Analysing the Compensatory Properties of the Outranking Approach PROMETHEE

Schär, Sebastian; Pohl, Erik; Geldermann, Jutta Article — Published Version Analysing the Compensatory Properties of the Outranking Approach PROMETHEE Journal of Multi‐Criteria Decision Analysis Provided in Cooperation with: John Wiley & Sons Suggested Citation: Schär, Sebastian; Pohl, Erik; Geldermann, Jutta (2025) : Analysing the Compensatory Properties of the Outranking Approach PROMETHEE, Journal of Multi‐Criteria Decision Analysis, ISSN 1099-1360, Wiley, Hoboken, NJ, Vol. 32, Iss. 2, https://doi.org/10.1002/mcda.70013 This Version is available at: https://hdl.handle.net/10419/323721 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ 1 of 21 Journal of MultiCriteria Decision Analysis, 2025; 32:e70013 https://doi.org/10.1002/mcda.70013 Journal of MultiCriteria Decision Analysis RESEARCH ARTICLE OPEN ACCESS Analysing the Compensatory Properties of the Outranking Approach PROMETHEE SebastianSchär1,2 | ErikPohl1 | JuttaGeldermann1 1Chair of Business Administration and Production Management, University of DuisburgEssen, Duisburg, Germany | 2Department of Environmental Social Sciences, Eawag: Swiss Federal Institute for Aquatic Science and Technology, Dübendorf,Switzerland Correspondence: Sebastian Schär ([email protected]) Received: 5 May 2025 | Revised: 5 May 2025 | Accepted: 12 May 2025 Funding: The authors received no specific funding for this work. Keywords: compensation| multiple criteria decision analysis| outranking| PROMETHEE ABSTRACT The PROMETHEE methods are increasingly applied in environmental and public policy decisionmaking due to their comprehensiveness and explainability. However, the literature contains differing statements regarding their compensatory properties. Compensation in multiple criteria decision aggregation procedures is commonly understood as allowing a gain in one criterion to offset a loss in another one. In certain domains, such as environmental or public policy decisionmaking, it may be undesirable, as some impacts may result in losses too severe to be counterbalanced by good performance on other criteria. Therefore, it may be necessary to limit the extent to which an aggregation procedure permits compensation or to explicitly control it as needed. Guidelines and detailed analytical tools, however, that help users and analysts to control compensation in the PROMETHEE methods remain scarce and often lack transparency. In this study, we analyse the compensatory behaviour of the PROMETHEE I and II methods and identify the key determinants for compensation in these methods. Based on these insights, we develop flow insensitivity intervals to assess the sensitivity of a given decision model towards compensatory effects and provide a set of general guidelines for controlling compensation in the PROMETHEE I and II methods for any given pair of criteria. The findings are illustrated at hand of an environmental management case study. By combining the guidelines with flow insensitivity intervals, users and analysts gain access to measures of varying granularity to evaluate and control compensation in a PROMETHEE decision model. 1 | Introduction An essential part of multiple criteria decision analysis (MCDA) is the formalised procedure to move from a decision model to a synthesis of the information that has been obtained about the different options as well as the objectives and preferences of everyone involved (Belton and Stewart2002). For this task, a rich set of different multiple criteria aggregation procedures (MCAP) has emerged from the scientific discourse. They are often differentiated into two schools of thought (Vansnick1990; Roy and Vanderpooten1996), analogous to the fundamental decision mechanisms developed by de Borda (1781) and Condorcet (1785). Frequently used approaches from the socalled American school of thought (Von Winterfeldt and Edwards 1986; Keeney 1992) are the multiple attribute value and utility theory (MAVT/MAUT) (Keeney1992; Keeney and Raiffa1993), while the Preference Ranking Organization METhod for Enrichment Evaluations (PROMETHEE) (Brans etal.1986) and the Elimination and Choice Translating Reality (ELECTRE) approach (Roy1991) are often applied outranking approaches from the French or European school of thought (Roy and Bouyssou1993). These two distinct schools are essentially based on different assumptions and axioms when establishing the set of synthesised This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2025 The Author(s). Journal of Multi-Criteria Decision Analysis published by John Wiley & Sons Ltd. 2 of 21 Journal of MultiCriteria Decision Analysis, 2025 information in the form of preference structures (see e.g., Vansnick 1986; Moulin 1988; Munda 2016; Roy 2016). This leads to a diverging interpretation of criteria weights and different aggregation properties (Munda2008; Figueira etal.2016; Brans and De Smet2016; Martel and Matarazzo2016). One aggregation property that distinguishes the various MCAPs is the possible occurrence of compensation. Compensation is commonly understood as allowing a performance gain in one objective or criterion to offset a performance loss on another one (Roy and Słowiński 2013). Due to their different axiomatic foundations, it is often argued that MCAP from the American school of thought are based on a compensatory aggregation logic, while outranking approaches generally consider this property as unwanted in the process of establishing preference relations (Vansnick1990; Roy and Słowiński 2013). Depending on the decision problem, compensatory effects could either be desired (e.g., for employee performance evaluations) or should be deliberately avoided (e.g., to ensure the concept of strong sustainability) (Cinelli et al. 2014). Especially in contexts where numerous facets of sustainability need consideration and counterbalancing effects between certain dimensions need to be avoided, or limited, the issue of compensation is of particular interest. A few selected examples comprise energy technology assessment (Diakoulaki etal.2005; Tsoutsos etal.2009; Oberschmidt etal.2010; Strantzali and Aravossis2016), production management (Hämäläinen 2004; Geldermann and Rentz 2001; Tong et al. 2022), environmental management (Kiker etal.2005; Huang etal.2011; Lienert etal.2015), and policy making (Ferretti2016; Salo and Hämäläinen2010). While there is scholarly work available that offers guidance on the choice of aggregation functions to steer compensatory effects in value theory based approaches (e.g., Langhans etal.2014; Cinelli etal.2014) provide a comprehensive body of research on the compensation degree of common MCDA methods, research that offers a detailed analysis of the compensation mechanisms of the PROMETHEE methods seems limited. Much more, we find equivocal statements regarding the compensatory behaviour of the PROMETHEE methods in the scientific discourse. In Cinelli etal.