Bicriteria scheduling problems involving job rejection,controllable processing times and rate-modifying activity

This paper addresses the bicriteria scheduling problems with simultaneous consideration of job rejection, controllable processing times and rate-modifying activity on a single machine. A job is either rejected, in which case a rejection penalty will be incurred, or accepted and processed on the machine. The rate-modifying activity is an activity on the machine that changes the processing times of the jobs scheduled after the activity. The processing time of a job scheduled after the rate-modifying activity decreases with a job-dependent factor. The processing time of each job can also be controlled by allocating extra resource which is either a linear or a convex function of the amount of a common continuously divisible resource allocated to the job. The objective is to determine the rejected job set, the accepted job sequence, the time (location) of the rate-modifying activity and the resource allocation that jointly find the trade-off between two criteria, where the first criterion is measured as the sum of total completion time and resource consumption cost while the second criterion is the total rejection cost. We consider four different models for treating the two criteria. The computational complexity status and solution procedures are provided for the problems under consideration.     

Due-date assignment and single-machine scheduling with generalised position-dependent deteriorating jobs and deteriorating multi-maintenance activities

This paper addresses a single-machine scheduling problem with simultaneous consideration of due-date assignment, generalised position-dependent deteriorating jobs, and deteriorating maintenance activities. It is assumed that the actual processing time of a job is a general non-decreasing function depending on the number of maintenance activities performed before it and its position in a sequence. Moreover, the machine may be subject to several maintenance activities up to a limit over the scheduling horizon. The maintenance activities do not necessarily restore the machine fully to its original perfect state and the duration of a maintenance activity depends on its start time. The objective is to find jointly the optimal job sequence, maintenance frequency and maintenance positions to minimise an objective function that includes the cost of due-date assignment, the cost of discarding jobs that cannot be completed by their due dates and the earliness of the scheduled jobs under the popular CON and SLK due-date assignment methods. We provide polynomial-time solution algorithms for various versions of the problem.     

Parallel-machine scheduling with controllable processing times and rate-modifying activities to minimise total cost involving total completion time and job compressions

In this paper, we consider unrelated parallel-machine scheduling involving controllable processing times and rate-modifying activities simultaneously. We assume that the actual processing time of a job can be compressed by allocating a greater amount of a common resource to process the job. We further assume that each machine may require a rate-modifying activity during the scheduling horizon. The objective is to determine the optimal job compressions, the optimal positions of the rate-modifying activities and the optimal schedule to minimise a total cost function that depends on the total completion time and total job compressions. If the number of machines is a given constant, we propose an efficient polynomial time algorithm to solve the problem.     

Machine scheduling in the presence of sequence-dependent setup times and a rate-modifying activity

Rate modifying activity (RM) is a type of maintenance after which the processing rate of the machine increases. RM is a very new topic in academic studies. However, it is very common in real world situations. In this paper, we study the integrated problem of assigning a common due-date to all jobs, scheduling the jobs and making decisions about the position of RM in a single machine environment in which the setup times are sequence dependent. The objective is minimising the summation of earliness costs, tardiness costs and due date related costs. This problem has never been studied in the literature with any arbitrary criterion. We construct a time-dependent travelling salesman problem (TDTSP) formulation for this problem. The position of the optimal common due date and some dominance properties for the position of RM are presented. A branch and bound (B&B) procedure is developed to solve the problem optimally. Numerical results justify the effectiveness of the B&B method for small problems. For larger problems, two robust metaheuristics are proposed.     

Minimizing total tardiness on a batch processing machine with incompatible job families

Motivated by an application in semiconductor manufacturing, we study the problem of minimizing total tardiness on a batch processing machine with incompatibl8e job families, where all jobs of the same family have identical processing times and jobs of different families cannot be processed together. We present a dynamic programming algorithm which has polynomial time complexity when the number of job families and the batch machine capacity are fixed. We also examine various heuristic solution procedures which can provide near optimal solutions in a reasonable amount of computation time.     

Control of a Batch Processing Machine Serving Compatible Job Families

We consider the control of a single batch processing machine with random arrivals, random processing times, and compatible job families (jobs from different families may be processed together in the same batch, with the processing time distribution of the entire batch determined by the job family in the batch having the greatest expected processing time). The objective is to minimize the long-run average time that jobs spend in the system. We present properties possessed by the optimal policies and discuss the structure of these policies. We next develop a simple heuristic scheduling policy to control the machine. Simulation results are provided to demonstrate the effectiveness of our heuristic over a wide range of problem instances.     

