Sensitivity analysis incorporating fuzzy evaluation for scaling constants of multiattribute utility functions

A multiattribute utility function can be represented by a function of single-attribute utility functions if the decision maker’s preference satisfies additive independence or mutually utility independence. Additive independence is a preference condition stronger than mutually utility independence, and the multiattribute utility function is in the additive form if the former condition is satisfied, otherwise it is in the multiplicative form. In this paper, we propose a method for sensitivity analysis of multiattribute utility functions in multiplicative form, taking into account the imprecision of the decision maker’s judgment in the procedures for determining scaling constants (attribute weights).     

Interdependent additive conjoint structures

This paper deals with the use of additive value models when attributes have overlapping dependencies. An interdependent additive formulation due to Fishburn (1967) is adapted to the axiomatic system defining an additive conjoint structure (Krantz et al., 1971). This is accomplished by replacing the independence condition defined within the additive conjoint structure with a conditional independence condition. The interdependent additive formulation is shown to be the appropriate form for value functions defined over attribute sets when two technical assumptions hold.     

A preference-based approach to spanning trees and shortest paths problems****

Comparison of solutions in combinatorial problems is often based on an additive cost function inducing a complete order on solutions. We investigate here a generalization of the problem, where preferences take the form of a quasi-transitive binary relation defined on the solutions space. We first propose preference-based search algorithms for two classical combinatorial problems, namely the preferred spanning trees problem (a generalization of the minimum spanning tree problem) and the preferred paths problem (a generalization of the shortest path problem). Then, we introduce a very useful axiom for preference relations called independence. Using this axiom, we establish admissibility results concerning our preference-based search algorithms. Finally, we address the problem of dealing with non-independent preference relations and provide different possible solutions for different particular problems (e.g. lower approximation of the set of preferred solutions for multicriteria spanning trees problems, or relaxation of the independence axiom for interval-valued preferred path problems).     

Additive conjoint measurement with ordered categories

Conjoint measurement studies binary relations defined on product sets and investigates the existence and uniqueness of, usually additive, numerical representations of such relations. It has proved to be quite a powerful tool to analyze and compare MCDM techniques designed to build a preference relation between multiattributed alternatives and has been an inspiring guide to many assessment protocols. The aim of this paper is to show that additive representations can be obtained on the basis of much poorer information than a preference relation. We will suppose here that the decision maker only specifies for each object if he/she finds it “attractive” (better than the status quo), “unattractive” (worse than the status quo) or “neutral” (equivalent to the status quo). We show how to build an additive representation, with tight uniqueness properties, using such an ordered partition of the set of objects. On a theoretical level, this paper shows that classical results of additive conjoint measurement can be extended to cover the case of ordered partitions and wishes to be a contribution to the growing literature on the foundations of sorting techniques in MCDM. On a more practical level, our results suggest an assessment strategy of an additive model on the basis of an ordered partition.     

Multivalent preference structures

This paper presents a valence approach for assessing multiattribute utility functions. Unlike the decomposition approach which uses independence axioms on whole attributes to obtain utility representations, the valence approach partitions the elements of each attribute into classes on the basis of equivalent conditional preference orders. These partitions generate multivalent utility independence axioms that lead to additive-multiplicative and quasi-additive representation theorems for multiaatribute utility functions defined over product sets of equivalence classes. Preference interdependencies are thereby reflected in these classes, so attribute interactions are readily interpreted and the functional forms of the representations are kept simple.     

A verification theorem of preference separability for additive value functions

This paper offers an operational verification theorem of preference separability for additive value functions. Additive covering of the attribute index set plays a key role in the derivation. Effective ways for utilizing the theorem to derive additive value functions in terms of orthogonal square designs and progressive ways to determine the form of value functions are also discussed.     

An axiomatic basis for multiattribute value functions defined on binary attributes

The insufficiency of solvability axioms for defining an additive conjoint structure in the case of binary attributes motivates an alternative approach to development of an axiomatic basis for multilinear, multiplicative and additive value functions defined on binary attributes. The alternative approach is to define four sufficiency conditions with respect to binary attributes with alternative combinations of these conditions allowing derivation of the multilinear, multiplicative and additive models.     

基于二元联系数距离的不确定语言多属性决策模型

研究了属性权重范围已知,方案主观偏好值为语言变量,决策信息为不确定语言决策矩阵的多属性决策问题.在给出不确定语言变量转换为二元联系数的公式以及二元联系数距离公式的基础上,将方案主观偏好语言评价值转换为二元联系数,将不确定语言决策矩阵转换为二元联系数决策矩阵,从而得到方案的二元联系数综合属性值,通过最小化方案的二元联系数综合属性值和主观偏好值之间距离,建立多目标优化模型,并将其转换为一个单目标规划模型计算出属性权重.然后,通过对方案的二元联系数综合属性值进行不确定性分析,得到各方案的排序总数,利用排序总数对方案进行排序择优.应用实例表明该决策方法可行有效.     

