Scheduling on parallel identical machines to minimize total tardiness

This paper focuses on the problem of scheduling n independent jobs on m identical parallel machines for the objective of minimizing total tardiness of the jobs. We develop dominance properties and lower bounds, and develop a branch and bound algorithm using these properties and lower bounds as well as upper bounds obtained from a heuristic algorithm. Computational experiments are performed on randomly generated test problems and results show that the algorithm solves problems with moderate sizes in a reasonable amount of computation time.     

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.     

Minimizing total tardiness in an unrelated parallel-machine scheduling problem

We consider a problem of scheduling n independent jobs on m unrelated parallel machines with the objective of minimizing total tardiness. Processing times of a job on different machines may be different on unrelated parallel-machine scheduling problems. We develop several dominance properties and lower bounds for the problem, and suggest a branch and bound algorithm using them. Results of computational experiments show that the suggested algorithm gives optimal solutions for problems with up to five machines and 20 jobs in a reasonable amount of CPU time.     

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.     

On the minimization of total weighted flow time with identical and uniform parallel machines

We consider the NP-hard problem of scheduling jobs on identical parallel machines to minimize total weighted flow time. We discuss the properties that characterize the structure of an optimal solution, present a lower bound and propose a branch and bound algorithm. The algorithm is superior to prior methods presented in the literature. We also extend the algorithm to uniform parallel machines and solve medium-sized problem instances.     

Minimizing maximum tardiness and number of tardy jobs on parallel machines subject to minimum flow-time

This paper considers the problems of scheduling jobs on parallel identical machines where an optimal schedule is defined as one that gives the smallest maximum tardiness (or the minimum number of tardy jobs) among the set of schedules with optimal total flow-time (the sum of the completion times of all jobs). We show that these problems are unary NP-Hard, develop lower bounds for these two secondary criteria problems, and describe heuristic algorithms for their solution. Results of a computational study show that the proposed heuristic algorithms are quite effective and efficient in solving these hierarchical criteria scheduling problems.     

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.     

Infinite split scheduling: a new lower bound of total weighted completion time on parallel machines with job release dates and unavailability periods

This paper addresses an identical parallel machine scheduling problem with job release dates and unavailability periods to minimize total weighted completion time. This problem is known to be NP-hard in the strong sense. We propose a new lower bound that can be computed in polynomial time. The test on more than 8 400 randomly generated instances shows a very significant improvement with respect to existing results for previously studied special cases: without unavailability constraints, unweighted version, or identical job release dates. For instance, the average improvement for the unweighted problem is as much as 20.43% for 2 machines, 53.03% for 7 machines and 66.70% for 15 machines. For some instances, the improvement can be even as much as 93%.     

Two meta-heuristics for parallel machine scheduling with job splitting to minimize total tardiness

Parallel machine scheduling is a popular research area due to its wide range of potential application areas. This paper focuses on the problem of scheduling n independent jobs to be processed on m identical parallel machines with the aim of minimizing the total tardiness of the jobs considering a job splitting property. It is assumed that a job can be split into sub-jobs and these sub-jobs can be processed independently on parallel machines. We present a mathematical model for this problem. The problem of total tardiness on identical parallel machines is NP-hard. Obtaining an optimal solution for this type of complex, large-sized problem in reasonable computational time by using an optimization solver is extremely difficult. We propose two meta-heuristics: Tabu search and simulated annealing. Computational results are compared on random generated problems with different sizes.     

Due-window assignment with identical jobs on parallel uniform machines

A scheduling problem with a common due-window, earliness and tardiness costs, and identical processing time jobs is studied. We focus on the setting of both (i) job-dependent earliness/tardiness job weights and (ii) parallel uniform machines. The objective is to find the job allocation to the machines and the job schedule, such that the total weighted earliness and tardiness cost is minimized. We study both cases of a non-restrictive (i.e. sufficiently late), and a restrictive due-window. For a given number of machines, the solutions of the problems studied here are obtained in polynomial time in the number of jobs.     

Optimal on-line flow time with resource augmentation

We study the problem of scheduling n jobs that arrive over time. We consider a non-preemptive setting on a single machine. The goal is to minimize the total flow time. We use extra resource competitive analysis: an optimal off-line algorithm which schedules jobs on a single machine is compared to a more powerful on-line algorithm that has ? machines. We design an algorithm of competitive ratio , where Δ is the maximum ratio between two job sizes, and provide a lower bound which shows that the algorithm is optimal up to a constant factor for any constant ?. The algorithm works for a hard version of the problem where the sizes of the smallest and the largest jobs are not known in advance, only Δ and n are known. This gives a trade-off between the resource augmentation and the competitive ratio.We also consider scheduling on parallel identical machines. In this case the optimal off-line algorithm has m machines and the on-line algorithm has ?m machines. We give a lower bound for this case. Next, we give lower bounds for algorithms using resource augmentation on the speed. Finally, we consider scheduling with hard deadlines, and scheduling so as to minimize the total completion time.     

