Mathematical Problems in Engineering

Volume 2015, Article ID 907034, 8 pages

http://dx.doi.org/10.1155/2015/907034

## An Ant Optimization Model for Unrelated Parallel Machine Scheduling with Energy Consumption and Total Tardiness

^{1}School of Computer Science, Guangdong Polytechnic Normal University, Guangzhou 510665, China^{2}XingFa Aluminum Holdings Limited, Foshan 528061, China^{3}School of Mechanical Engineering, Guangdong University of Technology, Guangzhou 510006, China

Received 8 October 2014; Accepted 9 January 2015

Academic Editor: Ben T. Nohara

Copyright © 2015 Peng Liang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This research considers an unrelated parallel machine scheduling problem with energy consumption and total tardiness. This problem is compounded by two challenges: differences of unrelated parallel machines energy consumption and interaction between job assignments and machine state operations. To begin with, we establish a mathematical model for this problem. Then an ant optimization algorithm based on ATC heuristic rule (ATC-ACO) is presented. Furthermore, optimal parameters of proposed algorithm are defined via Taguchi methods for generating test data. Finally, comparative experiments indicate the proposed ATC-ACO algorithm has better performance on minimizing energy consumption as well as total tardiness and the modified ATC heuristic rule is more effectively on reducing energy consumption.

#### 1. Introduction

In recent years, energy saving has been growing a great interest due to sequence of serious environmental impacts and rising energy cost [1–3]. In manufacturing industry, machine energy consumption can be characterized by power, process time, and state of machines [4]. In particular, a large amount of energy is wasted while keeping idle machine running (i.e., not processing jobs but still running machine) [5–7]. Research on Wichita, an aircraft small-part supplier, shows that at least 13% of total energy consumption can be saved by simply turning off machines while they are not processing any jobs [8]. Kordonowy [9] investigates the background runtime operations of machine and observes that more than 30% of input energy is consumed by background processes. What is more, Drake et al. [10] show that there is a significant amount of energy consumption while machine keeps on idling when no jobs are processed.

As a result, research on minimizing energy consumption with machine operation scheduling should be of benefit to energy saving and reducing carbon dioxide emissions. Only a few references consider the objective of energy consumption [4, 11]. Swaminathan and Chakrabarty [12] considered energy consumption in control systems to extend the life of batteries. Research on Tiwari et al. [13] proved that there is about 40% energy saving when proper power control software is used in microprocessor manufacturing. Mouzon and Yildirim [14] considered the problem of minimizing total energy consumption and total tardiness on signal machine. The total energy consumption is measured by summation of idle power and machine setup power. However, the key to save energy on single machine problem is to determine if the machine should be turned off or not during idle time. Yildirim and Mouzon [15] gave a math mathematical model for minimizing total energy consumption as well as max completion time on signal machine. A conventional genetic algorithm is adopted.

Actually, most of manufacturing systems are unrelated parallel machines. Furthermore, the manager should consider not only the energy consumption costs, but also the due dates of jobs. Ant colony optimization (ACO) algorithm has become more preferable to solve combinatorial optimization problems [16–18]. Yagmahan and Yenisey proposed a multiobjective ant colony system algorithm to solve a flow shop scheduling problem with respect to both of makespan and total flowtime [19]. Lin et al. [20] considered an ACO algorithm to solve the problem of scheduling unrelated parallel machines to minimize total weighted tardiness. Arnaout et al. [21] addressed the nonpreemptive unrelated parallel machine scheduling problem with machine dependent and sequence-dependent setup times via a modified ACO algorithm. The results showed that ACO outperformed the other algorithms. In this paper, we begin the research of minimizing energy consumption and total tardiness on unrelated parallel machines. The energy consumption of each machine is composed of power cost of machine setup (i.e., machine turning off and then turning on) and power wasted during machine idle period. The problem is formulated by a weighted summation of energy consumption and total tardiness. For solving this problem, we develop an ACO algorithm with ATC rules in which a machine reselection operation is applied.

After this introduction, we describe the problem in Section 2 and the mathematical model is presented in Section 3. The proposed ATC-ACO algorithm is set out in Section 4. Computation results and comparative analysis on 27 test problem configurations and 2187 experiments’ results are shown in Section 5. Finally, the main conclusions are included in Section 6.

#### 2. Problem Definition

In this section, a mathematical model is proposed for unrelated parallel machines with the objective of minimizing energy consumption and total tardiness, which is NP-hard, since minimizing energy consumption and total tardiness on single machine is proved to be NP-hard [14]. There are independent jobs that have to be processed on parallel machines. Each job can be processed by only one machine and each machine is continuously available. Each job arrives at time and has a process time on machine and a due date . The total tardiness is defined as , where represent the completion time of job . The machine characteristics are defined as follows. Machine consumes power while machine stands idle. Furthermore, machine consumes power when it is turned off and then turned on (i.e., a setup occurs). To solve this problem, total tardiness and energy consumption must be considered together. If there is a long idle period between two jobs, it may choose to turn off machine to save energy. It means that when the idle energy consumption is greater than machine setup energy consumption , the machine will be turned off to save energy. Finally, we conclude the breakeven duration is the ratio of machine setup energy consumption to machine idle energy consumption :

Unlike single machine scheduling framework proposed by Yildirim and Mouzon [15], unrelated parallel machines scheduling problem is much more complicated. Job assignment is affected not only by the processing time and tardiness cost, but also by the state of machine, which is illustrated on two-machine example in Figure 1. Assume six jobs denoted are scheduled on two machines denoted . The process time , release time , and due date are listed in Table 1. We use horsepower (hp) as the unit of power consumption. The setup energy is defined as hp and hp, idle power consumption is set to hp/sec and hp/sec, and tardiness cost is set to hp/sec, .