Decision-making in semi-democratic contexts

Decision-making in semi-democratic contexts Fiorenzo FRANCESCHINI Department of Management and Production Engineering - DIGEP, Politecnico di Torino, Italy Jos´e Luis GARC´ IA-LAPRESTA PRESAD Research Group, BORDA Research Unit, IMUVA, Departamento de Econom´ıa Aplicada, Universidad de Valladolid, Spain Abstract A general problem, which may concern practical contexts of different nature, is to aggregate multi-experts rankings on a set of alternatives into a single fused ranking. Aggregation should also take into account the experts’ importance, which may not necessarily be the same for all of them. We synthetically define this context as semi-democratic. The main aim of the paper is the analysis of the possible semi-democratic paradigms that can be conceived when the experts’ importance is not the same: (i) the importance is described by means of a weighting vector; (ii) the importance is expressed by a weak order on the set of experts; (iii) the importance is described by a weak order on the set of experts with additional information on the ordinal proximities among them. The three paradigms can be applied in different decision-making situations, where some experts perform multiple assignments. In this paper various situations are discussed and analyzed in detail. A series of examples, in the field of interior design of a new car, will complement the description. Keywords: group decision-making; semi-democratic decisions; qualitative scales; ordinal proximity measures; preferences; fusion techniques; aggregation. 1. Introduction A general problem, which may concern practical contexts of different nature, is to aggregate multi-experts rankings on a set of alternatives into a single fused ranking. Consider Table 1, where mdecision-making experts1formulate preference Email addresses: [email protected] (Fiorenzo FRANCESCHINI), [email protected] (Jos´e Luis GARC´ IA-LAPRESTA) 1By a decision-making expert we will refer to an abstract entity able to provide a decision: human beings, individual criteria in a multi-criteria decision process, software based intelligent agents on the Internet, etc. Preprint submitted to Information Fusion April 27, 2019 rankings among nalternatives of interest (x1, x2, x3, x4, etc.). Each ranking allows statements like x1x2,x1∼x2, where symbols and ∼mean “strictly preferred to” and “indifferent to”, respectively. The objective is to aggregate the mexperts’ rankings into a single fused one, which should reflect them as much as possible, even in the presence of divergent preferences. For this reason, the fused ranking can also be defined as consensus or compromise ranking (see Cook [10] and Herrera-Viedma et al. [25]). Aggregation should also take into account the experts’ importance, which is not necessarily equal for all of them. Inputs Output Experts Opinions Experts’ importance Social fused ranking e1x2(x1∼x3)x4weights x2(x1∼x3)x4 e2x3x2(x1∼x4) hierarchy · · · · · · hierarchy with ordinal proximity measures emx4x1x2x3 Table 1: Aggregation of multi-expert preference rankings into a single fused ranking. This decision-making problem is very diffused in a variety of real-life contexts, ranging from multi-criteria decision-aiding/making to social choice theory (see Arrow and Raynaud [3] and Greco et al. [23]). Two of the reasons for this diffusion are that (i) preference rankings are probably the most intuitive and effective way to represent preference judgments of alternatives, and (ii) they do not require a common reference scale – neither numeric, linguistic or ordinal – to be shared by the interacting agents (see Yager [45] and Chen et al. [8]). The scientific literature includes a large number of aggregation models, which have been analyzed extensively from the perspective of different axioms and properties (see Arrow [2], Fishburn [13], Saari [36], Cook [10] and Nurmi [34], among others). A long and lively debate has involved many scientists on the effects that the Arrow’s theorem can induce on practical decisions (see Arrow [2], Arrow and Raynaud [3], Franssen [19], See and Lewis [38], Keeney [28], Ladha et al. [29] and McComb et al. [31], among others). Some researchers have demonstrated the effectiveness of specific aggregation models, even though they do not