Two-agent Scheduling Problem Research Articles

Erratum to “Competitive Two-Agent Scheduling and Its Applications”

Erratum to “Competitive Two-Agent Scheduling and Its Applications” The paper “Competitive Two-Agent Scheduling and Its Applications” by Leung, Pinedo, and Wan appeared in Operations Research in 2010. Because competitive agent scheduling has become an active area of research, the paper has garnered its share of citations over the years. This particular direction of scheduling research has become so popular that the book Multiagent Scheduling: Models and Algorithms by Agnetis, Billaut, Gawiejnowicz, Pacciarelli, and Soukhal appeared in 2014. However, one of the results in the Leung et al. paper had a minor error in it. The paper considers a two-agent scheduling problem with a single machine, with each one of the two agents having their own set of jobs. The first agent has as the objective the minimization of the total tardiness of her set of jobs, and the second agent has to ensure that the total completion time of her set of jobs is less than or equal to Q. Leung et al. showed, through a dynamic programming formulation, that this problem can be solved in pseudopolynomial time and determined the time complexity of their algorithm. Their proof contains a minor error. The proper algorithm is still pseudopolynomial but of a slightly higher time complexity.

Integrated scheduling of production and distribution with two competing agents

This paper studies a two-agent scheduling problem on a single batch machine. Each job j has a processing time pj and a size sj . Jobs should be delivered to the agent as soon as possible. The objective is to minimize the service span of the first agent subject to an upper bound on the makespan of the other agent. We propose two approximation algorithms when jobs have identical processing times and in the general case. Then we conduct theoretical analyses to provide the provable guarantees on the performances of the algorithms. Finally, we conduct simulation analyses based on randomly generated instances to evaluate the average performances of the algorithm in the general case.

Two-Agent Single Machine Scheduling with Deteriorating Jobs and Rejection

We consider two two-agent scheduling problems with deteriorating jobs and rejection. Two agents A and B compete for the usage of a single machine. The actual processing time of job J j X is p j X = b j X a + b t , X ∈ A , B , where b j X is the normal processing time of J j X , a ≥ 0 , b ≥ 0 and t denotes the starting time of J j X . A job is either rejected by paying a rejection penalty, or accepted and processed on the machine. The objective is to minimize the sum of the completion time of the accepted A -jobs and total rejection penalty of the rejected A -jobs subject to an upper bound on the sum of the given objective function f B of the accepted B -jobs and total rejection penalty of the rejected B -jobs, where f B ∈ C max B , ∑ C j B . We give dynamic programming algorithms for them, respectively. When f B = C max B , we present a fully polynomial-time approximation scheme (FPTAS) for the case a = 0 and b = 1 . When f B = ∑ C j B , a fully polynomial-time approximation scheme is also presented.

Energy-aware optimization for the two-agent scheduling problem with fuzzy processing times

Energy-aware optimization for the two-agent scheduling problem with fuzzy processing times

A Lagrangian heuristics for balancing the average weighted completion times of two classes of jobs in a single-machine scheduling problem

• We introduce a new single-machine scheduling problem involving two classes of jobs. • The objective is to balance the average weighted completion times. • We show the NP-hardness of the problem. • We solve the problem by a Lagrangian relaxation heuristics. • Numerical results are presented on a set of randomly generated test problems. We tackle a new single-machine scheduling problem, whose objective is to balance the average weighted completion times of two classes of jobs. Because both the job sets contribute to the same objective function, this problem can be interpreted as a cooperative two-agent scheduling problem, in contraposition to the standard multiagent problems, which are of the competitive type since each class of job is involved only in optimizing its agent's criterion. Balancing the completion times of different sets of tasks finds application in many fields, such as in logistics for balancing the delivery times, in manufacturing for balancing the assembly lines and in services for balancing the waiting times of groups of people. To solve the problem, for which we show the NP-hardness, a Lagrangian heuristic algorithm is proposed. In particular, starting from a nonsmooth variant of the quadratic assignment problem, our approach is based on the Lagrangian relaxation of a linearized model and reduces to solve a finite sequence of successive linear assignment problems. Numerical results are presented on a set of randomly generated test problems, showing the efficiency of the proposed technique.

