MULTICRITERIA DECISION AIDING

MULTICRITERIA DECISION AIDING ANTOINE ROLLAND, Université LYON II CERRAL, 24 feb. 2014

PLAN 1 Introduction 2 MCDA Framework 3 utility functions 4 Outranking approach 5 Other methods A. Rolland MCDA 2 / 68

Introduction A. Rolland MCDA 3 / 68

DECISION MAKING Decision Making : the art of helping a decision maker to take a good decision A. Rolland MCDA 4 / 68

DECISION MAKING Decision Making : the art of helping a decision maker to take a good decision Is deciding difficult? A. Rolland MCDA 4 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide A. Rolland MCDA 5 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide Examples which master should I choose? classical problems : Knapsack Problem (KP), Minimum Spanning Tree Problem, Traveller Salesman Problem (TSP)... A. Rolland MCDA 5 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? A. Rolland MCDA 6 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? A. Rolland MCDA 7 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? A. Rolland MCDA 8 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? A. Rolland MCDA 9 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? (n 1)! possibilities! A. Rolland MCDA 10 / 68

EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? (n 1)! possibilities! 355687428096000 possibilities in Rhône-Alpes A. Rolland MCDA 11 / 68

COMBINATORIAL OPTIMIZATION finding the best solution into a finite set of objects without any possibility to look at all of them! A. Rolland MCDA 12 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide A. Rolland MCDA 13 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide Examples where are we going to drink beer this evening? classical problems : voting theory A. Rolland MCDA 13 / 68

EXAMPLE : VOTING FOR SWEETS Three friends want to choose sweets together. A. Rolland MCDA 14 / 68

EXAMPLE : VOTING FOR SWEETS 1 2 3 A. Rolland MCDA 15 / 68

EXAMPLE : VOTING FOR SWEETS 1 2 3 A. Rolland MCDA 16 / 68

EXAMPLE : VOTING FOR SWEETS 1 2 3 A. Rolland MCDA 17 / 68

EXAMPLE : VOTING FOR SWEETS 1 2 3 A. Rolland MCDA 18 / 68

EXAMPLE : VOTING FOR SWEETS 1 2? 3 A. Rolland MCDA 19 / 68

SOCIAL CHOICE finding the collective preferred solution knowing the preferences of every voter this solution sometimes should not exist! A. Rolland MCDA 20 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration A. Rolland MCDA 21 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration Examples Should I choose a bad movie with my favourite actor or a good movie without him? classical problems : multicriteria decision aiding A. Rolland MCDA 21 / 68

EXAMPLE : CHOOSING A CAMERA A. Rolland MCDA 22 / 68

EXAMPLE : CHOOSING A CAMERA A. Rolland MCDA 23 / 68

EXAMPLE : CHOOSING A CAMERA Mean Min Max σ Camera 1 14.66 8 17.2 2.6 Camera 2 14.26 8 18 3.2 A. Rolland MCDA 24 / 68

EXAMPLE : CHOOSING A CAMERA Mean Min Max σ Price Camera 1 14.66 8 17.2 2.6 600 Camera 2 14.26 8 18 3.2 800 A. Rolland MCDA 24 / 68

MULTICRITERIA finding the global preferred solution with possibly conflicting criteria A. Rolland MCDA 25 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration consequences are uncertain A. Rolland MCDA 26 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration consequences are uncertain Examples Should I take my umbrella? Expected Utility theory : basis of classical economic behaviour A. Rolland MCDA 26 / 68

EXAMPLE : UMBRELLA A. Rolland MCDA 27 / 68

EXAMPLE : UMBRELLA -2 5 4-5 Probability 0.3 0.7 A. Rolland MCDA 28 / 68

EXAMPLE : UMBRELLA Score -2 5 2.4 4-5 -2.3 Probability 0.3 0.7 A. Rolland MCDA 28 / 68

DECISION UNDER UNCERTAINTY finding the global preferred solution without knowing the exact consequences A. Rolland MCDA 29 / 68

DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide Combinatorial optimization there are several decision makers to decide Social Choice Theory there are several criteria to be taken into consideration Multicriteria decision Making consequences are uncertain Decision under uncertainty A. Rolland MCDA 30 / 68

FORMAL FRAMEWORK Social Choice Multicriteria Uncertainty Candidates Alternatives Actions Voters Criteria States of the nature Ranks Values Consequences (Number) (Weight) (Probability) Social Choice : individual preferences global preferences Multicriteria : preferences on criteria preferences on the alternatives Uncertainty : preferences on the consequences preferences on the actions A. Rolland MCDA 31 / 68

