Bi-criteria optimization problems for decision rules

We consider bi-criteria optimization problems for decision rules and rule systems relative to length and coverage. We study decision tables with many-valued decisions in which each row is associated with a set of decisions as well as single-valued decisions where each row has a single decision. Short rules are more understandable; rules covering more rows are more general. Both of these problems—minimization of length and maximization of coverage of rules are NP-hard. We create dynamic programming algorithms which can find the minimum length and the maximum coverage of rules, and can construct the set of Pareto optimal points for the corresponding bi-criteria optimization problem. This approach is applicable for medium-sized decision tables. However, the considered approach allows us to evaluate the quality of various heuristics for decision rule construction which are applicable for relatively big datasets. We can evaluate these heuristics from the point of view of (i) single-criterion—we can compare the length or coverage of rules constructed by heuristics; and (ii) bi-criteria—we can measure the distance of a point (length, coverage) corresponding to a heuristic from the set of Pareto optimal points. The presented results show that the best heuristics from the point of view of bi-criteria optimization are not always the best ones from the point of view of single-criterion optimization.     

Rough set-based logics for multicriteria decision analysis

In this paper, we propose some decision logic languages for rule representation in rough set-based multicriteria analysis. The semantic models of these logics are data tables, each of which is comprised of a finite set of objects described by a finite set of criteria/attributes. The domains of the criteria may have ordinal properties expressing preference scales, while the domains of the attributes may not. The validity, support, and confidence of a rule are defined via its satisfaction in the data table.     

Rough sets for pattern classification using pairwise-comparison-based tables

Rough set theory is a useful mathematical tool to deal with vagueness and uncertainty in available information. The results of a rough set approach are usually presented in the form of a set of decision rules derived from a decision table. Because using the original decision table is not the only way to implement a rough set approach, it could be interesting to investigate possible improvement in classification performance by replacing the original table with an alternative table obtained by pairwise comparisons among patterns. In this paper, a decision table based on pairwise comparisons is generated using the preference relation as in the Preference Ranking Organization Methods for Enrichment Evaluations (PROMETHEE) methods, to gauges the intensity of preference for one pattern over another pattern on each criterion before classification. The rough-set-based rule classifier (RSRC) provided by the well-known library for the Rough Set Exploration System (RSES) running under Windows as been successfully used to generate decision rules by using the pairwise-comparisons-based tables. Specifically, parameters related to the preference function on each criterion have been determined using a genetic-algorithm-based approach. Computer simulations involving several real-world data sets have revealed that of the proposed classification method performs well compared to other well-known classification methods and to RSRC using the original tables.     

Multi-criteria classification – A new scheme for application of dominance-based decision rules

We are considering the problem of multi-criteria classification. In this problem, a set of “if … then …” decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes. Given a sample of classification examples, called learning data set, the rules are induced from dominance-based rough approximations of preference-ordered decision classes, according to the Variable Consistency Dominance-based Rough Set Approach (VC-DRSA). The main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by (i) no rule, (ii) exactly one rule, (iii) several rules. The proposed classification scheme can be applied to both, learning data set (to restore the classification known from examples) and testing data set (to predict classification of new objects). A hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals.     

Rough set theory: a novel approach for extraction of robust decision rules based on incremental attributes

Rough set theory is a new data mining approach to manage vagueness. It is capable to discover important facts hidden in the data. Literature indicate the current rough set based approaches can’t guarantee that classification of a decision table is credible and it is not able to generate robust decision rules when new attributes are incrementally added in. In this study, an incremental attribute oriented rule-extraction algorithm is proposed to solve this deficiency commonly observed in the literature related to decision rule induction. The proposed approach considers incremental attributes based on the alternative rule extraction algorithm (AREA), which was presented for discovering preference-based rules according to the reducts with the maximum of strength index (SI), specifically the case that the desired reducts are not necessarily unique since several reducts could include the same value of SI. Using the AREA, an alternative rule can be defined as the rule which holds identical preference to the original decision rule and may be more attractive to a decision-maker than the original one. Through implementing the proposed approach, it can be effectively operating with new attributes to be added in the database/information systems. It is not required to re-compute the updated data set similar to the first step at the initial stage. The proposed algorithm also excludes these repetitive rules during the solution search stage since most of the rule induction approaches generate the repetitive rules. The proposed approach is capable to efficiently and effectively generate the complete, robust and non-repetitive decision rules. The rules derived from the data set provide an indication of how to effectively study this problem in further investigations.     

