Scheduling parallel machines with inclusive processing set restrictions and job release times

We consider the problem of scheduling a set of jobs with different release times on parallel machines so as to minimize the makespan of the schedule. The machines have the same processing speed, but each job is compatible with only a subset of those machines. The machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. We present an efficient algorithm for this problem with a worst-case performance ratio of 2. We also develop a polynomial time approximation scheme (PTAS) for the problem, as well as a fully polynomial time approximation scheme (FPTAS) for the case in which the number of machines is fixed.     

工件带简单线性恶化函数和共同交货期单机排序问题

本文研究单机排序问题,其中工件加工时间具有简单线性恶化函数.同时,所有工件均具有一个给定共同交货期.目标函数为最小化提前有奖延误受罚之和.在逆一致性条件下,给出了求解该排序问题的一个伪多项式时间动态规划算法.同时借助于几何舍入技巧,对求解这类排序问题给出了一个充分多项式时间的近似算法(FPTAS)。     

机器带不可用时间限制的简单线性恶化供应链排序问题

研究的单机供应链排序问题中, 机器有一个不可用时间限制, 工件的加工时间与恶化率及其开工时间有关, 且工件的加工不可恢复. 一个或多个完工工件可组成一个发送批由车辆发送给客户, 且在机器不可用时间限制之前完工的工件必须在限制开始之时或之前完成发送. 问题的目标是最小化总发送时间与总发送费用之和. 证明问题是NP-难的, 提出了伪多项式时间的动态规划算法. 进一步, 在确定问题目标函数值的上界及下界之后, 设计了一个完全多项式时间近似方案(FPTAS).     

工件可转包加工的排序问题研究

研究工件可以转包加工的单台机排序问题: 有n个工件, 在零时刻已经到达一个单台机处, 每个工件可以由加工者自有的单台机器加工或者转包给其他机器加工. 如果工件被转包加工, 那么其完工时间等于在自有机器上的加工时间, 而产生的加工费用与在自有机器上加工的费用不同. 假设被转包加工的工件的完工时间和加工费用与转包加工机器的总负载没有关系.目标函数是最小化工件最大完工时间与总加工费用的加权和. 该问题已经被证明是NP-难的. 最后给出该问题的伪多项式时间最优算法, 并且提出一个完全多项式时间近似方案(FPTAS).     

工件具有子工件工期的排序问题

研究了工件具有子工件工期的排序问题.需要在一台单机上加工若干个给定的工件.每个工件由若干个子工件组成,每个子工件都有各自的工期.只有当工件的每个子工件都按时完成,才能称该工件是按时完工工件,否则,称该工件产生延误.目标是最大化按时完工的工件个数.证明当每个工件都被分成两个子工件时,该问题是NP-难的,而且不存在完全多项式时间近似方案(fully polynomial time approximation scheme,简记为FPTAS).提出两个启发式算法,利用数值模拟比较它们的性能,并且将这两个启发式算法的解与最优解的上界进行比较.     

Customer order scheduling to minimize the number of late jobs

In the order scheduling problem, every job (order) consists of several tasks (product items), each of which will be processed on a dedicated machine. The completion time of a job is defined as the time at which all its tasks are finished. Minimizing the number of late jobs was known to be strongly NP-hard. In this note, we show that no FPTAS exists for the two-machine, common due date case, unless P = NP. We design a heuristic algorithm and analyze its performance ratio for the unweighted case. An LP-based approximation algorithm is presented for the general multicover problem. The algorithm can be applied to the weighted version of the order scheduling problem with a common due date.     

