Group Scheduling with Controllable Setup and Processing Times: Minimizing Total Weighted Completion Time

The following single machine scheduling problem is studied. A partition of a set of n jobs into g groups on the basis of group technology is given. The machine processes jobs of the same group contiguously, with a sequence independent setup time preceding the processing of each group. The setup times and the job processing times are controllable through the allocation of a continuously divisible or discrete resource to them. Each job uses the same amount of the resource. Each setup also uses the same amount of resource, which may be different from that for the jobs. Polynomial-time algorithms are constructed for variants of the problem of finding an optimal job sequence and resource values so as to minimize the total weighted job completion time, subject to given restrictions on resource consumption. The algorithms are based on a polynomial enumeration of the candidates for an optimal job sequence and solving the problem with a fixed job sequence by linear programming. This research was supported in part by The Hong Kong Polytechnic University under grant number G-T246 and the Research Grants Council of Hong Kong under grant number PolyU 5191/01E. In addition, the research of M.Y. Kovalyov was supported by INTAS under grant number 00-217.     

Single machine batch scheduling with resource dependent setup and processing times

Jobs are processed by 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 common for all batches. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource, which may be different from that for the jobs. Polynomial time algorithms are presented to find an optimal batch sequence and resource values such that either the total weighted resource consumption is minimized, subject to meeting job deadlines, or the maximum job lateness is minimized, subject to an upper bound on the total weighted resource consumption. The algorithms are based on linear programming formulations of the corresponding problems.     

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.     

Single Machine Batch Scheduling Problem with Resource Dependent Setup and Processing Time in the Presence of Fuzzy Due Date

We consider a batch scheduling problem on a single machine which processes jobs with resource dependent setup and processing time in the presence of fuzzy due-dates given as follows:1. There are n independent non-preemptive and simultaneously available jobs processed on a single machine in batches. Each job j has a processing time and a due-date.2. All jobs in a batch are completed together upon the completion of the last job in the batch. The batch processing time is equal to the sum of the processing times of its jobs. A common machine setup time is required before the processing of each batch.3. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource.4. The due-date of each job is flexible. That is, a membership function describing non-decreasing satisfaction degree about completion time of each job is defined.5. Under above setting, we find an optimal batch sequence and resource values such that the total weighted resource consumption is minimized subject to meeting the job due-dates, and minimal satisfaction degree about each due-date of each job is maximized. But usually we cannot optimize two objectives at a time. So we seek non-dominated pairs i.e. the batch sequence and resource value, after defining dominance between solutions.A polynomial algorithm is constructed based on linear programming formulations of the corresponding problems.     

Single-machine scheduling with deteriorating jobs and past-sequence-dependent setup times

In many realistic scheduling settings a job processed later consumes more time than the same job processed earlier – this is known as scheduling with deteriorating jobs. Most research on scheduling with deteriorating jobs assumes that the actual processing time of a job is an increasing function of its starting time. Thus a job processed late may incur an excessively long processing time. On the other hand, setup times occur in manufacturing situations where jobs are processed in batches whereby each batch incurs a setup time. This paper considers scheduling with deteriorating jobs in which the actual processing time of a job is a function of the logarithm of the total processing time of the jobs processed before it (to avoid the unrealistic situation where the jobs scheduled late will incur excessively long processing times) and the setup times are proportional to the actual processing times of the already scheduled jobs. Under the proposed model, we provide optimal solutions for some single-machine problems.     

需要安装时间的平行多功能机排序问题的启发式算法

考虑需要安装时间的平行多功能机排序问题。在该模型中,每个工件对应机器集合的一个子集,其只能在这个子集中的任一台机器上加工,称这个子集为该工件的加工集合;工件分组,同组工件具有相同的加工时间和加工集合,不同组中的工件在同一台机器上连续加工需要安装时间,目标函数为极小化最大完工时间。对该问题NP-难的一般情况设计启发式算法:首先按照特定的规则将所有工件组都整组地安排到各台机器上,然后通过在各机器间转移工件不断改进当前最大完工时间。通过与下界的比较检验算法的性能,大量的计算实验表明,算法是实用而有效的。     

Single machine group scheduling under decreasing linear deterioration

This paper considers single machine scheduling problems with group technology (GT) and deteriorating jobs. A sequence independent setup is required to process a job from a different group and jobs in each group are processed together. We consider the case of jobs whose processing times are a decreasing function of their starting time. The objectives of scheduling problems are to minimize the makespan and the total completion time, respectively. We also provide polynomial time algorithms to solve these problems.     