基于粗集的决策分析

粗糙集理论研究的重要内容是约简,目的在于获取优良的规则集合。本文描述了决策规则的多种指标,分析了他们体现的性质,并提出了规则集合的决策度量,从整体上体现了一个规则集合的性能,为多知识库决策奠定了基础。     

An adjustable approach to fuzzy soft set based decision making

Molodtsov’s soft set theory was originally proposed as a general mathematical tool for dealing with uncertainty. Recently, decision making based on (fuzzy) soft sets has found paramount importance. This paper aims to give deeper insights into decision making based on fuzzy soft sets. We discuss the validity of the Roy-Maji method and show its true limitations. We point out that the choice value designed for the crisp case is no longer fit to solve decision making problems involving fuzzy soft sets. By means of level soft sets, we present an adjustable approach to fuzzy soft set based decision making and give some illustrative examples. Moreover, the weighted fuzzy soft set is introduced and its application to decision making is also investigated.     

Self-consistency of decision rules for group decision making

The author treats, in this paper, a group of decision makers, where each of them already has preference on a given set of alternatives but the group as a whole does not have a decision rule to make their group decision, yet. Then, the author examines which decision rules are appropriate. As a criterion of “appropriateness” the author proposes the concepts of self-consistency and universal self-consistency of decision rules. Examining the existence of universally self-consistent decision rules in two cases: (1) decision situations with three decision makers and two alternatives, and (2) those with three decision makers and three alternatives, the author has found that all decision rules are universally self-consistent in the case (1), whereas all universally self-consistent decision rules have one and just one vetoer in the essential cases in (2). The result in the case (2) implies incompatibility of universal self-consistency with symmetry. An example of applications of the concept of self-consistency to a bankruptcy problem is also provided in this paper, where compatibility of self-consistency with symmetry in a particular decision situation is shown.     

Nonparametric predictive utility inference

We consider the natural combination of two strands of recent statistical research, i.e., that of decision making with uncertain utility and that of Nonparametric Predictive Inference (NPI). In doing so we present the idea of Nonparametric Predictive Utility Inference (NPUI), which is suggested as a possible strategy for the problem of utility induction in cases of extremely vague prior information. An example of the use of NPUI within a motivating sequential decision problem is also considered for two extreme selection criteria, i.e., a rule that is based on an attitude of extreme pessimism and a rule that is based on an attitude of extreme optimism.     

Rule Induction for Group Decisions with Statistical Data — An Example

An expert system was desired for a group decision-making process. A highly variable data set from previous groups' decisions was available to simulate past group decisions. This data set has much missing information and contains many possible errors. Classification and regression trees (CART) was selected for rule induction, and compared with multiple linear regression and discriminant analysis. We conclude that CART's decision rules can be used for rule induction. CART uses all available information and can predict observations with missing data. Errors in results from CART compare well with those from multiple linear regression and discriminant analysis. CART results are easier to understand.     

Optimal scale selection for multi-scale decision tables

Human beings often observe objects or deal with data hierarchically structured at different levels of granulations. In this paper, we study optimal scale selection in multi-scale decision tables from the perspective of granular computation. A multi-scale information table is an attribute-value system in which each object under each attribute is represented by different scales at different levels of granulations having a granular information transformation from a finer to a coarser labelled value. The concept of multi-scale information tables in the context of rough sets is introduced. Lower and upper approximations with reference to different levels of granulations in multi-scale information tables are defined and their properties are examined. Optimal scale selection with various requirements in multi-scale decision tables with the standard rough set model and a dual probabilistic rough set model are discussed respectively. Relationships among different notions of optimal scales in multi-scale decision tables are further analyzed.     

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.     

An interval set model for learning rules from incomplete information table

A novel interval set approach is proposed in this paper to induce classification rules from incomplete information table, in which an interval-set-based model to represent the uncertain concepts is presented. The extensions of the concepts in incomplete information table are represented by interval sets, which regulate the upper and lower bounds of the uncertain concepts. Interval set operations are discussed, and the connectives of concepts are represented by the operations on interval sets. Certain inclusion, possible inclusion, and weak inclusion relations between interval sets are presented, which are introduced to induce strong rules and weak rules from incomplete information table. The related properties of the inclusion relations are proved. It is concluded that the strong rules are always true whatever the missing values may be, while the weak rules may be true when missing values are replaced by some certain known values. Moreover, a confidence function is defined to evaluate the weak rule. The proposed approach presents a new view on rule induction from incomplete data based on interval set.     

