Oper Res Int J DOI 10.1007/s12351-016-0234-0 ORIGINAL PAPER
A web-based group decision support system for multicriteria ranking problems ´ lvarez Carrillo1,2 Juan Carlos Leyva Lo´pez1,2 • Pavel Anselmo A 2 • Diego Alonso Gaste´lum Chavira Jesu´s Jaime Solano Noriega3
•
Received: 1 March 2015 / Revised: 8 December 2015 / Accepted: 15 February 2016 Springer-Verlag Berlin Heidelberg 2016
Abstract In this paper, we present an implemented, web-based multicriteria group decision support system for solving multicriteria ranking problems by a collaborative group of decision-makers in sequential or parallel coordination mode and in a distributed and asynchronous environment. This system employs an order-based consensus model for collaborative groups that moves from consistency to consensus. The system is based on consensus measures and it has been designed to provide advice to the decision-makers to increase group consensus level while maintaining the individual consistency of each decision-maker. It is based on the use of fuzzy outranking relations to model individual and group preferences. For the exploitation of the models—formulated as a multiobjective optimization problem and solved with a multiobjective evolutionary algorithm—the system generates advice on how decision-makers should change their preferences to reach a ranking of alternatives with a high degree of consistency and consensus.
& Juan Carlos Leyva Lo´pez
[email protected] ´ lvarez Carrillo Pavel Anselmo A
[email protected] Diego Alonso Gaste´lum Chavira
[email protected] Jesu´s Jaime Solano Noriega
[email protected] 1
Universidad de Occidente, Blvd. Lola Beltra´n y Blvd. Rolando Arjona, Culiaca´n, Sinaloa, Mexico
2
Universidad Auto´noma de Sinaloa, Ciudad Universitaria, Culiaca´n, Sinaloa, Mexico
3
Universidad Auto´noma de Ciudad Jua´rez, Ave. Del Charro 450 Norte, Ciudad Jua´rez, Chihuahua, Mexico
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Keywords Group decision support systems Multicriteria decision aid Ranking problem Coordination modes ELECTRE methods
1 Introduction One of the difficulties of coordinating group decision-making processes with respect to using modeling techniques is the sharing understanding and communication of models between different decision-makers (DMs) (Chinneck and Greenberg 1999). Particularly in distributed group decision-making, conveying information among participants is more restricted than that in face-to-face meetings because of the lack of social cues (Mark and Bordetsky 1998). The coordination methods refer to the regulatory or meta-process that groups employ to manage the task activities (Rana et al. 1997). Coordination methods are dynamic processes that overlay structural designs and influence how they operate. The exchange of information among participants is the essence of all coordination methods. The group multicriteria decision analysis (MCDA) process is composed of defining the problem, developing criteria to assess the problem, allocating weights to the criteria, evaluating alternatives with respect to each criterion, arriving at a solution based on the MCDA evaluation and seeking consensus among participants to reach a group decision (Hwang and Lin 1987). Therefore, coordination in group MCDA processes can be characterized as deciding on the frequency of aggregations of individual preferences which reflects the degree to which the individuals in the group should be coupled in terms of procedures; and how the individual preferences should be aggregated in terms of algorithms. In group MCDA processes, the coordination methods are associated to the degree to which the individual activities are coupled. On the one hand, if the group is loosely coupled, the members may develop the criteria, allocate the weights, assess the alternatives and reach their own decision. Then the individual preferences of the decision are aggregated to reach a consensus on a group decision. In this type of coordination, individuals work in parallel without interacting with others during most of the group decision-making process. This is regarded as parallel coordination. On the other hand, sometimes group members need to stop and review their work to avoid exaggerating the degree of disagreement (Burke et al. 1999). In such a case, the group is tightly coupled, in which members have to interact with others to get consensus in the middle stages during the entire process. In an extreme case, the members can work out the criteria, weights, and assessment them, but have to reach consensus at each stage to get aggregated criteria, weights, and the final agreement on decision. This most tightly coupled process is referred to as sequential coordination. The consensus process deals with the achievement of the maximum degree of consensus or agreement between the set of decision-makers on the solution set of alternatives. Usually, this process is guided by the figure of a facilitator (Kacprzyk and Fedrizzi 1988) and it is carried out before the selection process. Clearly, the consensus process is an important step in solving group multicriteria decision-
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making problems because it aids to obtain solutions with high level of consensus among decision-makers, which is usually a desirable property. The aim of this paper is to present a new web based multicriteria group decision support system (MCGDSS) to deal with multicriteria ranking problems—how to facilitate ranking of a set of alternatives having evaluations in terms of several criteria in decreasing order of preferences—by a collaborative group of decisionmakers under distribute and asynchronous environment and to be conducted by a facilitator, where group members could work in a sequential or parallel coordination mode (Cao et al. 2004). The outranking and the multiobjective evolutionary approaches for decision aiding are particularly relevant to this MCGDSS. As a result, multicriteria outranking and multiobjective evolutionary methods were embedded in the implemented MCGDSS as useful decision-modeling tools for decision-makers to structure the group problems during the process and to reach a consensus on the final solution for the multicriteria ranking problem. This decision support system is based on the use of consensus and consistency measures which are interactively computed when the decision-makers provide their preferences. The decision support system uses both kinds of measures to offer advice to the decision-makers by means of easy to follow rules, thus providing a feedback mechanism to help decision-makers to change their preferences in order to obtain solutions with a high level of consensus. One of the main novelties in this contribution is the use of multiobjective evolutionary algorithms (Coello et al. 2002) and the multicriteria aggregation–disaggregation approach for group decision-making (Matsatsinis and Samaras 1997; Spyridakos 2012; Spyridakos and Yannacopoulos 2015) to identify the alternatives or intercriteria parameters that contribute less to a consensus level. The system also aims to help decision-makers to maintain a high consistency level in their preferences to avoid self contradiction and, when that is the case, to reduce as much as possible pairwise rank reversal (Mareschal et al. 2008) and incompleteness. This system is designed to help the facilitator to carry out his/her tasks during the different steps of the sequential or parallel process. The system has been fully implemented and the decision-makers can use it via a web interface which allows to carry out consensus processes in distributed environments. This means that the common essential condition of decision-makers to physically meet together is eliminated and therefore the decision-making process for group of decision-makers, living in different places, is facilitated. The rest of this paper is organized as follows. Section 2 presents the theoretical model on which the MCGDSS is based. In Sect. 3, the MCGDSS for group ranking problems is presented, and some of the technical details regarding its implementation and use are discussed. Finally, we present our conclusions in Sect. 4.
2 Theoretical model for the group decision support system In Herrera-Viedma et al. (2007) a consistency and consensus measures based theoretical model to guide the group decision-making consensus process with incomplete fuzzy preference relations was developed. This section briefly describes
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the extension of that model and the necessary restructuration and improvements needed in the group multicriteria decision-making consensus process to allow the use of fuzzy outranking relations, the use of parallel and sequential coordination modes as a new characterization of the degree to which the individual activities are coupled, the consistency and consensus measures based theoretical model, the consensus/consistency control and their use of a feedback mechanism to provide advice and to identify the DMs and the alternatives or particular intercriteria parameters that contribute less to the consensus level. 2.1 Multicriteria ranking problematic The objective of the multicriteria ranking problematic is to aid the DM through a ranking that is obtained by placing all alternatives of a set of alternatives A = {a1, a2, …, am} completely or partially ordered according to preferences. In a group multicriteria ranking problem a set of decision-makers D = {d1, d2, …, dk} have to rank the alternatives from a set of alternatives A in decreasing order of preferences. Preference relations are usually assumed to model decision-makers’ preferences. In the outranking approach literature the ‘‘true’’ preferences of a decision-maker are represented by crisp or fuzzy preference relations (Fodor and Roubens 1994). A ranking, then, can be perceived as ‘‘revealed’’ preferences. In adopting this problematic, one tries to use available information as much as possible to compare the elements of the set of alternatives A to each other to determine a ranking of the alternatives. Such a ranking is designed to help his/her think about the problem, to guide his/her discussions with other stakeholders and, more generally, to serve as a framework for approaching the next critical point of the decision process (Roy 1996). 2.2 Fuzzy outranking relations In MCDA, preference relations based on the outranking approach are usually assumed to model actors’ preferences. The outranking methods were originally developed to incorporate the fuzzy (imprecise and uncertain) nature of decisionmaking by using thresholds of indifference and preferences (Roy 1990). This feature is appropriate for helping to solve many group decision-aiding problems. Methods based on this approach are often presented as a combination of two phases: aggregation and exploitation. The aggregation process corresponds to the operation, which transforms marginal evaluations of separate criteria into a global outranking relationship between every pair of alternatives, which generally is neither transitive nor complete (Bouyssou and Vincke 1997). The outranking methods seek to build an outranking relation SA on a set of alternatives A. aiSAaj means that according to the global model of DM preferences, there are good reasons to consider that ‘‘ai is at least as good as aj’’ or ‘‘ai is not worse than aj’’. A fuzzy preference relation SrA on a set of alternatives A is a fuzzy set on the product set A 9 A, i.e., it is characterized by a membership function r: A 9 A ? [0, 1].
