Shop problems in scheduling

The shop problems in scheduling will be discussed in this thesis. The ones I\u27ll be discussing will be the flow shop, open shop, and job shop. The general idea of shop problems is that you\u27re given a set of jobs and a set of machines. Each job is predeterminely broken into parts and there are rules to how each part is executed on a machine. In this thesis, several shop problems and their algorithms will be introduced that I have researched. There are several examples and counter examples that I have constructed. Also I will discuss how an arbitrary problem that can be solved polynomially can be changed so that there are no polynomial algorithms that can solve it. Scheduling is used in computer science in the area of operating systems and it can be used in engineering. This is an important for a company when they want to run several jobs efficiently so that resources can be saved

General multiprocessor task scheduling.

Abstract: Most papers in the scheduling field assume that a job can be processed by only one machine at a time. Namely, they use a one-job-on-one-machine model. In many industry settings, this may not be an adequate model. Motivated by human resource planning, diagnosable microprocessor systems, berth allocation, and manufacturing systems that may require several resources simultaneously to process a job, we study the problem with a one-job-on-multiplemachine model. In our model, there are several alternatives that can be used to process a job. In each alternative, several machines need to process simultaneously the job assigned. Our purpose is to select an alternative for each job and then to schedule jobs to minimize the completion time of all jobs. In this paper, we provide a pseudopolynomial algorithm to solve optimally the two-machine problem, and a combination of a fully polynomial scheme and a heuristic to solve the three-machine problem. We then extend the results to a general m-machine problem. Our algorithms also provide an effective lower bounding scheme which lays the foundation for solving optimally the general m-machine problem. Furthermore, our algorithms can also be applied to solve a special case of the three-machine problem in pseudopolynomial time. Both pseudopolynomial algorithms (for two-machine and three-machine problems) are much more efficient than those in the literature. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 57-74, 1999 INTRODUCTION Due to the popularity of just-in-time and total quality management concepts, scheduling has played an important role in satisfying customer's expectation for on-time delivery. In the last four decades, many papers have been published in the scheduling area. There is a common assumption in the scheduling literature of a one-job-on-one-machine pattern. That is, at a given time, each job can be processed on only one machine. In many industry applications, this may not be an adequate model. Namely, a job may be processed simultaneously by several machines. For example, in semiconductor circuit design workforce planning, a design project is to be processed by m persons (a team of people). The project contains n tasks, and each task can be worked on by one of a set of alternatives, where each alternative contains one or more persons working simultaneously on that particular task. Task 1 can either be handled by person 1 and person 2 together, by person 1 and person 3 together, or just handled by person 1 alone. The processing time of each task depends on the group being assigned to handle the task. A group is formed when a set of people is working on a particular task, but a person may not belong to a fixed group all the time. Our goal is to assign these n tasks to m persons in order to minimize the project finishing time. Other applications can be found in (i) the berth allocation problems, where a large vessel may occupy several berths for loading and/or unloading, (ii) diagnosable microprocessor systems (Krawczyk and Kubale [20]), where a job must be performed on parallel processors in order to detect faults, (iii) manufacturing, where a job may need machines, tools, and people simultaneously, and (iv) scheduling a sequence of meetings where each meeting requires a certain group of people (Dobson and Karmarker [13]). In all the above examples, one job may need to be processed by several machines simultaneously. In literature