Single machine scheduling with general time-dependent deterioration, position-dependent learning and past-sequence-dependent setup times

The paper deals with single machine scheduling problems with setup time considerations where the actual processing time of a job is not only a non-decreasing function of the total normal processing times of the jobs already processed, but also a non-increasing function of the job’s position in the sequence. The setup times are proportional to the length of the already processed jobs, i.e., the setup times are past-sequence-dependent (p-s-d). We consider the following objective functions: the makespan, the total completion time, the sum of the δth (δ ≥ 0) power of job completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the δ th (δ ≥ 0) power of job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

Single-machine group scheduling with processing times dependent on position,starting time and allotted resource

This article considers scheduling problems on a single machine with learning effect, deteriorating jobs and resource allocation under group technology (GT) assumption. We assume that the actual processing time of a job depends on the job position, the group position, the starting time and the amount of resource allocated to them concurrently, and the actual setup times of groups depend on the group position and the amount of resource allocated to them concurrently. Two resource allocation functions are examined for minimizing the weighted sum of makespan and total resource cost. We prove that the problems have polynomial solutions under the condition that the number of jobs in each group are the same.     

需要安装时间的两台多功能机排序问题的计算复杂性

提出需要安装时间的多功能机排序问题,一般情况下,这是NP-困难的;主要研究只有两台机器时一些特殊情况下的计算复杂性.根据加工集合为机器全集的工件组数的不同,分别给出多项式时间算法和分枝定界算法.对各工件组的工件数和加工时间都相等的情况,给出一个多项式时间的最优算法-奇偶算法,从而证明此问题是多项式时间可解的.     

Approximation algorithms for two-machine flow shop scheduling with batch setup times

In many practical situations, batching of similar jobs to avoid setups is performed while constructing a schedule. This paper addresses the problem of non-preemptively scheduling independent jobs in a two-machine flow shop with the objective of minimizing the makespan. Jobs are grouped into batches. A sequence independent batch setup time on each machine is required before the first job is processed, and when a machine switches from processing a job in some batch to a job of another batch. Besides its practical interest, this problem is a direct generalization of the classical two-machine flow shop problem with no grouping of jobs, which can be solved optimally by Johnson's well-known algorithm. The problem under investigation is known to be NP-hard. We propose two O(n logn) time heuristic algorithms. The first heuristic, which creates a schedule with minimum total setup time by forcing all jobs in the same batch to be sequenced in adjacent positions, has a worst-case performance ratio of 3/2. By allowing each batch to be split into at most two sub-batches, a second heuristic is developed which has an improved worst-case performance ratio of 4/3. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Bi-criterion single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times

This paper addresses single-machine scheduling and due-window assignment with common flow allowances and resource-dependent processing times. Due-window assignment with common flow allowances means that each job has a job-dependent due window, the start time and finish time of which are equal to its actual processing time plus individual job-independent parameters shared by all the jobs, respectively. The processing time of each job can be controlled by extra resource allocation as a linear function of the amount of a common continuously divisible resource allocated to the job. Two criteria are considered, where one criterion is an integrated cost consisting of job earliness, weighted number of tardy jobs, and due-window assignment cost, while the other criterion is the resource consumption cost. Four different models are considered for treating the two criteria. It is shown that the problem under the model where the two criteria are integrated into a single criterion is polynomially solvable, while the problems under the other three models are all NP-hard and an optimal solution procedure is developed for them. Two polynomially solvable cases are also identified and investigated. Finally, numerical studies with randomly generated instances are conducted to assess the performance of the proposed algorithms.     

Single machine scheduling with a general exponential learning effect

In this paper we consider the scheduling problem with a general exponential learning effect and past-sequence-dependent (p-s-d) setup times. By the general exponential learning effect, we mean that the processing time of a job is defined by an exponent function of the total weighted normal processing time of the already processed jobs and its position in a sequence, where the weight is a position-dependent weight. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the δ ? 0th power of completion times, the total weighted completion time and the maximum lateness. We show that the makespan minimization problem, the total completion time minimization problem and the sum of the quadratic job completion times minimization problem can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

A comparison of heuristic algorithms for flow shop scheduling problems with setup times and limited batch size

In this paper, we propose different heuristic algorithms for flow shop scheduling problems, where the jobs are partitioned into groups or families. Jobs of the same group can be processed together in a batch but the maximal number of jobs in a batch is limited. A setup is necessary before starting the processing of a batch, where the setup time depends on the group of the jobs. In this paper, we consider the case when the processing time of a batch is given by the maximum of the processing times of the operations contained in the batch. As objective function we consider the makespan as well as the weighted sum of completion times of the jobs. For these problems, we propose and compare various constructive and iterative algorithms. We derive suitable neighbourhood structures for such problems with batch setup times and describe iterative algorithms that are based on different types of local search algorithms. Except for standard metaheuristics, we also apply multilevel procedures which use different neighbourhoods within the search. The algorithms developed have been tested in detail on a large collection of problems with up to 120 jobs.     

