The management of missing values in PROMETHEE methods

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1 The management of missing values in PROMETHEE methods Mémoire présenté en vue de l obtention du diplôme de Master en ingénieur civil électromécanicien, à finalité Gestion et technologies Núria Gens Fernández Directeur Professeur Yves De Smet Service CoDE SMG unit Année académique

2 Acknowledgements This thesis would not have been possible without the involvement of many people which have helped me either directly or indirectly. First of all, I would like to thank Professor Yves De Smet, my supervisor, for his valuable suggestions during the planning and development of this research work as well as for providing necessary information regarding the project. I would also like to extend my thanks to all the members of the SMG unit of the Computer and Decision Engineering (CoDE) department of the Université libre de Bruxelles for their constructive recommendations on this thesis. Thanks to my family, especially to my parents and my sisters, for accompanying me all the time and helping me so much throughout my study. And to Delfí for his support and his constant encouragement. I would like to mention as well all the people that I have meet here in Brussels for sharing their experience with me and all my friends in Barcelona for every friendly word they have sent me. The management of missing values in PROMETHEE methods Núria Gens 1

3 La gestion des valeurs manquantes dans les me thodes PROMETHEE Diplôme : Master en ingénieur civil électromécanicien, à finalité Gestion et technologies Mots clés : Aide multicritère à la décision, méthodes PROMETHEE, valeurs manquantes, méthodes statistiques, méthodes multicritère Résumé : La prise de décision est un processus qui implique l analyse de plusieurs alternatives. Quand la complexité et l importance des décisions augmentent, le besoin de formaliser le processus s avère nécessaire. L objectif de l'aide multicritère à la décision est d aider les décideurs placés devant de problèmes multicritères qui, par conséquent, doivent prendre en compte des critères multiples et contradictoires pour analyser les alternatives possibles. Plusieurs techniques ont été développées dans le contexte de l aide multicritère à la décision. Ce travail se concentre sur la méthode PROMETHEE. La méthode PROMETHEE I est utilisée pour obtenir un rangement partiel des alternatives et la méthode PROMETHEE II pour obtenir un rangement complet. Les bases de données qui comprennent les évaluations de chaque alternative dans chaque critère sont nécessaires dans la méthode PROMETHEE pour étudier les différentes options considérées dans les problèmes multicritères. Pourtant, il n est pas rare que les bases de données contiennent des valeurs manquantes. Le but de ce travail est d analyser la manière dont les valeurs manquantes devraient être gérées dans la méthode PROMETHEE. Le logiciel Visual PROMETHEE inclue déjà une procédure pour faire face au problème des valeurs manquantes. Cette technique est présentée dans ce mémoire et, de plus, d autres méthodes sont proposées. Toutes les techniques sont évaluées et comparées pour identifier celles qui agissent mieux quand les bases de données présentent des valeurs manquantes. Il faut souligner que les différentes méthodes présentées dans ce travail sont développées en utilisant MATLAB. Ce même logiciel est utilisé pour évaluer la précision des techniques. Pour évaluer et comparer les méthodes présentées, une méthodologie a été appliquée en utilisant quatre bases de données réelles. D abord, le rangement des alternatives d une base de données est déterminé en utilisant la méthode PROMETHEE II. Après la suppression d un ou de plusieurs valeurs de la base de données, le rangement des alternatives est effectué de nouveau en appliquant les différentes techniques exposées. Finalement, les deux résultats obtenus sont comparés en utilisant le coefficient de corrélation de Pearson et les coefficients de corrélation par rangs de Spearman et de Kendall. Le processus est effectué plusieurs fois et dans des conditions différentes. L étude des résultats a montré qu il n y a pas une méthode qui agisse mieux que les autres dans toutes les conditions considérées. Par conséquent, la procédure qui est implémentée dans le software Visual PROMETHEE semble être la plus appropriée dans les cas généraux, tout en étant la technique la plus simple à implémenter. Pourtant, quand les critères du problème de décision sont liés les uns aux autres, une des méthodes proposées conduit à de meilleurs résultats. Finalement, il est important de souligner que les deux techniques peuvent être combinées dans une seule base de données, en appliquant une méthode ou l autre en fonction de la valeur manquante considérée. Núria Gens, année académique The management of missing values in PROMETHEE methods Núria Gens 2

4 The management of missing values in PROMETHEE methods Keywords: Multicriteria Decision Aid, PROMETHEE methods, missing values, statistical methods, multicriteria methods Abstract: Decision making is a process where multiple alternatives need to be analysed. When the complexity and the importance of the decisions increase, formalisation of the process is required. The goal of Multicriteria Decision Aid is to help decision makers facing a multicriteria decision problem that involves taking into account multiple criteria, frequently in conflict, in order to analyse the possible alternatives. Several techniques have been developed in the field of Multicriteria Decision Aid. This thesis is focussed on the PROMETHEE method. The PROMETHEE I method is used to obtain a partial ranking of all the possible alternatives considered, while the PROMETHEE II is used to obtain a complete ranking. Databases with the evaluations of every alternative on every criterion are required in the PROMETHEE method to analyse the different alternatives considered in multicriteria problems. Nevertheless, it is not rare to find databases with some missing values. The aim of this thesis is to analyse how missing values should be managed in the PROMETHEE method. The software Visual PROMETHEE already includes a procedure for dealing with the problem concerning the missing values. The technique is presented in this thesis and other methods are proposed. All the techniques are evaluated and compared to each other to identify which ones perform better in the PROMETHEE method when the databases contain missing values. The methods presented in this study are developed using MATLAB and the same software is used to assess the performance of the techniques. In order to evaluate the methods presented in the thesis, a methodology has been applied using four real databases. First of all, the PROMETHEE II method is used to determine the ranking of the alternatives of a database. After the elimination of one or several values of the database, the ranking of the alternatives is determined again applying all the different techniques. Finally, both results are compared using the Pearson correlation coefficient, the Spearman rank correlation coefficient and the Kendall rank correlation coefficient. The process is performed multiple times and under different conditions. Examination of the results showed that no method performs better than the others in all the conditions considered. Therefore, the procedure implemented in the software Visual PROMETHEE, being the simplest technique to apply, seems to be the most appropriate in the general cases. However, one of the proposed methods led to better results when the criteria of the problem are strongly related to each other. It should be noted that both techniques can be combined in a database, applying one or the other method depending on the particular missing value considered. Núria Gens, academic year The management of missing values in PROMETHEE methods Núria Gens 3

