The biobjective undirected two-commodity minimum cost flow problem

We address the two-commodity minimum cost flow problem considering two objectives. We show that the biobjective undirected two-commodity minimum cost flow problem can be split into two standard biobjective minimum cost flow problems using the change of variables approach. This technique allows us to develop a method that finds all the efficient extreme points in the objective space for the two-commodity problem solving two biobjective minimum cost flow problems. In other words, we generalize the Hu's theorem for the biobjective undirected two-commodity minimum cost flow problem. In addition, we develop a parametric network simplex method to solve the biobjective problem.     

Optimal Two-commodity Flows with Non-linear Cost Functions

We consider networks in which two different commodities have to be transported across undirected arcs, subject to a shared capacity on the arcs. For each arc and commodity there is an associated non-linear cost that depends on the amount of the commodity transported across the arc. The aim is to minimize the sum of the costs over all arcs and commodities. Efficient algorithms for solving this problem for two types of objective functions will be presented: in the first the cost depends on the absolute value of the flow and in the second the cost is a quadratic function of the flow. Previous work on multi-commodity flow has concentrated on linear cost problems or tackled non-linear cost problems with Lagrangian relaxation methods and other more general techniques. The algorithms in this paper, on the other hand, provide a very efficient way of dealing with two types of non-linear two-commodity optimal flow problems.     

Theory and Methodology A bicriteria approach to the two-machine flow shop scheduling problem

In this paper we address the problem of minimizing makespan and sum of completion times simultaneously in a two-machine flow shop environment. We formulate the problem as a bicriteria scheduling problem, and develop a branch-and-bound procedure that iteratively solves restricted single objective scheduling problems until the set of efficient solutions is completely enumerated. We report computational results, and explore certain properties of the set of efficient solutions. We then discuss their implications for the Decision Maker.     

无容量限制的最小费用流问题

本文研究了无容量限制的带固定费用和可变费用的单物资和二物资的最小费用流问题,并分别给出了多项式算法.最后应用该算法,计算了一个二物资的最小费用流问题的实例.     

A computational comparison of flow formulations for the capacitated location-routing problem

In this paper we present a computational comparison of four different flow formulations for the capacitated location-routing problem. We introduce three new flow formulations for the problem, namely a two-index two-commodity flow formulation, a three-index vehicle-flow formulation and a three-index two-commodity flow formulation. We also consider an existing two-index vehicle-flow formulation and extend it by considering new families of valid inequalities and separation algorithms. We introduce new branch-and-cut algorithms for each of the formulations and compare them on a wide number of instances. Our results show that compact formulations can produce tight gaps and solve many instances quickly, whereas three-index formulations scale better in terms of computing time.     

Algorithm robust for the bicriteria discrete optimization problem

We apply Algorithm Robust to various problems in multiple objective discrete optimization. Algorithm Robust is a general procedure that is designed to solve bicriteria optimization problems. The algorithm performs a weight space search in which the weights are utilized in min-max type subproblems. In this paper, we experiment with Algorithm Robust on the bicriteria knapsack problem, the bicriteria assignment problem, and the bicriteria minimum cost network flow problem. We look at a heuristic variation that is based on controlling the weight space search and has an indirect control on the sample of efficient solutions generated. We then study another heuristic variation which generates samples of the efficient set with quality guarantees. We report results of computational experiments.     

Robust maximum weighted independent-set problems on interval graphs

We study the maximum weighted independent-set problem on interval graphs with uncertainty on the vertex weights. We use the absolute robustness criterion and the min–max regret criterion to evaluate solutions. For a discrete scenario set, we find that the problem is NP-hard for each of the robustness criteria; we also provide pseudo-polynomial time algorithms when there is a constant number of scenarios and show that the problem is strongly NP-hard when the set of scenarios is unbounded. When the scenario set is a Cartesian product, we prove that the problem is equivalent to a maximum weighted independent-set problem on the same interval graph but without uncertainty for the first objective function and that the scenario set can be reduced for the second objective function.     

Multicommodity flow models for spanning trees with hop constraints

In this paper we compare the linear programming relaxations of undirected and directed multicommodity flow formulations for the terminal layout problem with hop constraints. Hop constraints limit the number of hops (links) between the computer center and any terminal in the network. These constraints model delay constraints since a smaller number of hops decreases the maximum delay transmission time in the network. They also model reliability constraints because with a smaller number of hops there is a lower route loss probability. Hop constraints are easily modelled with the variables involved in multicommodity flow formulations. We give some empirical evidence showing that the linear programming relaxation of such formulations give sharp lower bounds for this hop constrained network design problem. On the other hand, these formulations lead to very large linear programming models. Therefore, for bounding purposes we also derive several lagrangean based procedures from a directed multicommodity flow formulation and present some computational results taken from a set of instances with up to 40 nodes.     