A Hybrid Heuristic for a Two-Agent Multi-Skill Resource-Constrained Scheduling Problem

This paper addresses an industrial case of the two-agent scheduling problem with a global objective function. Each agent manages one or several projects and competes with another agent for the use of common multi-skilled employees. There is a pool of employees, each of which can perform a set of skills with heterogeneous performance levels. The objectives of the two agents are both to minimize the total weighted tardiness of its tasks. Furthermore, We assume that some constraints (soft constraints) can be violated when there is no feasible schedule for the problem. Thus, the global objective function minimizes the constraint violations by reducing the undesirable deviations in the soft constraints from their respective goals. The overall objective is to find a schedule that minimizes both agents objective functions (local objectives) and the global objective function. We provide a mixed-integer goal programming (MIGP) formulation for the problem. In addition, we present a hybrid algorithm combining an exact procedure, a greedy heuristic, and a genetic algorithm to find an approximate Pareto solution set. We compare the performance of the hybrid algorithm against the corresponding MIGP formulation with simulated instances derived from real-world instances.

Online single-machine scheduling to minimize the linear combination of makespans of two agents

This paper considers an online two-agent scheduling problem. All jobs belong to two agents A and B, and each job is arriving over time, which means that no information of the job is known in advance before its arrival. The objective is to schedule all jobs on a single machine such that the linear combination of the makespans of the two agents, i.e. CmaxA+θCmaxB(θ≥1), is minimized. For the problem, we propose an online algorithm and show the competitive ratio of the algorithm is 1+2θ+θ22+2θ+θ2.

Nature-Inspired Metaheuristics for Two-Agent Scheduling with Due Date and Release Time

This paper delves into a two-agent scheduling problem in which two agents are competing for a single resource. Each agent has a set of jobs to be processed by a single machine. The processing time, release time, weight, and the due dates of each job are known in advance. Both agents have their objectives, which are conflicting in nature. The first agent tries to minimize the total completion time, while the second agent tries to minimize the number of tardy jobs. The two agents’ scheduling problem, an NP-hard problem, has a wide variety of applications ranging from the manufacturing industry to the cloud computing service provider. Due to the wide applicability, each variation of the problem requires a different algorithm, adapted according to the user’s requirements. This paper provides mathematical models, heuristic algorithms, and two nature-based metaheuristic algorithms to solve the problem. The algorithm’s performance was gauged against the optimal solution obtained from the AMPL-CPLEX solver for both solution quality and computational time. The outlined metaheuristics produce a solution that is comparable with a short computational time. The proposed metaheuristics even have a better solution than the CPLEX solver for medium-size problems, whereas the computation times are much less than the CPLEX solvers.

Algorithms for a two-agent single machine scheduling problem to minimize weighted number of tardy jobs

This paper considers a two-agent single machine scheduling problem, with a weighted number of tardy jobs as objectives. Each agent has a finite set of jobs to be processed on a single machine. Parameters like the processing time, the due date, and the weight of jobs are known in advance. The objective is to minimize the weighted number of tardy jobs of the first agent, subject to the upper bound of the weighted number of tardy jobs of the second agent. Two metaheuristics based on Particle Swarm Optimization and Tabu Search are introduced to solve this problem. To establish the performance of the proposed meta-heuristics, a dynamic programming based exact algorithm is developed. To test the validity of the proposed algorithm, a numerical experiment is performed on randomly generated problem instances to compare the performance of the proposed algorithms.

Proportionate Flow Shop Scheduling with Two Competing Agents to Minimize Weighted Late Work and Weighted Number of Late Jobs

We investigate a competitive two-agent scheduling problem in the setting of proportionate flow shop, where the job processing times are machine-independent. The scheduling criterion of one agent is to minimize its total weighted late work, and the scheduling criterion of the other agent is to minimize its total weighted number of late jobs. The goal is to find the Pareto-optimal curve (i.e., the set of all Pareto-optimal points) and identify a corresponding Pareto-optimal schedule for each Pareto-optimal point. An exact pseudo-polynomial-time algorithm and an [Formula: see text]-approximate Pareto-optimal curve are designed to solve the problem, respectively.

