Chapter 8

Real-Time Information-Driven Production Scheduling System

Abstract

During traditional manufacturing process, it’s hard to achieve real-time production scheduling because of the inefficient feedback of real-time information in the shop floor. To address the mentioned challenge, a framework of real-time information-driven production system is designed in this chapter. It’s also used to bridge the gap between production planning and control. The real-time information-driven production system consists of four modules. They are equipment agent, capability evaluation agent, real-time scheduling agent, and production execution monitor agent. Equipment agent is employed to capture and process real-time information in the shop floor. In the process planning stage, capability evaluation agent is used to accomplish the optimal tasks allocation according to the real-time utilization ratio of each machine. As the name suggests, real-time scheduling agent is responsible for the manufacturing tasks scheduling or rescheduling according to the traced real-time information. Production execution monitor agent aims to track and trace the real-time status of different manufacturing processes. The running mechanism of each agent is described in detail in this chapter.

Keywords

real-time information
production scheduling
running mechanism

8.1. Introduction

Production scheduling refers to the process of allocating right manufacturing resources to specific time periods to complete a set of manufacturing processes in the plan [1]. As an important manufacturing planning activity, it aims to tackle resource utilization and time span of the manufacturing operations. Agent-based manufacturing scheduling systems [2] provide a promising way for production scheduling. With the rapid developments of information communication technologies (e.g., wireless sensors, wireless sensor network), ubiquitous manufacturing (UM) has been proposed as one core manufacturing technology in advanced manufacturing systems [3,4]. Ubiquitious manufacturing makes the real-time status of manufacturing resources can be easily captured, then promote real-time production scheduling. Meanwhile, the unscheduled downtime in a UM system can reach a lowest level by using the captured real-time information, which provides one more effective way to deal with negative impact of production exceptions [5].
Though significant progress has been made in the field of real-time information capturing and processing, some research questions still need to be considered in how to implement real-time scheduling methods in real-life manufacturing floors.
1. During the process planning stage, real-time statuses of manufacturing resources are rarely considered when tasks are allocated to machines. The mentioned circumstance may lead to nonoptimal allocation of tasks and have negative influence on following production scheduling.
2. The scarcity of effective methods to capture and process real-time manufacturing information leads to the inefficient, inaccurate, and time-delay production monitor. Meanwhile, an effective mechanism has not been well developed to accomplish the real-time information integration between production scheduling stage and production execution exception monitor stage. This imposes more challenges on performing real-time production scheduling.
3. Nowadays, tasks in the manufacturing system show the features of large variety and small batch size. This phenomenon raises more requirements in the field of real-time production monitoring. It has to own the capabilities of dynamic decision making and adaptive control to tackle different production changes. Thus it is very important to develop real-time production scheduling architecture and method for the dynamic changing manufacturing environment.
To address previously mentioned questions, in this section, the promising advantages of advanced information communication technologies and multiagent are integrated to perform real-time production scheduling in a UM environment. The developed framework of real-time information-driven production scheduling (RIDPS) system aims to bridge the gap between production planning and execution during the whole manufacturing process. This chapter also aims to provide a way for enhancing the productivity and flexibility of manufacturing system through implementing the proposed real-time information-driven production system.
The rest of this chapter is organized as follows. Section 8.2 reviews related works in three different sides. The framework of RIDPS system is designed in Section 8.3. Multimodels, such as equipment agent (EA), capability evaluation agent (CEA), real-time scheduling agent (RSA), and production execution monitor agent (PEMA) are described in Sections 8.4–8.7, respectively. Section 8.8 explains the genetic algorithm (GA)-based production scheduling algorithm.

8.2. Related works

Three streams of works are relevant to this research. These include (1) agent technology and applications in manufacturing field, (2) real-time production scheduling, and (3) manufacturing information monitor technology.

