Makespan Minimization Research Articles

Complexity of scheduling few types of jobs on related and unrelated machines

Abstract The task of scheduling jobs to machines while minimizing the total makespan, the sum of weighted completion times, or a norm of the load vector are among the oldest and most fundamental tasks in combinatorial optimization. Since all of these problems are in general -hard, much attention has been given to the regime where there is only a small number k of job types, but possibly the number of jobs n is large; this is the few job types, high-multiplicity regime. Despite many positive results, the hardness boundary of this regime was not understood until now. We show that makespan minimization on uniformly related machines ( $$Q|HM|C_{\max }$$ Q | H M | C max ) is -hard already with 6 job types, and that the related Cutting Stock problem is -hard already with 8 item types. For the more general unrelated machines model ( $$R|HM|C_{\max }$$ R | H M | C max ), we show that if the largest job size $$p_{\max }$$ p max or the number of jobs n is polynomially bounded in the instance size |I|, there are algorithms with complexity $$|I|^{{{\,\mathrm{\textrm{poly}}\,}}(k)}$$ | I | poly ( k ) . Our main result is that this is unlikely to be improved because $$Q||C_{\max }$$ Q | | C max is $$\mathsf {W[1]}$$ W [ 1 ] -hard parameterized by k already when n, $$p_{\max }$$ p max , and the numbers describing the machine speeds are polynomial in |I|; the same holds for $$R||C_{\max }$$ R | | C max (without machine speeds) when the job sizes matrix has rank 2. Our positive and negative results also extend to the objectives $$\ell _2$$ ℓ 2 -norm minimization of the load vector and, partially, sum of weighted completion times $$\sum w_j C_j$$ ∑ w j C j . Along the way, we answer affirmatively the question whether makespan minimization on identical machines ( $$P||C_{\max }$$ P | | C max ) is fixed-parameter tractable parameterized by k, extending our understanding of this fundamental problem. Together with our hardness results for $$Q||C_{\max }$$ Q | | C max , this implies that the complexity of $$P|HM|C_{\max }$$ P | H M | C max is the only remaining open case.

Metaheuristics for multi-objective scheduling problems in industry 4.0 and 5.0: a state-of-the-arts survey

The advent of Industry 4.0 and the emerging Industry 5.0 have fundamentally transformed manufacturing systems, introducing unprecedented levels of complexity in production scheduling. This complexity is further amplified by the integration of cyber-physical systems, Internet of Things, Artificial Intelligence, and human-centric approaches, necessitating more sophisticated optimization methods. This paper aims to provide a more comprehensive perspective on the application of metaheuristic algorithms in shop scheduling problems within the context of Industry 4.0 and Industry 5.0. Through a systematic review of recent literature (2015–2024), we analyze and categorize various metaheuristic approaches, including Evolutionary Algorithms (EAs), swarm intelligence, and hybrid methods, that have been applied to address complex scheduling challenges in smart manufacturing environments. We specifically examine how these algorithms handle multiple competing objectives such as makespan minimization, energy efficiency, production costs, and human-machine collaboration, which are crucial in modern industrial settings. Our survey reveals several key findings: 1) hybrid metaheuristics demonstrate superior performance in handling multi-objective optimization compared to standalone algorithms; 2) bio-inspired algorithms show promising results in addressing complex scheduling and multi-objective manufacturing environments; 3) tri-objective and higher-order multi-objective optimization problems warrant further in-depth exploration; and 4) there is an emerging trend towards incorporating human factors and sustainability objectives in scheduling optimization, aligned with Industry 5.0 principles. Additionally, we identify research gaps and propose future research directions, particularly in areas such as real-time scheduling adaptation, human-centric optimization, and sustainability-aware scheduling algorithms. This comprehensive review provides insights for researchers and practitioners in the field of industrial scheduling, offering a structured understanding of current methodologies and future challenges in the evolution from Industry 4.0 to 5.0.

Flow Shop Scheduling with Shortening Jobs for Makespan Minimization

This paper deals with a two-machine flow shop problem with shortening jobs. A shortening job means that the job’s processing time is a decreasing function of its starting time. The aim is to find a sequence that minimizes the makespan of all the jobs. several dominance properties, some lower bounds, and an initial upper bound are derived, which are applied to propose a branch-and-bound algorithm to solve the problem. We also propose some heuristics and mathematical programming. Computational experiments are conducted to evaluate the performance of the proposed algorithms.

