Common Date Research Articles

Single Stage Minimum Absolute Lateness Problem with a Common Due Date on Non-identical Machines

Single Stage Minimum Absolute Lateness Problem with a Common Due Date on Non-identical Machines

Optimal single-machine sequencing and assignment of common due-dates

This paper considers an n-job one-machine sequencing problem with common due-dates. The objective is to determine the optimal common due-date value and the optimal job sequence that jointly minimize a cost function which is dependent on the individual job earliness and tardiness values. Using Kuhn-Tucker's optimality conditions for constrained convex programming problems, we show that for a given job sequence, the optimal due-date is a simple function of the number of jobs. This result allows separation of the due-date assignment problem from the job sequencing problem. A well-known theorem in algebra can be applied to solve the latter problem, which in turn yields the optimal solution to the overall problem.

Earliness-Tardiness Scheduling Problems, I: Weighted Deviation of Completion Times About a Common Due Date

This paper and its companion (Part II) concern the scheduling of jobs with cost penalties for both early and late completion. In Part I, we consider the problem of minimizing the weighted sum of earliness and tardiness of jobs scheduled on a single processor around a common due date, d. We assume that d is not early enough to constrain the scheduling decision. The weight of a job does not depend on whether the job is early or late, but weights may vary between jobs. We prove that the recognition version of this problem is NP-complete in the ordinary sense. We describe optimality conditions, and present a computationally efficient dynamic programming algorithm. When the weights are bounded by a polynomial function of the number of jobs, a fully polynomial approximation scheme is given. We also describe four special cases for which the problem is polynomially solvable. Part II provides similar results for the unweighted version of this problem, where d is arbitrary.

Earliness–Tardiness Scheduling Problems, II: Deviation of Completion Times About a Restrictive Common Due Date

A companion paper (Part I) considers the problem of minimizing the weighted earliness and tardiness of jobs scheduled on a single machine around a common due date, d, which is unrestrictively late. This paper (Part II) considers the problem of minimizing the unweighted earliness and tardiness of jobs, allowing the possibility that d is early enough to constrain the scheduling decision. We describe several optimality conditions. The recognition version of the problem is shown to be NP-complete in the ordinary sense, confirming a well known conjecture. Moreover, this complexity definition is precise, since we describe a dynamic programming algorithm which runs in pseudopolynomial time. This algorithm is also extremely efficient computationally, providing an improvement over earlier procedures, of almost two orders of magnitude in the size of instance that can be solved. Finally, we describe a special case of the problem which is polynomially solvable.

On the Multiple-machine Extension to a Common Due-date Assignment and Scheduling Problem

In a recent paper, Cheng considers the multiple-machine extension to a due-date assignment and scheduling problem. He implicitly assumes zero start times for all schedules (which does not ensure optimality), and overlooks the critical V-shape property for an optimal schedule (which leads him to over-estimate the effort needed to solve the problem). In this note, we correct these flaws, and also discuss the complexity of the problem and its special cases.

On the Multiple-Machine Extension to a Common Due-Date Assignment and Scheduling Problem

On the Multiple-Machine Extension to a Common Due-Date Assignment and Scheduling Problem

On the Multiple-machine Extension to a Common Due-date Assignment and Scheduling Problem

In a recent paper, Cheng considers the multiple-machine extension to a due-date assignment and scheduling problem. He implicitly assumes zero start times for all schedules (which does not ensure optimality), and overlooks the critical V-shape property for an optimal schedule (which leads him to over-estimate the effort needed to solve the problem). In this note, we correct these flaws, and also discuss the complexity of the problem and its special cases.

Optimal common due-date with limited completion time

T.C.E. Cheng [ Computers Ops Res. 15, 91–96 (1988)] recently presented an interesting variation of the common due-date scheduling problem. He considered the situation where jobs that only marginally miss the common due-date are not penalized. His major conclusion, as given in Lemma 1, is a sufficient but not necessary requirement for optimality. We prove that alternate optima exist.

Scheduling to minimize weighted earliness and tardiness about a common due-date

This paper examines the problem of sequencing a set of independent jobs on a single processor so as to minimize the sum of weighted absolute deviations of job completion times about a specified due-date. In contrast with prior research on this topic, no restriction is imposed on the due-date. The resulting problem is known to be computationally difficult. It is shown that the solution to this problem can be derived from solutions to different subproblems. Properties of these subproblems are identified and used to develop exact algorithm of pseudo-polynomial complexity. The same algorithms are modified to yield a fully polynomial approximation scheme with a constant performance guarantee. The paper also discusses interesting special cases which are amenable to easier solutions.

Heuristics for the Common Due Date Weighted Tardiness Problem

In this paper we consider the single machine “common due date weighted tardiness problem.” Initially the problem is related to other versions of the single machine total weighted tardiness problem. Several heuristics for the problem are discussed. Three “simple” heuristics are proposed and shown to have arbitrarily bad worst case performance. A fourth heuristic is then proposed and shown to have a worst case performance bound of 2.

COMMON DUE-DATE ASSIGNMENT AND SCHEDULING FOR A SINGLE PROCESSOR TO MINIMIZE THE 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 production facility. The objective is to determine the optimal common due-date value and job sequence that jointly minimize a cost function based on the assigned due-date and the number of tardy jobs. It is shown that the optimal due-date coincides with one of the job completion times and ordering jobs in non-decreasing processing times yields an optimal sequence which in general is not unique. An efficient (polynomial-bound) algorithm to find the optimal solution is presented and an illustrative example is provided.

