Identical Parallel Machines Research Articles

The Longest Processing Time rule for identical parallel machines revisited

We consider the $$P_m || C_{\max }$$ scheduling problem where the goal is to schedule n jobs on m identical parallel machines $$(m < n)$$ to minimize makespan. We revisit the famous Longest Processing Time (LPT) rule proposed by Graham in 1969. LPT requires to sort jobs in non-ascending order of processing times and then to assign one job at a time to the machine whose load is smallest so far. We provide new insights into LPT and discuss the approximation ratio of a modification of LPT that improves Graham’s bound from $$\left( \frac{4}{3} - \frac{1}{3m} \right) $$ to $$\left( \frac{4}{3} - \frac{1}{3(m-1)} \right) $$ for $$m \ge 3$$ and from $$\frac{7}{6}$$ to $$\frac{9}{8}$$ for $$m=2$$. We use linear programming to analyze the approximation ratio of our approach. This performance analysis can be seen as a valid alternative to formal proofs based on analytical derivation. Also, we derive from the proposed approach an $$O(n \log n)$$ time complexity heuristic. The heuristic splits the sorted job set in tuples of m consecutive jobs ($$1,\dots ,m; m+1,\dots ,2m;$$ etc.) and sorts the tuples in non-increasing order of the difference (slack) between largest job and smallest job in the tuple. Then, given this new ordering of the job set, list scheduling is applied. This approach strongly outperforms LPT on benchmark literature instances and is competitive with more involved approaches such as COMBINE and LDM.

MIP models and a matheuristic algorithm for an identical parallel machine scheduling problem under multiple copies of shared resources constraints

If parallel machines use shared resources during production, jobs on machines must wait until the required resources are available. If the shared resource is single, only one job can use it at a time, but if there are multiple copies of this resource, multiple jobs can be scheduled up to the number of copies at a time. For this reason, it is crucial to consider resource usage when scheduling this type of machine. In recent years, various studies have been carried out to address identical parallel machine scheduling problems. However, although shared resources in parallel machines are an important aspect of this problem, resources are rarely considered in these studies and, in fact, have not been studied for this particular aspect. In this study, an identical parallel machine scheduling problem with sequence-dependent setup times, machine eligibility restrictions and multiple copies of shared resources (IPMS-SMS) is considered. Mixed-integer programming (MIP) models and a model-based genetic algorithm (matheuristic) are proposed, and the objective function of the problem seeks to minimize the total weighted completion time. Randomly generated instances are solved using the proposed models and the matheuristic. Optimal schedules are obtained for almost all small problems using a mixed-integer programming model. However, better solutions are obtained for medium and large instances using the proposed matheuristic.

A Multifactory Integrated Production and Distribution Scheduling Problem with Parallel Machines and Immediate Shipments Solved by Improved Whale Optimization Algorithm

This paper discusses the integrated scheduling of production and distribution operations in a multifactory supply chain with make-to-order production system. For the production side of the supply chain, we considered distributed parallel-established factories with identical parallel machines available at each factory. We assumed that the factories could produce all customers’ orders with different production rates and costs. For the distribution side of the supply chain, we considered a limited number of homogeneous vehicles that immediately distribute the finalized orders to the customers. Then, a mixed-integer nonlinear programming model is developed to determine the detailed scheduling of production and distribution that minimizes the total costs of the supply chain including production, distribution, and late delivery costs. To solve the real-world scale problems, we developed a new whale optimization algorithm (WOA). Moreover, we conducted computational experiments by generating several test problems to evaluate the proposed algorithm. Statistical analysis showed that the proposed algorithm has better performance than traditional WOA for different scales of the problem. Moreover, it confirms the capability of the improved whale optimization algorithm (IWOA) to solve the medium-scale instances; however, the results indicate the better performance of genetic algorithm (GA) for the large-scale instances.

Enhanced arc-flow formulations to minimize weighted completion time on identical parallel machines

We consider the problem of scheduling a set of jobs on a set of identical parallel machines, with the aim of minimizing the total weighted completion time. The problem has been solved in the literature with a number of mathematical formulations, some of which require the implementation of tailored branch-and-price methods. In our work, we solve the problem instead by means of new arc-flow formulations, by first representing it on a capacitated network and then invoking a mixed integer linear model with a pseudo-polynomial number of variables and constraints. According to our computational tests, existing formulations from the literature can solve to proven optimality benchmark instances with up to 100 jobs, whereas our most performing arc-flow formulation solves all instances with up to 400 jobs and provides very low gap for larger instances with up to 1000 jobs.

