Precedence theorems and dynamic programming for the single-machine weighted tardiness problem

We tackle precedence-constrained sequencing on a single machine in order to minimize total weighted tardiness. Classic dynamic programming (DP) methods for this problem are limited in performance due to excessive memory requirements, particularly when the precedence network is not sufficiently dense. Over the last decades, a number of precedence theorems have been proposed, which distinguish dominant precedence constraints for a job pool that is initially without precedence relation. In this paper, we connect and extend the findings of the foregoing two strands of literature. We develop a framework for applying the precedence theorems to the precedence-constrained problem to tighten the search space, and we propose an exact DP algorithm that utilizes a new efficient memory management technique. Our procedure outperforms the state-of-the-art algorithm for instances with medium to high network density. We also empirically verify the computational gain of using different sets of precedence theorems.     

A column generation based decomposition algorithm for a parallel machine just-in-time scheduling problem

We propose a column generation based exact decomposition algorithm for the problem of scheduling n jobs with an unrestrictively large common due date on m identical parallel machines to minimize total weighted earliness and tardiness. We first formulate the problem as an integer program, then reformulate it, using Dantzig–Wolfe decomposition, as a set partitioning problem with side constraints. Based on this set partitioning formulation, a branch and bound exact solution algorithm is developed for the problem. In the branch and bound tree, each node is the linear relaxation problem of a set partitioning problem with side constraints. This linear relaxation problem is solved by column generation approach where columns represent partial schedules on single machines and are generated by solving two single machine subproblems. Our computational results show that this decomposition algorithm is capable of solving problems with up to 60 jobs in reasonable cpu time.     

Single machine total tardiness maximization problems: complexity and algorithms

In this paper, we consider some scheduling problems on a single machine, where weighted or unweighted total tardiness has to be maximized in contrast to usual minimization problems. These problems are theoretically important and have also practical interpretations. For the total weighted tardiness maximization problem, we present an NP-hardness proof and a pseudo-polynomial solution algorithm. For the unweighted total tardiness maximization problem with release dates, NP-hardness is proven. Complexity results for some other classical objective functions (e.g., the number of tardy jobs, total completion time) and various additional constraints (e.g., deadlines, weights and/or release dates of jobs may be given) are presented as well.     

Branch-and-bound algorithms for scheduling in permutation flowshops to minimize the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime,weighted tardiness and weighted earliness of jobs

The problem of scheduling in permutation flowshops is considered in this paper with the objectives of minimizing the sum of weighted flowtime/sum of weighted tardiness/sum of weighted flowtime and weighted tardiness/sum of weighted flowtime, weighted tardiness and weighted earliness of jobs, with each objective considered separately. Lower bounds on the given objective (corresponding to a node generated in the scheduling tree) are developed by solving an assignment problem. Branch-and-bound algorithms are developed to obtain the best permutation sequence in each case. Our algorithm incorporates a job-based lower bound (integrated with machine-based bounds) with respect to the weighted flowtime/weighted tardiness/weighted flowtime and weighted tardiness, and a machine-based lower bound with respect to the weighted earliness of jobs. The proposed algorithms are evaluated by solving many randomly generated problems of different problem sizes. The results of an extensive computational investigation for various problem sizes are presented. In addition, one of the proposed branch-and-bound algorithms is compared with a related existing branch-and-bound algorithm.     

Single-machine scheduling of proportionally deteriorating jobs by two agents

We consider a problem of scheduling a set of independent jobs by two agents on a single machine. Every agent has its own subset of jobs to be scheduled and uses its own optimality criterion. The processing time of each job proportionally deteriorates with respect to the starting time of the job. The problem is to find a schedule that minimizes the total tardiness of the first agent, provided that no tardy job is allowed for the second agent. We prove basic properties of the problem and give a lower bound on the optimal value of the total tardiness criterion. On the basis of these results, we propose a branch-and-bound algorithm and an evolutionary algorithm for the problem. Computational experiments show that the exact algorithm solves instances up to 50 jobs in a reasonably short time and that solutions obtained by the metaheuristic are close to optimal ones.     

An algorithm for single machine sequencing with release dates to minimize total weighted completion time

Each of n jobs is to be processed without interruption on a single machine which can handle only one job at a time. Each job becomes available for processing at its release date, requires a processing time and has a positive weight. Given a processing order of the jobs, the earliest completion time for each job can be computed. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times. In this paper a branch and bound algorithm for the problem is derived. Firstly a heuristic is presented which is used in calculating the lower bound. Then the lower bound is obtained by performing a Lagrangean relaxation of the release date constraints; the Lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimum solution of the relaxed problem thus yielding a lower bound. A method to increase the lower bound by deriving improved constraints to replace the original release date constraints is given. The algorithm, which includes several dominance rules, is tested on problems with up to fifty jobs. The computational results indicate that the version of the lower bound using improved constraints is superior to the original version.     

