Algorithms for discrete, non-linear and robust optimization problems with applications in scheduling and service operations

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2011.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 101-107).This thesis presents efficient algorithms that give optimal or near-optimal solutions for problems with non-linear objective functions that arise in discrete, continuous and robust optimization. First, we present a general framework for designing approximation schemes for combinatorial optimization problems in which the objective function is a combination of more than one function. Examples of such problems include those in which the objective function is a product or ratio of two or more linear functions, parallel machine scheduling problems with the makespan objective, robust versions of weighted multi-objective optimization problems, and assortment optimization problems with logit choice models. For many of these problems, we give the first fully polynomial time approximation scheme using our framework. Next, we present approximation schemes for optimizing a rather general class of non-linear functions of low rank over a polytope. In contrast to existing results in the literature, our approximation scheme does not require the assumption of quasi-concavity of the objective function. For the special case of minimizing a quasi-concave function of low-rank, we give an alternative algorithm which always returns a solution which is an extreme point of the polytope. This algorithm can also be used for combinatorial optimization problems where the objective is to minimize a quasi-concave function of low rank. We also give complexity-theoretic results with regards to the inapproximability of minimizing a concave function over a polytope. Finally, we consider the problem of appointment scheduling in a robust optimization framework. The appointment scheduling problem arises in many service operations, for example health care. For each job, we are given its minimum and maximum possible execution times. The objective is to find an appointment schedule for which the cost in the worst case scenario of the realization of the processing times of the jobs is minimized. We present a global balancing heuristic, which gives an easy to compute closed form optimal schedule when the underage costs of the jobs are non-decreasing. In addition, for the case where we have the flexibility of changing the order of execution of the jobs, we give simple heuristics to find a near-optimal sequence of the jobs.by Shashi Mittal.Ph.D

Task Packing: Efficient task scheduling in unbalanced parallel programs to maximize CPU utilization

Load imbalance in parallel systems can be generated by external factors to the currently running applications like operating system noise or the underlying hardware like a heterogeneous cluster. HPC applications working on irregular data structures can also have difficulties to balance their computations across the parallel tasks. In this article we extend, improve and evaluate more deeply the Task Packing mechanism proposed in a previous work. The main idea of the mechanism is to concentrate the idle cycles of unbalanced applications in such a way that one or more CPUs are freed from execution. To achieve this, CPUs are stressed with just useful work of the parallel application tasks, provided performance is not degraded. The packing is solved by an algorithm based on the Knapsack problem, in a minimum number of CPUs and using oversubscription. We design and implement a more efficient version of such mechanism. To that end, we perform the Task Packing “in place”, taking advantage of idle cycles generated at synchronization points of unbalanced applications. Evaluations are carried out on a heterogeneous platform using FT and miniFE benchmarks. Results showed that our proposal generates low overhead. In addition the amount of freed CPUs are related to a load imbalance metric which can be used as a prediction for it.Peer ReviewedPostprint (author's final draft

Approximationsalgorithmen für Packungs- und Scheduling-Probleme

Algorithms for solving optimization problems play a major role in the industry. For example in the logistics industry, route plans have to be optimized according to various criteria. However, many natural optimization problems are hard to solve. That is, for many optimization problems no algorithms with running time polynomial in the size of the instance are known. Furthermore, it is a widely accepted assumption that many optimization problems do not allow algorithms that solve the problem optimally in polynomial time. One way of overcoming this dilemma is using approximation algorithms. These algorithms have a polynomial running time, but their solutions are in general not optimal but rather close to an optimum. The main subject of this thesis is approximation algorithms for packing and scheduling problems: For the three-dimensional orthogonal knapsack problem (OKP-3) without rotations we present algorithms with approximation ratios arbitrarily close to 9, 8 and 7. For OKP-3 with 90 degree rotations around the z-axis or around all axes, we present algorithms with approximation ratios arbitrarily close to 6 and 5, respectively. Both for the malleable and for the non-malleable case of the non-preemptive parallel job scheduling problem in which the number of available machines is polynomially bounded in the number of jobs, we present polynomial time approximation schemes. For the cases in which additionally the machines allotted to each job have to be contiguous, we show the existence of approximation algorithms with ratio arbitrarily close to 1.5

Explicit Building-Block Multiobjective Genetic Algorithms: Theory, Analysis, and Developing

This dissertation research emphasizes explicit Building Block (BB) based MO EAs performance and detailed symbolic representation. An explicit BB-based MOEA for solving constrained and real-world MOPs is developed the Multiobjective Messy Genetic Algorithm II (MOMGA-II) which is designed to validate symbolic BB concepts. The MOMGA-II demonstrates that explicit BB-based MOEAs provide insight into solving difficult MOPs that is generally not realized through the use of implicit BB-based MOEA approaches. This insight is necessary to increase the effectiveness of all MOEA approaches. In order to increase MOEA computational efficiency parallelization of MOEAs is addressed. Communications between processors in a parallel MOEA implementation is extremely important, hence innovative migration and replacement schemes for use in parallel MOEAs are detailed and tested. These parallel concepts support the development of the first explicit BB-based parallel MOEA the pMOMGA-II. MOEA theory is also advanced through the derivation of the first MOEA population sizing theory. The multiobjective population sizing theory presented derives the MOEA population size necessary in order to achieve good results within a specified level of confidence. Just as in the single objective approach the MOEA population sizing theory presents a very conservative sizing estimate. Validated results illustrate insight into building block phenomena good efficiency excellent effectiveness and motivation for future research in the area of explicit BB-based MOEAs. Thus the generic results of this research effort have applicability that aid in solving many different MOPs

Interconnection Networks Embeddings and Efficient Parallel Computations.

