Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 9628935, 13 pages

https://doi.org/10.1155/2017/9628935

## Combining Extended Imperialist Competitive Algorithm with a Genetic Algorithm to Solve the Distributed Integration of Process Planning and Scheduling Problem

School of Information, Zhejiang University of Finance and Economics, No. 18 Xueyuan Street, Xiasha, Hangzhou 310018, China

Correspondence should be addressed to Shuai Zhang; moc.anis@419067sz

Received 30 April 2017; Accepted 1 November 2017; Published 20 November 2017

Academic Editor: Marco Mussetta

Copyright © 2017 Shuai Zhang 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

Distributed integration of process planning and scheduling (DIPPS) extends traditional integrated process planning and scheduling (IPPS) by considering the distributed features of manufacturing. In this study, we first establish a mathematical model which contains all constraints for the DIPPS problem. Then, the imperialist competitive algorithm (ICA) is extended to effectively solve the DIPPS problem by improving country structure, assimilation strategy, and adding resistance procedure. Next, the genetic algorithm (GA) is adapted to maintain the robustness of the plan and schedule after machine breakdown. Finally, we perform a two-stage experiment to prove the effectiveness and efficiency of extended ICA and GA in solving DIPPS problem with machine breakdown.

#### 1. Introduction

Given the increasing fluctuations of the manufacturing environment and the constant need to arrange the process integrally and dynamically, schemes that separately formulate the plan and schedule no longer accord with the increasingly complex requirements. Under these environments, a model that settles the plan and schedule simultaneously, that is, integrated process planning and scheduling (IPPS), becomes necessary. In summary, the objective of IPPS is to determine an optimal schedule with machine selection for operation and operation sequence for the jobs [1]. Beyond the former strategy, IPPS considers the whole process while enhancing it in terms of dealing with the dynamic environment and making the theory consistent with reality.

Although IPPS makes a greater degree of progress than the former approaches in arranging and programming the manufacturing contents, the absence of distributed feature raises a dilemma for this model. Against the backdrop of distributed production, several requirements such as raw material availability and transportation concerns have urged manufacturing companies to adopt distributed strategies. Therefore, we aim to extend the IPPS and develop a more advanced model to handle distributed planning and scheduling, that is, distributed integration of process planning and scheduling (DIPPS).

The advantage of DIPPS lies in its feature of coordinating planning and scheduling in a distributed environment. Thus, the DIPPS is more suitable for the current manufacturing environment than IPPS. The DIPPS is generalized as follows [2]: given jobs consisting of multiple alternative producing processes with different operations in optional manufacturing units (MUs) with distinct assembly techniques and equipment, we must determine the plans and schedules including units, process plans, and machines for each job by considering the objectives and constraints.

On the other hand, the robustness of plans and schedules is of great concern through the manufacturing process. Once machine breakdown occurs during the manufacturing, the previous plan and schedule are bound to fall short of their anticipated objectives as a result of the changed context. In this case, an updated plan and schedule that aim to sustain the constraints and objectives should be structured for the remaining jobs and operations.

To deal with the DIPPS problem proposed in this study, we combine an extended imperialist competitive algorithm (EICA) and a genetic algorithm (GA) together. Furthermore, the EICA is utilized to generate plans and schedules that are implemented before the emergency takes place, while GA is reserved to handle arrangements once a machine breakdown occurs. The traditional imperialist competitive algorithm (ICA) that simulates competition among empires has a strong global exploration capability in solving the NP-hard problems. In this study, we extend the ICA by improving its country structure, assimilation strategy, and adding a resistance procedure. The EICA has been proved as a more effective and efficient algorithm by comparing it with GA and traditional ICA in solving DIPPS problem, whereas for GA, as a mature and populated evolutionary algorithm, its capability in manufacturing practice has been studied and proved in our previous works [3, 4] and many other studies. Here, because of the advantage of its structural similarity with EICA and mature application in dealing with planning and scheduling, the GA can be easily formulated by implementing the structure from EICA; it therefore responds excellently to the breakdown emergency and alters the plan and schedule into satisfactory states.