8.2.1. Agent Technology and Applications in Manufacturing Field

Agent technology is an important subfield of artificial intelligence (AI), and it is one of the powerful technologies for the development of large-scale distributed systems to deal with the uncertainty in a dynamic environment [6–8]. A new inter- and intraagent cooperation approach was presented to improve the performance of multiagent or distributed manufacturing systems [7]. Sikora and Shaw described a model for the coordination and integration of enterprise information systems (EISs), which can model typical EISs as containing multiple agents with different functionalities [9]. An agent-based system for coordinated product development and manufacture was presented by Jia et al.; the system consists of two classes of agents, that is, managing agent as the core manager and functional agents with specific functionality [10]. An agent-based multicontract negotiation system was proposed to address the challenges of global manufacturing supply chain coordination [11]. Zhang et al. presented a novel gateway technology for the real-time management in UM environment; the agent-based smart objects management system was used to manage the multiple auto-ID sensors to capture the real-time status [4,12]. Farid and Ribeiro presented a multiagent system reference architecture for reconfigurable industrial systems based on a quantitative and formal design method, which was embedded in an traditional engineering design methodology called axiomatic design for large flexible engineering systems [13]. Trappey et al. proposed an agent-based cooperative mold production system, which aimed to sustain the collaborative and self-directed mold manufacturing outsourcing patterns [14]. Kumari et al. presented an autonomous self-adaptive multiagent system to aid small and medium enterprises to obtain the optimal decision so that the uncertainty in supply chain can be alleviated [15]. Wang et al. proposed a distributed multiagent reinforcement learning algorithm to solve the resource-constrained imperfect preventive maintenance question; the question was modeled as a semi-Markov decision process [16]. Vogel-Heuser et al. proposed a method that used the rapidly developing concept of cyber-physical systems for a case of manufacturing systems by methods of software agents [17]. Manupati et al. developed a multiobjective mathematical model in the background of networked manufacturing systems, and a mobile agent–based negotiation pattern was presented for solving the distributed problem [18]. Vidoni and Vecchietti designed a smart agent for ERP’s data structure analysis based on ANSI/ISA-95 standard; three open source ERPs: OpenERP, Dolibarr, and Adempiere were considered as the cases [19]. Ayhan et al. proposed a multiagent-based approach for changing management mode in industrial companies [20]. Bearzotti et al. designed a self-governing multiagent method to manage the supply chain event, which can perform autonomous corrective control activities to minimize the influence of deviations of the currently executing plan [21].

