Derivation of intuitionistic fuzzy weights based on intuitionistic fuzzy preference relations

This paper proposes linear goal programming models for deriving intuitionistic fuzzy weights from intuitionistic fuzzy preference relations. Novel definitions are put forward to define additive consistency and weak transitivity for intuitionistic fuzzy preference relations, followed by a study of their corresponding properties. For any given normalized intuitionistic fuzzy weight vector, a transformation formula is furnished to convert the weights into a consistent intuitionistic fuzzy preference relation. For any intuitionistic fuzzy preference relation, a linear goal programming model is developed to obtain its intuitionistic fuzzy weights by minimizing its deviation from the converted consistent intuitionistic fuzzy preference relation. This approach is then extended to group decision-making situations. Three numerical examples are provided to illustrate the validity and applicability of the proposed models.     

Determining common weights in data envelopment analysis based on the satisfaction degree

The traditional data envelopment analysis model allows the decision-making units (DMUs) to evaluate their maximum efficiency values using their most favourable weights. This kind of evaluation with total weight flexibility may prevent the DMUs from being fully ranked and make the evaluation results unacceptable to the DMUs. To solve these problems, first, we introduce the concept of satisfaction degree of a DMU in relation to a common set of weights. Then a common-weight evaluation approach, which contains a max–min model and two algorithms, is proposed based on the satisfaction degrees of the DMUs. The max–min model accompanied by our Algorithm 1 can generate for the DMUs a set of common weights that maximizes the least satisfaction degrees among the DMUs. Furthermore, our Algorithm 2 can ensure that the generated common set of weights is unique and that the final satisfaction degrees of the DMUs constitute a Pareto-optimal solution. All of these factors make the evaluation results more satisfied and acceptable by all the DMUs. Finally, results from the proposed approach are contrasted with those of some previous methods for two published examples: efficiency evaluation of 17 forest districts in Taiwan and R&D project selection.     

Multicriteria choice based on criteria importance methods with uncertain preference information

Multicriteria choice methods are developed by applying methods of criteria importance theory with uncertain information on criteria importance and with preferences varying along their scale. Formulas are given for computing importance coefficients and importance scale estimates that are “characteristic” representatives of the feasible set of these parameters. In the discrete case, an alternative with the highest probability of being optimal (for a uniform distribution of parameter value probabilities) can be used as the best one. It is shown how such alternatives can be found using the Monte Carlo method.     

A multi-objective optimization evolutionary algorithm incorporating preference information based on fuzzy logic

A multi-objective optimization evolutionary algorithm incorporating preference information interactively is proposed. A new nine grade evaluation method is used to quantify the linguistic preferences expressed by the decision maker (DM) so as to reduce his/her cognitive overload. When comparing individuals, the classical Pareto dominance relation is commonly used, but it has difficulty in dealing with problems involving large numbers of objectives in which it gives an unmanageable and large set of Pareto optimal solutions. In order to overcome this limitation, a new outranking relation called “strength superior” which is based on the preference information is constructed via a fuzzy inference system to help the algorithm find a few solutions located in the preferred regions, and the graphical user interface is used to realize the interaction between the DM and the algorithm. The computational complexity of the proposed algorithm is analyzed theoretically, and its ability to handle preference information is validated through simulation. The influence of parameters on the performance of the algorithm is discussed and comparisons to another preference guided multi-objective evolutionary algorithm indicate that the proposed algorithm is effective in solving high dimensional optimization problems.     

Pareto set reduction based on an arbitrary finite collection of numerical information on the preference relation

Doklady Mathematics -     

Multi-criteria semantic dominance: A linguistic decision aiding technique based on incomplete preference information

A linguistic decision aiding technique for multi-criteria decision is presented. We define a relation between alternatives as multi-criteria semantic dominance (MCSD). It adopts the similar ideal of the stochastic dominance by utilizing the partial information of the decision maker’s preference, which is only ordinal or partially cardinal. The MCSD rules based on three typical types of semanteme functions are introduced and proven. By using these rules, all the alternatives under consideration are divided into two mutually exclusive sets called efficient set and inefficient set. The decision maker who has such a semanteme function will never choose the alternative from the corresponding inefficient set as the optimal one. In such a way, when we analyze the linguistic decision information, the inherent fuzziness of preference can be handled and several controversial operations of the linguistic terms can be avoided. An example is also provided to illustrate the procedure of the proposed method.     