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When cardinality of A is small, the preference relation may be conveniently represented by the m 9 m matrix SrA = (rij), being rij = r(ai, aj) (Vi, j 2 {1, 2, …, m} interpreted as the degree or intensity of ai is at least as good as aj or ai outranks aj. We can induce a crisp outranking relation SkA for each cut level k, 0 B k B 1, associated with a given valued outranking relation SrA. From SkA, we can deduce the followings preference relations: Indifference ai IA aj $ rðai ; aj Þ k ^ rðaj ; ai Þ k Preference ai PA aj $ rðai ; aj Þ k ^ rðaj ; ai Þ k b Incomparability ai RA aj $ rðai ; aj Þ k b ^ rðaj ; ai Þ k b where k is a constant cutting level and b is threshold level indicating the minimum values a decision-maker (DM) may accept that ‘‘ai outranks aj’’ (aiSkAaj). The exploitation process addresses outranking relationship to clarify the decision through a partial or total preordering, reflecting some of the irreducible indifferences and incomparabilities (Fodor and Roubens 1994). ELECTRE III (Roy 1978) and PROMETHEE II (Brans and Vincke 1985) are representative methods of multicriteria decision aids that build and exploit a fuzzy outranking relationship. 2.3 Group MCDA process with sequential and parallel coordination modes Effective coordination is necessary to the achievement of distributed group decision-making tasks, since distributed decision-making activities are highly interdependent due to the range of expertise among members, on a range of subtasks, and so on (Wittenbaum et al. 1998). Because of this difficulty, even though individual productivity may be high, poor coordination can still result in process losses with serious consequences for group performance (Steiner 1972). Coordination methods are associated to the degree to which the individual activities are coupled. In the implemented system, we present two coordination methods in group MCDA processes from Cao and his colleagues’ (2004) notion. On the one hand, if the group is loosely coupled, the members may develop the criteria, assign the weights and thresholds values, assess the alternatives and reach their own decision. Then the individual preferences of the decision are aggregated to obtain a group decision. In this type of coordination, individuals work in parallel without interacting with others during most of the group decision-making process. This is regarded as parallel coordination. On the other hand, from time to time group members need to stop and review their work to avoid exaggerating the degree of disagreement (Burke et al. 1999). In such a case, the group is tightly coupled, in
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Problem definition Agreed definition
Alternative development Agreed alternatives
Sequential coordination mode
Parallel coordination mode
Criteria development
Criteria development Individual criteria
Group preference
Agreed criteria
Intercriteria parameters Intercriteria parameters
Individual criteria parameters
Agreed intercriteria parameters
Alternatives evaluation
Alternatives evaluation
Individual evaluations
Agreed evaluations
Collective solution
Individual preference
Consensus solution
Individual solutions Collective solutions
Agreed solution
Fig. 1 Group MCDA process with two coordination modes based in (Leyva and Alvarez 2013)
which members have to interact with others to get consensus in the middle stages during the whole process. In an extreme case, the members can work out the criteria, weights, threshold values, and assessment them, but have to reach consensus at each stage to get aggregated criteria, weights, threshold values, and the final agreement on decision. This most tightly coupled process is referred to as sequential coordination. The parallel and sequential coordination methods can be conceptualized as follows (Fig. 1). It should be noticed that both parallel and sequential methods are iterative, which means that at any stage group members may choose to go back to redo the previous work if there are problems with it. The next section discusses the parallel and sequential coordination methods in regard to the procedures and aggregation algorithms. 2.3.1 Parallel coordination In parallel coordinated groups, members approach the problem solution in parallel. Each member develops his or her own criteria, indicates the weight of each criterion and the threshold values, assesses the alternatives and arrives at his or her own solution to the problem. Then individual solutions are aggregated to form a group decision. Individual member may not necessarily interact with others during the decision-making process until he or she has to agree on a final group decision at the
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final stage. The procedures and aggregation methods of parallel coordination method implemented in this system are as follows, assuming there is a set of members in the group of decision-makers, and a set of alternatives to rank: 1.
2. 3. 4.
Each individual member develops his or her own family of criteria. The number of criteria developed by each individual may not necessarily be the same. According to Bouyssou (1990) the criteria family should be legible (containing sufficiently small number of criteria), operational, exhaustive (containing all points of view), monotonic and non-redundant (each criterion should be counted only once). These rules provide a coherent criterion family. Each individual member assign importance weights and thresholds to the criteria vector he or she developed. Each individual member assesses the alternatives to produce a personal ranking of the alternatives using the ELECTRE III method. Once the preferred rankings of alternatives from all the group members are available, they are presented to the group for discussion. If the group can arrive at a consensus, the agreed ranking of alternatives is regarded as the group decision. If not, an aggregated result can be achieved by using the ELECTRE III based method for group decision on ranking of alternatives. The multicriteria method proposes a best compromise. If the group is agreed upon the results of the global analysis, the best compromise can be adopted and the session can be closed. On the other hand, if for some reason some decision-makers do not agree with this compromise, the conflicts must be faced using a consensus procedure.
2.3.2 Sequential coordination Sequential coordination needs that consensus is sought throughout every stage of the decision-making process, from the development of criteria to the final consensus decision: The consensus may be reached by aggregating individual preferences and the group discussions. It is expected that the sequential coordination method imposes more restrictiveness on the group, as members have to frequently interact with others. As a result, it is likely that each individual will require more dedication to the group process compared with parallel coordinated groups. The procedure and aggregation methods related with sequential coordination method are below. 1. 2.
3.
Each individual member develops his or her own criteria. Note that the number of criteria developed by each individual may vary. The individual criteria are aggregated so that a coherent criteria family is achievable to present to members for discussion. The criteria family can be obtained from semantic analysis of individual criteria. The criteria family is presented to the members for agreement. Any disagreement may be reconciled through group discussion and polling. Using the Simos’ revised method (Figueira and Roy 2002) (or any other noncompensatory method for obtaining the criteria weighs), each individual
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4.
5.
member gives importance weights to the agreed set of criteria in terms of his or her own preference. The group’s weights on criteria are calculated by the Simos’ revised method (or any other non-compensatory method for obtaining the criteria weighs). Group members give importance weights to the agreed set of criteria in terms of his or her preferences. The proposed group’s weights are presented to the members for agreement. Any disagreement may be reconciled through group discussion. The members of the group assess the alternatives using the ELECTRE III method based on the agreed vectors of criteria and thresholds and weights. The group agrees on the aggregated result by discussion. If required, a voting procedure can be called on in order to reach group consensus.
2.4 Consensus model based on consensus measures and a consistency mechanism The model used to reach a consensus solution is an order-based model in which we consider the position of all of the alternatives in the ranking provided for each DM. In many decision-aiding situations of collaborative groups, the collective order obtained from the aggregation of individual orders may not be accepted by all of a group’s members; this scenario arises because the first collective result is produced by a mathematical analysis and without communication among the members of the group. Thus, based on consensus measures, the goal of achieving a consensus on the ranking of alternatives can be viewed as a dynamic process in which a facilitator, via an exchange of information and rational arguments, attempts to persuade individuals to adapt their preferences. In each round of an iterative and interactive procedure, the degree of the existing consensus and the proximity of individual orders to the collective temporary order are measured. The facilitator uses the degree of consensus to control the process. This scenario is repeated until the group becomes closer to a maximum consensus. A collective order is qualified as a consensus only when the group agreement level with respect to this order reaches a certain pre-established threshold. 2.4.1 Consistency mechanism In group MCDA processes, consensus among DMs is usually sought using the basic rationality principles that each DM presents. Thus, it is strongly recommended that the consistency criteria are first applied consistent with the rationality of each DM, and only afterwards should agreement of the DMs be obtained. Otherwise, if we were to secure consensus and only thereafter consistency, we could destroy consensus in favor of individual consistency, and the final solution might not be acceptable to the group of DMs. In ‘‘Appendix 1’’, a procedure is explained for reaching consistency based on multiobjective combinatorial optimization.