8.2.2. Real-Time Production Scheduling

The scheduling problem in shop floor represents a problem where the objective is to properly allocate available resources to tasks in order to optimize an objective function, which is usually related to time, like the makespan [22], total completion time [23], tardiness [24], maximum lateness, total throughput time [25], etc. Previous approaches mainly focus on the production scheduling before the production starts or only research under a theoretical environment. For example, Aghezzaf presented the production planning and warehouse management for supply networks with interfacility mold transfers [26]. Babayan and He presented an overall methodology of agent-based manufacturing systems scheduling, incorporating game theoretic analysis of agent cooperation, to solve the n-job 3-stage flexible flow shop scheduling problem [27]. Fang et al. provided a new mixed-integer linear programming model for scheduling a typical flow shop that combined the peak total power consumption and the associated carbon footprint with the makespan [28]. Luh and Chueh presented an innovative multimodal immune algorithm for discovering optimal solutions to job shop scheduling problems imitating the features of a biological immune system [29]. Giordani et al. presented a distributed multiagent production planning and scheduling framework for mobile robots [30]. Fan et al. considered the problem of integrated scheduling of production and delivery on a single machine, in which the availability constraint of the machine and jobs in processing may be disturbed [31]. Shishvan and Sattarvand discussed the long-term production planning problem of open pit mines by ant colony optimization method [32]. Cheng et al. considered an integrated scheduling of production and distribution so as to minimize the production and distribution costs, and an improved ant colony optimization method is used to solve the problem [33]. Arauzo et al. investigated a new method based on multiagent systems and a combinatorial auction mechanism was used to allocate resources for the projects tasks [34].
As to the real-time scheduling aspect, Anderson and Calandrino proposed a scheduling method for real-time systems, which is realized on multicore platforms, so that individual threads of multithreaded real-time tasks can be scheduled together [35]. Tabuada presented an event-triggered real-time scheduling method for stabilizing control tasks, where the real-time scheduler were regarded as a feedback controller that decided which task was performed at any given time [36]. Subramanian et al. analyzed real-time scheduling algorithms for coordinated aggregation of deferrable loads and storage. Three scheduling policies: earliest deadline first, least laxity first, and receding horizon control were discussed in the paper, and the performance of those algorithms were studied through simulations [37]. Buyurgan and Saygin proposed a multicriteria decision-making framework for real-time scheduling and part routing solutions by implementing pairwise comparison of possible future states of a manufacturing system [38]. Yan and Wang proposed a two-layer dynamic scheduling approach for the dynamic scheduling problem of a reentrant production line, in which all of the parts are assumed to have the same processing routes and need to be processed on every machine [39]. Du et al. presented a new framework for integrating, scheduling, and nonlinear control for continuous processes, and the approach for reducing dimensions of the problem and closed-loop process dynamics were considered [40]. Lee and Prabhu proposed a dynamic algorithm for distributed feedback control, which considered the functions of production and maintenance scheduling at the shop-floor level and machinery capacity control at the CNC level at the same time, while the two problems were usually considered in isolation in practice [41]. Luo et al. discussed the real-time scheduling problem for hybrid flow shop in UM environment; a multiperiod hierarchical scheduling method is presented in the paper [42]. Based on a DT model, Choi et al. presented a real-time scheduling method for reentrant hybrid flow shops [43]. Unlike the theoretical approach on reentrant hybrid flow shop scheduling, a real-time scheduling approach using a decision tree when selecting appropriate dispatching rules was creatively provided.

8.2.3. Manufacturing Information Monitor Technology

Thanks to the emerging advanced IoT technologies, more and more manufacturing enterprises began a widespread use of IoT technologies to implement and manage their business [44,45]. The IoT provides an IT infrastructure to facilitate the information exchange of “things and process” in a real-time and reliable way. Some scholars have explored IoT technologies practice in manufacturing information monitor.
Huang et al. proposed a theoretical wireless manufacturing (WM) framework [46]; WM relied substantively on wireless devices, such as radio frequency identification (RFID) or auto-ID sensors and wireless information networks for real-time monitoring of production data. Zhang et al. described a real-time information capturing and integration architecture of the Internet of manufacturing things [44], and a real-time manufacturing information integration service was proposed to fulfill the target of timely information exchange among EISs and manufacturing resources in shop-floor frontline. Based on RFID technology, Wang et al. proposed a digital warehouse management system in the tobacco industry [47]. By using RFID technology, the system enabled a plane warehouse to achieve visualized inventory management, automatic storage assignment, and high accuracy of inventory control. Combining RFID technology with ontologies, Grüninger et al. created smart objects in the context of manufacturing process to solve the problems of massive RFID tags interoperability [48]. Xu et al. focused on the research of closed-loop product information tracking and feedback in a wireless technology–enabled environment from the modeling point of view [49]. By using 2D barcode and RFID technologies, Lin et al. proposed a novel system called Mobile 2D Barcode/RFID-based Maintenance Management system to improve lab equipment and instrument maintenance management and provides a maintenance information sharing platform [50]. Subramaniam et al. presented an automated data collection and display structure for production lines; the system can generate an automated report, which stayed in place and the management only needed to act based on the results [51].

