Recent Results for Online Makespan Minimization

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.

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.

A heuristic algorithm is developed for the classical scheduling problem in which independent jobs are nonpreemptively scheduled on identical parallel machines with the objective of minimizing the makespan with load balancing. The algorithm uses iteratively the LPT procedure to be successful computational tools in many hard Combinatorial Optimization problems. We present an approximation algorithm based on balance the Hopfild with binary decision variables which are a kind of assignment variables for the classical deterministic scheduling problem of minimizing the makespan on identical machines. These problems are characterized by their complexity, known as NP-hard in most cases. Computational results indicate that the proposed algorithm is very competitive with respect to well-known constructive algorithms for a large number of benchmark.

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.

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.

MILP models to minimise makespan in additive manufacturing machine scheduling problems

Applying “Peeling Onion” approach for competitive analysis in online scheduling with rejection

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.

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

A new polynomial algorithm is developed for the classical multiprocessor scheduling problem in which independent jobs are nonpreemptively scheduled on identical parallel machines with the objective of minimizing the makespan, i.e. the latest job finishing time. The algorithm at first generates and merges a set of partial solutions in order to obtain a feasible solution for the multiprocessor scheduling problem. Then a set of bin packing problems are solved in order to improve the solution, by iteratively using a MultiFit type procedure on different job sets. The effectiveness of this approach is evaluated by solving a large number of benchmark instances. The results indicate that the proposed algorithm, called PSMF, is very competitive with well known constructive algorithms for a wide range of instances. Furthermore, PSMF is competitive with respect to some of the best heuristics for parallel machine scheduling problems only when it uses a 2-exchange procedure.

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.

This paper deals with the problem of scheduling dependent tasks with release time, deadline and due time on identical parallel machines. Based on decomposition-coordination methodology in large scale systems theory and Lagrangian relaxation technique, a multilayer hierarchical scheduling algorithm is presented. By making the most of structural properties of the precedence graph of partial ordering task set and introducing the concept of feasible start time interval , the proposed algorithm can be used for solving the scheduling problem under study effectively.

In this paper, we investigate the problem of semi-on-line scheduling n jobs on m identical parallel machines under the assumption that the ordering of the jobs by processing time is known and the jobs have arbitrary release times. Our aim is to minimize the maximum completion time. An ordinal algorithm is investigated and its worst case ratio is analyzed.