群体多指标决策的偏好集结方法

研究多个指标条件下,利用个体决策结果形成群体一致偏好的方法、假设个体有加性效用函数,将个体多指标效用函数表示成单个指标评价函数的加权和,群体指标评价函数表示成个体指标评价函数的加权和.通过协商指标权重、指标评价函数、支付意愿三个参数,成对个体达成双方一致.提出了(n-1)对个体之间达成双方一致,从而得出群体效用函数的决策方法,这种分析框架同样可以扩展到联盟协商一致中.     

Distributed preference relations for multiple attribute decision analysis

In this paper, we propose a new pairwise comparison approach called distributed preference relation (DPR) to simultaneously signify preferred, non-preferred, indifferent, and uncertain degrees of one alternative over another on a set of grades, which is more versatile for elicitation of preference information from a decision maker than multiplicative preference relation, fuzzy preference relation (FPR) and intuitionistic FPR. In a DPR matrix on a set of alternatives, each element is a distribution recording the preferred, non-preferred, indifferent, and uncertain degrees of one alternative over another using a set of grades. To facilitate the comparison of alternatives, we define a score matrix based on a DPR matrix using the given score values of the grades. Its additive consistency is constructed, analysed, and compared with the additive consistency of FPRs between alternatives. A method for comparing two interval numbers is then employed to create a possibility matrix from the score matrix, which can generate a ranking order of alternatives with possibility degrees. A problem of evaluating strategic emerging industries is investigated using the approach to demonstrate the application of a DPR matrix to modelling and analysing a multiple attribute decision analysis problem.     

一种基于三参数区间灰色语言变量的多属性群决策方法

针对属性值为三参数区间灰色语言变量的不确定型多属性决策问题进行了研究,本文将语言变量和三参数区间数融合, 提出了三参数区间灰色语言变量的概念, 定义了三参数区间灰色语言变量的运算规则和可能度公式,在此基础上建立了基于投影模型的三参数区间灰色语言变量的多属性群决策方法。最后,通过对移动银行服务质量评估案例验证本模型的有效性。     

基于联系数贴近度的区间数多属性决策方法

利用集对分析和联系数学理论对区间数多属性决策方法进行了研究.首先给出了联系数贴近度的定义和性质;然后在将区间数决策矩阵转化为联系数决策矩阵的基础上,依据传统的逼近理想解的排序方法(TOPSIS)的基本思路,基于联系数贴近度提出了一种区间数多属性决策新方法.该方法简洁直观,易于计算,无需对区间数进行排序;最后,通过算例表明了它的有效性和实用性.     

加型一致性毕达哥拉斯模糊偏好关系

在模糊偏好关系两种等价的加型一致性概念基础上,通过简单的数学证明,分析了区间值模糊偏好关系、直觉模糊偏好关系的相应的两种加型一致性并不是等价的.然后,在加型一致性直觉模糊偏好关系的启发下,构造了可以与毕达哥拉斯模糊偏好关系相互转换的两个区间值模糊偏好关系,并利用它们的加型一致性,定义了加型一致性毕达哥拉斯模糊偏好关系,并分析了其与杨艺等定义的加型一致性毕达哥拉斯模糊偏好关系的关系.其次,研究了加型一致性毕达哥拉斯模糊偏好关系的性质以及毕达哥拉斯模糊偏好关系的满意一致性,并给出满意一致性毕达哥拉斯模糊偏好关系下的方案优劣排序算法.最后,通过两个计算实例说明了排序算法可行有效.     

Bayesian rough set model: A further investigation

Bayesian rough set model (BRSM), as the hybrid development between rough set theory and Bayesian reasoning, can deal with many practical problems which could not be effectively handled by original rough set model. In this paper, the equivalence between two kinds of current attribute reduction models in BRSM for binary decision problems is proved. Furthermore, binary decision problems are extended to multi-decision problems in BRSM. Some monotonic measures of approximation quality for multi-decision problems are presented, with which attribute reduction models for multi-decision problems can be suitably constructed. What is more, the discernibility matrices associated with attribute reduction for binary decision and multi-decision problems are proposed, respectively. Based on them, the approaches to knowledge reduction in BRSM can be obtained which corresponds well to the original rough set methodology.     

Robust ordinal regression for value functions handling interacting criteria

We present a new method called UTA^GMS–INT for ranking a finite set of alternatives evaluated on multiple criteria. It belongs to the family of Robust Ordinal Regression (ROR) methods which build a set of preference models compatible with preference information elicited by the Decision Maker (DM). The preference model used by UTA^GMS–INT is a general additive value function augmented by two types of components corresponding to “bonus” or “penalty” values for positively or negatively interacting pairs of criteria, respectively. When calculating value of a particular alternative, a bonus is added to the additive component of the value function if a given pair of criteria is in a positive synergy for performances of this alternative on the two criteria. Similarly, a penalty is subtracted from the additive component of the value function if a given pair of criteria is in a negative synergy for performances of the considered alternative on the two criteria. The preference information elicited by the DM is composed of pairwise comparisons of some reference alternatives, as well as of comparisons of some pairs of reference alternatives with respect to intensity of preference, either comprehensively or on a particular criterion. In UTA^GMS–INT, ROR starts with identification of pairs of interacting criteria for given preference information by solving a mixed-integer linear program. Once the interacting pairs are validated by the DM, ROR continues calculations with the whole set of compatible value functions handling the interacting criteria, to get necessary and possible preference relations in the considered set of alternatives. A single representative value function can be calculated to attribute specific scores to alternatives. It also gives values to bonuses and penalties. UTA^GMS–INT handles quite general interactions among criteria and provides an interesting alternative to the Choquet integral.     

