Optimal Assignment of Total-work-content Due-dates and Sequencing in a Single-machine Shop

This paper considers the problem of optimal assignment of total-work-content due-dates to n jobs and of sequencing them on a single machine to minimize an objective function depending on the assigned due-date multiple value and maximum tardiness penalty. It is shown that both the earliest due-date and shortest processing time orders yield an optimal sequence. A simple analytical solution method is presented to find the optimal due-dates. After the theoretical treatment an illustrative example is presented for discussion.     

Minmax scheduling problems with common flow-allowance

In due-date assignment problems with a common flow-allowance, the due-date of a given job is defined as the sum of its processing time and a job-independent constant. We study flow-allowance on a single machine, with an objective function of a minmax type. The total cost of a given job consists of its earliness/tardiness and its flow-allowance cost components. Thus, we seek the job schedule and flow-allowance value that minimize the largest cost among all the jobs. Three extensions are considered: the case of general position-dependent processing times, the model containing an explicit cost for the due-dates, and the setting of due-windows. Properties of optimal schedules are fully analysed in all cases, and all the problems are shown to have polynomial time solutions.     

Minmax scheduling with job-classes and earliness–tardiness costs

We address scheduling problems with job-dependent due-dates and general (possibly nonlinear and asymmetric) earliness and tardiness costs. The number of distinct due-dates is substantially smaller than the number of jobs, thus jobs are partitioned to classes, where all jobs of a given class share a common due-date. We consider the settings of a single machine and parallel identical machines. Our objective is of a minmax type, i.e., we seek a schedule that minimizes the maximum earliness/tardiness cost among all jobs.     

Order acceptance and due-date quotation in low machine rates

In real manufacturing environments, some customer orders include multiple jobs. However, a single due-date should be assigned to each order. Further, machine processing rate is not constant at all times. In effect, in many manufacturing operations, the machine processing rate decreases to a subnormal level during time and needs a special type of maintenance to bring the normal state back. Due to this reduction in capacity, production schedulers may decide to reject some orders. In this paper, the novel extensive problem of selecting a subset of orders, assigning due-dates to selected orders, scheduling the selected orders and jobs within each one, and scheduling the rate-modifying maintenance is studied. The objective function is minimizing total cost of lost-sales of rejected orders together with due-date length and maximum of earliness and tardiness of selected orders. The problem is proved polynomial and an optimal approach is developed.     

Due-date assignment and single machine scheduling with deteriorating jobs

We study a scheduling problem with deteriorating jobs, that is, jobs whose processing times are an increasing function of their start times. We consider the case of a single machine and linear job-independent deterioration. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of due-date, earliness and tardiness penalties. We give an O(n log n) time algorithm to solve this problem.     

Optimal assignment of slack due-dates and sequencing in a single-machine shop

This paper considers the problem of optimal assignment of slack due-dates to n jobs and sequencing them on a single-machine to minimize a penalty function depending on the values of the assigned slack allowance and maximum job tardiness. It is shown that the earliest due-date order yields an optimal sequence and the optimal slack allowance is a simple function of the cost per unit slack allowance.     

A due-date assignment problem with learning effect and deteriorating jobs

In this paper we consider a single-machine scheduling problem with the effects of learning and deterioration. In this model, job processing times are defined by functions of their starting times and positions in the sequence. The problem is to determine an optimal combination of the due-date and schedule so as to minimize the sum of earliness, tardiness and due-date. We show that the problem remains polynomially solvable under the proposed model.     

Single-machine scheduling to minimize earliness and number of tardy jobs

This paper considers the problem of assigning a common due-date to a set of simultaneously available jobs and sequencing them on a single machine. The objective is to determine the optimal combination of the common due-date and job sequence that minimizes a cost function based on the assigned due-date, job earliness values, and number of tardy jobs. It is shown that the optimal due-date coincides with one of the job completion times. Conditions are derived to determine the optimal number of nontardy jobs. It is also shown that the optimal job sequence is one in which the nontardy jobs are arranged in nonincreasing order of processing times. An efficient algorithm of O(n logn) time complexity to find the optimal solution is presented and an illustrative example is provided. Finally, several extensions of the model are discussed.This research was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OPG0036424. The authors are thankful to two anonymous referees for their constructive comments.     

Algorithms for common due-date assignment and sequencing on a single machine with sequence-dependent setup times

This paper focuses on the single machine sequencing and common due-date assignment problem for the objective of minimizing the sum of the penalties associated with earliness, tardiness and due-date assignment. Unlike the previous research articles on this class of scheduling problem, we consider sequence-dependent setup times that make the problem much more difficult. To solve the problem, a branch and bound algorithm, which incorporates the method to obtain lower and upper bounds as well as a dominance property to reduce the search space, is suggested that gives the optimal solutions for small-sized instances. Heuristic algorithms are suggested to obtain solutions for large-sized problems within a reasonable computation time. The performances of both the optimal and heuristic algorithms, in computational experiments on randomly generated test instances, are reported.     

Minmax due-date assignment with a time window for acceptable lead-times

In a standard DIF due-date assignment model, customers may consider late due-dates as unacceptable, i.e., if a due-date is assigned later than a pre-specified lead time, the supplier is penalized. This note extends this setting by adding a lower bound on the acceptable lead-time, reflecting e.g., the time needed by the customer for preparation of storage space. Thus, in addition to the standard earliness/tardiness penalties of jobs, our model contains penalties for early and tardy due-dates. The objective is of a minmax type, i.e. we try to minimize the highest (job and due-date) cost. An efficient O(n) solution algorithm (where n is the number of jobs) is introduced.     

