An improved lower bound for a bi-criteria scheduling problem

For the bi-criteria scheduling problem of minimizing the sum of completion times and the sum of weighted completion times, min-sum of weighted completion times, we prove that there exists no constant β>1 such that (1+1/γ,β)-approximate schedules can be found for any γ>0. This result confirms a recently published conjecture.     

A memetic algorithm for minimizing the total weighted completion time on a single machine under linear deterioration

In this paper, we consider the problem of minimizing the total weighted completion time on a single machine. Jobs processing times are increasing linear function of start times. First, we present some new dominance properties for this NP-hard problem. And next, using these properties, we develop a memetic algorithm for the problem. The results of computational experiments show the good performance of the proposed algorithm.     

Reducibility among single machine weighted completion time scheduling problems

Various sum of weighted completion time problems are compared. The constraints considered include release date, deadline, and continuous machine processing. Relations between the problems are developed by examining the computational complexity of transforming one problem class into another. These results give indications of the relative computational effort required to solve different problem classes.     

Pareto optima for total weighted completion time and maximum lateness on a single machine

We consider the single-machine bicriterion scheduling problem of enumerating the Pareto-optimal sequences with respect to the total weighted completion time and the maximum lateness objectives. We show that the master sequence concept originally introduced for 1|r_j|∑w_jU_j by Dauzère-Pérès and Sevaux is also applicable to our problem and a large number of other sequencing problems. Our unified development is based on exploiting common order-theoretic structures present in all these problems. We also show that the master sequence implies the existence of global dominance orders for these scheduling problems. These dominance results were incorporated into a new branch and bound algorithm, which was able to enumerate all the Pareto optima for over 90% of the 1440 randomly generated problems with up to n=50 jobs. The identification of each Pareto optimum implicitly requires the optimal solution of a strongly NP-hard problem. The instances solved had hundreds of these Pareto solutions and to the best of our knowledge, this is the first algorithm capable of completely enumerating all Pareto sequences within reasonable time and space for a scheduling problem with such a large number of Pareto optima.     

Minimizing total weighted completion time in a two-machine flow shop scheduling under simple linear deterioration

In this paper, we address a two-machine flow shop scheduling problem under simple linear deterioration. By a simple linear deterioration function, we mean that the processing time of a job is a simple linear function of its execution start time. The objective is to find a sequence that minimizes total weighted completion time. Optimal schedules are obtained for some special cases. For the general case, several dominance properties and two lower bounds are derived to speed up the elimination process of a branch-and-bound algorithm. A heuristic algorithm is also proposed to overcome the inefficiency of the branch-and-bound algorithm. Computational analysis on randomly generated problems is conducted to evaluate the branch-and-bound algorithm and heuristic algorithm.     

NP-hardness of the single-variable-resource scheduling problem to minimize the total weighted completion time

Baker and Nuttle [K.R. Baker, H.L.W. Nuttle, Sequencing independent jobs with a single resource, Naval Research Logistics Quarterly 27 (1980) 499–510] studied the following single-variable-resource scheduling problem: sequencing n jobs for processing by a single resource to minimize a function of job completion times, when the availability of the resource varies over time. When the objective function to be minimized is the total weighted completion time, Baker and Nuttle conjectured that the problem is NP-hard. We show in this note that the conjecture is true.     

A tight lower bound for the completion time variance problem

We consider the completion time variance problem. Our main result is a tight lower bound for the mean completion time of an optimal sequence. This result can be applied to reduce the time required to solve the problem.     

Minimizing flow time variance in a single machine system using genetic algorithms

In this paper, we address an n-job, single machine scheduling problem with an objective to minimize the flow time variance. We propose heuristic procedure based on genetic algorithms with the potential to address more generalized objective function such as weighted flow time variance. The development and implementation of the algorithm is supported with literature review and statistical analysis of the results. Some general guidelines to select the parameter values of the genetic algorithm are also developed using an experimental design approach.     

