Two-stage assembly-type flowshop batch scheduling problem subject to a fixed job sequence

This paper discusses a two-stage assembly-type flowshop scheduling problem with batching considerations subject to a fixed job sequence. The two-stage assembly flowshop consists of m stage-1 parallel dedicated machines and a stage-2 assembly machine which processes the jobs in batches. Four regular performance metrics, namely, the total completion time, maximum lateness, total tardiness, and number of tardy jobs, are considered. The goal is to obtain an optimal batching decision for the predetermined job sequence at stage 2. This study presents a two-phase algorithm, which is developed by coupling a problem-transformation procedure with a dynamic program. The running time of the proposed algorithm is O(mn+n⁵), where n is the number of jobs.     

A note on parallel-machine due-window assignment

We consider a due-window assignment problem on identical parallel machines, where the jobs have equal processing times and job-dependent earliness-tardiness costs. We would like to determine a ‘due window’ during which the jobs can be completed at no cost and to obtain a job schedule in which the jobs are penalized if they finish before or after the due window. The objective is to minimize the total earliness and tardiness job penalty, plus the cost associated with the size of the due window. We present an algorithm that can solve this problem in O(n³) time, which is an improvement of the O(n⁴) solution procedure developed by Mosheiov and Sarig.     

Scheduling jobs with release dates,equal processing times,and inclusive processing set restrictions

We consider the problem of scheduling a given set of n jobs with equal processing times on m parallel machines so as to minimize the makespan. Each job has a given release date and is compatible to only a subset of the machines. The machines are ordered and indexed in such a way that a higher-indexed machine can process all the jobs that a lower-indexed machine can process. We present a solution procedure to solve this problem in O(n²+mnlogn) time. We also extend our results to the tree-hierarchical processing sets case and the uniform machine case.     

Solution of the NP-hard total tardiness minimization problem in scheduling theory

The classical NP-hard (in the ordinary sense) problem of scheduling jobs in order to minimize the total tardiness for a single machine 1‖ΣT _j is considered. An NP-hard instance of the problem is completely analyzed. A procedure for partitioning the initial set of jobs into subsets is proposed. Algorithms are constructed for finding an optimal schedule depending on the number of subsets. The complexity of the algorithms is O(n ²Σp _j ), where n is the number of jobs and p _j is the processing time of the jth job (j = 1, 2, …, n).     

Batch scheduling with controllable setup and processing times to minimize total completion time

We study a problem of scheduling n jobs on a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time dependent on the position of this batch in the batch sequence. Setup times and job processing times are continuously controllable, that is, they are real-valued variables within their lower and upper bounds. A deviation of a setup time or job processing time from its upper bound is called a compression. The problem is to find a job sequence, its partition into batches, and the values for setup times and job processing times such that (a) total job completion time is minimized, subject to an upper bound on total weighted setup time and job processing time compression, or (b) a linear combination of total job completion time, total setup time compression, and total job processing time compression is minimized. Properties of optimal solutions are established. If the lower and upper bounds on job processing times can be similarly ordered or the job sequence is fixed, then O(n³ log n) and O(n⁵) time algorithms are developed to solve cases (a) and (b), respectively. If all job processing times are fixed or all setup times are fixed, then more efficient algorithms can be devised to solve the problems.     

A due-window assignment problem with position-dependent processing times

We extend a classical single-machine due-window assignment problem to the case of position-dependent processing times. In addition to the standard job scheduling decisions, one has to assign a time interval (due-window), such that jobs completed within this interval are assumed to be on time and not penalized. The cost components are: total earliness, total tardiness and due-window location and size. We introduce an O(n³) solution algorithm, where n is the number of jobs. We also investigate several special cases, and examine numerically the sensitivity of the solution (schedule and due-window) to the different cost parameters.     