Single-machine scheduling with time windows and earliness/tardiness penalties

We study the single machine earliness/tardiness problem with arbitrary time windows (STW). We show that STW is NP-hard and then decompose it into the subproblems of finding a good job sequence and optimally inserting idle time into a given sequence. We propose heuristics for the sequencing subproblem by appropriately modifying heuristics originally developed for other single machine scheduling problems. Experimentation with randomly generated problems shows that one of the proposed heuristics is computationally efficient and capable of finding good solutions to problems of arbitrary size. We also propose an algorithm to optimally insert idle time into a given job sequence.     

公共交货期窗口的提前／延期惩罚

本讨论n个独立工件在一台机器上加工,而且加工时间服从正态分布的公共交货期窗口的提前/延期惩罚问题,在确定公共交货期窗口情况下,推导出工件的最优排序具有V型特征。     

Scheduling under a common due-data on parallel unrelated machines

Due-data determination problems have gained significant attention in recent years due to the industrial focus in the just-in-time philosophy. In this paper the problem of scheduling a set of independent jobs on parallel unrelated processors under a common due-date is examined. The common due-date is a decision variable. The objective is to allocate and sequence the jobs on the machines and to determine the optimal due-data, so that the total cost be minimised. This cost is composed of the due-date assignment, the total earliness and the total tardiness cost. As the problem is NP-hard, a polynomial time heuristic procedure, which provides efficient solutions, is developed. The procedure is illustrated by means of an example and is tested via two small size experiments.     

Due-date assignment and maintenance activity scheduling problem

In the scheduling problem addressed in this note we have to determine: (i) the job sequence, (ii) the (common) due-date, and (iii) the location of a rate modifying (maintenance) activity. Jobs scheduled before (after) the due-date are penalized according to their earliness (tardiness) value. The processing time of a job scheduled after the maintenance activity decreases by a job-dependent factor. The objective is minimum total earliness, tardiness and due-date cost. We introduce a polynomial (O(n⁴)) solution for the problem.     

Single Machine Scheduling with Flow Allowances

In this paper, a set of jobs is scheduled using the SLK due-date determination method, according to which all the jobs are given the same flow allowance. The single machine case is considered. The objective function is a cost function including three components, namely flow allowance and weighted earliness and tardiness. An analytical solution is given and an algorithm, which provides optimal solutions, is presented. Finally, the parallel machines case is discussed.     

Earliness and tardiness single machine scheduling with proportional weights

This paper presents the first efficient method to solve an earliness-tardiness single machine n job scheduling model with idle times permitted. In this model the earliness and tardiness penalties are proportional to the processing times of the jobs. A two stage decomposition process builds the optimal job arrangement as a sequence of single and multijob blocks. The arrangement of the jobs within each multijob block is unspecified since it depends on the start time of this block. This process is followed by a procedure that drastically reduces the number of candidate optimal n job sequences. Finally an available optimal timing algorithm is recommended and implemented to select the best schedule among those sequences. The solution procedure tested on a PC on 400 examples for n=40 and 50 proves to be very fast.     

Scheduling about a common due date with kob-dependent asymmetric earliness and tardiness penalties

This paper is concerned with the problem of scheduling n jobs with a common due date on a single machine so as to minimize the total cost arising from earliness and tardiness. A general model is examined, in which earliness penalty and tardiness penalty are, respectively, arbitrary non-decreasing functions. Moreover, the model includes two important features that commonly appear in practical problems, namely, 1) earliness and tardiness are penalized with different weights which are job-dependent, and 2) the earliness (or tardiness) penalty consists of two parts, one is a variable cost dependent on the length of earliness (or tardiness), while the other is a fixed cost incurred when a job is early (or tardy). This model provides a general and flexible performance measure for earliness/tardiness scheduling, which has not been addressed before. We establish a number of results on the characterizations of optimal and sub-optimal solutions, and propose two algorithms based on these results. The first algorithm can find, under an agreeable weight condition, an optimum in time O(n² P_n), and the second algorithm can generate a sub-optimum in time O(nP_n), where P_n is the sum of the processing times. Further, we derive an upper bound on the relative error of the sub-optimal solution and show that, under certain conditions, the error tends to zero as n increases. Computational results are also reported to demonstrate the effectiveness of the algorithms proposed.     

Single machine scheduling with randomly compressible processing times

This work is concerned with single machine scheduling with random compression of processing times. The objective is to find the optimal sequence to minimize the cost based on earliness, tardiness and compression. The analysis is carried out under a common due date. Both absolute derivation cost and squared derivation cost are considered. For both constrained problems and unconstrained problems, it is shown that an optimal schedule must be V-shaped. Remarks on common slack model is also provided.     

Single machine common flow allowance scheduling with controllable processing times

In this paper, we consider single machine SLK due date assignment scheduling problem in which job processing times are controllable variables with linear costs. The objective is to determine the optimal sequence, the optimal common flow allowance and the optimal processing time compressions to minimize a total penalty function based on earliness, tardiness, common flow allowance and compressions. We solve the problem by formulating it as an assignment problem which is polynomially solvable. For some special cases, we present an O(n logn) algorithm to obtain the optimal solution respectively.     

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.