A new algorithm for online uniform-machine scheduling to minimize the makespan

We consider the online scheduling problem with m−1, m?2, uniform machines each with a processing speed of 1, and one machine with a speed of s, 1?s?2, to minimize the makespan. The well-known list scheduling (LS) algorithm has a worst-case bound of [Y. Cho, S. Sahni, Bounds for list schedules on uniform processors, SIAM J. Comput. 9 (1980) 91-103]. An algorithm with a better competitive ratio was proposed in [R. Li, L. Shi, An on-line algorithm for some uniform processor scheduling, SIAM J. Comput. 27 (1998) 414-422]. It has a worst-case bound of 2.8795 for a big m and s=2. In this note we present a 2.45-competitive algorithm for m?4 and any s, 1?s?2.     

A two-stage hybrid flowshop scheduling problem with a function constraint and unrelated alternative machines

This article addresses a two-stage hybrid flowshop scheduling problem with unrelated alternative machines. The problem to be studied has m unrelated alternative machines at the first machine center followed by a second machine center with a common processing machine in the system. The objective is to minimize the makespan of the system. For the processing of any job, it is assumed that the operation can be partially substituted by other machines in the first center, depending on its machining constraints. Such scheduling problems occur in certain practical applications such as semiconductors, electronics manufacturing, airplane engine production, and petrochemical production. We demonstrate that the addressed problem is NP-hard and then provide some heuristic algorithms to solve the problem efficiently. The experimental results show that the combination of the modified Johnson's rule and the First-Fit rule provides the best solutions within all proposed heuristics.Scope and purpose     

Three insertion heuristics and a justification improvement heuristic for two-dimensional bin packing with guillotine cuts

The problem of packing two-dimensional items into two-dimensional bins is considered in which patterns of items allocated to bins must be guillotine-cuttable and item rotation might be allowed (2BP|?|G)$(2\mathrm{BP}|?|\mathrm{G})$. Three new constructive heuristics, namely, first-fit insertion heuristic, best-fit insertion heuristic, and critical-fit insertion heuristic, and a new justification improvement heuristic are proposed. All new heuristics use tree structures to represent guillotine-cuttable patterns of items and proceed by inserting one item at a time in a partial solution. Central to all heuristics are a new procedure for enumerating possible insertions and a new fitness criterion for choosing the best insertion. All new heuristics have quadratic worst-case computational complexity except for the critical-fit insertion heuristic which has a cubic worst-case computational complexity. The efficiency and effectiveness of the proposed heuristics is demonstrated by comparing their empirical performance on a standard benchmark data set against other published approaches.     

Notes on Max Flow Time Minimization with Controllable Processing Times

In a scheduling problem with controllable processing times the job processing time can be compressed through incurring an additional cost. We consider the identical parallel machines max flow time minimization problem with controllable processing times. We address the preemptive and non-preemptive version of the problem. For the preemptive case, a linear programming formulation is presented which solves the problem optimally in polynomial time. For the non-preemptive problem it is shown that the First In First Out (FIFO) heuristic has a tight worst-case performance of 3–2/m, when jobs processing times and costs are set as in some optimal preemptive schedule. Supported by Swiss National Science Foundation project 20-63733.00/1, Resource Allocation and Scheduling in Flexible Manufacturing Systems, and by the Metaheuristics Network, grant HPRN-CT-1999-00106.     

Lower Bounds on the Approximation Ratios of Leading Heuristics for the Single-Machine Total Tardiness Problem

The weakly NP-hard single-machine total tardiness scheduling problem has been extensively studied in the last decades. Various heuristics have been proposed to efficiently solve in practice a problem for which a fully polynomial time approximation scheme exists (though with complexity O(n ⁷/)). In this note, we show that all known constructive heuristics for the problem, namely AU, MDD, PSK, WI, COVERT, NBR, present arbitrarily bad approximation ratios. The same behavior is shown by the decomposition heuristics DEC/EDD, DEC/MDD, DEC/PSK, and DEC/WI.     

A new set of high-performing heuristics to minimise flowtime in permutation flowshops

This paper addresses the problem of scheduling jobs in a permutation flowshop with the objective of total completion time minimisation. Since this problem is known to be NP-hard, most research has focussed on obtaining procedures – heuristics – able to provide good, but not necessarily optimal, solutions with a reasonable computational effort. Therefore, a full set of heuristics efficiently balancing both aspects (quality of solutions and computational effort) has been developed. 12 out of these 14 efficient procedures are composite heuristics based on the LR heuristic by Liu and Reeves (2001), which is of complexity $n^{3}m$. In our paper, we propose a new heuristic of complexity $n^{2}m$ for the problem, which turns out to produce better results than LR. Furthermore, by replacing the heuristic LR by our proposal in the aforementioned composite heuristics, we obtain a new set of 17 efficient heuristics for the problem, with 15 of them incorporating our proposal. Additionally, we also discuss some issues related to the evaluation of efficient heuristics for the problem and propose an alternative indicator.     

