Elsevier Editorial System(tm) for Expert Systems With Applications Manuscript Draft Manuscript Number: ESWA-D-14-02232 Title: A consensus bipolar approach for group decision making problems Article Type: Full Length Article Keywords: group decision problem, consensus process, bipolarity, concordance, discordance, influence Corresponding Author: Dr. Yasmina Bouzarour-Amokrane, Ph.D. Corresponding Author's Institution: First Author: Yasmina Bouzarour-Amokrane, Ph.D. Order of Authors: Yasmina Bouzarour-Amokrane, Ph.D.; Ayeley Tchangani, Ph.D.; François Pérès, Pr.
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A consensus bipolar approach for group decision making problems Yasmina BOUZAROUR-AMOKRANE1, Ayeley TCHANGANI1,2, François PERES1 3Universite
De Toulouse / LGP 47, Avenue d'Azereix, BP 1629, 65016 Tarbes Cendex- France.
[email protected], francois.peres@enit 2Université
de Toulouse / IUT de Tarbes 1, rue Lauréamont, 65016 Tarbes cedex – France.
[email protected]
Abstract. This paper addresses the collaborative group decision making problems considering a consensus processes to achieve a common legitimate solution. The proposed resolution model is based on individual bipolar assessment. Each decision maker evaluates alternatives through selectability and rejectability measures which respectively represent the positive and negative aspects of alternatives considering objectives achievement. The impact of human behavior (influence, individualism, fear, caution, etc.) on decisional capacity is taking into account. The influence degrees exerted mutually by decision makers are modeled through concordance and discordance measures. The individualistic nature of decision makers is taking into account from the individualism degree. In order to achieve a common solution(s), consensus building models are proposed based on the satisficing game theory formalism for collective decision problems. Application example is given to illustrate the proposed concepts. Keywords: group decision problem, consensus process, bipolarity, concordance, discordance, influence.
1
Introduction
Nowadays, the increasing complexity of the socio-economic, engineering and environmental management make less possible decision by a single decision maker considering all aspects of a problem [1]. Therefore, the majority of decision problems are considered currently in the group decision process. This process is generally characterized by the existence of two or more persons (i) who have different perceptions, attitudes, motivations and personality,(ii) who recognize the existence of a common problem, and (iii) attempt to reach a collective decision [2]. Solve a decision problem often goes through the following phases: elicitation phase where different characteristics of the problem are defined by the decision group (objectives, alternatives, attributes, etc.), evaluation phase and a selection and recommendation phase. In the evaluation phase, the way information is managed can leads to two families of aggregation approaches, we speak of input and output aggregation [3] or common value tree for all decision makers and a value tree for each decision maker [4]. In the first case, aggregation is performed at the input when the decision group is invited to agree on a common set of 1
attributes, weights and other parameters, which amounts to solving a problem as a single decision making problem. In the second case, the individual evaluations are represented by individual value trees solved using standard process of decision support. The output aggregation is performed at the end. The present paper focuses on this second type of problem dealing with group decision making problem based on individual assessments. In the evaluation phase, decision makers can represent their assessments by preference order (where the alternatives are ranked from best to worst), a utility function (where the alternatives are represented by real value -physical or monetary value-), or a frequently used preference relation (where alternatives are evaluated by pairwise comparison) [5]. Depending on the nature of the data, the certainty of decision-makers, these preferences can be modeled by absolute evaluations when information is known or fuzzy evaluation in case of uncertainty. Currently, the fuzzy preferences are used in many areas due to the pressure, lack of knowledge and / or time. In general the fuzzy preferences can be modeled qualitatively by linguistic labels ([6],[7]) or quantitatively by numerical values representing the intensity of preferences ([8],[9]) or the degree of preference ([10],[11]). These fuzzy preference relations can be expressed by policy makers in different ways [12] ; interval preference relations ([13],[14]), triangular fuzzy preferences relations ([15]), intuitionistic fuzzy preference relations ([16],[17]), incomplete fuzzy preference relations ([18]), etc. There are also many studies that use multiplicative preference relations ([5],[19]), Such as incomplete multiplicative preferences ([20],[21]), interval multiplicative preferences ([18]), incomplete interval multiplicative preferences ([22]), triangular fuzzy multiplicative preferences ([18]), intuitionistic multiplicative preference relation ([23]), preferences multiplicative intuitionistic language ([12]), etc. These decision-makers evaluations are integrated into decision resolution procedures to reach an agreement on the selection of the best solutions. However, as a group decision members usually come from different horizons with different specialty areas and different levels of knowledge, each group member has distinct information and sharing in general a part of the objectives with other decision members [24]. This implies that individual assessments rarely meet ([25], [26]) and the divergence of opinions can generates conflict (disagreement) and / or agreement within the group decision making. The recommendation phase in this case usually requires the establishment of a consensus building process in order to reach the actors to a decision or common solutions [27]. To achieve a common accord, a variety of proposed models and resolution methods have been proposed in recent years. These approaches going from mechanist models of operational research to more sophisticated and soft computing oriented models that attempt to integrate human attitude (emotion, affect, fear, egoism, altruism, selfishness, etc.) [28]. The soft computing oriented models are used increasingly due to their ability to tolerate imprecision, uncertainty and partial truth in order to simulate human behavior with low cost [29], they allow to take into account the ambiguity in human thinking and uncertainty of the real world [30]. Consensus building processes The group consensus is usually considered as a total and final agreement between the decision members [3]. To reach a consensus, the researchers first proposed resolution procedures using group consensus functions that aggregate decision maker evaluations in a unique value representing the common opinion. Several aggregation methods have been proposed in the 2
