An efficient simple metaheuristic for minimizing the makespan in two-machine no-wait job shops

This paper examines the problem of scheduling two-machine no-wait job shops to minimize makespan. The problem is known to be strongly NP-hard. A two-phase heuristic is developed to solve the problem. Phase 1 of the heuristic transforms the problem into a no-wait flow shop problem and solves it using the well known Gilmore and Gomory algorithm. Phase 2 of the heuristic improves the solution obtained in phase 1 using a simple tabu search algorithm. Computational results show that the proposed heuristic performs extremely well in terms of both solution quality and computation time. It finds an optimal solution to about 90% of the problem instances and the average deviation from the lower bond for the other problem instances is infinitesimal.     

An improved heuristic for no-wait flow shop to minimize makespan

No-wait flow shop production has been widely applied in manufacturing, where no waiting time is allowed between intermediate operations. However, minimization of makespan for no-wait flow shop production is NP-hard. In this paper, we propose an average idle time (AIT) heuristic to minimize makespan in no-wait flow shop production. First, we take the current idle times and future idle times into consideration, proposing an initial sequence algorithm, and then use the insertion and neighborhood exchanging methods to further improve solutions. Compared with three existing best-known heuristics, our AIT heuristic can achieve the smallest deviations of 0.23% from optimum, based on Taillard’s benchmarks and 600 randomly generated instances, in the same computational complexity.     

Scheduling parallel jobs to minimize the makespan

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a prespecified, job-dependent number of machines when being processed. We prove that the makespan of any nonpreemptive list-schedule is within a factor of 2 of the optimal preemptive makespan. This gives the best-known approximation algorithms for both the preemptive and the nonpreemptive variant of the problem. We also show that no list-scheduling algorithm can achieve a better performance guarantee than 2 for the nonpreemptive problem, no matter which priority list is chosen. List-scheduling also works in the online setting where jobs arrive over time and the length of a job becomes known only when it completes; it therefore yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. We show that no list-scheduling algorithm has a constant competitive ratio. Still, we present the first online algorithm for scheduling parallel jobs with a constant competitive ratio in this context. We also prove a new information-theoretic lower bound of 2.25 for the competitive ratio of any deterministic online algorithm for this model. Moreover, we show that 6/5 is a lower bound for the competitive ratio of any deterministic online algorithm of the preemptive version of the model jobs arriving over time.     

Scheduling a capacitated batch-processing machine to minimize makespan

This paper aims at improving the utilization of a single batch-processing machine. The batch-processing machine can process a batch of jobs, as long as the number of jobs and the total size of all the jobs in a batch do not violate the machine's capacity. The processing time of the job and its size is known. The processing time of a batch is the longest processing time among all the jobs in the batch. The objective is to minimize the makespan. Since the problem under study is NP-hard, a Simulated Annealing (SA) approach is proposed. The effectiveness of our solution procedure in terms of solution quality and run time is evaluated through experiments. The results obtained from the SA approach were compared with a commercial solver called CPLEX. Our computational study demonstrates the effectiveness of our approach in solving problem instances with 20 or more jobs in a shorter run time with better solutions.     

Family scheduling with batch availability in flow shops to minimize makespan

This paper addresses a batch scheduling problem in flow shop production systems, where job families are formed based on setup similarities. In order to improve setup efficiency, we consider batching decisions in our solution procedure. Due to its high practical relevance, the batch availability assumption is also adopted in this study. In the presence of sequence-dependent setup times, it is proved that a permutation flow shop is generally not optimal. Therefore, our objective is to determine solutions with inconsistent batches, which essentially lead to non-permutation schedules, to minimize makespan. After examining structural properties, we develop a tabu search algorithm with multiple neighbourhood functions. Computational results confirm the remarkable benefits of batching decisions. Our algorithm also outperforms some well-known and well-performing approaches.     