Hybrid Whale Optimization‐Based Energy‐Efficient Lightweight Internet of Things Framework

ABSTRACTThe wireless intelligent computing paradigm has significantly provided services to various sectors in today's technology‐driven landscape. Despite its popularity, wireless intelligent computing faces challenges in addressing time‐sensitive tasks due to the physical distance between servers from users. Edge computing has been introduced for the internet of things (IoT) as an effective complement to enhance the wireless intelligent computing capacity for handling latency‐critical tasks. However, the limited resources of IoT and edge nodes can lead to suboptimal task management. In response to these challenges, we propose a lightweight approach that leverages a hybrid technique combining the whale optimization algorithm (WOA) with adaptive inertia weight and a genetic algorithm component. This method aims to enhance the efficiency of task offloading in a cloud‐edge computing environment. Experimental results demonstrate that the proposed strategy not only addresses the limitations of traditional methods but also achieves significant improvements, a 34% increase in makespan minimization, an 11% reduction in task rejection ratio, a 17% decrease in execution cost, and a 15% improvement in energy utilization compared to WOAs. The simulation results highlight the effectiveness of the proposed hybrid algorithm in enhancing quality of service (QoS) metrics for latency‐sensitive IoT applications.

Efficient arc-flow formulations for makespan minimisation on parallel machines with a common server

We consider the problem of scheduling non preemptively a set of jobs on parallel identical machines with prior setup operations on a single shared server, where the objective is to minimise the makespan. We develop an arc-flow formulation to the problem with two multigraphs, one for the machines and one for the server, with a same set of nodes representing points in time, and arcs associated with job execution, and with machines or server idleness. The resulting formulation, called Flow–Flow formulation (FFF), and its tuned version (FFT) are compared with the best existing model in the literature, a time-indexed variable formulation (F2), on benchmark instances with up to 200 jobs and 10 machines. Computational results showed that our Flow–Flow models outperformed F2 especially for instances with more than 50 jobs and optimally solved a majority of problems with 150 and 200 jobs for which F2 found only very few optimal solutions.

Ruzicka Indexive Throttled Deep Neural Learning for Resource-Efficient Load Balancing in a Cloud Environment

Cloud Computing (CC) is a prominent technology that permits users as well as organizations to access services based on their requirements. This computing method presents storage, deployment platforms, as well as suitable access to web services over the internet. Load balancing is a crucial factor for optimizing computing and storage. It aims to dispense workload across every virtual machine in a reasonable manner. Several load balancing techniques have been conventionally developed and are available in the literature. However, achieving efficient load balancing with minimal makespan and improved throughput remains a challenging issue. To enhance load balancing efficiency, a novel technique called Ruzicka Indexive Throttle Load Balanced Deep Neural Learning (RITLBDNL) is designed. The primary objective of RITLBDNL is to enhance throughput and minimize the makespan in the cloud. In the RITLBDNL technique, a deep neural learning model contains one input layer, two hidden layers, as well as one output layer to enhance load balancing performance. In the input layer, the number of cloud user tasks is collected and sent to hidden layer 1. In that layer, the load balancer in the cloud server analyzes the virtual machine resource status depending on energy, bandwidth, memory, and CPU using the Ruzicka Similarity Index. Then, it is classified VMs as overloaded, less loaded, or balanced. The analysis results are then transmitted to hidden layer 2, where Throttled Load Balancing is performed to dispense the workload of weighty loaded virtual machines to minimum loaded ones. The cloud server efficiently balances the workload between the virtual machines in higher throughput and lower response time and makespan for handling a huge number of incoming tasks. To evaluate experiments, the proposed technique is compared with other existing load balancing methods. The result shows that the proposed RITLBDNL provides better performance of higher load balancing efficiency of 7%, throughput of 46% lesser makespan of 41%, and response time of 28% than compared to conventional methods.

