Mathematical Problems in Engineering

Volume 2018, Article ID 7948693, 16 pages

https://doi.org/10.1155/2018/7948693

## A Mathematical Model for Multiworkshop IPPS Problem in Batch Production

Correspondence should be addressed to Li Ba; moc.361@ilabtuax

Received 20 September 2017; Revised 1 December 2017; Accepted 1 February 2018; Published 20 March 2018

Academic Editor: Dylan F. Jones

Copyright © 2018 Li Ba 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

Integrated Process Planning and Scheduling (IPPS) problem is an important issue in production scheduling. Actually, there exit many factors affecting scheduling results. Many types of workpieces are commonly manufactured in batch production. Moreover, due to differences among process methods, all processes of a workpiece may not be performed in the same workshop or even in the same factory. For making IPPS problem more in line with practical manufacturing, this paper addresses an IPPS problem with batches and limited vehicles (BV-IPPS). An equal batch splitting strategy is adopted. A model for BV-IPPS problem is established. Makespan is the objective to be minimized. For solving the complex problem, a particle swarm optimization (PSO) with a multilayer encoding structure is proposed. Each module of the algorithm is designed. Finally, case studies have been conducted to validate the model and algorithm.

#### 1. Introduction

Process planning and production scheduling are two indispensable subsystems in manufacturing systems. In traditional manufacturing, they are performed independently in series. A process planning subsystem determines the process route for each workpiece, and a scheduling subsystem allocates manufacturing resources according to the results from the process planning subsystem [1–3]. The independent and serial running mode of the two subsystems may lead to unrealistic process routes, uneven resource utilization, and bottlenecks in scheduling [1–3]. Integration of the two subsystems is an effective method to eliminate conflicts of resources, shorten finishing times of workpieces, and improve machine utilization [1–3]. The integration of process planning with scheduling is important for the development of manufacturing systems.

Integrated process planning and scheduling (IPPS) problem is a significant issue in the field of production scheduling. Existing research focusing on IPPS problems has typically considered only process stages. Few studies have considered batches in the IPPS problem even though many workpieces are processed in batch production. Batch splitting problem has been a significant issue in real production environments [4, 5]. Furthermore, there are some cases in which all processes cannot be accomplished in the same workshop due to their different processing characteristics. Because machines may be located in different workshops, transportation between them must be considered in scheduling which will influence the finishing time of the workpieces.

Batch splitting problem involves splitting lot numbers and splitting lot sizes for each workpiece. In fixed batch splitting, the lot numbers of all workpieces are constant, and the lot sizes of all sublots of a workpiece are equal. Equal batch splitting is typically used in production scheduling problems [4, 6]. The lot size of each sublot of a workpiece is equal, but the lot numbers of different types of workpieces are not equal (the lot number of each workpiece can be changed). Therefore, equal batch splitting is more flexible than fixed batch splitting. Equal batch splitting is adopted in this paper because fixed batch splitting may cause an imbalance between machine and load.

In conclusion, batch and transportation are simultaneously considered so that the IPPS problem better aligns with the real production environment. An IPPS problem considering equal batch splitting and limited vehicles (BV-IPPS) is proposed. The mathematical model of BV-IPPS is established to minimize the makespan. Given the complexity of the problem, a particle swarm optimization (PSO) is designed with a multilayer encoding structure. Finally, the model and algorithm are proven through a case study.

This paper initially provides a brief literature review related to IPPS problem in Section 2. A mathematical model for the problem is proposed in Section 3. A PSO with a multilayer encoding structure is designed in Section 4. The computational results are analysed in Section 5. Finally, the conclusions are provided in Section 6.

#### 2. Literature Review

The basic IPPS problem can be defined as follows [7]: “Given a set of parts that are to be processed on machines with operations including alternative manufacturing resources, select suitable manufacturing resources and sequences of operations to determine a schedule in which the precedence constraints among operations can be satisfied and the corresponding objectives can be achieved.”

Researchers began studying IPPS problem in the 1980s [8]. Chryssolouris et al. [9] proposed an approach to integrate process planning and scheduling. An increasing number of studies have focused on the IPPS problem in recent years.

Phanden et al. [1] determined that considering process planning and scheduling separately may cause many problems and limitations. They introduced three common approaches for integrating process planning and scheduling. Potential avenues for future IPPS work are discussed at the end of the paper.

*(**1) Integrated Approaches.* Saygin and Kilic [10] proposed a framework for integrating process planning and scheduling that consisted of machine tool selection, process plan selection, scheduling, and rescheduling modules. The framework was validated using many samples. Phanden et al. [11] established a makespan-targeted IPPS model composed of a process route selection module, scheduling module, analysis module, and process route modification module. A genetic algorithm (GA) was designed to solve the model. The availability of the model and algorithm was proven by a case study. Manupati et al. [12] proposed a mobile agent-based approach for integrating process planning and scheduling. A mathematical model for a biobjective IPPS problem with consideration of transportation was established. The approach was proven through many examples.