(2020, 2022), PROMETHEE I, II, TRI and V (Brans and De Smet2016; Figueira etal.2004; Brans and Mareschal1992) are classified as both not and partially compensatory. Further relevant works characterise the PROMETHEE methods as noncompensatory outranking methods (Pirlot1997; Greco etal.2021; Costa and Alves2021). In other literature it is stated that PROMETHEE methods ‘avoid fullcompensation’ (Prado etal.2012) or are either fully, partial or noncompensatory, depending on the selected method and its configuration (Bezerra etal.2021; Benoit and Rousseaux2003; Moghaddam et al. 2011; Guitouni and Martel 1998; Ishizaka and Resce2021). Only recently, Dejaegere and De Smet(2023) proposed the new PROMETHEE 𝛾 method, which can deliberately be used in totally compensatory or noncompensatory manner to circumvent unwanted compensatory behaviour of PROMETHEE I in certain situations. Additionally, they remark that ‘[…] it is not clear whether PROMETHEE is considered as noncompensatory or partially compensatory’ (Dejaegere and De Smet2023, 149). However, we are not aware of any work that discloses the preconditions for compensatory behaviour within the PROMETHEE methods and provides a comprehensive set of guidelines that allows applicants to steer it in the desired manner. To complement existing research, this article will provide the following main contributions: • The compensatory properties of the PROMETHEE methods are highlighted. In particular, the determinants for compensation in the PROMETHEE I and II methods are disclosed. • A measure to investigate the sensitivity of a given decision model that is aggregated according to PROMETHEE I or II towards compensatory effects is developed. In this way, it is possible to identify the criteria for which compensation can occur and to specify the extent to which compensatory effects are responsible for preserving a preference structure. • A set of guidelines for the design and parameterisation of a PROMETHEE model to control compensation in a finegrained resolution, ranging from full compensation to no compensation at all, is proposed. The remainder of the article is structured as follows. We first define the fundamentals of a multiple criteria decision problem and arrive at definitions for the notion of compensation (Section 2). In Section3, the PROMETHEE methods are introduced, with particular emphasis on PROMETHEE I and II. Following this, the compensatory behaviour of the PROMETHEE methods is characterised by using a differential calculus approach and showcased by a series of numerical examples (Section4). The results are discussed in Section5 and used to elaborate guidelines for applicants and analysts. Conclusions are drawn in Section6. 2 | Compensation in Multiple Criteria Aggregation Procedures Generally speaking, MCDA methods apply distinct aggregation procedures to synthesise preference information and evaluate a set of potential courses of action. In this section, the formal notation of these MCAPs is introduced and the notion of compensation is defined. 2.1 | Multiple Criteria Aggregation Procedures In a multiple criteria decision problem, a set of potential actions A ≔ { a 1 ,a 2 ,…,a i ,…,a m} is to be evaluated on a set of attributes or criteria G ≔ { g 1 ,g 2 ,…,g j ,…,g n} according to the preferences of one or more decisionmaker(s) (DM). A decision table then summarises the available alternatives, criteria and the performance of each alternative on each criterion, denoted as gj( a i) . An MCAP applies a specified mathematical procedure on the decision table to evaluate the formalised decision problem in order to produce a desired result which allows for the evaluation of alternatives in a comprehensive way (Roy 2016). 3 of 21 Depending on the desired type of result, decision problems are typically distinguished into four different types: Choice, sorting, ranking and description problematics (Roy2016). To reach the desired outcome, MCAPs establish preference structures which formalise the comparison of any pair of alternatives (ai,ax) ∈ A × A in clear or fuzzy language (Krantz etal.1971; Fishburn1999; Roubens and Vincke1985; Moretti etal.2016). A preference structure is a set of binary preference relations that describe a DM's attitudes towards a subset of ordered pairs from A×A , that is, alternatives, such that for each pair of alternatives exactly one relation holds (Moretti etal.2016). Accordingly, binary preference relations R are preference statements on pairs of A . They allow to express situations of preference or indifference, typically denoted ⟨P,I⟩ . In some cases, weak preference can also be expressed (Vincke1988). For a more extensive overview on the different types of preference structures that can be established, see for example Moretti etal.(2016). Each MCAP follows a different logic in the creation of preference structures. Approaches that maximise a value, utility or scoring function are often referred to as aggregating MCAPs (Vincke1992). They exploit an aggregated score to establish binary preference relations. In outranking methods, a preference structure is characterised by binary outranking relations, denoted S , that can or can not be transitive and complete, depending on the prevalence of incomparability between alternatives (Bouyssou 1996). The notion of incomparability goes back to the concordancediscordance principle, an essential axiomatic foundation of outranking approaches (Bouyssou and Pirlot2009; Figueira etal.2013). The concordancediscordance principle acknowledges that the existence of preference or indifference on a pair of alternatives is not always possible nor desired (Vincke1992). Therefore, an additional binary relation describing incomparability between alternatives ( J ) is introduced (Tsoukiàs and Vincke1995). In ⟨P,I,J⟩ preference structures, the relation P describes situations in which one alternative is clearly preferred over another, while the relations I and J both denote that neither alternative is preferred (Moretti etal.2016). For the reflexive and symmetric indifference relation I , the lack of preference is due to their equivalent valuation. The symmetric and irreflexive incomparability relation J , however, refers to situations in which a lack of information or conflicting evaluations does not allow for the expression of preference and ¬( a i Pa x) ,¬ ( a x Pa i) , and ¬( a i Ia x) simultaneously holds (Moretti etal.2016). The resulting preference structure can then be defined as follows Dejaegere and De Smet(2023): The outranking relation aiSax denotes the assertion that ‘ai is at least as good as ax’ or ‘ai outranks ax’. That means ai can be considered at least as good as ax since there is sufficient evidence supporting such a statement and no contradicting evidence, given the available information (Bouyssou and Vansnick1986). 