In this paper, we aim at generalising the evidential reasoning (ER) rule to establish a new maximum likelihood evidential reasoning (MAKER) framework for probabilistic inference from inputs to outputs in a system space, with their relationships characterised by imperfect data. The MAKER framework consists of three models: system state model (SSM), evidence acquisition model (EAM) and evidential reasoning model (ERM). SSM is introduced to describe system output in the form of ordinary probability distribution on singleton states of the system space to model randomness only, or more generally basic probability distribution on singleton states and their subsets, referred to as states for short, to depict both randomness and ambiguity explicitly. EAM is established to acquire evidence from a data source as system input in the form of basic probability distribution on the evidential elements of the data source, with each evidential element pointing to a state in the system space. ERM is created to combine pieces of acquired evidence, with each represented in the form of basic probability distribution on all the states and the powerset of the system space to facilitate an augmented probabilistic inference process where the trustworthiness of evidence is explicitly modelled alongside its randomness and ambiguity. Within the MAKER framework, the trustworthiness of evidence is defined in terms of its reliability and expected weight to measure the total degree of its support for all states. Interdependence between pairs of evidence is also measured explicitly. A general conjunctive MAKER rule and algorithm are then established to infer system output from multiple inputs by combining multiple pieces of evidence that have weights and reliabilities and are dependent on each other in general. Several special MAKER rules and algorithms are deduced to facilitate inference in special situations where evidence is exclusive or independent of each other. Specific conditions are identified and proven where the MAKER rule reduces to the ER rule, Dempster's rule and Bayes’ rule. A bi-objective nonlinear pre-emptive minimax optimisation model is built to make use of observed data for optimal learning of evidence weights and reliabilities by maximising the predicted likelihood of the true state for each observation. Two numerical examples are analysed to demonstrate the three constituent models of the MAKER framework, the MAKER rules and algorithms, and the optimal learning model. A case study for human well-being analysis is provided where data from a panel survey are used to show the potential applications of the MAKER framework for probabilistic reasoning and decision making under different types of uncertainty.

In artificial intelligence, it is crucial for pattern recognition systems to process data with uncertain information, necessitating uncertainty reasoning approaches such as evidence theory. As an orderable extension of evidence theory, random permutation set (RPS) theory has received increasing attention. However, RPS theory lacks a suitable generation method for the element order of permutation mass function (PMF) and an efficient determination method for the fusion order of permutation orthogonal sum (POS). To solve these two issues, this paper proposes a reasoning model for RPS theory, called random permutation set reasoning (RPSR). RPSR consists of three techniques, including RPS generation method (RPSGM), RPSR rule of combination, and ordered probability transformation (OPT). Specifically, RPSGM can construct RPS based on Gaussian discriminant model and weight analysis; RPSR rule incorporates POS with reliability vector, which can combine RPS sources with reliability in fusion order; OPT is used to convert RPS into a probability distribution for the final decision. Besides, numerical examples are provided to illustrate the proposed RPSR. Moreover, the proposed RPSR is applied to classification problems. An RPSR-based classification algorithm (RPSRCA) and its hyperparameter tuning method are presented. The results demonstrate the efficiency and stability of RPSRCA compared to existing classifiers.

The no-wait flow-shop scheduling problem (NFSSP) with sequence-dependent setup times (SDSTs) and release times (RTs) is applicable in many areas, such as steel production, food processing, and chemical processing. Estimation of the distribution algorithm (EDA) has recently been recognized as a prominent metaheuristic methodology in the field of evolutionary computation due to its excellent performance of global exploration. In this article, an innovative matrix-cube-based (i.e., 3-D) EDA (MCEDA) is first proposed to minimize the total earliness and tardiness (TET) of the NFSSP with SDSTs and RTs. This problem is NP-hard in the strong sense. First, a 3-D matrix cube is devised to learn the valuable information from promising solutions or excellent individuals. Second, an EDA model or probabilistic model based on the matrix cube and a special sampling method is presented to perform effective exploration in solution space and find promising regions. Third, based on a series of newly defined subneighborhoods, a new local search with both a speed-up scanning method and one search strategy is developed to execute exploitation from promising regions. Fourth, a speed-up evaluation method based on the problem’s property is designed to reduce the computational complexity for calculating criterion and accelerate the search process. Owing to the reasonable hybridization of exploration and exploitation, MCEDA can perform very efficient search in solution space. Extensive test results on instances of such a just-in-time problem first show that MCEDA can achieve better solution than state-of-the-art algorithms in obviously less computation time. Additional experiments on instances of various NFSSPs further confirm the efficiency and robustness of MCEDA.

