Meta-heuristics for dynamic lot sizing: A review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinatorial optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation, neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig–Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples.     

Rollout Algorithms for Stochastic Scheduling Problems

Stochastic scheduling problems are difficult stochastic control problems with combinatorial decision spaces. In this paper we focus on a class of stochastic scheduling problems, the quiz problem and its variations. We discuss the use of heuristics for their solution, and we propose rollout algorithms based on these heuristics which approximate the stochastic dynamic programming algorithm. We show how the rollout algorithms can be implemented efficiently, with considerable savings in computation over optimal algorithms. We delineate circumstances under which the rollout algorithms are guaranteed to perform better than the heuristics on which they are based. We also show computational results which suggest that the performance of the rollout policies is near-optimal, and is substantially better than the performance of their underlying heuristics.     

Biased random-key genetic algorithms for combinatorial optimization

Random-key genetic algorithms were introduced by Bean (ORSA J. Comput. 6:154–160, 1994) for solving sequencing problems in combinatorial optimization. Since then, they have been extended to handle a wide class of combinatorial optimization problems. This paper presents a tutorial on the implementation and use of biased random-key genetic algorithms for solving combinatorial optimization problems. Biased random-key genetic algorithms are a variant of random-key genetic algorithms, where one of the parents used for mating is biased to be of higher fitness than the other parent. After introducing the basics of biased random-key genetic algorithms, the paper discusses in some detail implementation issues, illustrating the ease in which sequential and parallel heuristics based on biased random-key genetic algorithms can be developed. A survey of applications that have recently appeared in the literature is also given.     

Combinatorial optimization with interaction costs: Complexity and solvable cases

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.     

经典一维装箱问题近似算法的研究进展

自20世纪70年代开始,随着计算复杂性理论的建立,近似算法逐渐成为组合优化的重要研究方向。作为第一批研究对象,装箱问题引起了组合优化领域学者的极大关注。装箱问题模型简单、拓展性强,广泛出现在各种带容量约束的资源分配问题中。除了在物流装载和材料切割等方面愈来愈重要的应用外,装箱算法的任何理论突破都关乎到整个组合优化领域的发展。直到今天,对装箱问题近似算法的研究仍如火如荼。本文主要针对一维模型,简述若干经典Fit算法的发展历程,分析基于线性规划松弛的近似方案的主要思路,总结当前的研究现状并对未来的研究提供一些参考建议。     

经典一维装箱问题近似算法的研究进展

自20世纪70年代开始,随着计算复杂性理论的建立,近似算法逐渐成为组合优化的重要研究方向。作为第一批研究对象,装箱问题引起了组合优化领域学者的极大关注。装箱问题模型简单、拓展性强,广泛出现在各种带容量约束的资源分配问题中。除了在物流装载和材料切割等方面愈来愈重要的应用外,装箱算法的任何理论突破都关乎到整个组合优化领域的发展。直到今天,对装箱问题近似算法的研究仍如火如荼。本文主要针对一维模型,简述若干经典Fit算法的发展历程,分析基于线性规划松弛的近似方案的主要思路,总结当前的研究现状并对未来的研究提供一些参考建议。     

On generation of permutations through decomposition of symmetric groups into cosets

A hardware-oriented algorithm for generating permutations is presented that takes as a theoretic base an iterative decomposition of the symmetric groupS _n into cosets. It generates permutations in a new order. Simple ranking and unranking algorithms are given. The construction of a permutation generator is proposed which contains a cellular permutation network as a main component. The application of the permutation generator for solving a class of combinatorial problems on parallel computers is suggested.     

Hybrid Rollout Approaches for the Job Shop Scheduling Problem

In this paper, we focus on heuristic approaches for solving the deterministic job shop scheduling problem. More specifically, a new priority dispatch rule and hybrid rollout algorithms are developed for approaching the problem under consideration. The proposed solution algorithms are tested on a set of instances taken from the literature and compared with other methods. The computational results validate the effectiveness of the developed solution approaches and show that the proposed rollout algorithms are competitive with respect to several state-of-art heuristics for solving the job shop scheduling problem. The author thanks Dr. Marco Mancini and Dr. Alessandro Tarasio for valuable suggestions about computational issues.     

