Online hierarchical scheduling: An approach using mathematical programming

This paper considers an online hierarchical scheduling problem on parallel identical machines. We are given a set of m machines and a sequence of jobs. Each machine has a different hierarchy, and each job also has a hierarchy associated with it. A job can be assigned to a machine only if its hierarchy is no less than that of the machine. The objective is to minimize makespan. Two models are studied in the paper. For the fractional model, we present an improved algorithm and lower bounds. Both algorithm and lower bounds are based on solutions of mathematical programming. For any given m, our algorithm is optimal by numerical calculation. For the integral model, we present both a general algorithm for any m, and an improved algorithm with better competitive ratios of 2.333 and 2.610 for m = 4 and 5, respectively.

MILP models to minimise makespan in additive manufacturing machine scheduling problems

Scheduling identical parallel machines with tooling constraints

We study the bicriteria scheduling of equal length jobs on uniform parallel machines. By introducing a new scheduling model, called single-machine scheduling with generated completion times (shortly, GCT-scheduling), we show that the scheduling of equal length jobs on uniform parallel machines can be polynomially transformed into the single-machine GCT-scheduling with a special setting of generated completion times. In the GCT-scheduling, a sequence of completion times is given in advance and the job scheduled at the i-th position will be assigned the i-th completion time. We present a comprehensive study on the complexities of the bicriteria single-machine GCT-scheduling problems with respect to various regular criteria. We then turn these complexity results into the forms of bicriteria scheduling of equal length jobs on uniform (or identical) parallel machines. Our research generalizes the existing results on bicriteria scheduling of equal length jobs in the literature. Particularly, one of our results solves the open problem posed by Sarin and Prakash (J Comb Optim 8:227–240, 2004), which asks for minimizing the total weighted completion time subject to the optimality of minimizing the total number of tardy jobs on identical parallel machines, and we show that this problem is solvable in polynomial time.

We consider the problem of scheduling orders on identical machines in parallel. Each order consists of one or more individual jobs. A job that belongs to an order can be processed by any one of the machines. Multiple machines can process the jobs of an order concurrently. No setup is required if a machine switches over from one job to another. Each order is released at time zero and has a positive weight. Preemptions are not allowed. The completion time of an order is the time at which all jobs of that order have been completed. The objective is to minimize the total weighted completion time of the orders. The problem is NP‐hard for any fixed number (≥2) of machines. Because of this, we focus our attention on two classes of heuristics, which we refer to as sequential two‐phase heuristics and dynamic two‐phase heuristics. We perform a worst case analysis as well as an empirical analysis of nine heuristics. Our analyses enable us to rank these heuristics according to their effectiveness, taking solution quality as well as running time into account. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006

Online makespan minimization is a classical problem in which a sequence of jobs $$\sigma = J_1, \ldots , J_n$$ź=J1,ź,Jn has to be scheduled on m identical parallel machines so as to minimize the maximum completion time of any job. In this paper we investigate the problem with an essentially new model of resource augmentation. More specifically, an online algorithm is allowed to build several schedules in parallel while processing $$\sigma $$ź. At the end of the scheduling process the best schedule is selected. This model can be viewed as providing an online algorithm with extra space, which is invested to maintain multiple solutions. The setting is of particular interest in parallel processing environments where each processor can maintain a single or a small set of solutions. As a main result we develop a $$(4/3+\varepsilon )$$(4/3+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that uses a constant number of schedules. The constant is equal to $$1/\varepsilon ^{O(\log (1/\varepsilon ))}$$1/źO(log(1/ź)). We also give a $$(1+\varepsilon )$$(1+ź)-competitive algorithm, for any $$0<\varepsilon \le 1$$0<ź≤1, that builds a polynomial number of $$(m/\varepsilon )^{O(\log (1/\varepsilon ) / \varepsilon )}$$(m/ź)O(log(1/ź)/ź) schedules. This value depends on m but is independent of the input $$\sigma $$ź. The performance guarantees are nearly best possible. We show that any algorithm that achieves a competitiveness smaller than 4 / 3 must construct $$\Omega (m)$$Ω(m) schedules. Our algorithms make use of novel guessing schemes that (1) predict the optimum makespan of a job sequence $$\sigma $$ź to within a factor of $$1+\varepsilon $$1+ź and (2) guess the job processing times and their frequencies in $$\sigma $$ź. In (2) we have to sparsify the universe of all guesses so as to reduce the number of schedules to a constant. We remark that the competitive ratios achieved using parallel schedules are considerably smaller than those in the standard problem without resource augmentation. Furthermore they are at least as good and in most cases better than the ratios obtained with other means of resource augmentation for makespan minimization.