This article investigates how to make use of imperfect data gathered from different sources for inference and decision making. Based on Bayesian inference and the principle of likelihood, a likelihood analysis method is proposed for acquisition of evidence from imperfect data to enable likelihood inference within the framework of the evidential reasoning (ER). The nature of this inference process is underpinned by the new necessary and sufficient conditions that when a piece of evidence is acquired from a data source it should be represented as a normalized likelihood distribution to capture the essential evidential meanings of data. While the explanation of sufficiency of the conditions is straightforward based on the principle of likelihood, their necessity needs to be established by following the principle of Bayesian inference. It is also revealed that the inference process enabled by the ER rule under the new conditions constitutes a likelihood inference process, which becomes equivalent to Bayesian inference when there is no ambiguity in data and a prior distribution can be obtained as a piece of independent evidence. Two examples in decision analysis under uncertainty and a case study about fault diagnosis for railway track maintenance management are examined to demonstrate the steps of implementation and potential applications of the likelihood inference process.

We propose to employ evidential reasoning (ER) rule to construct a clinical decision support system (CDSS) to aid physicians to predict the probability of intensive care unit (ICU) admission and in-hospital death for trauma patients once they arrive at a hospital. A generalized Bayesian rule is used to mine evidence from historical data. Evidence is profiled using a format of belief distribution, where the belief degrees of different trauma outcomes are assigned with derived probabilities linked to the corresponding outcomes. Inputs to the CDSS are clinical data of a patient, and output from the system is predicted belief degree of severe trauma, including ICU admission and in-hospital death. The inner logic of the CDSS is that pieces of evidence that match the clinical data of a patient are identified from the evidence base first, and then the ER rule-based evidence aggregation mechanism is utilized to combine the matched evidences to arrive at a prediction. The reliability, weight, and interdependence of clinical evidence are taken into account. Moreover, an evidence weight training module is constructed. The ER rule-based prediction model has superior performance compared with logistic regression and artificial neural network models. An innovative and pragmatic ER rule-based CDSS for trauma outcome prediction is contributed by this article. In the era of big data, this CDSS helps predict patient outcomes based on historical data and helps physicians in emergency departments make proper trauma management decisions.

Evidential reasoning (ER) rule has been widely used in addressing uncertainty, ignorance, and vagueness information. To explore its performance measure (PM), the perturbation analysis (PA) for the ER rule (ER rule-PA) is conducted, with perturbation taken into consideration. This article aims to analyze the robustness and stability of the ER rule, serving as theoretical basis and technical support for applied research and applications. The combination of two pieces of independent evidence is discussed, and perturbation is added to one piece of evidence. To represent the expected utility of evidence combination under perturbation, perturbation utility is introduced. The novel concept of perturbation coefficient is proposed to characterize the PM of the ER rule (ER rule-PM). The properties of perturbation coefficient are explored to demonstrate the impact of perturbation. The maximum permissible error (MPE) of perturbation coefficient is defined to characterize the acceptability of perturbation. A numerical study is examined to illustrate the implementation process of ER rule-PA. Moreover, a case study of reliability evaluation of aerospace relay is conducted to show the potential applications of ER rule-PA, which makes the proposed method more practical.

Automobile insurance fraud detection has become critically important for reducing the costs of insurance companies. The majority of insurance companies use expert knowledge to detect fraud. Experience-based knowledge are interpretable and re-usable but the simplistic way that this knowledge is used in practice, often leads to some degree of misjudgment. This paper aims to establish a unique Evidential Reasoning (ER) rule that combines independent evidence from both experience based indicators and probabilities of fraud obtained from historical data. Each piece of evidence is weighted and then combined conjunctively with the weights optimised using a maximum likelihood evidential reasoning (MAKER) framework for data-driven inferential modelling. Based on a real-world insurance claim dataset, our experimental results reveal that the proposed approach preserves the interpretability and usability of expert detection system, and anticipates the changes in fraud practices by tracking the trend of the weights of experience-based indicators. Furthermore, the experimental results show that the proposed approach outperforms a number of widely used machine learning models, such as logistic regression and random forests.