literature, simple average ([31]), geometric mean ([32]), Bayesian aggregation ([33]), aggregation using the analytic hierarchy process (AHP) (see eg [34],[35],[36],[37]), fuzzy set theory ([38],[39],[40],[41]), multi-criteria decision analysis methods ([42]), etc. The individual results of each alternative are obtained using tools as, utility functions that aggregate attributes evaluation considering their predefined weight in terms of a process of multi-criteria decision method [43] or, aggregation operator based on distances to the positive and negative ideal solutions by considering weight of attributes and / or decision-makers ([41], [44],[45],[46]). However, some researchers consider that unanimity is not required in the real decision problem and use consistency measures, also called degrees of 'soft' consensus ([47], [48], [49]). The consensus degrees are generally represented by consensus measures and proximity measures. The consensus measures indicate the degree of agreement between the expert opinions while proximity measures identify the distance between individual solutions and collective solution. Among the soft consensus models proposed in the literature, Herrera et al. [49] proposes to measure the consensus degree and linguistic distances. In [47], Herrera et al. propose a consensus model for group decision problem with fuzzy incomplete preference relations evaluation, they offer different preference structures based on two consensus criteria, a consensus measure which indicates the degree of agreement between the expert opinions and a proximity measure to evaluate the distance between individual solutions and collective solution. Tapia et al. [50] proposed a feedback mechanism in the soft consensus reaching process with fuzzy intervals for linguistic relationships. Consensus criteria (consensus and proximity measures) are calculated for each level of representation of preference relations. The fuzzy environment is also addressed by Khorshid [27] which provides a soft consensus model based on the flexible idea of coincidence between positive and negative ideal agreement. A consensus indicator is proposed to assess the degree of agreement of the group. To avoid changes of the evaluations by decision makers, Ben-Arieh et al. [51] have defined the scores assigned to decision makers in terms of their contribution to the group. Reaching a consensus is achieved by adjusting the weight of contribution to reach consensus and strengthen the joint decision without modification evaluations. The 'soft' consensus models have also been proposed for dynamic decision problems where the set of alternatives is considered dynamic. For example, in [52] Pérez et al. propose a new framework for linguistic group decision problems using the preference measures and expert levels of agreement to manage the change of some alternatives in a mobile decision system.
Considering group decision making problem, this paper propose a consensus building soft process based on proximity measures and bipolar consensus incorporating aspects of human behavior (positive affect, negative affect, selfish, prudence, etc.). The soft consensus is chosen because of its flexibility to achieve a maximum degree of agreement between the actors from individual preferences. The proposed processes avoids compensation may occur with a total aggregation and represents graphically the convergence of decision maker preferences. Considering a realistic context, human factors are incorporated into the model to represent the impact of human nature on the final decision. By considering social ties, proposed approach integrates the mutual influence of positive and negative interactions of the group members by considering individualism degree of each one. The weight of the decision makers which was the subject of several studies ([53],[54],[46],[41]) is also defined in this paper. Then an adapted 3
soft consensus process based on proximity measure and bipolar consensus measures is proposed to reach an agreement and select a common alternative for the group. Considering a bipolar framework, initial decision makers evaluations are represented by bipolar measures that express the degree of supportability and rejectability of alternatives. These bipolar evaluations can be obtained using BOCR analysis that allows a structured framework to evaluate the alternatives based on the benefit (b), opportunity (o), cost (c) and risk (r) factors. The (b), (o), (c), (r) factors are then aggregated on bipolar measures where (b) and (o) factors represent selectability measure and (c) and (r) represent rejectability measures. The satisficing game theory is used here as aggregation and recommendation tool. The remainder of the paper is organized as follow. Section 2 describe global framework of bipolar modeling of decision group problems when considering members interaction. The satisficing game theory used as aggregation tool is briefly described in this section. In section 3, presents two proposed consensus and selection process, an application example is given in section 4. Eventually section 5 provides an application example.
2 Bipolar group decision making framework modeling under interactions This section discusses the structuring and evaluation procedure based on bipolar analysis approach. Considering Multi-Attributes Multi-Objectives Group Decision Making problem (MAMOGDM probem), the general framework of the bipolar approach is presented and the satisficing game theory used as aggregation tool in individual evaluations is briefly described. The objective of proposed model is to provide a framework for integrating the human factor evaluation through parameters taking into account (positive/ negative) influence among decision makers considering the individualism of decision makers. 2.1
Proposed bipolar structuration of MAMOGDM problem
Let us consider a MAMOGDM problems characterized by the set of decision makers , , … . where each actor evaluate individually a common, finite and static set of alternatives noted , , … to achieve fixed objectives , , … . The alternative evaluation is done using a set of attributes (or criteria) noted , , … , for each objective . In the bipolar framework, alternatives are evaluated through bipolar ‘a priori’ meas ures. These measures noted /! , represent respectively the ‘supportability’ and the ‘rejectability’ degree accorded by the decision maker for an alternative without considering potential influence of his vicinity. These bipolar measures can be obtained using BOCR analysis that brings structured evaluation considering benefit, opportunity, cost and risk aspects. Each decision maker can evaluate alternatives through b, o, c, r factors. We noted by " , # , $ , % respectively the evaluations of (b), (o), (c), (r) factors given by the decision maker for the alternative with regard to achievement of the objective . These evaluations are aggregated for each decision maker over all objectives to represent each alternative with (b), (o), (c), (r) values noted " , # , $ , % . The bipolar measures are obtained by aggregation of (b), (o) 4
factors in the ‘selectability’ measure and (c), (r) factors in the ‘rejectability’ measure using the satisficing game theory introduced below. 2.2
Satisficing game theory
The satisficing game theory is based on the fact that decision makers in solving real problems do not necessarily seek the optimum solution -often costly in terms of time and money- but a satisfactory solution whose capabilities are estimated fairly good regarding to objectives achievement [55]. The satisficing game theory is based on this observation and provides adequate tools for the selection of satisficing alternatives. The concept of being good enough is suitable for our approach, where an alternative can be considered good enough when its supporting contribution exceeds the rejecting one. The satisficing game theory is characterized by the triplet 〈 , , ! 〉 where:
: is a set of alternatives (discret set) / ! : are mass functions defined in on the interval )0,1, where , ! measure the degree to which works towards success in achieving the decision maker’s goal and costs associated with this alternative respectively [56].
