Maximum Completion Time Research Articles

Scheduling Jobs with Release and Delivery Times Subject to Nested Eligibility Constraints

The problem of scheduling jobs with release and delivery time subject to machine eligibility constraints is considered. The eligible sets of the jobs are nested, and preemptions are not allowed. The goal is to minimize the maximum delivery completion time, i.e., the time by which all jobs are delivered. For the special case of equal release time, a 2-approximation algorithm is presented whose running time depends linearly on the number of jobs. For the general case of unequal release time, a polynomial time approximation scheme is derived.

A novel four-tier architecture for delay aware scheduling and load balancing in fog environment

Fog computing paradigm is located between IoT devices and cloud paradigm that aim is to minimize the latency in terms of task scheduling and load balancing. To deal with the huge amount of data sensing from different IoT devices, in this paper we propose a four tiers architecture for delay aware scheduling and Load Balancing in the fog environment. Tier-1 is the bottom tier which consists of IoT devices. In the second tier, the applications (workloads) are categorized into two categories: High Priority (HP) and Low Priority (LP) by router based on the Dual Fuzzy Logic Algorithm. Fuzzifier considers four input metrics: task size, arrival time, minimum execution time and maximum completion time. A task with high priority is transmitted to the third tier (fog tier). In the third tier, a novel fog structure has been invoked namely Artificial Fractals consists of nodes. The fog nodes are clustered using K-means++ clustering algorithm. Each fog node is carried out several actions such as scheduling, monitoring, and communication. To schedule tasks within the fog node, we propose the Earliest Deadline First (EDF) task scheduling algorithm. The current usage of fog node is determined by Artificial Neural Network (ANN). If an IoT device does not get the required resource then the request is forwarded to the cloud tier. Our proposed work has been validated over a real-time VSOT (Video Surveillance/Object Tracking) application using iFogSim and the performance is evaluated in terms of response time, scheduling time, load balancing rate, delay, and energy consumption.

Designing a Multi-Objective Mathematical Model for Flexible Job Shop Scheduling Problem With the Earliness/Tardiness Penalty

The scheduling of flexible job shop systems is one of the most important issues in the various fields of production and is currently being addressed by many researchers in the field of optimization issues. The present research includes flexible job shop scheduling problem (FJSSP) with multi-objective, minimizing maximum completion time (makespan), maximum machine workload, total machines workload and also, earliness/tardiness penalty with different constraints. In this research, the researcher is looking to design a mathematical model that can cover all the constraints and assumptions related to the problem. Therefore, the mathematical model was designed with multi-objective and different constraints with exact details and different assumptions that are consistent with the actual situation of the problem and implemented at Comex Company. What that distinguishes this research from other similar researches is the approach of multi-objective with different constraints, which, at the same time, it raises the complexity of the problem but the problem gets closer to the actual situation, with less research done. Finally, the results of the study showed that this mathematical model designed, as well as in the real environment which has the flexible job shop system, can be implemented within a reasonable time with the highest efficiency before the implementation of the model.

Two-machine open shop problem with agreement graph

This paper addresses the problem of scheduling jobs on a two-machine open shop subject to constraints given by an agreement graph G, such that jobs can be processed simultaneously on different machines if and only if they are represented by adjacent vertices in G. The problem of minimizing the maximum completion time (makespan) is strongly NP-hard for three distinct values of processing times and for G being a bipartite graph. We extend the study presented in [3] and [4] to the proportionate open shop system. We first prove the NP-hardness of the case of two values of processing times and more general agreement graphs which closes definitely the complexity status of the problem. Then, we present some restricted cases that can be solved in polynomial time. We also derive new complexity results of the open shop under resource constraints and of the partition into triangles problem.