An improved NEH heuristic to minimize makespan in permutation flow shops

For over 20 years the NEH heuristic of Nawaz, Enscore, and Ham [A heuristic algorithm for the m-machine, n-job flow-shop sequencing problem. Omega, The International Journal of Management Science 1983;11:91–5] has been commonly regarded as the best heuristic for solving the NP-hard problem of minimizing the makespan in permutation flow shops. The strength of NEH lies mainly in its priority order according to which jobs are selected to be scheduled during the insertion phase. Framinan et al. [Different initial sequences for the heuristic of Nawaz, Enscore and Ham to minimize makespan, idle time or flowtime in the static permutation flowshop problem. International Journal of Production Research 2003;41:121–48] presented the results of an extensive study to conclude that the NEH priority order is superior to 136 different orders examined. Based upon the concept of Johnson's algorithm, we propose a new priority order combined with a simple tie-breaking method that leads to a heuristic that outperforms NEH for all problem sizes.     

Flexible open shop scheduling problem to minimize makespan

This study investigates the static and dynamic versions of the flexible open shop scheduling problem with the goal of minimizing makespan. The asymptotic optimality of the general dense scheduling (GDS) algorithm is proven by the boundedness hypothesis. For large-scale problems, the GDS-based heuristic algorithms are presented to accelerate convergence. For moderate-scale problems, the differential evolution algorithm is employed to obtain high-quality solutions. A series of random experiments are conducted to demonstrate the effectiveness of the proposed algorithms.     

Heuristic for no-wait flow shops with makespan minimization based on total idle-time increments

No-wait flow shops with makespan minimization are classified as NP-hard. In this paper, the optimization objective is equivalently transformed to total idle-time minimization. The independence relationship between tasks is analyzed, and objective increment properties are established for the fundamental operators of the heuristics. The quality of the new schedules generated during a heuristic is judged only by objective increments and not by the traditional method, which computes and compares the objective of a whole schedule. Based on objective increments, the time complexity of the heuristic can be decreased by one order. A seed phase is presented to generate an initial solution according to the transformed objective. Construction and improvement phases are introduced by experimental analysis. The FCH （fast composite heuristic） is proposed and compared with the most effective algorithms currently available for the considered problem. Experimental results show that the effectiveness of the FCH is similar to that of the best methods but requires far less computation time. The FCH can also be efficient in real time scheduling and rescheduling for no-wait flow shops.     

The two-machine flowshop no-wait scheduling problem with a single server to minimize the total completion time

This work studies the scheduling problem where a set of jobs are available for processing in a no-wait and separate setup two-machine flow shop system with a single server. The no-wait constraint requires that the operations of a job have to be processed continuously without waiting between two machines. The setup time is incurred and attended by a single sever which can perform one setup at a time. The performance measure considered is the total completion time. The problem is strongly NP-hard. Optimal solutions for several restricted cases and some properties for general case are proposed. Both the heuristic and the branch and bound algorithms are established to tackle the problem. Computational experiments indicate that the heuristic and the branch and bound algorithm are superior to the existing ones in term of solution quality and number of branching nodes, respectively.     

两机无等待流水车间调度问题的性质

针对两机无等待流水车间调度问题,提出目标函数最大完工时间最小化的快速算法,并给出算法的复杂度。分析两机无等待流水车间调度问题的排列排序性质,证明了两机无等待流水车间调度问题的可行解只存在于排列排序中,排列排序的最优解一定是两机无等待流水车间调度问题的最优解。最后研究了同时包含普通工件和无等待工件的两机流水车间调度问题的复杂性,为进一步研究两机无等待流水车间调度问题提供了理论依据。     

Scheduling algorithms to minimize the number of tardy jobs in two-stage hybrid flow shops

This paper considers a two-stage hybrid flow shop scheduling problem for the objective of minimizing the number of tardy jobs. Each job is processed through the two production stages in series, each of which has multiple identical parallel machines. The problem is to determine the allocation of jobs to the parallel machines as well as the sequence of the jobs assigned to each machine. To solve the problem, a branch and bound algorithm, which incorporates the methods to obtain the lower and upper bounds as well as the dominance properties to reduce the search space, is suggested that gives the optimal solutions. In addition, two-phase heuristic algorithms are suggested to obtain good solutions for large-size problems within a reasonable amount of computation time. To show the performances of the optimal and heuristic algorithms suggested in this paper, computational experiments are done on a number of randomly generated test problems, and the test results are reported.     