An ILS heuristic for the ship scheduling problem: application in the oil industry

AbstractThis paper addresses a real‐life rescheduling problem of a pipe‐laying support vessel (PLSV) fleet in charge of subsea oil well connections. The short‐term schedule of these vessels is subject to uncertainties inherent to its operations, resulting in ships idleness or delays in oil production. The objective of this study is to develop methods to support a Brazilian oil and gas company in overcoming impacts caused by operational disruptions, while reaching its planned production level. The PLSV rescheduling problem was treated as an identical parallel machine scheduling problem, where the machines represent the vessels and the jobs are the activities for the subsea well connections. We propose a mathematical programming model and a method based on the iterated local search (ILS) metaheuristic to solve the problem. This paper contributes to this by considering simultaneously setup times, machine eligibility, release dates, due dates, and machine availability. Both methods were applied on 10 instances based on real PLSV data. Taking into account an objective function that measures the operational impact on schedules, the ILS provided an average improvement above 91% in schedules when compared to the initial solution provided by the studied company. The ILS outperformed a mathematical programming model for the problem, in eight instances, within a 30‐minute execution time limit, fitting to the company process.

Exact Algorithms for Distributionallyβ-Robust Machine Scheduling with Uncertain Processing Times

The β-robust machine scheduling has attracted increasing attention as an effective method to hedge against uncertainty. However, existing β-robust scheduling models rely on the normality assumption of uncertain parameters, and existing solution methods are based on branch and bound, which cannot solve problems of 45 jobs within 3,600 seconds. This paper proposes distributionally β-robust scheduling (DRS) models to handle uncertain processing times. The DRS models only require the lower bound, mean, and covariance information of processing times, and have the capability of handling both single and parallel machine problems. Another key contribution of this paper is to devise efficient parametric search (PS) methods for the DRS models. Specifically, we show that there exists a parameterized assignment problem (PAP), such that its optimal solutions are also optimal for the original problem. The proposed methods only need to perform a one-dimensional PS and solve a series of PAPs. We further propose a bidirectional PS to reduce the number of PAPs needed to be solved, and we design a speedup shortest augmentation path algorithm for these PAPs. Experimental results on both single and identical parallel machine problems show that the improved PS method outperforms existing algorithms by more than three orders of magnitude improvement in computation time for problems of 45 jobs, and it is able to solve problems of 500 jobs within 0.5 seconds.

A metaheuristic algorithm and simulation to study the effect of learning or tiredness on sequence-dependent setup times in a parallel machine scheduling problem

This work analyses the effects of learning or tiredness on the setup times in a scheduling problem with identical parallel machines. This problem involves setup times that depend on the sequence of jobs with the goal of minimizing the sum of total completion times. Due to the complexity of the problem and the assumption that high-quality solutions of the problem without effects are also high-quality solutions when these effects are considered, we firstly propose a metaheuristic algorithm aimed at finding high-quality and diverse solutions, ignoring the learning/tiredness issues. Then, we study the effects of learning or tiredness on the obtained solutions in a real-world scenario by using a multi-agent simulation approach. The computational experiments carried out demonstrate that the simulation model developed in this work is valid to handle randomness in a practical scenario, allowing to be adapted to different learning or tiredness effects. Furthermore, the computational experiments underscore the fact that the proposal can be used as a decision support tool aimed at estimating the amount of job to be assigned to the available machines on the basis of the operator profile.

An improved cuckoo search algorithm for scheduling jobs on identical parallel machines

Abstract In this paper, we propose an improved cuckoo search algorithm (ICSA) to minimize makespan for the identical parallel-machine scheduling problem. Starting with an initial population of schedules generated by using the longest processing time (LPT) rule and the job-interchange mechanism, we select the best schedule from this population and then execute the proposed ICSA. For the ICSA, we first propose a heuristic approach using a modulus operator to transform a continuous position in CSA into discrete schedule of jobs for generating a new cuckoo by Levy flights. Next, we present a heuristic procedure based on the pairwise exchange neighborhood to produce smart cuckoos in the proposed ICSA. We then conduct exhaustive computational experimentation on a large number of randomly generated well-known benchmark problems to show that the proposed ICSA produces better solutions than the six state-of-the-art existing algorithms.