8.3. Overall architecture of real-time information-driven production scheduling system

The overall architecture of RIDPS is shown in Fig. 8.1. The objective of RIDPS is to achieve the dynamic optimization of process tasks based on the real-time status of the equipment. By extending the IoT technologies into the traditional manufacturing field, the status of manufacturing resources can be sensed timely. Then, the tasks can be allocated to the equipment based on the real-time capability of machines. When production anomaly happens, such as machine broken, order canceling, and so on, the rescheduling can be executed according to the real-time production information.
image
Figure 8.1 Overall architecture of real-time information-driven production scheduling (RIDPS).
Four kinds of agents are designed in this chapter to achieve real-time production scheduling. They are briefly described as follows:
1. EA
It is used to actively acquire the real-time status of the equipment. This agent is the foundation of the real-time production scheduling. Some auto-ID devices are used to sense the raw production data, and then the captured data are integrated to reflect the meaningful equipment status.
2. CEA
It is responsible for evaluating the capability of the equipment according to the real-time status sensed by machine agent. Then, the process planning can choose an optimal equipment for each process of the tasks.
3. RSA
This agent is used to make an optimal plan, which can allocate tasks to the available equipment with the optimized objective function. First, the mathematic model is given; the objective function and constraints are designed based on the real-time production problem. Then, the intelligent algorithm is given to solve the scheduling or rescheduling problem.
4. PEMA
It aims to sensing the real-time data of the various kinds of manufacturing resources. During shop-floor frontline, production anomaly often happens; it is necessary to track and trace the anomaly in order to achieve the target for real-time production scheduling.

8.4. Equipment agent

EA is used to wrap the applications of equipment side and process the multisource real-time data captured from wireless devices (e.g., RFID). The functions of EA can be categorized into two aspects: (1) it can centrally manage the different kinds of auto-ID devices, which are deployed on the equipment side. These auto-ID devices can capture real-time manufacturing data based on a specific working logic; (2) EA can transform the captured manufacturing data into useful manufacturing information and provide the manufacturing information for other modules in the real-time production scheduling system.
Fig. 8.2 demonstrates the EA model. It consists of two components, namely data capturing and application service.
image
Figure 8.2 Machine agent model.
1. Data capturing
Auto-ID devices are deployed on the machine side. They are used to capture dynamic manufacturing data. Data capturing component aims to manage these auto-ID devices. It includes two modules.
The first module is named Definition and is auto driven. It is used to store various drivers of heterogeneous auto-ID devices and form a driver pool. This driver pool makes new deployed auto-ID devices present a mode of “plug and play.”
The second module is named standard data capturing module. This module aims to unify the data perception functions of different auto-ID devices so that the capturing data can be easily invoked with a standard way. For example, the perception function can be stated as follows, “readingData (Parameter [1], Parameter [i])” and “writingData (Parameter [1], Parameter [i]).”
2. Application services
This component is responsible for providing value-added manufacturing information for other modules in the real-time production scheduling system. It also includes two modules.
The first module is named reasoning module. It is used to improve the intelligence of EA. By this module, EA can learn which kind of resource is coming or leaving the equipment. In order to promote the decision making of EA, rule-based methods are installed in this module.
The second module is named real-time information processing. This module aims to process various captured real-time data. Comparing with the reasoning module, this module mainly focuses on getting useful real-time manufacturing information. For example, the “getMaterial()” will return detailed real-time information, such as the kinds of material, real-time quantity of required material, and so on.

8.5. Capability evaluation agent model

Capability evaluation agent (CEA) is responsible for optimally allocating processes of tasks to related manufacturing equipment according to real-time manufacturing information provided by EA. The model of CEA is demonstrated in Fig. 8.3. Assume that a manufacturing task includes “n” processes, and a group of machines in the shop floor can meet the manufacturing requirements of each process [i]. CEA aims to find an optimal machine from the candidate machines to complete process [i] through a bid mechanism. Traditional allocation methods rarely consider real-time workloads of machines and the changing manufacturing environment. The manufacturing may be invalid or nonoptimal during the execution process.
image
Figure 8.3 Capability evaluation agent (CEA) model.
In the model of CEA, capability ratio is employed to evaluate the capabilities of machines and find the optimal one for the process [i]. As said before, a bid mechanism is used to accomplish the selection of optimal machine. First, a group of candidate machines defined as g is formed. Then, the related EAs of candidate machines will bid to complete process [i] based on their real-time capabilities. Finally, the available capabilities of candidate machines will be calculated by CEA, and the optimal manufacturing machine will be selected according to objective function Eq. (8.1).