Thus, in the considered BOCR analysis framework, the bipolar measures using satisficing game theory are obtained with following equation.
!
- " . 1 / - # 1 ∑ - " . 1 / - #
1 / - $ . - % 2 ∑ 1 / - $ . - %
where 0 2 -3 2 1 :is risk aversion index of decision maker . It permits to consider the risk aversion of a decision maker on selection phase.
More -3 is close to 1, greater is risk aversion of decision maker who, pessimistic, will tend to give more importance to risk than cost in rejectability measure (equation (2)) and penalize opportunity in favor of benefit in selectability measure (equation (1)). Inversely, when the risk aversion index tends to 0, the decision maker is optimist. He will focus on opportunity to benefit in the selectability measure, and will overlook risk against cost in the rejectability measure. The satisficing game theory is characterized by the several sets that can be used on the recommendation phase. The satisficing set S5 ⊆ (at a boldness or caution index 7 of decision maker ) is the set of alternatives defined as following. 85 9: ∈ : < 7 ! = 3
7 is the caution index of decision maker that can be used to adjust the aspiration level: increase 7 allows reducing a satisficing set 85 (if too many alternatives are declared satisfic5
ing), on the contrary, decrease 7 allows increasing satisficing set (if 85 is empty for instance),
see figure 1.
ì
ì < 7 ì!
NO
ì!
Figure 1. graphical representation of satisficing set
But some satisficing alternative can be dominated by others alternatives presenting a higher selectability measure and a lower rejectability measure. To identify these alternatives an equilibrium set ? is defined by decision maker as follow. ? 9 ∈ : @ A = 4
where @ is the set of alternatives that are strictly better than (see figure 2). The set @ is defined with equations (5) @ ∪ ! 5
where E and F are defined by equations (6) and (7) below
G ∈ : ! G H ! I G < 6
! G ∈ : ! G 2 ! I G K 7 ì
ì
@ A
ì!
ì! Figure 2.graphic representation of non-dominated alternatives Thus, the satisficing equilibrium set ?E, is given as follow 6
?E, ? ∩ 85 8
Notice that the set ?E, constitutes a Pareto-equilibria set with an incomparability between a pair of alternatives and a trade-off process can be necessary for final choice in case of indecision. These alternatives are those laying on the portion of broken line green curve (above which there is no alternatives meaning the Pareto equilibrium) that is above the red line (separating satisficing and non-satisficing alternatives) of the following figure 3. ì
ì / 7 ì!
ì!
Figure 3. positions of alternatives in the plan F , E
The proposed bipolar evaluation approach is based on the individual assessment of alternatives potential considering their positive and negative aspects through a bipolar aggregation using the satisficing games theory tools. To provide a framework for integrating human factors evaluation, the following section proposes to model the influence between decision makers through concordance and discordance measures. The degree of individualism and the weight of decision makers will also be defined and integrated into the model resolution. 2.3
Influence modelisation procedure
A group decision is usually composed of persons whose perceptions, attitudes, motivations and personality are different. In a collaborative decision problem, decision makers are brought to collaborate in the evaluation phase. Depending on the personality of the actors, positive and / or negative influences can be exercised ; some persons in the group decision are exclusively individualistic and take into account only their point of view , while other persons say 'holistic' preferred to take into account the opinion of their vicinity according to the importance degree assigned to them. To model the influence related to each decision maker's opinion, this part proposes to define concordance and discordance degrees awarded by each decision maker to members of his neighborhood. These measures allow decision makers to express their level of agreement or disagreement overlooked 'influential’ actors. Indices of concordance and discordance are defined following ([57], [28]). Definition 1: let R 3 be the vicinity of decision maker , which represents a set of S decision makers whose opinion matters (influence) the decision maker in a positive or negative way. For each decision maker G belonging to a vicinity of de , we define respective7
U V ly T G , T G as relative concordance and discordance degrees accorded by decision maker to U opinion of decision maker G compared to other members of the vicinity, where ∑V5 WXY T G
V 1, ∑V5 W XY T G 1.
Several methods can be used to obtain concordance and discordance degrees, the Analytic Hierarchy Process (AHP) can be proposed for a relatively quick estimate. The AHP method is based on pairwise comparison by answering question of the form « what is the concordance (resp. discordance) degree accorded by the decision maker to the opinion of G compared to the opinion of GG » (where G , GG Z R 3 ). Based on this definition, we can consider that a decision maker is even more important in the community if other decision makers give him a good confidence. Thus, the degree of importance of decision maker can be defined as follows.
Definition 2: let T UG , T VG be respectively the relative concordance and discordance degrees accorded by the decision maker G to an opinion of . The importance degree of can be defined by equation (9) ∑ G S\ ]0, T UG / T VG ^
Θ
U V ∑
a 9∑ S\_0, T / T `=
9
Or, equation (10) which allows decision makers to have non-zero degrees of importance, unlike formulation given by equation (9) in which zero importance is possible. Expression (10) may be useful in the case of a holistic and collaborative group consists of decision makers sensitive to opinion of their neighborhood. Θ
1 ∑ G c h 1 . exp ]/∝ _ T UG / T VG `^
1 ∑
hh a c∑i c U V `^ 1 . exp ]/∝ _T / T
10
where α is a parameter setting. Considering the relative importance degree, the relative bipolar measures are then defined to include neighborhood effects in the final bipolar measures. In the BOCR analysis where decision makers evaluations are expressed by " , # , $ , % . The relative bipolar evaluations in this case can be expressed using equation (11) and (12). /Y
∑ G∈Y Θ G ] - " . _1 / - ` # ^ G
G
G
G
∑ ∑ GG ]∑ GG∈Y Θ GG ] - GG " GG . _1 / - GG ` # GG ^^
11
8
/Y !
∑ W∈Y Θ G ]_1 / - `$ . - % ^ G
G
G
G
∑ ∑ GG ]∑ GG∈Y ΘjGGk ]_1 / - GG `$ GG . - GG % GG ^^
12
where G - : risk aversion degree of decision maker i G ∈ R i , Θ G : relative importance degree of decision maker i G ∈ R i .
