multiple criteria decision support...the goals of multiple criteria decision analysis and multiple...
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Multiple Criteria Decision Support
Salvatore Corrente, José Rui Figueira, Salvatore Greco, andRoman Słowiński
ContentsIntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2An Introduction to MCDA: Notation, Problematics, and Main Approaches . . . . . . . . . . . . . . . . . . . 3
Multiple Attribute Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Outranking Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Decision Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Interaction Between Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Robust Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Robust Ordinal Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Stochastic Multicriteria Acceptability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Recent Developments and MCDA Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
S. Corrente (*)Department of Economics and Business, University of Catania, Catania, Italye-mail: [email protected]
J. R. FigueiraCEG-IST, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugale-mail: [email protected]
S. GrecoDepartment of Economics and Business, University of Catania, Catania, Italy
Portsmouth Business School, Centre of Operations Research and Logistics (CORL), University ofPortsmouth, Portsmouth, UKe-mail: [email protected]
R. SłowińskiInstitute of Computing Science, Poznań University of Technology, Poznań, Poland
Systems Research Institute, Polish Academy of Science, Warsaw, Polande-mail: [email protected]
© Springer Nature Switzerland AG 2020D. M. Kilgour, C. Eden (eds.), Handbook of Group Decision and Negotiation,https://doi.org/10.1007/978-3-030-12051-1_33-1
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AbstractMultiple criteria decision aiding (MCDA) methodologies aim at supportingcomplex decisions when many conflicting points of view have to be considered.In this chapter, after introducing the main principles, we present the basicapproaches and methodologies of MCDA, taking into account the most recentcontributions in the domain.
Introduction
What do the terms multiple criteria decision analysis and multiple criteria decisionaiding mean1? In the following, without seeking for being exhaustive, we reportsome statements by well-known researchers in the field, which help to characterizethe goals of multiple criteria decision analysis and multiple criteria decision aiding.
1. Bell 1979: Almost all the issues that decision makers face in actuality involvemultiple objectives that conflict in some measure with each other. In such issues,decisions that serve some objectives well will generally satisfy other objectivesless well than alternative decisions, which, however, would not be so satisfactoryfor the first group. The decision maker then must select from among the possibledecisions the one that somehow establishes the best mix of outcomes for hismultiple conflicting objectives. (. . .) Such problems include the use of energyresources, the management of the environment, the development of waterresources, and the expansion of regional development.
2. Keeney and Raiffa 1976: The theory of decision analysis is designed to help theindividual make a choice among a set of prespecified alternatives. (. . .) The aimof the analysis is to get your head straightened out.
3. Roy 2005: Decision aiding is the activity of the person who, through the use ofexplicit but not necessarily completely formalized models, helps obtain elementsof responses to the questions posed by a stakeholder in a decision process.
4. Belton and Stewart 2002: (. . .) we use the expression MCDA as an umbrella termto describe a collection of formal approaches which seek to take explicit accountof multiple criteria in helping individuals or groups explore decisions that matter.
5. Saaty 2005: The purpose of decision-making is to help people make decisionsaccording to their own understanding. (. . .) decision-making is the most frequentactivity of all people all the time (. . .)
This book is devoted to group decision-making, but it is apparent that any groupdecision is originally based on the decision of each single decision maker being acomponent of the group. The single decision makers are all facing, in general, amultiple criteria decision problem. For this reason, in this chapter, we are going to
1In this chapter, we shall use the acronymMCDA for indifferently referring to both multiple criteriadecision analysis and multiple criteria decision aiding.
2 S. Corrente et al.
describe the main ideas and the basic principles of multiple criteria decision aidingmethodologies. The concepts used here are useful also for methodologies related togroup decision introduced in chapters ▶ “MCDA Methods for Group DecisionProcesses: An Overview” and ▶ “A Group Multicriteria Approach”. Our chapteris therefore articulated in the following way: in the section “An Introduction toMCDA: Notation, Problematics and Main Approaches,” we give an introduction toMCDA presenting the notation used in the chapter, the main problematics dealt with,and the three aggregation methods being mostly used in the field, which are multipleattribute value theory (section “Multiple Attribute Value Theory”), outrankingmethods (section “Outranking Methods”), and decision rules (section “DecisionRules”). A discussion on interactions between criteria is provided in the section“Interaction Between Criteria.” In the section “Robust Recommendations,” we willfocus our attention on the robustness concerns and, in particular, on robust ordinalregression (section “Robust Ordinal Regression”) and stochastic multicriteriaacceptability analysis (section “Stochastic Multicriteria Acceptability Analysis”).Finally, section “Recent Developments and MCDA Applications” collects a briefsummary of recent developments and applications of MCDA.
An Introduction to MCDA: Notation, Problematics, and MainApproaches
In MCDA (see Greco et al. 2016) for an updated state-of-art survey on this topic), afinite set of alternatives A ¼ {a, b, c, . . .} is evaluated on the basis of a finite andcoherent family of evaluation criteria G ¼ {g1, . . ., gj, . . ., gn} where G is coherentif it satisfies the requirements of exhaustiveness, cohesiveness, and nonredundancy(Roy 1996):
• Exhaustiveness: All the relevant evaluation criteria are taken into account. This isto avoid situations in which two alternatives a and b have the same evaluation oncriteria from G but a and b are not considered indifferent.
• Cohesiveness: Given four alternatives a, b, c, d where a and b are indifferent, andc and d are obtained from a and b, respectively, by decreasing the performance ofa on one criterion, thus getting c, and increasing the performance of b on also onecriterion, thus getting d, then d should be considered at least as good as c.
• Nonredundancy: Removing one of the criteria from G, one of the two require-ments above is not satisfied anymore. This means that none of the aspects takeninto account in the considered problem is counted more than once.
The criteria in G are representing different aspects of evaluation of the alterna-tives considered in the decision problem the DM is faced with. Let Ej denote the setof all possible performance levels when considering alternative a on criterion gj. Inother words, Ej is the scale of criterion gj which has a quantitative or a qualitativenature. In general, in the first case, Ej � ℝ, while in the second, Ej is composed ofqualitative judgments that can assume linguistic forms such as those considered in
Multiple Criteria Decision Support 3
the chapter ▶ “Group Decisions with Linguistic Information: A Survey.” For exam-ple, considering an alternative being a sport car evaluated on two criteria, acceler-ation and comfort, the performances of each car on the acceleration are expressed ona quantitative scale (for example, 5 secs to pass from 0 to 100 km/h), while theperformances of each car on the comfort can be expressed in qualitative terms, suchas bad, medium, good, and so on.
In the following, by gj(a), we shall denote the quantitative or qualitative perfor-mance of a on gj, for all a � A, and for all gj � G.
