Fuzzy job shop scheduling with lot-sizing

This paper deals with a problem of determining lot-sizes of jobs in a real-world job shop-scheduling in the presence of uncertainty. The main issue discussed in this paper is lot-sizing of jobs. A fuzzy rule-based system is developed which determines the size of lots using the following premise variables: size of the job, the static slack of the job, workload on the shop floor, and the priority of the job. Both premise and conclusion variables are modelled as linguistic variables represented by using fuzzy sets (apart from the priority of the job which is a crisp value). The determined lots’ sizes are input to a fuzzy multi-objective genetic algorithm for job shop scheduling. Imprecise jobs’ processing times and due dates are modelled by using fuzzy sets. The objectives that are used to measure the quality of the generated schedules are average weighted tardiness of jobs, the number of tardy jobs, the total setup time, the total idle time of machines and the total flow time of jobs. The developed algorithm is analysed on real-world data obtained from a printing company.     

The design of simple subcontracting rules for make-to-order shops: An assessment by simulation

Subcontracting can be an important means of overcoming capacity shortages and of workload balancing, especially in make-to-order companies characterized by high variety, high demand variation and a job shop configuration. But there is a lack of simple, yet powerful subcontracting rules suitable for such contexts. The few existing rules were developed for single work center shops and neglect the actual subcontracting lead time, meaning some subcontracted jobs are destined to become tardy. This study uses Workload Control theory on matching required and available capacity over time to propose four new rules that address these shortcomings. The new rules are compared against four existing rules using an assembly job shop simulation model where the final, assembled product consists of several sub-assemblies that either flow through an internal job shop or are subcontracted. The best new rules stabilize the direct load queuing in front of a work center and significantly improve performance compared to the existing rules. For example, when the workload exceeds capacity by 10%, a 50% reduction in percentage tardy can be achieved. By examining how the workload behaves over time, we reveal that improvements come from selectively subcontracting the sub-assemblies that would otherwise cause overloads, thereby cutting off peaks in the workload.     

Flow shop scheduling problems with deteriorating jobs under dominating machines

This paper considers the general, no-wait and no-idle flow shop scheduling problems with deteriorating jobs. By a deteriorating job we mean that the processing time is an increasing function of its execution starting time. A linear deterioration function is assumed and some dominating relationships between machines can be satisfied. It is shown that for the problems to minimize the makespan or the weighted sum of completion time, polynomial algorithms still exist, although these problems are more complicated than the classical ones. When the objective is to minimize the maximum lateness, the solutions of a classical version may not hold.     

Scheduling linear deteriorating jobs with rejection on a single machine

We consider several single machine scheduling problems in which the processing time of a job is a linear function of its starting time and jobs can be rejected by paying penalties. The objectives are to minimize the makespan, the total weighted completion time and the maximum lateness/tardiness plus the total penalty of the rejected jobs. We show that these problems are NP-hard, and design algorithms based on dynamic programming (including pseudo-polynomial time optimal algorithms and fully polynomial time approximation schemes) to solve them.     

Lead time selection and the behaviour of work flow in job shops

Few studies in job shop scheduling consider dynamic shop load and work flow issues. However, these issues are of importance to management responsible for shop floor control. This research investigates relevant internal performance measures and their relationship to external measures. As well, a new dispatching mechanism that seeks to even out the distribution of jobs in queue is investigated. Results show that a shop load balance index, which takes both shop load levels and load variability into account, has a very strong relationship to the lead times required to maintain a desired level of delivery performance. It appears that good performance based on internal measures is consistent with good performance based on external measures.     

A tabu search and a genetic algorithm for solving a bicriteria general job shop scheduling problem

This paper deals with a general job shop scheduling problem with multiple constraints, coming from printing and boarding industry. The objective is the minimization of two criteria, the makespan and the maximum lateness, and we are interested in finding an approximation of the Pareto frontier. We propose a fast and elitist genetic algorithm based on NSGA-II for solving the problem. The initial population of this algorithm is either randomly generated or partially generated by using a tabu search algorithm, that minimizes a linear combination of the two criteria. Both the genetic and the tabu search algorithms are tested on benchmark instances from flexible job shop literature and computational results show the interest of both methods to obtain an efficient and effective resolution method.     

