円谷 友英

In a group decision making problem, the group decision as a consensus of the members is usually more emphasized than the individual decisions in the group. However, in our public or private life, we often make decisions individually with others' help. In other words, the individual decision is not always completely independent of the others but affected by the others to some extend. This paper proposes the approach to derive the individual decision from a group of individual judgments. We consider two viewpoints: the preciseness of the obtained individual decision to the corresponding given judgment and the similarity of the individual decisions. The latter is an intersection of the individual decisions, and it is positive and negative when there is a common of all and not, respectively. We formulate the problem to obtain the individual decisions from a group of individual judgments as an LP problem with two degrees of preciseness and similarity. Both degrees are given so as independently to be suitable for the decision maker's preference and the decision problem. Since they have a trade-off relation, there is a Pareto optimal state. The obtained individual decision is supported by the others and is a more suitable one to the situation among the possible decisions derived from the intuitive individual judgment.

Understanding the drivers of deforestation and forest degradation and the agents of such drivers is important for introducing appropriate policy interventions. Here, we identified drivers and agents of drivers through the analysis of local perceptions using questionnaire surveys, focus group discussions, and field observations. The Likert scale technique was employed for designing the questionnaire with scores ranging from 1 (strongly disagree) to 5 (strongly agree). We found nine direct drivers of forest deforestation and forest degradation, namely illegal logging (4.53 ± 0.60, ± is for standard deviation), commercial wood production (4.20 ± 0.71), land clearing for commercial agriculture (4.19 ± 1.15), charcoal production (3.60 ± 1.12), land clearing for subsistence agriculture (3.54 ± 0.75), new settlement and land migration (3.43 ± 0.81), natural disasters (3.31 ± 0.96), human-induced forest fires (3.25 ± 0.96), and fuelwood for domestic consumption (3.21 ± 0.77). We also found four main indirect drivers, namely lack of law enforcement, demand for timber, land tenure right, and population growth. Our analysis indicates that wood furniture makers, medium and large-scale agricultural investors, charcoal makers, land migrants, firewood collectors, and subsistent farmers were the agents of these drivers. Through focus group discussions, 12 activities were agreed upon and could be introduced to reduce these drivers. In addition to enforcing the laws, creating income-generating opportunities for locals along with the provision of environmental education could ensure long-term reduction of these drivers. The REDD+ project could be an option for creating local income opportunities, while reducing deforestation and forest degradation.

Climate-change mitigation projects are expected to improve local livelihoods in targeted areas. Several REDD+ projects aimed at reducing emissions from deforestation and forest degradation, conserving and enhancing forest carbon stocks, and sustainably managing forests have been implemented in Cambodia but few studies have examined the effects on local livelihoods before and during project implementation. Our study applies a sustainable livelihood framework to assess the livelihood assets of local communities in the Oddar Meanchey and Keo Seima REDD+ project sites in Cambodia before and during project implementation. Five capital assets, namely natural, physical, human, financial, and social capital, are assessed and scored on a 1-to-5 Likert scale. Data analysis collected through 252 interviews in Oddar Meanchey and Keo Seima reveals a slight increase in livelihood assets in both sites from project validation to implementation. Generally, the mean scores for local livelihood assets increased from 2.81 ± 0.07 (±is followed by the standard error) and 2.66 ± 0.06 to 3.07 ± 0.09 and 3.06 ± 0.08 in Oddar Meanchey and Keo Seima, respectively. Nevertheless, natural capital assets sharply declined from 3.50 and 3.32 to 2.09 and 2.25, respectively. Respondents mainly blamed illegal logging for the decline, suggesting that strict patrolling and enforcement must be implemented. Furthermore, the scarcity of carbon-credit buyers and the projects’ inability to generate carbon-based revenues has led to dissatisfaction among local communities, inducing avoidable illegal activities in pursuit of short-term benefits. A financial mechanism to ensure sufficient and sustained financial support regardless of carbon-market volatility is urgently needed.