Parallel computing in combinatorial optimization

This is a review of the literature on parallel computers and algorithms that is relevant for combinatorial optimization. We start by describing theoretical as well as realistic machine models for parallel computations. Next, we deal with the complexity theory for parallel computations and illustrate the resulting concepts by presenting a number of polylog parallel algorithms andP-completeness results. Finally, we discuss the use of parallelism in enumerative methods.     

Computation of general correlation coefficients for interval data

This paper provides a comprehensive analysis of computational problems concerning calculation of general correlation coefficients for interval data. Exact algorithms solving this task have unacceptable computational complexity for larger samples, therefore we concentrate on computational problems arising in approximate algorithms. General correlation coefficients for interval data are also given by intervals. We derive bounds on their lower and upper endpoints. Moreover, we propose a set of heuristic solutions and optimization methods for approximate computation. Extensive simulation experiments show that the heuristics yield very good solutions for strong dependencies. In other cases, global optimization using evolutionary algorithm performs best. A real data example of autocorrelation of cloud cover data confirms the applicability of the approach.     

Experiments with parallel algorithms for combinatorial problems

In the last decade many models for parallel computation have been proposed and many parallel algorithms have been developed. However, few of these models have been realized and most of these algorithms are supposed to run on idealized, unrealistic parallel machines.The parallel machines constructed so far all use a simple model of parallel computation. Therefore, not every existing parallel machine is equally well suited for each type of algorithm. The adaptation of a certain algorithm to a specific parallel architecture may severely increase the complexity of the algorithm or severely obscure its essence.Little is known about the performance of some standard combinatorial algorithms on existing parallel machines. In this paper we present computational results concerning the solution of knapsack, shortest paths and change-making problems by branch and bound, dynamic programming, and divide and conquer algorithms on the ICL-DAP (an SIMD computer), the Manchester dataflow machine and the CDC-CYBER-205 (a pipeline computer).     

Solving combinatorial problems with combined Min-Max-Min-Sum objective and applications

The purpose of this paper is to introduce and study a new class of combinatorial optimization problems in which the objective function is the algebraic sum of a bottleneck cost function (Min-Max) and a linear cost function (Min-Sum). General algorithms for solving such problems are described and general complexity results are derived. A number of examples of application involving matchings, paths and cutsets, matroid bases, and matroid intersection problems are examined, and the general complexity results are specialized to each of them. The interest of these various problems comes in particular from their strong relation to other important and difficult combinatorial problems such as: weighted edge coloring of a graph; optimum weighted covering with matroid bases; optimum weighted partitioning with matroid intersections, etc. Another important area of application of the algorithms given in the paper is bicriterion analysis involving a Min-Max criterion and a Min-Sum one.     

Solving large quadratic assignment problems on computational grids

The quadratic assignment problem (QAP) is among the hardest combinatorial optimization problems. Some instances of size n = 30 have remained unsolved for decades. The solution of these problems requires both improvements in mathematical programming algorithms and the utilization of powerful computational platforms. In this article we describe a novel approach to solve QAPs using a state-of-the-art branch-and-bound algorithm running on a federation of geographically distributed resources known as a computational grid. Solution of QAPs of unprecedented complexity, including the nug30, kra30b, and tho30 instances, is reported. Received: September 29, 2000 / Accepted: June 5, 2001?Published online October 2, 2001     

Circulant weighing matrices: a demanding challenge for parallel optimization metaheuristics

Circulant weighing matrices constitute a special type of combinatorial matrices that have attracted scientific interest for many years. The existence and determination of specific classes of circulant weighing matrices remains an active research area that involves both theoretical algebraic techniques as well as high-performance computational optimization approaches. The present work aims at investigating the potential of four established parallel metaheuristics as well as a special Algorithm Portfolio approach, on solving such problems. For this purpose, the algorithms are applied on a hard circulant weighing matrix existence problem. The obtained results are promising, offering insightful conclusions.     

