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A Bi-Objective Optimization Model to Eliciting Decision Maker’s Preferences for the PROMETHEE II Method Stefan Eppe, Yves De Smet, and Thomas St¨ utzle Computer & Decision Engineering (CoDE) Department Universit´e Libre de Bruxelles, Belgium (ULB) {stefan.eppe, yves.de.smet, stuetzle}@ulb.ac.be

Abstract. Eliciting the preferences of a decision maker is a crucial step when applying multi-criteria decision aid methods on real applications. Yet it remains an open research question, especially in the context of the Promethee methods. In this paper, we propose a bi-objective optimization model to tackle the preference elicitation problem. Its main advantage over the widely spread linear programming methods (traditionally proposed to address this question) is the simultaneous optimization of (1) the number of inconsistencies and (2) the robustness of the parameter values. We experimentally study our method for inferring the Promethee II preference parameters using the NSGA-II evolutionary multi-objective optimization algorithm. Results obtained on artificial datasets suggest that our method offers promising new perspectives in that field of research.

1

Introduction

To solve a multi-criteria decision aid problem, the preferences of a decision maker (DM) have to be formally represented by means of a model and its preference parameters (PP) [13]. Due to the often encountered difficulty for decision makers to provide values for these parameters, methods for inferring PP’s have been developed over the years [1, 3, 9, 10]. In this paper, we follow the aggregation/disaggregation approach [11] for preference elicitation: given a set A of actions, the DM is asked to provide holistic information about his preferences. She states her overall preference of one action over another rather than giving information at the preference parameter level, since the former seems to be a cognitively easier task. The inference of the decision maker’s (DM) preferences is a crucial step of multi-criteria decision aid, having great practical implications on the use of a particular MCDA method. In this paper, we work with the Promethee outranking method. To the best of our knowledge, only few works on preference elicitation exist for that method. Frikha et al. [8] propose a method for determining the criteria’s relative weights. They consider two sets of partial information provided by the DM: (i) ordinal preference between two actions, and (ii) a ranking of the relative weights. These

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Stefan Eppe, Yves De Smet, and Thomas St¨ utzle

are formalized as constraints of a first linear program (LP) that may admit multiple solutions. Then, for each criterion independently, an interval of weights that satisfies the first set of constraints is determined. Finally, a second LP is applied on the set of weight intervals to reduce the number of violations of the weights’ partial pre-order constraint. Sun and Han [14] propose a similar approach that also limits itself to determine the weights of the Promethee preference pa¨ rameters. These, too, are determined by resolving an LP. Finally, Ozerol and Karasakal [12] present three interactive ways of eliciting the parameters of the Promethee preference model for Promethee I and II. Although most methods for inferring a DM’s preferences found in the MCDA literature are based on the resolution of linear programs [1, 10], some recent works also explore the use of meta-heuristics to tackle that problem [4]. In particular, [6] uses the NSGA-II evolutionary multi-objective optimization (EMO) algorithm to elicit ELECTRE III preference parameters in the context of sorting problems. The goal of this work is to contribute to exploring the possible use of multiobjective optimization heuristics to elicit a decision maker’s preferences for the Promethee II outranking method. In addition to minimizing the constraint violations induced by a set of preference parameters (PP), we consider robustness of the elicited PP’s as a second objective. The experimental setup is described in detail in Sec. 2. Before going further in the description of our experimental setup, let us define the notation used in the following. We consider a set A = {a1 , . . . , an } of n = |A| potential actions to be evaluated over a set of m conflicting criteria. Each action is evaluated on a given criterion by means of an evaluation function fh : A → R : a → fh (a). Let F (a) = {f1 (a), . . . , fm (a)} be the evaluation vector associated to action a ∈ A. Let Ω be the set of all possible PP sets and let ω ∈ Ω be one particular PP set. Asking a DM to provide (partial) information about his preferences is equivalent to setting constraints on Ω 1 , each DM’s statement resulting in a constraint. We denote C = {c1 , . . . , ck } the set of k constraints. In this paper, we focus on the Promethee II outranking method [2], which provides the DM with a complete ranking over the set A of potential actions. The method defines the net flow Φ(a) associated to action a ∈ A as follows: Φ(a) =

m X X 1 wh (Ph (a, b) − Ph (b, a)) , n−1 b∈A\a h=1

where wh and Ph (a, b) are respectively the relative weight and the preference function (Fig. 1) for criteria h ∈ {1, . . . , m}. For any pair of actions (a, b) ∈ A×A, 1

The constraint can be direct or indirect, depending on the type of information provided. Direct constraints will have an explicit effect on the preference model’s possible parameter values (e.g., the relative weight of the first criterion is greater than 1 ), while indirect constraints will have an impact on the domain Ω (e.g., the first 2 action is better than the fifth one).

