An Integrated Scheduling Algorithm Based on a Process End Time-Driven and Long-Time Scheduling Strategy

Abstract

The integrated scheduling problem is a classical combinatorial optimization problem. The existing integrated scheduling algorithms generally adopt the short-time scheduling strategy that does not fully consider the impact of the degree of process parallelism on scheduling results. In order to further optimize the total processing time of a product and the utilization rate of a device, an integrated scheduling algorithm based on a process end time-driven and the long-time scheduling strategy is proposed. The proposed integrated scheduling algorithm sets up a separate candidate process queue for each device and determines the scheduling order for each scheduling queue on the premise of satisfying the constraint conditions of the process tree. Driven by the process end time, the algorithm finds schedulable processes for each device. If the schedulable process is unique, it is scheduled. Otherwise, if the schedulable process is not unique, the process with long-path and long-time is scheduled. In particular, the scheduling strategies of the scheduling queues of different devices are symmetric, and the constraint relationships between the processes in different queues are asymmetric. The case analysis results show that the proposed integrated scheduling algorithm is better than some existing algorithms in terms of the total processing time of a product and the average utilization rate of devices. Therefore, the proposed algorithm provides a new idea for processing the scheduling of a single complex product.

1. Introduction

With the arrival of the “Industry 4.0” era [1,2], intelligent manufacturing has become a research hotspot in the field of industrial production. In order to cope with the new challenges brought by “Industry 4.0”, the major developed and developing countries in the world have put forward their own coping strategies. For example, the United States proposed “Industrial Internet” [3], and China proposed “Made in China 2025” [4]. Under the background of this era, the production efficiency of manufacturing enterprises is the key factor that restricts the development of enterprises.

In order to improve the production efficiency of enterprises, the job-shop scheduling problem (JSSP) [5] has been a research hotspot in the field of manufacturing in the past few decades. JSSP is a classical combinatorial optimization problemand an NP-hard problem, which makes the resource utilization rate higher and the total product processing time shorter by reasonably arranging theprocessing sequence of the processes under the condition of limited resources and specific tasks [6]. For nearly half a century, in order to obtain better scheduling results, scholars have proposed various solutions to job-shop scheduling problems and published tens of thousands of papers. Here are some representative research results: Dagli and Sittisathanchai proposed a genetic algorithm and artificial neural networks-based job-shop scheduling algorithm in which a genetic algorithm was used as a search technique for an optimal schedule via a uniform randomly generated population of gene strings, and artificial neural network was used as a multi-criteria evaluatorto evaluate the fitness or performance of those stimulated gene strings [7]. For solving the fuzzy job-shop scheduling problems, Sakawa and Mori proposed a genetic algorithm, incorporating the concept of similarity among individuals into the genetic algorithms [8]. Aydin and Fogarty proposed a modular simulated annealing algorithm for a multi-agent system running on adistributed resource machine, which surpassed many existing methods [9]. Jia et al. proposed a job-shop scheduling algorithm by integrating the genetic algorithm and Gantt chart, which hasproved to be efficient in solving small-sized or medium-sized scheduling problems for a distributed manufacturing system [10]. In order to simultaneously minimize total processing time and total tardiness of jobs, Lei proposed a Pareto archive particle swarm optimization for the multi-objective job-shop scheduling problem, which combined the global best position selection and the crowding measure-based archive maintenance and could produce high-quality scheduling plans [11]. In order to solve the flexible job-shop problem with fuzzy processing time, Xu et al. proposed a teaching-learning-based optimization algorithm which adopted a bi-phase crossover scheme based on the teaching–learning mechanism and special local search operators to balance the exploration and exploitation capabilities [12]. Hart and Sim proposed a hyper-heuristic ensemble method for solving job-shop scheduling problems, which introduced a novel heuristic generator that evolves heuristics composed of linear sequences of dispatching rules and outperforms existing dispatching rules on a large set of new test instances [13]. Li et al. proposed a multi-objective low-carbon job-shop scheduling problem with variable processing speed constraint, in which an improved artificial bee colony algorithm was implemented to optimize objectives of minimizing the total processing time, total carbon emission and machine loading [14].Forthe flexible manufacturing scheduling problem, Xu and Chen proposed a Petri-Net-based scheduling algorithm by using estimate function and heuristic search [15]. For the No-wait flow-shop scheduling problem, Gao et al. proposed a matheuristic approach that combines exact and heuristic algorithms to minimize the total processing time [16]. In addition, Valenzuela-Alcaraz et al. proposed a cooperative coevolutionary algorithmin which the sequencing and the timetabling parts interact with each other to evolve quasi-optimal sequencing and timetabling decisions [17]. For the two-stage hybrid flow shop scheduling problem, Hidri and Elsherbeeny proposed a heuristic composed of two phases, in which a new family of lower bounds and an exact Branch and Bound algorithm were adopted to solve the hard test problems [18]. For the multi-objective complex job-shop scheduling problem, Tamssaouet et al. extended the batch-oblivious approach by considering unavailability periods and minimum time lags and by simultaneously optimizing multiple criteria that are relevant in the industrial context [19]. For green scheduling in flexible job shops, Wang et al. proposed a multi-objective cellular memetic optimization algorithm which combined the advantages of cellular structure for global exploration andvariable neighborhood search for local exploitation [20]. The various job-shop scheduling algorithms described above are only a small part of the JSSP solution. These algorithms have played an important role in improving the production efficiency of traditional manufacturing enterprises.