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Problem initiation Consensual preference process
Problem definition Alternatives development
Criteria development Alternatives evaluation Intercriteria parameters
Select DM with proximity measure wPAd,G < ρ
MCDA methods Construction of a valued outranking relation
Yes
Exploitation procedure
Consistency control process
Individual solution process
Group solution process
Multiobjective evolutionary algorithm
Change preferences
No numIter < maxIter
Restricted Pareto set Choose one ranking
No
Individual rankings
Enough Consistency Yes Individual decision ranking
Group MCDA methods
Collective preference relation
Collective ranking
Individual preferences ordered vector
No Global Yes consensus measure CA > α
Temporal collective preference ordered vector
Group decision Temporal collective preference ordered v ector is fina l consensus solution
End process MCGDM stage
Feedback mechanism
MCGDM process for group ranking problem
Generating ranking at individual and group level
Consensus process
Fig. 2 Diagram of the group multicriteria decision aiding consensus process based in (Leyva and Alvarez 2013)
2.4.1.1 Consistency control process Once the valued outranking relation for each DM is constructed, their associated ranking is derived from a multiobjective evolutionary algorithm (MOEA). The output of the MOEA is a small set of rankings with different level of consistency. The levels of consistency are given by the objective functions of the MOEA. If the selected ranking by the DM has enough consistency then it is considered the individual ranking of the DM. Otherwise, the valued outranking relation is exploited again (see Fig. 2). 2.4.2 Consensus measures We will refer to consensus as a measurable parameter of which the highest value corresponds to unanimity and the lowest value corresponds to complete disagreement.
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Initially, in any non-trivial group multicriteria ranking problem, the DMs disagree in their preferences so that consensus must be viewed as an iterative process, which means that agreement is obtained only after some rounds of consultation. In each round, we calculate two consensus parameters, a consensus measure and a proximity measure. The first parameter guides the consensus process, and the second parameter supports the group discussion phase of the consensus process. The main problem is how to make the individual positions converge and, therefore, how to support the DMs in obtaining and agreeing with a specific solution. To accomplish this goal, a consensus level a required for that solution is fixed in advance (a 2 [0, 1], a [ 0.5). When the consensus measure reaches this level, the decision-making session is finished and the solution is obtained. If that scenario does not occur, then the decision-makers’ preferences should be modified. This modification is accomplished in a group discussion session in which we use a proximity measure to propose a feedback process based on simple rules that supports the DMs in changing their preferences. The consensus model for this group multicriteria ranking problem, when group members work in parallel coordination mode, is presented in Fig. 2 and will be described in further detail in the following subsections. 2.4.2.1 Consensus and proximity measures The proposed model considers the position-weighted measure wPd,G A (Leyva and Alvarez 2015): mðm þ 1Þ þ 3m 2m m2 þ 4 m2 þ 2 2 m1 X n o X i;j ðOd ; OG Þ max pi ðOd Þ; pj ðOd Þ K
wPd;G A ¼1 m
2m2
ð1Þ
i¼1 j [ i
where Ki;j ðOd ; OG Þ ¼ 0 if i and j are in the same order in Od and OG, Ki;j ðOd ; OG Þ ¼ 1 if i and j are in the opposite order in Od and OG, pi ðOd Þ (or pj ðOd Þ) is the position weight of alternative i (or alternative j) in the individual order Od and b xc is the integer part of positive real number x, m is the cardinality of set A. This index measures the agreement level between an individual opinion and a collective opinion when both are expressed by rankings of a set of alternatives. It constitutes an interesting weighted version of the well-known Kendall’s ranks correlation index. The originality of the proposed index arises from the fact that it accounts for the position weights of the alternatives in an individual order to quantify the agreement level of the individual order with respect to a collective temporary order. Comparing Kendall’s index with the use of this new index, we obtain a faster convergence to a consensus solution. To use the index wPd,G A , it is first necessary to convert the difference of the ranking into numerical data. Thus, we use the equivalent representation of a ranking as a list of ranks Od = [od(1), …, od(m)], showing the position of alternative j in the ranking (Chiclana et al. 1998). Based on this view, an ordered vector of alternatives from best to worst is given.
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Each consensus parameter requires the use of index wPd,G A to obtain the level of agreement between the individual solution of DM d, Od = [od(1), …, od(m)], and the collective solution OG = [oG(1), …, oG(m)]. We define consensus indicators by comparing the positions’ weighted rank of alternatives in two preferences vectors, as follows: 1.
2.
3.
We use a multicriteria decision-aiding method (e.g., ELECTRE III or PROMETHEE II) to obtain individual rankings Rd for each DM; then, we use a group multicriteria decision-making aid (e.g., ELECTRE for groups, PROMETHEE for groups) to obtain collective ranking of alternatives RG. We calculate ordered vector of alternatives {Od; d = 1, 2, …, n} (from individual rankings Rd) and collective ordered solution OG (from the collective ranking RG), where n is the number of DMs in the group. We calculate the proximity measure of the dth decision-maker’s individual solution to the collective temporary solution, wPd,G A , by using expression (1).
When the proximity value associated with the dth decision-maker is close to 1, his contribution to consensus is high (positive), while if it is close to 0, then that decision-maker has a negative contribution to consensus. 4.
The consensus measure, CA, is calculated by aggregating the above consensus degrees for each DM.
We calculate the consensus degree of all of the DMs using the following expression: CA ¼
n X wPd;G A
d¼1
n
where n is the number of DMs in the group. 2.4.3 Consensus control mechanism In any rational group decision-aiding process, both consensus and consistency should be sought after, i.e., a solution with a high level of consensus is desirable. However, that solution also should be derived from consistently sufficient information (Chiclana et al. 1998). We propose the use of the consensus measure (CA) as a control parameter. When CA exceeds a minimum satisfaction threshold value a 2 [0, 1], the consensus-reaching process stops and the temporal collective preference ordered vector is the final consensus solution. This is the end of the decision process. Otherwise, the feedback mechanism is activated. Additionally, a parameter to control the maximum number of consensus rounds, maxIter, is used (independently of the current CA value) to avoid stagnation (see Fig. 3).
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Fig. 3 Controlling the consensus state
Feedback mechanism
Yes
numIter < maxIter
No
No Consensus Level
Global consensus measure Yes CA > α
Temporal collective preference ordered vector is final consensus solution End of process
2.4.4 Feedback mechanism When consensus measure CA has not reached the consensus level required, then the decision-makers’ rankings should be modified. As we said before, we are using proximity measures wPd,G A to build a feedback process so that DMs can adjust their preferences to achieve closer preferences among them. This feedback mechanism will be applied when the consensus level is not satisfactory and will be finished when a satisfactory consensus level is reached. The purpose of the feedback mechanism is to provide advice to the DMs using consensus criteria. An important aspect of this phase is that the recommendations for a particular DM are easy-to-follow rules expressed in the same format used by that used to represent performance values of the alternatives by the criteria or intercriteria parameters. Thus, the feedback mechanism consists of two submodules: Identification of the numerical values that need to be modified and generation of advice. For the first one, we use a two-step process to identify the DMs and the alternatives or particular intercriteria parameters that contribute less to the consensus level. Our procedure for generating an agreed collective solution is focused in the feedback process stage, proposing new alternative values or inter-criteria parameters for helping each DM to generate new individual solutions regarding individual and group preferences. Note that such a solution is possible because when a DM is attempting to solve a complex decision problem, there are values of alternatives or intercriteria parameters for which the DM is not sure of the correct value. This lack of clarity regarding the alternatives or parameter values naturally occurs in a multicriteria decision problem because at the beginning of the decision process, the DM does not have a clear understanding of the problem. The rules of this feedback process are easy to understand and to apply; they are expressed in the following form: ‘‘If proximity of the dth decision-maker’s individual solution to the collective temporary solution wPd,G is less than a A
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predefined threshold q 2 [0, 1] then the DM should change his or her preferences, where the DM freely decides whether to change his/her preferences, and then a new preferential model and ranking are generated based on the DM changing his/her preferences. It will be carried out as follows’’. 2.4.4.1 Process to identify the DMs and the alternatives or particular intercriteria parameters that contribute less to the consensus level Step 1 Identification of decision-makers: Identify members d whose proximity measure wPd,G is less than predetermined threshold q; then, these members A should change their preferences. Step 2 Step 2.1 Identification of alternatives: The set of alternatives that the above DMs should consider to change their evaluations on a particular criterion k is the one that reduces pairwise disagreements between the dth order and the collective temporary order, which is provided by the solution of the multiobjective optimization problem described in ‘‘Appendix 1’’. Step 2.2 Identification of intercriteria parameters: The set of intercriteria parameters that one DM should consider to change his or her determination on a particular criterion k is the one that reduces pairwise disagreements between the dth order and the collective temporary order to increase the value of proximity measure wPd,G A , which is provided by the solution of the multiobjective optimization problem described in ‘‘Appendix 1’’.