Organizations are interested in exploiting the data from the other organizations for better analyses. Therefore, the data related policies of organizations should be sensitive to the data privacy issue, which has been widely discussed recently. The present study is focused on inter-group data usage for a relative evaluation. This research is based on the data envelopment analysis (DEA), which is used to measure the efficiency of a decision making unit (DMU) relatively employed within a group. In DEA, establishing an efficient frontier consisting of efficient DMUs is essential. We can obtain the efficiency values of a DMU by projecting it to the efficient frontier, and including in the efficiency interval via the interval DEA. When the original data of multiple groups are not open to each other, the alternative is to exchange the information corresponding to the efficient frontiers to estimate the efficiency intervals of a DMU in such a manner that the alternative is in the other groups. Therefore, in this paper, we propose a method to replace the efficient frontier with a weight vector set, from which it is not possible to reconstruct the original data. Considering the weight vector sets of multiple groups, a DMU has three types of efficiency intervals: In its own group, in each of the other groups, and in the integrated group. They provide rich insights on the DMU from a broad perspective, and this encourages inter-group data usage. In this process, we focus on two types of information reduction: One is from the efficient frontier to the weight vector set, and the other is from a union of the groups to the integrated group.

Identification of the drivers and their agents of deforestation and forest degradation is important for effective implementation of the REDD+ activities. Here, we identified the direct and indirect drivers through the analysis of local perceptions. The mixed method was used for data collection and analysis. A survey of 215 families and four focus group discussions with 72 participants were conducted in seven community forests in Kampong Thom province, Cambodia. The Likert scale scoring was used to assess the level of acceptance by the locals. We found nine direct drivers and four main indirect drivers and six agents of forest deforestation and forest degradation.

A text written by non-native speakers makes readers feel unusual, even if there are no evident errors in the sense of basic grammar. We discuss such an intangible error in this paper. Therefore, the goal is to discover the similarities of writing among foreign languages by Japanese native speakers. We compare the original texts written in a foreign language and their revisions by a native speaker to identify the parts which s/he feels unusual and gives the options. To not rely on the subjective context and the uniqueness of a language, we replace a word into a part-of-speech by the morphological analysis. Some parts-of-speech are deleted from the original text by revising a text, and some parts-of-speech are added instead. We consider a set of different parts-of-speech, which is added or deleted by the revision, and find common trends with multiple languages from the numbers and ratios of sentences. These findings are not always familiar to the experts, and on the other hand, they may give new viewpoints to language learners and teachers to improve their activities.

In this study, we propose a method to assign the weights to the viewpoints for the assessment of an item. The weight vector is obtained from the pairwise comparison matrix of the item on the viewpoints given by a decision maker. The comparison matrix is incomplete in the sense of the missing comparisons and inconsistency among the comparisons. As for the missing comparisons, we assume the inclusion relation between the given comparison and the obtained interval weights as in Interval Analytic Hierarchy Process (AHP). As for the inconsistency, we assume the comparison matrices of the other items by the decision maker and find out his/her consistency degree. Moreover, the incompleteness motivates us to consider the other decision makers who give the comparison matrices of the item. In other words, all the given comparisons are reliable since a decision maker need not give the comparisons if s/he is not confident in it. It is reasonable that the weight vector of the item should be the core of those of all the decision makers. The weight vector of each decision with a higher consistency degree is more preferable. However, there is a trade-off among the consistency degree, the inclusion relation, and the core condition. We introduce a fuzzy approach, which simultaneously minimizes the relaxations of the inclusion relation and the core condition and maximizes the consistency degree, to obtain the weight vector of the target item.

In this study, we propose an approach to derive a group assessment of an item as its weight vector on multiple viewpoints. When there are a group of decision makers who give the judgments of the item as comparison matrices on the viewpoints, it is reasonable that the weight vector of the item is the core of those by all the decision makers. Each decision maker’s weight vector basically includes his/her given comparison matrix, which represents only a part of his/her thinking. Namely, there is an inclusion relation between a comparison and a ratio of the corresponding weights. In addition, there are items other than the target one. A decision maker gives the comparison matrices of some of the other items if s/he knows them, as well as the target one. It is natural that there is a correlation between the judgment of the target item to those of the others. The correlation is taken into consideration from the aspect of consistency of his/her judgments. We define a fuzzy degree of the consistency with all the comparison matrices s/he gives. As the consistency degree for a comparison matrix increases, it may become unable to satisfy the inclusion relation between the comparison matrix and the weight vector. Hence, we introduce a fuzzy degree of inclusion relation in order to relax it. There is a trade-off between them. Therefore, by maximizing both degrees we obtain the weight vector of the target item from the comparison matrices of the target item considering the consistency of each decision maker’s judgments. The proposed approach is applicable even in the case that a group of given comparison matrices is incomplete such that some comparisons in a comparison matrix are missing and/or the comparison matrices of some items are missing.