Two-agent scheduling on mixed batch machines to minimise the total weighted makespan

This paper studies a two-agent scheduling problem on mixed batch machines in parallel. A mixed batch machine can process several jobs simultaneously as a batch, as long as the number of jobs in the batch does not exceed the machine capacity. The processing time of a mixed batch is the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective is to minimise the weighted sum of two agents' makespans. We present four approximation algorithms based on two strategies: the machine-centric strategy and the agent-centric strategy. For each strategy, a full batch longest processing time (FBLPT) rule and a longest processing time greedy (LPTG) rule are used. We conduct theoretical analyses based on the worst-case performance ratio to provide the provable guarantees on the performances of the algorithms, and simulation analyses based on randomly generated instances to evaluate the average performances of the algorithms. Furthermore, we verify the consistency between the theoretical and simulation results. The algorithms using agent-centric strategy perform better than ones using machine-centric strategy. Finally, we provide managerial insights for the problem by analysing the technological parameters of batches, importance of agents, and demand seasonality.

Two-agent scheduling of unit processing time jobs to minimize total weighted completion time and total weighted number of tardy jobs

In this paper, we revisit a two-agent scheduling problem on a single machine with jobs of unit processing times. In this problem, we have two agents: agent A and agent B. Each agent has his/her own disjoint job set. The objective of agent A is to minimize the total weighted completion time, and the objective of agent B is to minimize the total weighted number of tardy jobs. The complexity of the decision version of the problem was proposed as open in Oron et al. (European Journal of Operational Research 244:86–99, 2015). We construct a reduction from a variant of Even-Odd Partition to show that this problem is NP-complete. Finally, we also propose a pseudo-polynomial algorithm and a dual FPTAS for the problem.

Two-agent scheduling problems under rejection budget constraints

We study two single machine scheduling problems with two competing agents and the option of rejecting jobs. We seek to minimize the scheduling criterion of one agent under the condition that the maximum completion time of the second agent does not exceed a given upper-bound. Moreover, we assume that jobs are either scheduled or rejected (incurring job-dependent penalties) subject to an upper bound on the total rejection cost. The scheduling objective for the first agent is either the makespan or the total completion time, while that of the second agent is the makespan. As both problems are known to be NP-hard, we develop efficient pseudo-polynomial dynamic programming algorithms to obtain optimal solutions. The algorithms are shown to efficiently solve medium to large size problems through an extensive numerical study.

Due date assignment and two-agent scheduling under multitasking environment

This paper addresses a two-agent scheduling problem with due date assignment under multitasking environment, in which the due dates of the jobs from the first agent are decision variables to be determined using the unrestricted (usually referred to as DIF) due date assignment method. Each agent requests the processing of its own set of jobs on a machine and wishes to minimize a certain scheduling criterion related to the completion times of its jobs only. Under multitasking, when a job (primary job) is processed, it is inevitably interrupted by other jobs (waiting jobs) that are available but unfinished, and the amount of time that each waiting job interrupting the primary job is a linear function of the remaining processing time of the waiting job. The overall objective is to determine the optimal primary job sequence along with the due dates of the jobs from the first agent as to minimize the weighted sum of the due date assignment cost and weighted number of late jobs from the first agent, while maintaining the total completion time of the jobs from the second agent not exceeding a given threshold. We show that the problem is $$\mathcal {NP}$$-hard, devise a pseudo-polynomial time dynamic programming algorithm, establishing that it is $$\mathcal {NP}$$-hard in the ordinary sense, and demonstrate that it admits a fully polynomial-time approximation scheme.