Coordination mechanisms for parallel machine scheduling

We consider coordination mechanisms for the distributed scheduling of n jobs on m parallel machines, where each agent holding a job selects a machine to process his/her own job. Without a central authority to construct a schedule, each agent acts selfishly to minimize his/her own disutility, which is either the completion time of the job or the congestion time (defined as the load of the machine on which the job is scheduled). However, the overall system performance is measured by a central objective which is quite different from the agents’ objective. In the literature, makespan is often considered as the central objective. We, however, investigate problems with other central objectives that minimize the total congestion time, the total completion time, the maximum tardiness, the total tardiness, and the number of tardy jobs. The performance deterioration of the central objective by a lack of central coordination, referred to as the price of anarchy, is typically measured by the maximum ratio of the objective function value of a Nash equilibrium schedule versus that of an optimal, coordinated schedule. In this paper we give bounds for the price of anarchy for the above objectives. For problems with due date related objectives, the price of anarchy may not be defined since the optimal value may be zero. In this case, we consider the maximum difference between the objective function value of an equilibrium schedule and the optimal value. We refer to this metric as the absolute price of anarchy and analyze its lower and upper bounds.     

A note on makespan minimization in two-stage flexible flow shops with uniform machines

We consider the two-stage flexible flow shop makespan minimization problem with uniform parallel machines. Soewandi and Elmaghraby [Soewandi, H., Elmaghraby, S., 2003. Sequencing on two-stage hybrid flowshops with uniform machines to minimize makespan. IIE Transaction 35, 467–477] developed a heuristic (S–E) and derived a machine speed-dependent worst-case ratio bound for it. We point out that this bound works well when the uniform machines have approximately equal speeds but is not indicative of the performance of the S–E heuristic when the machine speeds are in a wide range. Motivated by this observation, we propose an alternative tight machine-speed dependent worst-case bound for the S–E heuristic that works well when the machine speeds vary significantly. We then combine the two speed-dependent ratio bounds into a speed-independent bound. Our findings facilitate the narrowing of the gap between experimental performance and worst-case bound for the S–E heuristic.     

Online scheduling of parallel jobs on two machines is 2-competitive

We consider online scheduling of parallel jobs on parallel machines. For the problem with two machines and the objective of minimizing the makespan, we show that 2 is a tight lower bound on the competitive ratio. For the problem with m machines, we derive lower bounds using an ILP formulation.     

An exact algorithm for the precedence-constrained single-machine scheduling problem

This study proposes an efficient exact algorithm for the precedence-constrained single-machine scheduling problem to minimize total job completion cost where machine idle time is forbidden. The proposed algorithm is based on the SSDP (Successive Sublimation Dynamic Programming) method and is an extension of the authors’ previous algorithms for the problem without precedence constraints. In this method, a lower bound is computed by solving a Lagrangian relaxation of the original problem via dynamic programming and then it is improved successively by adding constraints to the relaxation until the gap between the lower and upper bounds vanishes. Numerical experiments will show that the algorithm can solve all instances with up to 50 jobs of the precedence-constrained total weighted tardiness and total weighted earliness–tardiness problems, and most instances with 100 jobs of the former problem.     

Lower bounds for the earliness–tardiness scheduling problem on parallel machines with distinct due dates

This paper addresses the parallel machine scheduling problem in which the jobs have distinct due dates with earliness and tardiness costs. New lower bounds are proposed for the problem, they can be classed into two families. First, two assignment-based lower bounds for the one-machine problem are generalized for the parallel machine case. Second, a time-indexed formulation of the problem is investigated in order to derive efficient lower bounds throught column generation or Lagrangean relaxation. A simple local search algorithm is also presented in order to derive an upper bound. Computational experiments compare these bounds for both the one machine and parallel machine problems and show that the gap between upper and lower bounds is about 1.5%.     

带机器准备时间的两台同型机复合半在线排序问题

本文研究了预知两种信息,带机器准备时间的两台同型平行机复合半在线排序问题,即已知所有工件加工时间总和和工件按加工时间非增顺序到达,目标为极小化最大机器完工时间的半在线排序模型.我们分析了它的下界,并给出了竞争比为7/6的最优算法.     

A Branch-and-Bound procedure to minimize total tardiness on one machine with arbitrary release dates

In this paper, we present a Branch-and-Bound procedure to minimize total tardiness on one machine with arbitrary release dates. We introduce new lower bounds and we generalize some well-known dominance properties. Our procedure handles instances as large as 500 jobs although some 60 jobs instances remain open. Computational results show that the proposed approach outperforms the best known procedures.     

Scheduling with earliness–tardiness penalties and parallel machines

This is a summary of the author’s PhD thesis supervised by Francis Sourd and Philippe Chrétienne and defended on 30 January 2007 at the Université Pierre et Marie Curie, Paris. The thesis is written in French and is available from the author upon request. This work is about scheduling on parallel machines in order to minimize the total sum of earliness and tardiness costs. To solve some variants of this problem we propose: an exact method based on continuous relaxations of convex reformulations derived from a 0–1 quadratic program; a heuristic algorithm that relies on a new exponential size neighborhood search; finally, a lower bound method based on a polynomial time solution of a preemptive scheduling problem for which the cost functions of the jobs have been changed into so called position costs functions. Partial funding provided by CONACyT (Mexican Council for Science&Technology).     

Due-date assignment on uniform machines

The classical weighted minsum scheduling and due-date assignment problem (with earliness, tardiness and due-date costs) was shown to be polynomially solvable on a single machine, more than two decades ago. Later, it was shown to have a polynomial time solution in the case of identical processing time jobs and parallel identical machines. We extend the latter setting to parallel uniform machines. We show that the two-machine case is solved in constant time. Furthermore, the problem remains polynomially solvable for a given (fixed) number of machines.