Aggregation for Atanassov's intuitionistic and interval valued fuzzy sets: the median operator

Atanassov's intuitionistic fuzzy sets (AIFS) and interval valued fuzzy sets (IVFS) are two generalizations of a fuzzy set, which are equivalent mathematically although different semantically. We analyze the median aggregation operator for AIFS and IVFS. Different mathematical theories have lead to different definitions of the median operator. We look at the median from various perspectives: as an instance of the intuitionistic ordered weighted averaging operator, as a Fermat point in a plane, as a minimizer of input disagreement, and as an operation on distributive lattices. We underline several connections between these approaches and summarize essential properties of the median in different representations.

Exploring distributive justice in health care

The allocation of resources to providers and the way in which the resources are then prioritised to specific service areas and patients remain the critical ethical decisions which determine the type of health system a community receives. Heath care providers will never be given enough resources to satisfy all the demands placed upon them by a community that is becoming increasingly informed and demanding. This paper discusses the matter of justice as it relates to the distribution of health resources. It translates the theoretical constructs of distribution into a practical situation that arose at The Geelong Hospital. It is important to emphasise that the aim of giving the example is not necessarily to provide the right answer but rather to assist in determining what ought to be the questions.

Minimum span frequency assignment problem on triangular lattices

This paper considers the Minimum Span Frequency Assignment Problem with Interference Graph on Triangular Grid (MSFAP-TG), a special case of the Minimum Span Frequency/Channel Assignment (MSFAP) for cellular systems and optical networks. The MSFAP-TG is interesting in its own right and thus worth studying. In this paper, we propose strong integer programming formulations for the MSFAP-TG and present polyhedral results on these formulations. In solving the MSFAP-TG, we implement these integer programs to obtain exact solutions. We also develop a heuristic for obtaining feasible solutions and upper bounds for the problems. With the use of these upper bounds, and a simple lower bound, the computation time of the exact algorithm can be improved substantially. The heuristic turns out to be quite good in terms of the quality of upper bounds and is extremely efficient in computation time. Last of all, we present new concepts for tackling large scale MSFAP-TGs.

Means on discrete product lattices

We investigate the problem of averaging values on lattices, and in particular on discrete product lattices. This problem arises in image processing when several color values given in RGB, HSL, or another coding scheme, need to be combined. We show how the arithmetic mean and the median can be constructed by minimizing appropriate penalties. We also discuss which of them coincide with the Cartesian product of the standard mean and median.

Distributive justice and the Nash bargaining solution

Suppes-Sen dominance or SS-proofness (SSP) is a commonly accepted criterion of impartiality in distributive justice. Mariotti (Review of Economic Studies, 66, 733–741, 1999) characterized the Nash bargaining solution using Nash’s (Econometrica, 18, 155–162, 1950) scale invariance (SI) axiom and SSP. In this article, we introduce equity dominance (E-dominance). Using the intersection of SS-dominance and E-dominance requirements, we obtain a weaker version of SSP (WSSP). In addition, we consider α − SSP, where α measures the degree of minimum acceptable inequity aversion; α − SSP is weaker than weak Pareto optimality (WPO) when α = 1. We then show that it is still possible to characterize the Nash solution using WSSP and SI only or using α -SSP, SI, and individual rationality (IR) only for any a Î [0,1)[01). Using the union of SS-dominance and E-dominance requirements, we obtain a stronger version of SSP (SSSP). It turns out that there is no bargaining solution that satisfies SSSP and SI, but the Egalitarian solution turns out to be the unique solution satisfying SSSP.

Image reduction using means on discrete product lattices

We investigate the problem of averaging values on lattices, and in particular on discrete product lattices. This problem arises in image processing when several color values given in RGB, HSL, or another coding scheme, need to be combined. We show how the arithmetic mean and the median can be constructed by minimizing appropriate penalties, and we discuss which of them coincide with the Cartesian product of the standard mean and median. We apply these functions in image processing. We present three algorithms for color image reduction based on minimizing penalty functions on discrete product lattices.

Penalty functions over a cartesian product of lattices

In this work we present the concept of penalty function over a Cartesian product of lattices. To build these mappings, we make use of restricted dissimilarity functions and distances between fuzzy sets. We also present an algorithm that extends the weighted voting method for a fuzzy preference relation.

Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers' preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.

Negations generated by bounded lattices t-norms

From the birth of fuzzy sets theory, several extensions have been proposed changing the possible membership values. Since fuzzy connectives such as t-norms and negations have an important role in theoretical as well as applied fuzzy logics, these connectives have been adapted for these generalized frameworks. Perhaps, an extension of fuzzy logic which generalizes the remaining extensions, proposed by Joseph Goguen in 1967, is to consider arbitrary bounded lattices for the values of the membership degrees. In this paper we extend the usual way of constructing fuzzy negations from t-norms for the bounded lattice t-norms and prove some properties of this construction.

(S,N)-implications on bounded lattices

Since the birth of the fuzzy sets theory several extensions have been proposed. For these extensions, different sets of membership functions were considered. Since fuzzy connectives, such as conjunctions, negations and implications, play an important role in the theory and applications of fuzzy logics, these connectives have also been extended. An extension of fuzzy logic, which generalizes the ones considered up to the present, was proposed by Joseph Goguen in 1967. In this extension, the membership values are drawn from arbitrary bounded lattices. The simplest and best studied class of fuzzy implications is the class of (S,N)-implications, and in this chapter we provide an extension of (S,N)-implications in the context of bounded lattice valued fuzzy logic, and we show that several properties of this class are preserved in this more general framework.

‘We haven’t done enough for White working-class children’: issues of distributive justice and ethnic identity politics

This article explores the politically contentious issue of White working-class student under-achievement within one particular school – a large and culturally diverse comprehensive secondary school in the greater London area. The article examines the equity philosophies and identity politics articulated by staff in their understanding of and approach to this under-achievement. It challenges the distributive logic and group identity politics that simplify, dislocate and distort equity issues. The article provides an illustration and theorising of remedies that offer potential in raising the educational performance of White working-class students – remedies not based on ethnic identity but rather on destabilising the patterns of economic distribution and social disengagement that compromise these students’ schooling attainment.

Service guarantee redress options in a hotel environment

Unconditional service guarantees are a popular marketing tool in the hotel industry worldwide. They promise total satisfaction and guests are free to invoke the guarantee whenever they are dissatisfied. While many hotels offer “money-back” compensation following guarantee invocation, others vary the payout depending on the severity of the service failure and still others will only compensate the customer if the problem leading to invocation of the guarantee cannot be fixed. To the researcher’s knowledge, the influence of compensation and fix (i.e., taking action to resolve the problem) on consumers’ perceptions of distributive justice has not been examined previously in a service guarantee context. This paper begins to address this gap by presenting a conceptual model and related propositions, arguing that redress (compensation and fix) is an important predictor of consumers’ perceptions of distributive justice, and that this relationship is moderated by service failure severity.

The influence of procedural and interactional justice, and disconfirmation on customers post recovery satisfaction evaluations

This study examines the influence of distributive and interactional justice and disconfirmation on customers’ postrecovery satisfaction evaluations, and in so doing, combines, for the first time, two existing instruments to operationalise the interactional justice construct. Using Structural Equation Modelling, the findings suggest that while both disconfirmation and justice are important predictors of satisfaction, distributive justice has the greatest influence. The research presented here reports on a section of a larger experiment-based study examining how customers’ postrecovery satisfaction evaluations are influenced by the way in which the organisation responds to the failure.

Educational administration and social justice

This paper argues that social justice is central to the pursuit of education and therefore should also be central to the practice of educational administration. Social justice in education, as elsewhere, demands both distributive justice (which remedies undeserved inequalities) and recognitional justice (which treats cultural differences with understanding and respect). But, given that cultures are always in the process of change, education is a key agency for negotiating cultural change through the exploration and negotiation of difference. Educational administration as a field can no longer escape the consideration of such issues as they are brought to the fore by the recognition of the failure of schools and school systems to ameliorate injustice in the distribution of resources and to recognise and celebrate difference as a means to social and cultural progress. We still need a model of educational administration centered around the problem of the justice and fairness of social and educational arrangements. Given the renewed interest in such issues, perhaps what was impossible twenty five years ago might now be achieved.