Connectedness of efficient solutions is a powerful property in multiple objective combinatorial optimization since it allows the construction of the complete efficient set using neighborhood search techniques. In this paper we show that, however, most of the classical multiple objective combinatorial optimization problems do not possess the connectedness property in general, including, among others, knapsack problems (and even several special cases of knapsack problems) and linear assignment problems. We also extend already known non-connectedness results for several optimization problems on graphs like shortest path, spanning tree and minimum cost flow problems. Different concepts of connectedness are discussed in a formal setting, and numerical tests are performed for different variants of the knapsack problem to analyze the likelihood with which non-connected adjacency graphs occur in randomly generated problem instances.
Multileaf Collimators (MLC) consist of (currently 20-100) pairs of movable metal leaves which are used to block radiation in Intensity Modulated Radiation Therapy (IMRT). The leaves modulate a uniform source of radiation to achieve given intensity profiles. The modulation process is modeled by the decomposition of a given non-negative integer matrix into a non-negative linear combination of matrices with the (strict) consecutive ones property.
Selection of new projects is one of the major decision making activities in any company. Given a set of potential projects to invest, a subset which matches the company's strategy and internal resources best has to be selected. In this paper, we propose a multicriteria model for portfolio selection of projects, where we take into consideration that each of the potential projects has several - usually conflicting - values.
Linear and integer programs are considered whose coefficient matrices can be partitioned into K consecutive ones matrices. Mimicking the special case of K=1 which is well-known to be equivalent to a network flow problem we show that these programs can be transformed to a generalized network flow problem which we call semi-simultaneous (se-sim) network flow problem. Feasibility conditions for se-sim flows are established and methods for finding initial feasible se-sim flows are derived. Optimal se-sim flows are characterized by a generalization of the negative cycle theorem for the minimum cost flow problem. The issue of improving a given flow is addressed both from a theoretical and practical point of view. The paper concludes with a summary and some suggestions for possible future work in this area.
Using covering problems (CoP) combined with binary search is a well-known and successful solution approach for solving continuous center problems. In this paper, we show that this is also true for center hub location problems in networks. We introduce and compare various formulations for hub covering problems (HCoP) and analyse the feasibility polyhedron of the most promising one. Computational results using benchmark instances are presented. These results show that the new solution approach performs better in most examples.
In StopLoc we consider the location of new stops along the edges of an existing public transportation network. Examples of StopLoc include the location of bus stops along some given bus routes or of railway stations along the tracks in a railway system. In order to measure the ''convenience'' of the location decision for potential customers in given demand facilities, two objectives are proposed. In the first one, we give an upper bound on reaching a closest station from any of the demand facilities and minimize the number of stations. In the second objective, we fix the number of new stations and minimize the sum of the distances between demand facilities and stations. The resulting two problems CovStopLoc and AccessStopLoc are solved by a reduction to a classical set covering and a restricted location problem, respectively. We implement the general ideas in two different environments - the plane, where demand facilities are represented by coordinates and in networks, where they are nodes of a graph.