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 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.     

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.     

Minmax earliness–tardiness costs with unit processing time jobs

The problem addressed in this paper is defined by M parallel identical machines, N jobs with identical (unit) processing time, job-dependent weights, and a common due-date for all jobs. The objective is of a minmax type, i.e. we are interested in minimizing the cost of the worst scheduled job. In the case of a non-restrictive (i.e., sufficiently large) common due-date, the problem is shown to have a solution that is polynomial in the number of jobs. The solution in the case of a restrictive due-date remains polynomial in the number of jobs, but is exponential in the number of machines. We introduce a lower bound on the optimal cost and an efficient heuristic. We show that the worst case relative error of the heuristic is bounded by 2 and that this bound is tight. We also prove that the heuristic is asymptotically optimal under very general assumptions. Finally, we provide an extensive numerical study demonstrating that in most cases the heuristic performs extremely well.     

Scheduling with controllable release dates and processing times: Total completion time minimization

The single machine scheduling problem with two types of controllable parameters, job processing times and release dates, is studied. It is assumed that the cost of compressing processing times and release dates from their initial values is a linear function of the compression amounts. The objective is to minimize the sum of the total completion time of the jobs and the total compression cost. For the problem with equal release date compression costs we construct a reduction to the assignment problem. We demonstrate that if in addition the jobs have equal processing time compression costs, then it can be solved in O(n²) time. The solution algorithm can be considered as a generalization of the algorithm that minimizes the makespan and total compression cost. The generalized version of the algorithm is also applicable to the problem with parallel machines and to a range of due-date scheduling problems with controllable processing times.     

Fast approximation algorithms for bi-criteria scheduling with machine assignment costs

We consider parallel machine scheduling problems where the processing of the jobs on the machines involves two types of objectives. The first type is one of two classical objective functions in scheduling theory: either the total completion time or the makespan. The second type involves an actual cost associated with the processing of a specific job on a given machine; each job-machine combination may have a different cost. Two bi-criteria scheduling problems are considered: (1) minimize the maximum machine cost subject to the total completion time being at its minimum, and (2) minimize the total machine cost subject to the makespan being at its minimum. Since both problems are strongly NP-hard, we propose fast heuristics and establish their worst-case performance bounds.     

Minimizing total completion time and total deviation of job completion times from a restrictive due-date

We study a single-machine scheduling problem with the objective of minimizing a linear combination of total job completion times and total deviation of job completion times from a common due-date. The due-date is assumed to be restrictive, i.e., it may be sufficiently small to have an impact on the optimal sequence. When more weight is allocated to total job completion times, the problem is shown to have a polynomial time solution. When more weight is allocated to total completion time deviations from the due-date, the problem is NP-hard in the ordinary sense. For the latter case, we introduce an efficient dynamic programming algorithm, which is shown numerically to perform well in all our tests.     

Sequencing and due-date determination in the stochastic single machine problem with earliness and tardiness costs

This paper studies the problem of simultaneous due-date determination and sequencing of a set of n jobs on a single machine where processing times are random variables and job earliness and tardiness costs are distinct. The objective is to determine the optimal sequence and the optimal due-dates which jointly minimize the expected total earliness and tardiness cost. We present an analytical approach to determine optimal due-dates, and propose two efficient heuristics of order O(n log n) to find candidates for the optimal sequence. It is demonstrated that variations in processing times increase cost and affect sequencing and due-date determination decisions. Our illustrative examples as well as computational results show that the proposed model produces optimal sequences and optimal due-dates that are significantly different from those provided by the classical deterministic single machine models. Furthermore, our computational experiments reveal that the proposed heuristics perform well in providing either optimal sequences or good candidates with low overcosts.     

Setting Job-Shop Due-Dates with Service-Level Constraints

The job-shop due-date assignment problem arises when a manager needs to ‘promise’ a delivery date to a customer. Previous methods yield due-dates which are either optimistic (unlikely to be achieved) or conservative (the promise will be met, but too easily, because the date given was very pessimistic). This paper investigates the due-date assignment problem with a customer ‘service-level’ constraint, the percentage of time that promised delivery dates are honoured. We formulate a rule to attain this service level, yet maintain as short a due-date lead time as possible. Unlike previous attempts, this due-date rule considers not only the job content and instantaneous shop congestion information, but also implicitly incorporates information on how the jobs will be scheduled (or ‘loaded’) once they are in the shop. We simulate a single-machine shop for various measures of performance under several dispatching priorities, comparing our due-date rule with one reported to yield satisfactory performance. Our rule meets all requirements and is found to be superior for most measures of performance.     

Simultaneous Job Scheduling and Resource Allocation on Parallel Machines

Most deterministic production scheduling models assume that the processing time of a job on a machine is fixed externally and known in advance of scheduling. However, in most realistic situations, apart from the machines, it requires additional resources to process jobs, and the processing time of a job is determined internally by the amount of the resources allocated. In these situations, both the cost associated with the job schedule and the cost of the resources allocated should be taken into account. Therefore, job scheduling and resource allocation should be carefully coordinated and optimized jointly in order to achieve an overall cost-effective schedule. In this paper, we study a parallel-machine scheduling model involving both job processing and resource allocation. The processing time of a job is non-increasing with the cost of the allocated resources. The objective is to minimize the total cost including the cost measured by a scheduling criterion and the cost of all allocated resources. We consider two particular problems of this model, one with the scheduling criterion being the total weighted completion time, and the other with that being the weighted number of tardy jobs. We develop a column generation based branch and bound method for finding optimal solutions for these NP-hard problems. The method first formulates the problems as set partitioning type formulations, and then solves the resulting formulations exactly by branch and bound. In the branch and bound, linear relaxations of the set partitioning type formulations are decomposed into master problems and single-machine subproblems and solved by a column generation approach. The algorithms developed based on this method are capable of solving the two problems with a medium size to optimality within a reasonable computational time.     