2.2 | Compensatory and NonCompensatory Multiple Criteria Aggregation Procedures The notion of compensation in MCAPs has been a subject of extensive study and the academic literature presents a set of definitions. Most commonly, the compensatory character of an MCAP is connected to its axiomatic foundations. Aggregating MCAPs interpret weights as tradeoffs between attributes and typically aggregate the performance of alternatives on these attributes in an additive manner, for example, via simple additive weighting (SAW) in the form ∑n j=1 wj⋅gj � ai � . Such MCAPs are considered inherently compensatory for two analogous reasons. First, the additive aggregation logic implies that a low performance on one attribute can be offset by a relatively good performance on another attribute with regards to the aggregated performance of an alternative (Langhans etal.2014). Furthermore, the weights that are attached to the different objectives are interpreted as tradeoffs between objectives (Moulin1988). Thus, they are equivalent to the objectives' substitution rates and a fully compensatory aggregation logic is inherently necessary (Munda2016). An illustrative example which derives this equivalence for the case of SAW is provided by (Munda2008, chapter4). The notion of compensation in this case can be defined as the property that the preference relations remain unaffected when a loss on one objective is accompanied by comparable gain on another one, adjusted for the weights (i.e., substitution rate) (Roy and Mousseau1996; Haag etal.2019). In outranking methods, the MCAP is not concerned with maximising an underlying value or utility function. Instead, pairwise performance assessments of alternatives are conducted on the set of criteria to establish preference structures, where the weights only represent the relative importance of each criterion (Figueira etal.2016). This leads to two fundamental differences compared to aggregating MCAPs: 1. The concordancediscordance principle confines compensatory effects in the construction of preference structures to the intracriterial comparisons of alternatives and only up to a specified threshold. Within the pairwise comparisons, small performance disadvantages of an alternative that do not surpass this threshold are completely disregarded. The threshold value defines the extent of what can be considered a ‘small’ difference in performance to distinguish between ‘sufficient’ and ‘insufficient’ evidence for preference (Pirlot1997). 2. The concordancediscordance principle is generally not concerned with performance substitution between the different criteria. The performance differences that exceed the specified thresholds in intracriterial comparisons are not able to affect the evaluation on another criterion without turning relative disadvantages into advantages and vice versa. It is thus irrelevant for the construction of preference relations by how much the performances of alternatives a iPax⇔aiSax∧¬ ( axSai ) aiIax⇔aiSax∧axSai ai Ja x ⇔¬ ( a i Sa x) ∧¬ ( a x Sa i) 4 of 21 Journal of MultiCriteria Decision Analysis, 2025 with regard to a given criterion differ if the threshold is exceeded, and any changes in performance across criteria will not be able to affect the preference structure (Figueira etal.2010). Since the preference structures in outranking methods are only affected by whether an alternative outranks another one and vice versa, they are generally classified as noncompensatory MCAPs (Fishburn1976; Bouyssou and Vansnick1986). At the same time, the concordancediscordance principle does not strictly imply that outranking relations are noncompensatory, given that small enough disadvantages are not considered in the construction of them (Dejaegere and De Smet 2023). Furthermore, not all outranking methods fully adopt the concordancediscordance principle. The PROMETHEE I and II methods are examples of outranking methods that are not based on the concordancediscordance principle, which leads to different compensatory properties. We characterise their compensatory behaviour after arriving at a set of working definitions. 2.3 | Definitions So far, no widely accepted and precise definition of compensation in MCAP could be identified in the literature. For the purpose of this work, compensation in MCAP is defined as the possibility for intercriterial performance substitution (Roy and Mousseau1996). Intercriterial performance substitution refers to the balancing or offsetting of a disadvantage on a criterion, in terms of the resulting preference structures, by a sufficient advantage on another criterion. Therefore, an MCAP is considered compensatory if a preference relation between two alternatives is altered by a change in performance on one criterion and can be reinstated by adjusting the alternatives' performance on another criterion (Roy and Słowiński2013). Definition 1. (Compensatory MCAP). Consider a decision problem with multiple criteria G ≔ {g1, … ,gn} and alternatives A ≔ { a 1 ,…,a m} , where gj( a i) denotes the performance of alternative ai on criterion gj . Let ai and ax then be two distinct alternatives from A with aiRax derived from the MCAP, where R is the binary preference relation obtained from the MCAP. We say that the MCAP is compensatory, if there are two criteria gj and gk , such that for a change of gj( a i) which alters the preference relation aiRax , we can find a corresponding change in gk( a i) that reinstates the initial preference relation aiRax . Definition 2. (Noncompensatory MCAP). Consider a decision problem with multiple criteria G ≔ {g1, … ,gn} and alternatives A ≔ { a 1 ,…,a m} , where gj( a i) denotes the performance of alternative ai on criterion gj . Let ai and ax then be two distinct alternatives from A with aiRax derived from the MCAP, where R is the binary preference relation obtained from the MCAP. We say that the MCAP is noncompensatory, for any two criteria gj and gk , and any change of gj( a i) which alters the preference relation aiRax , if there is no corresponding change in gk(ai) that reinstates the initial preference relation