With the increasing demand for personalized products, multi variety and small batch production scheduling has become a practical problem faced by many manufacturing enterprises. For the production of large quantities of the same products, the method of processing first and then assembling is generally adopted, and product manufacturing is completed efficiently through the effective scheduling of the processing and assembly lines. However, forsingle products with complex structures (such as a tree process diagram), if the method of processing first and then assembling is adopted for manufacturing, the internal relationship between processing and assembly of the product that can be processed in parallel will inevitably be split, which will affect the product manufacturing efficiency. Therefore, Xie et al. first proposed the third type of product manufacturing scheduling mode after product processing scheduling and product assembly scheduling, which processes processing and assembly together, referred to as integrated scheduling [21,22,23]. Specifically, Xie et al. proposed a time-selective integrated scheduling algorithm considering the compactness of serial processes (TSPC-ISA) [21]. The TSPC-ISA proposes the process sequence strategy which divides the processes into several process sequences with only serial relation based on the integral structure of the processing tree and determines the scheduling order according to the length of the process sequence. It also proposes the time-selective strategy which selects some legal processing time points for the scheduling process, obtains a set of trial scheduling schemes at eachtime point, and chooses the scheme with the least total time as the process scheduling scheme. Xie et al. also proposed an integrated scheduling algorithm based on events driven by machines’ idle (MIED-ISA) [22]. The MIED-ISA selects the processing process according to the machine idle event. If the schedulable process is unique, it schedules this process; otherwise, it schedules the process with the longest path length. If the processes with the longest path length are not unique, itschedules the process with the shortest processing time. Later, Xie et al. proposed a machine-driven integrated scheduling algorithm with rollback-preemptive (MDRP-ISA) [23]. The MDRP-ISA considersthe end of each process as a searchable event of a schedulable process. At this time, if a new schedulable process has preemptive ability, it generates a rollback event to reschedule. In case the rollback event is not generated, it schedules this process if the schedulable process is unique; otherwise, it schedules the process with long-path and short-time. In recent years, the integrated scheduling problem has also received the attention of other scholars. For the complex product scheduling problem with no-wait constraint, Guo et al. successively proposedan integrated scheduling algorithm based on a virtual component [24], and an integrated scheduling algorithm based on a reverse virtual component [25]. Wang et al. proposed an integrated scheduling algorithm for multi-device-processes with the strategy of exchanging adjacent parallel processes of the same device [26]. Wang et al. proposed an integrated scheduling algorithm for multiple complex products with due date constraints [27]. Zhang et al. proposed a multiple-devices-process integrated scheduling algorithm with a time-selective strategy for a process sequence [28]. Wang et al. proposed an improved integrated scheduling algorithm with a process sequence time-selective strategy [29]. Deng et al. proposed a distribution algorithm for solving the three-stage multi-objective integrated scheduling problem [30].

In addition to the integrated scheduling algorithm described above, there are also some integrated scheduling algorithms for specific problems. Zhuang et al. proposed the integrated scheduling algorithm of inbound and depart flights in airports [31]; Chen et al. proposed the integrated scheduling model of a mixed cross-operation for container terminals [32]; Fan et al. proposed the integrated scheduling of production and delivery on a single machine with availability constraints [33]; Yang et al. proposed the integrated scheduling method for AGV routing in automated container terminals [34]; Xu et al. proposed theintegrated scheduling optimization of a U-shaped automated container terminal under loading and unloading mode [35].

Among the above-described integrated scheduling algorithms, both process-based algorithms and device-based algorithms adopt the scheduling strategy of long-path and short-time in the scheduling process, which shows good scheduling performance for dealing with general integrated scheduling problems. However, in some specific scheduling tasks, the short-time strategy is not necessarily the best choice. In this paper, the limitations of the short-time strategy are analyzed by an example, and an integrated scheduling algorithm based on process end time-driven and the long-path and long-time scheduling strategy is proposed. In addition, the existing integrated scheduling algorithms basically set a unified scheduling queue for all the devices, so it is necessary to sort the schedulable processes according to specific strategies. This is essentially an asymmetric solution. The integrated scheduling algorithm proposed in this paper establishes an independent scheduling queue for each device. The scheduling strategies between these queues are consistent or symmetric, and the process constraints between queues are asymmetric. This design can simplify the scheduling process and reduce the complexity of the algorithm. The case analysis shows that the proposed algorithm not only makes full use of the idle time of equipment and improves the efficiency of equipment utilization, but also increases the parallelism of process processing on different equipment through the long-time strategy, reduces the total processing time of products and optimizes the scheduling result.

