Completion time variance minimisation on two identical parallel processors

The problem of scheduling jobs to minimise completion time variance (CTV) is a well-known problem in scheduling research. CTV is categorized as a non-regular performance measure and its value may decrease by increasing the job completion times. This objective is relevant in situations where providing uniform service to customers is important, and is in-line with just-in-time philosophy. The problem concerned in this paper is to schedule n jobs on two identical parallel machines to minimise CTV. We consider the unrestricted version of the problem. The problem is said to be restricted when a machine is not allowed to remain idle when jobs are available for processing. It may be necessary to delay the start of job processing on a machine in order to reduce the completion time deviations. This gives rise to the unrestricted version of the problem. We discuss several properties of an optimal schedule to the problem. In this paper, we develop a lower bound on CTV for a known partial schedule and propose a branch and bound algorithm to solve the problem. Optimal solutions are obtained and results are reported.     

A branch and bound algorithm for minimizing total completion time on a single batch machine with incompatible job families and dynamic arrivals

In this paper, we consider a single batch machine scheduling problem with incompatible job families and dynamic job arrivals. The objective is to minimize the total completion time. This problem is known to be strongly NP-hard. We present several dominance properties and two types of lower bounds, which are incorporated to construct a basic branch and bound algorithm. Furthermore, according to the characteristics of dynamic job arrivals, a decomposed branch and bound algorithm is proposed to improve the efficiency. The proposed algorithms are tested on a large set of randomly generated problem instances.     

针对经典排序问题的一种新算法的近似比分析

给定m台平行机(同型机),n个工件,寻找一种分配方案,使得把这n个工件分配到m台机器后,整体完工时间尽可能短,这个NP-难问题被称为经典排序问题。如果每个工件的加工时间满足一定的条件,则有望能在多项式时间内有效地得到最优的分配方案。Yue等对加工时间满足整除性质的经典排序问题考虑了一种新的算法,该算法总是能得到这种特殊情况的最优分配。该算法在多项式时间内能够得到最优分配,是对于一般的经典排序问题的近似算法。文章在此基础上,考虑该新算法在一般问题上的近似比。文中考虑了这个新算法的两种版本,分别得到了3/2和2-1/2 ^q(q∈Z⁺)的近似比。紧例子表明,文中对算法的两个版本的分析都是最优的。     

A non-delayed relax-and-cut algorithm for scheduling problems with parallel machines,due dates and sequence-dependent setup times

Consider the problem of scheduling a set of jobs to be processed exactly once, on any machine of a set of unrelated parallel machines, without preemption. Each job has a due date, weight, and, for each machine, an associated processing time and sequence-dependent setup time. The objective function considered is to minimize the total weighted tardiness of the jobs.This work proposes a non-delayed relax-and-cut algorithm, based on a Lagrangean relaxation of a time indexed formulation of the problem. A Lagrangean heuristic is also developed to obtain approximate solutions.Using the proposed methods, it is possible to obtain optimal solutions within reasonable time for some instances with up to 180 jobs and six machines. For the solutions for which it is not possible to prove optimality, interesting gaps are obtained.     

Minimizing makespan subject to minimum flowtime on two identical parallel machines

We consider the problem of scheduling jobs on two parallel identical machines where an optimal schedule is defined as one that gives the smallest makespan (the completion time of the last job) among the set of schedules with optimal total flowtime (the sum of the completion times of all jobs). We propose an algorithm to determine optimal schedules for the problem, and describe a modified multifit algorithm to find an approximate solution to the problem in polynomial computational time. Results of a computational study to compare the performance of the proposed algorithms with a known heuristic shows that the proposed heuristic and optimization algorithms are quite effective and efficient in solving the problem.Scope and purposeMultiple objective optimization problems are quite common in practice. However, while solving scheduling problems, optimization algorithms often consider only a single objective function. Consideration of multiple objectives makes even the simplest multi-machine scheduling problems NP-hard. Therefore, enumerative optimization techniques and heuristic solution procedures are required to solve multi-objective scheduling problems. This paper illustrates the development of an optimization algorithm and polynomially bounded heuristic solution procedures for the scheduling jobs on two identical parallel machines to hierarchically minimize the makespan subject to the optimality of the total flowtime.     

