The Coupled Task Scheduling Problem: Models and Solution Methods

University of Technology Sydney. Faculty of Science.The coupled task scheduling problem (CTSP) is studied in this thesis. The problem consists of scheduling a set of jobs on one or a set of machines, where each job consists of at least two tasks. The main characteristic of the problem is a fixed time-lag between the process of each two consecutive tasks of the same job, where its duration is fixed, i.e., the succeeding task cannot be started earlier or later than the time-lag is passed. The fixed time-lags were introduced to model radar tracking systems, and later extended to formulate problems in chemistry manufacturing systems and robotic cells. The motivation for studying the CTSP in this thesis is to model certain problems in healthcare scheduling with the same characteristics. One example is the scheduling of patients in a chemotherapy clinic, where each patient must undergo a number of consecutive treatments with time-lags in between. Meeting the fixed delays between the treatments of a patient is an important factor in gaining the best outcomes for them. To study the CTSP, a literature review is first conducted, followed by studying the problem in different scheduling environments, including the single-machine, parallel-machine, open-shop and flow-shop settings, where we propose several new complexity results and solution algorithms for different variants of the problem. Regarding the single-machine coupled task problem, a new mathematical formulation and two matheuristic algorithms are proposed for the classical problem, as well as a dynamic programming algorithm for a variant of the problem with time-dependent processing times. With regard to the parallel-machine environment, we first explore the complexity of the problem and propose NP-hardness proofs for certain cases, followed by approximation bounds for the two-machine problem. The latter result is then extended to the open-shop scheduling environment. The problem in the flow-shop environment is then extensively investigated under the permutation setting, and also under the case of ordered processing times. A set of publicly available hard data set and state-of-the-art algorithms are proposed for the ordered flow-shops. Then, flow-shop problem with coupled tasks is studied and polynomial-time algorithms are proposed for various settings of the problem, including the ordered processing times

Global burden of 288 causes of death and life expectancy decomposition in 204 countries and territories and 811 subnational locations, 1990–2021: a systematic analysis for the Global Burden of Disease Study 2021

BACKGROUND Regular, detailed reporting on population health by underlying cause of death is fundamental for public health decision making. Cause-specific estimates of mortality and the subsequent effects on life expectancy worldwide are valuable metrics to gauge progress in reducing mortality rates. These estimates are particularly important following large-scale mortality spikes, such as the COVID-19 pandemic. When systematically analysed, mortality rates and life expectancy allow comparisons of the consequences of causes of death globally and over time, providing a nuanced understanding of the effect of these causes on global populations. METHODS The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) 2021 cause-of-death analysis estimated mortality and years of life lost (YLLs) from 288 causes of death by age-sex-location-year in 204 countries and territories and 811 subnational locations for each year from 1990 until 2021. The analysis used 56 604 data sources, including data from vital registration and verbal autopsy as well as surveys, censuses, surveillance systems, and cancer registries, among others. As with previous GBD rounds, cause-specific death rates for most causes were estimated using the Cause of Death Ensemble model-a modelling tool developed for GBD to assess the out-of-sample predictive validity of different statistical models and covariate permutations and combine those results to produce cause-specific mortality estimates-with alternative strategies adapted to model causes with insufficient data, substantial changes in reporting over the study period, or unusual epidemiology. YLLs were computed as the product of the number of deaths for each cause-age-sex-location-year and the standard life expectancy at each age. As part of the modelling process, uncertainty intervals (UIs) were generated using the 2·5th and 97·5th percentiles from a 1000-draw distribution for each metric. We decomposed life expectancy by cause of death, location, and year to show cause-specific effects on life expectancy from 1990 to 2021. We also used the coefficient of variation and the fraction of population affected by 90% of deaths to highlight concentrations of mortality. Findings are reported in counts and age-standardised rates. Methodological improvements for cause-of-death estimates in GBD 2021 include the expansion of under-5-years age group to include four new age groups, enhanced methods to account for stochastic variation of sparse data, and the inclusion of COVID-19 and other pandemic-related mortality-which includes excess mortality associated with the pandemic, excluding COVID-19, lower respiratory infections, measles, malaria, and pertussis. For this analysis, 199 new country-years of vital registration cause-of-death data, 5 country-years of surveillance data, 21 country-years of verbal autopsy data, and 94 country-years of other data types were added to those used in previous GBD rounds. FINDINGS The leading causes of age-standardised deaths globally were the same in 2019 as they were in 1990; in descending order, these were, ischaemic heart disease, stroke, chronic obstructive pulmonary disease, and lower respiratory infections. In 2021, however, COVID-19 replaced stroke as the second-leading age-standardised cause of death, with 94·0 deaths (95% UI 89·2-100·0) per 100 000 population. The COVID-19 pandemic shifted the rankings of the leading five causes, lowering stroke to the third-leading and chronic obstructive pulmonary disease to the fourth-leading position. In 2021, the highest age-standardised death rates from COVID-19 occurred in sub-Saharan Africa (271·0 deaths [250·1-290·7] per 100 000 population) and Latin America and the Caribbean (195·4 deaths [182·1-211·4] per 100 000 population). The lowest age-standardised death rates from COVID-19 were in the high-income super-region (48·1 deaths [47·4-48·8] per 100 000 population) and southeast Asia, east Asia, and Oceania (23·2 deaths [16·3-37·2] per 100 000 population). Globally, life expectancy steadily improved between 1990 and 2019 for 18 of the 22 investigated causes. Decomposition of global and regional life expectancy showed the positive effect that reductions in deaths from enteric infections, lower respiratory infections, stroke, and neonatal deaths, among others have contributed to improved survival over the study period. However, a net reduction of 1·6 years occurred in global life expectancy between 2019 and 2021, primarily due to increased death rates from COVID-19 and other pandemic-related mortality. Life expectancy was highly variable between super-regions over the study period, with southeast Asia, east Asia, and Oceania gaining 8·3 years (6·7-9·9) overall, while having the smallest reduction in life expectancy due to COVID-19 (0·4 years). The largest reduction in life expectancy due to COVID-19 occurred in Latin America and the Caribbean (3·6 years). Additionally, 53 of the 288 causes of death were highly concentrated in locations with less than 50% of the global population as of 2021, and these causes of death became progressively more concentrated since 1990, when only 44 causes showed this pattern. The concentration phenomenon is discussed heuristically with respect to enteric and lower respiratory infections, malaria, HIV/AIDS, neonatal disorders, tuberculosis, and measles. INTERPRETATION Long-standing gains in life expectancy and reductions in many of the leading causes of death have been disrupted by the COVID-19 pandemic, the adverse effects of which were spread unevenly among populations. Despite the pandemic, there has been continued progress in combatting several notable causes of death, leading to improved global life expectancy over the study period. Each of the seven GBD super-regions showed an overall improvement from 1990 and 2021, obscuring the negative effect in the years of the pandemic. Additionally, our findings regarding regional variation in causes of death driving increases in life expectancy hold clear policy utility. Analyses of shifting mortality trends reveal that several causes, once widespread globally, are now increasingly concentrated geographically. These changes in mortality concentration, alongside further investigation of changing risks, interventions, and relevant policy, present an important opportunity to deepen our understanding of mortality-reduction strategies. Examining patterns in mortality concentration might reveal areas where successful public health interventions have been implemented. Translating these successes to locations where certain causes of death remain entrenched can inform policies that work to improve life expectancy for people everywhere. FUNDING Bill & Melinda Gates Foundation

The coupled task scheduling problem : an improved mathematical program and a new solution algorithm

The general single machine coupled task scheduling problem with the objective function of minimizing the makespan, which is strongly NP-hard, aims to schedule a set of coupled task jobs on one machine such that the completion time of the last job is minimized. We propose a new mixed-integer program (MIP) for the problem. We also propose a relax-and-solve (R&amp;S) matheuristic algorithm as the solution method. We show that the new MIP outperforms the available models and improves the quality of solutions. Also, the proposed MIP significantly improves the average gap to the best known feasible solution of an existing binary search algorithm. We show that our R&amp;S matheuristic produces new best solutions for almost 50% of the instances.</p

The gradual minimum covering location problem

The minimum covering location problem with distance constraints deals with locating a set of undesirable facilities on a geographical map, where there is a given minimum distance between any pair of located facilities. A covering radius is defined within which the population node is fully covered and beyond that it is not covered at all. This setting may not be applicable in practice because usually the coverage gradually decreases with an increase in the distance. Additionally, undesirable facilities may have a cooperative adverse impact on the nearby population. We introduce the gradual coverage to the problem that extends the classic definition of the coverage and is more suitable for modelling real-world applications. We name the problem the gradual minimum covering location problem with distance constraints (GMCLPDC). We propose a mixed-integer program for GMCLPDC where cooperation of facilities is also considered. We propose a threshold accepting heuristic as the solution method. We conduct computational experiments on instances with up to 10,000 nodes. The outcomes indicate that the heuristic delivers quality solutions and outperforms the solver Gurobi. We also show an application of our model in Sydney metropolitan area.</p

