A faster algorithm for a due date assignment problem with tardy jobs

doi:10.1016/j.orl.2009.10.013

Operations Research Letters

Volume 38, Issue 2, March 2010, Pages 127-128

https://doi.org/10.1016/j.orl.2009.10.013 Get rights and content

Abstract

The single-machine due date assignment problem with the weighted number of tardy jobs objective, (the TWNTD problem), and its generalization with resource allocation decisions and controllable job processing times have been solved in $O (n^{4})$ time by formulating and solving a series of assignment problems. In this note, a faster $O (n^{2})$ dynamic programming algorithm is proposed for the TWNTD problem and for its controllable processing times generalization in the case of a convex resource consumption function.

Introduction

The single-machine due date assignment problem with the weighted number of tardy jobs objective (the TWNTD problem) can be defined as follows: There are $n$ jobs available at time zero; job $j$ has a positive processing time $p_{j}$ and a positive weight $w_{j}$ . The objective is to determine the due date $d_{j}$ of each job $j$ and a job sequence so that the function $T C = a \sum_{j = 1}^{n} d_{j} + \sum_{j = 1}^{n} w_{j} U_{j}$ is minimized where $a$ is the positive unit due date assignment cost and $U_{j}$ is the tardiness indicator of job $j$ defined as follows: $U_{j} = 1$ if $C_{j} > d_{j}$ and $U_{j} = 0$ if $C_{j} \leq d_{j}$ where $C_{j}$ denotes the completion time of job $j$ .

Shabtay and Steiner [1] proposed an $O (n^{4})$ algorithm for the TWNTD problem by solving a series of assignment problems. In this note, we show that the TWNTD problem can be solved by dynamic programming (DP) in $O (n^{2})$ time by utilizing additional information about the ordering of the jobs in an optimal sequence. An extension of the $O (n^{2})$ DP algorithm to the more general resource allocation/scheduling problem with controllable job processing times and a convex resource allocation function is also presented.

Section snippets

A DP $O (n^{2})$ algorithm for the TWNTD problem

Shabtay and Steiner [1] proved some useful properties of an optimal TWNTD solution summarized next: Let $[j]$ denote the job occupying the $j$ position in an optimal TWNTD sequence $S^{*}$ ; $p_{[j]}$ , $d_{[j]}$ , $C_{[j]}$ , $w_{[j]}$ and $U_{[j]}$ are defined analogously. Let $E$ , $T$ denote the sets of the non-tardy (early) jobs and tardy jobs respectively in $S^{*}$ . According to Shabtay and Steiner [1], there is no-inserted idle time in $S^{*}$ and the jobs in $E$ are sequenced in the shortest-processing-time (SPT) order followed by the jobs

Extensions

Shabtay and Steiner [2] extended their $O (n^{4})$ algorithm for the TWNTD problem to the more general due date assignment problem with resource allocation decisions, controllable job processing times, and the weighted number of tardy jobs objective. In that case, the objective function (1) generalizes to $T C = a \sum_{j = 1}^{n} d_{j} + \sum_{j = 1}^{n} w_{j} U_{j} + γ C_{max} + \sum_{j = 1}^{n} u_{j} v_{j}$ where $u_{j}$ is the amount of resource allocated to job $j$ , $v_{j}$ is the cost of one unit of resource allocated to job $j$ , $C_{max}$ is the maximum job completion time

Acknowledgement

We would like to thank the Associate Editor for suggesting the Dynamic Programming Algorithm and for his/her other suggestions that helped us to improve an earlier version of this note.

References (2)

D. Shabtay et al.
Two due date assignment problems in scheduling a single machine
Operations Research Letters
(2006)
D. Shabtay et al.
Optimal due date assignment and resource allocation to minimize the weighted number of tardy jobs on a single machine
Manufacturing & Service Operations Management
(2007)

Cited by (15)