Some models for deriving the priority weights from interval fuzzy preference relations

Interval fuzzy preference relation is a useful tool to express decision maker’s uncertain preference information. How to derive the priority weights from an interval fuzzy preference relation is an interesting and important issue in decision making with interval fuzzy preference relation(s). In this paper, some new concepts such as additive consistent interval fuzzy preference relation, multiplicative consistent interval fuzzy preference relation, etc., are defined. Some simple and practical linear programming models for deriving the priority weights from various interval fuzzy preference relations are established, and two numerical examples are provided to illustrate the developed models.     

A method for estimating criteria weights from intuitionistic preference relations

An intuitionistic preference relation is a powerful means to express decision makers’information of intuitionistic preference over criteria in the process of multi-criteria decision making. In this paper, we first define the concept of its consistence and give the equivalent interval fuzzy preference relation of it. Then we develop a method for estimating criteria weights from it, and then extend the method to accommodate group decision making based on them And finally, we use some numerical examples to illustrate the feasibility and validity of the developed method.     

Smoothing preference kinks with information

We identify conditions under which receipt of information in the form of a (potentially) ambiguous signal leads to a smoother maximin expected utility (MEU) preference structure which translates behaviorally into a smaller no-trade price zone. Narrowing of the no-trade price zone depends critically on the rectangularity of the belief structure, which, in the context of an MEU model, is a requirement of dynamic consistency in Machina’s sense. Another important factor affecting the size of the no-trade price zone is the relative contribution of ambiguity in signals and ambiguity in posterior beliefs to the degree of prior ambiguity over market events.     

Ranking decision-making units using common weights in DEA

Data envelopment analysis (DEA) is a linear programming technique that is used to measure the relative efficiency of decision-making units (DMUs). Liu et al. (2008) [13] used common weights analysis (CWA) methodology to generate a CSW using linear programming. They classified the DMUs as CWA-efficient and CWA-inefficient DMUs and ranked the DMUs using CWA-ranking rules. The aim of this study is to show that the criteria used by Liu et al. are not theoretically strong enough to discriminate among the CWA-efficient DMUs with equal efficiency. Moreover, there is no guarantee that their proposed model can select one optimal solution from the alternative components. The optimal solution is considered to be the only unique optimal solution. This study shows that the proposal by Liu et al. is not generally correct. The claims made by the authors against the theorem proposed by Liu et al. are fully supported using two counter examples.     

Proposing a method for fixed cost allocation using DEA based on the efficiency invariance and common set of weights principles

One of the applications of data envelopment analysis is fixed costs allocation among homogenous decision making units. In this paper, we first prove that Beasley’s method (Eur J Oper Res 147(1):198–216, 2003), whose infeasibility has been claimed by Amirteimoori and Kordrostami (Appl Math Comput 171(1):136–151, 2005), always has a feasible solution and the efficiency invariance principle does not necessarily satisfy in Amirteimoori and Kordrostami’s method (Appl Math Comput 171(1):136–151, 2005). Hence, we present two equitable methods for fixed cost allocation based on the efficiency invariance and common set of weights principles such that, if possible, they help meet these two principles. In the first method, the costs are allocated to DMU in such a way that the efficiency score of DMUs does not change, and simultaneously this allocation has the minimum distance from the allocation that has been obtained with a common set of weights. However, in the second method, the costs are allocated in such a way that input and output of all units have a common set of weights and it has the minimum distance from the allocation that satisfies the efficiency invariance principle. Moreover, both methods, consider the satisfaction of each unit of the allocated cost. Finally, the proposed method is illustrated by two real world examples.     