aiRax . A change in the performance of an alternative ai on a criterion gj is henceforth denoted Δ g j( a i) . It is considered a ‘gain’ if it increases the performance difference to the next best alternative on the same criterion or reduces the gap to an alternative that performs better on this criterion. In turn, a change in performance that decreases the performance difference between alternative ai and a lower performing alternative on the same criterion is considered a ‘loss’. These definitions also apply to other initial preference structures, for example, if ai and ax are considered indifferent. Compensation can occur if any such performance changes alters a given preference relation between ai and any other alternative for a sufficiently large value of Δ g j( a i) . In addition, we define a total ‘noncompensatory MCAP’ since the given definition of noncompensatory MCAPs does not cover cases in which a preference relation can not be destroyed by the stated performance substitutions due to the nature of the aggregation process (Fishburn1976; Roy and Mousseau1996). Definition 3. (Totally noncompensatory MCAP). An MCAP is considered totally noncompensatory if the preference situation between two distinct pairs of alternatives (ai,ax) and ( a i ′,a x ′ ) is considered on an ordinal scale on each criterion gj (Roy and Mousseau1996). That is, the preference relation is independent from the difference in performance and there is no possibility of compensation relative to any criterion, so that holds for all alternatives a′ 𝚤 and a′ x that are deduced from ( ai , ax ) by changing their performance on any criterion without altering the ordinal order of alternatives on a criterion. Thus, in a totally noncompensatory MCAP there is no possibility for compensatory effects between different criteria as long as the preferential profile of alternatives is kept (Bouyssou1986). 3 | The PROMETHEE I and II Methods Given these definitions, outranking approaches following the concordancediscordance principle generally can be considered noncompensatory. However, PROMETHEE I and II are outranking methods that do not strictly follow the concordancediscordance principle. Instead, they establish a ranking of alternatives based on a set of valued outranking relations (Brans etal.1986). This means that a numeric value, describing the intensity of a preference relation, is attached to a pair of alternatives (Bouyssou and Pirlot2009). In PROMETHEE, the valued outranking relations are used to establish scores or outranking flows, which represent the performance of an alternative compared to the other alternatives (Brans and De Smet2016). To determine the outranking flows, the differences between performances of each pair of alternatives from A are computed for all criteria in the first step: (1) [ a i Ra x =a � 𝚤 Ra � x ∧a x Ra i =a � x Ra � 𝚤 ] (2) d j (a i ,a x ) = g j (a i ) − g j (a x ), ∀ a i ,a x∈ A j=1, …,n 5 of 21 By means of a preference function  it is then possible to calculate the individual preferences of the DM: A preference function is a monotonic function of dj( a i ,a x) and maps the intracriterial preferences of the DM, normalised to the interval [0, 1], while Equation(4) holds, so that the preference function for criteria to be minimised is as in Equation(5). In general, the shape and definition of a preference function can be selected by the DM. To reduce the cognitive load of modelling, a set of six nondecreasing preference functions has been established which are considered suitable for most contexts (Brans and De Smet2016). These functions and their parameters are depicted in TableA1 in the AppendixA. Depending on the shape of the preference function, additional threshold parameter values may be elicited from the DM. The indifference threshold qj delimits situations in which the difference in performance between two alternatives is too small to allow any statement of preference for either. Parameter pj models the threshold for a situation of strict preference. The inflection point 𝜎j of the Gaussian criterion (type VI) allows to model the DM's sensitivity towards performance differences in a nonlinear manner, as depicted in Table1. A lower value results in greater sensitivity to small differences in performance (single dotdash line ⋅− ), while a relatively high value means that preference sensitivity for large differences in performance between two alternatives is higher (double dotdash line ⋅⋅− ). The DMs intracriterial preferences are then aggregated to a global preference index for all pairs of alternatives as in Equation(6): A weighting coefficient wj≥0 denotes the relative importance of each criterion gj . The preference indexes can then be used to compute the outranking flow scores of an alternative: The positive flow score denoted 𝜙+ aggregates the evidence from all pairwise comparisons that reinforce a situation of preference for an alternative over all the other alternatives under consideration; the negative flow score 𝜙− in turn sums up all evidence speaking against such a statement (Linkov etal.2021, 9). In PROMETHEE I, ⟨P,I,J⟩ preference structures are established based on the positive and negative flow scores. Thus, the PROMETHEE I ranking of alternatives for a given decision problem is the intersection of the rankings obtained from the positive and negative flow scores and forms a partially ordered set (Brans and De Smet2016; Dejaegere and De Smet2023). The PROMETHEE II method yields a total ranking of alternatives and ⟨P,I⟩ preference structures by means of the PROMETHEE net flow: where is the single criterion flow of an alternative with respect to criterion gj∈G . (3)  j (a i ,a x ) =j (d j (a i ,a x )), ∀ a i ,a x∈ A j =1, …, n (4) j( a i ,a x) >0⇒ j( a x ,a i) = 0 (5) j( a i ,a x) =F j −d j( a i ,a x) (6) 𝜋( ai,ax ) = n ∑ j =1 wj⋅j ( ai,ax ) ∀ai,ax∈ A (7) 𝜙 + ( ai ) = 1 m−1 ∑ ax∈A 𝜋 ( ai,ax ) ∀ai∈ A 𝜙 − ( ai ) =1 m−1 ∑ ax∈A 𝜋 ( ax,ai ) ∀ai∈ A (8) a iPIax⇔ ⎧ ⎪ ⎨ ⎪ ⎩ 𝜙 +� ai � ≥𝜙 +� ax � ∧𝜙 −� ai � <𝜙 −� ax � or 𝜙+�ai�>𝜙+�ax�∧𝜙−�ai�≤𝜙−�ax � aiIIax⇔𝜙+�ai�=𝜙+�ax�∧𝜙−�ai�=𝜙−�ax� a iJIax⇔⎧ ⎪ ⎨ ⎪ ⎩ 𝜙+�ai�<𝜙+�ax�∧𝜙−�ai�<𝜙−�ax � or 𝜙+ � a i� >𝜙+ � a x� ∧𝜙− � a i� >𝜙− � a x� (9) 𝜙 net ( ai ) =𝜙+ ( ai ) −𝜙− ( ai ) = n ∑ j = 1 𝜙j ( ai ) ⋅w j (10) 𝜙 j ( ai ) = 1 m−1 ∑ a x ∈A [ j ( ai,ax ) −j(ax,ai) ] TABLE 1 | Decision table with three alternatives and four criteria used in the stereotypical cases. Criterion Performance Preference information Name A1 A2 A3 wj Polarity Function g1 g1( A 1) g1( A 2) g1( A 3) w1 Max 1 g2 g2(A1) g2(A2) g2(A3) w2 Max 2 g3 g3(A1) g3(A2) g3(A3) w3 Max 3 g4 g4( A 1) g4( A 2) g4( A 3) w4 Max 4 6 of 21 Journal of MultiCriteria Decision Analysis, 2025 The total ranking of all alternatives is then given by ordering them according to their net flow scores, while indifference may occur: The PROMETHEE method family also provides methods for sorting (Figueira etal.2005; De Smet2019) and choice problems (Brans and Mareschal1992). For a complete overview of PROMETHEE methods (Brans and De Smet2016; Belton and Stewart2002), or the encompassing survey paper on their development, extensions and future directions (Brans 2015), we kindly refer the reader to the respective literature. 