Abstract Intuitionistic multiplicative sets use an asymmetric, unbalanced scale to express information from positive, negative, and indeterminate information. They have been found capable of comprehensively and objectively representing a person's intuitive understanding and hence have attracted much attention. Distance techniques are widely used to measure the degree to which arguments deviate from one another. Several fuzzy set extensions have been developed, but little research has been conducted on measures of distance between intuitionistic multiplicative sets. In this paper, we start by presenting a variety of measures of the distance between intuitionistic multiplicative sets, including Hausdorff distance measures, weighted distance measures, ordered weighted distance measures, and continuous weighted distance measures. We then develop a distance‐based intuitionistic multiplicative‐technique for order preference by similarity to ideal solution method and a distance‐based intuitionistic multiplicative‐Vlsekriterijumska Optimizacija I Kompromisno Resenje method for handling multiple criteria decision‐making problems with intuitionistic multiplicative evaluation information. To demonstrate the practical application of these distance measures and the proposed methods, we provide a case study of hospital management of inpatient admission. The paper ends with comparative analyses of the two methods and some concluding remarks.

This paper proposes a new belief rule-based (BRB) expert system for fault diagnosis of marine diesel engines. The expert system is the first of its kind that consists of multiple concurrently activated BRB subsystems, in which each subsystem has its distinctive outputs and uses the evidential reasoning approach for inference. This novel modeling approach can be applied to identify fault modes that may co-exist. In essence, the group of BRB subsystems is used to model the nonlinear relationships between the fault features and the fault modes in marine diesel engines. The initial BRB expert system can be established by using expert experience and then optimized by using the data samples accumulated during the operation of marine diesel engines. Due to limitations in knowledge and data collected, ignorance is also considered in some BRB subsystems. The proposed BRB expert system is applied to abnormal wear detection for a kind of marine diesel engine. The performance of the BRB expert system is investigated in comparison with that of artificial neural network (ANN) models, support vector machine (SVM) models, and binary logistic regression model with fivefold cross-validation. The results show that the BRB expert system can be used for fault diagnosis of marine diesel engines in a probabilistic manner, which outperforms the ANN models, SVM models, and the binary logistic regression model in terms of accuracy and stability, and can effectively identify concurrent faults.

With the growing economic pressure, one lever for companies to be competitive is to efficiently monitor projects. To this goal, the different processes involved in the project must be carefully supervised and the project manager needs to be well-informed on their state and progress to take the best decisions thanks to accurate performances of indicators, process, and project. Another key factor of successful decision-making in engineering projects beyond taking into account information from the project manager on the project management (PM) is to consider information from the systems engineer on the product development. This paper proposes a process to monitor engineering projects relying on a set of project performance indicators that integrate the views of both project manager and systems engineer. This process includes three main activities: 1) defining project performance indicators related to the processes described by international guides and standards; 2) valuating and weighting these indicators by consulting project managers and systems engineers; and 3) constructing a hierarchical framework of indexes to support project monitoring. This proposal thus refers to the practices described in PM and systems engineering norms and improves them by involving project managers and systems engineers into decision-making, with the goal to take more coherent decisions.

The Evidential Reasoning (ER) approach is an inference method to analyse multi-criteria decision-making problems with uncertainties. However, due to the pressure of time, limit of knowledge and lack of data, such problems need to be assessed using both human judgments and limited data. In this case, it is suitable to provide a belief structure by means of linguistic terms, which we refer to as linguistic belief structure. Considering that different decision-makers may have different perceptions, the semantics of human judgments can also be different. In this paper, we introduce three linguistic scale functions to capture the semantics of linguistic belief degrees for constructing a complete linguistic information fusion method. We then extend the framework of the ER approach to multi-expert multi-criteria decision making. The method proposed in this study can fill the research gap of the ER approach for multi-expert multi-criteria decision-making problems with linguistic belief structures.

The demand is a fundamental variable in economic analysis that measures the needs of goods and services of the consumers. This paper presents a new approach for representing the demand by using aggregation systems that consider the attitudinal character of the consumers and their beliefs regarding the degree of importance of the different variables that may affect them. Several developments are introduced by using multiperson systems, different criterion, attributes, and states of nature. These developments are focused on the use of the weighted average, the ordered weighted average, and the ordered weighted averaging weighted average. Further aggregation systems are suggested by using the concept of the demand growth. The paper ends with an application of the new approach in a forecasting process that considers the attitude of the decision maker and its subjective beliefs.

Bibliometrics is a research field that analyzes the bibliographic material quantitatively. It provides efficient methodologies for classifying the information of a scientific discipline. This paper presents an overview of the most productive and influential authors and institutions in finance by using bibliometric indicators. The information is classified by using several global rankings that consider a wide range of indicators including number of papers, citations and the h-index. In general, the results are in accordance with the common knowledge and confirm the results obtained in previous studies providing updated information and more general representations. The USA is the most influential country in finance and the majority of influential authors and institutions are working there.

