An axiomatic approach to noncompensatory sorting methods in MCDM,I: The case of two categories

In the literature on MCDM, many methods have been proposed in order to sort alternatives evaluated on several attributes into ordered categories. Most of them were proposed on an ad hoc basis. The purpose of this paper is to contribute to a recent trend of research aiming at giving these methods sound theoretical foundations. Using tools from conjoint measurement, we provide an axiomatic analysis of the partitions of alternatives into two categories that can be obtained using what we call “noncompensatory sorting models”. These models have strong links with the pessimistic version of ELECTRE TRI. Our analysis allows to pinpoint what appears to be the main distinctive features of ELECTRE TRI when compared to other sorting methods. It also gives hints on the various methods that have been proposed to assess the parameters of ELECTRE TRI on the basis of assignment examples.     

Inferring an ELECTRE TRI Model from Assignment Examples

Given a finite set of alternatives, the sorting problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of parameters (weights, thresholds, category limits,...) in order to construct the Decision Maker's (DM) preference model. A direct elicitation of these parameters being rather difficult, we proceed to solve this problem in a way that requires from the DM much less cognitive effort. We elicit these parameters indirectly using holistic information given by the DM through assignment examples. We propose an interactive approach that infers the parameters of an ELECTRE TRI model from assignment examples. The determination of an ELECTRE TRI model that best restitutes the assignment examples is formulated through an optimization problem. The interactive aspect of this approach lies in the possibility given to the DM to revise his/her assignment examples and/or to give additional information before the optimization phase restarts.     

Using ELECTRE TRI outranking method to sort MOMILP nondominated solutions

Several interactive methods exist to identify nondominated solutions in a Multiple Objective Mixed Integer Linear Program. But what if the Decision Maker is also interested in sorting those solutions (assigning them to pre-established ordinal categories)? We propose an interactive “branch-and-bound like” technique to progressively build the nondominated set, combined with ELECTRE TRI method (Pessimistic procedure) to sort identified nondominated solutions. A disaggregation approach is considered in order to avoid direct definition of all ELECTRE TRI preference parameters. Weight-importance coefficients are inferred and category reference profiles are determined based on assignment examples provided by the Decision Maker. A computation tool was developed with a twofold purpose: support the Decision Maker involved in a decision process and provide a test bed for research purposes.     

An axiomatic approach to noncompensatory sorting methods in MCDM,II: More than two categories

A number of recent papers have investigated the foundations of methods allowing to sort multi-attributed alternatives between several ordered categories. This paper has a similar objective. Our analysis uses a general conjoint measurement framework, encompassing most sorting models used in MDCM, that was proposed in the literature. Within this framework, we provide an axiomatic analysis of what we call noncompensatory sorting models, with or without veto effects. These noncompensatory sorting models contain the pessimistic version of ELECTRE TRI as a particular case. Our analysis can be seen as an attempt to give a firm axiomatic basis to ELECTRE TRI, while emphasizing its specific feature, i.e., the rather poor information that this model uses on each attribute.     

Using assignment examples to infer weights for ELECTRE TRI method: Some experimental results

Given a finite set of alternatives A, the sorting (or assignment) problem consists in the assignment of each alternative to one of the pre-defined categories. In this paper, we are interested in multiple criteria sorting problems and, more precisely, in the existing method ELECTRE TRI. This method requires the elicitation of preferential parameters (weights, thresholds, category limits,…) in order to construct a preference model which the decision maker (DM) accepts as a working hypothesis in the decision aid study. A direct elicitation of these parameters requiring a high cognitive effort from the DM (V. Mosseau, R. Slowinski, Journal of Global Optimization 12 (2) (1998) 174), proposed an interactive aggregation–disaggregation approach that infers ELECTRE TRI parameters indirectly from holistic information, i.e., assignment examples. In this approach, the determination of ELECTRE TRI parameters that best restore the assignment examples is formulated through a nonlinear optimization program.In this paper, we consider the subproblem of the determination of the weights only (the thresholds and category limits being fixed). This subproblem leads to solve a linear program (rather than nonlinear in the global inference model). Numerical experiments were conducted so as to check the behaviour of this disaggregation tool. Results showed that this tool is able to infer weights that restores in a stable way the assignment examples and that it is able to identify “inconsistencies” in the assignment examples.     