satisfy some of the basic properties related to the Arrow’s theorem. For instance, Dym et al. [11] showed that, although the Borda aggregation model may not satisfy the Independence of Irrelevant Alternatives condition, this event rarely affect the most preferred alternatives. They concluded that Arrow’s theorem poses a considerable theoretical problem, but the practical implications are not so worrisome. Additional research has been carried out by See and Lewis [38], proposing a structured approach to avoid severe theoretical conflicts. Jacobs et al. [27] 2 recognized several additional issues related to the uncertainty, comparability and measurability concerned with aggregation models, both in the aggregation of preferences and performances. Franceschini and Maisano [16, 17] addressed the problem of the coherence between decision agent preferences and collective preference ranking. Other researchers focused their studies on proposing new approaches to manage linguistic distribution assessment in multi-attribute group decision making (see Keeney [28], de Andr´es et al. [1], Garc´ıa-Lapresta and P´erez-Rom´an [22], Yu et al. [46], Zhang et al. [49], Wu et al. [43] and Ure˜na et al. [40], among others). The literature is also rich in many practical applications in various fields. As an example, a small review is shown in Table 2 (see Greco et al. [23], ¨ On¨ut et al. [35], Yager [25], Griffin and Hauser [24], Franceschini et al. [14], Colomer [9], Saari [36] and Fishburn [13]). The main aim of the paper is the analysis of the possible paradigms that can be conceived when the experts’ importance is not the same. We synthetically define this context as semi-democratic. The term semi-democracy is used to refer to a context that shares both democratic and authoritarian features (see Møller and Skaaning [33]). In this specific framework the term is used to highlight that all experts participate (democracy) to the collective fused ranking, although they may have a different weight in the decision (semi or partial democracy). The modelling of these decision-making problems has only been partially explored in the literature. The complexity of the problem is due to the difficulty of modeling the different degree of importance of the decision experts. In some cases the numerical weight associated with each single expert is known, while in other cases only the expert hierarchy (but not their relative weights) is known. In some other situations it is possible to know, in addition to the expert hierarchy, also the proximities between the hierarchy levels. Most of the papers in the literature focus their attention on decision problems where the weights associated with the individual experts are known (see Arrow and Raynaud [3] and Greco et al. [23]); other papers concentrate on the technique for determining these weights (see Yue [47], Dubois et al. [12], Zhang and Guo [48], Mishra and Rani [32] and Hu et al. [26], among others). However, when we move on to less structured problems, where the hierarchy level between decision makers is expressed by more nuanced information i.e., only the hierarchical level of experts is known) or by a hierarchical proximity (i.e., only the hierarchical proximity of experts is known), the literature offers only few approaches for tackling this problem (see Yager [45]). By this paper we wish to provide a structured conceptual framework on the state of the art and on the potential future research areas for decision-making problems in semi-democratic contexts. In this manuscript various situations will be discussed and analyzed in detail. A series of examples, inspired by a real application in the field of interior design of a new car, will complement the description. The rest of the paper is organized as follows. Section 2 introduces Borda scores and ordinal proximity measures. Section 3 includes the paradigms considered in the paper to categorize semi-democratic contexts. Section 4 contains 3 Field Agents Alternatives Problem description Multicriteria decision aiding/making Qualitative/quantitative criteria Alternative locations Determination of the best location where to install a new manufacturing plant on the basis of several criteria such as road/railway infrastructure, electrical supply, labour cost, etc. (see Greco