Evolutionary algorithms for scheduling m-machine flow shop with lot streaming

This paper addresses the problem of making sequencing and scheduling decisions for n jobs–m-machines flow shops under lot sizing environment. Lot streaming (Lot sizing) is the process of creating sub lots to move the completed portion of a production sub lots to down stream machines. There is a scope for efficient algorithms for scheduling problems in m-machine flow shop with lot streaming. In recent years, much attention is given to heuristics and search techniques. Evolutionary algorithms that belong to search heuristics find more applications in recent research. Genetic algorithm (GA) and hybrid genetic algorithm (HEA) also known as hybrid evolutionary algorithm fall under evolutionary heuristics. On this concern this paper proposes two evolutionary algorithms namely, GA and HEA to evolve best sequence for makespan/total flow time criterion for m-machine flow shop involved with lot streaming and set-up time. The following two algorithms are used to evaluate the performance of the proposed GA and HEA: (i) Baker's algorithm (BA), an optimal solution procedure for two-machine flow shop problem with lot streaming and makespan objective criterion and (ii) simulated annealing algorithm (SA) for m-machine flow shop problem with lot streaming and makespan and total flow time criteria.     

Single machine batch scheduling problem with family setup times and release dates to minimize makespan

In this paper we consider the single machine batch scheduling problem with family setup times and release dates to minimize makespan. We show that this problem is strongly NP-hard, and give an time dynamic programming algorithm and an time dynamic programming algorithm for the problem, where n is the number of jobs, m is the number of families, k is the number of distinct release dates and P is the sum of the setup times of all the families and the processing times of all the jobs. We further give a heuristic with a performance ratio 2. We also give a polynomial-time approximation scheme (PTAS) for the problem.     

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.     

Resource constrained scheduling on multiple machines

This paper is devoted to Resource Constrained Scheduling. An instance for this problem is given by a set of n jobs with lengths and weights and a set of m machines with capacities. At every time each machine can run arbitrary many jobs in parallel but the total weight of these jobs must not exceed the capacity of the machine. Furthermore the m machines work in parallel and we wish to find a schedule that minimizes the makespan or the sum of completion times. Thus load balancing on a cluster of servers is a typical application for scheduling under resource constraints. For both measures the problem is -complete even for m=1. We show that the problem is approximable within factors of 2 (makespan) and 14.85 (sum of completion times) for arbitrary m.     

Scheduling unrelated parallel batch processing machines with non-identical job sizes and unequal ready times

This research analyzes the problem of scheduling a set of n jobs with arbitrary job sizes and non-zero ready times on a set of m unrelated parallel batch processing machines so as to minimize the makespan. Unrelated parallel machine is a generalization of the identical parallel processing machines and is closer to real-world production systems. Each machine can accommodate and process several jobs simultaneously as a batch as long as the machine capacity is not exceeded. The batch processing time and the batch ready time are respectively equal to the largest processing time and the largest ready time among all the jobs in the batch. Motivated by the computational complexity and the practical relevance of the problem, we present several heuristics based on first-fit and best-fit earliest job ready time rules. We also present a mixed integer programming model for the problem and a lower bound to evaluate the quality of the heuristics. The small computational effort of deterministic heuristics, which is valuable in some practical applications, is also one of the reasons that motivates this study. The results show that the heuristic proposed in this paper has a superior performance compared to the heuristics based on ideas proposed in the literature.     

No-wait scheduling in single-hop multi-channel LANs

An extended version of the multiprocessor machine scheduling problem with makespan objective, which arises from single-hop multi-channel LANs, is presented in this paper. In this scheduling model, each job consists of two operations, and each operation may be processed by anyone of a given set of machines, while in the job shop scheduling problem, such a set contains only one machine. For this NP-complete problem, we first analyze the performance ratios of two simple strategies, List Scheduling and Longest Processing Time First, for the off-line cases, which have performance ratios (5/2−1/2m) and (2+1/2(m+1)), respectively, with (m+1) being the number of the machines. We also show that the competitive ratio of the List Scheduling strategy for the on-line cases is (7/2−1/2m).     