2. Materials and Methods

2.1. Problem Description

The integrated scheduling of products unifies the processing and assembly processes as processing, unifies the processing and assembly equipment as equipment, and adopts the production mode of “processing while assembling” to complete product production. The goal of the integrated scheduling algorithm is to minimize the completion time of the product by arranging the reasonable execution sequence of the processes under the premise of specific equipment conditions and process requirements. When studying the integrated scheduling problem, the product processing process tree can be used to describe the processing time, processing equipment and the processing partial order relationship of each process of the product. The preconditions of the integrated scheduling problem include:

(1)

One process can only be processed on one machine.

(2)

One machine can only process one process at a time.

(3)

A process can only be started when all the immediately preceding processes are finished or there are no immediately preceding processes.

(4)

No process can be interrupted.

(5)

The processing completion time of the last process is the total processing time of the product.

It is assumed that the product consists of N processes and is processed on M sets of equipment. The total product processing time is expressed as:

$$\mathrm{T}=\max \left({\mathrm{ST}}_{\mathrm{n}}+{\mathrm{PT}}_{\mathrm{n}}\right), 0\le \mathrm{n}\le \mathrm{N}$$

(1)

${\mathrm{ST}}_{\mathrm{n}}$ represents the start processing time of process n; ${\mathrm{PT}}_{\mathrm{n}}$ represents the processing time of process n. Then, the objective function of product integrated scheduling is expressed as:

$$\mathrm{T}=\min \left[\max \left({\mathrm{ST}}_{\mathrm{n}}+{\mathrm{PT}}_{\mathrm{n}}\right)\right]$$

(2)

In order to facilitate the following algorithm description, the following definitions are given first:

Definition 1.

The schedulable process: a process that has no immediate process or whose immediate process has been processed.

Definition 2.

The unscheduled operation: a process whose immediate processes have not been processed.

Definition 3.

The equipment set: a set of all processing equipment in the scheduling system.

Definition 4.

The process set: a set of all processing processes in the scheduling system.

Definition 5.

The equipment candidate process queue: a queue of processwaiting to be scheduled on a certain equipment at a certain time.

Definition 6.

The constraint relation set: a set of constraint (partial order) relations between all processes in the scheduling system.

Definition 7.

The process end time queue: a queue composed of all process end times in the scheduling system.

Definition 8.

The schedulable process set: a set of schedulable processes on a certain equipment at a certain time.

Definition 9.

The path length: the sum of the processing times of all processes from the root node (including) to the current node (including) in the processing process tree of the product.

2.2. The Limitation of Short-Time Scheduling Strategy

The short-time scheduling strategy has been applied to many process-based and device-based integrated scheduling algorithms. Under the situation that the degree of parallelism between processes is not too high, the short-time strategy can achieve better scheduling performance. However, with the increase inthe degree of parallelism between processes, the short-time scheduling strategy may not be able to guarantee a smaller total product processing time.

The following is an example to analyze the limitations of the short-time scheduling strategy. It is assumed that the processing process tree of product A is as shown in Figure 1.If product A is scheduled according to the scheduling strategy of long-path and short-time, the Gantt chart of product scheduling is as shown in Figure 2. On the contrary, if product A is scheduled according to the scheduling strategy of long-path and long-time, the Gantt chart of product scheduling is as shown in Figure 3.

As shown in Figure 2, at time T0, the schedulable processes are P5 and P8, both of which have a path length of 22, and the process P5 with a short processing time is preferentially scheduled. After P5 processing, the schedulable processes are P4 and P8, which can be processing in parallel on M2 and M3 equipment, respectively. After P4 processing, only P3 can be scheduled, and P3 can be processed on M4 equipment. During P3 processing, P8 is finished. At this time, P7 can be scheduled to process on the M1 equipment, and there is no schedulable process on the M3 equipment, so it is temporarily idle. After P3 is finished, P7 is executing at this time. Only P2 can be scheduled on M3, and P2 is processing on M3 equipment. After P2 processing, P6 can be scheduled to be processed on M3. Finally, P1 is scheduled to be processed on M2. The total processing time of the product A scheduled according to the long-path and short-time strategy is 30.

As shown in Figure 3, at time T0, the schedulable processes are P5 and P8, both of which have a path length of 22, and the process P8 with a long processing time is preferentially scheduled; After P8 processing, the schedulable processes are P5 and P7, which can be processed in parallel on M3 and M1 equipment, respectively. At the time of P5 ending, P7 has also been processed. At this time, the schedulable processes are P4 and P6, which can be processed on M2 and M3 equipment, respectively. After P4 is completed, P3 and P2 can be scheduled to process on M4 and M3 equipment, respectively. Finally, P1 is scheduled to be executed on M2. The total processing time of the product A scheduled according to the long-path and long-time strategy is 29.

