 3 ) u 61 Very lowLowAverageHigh Very high Additional information: u 33  c 4 ; u 40  c 1 -c 3 ; u 61  c 1 -c 2

21. CUPUM'05 · 30-06-200521 Example Result with additional information: (c 2 ) u 1 (c 2 ) u 2 (c 2 ) u 3 (c 5 ) u 4 (c 1 ) u 5 (c 2 ) u 6 (c 4 ) u 7 (c 3 ) u 8 (c 3 ) u 9 (c 3 ) u 10 (c 2 ) u 11 (c 2 ) u 12 (c 3 ) u 13 (c 5 ) u 1 4 (c 3 ) u 15 (c 2 ) u 16 (c 3 ) u 17 (c 4 ) u 18 (c 3 ) u 19 (c 4 ) u 20 (c 3 ) u 21 (c 1 ) u 22 (c 2 ) u 23 (c 2 )(c 4 ) u 26 (c 4 ) u 27 (c 3 ) u 28 (c 4 ) u 29 (c 2 ) u 31 (c 2 ) u 32 (c 4 )u 24 u 25 (c 2 ) u 34 (c 4 ) u 35 (c 2 ) u 36 (c 3 ) u 37 u 30 (c 2 )(c 5 ) u 39 u 33 (c 3 )u 40 (c 1 ) u 41 (c 4 ) u 42 (c 1 ) (c 4 ) u 44 (c 4 ) u 38 (c 3 ) u 46 (c 3 ) u 47 (c 2 ) u 48 (c 4 ) u 49 (c 2 ) u 50 (c 2 ) u 51 u 43 (c 4 ) u 52 u 45 (c 4 ) u 53 (c 3 ) u 54 (c 3 ) u 55 (c 4 ) u 56 (c 4 ) u 57 (c 2 ) u 58 (c 2 )u 59 (c 1 ) u 60 (c 2 ) u 61 Very lowLowAverageHigh Very high

22. CUPUM'05 · 30-06-200522 Example Result after grouping: u4u4 u5u5 u 13 u 14 u 29 u 30 u6u6 u 12 u 28 u 31 u 35 u3u3 u7u7 u 11 u 15 u 27 u 32 u2u2 u 19 u 26 u 33 u 34 u8u8 u 10 u 16 u 20 u 25 u9u9 u 17 u 21 u 24 u1u1 u 18 u 22 u 23 Very lowLowAverageHigh Very high

23. CUPUM'05 · 30-06-200523 Applications: Generation of alternatives in Multicriteria Analysis → The problem: find a corridor for a tramway between an origin o and a destination d: ? od * 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}.

24. CUPUM'05 · 30-06-200524 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.

25. CUPUM'05 · 30-06-200525 Approach 1: Aggregation at the input level Multi-criteria classification for groups method Preference parameters inferences … Criterion Map for criterion g i and group 1 Composite Criterion Map for criterion g i Composite Intermediate Map Composite Decision Map Composite Intermediate Map Composite Criterion Map for criterion g 1 Composite Criterion Map for criterion g m Criterion Map for criterion g i and group K … Criterion Map for criterion g 1 and group 1 Criterion Map for criterion g 1 and group K … Criterion Map for criterion g m and group 1 Criterion Map for criterion g m and group K Additional information

26. CUPUM'05 · 30-06-200526 Approach 2: Aggregation at the output level Additional information … Criterion Map for criterion g 1 and group k Composite Decision Map Intermediate Map for group 1 Criterion Map for criterion g m and group k … Criterion Map for criterion g 1 and group 1 Criterion Map for criterion g m and group 1 … Criterion Map for criterion g 1 and group K Criterion Map for criterion g m and group K Intermediate Map for group 1 Decision Map for group 1 Additional information Intermediate Map for group k Decision Map for group k Additional information Intermediate Map for group K Decision Map for group K Multi-criteria classification Preference parameters inferences Multi-criteria classification Preference parameters inferences Multi-criteria classification Preference parameters inferences

27. CUPUM'05 · 30-06-200527 Conclusion → We have introduced the concept of decision map : Cartographic modellingDecision map an automatic processlargely 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 aggregationoutranking relations-based aggregation preference parameters are directly provided Preference parameters are indirectly provided → Perspective: Extend the inference model for groups