MCDA Framework A. Rolland MCDA 32 / 68

CRITERIA DEFINITION [ROYBOUYSSOU96] criterion= attribute with a complete binary preference relation (order, pre-order, interval order...) A criteria family should be : complete to describe the problem (exhaustivity) coherent with the global preferences as independant as possible (avoid redundancy) A. Rolland MCDA 33 / 68

PROBLEMS IN MULTICRITERIA DECISION THEORY [ROYBOUYSSOU96] Modelling decision problem [Tsoukias07] Choice Problem : one has to choose the best alternative(s). Ranking Problem : one has to rank the alternatives from the best to the worst. Sorting Problem : one has to sort the alternatives into pre-defined categories (ordered or not) A. Rolland MCDA 34 / 68

NOTATIONS Formal model : inputs a set of alternatives X = X 1... X n a representation of the preferences on the values of each criterion i N (utility function, qualitative preference relations i...) a representation of the importance of each criterion or set of criteria (weights, importance relation...) A. Rolland MCDA 35 / 68

TWO MAIN APPROACHES [GRABISCHPERNY03] x = (x 1,..., x n ) y = (y 1,..., y n ) a a(x), a(y) c c c(x 1, y 1 ),..., c(x n, y n ) c(y 1, x 1 ),..., c(y n, x n ) a P(x, y) quantitative approach aggregate then compare (scoring) qualitative approach compare then aggregate (outranking) A. Rolland MCDA 36 / 68

CHOOSING CAMERA A. Rolland MCDA 37 / 68

CHOOSING CAMERA crit. Camera 1 Camera 2 Camera 3 Nb Pixel 20m 12m 16m Sensibility 125-6400 80-12800 100-12800 Speed 30s-1/2000 15s-1/4000 30s-1/4000 Macro 10cm 15cm X Price 490 C 450 C 1200 C A. Rolland MCDA 38 / 68

Utility-based methods A. Rolland MCDA 39 / 68

additive aggregation function weighted mean non additive aggregation function maximin, minimax, minimin maximax OWA Choquet integral distances multi-objective optimization A. Rolland MCDA 40 / 68

HYPOTHESES values on different criteria are commensurable values on different criteria can compensate values of each alternative on the different criteria are well known a complete and transitive relation is expected as an output A. Rolland MCDA 41 / 68

UTILITY FUNCTIONS 1.2 1 0.8 0.6 0.4 0.2 G(X) 1.2 1 0.8 0.6 0.4 0.2 G(X) 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.2 1 0.8 0.6 0.4 0.2 G(X) 1.2 1 0.8 0.6 0.4 0.2 G(X) 0 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0.2 0.4 0.6 0.8 1 1.2 A. Rolland MCDA 42 / 68

CHOOSING CAMERA crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 A. Rolland MCDA 43 / 68

ADDITIVE AGGREGATION FUNCTION WEIGHTED SUM x y WS(x) WS(y) WS(x) = i w i f i (x) easy to understand and use do not favour compromise solutions (ex : A(18,3) ; B(3,18), C(10,10)) A. Rolland MCDA 44 / 68

ADDITIVE AGGREGATION FUNCTION crit. weight Camera 1 Camera 2 Camera 3 Nb Pixel 0.2 10 6 8 sensibility 0.3 3 10 9 Speed 0.1 5 5 10 Price 0.4 9 10 4 A. Rolland MCDA 45 / 68

ADDITIVE AGGREGATION FUNCTION crit. weight Camera 1 Camera 2 Camera 3 Nb Pixel 0.2 10 6 8 sensibility 0.3 3 10 9 Speed 0.1 5 5 10 Price 0.4 9 10 4 Score 7 8.7 6.9 A. Rolland MCDA 45 / 68

NON ADDITIVE AGGREGATION FUNCTION (1) MAX AND MIN maximin (pessimistic) x y min i maximax (optimistic) etc... (f i (x)) min(f i (y)) x y max(f i (x)) max(f i (y)) i i i crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 A. Rolland MCDA 46 / 68

NON ADDITIVE AGGREGATION FUNCTION (1) MAX AND MIN maximin (pessimistic) x y min i maximax (optimistic) etc... (f i (x)) min(f i (y)) x y max(f i (x)) max(f i (y)) i i i crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 Min 3 6 4 Max 9 10 10 A. Rolland MCDA 46 / 68

NON ADDITIVE AGGREGATION FUNCTION (2) OWA [YAGER98] x y OWA(x) OWA(y) OWA(x) = i w i f σ(i) (x) with f σ(1) (x) f σ(2) (x)... f σ(3) (x) weights are dedicated to the rank of the values and not to the criteria generalize all the position statistics (quartile, median...) A. Rolland MCDA 47 / 68

