Showing papers on "Single-machine scheduling published in 1998"

Common due window size and location determination in a single machine scheduling problem

Abstract: We consider a single machine static and deterministic scheduling problem in which jobs have a common due window. Jobs completed within the window incur no penalties, other jobs incur either earliness or tardiness penalties. The objective is to find the optimal size and location of the window as well as an optimal sequence to minimise a cost function based on earliness, tardiness, window size, and window location. We propose an O(n log n) algorithm to solve the problem.

An improvement of the Lagrangean relaxation approach for job shop scheduling: a dynamic programming method

Abstract: Concerns the use of Lagrangean relaxation for complex scheduling problems. The technique has been used to obtain near-optimal solutions for single machine and parallel machine problems. It consists of relaxing capacity constraints using Lagrange multipliers. The relaxed problem can be decomposed into independent job level subproblems. Luh et al. (1990, 1991) extended the technique to general job shop scheduling by introducing additional Lagrangean multipliers to relax precedence constraints, so that each job level relaxed subproblem can be further decomposed into a set of operation level subproblems which can easily be solved by enumeration. Unfortunately, the operation level subproblems exhibit solution oscillation from iteration to iteration and, in many cases, prevent convergence. Although several methods to prevent oscillation have been proposed, none is satisfactory. We propose an efficient pseudo-polynomial time dynamic programming algorithm. We show that, by extending the technique to job shop scheduling problems, the relaxation of the precedence constraints becomes unnecessary, and thus the oscillation problem vanishes. This algorithm significantly improves the efficiency of the Lagrangean relaxation approach to job-shop scheduling, and makes it possible to optimize "min-max" criteria by Lagrangean relaxation. These criteria have been neglected in the Lagrangean relaxation literature due to their indecomposability. Computational results are given to demonstrate the improvements due to this algorithm.

Bicriterion Single Machine Scheduling with Resource Dependent Processing Times

Abstract: A bicriterion problem of scheduling jobs on a single machine is studied. The processing time of each job is a linear decreasing function of the amount of a common discrete resource allocated to the job. A solution is specified by a sequence of the jobs and a resource allocation. The quality of a solution is measured by two criteria, F1 and F2. The first criterion is the maximal or total (weighted) resource consumption, and the second criterion is a regular scheduling criterion depending on the job completion times. Both criteria have to be minimized. General schemes for the construction of the Pareto set and the Pareto set $\epsilon$-approximation are presented. Computational complexities of problems to minimize F1 subject to F_2\le K$ and to minimize F2 subject to $F_1\le K$, where K is any number, are studied for various functions F1 and F2. Algorithms for solving these problems and for the construction of the Pareto set and the Pareto set $\epsilon$-approximation for the corresponding bicriterion problems are presented.

Discrete Lotsizing and Scheduling by Batch Sequencing

Abstract: The discrete lotsizing and scheduling problem for one machine with sequence-dependent setup times and setup costs is solved as a single machine scheduling problem, which we term the batch sequencing problem. The relationship between the lotsizing problem and the batch sequencing problem is analyzed. The batch sequencing problem is solved with a branch & bound algorithm which is accelerated by bounding and dominance rules. The algorithm is compared with recently published procedures for solving variants of the DLSP and is found to be more efficient if the number of items is not large.

Single-Machine Scheduling of Unit-Time Jobs with Earliness and Tardiness Penalties

Abstract: The problem of determining a schedule of jobs with unit-time lengths on a single machine that minimizes the total weighted earliness and tardiness penalties with respect to arbitrary rational due-dates is formulated as an integer programming problem. We show that if the penalties meet a certain criterion, called the Dominance Condition, then there exists an extremal optimal solution to the LP-relaxation that is integral, leading to a polynomial-time solution procedure. The general weighted symmetric penalty structure is one cost structure that satisfies the Dominance Condition; we point out other commonly found penalty structures that also fall into this category.

Sequencing jobs that require common resources on a single machine: A solvable case of the TSP

Abstract: In this paper a one-machine scheduling model is analyzed wheren different jobs are classified intoK groups depending on which additional resource they require. The change-over time from one job to another consists of the removal time or of the set-up time of the two jobs. It is sequence-dependent in the sense that the change-over time is determined by whether or not the two jobs belong to the same group. The objective is to minimize the makespan. This problem can be modeled as an asymmetric Traveling Salesman Problem (TSP) with a specially structured distance matrix. For this problem we give a polynomial time solution algorithm that runs in O(n logn) time. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Local search algorithms for a single-machine scheduling problem with positive and negative time-lags

Abstract: Positive and negative time-lags are general timing restrictions between the starting times of jobs which have been introduced in connection with the Metra potential method. Although very powerful, these relations have been considered only seldom in the literature since already for a single machine problem with positive and negative time-lags the problem of finding a feasible solution is ${\cal NP}$-complete. In this paper a local search approach for a single-machine scheduling problem with positive and negative time-lags and the objective to minimize the makespan is presented. Since the existence of a feasible initial solution for starting the search cannot be guaranteed, unfeasible solutions are incorporated into the search process. Computational results based on instances resulting from shop problems are reported.

