Elsevier

Applied Soft Computing

Volume 70, September 2018, Pages 147-156
Applied Soft Computing

Single machine scheduling with two-agent for total weighted completion time objectives

https://doi.org/10.1016/j.asoc.2018.05.027Get rights and content

Highlights

  • We studied a set of two-agent scheduling problem with a single machine.

  • The paper proposes two heuristics and one metaheuristic to solve the problem.

  • Lower bound has been developed for checking the effectiveness of the heuristics.

  • The efficiency of the heuristic is tested through randomly generated instances.

Abstract

This paper considers two-agent scheduling problem with a single machine which is responsible for processing jobs from two agents. The objective is to minimize the objective function of one agent, subject to an upper bound on the objective function of the other agent. The objectives considered in this paper are, (1) the minimization of total completion time and (2) the minimization of total weighted completion time. To solve these problems, one heuristic and an Ant Colony Optimization algorithm are proposed. The heuristic suggested in the paper are motivated by the Weighted Shortest Processing Time first (WSPT) rule. A numerical experiment is performed on randomly generated problem instances. The performance of the algorithm is evaluated by comparing it with the lower bound value of all three problems considered in the present paper.

Introduction

Scheduling problems for optimizing resource allocation have been studied for many years, as scheduling plays a critical role in many manufacturing and service industries. Furthermore, in manufacturing and service industries, the scheduling decision can have a significant impact on improving the productivity and efficiency of the production system.

During last two decades, two-agent scheduling problems have been used in many systems, such as railroad allocation systems [1], aircraft landing systems [2], telecommunication systems [3], and cloud computing systems [4] etc. Baker and Smith [5] first considered the two-agent scheduling problem. In this problem, two agents compete for limited production resources. Each agent has a set of jobs to be processed using a common processing source. Also, each agent may have different objectives at a given time. For instance, the objective functions could be the minimization of weighted tardiness and earliness [6], [7], [8], the minimization of total completion time [9], [10], [11] or the minimization of the number of tardiness jobs [12], [13].

The two-agent scheduling problem faces the dilemma of minimizing two conflicting objectives associated with two agents. In the present paper, the objectives of two-agents are in conflict, because the completion time of jobs for the two agents are mutually independent. To address the conflicting nature of objective functions, Suresh and Chaudhuri [14] suggested to two approaches: (1) assign weight to the objective of each agent and minimize the weighted objective function; (2) minimize the objective function of one agent while keeping the objective function of the other agent within a pre-specified level. T. E. Cheng et al. [15] named these two types of single machine with multi-agent scheduling problem as a “minimality model” and a “feasibility model,” respectively. The problems studied in the present paper is the feasibility model, in which the objective function of one agent is minimized, while keeping the objective function of the other agent within a pre-specified limit.

In the present paper, we study three two-agent scheduling problems with a total weighted completion time and total completion time objectives. In these problems, all jobs from two agents are processed by a common machine. For each problem, the processing time and weight of each job is given. The problem involves finding a schedule that minimizes the objective function of one agent, subject to an upper bound on the objective function of the other agent. In this paper, depending upon the combination of objectives, three problems are considered.

Problem 1

The objective here is to minimize the total weighted completion time of the one agent, subject to an upper bound on the total weighted completion time of the other agent.

Problem 2

The objective here is to minimize the total weighted completion time of the one agent, subject to an upper bound on the total completion time of the other agent.

Problem 3

The objective here is to minimize the total completion time of the one agent, subject to an upper bound on the total weighted completion time of the other agent.

The novelty of this paper is to introduce three new problems, develop a heuristic and a meta-heuristic, and introduce some benchmark problem instances for future research. The proposed ant-colony meta-heuristic solves the problem in two phases and accommodates the specific requirement of the considered problems.

The structure of the paper is as follows. Section 2 reviews the relevant literature and provides a general review of previous works. Section 3 offers a problem description and mathematical formulations of the two-agent scheduling problems. In Section 4, we present the proposed heuristic algorithm used for solving the scheduling problems studied in the present paper. In Section 5, a numerical experiment is preformed, and the corresponding results are reported. Section 6 provides a conclusion to this paper.

Section snippets

Literature review

The single machine with two agents scheduling problem was introduced by Baker and Smith [5]. They examined three basic objective functions: minimizing makespan, maximum tardiness, and weighted completion time. The various problems studied by Baker and Cole Smith [16] are considered to be NP-hard, as proved by Allesandro Agnetis et al. [9], [17].

There were a number of papers published after the work of Baker and Cole Smith [16] and Allesandro Agnetis et al. [9]. These papers studied various

Problem description and notation

This section provides a description of the problems considered in this paper. The jobs from two agents (agent A and agent B) are scheduled at a single machine which is responsible for processing a number of jobs that are supplied by two agents. The pre-emption of a job is not allowed. Each job has a given processing time and a weight associated with it. The problem considered in this paper involves finding a sequence for these jobs in such a way that the objective function of agent A is

The proposed algorithms

In this section, we introduce a heuristic and an Ant Colony Optimization (ACO) algorithm to solve the Problem 1. As mentioned before, these algorithms can also be used to solve all three problems.

Numerical results and analysis

In this section, numerical experiments and analysis are conducted in order to assess the performance of the proposed heuristics and Ant Colony Optimization algorithm. The branch-and-bound method of Agnetis et al. [9] is used to calculate the lower bound. Although, the branch-and-bound approach provides the optimal solution, it will stop during the execution. The lowest lower bound value from the open branch is reported as the lower bound value for the problem. We also reported the upper bound

Conclusion

The present paper considers single machine with two-agent scheduling problems. The two objectives considered in this paper are the minimization of the total weighted completion time and the minimization of total completion time. According to the different combinations of objective functions, three different problems are considered to minimize the objective function of agent A subject to an upper bound on the objective function of agent B. A heuristic and one meta-heuristic are proposed to solve

Acknowledgements

We are thankful to the reviewers for their valuable comments. This research is partially supported by NSERC discovery Grant 318689.

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