Data envelopment analysis with common weights: the compromise solution approach

A characteristic of data envelopment analysis (DEA) is to allow individual decision-making units (DMUs) to select the factor weights that are the most advantageous for them in calculating their efficiency scores. This flexibility in selecting the weights, on the other hand, deters the comparison among DMUs on a common base. In order to rank all the DMUs on the same scale, this paper proposes a compromise solution approach for generating common weights under the DEA framework. The efficiency scores calculated from the standard DEA model are regarded as the ideal solution for the DMUs to achieve. A common set of weights which produces the vector of efficiency scores for the DMUs closest to the ideal solution is sought. Based on the generalized measure of distance, a family of efficiency scores called ‘compromise solutions’ can be derived. The compromise solutions have the properties of unique solution and Pareto optimality not enjoyed by the solutions derived from the existing methods of common weights. An example of forest management illustrates that the compromise solution approach is able to generate a common set of weights, which not only differentiates efficient DMUs but also detects abnormal efficiency scores on a common base.     

Note on “Some models for deriving the priority weights from interval fuzzy preference relations”

Deriving accurate interval weights from interval fuzzy preference relations is key to successfully solving decision making problems. Xu and Chen (2008) proposed a number of linear programming models to derive interval weights, but the definitions for the additive consistent interval fuzzy preference relation and the linear programming model still need to be improved. In this paper, a numerical example is given to show how these definitions and models can be improved to increase accuracy. A new additive consistency definition for interval fuzzy preference relations is proposed and novel linear programming models are established to demonstrate the generation of interval weights from an interval fuzzy preference relation.     

Finding common RNA pseudoknot structures in polynomial time

This paper presents the first polynomial time algorithm for finding common RNA substructures that include pseudoknots and similar structures. While a more general problem is known to be NP-hard, this algorithm exploits special features of RNA structures to match RNA bonds correctly in polynomial time. Although the theoretical upper bound on the algorithm?s time and space usage is high, the data-driven nature of its computation enables it to avoid computing unnecessary cases, dramatically reducing the actual running time. The algorithm works well in practice, and has been tested on sample RNA structures that include pseudoknots and pseudoknot-like tertiary structures.     

Deriving priority weights from intuitionistic multiplicative preference relations under group decision-making settings

The intuitionistic multiplicative preference relation (IMPR), which takes into account both the ratio degree to which an alternative is preferred to another and the ratio degree to which an alternative is non-preferred to another, is a useful tool for decision makers to elicit their preference information using Saaty’s 1–9 scale. In this paper, we focus on group decision making with IMPRs. First, we analyze the flaws of the consistency definition of an IMPR in previous work and then propose a new definition to overcome the flaws. On this basis, a linear programming-based algorithm is developed to check and improve the consistency of an IMPR. Second, we discuss the relationships between an IMPR and a normalized intuitionistic multiplicative weight vector and develop two approaches to group decision making based on complete and incomplete IMPRs, respectively. Based on the proposed algorithm and approaches, a general framework for group decision making with IMPRs is proposed. Finally, some numerical examples are provided to demonstrate the proposed approaches. The results show that the proposed approaches can deal with group decision-making problems with IMPRs effectively.     

Finding common fixed points of a class of paracontractions

For a suitable set of not necessarily linear operators on a real Hilbert space, such that the set of common fixed points of the operators is nonempty and has an interior point, we construct an asynchronous parallel algorithm that leads to a common fixed point in a finite number of steps. This generalizes a result in [13]. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

Economic planning based on social preference functions

We argue that the most desirable social welfare functions for practical use (here sometimes called social preference functions) are those determined by Σ_ilog(u_i(x)?α) where u_i(x) is the utility of alternative x to individual i and α is the minimum standard of living decided upon by the society.     

Refined common knowledge logics or logics of common information

 In terms of formal deductive systems and multi-dimensional Kripke frames we study logical operations know, informed, common knowledge and common information. Based on [6] we introduce formal axiomatic systems for common information logics and prove that these systems are sound and complete. Analyzing the common information operation we show that it can be understood as greatest open fixed points for knowledge formulas. Using obtained results we explore monotonicity, omniscience problem, and inward monotonocity, describe their connections and give dividing examples. Also we find algorithms recognizing these properties for some particular cases. Received: 21 October 2000 / Published online: 2 September 2002 Key words or phrases: Multi-agent systems – Non-standard logic – Knowledge representation – Common knowledge – Common information – Fixed points, Kripke models – Modal logic     

On linear codes whose weights and length have a common divisor

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1<r<q=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r<q and whose dual minimum distance is at least 3, has length at least (r−1)q+(p−1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.