Yixun Lin

Parallel machine scheduling of machine-dependent jobs with unit-length

Abstract In this paper we study the parallel machine scheduling problem with unit-length jobs in which each job j is only allowed to be processed on a specified subset M j of machines. For the problem P|p j =1, M j |C max where the subset family { M j } is nested, Pinedo [Scheduling: Theory, Algorithms, and Systems, Prentice-Hall, Englewood Cliffs, NJ, 1995] established an least flexible job first algorithm. We further present polynomial algorithms for the problem with general subset family { M j } . As a generalization of the problem for nested family, we consider the problem for convex subset family { M j } and present an efficient algorithm for solving it.

Weight reduction problems with certain bottleneck objectives

Abstract This paper is concerned with bottleneck weight reduction problems (WRPs) stated as follows. We are given a finite set E , a class F of nonempty subsets of E , a weight w:E→ R + and a cost c:E→ R + . For each e ∈ E , c ( e ) stands for the cost of reducing weight w ( e ) by one unit. For each subset F∈ F , the bottleneck weight of F is w ( F )=min e ∈ F w ( e ). The weight of the family F is the maximum of w ( F ) for all F in F . The problem is to determine new weights x ( e )⩽ w ( e ) such that the weight of F is minimized under the constraint that the overall reduction cost does not exceed a given budget B . Similarly to capacity expansion problems, WRPs include NP -hard problems. A WRP can be formulated as a parametric optimization problem over all transversal sets T of the class F . This leads to (strongly) polynomial solution procedures for special systems F . In particular we outline a polynomial algorithm in the case when F is the class of all spanning trees in an undirected graph.

The Obnoxious Center Problem on a Tree

The obnoxious center problem in a graph G asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. We derive algorithms with linear running time for the cases when G is a path or a star, thus improving previous results of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377--396]. For subdivided stars we present an algorithm of running time O(n log n). For general trees, we improve an algorithm of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377--396] by a factor of log n. Moreover, a linear algorithm for the unweighted center problem on an arbitrary tree with neutral and obnoxious vertices is described.

Necessary and sufficient conditions of optimality for some classical scheduling problems

A scheduling problem is generally to order the jobs such that a certain objective function f(π) is minimized. For some classical scheduling problems, only sufficient conditions of optimal solutions are concerned in the literature. In this paper, we study the necessary and sufficient conditions by means of the concept of critical ordering (critical jobs and their relations). These results are meaningful in recognition and characterization of optimal solutions of scheduling problems.

A note on the single machine scheduling to minimize the number of tardy jobs with deadlines

It is known that the single machine scheduling problem of minimizing the number of tardy jobs is polynomially solvable. However, it becomes NP-hard if each job has a deadline. Recently, Huo et al. solved some special cases by a backwards scheduling approach. In this note we present a dual approach--forwards greedy algorithms which may have better running time. For example, in the case that the due dates, deadlines, and processing times are agreeable, the running time of the backwards scheduling algorithm is O(n2), while that of the forwards algorithm is O(nlogn).

Single machine preemptive scheduling with fixed jobs to minimize tardiness related criteria

Abstract We consider the single machine preemptive scheduling problem with some fixed jobs being previously given. The fixed jobs are already fixed in the schedule. The remaining jobs are to be assigned to the remaining time-slots of machine in such a way that they do not overlap each other and do not overlap with the fixed jobs. The objective is to minimize a tardiness related criterion. If the jobs are processed without preemption, it has been implied in the literature that this problem is strongly NP-hard. Suppose now that the jobs are processed preemptively, and that some specified free jobs must be on-time under the schedule. The considered problem will be denoted by 1|FB,pmtn, R on - time|F , where F is a tardiness related criterion. We show that the considered problem is polynomially equivalent to the problem 1| FB , pmtn | F . We also show that 1| FB , pmtn |∑ w j U j is polynomially equivalent to 1∥∑ w j U j . Consequently, the complexity of the problem 1|FB,pmtn, R on - time|F is classified according to different choices of F .

An Improved Algorithm for a Bicriteria Batching Scheduling Problem

This note is concerned with the bicriteria scheduling problem on a series-batching machine to minimize maximum cost and makespan. An O (n 5 ) algorithm has been established previously. Here is an improved algorithm which solves the problem in O (n 3 ) time.

Bicriteria scheduling for due date assignment with total weighted tardiness

In the due date assignment, the bicriteria scheduling models are motivated by the trade-off between the due date assignment cost and a performance criterion of the scheduling system. The bicriteria scheduling models related to the maximum tardiness and the weighted number of tardy jobs have been studied in the literature. In this paper we consider a new model with criteria of the due date assignment cost and the total weighted tardiness. The main results are polynomial-time algorithms for the linear combination version, the constraint version, and the Pareto optimization version of bicriteria scheduling.