The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

Combinatorial Optimization by Dynamic Contraction

A heuristic optimization methodology, Dynamic Contraction (DC), is introduced as an approach for solving a wide variety of hard combinatorial problems. Contraction is an operation that maps an instance of a problem to a smaller instance of the same problem. DC is an iterative improvement strategy that relies on contraction as a mechanism for escaping local minima. As a byproduct of contraction, efficiency is improved due to a reduction of problem size. Effectiveness of DC is shown through simple applications to two classical combinatorial problems: The graph bisection problem and the traveling salesman problem.     

Manufacturing cell formation using similarity coefficients and a parallel genetic TSP algorithm: Formulation and comparison

Many algorithms have been proposed to form manufacturing cells from component routings. However, many of these do not have the capability of solving large problems. We propose a procedure using similarity coefficients and a parallel genetic implementation of a TSP algorithm that is capable of solving large problems of up to 1000 parts and 1000 machines. In addition, we also compare our procedure with many existing procedures using nine well-known problems from the literature.

The results show that the proposed procedure compares well with the existing procedures and should be useful to practitioners and researchers.     

一些非线性发展方程的显式行波解

该文给出了一种构造非线性发展方程显式行波解的方法并用该方法得到了Hirota-Satsuma方程组,一类非线性常微分方程以及广义耦合标量场方程组的显式行波解.     

Explicit Traveling Wave Solutions and Their Dynamical Behaviors for the Coupled Higgs Field Equation

In this paper, we focus on the traveling wave solutions of the coupled Higgs field equation from the perspective of dynamical systems. Through the phase portraits, in addition to periodic wave solutions and solitary wave solutions, we also obtain explicit periodic singular wave solutions, singular wave solutions and kink wave solutions, which were not found in the previous works. The dynamical behavior of these solutions and their internal relations are revealed through asymptotic analysis. The results will help supplement the study of field equation.     

Thermodynamical approach to the traveling salesman problem: An efficient simulation algorithm

We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprisingly enough, using this simple algorithm, one can get very close to the optimal solution of the problem or even find the true optimum. We demonstrate this on several examples.We conjecture that the analogy with thermodynamics can offer a new insight into optimization problems and can suggest efficient algorithms for solving them.The author acknowledges stimulating discussions with J. Piút concerning the main ideas of the present paper. The author is also indebted to P. Brunovský, J. erný, M. Hamala, . Peko, . Znám, and R. Zajac for useful comments.     

Tight bounds for christofides' traveling salesman heuristic

Christofides [1] proposes a heuristic for the traveling salesman problem that runs in polynomial time. He shows that when the graphG = (V, E) is complete and the distance matrix defines a function onV × V that is metric, then the length of the Hamiltonian cycle produced by the heuristic is always smaller than 3/2 times the length of an optimal Hamiltonian cycle. The purpose of this note is to refine Christofides' worst-case analysis by providing a tight bound for everyn 3, wheren is the number of vertices of the graph. We also show that these bounds are still tight when the metric is restricted to rectilinear distances, or to Euclidean distances for alln 6.This work was supported, in part, by NSF Grant ENG 75-00568 to Cornell University. This work was done when the authors were affiliated with the Center for Operations Research and Econometrics, University of Louvain, Belgium.     

Computational analysis of glycolytic reaction in open spatial reactor

We consider the computational analysis of processes within the spatially-distributed model simulating the glycolytic reaction taking place in the one-side fed open chemical reactor. The main point of the simulation is the decomposition of the reaction–diffusion system into unidirectional reaction in a bulk supplied by feedback terms stated as boundary conditions on the lower boundary of the reactor, i.e. the unique plane where an exchange with an outer medium is possible within the real experimental situation. Analysis of the curvature of the reagents distribution curves proves kinematic character of the observed lateral waves corresponding to the picture of experimentally observed glycolytic traveling waves. At the same time, their origin relates to diffusion of the reagents in a vertical cross-section of the reactor. Study of the solutions for the concerned reaction–diffusion model in the case of stochastically different diffusion coefficients reveals the Turing structures.