Flexible flow shop scheduling problem to minimize makespan with renewable resources

This paper deals with a flexible flow shop (FFS) scheduling problem with unrelated parallel machines and renewable resource shared among the stages. The FFS scheduling problem is one of the most common manufacturing environment in which there is more than a machine in at least one production stage. In such a system, to decrease the processing times, additional renewable resources are assigned to the jobs or machines, and it can lead to decrease the total completion time. For this purpose, a mixed integer linear programming (MILP) model is proposed to minimize the maximum completion time (makespan) in an FFS environment. The proposed model is computationally intractable. Therefore, a particle swarm optimization (PSO) algorithm as well as a hybrid PSO and simulated annealing (SA) algorithm named SA-PSO, are developed to solve the model. Through numerical experiments on randomly generated test problems, the authors demonstrate that the hybrid SA-PSO algorithm outperforms the PSO, especially for large size test problems.

A meta-heuristic based on the Imperialist Competitive Algorithm (ICA) for solving Hybrid Flow Shop (HFS) scheduling problem with unrelated parallel machines

ABSTRACTThis paper addresses the hybrid flow shop (HFS) scheduling problem with unrelated parallel machines considering sequence and machine dependent setup times. The performance measurement used in this study is based on minimizing maximum completion time (makespan). Because of the complexity of the problem, a meta-heuristic based on the imperialist competitive algorithm (ICA) and the genetic algorithm (GA) has been used. To achieve reliable results, an experimental design has been developed to determine the incidence of the factors in the objective function which is applied in twelve instances. The efficiency of the meta-heuristic is compared with a model of mixed-integer programming validated in GAMS. The results of the study reveal that the meta-heuristic obtains equal or better solutions than those of the exact method for half of the instances.

Mathematical Modeling and Discrete Firefly Algorithm to Optimize Scheduling Problem with Release Date, Sequence-Dependent Setup Time, and Periodic Maintenance

An evolutionary discrete firefly algorithm (EDFA) is presented herein to solve a real-world manufacturing system problem of scheduling a set of jobs on a single machine subject to nonzero release date, sequence-dependent setup time, and periodic maintenance with the objective of minimizing the maximum completion time “makespan.” To evaluate the performance of the proposed EDFA, a new mixed-integer linear programming model is also proposed for small-sized instances. Furthermore, the parameters of the EDFA are regulated using full factorial analysis. Finally, numerical experiments are performed to demonstrate the efficiency and capability of the EDFA in solving the abovementioned problem.

A distributed permutation flowshop scheduling problem with the customer order constraint

In the classic distributed permutation flowshop scheduling problem (DPFSP), jobs are viewed as individual entities and processed independently. In many practical cases, however, a number of jobs actually come from the same customer order. Under this circumstance, it may be sensible to process jobs from the same customer order in a single factory to reduce transportation costs, paperwork and management burden. This study introduces the customer order constraint into the DPFSP. In our problem, a set of customer orders need to be manufactured in a number of factories and each order composed of some defined jobs should be processed in the same factory. The objective is to minimize the maximum completion time or makespan among factories. At first, we build a mathematical model to formulate this new problem. Then, given the NP-hardness of this problem, we present three heuristics exploring three rules for generating the seed job sequence as well as two rules for assigning orders to factories. Besides, we develop three meta-heuristics, namely, a variable neighborhood descent (ORVND), an artificial bee colony (ORABC) and an iterated greedy (ORIG). Effective mechanisms are suggested to improve the performance, including an order-insertion based neighborhood search and a greedy reinsertion strategy for ORVND, a two-level search scheme and a multi-neighbor scheme for ORABC, and the improved destruction, reconstruction and local search operations for ORIG. Finally, extensive experiments based on famous benchmarks are performed and numerical comparisons validate the high effectiveness of presented algorithms for the considered problem.

Scheduling identical jobs on uniform machines with a conflict graph

This paper addresses the problem of scheduling n identical jobs on a set of m parallel uniform machines. The jobs are subjected to conflicting constraints modelled by an undirected graph G, in which adjacent jobs are not allowed to be processed on the same machine. Minimising the maximum completion time in the schedule (makespan Cmax) is known to be NP-hard. We prove that when G is restricted to complete bipartite graphs the problem remains NP-hard for arbitrary number of machines, however, if m is fixed an optimal solution can be obtained in polynomial time. To solve the general case of the problem, we propose mixed-integer linear programming (MILP) formulations alongside with lower bounds and heuristic approaches. Furthermore, computational experiments are carried out to measure the performance of the proposed methods.