From the above analysis, it can be seen that the total processing time of product Ascheduled by the long-time scheduling strategy is smaller than that by short-time scheduling strategy. The reason is that, at the beginning of the scheduling, letting P5 with a shorter processing time execute first can advance the start execution time of subsequent P4, P3 and P2, but it will delay the start execution time of P8, P7 and p6. Moreover, due to the high parallelism between P7, P6 and P4 and P3, the impact of delayed execution of P7 and P6 is greater than that of delayed execution of P4 and P3, so the total processing time of product A increases instead. Therefore, the conclusion is that under the situation of highly parallel product processes, priority scheduling of short-time processes on the same equipment is not necessarily able to reduce the total processing time of products than priority scheduling of long-time processes; that is, the limitation of the short-time scheduling strategy in parallel scheduling.

2.3. Process End Time-Driven Scheduling Strategy

2.3.1. The Basic Idea of Process End Time-Driven Scheduling Strategy

Based on the above analysis, this paper proposes an integrated scheduling algorithm based on the process end time-driven and long-path and long-time scheduling strategy. The basic idea of the proposed algorithm is to design a candidate process queue for each device. When the end time of a process arrives, the algorithm selects a schedulable process in the candidate process queue of each device for scheduling by long-path and long-time scheduling strategy. Meanwhile, the end time of the scheduled process is calculated and added to the process end time queue. Specifically, if there is no schedulable process in the candidate process queue of an equipment at a certain time, the equipment is temporarily idle. On the other hand, if there is a unique schedulable process, the process is scheduled and the end time of the process is calculated. Alternatively, if the schedulable operation is not unique, the process with a long path length is preferentially scheduled, otherwise, if the path lengths of a plurality of schedulable processes are the same, the process with a long processing time is preferentially scheduled. Differently, if the processing times of all schedulable processes are the same, any one process can be selected for scheduling. Under the constraint conditions in the constraint relation set, the process scheduling on different equipment can be executed in parallel.

2.3.2. The Detailed Designof Process End Time-Driven Scheduling Strategy

Suppose there is a process set $\mathrm{P}=\left\{{\mathrm{P}}_1,{\mathrm{P}}_2,\dots ,{\mathrm{P}}_{\mathrm{N}}\right\}$ and a device set $\mathrm{D}=\left\{{\mathrm{D}}_1,{ \mathrm{D}}_2,\dots ,{\mathrm{D}}_{\mathrm{M}}\right\}$; the candidate process queue on device ${\mathrm{D}}_{\mathrm{m}}$ at t time is represented as ${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}\mathrm{t}}$, $1\le \mathrm{m}\le \mathrm{M}$, and the constraint relation set at t time is represented as ${\mathrm{CS}}_{\mathrm{t}}$ and the process end time queue is expressed as: $\mathrm{TQ}=\left\{{\mathrm{ET}}_1,{\mathrm{ET}}_2,\dots ,{\mathrm{ET}}_{\mathrm{N}}{|\mathrm{ET}}_{\mathrm{j}}\le {\mathrm{ET}}_{\mathrm{j}+1},1\le \mathrm{j}\le \mathrm{N}\right\}$.

Assuming that all devices are idle at time ${\mathrm{t}}_0$, and no process execution ends, in other words, $\mathrm{TQ}=\varnothing$. According to the process end time-driven and long-path and long-time scheduling strategy, the scheduling process of a product is as follows:

At time ${\mathrm{t}}_{\mathrm{i}}$, the scheduling algorithm searches the schedulable process in each equipment candidate process queue ${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_{\mathrm{i}}}$. If there is no schedulable process on a device ${\mathrm{D}}_{\mathrm{m}}$, the device is temporarily idle. In contrast, if there is a unique schedulable process ${\mathrm{P}}_{\mathrm{k}}\in {\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_{\mathrm{i}}}$ on ${\mathrm{D}}_{\mathrm{m}}$, ${\mathrm{P}}_{\mathrm{k}}$ is scheduled. However, if there are multiple schedulable processes on ${\mathrm{D}}_{\mathrm{m}}$ and there is a schedulable process ${\mathrm{P}}_{\mathrm{k}}\in {\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_{\mathrm{i}}}$ with the longest path length, ${\mathrm{P}}_{\mathrm{k}}$ is scheduled. On the other hand, if there are multiple schedulable processes on ${\mathrm{D}}_{\mathrm{m}}$ and their path lengths are the same, the schedulable process ${\mathrm{P}}_{\mathrm{k}}\in {\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_{\mathrm{i}}}$ with the longest processing time is selected for scheduling. Contrarily, if there are multiple schedulable processes with the same path length and the same processing time on ${\mathrm{D}}_{\mathrm{m}}$, any one of the schedulable processes ${\mathrm{P}}_{\mathrm{k}}\in {\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_{\mathrm{i}}}$ can be selected for scheduling. After determining the scheduled process ${\mathrm{P}}_{\mathrm{k}}$, its end timeis calculated according to Equation (3).