Scheduling the fabrication and assembly of components in a two-machine flowshop

In this paper we study the problem of scheduling the fabrication and assembly of components in a two-machine flowshop so as to minimize the makespan. Each job consists of a component unique to that job and a component common to all jobs. Both the unique and the common components are processed on the first machine. While the unique components are processed individually, the common components are processed in batches and a setup is needed to form each batch. The assembly operations of a job is performed on the second machine, and can only begin when both components for the job are available. We first show that the problem is NP-complete with either batch availability or item availability for the common components. We identify several properties of an optimal solution to the problem, and some polynomially solvable special cases.     

A method combining rules with genetic algorithm for minimizing makespan on a batch processing machine with preventive maintenance

This paper considers the problem of minimising makespan on a single batch processing machine with flexible periodic preventive maintenance. This problem combines two sub-problems, scheduling on a batch processing machine with jobs’ release dates considered and arranging the preventive maintenance activities on a batch processing machine. The preventive maintenance activities are flexible but the maximum continuous working time of the machine, which is allowed, is determined. A mathematical model for integrating flexible periodic preventive maintenance into batch processing machine problem is proposed, in which the grouping of jobs with incompatible job families, the starting time of batches and the preventive maintenance activities are optimised simultaneously. A method combining rules with the genetic algorithm is proposed to solve this model, in which a batching rule is proposed to group jobs with incompatible job families into batches and a modified genetic algorithm is proposed to schedule batches and arrange preventive maintenance activities. The computational results indicate the method is effective under practical problem sizes. In addition, the influences of jobs’ parameters on the performance of the method are analyzed, such as the number of jobs, the number of job families, jobs’ processing time and jobs’ release time.     

Minimizing makespan in a no-wait flowshop with two batch processing machines using estimation of distribution algorithm

This paper studies the problem of minimising makespan in a no-wait flowshop with two batch processing machines (comprised of a parallel batch processing machine and a serial batch processing machine), non-identical job sizes and unequal ready times. We propose a population-based evolutionary method named estimation of distribution algorithm (EDA). Firstly, the individuals in the population are coded into job sequences. Then, a probabilistic model is built to generate new population and an incremental learning method is developed to update the probabilistic model. Thirdly, the best-fit heuristic is used to group jobs into batches and a least idle/waiting time approach is proposed to sequence the batches on batch processing machines. In addition, some problem-dependent local search heuristics are incorporated into the EDA to further improve the searching quality. Computational simulation and comparisons with some existing algorithms demonstrate the effectiveness and robustness of the proposed algorithm. Furthermore, the effectiveness of embedding the local search method in the EDA is also evaluated.     

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We consider the problem of minimizing makespan Cmax on a single batch processing machine in the presence of dynamic job arrivals. The batch processing machine can process up to B jobs simultaneously. The processing time of a batch is given by the processing time of the longest job in the batch. We present polynomial and pseudopolynomial-time algorithms for several special cases, develop efficient heuristics for the general problem and evaluate their performance through extensive computational experiments. Our results indicate that several of the heuristics have an excellent average performance with a modest computational burden.     

Scheduling a batch processing machine with non-identical job sizes

The problem of scheduling batch processors is important in some industries and, at a more fundamental level, captures an element of complexity common to many practical scheduling problems. We describe a branch and bound procedure applicable to a batch processor model with arbitrary job processing times, job weights and job sizes. The scheduling objective is to minimize total weighted completion time. We find that the procedure returns optimal solutions to problems of up to 25 jobs in reasonable CPU time, and can be adapted for use as a heuristic for larger problems.     

Scheduling a no-wait flow shop containing unbounded batching machines

We study the problem of scheduling n jobs in a no-wait flow shop consisting of m batching machines. Each job has to be processed by all the machines. All jobs visit the machines in the same order. A job completed on an upstream machine should be immediately transferred to the downstream machine. Batching machines can process several jobs simultaneously in a batch so that all jobs of the same batch start and complete together. The processing time of a batch is equal to the maximum processing time of the jobs in this batch. We assume that the capacity of any batch is unbounded. The problem is to find an optimal batch schedule such that the maximum job completion time, that is the makespan, is minimized. For m = 2, we prove that there exists an optimal schedule with at most two batches and construct such a schedule in O(n log n) time. For m = 3, we prove that the number of batches can be limited to nine and give an example where all optimal schedules have seven batches. Furthermore, we prove that the best schedules with at most one, two and three batches are 3-, 2- and 3/2-approximate solutions, respectively. The first two bounds are tight for corresponding schedules. Finally, we suggest an assignment method that solves the problem with m machines and at most r batches in O(n^{m(r-2)+1+[m/r]}) time, if m and r are fixed. The method can be generalized to minimize an arbitrary maximum cost or total cost objective function.     