The just-in-time scheduling problem in a flow-shop scheduling system

We study the problem of maximizing the weighted number of just-in-time (JIT) jobs in a flow-shop scheduling system under four different scenarios. The first scenario is where the flow-shop includes only two machines and all the jobs have the same gain for being completed JIT. For this scenario, we provide an O(n³) time optimization algorithm which is faster than the best known algorithm in the literature. The second scenario is where the job processing times are machine-independent. For this scenario, the scheduling system is commonly referred to as a proportionate flow-shop. We show that in this case, the problem of maximizing the weighted number of JIT jobs is NP-hard in the ordinary sense for any arbitrary number of machines. Moreover, we provide a fully polynomial time approximation scheme (FPTAS) for its solution and a polynomial time algorithm to solve the special case for which all the jobs have the same gain for being completed JIT. The third scenario is where a set of identical jobs is to be produced for different customers. For this scenario, we provide an O(n³) time optimization algorithm which is independent of the number of machines. We also show that the time complexity can be reduced to O(n log n) if all the jobs have the same gain for being completed JIT. In the last scenario, we study the JIT scheduling problem on m machines with a no-wait restriction and provide an O(mn²) time optimization algorithm.     

Scheduling with step-improving processing times

We consider the scheduling problem of minimizing the makespan on a single machine with step-improving job processing times around a common critical date. For this problem we give an NP-hardness proof, a fast pseudo-polynomial time algorithm, an FPTAS, and an on-line algorithm with best possible competitive ratio.     

“Product Partition” and related problems of scheduling and systems reliability: Computational complexity and approximation

Problem Product Partition differs from the NP-complete problem Partition in that the addition operation is replaced by the multiplication operation. Furthermore it differs from the NP-complete problem Subset Product in that it does not contain the product value B in its input. We prove that problem Product Partition and several of its modifications are NP-complete in the strong sense. Our results imply the strong NP-hardness of a number of scheduling problems with start-time-dependent job processing times and a problem of designing a reliable system with a series–parallel structure. It should be noticed that the strong NP-hardness of the considered optimization problems does not preclude the existence of a fully polynomial time approximation scheme (FPTAS) for them. We present a simple FPTAS for one of these problems.     

Parallel-machine scheduling with simple linear deterioration to minimize total completion time

We consider the parallel-machine scheduling problem in which the processing time of a job is a simple linear increasing function of its starting time. The objective is to minimize the total completion time. We give a fully polynomial-time approximation scheme (FPTAS) for the case with m identical machines, where m is fixed. This study solves an open problem that has been posed in the literature for ten years.     

FPTAS for half-products minimization with scheduling applications

A special class of quadratic pseudo-boolean functions called “half-products” (HP) has recently been introduced. It has been shown that HP minimization, while NP-hard, admits a fully polynomial time approximation scheme (FPTAS). In this note, we provide a more efficient FPTAS. We further show how an FPTAS can also be derived for the general case where the HP function is augmented by a problem-dependent constant and can justifiably be assumed to be nonnegative. This leads to an FPTAS for certain partitioning type problems, including many from the field of scheduling.     

Approximation schemes for parallel machine scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to m parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which the unavailability is due to a priori assignment of some of the n jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that m is a constant, and that the jobs are non-resumable.For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even if w_i=p_i for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless P=NP.For the fixed job model, it has been shown that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless P=NP. In this paper, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.     

Single Machine Group Scheduling with Two Ordered Criteria

The problem of scheduling jobs on a single machine is considered. It is assumed that the jobs are classified into several groups and the jobs of the same group have to be processed contiguously. A sequence independent set-up time is incurred between each two consecutively scheduled groups. A schedule is specified by a sequence for the groups and a sequence for the jobs in each group. The quality of a schedule is measured by two critera ordered by their relative importance. The objective is to minimize the maximum cost, the secondary criterion, subject to the schedule is optimal with respect to total weighted completion time, the primary criterion. A polynomial time algorithm is presented to solve this bicriterion group scheduling problem. It is shown that this algorithm can also be modified to solve the single machine group scheduling problem with several ordered maximum cost criteria and arbitrary precedence constraints.     