In general, each criterion gj is associated with an increasing or a decreasingdirection of preference. On the one hand, gj has an increasing direction of preferenceif the greater the performance gj(a), the better is a on gj. On the other hand, gj has adecreasing direction of preference if the lower the performance gj(a), the better is aon gj. In the previous example, acceleration can be considered as a criterion having adecreasing direction of preference, while comfort can be considered as a criterionhaving increasing direction of preference. We used above “in general,” since therecould be criteria that have not a monotone direction of preference. In this case, weare in presence of an attribute. For example, let us assume that the DM has to choosethe locality where to spend his holidays next summer. To choose the best place, hewill take into account the average temperature of the considered places during thesummer time. Thinking about this attribute, the DM will probably prefer an averagetemperature of 35� to 25� or 45�. This means that this attribute has neither anincreasing direction of preference (in this case, 45� had been preferred to 35�) nora decreasing direction of preference (in this case, 25� had been preferred to 35�). Inwhat follows, we shall assume that criteria have an increasing or a decreasingdirection of preference. Although this is the most frequent assumption in MCDA,in recent years, many contributions have been presented taking into account non-monotone criteria (see, for example, Doumpos 2012; Ghaderi et al. 2017; Kadzińskiet al. 2020; Tehrani et al. 2012).
Three different decision aiding problem statements (problematics) are distin-guished in MCDA: choice, ranking, and sorting. In choice problems, one has tochoose the best alternative or the best subset of alternatives among those at hand byrejecting all the remaining alternatives, considered bad (see, for example, Bottero etal. 2019; Govindan et al. 2017; Malekmohammadi et al. 2011); in ranking problems,one has to rank-order all alternatives from the best to the worst with the possibility ofsome ties and incomparabilities (see, for example, Angilella et al. 2016a; Shanianand Savadogo 2006); finally, in sorting problems, one has to assign each alternativeto one or more contiguous classes that have been defined a priori and ordered fromthe best to the worst taking into account the preferences provided by the DM (see, forexample, Costa et al. 2018; Morais et al. 2014; Rocchi et al. 2018).
The starting point of any decision aiding problem is the construction of theperformance table.
4 S. Corrente et al.
Anyway, looking at the performance table above, the only objective informationthat one can see is the dominance relation in the set of alternatives, for which adominates b if and only if a is at least as good as b for all criteria and better for at leastone of them. For the sake of simplicity, and without loss of generality, we shallassume that all criteria have an increasing direction of preference. Consequently, weshall write that a dominates b if and only if gj(a)⩾ gj(b) for all j¼ 1, . . ., n, and thereexists at least one gj � G such that gj(a) > gj(b). Even if, as stated above, thedominance relation is a really objective information that can be obtained from theperformance table, it is very poor since, in general, when comparing a and b, a isbetter than b on some criteria, while b is better than a for the other criteria.Consequently, neither a dominates b nor the opposite situation occurs, thus thedominance relation leaves many alternatives noncomparable.
To deal with one of the three problem statements mentioned above, there is,therefore, the need to aggregate the performances of alternatives on the consideredevaluation criteria, aiming to get a comprehensive assessment of the alternatives athand, and to use this assessment for working out a recommendation proposed to theDM. In order to get a recommendation that would be convincing for the DM, theaggregation must take into account DM’s preferences which underline the relativeimportance of evaluation criteria in the comprehensive assessment. The aggregationthus leads to a comprehensive preference model that translates a value system of theDM at the current stage of the decision process.
Three different aggregation methods, very well known in MCDA literature, areassociated with the following three preference models, which are value functions(Keeney and Raiffa 1976), outranking relations (Roy 1996), and decision rules(Greco et al. 2001):
• A value function is a map U : A ! ℝ assigning to each alternative a � A a realnumber U(a) being representative of the comprehensive assessment of a in theconsidered problem; the greater the value of U(a), the better a is.
• An outranking relation S is a binary relation such that aSb iff a is at least as goodas b; in general, aSb iff a majority of criteria is in favor of this statement, and thereis not any criterion opposing too strongly to this outranking.
• Decision rules link a recommendation in the considered problem with the perfor-mances of alternatives on selected criteria; these are logical statements that put onthe condition side of a rule some threshold requirements on selected criteria, andon the decision side a recommendation concerning an alternative or a pair ofalternatives that satisfied the above requirements; for example, “if the consump-tion is at least 20 km/l and the price is at most 15,000 euros, then the car is
Multiple Criteria Decision Support 5
considered at least good”; the rules identify values that drive DM’s decisions –each rule is a scenario of a causal relationship between evaluations on a subset ofcriteria and a comprehensive judgment.
In the following sections, we are going to describe more in detail the threeaggregation methods and their associated preference models, putting an emphasison their main assumptions that differentiate them, and listing the most knownMCDA methods belonging to that aggregation family.
When choosing a particular method of MCDA for a real-world decision problem,an analyst, together with the DM, have to answer a series of interrelated questionsconcerning the nature of an expected recommendation, the character of availablepreference information, and the type of preference model (Roy and Słowiński 2013).
Multiple Attribute Value Theory
As already stated in the previous section, a value function assigns a real number toeach alternative in A representing its comprehensive assessment in the problem athand. The value function most used in the applications is the additive one.
U g1 að Þ, . . . , gn að Þð Þ ¼Xnj¼1
uj gj að Þ� �
ð1Þ
where uj(�) are nondecreasing functions of gj(a), called marginal value functions,such that if gj(a) < gj(b), then uj(gj(a)) O uj(gj(b)). In the following, for the sake ofsimplicity and without loss of generality, we shall write U(a) and uj(a), instead ofU(g1(a), . . ., gn(a)) and uj(gj(a)).
In its simplest form, this additive value function is reduced to a weighted sum.
U að Þ ¼Xnj¼1
wjgj að Þ ð2Þ
where wj> 0 for all criteria and wj/wi represents a trade-off between criteria gi and gj,that is, how much units one is willing to renounce on gi to have an increase of oneunit on gj (Belton and Stewart 2002). Let us observe that these trade-offs (orsubstitution rates) are therefore dependent on the scale on which the performancesof alternatives are defined.
A basic assumption, not always satisfied in the real applications, is that the set ofcriteria G is mutually preferentially independent (Keeney and Raiffa 1976; Wakker1989), meaning that each T � G is preferentially independent on G\T. Formally, T ispreferentially independent from G\T iff for all aT , bT �
Qgj � T
Ej and for all
cG∖T, dG∖T �Q
gj �G∖TEj,
6 S. Corrente et al.
aT , cG∖Tð Þ≿ bT , cG∖Tð Þ , aT , dG∖Tð Þ≿ bT , dG∖Tð Þwhere ≿ denotes a weak preference between alternatives, aT and bT denote partialevaluations of a and b, respectively, on criteria from T, while cG\T and dG\T denotepartial evaluations of some c and d, respectively, on criteria from G\T; (aT, cG\T) and(bT, cG\T) denote alternatives a and b, respectively, that got the same evaluations cG\Ton criteria from G\T; (aT, dG\T) and (bT, dG\T) are defined analogously. In practicalterms, T is preferentially independent of G\T if the preference between two alterna-tives (a and b) in not dependent on the common evaluation on the criteria from G\T(cG\T and dG\T) but on the evaluations on criteria from T only (aT and bT). Therefore,if T is preferentially independent of G\T and aT is preferred to bT, then the replacingof the common evaluations cG\T with the common evaluations dG\T does not invertthe preference of a over b. We shall see more in detail in section “InteractionBetween Criteria” that, in some cases, this assumption is not satisfied.
