In this paper we have raised, discussed, and solved the problems relating the Vargas's verification of consistency test for 3 × 3 comparison matrices in the Analytic Hierarchy Process (AHP). In the AHP community, assessing attribute weights by using a consistency ratio of 0.1 or less has been widely practiced. Though Vargas did study the problem, the method he adopted might lead to certain questionable results. By trying to clarify such a critical situation, we pointed out the problems arose form the solutions of two inequality systems within the proof process of Vargas. We proved that the solution for the first system was an interval, not a point as he had indicated. After we solved the first inequality system, we found that the criterion for the solution interval of the first system would satisfy the second inequality system. We have then justified the inherent problim in Vargas's verification procedure.

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This paper deals with the prediction of chaotic time series by using the multi-stage fuzzy inference system and its application to the analysis of operating flexibility. Multi-national corporation obtained by shifting manufacturing plants located in different countries is denoted as operating flexibility. Even though the operating flexibility is optimized by the stochastic dynamic programming under the known process of exchange rate, it is usually hard to explicitly predict the change. Therefore, the value of operating flexibility depends on the prediction of time series. Then, we utilize the prediction of exchange rate by using the multi-stage fuzzy inference system. Since we divide the fuzzy inference system and related input variables into several stages, the number of rules included in multi-stage fuzzy inference systems is remarkably smaller compared to conventional fuzzy inference systems. The weight included in inference rules are optimized by the backpropagation algorithm. We also propose a method to optimize the shape of membership function and the appropriate selection of input variables based upon the genetic algorithm (GA). The method is applied to the approximation of typical multi-dimensional chaotic dynamics. The simulation study for multi-dimensional chaotic dynamics shows that the inference system gives a better prediction. The prediction is then applied to estimate the exchange rates by using input variables consisting of economic indicators as well as the exchange rates. The result shows a better performance of the multi-stage fuzzy inference system than when conventional methods are used. Then, we can find effective operating flexibility for shifting the manufacturing plants depending on the predicted exchange rate.

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A large-scale project can be modeled as a project network in AOA form (arrows denote the activities and nodes denote the events). We study the case that activity durations are integer random variables. Given the project time (the deadline required to complete the project) constraint and budget constraint, we use some techniques in network analysis to develop two algorithms in order to generate all longer boundary duration vectors and shorter boundary duration vectors, respectively. Each feasible duration vector is between such longer and shorter boundary vectors. Whenever accidents happening in the project duration, the project manager can update the activity durations according to the longer and shorter boundary duration vectors without contradicting project time and budget constraints.

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This paper proposes a group consensus making method by using modified AHP (Analytic Hierarchy Process). Many application examples are reported as a decision-making support tool with AHP. However, AHP is a tool on condition of the only one decision-making person, and if it remains as it is, it is inapplicable to group decision-making. Therefore, it is important to establish a group AHP. It has proposed that each group member presents the pairwise comparison value between the evaluation items which self believes, and Saaty makes those geometric average value group pairwise comparison value. On the other hand, Yamada, Sugiyama and Yamaki present that each group member has proposed the pairwise comparison value of self to believe in a interval. This intervalis called the interval for assertion. In this paper, an applicable model is proposed when the problem of the group interval AHP method of Yamada et al. is solved. In this paper, by the introduction of the concept of a dissatisfied function, re-defined semantic of the interval for assertion. By using the dissatisfied function, it is impossible to give the rational standard which is the grade in which each member can permit the result of the group decision-making comes out. It is determined that the group decision-making will make all members' dissatisfied size below a certain limit value at this time. A weight vector is given as asolution of the optimization problem accompanied by a parameter, and the relation between dissatisfied limit value and the minimum residual of a model is given as a figure. Therefore, the group that uses this model can choose the own decision-making on intention referring to a figure.

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Grotschel, Lovasz and Schrijver introduced a convex set containing the stable set polytope of a graph. They proved that the set is a polytope if and only if the corresponding graph is perfect. In this paper, we give an alternative proof of the fact based on a new representation of the convex set described by infinitely many convex quadratic inequalities.

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This paper deals with the collapsing knapsack problem. In the literature, to solve the problem, a method incorporating a reduction from the problem to the 0-1 knapsack problem has been proposed. In this paper we show an alternative reduction which produces coefficients smaller than those by the previous. The improvement makes it possible to solve the resulting 0-1 knapsack problem faster than the previous. On our estimation in a case, the efficiency attains up to 150 times. We also show that the coefficients produced will be the smallest possible.

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In many network design problems, the best layout of components is searched considering construction cost and network reliability. Because of #P-completeness for computation of network reliability, most algorithms for the problems find approximate solutions. In this paper, we propose a complete enumeration method for the network design problems, which becomes a basis of exact algorithms. The detection of isomorphic networks appeared in the process of computation can reduce computational time compared with simple complete enumerations. We also discuss applications of the algorithm to other kinds of network design problems.

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We study a queueing system having a mixture of a special semi-Markov process (SSMP) and a Poisson process as the arrival process, where the Poisson arrival is regarded as interfering traffic. It is shown by numerical examples that the SSMP customers receive worse treatment than Poisson customers, i.e., the mean waiting time of SSMP customers is longer than that of Poisson customers. We also propose a model of Moving Picture Experts Group (MPEG) frame arrivals as an SSMP batch arrival process. This model captures two features of the MPEG coding scheme: (i) the frequency of appearance of the I-, B-, and P-frames in a Group of Pictures (GOP), and (ii) the distinct distributions for the size of the three types of frames. The mean and variance of waiting time of ATM cells generated from the MPEG frames are evaluated in the numerical examples drawn from some real video data.

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