Organizational Unit:

H. Milton Stewart School of Industrial and Systems Engineering

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https://hdl.handle.net/1853/70758

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Organizational Unit

College of Engineering

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Organizational Unit

William M. Keck Virtual Factory Lab (VFL)

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https://finding-aids.library.gatech.edu/agents/corporate_entities/1032

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Publication Search Results

Now showing 1 - 10 of 39

Decomposition Methods in Column Generation and Data-Driven Stochastic Optimization

(Georgia Institute of Technology, 2021-12-13) El Tonbari, Mohamed Ali El Moghazi

In this thesis, we are focused on tackling large-scale problems arising in two-stage stochastic optimization and the related Dantzig-Wolfe decomposition. We start with a deterministic setting, where we consider linear programs with a block-structure, but data cannot be stored centrally due to privacy concerns or decentralized storage of large datasets. The larger portion of the thesis is dedicated to the stochastic setting, where we study two-stage distributionally robust optimization under the Wasserstein ambiguity set to tackle problems with limited data. In Chapter 2, joint work with Shabbir Ahmed, we propose a fully distributed Dantzig-Wolfe decomposition (DWD) algorithm using the Alternating Direction Method of Multipliers (ADMM) method. DWD is a classical algorithm used to solve large-scale linear programs whose constraint matrix is a set of independent blocks coupled with a set of linking rows but requires to solve a master problem centrally, which can be undesirable or infeasible in certain cases due to privacy concerns or decentralized storage of data. To this end, we develop a consensus-based Dantzig-Wolfe decomposition algorithm where the master problem is solved in a distributed fashion. We detail the computational and algorithmic challenges of our method, provide bounds on the optimality gap and feasibility violation, and perform extensive computational experiments on instances of the cutting stock problem and synthetic instances using a Message Passing Interface (MPI) implementation, where we obtain high-quality solutions in reasonable time. In Chapter 3 and 4, we turn our focus to stochastic optimization, specifically applications where data is scarce and the underlying probability distribution is difficult to estimate. Chapter 3 is joint work with Anirudh Subramanyam and Kibaek Kim. Here, we consider two-stage conic DRO under the Wasserstein ambiguity set with zero-one uncertainties. We are motivated by problems arising in network optimization, where binary random variables represent failures of network components. We are interested in applications where such failures are rare and have a high impact, making it difficult to estimate failure probabilities. By using ideas from bilinear programming and penalty methods, we provide tractable approximations of our two-stage DRO model which can be iteratively improved using lift-and-project techniques. We illustrate the computational and out-of-sample performance of our method on the optimal power flow problem with random transmission line failures and a multi-commodity network design problem with random node failures. In Chapter 4, joint work with Alejandro Toriello and George Nemhauser, we study a two-stage model which arises in natural disaster management applications, where the first stage is a facility location problem, deciding where to open facilities and pre-allocate resources, and the second stage is a fixed-charge transportation problem, routing resources to affected areas after a disaster. We solve a two-stage DRO model under the Wasserstein set to deal with the lack of available data. The presence of binary variables in the second stage significantly complicates the problem. We develop an efficient column-and-constraint generation algorithm by leveraging the structure of our support set and second-stage value function, and show our results extend to the case where the second stage is a fixed-charge network flow problem. We provide a detailed discussion on our implementation, and end the chapter with computational experiments on synthetic instances and a case study of hurricane threats on the coastal states of the United States. We end the thesis with concluding remarks and potential directions for future research.
Scalable Approaches to Time-Dependent Path and Routing Problems

(Georgia Institute of Technology, 2021-07-07) He, Edward Yuhang

Many combinatorial optimization problems involve an aspect of time, whether due to changing conditions, updated information, or being under a continuous decision-making process. A popular domain for time-based decision making is logistics, where it is not only important to decide along which routes products need to be transported, but also when the transportation should occur. The timing for these problems is critical to take advantage of changing operational conditions such as traffic, capacity, and costs. Unfortunately, introducing time as an additional dimension significantly increases the size of models over their static counterparts. In fact, solving such models is often done via heuristics as existing solution methods are computationally intractable. This dissertation aims to provide insights into how scalable solution methodologies can be developed for such problems in path and routing problems.
Resource allocation optimization problems in the public sector

(Georgia Institute of Technology, 2020-04-25) Leonard, Taylor Joseph

This dissertation consists of three distinct, although conceptually related, public sector topics: the Transportation Security Agency (TSA), U.S. Customs and Border Patrol (CBP), and the Georgia Trauma Care Network Commission (GTCNC). The topics are unified in their mathematical modeling and mixed-integer programming solution strategies. In Chapter 2, we discuss strategies for solving large-scale integer programs to include column generation and the known heuristic of particle swarm optimization (PSO). In order to solve problems with an exponential number of decision variables, we employ Dantzig-Wolfe decomposition to take advantage of the special subproblem structures encountered in resource allocation problems. In each of the resource allocation problems presented, we concentrate on selecting an optimal portfolio of improvement measures. In most cases, the number of potential portfolios of investment is too large to be expressed explicitly or stored on a computer. We use column generation to effectively solve these problems to optimality, but are hindered by the solution time and large CPU requirement. We explore utilizing multi-swarm particle swarm optimization to solve the decomposition heuristically. We also explore integrating multi-swarm PSO into the column generation framework to solve the pricing problem for entering columns of negative reduced cost. In Chapter 3, we present a TSA problem to allocate security measures across all federally funded airports nationwide. This project establishes a quantitative construct for enterprise risk assessment and optimal resource allocation to achieve the best aviation security. We first analyze and model the various aviation transportation risks and establish their interdependencies. The mixed-integer program determines how best to invest any additional security measures for the best overall risk protection and return on investment. Our analysis involves cascading and inter-dependency modeling of the multi-tier risk taxonomy and overlaying security measurements. The model selects optimal security measure allocations for each airport with the objectives to minimize the probability of false clears, maximize the probability of threat detection, and maximize the risk posture (ability to mitigate risks) in aviation security. The risk assessment and optimal resource allocation construct are generalizable and are applied to the CBP problem. In Chapter 4, we optimize security measure investments to achieve the most cost-effective deterrence and detection capabilities for the CBP. A large-scale resource allocation integer program was successfully modeled that rapidly returns good Pareto optimal results. The model incorporates the utility of each measure, the probability of success, along with multiple objectives. To the best of our knowledge, our work presents the first mathematical model that optimizes security strategies for the CBP and is the first to introduce a utility factor to emphasize deterrence and detection impact. The model accommodates different resources, constraints, and various types of objectives. In Chapter 5, we analyze the emergency trauma network problem first by simulation. The simulation offers a framework of resource allocation for trauma systems and possible ways to evaluate the impact of the investments on the overall performance of the trauma system. The simulation works as an effective proof of concept to demonstrate that improvements to patient well-being can be measured and that alternative solutions can be analyzed. We then explore three different formulations to model the Emergency Trauma Network as a mixed-integer programming model. The first model is a Multi-Region, Multi-Depot, Multi-Trip Vehicle Routing Problem with Time Windows. This is a known expansion of the vehicle routing problem that has been extended to model the Georgia trauma network. We then adapt an Ambulance Routing Problem (ARP) to the previously mentioned VRP. There are no known ARPs of this magnitude/extension of a VRP. One of the primary differences is many ARPs are constructed for disaster scenarios versus day-to-day emergency trauma operations. The new ARP also implements more constraints based on trauma level limitations for patients and hospitals. Lastly, the Resource Allocation ARP is constructed to reflect the investment decisions presented in the simulation.
Topics in network optimization: The Steiner tree problem and semiconductor manufacturing

(Georgia Institute of Technology, 2020-01-06) Siebert Sandoval, Matias Ignacio

This thesis covers two very different topics related to problems defined on graphs. The first is a fundamental graph optimization problem called the Steiner tree problem. The second is an applied problem in semiconductor manufacturing. In the first part of this thesis, we propose a new approach to solve the Steiner tree problem when the number of terminal nodes is fixed. The Steiner tree problem is a classical network design problem. We are given a graph G = (V,E), a set of terminal nodes R, subset of V, and a non-negative cost vector for the edges in E. The problem is to find the minimum weight tree in G that spans all nodes in R. We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is integral so that it can be solved as a linear program. However, the number of IPs grows exponentially with the number of terminals in the Steiner tree. As a consequence, we are able to solve the Steiner tree problem by solving a polynomial number of LPs, when the number of terminals is fixed. To address the latter issue, we propose a local-search based algorithm to solve the problem for big instances. First, we propose a dynamic programming algorithm to solve each IP efficiently. Then, we provide a characterization of the neighborhood of each IP. Finally, we propose a simulated annealing framework to solve the problem. We present computational results for a large set of instances in the SteinLib library, comparing our proposed approach with state-of-the-art algorithms to solve the directed version of the Steiner tree problem. We study over 800 instances from the SteinLib library, and we show that the solution quality of the proposed approach surpasses the quality of the solution of the state-of-the-art algorithms. In the second part of this thesis, we study the semiconductor manufacturing problem, which is a highly complex and dynamic re-entrant process where wafers go through several processing steps, entering the same group of machines multiple times. We propose a fluid model lot dispatching policy that iteratively optimizes lot selection based on current work-in-progress (WIP) distribution of the entire system. A fluid model is an approximation technique used to model and study the dynamics of a stochastic queuing network framework. Furthermore, we propose to split the decision policies into two phases in order to include travel time information into the dispatching and targeting decisions. We provide simulation results for a prototype facility that show that our proposed policies outperform commonly used dispatching rules in throughput, machine utilization, and machine target accuracy.
Distributed integer programming

(Georgia Institute of Technology, 2017-08-15) Karabulut, Ezgi

In this thesis, we study distributed integer programming problems that involve multiple players with integer programming problems linked together with a common resource constraint. Our goal is to design decentralized algorithms that do not require a central processor to allocate the resource across the players to solve the overall problem. The algorithms that we design have optimality guarantees when applied to problems for which the marginal value of each additional resource is non-increasing. For problems that do not have this step-wise concave structure, we propose approximation algorithms and provide error bounds. We also perform experiments to evaluate the algorithms' average performance on problems without the desired structure. Finally, we consider the same problem in an online setting. We show that there exists no deterministic online algorithms for our problem that has the state of the art error bound. Therefore we propose a randomized decentralized online algorithm for our problem whose error bound matches the results in the literature.
Track layout accommodating dynamic routing in automated material handling systems

(Georgia Institute of Technology, 2016-05-10) Lee, Junho

Modern semiconductor fabrication facilities depend on automated material handling systems (AMHSs) to manage the processes and variations in time required to produce advanced semiconductor products. Unified AMHSs, which operate overhead hoist transport vehicles, support direct tool-to-tool transfers between different bays. AMHSs continue to use static routing, but dynamic routing schemes have been studied to improve traffic conditions. In this dissertation, we propose an approach integrating optimization and simulation to design a track layout of an AMHS employing dynamic routing. We present a network design problem accommodating alternative paths. Our formulation is based on a multi-commodity network design problem, and we add secondary flow variables to represent rerouted vehicles. The problem requires input data, especially a base graph and commodities, reflecting realistic traffic conditions. We use simulation to validate the design from the optimization problem and provide its input data. We obtain a solution design using a heuristic that combines optimization and simulation. Our computational results illustrate that the parameter that controls the uniqueness of alternative paths has a significant impact on the routing performance of vehicles.
Robust optimization with applications in maritime inventory routing

(Georgia Institute of Technology, 2015-02-20) Zhang, Chengliang

In recent years, the importance of incorporating uncertainty into planning models for logistics and transportation systems has been widely recognized in the Operations Research and transportation science communities. Maritime transportation, as a major mode of transport in the world, is subject to a wide range of disruptions at the strategic, tactical and operational levels. This thesis is mainly concerned with the development of robustness planning strategies that can mitigate the effects of some major types of disruptions for an important class of optimization problems in the shipping industry. Such problems arise in the creation and negotiation of long-term delivery contracts with customers who require on-time deliveries of high-value goods throughout the year. In this thesis, we consider the disruptions that can increase travel times between ports and ultimately affect one or more scheduled deliveries to the customers. Computational results show that our integrated solution procedure and robustness planning strategies can generate delivery plans that are both economical as well as robust against uncertain disruptions.
Congestion-aware dynamic routing in automated material handling systems

(Georgia Institute of Technology, 2014-08-20) Bartlett, Kelly K.

In semiconductor manufacturing, automated material handling systems (AMHSs) transport wafers through a complex re-entrant manufacturing process. In some systems, Overhead Hoist Transport (OHT) vehicles move throughout the facility on a ceiling-mounted track system, delivering wafers to machines and storage locations. To improve efficiency in such systems, this thesis proposes an adaptive dynamic routing approach that allows the system to self-regulate, reducing steady-state travel times by 4-6% and avoiding excessive congestion and deadlock. Our approach allows vehicles to be rerouted while in progress in response to changes in the location and severity of congestion as measured by edge traversal time estimates updated via exponential smoothing. Our proposed method is efficient enough to be used in a large system where several routing decisions are made each second. We also consider how the effectiveness of a AMHS layout differs between static and dynamic routing. We demonstrate that dynamic routing significantly reduces sensitivity to shortcut placement and allows an eight-fold increase in the number of shortcuts along the center loop. This reduces travel times by an additional 24%. To demonstrate the effectiveness of our proposed routing approach, we use a high-fidelity simulation of vehicle movement. To test the impact of routing methods on layout effectiveness, we developed an associated Excel-based automated layout generation tool that allows the efficient generation of thousands of candidate layouts. The user selects from among a set of modular templates to create a design and all simulation files are generated with the click of a button.
Topics in discrete optimization: models, complexity and algorithms