5 Table of contents Introduction...7 Chapter 1: Multicriteria Decision Aid Introduction Multicriteria Problems Illustrative example Multicriteria Decision Aid techniques PROMETHEE and GAIA methods Introduction Model Preference functions and generalised criteria Aggregated preference indexes Outranking flows Unicriterion net flow PROMETHEE I: partial ranking PROMETHEE II: complete ranking GAIA Analysis of the illustrative example Additional information of the problem Analysis Chapter 2: Statistical methods Introduction Databases ARWU CARS ECM EPI Evaluation of the different methods of managing the missing values Pearson correlation coefficient Spearman rank correlation coefficient Kendall rank correlation coefficient The management of missing values in PROMETHEE methods Núria Gens 4

6 Comparison between the coefficients Presentation of the statistical methods The mean The nearest neighbour The simple regression The regression The combinations Evaluation method and results of the statistical methods Comparison of the methods Analysis of the error caused by the replacement of the missing value Chapter 3: Multicriteria methods Introduction Presentation of the proposed multicriteria methods Maximum coincidences Maximum Kendall rank correlation Maximum Spearman rank correlation Evaluation method and results of the proposed multicriteria methods Comparison of the methods Rank reversal analysis Presentation of the Visual PROMETHEE method Evaluation method and results of the Visual PROMETHEE method Chapter 4: Example used to illustrate the performance of all the methods Introduction Comparison of all the methods with an example Presentation of the example Analysis of the database Presentation of the six cases Case Case Comparison of the Visual PROMETHEE method with the Mean method Case Case Comparison of the Simple Regression method with the Regression method. 77 The management of missing values in PROMETHEE methods Núria Gens 5

7 Case Case Conclusion Conclusions...81 References...83 Appendix MATLAB functions PROMETHEE II Coefficients One missing value Five and ten percent of missing values Analysis of the error Rank reversal Common sub functions Appendix Robustness of the results Appendix Boxplots of the results ARWU..101 CARS..104 ECM 107 EPI..110 The management of missing values in PROMETHEE methods Núria Gens 6

8 Introduction Decision making is a process where different options need to be identified and analysed. Although decisions are made continuously and frequently unconsciously, it is clear that some of them are more important than others. The need to use a formalized approach for solving the problem arises when the complexity and the importance of the decision increases. Decision makers generally need to take into account multiple criteria at the same time. Usually, there does not exist a solution that is better than all the others for all the criteria considered. Therefore, compromise solutions that are globally good for all the criteria must be identified. For instance, a company s director could need to choose between several providers according to the cost of their product, the delivery time and the quality of the article. Generally, no provider would be the best one considering every criterion. Consequently, a compromise solution must be achieved. The objective of Multicriteria Decision Aid is to help decision makers facing a multicriteria decision problem by providing them with valuable information about the different alternatives. One of the most relevant methods in this field is the PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) method. The PROMETHEE I method is used to obtain a partial ranking of all the possible alternatives considered, while the PROMETHEE II is used to obtain a complete ranking. Databases with the evaluations of every alternative in every criterion are required in order to analyse the different alternatives taking into account the multiple criteria. Nevertheless, it is not rare to find databases with some missing values. For instance, those missing values can exist due to the fact that some evaluations are not known or because there were errors in the transcription. However, when there are missing values in the evaluations, the PROMETHEE method cannot be applied. Therefore, missing values can be an important problem in multicriteria decision making. The aim of this thesis is to analyse how missing values should be managed in the PROMETHEE method. The problem concerning the databases with missing values in multicriteria problems was already considered and a specific procedure was proposed in this case [18]. In this thesis, several other ways of managing the missing values in the PROMETHEE method will be proposed and evaluated. Moreover, all the proposed techniques will be compared between them and also to the existing procedure just mentioned. The different techniques of managing the missing values will be developed using MATLAB and the same software will also be used to evaluate the accuracy of the methods. The MATLAB functions are included in the CD attached to this thesis. This thesis is divided in four chapters. In the first one, multicriteria problems and the techniques that exist in the field of Multicriteria Decision Aid will be defined and described. Special focus will be given to the description of the PROMETHEE method. A first group of techniques for managing the missing values is through statistical methods that are used to replace the missing values of the databases by evaluations determined using the The management of missing values in PROMETHEE methods Núria Gens 7

9 information included in the database. These techniques are described and evaluated in the second chapter. In the third chapter, another set of procedures that use multicriteria techniques to manage the missing values of the databases will be described in detail. They are called multicriteria methods. The accuracy of the procedures will be evaluated. Comparisons will be established between the multicriteria techniques and the statistical methods. Finally, the fourth chapter presents an example of multicriteria decision problem involving several cases where different values of the database are deleted. The different cases considered illustrate how the methods presented in this thesis perform in different scenarios. The thesis ends with a number of conclusions and some perspectives for further study. The management of missing values in PROMETHEE methods Núria Gens 8

10 Chapter 1: Multicriteria Decision Aid 1.1. Introduction Historically, the decision making process consisted of the determination of an optimal solution among all the possible ones. In fact, only one criterion was considered to evaluate the different alternatives. For example, an enterprise could organize the fabrication of its multiple products maximizing the economic benefit. For instance, a linear program would be formulated to solve the problem, which could consider the production hours and the time needed to manufacture each type of article, as well as the cost of the production, the price and the demand of each fabricated item [15]. Nevertheless, decision makers generally need to optimise multiple criteria at the same time [16]. Therefore, for these types of decisions, the concept of optimal solution is no longer valid, as it is rare that one particular solution is the best one for all the criteria considered. For instance, when individuals have to choose between different ways of travelling somewhere, it seems clear that the time spent undertaking the journey should be minimized. However, they don t only consider this aspect for each option, but also other criteria such as the monetary cost of the alternative and the comfort of the trip. Since there is normally no alternative optimizing all the criteria, a compromise solution should be chosen. This solution is different for each individual, because the decision maker s preferences strongly influence the final choice. It is clear, for instance, that the preferences of a student would be different from those of a person with reduced mobility and, therefore, the compromise solution would be different in each case. This is why, since the Sixties, the complexity of multicriteria decision problems has been considered and researchers look for compromise solutions that are globally good taking into account the different criteria instead of searching for an optimal one [20][21]. Frequently, several criteria are taken into account and it is common that the preferences for each criterion are in conflict. For instance, in the example of individuals that have to select a way of travelling somewhere, it is clear that the maximization of the comfort of the trip will be in conflict with the minimization of its price. The goal of Multicriteria Decision Aid (MCDA) is to support decision makers facing multicriteria decision problems by providing them with valuable information about the different alternatives [19]. Consequently, they are able to better understand the problem and the possible solutions and to find the alternative that suits their objective in the best possible way. The PROMETHEE (Preference Ranking Organization METHod for Enrichment Evaluations) and GAIA (Geometrical Analysis for Interactive Aid) methods are widely applied methods in MCDA [5][8]. In fact, they are used around the world in many different fields. M. Bezhadian et al. have compiled multiple applications of these methods in fields such as such as Business, Logistics and Transportation [2]. For instance, reference [2] includes an example where PROMETHEE methods were used in order to rank and select distribution centres for a firm in four areas of Belgium. The management of missing values in PROMETHEE methods Núria Gens 9