This paper proposes the method to derive the inner evaluation of a unit in a group of units in the sense of efficiency from the multiple perspectives. It aims to compare the multiple evaluations of a unit rather than the evaluations of the units. For a comprehensive analysis of a unit, its evaluations from the multiple perspectives are necessary, although it is not easy for us to compare them each other. Even from a specific perspective, we can have various evaluations depending on the viewpoints for the unit. In order to tackle these issues, first, we measure the efficiency of a unit from each perspective. We denote it as an efficiency interval considering various viewpoints based on interval data envelopment analysis. Then, we normalize the obtained multiple efficiency intervals with respect to the perspectives so that they can be compared. The normalized efficiency intervals are useful to know the characteristic of the unit in detail instead to rank the units.

In this study, our uncertain judgment on multiple items is denoted as a fuzzy weight vector. Its membership function is estimated from more than one interval weight vector. The interval weight vector is obtained from a crisp/interval comparison matrix by Interval Analytic Hierarchy Process (AHP). We redefine it as a closure of the crisp weight vectors which approximate the comparison matrix. The intuitively given comparison matrix is often imperfect so that there could be various approaches to approximate it. We propose two of them: upper and lower approximation models. The former is based on weight possibility and the weight vector with it includes the comparison matrix. The latter is based on comparison possibility and the comparison matrix with it includes the weight vector.

This study considers one of the decision problems, which is to assign numbers to multiple items based on our judgments. In this paper, a fuzzy weight vector is obtained from a group of the crisp comparison matrices which are given intuitively, since a fuzzy number is more suitable to represent our thinking than a crisp number. It is based on the idea that the result could be better if a comparison matrix is supplemented by the other comparison matrices on the same items. In this sense, the goal of this study is not to find a consensus, which is often discussed in group decision making literature. The proposed methods consider both relations among the comparisons in each comparison matrix and among the comparison matrices simultaneously so that there is no aggregation process. They derive an interval weight vector as a level set of fuzzy weight vector from a group of comparison matrices. Then, the membership function of the fuzzy weight vector is estimated from the obtained interval weight vectors.

Although writing is a tool for communication, the way one writer communicates a fact is not always the same as how another one does it. The written word is unique to the writer and reflects his or her preferred writing style. When something is written by a non-native speaker of language, native speakers and experts often feel slightly unusual, even if they can find no obvious errors. Moreover, they might revise the text based on their experience. On the other hand, the writer often feels slightly dissatisfied with the correction if it does not fit for his or her writing preference. It is difficult for the corrector to understand the writers' writing preference from the text, and it is also difficult for the writer to explain it explicitly since both writing and correcting a piece of text are based on one's subjectivity. The correction is unique to the text, so the inner evaluation of the text is important. This study proposes a method of deriving each writer's writing preference numerically from the expert's initial evaluation. In the process, the texts other than the target text are taken into consideration from the viewpoint that writing is a communication tool. The corrector may use the feedback from the proposed method to confirmhis or her intuitive judgments and to add some new viewpoints.

This study aims to obtain the target individual weight vector from a group of the comparison matrices which are intuitively given by a group of the experts. Each expert evaluates each individual on the multiple criteria and gives his/her intuitive judgments as a pairwise comparison matrix. A targeted individual is evaluated by a group of experts and each expert evaluates the other individuals as well as the target. Therefore, there are two kinds of the groups with respect to the target and each expert, respectively. The relations among the comparison matrices in the group are reflected by the inclusion relations of the corresponding weight vectors. As for the target, its final target weight vector is the core of its local target weight vectors by all the experts so that it is included in each of them. As for each expert, in order to reflect his/her evaluating tendency, the target weight vector by him/her is influenced by the local weights of the others by him/her so that it intersects with them. Then, we introduce fuzzy sets to relax two inclusion relations and formulate the LP problem.

