Building a binary outranking relation in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider multicriteria decision problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be uncertain and/or imprecise and is provided by one or several experts. We model this evaluation as a basic belief assignment (BBA). In order to compare the different pairs of alternatives according to each criterion, the concept of first belief dominance is proposed. Additionally, criteria weights are also expressed by means of a BBA. A model inspired by ELECTRE I is developed and illustrated by a pedagogical example.     

A ranking model in uncertain, imprecise and multi-experts contexts: The application of evidence theory

We consider ranking problems where the actions are evaluated on a set of ordinal criteria. The evaluation of each alternative with respect to each criterion may be imperfect and is provided by one or several experts. We model each imperfect evaluation as a basic belief assignment (BBA). In order to rank the BBAs characterizing the performances of the actions according to each criterion, a new concept called RBBD and based on the comparison of these BBAs to ideal or nadir BBAs is proposed. This is performed using belief distances that measure the dissimilarity of each BBA to the ideal or nadir BBAs. A model inspired by Xu et al.’s method is also proposed and illustrated by a pedagogical example.     

Concordant outranking with multiple criteria of ordinal significance

In this paper we address the problem of aggregating outranking statements from multiple preference criteria of ordinal significance. The concept of ordinal concordance of a global outranking situation is defined and an operational test for its presence is developed. Finally, we propose a new kind of robustness analysis for global outranking statements integrating classical dominance, ordinal and classical majority concordance in a same ordinal valued logical framework.Received: March 2004, Revised: October 2004, MSC classification: 90B50, 06A06, 03C80     

Finite approximations of data-based decision problems under imprecise probabilities

In decision theory under imprecise probabilities, discretizations are a crucial topic because many applications involve infinite sets whereas most procedures in the theory of imprecise probabilities can only be calculated for finite sets so far. The present paper develops a method for discretizing sample spaces in data-based decision theory under imprecise probabilities. The proposed method turns an original decision problem into a discretized decision problem. It is shown that any solution of the discretized decision problem approximately solves the original problem.In doing so, it is pointed out that the commonly used method of natural extension can be most instable. A way to avoid this instability is presented which is sufficient for the purpose of the paper.     

Rough set approach to multiple criteria classification with imprecise evaluations and assignments

Dominance-based Rough Set Approach (DRSA) has been introduced to deal with multiple criteria classification (also called multiple criteria sorting, or ordinal classification with monotonicity constraints), where assignments of objects may be inconsistent with respect to dominance principle. In this paper, we consider an extension of DRSA to the context of imprecise evaluations of objects on condition criteria and imprecise assignments of objects to decision classes. The imprecisions are given in the form of intervals of possible values. In order to solve the problem, we reformulate the dominance principle and introduce second-order rough approximations. The presented methodology preserves well-known properties of rough approximations, such as rough inclusion, complementarity, identity of boundaries and precisiation. Moreover, the meaning of the precisiation property is extended to the considered case. The paper presents also a way to reduce decision tables and to induce decision rules from rough approximations.     

An uncertainty interchange format with imprecise probabilities

This paper addresses the problem of exchanging uncertainty assessments in multi-agent systems. Since it is assumed that each agent might completely ignore the internal representation of its partners, a common interchange format is needed. We analyze the case of an interchange format defined by means of imprecise probabilities, pointing out the reasons of this choice. A core problem with the interchange format concerns transformations from imprecise probabilities into other formalisms (in particular, precise probabilities, possibilities, belief functions). We discuss this so far little investigated question, analyzing how previous proposals, mostly regarding special instances of imprecise probabilities, would fit into this problem. We then propose some general transformation procedures, which take also account of the fact that information can be partial, i.e. may concern an arbitrary (finite) set of events.     

A multicriteria perspective on reverse auctions

This is a summary of the main contributions of the author’s PhD thesis. The thesis is written in English. It was supervised by Philippe Vincke and was defended on 20 December 2005 at the Université Libre de Bruxelles. A copy is available from the author upon request. The aim of this work is to investigate the use of a partial relations for bids’ comparisons in the context of multi-attribute auctions. At first a theoretical framework is developed. Then a specific model referred to as the butterfly model is presented. Finally the concept of bidding niches is formalized and related to the aforementioned model.      

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.     

Multicriteria modelling for a repair contract problem based on utility and the ELECTRE I method

Each alternative for a repair contract implies a specific responsetime and related cost. The response time is associated withthe commitment of repair time, based on the contract. A decisionmaker chooses the best alternative taking into account the systemperformance and the cost of the contract. This problem has beenanalysed through a multicriteria decision model. This decisionmodel supports decision makers in the determination of the bestcombination of contracting conditions. The decision model proposedallows the decision maker to quantify the consequences of anaction taking into account two basic criteria: the cost of thecontract and the system performance. Two different decisionmodels have been built to support decision makers. These decisionmodels are based on different multicriteria approaches. Thefirst, reported in a previous paper, is based on the multiattributeutility theory (MAUT). The model presented in this paper isbased on the ELECTRE I method combined with utility functions.The paper presents the main theoretical aspects related to bothapproaches and practical implications related to model building.A numerical application is presented in order to illustratethe use of the decision model.     

