Theoretical and Computational Research in Various Scheduling Models

Nine manuscripts were published in this Special Issue on “Theoretical and Computational Research in Various Scheduling Models, 2021” of the MDPI Mathematics journal, covering a wide range of topics connected to the theory and applications of various scheduling models and their extensions/generalizations. These topics include a road network maintenance project, cost reduction of the subcontracted resources, a variant of the relocation problem, a network of activities with generally distributed durations through a Markov chain, idea on how to improve the return loading rate problem by integrating the sub-tour reversal approach with the method of the theory of constraints, an extended solution method for optimizing the bi-objective no-idle permutation flowshop scheduling problem, the burn-in (B/I) procedure, the Pareto-scheduling problem with two competing agents, and three preemptive Pareto-scheduling problems with two competing agents, among others. We hope that the book will be of interest to those working in the area of various scheduling problems and provide a bridge to facilitate the interaction between researchers and practitioners in scheduling questions. Although discrete mathematics is a common method to solve scheduling problems, the further development of this method is limited due to the lack of general principles, which poses a major challenge in this research field

A unified approach to truthful scheduling on related machines

We present a unified framework for designing deterministic monotone polynomial time approximation schemes (PTAS's) for a wide class of scheduling problems on uniformly related machines. This class includes (among others) minimizing the makespan, maximizing the minimum load, and minimizing the l_p norm of the machine loads vector. Previously, this kind of result was only known for the makespan objective. Monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it. The key idea of our novel method is to show that for goal functions that are sufficiently well-behaved functions of the machine loads, it is possible to compute in polynomial time a highly structured nearly optimal schedule. Monotone approximation schemes have an important role in the emerging area of algorithmic mechanism design. In the game-theoretical setting of these scheduling problems there is a social goal, which is one of the objective functions that we study. Each machine is controlled by a selfish single-parameter agent, where its private information is its cost of processing a unit sized job, which is also the inverse of the speed of its machine. Each agent wishes to maximize its own profit, defined as the payment it receives from the mechanism minus its cost for processing all jobs assigned to it, and places a bid which corresponds to its private information. For each one of the problems, we show that we can calculate payments that guarantee truthfulness in an efficient manner. Thus, there exists a dominant strategy where agents report their true speeds, and we show the existence of a truthful mechanism which can be implemented in polynomial time, where the social goal is approximated within a factor of 1+epsilon for every epsilon>0

Evaluating the Robustness of Resource Allocations Obtained through Performance Modeling with Stochastic Process Algebra

Recent developments in the field of parallel and distributed computing has led to a proliferation of solving large and computationally intensive mathematical, science, or engineering problems, that consist of several parallelizable parts and several non-parallelizable (sequential) parts. In a parallel and distributed computing environment, the performance goal is to optimize the execution of parallelizable parts of an application on concurrent processors. This requires efficient application scheduling and resource allocation for mapping applications to a set of suitable parallel processors such that the overall performance goal is achieved. However, such computational environments are often prone to unpredictable variations in application (problem and algorithm) and system characteristics. Therefore, a robustness study is required to guarantee a desired level of performance. Given an initial workload, a mapping of applications to resources is considered to be robust if that mapping optimizes execution performance and guarantees a desired level of performance in the presence of unpredictable perturbations at runtime. In this research, a stochastic process algebra, Performance Evaluation Process Algebra (PEPA), is used for obtaining resource allocations via a numerical analysis of performance modeling of the parallel execution of applications on parallel computing resources. The PEPA performance model is translated into an underlying mathematical Markov chain model for obtaining performance measures. Further, a robustness analysis of the allocation techniques is performed for finding a robustmapping from a set of initial mapping schemes. The numerical analysis of the performance models have confirmed similarity with the simulation results of earlier research available in existing literature. When compared to direct experiments and simulations, numerical models and the corresponding analyses are easier to reproduce, do not incur any setup or installation costs, do not impose any prerequisites for learning a simulation framework, and are not limited by the complexity of the underlying infrastructure or simulation libraries