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21
Choquetbased optimisation in multiobjective shortest path and spanning tree problems
"... This paper is devoted to the search of Choquetoptimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on pref ..."
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This paper is devoted to the search of Choquetoptimal solutions in finite graph problems with multiple objectives. The Choquet integral is one of the most sophisticated preference models used in decision theory for aggregating preferences on multiple objectives. We first present a condition on preferences (name hereafter preference for interior points) that characterizes preferences favouring compromise solutions, a natural attitude in various contexts such as multicriteria optimisation, robust optimisation and optimisation with multiple agents. Within Choquet expected utility theory, this condition amounts to using a submodular capacity and a convex utility function. Under these assumptions, we focus on the fast determination of Choquetoptimal paths and spanning trees. After investigating the complexity of these problems, we introduce a lower bound for the Choquet integral, computable in polynomial time. Then we propose different algorithms using this bound, either based on a controlled enumeration of solutions (ranking approach) or an implicit enumeration scheme (branch and bound). Finally, we provide numerical experiments that show the actual efficiency of the algorithms on multiple instances of different sizes.
Computational Decision Support: Regretbased Models for Optimization and Preference Elicitation
"... Decision making is a fundamental human, organizational, and societal activity, involving several key (and sometimes implicit) steps: the formulation of a set of options or decisions; information gathering to help assess the outcomes of these decisions and their likelihood; some assessment of the rel ..."
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Decision making is a fundamental human, organizational, and societal activity, involving several key (and sometimes implicit) steps: the formulation of a set of options or decisions; information gathering to help assess the outcomes of these decisions and their likelihood; some assessment of the relative utility or desirability of the possible outcomes; and an assessment of the tradeoffs involved
Approximating the minmax (regret) selecting items problem
, 2012
"... In this paper the problem of selecting p items out of n available to minimize the total cost is discussed. This problem is a special case of many important combinatorial optimization problems such as 01 knapsack, minimum assignment, single machine scheduling, minimum matroid base or resource alloc ..."
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Cited by 4 (3 self)
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In this paper the problem of selecting p items out of n available to minimize the total cost is discussed. This problem is a special case of many important combinatorial optimization problems such as 01 knapsack, minimum assignment, single machine scheduling, minimum matroid base or resource allocation. It is assumed that the item costs are uncertain and they are specified as a scenario set containing K distinct cost scenarios. In order to choose a solution the minmax and minmax regret criteria are applied. It is shown that both minmax and minmax regret problems are not approximable within any constant factor unless P=NP, which strengthens the results known up to date. In this paper a deterministic approximation algorithm with performance ratio of O(lnK) for the minmax version of the problem is also proposed.
Possibilistic Minmax Regret Sequencing Problems with Fuzzy Parameters
"... In this paper a class of sequencing problems with uncertain parameters is discussed. The uncertainty is modeled by using fuzzy intervals, whose membership functions are regarded as possibility distributions for the values of unknown parameters. It is shown how to use possibility theory to find robus ..."
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Cited by 1 (1 self)
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In this paper a class of sequencing problems with uncertain parameters is discussed. The uncertainty is modeled by using fuzzy intervals, whose membership functions are regarded as possibility distributions for the values of unknown parameters. It is shown how to use possibility theory to find robust solutions under fuzzy parameters this paper presents a general framework together with applications to some classical sequencing problems. First, the interval sequencing problems with the minmax regret criterion are discussed. The state of the art in this area is recalled. Next, the fuzzy sequencing problems, in which the classical intervals are replaced with fuzzy ones, are investigated. A possibilistic interpretation of such problems, solution concepts, and algorithms for computing a solution are described. In particular, it is shown that every fuzzy problem can be efficiently solved if a polynomial algorithm for the corresponding interval problem with the minmax regret criterion is known. Some methods of dealing with NPhard problems are also proposed and the efficiency of these methods is explored.
MinMax Problems on FactorGraphs
"... We study the minmax problem in factor graphs, which seeks the assignment that minimizes the maximum value over all factors. We reduce this problem to both minsum and sumproduct inference, and focus on the later. In this approach the minmax inference problem is reduced to a sequence of Constrain ..."
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Cited by 1 (1 self)
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We study the minmax problem in factor graphs, which seeks the assignment that minimizes the maximum value over all factors. We reduce this problem to both minsum and sumproduct inference, and focus on the later. In this approach the minmax inference problem is reduced to a sequence of Constraint Satisfaction Problems (CSP), which allows us to solve the problem by sampling from a uniform distribution over the set of solutions. We demonstrate how this scheme provides a message passing solution to several NPhard combinatorial problems, such as minmax clustering (a.k.a.Kclustering), asymmetric Kcenter clustering problem, Kpacking and the bottleneck traveling salesman problem. Furthermore, we theoretically relate the minmax reductions and several NP hard decision problems such as clique cover, setcover, maximum clique and Hamiltonian cycle, therefore also providing message passing solutions for these problems. Experimental results suggest that message passing often provides near optimal minmax solutions for moderate size instances. 1.
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"... Heuristic and exact algorithms for the interval minmax regret knapsack problem ..."
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Heuristic and exact algorithms for the interval minmax regret knapsack problem
MINMAX (REGRET) SCHEDULING PROBLEMS
, 2014
"... In this chapter a class of scheduling problems with uncertain parameters is discussed. The uncertainty is modeled by specifying a scenario set containing all possible vectors of the problem parameters which may occur. No additional information for the scenario set, such as a probability distribution ..."
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In this chapter a class of scheduling problems with uncertain parameters is discussed. The uncertainty is modeled by specifying a scenario set containing all possible vectors of the problem parameters which may occur. No additional information for the scenario set, such as a probability distribution, is provided. In order to choose a solution, the robust optimization framework is applied. The goal is to compute a schedule with the best worstcase performance over all scenarios. This performance is measured by the minmax and minmax regret criteria. The complexity of various minmax (regret) scheduling problems and some algorithms for solving them are described.
problems
"... General approximation schemes for minmax (regret) versions of some (pseudo)polynomial ..."
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General approximation schemes for minmax (regret) versions of some (pseudo)polynomial