These equations mean that relative bipolar measures of decision maker considers only the opinion or evaluation of his vicinity according to the importance of each one. The final bipolar measures (selectability and rejectability measures) are then obtained by equations (13) and (14). l . 1 / l
/Y
! l ! . 1 / l !
/Y
13
14
where /! represent a priori measures of alternative obtained by aggregating the corresponding factors. 0 2 l 2 1, is the individualism degree of decision maker . While l tends to 0, the decision maker is considered as ‘holistic’ (altruist) and gives a more importance to the global opinion represented by his vicinity. Inversely, if l tends to 1, the decision maker is ‘individualist’ and considers his opinion above his vicinity. When decision makers present their evaluations directly with ‘a priori’ bipolar ures /! , the relative bipolar measures can be obtained using the following equations. /Y V5 /Y V5 !
5G
5G
∑ G mn o G pq rs tnu G pv rs w 55 55 5 ∈x]u5 ^ 5G
G
∑sy∑ G yn o G pq rs tn u G pvz rs {{ 55 55 5 ∈x]u5 ^ 5G
5G
∑5G ∈x 5 mn o G pv rs tn u G pq rs w 55 55 5G
5G
∑sy∑ G yno G pv rs tn u G pq rs {{ 55 55 5 ∈x]u5^
15
16
For decision maker , these relative bipolar measures take into account the ‘a priori’ bipolar measures of his vicinity. In the selectablity measure, the decision maker considers also the ‘a priori’ rejectability measure of his vicinity R 3 according to their discordance degree. Inversely in relative rejectability measure, decision maker integers ‘a priori’ selectability measures of his vicinity R 3 . The finale bipolar measures can then obtained by the equations (13 and14). From the individual evaluation results, the selection of a solution or set of solutions approved by the group decision generally requires a consensus building process. The following section provides a process achieved consensus from local preferences represented by different sets obtained from the satisficing game theory.
9
3
Consensus and selection processes
The individual bipolar measures are used by decision makers to select the best alternative or the set of the best alternatives for each one. According to the knowledge and perceptions of actors, the decision-makers choices can be contradictory. To find a common satisficing alternative(s), a consensus processes is necessary. The reach consensus can be done in two ways: by aggregating final bipolar measures of decision makers into collective bipolar measures using equation 17 and 18. Then selected alternative(s) can be obtained from satisficing equilibrium set according to proposed selection criteria. The second way to reach a consensus can be done using local preferences. The local preferences are defined as the selected alternatives by each decision maker according to his bipolar measures. The local preferences can be expressed by selection of unique alternative considered as ‘the best’ one, or by the equilibrium satisficing set defined in the satisficing game theory (section 2.2). If selected alternative(s) is(are) the same for all decision makers, the common alternative can be chose easily, however, if selected alternatives are different a consensus process is necessary. In the following, we propose a consensus model for each case of unique selected alternative or selected alternatives set. U !U
∑ Θ 17 ∑ Θ ∑ Θ ! 18 ∑ Θ
where U and !U represent respectively, the collective selectability and rejectability measures. 3.1
Case of unique local preference (unique selected alternative)
In this case, each decision maker selects his ‘optimal’ or ‘more satisficing’ alternative noted ∗ (among a set of equilibrium satisficing for example). The selected alternative can be obtained by the following. ∗ arg max r∈? ,5 _ ` 19
where ] , ì! ^ is value function that can be take particular form depending on
the decision goal of decision maker , for example :
/ 7! 20
that gives the priority to alternatives with large difference between the selectability measure and the rejectability measure given the index of caution, or
! 21 that considers alternatives with the largest index of caution, or 10
y.
1
{ 22
!
that gives priority to alternatives with the largest selectability (respect. lowest rejectability) ; this later case is suitable when one of the measure is uniformly distributed over alternatives. If the selected alternative ∗ is the same for all decision makers, the final solution can be deduced easily. Conversely, if selected alternative ∗ varies from one person to other a reach consensus processes based on selection criteria is proposed to find a common acceptable alternative. According to the group nature, context and society where the decision problem is considered, the decisions rules can be changed. In his paper, Urfalino[58] addresses a question of decision rules choice according to periods and societies. These parameters will favor unanimity, consensus or majority voting in order to achieve legitimate collective decision. The group nature and social ties are also determinant parameters in the decision rules choice. These parameters can be introduced with the ‘individualism’ and ‘holism’ notions that play important role in the rules selection. Considering these notions, we propose some possible selection criteria in the following. 1. Select alternative chosen by decision maker who presents the more important importance degree according by the group. 2. Select alternative that presents the more important selectability m∗ arg max r ∈ ∩? ,5 _∑ Θ ` w s
3. Select m∗
alternative
that
arg minr ∈ ∩? ,5 _∑ Θ ! ` w s
presents
the
less
important
rejectability
4. Select the alternative that has the maximum benefit for all decision makers max r ∈ ∩? ,5 ]∑ Θ ] / ^^ s
5. Selecting a qualified majority: in some situations, the use of qualified majority decision is recommended in decision making. In this case, only the responses from a subset of makers are considered. We define ‘Qualified Majority’ as follows. Definition 3. a ‘Qualified Majority’ (QM) is a set of decision makers who present % of the group decision members and % of total importance. , , … S and ∑ a Θ < , ∑ a Θ 1 23
where , , are respectively possible rates that define ‘majority members’ and ‘global importance of the majority’, these rate can be defined by group decision or analyst. Consider the ‘qualified majority’ (QM), the selection of the final decision can be done using the following steps. 11
Step1. If decision makers have a common alternative (obtained from the previous selection criteria) then this alternative is selected. Step2. Otherwise, make adjustments to the caution index 7 of decision makers who show great caution successively in descending order until a common alternative.