$${\mathrm{ET}}_{{\mathrm{p}}_{\mathrm{k}}}={\mathrm{t}}_{\mathrm{i}}+{\mathrm{PT}}_{{\mathrm{p}}_{\mathrm{k}}}$$

(3)

Next, the process end time queue $\mathrm{TQ}$ is updated according to equation 4. Meanwhile, the time in $\mathrm{TQ}$ is sorted from small to large.

$$\mathrm{TQ}=\mathrm{TQ}+\left\{{\mathrm{ET}}_{{\mathrm{p}}_{\mathrm{k}}}\right\}$$

(4)

After oneprocessis completed, the constraint relation set ${\mathrm{CS}}_{\mathrm{t}}$ is updated. Next, for the next time ${\mathrm{t}}_{\mathrm{i}+1}\in \mathrm{TQ}$, the above process is repeated for scheduling until all the processes are completed. Finally, the end time of the last process is the total processing time of the product.

2.4. Algorithm Design and Complexity Analysis

2.4.1. Algorithm Design

According to the foregoing description, this paper proposed anintegrated scheduling algorithm based on process end time-driven and long-time scheduling strategy (PETD- ISA). The specific design of PETD-ISA is shown in Algorithm 1.

Algorithm 1 PETD-ISA
Input:	Processingprocess data of a product
Output:	Processing scheduling scheme and total processing time of the product
Procedure	PETD-ISA (processing process data of a product)
Step 1.	Set the process end time queue $\mathrm{TQ}=\left\{{\mathrm{t}}_0=0\right\}$$,{\mathrm{t}}_0$ represents the start processing time of the product;
Step 2.	Set device status ${\mathrm{DS}}_{\mathrm{m}}=\mathrm{IDLE}, 1\le \mathrm{m}\le \mathrm{M}$;
Step 3.	Calculate path lengths, $\mathrm{PL}=\left\{{\mathrm{PL}}_{\mathrm{n}}\right\}, 1\le \mathrm{n}\le \mathrm{N}$;
Step 4.	Set the process constraint relation set,$\mathrm{CS}={\mathrm{CS}}_{{\mathrm{t}}_0}$;
Step 5.	Set the candidate process queue for ${\mathrm{D}}_{\mathrm{m}}$, ${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}}={\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}{\mathrm{t}}_0}, 1\le \mathrm{m}\le \mathrm{M}$;
Step 6.	for ($\mathrm{t}={\mathrm{t}}_{\mathrm{i}}\in \mathrm{TQ}$)
Step 7.	In all candidate process queues ${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}\mathrm{t}}$, find the schedulable process set ${\mathrm{SPQ}}_{{\mathrm{D}}_{\mathrm{m}}\mathrm{t}}=\left\{{\mathrm{P}}_{\mathrm{k}1},{\mathrm{P}}_{\mathrm{k}2},\dots ,{\mathrm{P}}_{\mathrm{kl}}\right\}$ according to ${\mathrm{CS}}_{\mathrm{t}}$;
Step 8.	if $({\mathrm{SPQ}}_{{\mathrm{D}}_{\mathrm{m}}\mathrm{t}}=\varnothing )$ then ${\mathrm{DS}}_{\mathrm{m}}=\mathrm{IDLE}$;
Step 9.	else if (only one process ${\mathrm{P}}_{\mathrm{k}}$ in ${\mathrm{SPQ}}_{{\mathrm{D}}_{\mathrm{m}}\mathrm{t}}$) then schedule ${\mathrm{P}}_{\mathrm{k}}$;
Step 10.	else if (only one process ${\mathrm{P}}_{\mathrm{k}}$ with the longest path length) then schedule ${\mathrm{P}}_{\mathrm{k}}$;
Step 11.	else if (only one process ${\mathrm{P}}_{\mathrm{k}}$ with the longest processing time) then schedule ${\mathrm{P}}_{\mathrm{k}}$;
Step 12.	else randomly select a process ${\mathrm{P}}_{\mathrm{k}}$ with the longest processing time for scheduling.
Step 13.	Set the state of ${\mathrm{D}}_{\mathrm{m}}$ of the scheduled process ${\mathrm{P}}_{\mathrm{k}}$$, {\mathrm{DS}}_{\mathrm{m}}=\mathrm{BUSY}$;
Step 14.	Set the start processing time of ${\mathrm{P}}_{\mathrm{k}}$$, {\mathrm{ST}}_{{\mathrm{p}}_{\mathrm{k}}}={\mathrm{t}}_{\mathrm{i}}$;
Step 15.	Calculate the process end time of ${\mathrm{P}}_{\mathrm{k}}$$, {\mathrm{ET}}_{{\mathrm{p}}_{\mathrm{k}}}={\mathrm{t}}_{\mathrm{i}}+{\mathrm{PT}}_{{\mathrm{p}}_{\mathrm{k}}}$;
Step 16.	$\mathrm{TQ}=\mathrm{TQ}+\left\{{\mathrm{ET}}_{{\mathrm{p}}_{\mathrm{k}}}\right\}$, and sort $\mathrm{TQ}$ by time from small to large;
Step 17.	if (${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}}=\varnothing$ for all $\mathrm{m}, 1\le \mathrm{m}\le \mathrm{M}$) then jump to Step25;
Step 18.	else
Step 19.	Select next time $\mathrm{t}={\mathrm{t}}_{\mathrm{i}+1}$ from $\mathrm{TQ}$;
Step 20.	At time ${\mathrm{t}}_{\mathrm{i}+1}$, find a process ${\mathrm{P}}_{\mathrm{k}}$ whose ${\mathrm{ET}}_{{\mathrm{p}}_{\mathrm{k}}}={\mathrm{t}}_{\mathrm{i}+1}$;
Step 21.	Set the state of ${\mathrm{D}}_{\mathrm{m}}$ where ${\mathrm{P}}_{\mathrm{k}}$ is assinged: ${\mathrm{DS}}_{\mathrm{m}}=\mathrm{IDLE}$;
Step 22.	Update ${\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}}$$: {\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}}={\mathrm{PQ}}_{{\mathrm{D}}_{\mathrm{m}}}-\left\{{\mathrm{P}}_{\mathrm{k}}\right\}$;
Step 23.	Update ${\mathrm{CS}}_{\mathrm{t}}$$: {\mathrm{CS}}_{\mathrm{t}}={\mathrm{CS}}_{{\mathrm{t}}_{\mathrm{i}+1}}$;
Step 24.	Jump to Step 7;
Step 25.	Output the scheduling Gantt chart of the product according to process processing sequence;
Step 26.	Output the maximum process end time, ${\mathrm{T}}_{\max }=\max \left(\mathrm{T}\right)$, which is the total processing time ${\mathrm{T}}_{\mathrm{total}}$.
End procedure