A binary search algorithm for the general coupled task scheduling problem

The coupled task scheduling problem aims to schedule a set of jobs, each with at least two tasks and there is an exact delay period between two consecutive tasks, on a set of machines to optimize a performance criterion. We study the problem of scheduling a set of coupled jobs to be processed on a single machine with the objective of minimizing the makespan, which is known to be strongly NP-hard. We obtain competitive lower bounds for the problem through different procedures, including solving 0-1 knapsack problems. We obtain an upper bound by applying a heuristic algorithm. We then propose a binary search heuristic algorithm for the coupled task scheduling problem. We perform extensive computational experiments and show that the proposed method is able to obtain quality solutions. The results also indicate that the proposed solution method outperforms the standard exact solver Gurobi.</p

Coordinative production and maintenance scheduling problem with flexible maintenance time intervals

This study investigates the simultaneous scheduling of production and planning of maintenance activities in the flow shop scheduling environment. The problem is considered in a bi-objective form, minimizing the makespan as the production scheduling criterion and minimizing the system unavailability as the maintenance planning criterion. We propose the coordinative production and maintenance scheduling model in which the time interval between consecutive maintenance activities as well as the number of maintenance activities on each machine are assumed to be non-fixed. The coordinative model aims to find the best permutation of jobs as the production problem and to assign the maintenance activities into the schedule as the maintenance problem, simultaneously. Moreover, a special setting called single server maintenance is introduced and discussed. A bi-objective ant colony system algorithm is presented to solve the problem in focus, introducing some novel ideas. CDS and NEH heuristics are applied to define the heuristic information part of the proposed algorithm. Some experiments are carried out to select the appropriate heuristic method between CDS and NEH. Moreover, some experiments are performed using the well-known Taillard benchmark, comparing the performance of the proposed algorithm with another ant colony optimization algorithm. Computational experiments indicate the effectiveness of the proposed algorithm.</p

Scheduling coupled tasks on parallel identical machines

In many applications in the context of patient appointment scheduling there are recurring tasks with fixed delays between them. These tasks are commonly referred to as coupled-task jobs. In the coupled-task settings, each job consists of two tasks whereby the second task must start processing after an exact time lag following the completion of the first task. In this paper, we introduce the problem of scheduling a set of coupled-task jobs on parallel identical machines with the objective function of minimizing the makespan. We study the computational complexity of the general problem, as well as its special cases. We prove that the majority of these problems are (strongly) NP-hard. Nonetheless, we provide the optimal scheduling policy for two settings consisting of identical jobs. An important result of our work includes showing that the existence of a (2−ε)-approximation algorithm for the problem implies P=NP. The latter result improves a recently proposed bound for the open-shop counterpart as well

Flow-shop scheduling with exact delays to minimize makespan

The flow-shop scheduling problem with exact delays is generalization of no-wait flow-shop scheduling in which an exact delay exists between the consecutive tasks of each job. The problem with distinct delays to minimize the makespan is strongly NP-hard even for the two-machine case with unit execution time tasks. Providing polynomial-time solutions for special cases of the problem, we show that the two-machine permutation flow-shop case is solvable in O(n log n) time, while the case with more than two machines is strongly NP-hard. We also show that the multi-machine case with delays following the ordered structure possesses the pyramidal-shaped property and propose an O(n2)-time dynamic program to solve it. We further improve the time complexity of the solution algorithm to O(n log n) under certain conditions

Bi-objective single machine scheduling problem with stochastic processing times

In this study, a static single machine scheduling problem is investigated, where processing times are stochastic, due dates are deterministic and inserted idle time is allowed. Two objective functions are simultaneously taken into account, minimization of mean completion time and minimization of earliness and tardiness costs. A robust model is presented to tackle the problem, based on goal programming and a stochastic programming model named E-model. The proposed model not only obtains optimal operating systems, but also considers the variance of the objective functions and the correlation between them. Moreover, chance-constrained programming model is used to take into account the randomness in the constraints of the model. The model is presented with general distribution of processing times and the normal case is explored in experiments. Two sets of computational experiments are presented to test the efficiency of the proposed model. In the first set, the performance obtained by the bi-objective formulation is measured, where in the second set the performance obtained by incorporating robustness is measured. Results confirm the effectiveness of the proposed model, in both directions.</p

Four decades of research on the open-shop scheduling problem to minimize the makespan

One of the basic scheduling problems, the open-shop scheduling problem has a broad range of applications across different sectors. The problem concerns scheduling a set of jobs, each of which has a set of operations, on a set of different machines. Each machine can process at most one operation at a time and the job processing order on the machines is immaterial, i.e., it has no implication for the scheduling outcome. The aim is to determine a schedule, i.e., the completion times of the operations processed on the machines, such that a performance criterion is optimized. While research on the problem dates back to the 1970s, there have been reviving interests in the computational complexity of variants of the problem and solution methodologies in the past few years. Aiming to provide a complete road map for future research on the open-shop scheduling problem, we present an up-to-date and comprehensive review of studies on the problem that focuses on minimizing the makespan, and discuss potential research opportunities.</p