11 The PROMETHEE method is used to obtain a partial or complete ranking of all the possible actions and to observe their profiles considering multiple criteria. In turn, GAIA is a clear graphic visualisation of the multicriteria problem and the decision maker s preferences. This Thesis will be centred on these methods, which were developed at the beginning of the 1980s and have been widely studied since then [1][5][8]. It is important to underline that the result of solving a multicriteria decision problem depends not only on the evaluations of every alternative in every criterion but also on the decision maker, as he assigns the relevance of the criteria according to his preferences and decides how the evaluations should be interpreted. Consequently, a global compromise solution doesn t exist. This is why the decision maker has to provide all the supplementary information needed to the analyst. There are several multicriteria methods that focus on the same problem, but the information needed from the decision maker is different in each case. The PROMETHEE and GAIA methods require that the decision maker assigns relative weights to the criteria. The criteria with greater weights are the ones that are more important to the decision maker. Moreover, he also has to define for each criterion the preference degree between alternatives according to the difference between their evaluations. The information required from the decision maker will be described in detail in the following sections of this chapter. In this chapter, the multicriteria decision problems will be formally defined first. Then, an example of multicriteria problem will be presented, which will be used to illustrate some of the sections of this chapter. Then, the different methods existing in MCDA will be briefly described, focusing especially on the PROMETHEE and GAIA methods. Finally, the illustrative example will be analysed using those two methods Multicriteria Problems In this section, the multicriteria problems are formally described and the natural dominance relation between alternatives is presented in detail. Further explanations can be found in references [1], [5]. Let A be the finite set of n possible actions {a 1, a 2,, a n }. An action is the generic term that designates the object of the decision and, consequently, it can be an alternative, a candidate, a potential decision, etc. Let F be the set of all criteria {f 1 (a), f 2 (a),, f j (a),, f k (a)}, where f j (a i ) is the evaluation of the action a i according to the criterion f j. The basic data of a multicriteria problem can be represented by an evaluation table (table 1) that contains the evaluations of every alternative on every criterion. The acquisition of the evaluation table is a very important part of the problem s analysis, since it requires the definition of the possible actions and the criteria considered as well as the determination of the evaluations. In the example concerning the individuals that have to select a way of travelling somewhere, the actions considered would be all the manners in which people could arrive at their The management of missing values in PROMETHEE methods Núria Gens 10

12 destinations. For instance, an action could be driving, cycling, walking, etc. In addition, the set of criteria could consist of the time spent travelling, the monetary cost of the journey and the comfort of the trip. Finally, the evaluations of these criteria would be required to analyse the multicriteria problem. f 1 (.) f 2 (.) f j (.) f k (.) a 1 f 1 (a 1 ) f 2 (a 1 ) f j (a 1 ) f k (a 1 ) a 2 f 1 (a 2 ) f 2 (a 2 ) f j (a 2 ) f k (a 2 ) a i f 1 (a i ) f 2 (a i ) f j (a i ) f k (a i ) a n f 1 (a n ) f 2 (a n ) f j (a n ) f k (a n ) Table 1: Evaluation table Multicriteria problems are defined by the following expression: a, a,, a,, a (1.1) It is common that some of the criteria need to be maximized and other criteria have to be minimized. Nevertheless, it is assumed that the criteria have to be maximized because, if a given criterion had to be minimized, the symmetric values of the evaluations would be considered and the criterion would then need to be maximized. It is interesting to add that multicriteria decision problems frequently lead to ill posed mathematical problems because no alternative optimizes all the criteria at the same time. This is why the Multicriteria Decision Aid methods focus on detecting the best compromise solution between all possible actions. In order to be able to compare different options, the natural dominance relation between alternatives is defined: a b, j 1,, k adb h: a b a, b A a b, j 1,,k (1.2) h: a b arb h : a b where P stands for dominance, I for indifference and R for incomparability. In fact, these relations are very intuitive. Firstly, an action is better that another action if it is as good as the other on all the criteria and there is at least one criterion where the action is better than the other one. In addition, if two actions have the same evaluations in all the criteria, then they The management of missing values in PROMETHEE methods Núria Gens 11

13 are indifferent. Finally, if an action is better than another one for a criterion h and but it is worse for another criterion h, then the two actions are not comparable. This dominance relation has been criticized because of its weaknesses. Firstly, it should be noted that it does not take into account the different degrees of preference that the decision maker might have. For instance, in the example of the individual that needs to choose the way of travelling somewhere, the criterion time representing the time spent undertaking the journey could be evaluated by the minutes of the trip and should be minimized. Therefore, the evaluations of two actions a and b in the criterion time could be the following: Situation 1 Situation 2 Situation 3 time(a) 5min 5min 5min time(b) 50min 30min 6min Table 2: Evaluations of actions a and b on the criterion "time" It is clear that, in the first situation, the decision maker would strongly prefer the alternative a over b. Moreover, he would still prefer the option a in the second case, but both alternatives would probably seem indifferent to him in the third situation. Although it is obvious that the degree of preference of the decision maker between the two alternatives in the three situations would be different, the natural dominance relation between them would be apb in all the cases. In addition, it is important to underline the fact that the dominance relation that has been defined is poor to thoroughly analyse multicriteria problems due to the fact that it only determines the efficient solutions, which are the alternatives that are not dominated by any other. In other words, the efficient solutions are those actions that are not preferred by any other. Nevertheless, it is not always useful to identify those alternatives, as most of them are often efficient considering the evaluation table of a multicriteria problem. The limits of the dominance relation defined in (1.2) are illustrated in an example where four students are compared taking into account their marks in six different subjects. The students are the alternatives and the marks are the criteria of a multicriteria decision problem whose evaluation table is shown below: Mark 1 Mark 2 Mark 3 Mark 4 Mark 5 Mark 6 Mark 7 Student Student Student Student Table 3: Evaluation table of the example where four students are compared Considering the evaluation table shown in Figure 3 and the equations presented in (1.2), Student 3 would be dominated by Student 1. Since all the other alternatives would be incomparable to each other, the efficient solutions of the multicriteria problem would be: Student 1, Student 2 and Student 4. Nevertheless, if the evaluation table was analysed intuitively, Student 3 would probably be considered the second best student. Therefore, the second best option would have been excluded from the group of efficient solutions. The management of missing values in PROMETHEE methods Núria Gens 12

