In the knapsack sharing problem (KSP), formulated previously, we considered a game-theoretic situation in which two or more players (agents) compete for their share of capacity in a knapsack with their respective sets of items. As an extension of this problem, we formulate the extended knapsack sharing problem (XKSP). This is actually a family of KSP-like problems, and we present a dynamic programming-based (DP-based), pseudo-polynomial time algorithm to solve XKSP to optimality in a unified way. XKSP is shown to be {\cal NP}-hard, but due to the existence of this pseudo-polynomial time algorithm, it is only weakly {\cal NP}-hard. Next, we develop an algorithm to solve the problem approximately in polynomial time by decomposing it into a series of subproblems. Furthermore, we introduce a scaling factor into the DP computation to obtain a fully polynomial time approximation scheme (FPTAS) for XKSP with two agents. Extension to the case of more than two agents is discussed, together with a non-DP-based PTAS.

Variable selection is the process of finding variables relevant to a given dataset in model construction. One of the techniques for variable selection is exponentially evaluating many models with a goodness-of-fit (GOF) measure, for example, Akaike information criterion (AIC). The model with the lowest GOF value is considered as the best model. We proposed a mixed integer nonlinear programming approach to AIC minimization for linear regression and showed that the approach outperformed existing approaches in terms of computational time [13]. In this study, we apply the approach in [13] to AIC minimization for logistic regression and explain that a few of the techniques developed previously [13], for example, relaxation and a branching rule, can be used for the AIC minimization. The proposed approach requires solving relaxation problems, which are unconstrained convex problems. We apply an iterative method with an effective initial guess to solve these problems. We implement the proposed approach via SCIP, which is a noncommercial optimization software and a branch-and-bound framework. We compare the proposed approach with a piecewise linear approximation approach developed by Sato and others [16]. The results of computational experiments show that the proposed approach finds the model with the lowest AIC value if the number of candidates for variables is 45 or lower.

In revenue management, there are models which aim to maximize revenue by controlling policy for uncertain demands throughout a booking horizon. The models are called dynamic models. One of the applications of the dynamic models is reservation system which offers available seats for customers' requests. Recently, the system has allowed us to choose our booking seat position. However, the dynamic models in revenue management have not been included customers' selection behavior for seating position. This paper proposes choice-based seating position model with undistinguished multi-lines that is a dynamic model considered with the customers' selection behavior for seating positions. Approximate solutions for this model are calculated by Choice-based Deterministic Linear Programming (CDLP) and decomposition approximation which are used in choice-based network revenue management models. This paper suggests that CDLP is more effective than decomposition approximation for the choice-based seating position model, even through some reports in revenue management suggested that decomposition approximation could derive higher revenue than CDLP in their models.