ABSTRACT A HEURISTIC APPROACH FOR THE SINGLE MACHINE SCHEDULING TARDINESS PROBLEMS &Ouml / zbakir, Saffet Ilker M.Sc., Department of Industrial Engineering Supervisor : Prof. Dr. &Ouml / mer Kirca September 2011, 102 pages In this thesis, we study the single machine scheduling problem. Our general aim is to schedule a set of jobs to the machine with a goal to minimize tardiness value. The problem is studied for two objectives: minimizing total tardiness value and minimizing total weighted tardiness value. Solving optimally this problem is difficult, because both of the total tardiness problem and total weighted tardiness problem are NP-hard problems. Therefore, we construct a heuristic procedure for this problem. Our heuristic procedure is divided to two parts: construction part and improvement part. The construction heuristic is based on grouping the jobs, solving these groups and then fixing some particular number of jobs. Moreover, we used three type improvement heuristics. These are sliding forward method, sliding backward method and pairwise interchange method. Computational results are reported for problem size = 20, 40, 50 and 100 at total tardiness problem and for problem size = 20 and 40 at total weighted tardiness problem. Experiments are designed in order to investigate the effect of three factors which are problem size, tardiness factor and relative range of due dates on computational difficulties of the problems. Computational results show that the heuristic proposed in this thesis is robust to changes at these factors.

Hundreds of steel products need to be scheduled on a single or parallel machine in a steel plant every week. A good feasible schedule may save the company millions of dollars compared to a bad one. Single and parallel machine scheduling are also encountered often in many other industries, making it a crucial research topic for both the process system engineering and operations research communities. Single or parallel machine scheduling can be a challenging combinatorial optimization problem when a large number of jobs are to be scheduled. Each job has unique job characteristics, resulting in different setup times/costs depending on the processing sequence. They also have specific release dates to follow and due dates to meet. This work presents both an exact method using mixed-integer quadratic programming, and an approximate method with metaheuristics to solve real-world large-scale single/parallel machine scheduling problems faced in a steel plant. More than 1000 or 350 jobs are to be scheduled within a one-hour time limit in the single or parallel machine problem, respectively. The objective of the single machine scheduling is to minimize a combined total changeover, total earliness, and total tardiness cost, whereas the objective of the parallel machine scheduling is to minimize an objective function comprising the gaps between jobs before a critical time in a schedule, the total changeover cost, and the total tardiness cost. The exact method is developed to benchmark computation time for a small-scale single machine problem, but is not practical for solving the actual large-scale problem. A metaheuristic algorithm centered on variable neighborhood descent is developed to address the large-scale single machine scheduling with a sliding-window decomposition strategy. The algorithm is extended and modified to solve the large-scale parallel machine problem. Statistical tests, including Student's t-test and ANOVA, are conducted to determine efficient solution strategies and good parameters to be used in the metaheuristics. / Thesis / Master of Applied Science (MASc)

Majority of the studies in the scheduling literature is devoted to time based performance measures. In this thesis, we develop a model that considers monetary issues in single machine scheduling environments. We assume all the jobs should be completed by a common due date. An early revenue is earned if the completion time is before or on the due date, and a tardy revenue is gained if the job is completed after the due date. We consider restricted and unrestricted due date versions of the problem. Our objective is the maximization of the net present value of all revenues. We first investigate some special cases of the problem, and present polynomial time algorithms to solve them. Then, we develop branch and bound algorithms with lower and upper bounding mechanisms. Computational experiments have shown that the branch and bound algorithms can solve large-sized problems in reasonable times.