<p>本研究では，複数の評価者が複数の基準の下で，個別に評価した複数の対象のうち，ひとつを評価対象とし，その内部評価を導く．各評価者は，各対象を定められた評価基準の下で一対比較を行い，全対象に関する直感的な判断を与え，評価対象は，全評価者から評価される．したがって，評価対象に関する一対比較行列グループと各評価者に関する一対比較行列グループが考えられる．それぞれに固有のグループ内の相関に着目して，それを対応するウェイトベクトルの包含関係で表す．前者のグループ内の判断をまとめることで，評価対象の内部評価は得られるが，さらに，後者のグループにより，だれが評価したかを考慮して，評価者の特性を反映する．各グループ内の包含関係を順次緩和し，得られる区間ウェイトベクトルからファジィウェイトベクトルを導く．</p>

The organization is willing to take data from the other organizations for the better analysis, when the self-accumulated data only gives the limited findings for instance in case of focusing on a specific small group. The multi-accumulated data can be available for the analysis, if the different several organizations accumulated similar data independently. However, the organizations need to care for privacy since the original data could include a kind of personal information. Under such a privacy concern, this paper proposes the method for efficiency evaluation with multi-accumulated data based on Data Envelopment Analysis (DEA), which is a non-parametric technique to measure efficiency of a decision making unit (DMU) relatively in a group. The efficiency interval of a DMU in a group by referencing DMUs in another group as well as DMUs in its own group is obtained, even if the group cannot access to the original data of another group. Instead, the group takes the information of the efficient frontier of another group denoted as the weight set, from which the group cannot guess the original data of another group. As a result, three kinds of efficiency intervals for a DMU are obtained: the efficiency in its own group, that in another group, and that in the integrated group. Comparing them can give us a rich and useful information on the DMU from wide viewpoint.

An approach for the individual assessment under multiple criteria is proposed. The goal is to obtain the ratios of the evaluations of the criteria for a target unit by comparing it to the units in the group. A criterion is often abstract and it is difficult for us to evaluate the unit under it. While, there are some subcriteria which each criterion relates to. It is easy to evaluate the unit under them, if they are often objective. The first problem is to obtain the relation of the subcriteria to the criterion. It is denoted as the weights of the subcriteria, since the evaluations of the subcriteria are synthesized into that of the criterion. The weights are peculiar to the target unit. Furthermore, the weights for a criterion are varied depending on to which criterion it is compared, since the individual assessment considers the relative evaluations of the criteria. Therefore, the comparison of the evaluations of each pair of the criteria is obtained. Then, the second problem focuses on each unit and is to derive the ratios of the evaluations of the criteria from the comparisons of their pairs in the first problem.

In this paper, the approach to estimate a fuzzy weight vector from an interval comparison matrix is proposed. The interval comparison allows a decision maker to state his/her uncertain judgment as a range, instead of a crisp value. By increasing and decreasing its upper and lower bounds of the interval comparison by the inverse rates, the processed comparison matrices are derived from the given matrix. The membership function of the fuzzy weight is based on the certainty degrees of the interval weight vectors obtained from the processed matrices. The interval weight vector is defined as a closure of the normalized crisp weight vectors each of which is included in an interval comparison matrix. Its certainty degree is represented as the sum of the lower bounds of all the corresponding interval weights.

Different models to obtain the group weights of alternatives from a group of pairwise comparison matrices have been proposed. The interest of this paper is to induce the individual weights rather than the group weights. Therefore, we propose the model to obtain the individual weights that a decision maker alone cannot find. The individual weights consider the group weights, as well as his/her comparisons, since it is not sure how well the comparisons represent a decision maker's thinking. The degree of the group influence on individual revisions is based on the group reliability, which is the sum of the group weights of all alternatives. By agreement, the group weight should be the minimum of all decision makers' weights. In inducing the individual weights, this paper reconsiders the inconsistency among the comparisons, it stems from ignorance of the weight in a decision maker's mind. In addition to the weight surely assigned to each alternative, the unknown weight possibly shared by all alternatives is considered to reflect the inconsistency into it.

A decision problem is structured hierarchically by criteria and alternatives in AHP. The goal of this paper is to review criterion importance from the alternative evaluations and assign the crisp weights to the criteria. Usually, a decision maker gives the pairwise comparison matrix of the criteria and then the criterion importance is obtained. In addition, this paper uses some example alternatives to measure how much uncertain the decision maker’s thinking on the criterion is. First, the uncertainty is measured by the shared weights obtained from the comparison matrix of the alternatives under the criterion, since it is not sure which alternative has the weight. Then, the criterion importance weight is obtained from the optimistic viewpoint as the sum of the sure and possible weights from the comparison matrix of the criteria. It is based on the idea that the more uncertain a criterion is, the less the possible weight becomes. The model to derive the crisp weights of the criteria reflecting their uncertainties measured from the alternative evaluations is proposed.