Revisiting parametric multi-terminal problems: Maximum flows, minimum cuts and cut-tree computations

Given an undirected network, the multi-terminal network flows analysis consists in determining the all pairs maximum flow values. In this paper, we consider an undirected network in which some edge capacities are allowed to vary and we analyze the impact of such variations on the all pairs maximum flow values. We first provide an efficient algorithm for the single parametric capacity case, and then propose a generalization to the case of multiple parametric capacities. Moreover, we provide a study on Gomory–Hu cut-tree relationships.     

Relaxation analysis in linear vectorvalued maximization

As an analogy to the postoptimal sensitivity analysis in linear programming a theory of postefficient sensitivity analysis in linear vectorvalued maximization problems (LVMP), called relaxation analysis, is developed. Introducing parameters in the most coefficients of the given objective functions, the goal is to define an admissible region of the parameters such that the set E of all the efficient solutions of the initial LVMP does not change. Doing the same with the right hand side, the set of all admissible parameters is defined via an undirected graph which is generated by the initial LVMP.     

Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications

In this paper, we characterize the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with cone constraints in terms of the level-boundedness of the component functions of the objective on the perturbed sets of the original constraint set. This characterization is then applied to carry out the asymptotic analysis of a class of penalization methods. More specifically, under the assumption of nonemptiness and compactness of the weakly efficient solution set, we prove the existence of a path of weakly efficient solutions to the penalty problem and its convergence to a weakly efficient solution of the original problem. Furthermore, for any efficient point of the original problem, there exists a path of efficient solutions to the penalty problem whose function values (with respect to the objective function of the original problem) converge to this efficient point.     

A polynomial time approximation algorithm for the two-commodity splittable flow problem

We consider a generalization of the unsplittable maximum two-commodity flow problem on undirected graphs where each commodity ${i \in \{1, 2\}}$ can be split into a bounded number k _i of equally-sized chunks that can be routed on different paths. We show that in contrast to the single-commodity case this problem is NP-hard, and hard to approximate to within a factor of α > 1/2. We present a polynomial time 1/2-approximation algorithm for the case of uniform chunk size over both commodities and show that for even k _i and a mild cut condition it can be modified to yield an exact method. The uniform case can be used to derive a 1/4-approximation for the maximum concurrent (k ₁, k ₂)-splittable flow without chunk size restrictions for fixed demand ratios.     

Primal and dual algorithms for optimization over the efficient set

ABSTRACT

Optimization over the efficient set of a multi-objective optimization problem is a mathematical model for the problem of selecting a most preferred solution that arises in multiple criteria decision-making to account for trade-offs between objectives within the set of efficient solutions. In this paper, we consider a particular case of this problem, namely that of optimizing a linear function over the image of the efficient set in objective space of a convex multi-objective optimization problem. We present both primal and dual algorithms for this task. The algorithms are based on recent algorithms for solving convex multi-objective optimization problems in objective space with suitable modifications to exploit specific properties of the problem of optimization over the efficient set. We first present the algorithms for the case that the underlying problem is a multi-objective linear programme. We then extend them to be able to solve problems with an underlying convex multi-objective optimization problem. We compare the new algorithms with several state of the art algorithms from the literature on a set of randomly generated instances to demonstrate that they are considerably faster than the competitors.     

On Resource Allocation Problems with Interval-Scale Coefficients

In this paper we discuss the classical problem of the allocation of a single finite resource among many competing activities for the specific case where the coefficients of the objective function form an interval scale. In this case there is no longer a single optimal solution, but rather a set of efficient solutions. We recommend a technique equivalent to parametric objective function analysis to generate the set of efficient solutions to the problem.     

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.     

Routing open shop and flow shop scheduling problems

We consider a generalization of the classical open shop and flow shop scheduling problems where the jobs are located at the vertices of an undirected graph and the machines, initially located at the same vertex, have to travel along the graph to process the jobs. The objective is to minimize the makespan. In the tour-version the makespan means the time by which each machine has processed all jobs and returned to the initial location. While in the path-version the makespan represents the maximum completion time of the jobs. We present improved approximation algorithms for various cases of the open shop problem on a general graph, and the tour-version of the two-machine flow shop problem on a tree. Also, we prove that both versions of the latter problem are NP-hard, which answers an open question posed in the literature.     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

Connectedness of Efficient Solutions in Multiple Objective Combinatorial Optimization

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. However, we show that many classical multiple objective combinatorial optimization problems do not possess the connectedness property in general, including, among others, knapsack problems (and even several special cases) and linear assignment problems. We also extend 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 two variants of the knapsack problem to analyze the likelihood with which non-connected adjacency graphs occur in randomly generated instances.     

Weakly and properly nonessential objectives in multiobjective optimization problems

In this work we characterize objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be weakly nonessential (properly nonessential) are presented. We establish relations between weakly nonessential, properly nonessential and nonessential functions.