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Universal Shortest Paths
(2010)

We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.

It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions.

In this paper we give an overview on the system of rehabilitation clinics in Germany in general and the literature on patient scheduling applied to rehabilitation facilities in particular.
We apply a class-teacher model developed to this environment and then generalize it to meet some of the specific constraints of inpatient rehabilitation clinics. To this end we introduce a restricted edge coloring on undirected bipartite graphs which is called group-wise balanced. The problem considered is called patient-therapist-timetable problem with group-wise balanced constraints (PTTPgb). In order to specify weekly schedules further such that they produce a reasonable allocation to morning/afternoon (second level decision) and to the single periods (third level decision) we introduce (hierarchical PTTPgb). For the corresponding model, the hierarchical edge coloring problem, we present some first feasibility results.

Due to the increasing number of natural or man-made disasters, the application of operations research methods in evacuation planning has seen a rising interest in the research community. From the beginning, evacuation planning has been highly focused on car-based evacuation. Recently, also the evacuation of transit depended evacuees with the help of buses has been considered.
In this case study, we apply two such models and solution algorithms to evacuate a core part of the metropolitan capital city Kathmandu of Nepal as a hypothetical endangered region, where a large part of population is transit dependent. We discuss the computational results for evacuation time under a broad range of possible scenarios, and derive planning suggestions for practitioners.

Stop Location Design in Public Transportation Networks: Covering and Accessibility Objectives
(2006)

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.

Multifacility location problems arise in many real world applications. Often, the facilities can only be placed in feasible regions such as development or industrial areas. In this paper we show the existence of a finite dominating set (FDS) for the planar multifacility location problem with polyhedral gauges as distance functions, and polyhedral feasible regions, if the interacting facilities form a tree. As application we show how to solve the planar 2-hub location problem in polynomial time. This approach will yield an ε-approximation for the euclidean norm case polynomial in the input data and 1/ε.

A single facility problem in the plane is considered, where an optimal location has to be
identified for each of finitely many time-steps with respect to time-dependent weights and
demand points. It is shown that the median objective can be reduced to a special case of the
static multifacility median problem such that results from the latter can be used to tackle the
dynamic location problem. When using block norms as distance measure between facilities,
a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the
resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation
between dynamic location problems for T time periods and T-facility problems, this algorithm
can also be applied to the static 2-facility location problem.

Max ordering (MO) optimization is introduced as tool for modelling production
planning with unknown lot sizes and in scenario modelling. In MO optimization a feasible solution set \(X\) and, for each \(x\in X, Q\) individual objective functions \(f_1(x),\dots,f_Q(x)\) are given. The max ordering objective
\(g(x):=max\) {\(f_1(x),\dots,f_Q(x)\)} is then minimized over all \(x\in X\).
The paper discusses complexity results and describes exact and approximative
algorithms for the case where \(X\) is the solution set of combinatorial
optimization problems and network flow problems, respectively.

The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset \(W\) has to be selected such that given supplies can be transported to the nodes in \(W\). In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters.
We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status.
There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.

Scheduling and location models are often used to tackle problems in production, logistics, and supply chain management. Instead of treating these models independent of each other, as is usually done in the literature, we consider in this paper an integrated model in which the locations of machines define release times for jobs. Polynomial solution algorithms are presented for single machine problems in which the scheduling part can be solved by the earliest release time rule.

In this paper the multi terminal q-FlowLoc problem (q-MT-FlowLoc) is introduced. FlowLoc problems combine two well-known modeling tools: (dynamic) network flows and locational analysis. Since the q-MT-FlowLoc problem is NP-hard we give a mixed integer programming formulation and propose a heuristic which obtains a feasible solution by calculating a maximum flow in a special graph H. If this flow is also a minimum cost flow, various versions of the heuristic can be obtained by the use of different cost functions. The quality of this solutions is compared.

In this paper we discuss an earliest arrival flow problem of a network having arc travel times and capacities that vary with time over a finite time horizon T. We also consider the possibility to wait (or park) at a node before departingon outgoing arc. This waiting is bounded by the value of maximum waiting time and the node capacity which also vary with time.

We generalize the classical shortest path problem in two ways. We consider two - in general contradicting - objective functions and introduce a time dependency of the cost which is caused by a traversal time on each arc. The resulting problem, called time-dependent bicriteria shortest path problem (TdBiSP) has several interesting practical applications, but has not attained much attention in the literature.

The problem of finding an optimal location X* minimizing the maximum Euclidean distance to existing facilities is well solved by e.g. the Elzinga-Hearn algorithm. In practical situations X* will however often not be feasible. We therefore suggest in this note a polynomial algorithm which will find an optimal location X^F in a feasible subset F of the plane R^2

Finding "good" cycles in graphs is a problem of great interest in graph theory as well as in locational analysis. We show that the center and median problems are NP hard in general graphs. This result holds both for the variable cardinality case (i.e. all cycles of the graph are considered) and the fixed cardinality case (i.e. only cycles with a given cardinality p are feasible). Hence it is of interest to investigate special cases where the problem is solvable in polynomial time. In grid graphs, the variable cardinality case is, for instance, trivially solvable if the shape of the cycle can be chosen freely. If the shape is fixed to be a rectangle one can analyse rectangles in grid graphs with, in sequence, fixed dimension, fixed cardinality, and variable cardinality. In all cases a com plete characterization of the optimal cycles and closed form expressions of the optimal objective values are given, yielding polynomial time algorithms for all cases of center rectangle problems. Finally, it is shown that center cycles can be chosen as rectangles for small cardinalities such that the center cycle problem in grid graphs is in these cases completely solved.

In this paper we generalize the classical shortest path problem in two ways. We consider two objective functions and time-dependent data. The resulting problem, called the time-dependent bicriteria shortest path problem (TdBiSP), has several interesting practical applications, but has not gained much attention in the literature.

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.

We investigate two versions of multiple objective minimum spanning tree
problems defined on a network with vectorial weights. First, we want to minimize the maximum of Q linear objective functions taken over the set of all spanning trees (max linear spanning tree problem ML-ST). Secondly, we look for efficient spanning trees (multi criteria spanning tree problem MC-ST). Problem ML-ST is shown to be NP-complete. An exact algorithm which is based on ranking is presented. The procedure can also be used as an approximation scheme. For solving the bicriterion MC-ST, which in the worst case may have an exponential number of efficient trees, a two-phase procedure is presented. Based on the computation of extremal efficient spanning trees we use neighbourhood search to determine a sequence of solutions with the property that the distance
between two consecutive solutions is less than a given accuracy.

In this paper we consider the problem of decomposing a given integer matrix A into
a positive integer linear combination of consecutive-ones matrices with a bound on the
number of columns per matrix. This problem is of relevance in the realization stage
of intensity modulated radiation therapy (IMRT) using linear accelerators and multileaf
collimators with limited width. Constrained and unconstrained versions of the problem
with the objectives of minimizing beam-on time and decomposition cardinality are considered.
We introduce a new approach which can be used to find the minimum beam-on
time for both constrained and unconstrained versions of the problem. The decomposition
cardinality problem is shown to be NP-hard and an approach is proposed to solve the
lexicographic decomposition problem of minimizing the decomposition cardinality subject
to optimal beam-on time.

In this paper, theory and algorithms for solving the multiple objective minimum cost flow problem are reviewed. For both the continuous and integer case exact and approximation algorithms are presented. In addition, a section on compromise solutions summarizes corresponding results. The reference list consists of all papers known to the autheors which deal with the multiple objective minimum cost flow problem.