Bi-Objective Optimization Model to Eliciting PROMETHEE II Parameters

3

Ph (a, b) 1

0 qh

ph

d?h (a, b)

Fig. 1. Shape of a Promethee preference function type V, requiring the user to define, for each objective h, a indifference threshold qh , and a preference threshold ph . We have chosen to slightly modify the original definition of the preference function, replacing the difference dh (a, b) = fh (a) − fh (b), by a relative difference, defined as follows: h (b) d?h (a, b) = 1 f(fh (a)−f , i.e., we divide the difference by the mean value of both (a)+f (b)) 2

h

h

evaluations. For d?h (a, b) ∈ [0, qh ], both solutions a and b are considered indifferently; for a relative difference greater than ph , a strict preference (with value 1) of a over b is stated. Between the two thresholds, the preference evolves linearly with increasing evaluation difference.

we have one of the following relations: (i) the rank of action a is better than the rank of action b, iff Φ(a) > Φ(b); (ii) the rank of action b is better than the rank of action a, iff Φ(a) < Φ(b); (iii) action a has the same rank as action b, iff Φ(a) = Φ(b). Although six different types of preference functions are proposed [2], we will limit ourselves to the use of a relative version of the “V-shape” preference function P : A × A → R[0,1] (Fig. 1). For the sake of ease, we will sometimes write the Promethee II specific parameters explicitly: ω = {w1 , q1 , p1 , . . . , wm , qm , pm }, where wh , qh , and ph are respectively the relative weight, the indifference threshold, and the preference threshold associated to criterion h ∈ {1, . . . , m}. The preference parameters have to satPm isfy the following constraints: wh ≥ 0, ∀h ∈ {1, . . . , m}, w h=1 h = 1, and 0 ≤ qh ≤ ph , ∀ h ∈ {1, . . . , m}.

2

Experimental Setup

The work flow of our experimental study is schematically represented in Fig. 2: (1) For a given set of actions A, a reference preference parameter set ωref is chosen. (2) Based on A and ωref , in turn a set C = {c1 , . . . , ck } of constraints is generated. A fraction pcv of the constraints will be incompatible with ωref , in order to simulate inconsistencies in the information provided by the DM. (3) By means of an evolutionary multi-objective algorithm, the constraints are then used to optimize a population of parameter sets on two objectives: constraint violation and robustness. (4) The obtained population of parameter sets is clustered. (5) The clusters are analysed and compared with ωref . In the following paragraphs, we explain in more detail the different components of the proposed approach.

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Stefan Eppe, Yves De Smet, and Thomas St¨ utzle

Choose set of actions A Set reference preference params. ωref

Randomly generate constraints C

Optimize with NSGA-II

Compare with ref. parameters

Cluster set of parameters

Fig. 2. The work flow of our experimental study.

We consider non-dominated action sets of constant size (100 actions), ranging from 2 to 5 objectives. The use of non-dominated actions seems intuitively meaningful, but the impact of that choice on the elicitation process should be further investigated, since it does not necessarily correspond to real-life conditions. For our convenience, we have used approximations of the Pareto optimal frontier of multi-objective TSP instances that we already had.2 Nevertheless, the results presented in the following are in no way related to the TSP. The reference preference parameters ωref are chosen manually for this approach in order to be representative and allow us to draw some conclusions. We perform the optimization process exclusively on the weight parameters {w1 , . . . , wm }. Unless otherwise specified, we use the following values for the relative thresholds: qh = 0.02 and ph = 0.10, ∀h ∈ {1, . . . , m}. This means that the indifference threshold for all criteria is considered as 2% of the relative difference of two action’s evaluations (Fig. 1). The preference threshold is similarly set to 10% of the relative distance. We will consider constraints of the following form: Φ(a) − Φ(b) > ∆, where (a, b) ∈ A × A and ∆ ≥ 0. Constraints on the threshold parameters qh and ph , ∀h ∈ {1 . . . m} have not been considered in this work. We could address this issue in a future paper (e.g. stating that the indifference threshold of the third criterion has to be higher than a given value: q3 > 0.2). We have chosen to randomly generate a given number of constraints that will be consistent (i.e., compatible) with the reference preference parameters ωref . More specifically, given ωref and the action set A, the net flow Φωref (a) of each action a ∈ A is computed. Two distinct actions a and b are randomly chosen and a constraint is generated on their basis, that is compatible with their respective net flow values Φωref (a) and Φωref (b). For instance, if Φωref (a) > Φωref (b), the corresponding compatible constraint will be given by Φ(a) > Φ(b). A fraction of incompatible constraints will also be generated, with a probability that is defined 2

