Combinatorics and Optimizationhttp://hdl.handle.net/10012/99282019-02-23T01:37:35Z2019-02-23T01:37:35ZA computational study of practical issues arising in short-term scheduling of a multipurpose facilityStevenson, Zachariahhttp://hdl.handle.net/10012/144042019-01-24T03:32:02Z2019-01-23T00:00:00ZA computational study of practical issues arising in short-term scheduling of a multipurpose facility
Stevenson, Zachariah
This thesis focuses on two important considerations when solving short term scheduling problems for multipurpose facilities: deciding when rescheduling should be performed and choosing efficient time representations for the scheduling problems. This class of scheduling problems is of practical importance as it may be used for scheduling chemical production facilities, flexible manufacturing systems, and analytical services facilities, among others. In these cases, improving the efficiency of scheduling operations may lead to increased yield, or reduced makespan, resulting in greater profits or customer satisfaction. Therefore, efficiently solving these problems is of great practical interest. One aspect of real world implementations of these problems is the presence of uncertainty, such as in the form of new jobs arriving, or a machine breaking down. In these cases, one may want or need to reschedule operations subject to the new disturbance. An investigation into how often to perform these reschedulings is addressed in the first part of the thesis. When formulating these problems, one must also choose a time representation for executing scheduling operations over. A dynamic approach is proposed in the second part of the thesis which we show can potentially yield substantial computational savings when scheduling over large instances.
The first part of this thesis addresses the question of when to reschedule operations for a facility that receives new jobs on a daily basis. Through computational experiments that vary plant parameters, such as the load and the capacity of a facility, we investigate the effects these parameters have on plant performance under periodic rescheduling. These experiments are carried out using real data from an industrial-scale facility. The results show that choosing a suitable rescheduling policy depends on some key plant parameters. In particular, by modifying various parameters of the facility, the performance ranking of the various rescheduling policies may be reversed compared to the results obtained with nominal parameter values. This highlights the need to consider both facility characteristics and what the crucial objective of the facility is when selecting a rescheduling policy.
The second part of this thesis deals with the issue of deciding which timepoints to include in our model formulations. In general, adding more timepoints to the model will offer more flexibility to the solver and hence result in more accurate schedules. However, these extra timepoints will also increase the size of the model and accordingly the computational cost of solving the model. We propose an iterative framework to refine an initial coarse uniform discretization, by adding key timepoints that may be most beneficial, and removing timepoints which are unnecessary from the model. This framework is compared against existing static discretizations using computational experiments on an analytical services facility. The results of these experiments demonstrate that when problems are sufficiently large, our proposed dynamic method is able to achieve a better tradeoff between objective value and CPU time than the currently used discretizations in the literature.
2019-01-23T00:00:00ZOn the Excluded Minors for Dyadic MatroidsWong, Chung-Yinhttp://hdl.handle.net/10012/143672019-01-18T03:30:40Z2019-01-17T00:00:00ZOn the Excluded Minors for Dyadic Matroids
Wong, Chung-Yin
The study of the class of dyadic matroids, the matroids representable over both $GF(3)$ and $GF(5)$, is a natural step to finding the excluded minors for $GF(5)$-representability. In this thesis we characterize the ternary matroids $M$ that are excluded minors for dyadic matroids and contains a 3-separation. We will show that one side of the separation has size at most four, and that $M$ is obtained by adding at most four elements to another excluded minor $M'$. This reduces the problem of finding the excluded minors for dyadic matroids to the problem of finding the vertically 4-connected excluded minors for dyadic matroids.
2019-01-17T00:00:00ZSplit Cuts From Sparse DisjunctionsYang, Shenghaohttp://hdl.handle.net/10012/143632019-01-17T03:30:38Z2019-01-16T00:00:00ZSplit Cuts From Sparse Disjunctions
Yang, Shenghao
Cutting planes are one of the major techniques used in solving Mixed-Integer Linear Programming (MIP) models. Various types of cuts have long been exploited by MIP solvers, leading to state-of-the-art performance in practice. Among them, the class of split cuts, which includes Gomory Mixed Integer (GMI) and Mixed Integer Rounding (MIR) cuts from tableaux, are arguably the most effective class of general cutting planes within a branch-and-cut framework. Sparsity, on the other hand, is a common characteristic of real-world MIP problems, and it is an important part of why the simplex method works so well inside branch-and-cut. A natural question, therefore, is to determine how sparsity can be incorporated into split cuts and how effective are split cuts that exploit sparsity. In this thesis, we evaluate the strength of split cuts that arise from sparse split disjunctions. In particular, we implement an approximate separation routine that separates only split cuts whose split disjunctions are sparse. We also present a straightforward way to exploit sparsity structure that is implicit in the MIP formulation. We run computational experiments and conclude that, one possibility to produce good split cuts is to try sparse disjunctions and exploit such structure.
2019-01-16T00:00:00ZEdge State TransferChen, Qiutinghttp://hdl.handle.net/10012/143462019-01-12T03:30:18Z2019-01-11T00:00:00ZEdge State Transfer
Chen, Qiuting
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a.
This thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer.
We prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t.
We give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs.
We close this thesis by a list of open questions.
2019-01-11T00:00:00Z