In this paper we propose an improved algorithm to search optimal solutions to the single machine total weighted tardiness scheduling problem. We propose using longest common sequence to combine with the random key method. Numerical simulation shows that the scheme we proposed could improve the search efficiency of genetic algorithm in this problem for some cases.

abstract: This research by studies the computational performance of four different mixed integer programming (MIP) formulations for single machine scheduling problems with varying complexity. These formulations are based on (1) start and completion time variables, (2) time index variables, (3) linear ordering variables and (4) assignment and positional date variables. The objective functions that are studied in this paper are total weighted completion time, maximum lateness, number of tardy jobs and total weighted tardiness. Based on the computational results, discussion and recommendations are made on which MIP formulation might work best for these problems. The performances of these formulations very much depend on the objective function, number of jobs and the sum of the processing times of all the jobs. Two sets of inequalities are presented that can be used to improve the performance of the formulation with assignment and positional date variables. Further, this research is extend to single machine bicriteria scheduling problems in which jobs belong to either of two different disjoint sets, each set having its own performance measure. These problems have been referred to as interfering job sets in the scheduling literature and also been called multi-agent scheduling where each agent's objective function is to be minimized. In the first single machine interfering problem (P1), the criteria of minimizing total completion time and number of tardy jobs for the two sets of jobs is studied. A Forward SPT-EDD heuristic is presented that attempts to generate set of non-dominated solutions. The complexity of this specific problem is NP-hard. The computational efficiency of the heuristic is compared against the pseudo-polynomial algorithm proposed by Ng et al. [2006]. In the second single machine interfering job sets problem (P2), the criteria of minimizing total weighted completion time and maximum lateness is studied. This is an established NP-hard problem for which a Forward WSPT-EDD heuristic is presented that attempts to generate set of supported points and the solution quality is compared with MIP formulations. For both of these problems, all jobs are available at time zero and the jobs are not allowed to be preempted. / Dissertation/Thesis / Ph.D. Industrial Engineering 2012

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Este trabalho apresenta procedimentos que podem ser utilizados na programação da produção em um ambiente JIT. Esses procedimentos deveriam ser utilizados em sistemas clássicos de programação, onde a utilização do sistema kanban é inviável. O caso estudado se baseia em uma única máquina, com datas de entrega múltiplas e com penalidades distintas de adiantamento e de atraso para cada ordem. O objetivo a ser alcançado é a minimização do custo total. Para isso, é utilizado um procedimento de busca denominado beam search, para gerar as seqüências, e um algoritmo de inserção de ociosidade, para definir os programas. O algoritmo utilizado é uma generalização do algoritmo de GAREY et al. (1988) onde as penalidades são distintas para adiantamento e para atraso. O procedimento e o algoritmo são testados em várias condições sendo comparados com regras de despacho e com a função EXP-ET. Quando a função EXP-ET é utilizada com a possibilidade de inserção de ociosidade, o período de ociosidade ótimo é determinado. Assume-se que a dificuldade de solução do problema é dependente de dois parâmetros clássicos: fator de atraso médio e amplitude relativa das datas de entrega. Testes empíricos comparativos são realizados através de simulação computacional, onde se mede o tempo de solução e o valor alcançado pela função objetivo. Os resultados indicam que o desempenho dos vários procedimentos testados é altamente dependente dos dois parâmetros, mostrando que para a escolha de um procedimento apropriado, deve-se primeiramente conhecer o valor dos parâmetros. São fornecidos os resultados encontrados e os códigos computacionais utilizados no estudo. / This work presents some procedures which can be used in production scheduling problems in JIT environments. These procedures may be used in cases of classical production scheduling where the use of the kanban system is infeasible. The case studied is based on a single machine, with multiple due dates, and distinct earliness and tardiness penalties for each job. The objective function is to minimize total cost. A heuristic search procedure known as beam search is used to construct sequences of jobs, and an idleness insertion algorithm is used to obtain schedules. The algorithm used is a generalization of the GAREY et al. (1988) algorithm, where penalties are distinct for earliness and tardiness. The procedure and algorithm are tested in many conditions involving comparisons with dispatching rules and the EXP-ET function. When EXP-ET function is applied with possibility of idleness insertion, the optimal idleness period is provided. It was assumed that problem hardness is dependent on two classical parameters: average tardiness factor and relative range of due dates. Empirical comparative tests are conducted with computational simulation, where computational solution time and objective function value are evaluated. Results indicate that procedures performance is highly dependent on both parameters, showing that is necessary to know parameters values before choosing an appropriate procedure. The detailed results and computational code used in this study are also provided.

