Coalitional power structure without the pareto principle

In this paper we have extended Arrow's analysis to a framework where for any given profile of individual preference orderings the decision procedure specifies a non-trivial probability distribution over possible social orderings. We have demonstrated that if the social decision procedure satisfies certain probabilistic versions of weak independence of irrelevant alternatives, then it is characterized by a ‘power’ structure for all possible coalitions of individuals without assuming either the Pareto Principle or its antecedents. A generalised version of Arrow's impossibility theorem follows as a special case of our result. We have weakened Arrow's independence condition, and have shown the existence of a hierarchy of dictators without imposing the Pareto criterion.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.     

Multiperson decision-making based on multiplicative preference relations

A multiperson decision-making problem, where the information about the alternatives provided by the experts can be presented by means of different preference representation structures (preference orderings, utility functions and multiplicative preference relations) is studied. Assuming the multiplicative preference relation as the uniform element of the preference representation, a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives. In this decision model, several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations. The decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives, quantifier guided dominance degree and quantifier guided non-dominance degree. The consistency of the model is analysed to prove that it acts coherently.     

Design of group decisions II. In spaces of partial order fuzzy relations

For the case of initial data in the problem of group choice represented as fuzzy partial orderings two problems are solved: (1) design of a set group decisions which satisfy the Pareto unanimity principlle and stay ‘halfway’ between initial relations and (2) design of a unique group decision.     

A consistency and consensus-based method to group decision making with interval linguistic preference relations

Preference relations are a powerful tool to address decision-making problems. In some situations, because of the complexity of decision-making problems and the inherent uncertainty, the decision makers cannot express their preferences by using numerical values. Interval linguistic preference relations, which are more reliable and informative for the decision-makers’ preferences, are a good choice to cope with this issue. Just as with the other types of preference relations, the consistency and consensus analysis is very importance to ensure the reasonable ranking order by using interval linguistic preference relations. Considering this situation, this paper introduces a consistency concept for interval linguistic preference relations. To measure the consistency of interval linguistic preference relations, a consistency measure is defined. Then, a consistency-based programming model is built, by which the consistent linguistic preference relations with respect to each object can be obtained. To cope with the inconsistency case, two models for deriving the adjusted consistent linguistic preference relations are constructed. Then, a consistency-based programming model to estimate the missing values is built. After that, we present a group consensus index and present some of its desirable properties. Furthermore, a group consensus-based model to determine the weights of the decision makers with respect to each object is established. Finally, an approach to group decision making with interval linguistic preference relations is developed, which is based on the consistency and consensus analysis. Meanwhile, the associated numerical examples are offered to illustrate the application of the procedure.     

Interactive group decision-making procedure using weak strength of preference

In the paper, we present an interactive decision procedure for aggregating group members’ preferences which are specified in incomplete ways. A group consensus under incomplete information is not usually reached at a single step since less-specific preferences on attribute weights and performance scores make a clear selection of best alternative more difficult. To circumvent these difficulties, a measure, indicating the strength of preference between alternatives, is derived to help each of decision makers change his/her preference structure. To make preference changes based on the measure effective, we present a solution method to convert an intractable nonlinear programme into a linear one.     

Consistency analysis of triangular fuzzy reciprocal preference relations

In order to simulate the uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. Based on the reciprocity property, this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations. The new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by Buckley (1985). The properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail. In addition, the shortcomings of the proof procedure of the proposition given by Wang and Chen (2008) are pointed out. And the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations. Finally, using the (n − 1) restricted comparison ratios, a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed, and an algorithm is shown to make a consistent decision ranking. Numerical results are further calculated to illustrate the new definition and the obtained algorithm.     

An Interactive Multiple Criteria Approach to Ranking Alternatives

In this paper we are concerned with ranking various orderings of a set of alternatives to a composite order as a multiple criteria problem. The orderings (called preference orderings) can be real preference orderings or any natural orderings. The objective is to find the most preferred order of the decision maker using the preference orderings as criteria.In principle, the problem can be formulated as a multiple objective linear programming problem using the model of Bowman and Colantoni and then solved with the interactive method proposed by Zionts and Wallenius. However, the fact that we are dealing with integer variables prohibits us from applying this approach as such. We discuss the problem formulation and propose a modified approach to that of Zionts and Wallenius for solving the problem.     

METAPHORS AND OTHER LINGUISTIC POINTERS TO CHILDREN'S MENTAL REPRESENTATIONS

The mental representations that 6- and 7-year-old pupils form as a result of their interactions with their teacher's verbal, written, pictorial and concrete material representations has to be inferred from the language they use. In this study many pupils seem to have mental representations which capture surface characteristics of a particular teachers’ representation and use metaphoric language associated with that representation when describing their calculations. Pupils’ use of ‘you’ is characteristic of those who adopt a representation-specific procedure, whilst ‘if’ and ‘like’ are linguistic pointers to their use of generic examples to describe a procedure. Individual pupils show a preference for the same style of mental representation when describing images and procedures in mathematical and non-mathematical contexts.     

A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making

In decision making problems, there may be the cases where the decision makers express their judgements by using preference relations with incomplete information. Then one of the key issues is how to estimate the missing preference values. In this paper, we introduce an incomplete interval multiplicative preference relation and give the definitions of consistent and acceptable incomplete ones, respectively. Based on the consistency property of interval multiplicative preference relations, a goal programming model is proposed to complement the acceptable incomplete one. A new algorithm of obtaining the priority vector from incomplete interval multiplicative preference relations is given. The goal programming model is further applied to group decision-making (GDM) where the experts evaluate their preferences as acceptable incomplete interval multiplicative preference relations. An interval weighted geometric averaging (IWGA) operator is proposed to aggregate individual preference relations into a social one. Furthermore, the social interval multiplicative preference relation owns acceptable consistency when every individual one is acceptably consistent. Two numerical examples are carried out to show the efficiency of the proposed goal programming model and the algorithms.     

A method based on similarity measures for interactive group decision-making with intuitionistic fuzzy preference relations

In this paper, we study the group decision-making problem in which the preference information given by experts takes the form of intuitionistic fuzzy preference relations, and the information about experts’ weights is completely unknown. We first utilize the intuitionistic fuzzy weighted averaging operator to aggregate all individual intuitionistic fuzzy preference relations into a collective intuitionistic fuzzy preference relation. Then, based on the degree of similarity between the individual intuitionistic fuzzy preference relations and the collective one, we develop an approach to determine the experts’ weights. Furthermore, based on intuitionistic fuzzy preference relations, a practical interactive procedure for group decision-making is proposed, in which the similarity measures between the collective preference relation and intuitionistic fuzzy ideal solution are used to rank the given alternatives. Finally, an illustrative numerical example is given to verify the developed approach.     

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

Some issues in mathematical modelling of multiple criteria decision-making problems

Models for analyzing and solving multiple criteria decision-making (MCDM) problems are difficult to evaluate and compare, because they are intended for diverse orderings of a set of feasible alternatives. These models are based on a variety of assumptions about the decision maker's preferences and use different types of preference information. In this paper, a conceptual framework is developed for evaluating and comparing discrete alternative MCDM models available for a given decision situation. The procedure employed in the framework guides the user through an analysis of the decision situation making it possible for a decision maker or analyst to select the most appropriate MCDM model from among several alternative feasible models.     

Developing an implementation capacity: justifications from prior research

Implementing innovative operational research solutions into organizations can be messy. Pragmatic inquiry suggests the first step in dissolving a mess is to determine the mindset, or set of concepts, that will be used by decision makers to inform their day to day choice of activities. Van de Ven and Poole reviewed much of the organizational change literature and identified four reasons why change occurs. They labelled them, life cycle, evolution, teleology and dialectic. It would seem logical to suggest that any attempt to make sense of the mess of implementation needs to span these four reasons. However, these reasons need to be operationalized into a mindset for implementers. To do this, the management implementation literature was reviewed under each of these reasons. The result is a justification of four concepts that can create a mindset likely to improve the implementation capacity of organizations. This mindset is that organizations make greater use of ‘champions’ ‘continuous improvement’ ‘job rotation’ and ‘debate’ over alternative activities. Exactly how so, is for individual organizations to interpret.     

Group decision-making based on heterogeneous preference relations with self-confidence

Preference relations are very useful to express decision makers’ preferences over alternatives in the process of group decision-making. However, the multiple self-confidence levels are not considered in existing preference relations. In this study, we define the preference relation with self-confidence by taking multiple self-confidence levels into consideration, and we call it the preference relation with self-confidence. Furthermore, we present a two-stage linear programming model for estimating the collective preference vector for the group decision-making based on heterogeneous preference relations with self-confidence. Finally, numerical examples are used to illustrate the two-stage linear programming model, and a comparative analysis is carried out to show how self-confidence levels influence on the group decision-making results.     

Problems in the modelling of socio-cultural systems

The attempts made to apply explicit modelling procedures to investigate the behaviour of and change in past human societies are first briefly surveyed. The comparative lack of success of endeavours to model change in the system as a whole rather than in one specific field (or subsystem) are noted, and the difficulties inherent in the enterprise are reviewed.It is argued that one limitation of most recent models is their failure to tackle the elements of human society which are specifically human, including the ability to use symbols, and to formulate plans. Here the pitfalls of an ‘idealist’ approach—of claiming to reconstruct the specific thoughts of past individuals—are underlined. Yet it is precisely because contemporary archaeology has rightly eschewed idealist approaches that it has been over-cautious in dealing with the subject matter of many idealist arguments, namely communication and intelligently-formulated action.Some aspects of a suitable modelling procedure can now be suggested. The analogy offered by the geographers' concept of a mental map is a useful one, and offers the basis for the Thinks model. Comments are invited as to how the arguments can be developed in a manner which will be both explicit and instructive, and which will not require the imaginative and subjective leaps of ‘idealist’ thought.     

Social evaluation with variable population size: an alternative concept

This paper deals with some consequences of certain population changes on social welfare considerations. In particular, a general concept for expressing value judgments about how the addition of 1 further agent to some given society should affect social welfare and a replication invariance condition are introduced and discussed. An axiomatic framework for welfare orderings based on a generalized averaging procedure applied to individual utilities is provided.     

Scoring rule and majority agreements for large electorates with arbitrary preferences

This paper examines elections among three candidates when the electorate is large and voters can have any of the 26 nontrivial asymmetric binary relations on the candidates as their preference relations. Comparisons are made between rule-λ rankings based on rank-order ballots and simple majorities based on the preference relations. The rule-λ ranking is the decreasing point total order obtained when 1, λ and 0 points are assigned to the candidates ranked first, second and third on each voter's ballot, with 0 ? λ ? 1.Limit probabilities as the number of voters gets large are computed for events such as ‘the first-ranked rule-λ candidate has a majority over the second-ranked rule-λ candidate’ and ‘the rule-λ winner is the Condorcet candidate, given that there is a Condorcet candidate’. The probabilities are expressed as functions of λ and the distribution of voters over types of preference relations. In general, they are maximized at λ = 1/2 (Borda) and minimized at λ = 0 (plurality) and at λ = 1 for any fixed distribution of voters over preference types. The effects of more indifference and increased intransitivity in voter's preference relations are analyzed when λ is fixed.     

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.