NON ADDITIVE AGGREGATION FUNCTION (2) OWA [YAGER98] x y OWA(x) OWA(y) OWA(x) = i w i f σ(i) (x) avec f σ(1) (x) f σ(2) (x)... f σ(3) (x) Weights : (0.3 ;0.3 ;0.2 ;0.2) crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 A. Rolland MCDA 48 / 68

NON ADDITIVE AGGREGATION FUNCTION (2) OWA [YAGER98] x y OWA(x) OWA(y) OWA(x) = i w i f σ(i) (x) avec f σ(1) (x) f σ(2) (x)... f σ(3) (x) Weights : (0.3 ;0.3 ;0.2 ;0.2) crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 OWA 6.2 7.3 7.4 A. Rolland MCDA 48 / 68

NON ADDITIVE AGGREGATION FUNCTION (3) CHOQUET INTEGRAL[CHOQUET53] x y C(x) C(y) C(x) = µ(a i ) ( f σ(i) (x) f σ(i+1) (x) ) i with f σ(1) (x) f σ(2) (x)... f σ(3) (x) µ a measure on 2 N and A i = {1,..., i}. integral w.r.t. a non additive measure (capacity or fuzzy measure) able to model interactions between criteria include WS, OWA, etc... A. Rolland MCDA 49 / 68

NON ADDITIVE AGGREGATION FUNCTION (3) Example (4 criteria = 16 parameters) : µ({c 1 }) = 0.2 µ({c 2 }) = 0.1 µ({c 3 }) = 0.2 µ({c 4 }) = 0.1 µ({c 1, c 2 }) = 0.3 µ({c 1, c 3 }) = 0.6 µ({c 1, c 4 }) = 0.2 µ({c 2, c 3 }) = 0.6 µ({c 2, c 4 }) = 0.2 µ({c 3, c 4 }) = 0.3 µ({c 1, c 2, c 3 }) = 0.7 µ({c 1, c 2, c 4 }) = 0.4 µ({c 1, c 3, c 4 }) = 0.5 µ({c 2, c 3, c 4 }) = 0.4 µ({c 1, c 2, c 3, c 4 }) = 1 µ( ) = 0 crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 A. Rolland MCDA 50 / 68

NON ADDITIVE AGGREGATION FUNCTION (3) Example (4 criteria = 16 parameters) : µ({c 1 }) = 0.2 µ({c 2 }) = 0.1 µ({c 3 }) = 0.2 µ({c 4 }) = 0.1 µ({c 1, c 2 }) = 0.3 µ({c 1, c 3 }) = 0.6 µ({c 1, c 4 }) = 0.2 µ({c 2, c 3 }) = 0.6 µ({c 2, c 4 }) = 0.2 µ({c 3, c 4 }) = 0.3 µ({c 1, c 2, c 3 }) = 0.7 µ({c 1, c 2, c 4 }) = 0.4 µ({c 1, c 3, c 4 }) = 0.5 µ({c 2, c 3, c 4 }) = 0.4 µ({c 1, c 2, c 3, c 4 }) = 1 µ( ) = 0 crit. Camera 1 Camera 2 Camera 3 Nb Pixel 10 6 8 sensibility 3 10 9 speed 5 5 10 Price 9 10 4 Choquet Int. 5 6.2 7.6 A. Rolland MCDA 50 / 68

MULTICRITERIA OPTIMIZATION PRINCIPE x y d(x, z) d(y, z) with d(, ) a distance and z an ideal point Example : TOPSIS method [Hwang& Yoon81] computation of the ideal point and the anti-ideal point computation of d distance to the ideal point computation of d distance to the anti-ideal point computation of the global score : s = d d+d A. Rolland MCDA 51 / 68

MULTICRITERIA OPTIMIZATION PRINCIPE x y d(x, z) d(y, z) with d(, ) a distance and z an ideal point crit. Camera 1 Camera 2 Camera 3 Ideal Anti-Ideal Nb Pixel 10 6 8 10 6 sensibility 3 10 9 10 3 speed 5 5 10 10 5 Price 9 10 4 10 4 A. Rolland MCDA 52 / 68

Outranking approach A. Rolland MCDA 53 / 68

HYPOTHESIS a decision = a process of progressive construction of a preference relation incomparability between actions is enable propose a preference relation which is not a pre-order to enlighten the decision maker. A. Rolland MCDA 54 / 68

OUTRANKING RELATION PRINCIPE with C(x, y) = {i N x i i y i } x y C(x, y) N C(y, x) A. Rolland MCDA 55 / 68

ELECTRE METHOD[ROY68] OUTRANKING RELATION x outranks y (xsy) if C(x, y) > SC and j N, non y j V j x j xsy and non ysx : x is preferred to y (xpy or x y) xsy and ysx : x and y are indifferent (xiy or x y) non xsy and non ysx : x and y are incomparable (xry) Relation S gives a graph of preferences on X What do we do? Electre analyse a situation but not solve the problems! One can reduce the graph by merging cycles into one new alternative On can move the thresholds for a sensitivity analysis A. Rolland MCDA 56 / 68

