Online makespan scheduling with job migration on uniform machines

In the classic minimum makespan scheduling problem, we are given an input sequence of n jobs with sizes. A scheduling algorithm has to assign the jobs to m parallel machines. The objective is to minimize the makespan, which is the time it takes until all jobs are processed. In this paper, we consider online scheduling algorithms without preemption. However, we allow the online algorithm to reassign up to k jobs to different machines in the final assignment. For m identical machines, Albers and Hellwig (Algorithmica, 2017) give tight bounds on the competitive ratio in this model. The precise ratio depends on, and increases with, m. It lies between 4/3 and ~~ 1.4659. They show that k = O(m) is sufficient to achieve this bound and no k = o(n) can result in a better bound. We study m uniform machines, i.e., machines with different speeds, and show that this setting is strictly harder. For sufficiently large m, there is a delta = Theta(1) such that, for m machines with only two different machine speeds, no online algorithm can achieve a competitive ratio of less than 1.4659 + delta with k = o(n). We present a new algorithm for the uniform machine setting. Depending on the speeds of the machines, our scheduling algorithm achieves a competitive ratio that lies between 4/3 and ~~ 1.7992 with k = O(m). We also show that k = Omega(m) is necessary to achieve a competitive ratio below 2. Our algorithm is based on a subtle imbalance with respect to the completion times of the machines, complemented by a bicriteria approximation algorithm that minimizes the makespan and maximizes the average completion time for certain sets of machines

Minimizing schedule length on identical parallel machines: an exact algorithm

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent Univ., 1991.Thesis (Ph. D.) -- Bilkent University, 1991.Includes bibliographical references.The primary concern of this study is to investigate the combinatorial aspects of the single-stage identical parallel machine scheduling problem and to develop a computationally feasible branch and bound algorithm for its exact solution. Although there is a substantial amount of literature on this problem, most of the work in this area is on the development and performance analysis of approximation algorithms. The few optimizing algorithms proposed in the literature have major drawbacks from the computer implementation point of view. Even though the single-stage scheduling problem is known to be unary A/’P-hard, there is still a need to develop a computationally feasible optimizing algorithm that solves the problem in a reasonable time. Development of such an algorithm is necessary for solving the multi-stage parallel machine scheduling problems which are currently an almost untouched issue in the deterministic scheduling theory. In this study, a branch and bound algorithm for the single-stage identical parallel machine scheduling problem is proposed. Promising results were obtained in the empirical analysis of the performance of this algorithm. Furthermore, the procedure that is developed to determine tight bounds at a node of the enumeration tree, is an approximation algorithm that solves a special class of identical parallel machine scheduling problems of practical interest. This algorithm delivers a solution that is arbitrarily close to 4/3 times the optimum. To our knowledge this is the best result obtained for this problem so far.Akyel, H CemalPh.D

Data-driven optimization and analytics for operations management applications

Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2013.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 (pages 163-166).In this thesis, we study data-driven decision making in operation management contexts, with a focus on both theoretical and practical aspects. The first part of the thesis analyzes the well-known newsvendor model but under the assumption that, even though demand is stochastic, its probability distribution is not part of the input. Instead, the only information available is a set of independent samples drawn from the demand distribution. We analyze the well-known sample average approximation (SAA) approach, and obtain new tight analytical bounds on the accuracy of the SAA solution. Unlike previous work, these bounds match the empirical performance of SAA observed in extensive computational experiments. Our analysis reveals that a distribution's weighted mean spread (WMS) impacts SAA accuracy. Furthermore, we are able to derive distribution parametric free bound on SAA accuracy for log-concave distributions through an innovative optimization-based analysis which minimizes WMS over the distribution family. In the second part of the thesis, we use spread information to introduce new families of demand distributions under the minimax regret framework. We propose order policies that require only a distribution's mean and spread information. These policies have several attractive properties. First, they take the form of simple closed-form expressions. Second, we can quantify an upper bound on the resulting regret. Third, under an environment of high profit margins, they are provably near-optimal under mild technical assumptions on the failure rate of the demand distribution. And finally, the information that they require is easy to estimate with data. We show in extensive numerical simulations that when profit margins are high, even if the information in our policy is estimated from (sometimes few) samples, they often manage to capture at least 99% of the optimal expected profit. The third part of the thesis describes both applied and analytical work in collaboration with a large multi-state gas utility. We address a major operational resource allocation problem in which some of the jobs are scheduled and known in advance, and some are unpredictable and have to be addressed as they appear. We employ a novel decomposition approach that solves the problem in two phases. The first is a job scheduling phase, where regular jobs are scheduled over a time horizon. The second is a crew assignment phase, which assigns jobs to maintenance crews under a stochastic number of future emergencies. We propose heuristics for both phases using linear programming relaxation and list scheduling. Using our models, we develop a decision support tool for the utility which is currently being piloted in one of the company's sites. Based on the utility's data, we project that the tool will result in 55% reduction in overtime hours.by Joline Ann Villaranda Uichanco.Ph. D

Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective

Many packing, scheduling and covering problems that were previously considered by computer science literature in the context of various transportation and production problems, appear also suitable for describing and modeling various fundamental aspects in networks optimization such as routing, resource allocation, congestion control, etc. Various combinatorial problems were already studied from the game theoretic standpoint, and we attempt to complement to this body of research. Specifically, we consider the bin packing problem both in the classic and parametric versions, the job scheduling problem and the machine covering problem in various machine models. We suggest new interpretations of such problems in the context of modern networks and study these problems from a game theoretic perspective by modeling them as games, and then concerning various game theoretic concepts in these games by combining tools from game theory and the traditional combinatorial optimization. In the framework of this research we introduce and study models that were not considered before, and also improve upon previously known results.Comment: PhD thesi

