Integrated evidential reasoning approach in the presence of cardinal and ordinal preferences and its applications in software selection

A combination of cardinal and ordinal preferences in multiple-attribute decision making (MADM) demonstrates more reliability and flexibility compared with sole cardinal or ordinal preferences derived from a decision maker. This situation occurs particularly when the knowledge and experience of the decision maker, as well as the data regarding specific alternatives on certain attributes, are insufficient or incomplete. This paper proposes an integrated evidential reasoning (IER) approach to analyze uncertain MADM problems in the presence of cardinal and ordinal preferences. The decision maker provides complete or incomplete cardinal and ordinal preferences of each alternative on each attribute. Ordinal preferences are expressed as unknown distributed assessment vectors and integrated with cardinal preferences to form aggregated preferences of alternatives. Three optimization models considering cardinal and ordinal preferences are constructed to determine the minimum and maximum minimal satisfaction of alternatives, simultaneous maximum minimal satisfaction of alternatives, and simultaneous minimum minimal satisfaction of alternatives. The minimax regret rule, the maximax rule, and the maximin rule are employed respectively in the three models to generate three kinds of value functions of alternatives, which are aggregated to find solutions. The attribute weights in the three models can be precise or imprecise (i.e., characterized by six types of constraints). The IER approach is used to select the optimum software for product lifecycle management of a famous Chinese automobile manufacturing enterprise.     

Higher order dual Lagrange multiplier spaces¶for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach. Received: July 2002 / Accepted: November 2002     

Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers

Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretised? The most popular enforcement methods are the Lagrange multiplier method, the penalty method and combinations of these two. A frequently used discretisation method is the so called node-to-segment approach. However, this approach might lead to problems like jumps in contact forces, loss of convergence or failure to pass the patch test. Thus in the last few years, several segment-to-segment contact algorithms based on the mortar method were proposed.Combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations. On the other hand, a Lagrange multiplier mortar method requires additional unknowns. Hence, condensation of the Lagrange multipliers is desirable to preserve the initial size of the system of equations. This can be achieved by interpolating the Lagrange multipliers with so-called dual shape functions.Discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master elements, respectively. In current versions of dual Lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element (instead of the entire element) belongs to the contact area. We present a modified definition of the dual shape functions in such slave elements. The basic idea is to construct dual shape functions that fulfill the so-called biorthogonality condition within the contact area. This leads to consistent mortar matrices also in the boundary region. To avoid ill-conditioning of the stiffness matrix, the modified mortar matrices are weighted with appropriate weighting factors. In doing so, the corresponding modified Lagrange multiplier nodal values are of the same order as the unmodified ones. Various examples demonstrate the performance of the modified mortar contact algorithm.     

A two-slope achievement scalarizing function for interactive multiobjective optimization

The use of achievement (scalarizing) functions in interactive multiobjective optimization methods is very popular, as indicated by the large number of algorithmic and applied scientific papers that use this approach. Key parameters in this approach are the reference point, which expresses desirable objective function values for the decision maker, and weights. The role of the weights can range from purely normalizing to fully preferential parameters that indicate the relative importance given by the decision maker to the achievement of each reference value. Technically, the influence of the weights in the solution generated by the achievement scalarizing function is different, depending on whether the reference point is achievable or not. Besides, from a psychological point of view, decision makers also react in a different way, depending on the achievability of the reference point. For this reason, in this work, we introduce the formulation of a new achievement scalarizing function with two different weight vectors, one for achievable reference points, and the other one for unachievable reference points. The new achievement scalarizing function is designed so that an appropriate weight vector is used in each case, without having to carry out any a priori achievability test. It allows us to reflect the decision maker's preferences in a better way as a part of an interactive solution method, and this can cause a quicker convergence of the method. The computational efficiency of this new formulation is shown in several test examples using different reference points.     