The use of a value function provides a total-preorder of the considered alterna-tives.2 Indeed, given the value function U, a is strictly preferred to b, and we shallwrite a � b, iff U(a) > U(b), while a and b are indifferent, and we shall write a � b,iff U(a) ¼ U(b). Let us observe that a � b iff a ≿ b and not(b ≿ a), while a � b iffa ≿ b and b ≿ a.
Let us conclude this section by recalling that, a very well-known method in theliterature that can be included among the MAVT methods is the analytic hierarchyprocess (AHP; Saaty 1980) which application to group decision problems will bedescribed in the chapter ▶ “Group Decision Support Using the Analytic HierarchyProcess.”
Outranking Methods
Differently from MAVT where each alternative is assessed using a real number,outranking methods are based on a binary relation, here denoted by S, where aSb iffa is at least as good as b.
The main differences between the methods based on value functions andoutranking methods are the following:
• Methods based on value functions are compensatory, while outranking methodsare not; this means that in MAVT, a very bad performance on one criterion can becompensated by a very good performance on another criterion while this is not thecase in the outranking methods.
2A total-preoder on A is a reflexive and transitive binary relation on A such that for all a, b � A, aRb,or bRa. In particular, reflexive means that aRa for all a � A, while transitive means that if aRb andbRc, then aRc for all a, b, c � A.
Multiple Criteria Decision Support 7
• On the basis of the built value functions, a preference and an indifference relationare constructed in MAVT; in the outranking methods, in addition to the preferenceand indifference relations,3 an incomparability relation is also defined for which aand b are incomparable iff not(aSb) and not(bSa).
The most well-known families of outranking methods are ELECTRE (ELimina-tion Et Choix Trasuidant la REalité; see Figueira et al. (2013) for a full description ofthe ELECTRE methods and Govindan and Jepsen (2016) for a recent review ofELECTRE methods) and PROMETHEE (Preference Ranking OrganizationMEThod for Enrichment of Evaluations; see Brans and Vincke (1985) for thepaper introducing PROMETHEE methods and Behzadian et al. (2010) for a litera-ture review on their use in MCDA). We are now going to briefly describe methodsbelonging to ELECTRE and PROMETHEE families.
ELECTRE MethodsAll ELECTRE methods are based on the comparison of the reasons in favor and thereasons against the outranking of an alternative a over an alternative b. TheELECTRE methods differ by the way these reasons are taken into account and forthe different problems they are applied to.
The ELECTRE methods are based on the concept of quasi-criterion. This meansthat each criterion gj � G is associated with two different thresholds, indifference qjand preference threshold pj, where 0 O qj O pj. These thresholds are introduced totake into account an arbitrariness, imprecision, or lack of knowledge in defining theperformances of alternatives (Roy et al. 2014). Even if, in general, these thresholdsare dependent on the performances (qj(gj(a)) and pj(gj(a))), in the following, we shallassume, without loss of generality, that they are fixed. qj represents the maindifference between the performances of two alternatives on gj being compatiblewith their indifference on gj, while pj represents the lowest difference between theperformances of two alternatives on gj compatible with the preference of one overthe other on this criterion.
The construction of the outranking relation starts from the computation of thepartial concordance index cj(a, b) for each gj and for each pair of alternatives(a, b) � A � A. cj(a, b) � [0, 1] is a nonincreasing function of the differencegj(b) � gj(a) and it expresses how much gj is in favor of the outranking of a over b.The general definition of cj(a, b) is the following:
cjða, bÞ ¼ f
1 if gjðbÞ � gjðaÞOqj, ðaSjbÞ,pj � ½gjðbÞ � gjðaÞ
pj � qjif qj < gjðbÞ � gjðaÞ < pj, ðbQjaÞ,
0 if gjðbÞ � gjðaÞPpj, ðbPjaÞ:
3On one hand, a is preferred to b, and we shall write aPb, iff aSb but not(bSa); on the other hand, aand b are indifferent, and we shall write aIb, iff aSb and bSa.
8 S. Corrente et al.
If cj(a, b) ¼ 0, then gj is not in favor of the outranking of a over b. If, instead,cj(a, b) � ]0, 1[, gj is partially in favor of the same outranking. gj becomes stronglyin favor of the outranking of a over b iff cj(a, b) ¼ 1.
Adding up all the partial concordance indices, the comprehensive concordanceindex C(a, b) is obtained.
C a, bð Þ ¼Xnj¼1
wjcj a, bð Þ
where wj > 0 is an importance weight of criterion gj for all j ¼ 1, . . ., n, and, in
general,Pnj¼1
wj ¼ 1 . Let us observe that, differently from Eq. (2), the wj have the
meaning of a voting power (Roy 2005). This means that the value wj represents arelative importance of criterion gj inside the family of criteria G. As a consequence,they are not dependent on the scale on which the performances of alternatives aregiven. C(a, b) � [0, 1] and it represents how much the criteria in G are in favor ofthe outranking of a over b.
On the basis of C(a, b), the concordance test is therefore satisfied iff C(a, b) ⩾ λ,where λ � ]0.5,1], called cutting level, represents the minimum portion of criteriathat should be in favor of the outranking of a over b.
Close to the definition of an indifference and a preference threshold, the DM candefine a veto threshold vj. vj represents the lowest difference between the perfor-mances of two alternatives being incompatible with the preference of one over theother. This means that, even if the concordance test regarding alternatives a and b isverified, but the difference gj(b) � gj(a) exceeds vj, then a cannot outrank b. Fromthis definition, it is easy to observe why the ELECTRE methods are non-compen-satory methods, differently from those based on value functions.
Similarly to cj(a, b), that defines how much gj is in favor of the outranking of aover b for each gj, the discordance index dj(a, b) defines how much gj is against thehypothesis about outranking. dj(a, b) belongs to the interval [0, 1] and it is anondecreasing function of the difference gj(b) � gj(a) defined as follows:
dj a, bð Þ ¼
1 if gj bð Þ � gj að ÞPvj,
gj bð Þ � gj að Þh i
� pj
vj � pjif pj < gj bð Þ � gj að Þ < vj,
0 if gj bð Þ � gj að ÞOpj:
8>>>><>>>>:
If dj(a, b) ¼ 0, then gj is not against the outranking of a over b. If, instead,dj(a, b) � ]0, 1[, gj is partially against the same outranking. gj becomes stronglyagainst the outranking of a over b iff dj(a, b) ¼ 1.