Bi-objective Optimization Model for Integrated Preventive Maintenance and Flexible Job-shop Scheduling Problem

This paper develops an integrated model for flexible job-shop scheduling problem with the maintenance activities. Reliability models are used to perform the maintenance activities. This model involves two objectives: minimization of the maximum completion time for flexible job-shop production part and minimization of system unavailability for the PM (preventive maintenance) part. To aim the objectives, two decisions must be taken at the same time: assigning n jobs on m machines in order to minimize the maximum completion time and finding the appropriate times to perform PM activities to minimize the system unavailability. These objectives are obtained with considering dependent machine setup times for operations and release times for jobs. In advance, the maintenance activity numbers and PM intervals are not fixed. Two multi objective optimization methods are compared to find the pareto-optimal front in the flexible job-shop problem case. Promising the obtained results, a benchmark with a large number of test instances is employed.

Trade-off balancing between maximum and total completion times for no-wait flow shop production

We propose a trade-off balancing (TOB) heuristic in a no-wait flow shop to minimise the weighted sum of maximum completion time () and total completion time (TCT) based on machine idle times. We introduce a factorisation scheme to construct the initial sequence based on current and future idle times at the operational level. In addition, we propose a novel estimation method to establish the mathematical relationship between the objectives min() and min(TCT) at the production line level. To evaluate the performance of the TOB heuristic, computational experiments are conducted on the classic Taillard's benchmark and one-year historical data from University of Kentucky HealthCare (UKHC). The computational results show that minimisations of and TCT yield inconsistent scheduling sequences, and these two sequences are relatively uncorrelated. We also show that our TOB heuristic performs better than the best existing heuristics with the same computational complexity and generates stable performances in balancing trade-offs.

Performance Evaluation of Continuous and Discrete Particle Swarm Optimization in Job-Shop Scheduling Problems

The Particle Swarm Optimization (PSO) is an optimization method that was modeled based on the social behavior of organisms, such as bird flocks or swarms of bees. It was initially applied for cases defined over continuous spaces, but it can also be modified to solve problems in discrete spaces. Such problems include scheduling problems, where the Job-shop Scheduling Problem (JSP) is among the hardest combinatorial optimization problems. Although the JSP is a discrete problem, the continuous version of PSO has been able to handle the problem through a suitable mapping. Subsequently, its modified model, namely the discrete PSO, has also been proposed to solve it. In this paper, the performance of continuous and discrete PSO in solving JSP are evaluated and compared. The benchmark tests used are FT06 and FT10 problems available in the OR-library, where the goal is to minimize the maximum completion time of all jobs, i.e. the makespan. The experimental results show that the discrete PSO outperforms the continuous PSO for both benchmark problems.

Effective Neighborhood Generation Method in Search Algorithm for Flexible Job Shop Scheduling Problem

The flexible job shop scheduling problem (FJSSP) is an extension of the classical job shop scheduling problem (JSSP) that allocates jobs to resources while minimizing the maximum completion time of all the jobs. Machine assignment and job sequence are determined in the FJSSP. To efficiently solve the FJSSP, which is a non-deterministic polynomial-time hard problem, a heuristic method must be used. In previous studies, the FJSSP has been solved using neighborhood algorithms that employ various metaheuristic methods. These approaches constrain the neighborhood operation to jobs on a critical path and simultaneously change the machine assignment and job sequence. Branches on the critical path are easily generated in the FJSSP search processes; this branch structure can improve the efficiency of the FJSSP. This study investigates two neighborhood search algorithms used for changing the machine assignment and job sequence via a critical path. The first method changes the machine assignment and job sequence simultaneously, whereas the second method changes them independently. In this study, we propose an efficient neighborhood generating method using a branch block of critical path.