The supply represents the available number of products of a specific variable. This paper presents a new approach for studying the supply through the use of aggregation systems. Its main advantage is that it permits to represent the information in a more complete way. Thus, it is possible to develop forecasting methods that consider optimistic and pessimistic scenarios and those that are more expected to occur. A wide range of aggregation operators are used giving special focus on the ordered weighted average (OWA). Several generalizations that use it with the weighted average and the probability are presented including the OWA weighted average (OWAWA) and the probabilistic OWAWA (POWAWA) operator. A numerical example is also presented.

In this paper, a Genetic Algorithm (GA) search heuristic is proposed for validating the model-structure of Belief Rule-Based (BRB) methodologies. In order to ensure the balance between the model fit/ accuracy and the model complexity, the Akaike Information Criterion (AIC) is used in conjunction with the mentioned heuristic. The resulting framework is tested, using a model consisting of 3 inputs and one output, each of the 4 variables being allocated up to 5 referential values. The presented results illustrate the time-efficiency of the GA heuristic, as well as the penalty imposed by AIC on the number of parameters. The simplest model structure is indicated by AIC to be the optimal one. However, three additional model structures have been found to have AIC values which are moderately close to this optimum. An analysis of their coefficients of determination indicates a higher fit (than AIC optimum) on both testing sets and overall.

This paper proposes a new feature selection method using a mutual information-based criterion that measures the importance of a feature in a backward selection framework. It considers the dependency among many features and uses either one of two well-known probability density function estimation methods when computing the criterion. The proposed approach is compared with existing mutual information-based methods and another sophisticated filter method on many artificial and real-world problems. The numerical results show that the proposed method can effectively identify the important features in data sets having dependency among many features and is superior, in almost all cases, to the benchmark methods.

The evidential reasoning (ER) approach was developed to support multiple criteria decision analysis (MCDA). It is based on the Dampster's combination rule for criteria aggregation and belief function for treating ignorance. In the original ER approach, however, alternative ranking depends on the accurate estimation of a value function, which may be difficult in certain decision environments. In this paper, the link and difference between the ER algorithm and Dampster's combination rule are analysed first. A new alternative ranking method is then investigated as an integrated part of the enhanced ER approach.

In engineering practice, a system's behavior constantly changes over time. To predict the behavior of a complex engineering system, a model can be built and trained using historical data. This paper addresses the forecasting problems with a belief rule base (BRB) to trace and predict system performance in a more interpretable and transparent way. More precisely, it extends the BRB method to handle a system's behavior prediction, and a new prediction model based on BRB is presented, which can model and analyze prediction problems using not only numerical data but human judgmental information as well. The proposed forecasting model includes some unknown parameters that can be manually tuned and trained. To build an effective BRB forecasting model, a multiple-objective optimization model is provided to locally train the BRB prediction model by minimizing the mean square error (MSE). Finally, a practical case study is provided to illustrate the detailed implementation procedures and examine the feasibility of the proposed approach in engineering application. Furthermore, the comparative studies with other state-of-the-art prediction methods are carried out. It is shown that the proposed model is effective and can generate better prediction in terms of accuracy, as well as comprehensibility.

Tuning of the regularization parameter C is a well-known process in the implementation of a support vector machine (SVM) classifier. Such a tuning process uses an appropriate validation function whose value, evaluated over a validation set, has to be optimized for the determination of the optimal C. Unfortunately, most common validation functions are not smooth functions of C. This brief presents a method for obtaining the global optimal solution of these non-smooth validation functions. The method is guaranteed to find the global optimum and relies on the regularization solution path of SVM over a range of C values. When the solution path is available, the computation needed is minimal.

This paper presents a new wrapper-based feature selection method for support vector regression (SVR) using its probabilistic predictions. The method computes the importance of a feature by aggregating the difference, over the feature space, of the conditional density functions of the SVR prediction with and without the feature. As the exact computation of this importance measure is expensive, two approximations are proposed. The effectiveness of the measure using these approximations, in comparison to several other existing feature selection methods for SVR, is evaluated on both artificial and real-world problems. The result of the experiments show that the proposed method generally performs better than, or at least as well as, the existing methods, with notable advantage when the dataset is sparse.

Recently, a novel prediction model based on the evidential reasoning (ER) approach is developed to forecast reliability in engineering systems. In order to determine the parameters of the ER-based prediction model, some optimization models have been proposed to train the ER-based prediction model. However, these models are implemented in an offline fashion and thus it is very expensive to train and retrain them when new information is available. This correspondence paper is concerned with developing the recursive algorithms for updating the ER-based prediction model from the probability-based point of view. Using the recursive expectation maximization algorithm, two recursive algorithms are proposed for updating the parameters of the ER-based prediction model under judgmental and numerical outputs, respectively. As such, the proposed algorithms can be used to fine tune the ER-based prediction model online once new information becomes available. We verify the proposed method via a realistic example with missile reliability data.