*(**2) Improvements to Algorithms.* Petrović et al. [3] proposed a hybrid algorithm based on PSO and chaos theory. The advantages of the hybrid algorithm have been proven through many benchmarks by comparisons with other approaches. Xia et al. [13] proposed a dynamic IPPS problem with consideration of machine breakdown and new job arrival. A model for the problem was established. A hybrid GA with variable neighbourhood search was designed to solve the problem. Zhang and Wong [14] proposed a GA framework and integrated ant colony optimization (ACO) into the framework for solving IPPS problem. Lian et al. [15] proposed an imperialist competitive algorithm (ICA) to solve IPPS problem. Lian et al. [16] proposed a mathematical model for process planning problem with an objective of total cost minimization. An ICA was designed to solve the problem. Shao et al. [17] established an IPPS model with two objectives. Makespan and machine utilization are objectives which will be optimized. A modified genetic algorithm-based approach was designed for solving the problem. Li et al. [18] proposed an evolutionary algorithm-based approach for solving IPPS problem. Guo et al. [7] presented an advanced PSO approach to solve the IPPS problem. The advantages of PSO were proven by comparison with other intelligent algorithms. Seker et al. [19] proposed a hybrid heuristic algorithm based on a GA and artificial neural network (ANN) to solve IPPS problem.

*(**3) Other Research on IPPS Problem.* Zhang et al. [20] established an IPPS model with the total energy consumption as the objective. A genetic algorithm-based approach was proposed to solve the problem. Haddadzade et al. [2] proposed an IPPS problem that considered stochastic processing time. The Dijkstra algorithm and Monte Carlo sampling method were used to create examples, and a hybrid algorithm based on simulated annealing and tabu search was designed to solve the problem. Kis [21] proposed a particular job-shop scheduling problem in a chemical production environment. The process routes were directed acyclic graphs and consisted of several alternative subgraphs. A tabu search and GA were proposed to solve the problem. Li and McMahon [22] proposed a multiobjective IPPS problem. The target to be optimized was obtained by combining multiple objectives through linear weighting. A simulated annealing algorithm (SAA) was proposed to solve the problem. Moon and Seo [23] considered transportation in IPPS problem with the makespan as the objective, and an evolution algorithm was proposed to solve the problem.

Many existing studies on IPPS problem only involve the process stage; few consider batching, which may cause the IPPS problem to unrealistic. In real production, many workpieces are processed in batches. Batch splitting problem is a significant issue in real production environments [4, 5]. Furthermore, all processes may not able to be processed in the same workshop due to different process types. Transportation is another factor that must be considered in real production environments. References [12, 23] considered transport in an IPPS problem but did not consider the amounts and locations of transports, which may cause the methodology to not fit real production environments.

Recently, several studies have focused on production scheduling problems considering batch splitting. For example, batching has been considered in a job-shop scheduling problem (JSP) [4, 24], flow-shop scheduling problem (FSP) [25, 26], and parallel machine scheduling problem [27]. Research on production scheduling problems considering batch splitting has mainly focused on JSP and FSP. As manufacturing technology has improved over time, normal machine tools now coexist with numerically controlled machine tools and machining centres in many workshops [28]; however, because different machines have similar functions, multiple process routes are designed for the same workpiece to fully utilize different machines [1–3]. Designing multiple process routes for a workpiece is of great significance for improving the flexibility of scheduling and reducing resource conflicts [1–3]. IPPS problem involves parallel machines and alternative process routes simultaneously. Compared to JSP and FSP, IPPS problem has a larger solution space and is more complex. The scheduling result of IPPS problem is more in line with real production situations. IPPS problem has been identified as NP-hard [19].

Based on the above literature, this paper proposes an IPPS problem considering equal batch splitting and limited vehicles with the makespan as the objective to be minimized. A model for the problem is built, and a PSO with a multilayer encoding structure is proposed. Finally, the model and algorithm are validated through a case study.

#### 3. BV-IPPS Problem Formulation

##### 3.1. Problem Description

There are different workpieces that must be processed in a factory. The batches of different workpieces are different. Each workpiece can be processed through more than one process route, and each process can be processed by more than one machine. There are several vehicles for transporting workpieces. Under these conditions, the makespan is regarded as an objective. An optimal solution is obtained by selecting the process routes, machines, vehicles and sorting process sequences for each workpiece.

A simple example is shown to illustrate BV-IPPS problem. An order involves two types of jobs: and . A lot of and will be processed. is not divided. is divided into two sublots: and . There are six machines: –. There are two workshops: and . , and belong to . , and belong to . There are four process methods: , , and . is able to complete . and are able to complete . and are able to complete . is able to complete . There are three process routes: , and . is designed for . and are designed for . There are three vehicles for transmitting jobs: , and . Assuming is used for , is used for , and is used for . Processing flow of a scheduling solution is shown as Figure 1.