Overview: We study a classical scheduling problem that has been investigated for more than forty years. Consider a sequence of jobs σ = J 1, …, J n that has to be scheduled on m identical parallel machines. Each job J t has an individual processing time p t , 1 ≤ t ≤ n. Preemption of jobs is not allowed. The goal is to minimize the makespan, i.e. the maximum completion time of any job in the constructed schedule. In the offline variant of the problem all jobs of σ are known in advance. In the online variant the jobs arrive one by one. Each incoming job J t has to be assigned immediately to one of the machines without knowledge of any future jobs J t′, t′ > t.KeywordsCompetitive RatioOnline AlgorithmOnline ScheduleIdentical Parallel MachineMaximum Completion TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we investigate online scheduling problems on two parallel identical machines under a grade of service provision with makespan as the objective function. The jobs arrive over time and each job can be scheduled only when it has already been arrived. We consider three different versions: (i) the two machines cannot be idle at the same time until all arrived jobs have been processed; (ii) further to the first problem, jobs are processed on a first-come, first-serviced basis; (iii) each job must be assigned to one of the two machines as soon as it arrives. Respectively for three online scheduling problems, optimal online algorithms are proposed with competitive ratio $$ (\sqrt 5 + 1)/2 $$ , $$ (\sqrt 5 + 1)/2 $$ and $$ 5/3 $$ .

A common due-date assignment problem on parallel identical machines

We consider the on-line scheduling problem of jobs with precedence constraints on m parallel identical machines. Each job has a time processing requirement, and may depend on other jobs (has to be processed after them). A job arrives only after its predecessors have been completed. The cost of an algorithm is the time that the last job is completed. We show lower bounds on the competitive ratio of on-line algorithms for this problem in several versions. We prove a lower bound of 2 − 1/m on the competitive ratio of any deterministic algorithm (with or without preemption) and a lower bound of 2 − 2/(m + 1) on the competitive ratio of any randomized algorithm (with or without preemption). The lower bounds for the cases that preemption is allowed require arbitrarily long sequences. If we use only sequences of length O(m 2), we can show a lower bound of 2 − 2/(m + 1) on the competitive ratio of deterministic algorithms with preemption, and a lower bound of 2 − O(1/m) on the competitive ratio of any randomized algorithm with preemption. All the lower bounds hold even for sequences of unit jobs only. The best algorithm that is known for this problem is the well known List Scheduling algorithm of Graham. The algorithm is deterministic and does not use preemption. The competitive ratio of this algorithm is 2 − 1/m. Our randomized lower bounds are very close to this bound (a difference of 0(1/m)) and our deterministic lower bounds match this bound.

This paper addresses the problem of scheduling n jobs on a single machine and on m identical parallel machines to minimize the completion time variance of jobs. This problem of scheduling jobs on parallel machines is motivated by a case study in an automobile ancillary unit. First, a heuristic to solve the single-machine scheduling problem is proposed. The parallel-machine scheduling problem is solved in two phases: job-allocation phase and job-sequencing phase. Two heuristics are proposed in the job-allocation phase, whereas in the job-scheduling phase, the single-machine scheduling approach is used. In this paper, both versions of parallel-machine scheduling problem (restricted and unrestricted) are considered. A good upper bound is obtained using a genetic algorithm, to evaluate the performance of the proposed heuristics for the parallel-machine scheduling problem. An extensive computation evaluation of the proposed heuristics is presented for both single-machine scheduling problem and the parallel-machine scheduling problem (especially considering the case study), along with the comparison of performances with the existing heuristics in the literature.