Geometric Applications of a Randomized Optimization Technique

We propose a simple, general, randomized technique to reduce certain geometric optimization problems to their corresponding decision problems. These reductions increase the expected time complexity by only a constant factor and eliminate extra logarithmic factors in previous, often more complicated, deterministic approaches (such as parametric searching). Faster algorithms are thus obtained for a variety of problems in computational geometry: finding minimal k -point subsets, matching point sets under translation, computing rectilinear p -centers and discrete 1-centers, and solving linear programs with k violations. Received May 23, 1998, and in revised form March 29, 1999.     

Complexity and algorithms for convex network optimization and other nonlinear problems

Nonlinear optimization algorithms are rarely discussed from a complexity point of view. Even the concept of solving nonlinear problems on digital computers is not well defined. The focus here is on a complexity approach for designing and analyzing algorithms for nonlinear optimization problems providing optimal solutions with prespecified accuracy in the solution space. We delineate the complexity status of convex problems over network constraints, dual of flow constraints, dual of multi-commodity, constraints defined by a submodular rank function (a generalized allocation problem), tree networks, diagonal dominant matrices, and nonlinear Knapsack problem's constraint. All these problems, except for the latter in integers, have polynomial time algorithms which may be viewed within a unifying framework of a proximity-scaling technique or a threshold technique. The complexity of many of these algorithms is furthermore best possible in that it matches lower bounds on the complexity of the respective problems. In general nonseparable optimization problems are shown to be considerably more difficult than separable problems. We compare the complexity of continuous versus discrete nonlinear problems and list some major open problems in the area of nonlinear optimization. MSC classification: 90C30, 68Q25     

A special class of set covering problems

A class of set covering problems is being introduced. This class is obtained from reformulation of a well-known combinatorial problem of Erdös on the hypercube. An algorithmic method of solution to the problem is proposed. Max-flow algorithms are the main ingredients of our method.The computational results which will be presented here, improves the best existing bound related to the combinatorial problem. This, at the same time, provides a good approximate solution to the corresponding set covering problem of more than a thousand variables and constraints.Moreover, we show that our special class of problems can be recognized from the class of all set covering problems, by a polynomial algorithm with O (MN) complexity, where M and N are numbers of constraints and variables of a given instant, respectively.This research is supported by EPSCOR-NSF of Puerto Rico and by FIPI, the institutional grant of University of Puerto Rico.     

Heuristics and reduction methods for multiple constraints 0–1 linear programming problems

This work extends the efficient results relative to the 0–1 knapsack problem to the multiple inequality constraints 0–1 linear programming problems. The two crucial phases for the solving of this type of problems are presented: (i) Two linear expected time complexity greedy algorithms are proposed for the determination of a lower bound on the optimal value by using a cascade of surrogate relaxations of the original problem whose sizes are decreasing step by step. A comparative study with the best known heuristic methods is reported; it concerned the accuracy of the approximate solutions and the practical computational times. (ii) This greedy algorithm is inserted in an efficient reduction framework. Variables and constraints are eliminated by the conjunction of tests applied to Lagrangean and surrogate relaxations of the original problem. A lot of computational results are summarized by considering test problems of the literature.     

Molecular conformation on the CM-5 by parallel two-level simulated annealing

In this paper, we propose a new kind of simulated annealing algorithm calledtwo-level simulated annealing for solving certain class of hard combinatorial optimization problems. This two-level simulated annealing algorithm is less likely to get stuck at a non-global minimizer than conventional simulated annealing algorithms. We also propose a parallel version of our two-level simulated annealing algorithm and discuss its efficiency. This new technique is then applied to the Molecular Conformation problem in 3 dimensional Euclidean space. Extensive computational results on Thinking Machines CM-5 are presented. With the full Lennard-Jones potential function, we were able to get satisfactory results for problems for cluster sizes as large as 100,000. A peak rate of over 0.8 giga flop per second in 64-bit operations was sustained on a partition with 512 processing elements. To the best of our knowledge, ground states of Lennard-Jones clusters of size as large as these have never been reported before.Also a researcher at the Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN 55415     

Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives.