Combinatorics and Optimization

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This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 8 of 8

The complexity of some set-partitioning formulations for the vehicle routing problem with stochastic demands
(University of Waterloo, 2020-10-28) Gunter, Joshua; Fukasawa, Ricardo
The capacitated vehicle routing problem with stochastic demands (CVRPSD) is a variant of the deterministic capacitated vehicle routing problem (CVRP). The CVRP consists of planning routes for vehicles with a given capacity to deliver goods to a set of customers with known demands, with the goal being to find the cheapest such set of routes. In the CVRPSD, rather than being deterministic, customer demands are random variables from a given probability distribution. This creates the possibility of route failures, where the realized demand on a route is greater than the vehicle capacity. In this event, a recourse action following a pre-determined strategy must be taken. The goal is then to minimize the expected cost of the routes, i.e. the sum of the deterministic route lengths and expected additional costs incurred by recourse actions. A common approach when solving the CVRPSD is to formulate it as a 2-stage stochastic programming problem. In this framework, the first stage is planning a set of routes, while the second stage is computing the expected cost of route failures. While edge-based formulations were the dominant approach originally, the success of set-partitioning formulations for related problems such as the CVRP and the vehicle routing problem with time windows led to research into developing similar formulations for the CVRPSD. These formulations contain an exponential number of variables, necessitating the use of column generation to solve them. In column generation, an additional optimization problem known as the pricing problem needs to be repeatedly solved when solving the current LP relaxation through the branch-and-bound tree. The pricing problem for the deterministic CVRP is strongly NP-hard, and so to obtain tractable algorithms a relaxed version of the pricing problem which can be solved in pseudo-polynomial time is typically used. Similar methods for relaxing the pricing problem have been explored for the CVRPSD, but these make use of some simplifying assumptions for computing the expected cost of route failures, which needs to be done frequently when solving the pricing problem. In this thesis, we show that using these assumptions results in an ``approximate" pricing rather than ``exact" pricing, and present results on the hardness of performing exact pricing for set-partitioning formulations for the CVRPSD. Specifically, we show that when customer demands are given by a finite set of demand scenarios, exactly solving the pricing problem for the CVRPSD is strongly NP-hard. Additionally, we show that when customer demands are independent normal, under some assumptions there is a reduction from the Hamiltonian cycle problem to the pricing problem for the CVRPSD. This does not constitute a proof of strong NP-hardness due to the aforementioned assumptions required, but does suggest that even in this case the pricing problem may be harder than currently thought.
A computational study of practical issues arising in short-term scheduling of a multipurpose facility
(University of Waterloo, 2019-01-23) Stevenson, Zachariah; Fukasawa, Ricardo; Ricardez-Sandoval, Luis
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.
Computing with Multi-Row Intersection Cuts
(University of Waterloo, 2017-05-16) Xavier, Alinson Santos; Fukasawa, Ricardo
Cutting planes are one of the main techniques currently used to solve large-scale Mixed-Integer Linear Programming (MIP) models. Many important cuts used in practice, such as Gomory Mixed-Integer (GMI) cuts, are obtained by solving the linear relaxation of the MIP, extracting a single row of the simplex tableau, then applying integrality arguments to it. A natural extension, which has received renewed attention, is to consider cuts that can only be generated when considering multiple rows of the simplex tableau simultaneously. Although the theoretical importance of such multi-row cutting planes has been proved in a number of works, their effective use in practice remains a challenge. Since the entire class of multi-row cuts proves challenging to separate, one approach to obtain them is the following. First, the integral non-basic variables are fixed to zero. Then, a lattice-free set, which induces an intersection cut, is generated. Finally, the cut coefficients for the integral non-basic variables are computed by the so-called trivial lifting procedure. In this thesis, we address some computational aspects of this approach, and we make three novel contributions. In our first contribution, we describe a small subset of multi-row intersection cuts based on the infinity norm, which works for relaxations with arbitrary numbers of rows. We present an algorithm to generate them and run extensive computational experiments to evaluate their effectiveness. We conclude that these cuts yield benefits comparable to using the entire class of multi-row cuts, but at a small fraction of the computational cost. In our second contribution, we describe a practical method for performing the trivial lifting step on relaxations with two rows. Unlike previous methods, our method is applicable to intersection cuts derived from any lattice-free set, and, for maximal lattice-free sets, it is guaranteed to run in constant time. Computational experiments confirm that the algorithm is at least two orders of magnitudes faster than current alternatives. In our final contribution, we revisit single-row relaxations containing a single integral non-basic variable, with the goal of obtaining inequalities that are not dominated by GMI cuts. The novelty in our approach is that we use the framework of intersection cuts and trivial lifting, which allows us to obtain a geometric interpretation of our cuts, a fast algorithm for generating them, and an upper bound on their split rank. We run computational experiments and conclude that, for a few instances, they close considerably more gap than GMI cuts alone.
Craniosynostosis Surgery: A Study of Rearrangement
(University of Waterloo, 2020-12-22) Drygala, Marina; Koenemann, Jochen; Fukasawa, Ricardo