Serial-batching scheduling with two agents to minimize makespan and maximum cost

This paper considers three serial-batching scheduling problems with two agents to minimize the makespan of agent A and maximum cost (or maximum lateness) of agent B simultaneously. In these two-agent scheduling problems, jobs of different agents cannot be processed in a common batch, and the cost function of an agent depends only on its jobs. For each problem, we present a polynomial-time algorithm which can find all Pareto optimal solutions.

Multitasking scheduling problems with two competitive agents

This article addresses several two-agent scheduling problems in the presence of multitasking. Under multitasking, when a task is being executed, it is inevitably interrupted by the tasks that have not been finished, and the interruption time is proportional to the remaining processing time of the interrupting task. Each agent desires to minimize a certain cost function that depends only on its own tasks' completion times. Different cost functions are considered for both agents, including the maximum of regular cost, total completion time, and (weighted) number of tardy tasks. The purpose is to determine a feasible solution for all tasks of the two agents that minimizes the cost value of the first agent while maintaining the cost value of the other agent, not exceeding a given threshold value. Polynomial and pseudo-polynomial time algorithms are proposed to solve the setting, involving various combinations of cost functions.

Two-agent scheduling problems with the general position-dependent processing time

Two-agent scheduling problems with the general position-dependent processing time are studied in this paper. Under the constraint that the makespan of one agent is upper bounded, we show that the problem to minimize the other agent's makespan on a single machine is NP-hard. A polynomial time algorithm is proposed for the problem with learning or deteriorating effects. To minimize the total completion time of one agent's jobs on a single machine, we design a dynamic programming algorithm and an FPTAS. Moreover, we also provide dynamic programming algorithms and FPTASs for the problem to minimize one agent's makespan and the problem to minimize the total completion time of the other agent's jobs on parallel machines.

Two-agent scheduling on a single machine with release dates

This paper considers a two-agent scheduling problem, in which all jobs belong to two agents A and B, and each job has a release date. The objective is to schedule all jobs non-preemptively on a single machine such that the linear combination of the makespans of the two agents, i.e., CmaxA+θCmaxB(θ≥1), is minimized. It is known that the problem is NP-hard in the ordinary sense. For this case, we provide an approximation algorithm with worst-case ratio 1+θ1+θ+θ2. We also propose a Fully Polynomial Time Approximation Scheme (FPTAS) for the problem under study. Finally, computational experiments are conducted to verify the effectiveness of the approximation algorithm and the FPTAS.

A survey of due-date related single-machine with two-agent scheduling problem

In this paper, we consider due-date related single machine scheduling problems, where two agents compete for utilizing a common processing resource (i.e. a single machine). In this paper, we provide a detailed and a systemic literature review of the two-agent scheduling problem dealing with models with a given due date. We consider the following four main cases: 1) the earliness and tardiness, 2) the maximum lateness, 3) the number of tardy jobs and 4) the late work criteria. To do so, we classify due-date related, two-agent scheduling problems into two categories on the basis of the objective function setting, (i.e. the feasibility model and the minimality model). The feasibility model minimizes the objective function of one agent subject to an upper bound on the objective for the other agent. The minimality model assigns certain weights for two agents and as a result minimizes their weighted objectives. In the present paper, we list the computational complexities and proposed algorithms for the due-date related, two-agent scheduling problem, investigated in the literature since 2003.

The complexity of CO-agent scheduling to minimize the total completion time and total number of tardy jobs

In this paper, we revisit a two-agent scheduling problem on a single machine. In this problem, we have two competing agents A and B, which means that the job set of agent A and the job set of agent B are disjoint. The objective is to minimize the total completion time of agent A, under the constraint that the total number of tardy jobs of agent B is no larger than a given bound. The complexity of this problem was posed as open in Agnetis et al. (Oper Res 52:229–242, 2004). Leung et al. (Oper Res 58:458–469, 2010a, b. https://doi.org/10.1287/opre.1090.0744ec ) showed that the problem is binary NP-hard. However, their NP-hardness proof has a flaw. Here, we present a new NP-hardness proof for this problem. Our research shows that the problem is still NP-hard even if the jobs of agent A have a common processing time.