The two-stage assembly scheduling problem to minimize total completion time with setup times

We address the two-stage assembly scheduling problem where there are m machines at the first stage and an assembly machine at the second stage. The objective is to schedule the available n jobs so that total completion time of all n jobs is minimized. Setup times are treated as separate from processing times. This problem is NP-hard, and therefore we present a dominance relation and propose three heuristics. The heuristics are evaluated based on randomly generated data. One of the proposed heuristics is known to be the best heuristic for the case of zero setup times while another heuristic is known to perform well for such problems. A new version of the latter heuristic, which utilizes the dominance relation, is proposed and shown to perform much better than the other two heuristics.     

Lot-sizing scheduling with batch setup times

This paper is concerned with scheduling independent jobs on m parallel machines in such a way that the makespan is minimized. Each job j is allowed to split arbitrarily into several parts, which can be individually processed on any machine at any time. However, a setup for uninterrupted s_j time units is required before any split part of job j can be processed on any machine. The problem is strongly NP-hard if the number m of machines is variable and weakly NP-hard otherwise. We give a polynomial-time -approximation algorithm for the former case and a fully polynomial-time approximation scheme for the latter. AMS Subject Classifications: 68M20 · 90B35 · 90C59     

On-line Scheduling for Jobs with Arbitrary Release Times

This paper considers the problem of on-line scheduling a list of independent jobs in which each job has an arbitrary release time on m parallel identical machines. A tight bound is given for List Scheduling(LS) algorithm and a better algorithm is given for m2.AMS Subject Classifications: 90B35 (90C27).This research is supported by Singapore-MIT Alliance.     

Minimizing the total completion time in permutation flow shop with machine-dependent job deterioration rates

Deteriorating jobs scheduling problems can be found in many practical situations. Due to the essential complexity of the problem, most of the research focuses on the single-machine setting. In this paper, we address a total completion time scheduling problem in the m$m$-machine permutation flow shop. First of all, we propose a dominance rule and an efficient lower bound to speed up the searching for the optimal solution. Second, several deterioration patterns, which include an increasing, a decreasing, a constant, a V-shaped, and a Λ$\Lambda$-shaped one, are investigated in order to study the impact of the deterioration rates. Finally, the performance of some well-known heuristics under various deterioration patterns is evaluated.     

The 2-stage assembly flowshop scheduling problem with total completion time: Efficient constructive heuristic and metaheuristic

In this paper, we address the 2-stage assembly scheduling problem where there are m machines in the first stage to manufacture the components of a product and one assembly station (machine) in the second stage. The objective considered is the minimisation of the total completion time. Since the NP-hard nature of this problem is well-established, most previous research has focused on finding approximate solutions in reasonable computation time. In our paper, we first review and derive a number of problem properties and, based on these ideas, we develop a constructive heuristic that outperforms the existing constructive heuristics for the problem, providing solutions almost in real-time. Finally, for the cases where extremely high-quality solutions are required, a variable local search algorithm is proposed. The computational experience carried out shows that the algorithm outperforms the best existing metaheuristic for the problem. As a summary, the heuristics presented in the paper substantially modify the state-of-the-art of the approximate methods for the 2-stage assembly scheduling problem with total completion time objective.     

Balanced Scheduling toward Loss-Free Packet Queuing and Delay Fairness

In current networks, packet losses can occur if routers do not provide sufficiently large buffers. This paper studies how many buffers should be provided in a router to eliminate packet losses. We assume a network router has m incoming queues, each corresponding to a single traffic stream, and must schedule at any time on-line from which queue to take the next packet to send out. To exclude packet losses with a small amount of buffers, the maximum queue length must be kept low over the entire scheduling period. We call this new on-line problem the balanced scheduling problem (BSP). By competitive analysis, we measure the power of on-line scheduling algorithms to prevent packet losses. We show that a simple greedy algorithm is (log m)-competitive which is asymptotically optimal, while Round-Robin scheduling is not better than m-competitive, as actually is any deterministic on-line algorithm for BSP. We also give a polynomial time algorithm for solving off-line BSP optimally. We also study another on-line balancing problem that tries to balance the delay among the m traffic streams.     

Using Metric Spaces in Optimum Scheduling

The solution of an optimum problem of scheduling with n workpieces and m machine tools represents an optimum schedule of putting pieces on machines. In turn, the schedule is defined by an optimum collection of m permutations out of n objects, i.e., the vector permutation = (₁, ..., _m ), where each permutation _i (1 i m) points up the sequence of working of all pieces on the ith machine. In this case, to each admissible schedule there must correspond an integral point from the m-dimensional Euclidean space of permutations (or, which is practically the same, the permutation out of numbers {1, 2, ..., mn}. In an effort to seek an optimum schedule, use is made of the notion of a metric space in the set of admissible schedules and the justified methodology of the search for an optimum schedule. A few metric spaces are described and analyzed and their comparative effectiveness is investigated for the solution of a different-route problem of scheduling.