A speed-up procedure and new heuristics for the classical job shop scheduling problem: A computational evaluation

The speed-up procedure proposed for the permutation flowshop scheduling problem with makespan minimisation (commonly denoted as Taillard’s acceleration) remains, after 30 years, one of the most important and relevant studies in the scheduling literature. Since its proposal, this procedure has been included in countless approximate optimisation algorithms, and its use is mandatory for several scheduling problems. Unfortunately, despite the importance of such a procedure in solving scheduling problems, we are not aware of any related speed-up procedure proposed for the classical job-shop scheduling problem. First, this study aims to fill this gap by proposing a novel speed-up procedure for the job-shop scheduling problem with makespan minimisation, capable of reducing the complexity of insertion-based procedures n times. Second, to test its performance, the procedure is embedded in a critical-path-based local search method. Furthermore, we thirdly propose five constructive and composite heuristics to obtain high-quality solutions in short time intervals. The composite heuristics apply the previous procedure to reduce their computational efforts. Finally, to complete the study, we conduct an extensive computational evaluation on 243 test instances from eight distinct benchmarks. In this evaluation, 30 heuristics are re-implemented and compared under the same computer conditions. The results indicate the superiority of the proposed approaches compared to the competitive algorithms for the problem under study.

Online makespan minimization for MapReduce scheduling on multiple parallel machines

Abstract In this work, we investigate the online MapReduce processing problem on m m uniform parallel machines, aiming at minimizing the makespan. Each job consists of two sets of tasks, namely, the map tasks and the reduce tasks. A job’s map tasks can be arbitrarily split and processed on different machines simultaneously, while its reduce tasks can only be processed after all its map tasks have been completed. We assume that the reduce tasks are preemptive, but cannot be processed on different machines in parallel. We provide a new lower bound for this problem and present an online algorithm with a competitive ratio of 2 − 1 m 2-\frac{1}{m} ( m m is the number of machines) when the speeds of the machines are 1.

Scheduling Splitable Jobs on Identical Parallel Machines to Minimize Makespan using Mixed Integer Linear

The scheduling of parallel machines with and without a job-splitting property, deterministic demand,and sequence-independent setup time with the goal of minimizing makespan is examined in this work.For simultaneous processing by multiple machines, single-stage splitable jobs are broken into random(job) sections. When a job starts to be processed on a machine, an operator has to setup the machinefor an hour. By creating two Mixed Integer Linear Programming models, this work proposes amathematical programming strategy (MILP). A MILP model takes the job-splitting property intoaccount. Another model, however, does not include the job-splitting property. This study investigatesthe performance of the proposed models using Gurobi solver. These programs' numerical calculationsare based on actual problems in the Indonesian city of Bandung's plastics industry. On four identicalparallel injection molding machines, 318 jobs must be finished in 22 periods. The real schedulingmethod is contrasted with these two MILP models. The maximum workload imbalance, the maximumrelative percentage of imbalance, and the makespan of these three scheduling systems are used toevaluate their effectiveness. Without the job-splitting property, MILP can handle the real issue ofscheduling identical parallel machines on injection molding machines to reduce makespan, resultingin a 36% average decrease. The MILP model's job-splitting property can reduce makespan by anadditional 2.40%. The order of relative ranking is MILP with job-splitting property, MILP withoutjob-splitting property, and actual scheduling based on the makespan minimization, workloadimbalance, and relative percentage of imbalance.

Identical Parallel Machine Scheduling to Minimize Makespan Using Suggested Algorithm Method at XYZ Company

XYZ company produce the various shape of motor spare parts product. The company has threeidentical parallel spot welding machines that use a random method of production scheduling, basedon machine capacity without any sequence of jobs, and only use daily production targets given tooperators. Based on the data, the actual scheduling of the machines has a very large completion timedifference between each machine, or the machine loading is uneven. As a result, the makespanbecomes longer with a value of 440000 seconds (26 days). This research aims to minimize the existingmakespan by giving proposed scheduling, using the suggested algorithm method, which has a smallnumber of iterations and has an optimal result. The method begins with the longest processing timesequence rule which is used as the upper bound for the first iteration, then continued to calculate thelower bound and machine workload. The calculation stops at the 15th iteration because the completiontime value exceeds the lower and upper bound so that the optimal scheduling taken is scheduled inthe 14th iteration with a makespan value of 914412 seconds (16 days). The proposed scheduling canminimize the makespan from the actual schedule by 38%.