et al. [23]) Multicriteria decision aiding/making Qualitative/quantitative criteria Technology selection Machine tool selection (see ¨ On¨ut et al. [35]) Internet Different types of information concerning the user Data displayed on Internet sites Intelligent customization of data displayed on Internet sites, based on several types of information such as user’s country, websites visited previously, apps downloaded, etc. (see Yager [44]) Quality management Questionnaire/interview respondents Customer requirements Synthesis of customer requirements, which are evaluated by a sample of questionnaire/interview respondents (see Griffin and Hauser [24] and Franceschini et al. [14]) Voting theory Voters Candidates in an election Searching a reasonable mechanism for aggregating the opinions expressed by several voters on the candidates, in order to determine a winner or to rank all candidates in order of preference (see Colomer [9], Fishburn [13] and Saari [36]) Table 2: Examples of practical applications of the problem of interest. 4 a case study. Section 5 includes some concluding remarks. 2. Preliminaries Let E={e1, . . . , em}be a set of experts. With W(E) we denote the set of weak orders (or complete preorders) on E. Given S∈W(E), with and ∼ we denote the asymmetric and the symmetric parts of S, respectively. Given a set Y, with #Ywe denote the cardinality of Y. Definition 1. Given S∈W(E), let B:E−→ {0,0.5,1, . . . , m −1.5, m −1} be the mapping that assigns the Borda score of each expert ei∈E, defined as B(ei)=#{ej∈E|eiej}+1 2#{ej∈(E\ {ei})|ej∼ei}.(1) Example 1. Consider S∈W({e1, . . . , e5}) given by S e1 e2e3 e4e5 i.e., e1(e2∼e3)(e4∼e5). Then, we have B(e1) = 4, B(e2) = B(e3) = 2.5 and B(e4) = B(e5)=0.5. We now recall the notion of ordinal proximity measure, introduced by Garc´ıa- Lapresta and P´erez-Rom´an [22]. An ordinal proximity measure is a mapping that assigns an ordinal degree of proximity to each pair of linguistic terms of an ordered qualitative scale (OQS)L={l1, . . . , lg}, with l1<· · · < lgand g≥3. The mentioned ordinal degrees of proximity belong to a linear order ∆ = {δ1, . . . , δh}, with δ1 · · ·  δhand h≥3, being δ1and δhthe maximum and minimum degrees of proximity, respectively. It is important noticing that the elements of ∆ are not numbers. In fact, they are only abstract objects representing different degrees of proximity. Definition 2. ([22]) An ordinal proximity measure (OPM) on Lwith values in ∆ is a mapping π:L2−→ ∆, where π(lr, ls) = πrs represents the degree of proximity between lrand ls, satisfying the following conditions: 1. Exhaustiveness: For every δ∈∆, there exist lr, ls∈ L such that δ=πrs. 2. Symmetry:πsr =πrs, for all r, s ∈ {1, . . . , g}. 3. Maximum proximity:πrs =δ1⇔r=s, for all r, s ∈ {1, . . . , g}. 4. Monotonicity:πrs πrt and πst πrt, for all r, s, t ∈ {1, . . . , g}such that r < s < t. 5 Every OPM π:L2−→ ∆ can be represented by a g×gsymmetric matrix with coefficients in ∆, where the elements in the main diagonal are πrr =δ1, r= 1, . . . , g:      π11 · · · π1s· · · π1g · · · · · · · · · · · · · · · πr1· · · πrs · · · πrg · · · · · · · · · · · · · · · πg1· · · πgs · · · πgg       . This matrix is called the proximity matrix associated with π. Some procedures for generating OPMs in an OQS are introduced by Garc´ıa- Lapresta et al. [21]. 3. Paradigms to categorize semi-democratic contexts Let E={e1, . . . , em}be a set of experts that show their preferences on a set of alternatives X={x1, . . . , xn}through a profile of weak orders (R1, . . . , Rm)∈W(X)m. The aim is to generate a social weak order R∗∈W(X) representing individual preferences taking into account that the importance of experts may be different. According to the content of Table 3, we can identify three potential paradigms to categorize the concept of semi-democratic context for a set of experts: 1. A numerical weight wi∈[0,1] is assigned to each expert ei∈E. 2. The set of experts is categorized by a hierarchy, i.e., a weak order on the set of experts, S∈W(E). 3. The set of experts is categorized by a graduated hierarchy within an OQS equipped with an OPM, v1, . . . , vm∈ L. Inputs Output Experts Opinions Experts’ importance Social fused ranking e1R1∈W(X)w1, . . . , wm∈[0,1] R∗∈W(X) e2R2∈W(X)S∈W(E) · · · · · · v1, . . . , vm∈ L emRm∈W(X) Table 3: Aggregation of multi-expert preference rankings into a single fused ranking. 