14 1.3. Illustrative example An example of multicriteria decision problem is presented in this section in order to illustrate some of the ideas that are developed in the other parts of this chapter. The proposed example of multicriteria problem is based on an individual who is travelling to Barcelona and needs to book a room in a hotel. To select a good hotel according to his preferences, he uses a web page where multiple hotels around the world are presented and their characteristics are shown. However, it is not easy to choose the best option without thoroughly analysing the alternatives. It is clear that an optimal solution does not exist, since all the hotels present advantages and disadvantages. The individual would probably take into account the price of the stay and the minutes that he would spend walking if he would go from the hotel to the city centre. Moreover, he could also consider the score that is assigned to every hotel in the web page. This score is calculated considering the opinion of the users of the web page that have stayed there during their visit in Barcelona. Therefore, a multicriteria problem can be defined in order to select the best alternative. This example will be used to illustrate the phase of the PROMETHEE methods where the criteria need to be defined. Then, in the last section of this chapter, the example will be described in detail and thoroughly analysed in order to show how the analyses of multicriteria decision problems are developed Multicriteria Decision Aid techniques There exist different approaches in the Multicriteria Decision Aid field but they can be basically divided into three major categories: interactive, multiattribute and outranking methods. References [1][9][10] offer a more thorough description of these types of Multicriteria Decision Aid methods. An interactive method is a sequential process consisting of multiple iterations. Every iteration includes a calculation phase when a compromise solution is elaborated and a discussion phase when the decision maker considers the analyst proposition and develops his preferences progressively. The additional information obtained in the discussion stage is introduced into the model that is used in the calculation stage. The multiattribute methods transform multiple criteria into one common utility function that represents the overall performance of each alternative and needs to be maximized. Consequently, the utility function, which has to be constructed by the analyst, can be used to rank the alternatives from the best one to the worse one. Finally, the outranking methods, developed in France in the late sixties, are based on outranking relations among alternatives evaluated on several criteria. An outranking relation exists when it can be stated that one alternative is at least as good as another one given enough arguments. In fact, B. Roy defines the outranking relation as the relation that can be identified between two alternatives a and b such that a outranks b if there are enough arguments to declare that a is at least as good as b, while there is no essential reason to refute that assertion [10][17]. In most outranking methods, this relation is established by multiple The management of missing values in PROMETHEE methods Núria Gens 13

15 pairwise comparisons among the alternatives. The most popular families of this type of methods are the ELECTRE and the PROMETHEE methods. This thesis is focussed on the outranking method PROMETHEE, which is described in detail in the following sections PROMETHEE and GAIA methods Introduction The outranking relation methods are one of the most important categories in the field of Multicriteria Decision Aid and, as mentioned above, they are based on the outranking relation among the alternatives. The PROMETHEE method, developed by J.P. Brans and presented for the first time in 1982, is one of those methods and it is based on pairwise comparisons between actions on every criterion. These comparisons take into account preference functions that determine the degree of preference between the evaluations of each pair of actions. Therefore, the evaluations are enriched by the preference functions and the reliability of the analysis increases. Finally, a net flow is assigned to each action depending on how it is preferred to the others or not. The best actions are the ones with greater net flow. In particular, PROMETHEE I is used to obtain a partial ranking and PROMETHEE II is applied in order to obtain a complete ranking of all the feasible alternatives. The analysis is completed by the GAIA visual modelling that allows the decider to improve the understanding of multicriteria problems. Although PROMETHEE and GAIA methods can be used by individuals, it is more common that they are applied by groups of people that face complex problems that take into account multiple criteria. In the following sections, the PROMETHEE and GAIA methods will be described in detail, including all the necessary phases to solve a multicriteria problem applying these techniques. The detailed definition of these methods can also be consulted in [1],[5],[6],[8] Model Preference functions and generalised criteria Since the PROMETHEE methods are based on comparisons between pairs of actions for all the criteria, the difference between the evaluations of every pair of alternatives for every criterion f j must be calculated as: a, b a b (1.3) This difference is not only useful to determine which of the two actions is better than the other one for that criterion, but also to quantify the intensity of the preference. The greater the value of is, the stronger the preference of a over b is. However, it is important to remark that the value of the distance depends on the units of criterion j. To improve the comparisons between actions, the preference functions are defined. These functions let the decision maker describe how differences should be interpreted. For instance, if the distance is so small that the decision maker can neglect it, the two actions are considered as indifferent. Moreover, if the difference is greater than a certain value, a strict preference The management of missing values in PROMETHEE methods Núria Gens 14

16 can be defined between the two actions, independently of whether the distance increases even more. The preference functions are characterized in the following way: and they are defined as: a, b, a, b A (1.4) 0 a, b 1 (1.5) a, b 0, 0 a, b 0, 0 a, b 1, 0 a, b 1, 0 (1.6) The pair {f j, a, b} is called the generalized criterion associated with criterion f j for all j{1,,k} and it is composed by the criterion and its preference function. Each criterion must be associated with a generalized criterion in order to apply the PROMETHEE method. It seems logical that a, b is a non decreasing function that takes the value 0 when, 0. In order to facilitate the decision maker s task, six preference functions are proposed. The selection of the most adequate function for each criterion is carried out interactively by the analyst and the decision maker, who take into account the preference degrees and the differences observed. The type of preference function elected has to reflect the decision maker s attitude in relation to the distance than can exist between a pair of actions. It is important to underline that the possibility of choosing between different preference functions is an important help in the task of representing the decision maker s preferences in a more reliable way. Therefore, the decision process is enriched, giving more realism and fidelity to the solution of multicriteria decision problems. The different types of preference functions P that are commonly used are presented below, defined and plotted according to the difference d. Then, the illustrative example is used to show how the definition of the preference functions can be applied to a specific problem. Although some types are particular cases of others, they need to be defined separately because they clearly illustrate the decision maker s opinion on that criterion. It should be noted that the graphs of the preference functions have been extracted from reference [6]. The management of missing values in PROMETHEE methods Núria Gens 15

17 Type I: Usual P(d) = (1.7) Figure 1: Usual In this case, two actions a and b are only indifferent if a b. Otherwise, one is strictly preferred to the other. Therefore, the concept of preference is not enriched in this situation. Type II: U shape P(d) = 0 1 (1.8) When a U shape preference function is assigned to a criterion, a pair of actions a and b are indifferent while the difference between their evaluations ( a b) does not exceed a specified limit. For differences greater than that value, the preference is strict. The parameter q needs to be fixed, representing the maximum difference that is considered to be indifferent between two evaluations. In other words, q is a threshold of indifference. Type III: V shape Figure 2: U shape P(d) = (1.9) Figure 3: V shape The third type of preference function allows the decision maker to progressively prefer one action over another according to the distance existing between both actions. The preference degree increases linearly until the limit p, since for differences higher than that value the preference is already strict. The parameter p is determined by progressively decreasing its The management of missing values in PROMETHEE methods Núria Gens 16

18 value while the preference is considered to be strict (according to the decision maker s point of view). Therefore, p is a threshold of strict preference. Type IV: Level P(d) = 0 1 (1.10) Figure 4: Level In this case, two actions a and b are indifferent until the difference between their evaluations on that criterion reaches the limit q. Then, there is a weak preference if ( a b) is higher than q but lower than p and this is why the value ½ is assigned to the preference function. Finally, if the difference between the evaluations is greater than p, the preference of one action over the other is strict. It is clear that this type of preference function requires the definition of two parameters: q and p. It should be noted that the preference functions of type I and II are particular cases of this type of preference function. Type V: V shape with indifference P(d) = 0 1 (1.11) Figure 5: V shape with indifference The preference functions of type V are similar to those of type III but consider the indifference between two actions when the difference between their evaluations is smaller than a parameter q. Therefore, the determination of the parameters q and p is also needed. It should be underlined that the preference functions of type I, II and III are particular cases of this type of preference function. The management of missing values in PROMETHEE methods Núria Gens 17

19 Type VI : Gaussian P(d) = (1.12) Figure 6: Gaussian Finally, the Gaussian preference functions gradually increase according to the difference between actions. The parameter that has to be fixed in this case is s. When the distance between the actions is equal to s, the preference degree takes the value 0,39. In order to make the decision maker s task easier, the value of s can be determined between two fictional values q and p. If the decision maker wants to strengthen the preference degree when the distances are small, then s should be close to q. However, if he desires to soften the progression of the preference degree according to the distances, the parameter s should be near p [6]. The hotels example is useful to illustrate how the different types of preference functions could be applied in different situations. First of all, the individual could take into account the fact that the hotel offers free Wi Fi or not. Then, two types of evaluation would exist on this criterion W : Hotels that offer free Wi Fi: W(a)=1 Hotels that do not offer free Wi Fi: W(a)=0 where a is the alternative considered. In this case, the preference function used could be type I. Consequently, if the difference between the evaluations of two alternatives was greater than 0, the individual would strongly prefer one hotel over the other. In addition, the individual travelling to Barcelona could also consider the number of stars of each hotel. Nevertheless, it is possible that two hotels with only one star of distance seem indifferent to him. Consequently, the individual could select the preference function type II fixing a threshold of indifference of 1. Therefore, a hotel would be preferred to another when the difference between their numbers of stars would be greater than one. However, it should be noted that the degree of preference of a hotel with five stars over another with one star would be the same that the degree of preference of a hotel with three stars over another with one star. The model would be limited by this aspect and, therefore, the preference function The management of missing values in PROMETHEE methods Núria Gens 18

20 type IV would be more accurate to define the degree of preference between the numbers of stars. In order to enrich this criterion, the individual could choose the preference function type IV and fix a threshold of indifference of 1 and a threshold of strict preference of 3. Therefore, the degree of preference of a hotel with five stars over another with one star would be higher than the degree of preference of a hotel with three stars over another with one star, since the preference function would take the values 1 and ½ respectively. Given the criterion of the hotels example that takes into account the score that is assigned to every hotel in the web page, the individual could choose the preference function type III to reflect his point of view. For instance, he could consider a threshold of strict preference of 1. In this case, when a hotel with a score of 9 was compared to another with a score of 8, the preference function would take the value 1, while if it was compared to another with a score of 8,5, the preference function would be equal to ½. In this case, it would also be interesting to select the preference function type V, which would allow the individual to determine a threshold of indifference. Consequently, the decision maker would have the option of considering indifferent two hotels with very similar scores, such as 7,6 and 7,9. For instance, the threshold of indifference and the threshold of strict preference could be equal to 0,3 and 1. While a hotel with a score of 9,3 would be indifferent to another one with a 9, it would be strictly preferred to another one with a 8. Finally, type VI should be used if the individual would want that every little increase of the distance between two alternatives led to an increase in the preference degree between them. For example, this type of preference function could be applied to the criterion taking into account the price of the stay. If the individual assigned the value 50 to the parameter s, the degree of preference of a stay of 270 over a stay of 130 would be 0,98, while the degree of preference of a stay of 270 over a stay of 200 would be 0, Aggregated preference indexes In order to be able to analyse multicriteria problems with the PROMETHEE method, the decision maker is also required to assign the weight of relative importance to each criterion. These weights are non negative numbers that represent the relevance of the criteria according to the decision maker s point of view. The higher a weight is, the more significant the criterion is. It is also essential to mention that the weights of all criteria are normalized so that their sum is equal to one: 1 (1.13) The aggregated preference indexes π are the values that express the preference degree of one action over another considering all the criteria. They are defined as: a, b A π(a,b) = a, b (1.14) The management of missing values in PROMETHEE methods Núria Gens 19

