Capacitated,balanced, sequential location-allocation problems on chains and trees

This paper considers finite horizon, multiperiod, sequential, minisum location-allocation problems on chain graphs and tree networks. The demand has both deterministic and probabilistic components, and increases dynamically from period to period. The problem is to locate one additionalcapacitated facility in each of thep specified periods, and to determine the service allocations of the facilities, in order to optimally satisfy the demand on the network. In this context, two types of objective criteria or location strategies are addressed. The first is a myopic strategy in which the present period cost is minimized sequentially for each period, and the second is a discounted present worth strategy. For the chain graph, we analyze ap-facility problem under both these criteria, while for the tree graph, we analyze a 3-facility myopic problem, and a 2-facility discounted present worth problem. All these problems are nonconvex, and we specify a finite set of candidate solutions which may be compared in order to determine a global optimal solution.     

Solving a new stochastic multi-mode p-hub covering location problem considering risk by a novel multi-objective algorithm

In this paper, we develop a novel stochastic multi-objective multi-mode transportation model for hub covering location problem under uncertainty. The transportation time between each pair of nodes is an uncertain parameter and also is influenced by a risk factor in the network. We extend the traditional comprehensive hub location problem by considering two new objective functions. So, our multi-objective model includes (i) minimization of total current investment costs and (ii) minimization of maximum transportation time between each origin–destination pair in the network. Besides, a novel multi-objective imperialist competitive algorithm (MOICA) is proposed to obtain the Pareto-optimal solutions of the problem. The performance of the proposed solution algorithm is compared with two well-known meta-heuristics, namely, non-dominated sorting genetic algorithm (NSGA-II) and Pareto archive evolution strategy (PAES). Computational results show that MOICA outperforms the other meta-heuristics.     

On the computation of all supported efficient solutions in multi-objective integer network flow problems

This paper presents a new algorithm for identifying all supported non-dominated vectors (or outcomes) in the objective space, as well as the corresponding efficient solutions in the decision space, for multi-objective integer network flow problems. Identifying the set of supported non-dominated vectors is of the utmost importance for obtaining a first approximation of the whole set of non-dominated vectors. This approximation is crucial, for example, in two-phase methods that first compute the supported non-dominated vectors and then the unsupported non-dominated ones. Our approach is based on a negative-cycle algorithm used in single objective minimum cost flow problems, applied to a sequence of parametric problems. The proposed approach uses the connectedness property of the set of supported non-dominated vectors/efficient solutions to find all integer solutions in maximal non-dominated/efficient facets.     

The tricriterion shortest path problem with at least two bottleneck objective functions

The focus of this paper is on the tricriterion shortest path problem where two objective functions are of the bottleneck type, for example MinMax or MaxMin. The third objective function may be of the same kind or we may consider, for example, MinSum or MaxProd. Let p(n) be the complexity of a classical single objective algorithm responsible for this third function, where n is the number of nodes and m be the number of arcs of the graph. An O(m²p(n)) algorithm is presented that can generate the minimal complete set of Pareto-optimal solutions. Finding the maximal complete set is also possible. Optimality proofs are given and extensions for several special cases are presented. Computational experience for a set of randomly generated problems is reported.     

Shortest paths with ordinal weights

We investigate the single-source-single-destination “shortest” path problem in directed, acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated path vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated path vectors from source to sink.     

Scheduling interfering job sets on parallel machines

We consider bicriteria scheduling on identical parallel machines in a nontraditional context: jobs belong to two disjoint sets, and each set has a different criterion to be minimized. The jobs are all available at time zero and have to be scheduled (non-preemptively) on m parallel machines. The goal is to generate the set of all non-dominated solutions, so the decision maker can evaluate the tradeoffs and choose the schedule to be implemented. We consider the case where, for one of the two sets, the criterion to be minimized is makespan while for the other the total completion time needs to be minimized. Given that the problem is NP-hard, we propose an iterative SPT–LPT–SPT heuristic and a bicriteria genetic algorithm for the problem. Both approaches are designed to exploit the problem structure and generate a set of non-dominated solutions. In the genetic algorithm we use a special encoding scheme and also a unique strategy – based on the properties of a non-dominated solution – to ensure that all parts of the non-dominated front are explored. The heuristic and the genetic algorithm are compared with a time-indexed integer programming formulation for small and large instances. Results indicate that the both the heuristic and the genetic algorithm provide high solution quality and are computationally efficient. The heuristics proposed also have the potential to be generalized for the problem of interfering job sets involving other bicriteria pairs.     