An Enhanced tabu search algorithm to minimize a bi-criteria objective in batching and scheduling problems on unrelated-parallel machines with desired lower bounds on batch sizes

This paper addresses a sequence- and machine-dependent batch scheduling problem on a set of unrelated-parallel machines where the objective is to minimize a linear combination of total weighted completion time and total weighted tardiness. In particular, batch scheduling disregards the group technology assumptions by allowing for the possibility of splitting pre-determined groups of jobs into batches with respect to desired lower bounds on batch sizes. With regard to bounds on batch sizes, the MILP model is developed as two integrated batching and scheduling phases to present the problem. A benchmark of small-size instances on group scheduling shows the superior performance of batch scheduling up to 37% reduction in the objective function value. An efficient meta-heuristic algorithm based on tabu search with multi-level diversification and multi-tabu structure is developed at three levels, which moves back and forth between batching and scheduling phases. To eliminate searching in ineffective neighborhoods and thus enhance computational efficiency of search algorithms, several lemmas are proposed and proven. The results of applying lemmas reflect up to 40% reduction in computational times. Comparing the optimal solutions found by CPLEX and tabu search shows that the tabu search algorithm could find solutions, at least as good as CPLEX but in incredibly shorter computational time. In order to trigger the search algorithm, two different initial solution finding mechanisms have been developed and implemented. Also, due to lack of a qualified benchmark for unrelated-parallel machines, a comprehensive data generation mechanism has been developed in a way that it fairly reflects the real world situations encountered in practice. The machine availability times and job release times are considered to be dynamic and the run time of each job differs on different machines based upon the capability of the machines.     

Scheduling of a single crane in batch annealing process

This paper studies a single crane scheduling problem motivated by batch annealing process in the iron and steel industry. Each coil stack placed on fixed base needs to go through two-stage processing: heating and cooling. During each stage, limited special machines (furnace and cooler) must be operated by crane, respectively. Our problem is to assign the shared machines and schedule a single crane for minimizing the last coil stack completion time (makespan). A mixed integer linear programming (MILP) model is formulated by considering both machine and crane positions. We show that the problem is NP-hard in the strong sense. Some optimal properties on the problem are derived. A two-phase algorithm is constructed for the problem. In the first phase, a fully polynomial time approximation scheme (FPTAS) is developed for the assignment subproblem. In the second phase, a heuristics is proposed for the scheduling subproblem. From an absolute performance point of view, we analyze the quality of the two-phase algorithm. We also consider special cases where some properties or algorithms are developed. In order to further verify the performance of the two-phase algorithm, we develop a lower bound on the optimal objective function. Computational experiments on the randomly generated problem instances show that the algorithm is close to the lower bound within a reasonable computation time.     

A dominant class of schedules for malleable jobs in the problem to minimize the total weighted completion time

This paper is about scheduling parallel jobs, i.e. which can be executed on more than one machine at the same time. Malleable jobs is a special class of parallel jobs. The number of machines a malleable job is executed on may change during its execution.In this work, we consider the NP-hard problem of scheduling malleable jobs to minimize the total weighted completion time (or mean weighted flow time). For this problem, we introduce the class of “ascending” schedules in which, for each job, the number of machines assigned to it cannot decrease over time while this job is being processed.We prove that, under a natural assumption on the processing time functions of jobs, the set of ascending schedules is dominant for the problem. This result can be used to reduce the search space while looking for an optimal solution.     

Preemptive scheduling on uniformly related machines: minimizing the sum of the largest pair of job completion times

We revisit the classic problem of preemptive scheduling on m uniformly related machines. In this problem, jobs can be arbitrarily split into parts, under the constraint that every job is processed completely, and that the parts of a job are not assigned to run in parallel on different machines. We study a new objective which is motivated by fairness, where the goal is to minimize the sum of the two maximal job completion times. We design a polynomial time algorithm for computing an optimal solution. The algorithm can act on any set of machine speeds and any set of input jobs. The algorithm has several cases, many of which are very different from algorithms for makespan minimization (algorithms that minimize the maximum completion time of any job), and from algorithms that minimize the total completion time of all jobs.     

Approximation for scheduling on uniform nonsimultaneous parallel machines

We consider the problem of scheduling on uniform machines which may not start processing at the same time with the purpose of minimizing the maximum completion time. We propose using a variant of the MULTIFIT algorithm, LMULTIFIT, which generates schedules which end within 1.382 times the optimal maximum completion time for the general problem, and within \(\sqrt{6}/2\) times the optimal maximum completion time for problem instances with two machines. Both developments represent improvements over previous results. We prove that LMULTIFIT worst-case bounds for scheduling on simultaneous uniform machines are also LMULTIFIT worst-case approximation bounds for scheduling on nonsimultaneous uniform machines and show that worst-case approximation bounds of MULTIFIT variants for simultaneous uniform machines from previous literature also apply to LMULTIFIT. We also comment on how a PTAS for scheduling on a constant number of uniform machines with fixed jobs can be used to obtain a PTAS for scheduling on a constant number of uniform nonsimultaneous parallel machines.     