The remainder of this paper is organized as follows. In Section 2, the works relating to ICA and robustness are briefly introduced. In Section 3, a mathematical model for the DIPPS problem is established. In Section 4, the imperialist competitive algorithm is extended to effectively solve the DIPPS problem. In Section 5, the GA is adapted to deal with machine breakdown. In Section 6, a two-stage experiment is presented to prove the effectiveness and efficiency of EICA and GA to solve DIPPS problem in the case of a machine breakdown. Section 7 presents our conclusions.

#### 2. Related Work

##### 2.1. Evolutionary Algorithms

GA is an algorithm to search for the optimal solution by simulating the natural evolution process. And there are some well-known evolutionary algorithms inspired by GA, such as biogeography based optimization [5, 6] and genetic swarm optimization [7]. In our previous work, we used GA to optimize the DIPPS in fuzzy environment [8]. Besides the wide application of GA, there are some other evolutionary algorithms that have been used for solving optimization problem. For example, Rahmat-Samii et al. [9] used PSO for antenna design optimization. Dorronsoro et al. [10] used evolutionary algorithm to model and solve minimization problems. Grimaccia et al. [11] used social network optimization to design generators for vehicle energy harvesting.

In addition, ICA is a metaheuristic algorithm inspired by sociopolitical ideology and first proposed by Atashpaz-Gargari and Lucas [12]. Generally, there are always competitions when numerous countries exist. By means of war and conquest, some powerful countries, called imperialists, conquer and colonize others, forming empires. As time goes by, the imperialists assimilate their colonies and conquer colonies belonging to other ones. In contrast, weaker empires gradually lose their colonies to more powerful ones and eventually face extinction. At the end of competition, there is an ideal state in which the most powerful empire conquers all lands. By simulating the competition above, the ICA innovatively structures its procedure to solve a variety of outstanding problems. Through the last several years, several significant works [13–15] have sought to strengthen the global exploration power in order to broaden its application.

Since being proposed, the ICA has gained popularity and achieved significant performance in solving manufacturing planning and scheduling problems. To settle the optimization of process planning with various flexibilities, Lian et al. [16] utilized the ICA to find promising solutions with reasonable computational cost under the objective of minimizing total weighted sum of manufacturing cost. Shokrollahpour et al. [17] and Seidgar et al. [18] both exploited the ICA in assembly flow shop problem while, respectively, using the Taguchi method and neural network as their own tools in regulating the parameters. Additionally, in the no-wait two-stage hybrid flow shop, Moradinasab et al. [19] introduced a new procedure called global war in ICA to avoid the local optima. This step helps to transfuse some new empires in a certain extent and achieves desirable performance in the experiment. In addition, in the work of Zhou et al. [20], ICA was adopted to deal with the assembly sequence planning. Compared with GA and PSO, the ICA performs better in the experiment and the quality of result is less related to the initial populations. Moreover, Madani-Isfahani et al. [21] presented an ICA to solve a biobjective unrelated parallel machine scheduling problem where setup times are sequence dependent.

Despite the achievements in the specific domain of manufacturing arrangement, infrequent work has been done to settle IPPS problems, let alone for DIPPS problems. To the best of our knowledge, only Lian et al. [22] have applied the ICA to solve IPPS while omitting the disposition of robustness. The scarce adoption of ICA in this area is not contrary to our expectations. Because IPPS and even DIPPS problem have far more variables and constraints to deal with, they inevitably contain a high magnitude of information to manage. The complexity of simultaneous planning and scheduling also predisposes the process of measure searching to be handled delicately.

##### 2.2. Robustness

Under the constantly changing conditions of manufacturing, static and unchangeable plans and schedules are impractical. When the initial plans and schedules are put into effect, machine breakdown may take place and disturb the manufacturing procedure in an uncontrollable way that invalidates the former arrangement. To keep plans and schedules robust and flexible, some extra work is essential.

