Tomoe Entani

<p>Many local governments regularly survey their residents for administration evaluation and publish the results in their web pages. When we compare those results, we can find the characteristics of each local government. However, since each questionnaire consists of its own questions even if the issue is similar, it is difficult to associate one question by the local government to that by the other. We propose the system to classify the questions considering word order and meaning by Long Short Term Memory (LSTM) with word embedding.</p>

In Analytic Hierarchy Process (AHP), a decision problem is structured hierarchically with the criteria and alternatives. The propriety weight vector of the alternatives is obtained as the sums of their local weights considering the importance of the criteria. Once a decision maker finds out his/her preference as the importance of criteria, s/he can evaluate the other sets of alternatives. The preference for criteria is essential and independent of the alternatives. However, the criteria are intangible so that it sometimes difficult for a decision maker to give judgments on them directly. Hence, we propose a method to help a decision maker to find out his/her preference based on his/her experience. It is much easier for him/her to give the judgments on the alternatives that s/he knows well. The preference is denoted as a fuzzy vector of criteria from a possibilistic view. We also propose the model to obtain an interval score vector of new alternatives from the derived preference and their given local scores. Both models of deriving a preference vector and a score vector are based on interval inclusion relation.

<p>The analytic hierarchical process (AHP) is a method to obtain the priority weights of alterna- tives when a decision maker evaluates importance of the criteria and the alternatives under each criterion. It can handle an intuitive judgment by using a pairwise comparison technique to evaluate the criteria and alternatives. However, the criteria often relate to our feelings and value systems, and it is not easy for a decision maker to compare them and give judgments on them. Hence, we concern deriving preference, which is represented by the importance of criteria in this paper. We propose a method to obtain the im- portance of criteria through the alternatives indirectly that are familiar to the decision maker, instead of forcing him/her to evaluate them in the conventional AHP directly. Therefore, we rebuilt the hierarchical structure using the alternatives that he/she knows well and the criteria that represent his/her preferences. The decision maker does not only find the best alternative but also understands his/her preferences on ambiguous concepts.</p>

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In Interval AHP, a comparison is considered to consist of the real values in the corresponding interval weights. The widths of the interval weights are minimized so as to be close to the given comparisons as much as possible. In the same concept of Interval AHP, this study proposes the model to obtain fuzzy weights as the extension of the interval weights. In order to obtain fuzzy weights, Interval AHP is modified by focusing on the lower bounds of the interval weights from the viewpoint of belief function in evidence theory. As the given comparisons are more consistent, it means that they are based on the higher-level sets of fuzzy weights, which are denoted as interval weights.

The purpose of this paper is to reconsider the given comparisons, since they never represent the decision maker's judgments accurately. His/her judgment of two items may not be an exact value and s/he may not fix an exact wight of each item. In the conventional Interval AHP, reflecting the weight uncertainty, the weight of an item is denoted as an interval. This paper assumes that the weight of one of the items is certain in order to consider the comparison uncertainty. In other words, the item is assumed to be a standard in comparing the items. Then, a comparison can be estimated from the interval weights obtained by assuming one of two corresponding items as a standard. There are two estimations of the comparison and the interval which is included in both estimations is reliable so that the given comparison is replaced into the interval.

A group decision is often believed to be better than individual decision simply because of accumulating all members' knowledge, being able to check errors carefully and so on. The other reason is that the group facilitate creativity and develop new ideas by inspiring each other. For these advantages, the decision makers improve their given information and may revise their opinions referring to the others and the group's as in Delphi Method. While, this paper assumes the synergy effect of decision makers. Someone's consistency may be complemented by the others opinions and/or the integration may create the new value beyond any individual ones. Based on Interval AHP where uncertainty among comparisons are reflected into interval weights, the individual evaluations, their sum, and the group evaluations are examined.

This paper integrates individual opinions into a group opinion reflecting group members' reliabilities. By Interval AHP, the given information is the pairwise comparison matrix on alternatives and the individual opinion is obtained as interval weights of alternatives. Based on the idea that the individual who gives less uncertain information are more reliable, the individual importance is determined and reflected in the integration into group opinion. The reliability index of the given information is defined as inverse or negative of uncertainty index of the given comparison matrix, which is represented as the sum of widths of interval weights obtained by Interval AHP.

