Makespan Minimization Research Articles

New pseudo polynomial algorithms for a partial resource-constrained project scheduling problem

The resource-constrained project scheduling problem (RCPSP) aims to arrange activities to be processed by limit renewable resources. In real life, not all resources are constrained, nor do all activities face resource restrictions in a project. The RCPSP may emerge in parts of the project, which could be named as the partial RCPSP. In this paper, we consider a typical partial RCPSP that arranging $ K $ renewable resources to execute $ H $ parallel activities in a project for minimum makespan. In particular, we consider the problem under hypotheses that $ N=2 $ and $ N<H\le 2N $. We study the problem from the perspective of 'resource-unconstrained$ \rightarrow $resource-constrained', and discover the relationships between 'scheduling the $ H $ activities' and 'project makespan'. Based on the relationships, we redescribe the problem as new formats, which include only the $ H $ considered activities instead of all activities and emphasize the positions of theses activities on the activity-chains, and then present pseudo polynomial time algorithms for the problem under the hypotheses of $ N=2 $. We evaluate the performance of our algorithms, and present detailed computational results of instances that evaluate the efficiency and competitiveness of our procedure. For example, computational experiments test that about 0.008 seconds average is required for exactly disposing 300 parallel activities using our algorithms, while about 432 seconds average is required using general exact approaches. The proposed theories and methodologies can be extended to general partial RCPSP and some important types of RCPSP (such as reactive RCPSP). Furthermore, the obtained results may enlighten ideas and more effective theoretical tools for general RCPSP/MS-RCPSP.

A competitive constraint programming approach for the group shop scheduling problem

In this paper, we propose a competitive Constraint Programming (CP) approach to solve the Group Shop Scheduling Problem (GSSP) under the makespan minimization criteria. Our contribution is two-fold: we provide a flexible mathematical formulation to solve the GSSP that can be used without change to solve other closed-related scheduling problems such as the Open Shop Scheduling Problem (OSSP), Job Shop Scheduling Problem (JSSP), and Mixed Shop Scheduling Problem (MSSP); and we improve several lower bounds and upper bounds from 130 classical GSSP instances from the literature. We evaluate our approach by comparing the performance with competitive methods mainly based on metaheuristics, where we were able to prove optimality in more than 85% of the instances in competitive running time, with a relative percentage deviation lower than 3% on average. In contrast to metaheuristics approaches, our CP method does not require calibrations of multiple parameters, several replicates for each instance, and complex computational coding to be competitive in both, solution quality and computational running times.

Switched max-plus linear-dual inequalities for makespan minimization: the case study of an industrial bakery shop1

In this paper, an industrial bakery shop is modeled by switched max-plus linear-dual inequalities (SLDIs). SLDIs are timed discrete event systems suitable for describing flow shops with time-window constraints and switching operating modes, where each mode corresponds to a job type. We consider the scheduling problem of minimizing the makespan of the shop, and we show that the application of methods based on the max-plus algebra leads to a faster solution compared to standard techniques. The results of the paper are general, in the sense that they can be applied to any permutation flow shop with time-window constraints.

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.

Optimizing the Expected Maximum of Two Linear Functions Defined on a Multivariate Gaussian Distribution

We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random variables that are arbitrarily correlated, and we show that the problem is NP-hard even if the space of feasible solutions is unconstrained. We exploit a closed-form expression for the objective function from the literature to construct a cutting-plane algorithm for a highly nonlinear function, which includes the evaluation of the cumulative distribution function and probability density function of a standard normal random variable with decision variables as part of the arguments. To exhibit the model’s applicability, we consider two featured applications. The first is daily fantasy sports, where the algorithm identifies entries with positive returns during the 2018–2019 National Football League season. The second is a special case of makespan minimization for two parallel machines and jobs with uncertain processing times; for the special case where the jobs are uncorrelated, we prove the equivalence between its deterministic and stochastic versions and show that our algorithm can deliver a constant-factor approximation guarantee for the problem. The results of our computational evaluation involving synthetic and real-world data suggest that our discretization and upper bounding techniques lead to significant computational improvements and that the proposed algorithm outperforms suboptimal solutions approaches. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1259 .

On the complexity of open shop scheduling with time lags

The minimization of makespan in open shop with time lags has been to be shown NP-hard in the strong sense even for the case of two machines and unit-time operations. The minimization of total completion time, however, has remained open for that case though it has been it be shown NP-hard in the strong sense for weighted total completion time or for jobs with release dates. This note shows that the minimization of total completion time for two machines and unit-time operations is NP-hard in the strong sense which answers the long standing open question.

Two-agent scheduling on a bounded series-batch machine to minimize makespan and maximum cost

The paper studies simultaneous minimization of makespan of agent A and the maximum cost of agent B on a bounded series-batch machine, in which different agents’ jobs cannot be scheduled in the same batch and each agent’s objection function is only related to the completion times of his own jobs. In the paper, we present an algorithm that can obtain all Pareto optimal schedules of our problem in O ( n A + n B 5 ) time.

Multi-objective scheduling in the vegetable processing and packaging facility using metaheuristic based framework

This article solves the scheduling problem in vegetable processing and packaging facility, where multiple types of vegetable consignments need to be packed using dissimilar parallel machines. The problem considers the real-life complexities, such as arrival and discharge time of consignments and setup time for the pre-processing requirement between consecutive consignments. The objective is to identify optimal schedules with minimum total cost and minimum makespan by satisfying the associated constraints. This study proposes an optimization framework incorporating metaheuristic techniques and heuristic mechanism to determine optimal solution for the scheduling problem. The heuristic mechanism assists metaheuristic techniques in reducing constraint violations by rescheduling vegetable consignments assigned to the machine. The efficacy of the proposed framework is analyzed by solving a case study using multiple single and multi-objective metaheuristic techniques. All metaheuristic techniques are able to determine a better schedule for optimizing total cost and makespan using the proposed framework than their naïve metaheuristic technique on solving the single objective model. Differential Evolution with Multi-Population Based Ensemble of Mutation Strategies provided a better solution than other techniques for the single objective model of both objectives. Non-dominated Sorting Teaching Learning Based Optimization, with heuristic mechanism, provided strictly dominating solutions compared to the other techniques.

Modelling and optimization of distributed heterogeneous hybrid flow shop lot-streaming scheduling problem

More enterprises are facing rapid and changing market demand and small but frequent orders, and they extend the production capability of manufacturing systems to respond to these changes. This paper studies a distributed heterogeneous hybrid flow shop lot-streaming scheduling problem (DHHFLSP) with the minimization of makespan. In the DHHFLSP, the processing capacity of each factory is different and each job can be split into several sub-lots. These sub-lots are assigned to several non-identical factories. The mixed-integer linear programming model (MILP) of DHHFLSP is established. To solve the DHHFLSP, eighteen constructive heuristics and an iterated local search algorithm (ILS) are designed. In constructive heuristics, the jobs are sorted according to several time-based heuristic rules or they are divided into several groups according to bottleneck stages, and then these jobs are assigned into factories through two NEH-based job assignment rules. In the ILS, the NEH-based heuristic (NEHafter) plus Largest medium rule is used to generate an initial solution. Two greedy insertion operators with or without critical factories are adopted to generate the perturbation solutions. Four greedy local search operators are designed to make a deep search. The influence of the parameters and main components are investigated by a comprehensive analysis. The comparisons with several related algorithms on extensive testing instances demonstrate the effectiveness and efficiency of the ILS algorithm.

A makespan minimization problem for versatile developers in the game industry

Today, the development of a modern video game draws upon multiple areas of expertise. Moreover, its development cost could be as high as tens of millions of dollars. Consequently, we should carefully schedule its jobs so as not to increase the total cost. However, project leaders traditionally treat developers alike or even schedule all the jobs manually. In this study, we consider a versatile-developer scheduling problem. The objective is to minimize the makespan of a game project. We propose a branch-and-bound algorithm (B&B) to generate the optimal schedules for small problem instances. On the other hand, an imperialist competitive algorithm (ICA) is proposed to obtain approximate schedules for large problem instances. Lastly, computational experiments are conducted to show the performances of both algorithms. When the problem size is small (e.g., n ≤ 12), B&B can generate the optimal schedules within 5 s. For some large problem instances (e.g., n = 600), near-optimal schedules can be obtained by ICA within 10 min. The final results imply that both algorithms converge quickly and are of high solution quality.

Cost effective hybrid genetic algorithm for workflow scheduling in cloud

Cloud computing plays a significant role in everyone’s lifestyle by snugly linking communities, information, and trades across the globe. Due to its NP-hard nature, recognizing the optimal solution for workflow scheduling in the cloud is a challenging area. We proposed a hybrid meta-heuristic cost-effective load-balanced approach to schedule workflow in a heterogeneous environment. Our model is based on a genetic algorithm integrated with predict earliest finish time (PEFT) to minimize makespan. Instead of assigning the task randomly to a virtual machine, we apply a greedy strategy that assigns the task to the lowest-loaded virtual machine. After completing the mutation operation, we verify the dependency constraint instead of each crossover operation, which yields a better outcome. The proposed model incorporates the virtual machine’s performance variance as well as acquisition delay, which concedes the minimum makespan and computing cost. One of the most astounding aspects of our cost-effective hybrid genetic algorithm (CHGA) is its capacity to anticipate by creating an optimistic cost table (OCT) while maintaining quadratic time complexity. Based on the results of our meticulous experiments on some real-world workflow benchmarks and comprehensive analysis of some recently successful scheduling algorithms, we concluded that the performance of our CHGA is melodious. CHGA is 14.58188%, 11.40224%, 11.75306%, and 9.78841% cheaper than standard Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Cost Effective Genetic Algorithm(CEGA), and Cost-Effective Load-balanced Genetic Algorithm (CLGA), respectively.

Parallel solutions for preemptive makespan scheduling on two identical machines

We consider online preemptive scheduling of jobs arriving one by one, to be assigned to two identical machines, with the goal of makespan minimization. We study the effect of selecting the best solution out of two independent solutions constructed in parallel in an online fashion. Two cases are analyzed, where one case is purely online, and in the other one jobs are presented sorted by non-increasing sizes. We show that using two solutions rather than one improves the performance significantly, but that an optimal solution cannot be obtained for any constant number of solutions constructed in parallel. Our algorithms have the best possible competitive ratios out of algorithms for each one of the classes.

Mathematical Formulations for Asynchronous Parallel Disassembly Planning of End-of-Life Products

Disassembly is one of the most time-consuming and labor-intensive activities during the value recovery of end-of-life (EOL) products. The completion time (makespan) of disassembling EOL products is highly associated with the allocation of operators, especially in parallel disassembly. In this paper, asynchronous parallel disassembly planning (APDP), which avoids the necessity to synchronize disassembly tasks of manipulators during the parallel disassembly process, is studied to optimize the task assignment of manipulators for minimal makespan. We utilize four mixed integer linear programming (MILP) formulations to identify the optimal solutions. A set of different-sized instances are used to test and compare the performance of the proposed models, including some real-world cases. Finally, the proposed exact algorithm is further compared with the existing approach to solving APDP. Results indicate that a significant difference exists in terms of the computational efficiency of the MILP models, while three of four MILP formulations can efficiently achieve better solutions than that of the existing approach.

Multiple Mobile Robots Coordination in Shared Workspace for Task Makespan Minimization

In this paper we consider a system of multiple mobile robots (MMRS) and the process of their concurrent motion in a shared two-dimensional workspace. The goal is to plan the robot movement along given fixed paths so as to minimize the completion time of all the robots while ensuring that they never collide. Thus, the considered problem combines the problems of robot schedule optimization with collision and deadlock avoidance. The problem formulation is presented and its equivalent reformulation that does not depend explicitly on the geometry of the robot paths is proposed. An event-based solution representation is proposed, allowing for a discrete optimization approach. Two types of possible deadlocks are identified and deadlock avoidance procedures are discussed. We proposed two types of solving methods. First, we implemented two metaheuristics: the local-search-based taboo search as well as the population-based artificial bee colony. Next, we implemented 14 simple constructive algorithms, employing dispatch rules such as first-in first-out, shortest distance remaining first, and longest distance remaining first, among others. A set of problem instances for different numbers of robots is created and provided as a benchmark. The effectiveness of the solving methods is then evaluated by simulation using the generated instances. Both deterministic and lognormal-distributed uncertain robot travel times are considered. The results prove that the taboo search metaheuristic obtained the best results for both deterministic and uncertain cases, with only artificial bee colony and a few constructive algorithms managing to remain competitive. Detailed results as well as ideas to further improve proposed methods are discussed.

Fog-GMFA-DRL: Enhanced deep reinforcement learning with hybrid grey wolf and modified moth flame optimization to enhance the load balancing in the fog-IoT environment

Internet of Things (IoT) can facilitate a plethora of data transactions among various servers. In the IoT, fog servers are utilized to achieve effective data transactions from dynamic devices. However, load balancing is still a significant task researcher mainly focus on mitigating the load balancing issue. Some virtual machines may be overloaded when other virtual machines are idle due to a bad scheduling policy. Therefore, the proposed model is based on dynamic load balancing in a fog-IOT environment by utilizing a novel hybrid Grey Wolf Optimization (GWO) with the Modified Moth Flame algorithm (MMFA). In addition, the GMFA mainly helps to enhance Deep reinforcement learning (DRL). The performance of the actor-critic based deep reinforcement learning (DRL) approach is enhanced with the GMFA algorithm, and this combined strategy is named GMFA-DRL. RL offers several advantages regarding resource allocation issues, and simulations demonstrate that it performs better than reactive techniques. The proposed GMFA-DRL approach is implemented through the Python-based platform-Jupyter. The performance is evaluated using performance matrices such as Throughput, Latency, Makespan, Load Balancing Level (LBL), and Energy Consumption. The simulation results illustrate that the proposed model achieves high Throughput, low energy consumption, minimum Latency, minimum Makespan, and load balancing results. Therefore, the proposed approach can be proven more effective than the existing technique.

Enhanced hybrid multi-objective workflow scheduling approach based artificial bee colony in cloud computing

This paper presents a hybrid approach based Binary Artificial Bee Colony (BABC) and Pareto Dominance strategy for scheduling workflow applications considering different Quality of Services (QoS) requirements in cloud computing. The main purpose is to schedule a given application onto the available machines in the cloud environment with minimum makespan (i.e. schedule length) and processing cost while maximizing resource utilization without violating Service Level Agreement (SLA) among users and cloud providers. The proposed approach is called Enhanced Binary Artificial Bee Colony based Pareto Front (EBABC-PF). Our proposed approach starts by listing the tasks according to priority defined by Heterogeneous Earliest Finish Time (HEFT) algorithm, then gets an initial solution by applying Greedy Randomized Adaptive Search Procedure (GRASP) and finally schedules tasks onto machines by applying Enhanced Binary Artificial Bee Colony (BABC). Further, several modifications are considered with BABC to improve the local searching process by applying circular shift operator then mutation operator on the food sources of the population considering the improvement rate. The proposed approach is simulated and implemented in the WorkflowSim which extends the existing CloudSim tool. The performance of the proposed approach is compared with Heterogeneous Earliest Finish Time (HEFT) algorithm, Deadline Heterogeneous Earliest Finish Time (DHEFT), Non-dominated Sort Genetic Algorithm (NSGA-II) and standard Binary Artificial Bee Colony (BABC) algorithm using different sizes of tasks and various benchmark workflows. The results clearly demonstrate the efficiency of the proposed approach in terms of makespan, processing cost and resources utilization.

Parallel Scheduling of Machines, Tool Transporter and Tools in a Multi Machine FMS with Alternative Routing Using Flower Pollination Algorithm

Scheduling jobs and tools is a significant problem for manufacturing systems. Inefficient job scheduling and tool loading planning may result in the underutilization of capital-intensive machines and a high machine idle time. Tools are transferred by a tool transporter (TT) between machines. Therefore, efficient scheduling of jobs, TT, and the tools with alternate machines enables a manufacturing system to increase machines’ utilization and decrease their idle times. This paper addresses machines, TT, and tools concurrent scheduling with alternate machines and only one copy of every tool variety is made available where the tools are costly, in a multi-machine flexible manufacturing system, taking into account tool transfer times for makespan (MS) minimization. The tools are placed in a central tool magazine, which shares and serves them to many machines to cut down the price of duplicating the tools in each machine. The problem is to assign machines from alternate machines, tools, and corresponding TT trips, including the deadheading trip and loaded trip, to job operations for MS minimization. This paper uses a nonlinear mixed-integer programming framework to present the problem, and a flower pollination algorithm (FPA) is employed to solve it. The results show that FPA outperforms the Jaya algorithm, and the usage of alternate machines for the operations reduces the MS. Reduction in MS indicates an improvement in utilization of resources.

Concurrent Scheduling of Machines and AGVS in Multi-Machine FMS with Alternative Routing Using Symbiotic Organisms Search Algorithm

This paper addresses machines and automated guided vehicles (AGVs) simultaneous scheduling with alternative machines in a multi-machine flexible manufacturing system (FMS) to produce the best optimal sequences for the minimization of makespan (MKSN). This problem is highly complex to solve because it involves the selection of machines for job operations (jb-ons), the sequencing of jb-ons on the machines, the assignment of AGVs, and associated trips such as AGVs’ deadheaded trip and loaded trip times to jb-ons. This paper offers a nonlinear mixed integer programming (MIP) formulation for modeling the problem and the symbiotic organisms search algorithm (SOSA) for solving the problem. For verification, a manufacturing company’s industrial problem is employed. The results show that SOSA outperforms the existing methods and the Jaya algorithm, and using alternate machines for the operations can reduce the MKSN.

Hybrid meta‐heuristic algorithm for optimal virtual machine placement and migration in cloud computing

SummaryData sharing in cloud computing happens with multiple participants to freely distribute the group data, which focuses on advancing the effectiveness of work in cooperative backgrounds and has attained widespread benefits. The main intent of this article is to accomplish a virtual machines (VMs) placement and migration model using a hybrid meta‐heuristic concept. A new meta‐heuristic algorithm named DJ‐HA is developed for optimal VM placement and migration to reduce the count of active servers, and minimization of makespan, and energy consumption with a faster convergence rate in a cloud background. Then, the VM migration is done based on the multi‐objective function concerning energy consumption and makespan using the same hybrid DJ‐HA. From the result analysis, the energy consumption of the DJ‐HA is correspondingly secured at 4.3%, 3.5%, 31%, and 33% more advanced than PSO, GWO, DHOA, and JA, at the 100th iteration for Experiment 1. Accordingly, the cost function of the suggested DJ‐HA is secured at 88.8%, 89.4%, 33.3%, and 50% increased than PSO, GWO, DHOA, and JA at the 100th iteration for Experiment 4. Hence, it is proved that the suggested VM migration using DJ‐HA is enriched than the other conventional algorithms.

Improving makespan in dynamic task scheduling for cloud robotic systems with time window constraints

A scheduling method in a robotic network cloud system with minimal makespan is beneficial as the system can complete all the tasks assigned to it in the fastest way. Time window constraints on tasks are a natural way to order tasks. The makespan is the maximum amount of time between when the first processing unit starts executing its first task and when all processing units have completed their last scheduled task. Load balancing allocation and scheduling ensures that the time between when the first processing unit completes its scheduled tasks and when all other processing units complete their scheduled tasks is as short as possible. We propose a method to ensure that the time window constraints are met. We propose the grid of all tasks balancing algorithm (GTBA) for distributing and scheduling tasks with minimum makespan. The GTBA method is a combinatorial method that describes a way to distribute tasks among processing units and schedule them when a set of tasks arrives in such a way that the makespan is minimized, the loads of all processing units are nearly balanced, and it is ensured that the time window constraints are met. We prove the correctness of the proposed algorithm and present simulations illustrating the obtained results.