Este trabalho apresenta procedimentos que podem ser utilizados na programação da produção em um ambiente JIT. Esses procedimentos deveriam ser utilizados em sistemas clássicos de programação, onde a utilização do sistema kanban é inviável. O caso estudado se baseia em uma única máquina, com datas de entrega múltiplas e com penalidades distintas de adiantamento e de atraso para cada ordem. O objetivo a ser alcançado é a minimização do custo total. Para isso, é utilizado um procedimento de busca denominado beam search, para gerar as seqüências, e um algoritmo de inserção de ociosidade, para definir os programas. O algoritmo utilizado é uma generalização do algoritmo de GAREY et al. (1988) onde as penalidades são distintas para adiantamento e para atraso. O procedimento e o algoritmo são testados em várias condições sendo comparados com regras de despacho e com a função EXP-ET. Quando a função EXP-ET é utilizada com a possibilidade de inserção de ociosidade, o período de ociosidade ótimo é determinado. Assume-se que a dificuldade de solução do problema é dependente de dois parâmetros clássicos: fator de atraso médio e amplitude relativa das datas de entrega. Testes empíricos comparativos são realizados através de simulação computacional, onde se mede o tempo de solução e o valor alcançado pela função objetivo. Os resultados indicam que o desempenho dos vários procedimentos testados é altamente dependente dos dois parâmetros, mostrando que para a escolha de um procedimento apropriado, deve-se primeiramente conhecer o valor dos parâmetros. São fornecidos os resultados encontrados e os códigos computacionais utilizados no estudo. / This work presents some procedures which can be used in production scheduling problems in JIT environments. These procedures may be used in cases of classical production scheduling where the use of the kanban system is infeasible. The case studied is based on a single machine, with multiple due dates, and distinct earliness and tardiness penalties for each job. The objective function is to minimize total cost. A heuristic search procedure known as beam search is used to construct sequences of jobs, and an idleness insertion algorithm is used to obtain schedules. The algorithm used is a generalization of the GAREY et al. (1988) algorithm, where penalties are distinct for earliness and tardiness. The procedure and algorithm are tested in many conditions involving comparisons with dispatching rules and the EXP-ET function. When EXP-ET function is applied with possibility of idleness insertion, the optimal idleness period is provided. It was assumed that problem hardness is dependent on two classical parameters: average tardiness factor and relative range of due dates. Empirical comparative tests are conducted with computational simulation, where computational solution time and objective function value are evaluated. Results indicate that procedures performance is highly dependent on both parameters, showing that is necessary to know parameters values before choosing an appropriate procedure. The detailed results and computational code used in this study are also provided.

<p>This thesis contains two parts. In the first part, which contains Chapter 2 and Chapter 3, we consider scheduling problems with learning effect and time-dependent processing time on a single machine. In Chapter 2, we investigate the earliness-tardiness objective, as well as the objective without due date assignment consideration. By reducing them to a special linear assignment problem, we solve them in near-linear time. As a consequence, we improve the time complexity for some previous algorithms for scheduling problems with learning effect and/or time-dependent processing time. In Chapter 3, we investigate the total number of tardy jobs objective. By reducing them to a linear assignment problem, we solve them in polynomial time. For some important special cases, where there is only learning effect OR time-dependent processing time, we reduce the time complexity to quadratic time. In the second part, which contains Chapter 4 and Chapter 5, we investigate the bicriteria scheduling problems in a supply chain. We separate the objectives in two parts, where the delivery cost is one of them. We present efficient algorithms to identify all the Pareto-optimal solutions for various scenarios. In Chapter 4, we study the cases without due date assignment; while in Chapter 5 we study the cases with due date assignment consideration.</p>