We have taken solution sets of multi-objective TSP instances from [5], available on-line at http://iridia.ulb.ac.be/supp/IridiaSupp2011-006.

Bi-Objective Optimization Model to Eliciting PROMETHEE II Parameters

5

csvri (ω) 1

0

1 ∆ 2 i

∆i

Φω (ai ) − Φω (bi )

Fig. 3. Shape of the constraint violation rate function csvri (ω) associated with a given constraint ci ∈ C and a set of preference parameters ω. The constraint ci expresses the inequality Φω (ai ) − Φω (bi ) ≥ ∆i , linking together actions ai and bi ∈ A.

by the parameter pcv . For these, the previous inequality becomes Φ(a) < Φ(b) (for Φωref (a) > Φωref (b)). As already mentioned, we take a bi-objective optimization point of view on the preference elicitation problem. We hereafter define the objectives that we will consider for the optimization process. Global constraint violation rate (csvr) Each constraint ci ∈ C, with i ∈ {1 . . . k}, expresses an inequality relation between a pair of actions (ai , bi ) ∈ A×A by means of a minimal difference parameter ∆i . We define the violation rate of the i-th constraint as follows (Fig. 3):   ∆i − ( Φω (ai ) − Φω (bi ) ) csvri (ω) = ζ , 1 2 ∆i where ζ(x) = min (1, max (0, x)) is a help function that restrains the values of its argument x to the interval [0, 1]. Finally, the set of measures is aggregated on all constraints to compute a global constraint representing the average violation rate: csvr(ω) =

k 1X csvri (ω) k i=1

Example. Let us consider the first constraint given by Φ(a1 ) − Φ(a4 ) ≥ 0.3. We thus have ∆1 = 0.3. Let the pair of actual net flows (that directly depend on the associated preference parameter set ω1 ) be as follows: Φω1 (a1 ) = 0.2 and Φω1 (a4 ) = −0.1. Considering only one constraint for the sake of simplicity, the global constraint violation rate becomes   = 0. csvr(ω) = csvr1 (ω) = ζ 0.3−(0.2−(−0.1)) 1 0.3 2

For the given parameter set ω, the constraint is thus fully satisfied.

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Stefan Eppe, Yves De Smet, and Thomas St¨ utzle Parameter Population size Termination condition Probability of cross-over Probability of mutation

Value(s) npop 50 tmax 120 sec pxover 0.8 pmut 0.2

Table 1. Parameter values used for the NSGA-II algorithm.

Promethee II sampled sensitivity (p2ss) Given a preference parameter set ω, we compute its p2ss value by sampling a given number Np2ss of parameter ?s }, ∀s ∈ {1, . . . , Np2ss } “around” ω. Practically, we sets ω ?s = {ω1?s , . . . , ωm take Np2ss = 10 and we generate each parameter ωj?s , with j ∈ {1, . . . , m}, of the sample by randomly evaluating a normally distributed stochastic variable that is centred on the value ωj and has a relative standard deviation of ω 10%: ωj?s ∼ N (ωj , ( 10j )2 ). We define the sensitivity as the square root of the average square distance to the reference constraint violation csvr(ω): v u N u p2ss X u 2 1 t (csvr(ω ?s ) − csvr(ω)) p2ss (ω) = N p2ss s=1 As some first results have shown that the resulting set of preference parameters presented a clustered structure of sub-sets, we have decided to apply a clustering procedure (with regard to the weight parameters) on the set of results. Practically, we use the pamk function of R’s ’fpc’ package, performing a partitioning around medoids clustering with the number of clusters estimated by optimum average silhouette width. Finally, we compare the obtained results, i.e., a set of preference parameter sets, with the reference parameter set ωref . The quality of each solution is quantified by means of the following fitness measure: Correlation with the reference ranking τK We use Kendall’s τ to measure the distance between a ranking induced by a parameter set ωi and the reference parameter set ωref .