Reconsideration of the present value method

In investment decision-making, the net present value method is widely used as one of the best decision rules (techniques or criteria). At the same time, it is also used to evaluate decision alternatives for a long range period of time, in economics or even in control theory. Its theoretical validation as the best method for investment decision-making has been based on a basis such that the best technique (investment decision rule) will maximize shareholders' wealth which is measured by the present value of cash flows discounted at the opportunity cost of capital. Such a theoretical requirement as maximizing shareholders' wealth is very important for investment decision-makings. This requirement implies that an ordering relation of projects determined by the best investment rule must be order-isomorphic to that determined by the measure of shareholders' wealth. This order-isomorphism can be represented by necessary and sufficient conditions (or separate criteria). However, they are not suitable for comparing investment decision rules, because they are designed for selecting the best investment decision rule. At the same time, the other dominance of the net present value method over other investment rules is also found in its decision-theoretical aspects. Formulating the net present value method, internal rate of return method and simple sum method in an axiomatic fashion, the net present value method is compared with the other rules, and is shown to have enough clarity and simplicity in theory and practice.     

Group decision making methods based on intuitionistic fuzzy soft matrices

Soft set theory was originally proposed by Molodtsov as a general mathematical tool for dealing with uncertainty in 1999. Recently, researches of decision making based on soft sets have got some progress, but few people consider multi-experts situation. As such, this paper discusses multi-experts group decision making problems. Firstly, we give a concept of intuitionistic fuzzy soft matrix (IFSM) and prove some relevant properties of IFSM. Then, an adjustable approach is presented by means of median level soft set and p-quantile level soft set for dealing with decision making problems based on IFSM. Thirdly, we study aggregation methods of IFSM, give two kinds of aggregation operators and methods that how to determine experts’ weights under different situation with programming models, four corresponding algorithms have been proposed, too. Finally, a practical example has been demonstrated the reasonability and efficiency of these new algorithms.     

An adjustable approach to intuitionistic fuzzy soft sets based decision making

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. There has been some progress concerning practical applications of soft set theory, especially the use of soft sets in decision making. In this paper we generalize the adjustable approach to fuzzy soft sets based decision making. Concretely, we present an adjustable approach to intuitionistic fuzzy soft sets based decision making by using level soft sets of intuitionistic fuzzy soft sets and give some illustrative examples. The properties of level soft sets are presented and discussed. Moreover, we also introduce the weighted intuitionistic fuzzy soft sets and investigate its application to decision making.     

基于粗糙集的应急案例中概率规则挖掘方法

应急案例作为描述突发事件发生、发展及应对过程的文本,蕴含了潜在的规律与宝贵的经验。为了挖掘应急案例中各要素间潜在的关联关系,构建出基于粗糙集的应急案例中概率规则挖掘方法。首先,构建出应急案例知识五元组,描述应急案例共性特征,并将诸多应急案例信息组织成一张应急案例决策表;然后,应用遗传算法对应急案例决策表进行属性约简,进而获取概率规则;最后,以大兴安岭林区50起重特大火灾案例为例,阐述方法的具体执行过程,并通过两组测试实验证明了方法的可行性和有效性。该方法描述了应急案例的共性本体特征,具有较高的可重用性,有利于为决策者采取应急管理措施提供决策支持。     

Implementation of Backward Induction for Sequentially Adaptive Clinical Trials

In many clinical trials, patients are enrolled and data are collected sequentially, with interim decisions, including what treatment the next patient should receive and whether or not the trial should be terminated or continued, being based on the accruing data. This naturally leads to application of Bayesian sequential procedures for trial monitoring. This article discusses the implementation and computational tasks involved in the use of backward induction for making decisions during a clinical trial. An efficient method is presented for storing and retrieving decision tables that represent the decision trees characterizing all possible decisions made when implementing a clinical trial using backward induction. We address the general computational needs, and illustrate the ideas with a specific example of a two-arm trial with a binary outcome and a maximum sample size of 200 patients.