Interval AHP (Analytic Hierarchy Process) was proposed to obtain interval weights from a given pairwise comparison matrix showing relative importance between criteria. In this paper, Interval AHP is applied to group decision problems. Interval AHP is first revised suitably for comparing alternatives from the viewpoint that the interval weight vector shows the set of agreeable weight vectors for the decision maker. Under individual interval weight vectors obtained from individual pairwise comparison matrices, three approaches to obtaining a consensus interval weight vector are proposed. One is the perfect incorporation approach that obtains consensus interval weight vectors including all individual interval weight vectors. By this approach, we can count out indubitably inferior alternatives. The second is the common ground approach that obtains consensus interval weight vectors included in all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when all individual opinions are similar. The third is the partial incorporation approach that obtains consensus interval weight vectors intersecting all individual interval weight vectors. By this approach we can find agreeable group preference between alternatives when individual opinions are not similar. The usefulness of the proposed three approaches is demonstrated by simple numerical examples.

Interval AHP was proposed to express the decision maker's vague evaluations on criteria by interval weights from a given pairwise comparison matrix. It has been extended to group decision problems. Three complementary approaches have been proposed: the perfect incorporation approach for counting out indubitably inferior alternatives and the common ground and partial incorporation approaches for finding agreeable preference between alternatives. In this paper, we enhance those approaches by working out conceivable compromise and refinement. Compromise solution and refinement solution can be found by solving linear programming problems.

In Interval AHP, our uncertain judgments are denoted as interval weights by assuming a comparison as a ratio of the real values in the corresponding interval weights. Based on the same concept as Interval AHP, this study denotes uncertain judgments as fuzzy weights which are the extensions of the interval weights. In order to obtain the interval weight for estimating a fuzzy weight, Interval AHP is modified by focusing on the lower bounds of the interval weights similarly to the viewpoint of belief function in evidence theory. It is reasonable to maximize the lower bound since it represents the weight surely assigned to one of the alternatives. The sum of the lower bounds of all alternatives is considered as a membership value and then the fuzzy weight is estimated. The more consistent comparisons are given as a result of the higher-level sets of fuzzy weights in a decision maker's mind.

Group scenario summarization is a useful approach in group decision making. In order to construct intuitive summaries of AHP evaluation weights, this paper adopts several k-means-type clustering methods to AHP results. In criterion selection level, AHP weights on several criteria are summarized into interval weights for representing the tendencies of group preferences in each cluster. In alternative selection level, similarities among criteria are evaluated by comparing cluster tendencies in criterion-wise selections with the goal of merging familiar criteria. Through several comparative experiments, applicability of several clustering method such as noise rejection and k-member clustering is discussed.

In this paper, the proper judgment of a decision maker is estimated from the given comparisons based on Interval AHP. The given comparison cannot represent his/her judgment accurately, since s/he needs not to fix an exact weight to each item. Even if the weights are certain, s/he may not give an exact value to the comparison of two items by mistake. In the conventional Interval AHP, the former weight uncertainty is considered and the weights of the items is obtained as intervals. In order to consider the latter comparison uncertainty, this paper assumes that the weight of an item is constrained to be certain in the formulation of Interval AHP. The estimated comparison as the maximum interval ratio of the interval weights obtained by assuming that the weight of one of two corresponding items is certain is more reliable than the others. There are two reliable estimations of a comparison and the interval which is included in both of them is considered as the proper judgment.

本研究では，個人が与える代替案に関する一対比較行列に区間AHPを用いて，個人の考えを区間ウェイトとして求め，それらを近似することでグループの決定を導く．グループの決定と個人の考えは共に区間ウェイトで表され，それらの端点の誤差を個人の満足と不満として定義する．個人は，グループの決定が，自身の考えと重なっていれば満足し，齟齬があれば不満を抱く．したがって，グループの決定を，いずれの個人の考えとも共通する範囲（満足）があることを条件とし，この最大化と同時に，個人の考えが支持できない範囲（不満）の最小化を行うことにより求める．

In the group decision of this paper, it is assumed that the practice is entrusted to each decision maker. In such a decision problem, it is not necessary for a decision maker to obey the group decision completely, but necessary to consider it into his/her final decision. In this paper, when a group of decision makers give the comparisons of alternatives, their individual decisions are obtained as the interval weights of alternatives so as to have a common weight. The problem is formulated based on Interval AHP. By relaxing two conditions of the individual decisions for a consensus, a decision maker has to admit the modification of his/her initial judgments and/or the enlargement of his/her individual decision.