3.2
Case of the set of local preferences (expressed by satisficing equilibrium set)
In this part, local preferences are expressed according to the equilibrium satisficing set defined on satisficing game theory (see section 2.2). Considering that each decision maker identifie E his equilibrium satisficing set ?E5 , if ⋂
a ?5 ∅, then the selection of common alternative can be done for example with the following. 5 pq rs 5 pv rs
where ∏
a m
∗ arg max r ∈⋂ s
w for example.
,5 ?
_ ` 24
Conversely, if ⋂ a ? ∅, a reached consensus process is necessary. To deal with this, we propose two possible reaching consensus process; fist process is based on varying caution index of decision makers when second proposed process is based on the distance measures between individual preferences. These measures are commonly called ‘proximity measures’ when it comes to comparing individual decision maker opinions to global or collective opinion on the ‘soft’ consensus process. 3.2.1
E,
Consensus model based on caution index
One possible way to reach a consensus is to vary caution index of some decision makers to enlarge their satisficing equilibrium set and tend towards a non-empty intersection of these sets. To select decision makers who must modify their caution index, some selection criteria can be proposed as, importance of decision maker, satisficing equilibrium set dimensions
! rapport, etc. We propose to decrease caution index of decision makers according to -
-
The importance degree: decrease caution index of decision makers who presents the less E, important degree one by one until reach consensus ( ⋂
a ? ∅) The qualified majority (QM): if the qualified majority of decision maker present one or more common alternatives, according to the group nature relations and decisional context (unanimity or not, confidence or not, complementarity or not), select a more satisficing alternative among QM common alternatives or ask to the rest of group to enlarge their equilibrium satisficing set to achieve a consensus using importance degree as regulation factor for example. Satisficing equilibrium set dimension: decrease caution index of decision makers who present the smallest set of satisficing equilibrium, one by one until reach a consensus E, (⋂
a ? ∅). 12
-
Proximity limits: decision makers who present a large number of alternatives with
! tend to 7 r max rs ∈¡ m
5
pq rs
w are brought to reduce their index caution.
5 pv rs
In other words, decision makers consider alternative solutions with selectability measures much higher rejectability measures are brought to expand their range of solutions by reducing their caution indices one by one until reaching a consensus. This allows the decision makers to include in their satisficing equilibrium sets alternatives with better spreads. 3.2.2
Consensus model based on proximity measures
Supporting that a final agreement between decision makers is not necessary to resolve group decision problems, this section proposes to reach a common solution through a soft consensus process ([48], [49]) based on proximity and bipolar consensus measures defined from final bipolar measures. The proximity measures are used to assess the gap between decision makers’ evaluation on each alternative while bipolar consensus measures allow evaluating the gap between decision maker and the rest of the group considering bipolar measures of alternatives. These measures are part of a feedback mechanism proposed to guide decision makers in adjusting their evaluations to tend towards a single solution and reach a consensus. Unlike soft consensus process proposed in the literature ([47], [27]), the model presented here does not aim to converge all decision makers assessments but focuses on alternatives presenting initially the same evaluation trend (convergent). The targeted recommendations are given to decision-makers to correct their differences and to reach the common solution from selected alternatives [57]. Considering that the intersection of the equilibrium satisficing sets of decision makers is empE, ty ⋂
a ? ∅, reaching a consensus amounts to establishing an iterative process allowing to have several phases of consultation until an agreement. The objective is to identify alternatives on which decision-makers have convergent opinions and support them to adjust their evaluation in order to reach a final common solution based on initial convergent opinion. In each iteration, considering importance degree of decision makers, the proximity measures are calculated to determine convergent alternatives (that present the smallest distance compared to others), when bipolar consensus measures are measured to identify decision makers who present a large gap of supportability and/or rejectability evaluations for the preselected alternatives compared to the rest of the group. The main problem in this situation is the difficulty to determining the way to convergences individual preferences [50]. To achieve this convergence, boundary conditions (threshold) are fixed for each level (for proximity and bipolar consensus measures). This process is carried out in a discussion session led by a feedback mechanism that can guide decision makers in changing their opinions. The alternatives with confused evaluations will be readjusted by the decision makers who present a strong divergence. A discussion session is led by a feedback mechanism to lead decision makers towards a common legitimate solution. The parameters of the proposed reached consensus process are developed below. 13
3.2.2.1
Proximity and bipolar consensus measures
The proximity and bipolar consensus measures are defined as follow [57]. Definition 4. proximity measure is used to calculate the average distance between decision maker evaluations considering the alternative . It is obtained as follows, equation (25).
G ∑ ∑ G, G¤ ¢] s ^
_
`
.
G ]! s ^
£
25
where G G s Θ / Θ G , distance between the supportability measure of decision maker and G for the alternative , with regard to the importance of each decision maker.
! s Θ ! / Θ G ! , distance between the rejectability measure of decision maker and G for the alternative , with regard to the importance of each decision maker. G
_
`
G
!
! ¦ !
, binomial coefficient takes into account combinations with avoiding redundancy
distances (for example : s ). s
Definition 5. bipolar consensus measures are used to calculate the distance between a decision maker and the rest of the group considering supportability and rejectability evaluation for the alternative . The bipolar consensus measures are given by equation (26) and (27) respectively. G ∑ G , G¤ ] s ^ 26 s S/1 !s
∑ G, G¤ ]! s ^ S/1
G
27
where s , !s represent respectively selectability and rejectability consensus measures.
3.2.2.2
feedback mechanism
The feedback mechanism is able to assist the moderator in the process of reaching consensus or even replace it allowing experts to change their preferences in order to achieve a tolerated proximity degree. The main issues of this mechanism are: how to ensure that individual positions converge? And how to support a decision maker in obtaining and accepting a given solution? ([50], [59]). Literature generally represents the feedback process by identification and a recommendation phases. According to this structure, the following section presents a feedback process proposed in connection with the bipolar approach.
