Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of Global Optimizatio

Compact versus noncompact LP formulations for minimizing convex Choquet integrals

AbstractWe address here the problem of minimizing Choquet Integrals (also known as “Lovász Extensions”) over solution sets which can be either polyhedra or (mixed) integer sets. Typical applications of such problems concern the search of compromise solutions in multicriteria optimization. We focus here on the case where the Choquet Integrals to be minimized are convex, implying that the set functions (or “capacities”) underlying the Choquet Integrals considered are submodular. We first describe an approach based on a large scale LP formulation, and show how it can be handled via the so-called column-generation technique. We next investigate alternatives based on compact LP formulations, i.e. featuring a polynomial number of variables and constraints. Various potentially useful special cases corresponding to well-identified subclasses of underlying set functions are considered: quadratic and cubic submodular functions, and a more general class including set functions which, up to a sign, correspond to capacities which are both (k+1)−additive and k-monotone for k≥3. Computational experiments carried out on series of test instances, including transportation problems and knapsack problems, clearly confirm the superiority of compact formulations. As far as we know, these results represent the first systematic way of practically solving Choquet minimization problems on solution sets of significantly large dimensions

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal 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 Choquet-optimal 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.ouinonouirechercheInternationa

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal 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 Choquet-optimal 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

Choquet-based optimisation in multiobjective shortest path and spanning tree problems

This paper is devoted to the search of Choquet-optimal 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 Choquet-optimal 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.Multiobjective discrete optimisation Choquet integral Shortest path problem Minimum spanning tree problem Submodular capacity

Ordered Weighted Average optimization in Multiobjective Spanning Tree Problem

Rework adversely impacts the performance of building projects. In this study, data were analyzed from 788 construction incidents in 40 Spanish building projects to determine the effects of project and managerial characteristics on rework costs. Finally, regression analysis was used to understand the relationships among contributing factors and to develop a model for rework prediction. Interestingly, the rework prediction model showed that only the original contract value (OCV) and the project location in relation to the company’s headquarters contributed to the regression model. Project type, type of organization, type of contract, and original contract duration (OCD), which represents the magnitude and complexity of a project, were represented by the OCV. This model for rework prediction based on original project conditions enables strategies to be put in place prior to the start of construction, to minimize uncertainties, to reduce impacts on project cost and schedule, and, thus, to improve productivity.Peer ReviewedPostprint (author's final draft