(Georgia Institute of Technology, 2013-08-12) He, Qie

In this dissertation we examine several discrete optimization problems through the perspectives of modeling, complexity and algorithms. We first provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff. Under a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut. The analysis also suggests some guidelines on when type 1 triangle cuts are likely to be more effective than split cuts and vice versa. We next study a minimum concave cost network flow problem over a grid network. We give a polytime algorithm to solve this problem when the number of echelons is fixed. We show that the problem is NP-hard when the number of echelons is an input parameter. We also extend our result to grid networks with backward and upward arcs. Our result unifies the complexity results for several models in production planning and green recycling including the lot-sizing model, and gives the first polytime algorithm for some problems whose complexities were not known before. Finally, we examine how much complexity randomness will bring to a simple combinatorial optimization problem. We study a problem called the sell or hold problem (SHP). SHP is to sell k out of n indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. Although the deterministic version of SHP is trivial to solve, we show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number of scenarios in the second stage is constant. A max{1/2,k/n}-approximation algorithm is presented for the scenario-based SHP.
Lower bounds for integer programming problems

(Georgia Institute of Technology, 2013-05-13) Li, Yaxian

Solving real world problems with mixed integer programming (MIP) involves efforts in modeling and efficient algorithms. To solve a minimization MIP problem, a lower bound is needed in a branch-and-bound algorithm to evaluate the quality of a feasible solution and to improve the efficiency of the algorithm. This thesis develops a new MIP model and studies algorithms for obtaining lower bounds for MIP. The first part of the thesis is dedicated to a new production planning model with pricing decisions. To increase profit, a company can use pricing to influence its demand to increase revenue, decrease cost, or both. We present a model that uses pricing discounts to increase production and delivery flexibility, which helps to decrease costs. Although the revenue can be hurt by introducing pricing discounts, the total profit can be increased by properly choosing the discounts and production and delivery decisions. We further explore the idea with variations of the model and present the advantages of using flexibility to increase profit. The second part of the thesis focuses on solving integer programming(IP) problems by improving lower bounds. Specifically, we consider obtaining lower bounds for the multi- dimensional knapsack problem (MKP). Because MKP lacks special structures, it allows us to consider general methods for obtaining lower bounds for IP, which includes various relaxation algorithms. A problem relaxation is achieved by either enlarging the feasible region, or decreasing the value of the objective function on the feasible region. In addition, dual algorithms can also be used to obtain lower bounds, which work directly on solving the dual problems. We first present some characteristics of the value function of MKP and extend some properties from the knapsack problem to MKP. The properties of MKP allow some large scale problems to be reduced to smaller ones. In addition, the quality of corner relaxation bounds of MKP is considered. We explore conditions under which the corner relaxation is tight for MKP, such that relaxing some of the constraints does not affect the quality of the lower bounds. To evaluate the overall tightness of the corner relaxation, we also show the worst-case gap of the corner relaxation for MKP. To identify parameters that contribute the most to the hardness of MKP and further evaluate the quality of lower bounds obtained from various algorithms, we analyze the characteristics that impact the hardness of MKP with a series of computational tests and establish a testbed of instances for computational experiments in the thesis. Next, we examine the lower bounds obtained from various relaxation algorithms com- putationally. We study methods of choosing constraints for relaxations that produce high- quality lower bounds. We use information obtained from linear relaxations to choose con- straints to relax. However, for many hard instances, choosing the right constraints can be challenging, due to the inaccuracy of the LP information. We thus develop a dual heuristic algorithm that explores various constraints to be used in relaxations in the Branch-and- Bound algorithm. The algorithm uses lower bounds obtained from surrogate relaxations to improve the LP bounds, where the relaxed constraints may vary for different nodes. We also examine adaptively controlling the parameters of the algorithm to improve the performance. Finally, the thesis presents two problem-specific algorithms to obtain lower bounds for MKP: A subadditive lifting method is developed to construct subadditive dual solutions, which always provide valid lower bounds. In addition, since MKP can be reformulated as a shortest path problem, we present a shortest path algorithm that uses estimated distances by solving relaxations problems. The recursive structure of the graph is used to accelerate the algorithm. Computational results of the shortest path algorithm are given on the testbed instances.