Serial-batch scheduling – the special case of laser-cutting machines

The dissertation deals with a problem in the field of short-term production planning, namely the scheduling of laser-cutting machines. The object of decision is the grouping of production orders (batching) and the sequencing of these order groups on one or more machines (scheduling). This problem is also known in the literature as "batch scheduling problem" and belongs to the class of combinatorial optimization problems due to the interdependencies between the batching and the scheduling decisions. The concepts and methods used are mainly from production planning, operations research and machine learning

Algorithms for Scheduling Problems and Integer Programming

The first part of this thesis gives approximation results to scheduling problems. The classical makespan minimization problem on identical parallel machines asks for a distribution of a set of jobs to a set of machines such that the latest job completion time is minimized. For this strongly NP-complete problem we give a new EPTAS algorithm. In fact, it admits a practical implementation which beats the currently best approximation ratio of the MULTIFIT algorithm. A well-studied extension of the problem is the partition of the jobs into classes which impose a class-specific setup time on a machine whenever the processing switches to a job of a different class. For these so-called scheduling problems with batch setup times we present a 1.5-approximation algorithm for each of the three major settings. We achieve similar results for the likewise natural variant of many shared resources scheduling (MSRS) where instead of imposing a setup time each class is identified by a resource which can be occupied by at most one of its jobs at a time. For MSRS we present a 1.5-approximation and two EPTAS results. The second part provides results for fixed-priority uniprocessor real-time scheduling and variants of block-structured integer programming. We give a new approach to compute worst-case response times which admits a polynomial-time algorithm for harmonic periods even in the presence of task release jitters. In more detail, we prove a duality between Response Time Computation (RTC) and the Mixing Set problem. Furthermore, both problems can be expressed as block-structured integer programs which are closely related to simultaneous congruences. However, the setting of the famous Chinese Remainder Theorem is that each congruence has to have a certain remainder. We relax this setting such that the remainder of each congruence may lie in a given interval. We show that the smallest solution to these congruences can be computed in polynomial time if the set of divisors is harmonic

Minimizing Cumulative Batch Processing Time for an Industrial Oven Scheduling Problem

We introduce the Oven Scheduling Problem (OSP), a new parallel batch scheduling problem that arises in the area of electronic component manufacturing. Jobs need to be scheduled to one of several ovens and may be processed simultaneously in one batch if they have compatible requirements. The scheduling of jobs must respect several constraints concerning eligibility and availability of ovens, release dates of jobs, setup times between batches as well as oven capacities. Running the ovens is highly energy-intensive and thus the main objective, besides finishing jobs on time, is to minimize the cumulative batch processing time across all ovens. This objective distinguishes the OSP from other batch processing problems which typically minimize objectives related to makespan, tardiness or lateness. We propose to solve this NP-hard scheduling problem via constraint programming (CP) and integer linear programming (ILP) and present corresponding CP- and ILP-models. For an experimental evaluation, we introduce a multi-parameter random instance generator to provide a diverse set of problem instances. Using state-of-the-art solvers, we evaluate the quality and compare the performance of our CP- and ILP-models, which could find optimal solutions for many instances. Furthermore, using our models we are able to provide upper bounds for the whole benchmark set including large-scale instances

Non-identical parallel machines batch processing problem with release dates, due dates and variable maintenance activity to minimize total tardiness

[EN] Combination of job scheduling and maintenance activity has been widely investigated in the literature. We consider a non-identical parallel machines batch processing (BP) problem with release dates, due dates and variable maintenance activity to minimize total tardiness. An original mixed integer linear programming (MILP) model is formulated to provide an optimal solution. As the problem under investigation is known to be strongly NP-hard, two meta-heuristic approaches based on Simulated Annealing (SA) and Variable Neighborhood Search (VNS) are developed. A constructive heuristic method (H) is proposed to generate initial feasible solutions for the SA and VNS. In order to evaluate the results of the proposed solution approaches, a set of instances were randomly generated. Moreover, we compare the performance of our proposed approaches against four meta heuristic algorithms adopted from the literature. The obtained results indicate that the proposed solution methods have a competitive behaviour and they outperform the other meta-heuristics in most instances. Although in all cases, H + SA is the most performing algorithm compared to the others.Beldar, P.; Moghtader, M.; Giret Boggino, AS.; Ansaripoord, AH. (2022). Non-identical parallel machines batch processing problem with release dates, due dates and variable maintenance activity to minimize total tardiness. Computers & Industrial Engineering. 168:1-28. https://doi.org/10.1016/j.cie.2022.10813512816

A DECOMPOSITION-BASED HEURISTIC ALGORITHM FOR PARALLEL BATCH PROCESSING PROBLEM WITH TIME WINDOW CONSTRAINT

This study considers a parallel batch processing problem to minimize the makespan under constraints of arbitrary lot sizes, start time window and incompatible families. We first formulate the problem with a mixed-integer programming model. Due to the NP-hardness of the problem, we develop a decomposition-based heuristic to obtain a near-optimal solution for large-scale problems when computational time is a concern. A two-dimensional saving function is introduced to quantify the value of time and capacity space wasted. Computational experiments show that the proposed heuristic performs well and can deal with large-scale problems efficiently within a reasonable computational time. For the small-size problems, the percentage of achieving optimal solutions by the DH is 94.17%, which indicates that the proposed heuristic is very good in solving small-size problems. For large-scale problems, our proposed heuristic outperforms an existing heuristic from the literature in terms of solution quality

Parallel batching with multi-size jobs and incompatible job families

Parallel batch scheduling has many applications in the industrial sector, like in material and chemical treatments, mold manufacturing and so on. The number of jobs that can be processed on a machine mostly depends on the shape and size of the jobs and of the machine. This work investigates the problem of batching jobs with multiple sizes and multiple incompatible families. A flow formulation of the problem is exploited to solve it through two column generation-based heuristics. First, the column generation finds the optimal solution of the continuous relaxation, then two heuristics are proposed to move from the continuous to the integer solution of the problem: one is based on the price-and-branch heuristic, the other on a variable rounding procedure. Experiments with several combinations of parameters are provided to show the impact of the number of sizes and families on computation times and quality of solutions

Advances and Novel Approaches in Discrete Optimization

Discrete optimization is an important area of Applied Mathematics with a broad spectrum of applications in many fields. This book results from a Special Issue in the journal Mathematics entitled ‘Advances and Novel Approaches in Discrete Optimization’. It contains 17 articles covering a broad spectrum of subjects which have been selected from 43 submitted papers after a thorough refereeing process. Among other topics, it includes seven articles dealing with scheduling problems, e.g., online scheduling, batching, dual and inverse scheduling problems, or uncertain scheduling problems. Other subjects are graphs and applications, evacuation planning, the max-cut problem, capacitated lot-sizing, and packing algorithms