3 Web-based group decision support system Samaras et al. (2003) present a relevant decision support system that implements multicriteria decision analysis methods to resolve interesting problems to evaluation of the Athens Stock Exchange as portfolio management process; Also, Samaras and Matsatsinis (2004) provide an investment proposition adapted to the investment profile of the user-investor. Just as this system, many group decision support systems have been implemented to aid DMs to solve decision problems efficiently, more examples include the following: Interactive value management system (Fan and Shen 2011), Fuzzy information axiom (Cebi and Kahraman 2010), Decider (Ma et al. 2010), PSO-Fuzzy GDSS (Zhang et al. 2010), SADAGE (Leyva et al. 2008), TeamSpirit (Chen et al. 2007), IRIS (Damart et al. 2007), and the GDSS PROMETHEE (Macharis et al. 1998). Matsatsinis and Tzoannopoulos (2008) presented important developments and advances in multiple criteria group decision support through the usage of argumentation-based multi-agent systems. On the other hand, web-based applications are increasingly being used for group decision aiding and decision support environments because they offer many advantages. An example of these advantages is the possibility of accessing them from all over the world and, thus, providing the possibility of carrying out distributed decision-
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Proposed MCGDSS (2015)
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SADGAGE
M-MACBETH DSS
IVMS
Decider
Web intelligent
PSO-Fuzzy GDSS
RAGDSS
TeamSpirit
IRIS
VIP based GDSS
CyberGDSS
PROMETHEE-GDSS
WINGDSS
ELECTRE-III and ELECTRE-GD
MACBETH/Additive evaluation
Value management
Fuzzy aggregation method
Fuzzy relations
PSO-Fuzzy
Variable precision rough set, AHP
Weighted scoring
ELECTRE TRI/IRIS
VIP
SAW
PROMETHEE
AHP, MAVT
DIMCO
MAUT
ELECTRE, PROMETHEE
AHP
X
X
X
X
X
X
/
/
X
X
X
WB
X
X
X
X
X
X
X
X
X
X
X
X
X
/
AD
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
ET
X
X
X
X
X
X
X
X
X
X
PT
X
X
/
/
X
X
X
X
X
/
/
/
/
DiM
X
X
X
CM
X
X
X
X
X
X
X
X
X
X
X
X
CS
X
X
X
/
/
X
X
X
X
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C
AA
PA
AA
AA
AA
AA
AA
AA
AA
AA
AA
AA/ PA
AA
AA
AA
AA
PA
AA
PA
AA
A
MT MCDM technique, WB web based system, AD asynchronous and distributed environment, ET electronic tools, PT polling tools, DiM discussion model, CM coordination modes, CS consensus schemes, C consensus model, A application, AA academic application, PA professional application, X implemented, / partially implemented
Ma et al. (2010)
Fan and Shen (2011)
18
19
Zhang et al. (2010)
Xie et al. (2008)
15
Kacprzyk and Zadrozny (2010)
Chen et al. (2007)
14
17
Damart, Dias and Mousseau (2007)
13
16
Cao et al. (2004)
Macharis et al. (1998)
10
Dias and Clı´maco (2005)
Mustajoki and Ha¨ma¨la¨inen (1999)
9
11
Stanoulov (1995)
8
12
UNIDAS 2
Web-Hipre
Csa´ki et al. (1995)
7
JUDGES
Alicia and Sebastian
Carlsson et al. (1992)
Colson and Mareschal (1994)
5
Weighted criteria Weighted criteria
PLEXSYS 1989
AHP
Diverse
MT
GroupSystems ThinkTankTM
Team expert choice system
Co-oP
MCGDSS
6
Dennis et al. (1988)
Nunamaker et al. (1991)
Saaty (1989)
2
3
Bui and Jarke (1986)
1
4
Author
No.
Table 1 List of MCGDSS developed since 1980s up to date with its main features
J. C. Leyva Lo´pez et al.
A web-based group decision support system for…
making processes when DMs cannot meet together physically, see for instance (Leyva and Alvarez 2013; Leyva et al. 2011). In Table 1, we present a list of MCGDSS that have been developed since 1980s up to date. Some of them are discontinued (italicised rows); others have commercial purposes or applied in real problems. Most of the DSS implement a structured discussion; however some of them do not follow a formal discussion model (column DiM) for the resolution of the problem. In column DiM 7 DSS use a discussion model, 5 use it partially and 9 do not implement it. On another hand, we can stand out some features for GDSS: coordination modes, consensus schemes and consensus model. Coordination modes concern how individual activities are coupled and how individual preferences should be aggregated. (This feature is included in 3 DSS). Consensus schemes use of appropriate aggregation procedures for generating collective decisions for reducing the discordance among opinions (This feature is included in 11 DSS). Consensus model is a structured procedure to support in an objective way the reduction of disagreements and reach a high level of consensus solutions. This feature is implemented in 8 DSS and partially implemented in 2 DSS. These three main features are much related. All of them try to support DMs to reach better solutions; however they present different approaches, ones related with the activities of the individual and other with the aggregations procedures. The main advantages of the proposed MCGDSS is the inclusion of more models and tools for supporting DMs in the individual activities and aggregation preference to reduce disagreements and reach a high level of collective solution. In this section, we present a web-based MCGDSS named SADGAGE (by its acronym in Spanish: Sistema de Apoyo para la Toma de Decisisones en Grupo con Algoritmos Gene´ticos y ELECTRE) for solving multicriteria ranking problems, applying the theoretical consensus model presented in Sect. 2. It helps both the facilitator and decision-makers in their tasks (defining the problem, expressing alternatives and criteria, and so on) by means of a working agenda. The coordination modes functionality is an important contribution of the system because the discussion model allows facilitation and coordination of the decision process in two different working modes. On the one hand, the system provides group members with a facility to work in an individual manner and generates a ranking for each individual; later on, it generates a collective consensus ranking. On the other hand, the decision process can be coordinated in a consensual manner, in which the members must agree on every stage of the process and, therefore, obtain a collective ranking of alternatives. The system was programmed and fully implemented using the Visual Studio for Web in the.NET framework technology. The web site was developed with ASP.Net. The system has been designed in the form of different subsystems that interact with each other. Those different subsystems have been defined to separate the main logic of the system from data storage requirements and interface representation elements. Thus, it is possible to upgrade the system just by making changes in a particular subsystem. These subsystems are: • •
Group norm subsystem. Discussion subsystem.
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•
Multicriteria decision analysis subsystem.
The system SADGAGE is hosted at http://mcdss.udo.mx/sadgage. In the following, we describe the system’s subsystems, their interactions, and how users of the system (facilitator and decision-makers) are intended to interact with those subsystems. 3.1 System architecture The MCGDSS prototype was developed for the Internet platform; it runs within a web browser. The functional architecture incorporate the following features: use of a graphic interface, a group norm subsystem, a discussion subsystem and a multicriteria decision-aiding subsystem (see Fig. 4). Group norm subsystem. This subsystem provides the MCGDSS with functions to explicitly support definition and control of the decision-making process to be adopted for the meeting. With this capability, more objectivity and shorter meetings are expected. The group norm is the main component of this subsystem. It documents and controls the rules defined for conducting the meeting. A meeting starts when the facilitator presents the description of the decision problem to the group and defines the group norm. Three main aspects are defined in the group norm:
Web system User facilitator tool
User member tool
Group norm subsystem
Discussion Subsystem Storage module
Kaner model Decision zones Facilitation techniques
MCDA Subsystem Outranking methods Exploitation methods
Fig. 4 MCGDSS functional architecture
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Sequential coordination
Parallel coordination
A web-based group decision support system for…
Fig. 5 The Kaner model windows. This model is used for dividing the group ranking problem into a set of more specific issues
1. 2. 3.
Participants of the meeting definition. Discussion rules. Voting rules.
Discussion subsystem. In this MCGDSS, the Kaner model (1996) was adopted. This model is briefly detailed as follows. A group-ranking problem may be divided into a set of more specific issues, each one requiring a decision-making process. Each process consists of one or more zones. Four different zones come in the following temporal order: divergent (search for information); groan (discuss issues); convergent (attempt to reduce the number of solutions); and closure (select one solution by consensus or voting). Each zone can consist of one or more strategies (patterns) for handling the issue (see Fig. 5) (Leyva and Alvarez 2013). Multicriteria decision analysis subsystem. Following sequential or parallel coordination modes, a consensus would be sought throughout several stages of a multicriteria decision-making process, from problem formulation to multicriteria ranking determination. Consensus may be reached by applying aggregation methods at any appropriate stage. A procedure with a sequential coordination mode is carried out with the ELECTRE III-multiobjective evolutionary algorithm method; a procedure with a parallel coordination mode is carried out with the ELECTRE GDmultiobjective evolutionary algorithm method (see Fig. 6). 3.2 System features The prototype provides support for group MCDA process at three levels.