The flow chart of PETD-PAP-TWIS is shown in Figure 4.

2.4.2. Algorithm Complexity Analysis

It is assumed that the product is composed of N processes and processed on M devices. In step 1 of PETD-ISA, the time complexity for initialization is constant. In step 2, the time complexity of setting the device state is $\mathrm{O}\left(\mathrm{M}\right)$. In step 3, the time complexity for calculating the path length of each process is $\mathrm{O}\left(\mathrm{N}\right)$. In step 4, the time complexity fordetermining the process constraint relationship set is $\mathrm{O}\left(\mathrm{N}\right)$. In step 5, the time complexity fordetermining the candidate process queue of the device is $\mathrm{O}\left(\mathrm{N}\right)$. In steps 6–24, the scheduling process is executed at the end time of each process, and the maximum number of executions is N. In step 6, on average, the number of processes in each equipment candidate process queue is $\mathrm{N}/\mathrm{M}$, and the time complexity of finding schedulable processes and performing corresponding operations on each equipment is $\mathrm{O}\left(\mathrm{N}/\mathrm{M}\right)$, so in each scheduling process, the total time complexity is $\mathrm{O}\left(\mathrm{N}/\mathrm{M}\ast \mathrm{M}\ast \mathrm{N}\right)=\mathrm{O}\left({\mathrm{N}}^2\right)$. In step 23 and 24, the time complexity of the output scheduling scheme is $\mathrm{O}\left(\mathrm{N}\right)$. To sum up, the total time complexity of algorithm PETD-ISA is $\mathrm{O}\left({\mathrm{N}}^2\right)$.

3. Results

3.1. Performance Evaluation of PETD-ISA Algorithm

The integrated scheduling algorithm proposed in this paper (PETD-ISA) does not depend on any specific example. Here, in order to verify the effectiveness of PETD-ISA, we use product B as an example to compare the performance of PETD-ISA, the integrated scheduling algorithm based on event-driven by machines’ idle (MIED-ISA) [22], the machine-driven integrated scheduling algorithm with rollback-preemptive (MDRP-ISA) [23] and the time-selective integrated scheduling algorithm, considering the compactness of serial processes (TSPC-ISA) [21]. The processing process tree of product B is shown in Figure 5.

The Gantt chart of product B scheduled by TSPC-ISA, MIED-ISA, MDRP-ISA and PETD-ISA is shown in Figure 6, Figure 7, Figure 8 and Figure 9, respectively. In addition, the completion time and utilization rate of devices when different scheduling algorithms are used to process product B is shown in

Table 1. Completion time and utilization rate of devices when different scheduling algorithms are used to process product B.