Scheduling problems with position dependent job processing times: computational complexity results

In this paper, we analyse single machine scheduling problems with learning and aging effects to minimize one of the following objectives: the makespan with release dates, the maximum lateness and the number of late jobs. The phenomena of learning and aging are modeled by job processing times described by non-increasing (learning) or non-decreasing (aging) functions dependent on the number of previously processed jobs, i.e., a job position in a sequence. We prove that the considered problems are strongly NP-hard even if job processing times are described by simple linear functions dependent on a number of processed jobs. Additionally, we show a property of equivalence between problems with learning and aging models. We also prove that if the function describing decrease/increase of a job processing time is the same for each job then the problems with the considered objectives are polynomially solvable even if the function is arbitrary. Therefore, we determine the boundary between polynomially solvable and strongly NP-hard cases.     

Single-machine scheduling with deteriorating functions for job processing times

In many realistic scheduling settings a job processed later consumes more time than when it is processed earlier – this phenomenon is known as scheduling with deteriorating jobs. In the literature on deteriorating job scheduling problems, majority of the research assumed that the actual job processing time of a job is a function of its starting time. In this paper we consider a new deterioration model where the actual job processing time of a job is a function of the processing times of the jobs already processed. We show that the single-machine scheduling problems to minimize the makespan and total completion time remain polynomially solvable under the proposed model. In addition, we prove that the problems to minimize the total weighted completion time, maximum lateness, and maximum tardiness are polynomially solvable under certain agreeable conditions.     

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.     

Scheduling jobs with different,job-dependent earliness and tardiness penalties using the SLK method

The single machine job scheduling problem, where due dates are assigned using the SLK due date determination method, is examined assuming different penalties for the early and tardy jobs. These penalties are assumed to be job-dependent, proportional to the processing times of jobs raised to an integer, non-negative power. The objective function is the total weighted lateness. Several cases are examined and four algorithms providing the optimal sequences for these cases are presented. Examples are given and conclusions are drawn.     

A Model for Predicting the Performance of a Job Shop

This paper is concerned with presenting a method of modelling a job shop based on queueing theory. The model is very flexible and may be used to explore the relationships between the resources available in the shop (numbers and characteristics of machines, manpower levels and skills), the workload in the shop, the mode of operation of the shop (labour allocation and priority schemes) and the resulting level of congestion within the shop.A range of results is presented, based on the application of the model to a real environment, where good agreement with the observed performance was obtained.     

基于重新排序的退化工件最小化总延误时间问题

考虑了错位限制下的含有退化工件的重新排序问题,即工件的实际加工时间看作是工件开工时间的线性函数．重新排序就是在原始工件已经按照某种规则使目标函数达到最优时有一新工件集到达,新工件的安排使得原始工件重新排序进而产生错位．研究了最大序列错位和总序列错位限制下的退化工件最小化总延误时间问题,其最优排序的结构性质是使得原始工件集和新工件集中的工件是按加工率αj非减的序列排列,基于此通过分阶段排序和动态规划方法给出了两个问题的多项式时间的最优算法．     

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.     

带有退化效应和序列相关运输时间的排序问题

考虑带有退化效应和序列相关运输时间的单机排序问题. 工件的加工时间是其开工时间的简单线性增加函数. 当机器单个加工工件时, 极小化最大完工时间、(加权)总完工时间和总延迟问题被证明是多项式可解的, EDD序对于极小化最大延迟问题不是最优排序, 另外, 就交货期和退化率一致情形给出了一最优算法. 当机器可分批加工工件时, 分别就极小化最大完工时间和加权总完工时间问题提出了多项式时间最优算法.     