3.1. Assigning a numerical weight to each expert In some contexts experts may have recognized abilities and attributes and/or privileged positions of power, represented by weights (see Dubois et al. [12], Brans and Mareschal [6] and Greco et al. [23]). 6 The definition of the experts’ weights can be a very delicate issue. In some settings, the weight of an expert may be well defined; for example, the Gross National Product (GNP) or population size of a country represented by the member on an international committee can immediately be used as weights. In many other situations the definition of the weights is controversial, because there are no indisputable criteria that can be used for this operation. Weights are often imposed by decision-makers, according to political strategies (see Wang et al. [42]). For example, the scientific committee of a competitive examination for promotion of Faculty members may decide that scientific publications will account for 30% of the total performance, the international projects for 25%, the teaching activity for 35%, etc. The literature includes several techniques for the quantification of weights. For example, the AHP procedure uses the eigenvector method to derive a weight vector relating to experts (see Saaty [37]), while the method proposed by Martel and Ben Kh´elifa [30] determines the so-called “relative importance coefficient” of each expert, based on the combination of subjective and objective components. More specifically, the importance of experts is directly reported by means of a weighting vector w= (w1, . . . , wm)∈[0,1]msuch that w1+· · · +wm= 1 and 100 wi∈N, for every i∈ {1,...m}. The weighting scheme should follow the replication proposal given by Garc´ıa- Lapresta and Gonz´alez del Pozo [20]: the weak orders associated with the experts are replicated according to the corresponding percentages, 100 w1,. . . , 100 wm. In practice, it should be convenient to calculate the greatest common divisor (gcd) of percentages associated with the weights, and divide each percentage by the gcd. Then, the minimum number of replications of each profile is obtained: ti=100 wi gcd{100 w1,...,100 wm}, i ∈ {1, . . . , m}. If (t1, . . . , tm)/∈Nm, then these numbers have to be properly rounded. Thus, the experts’ weak orders of the original profile (R1, . . . , Rm)∈W(X)m are replicated accordingly:   t1 z }| { R1, . . . , R1,..., tm z }| { Rm, . . . , Rm ∈W(X)t1+···+tm. In some settings, weights are not available or cannot be defined on cardinal scales. In these cases, the importance hierarchy of agents may be expressed by a weak order (see Yager [45]). When the expert importance prioritization is doubtful, the formulation of rankings is certainly simpler and more intuitive than the formulation of weights (see Chen et al. [8]). 3.2. Hierarchy of experts In this subsection we will focus on a specific aggregation problem in which the experts’ importance is expressed through a weak order. This decision-making 7 context can be denominated as “ordinal semi-democratic”; the adjective “semidemocratic” indicates that agents do not necessarily have the same importance, while “ordinal” indicates that their rank is defined by a crude ranking. This makes the set of the possible solutions relatively wide, since they may range between the two extreme situations of (i) full dictatorship – in which the resulting fused ranking coincides with the preference ranking by the most important agent (dictator) – and (ii) full democracy – where the agents’ preference rankings are considered as equi-important. In spite of its practicality and adaptability to a large number of real contexts, this specific decision-making problem is almost completely ignored in the literature. Over ten years ago, Yager [45] proposed an algorithm (hereafter abbreviated as YA, which stands for Yager’s Algorithm) to address this problem in a relatively simple and fast way. Unfortunately, this algorithm has two important limitations: (i) the resulting fused ranking may sometimes not reflect the preference ranking for the majority of experts (see Wang [41]) and (ii) it is only applicable to linear orders, without incomparabilities and omissions of the alternatives of interest. The paper of Franceschini et al. [18] enhances the YA in order to overcome its limitations and adapt to less stringent preference rankings. In a formal way, the importance of experts is represented by means of a weak order S∈W(E). In this situation we can operate in two ways: 1. Direct method: adopting the YA algorithm or similar variants (Yager [45], Franceschini et al. [18]; see Section 4). 