we call this multiprocessor task scheduling (Drozdowski [15]) or a one-job-on-multiplemachine problem (Lee and Cai [24]). We are interested in the problem with a one-job-on-multiple-machine pattern. To describe the problem concisely we introduce the notation first. There are n jobs to be processed on m machines, and N(i) : the number of alternative machine sets to which job J i can be assigned, t i,I : the processing time of job J i if it is assigned to the processors in set I, where I is a set of machines (for example, t i,12 is the processing time of J i assigned to Processors 1 and 2). In this paper, job and task will be used interchangeably, and machine and processor will also be used interchangeably. For example, we may have four jobs to be processed by three machines. The alternative machine sets that can process each job are shown in the following matrices. J 1 can be processed by one of six alternatives (N(1) ϭ 6): {M 1 ,M 2 ,M 3 }, {M 1 ,M 2 }, {M 1 ,M 3 }, {M 2 ,M 3 }, {M 2 } or {M 3 }, with corresponding processing times t 1,123 ϭ 2, t 1,12 ϭ 3, t 1,13 ϭ 3, t 1,23 ϭ 3, t 1,2 ϭ 6, and t 1,3 ϭ 8. Therefore, if J 1 is processed by the first alternative: {M 1 ,M 2 ,M 3 } (represented by the first column in the first matrix where an entry of 1 in row j means that M j belongs to alternative 1), then the processing time is t 1,123 ϭ 2. Our purpose is to schedule jobs with a particular objective function. Vol. 46 (1999) In general, for the m-machine problem, the maximum possible number of alternative sets for Actually, for notational convenience we can always assume that ͪ for all i and let t iI ϭ ϱ for those I such that job i cannot be processed in parallel by processors with indices in I. In the above example, t 2,12 , t 2,13 , and t 2,3 are all equal to ϱ. In this paper, we are particularly interested in m ϭ 2 and 3. Hence for all i, we have N(i) ϭ 3 and 7 for m ϭ 2 and 3 respectively. There are two special cases that have been studied in the scheduling literature. In the first special case, N(i) ϭ 1 for all i. Namely, for each job a specifically fixed set of machines is assigned to it. We call this problem a fixed multiprocessor task scheduling problem. The second class of problems assumes that each job may require a fixed number of processors working simultaneously, yet the machines required are not specified. In the example above, if only t 1,12 , t 1,13 , t 1,23 , t 2,123 , t 3,1 , t 3,2 , t 3,3 , t 4,1 , t 4,2 , and t 4,3 are finite numbers, then it is equivalent to the problem where J 1 needs to be processed by two machines (any two machines simultaneously with processing time 3), J 2 by three machines simultaneously (with processing time 4) and jobs J 3 and J 4 by only one machine (with processing times 8 and 4, respectively). We call this problem a sized multiprocessor task scheduling problem. Blazewicz, Weglarz, and Drabowski To refer to the problem under study more precisely, we follow the standard notation used in scheduling literature. We use Pm͉set j ͉C max to denote the general problem of minimizing the makespan of multiprocessor tasks in the m-parallel-machine problem, where each job can be processed by a set of alternatives, and each alternative contains one or more machines simultaneously. Also, we use Pm͉fix j ͉C max to denote the first special case where the alternative assigned to each job has been fixed in advance. The paper is organized in the following way. In Section 1, we study the problem characteristics and provide some optimality properties. Section 2 discusses the two-machine problem. In particular, we provide a pseudopolynomial algorithm with running time O(nT 0 ) to solve the 59 Chen and Lee: General Multiprocessor Task Scheduling problem optimally. Section 3 discusses the three-machine problem. We provide a pseudopolynomial algorithm with running time O(nT 0 2 ) to find an effective lower bound for the optimal makespan and to solve optimally a special case of the problem. Both of our pseudopolynomial algorithms significantly improve previous results in the literature. We also provide a combination of a fully polynomial scheme and a heuristic method, with time complexity O(n 3 / 2 ) and error bound (3/2)(1 ϩ ), to solve the general three-machine problem. Section 4 extends our results to the m-machine problem. Finally, Section 5 concludes with a summary and a discussion of some future research topics