Single machine scheduling with exponential sum-of-logarithm-processing-times based learning effect

In this paper we consider the single machine scheduling problems with exponential sum-of-logarithm-processing-times based learning effect. By the exponential sum-of-logarithm-processing-times based learning effect, we mean that the processing time of a job is defined by an exponent function of the sum of the logarithm of the processing times of the jobs already processed. We consider the following objective functions: the makespan, the total completion time, the sum of the quadratic 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 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.     

分类数很大时的成组排序问题

成组排序具有深刻的实际应用背景,是近年来国外研究得较多的一个热点.已有的某些动态规划算法的复杂性随分类数的增长呈指数型增长趋势,本文用“归并”和解不超过四个新的子问题的方法把分类数较大时的问题转化为分类数较小时的相应问题,简化了问题的求解.     

Approximation algorithms for problems in scheduling with set-ups

In this paper, we present new approximation results for the offline problem of single machine scheduling with sequence-independent set-ups and item availability, where the jobs to be scheduled are independent (i.e., have no precedence constraints) and have a common release time.We present polynomial-time approximation algorithms for two versions of this problem. In the first version, the input includes a weight for each job, and the goal is to minimize the total weighted completion time. On any input, our algorithm produces a schedule whose total weighted completion time is within a factor 2 of optimal for that input.In the second version, the input includes a due date for each job, and the goal is to minimize the maximum lateness of any job. On any input, our algorithm produces a schedule with the following performance guarantee: the maximum lateness of a job is at most the maximum lateness of the optimal schedule on a machine that runs at half the speed of our machine.     

Controlling a Production Scheduling System

When using an automatic production scheduling system management require the system to respond to different overall policies, such as clear all arrears or minimize average lateness weighted by job importance. In many commercial scheduling packages there is no quantitative explanation of how to achieve this response. A successfully implemented scheduling system is introduced here where each policy or objective is obtained by selecting the appropriate criterion on which to sort the jobs. An analytical relationship between the policy and the sorting criterion is established. A new type of lateness penalty is developed heuristically, which is basically an exponential function. The sorting criterion to minimise this lateness penalty turns out to be simply sorting by least slack.     

Optimal methods for batch processing problem with makespan and maximum lateness objectives

In this paper we consider the problem of scheduling n jobs on a single batch processing machine in which jobs are ordered by two customers. Jobs belonging to different customers are processed based on their individual criteria. The considered criteria are minimizing makespan and maximum lateness. A batching machine is able to process up to b jobs simultaneously. The processing time of each batch is equal to the longest processing time of jobs in the batch. This kind of batch processing is called parallel batch processing. Optimal methods for three cases are developed: unbounded batch capacity, b > n, with compatible job groups and bounded batch capacity, b  n, with compatible and non compatible job groups. Each job group represents a different class of customers and the concept of being compatible means that jobs which are ordered by different customers are allowed to be processed in a same batch. We propose an optimal method for the problem with incompatible groups and unbounded batches. About the case when groups are incompatible and bounded batches, our proposed method is considered as optimal when the group with maximum lateness objective has identical processing times. We regard this method, however, as a heuristic when these processing times are different. When groups are compatible and batches are bounded we consider another problem by assuming the same processing times for the group which has the maximum lateness objective and propose an optimal method for this problem.     

Open shops with jobs overlap––revisited

In this communication we consider a two machine open shop in which a job requires processing on both machines. However, in contrast to the classical open shop, the two operations of any given job may overlap in time. The objective function under consideration is the minimization of the total completion time. This model has been considered before by Wagneur and Sriskandarajah [Eur. J. Oper. Res. 71 (1993) 366] and they presented a proof showing that minimizing the total completion time in a two machine open shop with jobs overlap is strongly NP-hard. Their proof is based on a reduction of the numerical matching with target sums (NMTS) problem; however, their proof is unfortunately not correct. In this communication we provide a counterexample that shows that their reduction does not hold. Our counterexample implies that the complexity status of the two machine open shop with job overlap remains open.