2. Indirect method: we can generate an “artificial” weighting vector based, for example, on the Borda scores obtained by the experts (Definition 1), wi=B(ei) B(e1) + · · · +B(em), i ∈ {1, . . . , m}, going back to the case analyzed in Subsection 3.1. It can be noticed that Sen [39] has already considered the Borda scores as weights of the objects in a ranking. In the Example 1, the following weighting vector w= (0.4,0.25,0.25,0.05,0.05) is obtained. Since gcd{100 w1,...,100 w5}= gcd{40,25,25,5,5}= 5, then R1,R2,R3, R4and R5should be replicated 40/5 = 8, 25/5 = 5, 25/5 = 5, 5/5 = 1 and 5/5 = 1 times, respectively:   8 z }| { R1, . . . , R1, 5 z }| { R2, . . . , R2, 5 z }| { R3, . . . , R3, R4, R5 ∈W(X)20. An equivalent approach consists of directly calculate the number of replications avoiding to obtain the weighting vector and the corresponding rounding problems. 8 Since B:E−→ {0,0.5,1, . . . , m −1.5, m −1}, i.e., B(ei) could be not integer, consider ti=2B(ei) gcd{2B(e1),...,2B(em)}, i ∈ {1, . . . , m}.(2) So, in Example 1 we directly obtain t1= 8, t2=t3= 5 and t4=t5= 1. Notice that the importance of expert e1is 8/5 = 1.6 times the importance of experts e2and e3; the importance of expert e1is 8/1 = 8 times the importance of experts e4and e5; and the importance of experts e2and e3is 5/1 = 5 times the importance of experts e4and e5. 3.3. Ordinal proximity measures This subsection introduces the third paradigm. In this case the experts’ importance is expressed again through a ranking, with an additional measure of the proximity (proximity graduation) of the ordinal semi-democratic hierarchy. A decision-maker evaluates the experts in an OQS L={l1, . . . , lg}equipped with an OPM π:L2−→ ∆ = {δ1, . . . , δh}, by assigning a linguistic term vi∈ L to each expert ei∈E. Let ρ: ∆ −→ Nbe the mapping defined as ρ(δr) = r. A score is given to each expert ei∈Ethrough the mapping S:E−→ R defined as S(ei) = h+ρ(π(vi, l1))−ρ(π(vi, lg))+ X vi>vj ρ(π(vi, vj))+ 1 2X vi=vj i6=j ρ(π(vi, vj)).(3) Since S(ei) could be not integer, following the same pattern that in (2), consider ti=2S(ei) gcd{2S(e1),...,2S(em)}, i ∈ {1, . . . , m}. The meaning of S(ei) in (3) is explained as follows. In order to all the scores S(e1), . . . , S(em) be positive, hpoints are initially assigned to each expert, just the number of ordinal degrees of proximity; π(vi, l1) measures the proximity between the assessment of eiand the lowest possible assessment, l1(the lower, the better); π(vi, lg) measures the proximity between the assessment of eiand the highest possible assessment, lg(now the higher, the better). Consequently, ρ(π(vi, l1)) −ρ(π(vi, lg)) is the number of steps for going from π(vi, l1) to π(vi, lg), being this difference positive whenever the assessment of eiis closer to lgthan to l1, and negative in the opposite case. Notice that h+ρ(π(vi, l1)) −ρ(π(vi, lg)) can be considered as the absolute part of the score S(ei), in the sense that it does not depend on the assessments obtained for other experts. However, the second part of S(ei), X vi>vj ρ(π(vi, vj)) + 1 2X vi=vj i6=j ρ(π(vi, vj)), 9 Alternatives x1x2x3x4x5 t1·B1(xi) 8 ·4 = 32 8 ·1.5 = 12 8 ·3 = 24 8 ·0 = 0 8 ·1.5 = 12 t2·B2(xi) 7 ·1 = 7 7 ·4 = 28 7 ·1 = 7 7 ·1 = 7 7 ·3 = 21 t3·B3(xi) 3 ·4 = 12 3 ·2.5 = 7.5 3 ·0 = 0 3 ·1 = 3 3 ·2.5=7.5 t4·B4(xi) 7 ·0.5=3.5 7 ·3 = 21 7 ·2 = 14 7 ·0.5=3.5 7 ·4 = 28 t5·B5(xi) 3 ·1.5=4.5 3 ·3 = 9 3 ·1.5=4.5 3 ·0 = 0 3 ·4 = 12 t6·B6(xi) 2 ·3.5 = 7 2 ·3.5 = 7 2 ·0.5 = 1 2 ·0.5 = 1 2 ·2=4 t7·B7(xi) 8 ·4 = 32 8 ·2 = 16 8 ·2 = 16 8 ·0 = 0 8 ·2 = 16 t8·B8(xi) 2 ·2.5 = 5 2 ·2.5 = 5 2 ·4 = 8 2 ·0 = 0 2 ·1=2 t9·B9(xi) 2 ·2.5 = 5 2 ·2.5 = 5 2 ·0.5 = 1 2 ·0.5 = 1 2 ·4=8 t10 ·B10(xi) 3 ·1 = 3 3 ·4 = 12 3 ·2 = 6 3 ·0 = 0 3 ·3=9 10 X k=1 tk·Bk(xi) 111 122.5 81.5 15.5 119.5 Table 6: Scores. (c) Generation of the fused ranking. Step (a). Based on the different classes we have the following hierarchy among