ON A GENERALIZED OPTIMAL COMMON DUE-DATE ASSIGNMENT PROBLEM

Abstract We consider a generalized optimal common due-date assignment problem and derive the optimality conditions for finding the optimal solution in this paper. We show that some versions of the common due-date determination problem can be treated as special cases of this generalized problem. For these special cases, we show that closed-form optimal solutions can be obtained from the optimality conditions. As for the solution of the generalized problem, we suggest an iterative solution procedure to help determine the optimal solution. Finally, we discuss the limiting behaviour of the optimal solution in a special situation and derive the asymptotic result

A Heuristic for Common Due-date Assignment and Job Scheduling on Parallel Machines

In this paper we consider the problem of scheduling a set of simultaneously available jobs on several parallel and identical machines. The problem is to find the optimal due-date, assuming this to be the same for all jobs. We also seek to sequence the jobs such that some are early and some are late so as to minimize a penalty function. For the single-machine problem, we present a simple proof of the well-known optimality result that the optimal due-date coincides with one of the job completion times. We show that the optimal job sequence for the single-machine problem can be easily determined. We prove that the same optimal due-date result can be generalized to the parallel-machine problem. However, determination of the optimal job sequence for such a problem is much more complex, and we present a simple heuristic to find an approximate solution. On the basis of a limited experiment, we observe that the heuristic is very effective in obtaining near-optimal solutions.

A Heuristic for Common Due-date Assignment and Job Scheduling on Parallel Machines

In this paper we consider the problem of scheduling a set of simultaneously available jobs on several parallel and identical machines. The problem is to find the optimal due-date, assuming this to be the same for all jobs. We also seek to sequence the jobs such that some are early and some are late so as to minimize a penalty function. For the single-machine problem, we present a simple proof of the well-known optimality result that the optimal due-date coincides with one of the job completion times. We show that the optimal job sequence for the single-machine problem can be easily determined. We prove that the same optimal due-date result can be generalized to the parallel-machine problem. However, determination of the optimal job sequence for such a problem is much more complex, and we present a simple heuristic to find an approximate solution. On the basis of a limited experiment, we observe that the heuristic is very effective in obtaining near-optimal solutions.

A Heuristic for Common Due-Date Assignment and Job Scheduling on Parallel Machines

A Heuristic for Common Due-Date Assignment and Job Scheduling on Parallel Machines

Note—A Note on the Minimization of Mean Squared Deviation of Completion Times About a Common Due Date

This paper points out a conceptual flaw present in a recent publication. It shows how the methodology can be modified to correct this flaw. It also presents improved bounds to determine the nature of a given problem.

Survey of scheduling research involving due date determination decisions

We attempt to present in this paper a critical review of a particular segment of scheduling research in which the due to date assignment decision is of primary interest. The literature is classified into the static and dynamic job shop situations. The static job shop is analyzed from two different perspectives: the due date is constrained to be greater than or equal to makespan, and the optimal due date and optimal sequence are to be determined when the method of assigning due dates is specified. The literature on dynamic job shops is also reviewed under two broad categories. First, we discuss all the literature concerned with comparative and investigative studies to identify the most desirable due date assignment method. Second, we discuss the literature dealing with determination of optimal due dates. We note that computer simulation and analytical methods are two common approaches for the second type of problems. We observe that while the static single-machine problem with constant or common due dates has been well researched, very little or no work has been done on the dynamic multi-machine problem with sophisticated due date assignment methods. Finally, we identify and suggest some worthwhile areas for future research.

An alternative proof of optimality for the common due-date assignment problem

For the n-job, one-machine scheduling problem with common due-dates, it is a well-known result that for any given job sequence there exists a job whose completion time is equal to the optimal value of the common due-date. In this note we offer an alternative proof of this optimal result using Kuhn-Tucker's optimality conditions for constrained convex programming problems.

A note on “minimizing the maximum deviation of job completion time about a common due-date”

A recent paper by Cheng analyzed the problem of simultaneous determination of a common due-date and a sequence of n jobs to minimize the maximum deviation of job completion time around the common due-date. It is assumed that all jobs are available at time zero and their processing times, t i's, are known in advance. The analysis in Cheng used an LP formulation and duality results to prove the theorems. This note provides an alternate simpler proof for the same results. We also give some results for the case when splitting and preemption are allowed. A recent paper by Cheng [1] gives an optimal solution to the problem of setting a common due-date and sequence of n jobs on a single machine. All jobs are available at time zero and their processing times, t i's, are known in advance. The objective is to determine the optimal common due-date value k ∗, and the sequence ( σ ) to minimize the maximum deviation of job completion time about the common due-date. We here give a much simpler proof of the results derived in [1]. The notations used here are the same as in [1]. Let C [1] denote the completion time (decision variable) and t [i] denote the given processing time of a job in the i th position of sequence σ.

Optimal common due-date with limited completion time deviation

Given a set of n jobs with deterministic processing times and the same ready times, the problem is to find the optimal common flow allowance k∗ for the common due-date assignment method, and the optimal job sequence σ∗ to minimize a penalty function of missing due-dates. It is assumed that penalty will not occur if the deviation of job completion from the due-date is sufficiently small. Three lemmas are presented and a numerical example is provided to illustrate the use of the results to determine the optimal solution to the due-date determination and sequencing problem.