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Fig. 6 MCDA windows. This system has implemented the ELECTRE III and ELECTRE for group decision methods
3.2.1 Individual activity support For an effective coordination, it is important to know the degree to which the individual activities are supported, whether it use the Sequential or Parallel Coordination Mode. In the individual level, each group member can input the data/ information group by group in dialogue boxes prompted by the system and contained in the Generate tool (Brainstorming). The grouped data/information is then stored in each participant’s document, which can be accessed by the facilitator through the Facilitation tool. User’s input of data/information may be organized and hierarchically displayed with the Organize tool and modified if needed. Group members can also assess the alternatives by calling embedded ELECTRE III model and feeding it in with their own decision preferences. The ELECTRE III model component accepts weights and intercriteria parameter of the participant over each
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criterion. He or she, alone or with the support of the facilitator, can determine these intercriteria parameters or make use of the tools embedded in the system for supporting the determination of the weights of the criteria in the ELECTRE type methods with the revised simos’ procedure. A rank of alternatives in decreasing order of preference based on participants’ individual preferences can also be displayed and analyzed. 3.2.2 Group activity support Within group MCDA process an effective coordination is necessary to the achievement of distributed group decision-making task, since distributed decision-making activities are highly interdependent. In this process, some steps need to reach a consensus between the group members. Once all participants’ preferences are available, an aggregation of these preferences then takes place as a starting point for generating group decision preference afterwards. This aggregation information is distributed to each group member for polling. The polling result is regarded as group’s aggregated preference if every participant agrees, or another round of polling may be needed until a final consensus is reached. In these cases, the group members use the voting tool. When a consensus has not reached then a feedback process is active so that DMs may adjust their preferences to achieve closer preferences among them. The feedback process use embedded tools for the identification of the numerical values that need to be modified. 3.2.3 Facilitation support The group MCDA process is guided by the figure of a facilitator, where group members could work in a sequential or parallel coordination mode. The system is designed to help the facilitator to carry out his/her tasks during the different steps of the process. Facilitator plays an important role in the group decision making process supported by the system prototype. He or she controls process agenda and monitors process status with the facilitation tool. The facilitation based upon a pre-defined agenda determines the progression from one segment of decision making process to the next. The facilitation support component allows the facilitator to trace the participation status and progress of each group member (see Fig. 7).
4 Example of application In this section we present an example of application of the SADGAGE software to solve a simple multicriteria group decision problem. The problem is that of selecting the best city for water supply investment from a set of six different cities of Sinaloa State, Mexico. SADGAGE software is used to support the DMs to generate a ranking agreement derived from individual results. It is important to know the priority of the six cities in Sinaloa, Mexico for receiving the water supply
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Fig. 7 Facilitation supports. Based on a predefined agenda, the facilitator controls the group MCDA process and monitors process status with a facilitation tool
investment because it must carried out in only one city at time. In this sense six cities are competing for the water supply resource and four DMs (each representing different interest) should prioritize them in order to know the ranking, which the resource would be used. The DMs should reach an agreement of the collective ranking generated. As a part of group decision-making, conflict can be generated because different points of view of each DM. Every group’s member can express his/her preferences and define criteria to analyze the cities. 4.1 Criteria description 4.1.1 Cost of investment Each city requires different amount of resource for the water supply project. This criterion specifies the investment amount. 4.1.2 Population It is the number of persons shall be beneficed by the project. It is expected the greatest number of people is beneficed by the project. 4.1.3 Quality of life This criterion represents the sanitary and hygienic conditions of the population. The life conditions index (LCI) is used to estimates sanitation conditions. Thus, a city with the lowest LCI has higher priority to be attended with a water supply project with respect a city with higher LCI because its sanitation condition is more critical.
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4.1.4 Tourism This criterion measures an economic aspect of the project. The evaluation was taken by the actors related with the condition of the city in respect to the tourism. The values attributed to each verbal concept are 0.00 (weak), 0.33 (regular), 0.67 (good) and 1.00 (very good). The evaluation was treated as a multicriteria ranking problem. The DMs assessed six cities. Table 2 shows the code of each city and criterion. Table 3(a) shows the performance matrix of cities for each criterion and Table 3(b) the inter-criteria parameters (weights, indifference and preference thresholds) in iteration 1. The next stage is constructing the preferential model of every DM. This stage can be performed with different outranking methods. For this example, the ELECTRE III method was used as an embedded module in SADGAGE system. We can see the preferential model of every DM in Table 4. The preferential model is exploited by a multi-objective evolutionary algorithm developed by (Leyva and Aguilera 2005). A raking of alternatives for every DM is generated. At this stage we have the individual ranking of every DM shown in Table 5. The collective ranking is shown in the last column of Table 5. This is a temporal ranking that reflects the group preferences in iteration 1. The group preferences were generated by an aggregation approach for groups which was strongly based on ELECTRE (Leyva and Ferna´ndez 2003). Both algorithms are embedded in the SADGAGE system. In Table 5, the disagreements between individual ranking and collective ranking are showed. Also we have a row showing the similarity (proximity) between the individuals’ rankings and the collective ranking. This value was calculated by the proximity index proposed by Leyva and Alvarez (2015). The consensus level is computed based in the individual proximity scores. We shall explain in few steps how these values are obtained by the collective judgment. The disagreements are seen as the rank reversal presented between two rankings in pairwise format. In Table 5, DM2 presents three disagreements with the temporal collective solution. The disagreements are presented in the next pairs from DM2’s ranking {(E B), (E D), (C A)} because collective ranking presents {(B E), (D E), (A C)} pairs. The proximity index is computed considering the disagreement and the disagreement intensity. It means, disagreements occurred near from top position are stronger penalized that disagreements occur near from bottom positions. In this Table 2 Alternatives and criteria for prioritizing cities in the water supply problem
Alternatives
Criteria
A
El fuerte
C1
Cost of investment
B
Guasave
C2
Population
C
Los mochis
C3
Quality of life
D
Navolato
C4
Tourism
E
Mazatlan
F
Culiacan
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J. C. Leyva Lo´pez et al. Table 3 Performance and inter-criteria parameters Cl
C2
C3
C4
(a) Performance of the alternatives A
689,376
97,536
0.476
0.00
B
1,348,254
285,912
0.420
0.33
C
2,000,485
416,299
0.600
0.00
D
824,500
135,603
0.450
0.33
E
2,200,352
438,434
0.478
0.67
F
4,000,146
858,638
0.500
1.00
C1 Min
C2 Max
C3 Min
C4 Max
(b) Parameters: weights, indifference and preference thresholds in iteration 1 DM1 w
0.30
0.25
0.25
0.20
q
4,00,000
10,000
0.05
0.33
p
9,00,000
20,000
0.1
0.67
DM2 w
0.10
0.30
0.20
0.40
q
4,70,000
0
0.025
0
p
10,00,000
50,000
0.6
0.33
DM3 w
0.10
0.30
0.40
0.20
q
3,00,000
30,000
0.05
0
p
6,00,000
1,25,000
0.07
0.33
DM4 w
0.40
0.15
0.30
0.15
q
2,50,000
10,000
0.01
0.33
p
5,50,000
30,000
0.03
0.67
sense, the proximity index shows disagreement intensity between individual and collective rankings. The next steps show the process to obtain the proximity value of DM3’s ranking. Let O3 = [6, 3, 5, 4, 2, 1] and OG = [5, 2, 6, 3, 4, 1] be the orders associated to the DM3 ranking R3 = [a6, a5, a2, a4, a3, a1] and the temporary collective ranking RG = [a6, a2, a4, a5, a1, a3] respectively with respect to the ranking [a1, a2, a3, a4, a5, a6], where a1 = A, a2 = B, a3 = C, a4 = D, a5 = E, a6 = F. For each alternative ai ; 1 i 6, the score of the alternatives are: s31 = 1, s32 = 4, s33 = 2, P6 3 s34 = 3, s35 = 5, s36 = 6, with i¼1 si ¼ 21. The relative importance of the 1 4 2 3 5 6 , d32 ¼ 21 , d33 ¼ 21 , d34 ¼ 21 , d35 ¼ 21 , d36 ¼ 21 . We have to alternatives are: d31 ¼ 21 PO3 ðiÞ 3 carry out the accumulated relative importance of pO3 ðiÞ ¼ j¼1 dj , resulting:
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A web-based group decision support system for… Table 4 Preferential model of every DM (fuzzy outranking relation) A1
A2
A3
A4
A5
A6
A1
A2
A3
A4
A5
A6
A1
1
0.75
0.75
1
0.55
0.55
A1
A2
0.84
1
0.92
0.93
0.86
1
0.29
0.7
0.37
0.3
0.3
0.55
A2
0.96
1
0.7
0.99
0.3
0.3
A3
0.45
0.6
1
0.45
A4
1
0.87
0.75
1
0.55
0.3
A3
0.87
0.51
1
0.46
0.43
1.27
0.74
0.55
A4
1
0.7
0.7
1
0.3
0.3
A5
0.7
0.69
1
0.7
1
A6
0.7
0.55
0.7
0.7
0.7
0.75
A5
0.9
0.92
1
0.9
1
0.3
1
A6
0.9
0.88
0.9
0.89
0.9
1
DM1
DM2
DM3
DM4
A1
1
0.38
0.7
0.77
0.5
0.5
A1
1
0.55
0.85
0.61
0.7
0.7
A2
0.9
1
0.7
0.93
0.5
0.5
A2
0.6
1
0.85
0.63
0.85
0.7
A3
0.5
0.3
1
0.3
0.4
0.1
A3
0.3
0.3
1
0.3
0.46
0.4
A4
1
0.7
0.7
1
0.5
0.5
A4
1
0.55
0.85
1
0.85
0.7
A5
0.9
0.74
1
0.9
1
0.5
A5
0.6
0.3
1
0.33
1
0.85
A6
0.9
0.5
0.9
0.9
0.9
1
A6
0.39
0.3
0.6
0.3
0.42
1
Table 5 Individuals and group ranking in iteration 1
Disagrees: the number of differences between two rankings seen in pairwise format
Position
DM1
DM2
DM3
DM4
Collective
1
D
F
F
B
F
2
A
E
E
D
B
3
B
B
B
A
D
4
E
D
D
E
E
5
F
C
C
C
A
6
C
A
A
F
C
Disagrees
7
3
3
6
Proximity
0.614
0.808
0.808
0.597
Consensus level (CA) 0.707
1þ4þ2þ3þ5þ6 1þ4þ2 ¼ 1; pO3 ð2Þ ¼ ¼ 0:333; 21 21 1þ4þ2þ3þ5 1þ4þ2þ3 ¼ 0:714; pO3 ð4Þ ¼ ¼ 0:476; ¼ 21 21 1 1þ4 ¼ 0:238; pO3 ð6Þ ¼ ¼ 0:047: ¼ 21 21
pO3 ð1Þ ¼ pO3 ð3Þ pO3 ð5Þ
After that, we proceed to carry out the computation of pOG ðiÞ , resulting: pOG ð1Þ ¼ 0:714;
pOG ð2Þ ¼ 0:238;
pOG ð3Þ ¼ 1;
pOG ð4Þ ¼ 0:333;
pOG ð5Þ ¼ 0:476;
pOG ð6Þ ¼ 0:047
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J. C. Leyva Lo´pez et al.