Algorithm	Completion Time of Devices/h				Utilization Rateof Devices
	M1	M2	M3	M4	M1	M2	M3	M4	AVG
TSPC-ISA	16	28	27	24	0.38	0.54	0.89	0.58	0.60
MIED-ISA	18	30	29	17	0.33	0.5	0.83	0.82	0.62
MDRP-ISA	18	30	29	17	0.33	0.5	0.83	0.82	0.62
PETD-ISA	12	28	27	24	0.5	0.54	0.89	0.58	0.63

.

As can be seen from Figure 6, Figure 7, Figure 8 and Figure 9, the total processing time of product B scheduled by TSPC-ISA, MIED-ISA, MDRP-ISA and PETD-ISA is 28, 30, 30 and 28, respectively. The results show that the algorithm proposed in this paper (PETD-ISA) is superior to MIED-ISA and MDRP-ISA in total processing time, and is the same as TSPC-ISA. In addition, as one can see from Table 1, the average utilization rate of devices is 0.60, 0.62, 0.62 and 0.63, respectively, when TSPC-ISA, MIED-ISA, MDRP-ISA and PETD-ISA is used to process product B. The results show that the PETD-ISA is better than other three algorithms in terms of the device utilization rate. Moreover, the implementation of PETD-ISA is much easier than TSPC-ISA, and the time complexity of the former is one order of magnitude lower than that of the latter. Considering various factors, the PETD-ISA algorithm is superior to the other three algorithms involved in the comparison.

3.2. Analysis of PETD-ISA Execution Process

In order to further illustrate the performance and working process of PETD-ISA, another more complex product example (product C) is used to compare PETD-ISA and TSPC-ISA [21]. Product Cconsists of 24 processes and is processed on 4 devices, whose processing process tree is shown in Figure 10.

The Gantt charts of product C scheduled by TSPC-ISA and PETD-ISA are shown in Figure 11 and Figure 12, respectively. In addition, the completion time and utilization rate of devices when different scheduling algorithms are used to process product C is shown in

Table 2. Completion time and utilization rate of devices when different scheduling algorithm are used to process product C.

Algorithm	Completion Time of Devices/h				Utilization Rateof Devices
	M1	M2	M3	M4	M1	M2	M3	M4	AVG
TSPC-ISA	26	24	23	18	0.85	0.79	0.74	0.72	0.78
PETD-ISA	26	24	21	13	0.85	0.79	0.81	1.0	0.86

. Moreover, the specific process of using PETD-ISA to schedule product C is shown in

Table 3. Specific process of using PETD-ISA to schedule product C.

Scheduling Time	Device	Candidate Process	Schedulable Process	Scheduled Process	End Time
T0	M1	P1(2)  P4(1)   P6(5)   P7(3)   P13(5)  P19(5)  P22(1)	P19(5)-18  P22(1)-21	P22(1)-21	T1
	M2	P2(3)  P3(1)   P10(4)  P12(2)  P14(3) P15(3)  P16(2)  P21(1)	P10(4)-15	P10(4)-15	T4
	M3	P8(3)  P9(3)   P11(2)  P18(2)  P20(4)  P24(3)	P24(3)-16	P24(3)-16	T3
	M4	P5(4)  P17(5)  P23(4)	P5(4)-17  P23(4)-20	P23(4)-20	T4
T1	M1	P1(2)  P4(1)   P6(5)   P7(3)   P13(5)  P19(5)	P19(5)-18	P19(5)-18	T6
T3	M3	P8(3)  P9(3)   P11(2)  P18(2)  P20(4)	-	-	-
T4	M2	P2(3)  P3(1)   P12(2)  P14(3)  P15(3)  P16(2)  P21(1)	P14(3)-16  P21(1)-20	P21(1)-20	T5
	M3	P8(3)  P9(3)   P11(2)  P18(2)  P20(4)	-	-	-
	M4	P5(4)  P17(5)	P5(4)-17	P5(4)-17	T8
T5	M2	P2(3)  P3(1)   P12(2)  P14(3)  P15(3)  P16(2)	P14(3)-16	P14(3)-16	T8
	M3	P8(3)  P9(3)   P11(2)  P18(2)  P20(4)	P20(4)-19	P20(4)-19	T9
T6	M1	P1(2)  P4(1)   P6(5)   P7(3)   P13(5)	P6(5)-11	P6(5)-11	T11
T8	M2	P2(3)  P3(1)   P12(2)  P15(3)  P16(2)	P15(3)-13	P15(3)-13	T11
	M4	P17(5)	P17(5)-13	P17(5)-13	T13
T9	M3	P8(3)  P9(3)   P11(2)  P18(2)	P9(3)-13	P9(3)-13	T12
T11	M1	P1(2)  P4(1)   P7(3)   P13(5)	P13(5)-15	P13(5)-15	T16
	M2	P2(3)  P3(1)   P12(2)   P16(2)	P2(3)-6	P2(3)-6	T14
T12	M3	P8(3)  P11(2)  P18(2)	P18(2)-10	P18(2)-10	T14
T13	M4	-	-	-	-
T14	M2	P3(1)   P12(2)  P16(2)	P12(2)-8	P12(2)-8	T16
	M3	P8(3)  P11(2)	-	-	-
T16	M1	P1(2)  P4(1)   P7(3)	-	-	-
	M2	P3(1)   P16(2)	P16(2)-10	P16(2)-10	T18
	M3	P8(3)  P11(2)	P8(3)-6	P8(3)-6	T19
T18	M1	P1(2)  P4(1)   P7(3)	-	-	-
	M2	P3(1)	-	-	-
T19	M1	P1(2)  P4(1)   P7(3)	P4(1)-3	P4(1)-3	T20
	M2	P3(1)	-	-	-
	M3	P11(2)	P11(2)-8	P11(2)-8	T21
T20	M1	P1(2)  P7(3)	P7(3)-6	P7(3)-6	T23
	M2	P3(1)	-	-	-
	M3	-	-	-	-
T23	M1	P1(2)	-	-	-
	M2	P3(1)	P3(1)-3	P3(1)-3	T24
T24	M1	P1(2)	P1(2)-2	P1(2)-2	T26
	M2	-	-	-	-
T26	-	-	-	-	-