Low-time complexity and low-cost binary particle swarm optimization algorithm for task scheduling and load balancing in cloud computing

With the increasing large number of cloud users, the number of tasks is growing exponentially. Scheduling and balancing these tasks amongst different heterogeneous virtual machines (VMs) under constraints such as, low makespan, high resource utilization rate, low execution cost and low scheduling time, become NP-hard optimization problem. So, due to the inefficiency of heuristic algorithms, many meta-heuristic algorithms, such as particle swarm optimization (PSO) have been introduced to solve the said problem. However, these algorithms do not guarantee that the optimal solution can be found, if they are not combined with other heuristic or meta-heuristic algorithms. Further, these algorithms have high time complexity, making them less useful in realistic scenarios. To solve the said NP-problem effectively, we propose an efficient binary version of PSO algorithm with low time complexity and low cost for scheduling and balancing tasks in cloud computing. Specifically, we define an objective function which calculates the maximum completion time difference among heterogeneous VMs subject to updating and optimization constraints introduced in this paper. Then, we devise a particle position updating with respect to load balancing strategy. The experimental results show that the proposed algorithm achieves task scheduling and load balancing better than existing meta-heuristic and heuristic algorithms.

An iterated local search and tabu search for two-parallel machine scheduling problem to minimize the maximum total completion time

The parallel machine scheduling problems exist in many real-life industrial examples. In order to effectively utilize the machines and decline the total waiting time of jobs assigned to each machine, the maximum total completion time per machine on a parallel machine is thus addressed. Due to the NP-hardness of the problem under consideration, in this paper the authors explore an optimal solution by a branch-and-bound method, incorporating several dominances and a lower bound. In addition, the authors also adopted an iterated local search (ILS) and a Tabu search to find its near-optimal solution for the problem. The computational results on randomly generated instances illustrate that the proposed Tabu search algorithm performs better than the two existing algorithms, and the ILS method for the small jobs, but the ILS method performs better than the two existing algorithms and Tabu search algorithm for the bigger jobs.

Multi-objective redundancy hardening with optimal task mapping for independent tasks on multi-cores

The rate of transient faults has increased significantly as the technology scales up. The tolerance of transient faults has become an important issue in the system design. Dual modular redundancy (DMR) and triple modular redundancy (TMR) are two commonly used techniques that can achieve fault detection and masking through executing redundant tasks. As DMR and TMR have different time and cost overheads, we must carefully determine which one should be used for each task (i.e., task hardening) to achieve the optimal system design. Furthermore, for multi-core systems, the system-level design includes the allocation of cores for the tasks (i.e., task mapping) as well. This paper aims at task hardening and mapping simultaneously for independent tasks on multi-cores with heterogeneous performances, in order to minimize the maximum completion time of all tasks (i.e., makespan). We demonstrate that once task hardening is given, task mapping of independent tasks can be achieved by employing min–max-weight perfect matching with a polynomial time complexity. Besides, as there is a trade-off between cost and time performance, we propose a multi-objective memetic algorithm (MOMA)-based task hardening method to obtain a set of solutions with different numbers of cores (i.e., costs), so the designer can choose different solutions according to different requirements. The key idea of the MOMA is to incorporate problem-specific knowledge into the global search of evolutionary algorithms. Our experimental studies have demonstrated the effectiveness of the proposed method and have shown that by combining the results of MOMA and MOEA we can provide a designer with a highly accurate set of solutions within a reasonable amount of time.

An elitist nondominated sorting hybrid algorithm for multi-objective flexible job-shop scheduling problem with sequence-dependent setups