有关单机两代理排序问题的两个结果

研究了两个单机两代理排序问题. 在第一个两代理排序问题中, 代理A的目标函数为极小化所有工件的加权完工时间总和, 代理B的目标函数为极小化最大工件费用. 在第二个两代理排序问题中, 代理A的目标函数为极小化所有工件的加权完工时间总和, 代理B的目标函数为极小化所有工件的最大完工时间. 证明了第一个问题是强NP-难的, 改进了已有的一般意义NP-难的结果; 对第二个问题给出了一个与现有的动态规划算法不同的动态规划算法.     

Minimizing due date related performance measures on two batch processing machines

This paper considers a scheduling problem in a two-machine flowshop of two batch processing machines. On each batch processing machine, jobs are processed in a batch, and each batch is allowed to contain jobs up to the maximum capacity of the associated machine. The scheduling problem is analyzed with respect to three due date related objectives including maximum tardiness, number of tardy jobs and total tardiness. In the analysis, several solution properties are characterized and based upon these properties, three efficient polynomial time algorithms are developed for minimizing the due date related measures.     

Resource-constrained job scheduling with recyclable resources

In resource-constrained project scheduling problems, resources are typically classified as being either renewable, non-renewable, or doubly-constrained. A new resource classification, recyclable, is introduced. Notation and a generalized problem formulation are developed for resource-constrained job scheduling problems where resources are recyclable. This foundation is then used for studying the single-machine scheduling problem with tooling constraints. For a given set of jobs, the problem is to find the job sequence, tool type quantities, and tool recycling schedule such that the sum of job completion times and quantity of tools allocated are both minimized. Two solution approaches are developed, and examples are used to compare and contrast the approaches. The results indicate that the ‘traditional’ job scheduling approach (i.e. schedule jobs first, then tools) can lead to sub-optimal solutions. Furthermore, by scheduling jobs and tools simultaneously, it may be possible to attain a given level of performance with fewer tools.     

Makespan minimization in machine dominated flowshop

In a flowshop scheduling problem, a set of jobs is processed by a set of machines. The jobs follow the same sequence in all machines. We study the flowshop scheduling problem under a new case of machine dominance that is often found in the manufacturing of computers and electronic devices. We provide a formula for makespan value for a given sequence, show that the makespan value depends only on certain jobs in the sequence, and present an algorithm that finds a sequence with minimum makespan. Numerical examples of the solution approaches are provided.     

Single machine scheduling with semi-resumable machine availability constraints

This paper considers the semi-resumable model of single machine scheduling with a non-availability period. The machine is not available for processing during a given time interval. A job cannot be completed before the non-availability period will have to partially restart after the machine has become available again. For the problem with objective of minimizing makespan, the tight worst-case ratio of algorithm LPT is given, and an FPTAS is also proposed. For the problem with objective of minimizing total weighted completion time, an approximation algorithm with worst-case ratio smaller than 2 is presented. Two special cases of the latter problem are also considered, and improved algorithms are given.     

A New Heuristic Method for the Permutation Flow Shop Scheduling Problem

A new heuristic method for the permutation flow shop scheduling problem is presented and compared with two other heuristics named NEH and SPIRIT. The new heuristic method is based on a property of the scheduling problem that provides an upper bound on the idle time of the last machine between any two adjacent jobs regardless of their position in the sequence of jobs. The results from computational experience have shown that the new heuristic outperforms, in solution quality, all others for problems having up to 50 jobs and 30 machines.     

Single-machine scheduling with deteriorating jobs and learning effects to minimize the makespan

This paper studies the single-machine scheduling problem with deteriorating jobs and learning considerations. The objective is to minimize the makespan. We first show that the schedule produced by the largest growth rate rule is unbounded for our model, although it is an optimal solution for the scheduling problem with deteriorating jobs and no learning. We then consider three special cases of the problem, each corresponding to a specific practical scheduling scenario. Based on the derived optimal properties, we develop an optimal algorithm for each of these cases. Finally, we consider a relaxed model of the second special case, and present a heuristic and analyze its worst-case performance bound.