Updating an optimal structured scheme

An optimal structured schedule at time t is considered for a set of jobs Z with given start and due date [d _i ,D _i ] volumes V_i (volume is defined as the number of homogeneous independent elementary operations of unit length that comprise the job), and penalty functions. The penalty for selecting an element of jobiZ at timet is _i (t). The schedule penalty is the total penalty of all the elements of all the jobs. An optimal schedule is a minimum-penalty schedule. We investigate the impact of changing the volume of a job from the setZ on the structure of the optimal schedule. Algorithms are proposed for handling the modified job set with both reduced and enlarged job volumes. These algorithms require ck computer operations, where k is the number of jobs in the original set, is the change in job volume (expressed by the number of units), andC is a constant.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 151–161, 1981.In conclusion, I would like to acknowledge the valuable attention of K. V. Shakhbazyan.     

On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion

Optimal schedules in the job shop problem with preemption and with the objective of minimizing an arbitrary regular function of operation completion times are studied. It is shown that for any instance of the problem there always exists an optimal schedule that meets several remarkable properties. Firstly, each changeover date coincides with the completion time of some operation, and so, the number of changeover dates is not greater than the total number of operations, while the total number of interruptions of the operations is no more than the number of operations minus the number of jobs. Secondly, every changeover date is “super-integral”, which means that it is equal to the total processing time of some subset of operations. And thirdly, the optimal schedule with these properties can be found by a simple greedy algorithm under properly defined priorities of operations on machines. It is also shown that for any instance of the job shop problem with preemption allowed there exists a finite set of its feasible schedules which contains at least one optimal schedule for any regular objective function (from the continuum set of regular functions).     

Open shop scheduling with some additional constraints

An open shop scheduling problem is presented; preemptions during processing of a job on a processorp is allowed but the job cannot be sent on another processorq before it is finished onp. A graph-theoretical model is described and a characterization is given for problems where schedules with such restricted preemptions useT time units whereT is the maximum of the processing times of the jobs and of the working times of the processors. The general case is shown to be NP-complete. We also consider the case where some constraints of simultaneity are present. Complexity of the problem is discussed and a solvable case is described.     

The dominance digraph as a solution to the two-machine flow-shop problem with interval processing times

The flow-shop minimum-length scheduling problem with n jobs processed on two machines is addressed where processing times are uncertain: lower and upper bounds for the random processing time are given before scheduling, but its probability distribution between these bounds is unknown. For such a problem, there often does not exist a dominant schedule that remains optimal for all possible realizations of the job processing times, and we look for a minimal set of schedules that is dominant. Such a minimal dominant set of schedules may be represented by a dominance digraph. We investigate useful properties of such a digraph.     

Asymptotic scheduling

We deal with the following scheduling problem: a finite set of jobs is given and each job consists in the execution of an infinite number of tasks. A task is a sequence of operations and each operation requires a specific machine. A machine can process only one operation at a time and preemption is not allowed. Performance measures of the processing system involve fixing a time horizon T, counting the number of tasks completed within T for each job and maximizing a specified function of these numbers to estimate the throughput of the schedule. Whilst computing the throughput for a given T is in general an extremely difficult problem, it is shown in this paper that the limit, as T tends to infinity, of the average throughput (i.e. the throughput divided by T) can be easily computed via Linear Programming under fairly mild conditions. This quantity, which may be called the asymptotic throughput, can be used to assess a bound on performance measures of real systems. Buffers play a crucial role and buffer sizes can be taken care of in assessing the system performance. Mathematics Subject Classification (2000):90B35, 90C05, 90C27, 90C90     