A fuzzy genetic algorithm with varying population size to solve an inventory model with credit-linked promotional demand in an imprecise planning horizon

A genetic algorithm (GA) with varying population size is developed where crossover probability is a function of parents’ age-type (young, middle-aged, old, etc.) and is obtained using a fuzzy rule base and possibility theory. It is an improved GA where a subset of better children is included with the parent population for next generation and size of this subset is a percentage of the size of its parent set. This GA is used to make managerial decision for an inventory model of a newly launched product. It is assumed that lifetime of the product is finite and imprecise (fuzzy) in nature. Here wholesaler/producer offers a delay period of payment to its retailers to capture the market. Due to this facility retailer also offers a fixed credit-period to its customers for some cycles to boost the demand. During these cycles demand of the item increases with time at a decreasing rate depending upon the duration of customers’ credit-period. Models are formulated for both the crisp and fuzzy inventory parameters to maximize the present value of total possible profit from the whole planning horizon under inflation and time value of money. Fuzzy models are transferred to deterministic ones following possibility/necessity measure on fuzzy goal and necessity measure on imprecise constraints. Finally optimal decision is made using above mentioned GA. Performance of the proposed GA on the model with respect to some other GAs are compared.     

A minimum distance estimator in an imprecise probability model - Computational aspects and applications

The article considers estimating a parameter θ in an imprecise probability model which consists of coherent upper previsions . After the definition of a minimum distance estimator in this setup and a summarization of its main properties, the focus lies on applications. It is shown that approximate minimum distances on the discretized sample space can be calculated by linear programming. After a discussion of some computational aspects, the estimator is applied in a simulation study consisting of two different models. Finally, the estimator is applied on a real data set in a linear regression model.     

Multicriteria Modelling of Repair Contract Based on Utility and ELECTRE I Method with Dependability and Service Quality Criteria

A DM faces a choice among several alternatives of repair contract for a system. Each alternative of a repair contract implies specific results regarding the following characteristics or criteria: response time, quality service, dependability and related cost. This problem has been analysed through a multicriteria decision model. The model is based on the ELECTRE method combined with utility functions. Main theoretical aspects and practical implications are presented, including a numerical application.     

A lexicographical compromise method for multiple criteria group decision problems with imprecise information

This paper addresses multiple criteria group decision making problems where each group member offers imprecise information on his/her preferences about the criteria. In particular we study the inclusion of this partial information in the decision problem when the individuals’ preferences do not provide a vector of common criteria weights and a compromise preference vector of weights has to be determined as part of the decision process in order to evaluate a finite set of alternatives. We present a method where the compromise is defined by the lexicographical minimization of the maximum disagreement between the value assigned to the alternatives by the group members and the evaluation induced by the compromise weights.     

A model of far-sighted electoral competition

A model is described in which candidates adopt positions in sequences of election contests against opponents randomly drawn from a large population. Symmetric steady-state equilibria of the model require rational selection of positions by all candidates against the aggregated behavior of the population, taking into account constraints which an individual's current selections impose on his future selections.     

Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral

Within the multicriteria aggregation–disaggregation framework, ordinal regression aims at inducing the parameters of a decision model, for example those of a utility function, which have to represent some holistic preference comparisons of a Decision Maker (DM). Usually, among the many utility functions representing the DM’s preference information, only one utility function is selected. Since such a choice is arbitrary to some extent, recently robust ordinal regression has been proposed with the purpose of taking into account all the sets of parameters compatible with the DM’s preference information. Until now, robust ordinal regression has been implemented to additive utility functions under the assumption of criteria independence. In this paper we propose a non-additive robust ordinal regression on a set of alternatives A, whose utility is evaluated in terms of the Choquet integral which permits to represent the interaction among criteria, modelled by the fuzzy measures, parameterizing our approach.     

A study of high-level managerial decision processes, with implications for MCDM research

We study six real-world major strategic decisions and discuss the role that analytic Multiple Criteria Decision Making (MCDM) models could play in helping decision makers structure and solve such problems. We have interviewed successful and well-educated managers who had access to quantitative decision models, but did not use them as part of their decision process. Our approach is a clinical one that takes a close look at the decision processes. We believe that the normative MCDM framework is oversimplified and does not always fit well with complex, real-world organizational decision processes. This may be one reason why decision tools are not used more widely for solving high-level decision problems. We believe that it would be worthwhile to revise some of the MCDM mainstream postulates and practices to make existing models and tools more suitable for practical purposes. The MCDM mainstream research has until today focused on the choice among alternatives. One should realize that MCDM models could also be used in creating alternatives, in assessing the importance of criteria, in providing the decision makers with “post-commitment support”, and as part of a devil's advocate approach.     

Sensitivity analysis for choice problems with partial preference relations

In this paper we consider choice problems under the assumption that the preferences of the decision maker are expressed in the form of a parametric partial weak order without assuming the existence of any value function. We investigate both the sensitivity (stability) of each non-dominated solution with respect to the changes of parameters of this order, and the sensitivity of the set of non-dominated solutions as a whole to similar changes. We show that this type of sensitivity analysis can be performed by employing techniques of linear programming.     

A stochastic decision model with vector-valued reward

In this note we consider a non-stationary stochastic decision model with vector-valued reward. Based on Pareto-optimality we define the maximal total reward as a set of vector valued total rewards, which have not a successor with respect to the underlying partially order relation. The principle of optimality is derived. Using the well-known von-Neumann-Morgenstern-property we formulate a Bellman-equation, which consists in a system of iterative set relations.