4 | Compensation in the PROMETHEE Methods Since the PROMETHEE methods do not strictly follow the concordancediscordance principle, they exhibit a varying compensation behaviour that can be either noncompensatory or explicitly allow for compensatory effects between criteria. Specifically, the valued outranking relations that the PROMETHEE methods produce allow for compensation based on the elicitation or selection of preference functions, preference parameters and criteria weights. The compensation behaviour of these methods thus depends on the choices made during the elicitation and modelling of the decision problem. The information that is required to construct these valued outranking relations, namely the outranking flows, also allows one to capture compensatory effects of the PROMETHEE methods and characterise determinants for compensation. This will be the focus of this section. 4.1 | Characterising the Compensation Behaviour of the PROMETHEE II Method The compensatory behaviour of the PROMETHEE methods is connected to the valued outranking relations that are produced to establish a ranking of alternatives. In order to characterise the compensation behaviour of the PROMETHEE methods, selected stereotypical cases are analysed. We limit the analysis of the compensatory behaviour to PROMETHEE I and II since they are the main methods of this family and are intended to be used for ranking problematics. The stereotypical cases comprise a set of three decision alternatives that are to be evaluated against a set of four cardinal criteria that are to be maximised, so that: • A = { A 1 ,A 2 ,A 3} , • G = { g 1 ,g 2 ,g 3 ,g 4} , and • gj∈[0,100] . The alternatives' performances, the chosen preference function for each criterion and the corresponding criteria weights are to be varied according to the studied case, leading to the generic decision table presented in Table1, which summarises the structure of the studied decision problems and the nomenclature adopted in this study. This forms the starting point for the analysis of the PROMETHEE methods' compensatory properties. Subsequently, a change in the performance of a selected alternative with regards to a specific criterion is introduced (formally denoted Δgj(Ai) ). In the cases presented here, this change represents a performance loss (or gain) whose effect on the outranking flows is to be compensated for by means of a performance increase (decrease) of the same alternative on another criterion (denoted Δgk(Ai) ). It is also analysed how the selection of preference functions, threshold parameters and criteria weights influence the compensatory behaviour. 4.1.1 | The Equivalence of PROMETHEE II to Additively Aggregating MCAPs Certain instances of additively aggregating MCAPs (e.g., SAW or MAVT/MAUT with linear value functions and an additive model) are equivalent to PROMETHEE II with type III preference functions and sufficiently large threshold parameters, as shown by Geldermann and Schöbel (2011) or Mareschal (2015). For these instances, both methods also exhibit comparable compensation behaviour, where a loss in performance on one criterion can be fully offset by a corresponding gain on another criterion. It may therefore be argued that the PROMETHEE II method is fundamentally compensatory according to the definition adopted in this work (see Section2.3). An exemplary model of PROMETHEE that corresponds to an MAVT model with linear value functions and an additive aggregation function is stated in Table 2. The criteria in the PROMETHEE model are modelled via the widely used linear preference function (type III) and the preference threshold is set to the performance difference between the best and worst performing alternative for each criterion ( p j=g max j −g min j ). Using the piecewise linear preference functions of type V with pertinent parameterisation is likewise possible in this example (Geldermann and Schöbel2011). (11) a iP II ax⇔𝜙 net( ai ) >𝜙 net( ax ) a i IIIa x ⇔𝜙net ( a i) =𝜙net ( a x) TABLE 2 | Decision table with three alternatives, four criteria and preference information to be used for the application of the PROMETHEE methods. Criterion Performance Preference information Name A1 A2 A3 wj Polarity Function pj g1 100 10 15 0.25 Max Type III 90 g2 090 70 0.25 Max Type III 90 g3 13 21 66 0.25 Max Type III 53 g4 78 100 42 0.25 Max Type III 58 7 of 21 The alternatives' global values and 𝜙net scores after aggregation according to MAVT and PROMETHEE II are shown in Figure1. They are indicated by the square marks connected by a solid line. According to both methods, the total ranking of alternatives is A2≻A3≻A1 , while ≻ denotes a situation of preference. Furthermore, Figure 1 displays the equivalent compensation behaviour of both methods. One can identify a performance gain of A3 on the first criterion that dissolves the preference relation between A2 and A3 and yields indifference between these alternatives for both methods. In this case, such a Δg1(A3) takes the value of 28.6. The global values, as well as the net flows of A2 and A3 , denoted V′ and 𝜙′ net are equal after the performance increase of A3 on criterion g1 . This is visualised by the two horizontally aligned dotted lines. Given equal criteria weights, a corresponding performance loss on the second criterion, Δ g 2( A 3) =− 28.6 , reestablishes the original preference relations as well as the exact scores for V(A3) and 𝜙net(A3) . This behaviour can be generalised, so that the extent of required performance gain on criterion g2 to compensate for a loss on g1 is It depends on the ratio of the criteria weights, as well as the preference thresholds of both of the involved criteria to account for different units of measurement. Furthermore, the condition that the linear preference function (type III) is used and parameterised so that it is equally shaped to a linear value function must hold: We do not recommend to design a type III preference function according to Equation (13) in practical application and further discussing it in Section5.3. The