具有不可用区间且工件可拒绝下的单机重新排序问题的近似方案

本文考虑了机器具有不可用区间且工件可拒绝下的单机重新排序问题,在该问题中,给定一个工件集需在一台机器上加工,每个工件有自己的加工时间和权重,且对该工件集目标函数为极小化总加权完工时间的排序计划已给定,根据该排序计划中每个工件的完工时间已确定每个工件的承诺交付时间。然而,在工件正式开始加工前,原计划用于加工的某段时间区间因临时用于检修机器而导致机器在该时间区间不再可用,需要对工件重新排序。为了确保在新的重新排序中,工件的延误成本不致太大,决策者可以选择拒绝部分工件,但需支付相应的拒绝费用。任务是确定接受工件集和拒绝工件集,并将接受的工件在考虑机器具有不可用区间的条件下重新排序使得接受工件集的总加权完工时间,总拒绝费用及赋权最大延误之和最小。该问题是NP-困难的,对此给出了伪多项式时间动态规划精确算法,利用稀疏技术设计了完全多项式时间近似方案。     

New steps in the amazing world of sequences and schedules

In this paper we consider classical shop problems:n jobs have to be processed onm machines. The processing timep _i,j of jobi on machinej is given for all operations (i, j). Each machine can process at most one job at a time and each job can be processed at most on one machine at a given time. The machine orders are fixed (job-shop) or arbitrary (open-shop). We have to determine a feasible combination of machine and job orders, a so-called sequence, which minimizes the makespan. We introduce a partial order on the set of sequences with the property that there exists at least one optimal sequence in the set of minimal elements of this partial order independent of the given processing times. The set of minimal elements (set of irreducible sequences) can be in detail described in the case of the two machine open-shop problem. The cardinality is calculated. We will show which sequences are generated by the well-known polynomial algorithms for the construction of optimal schedules. Furthermore, we investigate the problemO∥C _max on an operation set with spanning tree structure. Supported by Deutsche Forschungsgemeinschaft, Project ScheMA     

Preemptive scheduling of equal length jobs with release dates on two uniform parallel machines

We consider a problem of scheduling n jobs on two uniform parallel machines. For each job we are given its release date when the job becomes available for processing. All jobs have equal processing requirements. Preemptions are allowed. The objective is to find a schedule minimizing total completion time. We suggest an O(n³) algorithm to solve this problem.     

工件延误和可拒绝下的单机重新排序问题的近似方案

研究工件延误产生干扰且延误工件可拒绝下的单机重新排序问题.在该问题中,给定计划在零时刻到达的一个工件集需在一台机器上加工,工件集中的每个工件有它的加工时间和权重,在工件正式开始加工前,按照最短赋权加工时间优先的初始排序已经给定,目标函数是极小化赋权完工时间和,据此每个工件的承诺交付截止时间也给定.然而,在工件正式开始加...     

Improved Bounds for Acyclic Job Shop Scheduling

In acyclic job shop scheduling problems there are n jobs and m machines. Each job is composed of a sequence of operations to be performed on different machines. A legal schedule is one in which within each job, operations are carried out in order, and each machine performs at most one operation in any unit of time. If D denotes the length of the longest job, and C denotes the number of time units requested by all jobs on the most loaded machine, then clearly lb = max[C,D] is a lower bound on the length of the shortest legal schedule. A celebrated result of Leighton, Maggs, and Rao shows that if all operations are of unit length, then there always is a legal schedule of length O(lb), independent of n and m. For the case that operations may have different lengths, Shmoys, Stein and Wein showed that there always is a legal schedule of length , where the notation is used to suppress terms. We improve the upper bound to . We also show that our new upper bound is essentially best possible, by proving the existence of instances of acyclic job shop scheduling for which the shortest legal schedule is of length . This resolves (negatively) a known open problem of whether the linear upper bound of Leighton, Maggs, and Rao applies to arbitrary job shop scheduling instances (without the restriction to acyclicity and unit length operations). Received June 30, 1998 RID="*" ID="*" Incumbent of the Joseph and Celia Reskin Career Development Chair RID="†" ID="†" Research was done while staying at the Weizmann Institute, supported by a scholarship from the Minerva foundation.     