In order to determine the parameters of belief-rule-base (BRB) accurately, several optimization methods have been proposed for training BRB, on the basis of a generic rule-base inference methodology using the evidential reasoning (RIMER) approach. These optimization methods are implemented offline, and such are not suitable for training BRB in a dynamic fashion. In this paper, two recursive algorithms are proposed to update BRB online that can simulate dynamic systems. The main feature of the proposed algorithms is that only partial input and output information is required, which can be incomplete or vague, numerical or judgmental, or mixed. If the internal structure of a BRB is initially decided using expert judgments, domain-specific knowledge and/or commonsense rules, the proposed algorithms can be used to fine-tune the initial BRB online, once input and output datasets become available. Using the proposed algorithms, there is no need to collect a complete set of data before a BRB can be trained, which is necessary if the BRB is used to simulate a dynamic system. A numerical example and a case study are reported to demonstrate the potential of the algorithms for online fault diagnosis.

This paper presents a novel wrapper-based feature selection method for Support Vector Regression (SVR) using its probabilistic predictions. The method computes the importance of a feature by aggregating the difference, over the feature space, of the conditional density functions of the SVR prediction with and without the feature. As the exact computation of this importance measure is expensive, two approximations are proposed. The effectiveness of the measure using these approximations, in comparison to several other existing feature selection methods for SVR, is evaluated on both artificial and real-world problems. The result of the experiment shows that the proposed method generally performs better, and at least as well as the existing methods, with notable advantage when the data set is sparse.

In a multiple-criteria decision analysis (MCDA) problem, qualitative information with subjective judgments of ambiguity is often provided by people, together with quantitative data that may also be imprecise or incomplete. There are several uncertainties that may be considered in an MCDA problem, such as fuzziness and ambiguity. The evidential reasoning (ER) approach is well suited for dealing with such MCDA problems and can generate comprehensive distributed assessments for different alternatives. Many researches in dealing with imprecise or uncertain belief structures have been conducted on the ER approach. In this paper, both triangular fuzzy weights of criteria and fuzzy utilities assigned to evaluation grades are introduced to the ER approach, which may be incurred in several circumstances such as group decision-making situation. The Hadamard multiplicative combination of judgment matrix is extended for the aggregation of triangular fuzzy judgment matrices, the result of which is applied as the fuzzy weights used in the fuzzy ER approach. The consistency of the aggregated triangular fuzzy judgment matrix is also proved. Several pairs of ER-based programming models are designed to generate the total fuzzy belief degrees and the overall expected fuzzy utilities for the comparison of alternatives. A numerical example is conducted to show the effectiveness of the proposed approach. © 2009 Wiley Periodicals, Inc.

Abstract: A critical issue in the clinical decision support system (CDSS) research area is how to represent and reason with both uncertain medical domain knowledge and clinical symptoms to arrive at accurate conclusions. Although a number of methods and tools have been developed in the past two decades for modelling clinical guidelines, few of those modelling methods have capabilities of handling the uncertainties that exist in almost every stage of a clinical decision‐making process. This paper describes how to apply a recently developed generic rule‐base inference methodology using the evidential reasoning approach (RIMER) to model clinical guidelines and the clinical inference process in a CDSS. In RIMER, a rule base is designed with belief degrees embedded in all possible consequents of a rule. Such a rule base is capable of capturing vagueness, incompleteness and non‐linear causal relationships, while traditional IF–THEN rules can be represented as a special case. Inference in such a rule base is implemented using the evidential reasoning approach which has the capability of handling different types and degrees of uncertainty in both medical domain knowledge and clinical symptoms. A case study demonstrates that employing RIMER in developing a guideline‐based CDSS is a valid novel approach.

Many multiple attribute decision analysis (MADA) problems are characterized by both quantitative and qualitative attributes with various types of uncertainties. Incompleteness (or ignorance) and vagueness (or fuzziness) are among the most common uncertainties in decision analysis. The evidential reasoning (ER) and the interval grade ER (IER) approaches have been developed in recent years to support the solution of MADA problems with interval uncertainties and local ignorance in decision analysis. In this paper, the ER approach is enhanced to deal with both interval uncertainty and fuzzy beliefs in assessing alternatives on an attribute. In this newly developed fuzzy IER (FIER) approach, local ignorance and grade fuzziness are modeled under the integrated framework of a distributed fuzzy belief structure, leading to a fuzzy belief decision matrix. A numerical example is provided to illustrate the detailed implementation process of the FIER approach and its validity and applicability.