generalised proof for Equation(12) can be found in the AppendixA. Despite the adjusted performances, the 𝜙net scores for all other alternatives also remain constant after applying performance changes of Δ g 1( A 3) = 28.6 and Δ g 2( A 3) =− 28.6 , as the orange bar in the three panels on the right hand side of Figure2 shows. Given the methods pairwisecomparison logic to produce a set of valued outranking relations this is not necessarily expected. The compensating effect that is responsible for this circumstance becomes visible upon investigation of corresponding single criterion flows. The bottom right panel shows the single criterion flows for Alternative 3. Compared to the single criterion flows before the performance adjustments, which is indicated by the shaded area, the relative weakness of A3 regarding criterion g1 is reduced while the contrary happens for the performance on criterion g2 . In a noncompensatory setting, this change in single criterion flows would not happen. Logically, the single criterion flows of A1 and A2 are also changing on the two performanceadjusted criteria due to the PROMETHEE calculation procedure. This means that across all three alternatives, only the single criterion flows regarding the first two criteria are changing. All other single criterion flows remain constant. The corresponding effect of performance substitution for the MAVT model with additive aggregation function and other value aggregation methods is extensively studied by Langhans etal.(2014). 4.1.2 | NonCompensatory Modelling in PROMETHEE II However, there are also modelling variants of PROMETHEE II where compensatory effects are largely or even completely avoided. If compensation needs to be avoided, this can be done quite intuitively by changing the preference function of the affected criteria. The results of switching to a type I preference function for criterion g1 and g2 for the stated decision problem are given in Table3. Following the definition of this preference function, the performance alterations Δg1(A3) = 28.6 and Δg2(A3) =− 28.6 do neither affect the PROMETHEE net flows nor the single criterion flows. Thus, compensatory effects can not occur when confining to this preference function type. Due to the calculation logic of the outranking flows and the piecewise definition of the remaining preference function types, (12) Δ g2 ( Ai ) = p 2⋅ w 1 p1 ⋅ w2 ⋅Δg1 ( Ai ) (13)  j � dj � = ⎧ ⎪ ⎨ ⎪ ⎩ 0dj≤0 dj gmax j −gmin j 0≤gmax j−g min j FIGURE 1 | Global values and PROMETHEE net flows for decision problem in Table2 and effects of changing the performance of A3 on g1 . 8 of 21 Journal of MultiCriteria Decision Analysis, 2025 FIGURE 2 | PROMETHEE single criterion flows and net flows for the decision problem in Table2 (panels on the left half). The three panels on the right side highlight the compensatory effects when decreasing the performance of A3 on g1 and compensating for this loss by a gain on criterion g2 . Although the PROMETHEE net flows of A3 remain similar since the loss is compensated, the compensation becomes observable in the changed single criterion flows for the two affected criteria ( 𝜙1 and 𝜙2 in this instance). TABLE 3 | PROMETHEE single criterion flows and net flows for the decision problem in Table2 before and after altering the performance of A3 on criterion g1 and g2 by −28.6 and 28.6 points. Criteria performance Alternative PROMETHEE flows 𝝓1(ai) 𝝓2(ai) 𝝓3(ai) 𝝓4(ai) 𝝓net(ai) g1( A 3) = 70; g 2( A 3) = 15 A1 0.25 −0.25 −0.144 0.03 −0.114 A2 −0.25 0.25 −0.087 0.172 0.085 A3 0 0 0.231 −0.203 0.029 g1( A 3) = 43.6; g2( A 3) = 41.4 A1 0.25 −0.25 −0.144 0.03 −0.114 A2 −0.25 0.25 −0.087 0.172 0.085 A3 0 0 0.231 −0.203 0.029 Note: Criteria g1 and g2 are modelled via the usual preference function of type I. The linear preference function and parameterisation is kept for g3 and g4 . The net flows (bold) of an alternative are the sum of the single criterion flows. 15 of 21 possible compensatory effects and may not adequately map the preferences of the DM. In addition, there is even a risk to allow for full compensation on criteria where the alternatives perform relatively similar. The following example will highlight the benefits and drawbacks of this approach to control compensation. 5.2.2 | Compensatory Properties of the Type I Preference Function As noted above, changing to a type I preference function does not necessarily eliminate the issue of compensation in a PROMETHEE model. It does, however, substantially reduce the sensitivity of it towards compensatory effects. Consider the same decision problem as in Table8 with the only modifications being (i) criteria g1 and g9 are both of type I and (ii) all criteria weights are equal. These changes lead to larger flow insensitivity intervals compared to the initial model, as Table9 shows. An analogous pattern could be observed if other criteria than g1 or g9 were changed to type I. At the same time, changing to a type I preference function can also create unwanted instances of full compensation. This happens when the alternatives' performances markedly change beyond the flow insensitivity intervals. Figure6 displays the single criterion and net flows when the performance of A4 on criteria g1 and g9 changes beyond the flow insensitivity intervals displayed in Table9. In a first step, g9( A 4) is increased by 0.2 units. G9 is an environmental criterion and to be minimised, so that this change reflects a performance loss. The striped bars indicate the new flows after this change is applied. The performance loss of A4 on the environmental criterion g9 leads to a rank change so that A1 becomes the preferred alternative. In a second step, the performance score of A4 on the technical criterion g1 is reduced by 31.4 units. Since g1 is also to be minimised, it reflects a performance gain. The effects that this change has are visualised by the dotted bars in the figure. The introduced performance gain on g1 fully compensates the loss in flows on g9 and reinstates the initial net flows and rank order. The single criterion flows on all other criteria are not affected by the performance changes and thus not displayed. This example highlights that the previously introduced guidelines offer starting points for considering the issue of compensation in the PROMETHEE modelling process. In addition, the flow insensitivity intervals offer a more detailed means to analyse the occurrence and extent of compensation in a targeted manner. They are particularly potent if additional information on the uncertainty in the performance scores is known. In this case, preference thresholds