极小化时间表长的无等待流水车间调度

针对以时间表长最小为目标函数的无等待流水车间（No-Wait Flow Shop,NWFS）调度问题,提出了一个混合禁忌搜索算法（Hybrid Taboo Search,HTS）,以启发式算法产生的解作为初始解,通过禁忌搜索进一步提高解的质量。大量随机产生实例的实验结果表明：提出的HTS算法在总体性能上优于经典的RAJ、VNS和GASA算法,因此该算法具有可行性和优越性。     

Scheduling linear deteriorating jobs to minimize makespan with an availability constraint on a single machine

The scheduling problem with deteriorating jobs to minimize the makespan on a single machine where the facility has an availability constraint is studied in this paper. By a deteriorating job we mean that the processing time for the job is a function of its starting time. Even with the introduction of the availability to a facility, the linear deteriorating model can be solved using the 0-1 integer programming technique if the actual job processing time is proportional to the starting time.     

Single-machine scheduling with periodic maintenance to minimize makespan

We consider a single-machine scheduling problem with periodic maintenance activities. Although the scheduling problem with maintenance has attracted researchers’ attention, most of past studies considered only one maintenance period. In this research several maintenance periods are considered where each maintenance activity is scheduled after a periodic time interval. The objective is to find a schedule that minimizes the makespan, subject to periodic maintenance and nonresumable jobs. We first prove that the worst-case ratio of the classical LPT   algorithm is 2. Then we show that there is no polynomial time approximation algorithm with a worst-case ratio less than 2 unless P=NP$P=\text{<italic>NP</italic>}$, which implies that the LPT algorithm is the best possible.     

Scheduling resumable simple linear deteriorating jobs on a single machine with an availability constraint to minimize makespan

We consider a single-machine scheduling problem in which the processing time of each job is a simple linear deteriorating function of its waiting time. The machine is subject to an availability constraint. Jobs interrupted by machine unavailability can resume their processing. The objective is to minimize the makespan. We first show that the problem can be solved optimally by 0–1 integer programming. We then prove that the problem is NP-hard in the ordinary sense and there exists a fully polynomial time approximation scheme for it.     

Single-machine scheduling with periodic maintenance to minimize makespan revisited

In this paper, a single-machine scheduling problem with periodic maintenance and non-preemptive jobs is considered. The objective is to minimize the makespan. It shows that the classical list scheduling (LS) algorithm, the longest processing time first (LPT) algorithm, and the modified longest processing time first (MLPT) algorithm all have the same worst-case ratio and the same computational complexity for the considered problem. To compare the performances of three considered algorithms in detail, the worst-case ratios of algorithms are formed as single-variable functions of the total number of maintenance activities needed in the optimal schedule. Analysis results show that the bound associated with the LS algorithm is always dominated by the bound associated with the LPT algorithm, and the latter is always dominated by the bound associated with the MLPT algorithm.     

Heuristics for two-machine no-wait flowshop scheduling with an availability constraint

In this paper we study the two-machine no-wait flowshop problem with an availability constraint. The problem has been shown to be NP-hard, and some heuristics with a worst-case error bound of 2 have been developed for it. We provide two improved heuristics for the problem, and show that each has a worst-case error bound of 5/3.     

A two-machine flowshop makespan scheduling problem with deteriorating jobs

In traditional scheduling problems, the job processing times are assumed to be known and fixed from the first job to be processed to the last job to be completed. However, in many realistic situations, a job will consume more time than it would have consumed if it had begun earlier. This phenomenon is known as deteriorating jobs. In the science literature, the deteriorating job scheduling problems are relatively unexplored in the flowshop settings. In this paper, we study a two-machine flowshop makespan scheduling problem in which job processing times vary as time passes, i.e. they are assumed as increasing functions of their starting times. First, an exact algorithm is established to solve most of the problems of up to 32 jobs in a reasonable amount of time. Then, three heuristic algorithms are provided to derive the near-optimal solutions. A simulation study is conducted to evaluate the performances of the proposed algorithms. In addition, the impact of the deterioration rate is also investigated.