Systematic decision process for intelligent decision making

In this paper, Systematic decision process (SDP) for solving Multiple Criteria Decision Making problems with application for manufacturing location selection is introduced. SDP is a comprehensive approach which is based on eliciting strength of preferences for assessing additive utility functions. SDP consists of three steps: I. assessing weights, II. assessing qualitative criteria, and III. ranking alternatives using the assessed additive utility function. Strengths of preferences can be expressed by using either qualitative or numerical ratings. If the decision maker is inconsistent in his/her responses, such inconsistencies are identified by the method. It is shown that the method has advantages in terms of simplicity and accuracy compared to existing methods such as Analytical Hierarchy Process. Furthermore, a quadratic optimization method for assessing weights of additive utility function by use of pair comparison of actual alternatives is developed. Computational experiments are provided.     

Expert preference system Computer assisted system for developing expert utility functions

The Expert Preference System is a computer assisted procedure for developing a decision maker's preference utility function. The system will elicit proper preferences for different levels of criteria from the decision maker by supplying useful information and by checking inconsistencies in the responses. The system will then determine the mathematical function that will best predict and describe the decision maker's preference utility function.     

A practical measure of the information in a logical theory

The information in data depends on the subjective value system that the receiver of the data uses to interpret them. This paper looks at the information in a theory of first order logic (a knowledge base) from the perspective of a decision maker for whom the validation of formulae (facts and rules) have varying importance. The decision maker's preferences and prior knowledge are both incorporated into the information measure. The value of information is determined by what it conveys about the formulae of importance to the decision maker. The information measure is applied as a heuristic in commonsense reasoning; in relevance assessment ; and as a preference function in belief revision.     

Multi-criteria inventory classification with reference items

The ABC method is a well-known approach to classify inventory items into ordered categories, such as A, B and C. As emphasized in the literature, it is reasonable to evaluate the inventory classification problem in the multi-criteria context. From this point of view, it corresponds to a sorting problem where categories are ordered. Here, one important issue is that the weights of the criteria and categorization preferences can change from industry to industry. This requires the analysis of the problem in a specific framework where the decision maker (expert)’s preferences are considered. In this study, the preferences of the decision maker are incorporated into the decision making process in terms of reference items into each class. We apply two utility functions based sorting methods to the problem. We perform an experiment and compare results with other algorithms from the literature.     

Urban Transport Network Regulation and Evaluation: A Fuzzy Evolutionary Approach

The real-time regulation of urban collective transport traffic is a very delicate problem, particularly in case of appearance of simultaneous disturbances (vehicle's breakdown, strike, demonstration, etc.). Indeed, a regulator (a decision maker) has to carry out difficult tasks that are often inaccessible at the human scale and involve the assistance of a decision support system (DSS). In this paper, we present the main part of this DSS, i.e., the generator and the evaluator of the decision strategies for disrupted transport network regulation. The proposed module is based on a hybrid approach using a fuzzy evaluation method and evolutionary algorithms. It treats the online regulation problem as an optimization one and provides the regulator with evaluated and classified effective decisions by taking into account his/her preferences.     

An interactive sorting method for additive utility functions

In this paper, we consider the problem of placing alternatives that are defined by multiple criteria into preference-ordered categories. We consider a method that estimates an additive utility function and demonstrate that it may misclassify many alternatives even when substantial preference information is obtained from the decision maker (DM) to estimate the function. To resolve this difficulty, we develop an interactive approach. Our approach occasionally requires the DM to place some reference alternatives into categories during the solution process and uses this information to categorize other alternatives. The approach guarantees to place all alternatives correctly for a DM whose preferences are consistent with any additive utility function. We demonstrate that the approach works well using data derived from ranking global MBA programs as well as on several randomly generated problems.     

Fuzzy modeling for intelligent decision making under uncertainty

We consider here the problem of decision making under uncertainty. We suggest an approach for the construction of decision functions which allow for the inclusion of probabilistic information as well as for the inclusion of information about the decision maker's attitude and preferences. Use is made of the fuzzy modeling technology to construct these functions from specifications provided by the decision maker.     

Entropy methods for adaptive utility elicitation

This paper presents an optimal question-selection algorithm to elicit von Neumann and Morgenstern utility values for a set of ordered prospects of a decision situation. The approach uses information theory and entropy-coding principles to select the minimum expected number of questions needed for utility elicitation. At each stage of the questionnaire, we use the question that will provide the largest reduction in the entropy of the joint distribution of the utility values. The algorithm uses questions that require binary responses, which are easier to provide than numeric values, and uses an adaptive question-selection scheme where each new question depends on the previous response obtained from the decision maker. We present a geometric interpretation for utility elicitation and work through a full example to illustrate the approach.     