The position weight are calculated by pi ðO3 Þ ¼
pOG ðiÞ pO3 ðiÞ OG ðiÞO3 ðiÞ ,
resulting:
0:761 1 0:238 0:333 ¼ 0:285; p2 ðO3 Þ ¼ ¼ 0:095 56 23 1 0:761 0:333 0:476 p3 ðO3 Þ ¼ ¼ 0:285; p4 ðO3 Þ ¼ ¼ 0:142 65 34 0:476 0:047 p5 ðO3 Þ ¼ ¼ 0:119; p6 ðO3 Þ ¼ 1: 41 p1 ðO3 Þ ¼
Now, we are in conditions for calculating the position-weighted version of the Kendall distance (KpðO3 ; OG Þ), as follows. For calculating the position-weighted version of the Kendall distance, we identify i;j ðO3 ; OG Þ between O3 and OG: O35 \O32 and the pairwise disagreements K G G 3 3 G 3 3 G G O5 [ O2 ; O5 \O4 and O5 [ OG 4 ; O3 \O1 and O3 [ O1 . Then, the positionweighted version of the Kendall distance is: KpðO3 ; OG Þ ¼ maxf0:119; 0:095g þ maxf0:119; 0:142g þ maxf0:285; 0:285g ¼ 0:547; and finally the proximity index value is: wP3;G A ¼1
42 0:547 ¼ 0:808: 120
For ranking associated to the DM1, DM2, and DM4, the proximity index values 2,G are computed in the same form, obtaining wP1,G A = 0.614, wPA = 0.808, 4,G wPA = 0.597, respectively. Consensus index is computed with the next equation. CA ¼
n X wPd;G A
d¼1
n
;
CA ¼
0:614 þ 0:808 þ 0:808 þ 0:597 ¼ 0:707: 4
In iteration 1, the rankings of the DM1 and DM4 with proximity value 0.614 and 0.597, respectively, present greater difference to the collective ranking than DM2 (0.808) and DM3 (0.808). The similarity values obtained in this iteration gain a consensus level (CA) of 0.707. For this procedure a required level of consensus is a = 0.75 (CA [ a). In this case the consensus level is not reached (0.707 \ a), it means that a second iteration is needed with the possibility of changing their preferences through a modifying parameters stage. DM1 and DM4 were agreed in setting up some parameters in the parameters modification stage. To support this stage we use the inferring parameter model (within the feedback stage) (see Alvarez et al. 2015) that uses as input a set of similar preference between individual and group solutions. The model concerns number of parameters’ changes (f1), number of disagreements (f2) and agrees of the first ranking that still remaining (f3); those objectives are evaluated in the new parameters proposal (see ‘‘Appendix 2’’ for a further explanation of these objective functions). For this example only two proposals are showed as a result of the feedback stage in the Tables 6 and 7. The new parameters suggested for the DM1
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A web-based group decision support system for… Table 6 Inter-criteria parameters proposals for improving the DM1’ agreement DM1 no.
f1
f2
f3
e
Proposal
Ranking
Proximity
1a
7
4/15
-
1.00
q1 = 3,33,817, p2 = 26,280, p3 = 0.124221, q4 = 0.14203, p4 = 0.223101, w2 = 0.30, w3 = 0.20
EFB DCA
0.750
1.00
q1 = 3,95,266, w2 = 0.379, p2 = 22,710, w3 = 0.121, q3 = 0, p3 = 0.124221, q4 = 0.14203, p4 = 0.776349
DBA EFC
0.700
8/8 2
8
1/15
8/8
a
Proposal selected by the DM1
Table 7 Inter-criteria parameters proposals for improving the DM4’ agreement DM4 no.
f1
f2
f3
e
Proposal
Ranking
Proximity
1
5
5/15
-
1.00
q1 = 3,28,037, p1 = 3,69,188, q2 = 16,454, q4 = 0.258743, p4 = 0.383351
DBA EFC
0.612
1.00
q1 = 3,28,037, p1 = 3,69,188, q2 = 20,000, w3 = 0.171, w4 = 0.279, q4 = 0.258743, p4 = 0.383351
DEB FAC
0.716
9/9 2a
7
1/15
9/9
a
Proposal selected by the DM4
and DM4 implies some changes in the weights, indifference and preference thresholds. Columns 2, 3 and 4 are the evaluation of f1, f2, f3, respectively. Column 5 is the epsilon value used to specify the neighborhood in which the parameter may vary. Column 6 presents the proposed parameters and column 7 presents the new generated ranking. The last column is the proximity index that the new ranking presents with the collective ranking of iteration 1. In iteration 2, the new inter-criteria parameters suggested by inferring model are showed in Tables 6 and 7 to the DM1 and DM4, respectively to support the modifying parameters stage. The proposal selected by DM1 and DM4 are used to generate new individual’s preference models and rankings, the DM2 and DM3 remain their previous ranking. In Table 8, a new collective ranking is generated with better consensus level because the proximity of DM1 and DM2’ ranking was improved in 0.993 and 0.660, respectively. This new proximity between rankings generates a better consensus level (CA = 0.894) greater than required (CA [ a). In this application, the procedure finished with the consensus level obtained by iteration 2.
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J. C. Leyva Lo´pez et al. Table 8 Individuals and group ranking in iteration 2
Position
DM1
DM2
DM3
DM4
Collective
1
E
F
F
D
F
2
F
E
E
E
E
3
B
B
B
B
B
4
D
D
D
F
D
5
C
C
C
A
C
6
A
A
A
C
A
Disagrees
1
0
0
6
Proximity
0.993
1.000
1.000
0.640
Consensus level (CA) 0.894
5 Conclusions and future works We have presented a web-based group decision support system for solving multicriteria ranking problems in a distributed and asynchronous environment and in a sequential or parallel coordination mode. We employed an order-based consensus model in a group MCDA for collaborative groups working in parallel or sequential coordination mode that moves from consistency to consensus. It is based on the use of fuzzy outranking relations to model individual and group preferences. This model includes a process for reaching consistency based on multiobjective combinatorial optimization. We define consensus and proximity measures; the first guides the consensus process, and the second supports the group discussion phase of the consensus process. In particular, exploitation of the model, which is based on multiobjective combinatorial optimization, generates advice on how DMs should change their preferences to reach a ranking of alternatives with high degrees of consistency and consensus. The system aims were to facilitate DMs expression of their preferences on alternatives in the multicriteria ranking problem while maintaining their consistency and to provide easy-to-understand recommendations in the form of simple rules to help the DMs converge to a solution for the multicriteria ranking problem with a high level of consensus. The proposed MCGDSS provides support for group MCDA process. It makes use of tools embedded in the system for supporting DMs in the individual and group activities to reduce disagreements and reach a high level of collective solution. In future works, we will improve and adapt the system to allow DMs to use the system from almost any mobile device. Acknowledgments This paper has been developed with the partial financing of the National Council for Science and Technology (CONACyT) of Mexico funds (Inovapyme-Projects: 179729, 217534). Compliance with ethical standards ´ lvarez Carrillo has received research grants from National Conflict of interest Author Pavel Anselmo A Council for Science and Technology (CONACyT) of Mexico funds (Inovapyme-Project: 179729). Author Diego Alonso Gastelum Chavira has received research grants from National Council for Science and
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A web-based group decision support system for… Technology (CONACyT) of Mexico funds (Inovapyme-Project: 217534). Author Juan Carlos Leyva Lopez works at Universidad de Occidente and Universidad Autonoma de Sinaloa. Author Pavel Anselmo Alvarez Carrillo works at Universidad de Occidente and Universidad Autonoma de Sinaloa. The software is registered with the National Institute of Copyrights (INDAUTOR) of Mexico.