Optimal Value and Post Optimal Solution in a Transportation Problem

In this work, we analyse the transportation problem of a real-life situation by obtaining the optimal feasible solutions, thus carrying out the sensitivity analysis of the problem. The work utilises the data obtained from the Asejire and Ikeja plants of Coca-Cola company, aiming to aid decision-making regarding the best possible options to satisfy customers at the barest minimum cost of transportation. Rerunning the optimization of a problem is an expensive scheme for gathering and obtaining enough data required for a problem. Thus, to minimize the transportation cost, the sensitivity analysis of parameters is a good tool to determine the behaviour of some input parameters where the values of these parameters are varied arbitrarily such that optimal results are verified. Maple 18 Software is used to solve the problem and the result obtained is compared with the values evaluated from northwest corner method, least cost method and Vogel''s approximation method. The study critically shows how a little change in a unit or more of any model parameter affects the expected results.     

An algorithm for ordinal sorting based on ELECTRE with categories defined by examples

This work proposes a Progressive Assisted Sorting Algorithm (PASA) based on a multicriteria evaluation ELECTRE-type method. The purpose of the PASA is to aid a decision maker to progressively sort a set of alternatives into a set of categories, which we considered are ordered (ordinal sorting), following a consistency principle. We consider the principle that if an alternative outranks (is as good as) a second one, then it must belong to the same category or to a better category. The set of alternatives already sorted by the decision maker will implicitly define the categories, and will constrain the range of categories where other alternatives may be sorted. We show how the same idea may be used in an aggregation/disaggregation approach, considering some parameters of ELECTRE are not fixed a priori, but are constrained only by the examples provided. In this context, we establish a “convex-shape property” stating that the range of possible categories for an alternative is always an interval of categories. A discussion contrasting this approach with ELECTRE TRI is included in the conclusions.     

A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment

The purpose of this paper is to design a new extension of the ELECTRE, known as the elimination and choice translating reality method, for multi-criteria group decision-making problems based on intuitionistic fuzzy sets. This method is widely utilized when a set of alternatives should be identified and evaluated with respect to a set of conflicting criteria by reflecting decision makers’ (DMs’) preferences. However, handling the exact data and numerical measure is difficult to be precisely focused because the DMs’ judgments are often vague in real-life decision problems and applications. A more realistic and practical approach can be to use linguistic variables expressed in intuitionistic fuzzy numbers instead of numerical data to model DMs’ judgments and to describe the inputs in the ELECTRE method. The proposed intuitionsitic fuzzy ELECTRE utilizes the truth-membership function and non-truth-membership function to indicate the degrees of satisfiability and non-satisfiability of each alternative with respect to each criterion and the relative importance of each criterion, respectively. Then, a new discordance intuitionistic index is introduced, which is extended from the concept of the fuzzy distance measure. Outranking relations are defined by pairwise comparisons and a decision graph is depicted to determine which alternative is preferable, incomparable or indifferent in the intuitionistic fuzzy environment. Finally, a comprehensive sensitivity analysis is employed to further study regarding the impact of threshold values on the final evaluation, and a comparative analysis is demonstrated with an application example in flexible manufacturing systems between the proposed ELECTRE method and the existing intuitionistic fuzzy technique for order preference by similarity to ideal solution (IF-TOPSIS) method.     

Stability and Hopf bifurcation analysis on a four-neuron BAM neural network with distributed delays

In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained.     

An evolutionary approach to construction of outranking models for multicriteria classification: The case of the ELECTRE TRI method

Outranking methods constitute an important class of multicriteria classification models. Often, however, their implementation is cumbersome, due to the large number of parameters that the decision maker must specify. Past studies tried to address this issue using linear and nonlinear programming, to elicit the necessary preferential information from assignment examples. In this study, an evolutionary approach, based on the differential evolution algorithm, is proposed in the context of the ELECTRE TRI method. Computational results are given to test the effectiveness of the methodology and the quality of the obtained models.     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

Stability analysis of combustion waves for competitive exothermic reactions using Evans function

We consider travelling wave solutions of a reaction-diffusion system corresponding to an adiabatic two-step competitive exothermic reaction scheme. In such a scheme, a combustion process is assumed to be lumped into two different exothermic reactions. Although the rate constants of the reactions are distinct, both reactions occur simultaneously and feed on the same reactant. The travelling wave solutions are obtained via the shooting-relaxation method. The linear stability analysis is conducted using the Evans function technique and the compound matrix method. Further, threshold values of parameters corresponding to Hopf points are established. It is shown that the system exhibits pulsating behaviour when the parameter values are greater than the threshold values. The onset of instability is found for a broad range of parameter values. Two different numerical methods are then used to obtain solutions from the governing partial differential equations to validate the results.     