It is reasonable to adopt the majority or average of individual opinions as the group opinion, since it is supported by many members. However, it happens that the minority is better than the majority or a unique opinion stimulates the discussion and helps to improve the individual and group opinions. Based on this idea, the final group opinion in this paper takes outliers positively into account. Instead of excluding unique opinion, the relative relation of an individual to the group is considered as his/her outlier degree and it is introduced as his/her weight in the group. The outlier degree is measured by the increase of group compromise, which is sum of differences between individuals and group opinions, by adding an individual to the group. When he/she stands far from the group, he/she is paid more attention than general individuals in integration. Then, the group opinion is obtained as the sum of weighted individual opinions or by minimizing weighted compromises. The effect of individual weights is more by the former than by the latter.

When the decision maker gives a pairwise comparison matrix on alternatives, their priority weights are obtained by Analytic Hierarchy Process (AHP). In a group decision making, we assume that each group member gives his/her own pairwise comparison matrix. In this paper, a method for obtaining the interval priority weights of alternatives as the collective evaluation from the given many matrices is investigated. We basically presume that the individual matrices are the possible realizations of the collective interval priority weights in order to reflect all group members' opinions and their widths are minimized. However, as is often in the real world, the individual opinions are not always similar so that the collective intervals might be extremely wide. To avoid this, we relax the presumption by introducing the deviations for the individual intervals from the collective ones. Then the problem is formulated as the biobjective programming problem which minimizes the widths of the collective intervals and the deviations from individual intervals.

In this paper, we deal with interval probabilities which are the extension of the conventional probabilities. In case of interval probabilities, the uncertainty of information can be measured by two ways; one is the interval entropy which is the extension of the entropy of the conventional probabilities and the other is the amount of ignorance. The interval probabilities can be identified as the Belief and Plausibility functions by the basic probability assignments in the theory of evidence. However, it is difficult for a decision maker to assign basic probabilities for all the subsets of a set of elements. In order to reduce the judgements given by a decision maker, we use the pairwise comparisons for probabilities of pairs of elements to identify them.

Analytic Hierarchy Process (AHP) is proposed to give priority weights with respect to many alternatives with many criteria. In the conventional AHP, local weights of alternatives under each criterion are obtained from the pairwise comparisons given by a decision maker. By extending the weights from crisp to interval to reflect inconsistency of the given comparisons, Interval AHP has been proposed. The global weight of an alternative is assumed as the weighted sum of local weights. The weights represent importance of criteria in evaluation so that they can be determined for each alternative from various viewpoints. We propose the interval global weights whose bounds are from the optimistic and pessimistic evaluations. Although their widths represent the possibilities of global weights, too large width from too optimistic and/or pessimistic evaluations can not be a meaningful for a final decision. Then, the obtained interval global weight should be normalized relatively so as to reduce redundancy.

We propose a new approach for obtaining two interval efficiency values with interval data as an extension of DEA. We deal with interval data that can reflect uncertainty in real situations. The two interval efficiency values are obtained from the optimistic and pessimistic viewpoints. Their upper and lower bounds are obtained by two different extreme values in the given interval data respectively. Thus, we formulate four types of efficiency values from two viewpoints with two extreme values in the given interval data. Our emphasis is to obtain two interval efficiency values reflecting uncertainty of the given data. Thus our approach can be described as a kind of interval data analysis. A numerical example is shown to illustrate our proposed approach.

The exponential possibility regression is applied to estimate interval efficiency values. Interval efficiency value is obtained by interval DEA. It shows the possible efficiency and its upper and lower limits are obtained from the opimistic and pessimistic viewpoints for an object to be analyzed. The interval efficiency value for the object is calculated relatively by using all given inputs and outputs without assuming an identical single model for all DMUs. We analyze the interval efficiency values by regression analysis so that the identical model can be obtained. The obtained linear model in regression analysis can represent the relation between the given inputs and outputs and the obtained interval efficiency values in interval DEA, When new objects are added, we can estimate their interval efficiency values using the models obtained by regression analysis.

The exponential possibility regression is applied to estimate interval efficiency values. Interval efficiency value is obtained by interval DEA. It shows the possible efficiency and its upper and lower limits are obtained from the opimistic and pessimistic viewpoints for an object to be analyzed. The interval efficiency value for the object is calculated relatively by using all given inputs and outputs without assuming an identical single model for all DMUs. We analyze the interval efficiency values by regression analysis so that the identical model can be obtained. The obtained linear model in regression analysis can represent the relation between the given inputs and outputs and the obtained interval efficiency values in interval DEA, When new objects are added, we can estimate their interval efficiency values using the models obtained by regression analysis.