Mathematical Problems in Engineering

Volume 2018, Article ID 5914360, 13 pages

https://doi.org/10.1155/2018/5914360

## A New Imperialist Competitive Algorithm for Multiobjective Low Carbon Parallel Machines Scheduling

School of Automation, Wuhan University of Technology, Wuhan 430070, China

Correspondence should be addressed to Qingyong Zhang; nc.ude.tuhw@gnahzyq

Received 22 October 2017; Revised 6 March 2018; Accepted 11 March 2018; Published 16 April 2018

Academic Editor: Erik Cuevas

Copyright © 2018 Zixiao Pan 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 paper considers low carbon parallel machines scheduling problem (PMSP), in which total tardiness is regarded as key objective and total energy consumption is a non-key one. A lexicographical method is used to compare solutions and a novel imperialist competitive algorithm (ICA) is presented, in which a new strategy for initial empires is adopted. Some new improvements are also added in ICA to obtain high quality solutions, which are adaptive assimilation, adaptive revolution, imperialist innovation, and alliance and the novel way of imperialist competition. Extensive experiments are conducted to test the search performance of ICA by comparing it with methods from literature. Computational results show the promising advantages of ICA on low carbon PMSP.

#### 1. Introduction

Parallel machine scheduling problem (PMSP) is the typical problem in the actual manufacturing processing [1]. Allocating the manufacturing resources and scheduling the production tasks must be designed to achieve the specified performance. In the traditional PMSP, objectives such as total tardiness have extensively been investigated; however, energy consumption and carbon emission are seldom considered [2, 3]. If there is no consideration on energy related objective, then the corresponding research results are difficult to meet the increasingly stringent requirements of energy conservation and environmental protection on manufacturing activities in China. Thus, it is necessary to focus on low carbon PMSP to improve the traditional scheduling performance and reduce carbon emissions and energy consumption.

In recent years, few papers discussed low carbon PMSP. Several heuristic algorithms has been proposed by Li et al. [4] to minimize makespan or the total completion time under the prerequisite that the total of energy cost and disposal cost is not more than a given threshold. Che et al. [5] gave a mixed integer programming model and two-stage heuristic method on unrelated parallel machine scheduling problem with different power cost. Wang et al. [6] proposed a two-stage algorithm to determine the cutting speed and minimize the maximum completion time of jobs under the given power load peak demand. Liang et al. [3] presented an ant colony optimization (ACO) algorithm to optimize the weighted sum of energy consumption and total tardiness; Li et al. [2] described the model of the problem and proposed 10 kinds of heuristic algorithms.

Meanwhile, many works have been done on multiobjective PMSP by using various metaheuristics. Lin et al. [7] proposed tabu-enhanced iterated Pareto greedy algorithm to minimize total tardiness, makespan, and total weighted completion time. Ying [8] gave a multiobjective multipoint simulated annealing for PMSP with total tardiness, total weighted completion time, and maximum completion time. Li et al. [9] provided a hybrid nondominated sorting genetic algorithm-II (NSGA-II [10]) with fuzzy logic controller and an exact method. Pakzad-Moghaddam [11] presented a Lévy flight embedded particle swarm optimization for PMSP with learning and adapting. Afzalirad and Rezaeian [12] reported a multiobjective ant colony optimization to minimize mean weighted flow time and mean weighted tardiness. Zarandi and Kayvanfar [13] applied two multiobjective metaheuristics for PMSP with total cost of tardiness and maximum completion time.

As mentioned above, low carbon PMSP is seldom studied. The major existing methods for low carbon PMSP are heuristic algorithms. Although metaheuristics have been extensively used to solve multiobjective PMSP without energy related objective, they are hardly applied to optimize low carbon scheduling on parallel machines. To the best of our knowledge, only ACO is found to solve PMSP with energy consumption and other metaheuristics such as imperialist competitive algorithm (ICA) are not used. In addition, the literature on multiobjective PMSP hardly considers the relative importance of objectives. In fact, objectives should be treated differently in many actual manufacturing cases. Take make-to-order production as example, the delay delivery of orders often leads to losses in profits and market reputation; when manufacturers face the high pressure of on-time delivery, it is natural for them to regard on-time delivery as key objective and energy consumption as non-key one, so it is necessary to focus on low carbon PMSP with the consideration on the relative importance of objectives.

ICA is a metaheuristic that mimics sociopolitical imperialist competition for global optimal solution [14]. ICA starts with an initial population, which is a group of countries. The countries with best cost are regarded as imperialists and the rest of countries are called colonies. ICA consists of assimilation of colonies, revolution of colonies, and imperialistic competition among empires and so on. ICA has been successfully applied to solve various optimization problems such as the traveling salesman problem [15], skeletal structure optimization [16], facility layout [17, 18], dynamic economic dispatch scheduling [19], assembly line balancing [20–22], artificial neural network optimization [23], supply chain network design [24], and scheduling [25–34]. However, as stated above, ICA is not used to solve the problem of low carbon parallel machine scheduling. ICA is an effective global optimization method and has strong local search ability with flexible structure [35]. These features make ICA have some advantages in solving low carbon PMSP; for example, it is not necessary to improve local search ability like GA. These things on ICA motivate us to apply ICA to solve low carbon PMSP.

In this study, PMSP with total tardiness (key objective) and total energy consumption (non-key objective) is considered and a novel ICA is proposed for minimizing total tardiness with the consideration on total energy consumption. The lexicographical method is adopted to compare solutions. In ICA, some new strategies are applied to generate high quality solutions, which include the novel method for initial empires, adaptive assimilation, adaptive revolution, imperialist innovation, and alliance and the new imperialist competition. A number of experiments are conducted using many instances. Computational results show the effectiveness and advantages of the new ICA on the low carbon PMSP.