This paper presents a new wrapper-based feature selection method for multilayer perceptron (MLP) neural networks. It uses a feature ranking criterion to measure the importance of a feature by computing the aggregate difference, over the feature space, of the probabilistic outputs of the MLP with and without the feature. Thus, a score of importance with respect to every feature can be provided using this criterion. Based on the numerical experiments on several artificial and real-world data sets, the proposed method performs, in general, better than several selected feature selection methods for MLP, particularly when the data set is sparse or has many redundant features. In addition, as a wrapper-based approach, the computational cost for the proposed method is modest.

In this paper, we propose a new interactive method for multiobjective programming (MOP) called the PROJECT method. Interactive methods in MOP are techniques that can help the decision maker (DM) to generate the most preferred solution from a set of efficient solutions. An interactive method should be capable of capturing the preferences of the DM in a pragmatic and comprehensive way. In certain decision situations, it may be easier and more reliable for DMs to follow an interactive process for providing local tradeoffs than other kinds of preferential information like aspiration levels, objective function classification, etc. The proposed PROJECT method belongs to the class of interactive local tradeoff methods. It is based on the projection of utility function gradients onto the tangent hyperplane of an efficient set and on a new local search procedure that inherits the advantages of the reference-point method to search for the best compromise solution within a local region. Most of the interactive methods based on local tradeoffs assume convexity conditions in a MOP problem, which is too restrictive in many real-life applications. The use of a reference-point procedure makes it possible to generate any efficient solutions, even the nonsupported solutions or efficient solutions located in the nonconvex part of the efficient frontier of a nonconvex MOP problem. The convergence of the proposed method is investigated. A nonlinear example is examined using the new method, as well as a case study on efficiency analysis with value judgements. The proposed PROJECT method is coded in Microsoft Visual C++ and incorporated into the software PROMOIN (Interactive MOP).

This paper presents a new wrapper-based feature selection method for multi-layer perceptrons (MLP) neural networks. It uses a feature ranking criterion to measure the importance of a feature by computing the aggregate difference, over the feature space, of the probabilistic outputs of the MLP with and without the feature. Thus, a score of importance with respect to every feature can be provided using this criterion. The proposed criterion has inexpensive evaluation. Based on the numerical experiment on several artificial and real-world data sets, the proposed method performs at least as well, if not better, than several existing feature selection methods for MLP.

A belief rule-base inference methodology using the evidential reasoning approach (RIMER) has been developed recently, where a new belief rule representation scheme is proposed to extend traditional IF-THEN rules. The belief rule expression matrix in RIMER provides a compact framework for representing expert knowledge. However, it is difficult to accurately determine the parameters of a belief rule base (BRB) entirely subjectively, particularly, for a large-scale BRB with hundreds or even thousands of rules. In addition, a change in rule weight or attribute weight may lead to changes in the performance of a BRB. As such, there is a need to develop a supporting mechanism that can be used to train, in a locally optimal way, a BRB that is initially built using expert knowledge. In this paper, several new optimization models for locally training a BRB are developed. The new models are either single- or multiple-objective nonlinear optimization problems. The main feature of these new models is that only partial input and output information is required, which can be either incomplete or vague, either numerical or judgmental, or mixed. The models can be used to fine tune a BRB whose internal structure is initially decided by experts' domain-specific knowledge or common sense judgments. As such, a wide range of knowledge representation schemes can be handled, thereby facilitating the construction of various types of BRB systems. Conclusions drawn from such a trained BRB with partially built-in expert knowledge can simulate real situations in a meaningful, consistent, and locally optimal way. A numerical study for a hierarchical rule base is examined to demonstrate how the new models can be implemented as well as their potential applications.

Before implementing a design of a large engineering system different design proposals are evaluated and ranked according to different criteria, such as, safety, cost and technical performance. The experts' knowledge about these criteria is usually vague and/or incomplete, and their nature may be quantitative or qualitative. Therefore the preference modelling for the criteria could imply the use of different types of information such as numerical and/or linguistic (non-homogeneous framework). However, in most of evaluation processes the experts are forced to provide their scores in the same expression domain and in the same scale. The aim of this paper is to propose an evaluation model based on a multi-criteria decision analysis that offers to the experts the possibility of expressing their knowledge in a non-homogeneous evaluation framework, such that the experts can provide their assessments within different domains and scales according to their knowledge and the nature of the criteria. To do so, we propose the use of the fuzzy logic and the fuzzy linguistic approach in order to manage the uncertainty related to the information provided by the experts.