Solving the project scheduling problem with dependent resource constraints

ABSTRACT Most resource-constrained project scheduling problems consider resources that are independent of the job processing sequence. However, solving project scheduling problems in some industrial manufacturing is challenging because the required resources depend on the job processing sequence in existing scheduling approaches. To address this issue, this paper proposes a dependent resource-constrained project scheduling problem formulated with two types of models: a conceptual model and a binary integer programming model. The performance of four proposed algorithms – binary integer programming, reduced subproblem of the binary integer programming, job sequence assignment, and job as early as possible – is tested on 68 combinations of numerical simulation experiments. The computational results show that the job as early as possible algorithm can find a better approximate solution for solving real-world problems. This achievement aids project managers in generating activity schedules and finding the approximate minimum makespan for a project with dependent resource constraints.

Minimising the makespan on parallel identical machines with log-linear position-dependent processing times

This article studies the minimum makespan scheduling problem on parallel identical machines with the log-linear learning/ageing effect, also known as polynomial positional learning/ageing effect. To develop a Fully Polynomial Time Approximation Scheme to the problem, we start with an intermediate artificial variant that rounds the values to integers and restricts the solutions to instances sorted in the shortest/longest processing time order. To this end, we propose a dynamic programming algorithm and show that the difference between its returned value and the minimum makespan of the original problem is independent of the processing times. This then leads to an algorithm with provable guaranteed ϵ-additive approximation and pseudo-polynomial running time algorithm, resulting in the desired fully polynomial time approximation solution to the original, not restricted problem.

Integration of MILP and discrete-event simulation for flow shop scheduling using Benders cuts

Optimization-based scheduling in the chemical industry is highly beneficial but also highly difficult due to its combinatorial complexity. Different modeling and optimization techniques exist, each with individual strengths. We propose Benders decomposition to integrate mixed-integer linear programming (MILP) and discrete-event simulation (DES) to solve flow shop scheduling problems with makespan minimization objective. The basic idea is to generate valid Benders cuts based on sensitivity information of the DES sub problem, which can be found in the critical paths of DES solutions. For scaled literature flow shops, our approach requires at least an order of magnitude fewer iterations than a genetic algorithm and provides optimality gap information. For a real-world case study, our approach finds good solutions very quickly, making it a powerful alternative to established methods. We conclude that the Benders-DES algorithm is a promising approach to combine rigorous MILP optimization capabilities with high-fidelity DES modeling capabilities.

A Combined Discrete Event Simulation and Factorial Design Experiment for the Scheduling Problem in a Hybrid Flow Shop

Production plants have always been confronted with the problem of scheduling, as this has a direct impact on production time and therefore on production costs. This is especially true in today's world, where it is necessary to produce a quality product at a low price and in a short time, while at the same time responding flexibly to customer demands. Due to the complexity and for economic reasons, testing different variants in a real production environment is insufficient and ineffective. For this reason, this paper proposes a new method that combines discrete event simulation and factorial design experiment to find the optimal schedule in hybrid flow shop. It was tested with the goal of achieving the minimum makespan. The results show that this method makes it possible to find improved solution very quickly. Compared to the original production schedule, the makespan could be reduced by 3 hours and 31 minutes which is reduction of 16.3%. The proposed methodology can be used in many discrete and process production plants.

A novel and efficient mathematical optimization model for multi-stage assembly flow shop considering post-processing

ABSTRACT We addressed the multistage assembly flow shop problem with post-processing and makespan minimization, a production environment commonly encountered in diverse industries such as automotive, dental, medical equipment, and clothing manufacturing. In this context, we presented an innovative mixed-integer linear programming model with a position-based strategy. Our proposed formulation demonstrated remarkable efficiency when compared to the model of the literature. It achieved optimal solutions in 77.16% of instances, with an average optimality gap of 10.59%. This study constitutes a significant contribution to the efficient resolution of a practical and frequently encountered scheduling problem that has received relatively limited attention in existing literature. The findings highlight the crucial role of mathematical optimization models as valuable decision-making tools for scheduling within the addressed production system.