The ELECTRE methods take into account the reasons against the outranking ofan alternative over another, in two different possible ways. On one hand, theconcordance and the non-discordance test are performed separately. In particular,
Multiple Criteria Decision Support 9
the non-discordance test is passed iff gj(b) � gj(a) < vj for all gj � G.4 On the other
hand, the reasons in favor and against the outranking of a over b are put togetherdefining the so-called credibility index σ(a, b).
σ a, bð Þ ¼Y
gj �G :
dj a, bð Þ > C a, bð Þ
1� dj a, bð Þ1� C a, bð Þ :
σ(a, b) ⩾ λ means that the credibility of outranking reached a necessary cuttinglevel λ to state that a outranks b. σ(a, b) � [0, 1] takes into account simultaneouslythe reasons in favor (C(a, b)) and the reasons against (dj(a, b)) the outranking of aover b. In particular, σ(a, b)¼ C(a, b) if no criterion is opposing to the outranking ofa over b, while C(a, b) is reduced by multiplying it for 1�dj a, bð Þ
1�C a, bð Þ if gj is such that
dj(a, b) > C(a, b).On the basis of the concepts previously defined, two main different outranking
relations can be considered:
O1. aS1b iff C(a, b)⩾ λ and gj(b)� gj(a)< vj, for all gj � G, which is the definitionof the outranking relation used in the ELECTRE IS method.
O2. aS2b iff σ(a, b)⩾ λ, which is the definition of the outranking relation used in theELECTRE III method.
PROMETHEE MethodsSimilarly to the ELECTRE methods, the PROMETHEE methods provide recom-mendations on the considered problem by building two or three binary relations onthe basis of the computations of different flows in a graph representing an outrankingrelation.
First of all, for each criterion gj � G, the PROMETHEEmethods build a functionπj : A� A! [0, 1], where πj(a, b) measures the degree of the preference of a over bon gj. πj(a, b) is a nondecreasing function of the difference gj(a) � gj(b) and sixdifferent functions have been proposed in Brans and Vincke (1985), with the mostused one being the V-shape function defined as follows:
πj a, bð Þ ¼
1 if gj að Þ � gj bð ÞPpj,
gj að Þ � gj bð Þh i
� qj
pj � qjif qj < gj að Þ � gj bð Þ < pj,
0 if gj að Þ � gj bð ÞOqj,
8>>>><>>>>:
4This is equivalent to say that dj(a, b) < 1 for all gj � G.
10 S. Corrente et al.
where the indifference and preference thresholds have an analogous meaning as inELECTRE methods. As for the ELECTRE methods, the partial preference indicesπj(a, b) are therefore aggregated by mean of the following weighted sum:
π a, bð Þ ¼Xnj¼1
wjπj a, bð Þ
where wj represents a relative importance of gj and it is interpreted as in theELECTRE methods. π(a, b) � [0, 1] denotes the degree of outranking of a overb. Using π(a, b), an outranking graph can be drawn, where nodes represent thealternatives and directed arcs between them are valued by a degree of outranking.This graph is a base for calculation of a positive, a negative, and a net flow for eachalternative a � A:
• The positive flow, ϕ+(a), represents how much, in average, a is preferred to allother alternatives in A; it is formally computed as:
ϕþ að Þ ¼ 1
j A j �1
Xb�A∖ af g
π a, bð Þ: ð3Þ
• The negative flow, ϕ�(a), represents how much, in average, all alternatives in Aare preferred to a; it is formally computed as:
ϕ� að Þ ¼ 1
j A j �1
Xb�A∖ af g
π b, að Þ: ð4Þ
• The net flow, ϕ(a), is a balance between the two flows defined above and itrepresents the comprehensive assessment of a taking into account both how mucha is preferred to the other alternatives and how much the other alternatives arepreferred to a; this is simply computed as:
ϕ að Þ ¼ ϕþ að Þ � ϕ� að Þ: ð5Þ
On the basis of the computed flows, several PROMETHEE methods have beendefined. In the following, we shall recall the PROMETHEE I and II only, being themost popular ones. PROMETHEE I defines preference PI, indifference II, andincomparability relation RI computed as follows:
• aPIb iff ϕ+(a)⩾ ϕ+(b), ϕ�(a)O ϕ�(b) and at least one of the inequalities is strict• aIIb iff ϕ+(a) ¼ ϕ+(b) and ϕ�(a) ¼ ϕ�(b)• aRIb iff not(aPIb), not(bPIa) and not(aIIb)
PROMETHEE I provides, therefore, a partial preorder on the set of alternatives A.
Multiple Criteria Decision Support 11
PROMETHEE II, instead, provides a total order of the alternatives in A definingpreference PII and indifference relation III on the basis of the net flows. In particular:
• aPIIb iff ϕ(a) > ϕ(b)• aIIIb iff ϕ(a) ¼ ϕ(b)
Decision Rules
Decision rule model is an MCDA preference model based on the dominance-basedrough set approach (DRSA) proposed in Greco et al. (2001). The basic preferenceinformation supplied by the DM is composed of pairwise preference comparisons ofalternatives or classification of alternatives in preference ordered classes. On thebasis of this information, some decision rules explaining the preference informationare induced. The two most typical syntactical forms of the induced decision rules arethe following:
• “if the strength of the preference of alternative x over alternative y is at least πi1 oncriterion gi1 and at least πi2 on criterion gi2 and . . . and at least πir on criterion gir ,then x is at least as good as y,” with gi1 , gi2 , . . . , gir �G
• “if alternative x has an evaluation not worse than li1 on criterion gi1 and not worsethan li2 on criterion gi2 . . . and not worse than lir on criterion gir, then x is assignedto a class not worse than Cs,” with gi1 , gi2 , . . . , gir �G, C1, C2, . . ., Cp is a set ofincreasing preferentially ordered classes and Cs � {C1, C2, . . ., Cp}
Two examples of decision rules are the following:
R1 “if student S1 is at least weakly preferred in Mathematics and strongly pre-ferred or more in Physics over student S2, then S1 is at least as good as S2”,
R2 “if student S has an evaluation at least medium in Literature and good or betterin Philosophy, then the student S is comprehensively at least medium”.
Each one of the induced decision rules is associated with the correspondingpieces of preference information, that is, pairs of alternatives for which there is thepreference relation suggested by a pairwise comparison rule or alternatives assignedto the class by a classification rule. For example, rule R1 will be associated with theset of pairs of students (S1, S2) such that S1 is at least weakly preferred in Mathe-matics and strongly preferred or more in Physics over S2, and S1 is at least as good asS2. Analogously, rule R2 will be associated with students S having an evaluation atleast medium in Literature and good or better in Philosophy, being comprehensivelyat least medium.