Note: P19 (5)-18 in the table indicates that the execution time of process P19 is 5, and the path length in the processing process tree is 18.

. It can be seen from Figure 11 and Figure 12 that the total processing time of product C scheduled by TSPC-ISA and PETD-ISA is the same, which is 26. In fact, for product C, 26 is the minimum total processing time. Moreover, as one can see from Table 2, the average utilization rate of devices is 0.78 and 0.86 when TSPC-ISA and PETD-ISA is used to schedule product C, respectively. This again illustrates the effectiveness of PETD-ISA for the general integrated scheduling problem.

4. Discussion

Job-shop scheduling problem is a classical combinatorial optimization problem and also a NP hard problem. Over the years, scholars have carried out in-depth research on the job-shop scheduling problem and proposed a variety of solutions. Integrated scheduling is an extended field of job-shop scheduling, which was first proposed by Xie et al. [22,23]. For the production scheduling of a single complex product, Xie et al. proposed the concept of integrated scheduling that processes processing and assembly together and designed a series of integrated scheduling algorithms for different scheduling problems [21,28,29,30,31].

In the existing integrated scheduling problem solutions, the scheduling strategy of short-time is usually adopted. When it is necessary to schedule processes, priority is given to schedulable processes with short processing time. However, the short-time scheduling strategy is only suitable for scheduling tasks with a low degree of parallelism between processes. For scheduling tasks with a high degree of parallelism between processes, the long-time scheduling strategy may obtain shorter product completion time. This paper discusses this through case analysis, and the example in Section 2.2 verifies our point of view.

In addition, the existing integrated scheduling algorithms usually only set up one candidate process queue, which makes the scheduling algorithm more complex. By setting up a candidate processes queue for each processing equipment, it is not only more convenient to implement the process end time-driven scheduling strategy, but also tosimplify the implementation of the algorithm and reduce the complexity of the algorithm.

Moreover, the process end time-driven scheduling strategy can provide unified scheduling instructions for different processing equipment to reduce equipment idle timeand improve equipment utilization. Based on the reasons above, this paper proposed a new integrated scheduling algorithm based on a process end time-driven and long-time scheduling strategy. We proposed a scheduling strategy driven by process end time, which sets up a candidate process sequence for each device and makesthe next scheduling decision at the end of each execution process. We also adopt thelong-time scheduling strategy to select appropriate processes for priority processing. The example analysis showed that the performance of the integrated scheduling algorithm proposed in this paper was better than several existing integrated scheduling algorithms in terms of the total processing time of a product and the algorithm complexity. In a word, the research results of this paper provide new ideas for solving integrated scheduling problems.

5. Conclusions

The integrated scheduling algorithm based on a process end time-driven and long-time scheduling strategy is proposed in this work. Firstly, the short-time and the long-time scheduling strategy are discussed, and the case analysis results show that the long-time strategy may be better than the short-time strategy in optimizing the total processing time for products with a high degree of process parallelism. Secondly, a scheduling strategy driven by process end time is implemented, which designs a candidate process sequence for each device and makes the next scheduling decision at the end of each execution process. Finally, the comparison analysis shows that the proposed algorithm reduces the total processing time of products and the complexity of the algorithm compared with the existing integrated scheduling algorithms.

In future work, the process end time-driven scheduling strategy and the long-time strategy will be further studied and applied to more integrated scheduling problem solutions, such as multi-workshop integrated scheduling and flexible equipment integrated scheduling.