Optimal algorithms for the batch scheduling problem in OBS networks

This paper introduces optimal batch scheduling algorithms for the problem of scheduling batches of bursts in optical burst switching networks. The problem is modelled as a job scheduling problem with identical machines. The consideration of previously scheduled bursts in the scheduling allows such modeling. Two optimal algorithms with polynomial time complexity are derived and evaluated. Results show that the algorithm that allows re-scheduling of previously scheduled bursts leads to preferred solutions.Moreover, an extended version of the JET reservation protocol is proposed for efficient handling of batches of bursts. Results obtained via simulation prove the superior performance of the BATCHOPT algorithm.     

Minimizing Makespan and Preemption Costs on a System of Uniform Machines

It is well known that for preemptive scheduling on uniform machines there exist polynomial time exact algorithms, whereas for non-preemptive scheduling there are probably no such algorithms. However, it is not clear how many preemptions (in total, or per job) suffice in order to guarantee an optimal polynomial time algorithm. In this paper we investigate exactly this hardness gap, formalized as two variants of the classic preemptive scheduling problem. In generalized multiprocessor scheduling (GMS) we have a job-wise or total bound on the number of preemptions throughout a feasible schedule. We need to find a schedule that satisfies the preemption constraints, such that the maximum job completion time is minimized. In minimum preemptions scheduling (MPS) the only feasible schedules are preemptive schedules with the smallest possible makespan. The goal is to find a feasible schedule that minimizes the overall number of preemptions. Both problems are NP-hard, even for two machines and zero preemptions. For GMS, we develop polynomial time approximation schemes, distinguishing between the cases where the number of machines is fixed, or given as part of the input. Our scheme for a fixed number of machines has linear running time, and can be applied also for instances where jobs have release dates, and for instances with arbitrary preemption costs. For MPS, we derive matching lower and upper bounds on the number of preemptions required by any optimal schedule. Our results for MPS hold for any instance in which a job, J_j, can be processed simultaneously by ρ_j machines, for some ρ_j ≥ 1.     

求解工件车间调度问题的一种新的邻域搜索算法

该文提出了一种新的求解工件车间调度(job shop scheduling)问题的邻域搜索算法．问题的目标是：在满足约束条件的前提下使得调度的makespan尽可能地小．定义了一种新的优先分配规则以生成初始解;定义了一种新的邻域结构;将邻域搜索跟单机调度结合在一起;提出了跳坑策略以跳出局部最优解并且将搜索引向有希望的方向．计算了当前国际文献中的一组共58个benchmark问题实例,算法的优度高于当前国外学者提出的两种著名的先进算法．其中对18个10工件10机器的实例,包括最著名的难解实例ft10,在可接受的时间内都找到了最优解．这些实例是当前文献中报导的所有规模为10工件10机器的实例．     

Integrating employee timetabling with scheduling of machines and transporters in a job-shop environment: A mathematical formulation and an Anarchic Society Optimization algorithm

This paper addresses a ternary-integration scheduling problem that incorporates employee timetabling into the scheduling of machines and transporters in a job-shop environment with a finite number of heterogeneous transporters where the objective is to minimize the completion time of all jobs. The problem is first formulated as a mixed-integer linear programming model. Then, an Anarchic Society Optimization (ASO) algorithm is developed to solve large-sized instances of the problem. The formulation is used to solve small-sized instances and to evaluate the quality of the solutions obtained for instances with larger sizes. A comprehensive numerical study is carried out to assess the performance of the proposed ASO algorithm. The algorithm is compared with three alternative metaheuristic algorithms. It is also compared with several algorithms developed in the literature for the integrated scheduling of machines and transporters. Moreover, the algorithms are tested on a set of adapted benchmark instances for an integrated problem of machine scheduling and employee timetabling. The numerical analysis suggests that the ASO algorithm is both effective and efficient in solving large-sized instances of the proposed integrated job-shop scheduling problem.     

Minimizing total completion time in a two-stage hybrid flow shop with dedicated machines at the first stage

This paper considers a two-stage hybrid flow shop scheduling with dedicated machines at stage 1 with the objective of minimizing the total completion time. There exist two machines at stage 1 and one machine at stage 2. Each job must be processed on one of the two dedicated machines at stage 1 depending on the job type; subsequently, the job is processed on the single machine at stage 2.First, we introduce the problem and establish the complexity of the problem. For a special case in which the processing times on the machine at stage 2 are identical, an optimal solution is presented; for three special cases, we show that the decision version is unary NP-complete. For the general case, two simple and intuitive heuristics are introduced, and a worst case bound on the relative error is found for each of the heuristics. Finally, we empirically evaluate the heuristics, including an optimal algorithm for a special case.     