3

Results

The aim of the tests that are described below is to provide some global insight into the behaviour of the proposed approach. Further investigations should be carried out in order to gain better knowledge, both on a larger set of randomly generated instances and on real case studies. In the following, main parameters of the experimental setup are systematically tested. We assume that the parameters of the tests, i.e., instance size, number of objectives, number of constraints,

Bi-Objective Optimization Model to Eliciting PROMETHEE II Parameters Parameter Size of the action set Number of criteria of the action set

n m

Number of constraints Constraint violation rate Scalar weight parameter

k pcv w

7

Value(s) 100 2, 3, 4, 5 2, 10, 20, 30, 40, 50 0, 0.05, 0.10, 0.20, 0.30 0.10, 0.20, 0.30, 0.40, 0.50

PROMETHEE II Sampled Sensitivity (p2ss)

Table 2. This table provides the parameter values used for the experiments. For each parameter, the value in bold represents its default value, i.e., the value that is taken in the experiments, if no other is explicitly mentioned.

0.06 pcv = 0.00 0.05 0.10 0.20 0.30

a

0.04

0.02

b 0 0

0.1

0.2

0.3

Constraint Set Violation Rate (csvr)

Fig. 4. This plot represents the approximated Pareto frontiers in the objective space, for 20 constraints and several values of the constraint violation rate pcv , i.e., the proportion of inconsistent constraints with respect to the total number of constraints. As expected, increasing the value of pcv has the effect of deteriorating the quality of the solution set both in terms of constraint violation rate and Promethee II sampled sensitivity.

etc., are independent from each other, so that we can study the impact each of them has on the results of the proposed model. The values used for the parameters of the experiments are given in Table 2. In the following, we only present

Stefan Eppe, Yves De Smet, and Thomas St¨ utzle

PROMETHEE II Sampled Sensitivity (p2ss)

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0.16 w = 0.10 0.20 0.30 0.40 0.50

0.12

0.08

w = 0.50 w = 0.30 w = 0.20

0.04

w = 0.40 w = 0.10

0 0

0.1

0.2

0.3

0.4

Constraint Set Violation Rate (csvr)

Fig. 5. Approximations of the Pareto optimal frontier are shown for different values of the reference weight parameters w = {0.1, . . . , 0.5} for an action set with two criteria. The weights of the reference preference model ωref are given by w1 = w and w2 = 1−w.

the most noticeable results. Figure 4 shows the effect of changing the proportion of incompatible constraints with respect to the total number of constraints. As expected, higher values of the constraint incompatibility ratios induce worse results on both objectives (csvr and p2ss). Thus, the more consistent the information provided by the decision maker, the higher the possibility for the algorithm to reach stable sets of parameters that do respect the constraints.3 The second and more noteworthy observation that can be made on that plot is related to the advantage of using a multi-objective optimization approach for the elicitation problem. Indeed, as can be seen, optimizing only the constraint violation rate (csvr) would have led to solutions with comparatively poor performances with regard to sensitivity (area marked with an a on the plot). This would imply that small 3

We investigate the impact of inconsistencies in partial preferential information provided by the DM. We would like to stress that the way we randomly generate inconsistent constraints (with respect to the reference preference parameters ωref ) induces a specific type of inconsistencies. Other types should be studied in more depth in a future work.

PROMETHEE II Sampled Sensitivity (p2ss)

Bi-Objective Optimization Model to Eliciting PROMETHEE II Parameters

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0.02 w = 0.30 : Cluster Cluster w = 0.40 : Cluster Cluster

0.015

1 2 1 2

0.01

0.005

0 0

0.1

0.2

0.3

Constraint Set Violation Rate (csvr)

Fig. 6. Results of the clustering applied on two different reference parameter sets (for a action set with two criteria), that are characterized by respective weight parameters w = 0.30 and 0.40. For each set, 2 clusters have been automatically identified. The proximity of the centroid parameter set of each cluster to the reference parameter set is measured by means of Kendall’s τ (Fig. 7) to compare the clusters for each weight parameter. The filled symbol (cluster 1) corresponds to the better cluster, i.e., the one that best fits the reference weights.

changes to csvr-well-performing preference parameters might induce important alteration of the constraint violation rate. However, due to the steepness of the approximated Pareto frontier for low values of csvr the DM is able to select much more robust solutions at a relatively small cost on the csvr objective (area b). For action sets that are evaluated on two criteria4 , we also observe the effects of varying the value of the weight preference parameter w, where w1 = w and w2 = 1 − w. As shown in Fig. 5, the underlying weight parameter w has an impact on the quality of the resulting Pareto set of approximations. It suggests that the achievable quality for each objective (i.e., csvr and p2ss) is related to the “distance” from an equally weighted set of criteria (w = 0.5): lowering the values of w makes it harder for the algorithm to optimize on the constraint vio4

Similar results have been observed for higher number of criteria.