To obtain a greater performance, many processors are allowed to cooperate to solve a single problem. These processors communicate via an interconnection network or a bus. The most essential function of the underlying interconnection network is the efficient interchanging of messages between processes in different processors. Parallel machines based on the hypercube topology have gained a great respect in parallel computation because of its many attractive properties. Many versions of the hypercube have been introduced by many researchers mainly to enhance communications. The twisted hypercube is one of the most attractive versions of the hypercube. It preserves the important features of the hypercube and reduces its diameter by a factor of two. This dissertation investigates relations and transformations between various interconnection networks and the twisted hypercube and explore its efficiency in parallel computation. The capability of the twisted hypercube to simulate complete binary trees, complete quad trees, and rings is demonstrated and compared with the hypercube. Finally, the fault-tolerance of the twisted hypercube is investigated. We present optimal algorithms to simulate rings in a faulty twisted hypercube environment and compare that with the hypercube

The dimension of C1C^1 splines of arbitrary degree on a tetrahedral partition

We consider the linear space of piecewise polynomials in three variables which are globally smooth, i.e., trivariate $C^1$ splines. The splines are defined on a uniform tetrahedral partition $\Delta$, which is a natural generalization of the four-directional mesh. By using Bernstein-B{\´e}zier techniques, we establish formulae for the dimension of the $C^1$ splines of arbitrary degree

Stochastic Fractal Based Multiobjective Fruit Fly Optimization

The fruit fly optimization algorithm (FOA) is a global optimization algorithm inspired by the foraging behavior of a fruit fly swarm. In this study, a novel stochastic fractal model based fruit fly optimization algorithm is proposed for multiobjective optimization. A food source generating method based on a stochastic fractal with an adaptive parameter updating strategy is introduced to improve the convergence performance of the fruit fly optimization algorithm. To deal with multiobjective optimization problems, the Pareto domination concept is integrated into the selection process of fruit fly optimization and a novel multiobjective fruit fly optimization algorithm is then developed. Similarly to most of other multiobjective evolutionary algorithms (MOEAs), an external elitist archive is utilized to preserve the nondominated solutions found so far during the evolution, and a normalized nearest neighbor distance based density estimation strategy is adopted to keep the diversity of the external elitist archive. Eighteen benchmarks are used to test the performance of the stochastic fractal based multiobjective fruit fly optimization algorithm (SFMOFOA). Numerical results show that the SFMOFOA is able to well converge to the Pareto fronts of the test benchmarks with good distributions. Compared with four state-of-the-art methods, namely, the non-dominated sorting generic algorithm (NSGA-II), the strength Pareto evolutionary algorithm (SPEA2), multi-objective particle swarm optimization (MOPSO), and multiobjective self-adaptive differential evolution (MOSADE), the proposed SFMOFOA has better or competitive multiobjective optimization performance

Stochastic Fractal Based Multiobjective Fruit Fly Optimization

The fruit fly optimization algorithm (FOA) is a global optimization algorithm inspired by the foraging behavior of a fruit fly swarm. In this study, a novel stochastic fractal model based fruit fly optimization algorithm is proposed for multiobjective optimization. A food source generating method based on a stochastic fractal with an adaptive parameter updating strategy is introduced to improve the convergence performance of the fruit fly optimization algorithm. To deal with multiobjective optimization problems, the Pareto domination concept is integrated into the selection process of fruit fly optimization and a novel multiobjective fruit fly optimization algorithm is then developed. Similarly to most of other multiobjective evolutionary algorithms (MOEAs), an external elitist archive is utilized to preserve the nondominated solutions found so far during the evolution, and a normalized nearest neighbor distance based density estimation strategy is adopted to keep the diversity of the external elitist archive. Eighteen benchmarks are used to test the performance of the stochastic fractal based multiobjective fruit fly optimization algorithm (SFMOFOA). Numerical results show that the SFMOFOA is able to well converge to the Pareto fronts of the test benchmarks with good distributions. Compared with four state-of-the-art methods, namely, the non-dominated sorting generic algorithm (NSGA-II), the strength Pareto evolutionary algorithm (SPEA2), multi-objective particle swarm optimization (MOPSO), and multiobjective self-adaptive differential evolution (MOSADE), the proposed SFMOFOA has better or competitive multiobjective optimization performance