Complexity of Sets Obtained as Values of Propositional Formulas

Interpretation of logical connectives as operations on sets of binary strings is considered; the complexity of a set is defined as the minimum of Kolmogorov complexities of its elements. It is readily seen that the complexity of a set obtained by the application of logical operations does not exceed the complexity of the conjunction of their arguments (up to an additive constant). In this paper, it is shown that the complexity of a set obtained by a formula Φ is small (bounded by a constant) if Φ is deducible in the logic of weak excluded middle, and attains the specified upper bound otherwise.     

Rough sets theory for multicriteria decision analysis

The original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation. It operates on a data table composed of a set U of objects (actions) described by a set Q of attributes. Its basic notions are: indiscernibility relation on U, lower and upper approximation of either a subset or a partition of U, dependence and reduction of attributes from Q, and decision rules derived from lower approximations and boundaries of subsets identified with decision classes. The original rough set idea is failing, however, when preference-orders of attribute domains (criteria) are to be taken into account. Precisely, it cannot handle inconsistencies following from violation of the dominance principle. This inconsistency is characteristic for preferential information used in multicriteria decision analysis (MCDA) problems, like sorting, choice or ranking. In order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary. The main change is the substitution of the indiscernibility relation by a dominance relation, which permits approximation of ordered sets in multicriteria sorting. To approximate preference relations in multicriteria choice and ranking problems, another change is necessary: substitution of the data table by a pairwise comparison table, where each row corresponds to a pair of objects described by binary relations on particular criteria. In all those MCDA problems, the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model. It is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax. In order to workout a recommendation in one of the MCDA problems, we propose exploitation procedures of the set of decision rules. Finally, some other recently obtained results are given: rough approximations by means of similarity relations, rough set handling of missing data, comparison of the rough set model with Sugeno and Choquet integrals, and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive.     

Valued preference-based instantiation of argumentation frameworks with varied strength defeats

A Dung-style argumentation framework aims at representing conflicts among elements called arguments. The basic ingredients of this framework is a set of arguments and a Boolean abstract (i.e., its origin is not known) binary defeat relation. Preference-based argumentation frameworks are instantiations of Dung's framework in which the defeat relation is derived from an attack relation and a preference relation over the arguments. Recently, Dung's framework has been extended in order to consider the strength of the defeat relation, i.e., to quantify the degree to which an argument defeats another argument. In this paper, we instantiate this extended framework by a preference-based argumentation framework with a valued preference relation. As particular cases, the latter can be derived from a weight function over the arguments or a Boolean preference relation. We show under some reasonable conditions that there are “less situations” in which a defense between arguments holds with a valued preference relation compared to a Boolean preference relation. Finally, we provide some conditions that the valued preference relation shall satisfy when it is derived from a weight function.     

An axiomatic analysis of concordance–discordance relations

Outranking methods propose an original way to build a preference relation between alternatives evaluated on several attributes that has a definite ordinal flavor. Indeed, most of them appeal the concordance/non-discordance principle that leads to declaring that an alternative is “superior” to another, if the coalition of attributes supporting this proposition is “sufficiently important” (concordance condition) and if there is no attribute that “strongly rejects” it (non-discordance condition). Such a way of comparing alternatives is rather natural. However, it is well known that it may produce binary relations that do not possess any remarkable property of transitivity or completeness. This explains why the axiomatic foundations of outranking methods have not been much investigated, which is often seen as one of their important weaknesses. This paper uses conjoint measurement techniques to obtain an axiomatic characterization of preference relations that can be obtained on the basis of the concordance/non-discordance principle. It emphasizes their main distinctive feature, i.e. their very crude way to distinguish various levels of preference differences on each attribute. We focus on outranking methods, such as ELECTRE I, that produce a reflexive relation, interpreted as an “at least as good as” preference relation. The results in this paper may be seen as an attempt to give such outranking methods a sound axiomatic foundation based on conjoint measurement.     

一种考虑决策者风险偏好的区间数多属性决策方法

针对属性值以区间数形式给出的多属性决策问题,提出了一种决策分析方法。在本文中,首先描述了属性值为区间数形式的多属性决策问题;然后通过引入决策者的风险偏好因子将区间数决策信息映射为实数值决策信息,并依据属性值与属性均值绝对偏差的大小确定了属性的权重,在此基础上依据所得权重给出了基于加权和法的方案排序方法,通过对风险偏好因子的不同取值还可进行方案排序的灵敏度分析。最后,通过一个算例说明了本文给出方法的可行性和有效性。