The linear ordering problem is, given a nonnegative square matrix, to find a simultaneous permutation of the row and column indices such that the sum of the elements below the main diagonal of the permuted matrix is minimized. The linear ordering problem is an NP-hard optimization problem with a wide variety of applications, including triangulation of input-output tables and aggregation of individual preferences. In the context of the preference aggregation, Truchon (1998) showed that every optimal solution of a linear ordering problem satisfies a property called the extended Condorcet criterion, which states that there exists an ordered partition of the row and column index set of the given matrix such that the relative order of every pair of indices belonging to different components of the partition is fixed for all optimal solutions. Such fixing is desirable for investigating the whole set of the optimal solutions since we can reduce the computational time for enumerating them. In this study, we show that the optimal solutions of the linear ordering problem have an even stronger property than the extended Condorcet criterion, which enables us to have more pairs of indices whose relative orders are fixed for all optimal solutions.

When analyzing solutions for bicooperative games as well as classical cooperative games, traditional approaches regard both of games and payoff vectors as linear spaces, but in this paper, we present another approach to analyze solutions from the viewpoint of representation of the symmetric group. First, we regard the space of games as a representation of the symmetric group. Then, by using tools of representation theory, we obtain a decomposition of the space and specify useful subrepresentations. Exploiting this decomposition, we show an explicit formula of linear symmetric solutions. Additionally, we also show expressions of linear symmetric solutions restricted by parts of the axioms of the Shapley value for bicooperative games.

In this study, we investigate multi-objective decision making problems with respect to a variable domination structure. In order to solve such problems, we introduce two types of solutions from vector optimization problems with respect to a variable domination structure; afterward, we characterize them. These solution concepts are helpful in multi-objective decision making problems where different preferences or restrictions of objective functions for different alternatives are at hand; or where different preferences of objective functions with respect to different objective functions are assumed. These solution concepts are proposed for multi-objective location problems where there are different preferences of objective functions at each location; our results are applied for selecting a location to establish an outlet store.