14
Identification phase. The identification phase evaluates the closeness of decision maker assessments for each alternative and compares them to tolerances threshold fixed by the decision group or by the moderator. The alternatives which represent a convergent opinion among group decision members (manifested by a smallest proximity measure) considered more interesting, are selected. Then, bipolar consensus measures are used to identify decision makers who present a significant deviation compared the rest of the group according selected alternatives. From these measurements, evaluations of alternatives can be modified using the following steps. (a) Identification of alternatives whose proximity measure respects the condition 1 2 T , where T is the tolerance threshold representing the permissible difference between alternatives (average distances on the set of alternatives can be considered as tolerance, for example). This allows excluding alternatives that may create conflicts and focus on alternatives already having a certain convergence.
(b) For each alternative that respect condition (1), identification of decision makers that don’t respect one or both following conditions: 2 is 2 T , 3 !is 2 T! , where T /T! are a tolerance thresholds of selectability/rejectability measure respectively. These conditions allow identifying decision makers who present a divergent opinion regarding a rest of group. The average of all alternatives can be proposed as threshold. Recommendation phase. In this phase, recommendations are given to decision makers do not fulfill the conditions (2) and (3) in order to change their assessments concerning alternatives that meet the condition (1). The recommendations are based on the following rules: -
for s K T , the decision maker has a large gap related to its selectability measure
which means that the selectability measure given by concerning alternative , moves
away from the evaluations given by other actors. To know the divergence direction and thus know if the considered alternative , has more importante selecability measure (positive divergence) or a low measure (negative divergence) compared to other actors, the following equation is proposed. :§s
∑ G, G¤ ] s ^ S/1
G
28
If :§s > 0, the alternative has a good selectability measure, no change is required. Other
wise (:§ s H 0), the selectability measure is smaller than the measures average, an increase selectability measure is recommended for considered alternative .
The same recommendations are implemented for rejectability measure.
15
-
for ! s < T! , the decision maker has a strong divergence from the rest of the group. The
divergence direction indicates whether the alternative has a low rejectability (positive divergence) or a significant rejectability (negative divergence). The divergence direction of rejectability is given by equation (29). :§! s
∑ G, G¤ ]! s ^ S/1
G
29
If :§! s > 0, the alternative has a low rejectability measure, no change is required. Other-
wise (:§!s H 0), the rejectability measure is greater than the measures average, reduction of
rejectability measure is recommended for considered alternative .
Once modifications realized, the equilibrium satisficing sets for each decision maker are reconE, structed. The iterative process is stopped when ⋂
a ? ∅ . If obtained solution satisfies group decision maker, the process can be stopped. Else, other iteration can be proposed. The next section presents an application example to illustrate proposed approach.
4
Application example
In this section, we use proposed group decision making bipolar analysis approach to resolve real size wind farm implantation problem adapted from [60]. Install a wind farm involves various actors in society such as; wind specialist, local administration and public authority. In France for example, the selection of potential implantation sites (selection of alternatives) returns generally to the local administration, which taking into account previous studies, will retain potential installation areas approved by the prefect. A more sophisticated study is then performed to select the site chosen for the implementation, bipolar collective approach presented in this chapter can be used for analyze study. To illustrate the developed bipolar model, this section adapts original problem data of the of installing a wind farm [60], considering a decision committee composed of three entities : wind specialist, local administration and public authority denoted respectively , , ¨ . The goal is to select a location for a wind farm installation to achieve a set of objectives related to socioeconomic, performance and operability notions. Five potential sites are under consideration. The importance degrees of socio-economic, performance and operability objectives are given by decision makers , , ¨ respectively as follows: T ,© )0.1 0.8 0.1,, T ,© )0.8 0.1 0.1,, T ¨,© )0.1 0.1 0.8, . It is assumed that specialists wind (entity d ) give more importance to the performance objective, their first goal is to establish a production site with a good yield. To manage the budget and preserve the municipal property, local administration (d ) promotes socio-economics aspects while public authority (d¨ ) are supposed to pay more attention to the operability of the future site. The bipolar evaluation of alternatives is carried out through BOCR analysis that consists in evaluating alternatives over benefit, opportunity, cost and risk factors through a distribution of attributes on these factors [61]. The results are summarized in Table 1 and 2, representing respectively the evaluation results of the BOCR analysis and 'a priori' bipolar measures obtained for a risk aversion given by the vector - )0.3 0.5 0.7,. It is assumed in this case that 16
the wind specialists (d1) have a low risk aversion against a medium risk aversion for local administration (d2) and an important caution about the public authority (d3). Table 1. evaluation results of BOCR analysis
decision makers ¨
Factors
a1
a2
a3
a4
a5
B O C R B O C R B O C R
0.1892 0.1965 0.2072 0.1976 0.1924 0.1964 0.1618 0.2000 0.2137 0.1981 0.1902 0.2078
0.2284 0.1990 0.1814 0.2064 0.1900 0.1980 0.2172 0.2043 0.1519 0.1945 0.1829 0.1955
0.1498 0.1980 0.2132 0.1986 0.2347 0.1898 0.1507 0.2045 0.1957 0.2026 0.1891 0.2115
0.2250 0.2020 0.2298 0.2058 0.1581 0.2103 0.2216 0.1894 0.2313 0.2005 0.1792 0.1994
0.2076 0.2045 0.1684 0.1915 0.2249 0.2055 0.2487 0.2018 0.2073 0.2043 0.2587 0.1857
Table 2. ‘a poriori’ bipolar measures
decision makers
¨
a1
a2
a3
a4
a5
Rejectability measure (! )
0.1943
0.2078
0.1835
0.2089
0.2055
0.2043
0.1889
0.2088
0.2226
0.1753
0.1944
0.1940
0.2122
0.1842
0.2152
0.1809
0.2108
0.1776
0.2055
0.2252
Rejectability measure (!¨ )
0.2090
0.1647
0.1978
0.2221
0.2064
0.2025
0.1917
0.2048
0.1933
0.2076
Bipolar measures
Selectability measure ( ) Selectability measure ( ) Rejectability measure (! ) Selectability measure ( ¨ )
To represent the social link and the potential influences between decision makers, the proposed approach suggests modeling the interactions between decision makers through U V concordance and discordance measures notes respectively T G /T G and represented by the following matrices. / U T G ª0.7 0.2
/ 0.7 0.1 0.9 V / / 0.3« T G ª0.2 0.6 0.4 0.8 /
0.3 0.8« /
These matrices show for example that decision maker gives good confidence in decision maker through a good degree of concordance and a low degree of discordance. Considering the concordance and discordance given above, the importance degree Θ of each decision maker is deduced from equation (10) is given by the following vector. Θ )0.3334 0.3285 0.3381, 30
The relatives measures taking into account the vicinity can be obtained using equations (11), (12) and bipolar final measures from equations (13), (14). Considering that the individualism 17
degree is medium (l i )0.5 0.5 0.5,), the results are respectively presented in Table 3 and Table 4. Table 3. relative bipolar measures
Decision makers
¨
Relative bipolar measures ¬ /Y V¬ ! /Y V /Y V ! ¨/Y V® ¨/Y V® !