Makespan minimization on identical machines is a fundamental problem in online scheduling. The goal is to assign a sequence of jobs to m identical parallel machines so as to minimize the maximum completion time of any job. Already in the 1960s, Graham showed that Greedy is (2-1/m)-competitive. The best deterministic online algorithm currently known achieves a competitive ratio of 1.9201. No deterministic online strategy can obtain a competitiveness smaller than 1.88. In this paper, we study online makespan minimization in the popular random-order model, where the jobs of a given input arrive as a random permutation. It is known that Greedy does not attain a competitive factor asymptotically smaller than 2 in this setting. We present the first improved performance guarantees. Specifically, we develop a deterministic online algorithm that achieves a competitive ratio of 1.8478. The result relies on a new analysis approach. We identify a set of properties that a random permutation of the input jobs satisfies with high probability. Then we conduct a worst-case analysis of our algorithm, for the respective class of permutations. The analysis implies that the stated competitiveness holds not only in expectation but with high probability. Moreover, it provides mathematical evidence that job sequences leading to higher performance ratios are extremely rare, pathological inputs. We complement the results by lower bounds, for the random-order model. We show that no deterministic online algorithm can achieve a competitive ratio smaller than 4/3. Moreover, no deterministic online algorithm can attain a competitiveness smaller than 3/2 with high probability.

This work is an extension of previous works [1], [2], [3] about scheduling jobs on identical parallel machines with constraints of unavailabilities to minimize the makespan. We are interested in this paper on the case with two identical parallel machines such that each machine is unavailable for one unavailability period such that the preemption of jobs is not allowed. We are divided this problem on five cases and for each of them we are proposed a heuristic to minimize the makespan. The obtained results, compared with those of Chien and Liao [13], show that our heuristics quit efficient in solving large problems.

This paper investigates an online hierarchical scheduling problem on m parallel identical machines. Our goal is to minimize the total completion time of all jobs. Each job has a unit processing time and a hierarchy. The job with a lower hierarchy can only be processed on the first machine and the job with a higher hierarchy can be processed on any one of m machines. We first show that the lower bound of this problem is at least $$1+\min \{\frac{1}{m}, \max \{\frac{2}{\lceil x\rceil +\frac{x}{\lceil x\rceil }+3}, \frac{2}{\lfloor x\rfloor +\frac{x}{\lfloor x\rfloor }+3}\}$$1+min{1m,max{2źxź+xźxź+3,2źxź+xźxź+3}, where $$x=\sqrt{2m+4}$$x=2m+4. We then present a greedy algorithm with tight competitive ratio of $$1+\frac{2(m-1)}{m(\sqrt{4m-3}+1)}$$1+2(m-1)m(4m-3+1). The competitive ratio is obtained in a way of analyzing the structure of the instance in the worst case, which is different from the most common method of competitive analysis. In particular, when $$m=2$$m=2, we propose an optimal online algorithm with competitive ratio of $$16$$16$$/$$/$$13$$13, which complements the previous result which provided an asymptotically optimal algorithm with competitive ratio of 1.1573 for the case where the number of jobs n is infinite, i.e., $$n\rightarrow \infty $$nźź.

In this paper, we propose an innovation strategy in the industry (case of the scheduling problem on two parallel identical machines), with the objective of minimizing the weighted sum of the end dates of jobs, this problem is NP-hard. In this frame, we suggested a novel heuristics: (H1), (H2), (H3), with two types of neighborhood (neighborhood by SWAP and neighborhood by INSERT). Next, we analyze the incorporation of three diversification times (T1), (T2), and (T3) with the aim of exploring unvisited regions of the solution space. It must be noted that job movement can be within one zone or between different zones. Computational tests are performed on 6 problems with up to 2 machines and 500 jobs.