This article presents the mathematical background of general interactive systems. The first principle of designing a large system is “divide and rule,” which implies that we could possibly reduce human error if we divided a large system in smaller subsystems. Interactive systems are, however, often composed of many subsystems that are organically connected to one another and thus extremely difficult to divide. In other words, we cannot apply a traditional mechanism of mathematical functions to the programming of interactive systems. We, however, can overcome this difficulty by applying a framework of category theory to the programming, but this requires highly abstract mathematics, which is not very popular. In this article we introduce the fundamental idea of the category theory using only λ-calculus, and then demonstrate how it can be used in the practical design of an interactive system. Finally, we mention how this discussion relates to Kleisli category in mathematics.
In the steel industry, the role of the slab yard which is the intermediate process between the steelmaking and the rolling is becoming significant. By promoting the logistics from steelmaking to rolling, we can reduce the cost of fuel to reheating and improve the productivity. In the yard slabs are expected to be sorted into piles as little transport as possible and as high as possible. It is the slab stacking problem (SSP). In the previous paper, we formulated SSP with the 0-1 programming. It turned out that this formulation gets lack of memory with large scale problem. For settling this problem, in this paper, we apply the column generation method to SSP. But, since SSP is 0-1 integer problem, the optimal error remains. So we developed the method of local search for the neighborhood of the generating columns using the dual optimal solution to obtain the exact solution. Finally, we show the effectiveness of this method with the computational experiment using the real operational data.