/Y V
a1
a2
a3
a4
a5
0.1976 0.1996 0.2008 0.2049 0.1947 0.1900
0.1861 0.1893 0.1929 0.1815 0.1972 0.2043
0.1926 0.2050 0.1968 0.2013 0.2014 0.1894
0.2102 0.1951 0.2060 0.2166 0.2026 0.2040
0.2135 0.2109 0.2035 0.1957 0.2041 0.2122
a4
a5
Table 4. final bipolar measures
Decision makers
¨
Final bipolar measures
Selectability measure ( ) Rejectability measure (! ) Selectability measure ( ) Rejectability measure (! ) Selectability measure ( ¨ ) Rejectability measure (μ¨° )
a1 0.1960 0.2020 0.1976 0.1929 0.2019 0.1963
a2 0.1969 0.1891 0.1935 0.1961 0.1809 0.1980
a3 0.1881 0.2069 0.2045 0.1894 0.1996 0.1971
0.2095 0.2089 0.1951 0.2111 0.2123 0.1987
0.2095 0.1931 0.2093 0.2105 0.2053 0.2099
The graphical representation of the evaluation results (table 4) in the plane (μ° , μ± is given by the following figure 4. The index of caution 7 is assumed to be 1 for each decision maker . The satisficing equilibrium sets ?E, of actors are as follows: ?E, ² , , ³ , ? E, ¨ and ?E,¨ , ³ . E, In this case there is no common solution _⋂¨ a ? ∅`, we propose then to seek a consensus using the proposed models. By varying the caution index 7 first, satisficing equilibrium sets of decision makers will be resized to make their non-empty intersection _⋂¨ a ?E, ∅`. The process reached consensus based on proximity and consensus measures, is proposed in a second time.
18
0.22
Selectability measure
d1 0.21
0.2
d2 d3 q
1
q
2
q3
a4
a5 a3 a2
a1
a4
a1 a3 a1
a2
0.19
0.18 0.18
a5 a5
a4 a3
a2 0.185
0.19
0.195
0.2
0.205
0.21
0.215
0.22
Rejectability measure
Figure 4. graphic representation of final bipolar measures for each decision maker
Consensus building process based on caution index variation The proposed model achieved with this consensus approach allows decision makers to avoid changing their evaluations. Only a concession on the caution index is required based on a selection criteria proposed in section 3.2.1. In this example, we note that the equilibrium satisficing sets of qualified majority of 67% composed of decision makers , ¨ which consider the alternative ³ as a final solution _ ?E, ∩ ?E,¨ ³ `. Depending on the relationships nature and group decision-making context, the decision maker can accept selecting the alternative ³ unsatisfactory for him without requiring modification. Otherwise, decision maker will have to change its caution index to expand its satisficing equilibrium set until include alternative ³ . The caution index variation may also be according to the importance degree of decision makers. In this case, the decision makers classification deduced from equation (26), (d¨ ≽ ≽ indicates that the decision maker must reduce its caution index first. If the intersection is still empty after changing, the decision maker changes its index of caution, and so on until a common solution is obtained. Considering that the caution index of decision maker is given by q 0.95 this modification E, allows to enlarge its satisficing equilibrium set to ?¶.·² ¨ , ² , however the intersection E, ¨ ⋂ a ? remains empty. The decision maker must change its caution index, considering that caution index becomes q 0.95 , the satisficing equilibrium set of d remains E, ² , , ³ , the alternative is satisficing but dominated and then excluded identical ?¶.·² E, from satisficing equilibrium set ?¶.·² . The intersection at this level is still empty. It then asks the decision maker to modify its caution index. Considering that 7¨ 0.95, the satisficing E,¨ equilibrium set of ¨ is given by ?¶.·² , ³ , the alternative ² is satisficing but dominated by a4. In this case, a new iteration is realized, decision maker must reduce its caution index E, more. Considering that 7 0,91, the satisficing equilibrium set is equal to ?¶.· ² , ¨ , where the alternative ³ is satisficing but dominated. It is noted that all the remaining alternatives can be satisfactory, but are still dominated. In this case, since the adjustments of assessments are not accepted, a possible solution amounts to consider the intersection of the satisficing equilibrium set of the most important decision maker with the satisficing sets of the rest of the group regardless of dominance for the rest of the group. This involves the selection
19
of a qualified 'more satisfactory' solution by the largest maker. The solution in this case is the E, E, result of the following intersection ? E,¨ ∩ 8¶.·² ∩ 8¶.· ³ . Consensus building process based on distance measures The second proposed process is based on distance measures that allow the analyst to identify alternatives and decision makers with strong differences. Using a feedback mechanism, decision makers have the ability to change their assessments based on recommendations made by the analyst during the discussion sessions, with the aim to converge to a common solution. The first phase of the feedback mechanism uses proximity measures and bipolar consensus to identify respectively, divergent alternatives and decision makers with assessments that deviate from those of other decision makers. The second phase allows the analyst to make targeted recommendations to divergent decision makers. Identification phase. The identification of the alternatives with strong differences consists in calculating proximity measures defined by equation (25) from the final bipolar measures (table 4). The obtained results are given by the following vector )0.0077 0.0079 0.0119 0.0113 0.0090,. Assuming that the average distances on the set of alternatives is the tolerance threshold, the proximity distance must not exceed 0.0096 ( 2 0.0096). We can deduce that the alternative a3 and a4 have widely divergent and should therefore be discarded. Assuming that the thresholds ù , ù! were obtained from averages of bipolar distances on the set of alternatives, the following table 5 shows the gaps observed at the actor level for each alternative. Table 5. bipolar consensus measures
D a1 a2 a3 a4 a5
0,0017 0,0033 0,0046 0,0039 0,0007
is
0,0019 0,0022 0,0024 0,0067 0,0009
¨ 0,0031 0,0034 0,0026 0,0048 0,0005
0,0025 0,0026 0,0045 0,0014 0,0057
!is
0,0035 0,002 0,0056 0,0012 0,0033
¨ 0,002 0,0032 0,0034 0,0023 0,0042
Recommandation phase. Table 5 shows that decision maker presents a deviation from the average concerning selectability measure of alternative . The meaning of the divergence is positive (:§E 0.003), the selectability measure is important and cannot be modified.