Max{ACRmm∈g}

image(8.1)
where, ACRm=ACm/TCm×100%.image
Here, ACRm is defined as the available capability ratio of machine “m”. ACm represents the real-time available capability of machine “m”; it is changing dynamically with allocated tasks queue. TCm indicates the total capability of machine “m”; it is constant.

8.6. Real-time scheduling agent model

Fig. 8.4 presents the overview of RSA; it assigns the processes of tasks to the equipment based on the real-time manufacturing information from CEA and PMA. After the scheduling, the EA can obtain the task queue, which can be represented as {i, j, k, BT, CT}. Here, {i, j, k, BT, CT} denotes the equipment “k” which is allocated to finish the jth process of task “i” with the begin time “BT” and completed time “CT.”
image
Figure 8.4 Real-time scheduling agent (RSA) model.
Three modules, namely problem construction modules, solving module, and rescheduling module are designed in RSA.
1. Problem construction
Before the mathematic formulation are provided, the corresponding notations are given at first. The details can be shown in Table 8.1.

Table 8.1

Notations

Notations Description
E = {E1, …, Ek, …, Em} Set of equipment, m denotes the total number of equipment
T = {T1, …, Ti, …, Tn} Set of tasks, n represents the total number of tasks
TPi={tp1,tp2,…,tpNi} image Set of processes of task “i,” Ni denotes the total number of processes of task “i”
(Ti, TPj, Ek) The process “j” of task “i” will be processed at equipment “k”
BT(Ti, Pj, Ek) Begin time of (Ti, TPj, Ek)
PT(Ti, Pj, Ek) Process time of (Ti, TPj, Ek)
C = {C1, …, Ci, …, Cn} Set of complete time of task “i”
D = {D1, …, Di, …, Dn} Set of due date of task “i”

According to the notations, the mathematic model is constructed as follows.
Objective function:
Min∑i=1nμi⋅max(0,Ci−Di)
image(8.2)
Subject to:
BT(Ti,TPj+1,Ea)−BT(Ti,TPj,Eb)≥PT(Ti,TPj,Eb)
image(8.3)
BT(Tx,TPc,Ek)−BT(Ty,TPd,Ek)≥PT(Ty,TPd,Ek)    orBT(Ty,TPd,Ek)−BT(Tx,TPc,Ek)≥PT(Tx,TPc,Ek)
image(8.4)
i,x,y∈[1,n] j∈[1,Ni], c∈[1,Nx], k,a,b∈[1,m]
image
Eq. (8.2) represents the objective function, which takes the maximal total tardiness cost of all the tasks into consideration. Eq. (8.3) denotes processes order constraint, which means that the later process of task need to begin after the prior one. Eq. (8.4) denotes resource constraint, that is, the process of different tasks cannot be processed at the same equipment at the same time.
2. GA-based solving module
This module aims to find an optimal production queue based on established mathematic problem by using the intelligent algorithm. Since GAs are adaptive methods that are widely studied and used for solving optimization problems in manufacturing fields, they are adopted to solve the aforementioned problem. More details of the GA-based solving method are described in Section 8.8.
3. Rescheduling module
As discussed before, production anomaly is unavoidable, rescheduling is essential in maximizing the production capability. During the execution stage, the PMA will feedback the production information on time. The total deviation ratio of the machines at time (t) will be computed as shown in Eq. (8.5)
∆(t)=∑k=1mw1×ATt(i,j)+w2×LTt(i,j)+ETt(k)DTt(k)×100%
image(8.5)
where, w1 and w2 are weights, generally, w1 is less than 1 and w2 is greater than 1; ATt(j) is the ahead time of the jth process of task (i) assigned to equipment (k) at time (t) contrast to the previous plan; LTt(j) is the tardiness time of the jth process of task (i) assigned to equipment (k) at time (t) contrast to the previous plan; ETt(k) is the anomaly handling time of the equipment (k); DTt(k) is the duration time of equipment (k). In the previous plan, let T be the completed time of the last process assigned to equipment (k), then DTt(k) can be obtained by subtracting current time (t) from T.
According to the calculation of ∆(t), the rescheduling approach can be executed. Two manners, namely local rescheduling and global rescheduling, are presented in this module. On one hand, if ∆(t) ≤ λ, the local rescheduling manner is triggered. It is applied to requeuing a small quantity of the tasks while other tasks’ queue is not altered. It deals with the anomalies, such as the queue anomaly of an equipment. On the other hand, if ∆(t) > λ, the global rescheduling manner is triggered. It is applied to requeuing all the tasks from the anomaly time (t) by using the procedure designed in GA-based solving module. Here, λ is a constant and is default set as 10%, and it can vary based on different managers.

