João C. N. Clímaco

An aggregation/disaggregation approach to obtain robust conclusions with ELECTRE TRI

ELECTRE TRI is a well-known method to assign a set of alternatives to a set of predefined categories, considering multiple criteria. Using this method requires setting many parameters, which is often a difficult task. We consider the case where the decision makers (DMs) in the decision process are unsure of which values should each parameter take, which may result from uncertain, imprecise or inaccurately determined information, as well as from lack of consensus among them. This paper discusses the synergy between two approaches developed independently to deal with this difficulty. The first approach infers the value of parameters from assignment examples provided by the DMs, as an elicitation aid. Each assignment example originates mathematical constraints that the parameter values should satisfy. The second approach considers a set of constraints on the parameter values reflecting the imprecise information that the DMs are able to provide. Then, it computes the best and worst categories for each alternative compatible with constraints, in order to present robust conclusions. Both approaches avoid asking for precise values for the parameters. Rather, they proceed to solve the problem in a way that requires from the DMs much less effort. By integrating these two approaches, this paper proposes a new interactive approach, where the insight obtained during robustness analyses guides the DMs during the elicitation phase.

A bicriterion shortest path algorithm

Abstract Among the network models, one of the more popular is the so called shortest path problem. This model is used whenever it is intended to minimize a linear function which represents a distance between a predetermined pair of nodes in a given network. Often a single objective function is not sufficient to completely characterize some real-world problems. For instance, in a road network two parameters - as cost and time - can be assigned to each arc. Clearly the fastest path may be too costly. Nevertheless the decision-maker must choose one solution, possibly not the best for both criteria. In this paper we present an algorithm for this problem. With this algorithm a special set of paths (the set of Pareto optimal paths) is determined. One objective for any Pareto optimal path can not be improved without worsening the other one.

Resolving inconsistencies among constraints on the parameters of an MCDA model

We consider a framework where decision makers (DMs) interactively define a multicriteria evaluation model by providing imprecise information (i.e., a linear system of constraints to the models parameters) and by analyzing the consequences of the information provided. DMs may introduce new constraints explicitly or implicitly (results that the model should yield). If a new constraint is incompatible with the previous ones, then the system becomes inconsistent and the DMs must choose between removing the new constraint or removing some of the older ones. We address the problem of identifying subsets of constraints which, when removed, lead to a consistent system. Identifying such subsets would indicate the reason for the inconsistent information given by DMs. There may exist several possibilities for the DMs to resolve the inconsistency. We present two algorithms to identify such possibilities, one using {0,1} mixed integer linear programming and the other one using linear programming. Both approaches are based on the knowledge that the system was consistent prior to introducing the last constraint. The output of these algorithms helps the DM to identify the conflicting pieces of information in a set of statements he/she asserted. The relevance of these algorithms for MCDA is illustrated by an application to an aggregation/disaggregation procedure for the Electre Tri method. � 2002 Elsevier Science B.V. All rights reserved.

Capacitated single allocation hub location problem-A bi-criteria approach

A different approach to the capacitated single allocation hub location problem is presented. Instead of using capacity constraints to limit the amount of flow that can be received by the hubs, we introduce a second objective function to the model (besides the traditional cost minimizing function), that tries to minimize the time to process the flow entering the hubs. Two bi-criteria single allocation hub location problems are presented: in a first model, total time is considered as the second criteria and, in a second model, the maximum service time for the hubs is minimized. To generate non-dominated solutions an interactive decision-aid approach developed for bi-criteria integer linear programming problems is used. Both bi-criteria models are tested on a set of instances, analyzing the corresponding non-dominated solutions set and studying the reasonableness of the hubs flow charge for these non-dominated solutions. The increased information provided by the non-dominated solutions of the bi-criteria model when compared to the unique solution given by the capacitated hub location model is highlighted.

