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.     

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.     

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.     

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 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).     

Minimizing the number of early jobs on a proportionate flowshop

A proportionate flowshop is a special case of the classical flowshop, where the job processing times are machine-independent. We study the problem of minimizing the number of early jobs in this machine setting. This objective function has hardly been investigated on a single machine, and never on a flowshop. We introduce an efficient iterative solution algorithm. In each iteration, a single job is moved to the first position (and is added to the set of early jobs), and the remaining jobs are rescheduled such that the maximum earliness is minimized. The algorithm guarantees an optimal solution in O(n³) time, where n is the number of jobs.     

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.     

Three-machine shop scheduling with partially ordered processing routes

This paper considers the problem of sequencing n jobs in a three-machine shop with the objective of minimising the maximum completion time. The shop consists of three machines, M₁,M₂ and M_{3}. A job is first processed on M₁ and then is assigned either the route (M₂,M_{3}) or the route (M_{3},M₂). Thus, for our model the processing route is given by a partial order of machines, as opposed to the linear order of machines for a job shop, or to an arbitrary sequence of machines for an open shop. The main result is on O(nlog n) time heuristic, which generates a schedule with the makespan that is at most 5/3 times the optimum value.     

Optimal Due-Date Determination and Sequencing of <Emphasis Type="Italic">n</Emphasis> Jobs on a Single Machine

Given a set of n jobs with deterministic processing times and the same ready times, the problem is to find the optimal processing-time multiple k* for the T.W.K. due-date assignment method, and the optimal sequence σ* to minimize the total amount of missed due-dates. It is found that k* is a constant for a given job set and σ* should be in S.P.T. sequence. After the theoretical treatment, a numerical example is given for discussion. The optimal results can readily be extended to situations in which the processing times are random variables with known means and having the same coefficient of variation. From a practical point of view, the main merit of this paper is that it demonstrates how, under certain production environments in which completion times of the jobs can be anticipated, to determine the optimal due-dates and obtain the optimal sequence.     

The equal allocation policy in open shop batch scheduling

We study the optimality of the very practical policy of equal allocation of jobs to batches in batch scheduling problems on an m-machine open shop. The objective is minimum makespan. We assume unit processing time jobs, machine-dependent setup times and batch availability. We show that equal allocation is optimal for a two-machine and a three-machine open shop. Although, this policy is not necessarily optimal for larger size open shops, it is shown numerically to produce very close-to-optimal schedules.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Single Stage Minimum Absolute Lateness Problem with a Common Due Date on Non-Identical Machines

In this paper we consider scheduling n single operation jobs with a common due date on m non-identical machines (in parallel) so as to minimize the sum of the absolute lateness. We reduce the problem to a transportation problem that can be solved by a polynomial time algorithm. Furthermore, we consider the problem in the case of identical machines and we give a heuristic algorithm to minimize makespan among all schedules that minimize the absolute lateness problem.     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

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.     

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.     

Parallel-machine scheduling with release dates and rejection

In this paper, we consider the parallel-machine scheduling problem with release dates and rejection. A job is either rejected, in which case a rejection penalty has to be paid, or accepted and processed on one of the m identical parallel machines. The objective is to minimize the sum of the makespan of the accepted jobs and the total rejection penalty of the rejected jobs. When m is a fixed constant, we provide a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme for the problem. When m is arbitrary, we present a 2-approximation algorithm for the problem.     

On the Development of a Mixed-Integer Linear Programming Model for the Flowshop Sequencing Problem

This paper describes the development of a mixed-integer linear programming (MILP) model for the standard N-job, M-machine flowshop sequencing problem. Based on an earlier all-integer model developed by Wagner, this MILP model has been used to solve optimally problems with as many as 25 jobs and as many as 10 machines. Variants of the standard flowshop model, including a variety of performance measures, are also presented. Computational experience involving the successful solution of over 175 flowshop problems is discussed, and suggestions for future research projects are offered.     

Ensemble of a subset of <Emphasis Type="Italic">k</Emphasis>NN classifiers

Combining multiple classifiers, known as ensemble methods, can give substantial improvement in prediction performance of learning algorithms especially in the presence of non-informative features in the data sets. We propose an ensemble of subset of kNN classifiers, ESkNN, for classification task in two steps. Firstly, we choose classifiers based upon their individual performance using the out-of-sample accuracy. The selected classifiers are then combined sequentially starting from the best model and assessed for collective performance on a validation data set. We use bench mark data sets with their original and some added non-informative features for the evaluation of our method. The results are compared with usual kNN, bagged kNN, random kNN, multiple feature subset method, random forest and support vector machines. Our experimental comparisons on benchmark classification problems and simulated data sets reveal that the proposed ensemble gives better classification performance than the usual kNN and its ensembles, and performs comparable to random forest and support vector machines.     

Λ-Shaped Policies to Schedule Deteriorating Jobs

We study a problem of scheduling deteriorating jobs, i.e. jobs whose processing times are an increasing function of their starting times. We consider the case of a single machine and linear job-independent deterioration. The objective is to minimize the sum of weighted completion times, with weights proportional to the basic processing times. The optimal schedule is shown to be Λ-shaped, i.e. the sequence of the basic processing times has a single local maximum. Moreover, we show that the problem is solved in O(N log N) time. In the last section we test heuristics for the case of general weights.     

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.