21 Frequently, a is better than b in some of the criteria and b is better than a in other criteria. As a consequence, both π(a,b) and π(b,a) are usually positive. The aggregated preference indexes have to be interpreted considering that: πa, b 0 weakglobal preference of aoverb πa, b 1 strong global preference of aoverb (1.15) Finally, it is interesting to underline the proprieties of this index: πa, a 0 0 πa, b 1 a, b A 0 πb, a 1 0 πa, b πb, a 1 (1.16) Outranking flows The PROMETHEE method is based on the quantification of how an action a outranks all the other actions and how this action a is outranked by all the others. This is why the positive φ + (a) and the negative φ (a) outranking flows are defined. The first measure expresses the outranking capacity of the action a with respect to all the other actions while the second one represents how a is outranked by the others. In other words, the positive flow shows the power of an action and the negative flow indicates its weakness. It is clear that the higher the positive outranking flow is, the better the alternative is. Correspondingly, the greater the negative outranking flow is, the worse the alternative is. The values of the two measures are calculated as the sum of all the aggregated preference indexes divided by the number of actions that a is compared to (n 1). Consequently, the values are normalized between 0 and 1. The positive outranking flow: The negative outranking flow: φ + (a)= φ (a)= πa, b (1.17) πb, a (1.18) The net outranking flow φ(a) of the action a is defined as the difference between the positive outranking flow and the negative outranking flow: φ(a)= φ + (a) φ (a) (1.19) 1Φ1 Φx0 (1.20) The higher this measure is, the better the action a is. If the net flow is positive, then the alternative is more outranking the others on all the criteria, but when it is negative the alternative is more outranked. The management of missing values in PROMETHEE methods Núria Gens 20

22 It is clear that the net flows of two actions should express similar preferences to the pair wise preference degrees between two actions, since the net flows are used to rank the actions. In fact, it is demonstrated that the net preference flow is a centred score that minimizes the sum of the square deviations from the pair wise comparisons of the actions [3]. A numerical score s i (i=1,, n) defined on A is considered in order to rank the actions of A, being better the actions with greater score. Since the comparison of two actions considering the score should be similar to the comparison taking into account the pair wise preferences, the expression (1.21) should be minimized. It should be noted that if the expression was minimized, the translations of the score would also minimize it. Moreover, s i is defined as a centred numerical score. The restriction originated by this fact is: (1.21) 0 (1.22) The Lagrange function to find the score s i is: L(s 1,, s n, λ) = (1.23) The derivative of L(s 1,, s n, λ) with respect to λ leads to the condition that s i is a centred numerical score. It is interesting to develop the derivative of L(s 1,, s n, λ) with respect to s i : (1.24) (1.25) 04 1 (1.26) 04 (1.27) Since the equation is considered for all i=1,, n, λ needs to be equal to zero. Therefore, the final result is: The management of missing values in PROMETHEE methods Núria Gens 21

23 1 1 (1.28) The expression shows that the score s i considered is proportional to the net flow. Therefore, it is demonstrated that the net flow is a centred score that minimizes the sum of the squared deviations from the pair wise comparisons between actions Unicriterion net flow Taking into account the determination of the positive and the negative outranking flows and of the aggregated indices, the following expressions are obtained: a 1 1 πa,b 1 1 a, b a 1 1 πb,a 1 1, a (1.29) (1.30) Therefore, we can describe the net flow as expressed below: It is clear that φ(a)= φ + (a) φ (a)=, b, a (1.31) if an unicriterion net flow φ j (a) is defined as φ j (a)= φ(a)=, a (1.32), b, a (1.33) The unicriterion net flow φ j (a)[ 1, 1] is obtained when only one criterion is considered and it shows how an action is outranking or outranked by the others when criterion j is taken into account. In fact, the value is determined as if the net flow was calculated assigning the total weight to that single criterion. The profile of an alternative is the representation of its unicriterion net flows and it is useful for observing how the action performs on every criteria PROMETHEE I: partial ranking The PROMETHEE I partial ranking is based on the positive and negative outranking flows. Frequently, the rankings induced by both measures are different. The partial PROMETHEE I ranking is the intersection between them. It is a partial ranking due to the fact that if there is no preference or indifference relation between two alternatives they are considered incomparable. First of all, it is essential to define the expressions (S +,I + ) and (S,I ) as the two pre orders resulting from the positive and negative flows. The management of missing values in PROMETHEE methods Núria Gens 22

24 (1.34) (1.35) The partial PROMETHEE I ranking is defined as follows: a b a b a b in all the other cases (1.36) where P (1) stands for preference, I (1) for indifference and R (1) for incomparability in PROMETHEE I. If a b, it means that a is preferred to b, given that the action a has both a higher power and a lower weakness compared to b. If a b, it is certain that the positive outranking flow and the negative outranking flow of both alternatives are equal. Finally, if a b, a higher power of an action is related to a lower weakness of the other. In other words, the information provided by the outranking flows is contradictory PROMETHEE II: complete ranking It is not rare that the decision maker requests a complete ranking of all the possible alternatives. The PROMETHEE II offers a complete ranking considering the net outranking flow of the different actions: (1.37) where P (2) stands for preference and I (2) for indifference in PROMETHEE II. Clearly, the greater the net flow is, the better the alternative is. It is important to underline that, in this method, there are no incomparabilities between actions. It should be underlined that the complete rankings obtained with PROMETHEE II are compatible with the partial rankings obtained with PROMETHEE I. This means that when a b, then GAIA The PROMETHEE method can be complemented by a visual analysis called GAIA that offers the decision maker a clear graph that illustrates the multicriteria problem. Therefore, the comprehension of the problem is improved, which is obviously advantageous to achieve a good decision. If the actions of the problem are symbolized by points whose coordinates depend on their evaluations, it is clear that a multidimensional space is needed in order to represent the alternatives taking into account the different criteria. The goal of GAIA is to detect the best view point so that it includes as much information as possible. The analysis is based on the The management of missing values in PROMETHEE methods Núria Gens 23

25 unicriterion net flows, which are much more accurate describing how the decision maker perceives the different criteria than the evaluations of the actions. Moreover, these net flows are independent from the units of the criteria. In particular, all the alternatives considered in the multicriteria problem are represented in a k dimensional space by a point whose coordinates are the unicriterion net flows. In fact, each action is defined by a point α i : α i =( φ 1 (a i ), φ 2 (a i ),, φ j (a i ),, φ k (a i )) i=1,2,,n (1.38) Consequently, the actions can be represented by n points in the k dimensional space. These points representing the actions and the axes representing the criteria are projected on a plane called GAIA plane, which is the plane that preserves the maximum possible information when the projection is done. The plane is determined using the Principal Component Analysis (PCA), which is explained in detail in [11]. The matrix φ that contains all the unicriterion net flows is: φ= φ j (a i ) a i A; j{1,2,,k} (1.39) First of all, the goal of projecting the points on a direction defined by a unit vector u centred at the origin is considered. In order to preserve the maximum information about the points, the points should be as near to the projection vector as possible. Therefore, the vector u should minimize the square sum of the distances between the points α i and their projections p i : which is equal to maximize the expression Figure 7. min α p (1.40) Op. This equivalence can be observed in α 0 u Figure 7: Projection of the point α i on the vector u Moreover, it is clear that Op = α i u, since it is the projection of the point α i on the vector u. Consequently, the maximization of Op is equivalent to: p 10 (1.41) where C is the covariance matrix of the φ j. The matrix C is calculated as: C= φ φ (1.42) The management of missing values in PROMETHEE methods Núria Gens 24

