A high-frequency induction heating apparatus was used for the heat treatment of a commercial 6061 aluminum alloy bar with the objective of improving the mechanical properties and productivity. Heating states of the 6061 alloy bar were examined in terms of temperature distribution, heating rate, overheating and temperature fluctuation; moreover, the mechanical properties of the alloy after heat treatment were also investigated. The results of this study are as follows. When the 6061 alloy bar was rapidly heated to the heat treatment temperature using the induction heating apparatus, temperature distribution and overheating of the sample were small as well as the temperature fluctuations in the holding process. A rapid heating rate of about 21°C/s heated the sample to the heat treatment temperature of 560°C in 26 s. The sample showed equivalent or superior mechanical properties compared with a sample heated by conventional electric furnace. Temperature and time of the heat treatment process greatly influenced the mechanical properties of the 6061 alloy, while there was no significant difference in mechanical properties of the sample heat-treated at various heating rates.

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The energy portfolio and production planning problem for multiple companies under energy constraints is formulated as a mixed integer nonlinear programming problem. A new Lagrangian decomposition and coordination approach is proposed to solve the problem effectively. In this paper, we propose efficient computation algorithms for lower bound and upper bound. The lower bound is computed by relaxing the nonlinear term in the objective function. The upper bound is derived by Lagrangian relax and fix heuristic that successively fixes the solution of subproblems to create a feasible solution. Computational results show that the proposed method can effectively solve the problem compared with conventional Lagrangian decomposition and coordination method.

Rough set approaches provide useful tools to induce minimal decision rules from given data. Acquired minimal rules are typically used to build a classifier. However, minimal rules are sometimes used for design knowledge. Specifically, if a new object is designed to satisfy the condition of a minimal rule, it can be classified into a class suggested by the rule. Although we are interested in the goodness of the set of obtained minimal decision rules for the purpose of building a classifier, we are more interested in the goodness of an individual minimal decision rule for design knowledge. In this study, we propose robustness measures as a new type of evaluation index for decision rules. The measure evaluates the extent to which interestingness is preserved after the some conditions are removed. Four numerical experiments are conducted to examine the usefulness of robusetness measures. Decision rules selected by robustness scores are compared with those selected by recall, which is the well-known measure to select good rules. Our results reveal that a different aspect of the goodness of a rule is evaluated by the robustness measure and thus, the robustness measure acts as an independent and complementary index of recall.

In rough set approaches, decision rules are induced from a given data table showing the relation between attribute values and classes of objects. The induced decision rules are used for the classification of new objects by their attribute values. However, some of new objects do not match any decision rule conditions because the given data table does not always include all possible patterns. In those cases, no estimated classes are obtained. Classes of such new objects are estimated by using partially matched decision rules. In this paper, to raise the classification accuracy, we propose to add supplementary rules which can work well for the mismatched new objects in the class estimation. We define the supplementary rules and propose a method for inducing them. We examine the performance of the classifier with supplementary rules by comparisons with the classifier without supplementary rules.