IDENTICAL PARALLEL MACHINES SCHEDULING USING GENETIC ALGORITHM

Minimizing the scheduling production time consider one of the most important factors forcompanies which their objectives is achieve the maximum profits. This paper studies theidentical parallel machine scheduling problem which involves the assignment numbers ofjob (N) to set of identical parallel machine (M) in order to minimize the makespan(maximum completion time of all job). There are numerous troubles in solving the largesize of “parallel machine scheduling” problem with an excessive jobs and machines, sothe genetic algorithm was proposed in this paper which is consider an efficient algorithmthat fits larger size of identical “parallel machine scheduling” for minimizing themakespan. Most studies in the scheduling field suppose setup time is insignificant orincluded in the processing time, in this paper both the sequence independent setup timesand processing time were considered. The solutions of algorithms are coding in(MATLAB). A numerical example of (11) jobs are schedule on (3) machines todemonstrative the effectiveness of algorithm solution. The result show the algorithm caneffectively solve large size of scheduling problem and given the best schedule withminimum makespan.

Online Parallel-Machine Scheduling in KRT Environment to Minimize Total Weighted Completion Time

This paper studies online scheduling on [Formula: see text] identical parallel machines under the KRT environment, where jobs arrive over time and “KRT” means that in the online setting no jobs can be released when all of the machines are busy. The goal is to determine a feasible schedule to minimize the total of weighted completion times. When [Formula: see text], we prove that WSPT is an optimal online algorithm. When [Formula: see text], we first present a lower bound [Formula: see text], and then show that WSPT is a 2-competitive online algorithm for the case [Formula: see text]. For the case in which [Formula: see text] and all jobs have equal processing times, we provide a best possible online algorithm with a competitive ratio of [Formula: see text].

Approximating total weighted completion time on identical parallel machines with precedence constraints and release dates

We present a (2+2ln2+ε)-approximation algorithm for the classical nonpreemptive scheduling problem to minimize the total weighted completion time of jobs on identical parallel machines subject to release dates and precedence constraints, improving upon the previously best known 4-approximation algorithm from 1998. The result carries over to the more general problem with precedence delays and generalizes a recent result by Li (2017) for the problem without release dates or delays.

Parallel dedicated machine scheduling with conflict graphs

Abstract This paper investigates a variant of parallel-machine scheduling problems with conflict constraints. A set of identical parallel machines is available for processing a set of jobs subject to conflict constraints, which specify pairs of jobs that are mutually disjoint due to resource availability. Jobs conflicting to each other cannot be processed simultaneously. The scheduling problem is to construct a feasible schedule that optimizes the considered managerial performance measures. This paper discusses the specific two-machine setting where each machine has a designated set of jobs to process. We give NP-hardness proofs for the case with the presence of a fixed processing sequence on one of the machines. Polynomial-time dynamic programming algorithms are proposed to produce optimal schedules for the case where the processing sequences on both machines are known and fixed a priori.

Distributionally robust scheduling on parallel machines under moment uncertainty

This paper investigates a distributionally robust scheduling problem on identical parallel machines, where job processing times are stochastic without any exact distributional form. Based on a distributional set specified by the support and estimated moments information, we present a min-max distributionally robust model, which minimizes the worst-case expected total flow time out of all probability distributions in this set. Our model doesn’t require exact probability distributions which are the basis for many stochastic programming models, and utilizes more information compared to the interval-based robust optimization models. Although this problem originates from the manufacturing environment, it can be applied to many other fields when the machines and jobs are endowed with different meanings. By optimizing the inner maximization subproblem, the min-max formulation is reduced to an integer second-order cone program. We propose an exact algorithm to solve this problem via exploring all the solutions that satisfy the necessary optimality conditions. Computational experiments demonstrate the high efficiency of this algorithm since problem instances with 100 jobs are optimized in a few seconds. In addition, simulation results convincingly show that the proposed distributionally robust model can hedge against the bias of estimated moments and enhance the robustness of production systems.

Scheduling on Parallel Machines with Mold Constraints

This paper deals with makespan minimization scheduling problem on two identical parallel machines with mold constraint. The mold constraint in this problem is described as a kind of resource-constrained. In the production of wafer fabrication, two jobs with same mold cannot be processed on two parallel machines at the same time. This problem is quite common in practice, but there are few literatures relevant to this problem. Since the problem is NP-hard, a mathematical model is discussed to describe and solve the problem and a heuristic is proposed. Computational results show that the proposed heuristic can efficiently obtain good solutions for medium and large size problems.

A note on posterior tight worst-case bounds for longest processing time schedules

This note proposes and analyzes a posterior tight worst-case bound for the longest processing time (LPT) heuristic for scheduling independent jobs on identical parallel machines with the objective of minimizing the makespan. It makes natural remarks on the well-known posterior worst-case bounds, and shows that the proposed bound can complement the well-known posterior bounds to synergistically achieve a better posterior worst-case bound for the LPT heuristic. Moreover, it gives some insight on LPT asymptotical optimality.