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Stefan Eppe, Yves De Smet, and Thomas St¨ utzle

1

Kendall’s τ

0.95 0.9 0.85 0.8 0.75 0.7 0.1

0.2

0.3

0.4

0.5

Weight parameter (w)

Fig. 7. Kendall’s τ represented for different reference parameter weights w ∈ {0.1, . . . , 0.5}. For each weight, the mean values of all clusters are shown. For w = 0.30, for instance, the upper circle represents the first (best) cluster, and the lower one represents the other cluster of one same solution set.

lation objective csvr. On the other hand, having a underlying preference model with a low value of w seems to decrease the sampled sensitivity p2ss, making the model more robust to changes on parameter values. It should be noted that for w = 0.5 there appears to be an exception in the central area of the Pareto frontier. This effect has not been studied yet. In this experimental study, we will compare the obtained results with the reference parameters ωref . To that purpose, we partition the set of obtained preference parameters based on their weights into a reduced number of clusters. The clustering is thus performed in the solution space (on the weights) and represented in the objective space (csvr - p2ss). Figure 6 shows the partition of the resulting set for a specific instance, for two different weights of the reference preference parameters: (1) ωref = 0.30 and (2) ωref = 0.40. Both cases suggest that there is a strong relationship between ωref and the objective values (csvr and p2ss). Indeed, in each case, two separated clusters are detected: cluster 1, with elements that are characterized by relatively small csvr values and a relatively large dispersion of p2ss values; cluster 2, with elements that have relatively small p2ss values and a relatively higher dispersion of csvr values. In both cases, too, the centroid associated to cluster 1 has a weight vector that is closer, based on an Euclidean distance, the the weight vector of ωref than the centroid of cluster 2.

Bi-Objective Optimization Model to Eliciting PROMETHEE II Parameters

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Although this has to be verified through more extensive tests, this result could suggest a reasonable criterion for deciding which cluster to choose from the set of clusters, and therefore provide the DM with a sensible set of parameters that is associated to that cluster. Finally, in order to assess the quality of the result with respect to the reference parameter set, we plot (Fig. 7) the values of Kendall’s τ for each cluster that has been determined for a range of reference weight parameters w ∈ {0.1, 0.2, 0.3, 0.4, 0.5}. For each weight w, we plot Kendall’s τ for each cluster’s medoid (compared to the reference parameter set ωref ). We first observe that we have between 2 and 6 clusters depending on the considered weight. Although the results worsen (slightly, except for w = 0.5), the best values that correspond to the previously identified “best” clusters remain very high: The ranking induced by the reference parameter set are reproduced to a large extent. These results are encouraging further investigations, because they tend to show that our approach converges to good results (which should still be quantitatively measured by comparing with other existing methods).

4

Conclusion

Eliciting DM’s preferences is a crucial step of multi-criteria decision aid that is commonly tackled in the MCDA community by solving linear problems. As some other recent papers, we explore an alternative bi-objective optimization based approach to solve it. Its main distinctive feature is to explicitly integrate the sensitivity of the solution as an objective to be optimized. Although this aspect has not been explored yet, our approach should also be able, without any change, to integrate constraints that are more complicated than linear ones. Finally, and although we have focused on the Promethee II outranking method in this paper, we believe that the approach could potentially be extended to a wider range of MCDA methodologies. Future directions for this work should include a more in-depth analysis of our approach, as well as an extension to real, interactive elicitation procedures. A further goal could also be to determine additional objectives that would allow eliciting the threshold values of the Promethee preference model. Finally, investigating other ways of expressing robustness would probably yield interesting new paths for the future. Acknowledgments. Stefan Eppe acknowledges support from the META-X Arc project, funded by the Scientific Research Directorate of the French Community of Belgium.

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