On maximal independent sets of vertices in claw-free graphs

A graph is claw-free if: whenever three (distinct) vertices are joined to a single vertex, those three vertices are a nonindependent (nonstable) set. Given a finite claw-free graph with real numbers (weights) assigned to the vertices, we exhibit an algorithm for producing an independent set of vertices of maximum total weight. This algorithm is “efficient” in the sense of J. Edmonds, that is to say, the number of computational steps required is of polynomial (not exponential or factorial) order in n, the number of vertices of the graph. This problem was solved earlier by Edmonds for the special case of “edge-graphs”; our solution is by reducing the more general problem to the earlier-solved special case. Separate attention is given to the case in which all weights are (+1) and thus an independent set is sought which is maximal in the sense of its cardinality.     

An efficient algorithm for the Steiner Tree Problem with revenue,bottleneck and hop objective functions

This paper considers a tricriteria Steiner Tree Problem arising in the design of telecommunication networks. The objective functions consist of maximizing the revenue and of minimizing the maximal distance between each pair of interconnected nodes, as well as the maximal number of arcs between the root and each node. A polynomial algorithm is developed for the generation of a minimal complete set of Pareto-optimal Steiner trees. Optimality proofs are given and computational experience on a set of randomly generated problems is reported.     

Cone convexity,cone extreme points,and nondominated solutions in decision problems with multiobjectives

Although there is no universally accepted solution concept for decision problems with multiple noncommensurable objectives, one would agree that agood solution must not be dominated by the other feasible alternatives. Here, we propose a structure of domination over the objective space and explore the geometry of the set of all nondominated solutions. Two methods for locating the set of all nondominated solutions through ordinary mathematical programming are introduced. In order to achieve our main results, we have introduced the new concepts of cone convexity and cone extreme point, and we have explored their main properties. Some relevant results on polar cones and polyhedral cones are also derived. Throughout the paper, we also pay attention to an important special case of nondominated solutions, that is, Pareto-optimal solutions. The geometry of the set of all Pareto solutions and methods for locating it are also studied. At the end, we provide an example to show how we can locate the set of all nondominated solutions through a derived decomposition theorem.     

最小最大后悔准则下新增设施选址策略研究

基于新增设施选址问题,考虑网络节点权重不确定性,以设施中最大负荷量最小为目标,提出最小最大后悔准则下的新增设施选址问题。在网络节点权重确定时,通过证明将网络图中无穷多个备选点离散为有限个设施候选点,设计了时间复杂度为O(mn²)的多项式算法;在节点权重为区间值时,通过分析最大后悔值对应的最坏情境权重结构,进而确定最大后悔值最小的选址,提出时间复杂度为O(2ⁿm²n³)的求解算法;最后给出数值算例。     

Approximation of convex bodies by multiple objective optimization and an application in reachable sets

In this paper, we focus on approximating convex compact bodies. For a convex body described as the feasible set in objective space of a multiple objective programme, we show that finding it is equivalent to finding the non-dominated set of a multiple objective programme. This equivalence implies that convex bodies can be approximated using multiple objective optimization algorithms. Therefore, we propose a revised outer approximation algorithm for convex multiple objective programming problems to approximate convex bodies. Finally, we apply the algorithm to solve reachable sets of control systems and use numerical examples to show the effectiveness of the algorithm.     

改进的多目标规划遗传算法

本讨论了[1]中多目标规划遗传算法存在的缺陷，并提出了相应改进策略．这些策略包括：引进精粹策略，杂交限制，终止条件，个体表示改进等方面，利用这些策略使算法能克服终止准则和小生境聚集的缺陷，使得算法能更快的收敛到Pareto最优解集同时又有好有分布的Pareto最优解集．     

Algorithms for clique-independent sets on subclasses of circular-arc graphs

A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose polynomial time algorithms for finding the maximum cardinality and weight of a clique-independent set of a -free CA graph. Also, we apply the algorithms to the special case of an HCA graph. The complexity of the proposed algorithm for the cardinality problem in HCA graphs is O(n). This represents an improvement over the existing algorithm by Guruswami and Pandu Rangan, whose complexity is O(n²). These algorithms suppose that an HCA model of the graph is given.     

A method for finding well-dispersed subsets of non-dominated vectors for multiple objective mixed integer linear programs

We present an algorithm for generating a subset of non-dominated vectors of multiple objective mixed integer linear programming. Starting from an initial non-dominated vector, the procedure finds at each iteration a new one that maximizes the infinity-norm distance from the set dominated by the previously found solutions. When all variables are integer, it can generate the whole set of non-dominated vectors.     