In this paper, a generic rule-base inference methodology using the evidential reasoning (RIMER) approach is proposed. Existing knowledge-base structures are first examined, and knowledge representation schemes under uncertainty are then briefly analyzed. Based on this analysis, a new knowledge representation scheme in a rule base is proposed using a belief structure. In this scheme, a rule base is designed with belief degrees embedded in all possible consequents of a rule. Such a rule base is capable of capturing vagueness, incompleteness, and nonlinear causal relationships, while traditional if-then rules can be represented as a special case. Other knowledge representation parameters such as the weights of both attributes and rules are also investigated in the scheme. In an established rule base, an input to an antecedent attribute is transformed into a belief distribution. Subsequently, inference in such a rule base is implemented using the evidential reasoning (ER) approach. The scheme is further extended to inference in hierarchical rule bases. A numerical study is provided to illustrate the potential applications of the proposed methodology.

Before implementing a design of a large engineering system different design proposals are evaluated. The information used by experts to evaluate different options may be vague and/or incomplete. Although different probabilistic tools and techniques have been used to deal with these kinds of problems, it seems better to use the fuzzy linguistic approach to model vagueness and the Dempster-Shafter theory of evidence for modeling incompleteness and ignorance. In the evaluation of alternative designs, different criteria can be considered. In this article an evaluation process is developed in terms of Safety and Cost analysis. Both criteria involve uncertainty, vagueness, and ignorance due to their nature. Therefore, we propose an evaluation process defined in a linguistic framework where both criteria will be conducted in different utility spaces, i.e., in a multigranular linguistic domain. Once the evaluation framework has been defined, we present an evaluation process based on a Multi-Expert Multi-Criteria decision model that will be able to deal with multigranular linguistic information without loss of information in order to evaluate different design options for an engineering system in a precise manner. Accordingly, we propose the use of a multigranular linguistic model based on the Linguistic Hierarchies presented by Herrera and Martínez (“A model based on linguistic 2-tuples for dealing with multigranularity hierarchical linguistic contexts in multi-expert decision-making.” IEEE Trans Syst Man Cybern B 2001;31(2):227–234). © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1161–1194, 2005.

In this paper a generic Rule-base Inference Methodology using the Evidential Reasoning approach (RIMER) is proposed. A new knowledge representation scheme in a rule-base is proposed using a belief structure and fuzzy set theory. In this scheme, a rule-base is designed on the basis of the belief structure with belief degrees embedded in all possible consequents to capture vagueness, incompleteness and nonlinear causal relationships, whilst traditional IF-THEN rules can be represented as a special case. In an established rule-base, an input to an antecedent attribute is transformed into a belief distribution. Subsequently, inference in such a rule-base is implemented using the evidential reasoning approach. The scheme is further extended to inference in hierarchical rule bases. A numerical study is provided to illustrate the potential applications of the proposed methodology.

In multiobjective optimization, tradeoff analysis plays an important role in determining the best search direction to reach a most preferred solution. This paper presents a new explicit interactive tradeoff analysis method based on the identification of normal vectors on a noninferior frontier. The interactive process is implemented using a weighted minimax formulation by regulating the relative weights of objectives in a systematic manner. It is proved under a mild condition that a normal vector can be identified using the weights and Kuhn-Tucker (K-T) multipliers in the minimax formulation. Utility gradients can be estimated using local preference information such as marginal rates of substitution. The projection of a utility gradient onto a tangent plane of the noninferior frontier provides a descent direction of disutility and thereby a desirable tradeoff direction, along which tradeoff step sizes can be decided by the decision maker using an explicit tradeoff table. Necessary optimality conditions are established in terms of normal vectors and utility gradients, which can be used to guide the elicitation of local preferences and also to terminate an interactive process in a rigorous yet flexible way. This method is applicable to both linear and nonlinear (either convex or nonconvex) multiobjective optimization problems. Numerical examples are provided to illustrate the theoretical results of the paper and the implementation of the proposed interactive decision analysis process.

In multiple attribute decision analysis (MADA), one often needs to deal with both numerical data and qualitative information with uncertainty. It is essential to properly represent and use uncertain information to conduct rational decision analysis. Based on a multilevel evaluation framework, an evidential reasoning (ER) approach has been developed for supporting such decision analysis, the kernel of which is an ER algorithm developed on the basis of the framework and the evidence combination rule of the Dempster-Shafer (D-S) theory. The approach has been applied to engineering design selection, organizational self-assessment, safety and risk assessment, and supplier assessment. In this paper, the fundamental features of the ER approach are investigated. New schemes for weight normalization and basic probability assignments are proposed. The original ER approach is further developed to enhance the process of aggregating attributes with uncertainty. Utility intervals are proposed to describe the impact of ignorance on decision analysis. Several properties of the new ER approach are explored, which lay the theoretical foundation of the ER approach. A numerical example of a motorcycle evaluation problem is examined using the ER approach. Computation steps and analysis results are provided in order to demonstrate its implementation process.