8.7. Production execution monitor agent model

Fig. 8.5 shows the model of PEMA. It is responsible for analyzing the real-time information from EAs and sending real-time manufacturing information to RSA.
image
Figure 8.5 Production execution monitor agent (PEMA) model.
The working logic of PEMA is composed of three stages. At the first stage, data source service is invoked to obtain essential information [e.g., bill of material (BOM)] about related production task. The essential information is stored in the up-level enterprise information management systems. At the second stage, a new work in progress (WIP) instance is created according to the obtained manufacturing information and information schema (wipML). The information about manufacturing BOM, which can be captured by EAs, is included in the new instance. A binding model is employed to link the information nodes of manufacturing BOM to related EAs. At the third stage, critical event (CE) structure is used to process the captured manufacturing information from EAs during manufacturing execution process. To meet the requirements of each stage, two components are designed in the PEMA. They are data source service and WIP tracking and tracing, respectively. They are described in detail as follows.
1. Data source service
Data source service is used to offer a mechanism for the information sharing and integration between EISs and manufacturing execution level. However, the heterogeneity of EISs proposes high challenge to share and integrate the information between these manufacturing elements. To address this problem, Business to Manufacturing Markup Language (B2MML) is employed here to provide standard information schemas for EISs and manufacturing execution level.
Generally, parameters of data source of the EISs that decision makers want to know or acquire are input in the data source service, and the shared and integrated information which is based on B2MML schemas is the output.
2. WIP tracking and tracing
This component is used to configure distributed EAs based on the relationships between them and to obtain real-time information of WIP in the manufacturing shop floor.
CE structure is adopted in this component. On one hand, it aims to get more useful and actionable information based on the initial information in manufacturing shop floor. On the other hand, it is used as a tool for controlling the event-driven information systems. After getting real-time production information, an instance is created. According to created instance, CE is formed. Then the real-time information of CE can be obtained. Based on the real-time information of CE, users can monitor and control the manufacturing process in the shop floor.

8.8. GA-based production scheduling algorithm

This section proposes the GA-based production scheduling algorithm. The GA is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms, and it is commonly used to generate high-quality solutions to optimization problems. Three elements are important in the algorithm, namely, gene and chromosome, fitness function, and iteration strategy.
1. Gene and chromosome
At each generation, every chromosome represents a plan for the given production schedule problem. Before the GA can be run, a suitable encoding schema of the problem must be set. The goal is to encode a set of genes (parameters) and to join them together in order to form a chromosome.
In this chapter, the genes and chromosomes are constructed according to the integer-based pattern. An integer i (1 ≤ i ≤ n) is used to form a gene, where, n is the total number of tasks. The integer will be repeated Ni times for task (i) in the chromosome, and the occurrence order of the integer “i” reflects the processing order. By queuing the different genes and connected by char “-,” the chromosome is formed. Thus, the length of the chromosome is determined by the total number of the processes of all the tasks.
2. Fitness function
A fitness function is set to assign a figure of merit to each chromosome. In this chapter, the objective function in Section 8.6 is applied to be the fitness function to estimate the chromosome.
3. Evolution strategy