Solutions to the Constrained Flowshop Sequencing Problem

If the flowshop sequencing problem is constrained so that job profiles are maintained independently of other work on the shop, then a range of new problem solutions are possible. Using geometrical relationships between the cumulative process times, a slope matching method has been devised which generally provides better results than Palmer's method and Gupta's algorithm. Secondly, it has been possible to reformulate the flowshop sequencing problem in terms of the travelling salesman problem. This has enabled the performance of the slope matching, Palmer and Gupta methods to be compared against a non-heuristic solution. Finally, provided that expressing a job in terms of the slopes of the cumulative start and end times is satisfactory, this enables in n-job, M-machine flowshop to be converted into an equivalent n-job 2 machine flowshop to which the use of Johnson's method will provide optimal sequences.     

A note on scheduling jobs with equal processing times and inclusive processing set restrictions

We consider the problem of scheduling n jobs on m parallel machines with inclusive processing set restrictions. Each job has a given release date, and all jobs have equal processing times. The objective is to minimize the makespan of the schedule. Li and Li (2015) have developed an O(n²+mn log?n) time algorithm for this problem. In this note, we present a modified algorithm with an improved time complexity of O(min{m, log?n} ? n log?n).     

The Systems Approach to Hospitals

By exploiting the relationship between scheduling and sorting, this paper describes a functional heuristic algorithm for seeking a quick and approximate solution to the n-job, M-machine flowshop scheduling problem under the assumption that all jobs are processed on all machines in the same order and no passing of jobs is permitted. The proposed functional heuristic algorithm can be executed by hand for reasonably large size problems and yields solutions which are closer to optimal solutions than those obtained by Palmer's slope index algorithm.     

Two-stage flexible flow shop scheduling subject to fixed job sequences

This paper investigates the scheduling problem in a two-stage flexible flow shop, which consists of m stage-1 parallel dedicated machines and a stage-2 bottleneck machine, subject to the condition that n _l jobs per type l∈{1, …, m} are processed in a fixed sequence. Four regular performance metrics, including the total completion time, the maximum lateness, the total tardiness, and the number of tardy jobs, are considered. For each considered objective function, we aim to determine an optimal interleaving processing sequence of all jobs coupled with their starting times on the stage-2 bottleneck machine. The problem under study is proved to be strongly NP-hard. An O(m²Π_l=1 ^m n _l ²) dynamic programming algorithm coupled with numerical experiments is presented.     

A graph-theoretic approach to interval scheduling on dedicated unrelated parallel machines

We study the problem of scheduling n non-preemptive jobs on m unrelated parallel machines. Each machine can process a specified subset of the jobs. If a job is assigned to a machine, then it occupies a specified time interval on the machine. Each assignment of a job to a machine yields a value. The objective is to find a subset of the jobs and their feasible assignments to the machines such that the total value is maximized. The problem is NP-hard in the strong sense. We reduce the problem to finding a maximum weight clique in a graph and survey available solution methods. Furthermore, based on the peculiar properties of graphs, we propose an exact solution algorithm and five heuristics. We conduct computer experiments to assess the performance of our and other existing heuristics. The computational results show that our heuristics outperform the existing heuristics.     

Scheduling an unbounded batching machine with family jobs and setup times

We consider the problem of scheduling n jobs on an unbounded batching machine that can process any number of jobs belonging to the same family simultaneously in the same batch. All jobs in the same batch complete at the same time. Jobs belonging to different families cannot be processed in the same batch, and setup times are required to switch between batches that process jobs from different families. For a fixed number of families m, we present a generic forward dynamic programming algorithm that solves the problem of minimizing an arbitrary regular cost function in pseudopolynomial time. We also present a polynomial-time backward dynamic programming algorithm with time complexity O (mn(n/m+1) ^m ) for minimizing any additive regular minsum objective function and any incremental regular minmax objective function. The effectiveness of our dynamic programming algorithm is shown by computational experiments based on the scheduling of the heat-treating process in a steel manufacturing plant.     