Parallel machine scheduling with a common due window

In this paper, we consider a machine scheduling problem where jobs should be completed at times as close as possible to their respective due dates, and hence both earliness and tardiness should be penalized. Specifically, we consider the problem with a set of independent jobs to be processed on several identical parallel machines. All the jobs have a given common due window. If a job is completed within the due window, then there is no penalty. Otherwise, there is either a job-dependent earliness penalty or a job-dependent tardiness penalty depending on whether the job is completed before or after the due window. The objective is to find an optimal schedule with minimum total earliness–tardiness penalty. The problem is known to be NP-hard. We propose a branch and bound algorithm for finding an optimal schedule of the problem. The algorithm is based on the column generation approach in which the problem is first formulated as a set partitioning type formulation and then in each branch and bound iteration the linear relaxation of this formulation is solved by the standard column generation procedure. Our computational experiments show that this algorithm is capable of solving problems with up to 40 jobs and any number of machines within a reasonable computational time.     

Minimizing Weighted Number of Early and Tardy Jobs with a Common Due Window Involving Location Penalty

This paper studies a single machine scheduling problem to minimize the weighted number of early and tardy jobs with a common due window. There are n non-preemptive and simultaneously available jobs. Each job will incur an early (tardy) penalty if it is early (tardy) with respect to the common due window under a given schedule. The window size is a given parameter but the window location is a decision variable. The objective of the problem is to find a schedule that minimizes the weighted number of early and tardy jobs and the location penalty. We show that the problem is NP-complete in the ordinary sense and develop a dynamic programming based pseudo-polynomial algorithm. We conduct computational experiments, the results of which show that the performance of the dynamic algorithm is very good in terms of memory requirement and CPU time. We also provide polynomial time algorithms for two special cases.     

The unbounded parallel-batch scheduling with rejection

In this paper, we consider the unbounded parallel-batch scheduling with rejection. A job is either rejected, in which case a certain penalty has to be paid, or accepted and processed in batches on a machine. The processing time of a batch is defined as the longest processing time of the jobs contained in it. Four problems are considered: (1) to minimize the sum of the total completion time of the accepted jobs and the total rejection penalty of the rejected jobs; (2) to minimize the total completion time of the accepted jobs subject to an upper bound on the total rejection penalty of the rejected jobs; (3) to minimize the total rejection penalty of the rejected jobs subject to an upper bound on the total completion time of the accepted jobs; (4) to find the set of all the Pareto optimal schedules. We provide a polynomial-time algorithm for the first problem. Furthermore, we show that all the other three problems are binary NP-hard and present a pseudo-polynomial-time algorithm and a fully polynomial-time approximation scheme for them.     

Asymptotically Optimal Algorithms for Job Shop Scheduling and Packet Routing

We propose asymptotically optimal algorithms for the job shop scheduling and packet routing problems. We propose a fluid relaxation for the job shop scheduling problem in which we replace discrete jobs with the flow of a continuous fluid. We compute an optimal solution of the fluid relaxation in closed form, obtain a lower bound C_max to the job shop scheduling problem, and construct a feasible schedule from the fluid relaxation with objective value at most where the constant in the O( · ) notation is independent of the number of jobs, but it depends on the processing time of the jobs, thus producing an asymptotically optimal schedule as the total number of jobs tends to infinity. If the initially present jobs increase proportionally, then our algorithm produces a schedule with value at most C_max + O(1). For the packet routing problem with fixed paths the previous algorithm applies directly. For the general packet routing problem we propose a linear programming relaxation that provides a lower bound C_max and an asymptotically optimal algorithm that uses the optimal solution of the relaxation with objective value at most Unlike asymptotically optimal algorithms that rely on probabilistic assumptions, our proposed algorithms make no probabilistic assumptions and they are asymptotically optimal for all instances with a large number of jobs (packets). In computational experiments our algorithms produce schedules which are within 1% of optimality even for moderately sized problems.     