Bi-criteria minimization for the permutation flowshop scheduling problem with machine-based learning effects

In traditional scheduling problems, the processing time for the given job is assumed to be a constant regardless of whether the job is scheduled earlier or later. However, the phenomenon named “learning effect” has extensively been studied recently, in which job processing times decline as workers gain more experience. This paper discusses a bi-criteria scheduling problem in an m-machine permutation flowshop environment with varied learning effects on different machines. The objective of this paper is to minimize the weighted sum of the total completion time and the makespan. A dominance criterion and a lower bound are proposed to accelerate the branch-and-bound algorithm for deriving the optimal solution. In addition, the near-optimal solutions are derived by adapting two well-known heuristic algorithms. The computational experiments reveal that the proposed branch-and-bound algorithm can effectively deal with problems with up to 16 jobs, and the proposed heuristic algorithms can yield accurate near-optimal solutions.     

An effective hybrid discrete differential evolution algorithm for the flow shop scheduling with intermediate buffers

In this paper, an effective hybrid discrete differential evolution (HDDE) algorithm is proposed to minimize the maximum completion time (makespan) for a flow shop scheduling problem with intermediate buffers located between two consecutive machines. Different from traditional differential evolution algorithms, the proposed HDDE algorithm adopted job permutation to represent individuals and applies job-permutation-based mutation and crossover operations to generate new candidate solutions. Moreover, a one-to-one selection scheme with probabilistic jumping is used to determine whether the candidates will become members of the target population in next generation. In addition, an efficient local search algorithm based on both insert and swap neighborhood structures is presented and embedded in the HDDE algorithm to enhance the algorithm’s local searching ability. Computational simulations and comparisons based on the well-known benchmark instances are provided. It shows that the proposed HDDE algorithm is not only capable to generate better results than the existing hybrid genetic algorithm and hybrid particle swarm optimization algorithm, but outperforms two recently proposed discrete differential evolution (DDE) algorithms as well. Especially, the HDDE algorithm is able to achieve excellent results for large-scale problems with up to 500 jobs and 20 machines.     

The Machine Duplication Problem in a Job Shop with Two Jobs

This paper addresses a problem related to the classical job shop scheduling problem with two jobs. The problem consists in concurrently determining the best subset of machines to be duplicated and the optimal scheduling of the operations in order to minimize completion time. Such a problem arises in the tool management for a class of flexible manufacturing cells. The job shop with two jobs is first reviewed, the application of the classical search algorithm A* to this problem is discussed and its performance compared with a previous approach. The complexity of the machine duplication problem is then analysed. The problem is proved to be in general NP-hard in the strong sense, but in a class of special cases, relevant from the applications viewpoint, it can be solved in polynomial time by a dynamic programming algorithm. A heuristic based on such an algorithm and on A* is proposed for the general problem; the results are satisfactory in terms of both efficiency and quality of the solution.     

A metaheuristic for the fixed job scheduling problem under spread time constraints

In a scheduling problem, a job is said to be fixed when its due date corresponds exactly to its release date plus its processing time. This paper addresses the fixed job scheduling problem where processors are subject to spread time constraints, i.e., the amount of time spent between the starting time of the first job on a processor and the completion time of the last job on the same processor should not exceed a given duration. Existing exact approaches are tested on medium size instances. As large instances are clearly intractable with these approaches, a greedy heuristic and a grouping genetic algorithm are proposed. Computational results show the effectiveness of the proposed heuristics.     

A novel MILP model for job shop scheduling problem with mobile robots

The mobile robot is the essential equipment for automated logistics in the intelligent workshop, but the literature on shop scheduling rarely considers transport resources. This paper studies the integrated scheduling of machines and mobile robots, which can facilitate the efficiency of production systems. For the job shop scheduling problem with mobile robots (JSPMR), the existing mathematical models are too complex to obtain the optimal solution in an efficient time. Therefore, a novel mixed integer linear programming (MILP) model is proposed to minimize the makespan. Firstly, in view of the property of the problem, a disjunctive graph model is modified to describe the relationship between transport and processing tasks. Secondly, a more accurate and simplified MILP is proposed based on the modified disjunctive graph model. Two related proofs are given to prove the proposed model satisfies all special situations. Thirdly, the proposed MILP is tested on the well-known benchmark, including 82 instances. The proposed model is the first MILP model to obtain optimal solutions for all instances. Finally, 40 larger-scale instances are presented based on a real-world engineering case and used to validate the performance of models further. The comparison results verify the effectiveness and superior computational performance of the proposed model.