-
-
Decision maker presents a divergence regarding the rejectability measures of alternatives a1 and a5. The divergence direction of the rejectability measures is given by :§!¬ 0.0035 and :§!¹ /0.0015 . The negative divergence of alternative a5 leads to a recommendation to reduce its rejectability measure. Alternative a1 which has a low rejectability is spared.
Decision maker ¨ presents a strong rejectability measure on alternative a5 compared to the rest of the group. The negative divergence direction :§!¨¹ /0.0042 implies a recommendation of reducing this measure.
The reduction of rejectability measures of alternatives a5 by decision makers d2 and d3 to ì! ² = 0.2005 and ì¨! ² = 0.1980 respectively allows for the following graphical 20
representation (see figure 5). The satisficing equilibrium sets _ ?E, ` of each decision makers i are deduced as follows; ?E, ² , , ³ , ?E, ¨ , ² and ?E,¨ , ³ , ² . The solution obtained by the intersection of the sets is the alternative a5 _⋂¨ a ?E, ² `. 0.22
Selectability measure
0.215
d1 d2
0.21
d3
0.205
q1
0.2
q
2
q3
0.195
a4
a5 a3 a1
a2
a4
a5
a5 a1 a3
a1
a2
a4
0.19
a3
0.185 0.18 0.18
a2 0.185
0.19
0.195 0.2 0.205 Rejectability measure
0.21
0.215
0.22
Figure 5. graphic representation of final bipolar measures (iteration1)
In the example discussed here, the integration of positive and negative influences of decision makers in the model and the relatively small number of decision makers allowed reaching a consensus quickly after a single recommendation step. The individualism average rate considered for all decision makers allows also to nuance the individual assessments and reduce differences that a high degree of individualism could make appear as shown in figure 6 for individualism degrees l equal to 0.9 for each decision maker. A sensitivity analysis can be performed by varying the caution index and/or individualism degree to test different possible scenarios and stability of recommended solutions. 0.23
Selectability measure
0.22 0.21 0.2 0.19
d1
a4
d2 d3 q1 q
a3 a5
a1
a2 a1
0.18 0.17 0.16 0.16
a4
a5
a3 a1 a 2 a4 a3
2
q3
a5
a2 0.17
0.18
0.19 0.2 Rejectability measure
0.21
0.22
0.23
Figure 6. graphical representation of final bipolar measures for º )0.9 0.9 0.9,
5
Conclusion and perspectives
This paper has addressed the group decision problem involving a group of decision makers to select one or more alternatives from a common set of potential solutions. Considering 21
collaborative decision problem based on individual evaluation, interactions and potential influences mutually exercised by decision-makers have been integrated in assessment model through concordance and discordance measures. These measures have been defined to model respectively the impact of positive and negative influence that decision makers could have on their vicinity. From individual assessment results, bipolar measures have been generated in order to integrate the potential influence of the vicinity on the final result bipolar measures of each decision maker. The individualism degree of decision makers is also taken into account to represent the individual nature of actors and their impact on the final result. A consensus building processes is then proposed from final bipolar measures in order to guide decision makers toward a common solution. Based on the formalism of the satisficing game theory, common solutions were represented by the result of the intersection of satisficing equilibrium sets of decision makers. The first proposed process for achieving consensus is based on the caution index variation, considering that the adjustments and changes in individual assessments were not admitted. The decision makers had to reduce their caution index degree based on a set of selection criteria. This readjustment has expanded the satisficing equilibrium sets and increase the chances of obtaining an non-empty sets intersection. An iterative process of reaching consensus has also been proposed to find common solutions where individual assessments adjustments are possible. Based on proximity and bipolar consensus measures, the proposed feedback process aimed to identify at first alternative with widely divergent and, in a second step, decision makers showing a significant difference in the bipolar valuation considering the rest of the group. Once the identification phase realized, targeted recommendations were addressed to decision makers with significant differences regarding alternatives to converging trend. An example application is used to illustrate the proposed approach. The results obtained through the application of the proposed approach highlighted the applicability and interest of our approach. This effectiveness will have to be confirmed in a real context. The development of a software tool for decision support based on the bipolar approach is envisaged as well as the development of the bipolar evaluation approach for the resolution of strategic game problems.
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26
*Highlights (for review)
Highlights.
We addressed the multi-criteria multi-objectives decision making problem using a soft consensus approach.
Considering collaborative decision problem based on individual evaluation, the proposed approach take into account interactions and potential influences mutually exercised by decision-makers.
The impact of human behavior (influence, individualism, fear, caution, etc.) on decisional capacity is considered in the valuation model.
A soft consensus process based on distance evaluation and feedback mechanism is proposed to achieve a common solution. The satisficing game theory formalism is proposed as recommendation tool.