The remainder of the paper is organized as follows. Problem under study and introduction to ICA are described in Sections 2 and 3. ICA for the problem is reported in Section 4. Numerical test experiments on ICA are shown in Section 5 and the conclusions are summarized in the final section and some topics of the future research are provided.

#### 2. Problem Description

Low carbon PMSP is described as follows. There are independent jobs processed on unrelated parallel machines. Each job is available at time zero, has a processing time on machine , , and a due date . indicates energy consumption of each machine per unit time. At any time, each machine can process at most one job and each job can be processed on at most one machine.

Low carbon PMSP is composed of machine assignment subproblem and scheduling subproblem. The former is to decide an appropriate machine for each job and the latter is to obtain the processing sequence of jobs on each machine.

The goal of the problem is to minimize the following two objectives simultaneously:where is total tardiness and is total energy consumption, represents the completion time of job , is the total energy consumption of machine , and is the 0-1 variable. If job is processed on machine , is 1; otherwise is equal to 0.

In the literature on multiobjective PMSP, all objectives are often set to have the same importance. In fact, the importance of objectives is not different in many real-life manufacturing situations. For example, in make-to-order production environment, on-time delivery is vital for manufacturers. The delayed delivery will cause loss in revenue, customers, and reputation of company and so on, so it is essential to regard total tardiness as key objective and total energy consumption as non-key one to implement as many on-time deliveries for customers as possible.

The lexicographical method is often utilized to deal with multiple objectives with different importance, which gives key objectives higher priority than non-key ones. When the lexicographical approach is applied, if or and , then , which denotes that substitutes for . The above conditions mean three cases: (1) , ; (2) , ; (3) , ; that is, once , cannot be replaced with even if . In this study, the lexicographical method is used to compare solutions.

#### 3. Introduction to ICA

ICA is a population-based metaheuristic. Each individual of population represents a country and some best countries are selected as imperialists in the initialization. After initial empire is built by using an imperialist and colonies, new solutions are generated by the assimilation and revolution of colonies, the exchange of imperialist and colony if possible, and imperialist competition.

The procedure of ICA [14] is shown as follows.(1)Initialization: generate an initial population .(2)Construct initial empire: compute the cost for each individual; sort in descending order for all solutions; select best solutions from as imperialists; and assign remaining countries to the imperialists.(3)For each empire, execute assimilation of colonies, perform revolution of some colonies, and exchange position of colony and imperialist if possible.(4)Achieve imperialist competition.(5)Eliminate the empire without any countries.(6)If the termination criterion is not met, go to step (3); otherwise, stop the search.

The cost of a country is calculated according to objective function. The better a solution is, the less its cost is. best solutions with the lowest cost are chosen as imperialists. The rest of countries are set to be colonies. There are colonies; . The initial empires are formed by assigning colonies to imperialists according to the power of imperialistwhere is the power of imperialist and indicates the normalized cost, where is the cost of imperialist .

The number of initial colonies possessed by imperialist is calculated as , where is a function that gives the nearest integer of a fractional number. is the set of colonies of imperialist .

In the assimilation process, a colony in each empire moves along with direction toward its imperialist. The moving distance is a random number gotten by random distribution in interval , where and is distance among colony and imperialist. Setting causes the colony to move toward the imperialist direction. However, imperialist cannot absorb their colonies in direct movement, resulting in a deviation from the direct line. The deviation is represented by , which follows uniform distribution in , where is an arbitrary parameter.

Revolution is about a change in position of some colonies because of an unexpected changes in their characteristics. For example, by changing the language or religion of a colony, its characteristics will be changed and, accordingly, its position will be changed [14]. The revolution in ICA is similar to mutation in GA, which increases exploration and prevents the early convergence to local optima.

After assimilation and revolution are done in an empire, the cost of each colony is compared with that of its imperialist and a colony is swapped with the imperialist if the colony has less cost than the imperialist.

Imperialist competition is an important step based on the total power of an empire. Let be the total cost of empire ; we first calculate for each empire bywhere is a positive number between 0 and 1 and close to 0.

We then compute the normalized total cost of empire and the power of empire by

After a vector is defined, the weakest colony from the weakest empire is assigned to the empire with the largest index, where denotes a random number following uniform distribution in .

#### 4. A Novel ICA for Low Carbon Parallel Machines Scheduling Problem

In general, imperialist is just updated with the evolved colony and seldom changed in other way; moreover, the search on imperialist and the information exchange between imperialists are seldom considered in the existing ICA. Imperialist is good solution in population and the local and global search on these good solutions are easy to produce high quality solution and improve search efficiency, so it is necessary to introduce the innovation and alliance of imperialist to simulate local search and global search of imperialist. Two objectives of the problem have different importance; some new strategies are applied to form initial empires and do imperialist competition to adapt to the above characteristic. A novel ICA is proposed based on the above ideas. In the following, we describe the detailed steps of ICA.

##### 4.1. Coding and Decoding

For PMSP, each job is required to allocate a machine and processing sequence is needed to decide for all jobs on each machine. In this study, two-dimensional coding method is applied to describe a solution of low carbon PMSP. Each solution is represented as two strings.where is used to decide processing sequence of jobs, , the second string is for machine assignment, and is the machine allocated to job .

Figure 1 shows a solution of low carbon PMSP with 10 jobs and 5 machines. It can be seen that jobs 2, 6, and 7 are processed on machine and their processing sequence is 2-7-6.