Group decision making is an important issue in many societies and various decision support frameworks have been developed. However, it is mostly difficult to put together various feelings of many decision makers into a solo decision because they often make their decision based on different scenarios. This paper introduces a noise clustering-based approach for estimating intrinsic scenarios in group decision making in Delphi method and Analytic Hierarchy Process (AHP) contexts. In several experiments, it is demonstrated that the noise clustering-based approach is useful for putting together the decisions of same scenarios in multi-scenario situations. © 2013 IEEE.

In the well-known multiple criteria approach, AHP (Analytic hierarchy process), an expert is required to give the comparisons on criteria and those on alternatives under all criteria and then the decision problem is explored completely. However, it is time-consuming task for an expert to complete the required comparisons since the numbers of comparisons increase drastically as the numbers of criteria and alternatives. The purpose of this paper is to reduce such burdens of experts by using Interval AHP. In case of a group of experts, some find the criteria that are more importance than the others. Then, it is enough for an expert to evaluate under some assigned criteria instead of all criteria. Furthermore, under the assigned criteria he/she does not necessarily make to give the comparisons without confidence in Interval AHP, since it is formulated as a LP problem. The proposed method helps to practice in real situations by reducing some burdens on experts.

The decision problem in this paper is to induce the preference range of a group, for which Interval AHP is suitable because of an interval weight. Three crucial factors, namely, constructive conflict, consideration, and closure, are considered in decision making with respect to Interval AHP. Constructive conflict is encouraged by giving and modifying judgments independently. For fair consideration, the group decision is specified from the possible aggregation of all of the judgments, which could go beyond any of them individually. To support an appropriate closure, the judgment is scrutinized from several viewpoints in reconsideration and it avoids from inconsistent and opportunistic modification. (C) 2013 The Authors. Published by Elsevier B.V.

In a multi-attribute decision making problem, indigenous values are assigned to attributes based on a decision maker's subjective judgments. The given judgments are often uncertain, because of the uncertainty of situations and intuitiveness of human judgments. In order to reflect the uncertainty in the assigned values, they are denoted as intervals whose widths represent the possibilities of attributes. Since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes, he/she gives a pairwise comparison matrix by comparing two attributes at one occasion. The given matrix contains two kinds of uncertainty, one is inconsistency among comparisons and the other is incompleteness of comparisons. This paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix. At first, the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability, sum or maximum of widths, or ignorance. Then, considering that too uncertain information is not useful, the intervals of attributes are obtained by minimizing their uncertainty indexes. © 2012 Elsevier B.V. All rights reserved.

The group decision in this study follows the structure of AHP, where two kinds of criteria-comparisons and alternatives-comparisons are given by a member. Reflecting the difference of the comparisons, two models to aggregate individual decisions into a group decision are proposed and compared. The grouping process is achieved by hierarchical clustering, in which an individual is merged into the nearest cluster one by one. The similarity of individuals is measured by the uncertainty of the group decision, since it tends to be uncertain in case of different thinking individuals. The uncertainty is quantified by Interval AHP, which uses interval weights to reflect the uncertainty of the decision problem. Then, based on the increase of uncertainty by each step, the sub-groups are noticed. In order for individuals to recognize their standpoints and reconsider their judgments if necessary, the group decision in progress is open to them. © 2012 IEEE.

In this paper, the decision problem follows Interval AHP assuming a group of decision makers. By Interval AHP, the interval weights are assigned to alternatives from the given pairwise comparison matrix on alternatives based on the idea that the less uncertain weights are the better decision. The group decision is modeled on the concept of community of practice, where the group does not only make decision but also practice it. Therefore, the members' opinions are aggregated so as to be optimal from the global group viewpoints and their commitments are improved from the local individual viewpoints. For the optimal decision, the similarity between members is measured based on the uncertainty of the group opinion and the members who have similar opinions are grouped step by step. In order to improve commitment, these grouping process are open to the members. In addition, for the acceptable decision, the deviations of a member from the other members and sub-groups are measured based on the minimum distances of their opinions. The deviations are hints for a member to recognize his/her standpoint and to reconsider his/her judgments if necessary. © 2012 IEEE.