Fast algorithms for two scheduling problems

The thesis deals with problems from two distint areas of scheduling theory. In the first part we consider the preemptive Sum Multicoloring (pSMC) problem. In an instance of pSMC, pairwise conflicting jobs are represented by a conflict graph, and the time demands of jobs are given by integer weights on the nodes. The goal is to schedule the jobs in such a way that the sum of their finish times is minimized. We give the first polynomial algorithm for pSMC on paths and cycles, running in time O(min(n², n log p)), where n is the number of nodes and p is the largest time demand. This answers a question raised by Halldórsson et al. [51] about the hardness of this problem. Our result identifies a gap between binary-tree conflict graphs - where the question is NP-hard - and paths. In the second part of the thesis we consider the problem of scheduling n jobs on m machines of different speeds s.t. the makespan is minimized (Q||C_max). We provide a fast and simple, deterministic monotone 2.8-approximation algorithm for Q||C_max. Monotonicity is relevant in the context of truthful mechanisms: when each machine speed is only known to the machine itself, we need to motivate that machines "declare" their true speeds to the scheduling mechanism. So far the best deterministic truthful mechanism that is polynomial in n and m; was a 5-approximation by Andelman et al. [3]. A randomized 2-approximation method, satisfying a weaker definition of truthfulness, was given by Archer and Tardos [4, 5]. As a core result, we prove the conjecture of Auletta et al. [8], that the greedy list scheduling algorithm Lpt is monotone if machine speeds are all integer powers of two (2-divisible machines). Proving the worst case bound of 2.8 involves studying the approximation ratio of Lpt on 2-divisible machines. As a side result, we obtain a tight bound of (sqrt(3) + 1)/2 ~= 1.3660 for the "one fast machine" case, i.e., when m - 1 machine speeds are equal, and there is only one faster machine. In this special case the best previous lower and upper bounds were 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m), shown in a classic paper by Gonzalez et al. [42]. Moreover, the authors of [42] conjectured the bound 4/3 to be tight. Thus, the results of the thesis answer three open questions in scheduling theory.In dieser Arbeit befassen wir uns mit Problemen aus zwei verschiedenen Teilgebieten der Scheduling-Theorie. Im ersten Teil betrachten wir das sog. preemptive Sum Multicoloring (pSMC) Problem. In einer Eingabe für pSMC werden paarweise Konflikte zwischen Jobs durch einen Konfliktgraphen repräsentiert; der Zeitbedarf eines Jobs ist durch ein ganzzahliges, positives Gewicht in seinem jeweiligen Knoten gegeben. Die Aufgabe besteht darin, die Jobs so den Maschinen zuzuweisen, dass die Summe ihrer Maschinenlaufzeiten minimiert wird. Wir liefern den ersten Algorithmus für pSMC auf Pfaden und Kreisen mit polynomieller Laufzeit; er benötigt O(min(n², n log p)) Zeit, wobei n die Anzahl der Jobs und p die maximale Zeitanforderung darstellen. Dies liefert eine Antwort auf die von Halldórsson et al. [51] aufgeworfene Frage der Komplexitätsklasse von pSMC. Unser Resultat identifiziert eine Diskrepanz zwischen der Komplexität auf binären Bäumen - für diese ist das Problem NP-schwer - und Pfaden. Im zweiten Teil dieser Arbeit betrachten wir das Problem, n Jobs auf m Maschinen mit unterschiedlichen Geschwindigkeiten so zu verteilen, dass der Makespan minimiert wird (Q||C_max). Wir präsentieren einen einfachen deterministischen monotonen Algorithmus mit Approximationsgüte 2.8 für Q||C_max. Monotonie ist relevant im Zusammenhang mit truthful Mechanismen: wenn die Geschwindigkeiten der Maschinen nur diesen selbst bekannt sind, müssen sie motiviert werden, dem Scheduling Mechanismus ihre tatsächlichen Geschwindigkeiten offenzulegen. Der beste bisherige deterministische truthful Mechanismus mit polynomieller Laufzeit in n und m von Andelman et al. [3] erreicht Approximationsgüte fünf. Eine randomisierte Methode mit ApproximationsgÄute zwei, die jedoch nur eine schwächere Definition von truthful Mechanismen unterstützt, wurde von Archer und Tardos [4, 5] entwickelt. Als ein zentrales Ergebnis beweisen wir die Vermutung von Auletta et al. [8], dass der greedy list-scheduling Algorithmus Lpt monoton ist, falls alle Maschinengeschwindigkeiten ganze Potenzen von zwei sind (2-divisible Maschinen). Der Beweis der obigen Approximationsschranke von 2.8 benutzt die Approximationsgüte von Lpt auf 2-divisible Maschinen. Als Nebenresultat erhalten wir eine scharfe Schranke von (sqrt(3) + 1)/2 ~= 1.3660 für den Fall "einer schnellen Maschine", d.h. m - 1 Maschinen haben identische Geschwindigkeiten und es gibt nur eine schnellere Maschine. Die bisherigen besten unteren und oberen Schranken für diesen Spezialfall waren 4/3 - epsilon < Lpt/Opt <= 3/2 - 1/(2m). Letztere wurden 1977 von Gonzalez, Ibara und Sahni [42] bewiesen, die mutmaßten, dass die tatächliche obere Schranke bei 4=3 läge. Alles in allem, liefert diese Arbeit Antworten auf drei offene Fragen im Bereich der Scheduling-Theorie