In this paper, an elitist nondominated sorting hybrid algorithm, namely ENSHA, is proposed to solve the multi-objective flexible job-shop scheduling problem (MOFJSSP) with sequence-dependent setup times/costs (MOFJSSP_SDST/C). The objectives to be minimized are the maximal completion time (i.e., makespan) and the total setup costs (TSC). The makespan is an efficiency-focused objective whereas the TSC is an economic focused one. Existing works mainly consider the efficiency-focused multiple criteria. The main highlights of this paper are threefold, i.e., the operation-based sequence model, the problem-dependent job assignment rules and the novel evolutionary framework of ENSHA. For the operation-based sequence model, this is the first time that the sequence model of MOFJSSPs has been proposed and the TSC has been treated as an independent objective in MOFJSSPs. For the job assignment rules, the solution representation is first proposed, and then three job assignment rules are specifically designed to decode solutions or sequences into feasible scheduling schemes. For the novel evolutionary framework, it works with two populations, i.e., the main population (MP) and the auxiliary population (AP). First, ENSHA adopts the elitist nondominated sorting method for evolving MP to maintain high-quality solutions regarding both the convergence and diversity. Next, a machine learning strategy based on the estimation of distribution algorithm (EDA) is proposed to learn the valuable information from nondominated solutions in MP for building a probabilistic model. This model is then used to generate the offspring of AP. Furthermore, a simple yet effective cooperation-based refinement mechanism is raised to combine MP and AP, so as to generate MP of the next generation. Finally, experimental results on 39 benchmark instances and a real-life case study demonstrate the effectiveness and application values of the proposed ENSHA.

Flexible job-shop scheduling with tolerated time interval and limited starting time interval based on hybrid discrete PSO-SA: An application from a casting workshop

In this paper, we addressed two significant characteristics in practical casting production, namely tolerated time interval (TTI) and limited starting time interval (LimSTI). With the consideration of TTI and LimSTI, a multi-objective flexible job-shop scheduling model is constructed to minimize total overtime of TTI, total tardiness and maximum completion time. To solve this model, we present a hybrid discrete particle swarm optimization integrated with simulated annealing (HDPSO-SA) algorithm which is decomposed into global and local search phases. The global search engine based on discrete particle swarm optimization includes two enhancements: a new initialization method to improve the quality of initial population and a novel gBest selection approach based on extreme difference to speed up the convergence of algorithm. The local search engine is based on simulated annealing algorithm, where four neighborhood structures are designed under two different local search strategies to help the proposed algorithm jump over the trap of local optimal solution. Finally, computational results of a real-world case and simulation data expanded from benchmark problems indicate that our proposed algorithm is significant in terms of the quality of non-dominated solutions compared to other algorithms.

Energy-Efficient Dynamic Computation Offloading and Cooperative Task Scheduling in Mobile Cloud Computing

Mobile cloud computing (MCC) as an emerging and prospective computing paradigm, can significantly enhance computation capability and save energy for smart mobile devices (SMDs) by offloading computation-intensive tasks from resource-constrained SMDs onto resource-rich cloud. However, how to achieve energy-efficient computation offloading under hard constraint for application completion time remains a challenge. To address such a challenge, in this paper, we provide an energy-efficient dynamic offloading and resource scheduling (eDors) policy to reduce energy consumption and shorten application completion time. We first formulate the eDors problem into an energy-efficiency cost (EEC) minimization problem while satisfying task-dependency requirement and completion time deadline constraint. We then propose a distributed eDors algorithm consisting of three subalgorithms of computation offloading selection, clock frequency control, and transmission power allocation. Next, we show that computation offloading selection depends on not only the computing workload of a task, but also the maximum completion time of its immediate predecessors and the clock frequency and transmission power of the mobile device. Finally, we provide experimental results in a real testbed and demonstrate that the eDors algorithm can effectively reduce EEC by optimally adjusting CPU clock frequency of SMDs in local computing, and adapting the transmission power for wireless channel conditions in cloud computing.

Single-machine scheduling with deteriorating effects and machine maintenance

In this paper, the single-machine scheduling problems with deteriorating effects and a machine maintenance are studied. In this circumstance, the deterioration rates of the jobs during the machining process are the same which reduces the production efficiency. The actual processing time of the job is a linearly increasing function of the starting time. In this process, the machine only performs a maintenance activity, and the maintenance time is a fixed value. After the maintenance work is completed, the machine will be restored to the initial state, and the deterioration of the job will be start again. The goal is to determine the optimal schedule in order to minimise the maximum completion time (i.e. the makespan) and the sum of job completion times. We prove that both problems are polynomial time solvable, and we also provide the corresponding algorithms.