The efficiency of the Interval Data Envelopment Analysis (Interval DEA), we proposed, obtains its bounds from optimistic and pessimistic viewpoints. Intervals represent the uncertainty of given input-output data and the intuitive evaluation of decision makers. The partial order relation that intervals give elements may be complex, especially when elements are numerous. The efficiency measurement we propose combining optimistic and pessimistic efficiency in Interval DEA is comparable because both represent the difference of the analyzed Decision Making Unit (DMU) from the most efficient one. The efficiency measurement is defined as their minimum and determined mainly by pessimistic efficiency. Optimistic efficiency is considered if it is inadequate compared to pessimistic efficiency. Pessimistic efficiency based evaluation resembles natural evaluation and DMUs are arranged linearly.

This paper investigates a group decision problem focusing on outliers in a group. The group decision in this paper is considered as approximation of all individual opinions, which are given normalized crisp or interval values. The difference of an individual opinion from the group opinion represents his/her compromise in order to reach a consensus so that it should be minimized. The first step is to detect outliers in a group and assign outlier degree to individual. The outlier degree is distance of an individual from the group of others and defined as the reduction of compromise by excluding him/her from the group. The outlier degree of a unique individual is more than those of other general individuals. Similarly to outlier degree, inlier degree is closeness of others to an individual and defined as the compromise of the group without him/her. The second step is to approximate individual opinions so as to be a group opinion. The weighted sum of compromises of all individuals is minimized introducing the outlier degrees as the weights of individuals. In this model, outliers are considered positively into account for a group opinion. Therefore, based on compromises of individuals, the models to define outlier degree and to obtain group opinion by approximating individual opinions are proposed in this paper.

AHPを用いると，決定者の一対比較行列から，その決定者の評価を表す代替案優先度が求められる．本研究では，メンバそれぞれの一対比較行列が与えられた場合に，これらからグループの総意として各代替案の優先度を区間値で定める方法を提案する．グループの総意としての区間値優先度は，各メンバが与えた一対比較値をその範囲内の値を用いて実現できることが望ましい．しかし，メンバ間の意見相違を鑑みれば，この条件は厳し過ぎて，極端に幅の広い区間値優先度しか得られないことがある．そこで，各メンバは，グループの総意を表す区間値優先度から幾分かの差異を許容するとして，この条件を緩和する．これより，区間値優先度を求める問題は，その区間の幅と各メンバが許容する優先度との差異をともに最小化する2目的の問題として定式化でき，各目的に重みを与えることで容易に計算できる．

For encouraging communication in a group decision making, this paper proposes methods to aggregate individual preferences. The individual preferences are denoted as the interval priority weights of alternatives by Interval Analytic Hierarchy Process (Interval AHP). It is proposed to handle subjective judgments since the induced results are intervals reflecting uncertainty of given information. When each decision maker gives the judgments on alternatives, the priority weights of alternatives are obtained. In the sense of reducing communication barriers, such information helps group members to realize their own preferences and the others' opinions. Then, they are aggregated based on the concept of the interval regression analysis with interval output data, where two inclusion relations between the estimations and the observations are assumed. From the possibility view, the least upper approximation model is determined so as to include all observations. While, from the necessity view, the greatest lower approximation model is determined so as to be included in all observations. The former possible aggregations are acceptable for each group member and the latter necessary ones are useful for the supervisor at the upper level of decision making.

In this paper, group decision making is discussed from the view of aggregating the members' opinions. At first, the individual preferences are obtained by Interval Analyti Hierarchy Process (Interval AHP), which can handle subjective judgments and reflect uncertainty of information. The interval priority weights of alternatives are induced from the pairwise comparison matrix given by a decision maker easily. They help decision makers to reveal the differences and similarities between their own opinions and others concretely, and encourage them to reduce their communication barriers. Then, the individually obtained preferences are aggregated so as to reach the final group decision. The aggregation models from two viewpoints, such as decision makers and alternatives, are proposed. The former model emphasizes the members' consensus by reducing their dissatisfaction for the aggregation. The latter model emphasizes estimating the plausible values of alternatives by taking the essential part of many decision makers' opinions on each alternative. © 2010 IEEE.

