Partial Solutions and MultiFit Algorithm for Multiprocessor Scheduling

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.

MILP models to minimise makespan in additive manufacturing machine scheduling problems

The Wafer Probing Scheduling Problem (WPSP) is a practical generalization of the parallel-machine scheduling problem, which has many real-world applications, particularly, in the Integrated Circuit (IC) manufacturing industry. In the wafer probing factories, the jobs are clustered by their product types, which must be processed on groups of identical parallel machines and be completed before the due dates. The job processing time depends on the product type, and the machine setup time is sequentially dependent on the orders of jobs processed. Since the wafer probing scheduling problem involves constraints on job clusters, job-cluster dependent processing time, due dates, machine capacity, and sequentially dependent setup time, it is more difficult to solve than the classical parallel-machine scheduling problem. In this paper, we consider the WPSP and formulate the WPSP as an integer programming problem to minimize the total machine workload. We demonstrate the applicability of the integer programming model by solving a real-world example taken from a wafer probing shop floor in an IC manufacturing factory.

Preemptive scheduling problems on parallel machines are classic problems. Given the goal of minimizing the makespan, they are polynomially solvable even for the most general model of unrelated machines. In these problems, a set of jobs is to be assigned to be executed on a set of m machines. A job can be split into parts arbitrarily and these parts are to be assigned to time slots on the machines without parallelism, that is, for every job, at most one of its parts can be processed at each time.Motivated by sensitivity analysis and online algorithms, we investigate the problem of designing robust algorithms for constructing preemptive schedules. Robust algorithms receive one piece of input at a time. They may change a small portion of the solution as an additional part of the input is revealed. The capacity of change is based on the size of the new input. For scheduling problems, the maximum ratio between the total size of the jobs (or parts of jobs) which may be re-scheduled upon the arrival of a new job j, and the size of j, is called migration factor.We design a strongly optimal algorithm with the migration factor \(1-\frac 1m\) for identical machines. Such algorithms avoid idle time and create solutions where the (non-increasingly) sorted vector of completion times of the machines is minimal lexicographically. In the case of identical machines this results not only in makespan minimization, but the created solution is also optimal with respect to any ℓ p norm (for p > 1). We show that an algorithm of a smaller migration factor cannot be optimal with respect to makespan or any other norm, thus the result is best possible in this sense as well. We further show that neither uniformly related machines nor identical machines with restricted assignment admit an optimal algorithm with a constant migration factor. This lower bound holds both for makespan minimization and for any ℓ p norm. Finally, we analyze the case of two machines and show that in this case it is still possible to maintain an optimal schedule with a small migration factor in the cases of two uniformly related machines and two identical machines with restricted assignment.

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.

In this paper we present algorithms for the problem of scheduling n independent jobs on m identical machines. As a generalization of the classical multiprocessor scheduling problem each machine is available only at a machine dependent release time. Two objective functions are considered. To minimize the makespan, we develop a dual approximation algorithm with a worst case bound of 5/4. For the problem of maximizing the minimum completion time, we develop an algorithm, such that the minimum completion time in the schedule produced by this algorithm is at least 2/3 times the minimum completion time in the optimum schedule. The paper closes with some numerical results.

A branch and bound algorithm for an identical parallel machine scheduling problem with a job splitting property

SYNOPTIC ABSTRACTIn this article, we consider a parallel machine scheduling problem that integrates partial maintenance activities due to job-dependent machine deteriorations. In this problem, we are given identical parallel machines, each with an initial maintenance level which is a non-negative integer value to represent its maintenance state, and a set of jobs each with a non-preemptive processing time and a machine deterioration. For each job, its machine deterioration qualifies the decrement of the machine maintenance level after processing it. When each machine processes jobs, one should guarantee a non-negative maintenance level at any time point. Thus, whenever necessary, a maintenance activity must be performed to restore the machine maintenance level. The goal of this problem is to schedule the jobs and the maintenance activities, such that the makespan achieves the minimum. We take the partial maintenance activity pattern, in which each activity can be performed to increase the machine maintenance level to any level not exceeding the maximum. To tackle this problem, we first model it by an integer programming formulation, and then propose an LP-based approximation algorithm with an instance-dependent approximation factor and show the approximability lower bound. Finally, we present a fast heuristic algorithm followed by a numerical example to show how this algorithm works.

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

Identical parallel machine scheduling problem for minimizing the makespan is a very important production scheduling problem which has been proven to be NP-hard. The problem further compounds with additional constraints. Genetic algorithms (GA) have shown great advantages in solving the combinatorial optimization problem in view of its characteristic that has high efficiency and that is fit for practical application. In this chapter we present a spreadsheet based GA approach for minimizing the makespan for scheduling of a set of tasks for identical parallel machines and worker assignment to machines. The results obtained from the proposed approach are compared with two sets of benchmark problems consisting of 100 problems each. It has been demonstrated that the performance of proposed approach is superior to the results that have been obtained earlier. The proposed approach produces optimal solution for almost 95% of the problems demonstrating the effectiveness of the proposed approach. An empirical analysis of GA parameters has also been carried out to see the effect on the performance of the proposed algorithm.

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.

This paper deals with a multiobjective parallel machines scheduling problem. It consists in scheduling n independent jobs on m identical parallel machines. The job data such as processing times, release dates, due dates and sequence dependent setup times are considered. The goal is to optimize two different objectives: the makespan and the total tardiness. A mixed integer linear program is proposed to model the studied problem. As this problem is NP-hard in the strong sense, a metaheuristic method which is the second version of the non dominated sorting genetic algorithm (NSGA-II) is proposed to solve this problem. Since the parameters setting of a genetic algorithm is difficult, a fuzzy logic controller coupled with the NSGA-II (FLC-NSGA-II) is therefore proposed. The role of the fuzzy logic is to better set the crossover and the mutation probabilities in order to update the search ability. After that, an exact method based on the two phase method is also developed. We have used four measuring criteria to compare these methods. The experimental results show the advantages and the efficiency of FLC-NSGA-II.

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

A composite algorithm is developed for the classical problem of scheduling independent jobs on identical parallel machines with the objective of minimizing the makespan. The algorithm at first obtains a family of initial partial solutions and combines these partial solutions until a feasible solution is generated. Then local search procedures are used for improving the solution. The effectiveness of this approach is evaluated through extensive computational comparisons with recent improvement heuristics for different classes of benchmark instances.

A common due-date assignment problem on parallel identical machines