experts: (e1∼e7)(e2∼e4)(e3∼e5∼e10)(e6∼e8∼e9). According to the four classes of importance, the preference rankings of Table 5 are reorganized in Table 7. Classes are strictly decreasing in terms of importance. Each cell element originates from the level-by-level union of the preferences related to each single expert. Importance class A B C D Experts {e1, e7} {e2, e4} {e3, e5, e10} {e6, e8, e9} Reorganized x1x1x2x5x1x2x5x1x2x3x5 preferences x2x2x3x5x2x5x2x2x5x5x1x1x2x2x5 x2x4x5x1x3x3x4x1x3x3x4x3x3x4x4x5 x4x1x4x1x3x4x4 x4 Table 7: Reorganized preferences according to the four classes of experts. Step (b). The second step of the method concerns the construction of the reading sequence. The reading sequence represents the ordered path followed by the algorithm to allocate the alternative positions (see Table 16 8). The logic of the sequence is to read the most preferred alternative first (Franceschini et al. [15]). Importance class A B C D Experts {e1, e7} {e2, e4} {e3, e5, e10} {e6, e8, e9} 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Table 8: Reading sequence number (S) related to the reorganized vectors in Table 7. Step (c). This last step generates the final fused ranking. A step-by-step application of the Ordinal Prioritization Method is illustrated in Table 9. A detailed description of the method is reported in Franceschini et al. [15]. Data is related to the example of Tables 7 and 8. The first three columns are related to the reading sequence: Sis the sequence number, jdenotes the importance class selected, while the column Element (I) is the set of alternatives taken, step by step, from the table of the reorganized preferences (Table 7). The subsequent columns refer to the construction of the total ranking. Eis the set of alternatives included in the gradual ranking and Ris the set of alternatives not yet included in the gradual ranking (residual elements). We remark that an alternative is added to the total ranking only when the number of occurrences is greater than or equal to Tk(occurrence threshold). Tkis defined by the algorithm designer. It is worth noting that greater values of Tkassign less significance to the ranking of experts. In this case study we fix Tk= 3 for all the alternatives. By this approach the final ranking of the alternatives is: x1(x2∼x5)x3x4. 2. We can generate an “artificial” weighting vector based, for example, on the Borda scores obtained by the experts, coming back again to Subsection 3.1 Taking into account Eq. (1), we have B(e1) = B(e7)=8.5, B(e2) = B(e4)=6.5, B(e3) = B(e5) = B(e10) = 4 , B(e6) = B(e8) = B(e9)=1. Hence, t1=t7= 17 , t2=t4= 13 , t3=t5=t10 = 8 , t6=t8=t9= 2. Consequently, R1,R2,R3,R4,R5,R6,R7,R8,R9and R10 should be replicated 17, 13, 8, 13, 8, 2, 17, 2, 2 and 8 times, respectively. 17 Ocurrencies (Ok) S j Element (I)E x1x2x3x4x5Residual elements (R) Gradual ranking 0{x1, x2, x3, x4, x5} 1 A {x1, x1}20000 {x1, x2, x3, x4, x5} 2 B {x2, x5}21001 {x1, x2, x3, x4, x5} 3 C {x1, x2, x5} {x1}32002 {x2, x3, x4, x5}x1 4 D {x1, x2, x3, x5} {x2, x5}43103 {x3, x4}x1(x2∼x5) 5 A {x2, x2, x3, x5}45204 {x3, x4}x1(x2∼x5) 6 B {x2, x5}46205 {x3, x4}x1(x2∼x5) 7 C {x2, x2, x5, x5}48207 {x3, x4}x1(x2∼x5) 8 D {x1, x1, x2, x2, x5}6 10 2 0 8 {x3, x4}x1(x2∼x5) 9 A {x2, x4, x5}6 11 2 1 9 {x3, x4}x1(x2∼x5) 10 B {x1, x3, x3, x4} {x3}7 11 4 2 9 {x4}x1(x2∼x5)x3 11 C {x1, x3, x3, x4} {x4}8 11 6 3 9 x1(x2∼x5)x3x4 End Table 9: Step-by-step application of the Ordinal Prioritization Method (Franceschini et al. [15]). 18 Then, we have the following profile:   17 z }| { R1, . . . , R1, 13 z }| { R2, . . . , R2, 8 z }| { R3, . . . , R3, 13 z }| { R4, . . . , R4, 8 z }| { R5, . . . , R5, 2 z }| { R6, . . . , R6, 17 z }| { R7, . . . , R7, 2 z }| { R8, . . . , R8, 2 z }| { R9, . . . , R9, 8 z }| { R10, . . . , R10 ∈W(X)90. If we apply the Borda rule to this profile, we obtain the following total scores for x1,x2,x3,x4and x5, respectively: 224.5, 243.5, 162, 29.5 and 240.5. Then, the final ranking of the alternatives is x2x5x1 x3x4. 