26 Finally, the system resulting from derivating the Lagrange function associated with the problem (L(u,λ)=u Cu λ(u u 1)) is: λu 1 (1.43) Therefore, λ is a eigenvalue of matrix C and u is the eigenvector associated with λ. The solution is then a unitary eigenvector with the greatest eigenvalue of matrix C. Considering also the eigenvector v associated with the second higher eigenvalue, u and v define the best plane where the points can be projected. In order to illustrate all the information that the GAIA plane can offer, an example is presented below. It is the result of analysing the characteristics of eight different mobile phone models minimizing their price ( ) and maximizing their processor speed (GHz), their internal memory (GB) and the quality of their camera (Mpx). The evaluation table of the multicriteria problem is shown in table 4. All the criteria have the same weight and the preference function type Usual is assigned to all of them. Price( ) Processor speed (GHz) Internal memory (GB) Sony Xperia M (1) Samsung Galaxy Ace 2 (2) 199 0,8 4 5 Sony Xperia L (3) Samsung Galaxy S III mini (4) Samsung Galaxy S III (5) 409 1, Samsung Galaxy Core (6) 229 1,2 8 5 Camera (Mpx) Sony Xperia E (7) ,2 Samsung Galaxy Trend Plus (8) 169 1,2 4 5 Table 4: Evaluation table of the example where eight mobile phone models are compared The software used to analyse this problem is called D Sight. D Sight is a software, developed at the Université libre de Bruxelles (ULB), that implements the PROMETHEE and GAIA methods in order to analyse multicriteria problems. The goal of D Sight is to help decision makers to better understand their problems and to take better decisions. This software includes features such as the projections of all the alternatives against the criteria axes or against the decision stick or the possibility of doing sensitivity analysis. The characteristics of the software are described in [7]. The management of missing values in PROMETHEE methods Núria Gens 25

27 Figure 8: GAIA plane First of all, it is necessary to remark that the delta value (88,6%) indicated in Figure 8 shows the reliability of the representation, since it depends on the amount of information that is conserved by the plane. It is also interesting to mention that the orientation of the axes (represented in green) shows which criteria are compatible and which aren t. In the case of Figure 8, the Price seems to be in conflict with the Internal Memory and Camera, since they are pointing in very different directions. Therefore, it is not easy to find a mobile phone model that performs well on the criterion Price and also on either of the two other criteria. However, the criterion Price seems to be perpendicular to the criterion Processor Speed, which indicates that they are independent. Moreover, the length of the axes is independent from the weight of a criterion, but it indicates its importance according to the way in which it distinguishes the alternatives. The criteria that are not as relevant for differentiating between the different alternatives as the others are shorter. However, in this case all the criteria axes have a good size, indicating that they represent a significant variation between the different evaluations. GAIA also allows the decision maker to distinguish the profiles of the alternatives (represented in blue), since it is possible to observe what are the best and the worse performances of an action according to the different criteria. For instance, it is clear that the mobile phone number 7 is very good considering the criterion Price, but it is not considering the other criteria. In addition, the decision maker is able to discover the alternatives that are similar to each other and those that have extremely different profiles. Another important aspect of the graph in Figure 8 is the position of the decision stick (represented in red), which illustrates the importance of the criteria according to the decision The management of missing values in PROMETHEE methods Núria Gens 26

28 maker s point of view. In this case, it is noticeable that the decision maker prioritizes the mobile phone s performance over its price. Obviously, the direction of the decision stick changes if the weights of the criteria are modified. On the one hand, if the decision stick is large, the decision maker should select the alternatives that are farther in its direction. On the other hand, if the decision stick is short, it means that the criteria are strongly conflicting between them and a good compromise solution would be close to the origin. Furthermore, when the actions are projected on the decision stick, their positions stand for the PROMETHEE II ranking. Nevertheless, it should be noted that the ranking obtained by projecting the alternatives on the decision axis may not always be completely correct. This is due to the fact that the k dimensional problem is represented in a two dimensional representation. In the example, the PROMETHEE II complete ranking obtained is 5,3,6,8,4,1,2 and 7, which is different from the order of the projections of the alternatives shown in Figure 9. Figure 9: Alternatives projected on the decision stick in GAIA plane 1.6. Analysis of the illustrative example Additional information of the problem In order to illustrate the PROMETHEE and GAIA methods, a multicriteria decision problem is analysed applying these techniques. The study of the alternatives will be performed using the software D sight. The proposed example of multicriteria decision problem is the illustrative example that has been presented section 1.3 and is based on a decision maker that needs to select a hotel in Barcelona. The individual considers ten options which are evaluated according to three criteria. It is necessary to define the preference function and the weight of the three criteria that the decision maker decides to include in the analysis: Centre The decision maker decides to assign 45% of the total weight to the most important criterion according to his point of view: the distance between the hotel and the city centre. In fact, he desires to minimize the time spent walking from one place to the other. The preference function used in this case is V shape with indifference with The management of missing values in PROMETHEE methods Núria Gens 27