In many decision situations, it is inevitable to deal with both quantitative and qualitative information under uncertainty. Evidence-based reasoning within a multiple criteria decision analysis framework provides an alternative way of handling such information systematically and consistently. In this paper, the evidential reasoning (ER) approach is introduced, which is based on a recursive ER algorithm that, in essence, constitutes a nonlinear information aggregation process. To facilitate the application of the ER approach and as an indispensable part of its development, the nonlinear features of the ER information aggregation process need to be thoroughly investigated and properly understood. This forms the theme of this paper where the nonlinear features are explored by examining typical reasoning patterns in aggregating harmonic, quasi-harmonic, and contradictory decision information. This analytical investigation provides insights into the recursive nature of the ER approach as well as valuable experience that could be useful to other researchers and practitioners interested in developing and applying operation research/artificial intelligence (OR/AI)-based approaches for decision analysis under uncertainty. The analytical study is complemented by the numerical studies of two application examples. The analysis of a quality assessment problem for motor engines is aimed to show the step-by-step process of implementing the ER approach and to illustrate its nonlinear features in a real-life decision situation. The study of a more complex assessment problem in ship design is intended to demonstrate the potential of the ER approach and its supporting software for dealing with general decision problems.

The paper presents a new genetic algorithm (GA)-based discrete dynamic programming (DDP) approach for generating static schedules in a flexible manufacturing system (FMS) environment. This GA-DDP approach adopts a sequence-dependent schedule generation strategy, where a GA is employed to generate feasible job sequences and a series of discrete dynamic programs are constructed to generate legal schedules for a given sequence of jobs. In formulating the GA, different performance criteria could be easily included. The developed DDF algorithm is capable of identifying locally optimized partial schedules and shares the computation efficiency of dynamic programming. The algorithm is designed In such a way that it does not suffer from the state explosion problem inherent in pure dynamic programming approaches in FMS scheduling. Numerical examples are reported to illustrate the approach.

A new evidential reasoning based approach is proposed that may be used to deal with uncertain decision knowledge in multiple-attribute decision making (MADM) problems with both quantitative and qualitative attributes. This approach is based on an evaluation analysis model and the evidence combination rule of the Dempster-Shafer theory. It is akin to a preference modeling approach, comprising an evidential reasoning framework for evaluation and quantification of qualitative attributes. Two operational algorithms have been developed within this approach for combining multiple uncertain subjective judgments. Based on this approach and a traditional MADM method, a decision making procedure is proposed to rank alternatives in MADM problems with uncertainty. A numerical example is discussed to demonstrate the implementation of the proposed approach. A multiple-attribute motor cycle evaluation problem is then presented to illustrate the hybrid decision making procedure.>

Based on an evidential reasoning framework, a general multilevel evaluation process is developed in this paper for dealing with a multiple attribute decision making (MADM) problem with both quantitative and qualitative attributes. In this new process, a qualitative attribute may be evaluated by uncertain subjective judgments through multiple levels of factors and each of the judgments may be assigned by single or multiple experts in any rational way within the evidential reasoning framework. The qualitative attributes can then be quantified by means of general evaluation analysis and evidential reasoning. A few evaluation analysis models and the corresponding evidential reasoning algorithms are explored for parallel combination and hierarchical propagation of factor evaluations. With all the qualitative attributes being quantified by this rational process, the MADM problem represented by an extended decision matrix is then transformed into an ordinary decision matrix, which can be dealt with using a traditional MADM method. This new general evaluation process and the hybrid decision making procedure are demonstrated using a multiple attribute motor cycle evaluation problem with uncertainty.>

The interactive step tradeoff method (ISTM) is composed of three basic steps. First, an efficient solution and the corresponding local tradeoff information are provided by the analyst. Then, the decision maker determines the preference direction and step size. Again the analyst looks for a new efficient solution according to the preference information; the new solution should dominate the previous one. In ISTM, the efficient solution and the local tradeoff information, the current values of objective functions, and the tradeoff rates between them are obtained by solving an auxiliary problem. The auxiliary problem is defined, and relationships between the optimal solutions of the auxiliary problem and the efficient solutions of the original problem are explored. The relationships between the Kuhn-Tucker multipliers (or simplex multipliers) of the auxiliary problems and the tradeoff rates are analyzed. The particular steps of the ISTM algorithm are given. An example is discussed to illustrate the use of the algorithm.>