Kim Allan Andersen

A label correcting approach for solving bicriterion shortest-path problems

Abstract This article contributes with a very fast algorithm for solving the bicriterion shortest-path problem. By imposing some simple domination conditions, we reduce the number of iterations needed to find all the efficient (Pareto optimal) paths in the network. We have implemented the algorithm and tested it with the Label Correcting algorithm. We have also made a theoretical argument of the performance of all the existing algorithms, in order to rank them by performance. Included is a discussion on the structure of random generated networks, generated with two different methods, and of the characteristics of these networks. Scope and purpose Consider a network through which you have to send some commodities from a source node s to a terminal node t in the cheapest way. If you have only a single attribute on each link, the problem is easily solved with e.g. Dijkstras shortest-path algorithm. If each link is assigned two attributes, e.g. cost and travel time, the problem of finding all efficient s − t paths is much harder. The problem often occurs as a subproblem in transportation models and combinatorial optimization. We have examined the existing algorithms for the bicriterion shortest-path (BSP) problem, and made a ranking based on their computational complexity. By imposing preprocessing conditions on the fastest algorithm, we have been able to reduce cpu-times considerably. The structure of random networks turned out to be important for the computational results, and has therefore been investigated. A new program for generating random networks is presented.

Finding the K shortest hyperpaths

The K shortest paths problem has been extensively studied for many years. Efficient methods have been devised, and many practical applications are known. Shortest hyperpath models have been proposed for several problems in different areas, for example in relation with routing in dynamic networks. However, the K shortest hyperpaths problem has not yet been investigated. In this paper we present procedures for finding the K shortest hyperpaths in a directed hypergraph. This is done by extending existing algorithms for K shortest loopless paths. Computational experiments on the proposed procedures are performed, and applications in transportation, planning and combinatorial optimization are discussed.

On bicriterion minimal spanning trees: an approximation

Abstract In this paper we focus on the problem of computing the set of efficient spanning trees for a given network where each arc carries two attributes. This problem is N P -complete. We discuss two heuristics, namely, neighbourhood search (which is a well-known method) and adjacent search, a new method. They both approximate the set of efficient spanning trees. The difference lies in which kind of spanning trees are generated in each iteration. For neighbourhood search, all spanning trees which are adjacent to at least one spanning tree in the current approximation set are considered. Adjacent search is similar to neighbourhood search except that only spanning trees which are adjacent to at least two spanning trees in the current approximation set are considered. Based on computational results it is concluded that adjacent search is a reasonable alternative to neighbourhood search, especially for large problems.

A Branch and Bound Algorithm for a Class of Biobjective Mixed Integer Programs

Most real-world optimization problems are multiobjective by nature, involving noncomparable objectives. Many of these problems can be formulated in terms of a set of linear objective functions that should be simultaneously optimized over a class of linear constraints. Often there is the complicating factor that some of the variables are required to be integral. The resulting class of problems is named multiobjective mixed integer programming (MOMIP) problems. Solving these kinds of optimization problems exactly requires a method that can generate the whole set of nondominated points (the Pareto-optimal front). In this paper, we first give a survey of the newly developed branch and bound methods for solving MOMIP problems. After that, we propose a new branch and bound method for solving a subclass of MOMIP problems, where only two objectives are allowed, the integer variables are binary, and one of the two objectives has only integer variables. The proposed method is able to find the full set of nondominat...

Capacitated Network Design with Uncertain Demand

We consider a capacity-expansion problem arising in the design of telecommunication networks. The problem is to install capacity on links of the network so as to meet customer demand while minimizing total costs incurred. When studying this and related problems it is customary to assume that point-to-point demands are given. This will not bethe case in practice, however, since future demand is generally unknown and the decision must be based on uncertain forecasts. We develop a stochastic integer programming formulation of the problem and propose an L-shaped solution procedure based on well-known cutting-plane procedures for the deterministic problem. The algorithm was tested on two sets of real-life problem instances and we present results of our computational experiments.

The bicriterion semi-obnoxious location (BSL) problem solved by an [epsilon]-approximation

Abstract Locating an obnoxious (undesirable) facility is often modeled by the maximin or maxisum problem. But the obnoxious facility is often placed unrealistically far away from the demand points (nodes), resulting in prohibitively high transportation cost/time. One solution is to model the problem as a semi-obnoxious location problem. Here we model the problem as a bicriterion problem, not in advance determining the importance of the obnoxious objective compared to the cost/time objective. We consider this model for both the planar and the network case. The two problems are solved by an approximation algorithm, and the models are briefly compared by means of a real-life example.

Multiperiod capacity expansion of a telecommunications connection with uncertain demand

We consider multiperiod capacity expansion of a telecommunications connection with uncertain demand. We present a new preprocessing rule which drastically reduces computation time of an existing algorithm for a two-stage formulation of the problem. Also, an alternative multistage formulation of the problem is given and we elaborate a recursive solution procedure. Both algorithms have been implemented in C++, and we present the results of a series of computational experiments, demonstrating the effect of the new preprocessing rule and the practicability of the multistage procedure.

Finding the K shortest hyperpaths using reoptimization

We present some reoptimization techniques for computing (shortest) hyperpath weights in a directed hypergraph. These techniques are exploited to improve the worst-case computational complexity (as well as the practical performance) of an algorithm finding the K shortest hyperpaths in acyclic hypergraphs.

An algorithm for ranking assignments using reoptimization

We consider the problem of ranking assignments according to cost in the classical linear assignment problem. An algorithm partitioning the set of possible assignments, as suggested by Murty, is presented where, for each partition, the optimal assignment is calculated using a new reoptimization technique. Its computational performance is compared with all available implementations of algorithms with the same time complexity. The results are encouraging.

Applying the minimax criterion in stochastic recourse programs

We consider an optimization problem in which some uncertain parameters are replaced by random variables. The minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data. Only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions. In this paper, we establish a number of stability results for the minimax stochastic program, justifying in particular the approach of restricting attention to probability measures with support in some known finite set. Following this approach, we elaborate solution procedures for the minimax problem in the setting of two-stage stochastic recourse models, considering the linear recourse case as well as the integer recourse case. Since the solution procedures are modifications of well-known algorithms, their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments.