Optimal Makespan Research Articles

Improved Anonymous Multi-Agent Path Finding Algorithm

We consider an Anonymous Multi-Agent Path-Finding (AMAPF) problem where the set of agents is confined to a graph, a set of goal vertices is given and each of these vertices has to be reached by some agent. The problem is to find an assignment of the goals to the agents as well as the collision-free paths, and we are interested in finding the solution with the optimal makespan. A well-established approach to solve this problem is to reduce it to a special type of a graph search problem, i.e. to the problem of finding a maximum flow on an auxiliary graph induced by the input one. The size of the former graph may be very large and the search on it may become a bottleneck. To this end, we suggest a specific search algorithm that leverages the idea of exploring the search space not through considering separate search states but rather bulks of them simultaneously. That is, we implicitly compress, store and expand bulks of the search states as single states, which results in high reduction in runtime and memory. Empirically, the resultant AMAPF solver demonstrates superior performance compared to the state-of-the-art competitor and is able to solve all publicly available MAPF instances from the well-known MovingAI benchmark in less than 30 seconds.

Checkpointing Strategies to Tolerate Non-Memoryless Failures on HPC Platforms

This article studies checkpointing strategies for parallel applications subject to failures. The optimal strategy to minimize total execution time, or makespan, is well known when failure IATs obey an Exponential distribution, but it is unknown for non-memoryless failure distributions. We explain why the latter fact is misunderstood in recent literature. We propose a general strategy that maximizes the expected efficiency until the next failure, and we show that this strategy achieves an asymptotically optimal makespan, thereby establishing the first optimality result for arbitrary failure distributions. Through extensive simulations, we show that the new strategy is always at least as good as the Young/Daly strategy for various failure distributions. For distributions with high infant mortality (such as LogNormal with shape parameter k =2.51 or Weibull with shape parameter 0.5), the execution time is divided by a factor of 1.9 on average, and up to a factor 4.2 for recently deployed platforms.

The product–mold–machine manufacturing problem: Complexity, MILP models and constructive heuristics

This study addresses the product–mold–machine manufacturing problem, a problem intrinsic to the fabrication processes in industries such as tires, plastics, ceramics, and glass. Each product possesses distinct shape and mechanical attributes, resulting from processing its corresponding mold on a specific machine. Uniquely, machines in this problem can handle multiple molds concurrently, implying that their processing time is dictated by the longest mold processing time within the set being used. The goal is to devise a production strategy that reduces the overall periods required to satisfy each product’s demand. This has to be achieved under constraints like limited availability of machines and molds, changeover times, and incompatibility constraints. We establish the problem’s NP-complete nature in the strong sense. Further, we introduce two mixed-integer linear programming (MILP) models and two polynomial-time constructive heuristics. These heuristics offer upper bounds for the optimal makespan and present easily implementable feasible solutions. Our numerical tests, conducted on real-world data, evaluate the efficacy of different solution strategies, specifically those using the preliminary insights from our heuristics to guide the mathematical formulations. The outcomes strongly advocate for leveraging these feasible solutions as starting points when solving the MILP models. Additionally, the notable efficiency and solution quality of our heuristics spotlight them as potent alternatives, especially in settings where traditional optimization tools might be absent.

Two parallel-machine scheduling with maximum waiting time for an emergency job

In modern manufacturing and service industries, urgent orders and service tasks are common, and the speed of handling such urgent tasks is an important indicator of production and service efficiency. In this study, we consider scheduling jobs on two parallel machines with the random arrival of an emergency job. The objective is to minimise the makespan, subject to a given maximum waiting time of the emergency job. We first show that the worst-case ratio of the existing algorithm LPT l -SPT m − l is at least 3/2 when m = 2. We then analyse some properties of the optimal schedule and derive lower bounds on the optimal makespan. Finally, we present an improved approximation algorithm with a tight worst-case ratio of 4/3. We also provide numerical results showing that our proposed algorithm outperforms algorithm LPT l -SPT m − l .

Hybrid metaheuristic model based performance-aware optimization for map reduce scheduling

Because of the rapid rise in the count of corporations using cloud-dependent infrastructure as the foundation for big data storing and analysis. The fundamental difficulty in scheduling big data services in cloud-dependent systems is ensuring the shortest possible makespan while simultaneously reducing the number of resources being used. We have created a new, secure map reduce scheduling method that functions as follows. Initially, the cloud architecture is designed and the tasks are generated. In the pre-processing phase, the huge set of tasks was processed by the map-reduce scheduling framework. Afterward, the optimal task scheduling task is conducted which utilizes a hybrid algorithm named Tunicate Combined Moth Flame Algorithm (TCMFA), which provides better task scheduling via providing optimal makespan, execution time, and security. This proposed TCMFA is the hybridization of both Moth Flame Optimization (MFO) and Tunicate Swarm Algorithm (TSA). The error rate of the TCMFA gets reduced to 320 approximately over other conventional methods such as RSA, ACO, GHO, BTS, OWPSO, BES, PRO, SOA, COOT, TSA & MFO which proves the accuracy of our TCMFA and makes it more efficient and secure for optimal map-reduce scheduling.

A flexible algorithm to offload DAG applications for edge computing

Multi-access Edge Computing (MEC) is an enabling technology to leverage new network applications, such as virtual/augmented reality, by providing faster task processing at the network edge. This is done by deploying servers closer to the end users to run the network applications. These applications are often intensive in terms of task processing, memory usage, and communication; thus mobile devices may take a long time or even not be able to run them efficiently. By transferring (offloading) the execution of these applications to the servers at the network edge, it is possible to achieve a lower completion time (makespan) and meet application requirements. However, offloading multiple entire applications to the edge server can overwhelm its hardware and communication channel, as well as underutilize the mobile devices’ hardware. In this paper, network applications are modeled as Directed Acyclic Graphs (DAGs) and partitioned into tasks, and only part of these tasks are offloaded to the edge server. This is the DAG application partitioning and offloading problem, which is known to be NP-hard. To approximate its solution, this paper proposes the FlexDO algorithm. FlexDO combines a greedy phase with a permutation phase to find a set of offloading decisions, and then chooses the one that achieves the shortest makespan. FlexDO is compared with a proposal from the literature and two baseline solutions, considering realistic DAG applications extracted from the Alibaba Cluster Trace Program. FlexDO results are consistently only 3.9% to 8.9% above the optimal makespan in all test scenarios, which include different levels of CPU availability, a multi-user case, and different communication channel transmission rates. FlexDO outperforms both baseline solutions by a wide margin, and is three times closer to the optimal makespan than its competitor. It achieves up to 87% similarity with the optimal decision and does not burden the edge server.

An integrated hybrid metaheuristic model for the constrained scheduling problem

Several problems in the domains of project management (PM) and operations research (OP) can be classified as optimization problems which are classically NP-Hard. One such highly important problem is the resource constrained project scheduling problem (RCPSP). The main aim of this problem is to find a schedule of the lowest and optimum makespan to complete a project, which involves resource as well as precedence constraints. But, being classically NP-Hard, the RCPSP requires exponential computational resources as the problem complexity increases. Thus, approximate techniques like computational intelligence (CI) based approaches provide better chances of finding near optimal solutions. This paper presents the usage of a hybrid technique using the phases of teaching learning-based optimization (TLBO) metaheuristic integrated with operators like crossover and mutation from the genetic algorithm (GA). An integrated hybrid using TLBO and 2-point crossover is applied in the teacher and learner phases to the discrete RCPSP problem. Further, to diversify the population, and enhance global search, the mutation operator is applied. The proposed model is extensively tested on well-known benchmark test instances and has been compared with other seminal works. The encouraging results make evident the efficiency of the provided solution for the RCPSP problem of varying magnitudes.

Solving Restricted Preemptive Scheduling on Parallel Machines with SAT and PMS

Restricted preemption plays a crucial role in reducing total completion time while controlling preemption overhead. A typical version of restricted preemptive models is k-restricted preemptive scheduling, where preemption is only allowed after a task has been continuously processed for at least k units of time. Though solving this problem of minimizing the makespan on parallel machines is NP-hard in general, it is of vital importance to obtain the optimal solution for small-sized problems, as well as for evaluation of heuristics. This paper proposes optimal strategies to the aforementioned problem. Motivated by the dramatic speed-up of Boolean Satisfiability (SAT) solvers, we make the first step towards a study of applying a SAT solver to the k-restricted scheduling problem. We set out to encode the scheduling problem into propositional Boolean logic and determine the optimal makespan by repeatedly calling an off-the-shelf SAT solver. Moreover, we move one step further by encoding the problem into Partial Maximum Satisfiability (PMS), which is an optimized version of SAT, so that the explicit successive calls of the solver can be eliminated. The optimal makespan of the problem and the performance of the proposed methods are studied experimentally. Furthermore, an existing heuristic algorithm is evaluated by the optimization methods.

Mathematical models and benchmarking for the fuzzy job shop scheduling problem

The fuzzy job shop scheduling problem with makespan minimisation has received considerable attention over the last decade. Different sets of benchmark instances have been made available, and many metaheuristic solutions and corresponding upper bounds of the optimal makespan have been given for these instances in different publications. However, unlike the deterministic case, very little work has been invested in proposing and solving mathematical models for the fuzzy problem. This has resulted both in a lack of a good characterisation of the hardness of existing benchmark instances and in the absence of reliable lower and upper bounds for the makespan. In consequence, it is difficult, if not impossible to properly assess and compare new proposals of exact or approximate solving methods, thus hindering progress in this field. In this work we intend to fill this gap by proposing and solving two mathematical models, a mixed integer linear programming model and a constraint programming model. A thorough analysis on the scalability of solving these mathematical models with commercial solvers is carried out. A state-of-the-art metaheuristic algorithm from the literature is also used as reference point for a better understanding of the results. Using solvers of different nature allows us to improve known upper and lower bounds for all existing instances, and certify optimality for many of them for the first time. It also enables us to structurally characterise the instances’ hardness beyond their size.

Modeling and optimization of bakery production scheduling to minimize makespan and oven idle time

Makespan dominates the manufacturing expenses in bakery production. The high energy consumption of ovens also has a substantial impact, which bakers may overlook. Bakers leave ovens running until the final product is baked, allowing them to consume energy even when not in use. It results in energy waste, increased manufacturing costs, and CO2 emissions. This paper investigates three manufacturing lines from small and medium-sized bakeries to find optimum makespan and ovens’ idle time (OIDT). A hybrid no-wait flow shop scheduling model considering the constraints that are most common in bakeries is proposed. To find optimal solutions, non-dominated sorting genetic algorithm (NSGA-II), strength Pareto evolutionary algorithm (SPEA2), generalized differential evolution (GDE3), improved multi-objective particle swarm optimization (OMOPSO), and speed-constrained multi-objective particle swarm optimization (SMPSO) were used. The experimental results show that the shortest makespan does not always imply the lowest OIDT. Even the optimized solutions have up to 231 min of excess OIDT, while the makespan is the shortest. Pareto solutions provide promising trade-offs between makespan and OIDT, with the best-case scenario reducing OIDT by 1348 min while increasing makespan only by 61 min from the minimum possible makespan. NSGA-II outperforms all other algorithms in obtaining a high number of good-quality solutions and a small number of poor-quality solutions, followed by SPEA2 and GDE3. In contrast, OMOPSO and SMPSO deliver the worst solutions, which become pronounced as the problem complexity grows.

Equitable Allocation of Operations and Makespan Minimization for Autonomous Agents

We study the problem of allocating a set of indivisible operations to a set of agents in a fair and efficient manner while also minimizing the makespan. We first present the Operation Trading Algorithm that generates allocations satisfying the DEQx (Duplicated Equitability up to any operation) fairness criterion while also guaranteeing an upper bound of 2 on the makespan for identical agents. The pairwise approach used in this algorithm has the added advantages of being decentralizable and robust. We then define a relaxed version of the DEQ1 (Duplicated Equitability upto some operation) fairness criterion called partial-DEQ1. A market based algorithm is presented to achieve partial-DEQ1 along with Pareto Optimality. Following this, it is shown that the algorithm also guarantees an upper bound of 1.618 on the makespan for 2 non-identical agents. Parametric Pruning further improves the upper bound to 1.5 which is theoretically the best possible upper bound. To the best of our knowledge, these are the first algorithms designed to achieve the mentioned fairness criteria. The algorithms additionally guarantee upper bounds on the makespan. Finally, we show the efficacy of the algorithms in generating allocations with near optimal makespans by numerically evaluating the algorithms on random instances. Note to Practitioners—This paper provides fast algorithms that can be used for task allocation among any number of agents. The algorithms generate allocations that satisfy fairness criteria along with minimizing the makespan, thus achieving two practically important criteria simultaneously. Additionally, some of the algorithms introduced can be implemented in a reactive manner thus making it easy to deal with sudden changes. The algorithms can also be implemented in a decentralized manner in environments where maintaining a centralized situational awareness is difficult due to communication and machine failures.

Optimization of service scheduling problem for overlapping tower cranes with cooperative coevolutionary genetic algorithm

PurposeIn regarding to operational efficiency and safety improvements, multiple tower crane service scheduling problem is one of the main problems related to tower crane operation but receives limited attention. The current work presents an optimization model for scheduling multiple tower cranes' service with overlapping areas while achieving collision-free between cranes.Design/methodology/approachThe cooperative coevolutionary genetic algorithm (CCGA) was proposed to solve this model. Considering the possible types of cross-tasks, through effectively allocating overlapping area tasks to each crane and then prioritizing the assigned tasks for each crane, the makespan of tower cranes was minimized and the crane collision avoidance was achieved by only allowing one crane entering the overlapping area at one time. A case study of the mega project Daxing International Airport has been investigated to evaluate the performance of the proposed algorithm.FindingsThe computational results showed that the CCGA algorithm outperforms two compared algorithms in terms of the optimal makespan and the CPU time. Also, the convergence of CCGA was discussed and compared, which was better than that of traditional genetic algorithm (TGA) for small-sized set (50 tasks) and was almost the same as TGA for large-sized sets.Originality/valueThis paper can provide new perspectives on multiple tower crane service sequencing problem. The proposed model and algorithm can be applied directly to enhance the operational efficiency of tower cranes on construction site.

Streaming algorithms for multitasking scheduling with shared processing

In this paper, we design the first streaming algorithms for the problem of multitasking scheduling on parallel machines with shared processing. In one pass, our streaming approximation schemes can provide an approximate value of the optimal makespan. If the jobs can be read in two passes, the algorithm can find the schedule with the approximate value. This work not only provides an algorithmic big data solution for the studied problem, but also gives an insight into the design of streaming algorithms for other problems in the area of scheduling.

A Statistical Comparison of Metaheuristics for Unrelated Parallel Machine Scheduling Problems with Setup Times

Manufacturing scheduling aims to optimize one or more performance measures by allocating a set of resources to a set of jobs or tasks over a given period of time. It is an area that considers a very important decision-making process for manufacturing and production systems. In this paper, the unrelated parallel machine scheduling problem with machine-dependent and job-sequence-dependent setup times is addressed. This problem involves the scheduling of tasks on unrelated machines with setup times in order to minimize the makespan. The genetic algorithm is used to solve small and large instances of this problem when processing and setup times are balanced (Balanced problems), when processing times are dominant (Dominant P problems), and when setup times are dominant (Dominant S problems). For small instances, most of the values achieved the optimal makespan value, and, when compared to the metaheuristic ant colony optimization (ACOII) algorithm referred to in the literature, it was found that there were no significant differences between the two methods. However, in terms of large instances, there were significant differences between the optimal makespan obtained by the two methods, revealing overall better performance by the genetic algorithm for Dominant S and Dominant P problems.

Speed-robust scheduling: sand, bricks, and rocks

The speed-robust scheduling problem is a two-stage problem where, given m machines, jobs must be grouped into at most m bags while the processing speeds of the machines are unknown. After the speeds are revealed, the grouped jobs must be assigned to the machines without being separated. To evaluate the performance of algorithms, we determine upper bounds on the worst-case ratio of the algorithm’s makespan and the optimal makespan given full information. We refer to this ratio as the robustness factor. We give an algorithm with a robustness factor $$2-\frac{1}{m}$$ for the most general setting and improve this to 1.8 for equal-size jobs. For the special case of infinitesimal jobs, we give an algorithm with an optimal robustness factor equal to $$\frac{e}{e-1}\approx 1.58$$ . The particular machine environment in which all machines have either speed 0 or 1 was studied before by Stein and Zhong (ACM Trans Algorithms 16(1):1-20, 2020. https://doi.org/10.1145/3340320 ). For this setting, we provide an algorithm for scheduling infinitesimal jobs with an optimal robustness factor of $$\frac{1+\sqrt{2}}{2}\approx 1.207$$ . It lays the foundation for an algorithm matching the lower bound of $$\frac{4}{3}$$ for equal-size jobs.

Malleable scheduling beyond identical machines

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S with respect to j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than \(\frac{e}{e-1}\), unless \(P = NP\). On the positive side, we present polynomial-time algorithms with approximation ratios \(\frac{2e}{e-1}\) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of \(1+\varphi \) for unrelated speeds (\(\varphi \) is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the \(L_p\) norm of their speeds.

Improved MILP formulation equipped with valid inequalities for scheduling a batch processing machine with non-identical job sizes

We consider the problem of minimizing makespan on a single batch processing machine with a limited capacity. The machine can simultaneously process a group (batch) of jobs, each of them having different sizes and processing times. The processing time of a batch is the longest processing time of all jobs in the batch. We analyze the strength of the LP-relaxations of two mixed integer linear programming (MILP) formulations: a straightforward formulation and an advanced formulation. We prove that while the LP-relaxation of the straightforward formulation can perform arbitrarily bad in providing an LP bound on the optimal makespan, the LP-relaxation of the advanced formulation have some merit in providing a relatively good bound of makespan. With a detailed analysis of the problem, we prove a number of properties of the optimal solution and derive their respective valid inequalities (VI). These VIs are exploited in a novel MILP formulation for the problem. Extensive computations reveal that, with the help of the VIs, the proposed MILP formulation can solve large instances of the problem.

Genetic Algorithm Application for Permutation Flow Shop Scheduling Problems

In this paper, permutation flow shop scheduling problems (PFSS) are investigated with a genetic algorithm. PFSS problem is a special type of flow shop scheduling problem. In a PFSS problem, there are n jobs to be processed on m machines in series. Each job has to follow the same machine order and each machine must process jobs in the same job order. The most common performance criterion in the literature is the makespan for permutation scheduling problems. In this paper, a genetic algorithm is applied to minimize the makespan. Taillard’s instances including 20, 50, and 100 jobs with 5, 10, and 20 machines are used to define the efficiency of the proposed GA by considering lower bounds or optimal makespan values of instances. Furthermore, a sensitivity analysis is made for the parameters of the proposed GA and the sensitivity analysis shows that crossover probability does not affect solution quality and elapsed time. Supplementary to the parameter tuning of the proposed GA, we compare our GA with an existing GA in the literature for PFSS problems and our experimental study reveals that our proposed and well-tuned GA outperforms the existing GA for PFSS problems when the objective is to minimize the makespan.

Reinforcement learning based path planning of multiple agents of SwarmItFIX robot for fixturing operation in sheetmetal milling process

SwarmItFIX (self-reconfigurable intelligent swarm fixtures) is a multi-agent setup mainly used as a robotic fixture for large Sheet metal machining operations. A Constraint Satisfaction Problem (CSP) based planning model is utilized currently for computing the locomotion sequence of multiple agents of the SwarmItFIX. But the SwarmItFIX faces several challenges with the current planner as it fails on several occasions. Moreover, the current planner computes only the goal positions of the base agent, not the path. To overcome these issues, a novel hierarchical planner is proposed, which employs Monte Carlo and SARSA TD based model-free Reinforcement Learning (RL) algorithms for the computation of locomotion sequences of head and base agents, respectively. These methods hold two distinct features when compared with the existing methods (i) the transition model is not required for obtaining the locomotion sequence of the computational agent, and (ii) the state-space of the computational agent become scalable. The obtained results show that the proposed planner is capable of delivering optimal makespan for effective fixturing during the sheet metal milling process.

A novel genetic algorithm based system for the scheduling of medical treatments

The manual scheduling of medical treatment in a health centre is a complex, time consuming, and error prone task. Furthermore, there is no guarantee a manually generated schedule maximises the operational efficiency of the centre. Scheduling problems have seen extensive research across several domains. The current work presents a novel genetic algorithm for the scheduling of repetitive Transcranial Magnetic Stimulation (rTMS) appointments. The proposed List Scheduling Wildcard Tournament Genetic Algorithm (LSWT-GA) combines an innovative survivor selection policy with heuristic population initialisation. The algorithm aims to optimise the operational efficiency of a medical centre through efficient rTMS appointment scheduling. Additionally, the algorithm has the capacity to consider patient priority. Empirical experiments were conducted to evaluate the performance of the proposed algorithm, using a synthetic data set specifically developed to simulate the medical treatment scheduling problem. The experimental results showed the LSWT-GA algorithm outperforms other algorithms, obtaining the optimal makespan more frequently than a List Scheduling Genetic Algorithm (LS-GA) using traditional survivor selection policies and a standard genetic algorithm using random population initialisation (Random-GA). In addition to the novel genetic algorithm, LSWT-GA, the paper also makes a theoretical contribution by evaluating the run time of the LSWT-GA for makespan minimisation. The proposed algorithm and related findings can be applied directly to the administration systems in medical and healthcare centres and helps improve the deployment of medical resources for better treatment effect.