Optimal Release Times in Single-Stage Manufacturing Systems with Blocking: Optimal Control Perspective

This paper concerns a due-date matching problem in a single-stage manufacturing system. Given a finite sequence of jobs and their service order, and given the delivery due date of each job, the problem is to choose the jobs release (arrival) times so as to match as closely as possible their completion times to their respective due dates. The system is modelled as a deterministic single-server FIFO queue with an output buffer for storing jobs whose service is completed prior to their due dates. The output buffer has a finite capacity; when it is full, the server is being blocked. Associated with each job there is a convex cost function penalizing its earliness as well as tardiness. The due-date matching problem is cast as an optimal control problem, whose objective is to minimize the sum of the above cost functions by the choice of the jobs arrival (release) times. Time-box upper-bound and lower-bound constraints are imposed on the jobs output (delivery) times. The optimal-control setting brings to bear on the development of fast and efficient algorithms having intuitive geometric appeal and potential for online implementation.Communicated by W. B. GongResearch supported in part by the National Science Foundation under Grant ECS-9979693 and by the Georgia Tech Manufacturing Research Center under Grant B01-D06.     

Due date assignment for multistage assembly systems

This paper is concerned with the study of the constant due-date assignment policy in a multistage assembly system. The multistage assembly system is modeled as an open queueing network. It is assumed that the product order arrives according to a Poisson process. In each service station, there is either one or infinite machine with exponentially distributed processing time. The transport times between every pair of service stations are independent random variables with generalized Erlang distributions. It is assumed that each product has a penalty cost that is some linear function of its due-date and its actual completion time. The due date is found by adding a constant to the time that the order arrives. This constant value is the constant lead time that a product might expect between time of placing the order and time of delivery. By applying the longest path analysis in queueing networks, we obtain the distribution function of manufacturing lead time. Then, the optimal constant lead time is computed by minimizing the expected aggregate cost per product. Finally, the results are verified by Monte Carlo simulation.     

Optimal due-date assignment problem with learning effect and resource-dependent processing times

In this paper, we consider a single-machine earliness-tardiness scheduling problem with due-date assignment, in which the processing time of a job is a function of its position in a sequence and its resource allocation. The due date assignment methods studied include the common due date, and the slack due date, which reflects equal waiting time allowance for the jobs. For each combination of due date assignment method and processing time function, we provide a polynomial-time algorithm to find the optimal job sequence, due date values, and resource allocations that minimize an integrated objective function, which includes earliness, tardiness, due date assignment, and total resource consumption costs.     

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.     

A Generalized Model of Optimal Due-Date Assignment by Linear Programming

A linear-programming model to find the optimal ‘CON due-date’ is considered for n independent jobs to be processed on a single machine. The term ‘CON due-date’ stands for constant-allowance due-date, where each job receives exactly the same due-date. The measure of performance considered is a more generalized version of similar problems studied earlier. Duality theory is used to obtain an optimal solution. Some earlier studies are shown to be special cases of the model studied in this paper. Numerical examples are presented for better understanding.     

Minimizing weighted earliness–tardiness and due-date cost with unit processing-time jobs

The classical single-machine scheduling and due-date assignment problem of Panwalker et al. [Panwalker, S.S., Smith, M.L., Seidmann, A., 1982. Common due date assignment to minimize total penalty for the one machine scheduling problem. Operations Research 30(2) (1982) 391–399] is the following: All n jobs share a common due-date, which is to be determined. Jobs completed prior to or after the due-date are penalized according to a cost function which is linear and job-independent. The objective is to minimize the total earliness–tardiness and due-date cost. We study a generalized version of this problem in which: (i) the earliness and tardiness costs are allowed to be job dependent and asymmetric and (ii) jobs are processed on parallel identical machines. We focus on the case of unit processing-time jobs. The problem is shown to be solved in polynomial (O(n⁴)) time. Then we study the special case with no due-date cost (a classical problem known in the literature as TWET). We introduce an O(n³) solution for this case. Finally, we study the minmax version of the problem, (i.e., the objective is to minimize the largest cost incurred by any of the jobs), which is shown to be solved in polynomial time as well.     

Single machine scheduling with truncated job-dependent learning effect

In this paper we consider the single machine scheduling problem with truncated job-dependent learning effect. By the truncated job-dependent learning effect, we mean that the actual job processing time is a function which depends not only on the job-dependent learning effect (i.e., the learning in the production process of some jobs to be faster than that of others) but also on a control parameter. The objectives are to minimize the makespan, the total completion time, the total absolute deviation of completion time, the earliness, tardiness and common (slack) due-date penalty, respectively. Several polynomial time algorithms are proposed to optimally solve the problems with the above objective functions.     

Learning effect and deteriorating jobs in the single machine scheduling problems

This paper studies the single machine scheduling problems with learning effect and deteriorating jobs simultaneously. In this model, the processing times of jobs are defined as functions of their starting times and positions in a sequence. It is shown that even with the introduction of learning effect and deteriorating jobs to job processing times, the makespan, the total completion time and the sum of the k$k$th power of completion times minimization problems remain polynomially solvable, respectively. But for the following objective functions: the total weighted completion time and the maximum lateness, this paper proves that the shortest weighted processing time first (WSPT) rule and the earliest due-date first (EDD) rule can construct the optimal sequence under some special cases, respectively.     

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.     

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.