A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process

One of the largest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The SCC scheduling is a complex hybrid flowshop (HFS) scheduling problem with the following features: job grouping and precedence constraints, no idle time within the same group of jobs and setup time constraints on the casters. This paper first models the scheduling problem as a mixed-integer programming (MIP) problem with the objective of minimizing the total weighted earliness/tardiness penalties and job waiting. Next, a Lagrangian relaxation (LR) approach relaxing the machine capacity constraints is presented to solve the MIP problem, which decomposes the relaxed problem into two tractable subproblems by separating the continuous variables from the integer ones. Additionally, two methods, i.e., the boundedness detection method and time horizon method, are explored to handle the unboundedness of the decomposed subproblems in iterations. Furthermore, an improved subgradient level algorithm with global convergence is developed to solve the Lagrangian dual (LD) problem. The computational results and comparisons demonstrate that the proposed LR approach outperforms the conventional LR approaches in terms of solution quality, with a significantly shorter running time being observed.     

A general approach for optimizing regular criteria in the job-shop scheduling problem

Even though a very large number of solution methods has been developed for the job-shop scheduling problem, a majority has been designed for the makespan criterion. In this paper, we propose a general approach for optimizing any regular criterion in the job-shop scheduling problem. The approach is a local search method that uses a disjunctive graph model and neighborhoods generated by swapping critical arcs. The connectivity property of the neighborhood structure is proved and a novel efficient method for evaluating moves is presented. Besides its generality, another prominent advantage of the proposed approach is its simple implementation that only requires to tune the range of one parameter. Extensive computational experiments carried out on various criteria (makespan, total weighted flow time, total weighted tardiness, weighted sum of tardy jobs, maximum tardiness) show the efficiency of the proposed approach. Best results were obtained for some problem instances taken from the literature.     

Sequencing with general precedence constraints

Consider a set of jobs where an arbitrary precedence relationship exists among the jobs and a cost is associated with every permutation of the jobs. The objective is to find a minimum-cost permutation which is consistent with the precedence relations. A class of problems is studied which includes the least-cost fault detection problem, the one-machine total weighted completion time problem, and the two-machine maximum flow-time problem.Transformations are developed which systematically reduce the size of the general precedence-constrained problem. This process continues until either the problem is solved or no further reductions are possible. The worst-case effectiveness of these transformations is analyzed in detail. These results generalize the majority of work previously done on efficient sequencing with precedence constraints.     

A branch and bound algorithm to minimize total tardiness of jobs in a two identical-parallel-machine scheduling problem with a machine availability constraint

This research focuses on the problem of scheduling jobs on two identical parallel machines that are not continuously available with the objective of minimizing total tardiness. After processing a given number of jobs, each machine requires a preventive maintenance task, during which the machine cannot process jobs. We present dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as an upper bound obtained from a heuristic algorithm. Performance of the algorithm is evaluated through a series of computational experiments on randomly generated instances and results are reported.     

An algorithm for single machine sequencing with deadlines to minimize total weighted completion time

Each of n jobs is to be processed without interruption on a single machine. Each job becomes available for processing at time zero, has a deadline by which it must be completed and has a positive weight. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times. In this paper a branch and bound algorithm for the problem is presented which incorporates lower bounds that are obtained using a new technique called the multiplier adjustment method. Firstly several dominance conditions are derived. Then a heuristic is described and sufficient conditions for its optimality are given. The lower bound is obtained by performing a Lagrangean relaxation of the deadline constraints; the Lagrange multipliers are chosen so that the sequence generated by the heuristic is an optimal solution of the relaxed problem, thus yielding a lower bound. The algorithm is tested on problems with up to fifty jobs.     

具有不可用区间且工件可拒绝下的单机重新排序问题的近似方案

本文考虑了机器具有不可用区间且工件可拒绝下的单机重新排序问题,在该问题中,给定一个工件集需在一台机器上加工,每个工件有自己的加工时间和权重,且对该工件集目标函数为极小化总加权完工时间的排序计划已给定,根据该排序计划中每个工件的完工时间已确定每个工件的承诺交付时间。然而,在工件正式开始加工前,原计划用于加工的某段时间区间因临时用于检修机器而导致机器在该时间区间不再可用,需要对工件重新排序。为了确保在新的重新排序中,工件的延误成本不致太大,决策者可以选择拒绝部分工件,但需支付相应的拒绝费用。任务是确定接受工件集和拒绝工件集,并将接受的工件在考虑机器具有不可用区间的条件下重新排序使得接受工件集的总加权完工时间,总拒绝费用及赋权最大延误之和最小。该问题是NP-困难的,对此给出了伪多项式时间动态规划精确算法,利用稀疏技术设计了完全多项式时间近似方案。     

A new dominance rule to minimize total weighted tardiness with unequal release dates

We present a new dominance rule by considering the time-dependent orderings between each pair of jobs for the single machine total weighted tardiness problem with release dates. The proposed dominance rule provides a sufficient condition for local optimality. Therefore, if any sequence violates the dominance rule then switching the violating jobs either lowers the total weighted tardiness or leaves it unchanged. We introduce an algorithm based on the dominance rule, which is compared to a number of competing heuristics for a set of randomly generated problems. Our computational results indicate that the proposed algorithm dominates the competing algorithms in all runs, therefore it can improve the upper bounding scheme in any enumerative algorithm. The proposed time-dependent local dominance rule is also implemented in two local search algorithms to guide these algorithms to the areas that will most likely contain the good solutions.     