The set of decision rules induced from the preference information supplied by theDM, constitute a preference model that after being discussed with the DM andaccepted by him, can be applied on the alternatives of the decision problem at
12 S. Corrente et al.
hand. This means that if rule R1 is accepted by the DM, each time that a student S1 isat least weakly preferred in Mathematics and strongly preferred or more in Physicsover S2, then S1 will be considered at least as good as S2. Analogously, if rule R2 isaccepted, each student S having an evaluation at least medium in Literature and goodor better in Philosophy will be considered comprehensively at least medium. Theadvantage of this approach is that the decision model is transparent and easilyunderstandable by the DM that can find the arguments supporting the recommendeddecision in the same rule defining the preference relation or the classification.
For tutorials and surveys on the use of DRSA in MCDA, one can refer toSłowiński et al. (2014, 2015).
Interaction Between Criteria
As observed in the section “Multiple Attribute Value Theory,” the use of an additivevalue function assumes that the criteria from set G are mutually preferentiallyindependent. Let us consider a problem in which the dean of a scientifically orientedhigh school has to evaluate four students with respect to three subjects, such asMathematics (M), Physics (P), and Literature (L) (Grabisch 1996). The marks ofthese four students on the three subjects are given in the table below using a 20-pointscale:
When comparing a and b, the dean states that b is preferred to a, while whencomparing c and d, he expresses his preference for c over d. These preferences can bejustified in the following way. On one hand, since a and b have both high marks onMathematics and Physics, then the dean prefers the student presenting a higher markon Literature. On the other hand, since the performances of c and d on Physics arenot very good and for the dean the scientific subjects are really important, then heprefers c over d since c has a better mark than d on Mathematics. Let us try torepresent these preferences by using the value function from Eq. (1). On one hand,the preference of b over a is translated into the constraint:
uM 15ð Þ þ uP 13ð Þ þ uL 10ð Þ < uM 12ð Þ þ uP 13ð Þ þ uL 12ð Þ ð6Þwhile the preference of c over d is translated into the constraint:
uM 15ð Þ þ uP 5ð Þ þ uL 10ð Þ > uM 12ð Þ þ uP 5ð Þ þ uL 12ð Þ: ð7Þ
Multiple Criteria Decision Support 13
The two inequalities are obviously in contradiction since, from Eq. (6) we get thatuM(15) + uL(10) < uM(12) + uL(12), while from Eq. (7), we get that uM(15) + uL(10)> uM(12) + uL(12). This means that an additive value function is not able to representthese preferences since, observing carefully the students’ marks shown in the table,one can see that the criteria {M, P, L} are not mutually preferentially independentsince {M, L} is not preferentially independent of P. Indeed, considering only criteriaM and L, a has the same marks as c and b has the same marks as d. Therefore,observing that a and b have the same mark on P (13) and c and d have the same markon P (5), if {M, L} was preferentially independent of P, then the preference of b overa should imply the preference of d over c, which is not true in this case since b ispreferred to a but c is preferred to d.
In this case, we can say that the three criteria present a certain degree ofinteraction. On one hand, two criteria are positively interacting if a good perfor-mance on one criterion does not imply, in general, a good performance on the other.Consequently, one would like to give a bonus to an alternative presenting goodperformances on both criteria. On the other hand, two criteria are negativelyinteracting if a good performance on one criterion implies a good performance onthe other too. In this case, one would therefore give a malus to an alternativepresenting good performances on both criteria.
The mentioned interaction between criteria is taken into account in different waysin the literature. In the following, we are going to briefly recall the most knownmethods:
• Multilinear value functions (Belton and Stewart 2002; Keeney and Raiffa 1976):
U að Þ ¼Xnj¼1
uj að Þ þXnj¼1
Xj<iOn
uj að Þui að Þ þ . . .þ u1 að Þu2 að Þ� � �un að Þ:
An interesting opinion on this model was given by Stewart (2005):
the large number of parameters which have to fit decision maker preferences isprohibitive in most real world applications. . . . In exploring the literature, it is difficultto find many reported applications even of the multiplicative model, let alone themultilinear model.
• Augmented value functions (Greco et al. 2014):
UintðaÞ ¼Xnj¼1
ujðgjðaÞÞ þXi, j�G
synþi,jðgiðaÞ, gjðaÞÞ �Xi, j�G
syn�i,jðgiðaÞ, gjðaÞÞ,
14 S. Corrente et al.
where synþij , syn�ij : Ei � Ej ! ½0, 1 are nondecreasing in both its arguments.
synþi,j gi að Þ, gj að Þ� �
represents a bonus that should be added toPnj¼1
ujðgjðaÞÞ if
criteria gi and gj present a positive interaction, while syn�i,j gi að Þ, gj að Þ� �
repre-
sents a malus that should be subtracted fromPnj¼1
ujðgjðaÞÞ if criteria gi and gj are
negatively interacting.• The nonadditive integrals (Grabisch and Labreuche 2010) and, in particular, the
Choquet integral preference model (Choquet 1953; Grabisch 1996):
Chμ að Þ ¼Xnj¼1
g jð Þ að Þ � g j�1ð Þ að Þh i
� μ g jð Þ, . . . , g nð Þn o� �
where (�) is a permutation of the indices of criteria, such that0 ¼ g(0)(a) O g(1)(a) O . . . O g(n)(a) and μ : 2G ! [0, 1] is a capacity being aset function assigning to each subset of criteria of G a number in [0, 1] such thatthe monotonicity (μ(A) O μ(B) for all A � B � G) and normalization (μ(∅) ¼ 0and μ(G) ¼ 1) constraints are satisfied. In this context, the importance of acriterion gj is not dependent on itself only but also on its contribution to allpossible coalitions of criteria. For this reason, the Shapley (Shapley 1953) and theMurofushi (Murofushi and Soneda 1993) indices are defined. The Shapley indexφ({gj}) and the Murofushi index φ({gi, gj}) are computed as follows:
φ gj
n o� �¼
XT�G∖ gjf g
jG∖Tj�1ð Þ! j T j !j G j ! μ T [ gj
n o� �� μ Tð Þ
h i
and
φ gi, gj
n o� �¼
XT�G∖ gi ,gjf g
jG� Tj�2ð Þ! j T j !jGj�1ð Þ! τ T, gi, gj
� �
where τ(T, gi, gj) ¼ μ(T [ {gi, gj}) � μ(T [ {gi}) � μ(T [ {gj}) + μ(T ).To make the representation easier, in general, the Möbius transform of μ (Rota
1964) and k additive capacities (Grabisch 1997) are considered. The interestedreader refers, for example, to Angilella et al. (2016b) for more details on these twoconcepts.