ELECTRE TRI for Groups with Imprecise Information on Parameter Values

ELECTRE TRI is a well-known method to assign actions to predefined ordered categories, considering multiple criteria. Using this method requires setting many parameters, which is often a difficult task. We consider the case where a group of Decision Makers (DMs) is unsure of which values each parameter should take, which may result from insufficient, imprecise or contradictory information, as well as from lack of consensus among the group members. In a framework where DMs provide constraints bounding and interrelating the parameter values, rather than fixing precise figures, we discuss the problem of finding the best and worst category that each action may attain.

Additive aggregation with variable interdependent parameters: the VIP Analysis software

We consider the aggregation of multicriteria performances by means of an additive value function under imprecise information. The problem addressed here is the way an analysis may be conducted when the decision makers are not able to (or do not wish to) fix precise values for the importance parameters. These parameters can be seen as interdependent variables that may take several values subject to constraints. Firstly, we briefly classify some existing approaches to deal with this problem. We argue that they complement each other, each one having its merits and shortcomings. Then, we present a new decision support software—VIP analysis—which incorporates approaches belonging to different classes. It proposes a methodology of analysis based on the progressive reduction of the number of alternatives, introducing a concept of tolerance that lets the decision makers use some of the approaches in a more flexible manner.

An interactive bi-objective shortest path approach: searching for unsupported nondominated solutions

Abstract In many network routing problems several conflicting objectives must be considered. Even for the bi-objective shortest path problem, generating and presenting the whole set of nondominated solutions (paths) to a decision maker, in general, is not effective because the number of these paths can be very large. Interactive procedures are adequate to overcome these drawbacks. Current et al. [1] proposed an interactive approach based on a NISE-like procedure to search for nondominated supported solutions and using auxiliar constrained shortest path problems to carry out the search inside the duality gaps. In this paper we propose a new interactive approach to search for unsupported nondominated solutions (lying inside duality gaps) based on a k-shortest path procedure. Both approaches are compared. Scope and purpose Network routing problems are generally multidimensional in nature, and in many cases the explicit consideration of multiple objectives is adequate. Objectives related to cost, time, accessibility, environmental impact, reliability and risk are appropriated for selecting the most satisfactory (“best compromise”) route in many problems. In general there is no single optimal solution in a multiobjective problem but rather, a set of nondominated solutions from which the decision maker must select the most satisfactory. However, generating and presenting the whole set of nondominated paths to a decision maker, in general, is not effective because the number of these paths can be very large. Interactive procedures are adequate to overcome these drawbacks. This paper introduces an interactive procedure to assist the decision maker in identifying the “best compromise” solution for the bi-objective shortest path problem. The procedure incorporates an efficient k-shortest path algorithm to identify nondominated solutions lying inside duality gaps. Test problem results indicate that the procedure can be readily executed on a PC for large-scale instances of problems.

Internet packet routing: Application of a K-quickest path algorithm

This paper describes a study on the application of an algorithm to rank the K-quickest paths to the routing of data packets in Internet networks. For this purpose an experimental framework was developed by considering two types of random generated networks. To simulate values of the IP packet sizes, a truncated Pareto distribution was defined, having in mind to reflect a key feature of Internet traffic, namely its self-similar stochastic nature. Results concerning the average CPU times of the algorithm for the different sets of experiments will be presented and discussed.

Solving bicriteria 0-1 knapsack problems using a labeling algorithm

This paper examines the performances of a new labeling algorithm to find all the efficient paths (or non-dominated evaluation vectors) of the bicriteria 0-1 knapsack problem. To our knowledge this is the first time a bicriteria 0-1 knapsack is solved taking advantage of its previous conversion into a bicriteria shortest path problem over an acyclic network. Computational experiments and results are also presented regarding bicriteria instances of up to 900 items. The algorithm is very efficient for the hard bicriteria 0-1 knapsack instances considered in the paper.

An algorithm for ranking quickest simple paths

In this paper, an algorithm for ranking loopless paths in undirected networks, according to the transmission time, is presented. It is shown that the worst-case computational time complexity of the algorithm presented is O(Kr(m+n logn)), which is also the best-known complexity to solve this problem. The worst-case memory complexity is O(Kn), which improves the existing algorithms. Finally, comparative computational results, with other algorithms for the same problem, are reported.