A Case-Based Micro Interactive Genetic Algorithm (CBMIGA) for interactive learning and search: Methodology and application to groundwater monitoring design

Interactive optimization algorithms use real–time interaction to include decision maker preferences based on the subjective quality of evolving solutions. In water resources management problems where numerous qualitative criteria exist, use of such interactive optimization methods can facilitate in the search for comprehensive and meaningful solutions for the decision maker. The decision makers using such a system are, however, likely to go through their own learning process as they view new solutions and gain knowledge about the design space. This leads to temporal changes (nonstationarity) in their preferences that can impair the performance of interactive optimization algorithms. This paper proposes a new interactive optimization algorithm – Case-Based Micro Interactive Genetic Algorithm – that uses a case-based memory and case-based reasoning to manage the effects of nonstationarity in decision maker’s preferences within the search process without impairing the performance of the search algorithm. This paper focuses on exploring the advantages of such an approach within the domain of groundwater monitoring design, though it is applicable to many other problems. The methodology is tested under non-stationary preference conditions using simulated and real human decision makers, and it is also compared with a non-interactive genetic algorithm and a previous version of the interactive genetic algorithm.     

Approaches to Pythagorean Fuzzy Stochastic Multi‐criteria Decision Making Based on Prospect Theory and Regret Theory with New Distance Measure and Score Function

In this paper, we initiate a new axiomatic definition of Pythagorean fuzzy distance measure, which is expressed by Pythagorean fuzzy number that will reduce the information loss and remain more original information. Then, the objective weights of various criteria are determined via grey system theory. Combining objective weights with subjective weights, we present the combined weights, which can reflect both the subjective considerations of the decision maker and the objective information. Meanwhile, a novel score function is proposed. Later, we present two algorithms to solve stochastic multicriteria decision making problem, which takes prospect preference and regret aversion of decision makers into consideration in the decision process. Finally, the effectiveness and feasibility of approach is demonstrated by a numerical example.     

Learning a decision maker's utility function from (possibly) inconsistent behavior

When modeling a decision problem using the influence diagram framework, the quantitative part rests on two principal components: probabilities for representing the decision maker's uncertainty about the domain and utilities for representing preferences. Over the last decade, several methods have been developed for learning the probabilities from a database. However, methods for learning the utilities have only received limited attention in the computer science community.

A promising approach for learning a decision maker's utility function is to take outset in the decision maker's observed behavioral patterns, and then find a utility function which (together with a domain model) can explain this behavior. That is, it is assumed that decision maker's preferences are reflected in the behavior. Standard learning algorithms also assume that the decision maker is behavioral consistent, i.e., given a model of the decision problem, there exists a utility function which can account for all the observed behavior. Unfortunately, this assumption is rarely valid in real-world decision problems, and in these situations existing learning methods may only identify a trivial utility function. In this paper we relax this consistency assumption, and propose two algorithms for learning a decision maker's utility function from possibly inconsistent behavior; inconsistent behavior is interpreted as random deviations from an underlying (true) utility function. The main difference between the two algorithms is that the first facilitates a form of batch learning whereas the second focuses on adaptation and is particularly well-suited for scenarios where the DM's preferences change over time. Empirical results demonstrate the tractability of the algorithms, and they also show that the algorithms converge toward the true utility function for even very small sets of observations.     

A robust extraction algorithm for biomedical signals from noisy mixtures

Blind source extraction (BSE) is widely used to solve signal mixture problems where there are only a few desired signals. To improve signal extraction performance and expand its application, we develop an adaptive BSE algorithm with an additive noise model. We first present an improved normalized kurtosis as an objective function, which caters for the effect of noise. By combining the objective function and Lagrange multiplier method, we further propose a robust algorithm that can extract the desired signal as the first output signal. Simulations on both synthetic and real biomedical signals demonstrate that such combination improves the extraction performance and has better robustness to the estimation error of normalized kurtosis value in the presence of noise.     

