Flowshop scheduling problem to minimize total completion time with random and bounded processing times

The flowshop scheduling problems with n jobs processed on two or three machines, and with two jobs processed on k machines are addressed where jobs have random and bounded processing times. The probability distributions of random processing times are unknown, and only the lower and upper bounds of processing times are given before scheduling. In such cases, there may not exist a unique schedule that remains optimal for all feasible realizations of the processing times, and therefore, a set of schedules has to be considered which dominates all other schedules for the given criterion. We obtain sufficient conditions when transposition of two jobs minimizes total completion time for the cases of two and three machines. The geometrical approach is utilized for flowshop problem with two jobs and k machines.     

A note on scheduling alternative operations in two-machine flowshops

In this note, we consider a class of problems for scheduling a set of jobs each of which consists of two consecutive operations. The jobs are to be processed in a two-machine flowshop in which either or both machines are versatile. A versatile machine can perform both operations of a job. The objective is to minimise the makespan. While the whole class of problems has been shown to be NP-complete, we provide a general pseudopolynomial dynamic programming scheme which solves the problems analytically. This also establishes that the problems are only NP-complete in the ordinary sense. The solution scheme can be modified to solve another class of similar two-machine flowshop scheduling problems.     

A branch-and-price algorithm for scheduling parallel machines with sequence dependent setup times

We consider the problem of scheduling n independent jobs on m unrelated parallel machines with sequence-dependent setup times and availability dates for the machines and release dates for the jobs to minimize a regular additive cost function. In this work, we develop a new branch-and-price optimization algorithm for the solution of this general class of parallel machines scheduling problems. A new column generation accelerating method, termed “primal box”, and a specific branching variable selection rule that significantly reduces the number of explored nodes are proposed. The computational results show that the approach solves problems of large size to optimality within reasonable computational time.     

Interval Scheduling on identical machines

Interval Scheduling problems (IS) address the situation where jobs with fixed start and fixed end times are to be processed on parallel identical machines. The optimization criteria of interest are the maximization of the number of jobs completed and, in case weights are associated with jobs, the subset of jobs with maximal total weight. We present polynomial solutions to several IS problems and study computational complexity issues in the situation where bounds are imposed on the total operating time of the machines. With this constraint, we show that tractability is achieved again when job preemption is allowed.     

Approximation schemes for parallel machine scheduling with availability constraints

We investigate the problems of scheduling n weighted jobs to m parallel machines with availability constraints. We consider two different models of availability constraints: the preventive model, in which the unavailability is due to preventive machine maintenance, and the fixed job model, in which the unavailability is due to a priori assignment of some of the n jobs to certain machines at certain times. Both models have applications such as turnaround scheduling or overlay computing. In both models, the objective is to minimize the total weighted completion time. We assume that m is a constant, and that the jobs are non-resumable.For the preventive model, it has been shown that there is no approximation algorithm if all machines have unavailable intervals even if w_i=p_i for all jobs. In this paper, we assume that there is one machine that is permanently available and that the processing time of each job is equal to its weight for all jobs. We develop the first polynomial-time approximation scheme (PTAS) when there is a constant number of unavailable intervals. One main feature of our algorithm is that the classification of large and small jobs is with respect to each individual interval, and thus not fixed. This classification allows us (1) to enumerate the assignments of large jobs efficiently; and (2) to move small jobs around without increasing the objective value too much, and thus derive our PTAS. Next, we show that there is no fully polynomial-time approximation scheme (FPTAS) in this case unless P=NP.For the fixed job model, it has been shown that if job weights are arbitrary then there is no constant approximation for a single machine with 2 fixed jobs or for two machines with one fixed job on each machine, unless P=NP. In this paper, we assume that the weight of a job is the same as its processing time for all jobs. We show that the PTAS for the preventive model can be extended to solve this problem when the number of fixed jobs and the number of machines are both constants.     

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.     

On a methodology for discrete–continuous scheduling

Discrete–continuous problems of scheduling nonpreemptable jobs on parallel machines are considered. The problems arise e.g. when jobs are assigned to multiple parallel processors driven by a common electric, hydraulic or pneumatic power source. Existing models have assumed job processing rates as a function of the number of jobs currently being processed, or equivalently the number of machines currently in operation. In this paper a more general model is proposed in which processing rates of a job assigned to a machine depend on the amount of a continuous, i.e. continuously divisible resource (e.g. power) allotted to this job at a time. Thus the problem consists of two interrelated subproblems: (i) to sequence jobs on machines, and (ii) to allocate the continuous resource among jobs already sequenced. We provide a comprehensive analysis of the problem. This includes properties of optimal schedules, efficiently (in particular analytically) solvable cases, formulations of the possibly simplest mathematical programming problems for finding optimal schedules in the general case, heuristics and the worst-case analysis. Although our objective function in this paper is to minimize makespan of a set of independent jobs, the presented methodology can be applied to other criteria, precedence-related jobs, and many resource types (apart from, or instead of machines).     