Flowshop scheduling problem to minimize total completion time with random and bounded processing times

The flowshop scheduling problems with n jobs processed on two or three machines, and with two jobs processed on k machines are addressed where jobs have random and bounded processing times. The probability distributions of random processing times are unknown, and only the lower and upper bounds of processing times are given before scheduling. In such cases, there may not exist a unique schedule that remains optimal for all feasible realizations of the processing times, and therefore, a set of schedules has to be considered which dominates all other schedules for the given criterion. We obtain sufficient conditions when transposition of two jobs minimizes total completion time for the cases of two and three machines. The geometrical approach is utilized for flowshop problem with two jobs and k machines.     

Common due window size and location determination in a single machine scheduling problem

We consider a single machine static and deterministic scheduling problem in which jobs have a common due window. Jobs completed within the window incur no penalties, other jobs incur either earliness or tardiness penalties. The objective is to find the optimal size and location of the window as well as an optimal sequence to minimise a cost function based on earliness, tardiness, window size, and window location. We propose an O(n log n) algorithm to solve the problem.     

Balancing Workloads and Minimizing Set-up Costs in the Parallel Processing Shop

This paper presents an optimal scheduling algorithm for minimizing set-up costs in the parallel processing shop while meeting workload balancing restrictions.There are M independent batch type jobs which have sequence dependent set-up costs and N parallel processing machines. Each of the M jobs must be processed on exactly one of the N available machines. It is desirable to minimize total changeover costs with the restriction that each machine workload assignment T _n be within P units of the average machine assignment. The paper describes a static problem in which all jobs are available at time zero. The sequence dependent change over costs are identical for each machine. An extension of the algorithm handles nonidentical processor problems.A combinatorial programming approach to the problem is used. For the special case of identical processors, the problem can be treated as a multi-salesman travelling salesman problem. A general branch and bound algorithm and numerical results are given.     

Online batch scheduling with kind release times and incompatible families to minimize makespan

This paper studies online scheduling of jobs with incompatible families on a single unbounded batch machine under the KRT environment, where jobs arrive over time and “KRT” means that in the online setting no jobs can be released when the machine is busy. The goal is to determine a feasible schedule to minimize the makespan. We provide a best online algorithm with a competitive ratio of \(1+\frac{\sqrt{f^{2}-f+1}-1}{f}\), where f is the number of job families.     

Specially Structured Precedence Constraints in Three-Dimensional Bottleneck Assignment Problems

A three-dimensional, time-minimizing (bottleneck) assignment problem consists of assigning n jobs to n workers to be performed on n machines under different forms of feasibility conditions so that the different functions of the individual times taken by a worker to finish a job on a given machine are minimized. The usual assumption made in such a problem is that all the jobs can be commenced simultaneously. In this paper, two specially structured precedence constraints on jobs are considered, which necessitate modifications in this assumption. Further, the main purpose here is to develop branch-and-bound-type algorithms for solving the corresponding problems and to illustrate them by a numerical example.     

Due-date assignment with asymmetric earliness–tardiness cost

The problem of minimizing the maximal weighted absolute lateness (MWAL) is known to be NP-hard. The due-date assignment part of MWAL for a given sequence has been shown in the literature to be solved on a single machine in O(n²) time. In this paper, we study a more general version of the problem with asymmetric cost (nonidentical earliness and tardiness weights). We introduce a linear-programming-based O(n) solution for this case. We also extend our proposed solution procedure to other machine settings such as flow-shop and parallel machines.     

Due-date assignment and single machine scheduling with deteriorating jobs

We study a scheduling problem with deteriorating jobs, that is, jobs whose processing times are an increasing function of their start times. We consider the case of a single machine and linear job-independent deterioration. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of due-date, earliness and tardiness penalties. We give an O(n log n) time algorithm to solve this problem.     

Minimizing a Quadratic Cost Function of Waiting Times in Single-Machine Scheduling

This paper deals with scheduling n jobs on a single machine in order to minimize the weighted sum of squared waiting times of the jobs. We present a powerful decomposition mechanism, based on a precedence relation concept, that easily handles problems of the size n = 50 and 100 where the processing times and penalties are independently drawn from a uniform distribution. This mechanism is incorporated along with new branching rules in a branch-and-bound scheme that efficiently handles tough problems of the size 20 and 50.