The complexity of single machine scheduling with two distinct deadlines and identical decreasing rates of processing times

Abstract: The makespan problem on a single machine for a set of tasks with two distinct deadlines and identical decreasing rates of processing times is considered in this paper. Ho et al. [1] have proposed a model of a task system in which the processing time of a task decreases with its starting time. When the decreasing rate is identical, the computational complexity of the makespan problem with two distinct deadlines is posed as an open problem. In this paper we show that the problem is NP-complete. It follows that both the corresponding flow time problem and maximum lateness problem are also NP-complete.

Approximation algorithms for scheduling problems

Abstract: This thesis describes efficient approximation algorithms for some NP-Hard deterministic machine scheduling and related problems. An approximation algorithm for an NP-Hard optimization problem is a polynomial time algorithm which, given any instance of the problem, returns a solution whose value is within some guaranteed multiplicative factor of the optimal solution value for that instance. The factor is called the approximation ratio of the algorithm. A typical problem in machine scheduling consists of a set of jobs that are to be executed on a set of available machines subject to a variety of constraints. Two common objectives are minimizing makespan (the time to complete all jobs) and minimizing average completion time. Constraints that we study include precedence constraints and release dates. Brief descriptions of the problems we study and highlights of our results follow. (1) Minimizing average completion time and its weighted generalization for single and parallel machine problems. We introduce new techniques that either improve earlier results and/or result in simple and efficient approximation algorithms. For single machine scheduling we obtain an e/$(e - 1)$ approximation with release dates only and a 2 approximation with precedence constraints only. We give an algorithm that converts an x approximate single machine schedule into a $(2x + 2)$ approximate parallel machine schedule. The conversion algorithm is simple and yields efficient algorithms for several variants. (2) Minimizing makespan on machines with different speeds when jobs have precedence constraints. We obtain an $O(\log m)$ approximation (m is the number of machines) in $O(n\sp3)$ time. (3) Scheduling pipelined operator trees. This belongs to a class of new scheduling problems that arise from query optimization in parallel databases. The novel aspect consists of modeling communication costs between operators in a task system representing a query execution plan. We obtain a PTAS for this problem. We also obtain simpler $O(n \log n)$ time algorithms that have ratios of 3.56 and 2.58 respectively. (4) Multi-dimensional generalizations of three well known problems in combinatorial optimization: multi-processor scheduling, bin packing, and the knapsack problems. We obtain several approximability and inapproximability results that include a PTAS for vector scheduling when the dimension is fixed.

Decomposition in single-machine scheduling

Abstract: This paper discusses the recent research on decomposition techniques in single-machine scheduling A variety of orderings between adjacent and nonadjacent jobs in an optimal scheduling are presented A list of decomposition rules is given that enable one to solve large size instances of six single-machine models A partition technique is also developed to determine the optimal completion times of a general earliness-tardiness model for a given arrangement of jobs

Single machine scheduling problem with fuzzy due date and processing time

Abstract: This paper considers a single machine scheduling problem with fuzzy due date and fuzzy processing time. There are n jobs J 1, J 2, …, Jn to be processed on a single machine. Associated with each job Jj , there is a membership function of due date which describes the degree of satisfaction with respect to the completion time of Jj . The due date is structured similar to a trapezoidal fuzzy set, and the processing time is described as a triangular fuzzy set. The objective is to maximize the minimum grade of satisfaction over given jobs. In this paper, polynomial‐time algorithms with O(n2 logG) are proposed to solve the scheduling problem and are illustrated by means of an example.

A scheduling problem for a finite buffer capacity pull manufacturing system

Abstract: Studies a scheduling problem for a pull manufacturing system, with finite inventory/backlog space and constant demand rate. The system is a single machine, multi part-type one, and the control objective is that of minimizing an infinite horizon undiscounted cost in the inventory and backlog level, and demand losses. No set-up times and costs are considered. The analysis is made in the framework of continuous flow approximation. The optimal policy is derived analytically and compared with the solution of an infinite buffer capacity problem.

Robustness of single machine scheduling problems to earliness and tardiness penalty errors

Abstract: The Early/Tardy Machine Scheduling Problem is often proposed as a model of the situation faced by just-in-time manufacturers, whereby the objective is to minimize the weighted sum of early and tardy penalties for each job, and each penalty is proportional to the amount of time the job is early or tardy. Relatively little attention has been given in the literature to either the determination of these penalties or the effect that errors in estimating these penalties have on the final schedule. In this paper, we examine the sensitivity of the final schedule to errors in estimating these penalties. Results indicate that for some classes of problems, very large estimation errors can be sustained without compromising solution quality, whereas other problem classes require more careful evaluation of these penalties. Fortunately, the different classes of problem situations are relatively easy to identify.