ELECTRE METHOD [ROY68] crit. Camera 1 Camera 2 Camera 3 Nb Pixel 20m 12m 16m Sensibility 125-6400 80-12800 100-12800 Speed 30s-1/2000 15s-1/4000 30s-1/4000 Macro 10cm 15cm X Price 490 C 450 C 1200 C A. Rolland MCDA 57 / 68

ELECTRE METHOD [ROY68] crit. Camera 1 Camera 2 Camera 3 Nb Pixel 20m 12m 16m Sensibility 125-6400 80-12800 100-12800 Speed 30s-1/2000 15s-1/4000 30s-1/4000 Macro 10cm 15cm X Price 490 C 450 C 1200 C Camera 1 Camera 2 Camera 3 Camera 1 0.4 0.4 Camera 2 0.6 0.4 Camera 3 0.4 0.4 avec w 1 = w 2 = w 3 = w 4 = w 5 = 0.2 A. Rolland MCDA 57 / 68

EXAMPLE : PROMETHEE [BRANSETAL84] compare alternatives with preference intensity P i (x, y) = p(g i (x) g i (y)) P(x, y) = 0 no preference of x on y P(x, y) = 1 strong preference of x on y preference indicator π(a, b) = i ω ip i (a, b) and π(b, a) flux computation Φ + (x) = y π(x, y) and Φ (x) = y π(y, x) flux aggregation A. Rolland MCDA 58 / 68

DECISION RULES [SLOWINSKYETAL01] Sorting problem use dominance and Rough sets is well adapted to imprecise or incomplete data easy to understand by DM PRINCIPE If x dominates y then x should be classified in a category as least as good as y one. A. Rolland MCDA 59 / 68

Elicitation A. Rolland MCDA 60 / 68

MODELS : WHAT FOR? PRESCRIPTIVE APPROACH To help a decision maker by the proposal of a solution obtained by a model DESCRIPTIVE APPROACH To describe a decision maker s preferences by the chosen model. ELICITATION The elicitation of the decision maker s preferences consists in obtaining parameters of a decisional model which explain the past decisions in order to help in the future decisions. A. Rolland MCDA 61 / 68

ELICITATION OF THE PARAMETERS OPTION 1 : EXPLICIT ELICITATION explain the model to the decision maker let him choose the parameters OPTION 2 : IMPLICIT ELICITATION present some (fictitious) alternatives and ask the decision maker to compare them deduct the parameters with optimization program linked to machine learning A. Rolland MCDA 62 / 68

ELICITATION OF THE PARAMETERS OPTION 2 : IMPLICIT ELICITATION present some (fictitious) alternatives and ask the decision maker to compare them deduct the parameters with optimization program For a score approach, need to find both : values of the utility functions values of the trade-off between criteria. A. Rolland MCDA 63 / 68

AHP [SAATY71, SAATY80] A method to determine the criteria weights (for a weighted sum) use of comparison of alternatives and criteria should include group decision PRINCIPE Divide the (complex) problem into a hierarchical structure Compare the criteria importance : from 1 (indifference) to 9 (extreme preference) Compare the alternatives Synthesise the comparisons (mean) to obtain a ranking Coherence of judgements A. Rolland MCDA 64 / 68

Conclusion A. Rolland MCDA 65 / 68

THINGS TO REMEMBER no miracle! by some each method has its own properties, desirable... or not! A. Rolland MCDA 66 / 68

THINGS TO REMEMBER no miracle! by some each method has its own properties, desirable... or not! CHALLENGES axiomatic approach big data Elicitation of parameters (preference learning) A. Rolland MCDA 66 / 68

BIBLIOGRAPHY Ph. Vincke. Multicriteria Decision-Aid. J. Wiley, New York, 1992 B. Roy, Multicriteria Methodology for Decision Aiding, Kluwer Academic Publisher, 1996 D. Bouyssou, D. Dubois, M. Pirlot and H. Prade (Edts), Decision-making Process Concepts and Methods, ISTE & Wiley, 2009 (3 volumes) M. Ehrgott. Multicriteria Optimization. Second edition. Springer, Berlin, 2005. A. Rolland MCDA 67 / 68

BIBLIOGRAPHY Fishburn Utility theory for Decision Making, 1970, Wiley Keeney-Raiffa Decisions with multiple objectives ; preferences and trade-off, 1976, Wiley Marichal, Aggregation Operators for Multicriteria Decision Aid, Institute of Mathematics, University of Liège, 1998 M. Ehrgott. Multicriteria Optimization. Second edition. Springer, Berlin, 2005. A. Rolland MCDA 68 / 68