Minmax scheduling with job-classes and earliness–tardiness costs

We address scheduling problems with job-dependent due-dates and general (possibly nonlinear and asymmetric) earliness and tardiness costs. The number of distinct due-dates is substantially smaller than the number of jobs, thus jobs are partitioned to classes, where all jobs of a given class share a common due-date. We consider the settings of a single machine and parallel identical machines. Our objective is of a minmax type, i.e., we seek a schedule that minimizes the maximum earliness/tardiness cost among all jobs.     

Specially Structured Precedence Constraints in Three-Dimensional Bottleneck Assignment Problems

A three-dimensional, time-minimizing (bottleneck) assignment problem consists of assigning n jobs to n workers to be performed on n machines under different forms of feasibility conditions so that the different functions of the individual times taken by a worker to finish a job on a given machine are minimized. The usual assumption made in such a problem is that all the jobs can be commenced simultaneously. In this paper, two specially structured precedence constraints on jobs are considered, which necessitate modifications in this assumption. Further, the main purpose here is to develop branch-and-bound-type algorithms for solving the corresponding problems and to illustrate them by a numerical example.     

Nonpreemptive flowshop scheduling with machine dominance

Flowshop scheduling deals with processing a set of jobs through a set of machines, where all jobs have to pass among machines in the same order. With the exception of minimizing a makespan on two machines, almost all other flowshop problems in a general setup are known to be computationally intractable. In this paper we study special cases of flowshop defined by additional machine dominance constraints. These constraints impose certain relations among the job processing times on different machines and make the studied problems tractable.     

The Systems Approach to Hospitals

By exploiting the relationship between scheduling and sorting, this paper describes a functional heuristic algorithm for seeking a quick and approximate solution to the n-job, M-machine flowshop scheduling problem under the assumption that all jobs are processed on all machines in the same order and no passing of jobs is permitted. The proposed functional heuristic algorithm can be executed by hand for reasonably large size problems and yields solutions which are closer to optimal solutions than those obtained by Palmer's slope index algorithm.     

A bicriteria parallel machine scheduling with a learning effect of setup and removal times

In studies on scheduling problems, generally setup times and removal times of jobs have been neglected or by including those into processing times. However, in some production systems, setup times and removal times are very important such that they should be considered independent from processing times. Since, in general jobs are done according to automatic machine processes in production systems processing times do not differ according to process sequence. But, since human factor becomes influential when setup times and removal times are taken into consideration, setup times will be decreasing by repeating setup processes frequently. This fact is defined with learning effect in scheduling literature. In this study, a bicriteria m-identical parallel machines scheduling problem with a learning effect of setup times and removal times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and total tardiness. A mathematical programming model is developed for the problem which belongs to NP-hard class. Results of computational tests show that the proposed model is effective in solving problems with up to 15 jobs and five machines. We also proposed three heuristic approaches for solving large jobs problems. According to the best of our knowledge, no work exists on the minimization of the weighted sum of total completion time and total tardiness with a learning effect of setup times and removal times.     

An efficient polynomial time approximation scheme for load balancing on uniformly related machines

We consider basic problems of non-preemptive scheduling on uniformly related machines. For a given schedule, defined by a partition of the jobs into m subsets corresponding to the m machines, \(C_i\) denotes the completion time of machine i. Our goal is to find a schedule that minimizes or maximizes \(\sum _{i=1}^m C_i^p\) for a fixed value of p such that \(0 . For \(p>1\) the minimization problem is equivalent to the well-known problem of minimizing the \(\ell _p\) norm of the vector of the completion times of the machines, and for \(0 , the maximization problem is of interest. Our main result is an efficient polynomial time approximation scheme (EPTAS) for each one of these problems. Our schemes use a non-standard application of the so-called shifting technique. We focus on the work (total size of jobs) assigned to each machine and introduce intervals of work that are forbidden. These intervals are defined so that the resulting effect on the goal function is sufficiently small. This allows the partition of the problem into sub-problems (with subsets of machines and jobs) whose solutions are combined into the final solution using dynamic programming. Our results are the first EPTAS’s for this natural class of load balancing problems.     