Convex cone-based partial order for multiple criteria alternatives

In this paper, we consider the problem of finding a preference-based strict partial order for a finite set of multiple criteria alternatives. We develop an approach based on information provided by the decision maker in the form of pairwise comparisons. We assume that the decision maker's value function is not explicitly known, but it has a quasi-concave form. Based on this assumption, we construct convex cones providing additional preference information to partially order the set of alternatives. We also extend the information obtained from the quasi-concavity of the value function to derive heuristic information that enriches the strict partial order. This approach can as such be used to partially rank multiple criteria alternatives and as a supplementary method to incorporate preference information in, e.g. Data Envelopment Analysis and Evolutionary Multi-Objective Optimization.     

A novel approach for combination of individual and group decisions based on fuzzy best-worst method

The process of decision-making in an enterprise may either keep the business on track or derail it. Thus, a senior decision maker often use a group of experts as the supportive team to ensure appropriate decisions. The experts often have different expertise level regarding their knowledge, talent, proficiency, and experience. In this study, we first extend the best-worst method based on the linguistic preferences of decision-makers about importance of attributes. These preferences are converted into triangular fuzzy numbers to be utilized in the linear programming model. That is, in contrast with the original best-worst method in which the preferences towards the attributes are crisp, fuzzy preferences are considered in the proposed method to reflect the imprecise comments of experts. Second, we propose a novel group decision making approach based on the fuzzy best-worst method to combine the opinion of senior decision-maker and the opinions of the experts. Indeed, our model helps the senior decision-maker to make a significant trade-off between democratic and autocratic decision-making styles. From sensitivity analyses on two numerical examples, we show that, when there is conflict between senior decision-maker and group of decision-makers, the consistency of group decision-making (democracy) will increase as it tends to individual decision-making (autocracy).     

基于模糊逻辑NSGA-Ⅲ的开关磁阻发电机多目标优化算法

低资源型的汉越神经机器翻译中,数据稀疏问题是影响翻译性能的主要原因,目前缓解该问题的途径之一是通过语料扩充方法生成伪平行数据,并用于机器翻译模型的训练,伪平行数据生成方法主要有基于词的替换、单语数据回译和枢轴翻译3种.目前的研究集中于3种方法的单独使用,缺少方法间融合利用方面的研究工作,针对此问题,提出了融入双语词典的正反向枢轴方法,利用英语作为枢轴语言,在汉到英到越正向枢轴的基础上,融入利用稀有词构建的汉-英和英-越双语词典,将汉语单语数据通过模型翻译成英语数据,再利用英-越模型将其翻译成越南语数据,其次进行越到英到汉反向枢轴翻译将越南语单语数据翻译为汉语,以此在2个方向上生成汉越伪平行数据,并利用语言模型对生成的伪平行数据进行筛选.汉-越翻译任务上的实验结果表明,提出的融入双语词典的正反向枢轴方法,能够产生更优的伪平行语料,进而显著提升汉越神经机器翻译任务的性能.     

Modified augmented Lagrangian coordination and alternating direction method of multipliers with parallelization in non-hierarchical analytical target cascading

Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian with Alternating Direction method of multipliers (AL-AD), one of efficient ATC coordination methods, has been widely used in both hierarchical and non-hierarchical ATC and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of distributed coordination which consists of several non-hierarchical subproblems is that it can solve subsystem problems in parallel and thus reduce computational time. Therefore, previous studies have proposed an augmented Lagrangian coordination strategy for parallelization by eliminating interactions among subproblems. The parallelization is achieved by introducing a master problem and support variables or by approximating a quadratic penalty function to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study shows that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier. Therefore, the Lagrange multiplier may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the augmented Lagrangian coordination approach. To solve this problem, we propose a modified AL-AD with parallelization in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the Lagrange multiplier and also maintains penalty weight at an appropriate level not to cause oscillation. Without approximation or introduction of an artificial master problem, the modified AL-AD with parallelization can achieve similar accuracy and convergence with much less computational cost compared with conventional AL-AD with sequential solving.