4.3. Ordinal proximity measures Under the approach of Subsection 3.3, we will consider three different OPMs. Once the number of replications of each weak order are obtained, following the procedure illustrated in Example 3, we apply the Borda rule to the corresponding profiles. (a) With the OPM with associated proximity matrix A323 =     δ1δ3δ4δ5 δ1δ2δ4 δ1δ3 δ1    , that can be visualized in Fig. 4, the final ranking of the alternatives is: x1x2x5x3x4. l1l2l3l4 Figure 4: Ordinal proximity measure with associated matrix A323. (b) With the OPM with associated proximity matrix A232 =     δ1δ2δ4δ5 δ1δ3δ4 δ1δ2 δ1    , that can be visualized in Fig. 2, the final ranking of the alternatives is: x2x5x1x3x4. 19 (c) With the OPM with associated proximity matrix A423 =     δ1δ4δ6δ7 δ1δ2δ5 δ1δ3 δ1    , that can be visualized in Fig. 5, the final ranking of the alternatives is: x1x2x5x3x4. l1l2l3l4 Figure 5: Ordinal proximity measure with associated matrix A423. Taking into account the opinions of the ten customers on the five interior designs of a new car included in Table 5, the outcomes obtained under the approaches introduced in Subsections 3.1, 3.2 and 3.3, developed for the case study in Subsections 4.1, 4.2 and 4.3, respectively, are summarized in Table 10. Approach Case study Subcase Preference ranking Subsection Subsection 3.1 4.1 x2x5x1x3x4 3.2 4.2 1 x1(x2∼x5)x3x4 3.2 4.2 2 x2x5x1x3x4 3.3 4.3 A323 x1x2x5x3x4 3.3 4.3 A232 x2x5x1x3x4 3.3 4.3 A423 x1x2x5x3x4 Table 10: Summary. The winner is x1or x2, depending on the case, but when x2is the winner, x1is always in the third position. The fourth and the fifth positions are always for x3and x4, respectively. Taking into account the six cases considered, x1,x2,x3,x4and x5have average positions 2, 1.58, 3, 4 and 2.42, respectively. Thus, on average, the final ranking is x2x1x5x3x4, that it does not coincide with any of the outcomes obtained in the six cases. Notice that this ranking is the same than the one obtained when applying the Borda rule to the six preference rankings of Table 10. This is due to the fact that the Borda rule ranks the alternatives according to their average positions. 5. Conclusions The proposed method allow to aggregate multi-experts rankings of different alternatives into a single fused ranking according to different semi-democratic 20 paradigms: (i) the importance of experts is directly reported by means of a weighing vector; (ii) the importance of experts is expressed by a weak order on the set of expert; (iii) the importance of experts is described by a weak order with ordinal proximity measures on the set of expert. The three paradigms can be applied in different decision-making situations, where some experts perform multiple assignments. The results obtained in the case study highlight the following aspects: •Different methods lead to different rankings of the alternatives, even if sometimes they appear to coincide. •There is a general agreement between the methods for top and bottom positions in the rankings. •The use of one method or another depends on the quality of information available from the different semi-democratic decision making contexts. It is important to mention some advantages of the methods proposed in this paper with respect to other proposals: •Their simplicity and the greater adherence of data properties to real situations. •Weights are treated in a purely ordinal way by replicating experts’ opinions according to the proportions between weights. •The use of minimal structures for representing hierarchies on the set of experts (weak orders and ordinal proximity measures). For instance, taking a particular membership function is a much stronger hypothesis than considering a weak order between experts. The main contribution of this paper is to provide a general overview of the state-of-art of the methods able to tackle decision-making problems in semidemocratic contexts. In our analysis it was assumed that the preference rankings of experts are complete; i.e., all experts are able to rank all the alternatives of interest, without omitting any of them. The analysis does not consider the (possible) uncertainty in expert rankings, and/or preference rankings with incomparability between some alternatives. Regarding the future, we plan to extend the analysis to situations where experts are not able to provide complete weak rankings, but only partial preference rankings, uncertainty rankings or even rankings with some forms of incomparability between alternatives to be evaluated. Acknowledgments. 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