This paper discusses an implicit reformulation of a class of MPEC (mathematical program with equilibrium constraints) problems. We particularly focus on an MPEC problem arising from the robust optimization of elastic structures subjected to the uncertain external load. We first review the relation between the worst-case detection and robust constraint satisfaction of the structural responses, and then derive an MPEC formulation of the robust structural optimization. Since a standard constraint qualification is not satisfied at any feasible solution of an MPEC problem, we propose a reformulation based on the smoothed Fischer-Burmeister function, in which the smoothing parameter is treated as an independent variable. Numerical examples of robust structural optimization are presented to demonstrate that the presented formulation can be solved by using a standard nonlinear programming approach.

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We propose a pricing method by mathematical programming for swing options with typical constraints on a lattice model. We show that the problem of pricing typical swing options has a particular optimal solution such that there are only seven kinds of changed amounts in the solution. Using the solution, we formulate the pricing problem as a linear program. The solution can be applied to the methods of Jaillet, Ronn and Tompaidis (2004), and Barrera-Esteve et al. (2006) for improving time complexity. Another feature of our method is the capability to price swing options in an incomplete market. In an incomplete market, the price of a swing option is defined as an upper and a lower bound of arbitrage-free prices. We formulate the problem of finding an upper bound as a linear program. For a lower bound, we give a bilinear programming formulation.

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Following [23], the customers for the CD market are classified into twelve segments along two axes: "Artist Loyalty" and "Market Sensitivity". The purchasing behavior of individual customers is formulated as a Markov chain in discrete time. The transition structure of the customers across the twelve segments is common. However, the market segmentation is reflected by altering transition probabilities in terms of eight different parameters. In [23], repeated vector-matrix multiplications are employed for evaluating necessary probabilistic entities. In this paper, through the spectral analysis of the transition probability matrix, the state probability vector of the underlying Markov chain is obtained in a closed form, which enables one to apply a nonlinear optimization solver for estimating the eight parameters involved so as to minimize the Euclidian distance between the expected number of CDs sold over 20 weeks calculated from the Markov model and the actual POS data. Furthermore, several new indicators are evaluated for capturing the brand characteristics of individual music artists. Five leading Japanese music artists are actually analyzed, suggesting new effective marketing strategies for each of them. The proposed approach can be applied to any market in which products are likely to be purchased only once.

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A quick decision on implementation of countermeasures for quality problems is important for reducing the quality cost of electronic products. At the beginning of the production period, the claim rate is uncertain owing to the lack of sufficient field claim data. Therefore, a quick decision on implementation of countermeasures may lead to misjudgment. On the other hand, until the end of the production period, there is an option to postpone the decision. When the option to postpone the decision is exercised, increased field claim data will reduce the uncertainty of the claim rate. Then the probability of misjudgment will decrease. However, postponing the decision may increase the number of inferior products before the countermeasures. In order to support the decision on implementation of countermeasures for quality problems under uncertainty, we propose a new method that combines Bayesian updating and real options. We evaluate the value of the option to postpone the decision qualitatively and show the possibility of reducing the quality cost.

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