Distributionally robust fixed interval scheduling on parallel identical machines under uncertain finishing times

We deal with fixed interval scheduling (FIS) problems on parallel identical machines where the job starting times are given but the finishing times are subject to uncertainty. In the operational problem, we construct a schedule with the highest worst-case probability that it remains feasible, whereas in the tactical problem we are looking for the minimum number of machines to process all jobs given a minimum level for the worst-case probability that the schedule is feasible. Our ambiguity set contains joint delay distributions with a given copula dependence, where a proportion of marginal distributions is stressed and the rest are left unchanged. We derive a trackable reformulation and propose an efficient decomposition algorithm for the operational problem. The algorithm for the tactical FIS is based on solving a sequence of the operational problems. The algorithms are compared on simulated FIS instances in the numerical part.

Scheduling parallel identical machines to minimize makespan:A parallel approximation algorithm

Approximation algorithms for scheduling parallel machines have been studied for decades, leading to significant progress in terms of their approximation guarantees. The algorithms that provide near optimal performance are not feasible to use in practice due to their huge execution time requirements, thus underscoring the importance of developing efficient parallel approximation algorithms with near-optimal performance guarantees that are suitable for execution on current parallel systems, such as multi-core systems. We present the design and analysis of a parallel approximation algorithm for the problem of scheduling jobs on parallel identical machines to minimize makespan. The design of the parallel approximation algorithm is based on the best existing polynomial-time approximation scheme (PTAS) for the problem. To the best of our knowledge, this is the first practical parallel approximation algorithm for the minimum makespan scheduling problem that maintains the approximation guarantees of the sequential PTAS and it is specifically designed for execution on shared-memory parallel machines. We implement and run the algorithm on a large multi-core system and perform an extensive experimental analysis on data generated from realistic probability distributions. The results show that our proposed parallel approximation algorithm achieves significant speedup with respect to both the sequential PTAS and the CPLEX-based solver that solves the mixed integer program formulation of the problem.

Integrative Programming of Production and Preventive Maintenance Tasks in an Environment of Identical Parallel Machines: Proposal of a Construction Heuristic

Abstract The article presents a constructive heuristic for the integrated programming of production orders and preventive maintenance (PM) interventions in an environment of identical parallel machines, aimed at minimizing the completion time of the last scheduled job (makespan). Unlike other proposals found in the literature, the one here considers that the time between preventive interventions is not previously known, but instead, this is assumed as one of the decision variables of the system in the beginning of each of these interventions; in addition, the proposal also considers the possibility that not all jobs have the same release time. The design of the heuristic algorithm is based on the general approach of the Longest Processing Time dispatch rule, with a slight modification in its conception to consider the random character of the equipment failures and their impact on the execution time of the tasks. Consequently, a simple dispatch rule called rj - LPT was also proposed to comply with one of the steps of the heuristic, as well as an algorithm to determine the minimum dimension for the makespan, which constitutes the basis of the stop criterion linked to the initial solution.

Online scheduling with chain precedence constraints of equal-length jobs on parallel machines to minimize makespan

We study the online scheduling problem on m identical parallel machines to minimize makespan, i.e., the maximum completion time of the jobs, where m is given in advance and the jobs arrive online over time. We assume that the jobs, which arrive at some nonnegative real times, are of equal-length and are restricted by chain precedence constraints. Moreover, the jobs arriving at distinct times are independent, and so, only the jobs arriving at a common time are restricted by the chain precedence constraints. In the literature, a best possible online algorithm of a competitive ratio 1.3028 is given for the case $$m=2$$ . But the problem is unaddressed for $$m\ge 3$$ . In this paper, we present a best possible online algorithm for the problem with $$m\ge 3$$ , where the algorithm has a competitive ratio of 1.3028 for $$3\le m\le 5$$ and 1.3146 for $$m\ge 6$$ .

Bi-objective identical parallel machine scheduling to minimize total energy consumption and makespan

Abstract Currently, energy consumption reduction is playing a more and more important role in production and manufacturing, especially for energy-intensive industries. An optimal production scheduling can help reduce unnecessary energy consumption. This paper considers an identical parallel machine scheduling problem to minimize simultaneously two objectives: the total energy consumption (TEC) and the makespan. To tackle this NP-hard problem, an augmented ɛ -constraint method is applied to obtain an optimal Pareto front for small-scale instances. For medium- and large-scale instances, a constructive heuristic method with a local search strategy is proposed and the NSGA-II algorithm is applied to obtain good approximate Pareto fronts. Extensive computational experiments on randomly generated data and a real-world case study are conducted. The result shows the efficiency and effectiveness of the proposed methods.