Appendix 1: A procedure for reaching consistency based on multiobjective combinatorial optimization Let A = {a1, a2, …, am} be the set of decision alternatives or potential actions, and let us consider a fuzzy outranking relationship SrA defined on A 9 A; this means that we associate with each ordered pair (ai, aj) 2 A 9 A a real number r(ai, aj)(0 B r(ai, aj) B 1) that reflects the degree of strength of the arguments favoring the crisp outranking aiSAaj. The exploitation phase transforms the global information included in SrA into a global ranking of the elements of A (Fodor and Roubens 1994). The main difficulty in the exploitation phase of all outranking methods consists of finding reasonable means of dealing with non-transitivities without losing too much of the content of the original outranking relationship (Leyva and Araoz 2013). In methods based on score functions, non-consistent situations could occur when the prescription is constructed. The most significant situation is the following: suppose that ai and aj are two actions such that r(ai, aj) C k and r(aj, ai) B k - b(b [ 0). If k C c and b C t (c and t representing consensus and threshold levels, respectively, which are usually given by the decision-maker), we should accept that ‘‘ai outranks aj’’ (aiSkAaj) and that ‘‘aj does not outrank ai’’ ðaj SkA ai Þ. In this case, the global preference model captured in the outranking relationship provides a presumed preference favoring ai. A score function or other similar method could lead, however, to a final ordering in which aj is ranked before, giving a pairwise rank reversal effect (Mareschal et al. 2008; Roy and Bouyssou 1993). Methods based on score functions do not have a mechanism to detect and minimize this type of irregularity. Leyva and Aguilera (2005) and Leyva and Araoz (2013) propose a mechanism showing an approach to minimizing inconsistency—in terms of pairwise rank reversal and incompleteness. In this paper, we extend this mechanism to the group multicriteria ranking problem under fuzzy outranking relations SrA. We give a characterization of fuzzy consistency based on the crisp consistency in a family of nested crisp outranking relations SkA (SkA = {(a, b) 2 A 9 A: r(a, b) C k}, k 2 [k0, 1]); these crisp relations correspond to k-cuts of SrA, where the cutting level k represents the minimum value for SrA such that aSkAb is true. This characterization facilitates the verification of pairwise rank reversal and completeness in the case of fuzzy outranking relations. Using this characterization, we describe in the following a procedure that minimizes inconsistencies—in terms of pairwise rank reversal and incompleteness—between fuzzy outranking relations and the associated rankings.
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Procedure A potential solution of this type of ranking problem could be represented as an _ ordinal representation. In general, a potential solution p is a ranking of the set of decision alternatives by decreasing order of preference. These alternatives are joined together to form a ranking. The ranking is represented as the permutation of an m-ary alphabet where m is the number of alternatives in the decision problem. In such a representation, each alternative is coded into m-ary form. Alternatives are then linked together to produce one long m-ary permutation. An alternative coded with value aki in the ith entry of the permutation means that the alternative coded with value aki is ranked in the ith place of the ranking and aki is preferred to akj if i \ j, where aki 2 A ¼ fa1 ; a2 ; . . .; am g, i = 1, 2, …, m, and [k1, k2, …, km] is a permutation of [1, 2, …, m]. _ Each potential solution p in the space of permutations is associated with a number k (0 B k B 1), which is connected with the credibility level of a crisp outranking relationship defined on the set of alternatives. We define the objective _ _ functions f and u of a permutation p with credibility level k as follows: Let p ¼ ak1 ak2 . . . akm be the schematic representation of a potential solution (permutation) of the ranking problem, and suppose that given aki and akj , two alternatives such that rðaki ; akj Þ k and rðakj ; aki Þ k b (b [ 0, representing a threshold level), we accept that ‘‘aki outranks akj ’’ ðaki SkA akj Þ and ‘‘akj does not outrank aki ’’ ðakj nSkA aki Þ. In this case, in the crisp outranking relationship generated by k, SkA, a presumed preference favoring aki , holds. Then: _ f ðpÞ ¼ ðaki ; akj Þ : aki nSkA akj and akj nSkA aki ; ð2Þ i ¼ 1; 2; . . .; m 1; j ¼ 2; 3; . . .; m; i\jgj _
where [k1, k2, …, km] is a permutation of [1, 2, …, m] and f ðpÞ is the number of _ incomparabilities between pairs of actions ðaki ; akj Þ in the permutation p ¼ ak1 ak2 . . . akm in the sense of the crisp relationship SkA. Note that the quality of _ potential solution p increases with decreasing f score. _ The objective function u of a permutation p measures the amount of inconsistency (in relative terms); we chose to define it as follows: _ uðpÞ ¼ ðaki ; akj Þ : aki SkA akj and akj nSkA aki ; ð3Þ i ¼ 1; 2; . . .; m; j ¼ 1; 2; . . .; m; i [ jgj _
_
uðpÞ is the number of preferences between alternatives in permutation p that are not _ ‘‘well-ordered’’ in the sense of SkA. uðpÞ measures the amount of pairwise rank _ reversals between p and SkA. _ _ A permutation p is consistent with respect SkA if uðpÞ ¼ 0 and inconsistent if _ uðpÞ [ 0. Defining the objective function u as taking the zero minimum value if and _ only if the solution is consistent seems a natural approach. Each permutation p can
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then be represented by a triad of values f, u, and k. We are interested in the following: 1.
2. 3.
Permutations whose objective function u value is equal to zero. This assures us that the ranking represented by the permutation is transitive; this is one of two characteristics that should be exhibited by all recommendations (solutions) of ranking problems (Vanderpooten 1990). Permutations whose objective function f value is equal (or near) to zero. This objective improves the comparability of S on A. Permutations whose credibility level k is near to 1. This indicates that the ranking represented by the permutation with credibility level k is more trustworthy whenever the objective functions u and f values are zero or near to zero. In practice, the requirement connected to function f does not permit k values to approach 1 because in this case we could have many incomparable genes.
We want to solve the multiobjective combinatorial optimization problem: _
MinðuðpÞ; Subject to _
_
Minðf ðpÞÞ;
p 2 PermðAÞ;
k 2 ½0; 1 ;
MaxðkÞ ð4Þ
k k0
ðwhere PermðAÞ is the set of permutations of A and k0 is a minimum level of credibilityÞ Because of the structure of the multiobjective optimization problem (characterized by representation of a potential solution as a permutation of the elements of A, and the multiobjective and combinatorial nature), we use the evolutionary algorithm proposed in Leyva and Aguilera (2005) to exploit the fuzzy outranking relationship and to obtain a recommendation in the form of the most consistent ranking possible.
Appendix 2: Two procedures for the feedback mechanism based on multiobjective combinatorial optimization In this appendix, we present two MOEAs based on a posterior articulation of preferences. These MOEAs are able to detect either the set of alternatives or the set of intercriteria parameters that a DM should consider to change their evaluation of a particular subset of criteria fgk1 ; gk2 ; . . .; gkl g, with the purpose of reducing pairwise disagreements between the dth order and the collective temporary order. The algorithms borrow fundamental elements from NSGA II (Deb et al. 2002). In the following subsections, after some basic definitions, we present in further detail fundamental aspects of the algorithms.