Cranial vault remodeling, a form of skull surgery is currently performed according to intuition developed over years of experience. The problem of how to best perform this surgery has not yet thoroughly been studied by computer scientists. In this thesis we provide a mathematical formulation of a simplified version of cranial vault remodeling surgery, and prove NP-hardness for our formulation. We then provide several heuristics and test their performance in an experimental study.
Mixed Integer Programming Approaches for Group Decision Making
(University of Waterloo, 2022-10-26) Iam, Hoi Cheong; Fukasawa, Ricardo; Naoum-Sawaya, Joe
Group decision making problems are everywhere in our day-to-day lives and have great influence on the daily operation of companies and institutions. With the recent advances in computational technology, it's not surprising that some companies would want to harvest that power to aid their decision-making procedures. Ethelo, the company that we partnered with in this project, developed an online platform that aids decision-making procedures by formulating the decision-making problem as a mixed integer nonlinear program (MINLP), providing feedback by solving the MINLP in real-time, and allowing the general public to contribute their opinions. Since an interactive component is involved, it is the goal of this thesis to attempt to reduce the solve time of their MINLP by applying tools from Operational Research. The main contribution in this thesis is threefold: first, we noticed that a big proportion of the MINLPs can be easily reposed as linear integer programs, and that a runtime reduction of at least 87.9\% can be achieved by simply redirecting them to a linear solver. Second, we identified a knapsack-like polyhedral structure that, to the best of our knowledge, has not been studied before, and derived a sufficient condition to identify the cases for which all valid cuts can be derived by considering other knapsack or covering problems. Finally, for the more general case where the objective function is nonlinear and not continuous, we derived a few different formulations to get to different approximations of the nonlinear model, and tested all of the approximations computationally.
Split Cuts From Sparse Disjunctions
(University of Waterloo, 2019-01-16) Yang, Shenghao; Fukasawa, Ricardo; Poirrier, Laurent
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.
A Study of Time Representation in a Class of Short Term Scheduling Problems
(University of Waterloo, 2016-08-17) Lagzi, Saman; Fukasawa, Ricardo; Ricardez-Sandoval, Luis
The problem of scheduling operations has received significant attention from academia and industrial practitioners in the past few decades. A key decision in various scheduling operations problems is when to perform an operation and thus the quality of the final schedule can be seriously affected by the choice of how to model the times at which such a decision may take place. The two most commonly used approaches for modeling these times are: Discrete time approaches, which pre-specify a finite set of time points when any decision may be taken, and continuous time approaches, in which the optimization model determines, through the use of continuous decision variables, at which point in time the operation will be performed. The focus of this thesis will be to study the benefits and limitations of each of these approaches within the context of an analytical services facility. Such a facility receives a large number of samples that need to be analyzed/processed through a specific sequence of limited resources/machines before its analysis is completed. The results of these analyses form a basis for many of the decisions made in their client industries (e.g. oil and mining), which in turn indicates the economic importance of the analytical services sector. The operations of such facilities have several particular conditions that need to be modeled and a particularly important one is called multitasking. If analyzing each type of samples is regarded as a task, then the machines in such facilities have the ability to perform multiple tasks at the same time as they are able to analyze different types of samples together at the same (as long as their capacity is not overloaded). The above mentioned study will be performed through an empirical comparison of the discrete and continuous approaches that take into account all the conditions in such facilities, including multitasking. While discrete and continuous approaches have often been independently employed, few studies have considered a comparison between them [37, 28, 39]. In addition, none of these studies consider the operational conditions that are present in short-term scheduling of operations in an analytical services facility. Since the continuous time formulations in the literature are not capable of accounting for multitasking, this thesis presents a novel continuous time mixed-integer linear programming (MILP) formulation that is capable of accommodating such feature and several other operational constraints present at analytical services facilities. The performance of the presented formulation is studied in comparison with a singletasking formulation. The results show that, while the multitasking formulation is not more costly in terms of solution time, it is capable of producing significantly better solutions. Furthermore, this thesis extends the idea of flexible time discretization for discrete time formulations, previously proposed by Velez and Maravelias [40], to be able to account for the operational constraints of an analytical services facility.
Two combinatorial problems from craniosynostosis
(University of Waterloo, 2022-09-08) Rundstrom, Mathieu Erik Charles; Koenemann, Jochen; Fukasawa, Ricardo
The art of cranial vault remodelling surgery is amazing and fascinating, but remains much of that, an art, to this day. In this thesis, we provide two mathematical approaches to tackle cranial vault remodelling surgery, in the hopes of bringing some insights from the perspective of combinatorial optimization. First, an integer programming formulation which we analyze experimentally, and second, a more theoretical approach using the local ratio.

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