During each iteration, parents must be selected to generate new offspring. Generally, parents are randomly selected from the population, using a scheme which is created according to the fitness function. Three kinds of evolution mechanisms, that is, selection, crossover, and mutation, are used in GA. Normally, the mechanism “selection” is easy and has well-behaved offspring compared with the other two. Thus, only crossover and mutation mechanisms are presented.

Crossover mechanism is applied to create offspring by recombining the randomly selected parents. Here, we use the multipoint crossover mechanism to change the task order without violating the validity of the produced chromosome. The proposed crossover mechanism contains five main steps.

Step 1: Randomly select two parents, for example, P1: (3-1-3-2-1-2); P2: (1-2-3-1-2-3).
Step 2: Choose two positions to generate two crossover sections for the two parents randomly. Such as, if points 2 and 3 are chosen, the selected crossover sections are (1-3) in P1 and (2-3) in P2.
Step 3: For each parent, adopt character “0” to substitute the genes in the crossover section of another parent. For example, for P1, use character “0” to substitute the genes in the crossover section of P2, that is, the genes “2” and “3.” Thus, a new parent P1′: (0-1-3-0-1-2) can be acquired.
Step 4: Move the characters “0,” so that they can arrive the cross-section. For example, after this step, P1′ (0-1-3-0-1-2) will be converted to P1″ (1-0-0-3-1-2).
Step 5: Replace the characters “0” with the crossover section of another parent to create offspring so that the characters “0” in P1″ (1-0-0-3-1-2) will be substituted by the crossover section of P2, that is, (2-3). Thus, the new offspring (1-2-3-3-1-2) is created.

Mutation mechanism aims to keep the variety in each population. Only a single chromosome can be altered in the mutation mechanism; the offspring will be generated by changing one or more genes. Contrast to the normal mutation mechanism, an original mutation mechanism is provided to avoid the absence of good results. A performance function is designed to assess each gene of the chromosome, and two points of the inferior genes are randomly selected as the mutation points. The performance function is given in Eq. (8.6).

PVi=Btj−Ctj+Btj−CtjCtkj−Btj

image(8.6)
where, Pi denotes the performance of gene (i), j represents the equipment ID, k denotes the total number of genes allocated to equipment (j), Btijimage means the begin time of gene (i) assigned at equipment (j), Ctijimage represents the completed time of gene (i) at equipment (j), Bti+1jimage and Cti−1jimage represent the begin time of the later process and the completed time of prior process of gene (i) assigned at equipment (j). Clearly, the lesser the value of Pi, the better the performance of gene (i).

The presented mutation mechanism contains four main steps, as follows.

Step 1: Choose a parent P1, and use Eq. (8.7) to calculate the performance value for each gene of P1. And store the values in a matrix

M={(g1,pv1),...,(gi,pvi),...,(gm,pvm)}

image(8.7)
where gi represents the jth gene of P1 and pvi means the performance value.
Step 2: Permute M based on the rising value of pvi and obtain M′. In M′, the later positions mean the related genes have worse performance. Let p′ be a gauge for evaluating the performance of genes. If pvi > p′, the corresponding gene gi in P1 is inferior, and the inferior genes can be stored as IG = {g1, …, gk, …, gl}.
Step 3: Randomly choose two genes from IG, for example, gi and gj (i ≠ j). These genes are the substituted genes for P1.
Step 4: Alter the selected genes (gi and gj) of P1 and obtain a new chromosome. That is the mutated offspring.

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