Single-machine scheduling with common due-window assignment for deteriorating jobs

In this paper, we consider a single-machine common due-window assignment scheduling problem with deteriorating jobs. Jobs’ processing times are defined by function of their starting times and job-dependent deterioration rates that are related to jobs and are not all equal. The objective is to determine an optimal combination of sequence and common due-window location so as to minimize the weighted sum of earliness, tardiness and due-window location penalties. We propose an O(n² log n) time algorithm to solve the problem and discuss several instances to illustrate it.     

Parallel machine scheduling with a common due window

In this paper, we consider a machine scheduling problem where jobs should be completed at times as close as possible to their respective due dates, and hence both earliness and tardiness should be penalized. Specifically, we consider the problem with a set of independent jobs to be processed on several identical parallel machines. All the jobs have a given common due window. If a job is completed within the due window, then there is no penalty. Otherwise, there is either a job-dependent earliness penalty or a job-dependent tardiness penalty depending on whether the job is completed before or after the due window. The objective is to find an optimal schedule with minimum total earliness–tardiness penalty. The problem is known to be NP-hard. We propose a branch and bound algorithm for finding an optimal schedule of the problem. The algorithm is based on the column generation approach in which the problem is first formulated as a set partitioning type formulation and then in each branch and bound iteration the linear relaxation of this formulation is solved by the standard column generation procedure. Our computational experiments show that this algorithm is capable of solving problems with up to 40 jobs and any number of machines within a reasonable computational time.     

工件延误和可拒绝下的单机重新排序问题的近似方案

研究工件延误产生干扰且延误工件可拒绝下的单机重新排序问题。在该问题中,给定计划在零时刻到达的一个工件集需在一台机器上加工,工件集中的每个工件有它的加工时间和权重,在工件正式开始加工前,按照最短赋权加工时间优先的初始排序已经给定,目标函数是极小化赋权完工时间和,据此每个工件的承诺交付截止时间也给定。然而,在工件正式开始加工时,工件集中的部分工件由于延误不能按时到达,这对初始排序的执行产生了干扰,所以需要对初始排序进行调整,即重新排序。为了保证服务水平,允许对延误工件拒绝加工,但需支付相应的拒绝费用。调整后的重新排序的目标是在保证接受工件集中工件的最大延误不超过给定的上界的约束下,使得接受工件集的赋权完工时间和,拒绝工件集的拒绝费用和以及接受工件集中工件的最大延误的赋权惩罚费用之和达到极小。对该问题,设计了一个伪多项式时间动态规划精确算法,并利用稀疏技术得到了一个完全多项式时间近似方案。     

A fast ant-colony algorithm for single-machine scheduling to minimize the sum of weighted tardiness of jobs

The problem of scheduling on a single machine is considered in this paper with the objective of minimizing the sum of weighted tardiness of jobs. A new ant-colony optimization (ACO) algorithm, called fast ACO (FACO), is proposed and analysed for solving the single-machine scheduling problem. By considering the benchmark problems available in the literature for analysing the performance of algorithms for scheduling on a single machine with the consideration of weighted tardiness of jobs, we validate the appropriateness of the proposed local-search schemes and parameter settings used in the FACO. We also present a comparison of the requirements of CPU time for solving the single-machine total-weighted tardiness problem by the FACO and the existing algorithms.     

Maximizing set function formulation of two scheduling problems

In this note we consider two problems: (1) Schedulingn jobs non-preemptively on a single machine to minimize total weighted earliness and tardiness (WET). (2) Schedulingn jobs nonpreemptively on two parallel identical processors to minimize weighted mean flow time (WMFT). A new approach for these problems is presented. The approach is based on a problem of maximizing a submodular set function. Heuristic algorithm for the problems also is presented.     

Complexity results for scheduling chains on a single machine

We investigate the computational complexity of deterministic sequencing problems in which unit-time jobs have to be scheduled on a single machine subject to chain-like precedence constraints. NP-hardness is established for the cases in which the number of late jobs or the total weighted tardiness is to be minimized, and for several related problems involving the total weighted completion time criterion.     

An Exact Procedure for the Resource-Constrained Weighted Earliness–Tardiness Project Scheduling Problem

In this paper we study the resource-constrained project scheduling problem with weighted earliness–tardinesss penalty costs. Project activities are assumed to have a known deterministic due date, a unit earliness as well as a unit tardiness penalty cost and constant renewable resource requirements. The objective is to schedule the activities in order to minimize the total weighted earliness–tardinesss penalty cost of the project subject to the finish–start precedence constraints and the constant renewable resource availability constraints. With these features the problem becomes highly attractive in just-in-time environments.We introduce a depth-first branch-and-bound algorithm which makes use of extra precedence relations to resolve resource conflicts and relies on a fast recursive search algorithm for the unconstrained weighted earliness–tardinesss problem to compute lower bounds. The procedure has been coded in Visual C++, version 4.0 under Windows NT. Both the recursive search algorithm and the branch-and-bound procedure have been validated on a randomly generated problem set.