• ELECTRE methods with interactions (Figueira et al. 2009a): the interactionbetween criteria is taken into account by redefining the concordance indexC(a, b) in the following way:
Multiple Criteria Decision Support 15
Cða,bÞ ¼ 1
Kða,bÞ� X
gj � CðbPaÞwjcjða,bÞ þ
Xfgi,gjg� Lða,bÞ
wijZ�ciða,bÞ,cjða,bÞ
��
�X
ðgi,ghÞ�Oða,bÞw0ihZ
�ciða,bÞ,chðb,aÞ
��
where:
– K a, bð Þ ¼ Pnj¼1
wj þP
gi , gjf g� L a, bð ÞwijZ ci a, bð Þ, cj a, bð Þ� ��
Pgi , ghð Þ�O a, bð Þ
w0ihZ ci a, bð Þ, ch b, að Þð Þ:
– C(aTb) denotes the set of criteria for which aTb and T � {S, Q, P} andC aTbð Þ is the complement of C(aTb) in G.
– Lða, bÞ ¼ ffgi, gjg � G : gi, gj � �CðbPaÞg.– O a, bð Þ ¼ gi, gj
� ��G� G : gi �C bPað Þ and gh �C bPað Þ
n o; it is the set
of ordered pairs of criteria (gi, gj) such that gi is not in favor of the preferenceof b over a, while gh is in favor of the same preference.
– wij is a positive value that should be added (subtracted) in case criteria gi and gjare positively (negatively) interacting.
– w0ih is a positive number representing the antagonistic effect exercised by gh
over gi when gi is in favor of the outranking of a over b and gh is against thesame outranking.
• The bipolar PROMETHEE methods (Corrente et al. 2014a): for each criteriongj � G, a bipolar preference function πBj a, bð Þ is defined.
πBj a, bð Þ ¼ πj a, bð Þ if πj a, bð Þ > 0,
�πj b, að Þ if πj a, bð Þ ¼ 0,
�
where πBj a, bð Þ represents the bipolar preference of a over b on gj: if πBj a, bð Þ > 0,
then a is preferred to b on gj, while, if πBj a, bð Þ < 0, b is preferred to a on gj. Iff
πBj a, bð Þ ¼ 0 none of the two alternatives is preferred over the other on gj.
The bipolar vector πBða, bÞ ¼ ðπB1 ða, bÞ, . . . , πBn ða, bÞÞ is therefore aggregatedby using the bipolar Choquet integral (Grabisch and Labreuche 2005a, b).
Robust Recommendations
All the methods described in the previous sections involve definitions of severalparameters. For example, the marginal value functions uj(�) and the substitutionratios wj/wi in the additive value functions, or the importance coefficients and thethresholds in the outranking methods. These parameters can be obtained by using a
16 S. Corrente et al.
direct or an indirect preference information (Jacquet-Lagrèze and Siskos 2001) (seealso chapter ▶ “Multiple Criteria Group Decisions with Partial Information AboutPreference”). The DM provides a direct preference information if he gives directlyvalues to all parameters involved in the model. In general, this way of givingpreference information implies a great cognitive effort from the part of the DMwho has to specify a huge number of parameters of which, in most of the cases, hedoes not know or understand the meaning. For this reason, the indirect preferenceinformation is preferred in practice. The DM provides an indirect preference infor-mation if he expresses his preferences through statements, like comparisons betweensome reference alternatives (a is preferred to b or a is indifferent to b) or comparisonbetween criteria with respect to their importance (for example, gi is more importantthan gj or gi is as important as gj) or the statements on the possible interactionsexisting between them (for example, gi and gj are positively interacting or giexercises an antagonistic effect over gj). From this preference information, param-eters compatible with these statements can be inferred. By applying the indirectpreference information one aims, therefore, to discover an instance of the preferencemodel compatible with the information provided by the DM. This technique isknown in MCDA under the name of ordinal regression and many contributionshave applied this technique to different preference models. In Jacquet-Lagrèze andSiskos (1982), the authors proposed the UTA method. The underlying preferencemodel is an additive value function as that one in Eq. (1). In UTA, each uj is a
piecewise linear value function defined by the utility uj xkj
� �of the breakpoints xkj
defining the partition of the interval [αj, βj] that contains the evaluations of thealternatives on criterion gj. The DM provides a partial preorder regarding a subset ofalternatives, AR � A, and this preorder is translated to inequality constraints. Forexample, if a is preferred to b, then U(a) > U(b), while the indifference between aand b is translated into the constraint U(a) ¼ U(b). To check for the existence of aninstance of a value function compatible with these preferences, an LP problem has tobe solved.
The same approach is used for nonadditive integrals and, in particular, for theChoquet integral preference model (Marichal and Roubens 2000). In this case, theDM provides a partial preorder on the set of alternatives as in Jacquet-Lagrèze andSiskos (1982) together with some preferences regarding importance of criteria andinteraction between criteria. Again, an instance of the capacity compatible with thepreferences provided by the DM is obtained solving an LP problem.
The indirect preference information is also implemented by Figueira and Roy(2002) in a revised version of the Simos method (Simos 1990a, b) called deck ofcards method (DCM). The DCM, known also as SRF method, proposes a procedureto assign a value to the weights used in the outranking methods, starting frompreferences provided by the DM in very natural terms. The method is composedof three steps: (i) rank-order all the criteria from the least important to the mostimportant with the possibility of some ex-aequo; (ii) add some blank cards betweensuccessive subsets of criteria to increase the difference between their importance;and (iii) provide a ratio z between the weight of the most important criterion and the
Multiple Criteria Decision Support 17
least important one. From these preferences, the analyst supporting the DM istherefore able to assign a single value to each criterion.
All above methods using the indirect preference information provided by the DMaim at discovering a single instance of the considered preference model compatiblewith this preference. In general, however, there could exist more than one instance ofthe preference model compatible with the preferences provided by the DM (in thefollowing, a compatible model). All compatible models give the same recommen-dations on the reference alternatives, that is, the alternatives in AR, but they couldprovide different recommendations on the non-reference alternatives.
To stress this point, let us provide a very simple example involving four alterna-tives evaluated on two criteria with performances given in the table below.
Let us assume that the DM prefers a to b and that we translate this preferenceusing a weighted sum as a preference model. It is straightforward to observe that:
10w1 þ 6w2 > 7w1 þ 9w2 , 3w1 > 3w2 , w1 > w2:
Considering a vector w ¼ (w1, w2), such that w1 > w2, is enough to represent thispreference. Let us consider, therefore,w(1)¼ (0.6,0.4) andw(2)¼ (0.7,0.3). Both of themrepresent the preference of the DM but, comparing the other two alternatives, that is cand d, one can observe that using w(1), d is preferred to c (U(1)(d)¼ 7.4> U(1)(c)¼ 7),while using w(2), c is preferred to d (U(2)(c) ¼ 7.5 > U(2)(d) ¼ 6.3).