Single-machine due window assignment and scheduling with a common flow allowance and controllable job processing time

In this paper, we consider single-machine due window assignment and scheduling with a common flow allowance and controllable job processing times, subject to unlimited or limited resource availability. Due window assignment with a common flow allowance means that each job has a job-dependent due window, the starting time and completion time of which are equal to its actual processing time plus the job-independent parameters q₁ and q₂, respectively, which are common to all the jobs. The processing time of each job is either a linear or a convex function of the amount of a common continuously divisible resource allocated to the job. We study five versions of the problem that differ in terms of the objective function and processing time function being used. We provide structural properties of the optimal schedules and polynomial-time solution algorithms for the considered problems.     

Job sequencing rules for minimizing the expected makespan in flexible machines

We consider scheduling of a deteriorating flexible machine that is capable of processing a number of diverse jobs with negligible setup times between jobs. Specifically, we develop rules for sequencing N jobs on such a machine such that its expected makespan (sum of all job processing times and machine down-time) is minimized. Using the Weibull distribution to characterize machine failures in our model, we permit different jobs to contribute to machine deterioration (and hence its failure) at different failure rates, and do not require these rates to remain constant with machine-use time. We validate the effectiveness of these job sequencing rules for different cases, using extensive simulation tests.     

Scheduling in a contaminated area: A model and polynomial algorithms

There are n jobs to be scheduled in a contaminated area. The jobs can be rescue, de-activation or cleaning works to be executed by a single worker in an area contaminated with radio-active or chemical materials. Precedence relations can be given on the set of jobs. An execution of each job can be preempted. However, the length of the minimal uninterrupted work period is given and it is the same for all jobs. Each work period for a job should be accompanied by a rest period whose length depends on the start time of the work period and its length. We focus on a short term planning problem. We show that this problem can be modelled by a scheduling problem with start time dependent job processing times. The dependency functions are exponentially decreasing ones. We also construct two polynomial time algorithms for the both cases—with and without precedence constraints.     

Single machine past-sequence-dependent setup times scheduling with general position-dependent and time-dependent learning effects

In this paper we consider the single machine past-sequence-dependent (p-s-d) setup times scheduling problems with general position-dependent and time-dependent learning effects. By the general position-dependent and time-dependent learning effects, we mean that the actual processing time of a job is not only a function of the total normal processing times of the jobs already processed, but also a function of the job’s scheduled position. The setup times are proportional to the length of the already processed jobs. We consider the following objective functions: the makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times, the total lateness, the total weighted completion time, the maximum lateness, the maximum tardiness and the number of tardy jobs. We show that the problems of makespan, the total completion time, the sum of the θth (θ ? 0) power of job completion times and the total lateness can be solved by the smallest (normal) processing time first (SPT) rule, respectively. We also show that the total weighted completion time minimization problem, the maximum lateness minimization problem, maximum tardiness minimization problem and the number of tardy jobs minimization problem can be solved in polynomial time under certain conditions.     

Single machine scheduling problem with multiple due windows assignment in a group technology

We consider group scheduling problem on a single machine with multiple due windows assignment. Jobs are divided into groups in advance according to their processing similarities, and all jobs of the same group are required to be processed contiguously on the machine in order to achieve production efficiency and save time/money resource. A sequence-independent setup time precedes the processing of each group. The goal is to determine the optimal sequence for both groups and jobs, together with an optimal combination of the due windows assignment strategy so as to minimize the total of earliness, tardiness and due windows related costs. We give an \(O(n\log n)\) time algorithm for the problem.     

A note on single-machine scheduling with decreasing time-dependent job processing times

In this note we consider some single-machine scheduling problems with decreasing time-dependent job processing times. Decreasing time-dependent job processing times means that its processing time is a non-increasing function of its execution start time. We present polynomial solutions for the sum of squared completion times minimization problem, and the sum of earliness penalties minimization problem subject to no tardy jobs, respectively. We also study two resource constrained scheduling problems under the same decreasing time-dependent job processing times model and present algorithms to find their optimal solutions.