Artificial immune system for static and dynamic production scheduling problems

Over many decades, a large number of complex optimization problems have brought researchers' attention to consider in-depth research on optimization. Production scheduling problem is one of the optimization problems that has been the focus of researchers since the 60s. The main problem in production scheduling is to allocate the machines to perform the tasks. Job Shop Scheduling Problem (JSSP) and Flexible Job Shop Scheduling Problem (FJSSP) are two of the areas in production scheduling problems for these machines. One of the main objectives in solving JSSP and FJSSP is to obtain the best solution with minimum total completion processing time. Thus, this thesis developed algorithms for single and hybrid methods to solve JSSP and FJSSP in static and dynamic environments. In a static environment, no change is needed for the produced solution but changes to the solution are needed. On the other hand, in a dynamic environment, there are many real time events such as random arrival of jobs or machine breakdown requiring solutions. To solve these problems for static and dynamic environments, the single and hybrid methods were introduced. Single method utilizes Artificial Immune System (AIS), whereas AIS and Variable Neighbourhood Descent (VND) are used in the hybrid method. Clonal Selection Principle (CSP) algorithm in the AIS was used in the proposed single and hybrid methods. In addition, to evaluate the significance of the proposed methods, experiments and One-Way ANOVA tests were conducted. The findings showed that the hybrid method was proven to give better performance compared to single method in producing optimized solution and reduced solution generating time. The main contribution of this thesis is the development of an algorithm used in the single and hybrid methods to solve JSSP and FJSSP in static and dynamic environment

A Taxonomy of Polynomially Solvable Shop Problems with Limited Number of Machines or Jobs

Among shop scheduling problems, job shop and mixed shop are one of the most general models encompassing open shop and flow shop. Many job shop problems are NP hard, but there are numerous cases, which possess polynomial solutions when the number of jobs or the number of machines (or both) is limited. This thesis gives an overview of methods and algorithms for solving - in polynomial time - such special shop problems, including open, flow, job shop and mixed shop. The tools used include Monge interchange, dynamic programming, greedy techniques and sweep line algorithms and the primary focus of this thesis is to give a taxonomy of such problems with their solutions. Additionally the thesis outlines a neighborhood search technique which uses the disjunctive graph model and which can be applied as a heuristic for a wide range of NP-hard shop problems

Melhoria do planeamento programação e controlo de produção (ppcp) de uma empresa de calçado

Atualmente o sistema produtivo do tipo job shop é muito comum nas PMEs (Pequenas e Médias Empresas). Estas empresas trabalham por encomenda. Produzem grande variedade de modelos, e em pequenas quantidades. Os prazos de entrega são um fator de elevada importância, pois os clientes exigem um produto de qualidade no tempo certo. O presente trabalho, pretende criar uma ferramenta de programação da produção para a secção da costura, usando dados reais da empresa, que tem uma implantação do tipo job shop com máquinas multi-operação (Multi-Purpose -Machines Job Shop). No final, são reunidas as principais conclusões e perspetivados futuros desenvolvimentos. Os resultados obtidos comprovam que o algoritmo desenvolvido, com base no algoritmo de Giffler & Thompson, consegue obter com grande precisão e de forma rápida o escalonamento / balanceamento da secção da costura. Com a ferramenta criada, a empresa otimiza a programação da secção da costura e fornece informação importante á gestão da produção, possibilitando uma melhoria do planeamento da empresa.Currently the production system job shop like is very common in SMEs (Small and Medium Enterprises). These companies work to order. Produce a large variety of models, in small quantities. Delivery times are a key factor of great importance, as the customers demand a quality product on time. The present work aims to create a programming tool for the seam production section, by using actual data from the company, which has a job shop type plant with multi-operation machines (Multi-Purpose - Machines Job Shop). In the end, are brought together the main conclusions and future foreseen developments. The results show that the algorithm developed, on the basis of the Giffler & Thompson algorithm, proved to achieve very accurately and quickly scaling / balancing of the stitching section. With the new developed tool, the company optimized the seam programming section and provides important information to the production management, enabling an improvement in the planning of the whole company

Particle Swarm Optimization

Particle swarm optimization (PSO) is a population based stochastic optimization technique influenced by the social behavior of bird flocking or fish schooling.PSO shares many similarities with evolutionary computation techniques such as Genetic Algorithms (GA). The system is initialized with a population of random solutions and searches for optima by updating generations. However, unlike GA, PSO has no evolution operators such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. This book represents the contributions of the top researchers in this field and will serve as a valuable tool for professionals in this interdisciplinary field