29 threshold of indifference equal to 3 minutes and a threshold of strict preference equal to 10 minutes. Price The price of the stay is also a significant characteristic when choosing a hotel and this is why 35% of the total weight is attributed to this criterion. It is important to mention that the individual is staying two nights in Barcelona. Consequently, the preference function applied in this criterion is also V shape with indifference with a threshold of indifference equal to 10 and a threshold of strict preference equal to 50. Score The last attribute that is taken into account is the score given to the hotel. This criterion is also significant because it shows the satisfaction of the clients that have already stayed in that hotel on a scale of 0 to 10. Consequently, it constitutes 20% of the total weight. A V shape with indifference preference function is used considering a threshold of indifference equal to 0,3 and a threshold of strict preference equal to 1. These values seem logical examining the evaluation table that is shown below. It seems reasonable that, for example, a hotel with a score of 7,6 would not be preferred over another one with a 7,9 but a score of 9 would be much more desirable than an evaluation of 7, Analysis Ten alternatives have been considered in this multicriteria decision problem and the evaluation table where the evaluations of every alternative are presented is shown in Table 5. Name Score Price ( ) Centre (min) NH les corts 7, La Ciudadela 8, Granvia Hilton barcelona 8, Holiday Inn Express Barcelona City 7, Hotel Sagrada Familia 7, Hotel Canton 8,3 129,5 15 Chic & Basic Zoo 8,2 142,8 20 Princesa Sofia Gran Hotel 8,3 268,4 57 Arenas atiram hotels 7 193,14 55 Table 5: Evaluation table of the example where ten hotels are compared In order to make all the graphs imported from D Sight more clear and understandable, the names of the hotels have been replaced by shorter names. Short Name NHLC LC G HB HIEBC Name NH Les Corts La Ciudadela Granvia Hilton Barcelona Holiday Inn Express Barcelona City The management of missing values in PROMETHEE methods Núria Gens 28

30 HSF HC C&BZ PSGH AAH Hotel Sagrada Familia Hotel Canton Chic & Basic Zoo Princesa Sofia Gran Hotel Arenas Atiram Hotels Table 6: Short names assigned to the hotels All the parameters and evaluations that have been presented lead to the representation of the GAIA plane shown in Figure 10. Figure 10: GAIA representation Firstly, it should be noted that the delta value is equal to 96%, which is a very high value. Consequently, the information lost due to the projection of the three dimensional problem into a two dimensional space is limited. In fact, it is clear that the information lost by projecting a three dimensional problem into a two dimensional space is generally small. It may seem illogical that the price of the hotels and the distance to the city centre are not in strong conflict between each other, but it is clear that the price takes into account a lot more characteristics apart from the hotel s situation such at its luxury. Concerning the positions of the alternatives, the hotel G called Granvia appears to be strongly differentiated from the others. Indeed, it s the best alternative according to the criteria Score and Centre but is also the worst considering its price. Consequently, even though the hotel seems a really good place to stay during the visit to Barcelona, the decision maker might be dissatisfied by this choice because it might be too expensive. The GAIA representation also reveals that some of the alternatives perform much better than others taking into account all the criteria. It looks as if the alternatives HB, HIEBC, PSGH and The management of missing values in PROMETHEE methods Núria Gens 29

31 AAH are worse than the others. In fact, the representation of PROMETHEE I and PROMETHEE II rankings displayed in Figure 11 confirms this idea. Figure 11:PROMETHEE I and PROMETHEE II representation The alternatives are represented in Figure 11 using their negative and positive flows as axes. Considering the rules of PROMETHEE I, when their projection lines cross, the alternatives are incomparable. However, an alternative a is preferred to b if a is above b without the lines crossing. Therefore, it is clear that the most preferred hotels according to PROMETHEE I are those whose short names are: HC, C&BZ, LC, G and HSF. In fact, all the others are dominated by these alternatives. It should be noted that, although it is not possible to do a complete ranking considering the rules of PROMETHEE I in this example, it is clear that HC dominates all the other alternatives and, consequently, Hotel Canton is a really good hotel for the decision maker s stay in Barcelona. If the individual requires a complete ranking of the alternatives, the vertical axis of Figure 11 represents the net flow score, since it is calculated substracting the Phi axis from the Phi + axis. Therefore, the PROMETHEE II ranking is: HC, C&BZ, LC, G, HSF, NHLC, HIEBC, HB, AAH, PSGH In fact, this ranking is also obtained when the points representing the alternatives in the GAIA plane are projected to the decision stick, as it is observable in Figure 12. The management of missing values in PROMETHEE methods Núria Gens 30

32 Figure 12: Projection of the alternatives on the decision stick To complete the analysis of the alternatives, it seems reasonable to compare the profiles of HC and G. These two options should be studied because, although both rankings place HC (Hotel Canton) in the first position, it is noticeable that G (Granvia) performs really well in two of the three criteria. Score Centre Price Figure 13: Profiles of alternatives "G" (Alternative 3) and "HC" (Alternative 7) According to Figure 13, Granvia is better than Hotel Canton in two of the criteria, but much worse in Price as it is very weak on this criterion. Therefore, Hotel Canton seems the best The management of missing values in PROMETHEE methods Núria Gens 31

33 solution considering the decision maker s preferences. However, after the analysis of the alternatives, he is in a good position to select the option that suits him more. It is also important to perform a sensitivity analysis in order to complete the study of the multicriteria decision problem. D Sight includes the tool stability intervals, which shows for each criterion the interval in which the weights can be modified without affecting the ranking. When a weight is varied, the other weights are proportionally changed according to the initial ratios. The stability level considered is one, which means that the intervals stand for the weight values that let the first alternative of the ranking stay in this position, although the other hotels might have changed their positions in the ranking. However, it should be noted that the software allows its users to calculate the stability intervals in all the levels up to (n 1). It is clear that the higher the level is, the smaller the intervals are. The stability intervals obtained using D Sight are the following: Criteria Min Weight Value Max Weight Score 0,00% 20,00% 43,64% Price 20,59% 35,00% 100,00% Centre 0,00% 45,00% 72,44% Table 7: Stability intervals It can be observed that the weight values are robust, since the weights can be modified considerably without changing the first alternative of the ranking. Nevertheless, it is interesting to investigate what is the result of diminishing the weight of the criterion Price until 20%, due to the fact that it is the limit closer to the real value of the percentage. Therefore, the sensitivity analysis is completed with the Walking Weights, another tool in D Sight that allows the decision maker to modify the weights of the different criteria and see how the scores of the different alternatives and the ranking would be modified in real time. The management of missing values in PROMETHEE methods Núria Gens 32

34 Figure 14: Walking Weights In Figure 14, the result of diminishing the weight of Price until 20% is shown, as well as the PROMETHEE II ranking in the new situation. It can be noticed that the first option has changed, since 20% is outside the stability interval of the criterion Price. The first alternative in this case is G, which stands for Granvia, the other hotel that is considered as an interesting option for the decision maker. To conclude the analysis of the multicriteria decision problem, it should be noted that the stability analysis shows the accuracy of the study, as it demonstrates that Hotel Canton is the best option for the decision maker according to his preferences and also presents the hotel Granvia as an interesting option if the price of the stay loses importance. The management of missing values in PROMETHEE methods Núria Gens 33