A k-product uncapacitated facility location problem

A k-product uncapacitated facility location problem can be described as follows. There is a set of demand points where clients are located and a set of potential sites where facilities of unlimited capacities can be set up. There are k different kinds of products. Each client needs to be supplied with k kinds of products by a set of k different facilities and each facility can be set up to supply only a distinct product with a non-negative fixed cost determined by the product it intends to supply. There is a non-negative cost of shipping goods between each pair of locations. These costs are assumed to be symmetric and satisfy the triangle inequality. The problem is to select a set of facilities to be set up and their designated products and to find an assignment for each client to a set of k   facilities so that the sum of the setup costs and the shipping costs is minimized. In this paper, an approximation algorithm within a factor of 2k+1$2k+1$ of the optimum cost is presented. Assuming that fixed setup costs are zero, we give a 2k-1$2k-1$ approximation algorithm for the problem. In addition we show that for the case k=2$k=2$, the problem is NP-complete when the cost structure is general and there is a 2-approximation algorithm when the costs are symmetric and satisfy the triangle inequality. The algorithm is shown to produce an optimal solution if the 2-product uncapacitated facility location problem with no fixed costs happens to fall on a tree graph.     

Global Rigidity: The Effect of Coning

Recent results have confirmed that the global rigidity of bar-and-joint frameworks on a graph G is a generic property in Euclidean spaces of all dimensions. Although it is not known if there is a deterministic algorithm that runs in polynomial time and space, to decide if a graph is generically globally rigid, there is an algorithm (Gortler et al. in Characterizing generic global rigidity, arXiv:, 2007) running in polynomial time and space that will decide with no false positives and only has false negatives with low probability. When there is a framework that is infinitesimally rigid with a stress matrix of maximal rank, we describe it as a certificate which guarantees that the graph is generically globally rigid, although this framework, itself, may not be globally rigid. We present a set of examples which clarify a number of aspects of global rigidity.     

Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G=(V,E) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P of source-sink pairs of vertices of G, an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source-sink pairs of P. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that P consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s₁ and s₂ to, respectively, a given vertex t₁, and each vertex of a given subset of vertices T₂.     

Factoring and recognition of read-once functions using cographs and normality and the readability of functions associated with partial k-trees

An approach for factoring general boolean functions was described in Golumbic and Mintz [Factoring logic functions using graph partitioning, in: Proceedings of IEEE/ACM International Conference on Computer Aided Design, November 1999, pp. 195-198] and Mintz and Golumbic [Factoring Boolean functions using graph partitioning, Discrete Appl. Math. 149 (2005) 131-153] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring read-once functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality.For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2^k function and a read 2^k formula for F can be obtained in polynomial time.     

The complete optimal stars-clustering-tree problem

We consider the following complete optimal stars-clustering-tree problem: Given a complete graph G=(V,E) with a weight on every edge and a collection of subsets of V, we want to find a minimum weight spanning tree T such that each subset of the vertices in the collection induces a complete star in T. One motivation for this problem is to construct a minimum cost (weight) communication tree network for a collection of (not necessarily disjoint) groups of customers such that each group induces a complete star. As a result the network will provide a “group broadcast” property, “group fault tolerance” and “group privacy”. We present another motivation from database systems with replications. For the case where the intersection graph of the subsets is connected we present a structure theorem that describes all feasible solutions. Based on it we provide a polynomial algorithm for finding an optimal solution. For the case where each subset induces a complete star minus at most k leaves we prove that the problem is NP-hard.     

Solving multi-objective multicast routing problems by evolutionary multi-objective simulated annealing algorithms with variable neighbourhoods

This paper presents the investigation of an evolutionary multi-objective simulated annealing (EMOSA) algorithm with variable neighbourhoods to solve the multi-objective multicast routing problems in telecommunications. The hybrid algorithm aims to carry out a more flexible and adaptive exploration in the complex search space by using features of the variable neighbourhood search to find more non-dominated solutions in the Pareto front. Different neighbourhood strictures have been designed with regard to the set of objectives, aiming to drive the search towards optimising all objectives simultaneously. A large number of simulations have been carried out on benchmark instances and random networks with real world features including cost, delay and link utilisations. Experimental results demonstrate that the proposed EMOSA algorithm with variable neighbourhoods is able to find high-quality non-dominated solutions for the problems tested. In particular, the neighbourhood structures that are specifically designed for each objective significantly improved the performance of the proposed algorithm compared with variants of the algorithm with a single neighbourhood.