can be modified considering this TABLE 9 | Flow insensitivity intervals for an environmental management decision problem with adapted preference functions. Criterion g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11 Function type IIV IV III III IV IV V I IV IV Δ g j( A 1) [−937.4,9.8) {} [−1, 0] {} {} {} {} [−70,4.2) (−0.02,∞) {} {} Δgj(A2) (−9.8,49.6) {} [−1, 0] (−0.12, ∞) (−280.36, ∞) {} {} (−2.2,5) (−0.22,0.02) {} {} Δ g j( A3 ) (−49.6,31.4) {} [0, 2] {} {} {} {} (−1.2,2.2) (−0.2,0.22) {} {} Δ g j( A4 ) (−31.4,∞) {} [−1, 0] [−1.34,0.06) {} {} {} (−4.2,1.2) [−3.47,0.2) {} {} Note: Boldfaced intervals indicate changes. FIGURE 6 | PROMETHEE single criterion flows and net flows for the decision problem in Table9. Criteria g1 and g9 modelled via type I preference function. Initial net flows are indicated by solid bars. Striped bars (performance of A4 on g9 changed beyond insensitivity boundaries) indicate the change in net flows and ranking between A1 and A4 . Dotted bars (performance of A4 on g1 changed in opposite polarity) showcase full compensation between both criteria. 16 of 21 Journal of MultiCriteria Decision Analysis, 2025 uncertainty to circumvent unwanted compensatory effects in a targeted and granular manner. 5.3 | Discussion The previous analysis is based on a number of deliberate assumptions regarding the definition of compensation and the nuances in which it can occur. We now discuss these assumptions and the practical implications that can be drawn from them. For the purpose of this work, the general notion of compensation in an MCAP has been defined as the property that preference relations, which are dissolved by changes in an alternative's performance, can be reinstated by a performance change on any other criterion for the same alternative (see Section2.3). A definition analogous to other ranking approaches, such as MAUT/MAVT, could refer to the net flows instead of the ranking of alternatives derived from comparing their net flows. Adapted to the PROMETHEE methods, compensation would then be defined as the possibility of net flow substitution between criteria. According to this revised definition, an instance of PROMETHEE is fully compensatory if there exists a pair of performance score changes for a given alternative that maintains its net flow. However, such a definition may not be a reasonable choice in light of the method's different axiomatic foundations compared to the MAUT/MAVT (see Section2.1). Additionally, the actual net flows are typically not used when interpreting a ranking that has been conducted according to PROMETHEE II, as often done for value theorybased approaches (Coquelet etal.2024). We therefore argue that the broad definition of a compensating MCAP adopted in this work, based on the fundamental ability of altering or preserving preference structures (Section2.3) is more reasonable.1 This fundamental definition of compensation can then be operationalised for the PROMETHEE methods by means of the single criterion flows. It allows for nuance between partial and full compensation based on the ability of single criterion flow substitution and is of higher use for practical application of PROMETHEE I and II. In this regard, we presented a fully compensatory case of PROMETHEE II in Section4.1.1. It is based on a set of artificial assumptions that appear not very realistic for practical application. We especially do not recommend setting the preference threshold of a type III preference functions in such a way since we believe that it contradicts the conceptual foundations of the PROMETHEE methods. We only use it here to showcase that there can be instances of compensation in the PROMETHEE methods. More precisely, this case shows that PROMETHEE II can be fully compensatory, according to how we define the notion of compensation in MCAP. It provides contradicting evidence to some statements in the literature on the compensatory characteristics of the PROMETHEE methods. The theoretical insights gained from this analysis lead to several practical considerations, especially in managing compensatory effects within PROMETHEE models and facilitating communication with DMs. As highlighted in the case study (Section5.2), the type I function can be an approach to circumvent compensation between criteria. However, it requires careful attention since there may be performance changes that introduce full compensation into a relationship of two criteria (see Section5.2.2). Resorting to a type I preference function also comes at the expense of being unable to model the DMs preferences in finer resolution, which is a distinct feature of the PROMETHEE methods compared to other outranking methods. The type I criterion is often criticised because it does not take into account imperfect information, ambiguity or other factors that might cause a DM to hesitate in expressing a clear preference for one action over another (Brans etal.1986; Roy etal.2014). In turn, when the DM requests another preference modelling, for example, via the type III preference function, but simultaneously desires no compensation for this criterion, careful communication and adjustments are required. To reconcile their preference for no compensation, it would be necessary to switch to other functions and introduce preference or indifference thresholds that model this purpose. This requires a dialogue with the DM and a casespecific and deliberate tradeoff between modelling choices. 6 | Conclusion In this work, we conducted an indepth analysis of the compensatory properties of the PROMETHEE outranking methods. Our focus has been on characterising the determinants of compensation within PROMETHEE I and II. Compensatory effects can preserve a preference structure when performance on multiple objectives changes at the same time. Since this behaviour may be undesired for certain objectives, for example, to enforce the principle of strong sustainability, a preservation of the ranking due to compensatory mechanisms may not reflect the DMs actual preferences. In particular, we have revealed how the choice of preference functions and elicitation of their parameters can allow for instances of full or partial compensation regarding the derived preference structure. Ultimately, there are instances where PROMETHEE II and also PROMETHEE I can be compensatory for any type of preference function. The findings contribute to the existing research by disclosing determinants for compensation in PROMETHEE I and II and offering mechanisms of different granularity to capture and control compensatory effects in these methods. We have introduced a compensation sensitivity analysis as a novel tool to investigate the sensitivity of a PROMETHEE decision model towards compensatory effects and highlighted different approaches to control compensation. Flow insensitivity intervals help decisionmakers (DMs) and analysts to identify components of a decision model that may trigger compensatory effects and conduct appropriate changes. Specifically, they allow determining the changes in performance scores that will affect the outranking flows and thus can compensate for changes on other criteria. This information can be utilised to control compensatory effects in a finegrained resolution by means of adjusting the preference functions or the threshold parameters. An environmental management case study demonstrates that controlling compensation in a PROMETHEE model can have a substantial impact on the final ranking. By deliberately controlling for compensation, it is possible to avoid unintended shifts in the ranking of alternatives that might happen due to compensatory effects between a particular pair of criteria. Controlling 17 of 21 for compensation is enabled by referring to the set of general guidelines and by using the results of the compensation sensitivity analysis developed in this study. It allows one to take the aspiration of a noncompensatory decision model into account in the design and parameterisation of a PROMETHEE model. The question of how to deal with compensatory properties in MCAP and the recognition of incomparability has contributed markedly to the emergence of some outranking methods. As highlighted by Dejaegere and De Smet (2023), compensatory properties could be linked to the existence of incomparability. In further research, a detailed analysis of the links between compensation and (in)comparability in PROMETHEE I, II could provide interesting insights. In particular, the investigation of whether and when compensation in PROMETHEE II leads to incomparability in PROMETHEE I and comparisons with the newly proposed PROMETHEE 𝛾 method. Another central characteristic of outranking procedures is the logic of pairwise comparisons and the muchdiscussed rank reversal phenomenon, which is strongly related to it. The relations between rank reversal and compensation, however, are not yet disclosed. Future research could also explore further applications of the compensation sensitivity analysis across different problem contexts and settings. An analytical study to explore the characteristics of a decision problem, in terms of the number of alternatives and criteria, could be an interesting point of departure. In addition, an investigation into what extent and how the presented findings on the compensation behaviour of the PROMETHEE methods can systematically inform the preference elicitation process also poses an intriguing avenue of research. Acknowledgements The authors are grateful to David Strzalka for his Master's thesis at the University of DuisburgEssen, which provided additional insights into the fully compensatory case presented in this work. We also thank our student assistant, Matthias Gerulat, who helped us with the formatting of the manuscript and the design of figures. We kindly acknowledge support by the Open Access Publication Fund of the University of DuisburgEssen. Open Access funding enabled and organized by Projekt DEAL. Conflicts of Interest The authors declare no conflicts of interest. Data Availability Statement The data that support the findings of this study are available in the Supporting Information of this article. Endnotes 1 Even though, putting increased attention on the actual flow scores that PROMETHEE yields could also yield important insights in the analysis of results (Dejaegere etal.2022; Dejaegere and De Smet2023). References Belton, V., and T. Stewart. 2002. Multiple Criteria Decision Analysis: An Integrated Approach. Springer Science and Business Media. https:// doi. org/ 10. 1007/ 9781461514954. Benoit, V., and P. Rousseaux. 2003. “Aid for Aggregating the Impacts in Life Cycle Assessment.” International Journal of Life Cycle Assessment 8: 74–82. https:// doi. org/ 10. 1007/ BF029 78430 . Bezerra, P. R. S., F. 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The PROMETHEE net flow of an alternative is determined according to Equations(9) and (10). Any value of Δgj(ai) ≠ 0 then alters the PROMETHEE net flow of an Alternative by Δ𝜙net(ai) , while The PROMETHEE net flow of an alternative after the performance on two criteria, criterion gj and any other criterion gk , has been altered is denoted 𝜙net( a i)′ . The performance change that is required on any other criterion gk to reinstate the original PROMETHEE net flow and thus verify that (A1) Δ gk ( ai ) =− p k⋅ w j p j ⋅w k ⋅Δgj ( ai ) ��� Δ𝜙net�ai�� �� = � ��� ������ 𝜙net�ai�− ⎡ ⎢ ⎢⎢⎢⎣ 1 m−1 n � j=1� ax∈A ⎡ ⎢ ⎢⎢⎢⎣ Δj�ai,ax�−Δj�ax,ai� ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ g j( a i) =g j( a i) +Δg j( a i) ⎤ ⎥ ⎥⎥⎥⎦ ⋅wj ⎤ ⎥ ⎥⎥⎥⎦ ���������� 𝜙 net ( ai ) � ! =𝜙net ( ai ) +Δ𝜙net ( ai ) ⏟⏞⏞⏟⏞⏞⏟ =0 20 of 21 Journal of MultiCriteria Decision Analysis, 2025 is then given by Under the condition that Δ 𝜙 net( a i) = 0 , we then have that 𝜙 net (ai) � =𝜙+(ai) � −𝜙−(ai) � =1 m−1∑ ax∈A, i≠x 𝜋(ai,ax)−1 m−1∑ ax∈A, i≠x 𝜋(ax,ai) =1 m−1∑ ax∈A, i≠x n ∑ j=1 j(dj(ai,ax))⋅wj−1 m−1∑ ax∈A, i≠x n ∑ j=1 j(dj(ax,ai))⋅wj =1 m−1 ⋅∑ ax∈A[gj(ai)+Δgj(ai)−gj(ax) pj ⋅wj+gk(ai)+Δgk(ai)−gk(ax) pk ⋅wk+…gn(ai)−gn(ax) pn ⋅wn ] − 1 m−1 ⋅ ∑ ax ∈ A[ gj ( ax ) −gj ( ai ) +Δgj ( ai ) pj ⋅wj+gk ( ax ) −gk ( ai ) +Δgk ( ai ) pk ⋅wk+…gn ( ax ) −gn ( ai ) pn ⋅wn ]. Δ𝜙 net( ai ) =𝜙 net( ai ) −𝜙 net( ai )� = 1 m−1 ⋅[wj⋅(gj(ai)−gj(ax)) pj +wk⋅(gk(ai)−gk(ax)) pk −wj⋅(gj(ai)+Δgj(ai)−gj(ax)) pj +wk⋅(gk(ai)+Δgk(ai)−gk(ax)) pk ] =pk⋅wj(gj(ai)−gj(ax)−gj(ai)+gj(ax)+Δgj(ai)) −pj⋅wk(gk(ax)−gk(ai)−gk(ax)+gk(ai)+Δgk(ai)) =pk⋅wi⋅Δgj(ai)−pj⋅wk⋅Δgk(ai) =Δgk ( ai ) =−pk⋅wj p j ⋅w k ⋅Δgj ( ai ) 21 of 21 TABLE A1 | Overview on the six generalised preference functions in the PROMETHEE methods (Brans and De Smet2016). Criterion Graph Definition Parameter Type I: Usual criterion  j ( dj ) = { 0dj≤ 0 1d j > 0 — Type II: Usual criterion with indifference area  j ( dj ) = { 0dj≤q j 1d j >q j qj Type III: Criterion with linear preference  j�dj�= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0dj≤0 dj pj 0≤dj≤p j 1dj>pj pj Type IV: Level criterion  j�dj�= ⎧ ⎪ ⎨ ⎪ ⎩ 0d j≤ q j 1 2qj<dj≤p j 1d j >p j pj,qj Type V: Criterion with linear preference and indifference area  j�dj�= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0dj≤qj dj−qj pj−qj qj<dj≤p j 1dj>pj pj,qj Type VI: Gaussian criterion  j � dj � = ⎧ ⎪ ⎨ ⎪ ⎩ 0dj≤ 0 1−e− d2 j 2𝜎2 jdj> 0 𝜎j