Iterative algorithms for batching and scheduling to minimise the total job tardiness in two-stage hybrid flow shops

This study considers the batching and scheduling problem in two-stage hybrid flow shops in which each job with a distinct due-date is processed through two serial production stages, each of which has identical machines in parallel. Under the fundamental trade-off that large batch sizes with less frequent changeovers may reduce setup costs and hence increase machine utilisation, while small batch sizes may reduce job flow times and hence improve scheduling performance, the problem is to determine the number of batches, the batch compositions, the allocation of batches to the parallel machines at each stage, and the sequence of the batches allocated to each machine for the objective of minimising the total job tardiness. A mixed integer programming model is developed for the reduced problem in which the number of batches is given, and then, three iterative algorithms are proposed in which batching and scheduling are done repeatedly until a good solution is obtained. To show the performance of the algorithms, computational experiments were done on a number of test instances, and the results are reported. In particular, we show that the number of batches decreases as the ratio of the batch setup time to the job processing time increases.     

An improved approximation algorithm for the two-machine open shop scheduling problem with family setup times

We consider the problem of scheduling families of jobs in a two-machine open shop so as to minimize the makespan. The jobs of each family can be partitioned into batches and a family setup time on each machine is required before the first job is processed, and when a machine switches from processing a job of some family to a job of another family. For this NP-hard problem the literature contains (5/4)-approximation algorithms that cannot be improved on using the class of group technology algorithms in which each family is kept as a single batch. We demonstrate that there is no advantage in splitting a family more than once. We present an algorithm that splits one family at most once on a machine and delivers a worst-case performance ratio of 6/5.     

A hybrid VNS-HS algorithm for a supply chain scheduling problem with deteriorating jobs

This paper investigates a coordinated scheduling problem in a two stage supply chain where parallel-batching machine, deteriorating jobs and transportation coordination are considered simultaneously. During the production stage, jobs are processed by suppliers and there exists one parallel-batching machine in each supplier. The actual processing time of a job depends on its starting time and normal processing time. The normal processing time of a batch is equal to the largest normal processing time among all jobs in its batch. During the transportation stage, the jobs are then delivered to the manufacturer. Since suppliers are distributed in different locations, the transportation time between each supplier and the manufacturer is different. Based on some structural properties of the studied problem, an optimal algorithm for minimising makespan on a single supplier is presented. This supply chain scheduling problem is proved to be NP-hard, and a hybrid VNS-HS algorithm combining variable neighbourhood search (VNS) with harmony search (HS) is proposed to find a good solution in reasonable time. Finally, some computational experiments are conducted and the results demonstrate the effectiveness and efficiency of the proposed VNS-HS.     

Determination of an optimal common due date and optimal sequence in a single machine job shop

In this paper we consider an n jobs one machine sequencing problem in which all jobs have a common due date and a deviation in its completion time occurs when a job is completed before or after the common due date. The objective is to find an optimal value of this common due date and a corresponding optimal sequence such that the mean absolute deviation of the completion times of the jobs in the optimal sequence from the corresponding optimal common due date is at its global minimum. Starting with an arbitrary sequence we relate the problem to a generalized linear goal program from which some basic results are proved using elementary properties of linear equations and a linear goal programming problem. Using these results and the idea of sensitivity analysis in linear programming, an algorithm is developed that determines the optimal due date and the corresponding optimal sequence yielding the global minimum value of the mean absolute deviation of the completion times of the jobs in the optimal sequence from the corresponding optimal common due date. In the end a numerical example to explain the algorithm is provided.     

Due date assignment in single machine with stochastic processing times

This paper considers two different due date assignment and sequencing problems in single machine where the processing times of jobs are random variables. The first problem is to minimise the maximum due date so that all jobs are stochastically on time. It is shown that sequencing the jobs in decreasing service level (DSL) order optimally solves the problem. The results are then extended for two special cases of flow shop problem. The other problem is to minimise a total cost function which is a linear combination of three penalties: penalty on job earliness, penalty on job tardiness, and penalty associated with long due date assignment. The assignment of a common due date and distinct due dates are investigated for this problem. It is shown that the optimal sequence for the case of common due date is V-shaped.     

Batch scheduling problem for a machinery factory with fixed-position layout

We consider a batch scheduling problem for a two-stage flow shop with fixed-position layout. In the first stage, a fixed number of jobs are assembled on a batch machine with a family batch setup time and a common processing time. In the second stage, the assembled jobs are individually performed for system integration on a discrete machine. The finished job is immediately packed and shipped if the payment has been made; otherwise, it is moved to a temporary storage area, incurring additional removal time. This study develops a mixed integer programming (MIP) to solve the problem of minimising the total completion time and proposes two heuristics for large-size problems. Computational results show that the proposed methods can be applied to resolve real-world problems similar to those in this study.