Tomoe Entani

We improve the efficiency interval of a DMU by adjusting its given inputs and outputs. The Interval DEA model has been formulated to obtain an efficiency interval consisting of evaluations from both the optimistic and pessimistic viewpoints. DMUs which are not rated as efficient in the conventional sense are improved so that their lower bounds become as large as possible under the condition that their upper bounds attain the maximum value one. The adjusted inputs and outputs keep each other balanced by improving the lower bound of efficiency interval, since the lower bound becomes small if all the inputs and outputs are not proportioned. In order to improve the lower bound of efficiency interval, different target points are defined for different DMUs. The target point can be regarded as a kind of benchmark for the DMU. First, a new approach to improvement by adjusting only outputs or inputs is proposed. Then, the combined approach to improvement by adjusting both inputs and outputs simultaneously is proposed. Lastly, numerical examples are shown to illustrate our proposed approaches. © 2004 Elsevier B.V. All rights reserved.

Even if the given data are crisp, there exists uncertainty in decision making process and inconsistency based on human judgements. The purpose of this paper is to obtain the evaluations which reflect such an uncertainty and inconsistency of the given information. Based on the idea that intervals are more suitable than crisp values to represent evaluations in uncertain situations, we introduce this interval analysis concept into two well-known decision making models, DEA and AHP. In the conventional DEA, the relative efficiency values are measured and in the proposed interval DEA, the efficiency values are defined as intervals considering various viewpoints of evaluations. In the conventional AHP, the priority weights of alternatives are obtained and in the proposed interval AHP, the priority weights are also defined as intervals reflecting the inconsistency among the given judgements. © Springer-Verlag Berlin Heidelberg 2006.

Analytic Hierarchical Process(AHP) is proposed to give the priority weight with respect to many items. The priority weights are obtained from the pairwise comparison matrix whose elements are given by a decision maker as crisp values. We extend the crisp pairwise comparisons to fuzzy ones based on uncertainty of human judgement. To give uncertain information as a fuzzy value is more rational than as a crisp value. We assume that the item's weight is a fuzzy value, since the comparisons are based on human intuition so that they must be inconsistent each other. We propose a new AHP, where the item's weight is given as a fuzzy value, in order to deal with inconsistency in the given matrix. The purpose is to obtain fuzzy weights so as to include all the given fuzzy pairwise comparisons, in the similar way to the upper approximation in interval regression analysis. © 2006 Springer.

The decision problem in Analytic Hierarchy Process (AHP) is structured hierarchically as several criteria and alternatives. It is proposed to determine the global weights of alternatives considering the referenced priority weights of criteria and local weights of alternatives. We assume them as intervals, since they are obtained from the corresponding pairwise comparison matrices given by a decision maker based on his/her intuition. The width represents the possibility of each weight reflecting the inconsistency of the given comparisons. Then, the global weights calculated with them should be also intervals and such intervals tend to contain redundant parts. We propose the models to modify the intervals so as to be normalized keeping their possibilities. Instead of crisp normalization, the interval probability fills the role of interval normalization. The modified interval global weights reflect a decision maker's uncertain judgments as intervals without redundancy.

Analytic hierarchy process (AHP) is proposed to give priority weights with respect to many items. The priority weights are obtained from the comparison matrix whose elements are given by a decision maker as crisp numbers. We assume that the item's weights are intervals, because a comparison matrix based on human intuition must be inconsistent. Three models depending on the objective functions are proposed in this paper. In each model, the interval weights are determined so as to include the given comparisons. This concept is similar to the upper approximation in interval regression analysis.

We improve the efficiency of a DMU considering its interval efficiency value based on DEA by adjusting its outputs. The DEA model to obtain an interval efficiency value which consists of evaluations from both the optimistic and pessimistic viewpoints has been formulated. DMUs which are not rated as efficient in the conventional sense are improved by adjusting outputs so that their lower bounds become as large as possible in the production possibility set and their upper bounds attain the maximum value one. The lower bound of interval efficiency value becomes small when the DMU's inputs and outputs are not balanced. Therefore considering the lower bounds in improvement, the adjusted outputs can keep the balance each other. A new approach to improvement by adjusting only outputs without decreasing the given outputs anymore is proposed in this paper. By extending the proposed approach, it is suggested that the approaches to improvement by adjusting only inputs and both inputs and outputs simultaneously can be obtained. Numerical examples are shown to illustrate our proposed approaches.

In this paper, multi-source possibilistic information is represented by a set of possibilistic constraints to characterize decision variables from different information aspects. Possibilistic linear programming is used to integrate multi-source possibilistic information into the upper and the lower possibility distributions of decision vector.

In this paper, we propose a formulation of DEA with constraints of weight intervals given by a group of decision makers. The importance grades of items are considered as probability measures whose sum is equal to one. In order to make weights in DEA represent importance grades given by decision makers, all the given data are divided by respective input/output data. The efficiency values obtained from the normalized data and the original data are the same. In DEA, the product of data and weight is considered as the importance grade of each item rather than the weight itself. By the normalization, the product of data and weight becomes equal to the weight itself. Therefore, the optimal weights obtained from the normalized data naturally represent the importance grades of decision makers. The sum of the optimal weights for inputs obtained in DEA becomes one and the sum of the optimal weights for outputs comes out to be the efficiency value in DEA. Assuming a group of decision makers, the interval importance grade is formed from many decision makers' importance grades, which can be obtained through AHP. If the given information with respect to the evaluation contains partial ignorance, the importance grades are assumed as semi-mobile probability mass in the setting of Dempster-Shafer theory. Both of these probabilistic weights are easily included as weight constraints in CCR model of DEA with normalized data.