Abstract Given a directed temporal graph, a start node , and objectives, the task in the single‐source multiobjective temporal shortest path problem (SSMTSPP) consists of computing the set of nondominated images of temporal ‐‐paths for each node as well as one corresponding efficient path for each of these images. This problem generalizes both the multiobjective shortest path problem in static graphs and the single‐objective temporal shortest path problem. In this article, we provide a general label setting algorithm for the SSMTSPP that can handle a large variety of different objectives. The only condition imposed on the objectives is a monotonicity property that generalizes the nonnegativity of the arc costs required for the well‐known label setting algorithm for solving the static single‐source shortest path problem in both the single objective and the multiobjective case. Our analysis of the presented algorithm shows that its worst‐case running time is polynomial in the sum of the input size of the problem instance and the number of nondominated images, which implies that it runs in polynomial time as long as the number of nondominated images is polynomial in the instance size (i.e., for all tractable versions of the problem). To complement this result, we provide a complete classification into tractable and intractable problems for all SSMTSPPs involving a large variety of objectives. In particular, using our general analysis, this provides a large range of specific SSMTSPPs for which our general label setting algorithm runs in polynomial time.

Optimizing over the efficient set of a discrete multi-objective problem is a challenging issue. The main reason is that, unlike when optimizing over the feasible set, the efficient set is implicitly characterized. Therefore, methods designed for this purpose iteratively generate efficient solutions by solving appropriate single-objective problems. However, the number of efficient solutions can be quite large and the problems to be solved can be difficult practically. Thus, the challenge is both to minimize the number of iterations and to reduce the difficulty of the problems to be solved at each iteration. In this paper, a new enumeration scheme is proposed. By introducing some constraints and optimizing over projections of the search region, potentially large parts of the search space can be discarded, drastically reducing the number of iterations. Moreover, the single-objective programs to be solved can be guaranteed to be feasible, and a starting solution can be provided allowing warm start resolutions. This results in a fast algorithm that is simple to implement. Experimental computations on two standard multi-objective instance families show that our approach seems to perform significantly faster than the state of the art algorithm.

Abstract We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

In this paper, we propose a generic algorithm to compute exactly the set of nondominated points for multiobjective discrete optimization problems. Our algorithm extends the ε-constraint method, originally designed for the biobjective case only, to solve problems with two or more objectives. For this purpose, our algorithm splits the search space into zones that can be investigated separately by solving an integer program. We also propose refinements, which provide extra information on several zones, allowing us to detect, and discard, empty parts of the search space without checking them by solving the associated integer programs. This results in a limited number of calls to the integer solver. Moreover, we can provide a feasible starting solution before solving every program, which significantly reduces the time spent for each resolution. The resulting algorithm is fast and simple to implement. It is compared with previous state-of-the-art algorithms and is seen to outperform them significantly on the experimented problem instances.

Abstract Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $$(1+\varepsilon , \dots , 1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximate Pareto set (also called an $$\varepsilon $$ ε -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate $$\varepsilon $$ ε -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact $$\varepsilon $$ ε -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

This paper deals with the problem of capacity expansion of a network under independent uncertain demands defined by interval sets. In this context, decisions about capacity expansion must be made before the demands are revealed. Standard robust models require the definition of an uncertainty domain and look for the minimum cost solution able to satisfy any demand within this domain. We propose, justify, and illustrate an alternative robust model based on a bi‐objective formulation. Therefore, in addition to the cost criterion, we consider a second criterion, related to the Quality of Service, which measures the ability of a solution to handle any demand. The decision‐maker can be interested in efficient solutions offering a compromise between these criteria. We study the complexity of the enumeration of the corresponding nondominated set, and propose exact and approximation algorithms. © 2016 Wiley Periodicals, Inc. NETWORKS, Vol. 68(3), 185–199 2016

Multi-attribute auctions allow negotiations over multiple attributes besides price. Multiple criteria English reverse auction mechanisms differ regarding the aggregation model used to represent the buyer's preferences and the feedback information pro

Research in Information Retrieval usually shows performanceimprovement when many sources of evidence are combined to produce a ranking of documents (e.g., texts, pictures, sounds, etc.). In this paper, we focus on the rank aggregation problem, also called data fusion problem, where rankings of documents, searched into the same collection and provided by multiple methods, are combined in order to produce a new ranking. In this context, we propose a rank aggregation method within a multiple criteria framework using aggregation mechanisms based on decision rules identifying positive and negative reasons for judging whether a document should get a better rank than another. We show that the proposed method deals well with the Information Retrieval distinctive features. Experimental results are reported showing that the suggested method performs better than the well-known CombSUM and CombMNZ operators.

This paper discusses induction of decision rules from data tables representing information about a set of objects described by a set of attributes. If the input data contains inconsistencies, rough sets theory can be used to handle them. The most popular perspectives of rule induction are classification and knowledge discovery. The evaluation of decision rules is quite different depending on the perspective. Criteria for evaluating the quality of a set of rules are presented and discussed. The degree of conflict and the possibility of achieving a satisfying compromise between criteria relevant to classification and criteria relevant to discovery are then analyzed. For this purpose, we performed an extensive experimental study on several well-known data sets where we compared two different approaches: (1) the popular rough set based rule induction algorithm LEM2 generating classification rules, (2) our own algorithm Explore—specific for discovery perspective. © 2001 John Wiley & Sons, Inc.

This paper proposes new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the concept of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.