Single machine scheduling with due date assignment to minimize the total weighted lead time penalty and late work
2023, Omega (United Kingdom)
We study a single machine scheduling problem with due date assignment for minimizing the total weighted lead time penalty and late work, where the lead time penalty is the excess of an assigned due date over a given lead time, and the late work is the part of a job executed after the assigned due date. For the problem with common due date assignment, we show that it is solvable in $O (n \log n)$ time. For the problem with unrestricted due date assignment, we determine an optimal due date assignment for a given schedule, and transform the problem to an equivalent problem for minimizing the total
-shape function, where
depicts the shape of the penalty function for a job in relation to its completion time. Furthermore, we show that the problem with unrestricted due date assignment is unary NP-hard by proving the unary NP-hardness of the equivalent problem, and study two special cases. For the case with identical lead times, we show that it is binary NP-hard by providing a dynamic programming algorithm. While for the case with identical processing times, we prove that it is solvable in $O (n^{3})$ time. Finally, we study the problem of minimizing the total
-shape function, and show that its two special cases are binary NP-hard by proving that they are pseudo-polynomially solvable.
A classification of dynamic programming formulations for offline deterministic single-machine scheduling problems
2023, European Journal of Operational Research
We review dynamic programming (DP) algorithms utilized to solve offline deterministic single-machine scheduling problems. We classify DP algorithms based on problem properties and provide insights on how these properties facilitate the use of specific types of DP algorithms. These properties center on whether jobs in a schedule can be naturally partitioned into subsets or whether a complete schedule is the outcome of blending distinct subsequences and/or whether the overall scheduling objective is a compromise of conflicting objectives. We propose generalizations of existing DP algorithms so they can be applied to more general problems such as proportionate flow shops. In some cases, we show how the running time of a DP algorithm can be improved. We also survey models where a DP formulation is part of a hybrid enumerative algorithm. A discussion on how pseudo-polynomial DP algorithms can be converted to fully polynomial time approximation schemes is also presented. We conclude our review with a timeline of DP algorithmic development during the last 50 years.
Two-agent single-machine scheduling with unrestricted due date assignment
2015, Computers and Industrial Engineering
We address two scheduling problems arising when two agents (agents A and B), each with a set of jobs, compete to perform their respective jobs on a common machine, where the due dates of agent A’s jobs are decision variables to be determined by the scheduler. Specifically, the objective is to determine the optimal due dates for agent A’s jobs and the job sequence for both agents’ jobs simultaneously to minimize the total cost associated with the due date assignment and weighted number of tardy jobs of agent A, while keeping the maximum of regular functions (associated with each B-job) or the number of tardy jobs of agent B below or at a fixed threshold. We prove that both problems are $NP$ -hard in the strong sense and develop polynomial or pseudo-polynomial solutions for some important special cases.
Four single-machine scheduling problems involving due date determination decisions
2013, Information Sciences
This paper considers single-machine scheduling problems with simultaneous consideration of due date assignment, past-sequence-dependent (p-s-d) delivery times, and position-dependent learning effects. By p-s-d delivery times, we mean that the delivery time of a job is proportional to the job’s waiting time. Specifically, we study four variants of the problem: (i) the variant of total earliness and tardiness with common due date assignment (referred to as TETDC), (ii) the variant of total earliness and weighted number of tardy jobs with CON due date assignment (referred to as TEWNTDC), (iii) the variant of total earliness and weighted number of tardy jobs with slack due date assignment (referred to as TEWNTDS), and (iv) the variant of weighted number of tardy jobs with different due date assignment (referred to as WNTDD). We derive the structural properties of the optimal schedules and show that the variants TETDC, TEWNTDC and TEWNTDS are all polynomially solvable. Although the complexity status of the variant WNTDD is still open, we show that two special cases of it are polynomially solvable.
Due-window assignment problems with unit-time jobs
2013, Applied Mathematics and Computation
Citation Excerpt :
Meeting promised delivery dates or due-dates is clearly one of management’s primary objectives. We refer the readers to the survey paper of Gordon et al. [1], and to more recent papers such as: Mosheiov and Yovel [2], Liao and Cheng [3], Baykasoglu et al. [4], Li et al. [5], Nearchou [6], Shabtay [7], Gordon and Strusevich [8], Gordon and Tarasevich [9], Mosheiov and Sarig [10], and Koulamas [11]. In recent years, several papers focused on due-window assignment, where the assumption is that jobs completed within a given time interval (rather than a time point) are not penalized, whereas the remaining jobs are penalized according to their earliness/tardiness.
We study a class of due-window assignment problems. The objective is to find the job sequence and the window starting time and size, such that the total cost of earliness, tardiness and due-window is minimized. The study assumes unit-time jobs, and considers settings of a single machine and of parallel identical machines. Both the due-window starting time and size are decision variables. For the single machine setting, we study a complete set of problems consisting of all possible combinations of the decision variables and four cost factors (earliness, tardiness, due-window size and due-window starting time). For parallel identical machines, we study a complete set of problems consisting of all possible combinations of the decision variables and three cost factors (earliness, tardiness and due-window size). All the problems are shown to be solvable in no more than O(n³) time, where n is the number of jobs.
A unified solution approach for the due date assignment problem with tardy jobs
2011, International Journal of Production Economics
Citation Excerpt :
This structure facilitates the solution of either problem in O(n2) by dynamic programming (DP). This observation, to be further analyzed in the next section, was first made by Engels et al. (2003) in the context of the scheduling problem with job rejection and also by Koulamas (2010) in the context of the due date assignment problem. The objective of this paper is to show that the DP algorithms of Engels et al. (2003) and Koulamas (2010) can be applied to a number of other due date assignment problems with the weighted number of tardy jobs objective.
We analyze a number of due date assignment problems with the weighted number of tardy jobs objective and show that these problems can be solved in O(n²) time by dynamic programming. We show that the effects of learning or the effects of past-sequence-dependent setup times can be incorporated into the problem formulation at no additional computational cost. We also show that some single-machine due date assignment problems can be extended to an identical parallel machine setting. Finally, we improve the complexity of the solution algorithms for two other due date assignment problems.

View all citing articles on Scopus

View full text