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Basic definitions Definition 1 Let A = {a1, a2, …, am} be a set of m alternatives and G = {g1, g2, …, gn} be a set of n decision criteria defined on A. Without loss of generality, we can consider the first t criteria Go = {g1, g2, …, gt} as objective criteria and the rest n - t criteria Gs = {gt?1, gt?2, …, gn} as subjective criteria. The performance matrix, denoted by Md, of the dth DM, is the matrix of order m 9 n where the (i, j) entrance of Md is gj(ai). Definition 2 The performance profile Q(ai) associated with action ai 2 A is an ntupla defined as follows: Qðai Þ ¼ ha1i ; a2i ; . . .; asi ; . . .; ani i
where; asi ¼ gs ðai Þ:
Definition 3 Let Kd be the set of pairwise disagreements between the dth order Od and the collective temporary order OG defined as follows: ðai ; aj Þ 2 A Aji\j; ðOd ðai Þ\Od ðaj Þ ^ OG ðai Þ [ OG ðaj ÞÞ_ d d G K ðO ; O Þ ¼ ðOd ðai Þ [ Od ðaj Þ ^ OG ðai Þ\OG ðaj ÞÞ The first complement of Kd, Kd,C first, is the set of pairwise agreements between the first dth order Od and the collective temporary order OG and is defined as follows: ðai ; aj Þ 2 A Aji\j; ðOd ðai Þ\Od ðaj Þ ^ OG ðai Þ\OG ðaj ÞÞ_ d;C d G Kfirst ðO ; O Þ ¼ ðOd ðai Þ [ Od ðaj Þ ^ OG ðai Þ [ OG ðaj ÞÞ d;C Note that K d þ Kfirst ¼ ðm 1Þ!. Definition 4 Let KdM be the set of marginal pairwise disagreements between the dth order Od and the collective temporary order OG defined as follows: n
d ¼ asi ; asj 2 Qðai Þs Q aj s j ai ; aj 2 K d ; KM i; j ¼ 1; 2; . . .; m; i 6¼ j; s ¼ 1; 2; . . .; ng; where Q(ai)|s is the sth projection map of Q(ai). Definition 5 An alternative ai is feasible (firm) over aj on a criterion gk in the sense of a marginal strict preference relationship, denoted aiPfkaj or ajPfkai, if the kth criterion is an objective criterion or the kth criterion is a subjective criterion and the dth DM is completely sure about this asseveration and will not change his/her preference during the consensus process. Let Xk be the set of pairs (aki , akj ) such that the alternative ai is feasible over aj on criterion gk, i, j = 1, 2, …, m, k = 1, 2, …, n, i.e.: n
o aki ; akj jai Pfk aj _ aj Pfk ai ; i; j ¼ 1; 2; . . .; m ; k ¼ 1; 2; . . .; n: Xk ¼ Let X be the set of pairs (aki , akj ) such that the alternative ai is feasible over aj, i, j = 1, 2,…, m, on at least one k = 1, 2, …, n i.e.:
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X¼
n [
Xk
k¼1
Let Xs be the set of pair (aki , akj ) such that the alternative ai is feasible over aj, i, j = 1, 2, …, m, on at least one subjective criterion gk, k = t ? 1, t ? 2, …, n i.e.: Xs ¼
n [
Xk :
k¼tþ1
Definition 6 Let In_KdM be the set of marginal pairwise disagreements between the dth order and the collective temporary order that are not feasible (or infeasible). It is defined as follows: d d In KM ¼ KM X:
Definition 7 Let Fea_KdM be the set of marginal pairwise disagreements between the dth order and the collective temporary order that are subjectively feasible. It is defined as follows: d d Fea KM ¼ KM \ Xs :
The preferences will be changed using the following three rules: Rule 1 If Od(aj) - OG(aj) [ 0, then increase evaluations associated to alternative aj. Rule 2 If Od(aj) - OG(aj) = 0, do not change evaluations associated to alternative aj. Rule 3 If Od(aj) - OG(aj) \ 0, then decrease evaluations associated to alternative aj.
A multiobjective evolutionary algorithm for identification of alternatives We use a value-encoding scheme to represent a potential solution. In value encoding, each chromosome is represented as a string of some values. The values can be integer, real number, character or some object. Let p~ ¼ Qp1 p2 . . . pmn be the schematic representation of an individual’s chromosome, p~ 2 mn i¼1 Ci , where Ci is the set of values that pi can take. Objective functions f1, and f2 The fitness of an individual is calculated according to a given fitness procedure. The approach for defining an individual’s fitness involves the non-dominated solutions in a form similar to NSGA II (Deb et al. 2002). We define the objective function f1 of an individual p~ as follows.
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Let p~ ¼ p1 p2 . . . pmn be the schematic representation of an individual’s chromosome. Let p~R ¼ r1 r2 . . . rmn be a reference individual representing the original performance matrix: r1 = g1(a1), r2 = g2(a1),…, rn = gn(a1), rn?1 = g1(a2), rn?2 = g2(a2),…, r2n = gn(a2), …, r(m-1)n?1 = g1(am), r(m-1)n?2 = g2(am), …, rmn = gn(am). Then: pÞ ¼ jfðpi ; ri Þj f1 ð~
pi 6¼ ri
i ¼ 1; 2; . . .; mngj
pÞ is the number of modified marginal evaluations of alternatives. Note that the f1 ð~ quality of solution increases with decreasing f1 score. With this objective function, we want to preserve, as much as possible, the original performance matrix. The objective function f2 of an individual p~ measures the amount of unfeasibility and subjective feasibility (in relative terms); we chose to define it as follows: f2 ð~ pÞ ¼ In K d þ Fea K d M
M
f2 ð~ pÞ is the number of infeasible marginal pairwise disagreements between the dth order and the collective temporary order plus the number of subjective feasible marginal pairwise disagreements between the dth order and the collective temporary order. With this objective function we want to reduce pairwise disagreements between the dth order and the collective temporary order to increase the value of proximity measure wPd,G A . We include |Fea_KdM| in the definition of f2 because we can modify the values of (aki , akj ) 2 Fea_KdM but always taking care to keep the original preferences given by the dth decision-maker, for the reason that an original preference (ak*, ak**) Fea_KdM, may be lost, where *, or ** can be i or j. An individual p~ is feasible if f2 ð~ pÞ ¼ 0 and infeasible if f2 ð~ pÞ [ 0. Defining the objective function f2 as taking the zero minimum value if and only if the solution is feasible seems a natural approach. We are interested in the following: 1.
2.
Individuals whose objective function f1 value is close to zero. This assures us that the ordering represented by the individual is almost equal to the original performance matrix; this is one characteristic always appreciated for all rational decision-makers. Individuals whose objective function f2 value is close to zero. This objective improves feasibility and reduces the disagreements between the two orders.
Then, we use an evolutionary search to solve the multiobjective combinatorial problem: Minðf1 ð~ pÞÞ; Subject to mn Y Ci : p~ 2 i¼1
123
Minðf2 ð~ pÞÞ
A web-based group decision support system for…
A multiobjective evolutionary algorithm for identification of intercriteria parameters We use a value-encoding scheme to represent a potential solution p~. Let p~ ¼ p1 p2 . . . p4n be the schematic representation of an individual’s chromosome. Q p~ 2 4n i¼1 Ci , where Ci is the set of values that pi can takes. This set of values is dependent of the problem. In Alvarez et al. (2015) is carried out a case study using this algorithm. Objective functions f1, f2, and f3 The fitness of an individual is calculated according to a given fitness procedure. The approach for defining an individual’s fitness involves non-dominated solutions in a form similar to NSGA II (Deb et al. 2002). We define the objective function f1 of an individual p~ as follows. Let p~ ¼ p1 p2 . . . p4n be the schematic representation of an individual’s chromosome. Let p~R ¼ r1 r2 . . . r4n be a reference individual representing the original intercriteria parameters: r1 = w1, r2 = q1, r3 = p1, r4 = v1, r5 = w2, r6 = q2, …, r4n-2 = qn, r4n-1 = pn, …, r4n = vn. Next, we have the following: pÞ ¼ jfðpi ; ri Þj f1 ð~
pi 6¼ ri
i ¼ 1; 2; . . .; 4ngj
pÞ is the number of modified intercriteria parameters. Note that the quality of f1 ð~ solution increases with decreasing f1 score. With this objective function, we want to preserve, as much as possible, the original intercriteria parameters. The objective function f2 of an individual p~ measures the amount of pairwise disagreements between the dth order Od and the collective temporary order OG; we chose to define it as follows: f2 ð~ pÞ ¼ K d f2 ð~ pÞ is the number of pairwise disagreements between the dth order and the collective temporary order. With this objective function, we want to reduce pairwise disagreements between the dth order and the collective temporary order to increase the value of proximity measure wPd,G A . The objective function f3 of an individual p~ measures the amount of pairwise agreements between the first dth order Od and the collective temporary order OG; we chose to define it as follows: d;C f3 ð~ pÞ ¼ Kfirst f3 ð~ pÞ is the number of pairwise agreements between the first dth order and the collective temporary order. With this objective function, we want to maximize the original pairwise agreements between the dth order and the collective temporary order to preserve the rationality and values system of the DM.
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We are interested in: 1.
2. 3.
Individuals whose objective function f1 value tend to zero. This assures us that the ordering represented by the individual is almost equal to the original set of intercriteria parameters; this is one characteristic always appreciated for all rational decision-makers. Individuals whose objective function f2 value tend to zero. This objective improves feasibility and reduces the disagreements between the two orders. Individuals whose objective function f3 tend to (m - 1)! This assure us that the ordering represented by the individual increases the value of the proximity measure wPd,G A .
Thus, we use an evolutionary search for solving the multiobjective combinatorial problem: Minðf1 ð~ pÞÞ;
Minðf2 ð~ pÞÞ;
Maxðf3 ð~ pÞÞ
Subject to p~ 2
4n Y
Ci :
i¼1
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