Sensitivity analysis for the ELECTRE multicriteria method

In this paper, we develop sensitivity ranges for parameters used in the ELECTRE I multicriteria decision method. Parameters studied include criteria weights as well as significance thresholds. For these parameters, intervals are determined in which parameter changes will not affect the set of non-outranked alternatives. The approach developed is illustrated by a numerical example and possible applications are discussed.

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Zusammenfassung In dieser Arbeit wird ein Ansatz zur Sensitivitätsanalyse für das Multikriteria-Entscheidungsverfahren ELECTRE I vorgestellt. Für die Gewichte der Kriterien sowie die im Verfahren verwendeten Signifikanzschwellen werden jeweils Intervalle konstruiert, innerhalb derer Parameteränderungen keine Änderungen im Ergebnis des Verfahrens bewirken. Die entwickelte Sensitivitätsanalyse wird anhand eines Beispiels illustriert und mögliche Anwendungen des Konzeptes werden aufgezeigt.

     

A data model for algorithmic multiple criteria decision analysis

Various software tools implementing multiple criteria decision analysis (MCDA) methods have appeared over the last decades. Although MCDA methods share common features, most of the implementing software have been developed independently from scratch. Majority of the tools have a proprietary storage format and exchanging data among software is cumbersome. Common data exchange standard would be useful for an analyst wanting to apply different methods on the same problem. The Decision Deck project has proposed to build components implementing MCDA methods in a reusable and interchangeable manner. A key element in this scheme is the XMCDA standard, a proposal that aims to standardize an XML encoding of common structures appearing in MCDA models, such as criteria and performance evaluations. Although XMCDA allows to present most data structures for MCDA models, it almost completely lacks data integrity checks. In this paper we present a new comprehensive data model for MCDA problems, implemented as an XML schema. The data model includes types that are sufficient to represent multi-attribute value/utility models, ELECTRE III/TRI models, and their stochastic extensions, and AHP. We also discuss use of the data model in algorithmic MCDA.     

A multicriteria model for risk sorting of natural gas pipelines based on ELECTRE TRI integrating Utility Theory

This paper proposes a multicriteria model for assessing risk in natural gas pipelines, and for classifying sections of pipeline into risk categories. The model integrates Utility Theory and the ELECTRE TRI method. It aims to help transmission and distribution companies, when engaged in risk management and decision-making, to consider the multiple dimensions of risk that may arise from pipeline accidents. Pipeline hazard scenarios are presented, and it is argued that the assessment of risk in natural gas pipelines should not be based solely on probabilities of human fatalities, but should involve a wider perspective that simultaneously takes into consideration the human, environmental and financial dimensions of impacts of pipeline accidents. Finally, in order to verify the effectiveness of the model set out, a numerical application based on a real case study is presented.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

逐步回归方法的拓广研究

本文在运用泛函分析的观点详细剖析逐步回归方法的基础上,提出了一种改进的计算过程,可使逐步回归方法得以广泛地用于静态、动态和分布参数等系统的模型辨识。     

Robust control design for a class of mismatched uncertain nonlinear systems

We consider the robust control design problem for a class of nonlinear uncertain systems. The uncertainty in the system may be due to parameter variations and/or nonlinearity. It may be (possibly fast) time-varying. The system does not satisfy the so-called matching condition. Under a state transformation, which is based on the possible bound of the uncertainty, a robust control scheme can be designed. The control renders the uncertain system practically stable. Furthermore, the uniform ultimate boundedness ball and uniform stability ball can be made arbitrarily small by suitable choice of design parameters.     

Robust control design for a class of mismatched uncertain nonlinear systems

We consider the robust control design problem for a class of nonlinear uncertain systems. The uncertainty in the system may be due to parameter variations and/or nonlinearity. It may be possibly fast, time-varying. The system does not satisfy the so-called matching condition. Under a state transformation, which is based on the possible bound of the uncertainty, a robust control scheme can be designed. The control renders the original uncertain system practically stable. Furthermore, the uniform ultimate boundedness ball and uniform stability ball of the original system can be made arbitrarily small by suitable choice of design parameters.     

Multicriterion analysis of a vegetation management problem using ELECTRE II

A brief review of the ELECTRE II technique based on a strong and a weak ordinal outranking relationship is given. This technique, which is applied to alternative vegetation management schemes, leads to a complete ranking of the alternatives by means of three thresholds for concordance conditions and two thresholds for discordance conditions. Specifically, six alternative schemes are evaluated with respect to seven criteria, leading to the determination of a preferred system. A sensitivity analysis indicates that the ELECTRE II ranking is fairly robust with respect to parameter changes for the conditions of the case study.