A lower bound for the blow-up time to a damped semilinear wave equation

This paper studies the blow-up property of weak solutions to an initial and boundary value problem for a nonlinear viscoelastic hyperbolic equation with nonlinear sources. A lower bound for the blow-up time is given.     

Two simple constant ratio approximation algorithms for minimizing the total weighted completion time on a single machine with a fixed non-availability interval

In this note, we consider the scheduling problem of minimizing the sum of the weighted completion times on a single machine with one non-availability interval on the machine under the non-resumable scenario. Together with a recent 2-approximation algorithm designed by Kacem [I. Kacem, Approximation algorithm for the weighted flow-time minimization on a single machine with a fixed non-availability interval, Computers & Industrial Engineering 54 (2008) 401–410], this paper is the first successful attempt to develop a constant ratio approximation algorithm for this problem. We present two approaches to designing such an algorithm. Our best algorithm guarantees a worst-case performance ratio of 2+ε$2+\epsilon$.     

Two-machine flow-shop scheduling with equal processing time on the second machine for minimizing total weighted completion time

We consider the classical two-machine flow-shop scheduling for minimizing the total weighted completion time. For this problem, the computational complexity of a version in which the jobs have a common processing time on the second machine, has not been addressed. We show that the problem is unary NP-hard, answering an open problem posed in Zhu et al. (2016). Then we present an approximation algorithm for the problem with worst-case performance ratio at most 2.     

Online weighted flow time and deadline scheduling

In this paper we study some aspects of weighted flow time. We first show that the online algorithm Highest Density First is an O(1)-speed O(1)-approximation algorithm for P|r_i,pmtn|∑w_iF_i. We then consider a related Deadline Scheduling Problem that involves minimizing the weight of the jobs unfinished by some unknown deadline D on a uniprocessor. We show that any c-competitive online algorithm for weighted flow time must also be c-competitive for deadline scheduling. We then give an O(1)-competitive algorithm for deadline scheduling.     

Minimizing flow time in cyclic schedules for identical jobs with acyclic precedence: the bottleneck lower bound

This paper studies minimizing the flow time of a cyclic schedule for repeated identical jobs, where one job is started/completed in each cycle, subject to the schedule achieving maximum throughput. We propose a branch and bound method for a single machine problem, and use this method to derive an improved lower bound for the multiple machine problem.     

Approximating total weighted completion time on identical parallel machines with precedence constraints and release dates

We present a $(2+2ln2+\epsilon )$-approximation algorithm for the classical nonpreemptive scheduling problem to minimize the total weighted completion time of jobs on identical parallel machines subject to release dates and precedence constraints, improving upon the previously best known 4-approximation algorithm from 1998. The result carries over to the more general problem with precedence delays and generalizes a recent result by Li (2017) for the problem without release dates or delays.     

A branch and bound algorithm for the one-machine scheduling problem with minimum and maximum time lags

We consider the one-machine scheduling problem with minimum and maximum time lags while minimizing the makespan. This problem typically arises in a manufacturing environment where the next job has to be carried out within a specific time range after the completion of the immediately preceding job. We describe a branch and bound algorithm, based on the input and output of a clique and the relevant propositions, for finding the optimal waiting times. The computational experiments give promising results, showing whether a given instance is feasible or infeasible. With the proposed branch and bound algorithm we can either find an optimal schedule or establish the infeasibility within an acceptable run time.     

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.     

Improving the preemptive bound for the single machine dynamic maximum lateness problem

The preemptive lower bound is known to be the main available bound for the single machine dynamic maximum lateness problem. We propose an O(nlogn) time improvement to this bound which reduces by more than 80% the average the gap between the preemptive solution value and the optimal solution value.     

An improved branch and bound algorithm for single machine scheduling with deadlines to minimize total weighted completion time

We consider a scheduling problem in which n jobs with distinct deadlines are to be scheduled on a single machine. The objective is to find a feasible job sequence that minimizes the total weighted completion time. We present an efficient branch-and-bound algorithm that fully exploits the principle of optimality. Favorable numerical results are also reported on an extensive set of problem instances of 20-120 jobs.