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).     

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.     

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).     

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.     

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.     

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.     

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.     

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.     

Radar placement along banks of river

In this paper, we consider the Radar Placement and Power Assignment problem (RPPA) along a river. In this problem, a set of crucial points in the river are required to be monitored by a set of radars which are placed along the two banks. The goal is to choose the locations for the radars and assign powers to them such that all the crucial points are monitored and the total power is minimized. If each crucial point is required to be monitored by at least k radars, the problem is a k-Coverage RPPA problem (k-CRPPA). Under the assumption that the river is sufficiently smooth, one may focus on the RPPA problem along a strip (RPPAS). In this paper, we present an O(n ⁹) dynamic programming algorithm for the RPPAS, where n is the number of crucial points to be monitored. In the special case where radars are placed only along the upper bank, we present an O(kn ⁵) dynamic programming algorithm for the k-CRPPAS. For the special case that the power is linearly dependent on the radius, we present an O(n log n)-time \({2\sqrt 2}\)-approximation algorithm for the RPPAS.     

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.     

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.     

Mappings of preserving <Emphasis Type="Italic">n</Emphasis>-distance one in <Emphasis Type="Italic">n</Emphasis>-normed spaces

We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\). Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.     

Minimizing the Flow-time Variance in Single-machine Systems

In this paper we deal with an n-job, single-machine scheduling problem. All jobs are available from the start, and the objective is to minimize the variance of job flow-times. A heuristic procedure which is based on the complementary pair-exchange principle is proposed. It has been concluded that this heuristic procedure provides improved results (in terms of objective-function value) when compared with other heuristics. Our heuristic procedure has the complexity of O(n log 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.     

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.     

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.     

The connected <Emphasis Type="Italic">p</Emphasis>-median problem on block graphs

In this paper, we study a variant of the p-median problem on block graphs G in which the p-median is asked to be connected, and this problem is called the connected p-median problem. We first show that the connected p-median problem is NP-hard on block graphs with multiple edge weights. Then, we propose an O(n)-time algorithm for solving the problem on unit-edge-weighted block graphs, where n is the number of vertices in G.     

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.     

Homogenization of a variational inequality for the <Emphasis Type="Italic">p</Emphasis>-Laplacian in perforated media with nonlinear restrictions for the flux on the boundary of isoperimetric perforations: <Emphasis Type="Italic">p</Emphasis> equal to the dimension of the space

We address the homogenization of a variational inequality posed in perforated media issue from a unilateral problem for the p-Laplacian. We consider the n-Laplace operator in a perforated domain of ?ⁿ, n ≥ 3, with restrictions for the solution and its flux (the flux associated with the n-Laplacian) on the boundary of the perforations which are assumed to be isoperimetric. The solution is assumed to be positive on the boundary of the holes and the flux is bounded from above by a negative, nonlinear monotone function multiplied by a large parameter. A certain non periodical distribution of the perforations is allowed while the assumption that their size is much smaller than the periodicity scale is performed. We make it clear that in the average constants of the problem, the perimeter of the perforations appears for any shape.     

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.