Single machine batch scheduling to minimize the weighted number of late jobs

The single machine batch scheduling problem to minimize the weighted number of late jobs is studied. In this problem,n jobs have to be processed on a single machine. Each job has a processing time, a due date and a weight. Jobs may be combined to form batches containing contiguously scheduled jobs. For each batch, a constant set-up time is needed before the first job of this batch is processed. The completion time of each job in the batch coincides with the completion time of the last job in this batch. A job is late if it is completed after its due date. A schedule specifies the sequence of jobs and the size of each batch, i.e. the number of jobs it contains. The objective is to find a schedule which minimizes the weighted number of late jobs. This problem isNP-hard even if all due dates are equal. For the general case, we present a dynamic programming algorithm which solves the problem with equal weights inO(n ³) time. We formulate a certain scaled problem and show that our dynamic programming algorithm applied to this scaled problem provides a fully polynomial approximation scheme for the original problem. Each algorithm of this scheme has a time requirement ofO(n ³/ +n ³ logn). A side result is anO(n logn) algorithm for the problem of minimizing the maximum weight of late jobs.Supported by INTAS Project 93-257.     

Two machine preemptive scheduling problem with release dates,equal processing times and precedence constraints

We consider a scheduling problem with two identical parallel machines and n jobs. For each job we are given its release date when job becomes available for processing. All jobs have equal processing times. Preemptions are allowed. There are precedence constraints between jobs which are given by a (di)graph consisting of a set of outtrees and a number of isolated vertices. The objective is to find a schedule minimizing mean flow time. We suggest an O(n²) algorithm to solve this problem.The suggested algorithm also can be used to solve the related two-machine open shop problem with integer release dates, unit processing times and analogous precedence constraints.     

A priority list based heuristic for the job shop problem

A priority list for the job shop scheduling problem is defined to be any permutation of a set of symbols where the symbol for each job appears as many times as the number of its operations. Every priority list can be associated in a natural way with a feasible schedule, and every feasible schedule arises in this way. Priority lists are therefore a representation of feasible schedules that avoid the problems normally associated with schedule infeasibility. As a result, the three ingredients of local search heuristics, namely picking initial starting schedules, constructing search neighbourhoods and computing makespans, become faster and easier when performed in the space of priority lists rather than in the space of feasible schedules. As an illustration of their usefulness, a priority list based simulated annealing heuristic is presented, which, although simple, is competitive with the current leading schedule based simulated annealing and tabu search heuristics.     

A polynomial algorithm for a two-machine no-wait job-shop scheduling problem

This paper considers the no-wait scheduling of n jobs, where each job is a chain of unit processing time operations to be processed alternately on two machines. The objective is to minimize the mean flow time. We propose an O(n⁶)-time algorithm to produce an optimal schedule. It is also shown that if zero processing time operations are allowed, then the problem is NP-hard in the strong sense.     

工件存在不同交货宽容期的最大加权超前延误问题

本文考虑下述排序问题：有n个工件需在同一台机器上加工,对各工件有一宽容交货期,若一工件在其宽容期前完工则受加权超前惩罚,若在其宽容期后完工则受加权延误惩罚,要求适当安排一加工方式使最大惩罚最小,文中相应某指定工件需准时完工的上述问题证得了Np-hard性,给出了最优算法,并作了一些讨论。     

Single machine group scheduling with ordered criteria

The single machine group scheduling problem is considered. Jobs are classified into several groups on the basis of group technology, i.e. jobs of the same group have to be processed jointly. A machine set-up time independent of the group sequence is needed between each two consecutive groups. A schedule specifies the sequence of groups and the sequence of jobs in each group. The quality of a schedule is measured by the criteriaF ₁, ...,F _m ordered by their relative importance. The objective is to minimize the least important criterionF _m subject to the schedule being optimal with respect to the more important criterionF _m–1 which is minimized on the set of schedules minimizing criterionF _m–2 and so on. The most important criterion isF ₁, which is minimized on the set of all feasible schedules. An approach to solve this multicriterion problem in polynomial time is presented if functionsF ₁, ...,F _m have special properties. The total weighted completion time and the total weighted exponential time are the examples of functionsF ₁, ...,F _m–1 and the maximum cost is an example of functionF _m for which our approach can be applied.The research of the authors was partially supported by a KBN Grant No. 3 P 406 003 05, the Fundamental Research Fund of Belarus, Project N 60-242, and the Deutsche Forschungsgemeinschaft, Project Schema, respectively. The paper was completed while the first author was visiting the University of Melbourne.