Efficiency and energy consumption of the automated container yard with twin rail-mounted gantry cranes considering crane scheduling strategies and handshake area designs

This study investigates the effects of twin rail-mounted gantry crane (RMG) scheduling strategies and handshake area designs on container yard efficiency and energy consumption. A handshake area is set for temporary storage of containers within each block, enabling the cooperation and interference avoidance of twin RMGs. Considering the complexity and dynamics, we establish a simulation model for the twin RMG operations with agent-based and discrete event simulation modeling. Container relocation, microscopic movements of RMGs, and stochastic arrivals of automated guided vehicles (AGVs) and external trucks (ETs) are simulated, and the corresponding efficiency and energy consumption are quantitatively estimated. By testing the six RMG scheduling strategies and thirteen handshake area designs, the results show that the 2-OPT RMG scheduling strategy (an optimization heuristic that attempts to find the schedule with the minimum makespan) outperforms the other strategies in terms of efficiency. However, the optimal RMG strategy for minimal energy consumption depends on the ratio of landside and seaside requests and handshake area settings. Furthermore, the location of the handshake area significantly affects the efficiency and power consumption of AGVs and RMGs. This study can provide decision support to improve the efficiency and reduce energy consumption of the twin RMGs by selecting a specified combination of the RMG scheduling strategy and the handshake area design.

Scheduling stochastic distributed flexible job shops using an multi-objective evolutionary algorithm with simulation evaluation

The trend of reverse globalisation prompts manufacturing enterprises to adopt distributed structures with multiple factories for improving production efficiency, meeting customer requirements, and responding disturbance events. This study focuses on scheduling a distributed flexible job shop with random job processing time to achieve minimal makespan and minimal total tardiness. First, a stochastic programming model is established to formulate the concerned problems. Second, in accordance with the natures of two objectives and randomness, an evolutionary algorithm incorporating an evaluation method is designed. In it, population-based and external archive-based search processes are developed for searching candidate solutions, and the evaluation method integrates stochastic simulation and discrete event simulation to calculate objective values of acquired solutions. Finally, a mathematical optimisation solver, CPLEX, is employed to validate the developed model and optimisation approach. A set of cases is solved to verify the performance of the proposed method. The comparisons and discussions show the superiority of the proposed method for handling the problems under study.

New bounds for single-machine time-dependent scheduling with uniform deterioration

We consider the single-machine time-dependent scheduling problem with linearly deteriorating jobs arriving over time. Each job i is associated with a release time ri and a processing time pi(si)=αi+βisi, where αi,βi>0 are parameters and si is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. We develop new bounds and approximation results for the special case of the problems with uniform deterioration, i.e. βi=β, for each i. The main contribution is a O(1+1/β)-approximation algorithm for the makespan problem and a O(1+1/β2) approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with β=O(1/n) and β=Ω(n), where n is the number of jobs. Our analysis is based on an approach for comparing computed and optimal schedules via bounding pseudomatchings.

Synchronisation of operations and maintenance in distributed unrelated parallel machine systems: a hierarchical optimisation framework

ABSTRACT Synchronisation of operations and maintenance means that production and maintenance departments work together effectively to achieve integrated optimisation of production scheduling and maintenance activities. This paper addresses such an integrated optimisation problem towards distributed unrelated parallel machine systems, where deteriorating processing time and machine failures are considered. For the trade-off between production and maintenance to obtain a minimal makespan in the worst-case scenario, a hierarchical optimisation framework with reinforcement learning-based encoder and knowledge-based intelligent optimiser (HOFRK) is developed. Experimental results show that HOFRK can be treated as an excellent optimiser for enterprise information systems to solve the proposed problem.

Makespan minimization for workflows with multiple privacy levels

With the advanced development of Metaverse, various service requests are required to be processed as soon as possible which contains a series of tasks with topology structure (workflow) and different privacy constraints. Due to the diversity and complex hierarchical structure of resources, how to rent suitable resources from cloud, edge, and local nodes to minimize the makespan of workflows is currently one of the key issues in service computing scenarios. Due to the limited computing resources of local devices, tasks need to be offloaded to edge or cloud servers. Based on task privacy constraints, it is challenging to determine the appropriate offloading nodes for each task in the workflow. In this paper, for workflows with multi-privacy level tasks, a privacy-preserving workflow scheduling algorithm based on clustering (PWCSA) is proposed to minimize the workflow makespan with limited bandwidth. Multiple tasks of different privacy levels are clustered into one task pipeline to reduce the data transmission time caused by limited bandwidth, thereby reducing workflow makespan. After statistically calibrating algorithm parameters over a comprehensive set of workflow instances, the proposed algorithms are compared with the state-of-the-art algorithms over five workflow-type instances. The results indicate that the proposed PWCSA algorithm outperforms other algorithms with acceptable computational time. Through a number of workflow instances, the system parameters are verified by analysis of variance methods.