Birnbaum‐Saunders distribution: A review of models,analysis, and applications

Birnbaum and Saunders introduced a two‐parameter lifetime distribution to model the fatigue life of a metal, subject to cyclic stress. Since then, extensive work has been done on this model providing different interpretations, constructions, generalizations, inferential methods, and extensions to bivariate, multivariate, and matrix‐variate cases. More than 200 papers and one research monograph have already appeared describing all these aspects and developments. In this paper, we provide a detailed review of all these developments and, at the same time, indicate several open problems that could be considered for further research.     

Measurement of preferences with self-explicated approaches: A classification and merge of trade-off- and non-trade-off-based evaluation types

Self-explicated approaches are popular preference measurement approaches for products with many attributes. This article classifies previous self-explicated approaches according to their evaluation types, i.e. trade-off- versus non-trade-off-based, and outlines their advantages and disadvantages. In addition, it proposes a new method, the presorted adaptive self-explicated approach that is based on Netzer and Srinivasan’s (2011) adaptive self-explicated approach and that combines trade-off- and non-trade-off-based evaluation types. Two empirical studies compare this new method with the most popular existing self-explicated approaches, including the adaptive self-explicated approach and paired comparison preference measurement. The new method overcomes the insufficient discrimination between importance weights, as usually found in non-trade-off-based evaluation types; discourages respondents’ simplification strategies, as are frequently encountered in trade-off evaluation types; is easy to implement; and yields high predictive validity compared with other popular self-explicated approaches.     

A new distance measure including the weak preference relation: Application to the multiple criteria aggregation procedure for mixed evaluations

We introduce a new distance measure between two preorders that captures indifference, strict preference, weak preference and incomparability relations. This measure is the first to capture weak preference relations. We illustrate how this distance measure affords decision makers greater modeling power to capture their preferences, or uncertainty and ambiguity around them, by using our proposed distance measure in a multiple criteria aggregation procedure for mixed evaluations.     

Accept & reject statement-based uncertainty models

We develop a framework for modelling and reasoning with uncertainty based on accept and reject statements about gambles. It generalises the frameworks found in the literature based on statements of acceptability, desirability, or favourability and clarifies their relative position. Next to the statement-based formulation, we also provide a translation in terms of preference relations, discuss—as a bridge to existing frameworks—a number of simplified variants, and show the relationship with prevision-based uncertainty models. We furthermore provide an application to modelling symmetry judgements.     

Stochastic preference analysis in numerical preference relations

Numerical preference relations (NPRs) consisting of numerical judgments can be considered as a general form of the existing preference relations, such as multiplicative preference relations (MPRs), fuzzy preference relations (FPRs), interval MPRs (IV-MPRs) and interval FPRs (IV-FPRs). On the basis of NPRs, we develop a stochastic preference analysis (SPA) method to aid the decision makers (DMs) in decision making. The numerical judgments in NPRs can also be characterized by different probability distributions in accordance with practice. By exploring the judgment space of NPRs, SPA produces several outcomes including the rank acceptability index, the expected priority vector, the expected rank and the confidence factor. The outcomes are obtained by Monte Carlo simulation with at least 95% confidence degree. Based on the outcomes, the DMs can choose some of them which they find most useful to make reliable decisions.     

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G. A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S. Let k≥2 be an integer. A set X of vertices of G is k-Steiner convex if it contains the Steiner interval of every set of k vertices in X. A vertex x∈X is an extreme vertex of X if X?{x} is also k-Steiner convex. We call such vertices k-Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.     

How people make friends in social networking sites—A microscopic perspective

We study the detailed growth of a social networking site with full temporal information by examining the creation process of each friendship relation that can collectively lead to the macroscopic properties of the network. We first study the reciprocal behavior of users, and find that link requests are quickly responded to and that the distribution of reciprocation intervals decays in an exponential form. The degrees of inviters/accepters are slightly negatively correlative with reciprocation time. In addition, the temporal feature of the online community shows that the distributions of intervals of user behaviors, such as sending or accepting link requests, follow a power law with a universal exponent, and peaks emerge for intervals of an integral day. We finally study the preferential selection and linking phenomena of the social networking site and find that, for the former, a linear preference holds for preferential sending and reception, and for the latter, a linear preference also holds for preferential acceptance, creation, and attachment. Based on the linearly preferential linking, we put forward an analyzable network model which can reproduce the degree distribution of the network. The research framework presented in the paper could provide a potential insight into how the micro-motives of users lead to the global structure of online social networks.     

Mathematical programming models for revenue management under customer choice

We study a network airline revenue management problem with discrete customer choice behavior. We discuss a choice model based on the concept of preference orders, in which customers can be grouped according to a list of options in decreasing order of preference. If a customer’s preferred option is not available, the customer moves to the next choice on the list with some probability. If that option is not available, the customer moves to the third choice on the list with some probability, and so forth until either the customer has no other choice but to leave or his/her request is accepted. Using this choice model as an input, we propose some mathematical programs to determine seat allocations. We also propose a post-optimization heuristic to refine the allocation suggested by the optimization model. Simulation results are presented to illustrate the effectiveness of our method, including comparisons with other models.     

An integrated operation module for individual risk management

In this study, we used the cumulative prospect theory to propose the individual risk management process (IRM) which includes risk analysis and risk response stages. According to an individual’s preferential structure, the process has been developed into an operational module which includes two sub-modules. From this, the individual’s risk level for the confronted risk can be identified from the risk analysis, while the response strategies can be assessed at the risk response stage. Therefore, optimal response strategies can be recommended based on individual risk tolerance levels.     

Ordered sets with interval representation and (m,n)-Ferrers relation

Semiorders may form the simplest class of ordered sets with a not necessarily transitive indifference relation. Their generalization has given birth to many other classes of ordered sets, each of them characterized by an interval representation, by the properties of its relations or by forbidden configurations. In this paper, we are interested in preference structures having an interval representation. For this purpose, we propose a general framework which makes use of n-point intervals and allows a systematic analysis of such structures. The case of 3-point intervals shows us that our framework generalizes the classification of Fishburn by defining new structures. Especially we define three classes of ordered sets having a non-transitive indifference relation. A simple generalization of these structures provides three ordered sets that we call “d-weak orders”, “d-interval orders” and “triangle orders”. We prove that these structures have an interval representation. We also establish some links between the relational and the forbidden mode by generalizing the definition of a Ferrers relation.