Backtracking and its Amoebic Properties in One-dimensional Machine Location Problems

This paper studies the assignment of M unique machines to M equally spaced locations along a linear material handling track with the objective of minimizing the cost of (jobs) backtracking (i.e. moving upstream). Because of the arrangement of machines and restrictions imposed by the sequence of operations for each job, some jobs may have to backtrack to complete required processing on different machines. This problem is formulated as a quadratic assignment problem. An optimal solution to a problem with large M is computationally intractable. The backtracking distance matrix in problems involving equally-spaced machine locations in one dimension is seen to possess some unique characteristics called amoebic properties. Ten amoebic properties have been identified and exploited to devise a heuristic and a lower bound on the optimal solution. Results which describe the performance of the heuristic and the lower bound are presented.     

机器具有优势关系下的工件加工时间可控的流水作业排序问题

本文主要研究机器具有优势关系下的工件加工时间可控的流水作业排序问题.我们主要对以下两种情形进行了讨论:工件加工时间为线性恶化和线性学习.对于每一种加工模型,我们分别研究了几类不同的优势机器,并且对每种情况均给出了多项式时间算法.     

A bicriteria m-machine flowshop scheduling with sequence-dependent setup times

In this study, a bicriteria m-machine flowshop scheduling with sequence-dependent setup times is considered. The objective function of the problem is minimization of the weighted sum of total completion time and makespan. Only small size problems with up to 6 machines and 18 jobs can be solved by the proposed integer programming model. Also the model is tested on an example. We also proposed three heuristic approaches for solving large jobs problems. To solve the large sizes problems up to 100 jobs and 10 machines, special heuristics methods is used. Results of computational tests show that the proposed model is effective in solving problems.     

Parallel machine scheduling with high multiplicity

In high-multiplicity scheduling problems, identical jobs are encoded in the efficient format of describing one of the jobs and the number of identical jobs. Similarly, identical machines are efficiently encoded in the same manner. We investigate parallel-machine, high-multiplicity problems, where there are three possible machine speed structures: identical, proportional, or unrelated. For the objectives of minimizing the sum of job completion times and minimizing the makespan, we consider both nonpreemptive and preemptive problems. For some problems, we develop polynomial time algorithms. For several problems, we demonstrate that the recognition versions can be solved in polynomial time, while the optimization versions require pseudo-polynomial time. We also show that changing from standard binary encoding to high-multiplicity encoding does not affect the complexity class of NP-complete problems. Received: April 1996 / Accepted: July 2000?Published online January 17, 2001     

Resolution of a scheduling problem in a flowshop robotic cell

We develop in this paper a generic and precise identification of a scheduling problem in a flexible manufacturing system. We consider a flowshop robotic cell that processes several jobs. We assume that there is no intermediate buffer between machines. So, jobs may be blocked when downstream machines are busy. We present an integer programming model to determine the sequence of jobs that minimizes the makespan criterion. In order to solve large size problems, we propose a genetic algorithm (GA). Finally, computational experiments are proposed in order to compare the makespan returned by the GA to a lower bound.     

带有拒绝的单机和同型机排序问题

研究了带有拒绝的单机和同型机排序问题. 对于单机情形, 工件的惩罚费用是对应加工时间的\alpha倍.如果工件有到达时间, 目标为最小化时间表长与惩罚费用之和, 证明了这个问题是可解的.如果所有工件在零时刻到达, 目标为最小化总完工时间与惩罚费用之和, 也证明了该问题是可解的.对于同型机排序问题, 研究了工件分两批在线实时到达的情形, 目标为最小化时间表长与惩罚费用之和.针对机器台数2和m, 分别给出了竞争比为2和4-2/m的在线算法.     

Time-Cost Trade-Off Analysis in Some Constrained Assignment Problems

Several variations of two-dimensional (workers x jobs) and three-dimensional (workers x jobs x machines) time- as well as cost-minimizing assignment problems, which arise owing to (i) precedence relations of some form among the jobs or (ii) capacity restrictions on workers/machines imposed by the requirement that the surplus resources have to be fully employed, have been considered in the literature. In this paper, an algorithm is presented for time-cost trade-off analysis which is applicable to any general pair of such constrained problems. The algorithm is also illustrated by a numerical example.