Among the methods applied for replanning and rescheduling, right-shifting is the most convenient way. This corresponds to waiting for the breakdown to be fixed and then carrying on with the work [23]. For instance, Liu et al. [24] used right-shift rescheduling to retain the same sequence of all remaining jobs as that of the predictive schedule. Although this method saves quite a lot of follow-up work, it loses the optimality of planning and scheduling at the same time. Therefore, other means have been figured out. Saygin and Kilic [25] adopted a step-by-step manner by dividing the whole scheduling period into shorter periods and proceeding by overlapping the schedule of each period on the previous one to handle the effect of changes like breakdowns. Jensen [23] proposed a new way of creating robust and flexible solutions for job-shop scheduling problems by busing a robustness measure based on a neighborhood for schedules. Additionally, Hasan et al. [26] used shifted gap-reduction instead of right-shifting in order to minimize the effect of interruptions in job-shop scheduling problem.

When selecting methods for dealing with machine breakdown in scheduling and planning, the focal points should be targeted at convenience and optimization where the former point pays attention to the adjustment time of replanning and rescheduling while the latter one is concerned with the optimality of replanning and rescheduling. In this study, GA is associated with solving the machine breakdown. Because of the similarity of EICA and GA in representation, GA can be structured and put into work promptly once breakdown takes place, and GA is more effective than ICA in solving the problem with small solution space. In addition, with abundant verification preformed in previous works for IPPS problems, it has a positive reputation for strong and reliable performance.

#### 3. The Mathematical Model for the DIPPS Problem

As defined in the Introduction, DIPPS aims to determine an appropriate manufacturing unit (MU) while selecting the process plans and schedules for jobs. The so-called MUs are some geographically dispersed units contained in an integral factory system that have the capability to operate independently. In the DIPPS problem on which we focus in this study, each MU can process all types of jobs that need to be treated. However, because of the multifarious assembly techniques and equipment necessitated by differences in construction year and purposes among MUs, the respective optional process plans and machines are totally different. Based on this situation, the arrangement should be settled cautiously. Furthermore, owing to geographical dispersion, the transportation time for the finished work from MUs to the central factory is also a significant concern in the course of scheming.

To examine the optimality and rationality of the plan and schedule, we formulate the relevant objectives and constraints; before formulating them, the corresponding indexes are expounded as follows: : the quantity of MU : the quantity of job : the quantity of plans of the th job in the th MU : the quantity of operations of the th plan of th job in the th MU : the quantity of machines in the th MU : the th operation of the th plan of the th job processed by the th machine in the th MU : the initial time of : the initial time of any other operation processed on the same machine as : the operating time of , which contains the setup time : the completion time of : the completion time of the th job in the th MU : the transportation time of the th job from the th MU to central factory : the binary variable, with 1 representing that the th job assigned to the th MU and 0 otherwise.

In order to evaluate the excellence of methods for the DIPPS, an objective is formulated.

*Objective*. Minimize the total makespan ; that is, minimize the period of time from the very beginning of processing to the end of transportation:Meanwhile, the following constraints are required for the sake of rationality.

*Constraint 1*. Only one job can be assigned to a single MU:

*Constraint 2. *Once an operation is under processing, no other operations can cut in:

*Constraints 3*. Parallel processing for more than one job is not allowed on any machine:

*Constraint 4*. The operating sequence of operations of a specific job cannot be altered once determined:

In this study, with a goal of simplifying the resolving process, a simplified example containing two independent MUs and three jobs is structured. For the purpose of representing alternative plans and machines for different jobs and operations in diverse MUs, a directed acyclic graph (DAG) is adopted here. The traditional DAG contains vertices and directed edges and is applied throughout mathematics, computer science, and engineering with the capability to clearly represent processes. In this study, we extend the DAG with more features to illustrate the information visually and prepare for the subsequent calculation.

Specifically, Figure 1 exhibits alternative plans and schedules for different jobs through different MUs. Take the first DAG in Figure 1 which shows Job 1 alternative plans and schedules in MU 1; for example, each vertex and directed edge stands for the operation and the processing trend, respectively. The black dots named “branch points” are attached in the DAG to represent the branch information that will be used in the construction of EICA. To clarify the start and end of the plan, two virtual operations without any operating time, labeled “” and “,” are set. With regard to valid operations, that is, the vertices except the starting and the ending ones, each of them represents the optional machine number and respective operating times by data sets. Tracking along with the directed edges from the “” operation to the “” operation without any backtracking, one specific plan with a certain sequence and corresponding operations is determined.