This simple example proves that the choice of the compatible model will affectthe final recommendations. Therefore, more robust recommendations could beprovided by taking into account not only one compatible instance of the preferencemodel but all of them simultaneously.
Robust Ordinal Regression
The robust ordinal regression (ROR) (see Greco et al. (2008) for the paper intro-ducing ROR and Corrente et al. (2013a, 2014c) for two recent surveys on ROR)takes into account all instances of the considered preference model by defining anecessary and a possible preference relation on A. In particular, a is necessarilypreferred to b, and we shall write a≿Nb, iff a is at least as good as b for all compatiblemodels, while a is possibly preferred to b, and we shall write a≿Pb, iff a is at least asgood as b for at least one compatible model. ≿N and ≿P satisfy a set of propertiesamong which inclusion (≿N � ≿P) and completeness (for all a, b � A, a≿Nb or
18 S. Corrente et al.
b≿ba), while the basic axioms on ROR have been studied in Giarlotta and Greco(2013).
ROR has been already applied to all preference models described above and, inparticular, to value functions (Corrente et al. 2012; Figueira et al. 2009b; Greco et al.2008, 2010), ELECTRE methods (Corrente et al. 2013b; Greco et al. 2011a),PROMETHEE methods (Corrente et al. 2013b; Kadziński et al. 2012a), Choquetintegral (Angilella et al. 2010b, 2016b), and decision rules (Kadziński et al. 2015,2016).
Depending on the underlying preference model used to represent the preferencesprovided by the DM, the necessary and possible preference relations are computedby solving LP or MILP problems. In both cases, the concept is however the same. Inthe following, we are going to describe how these relations are computed if thepreference model is an additive value function. Denoting by EAR
the set containingthe constraints translating the preferences given by the DM and the technicalconstraints depending on the considered preference model, the two preferencerelations are obtained by solving the following programming problems for eachpair of alternatives (a, b) � A � A:
eN a, bð Þ ¼ max e, subject to
U bð ÞPU að Þ þ e,
EAR
)EN a, bð Þ
eP a, bð Þ ¼ max e, subject to
U að ÞPU bð Þ,
EAR
)EP a, bð Þ
On one hand, we shall conclude that a is necessarily preferred to b iff EN(a, b) isinfeasible or eN(a, b) O 0. On the other hand, we shall conclude that a is possiblypreferred to b if EP(a, b) is feasible and eP(a, b) > 0. Of course, in consequence ofthe properties holding for the two relations, not all programming problems have to besolved but just some of them. More details with respect to this aspect can be found inCorrente et al. (2016a).
The necessary preference relation provides a partial preorder of the alternativessince it is a reflexive and transitive binary relation. Therefore, it is possible that somepairs of alternatives are not comparable with respect to this preference relation sinceneither a is necessarily preferred to b nor b is necessarily preferred to a. In some real-world problems, however, it is often required to get a complete order of thealternatives and, therefore, the results obtained by taking into account the wholeset of compatible models have to be aggregated to provide a conclusive recommen-dation. For such a reason, among the many compatible models, the most represen-tative one can be selected. This model is the one, among those compatible,maximizing the difference between the alternatives (a, b) � A � A for whicha≿Nb but not(b≿Na) and minimizing the difference between the alternatives (a,b) � A� A for which not(a≿Nb) and not(b≿Na). The most representative model hasbeen defined for value functions (Corrente et al. 2012; Greco et al. 2011b; Kadzińskiet al. 2012a, 2013), outranking methods (Kadziński et al. 2012b), and Choquetintegral (Angilella et al. 2010a).
Multiple Criteria Decision Support 19
Stochastic Multicriteria Acceptability Analysis
As already observed in the previous section, the ROR provides extreme informationon each pair of alternatives (a, b) � A � A. Indeed, the necessary and possiblepreference relations tell us only if all models agree on the fact that a is at least asgood as b (a≿Nb) or if at least one model agrees on the fact that a is at least as goodas b (a≿Pb). Anyway, in most of the cases, the alternatives are incomparable withrespect to the necessary preference relation since there are compatible models forwhich a is at least as good as b (a≿Pb) and compatible models for which b is at leastas good as a (b≿Pa). In these cases, nothing can be concluded regarding thecomparison between the two alternatives. Therefore, to have more information onthem, one should “count” the number of compatible models for which a is preferredto b and the number of compatible models for which the opposite is true.
The stochastic multicriteria5 acceptability analysis (SMAA) (see Lahdelma et al.(1998) for the first paper on SMAA and Lahdelma and Salminen (2016), Pelissari etal. (2019), and Tervonen and Figueira (2008) for three surveys on the methodology),alike the ROR, takes into account simultaneously all the models compatible with thepreferences provided by the DM, however, in a different way. SMAA providesinformation in probabilistic terms, that is, the frequency with which a certainalternative is in a ranking position or the frequency with which an alternative ispreferred to another one when a big sample of compatible preference models isconsidered. In particular, three different indices can be defined in SMAA:
• The rank acceptability index bk(a): It is the frequency with which an alternativereaches a certain ranking position; of course, it can therefore be considered if theunderlying preference model produces a total ranking of the alternatives such as avalue function (Lahdelma et al. 1998), the Choquet integral (Angilella et al. 2015,2016b), or the PROMETHEE II method (Corrente et al. 2014b).
• The pairwise winning index p(a, b) (Leskinen et al. 2006): It is the frequency withwhich the alternative a is preferred to the alternative b; it can be computed notonly in case the model provides a total order of the alternatives but also in case itprovides a partial preorder as for the ELECTRE III method (Corrente et al. 2017)or for the PROMETHEE I method (Corrente et al. 2014b).
• The central weight vector w1(a): It is computed only for alternatives a such thatb1(a) > 0 and, therefore, alternatives that can be the first for at least onecompatible model. It is formally computed as a barycenter of the space ofcompatible models giving to a the first position and it represents, therefore, the“typical” preference giving to a the best possible position.
On the basis of the mentioned indices and, in particular, of the rank acceptabilityindices for each alternative, one can also compute the best and worst reachablepositions, the ranking positions presenting the highest frequencies, or the cumulative
5Multiobjective and multiattribute are used as well.
20 S. Corrente et al.
rank acceptability indices giving for each a � A and each k ¼ 1, . . ., j Aj thefrequency with which a reaches at least the position k (bOk(a)) or at most the positionk (b⩾(a)).