Minimizing earliness and tardiness of job completions about a fuzzy due date

Abstract: We examine the problem of scheduling n jobs on a single machine. Each job i is associated with a weight w/sub i/ and a processing time p/sub i/, and the objective of the problem is to minimize the weighted earliness and tardiness of job completions from a common due date D, where D is a fuzzy number, governed by a triangular membership function. A fuzzy distance function is introduced first to measure the deviations of job completions from the fuzzy due date. Then, a property of an optimal schedule is obtained, and a pseudo-polynomial algorithm that can find the optimal schedule under a condition is derived. Numerical results are also reported to show the effectiveness of the algorithm in general cases where the condition is not satisfied.

Single machine scheduling with controllable processing times and compression costs (part i: equal times and costs)

Abstract: Most papers in scheduling research have treated individual job processing times as fixed parameters. However, in many practical situations, a manager may control processing time by reallocating resources. In this paper, authors consider a machine scheduling problem with controllable processing times. In the first part of this paper, a special case where the processing times and compression costs are uniform among jobs is discussed. Theoretical results are derived that aid in developing an O(n2) algorithm to slove the problem optimally. In the second part of this paper, authors generalize the discussion to general case. An effective heuristic to the general problem will be presented.

On-line capacitated single machine scheduling and its heuristics

Abstract: Capacitated single machine scheduling (CSMS) problem is a variant of the classical bin-packing problems. In this paper, we mainly study its on-line scheduling and apply some heuristics for bin-packing like next-fit (NF), next-fit-decreasing (NFD), next-fit-increasing (NFI), first-fit (FF) and first-fit-decreasing (FFD), to CSMS. There are four sections in this paper. Data constraints are given in Section 1. Sections 2 and 3 respectively present the worst case ratios when NF, NFD, NFI, FF, FFD are applied. Finally conclusions are given in section 4.

Single machine scheduling with controllable processing times and compression costs:proof of theorems

Abstract: This report is virtually the appendix part of the author’s previous paper which includes the proofs for the theorems and lemmas.

Application of the genetic algorithm for global scheduling and a single machine scheduling problem with periodic maintenance and semiresumable jobs

Abstract: This thesis considers two different scheduling problems in industrial settings. The first problem consists of minimizing the total completion time of jobs scheduled on a single machine that must undergo periodic maintenance. Additionally, if a job is not processed until completion before the machine is stopped for maintenance, an additional setup is necessary before processing on the job may be resumed. In this thesis, this problem is proved to be NP-complete in the strong sense. Additionally, a special case of the problem is presented where only two production periods and one maintenance period may occur. This special case is proved to be NP-hard, and a pseudopolynomial time dynamic programming algorithm to solve the special case is presented. The second problem considered here is the job shop scheduling problem where the objective is to minimize the makespan. Local search techniques which have been applied to this problem are discussed with the emphasis being on genetic algorithms. A genetic algorithm and a scheduling model for an actual industrial job shop are developed and combined to provide a search algorithm which finds good schedules for the job shop. This algorithm is compared to the scheduling procedure currently in use by the management of the modeled shop, and results are presented.

Two Machine Flow Shop Scheduling to Minimize Functions of the Infeasabilities

Abstract: We study the following version of the two machine flow shop problem. All jobs are completed by a deadline which is less than the optimal makespan. For each job the infeasibility is defined as the length of the time interval for which the processing of this job on the first machine is not completed but its processing on the second machine has already started. A given function of the infeasibilities is to be minimized. We prove that this problem restricted to permutation solutions is equivalent to a single machine scheduling problem, where an analogous function of tardiness is to be minimized. A number of complexity results for this problem not restricted to permutation solutions is also presented.

Discovering Heuristic Strategy for Solving Scheduling Problems

Abstract: Optimally solving a scheduling problem is computationally intractable, hence varieties of techniques and methods have been explored to obtain sub-optimal solutions to such problems. In this paper we introduce a technique that effectively combines appropriate heuristics so as to find a quality solution when used with a hill climbing algorithm. We will demonstrate the proposed approach by applying it to solve some single machine scheduling problems. We compare the quality of the solutions with the ones obtained by using randomized techniques.

Stochastic single machine scheduling with a general objective function

Abstract: We consider the problem of scheduling n jobs on a single machine which is subject to stochastic breakdowns to minimize the expectation of the linear combination of three functions of job completion times: (i) the weighted sum of the squares, (ii) the square of the weighted mean and (iii) the weighted mean. Many regular and irregular objective functions are the particular cases of this general objective function. The deterministic equivalent objective function of this problem is derived when the counting process N(t) describing the number of the machine breakdowns is a generalized Poisson process. For the two cases: (a) the processing times of the jobs are equal and (b) the weights of the jobs are proportional to their processing times, several properties of the optimal sequences of the problem are developed.