The group decision making is discussed from the view of Analytic Hierarchy Process (AHP), by which the priority weights of alternatives are induced from the pairwise comparison matrix given by a decision maker. This paper uses Interval AHP which induces interval priority weights to reflect uncertainty of the given information. In the problem setting of the group decision making, all group members individually give matrices. Two different ways to aggregate individual opinions have been proposed. One is to aggregate the given information about the comparisons and the other is to aggregate the induced priority weights. This paper proposes straightforward approaches without aggregation steps. The priority weights of alternative as overall evaluations of the group are obtained directly from all the given matrices. They are obtained based on strict or rough inclusion relation between the given comparisons and their approximations consisting of the collective interval priority weights.

This paper investigates a decision problem of evaluating alternatives under multi-criteria. For several terms, the individual evaluations of each alternative under each criterion are recorded repeatedly. Then, as a final decision the collective evaluation over all terms and all criteria is obtained. In order to deal each term and criterion independently, the aggregations of individual evaluations are done by the approximation, differently from by the summation. The first step is to aggregate evaluations over all terms. From the possibility view, the aggregation is obtained as the upper approximation of individual evaluations so that the varieties depending on terms are reflected in an interval. The second step is to aggregate the obtained interval evaluations over all criteria. From the core evaluations view, the aggregations are based on the lower approximation. Depending on the viewpoints of criteria and alternatives, two aggregation models are proposed.

In Interval DEA (Data Envelopment Analysis), efficiency in has been proposed and its bounds are obtained from the optimistic and pessimistic viewpoints, respectively. Intervals are suitable to represent, uncertainty of the given input-output data and decision makers&apos; intuitive evaluations. Although the intervals give elements a partial order relation, it is sometimes complex, especially in case of many elements. The efficiency measurement combining optimistic and pessimistic efficiencies in Interval DEA is proposed. They are compared front the view that both of them represent the difference of the analyzed DMU (Decision Making Unit) from the most efficient one. The proposed efficiency measurement is mainly determined by the pessimistic efficiency. The optimistic one is considered if it is inadequate comparing to the pessimistic one. Such a pessimistic efficiency based evaluation is more similar to our natural evaluation and DMUs are arranged as a linear order.

This paper considers Interval Analytic Hierarchy Process (Interval AHP) in group decision making for encouraging communication. Interval AHP is suitable method to handle subjective judgments since the induced results are intervals which can include uncertainty of given information. The decision makers&apos; opinions can be aggregated at some stages of decision making process to reach consensus. One of the approaches is to aggregate the given judgments considering outliers and from them the group preferences are obtained. By the other approach, first the individual preferences are obtained from the respective decision maker&apos;s judgments and then they are aggregated. In the sense of reducing communication barriers, obtaining individual priority weights of alternatives beforehand helps decision makers realize their own and the others&apos; opinions. The judgments and preferences can be aggregated based on the possibility view or by introducing importance weights of decision makings.

In analytic hierarchy process (AHP) structured hierarchically as several criteria and alternatives, the priority of an alternative is obtained by using the pairwise comparisons based on a decision maker's intuition. Thus, the given comparisons are uncertain and inconsistent with each other. We use the interval approach for obtaining interval evaluations which are suitable for handling uncertain data. Since the given comparisons are ratio measures and too large intervals are not useful information, the intervals should be normalized and their redundancy should be reduced. We introduce interval probability which fills the role of interval normalization instead of crisp normalization in the estimations at each hierarchy. Then, as a final decision, the interval global weights reflecting the decision maker's uncertain judgements as their widths without redundancy are obtained. (C) 2007 Elsevier B.V. All rights reserved.

Interval probabilities have been proposed as one of non-additive measures. The frame of interval probabilities is similar to evidence theory proposed by Dempster and Shafer and they can be regarded as evidences on a finite set. The interval probability is suitable to represent ignorance on the given phenomenon so that it can be used as a kind of subjective probability. We show how to obtain the evidence by a pairwise comparison matrix on a finite set. The pariwise comparisons are usually inconsistent each other since they are given based on human judgements. The interval probabilities from them are determined so as to include such inconsistency. In case of two evidences whose prior and conditional probabilities are obtained as intervals, the marginal and posterior probabilities are also calculated as interval probabilities from the view of possibility. The illustrative numerical example is given in this paper. © 2007 IEEE.