The formal definition of the introduced indices depends on the assumed prefer-ence model. Anyway, all of them are computed by solving multidimensional inte-grals that can be approximated by Monte Carlo simulations. Indeed, the preferencesprovided by the DM define a space of compatible models for which several instancesof the assumed preference model have to be sampled. If the space of compatiblemodels is defined by linear inequalities and equality constrains and, therefore, itconstitutes a convex space, one can sample instances of its elements by using the hit-and-run (HAR) method introduced in Smith (1984) but then applied to MCDA inTervonen et al. (2013) and Van Valkenhoef et al. (2014) (see Corrente et al. (2019)for a very detailed description of the HAR method in this case).
Let us conclude this section by mentioning that ROR and SMAA have been puttogether in Kadziński and Tervonen (2013a, b), while procedure to aggregate therank acceptability indices and the pairwise winning indices have been proposed inKadziński and Michalski (2016).
Recent Developments and MCDA Applications
In this section, we shall briefly recall two recent developments and we shall list someresearch areas in which MCDA has been fruitfully applied.
MCHP: In MCDA problems, it is assumed, in general, that the evaluation criteriaare all at the same level. This is not the case, however, in several MCDA applicationswhere it is possible to observe a root criterion, being the objective of the problem,some macrocriteria, being the main aspects that need to be taken into account in theproblem at hand, some other criteria descending from the macrocriteria, until thebottom of the hierarchy where there are so called elementary criteria, being thecriteria on which the performances of the alternatives are directly given. To deal withsuch problems where the evaluation criteria are structured in a hierarchical way, themultiple criteria hierarchy process (MCHP) (Corrente et al. 2012) has been pro-posed. It gives important advantages from the input and from the output point ofview with respect to the classical MCDA methods where all evaluation criteria areconsidered at the same level:
1. From the input point of view, considering the indirect way of providing prefer-ence information, the DM can supply information not only comprehensively, thatis taking into account simultaneously all the criteria, but also partially, that isconsidering only a subset of criteria corresponding to particular node in thehierarchy tree. This is beneficial since the DM could be a bit confused whenbeing forced to provide a full information that summarizes all the aspects of thealternatives, but he could be more confident in providing a preference informationon only those aspects he knows better.
Multiple Criteria Decision Support 21
2. From the output point of view, together with the recommendations in which allaspects are taken into account simultaneously, when applying the MCHP the DMcan learn if a is preferred to b on g1 or if b is preferred to a on g2 and so on. Thisgives the DM the possibility to get a better insight to the problem he is copingwith because in this way he learns which are the weak and strong points of eachalternative.
The MCHP has been applied to all mentioned preference models considering alsothe ROR and the SMAA methodologies (Angilella et al. 2016b; Arcidiacono et al.2018; Corrente et al. 2013b, 2016b, 2017).
IEMO: In multiobjective optimization, one aims to optimize simultaneously a setof objective functions f1(x), . . ., fn(x) under some constraints c1(x) ⩾ d1, . . .,cm(x)⩾ dm. The goal is to find the best vector of variables x optimizing the objectivefunctions and satisfying all considered constraints. This problem is a particular caseof a multiple criteria choice problem where objective functions are evaluationcriteria and the set of constraints is defining a set of possible alternatives. As alreadyobserved in the introductory section, since the objective functions (criteria) are inconflict, there is not any x optimizing simultaneously all the objective functions. Thebest one can hope is therefore finding the Pareto set, that is the set composed of allnon-dominated x. In recent years, to compute the whole Pareto set or its represen-tative approximation, the evolutionary algorithms have been successfully applied.They mimic the evolution of the populations in biology and try to approximate thePareto front by means of a population of solutions x. The best known of theseevolutionary algorithms is NSGA-II (Deb et al. 2002). However, even if the DMwould know the Pareto set, he should finally choose in this set the best solution x or asubset composed of all satisfactory solutions with respect to his preferences. For thisreason, two extreme approaches can be taken into account: (i) defining a priori avalue function representing the preferences of the DM and substituting all theconsidered objective functions, that is, U(x) ¼ U( f1(x), . . ., fn(x)). In this way, themultiobjective problem is reduced to a single objective problem where one aims tofind x maximizing the utility function of the DM; and (ii) build the whole Paretofront and then deciding among these solutions the preferred one by specifying, forexample, trade-offs between the different objective functions. These two approachesare equally impractical since the first assumes that the preferences of the DM can beformalized a priori by a value function, while, the second one assumes that the DM isable to provide his preferences on a set composed of many solutions described byvector evaluations (Branke et al. 2008).
Interactive evolutionary multiobjective optimization (IEMO) methods representan intermediate approach since they give to the DM the possibility of including hispreferences in the search of the solution space, i.e., in the evolution of the populationof solutions. This permits, in consequence, to focus the search on the most appealingpart of the Pareto front with respect to his preferences. Several methods integratingMCDA and evolutionary algorithms have been proposed recently. See, for example,Branke et al. (2015, 2016), Greenwood et al. (1997), and Phelps and Köksalan(2003).
22 S. Corrente et al.
MCDA methods have been very successful in handing many complex real-worlddecision problems and it is impossible to give here an exhaustive inventory of allapplications. A very partial list of fields where MCDA has been successfully appliedis the following:
• Economics and finance (Doumpos and Zopounidis 2014; Zavadskas and Turskis2011)
• Energy planning (Diakoulaki et al. 2005; Wang et al. 2009)• Engineering (Bertola et al. 2019; Rogers et al. 2013; Zavadskas et al. 2015a, b)• Environmental problems (Cegan et al. 2017; Huang et al. 2011; Kiker et al. 2005;
Linkov and Moberg 2011; Malczewski 1999; Malczewski and Rinner 2016)• Medicine (Diaby et al. 2013; Thokala et al. 2016)• Natural resource management (Mendoza and Martins 2006)
Looking at the future perspectives, we believe that MCDA has to proceed in adirection in which the information required from the DM and supplied to the DM, aswell as the decision model adopted are “as simple as possible, but not simpler”(Arcidiacono et al. 2020). Indeed, on one hand, the DM has to be given thepossibility to express his preferences with the desired degree of detail in all theirrichness of contents, while, on the other hand, the DM must have the possibility ofunderstanding all the aspects of the final recommendation including its pros and conswith respect to other decisions.
Cross-References
▶A Group Multicriteria Approach▶Group Decision Support Using the Analytic Hierarchy Process▶Group Decisions with Linguistic Information: A Survey▶MCDA Methods for Group Decision Processes: An Overview▶Multiple Criteria Group Decisions with Partial Information About Preference
Acknowledgments Salvatore Corrente and Salvatore Greco gratefully acknowledge the fundingby the research project “Data analytics for entrepreneurial ecosystems, sustainable development andwellbeing indices” of the Department of Economics and Business of the University of Catania. JoséRui Figueira acknowledges the support from the hSNS FCT – Research Project (PTDC/EGE-OGE/30546/2017) and the FCT grant SFRH/BSAB/139892/2018 under POCH Program. The research ofRoman Słowiński has been partially supported by the statutory funds of Poznan University ofTechnology.
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