Symmetric Two-Workshop Heuristic Integrated Scheduling Algorithm Based on Process Tree Cyclic Decomposition

Abstract

The existing research on the two-workshop integrated scheduling problem with symmetrical resources does not consider the complex product attribute structure and the objective situation of plant equipment resources. This results in the prolongation of the product makespan and the reduction of the utilization rate of the general equipment in the workshop. To solve the above problems, a two-workshop integrated scheduling algorithm based on process tree cyclic decomposition (STHIS-PTCD) was proposed. First, a workshop scheduling scheme based on the sub-tree cyclic decomposition strategy was proposed to improve the closeness of continuous processing further. Second, an operation allocation scheme based on the principle of workshop processing balance was presented. On the basis of ensuring the advantages of parallel processing, it also effectively reduces the idle time of equipment resources and then optimizes the overall effect of the integrated scheduling of both workshops. Through the comparison and analysis of all the existing resource-symmetric two-workshop integrated scheduling algorithms, the scheduling effect of the proposed algorithm is the best.

1. Introduction

As an important production mode [1], distributed manufacturing has always been considered by experts and scholars with regards to its production efficiency, economic benefits, and social benefits. However, due to the large differences in production scale, production capacity, and regional differences among various factories, the distributed scheduling problem is mainly related to the study of the problem of job allocation and scheduling. These considerations achieve a better scheduling effect.

Aiming at the distributed scheduling problem of flow-shop and job-shop in traditional industry, many experts and scholars have carried out in-depth research from different research angles [2,3,4,5,6,7,8,9,10,11]. Zhao et al. proposed a pure reactive scheduling method for updating scheduling strategies to deal with the interference of uncertainty in the arrival of new jobs on the shop floor [2]. Delgoshaei et al. proposed a fuzzy-weighted NSGA-II (FW-NSGA-II) to address the developed non-linear fuzzy multi-objective dual resource-constrained scheduling problem [3]. Song et al. used the genetic programming hyperheuristic algorithm to solve a class of distributed assembly permutation flow shop scheduling problems related to time and sequence [4]. Chen et al. solved the distributed flow shop scheduling problem by constructing a heuristic algorithm and iterative greedy algorithm [5]. Zhang et al. proposed an improved heuristic Kalman algorithm to solve the dynamic job-shop scheduling problem [6]. Olmo A D et al. presented an indirect monitoring procedure and its application in AISI 1045 high-speed broaching for the unattended machine tool scheduling problem [7].

For multi-variety, small-batch, and complex products, different from the distributed job scheduling problem, this paper studies the distributed integrated scheduling problem. It is a scheduling mode for part processing and assembly synchronically [12,13,14,15,16,17,18] for the tree structure products with constraints between operations.

At present, in the existing resource-symmetric two-shop integrated scheduling problem, the research results include the balanced processing algorithm of two-shop scheduling processes [13], the allied-critical path algorithm [14], the neighborhood rendering algorithm [15], and the timing-selected algorithm [16]. Through improving horizontal parallel processing of schedulable operations, adopting the long-path method and the long-time method for key operations, neighborhood rendering strategy, and workshop selection strategy of the same equipment, these research results achieved good scheduling effect. However, the following major problems still remain.

• The integrated scheduling problem is proved to be a NP-hard problem, and the intensity of continuous processing can be further improved in the existing two-workshop integrated scheduling algorithm.
• On the basis of minimizing the time cost, the integrated scheduling optimization objective fails to fully and comprehensively consider the complex product properties and the comprehensive utilization of the objective situation of workshop equipment resources, thus increasing the migration frequency.
• The equipment resources in the workshop have the problem of low utilization rate or even being idle.

In order to solve the above common problems, this paper proposed a two-workshop integrated scheduling algorithm based on process tree cycle decomposition. It consists of the following three strategies.

• Sub-tree cycle decomposition strategy (SCDS). It decomposed the process tree into several operation strings, which could be scheduled together and reduced the number of operation migrations.
• Operation weight-based scheduling strategy (OWSS). It comprehensively considers the factors of equipment priority and operation constraint degree, and it establishes scheduling operation set in descending order of weight value, which improves the tightness of operation in longitudinal processing.
• Balanced workshop allocation strategy (BWAS). It is based on reducing the processing time difference between the two workshops to allocate the process set to be scheduled, so as to improve the utilization rate of the two vehicles.

2. Problem Description

Definition 1.

Symmetric two-workshop. Two different workshops with the same equipment resource and layout.

Definition 2.

Operation migration. If an operation is not processed in the same workshop as its immediate predecessor, then an operation migration is said to have occurred.

Definition 3.

Integrated scheduling. A scheduling method that schedules the operations of jobs and assembly simultaneously.

Definition 4.

Makespan. The completion time of the product is also the maximum completion time of the two workshops.

The symmetric two-workshop integrated scheduling problem can be described as: a single complex tree-structured product with n operations that can be produced by two workshops, f₁ and f₂, with the same equipment resource and layout. Each workshop contains m machines. The specific requirements are as follows:

(1) The processing structure of the complex product is a treelike structure, which can be represented by a processing tree.

(2) Each node of the tree represents an operation with three known information: operation number, machine type, and processing time.

(3) There are no machines of the same type in the same workshop.

(4) Each operation can only be processed on its specific type of machine in any workshop.

(5) An operation can only be processed after its immediate predecessors have completed.

The objective of our study is to minimize the makespan of products. The objective function is as follows.

Min max (C_m)

(1)

The completion time of operation i can be calculated with Equation (2):

$$C_i=S_i+{\displaystyle {\sum }_{k=1}^2{X}_{i,k}}+P_i$$

(2)

where $X_{i,k}=\left\{\begin{array}{l}1 \mathrm{If} \mathrm{operation} i \mathrm{is} \mathrm{processed} \mathrm{in} \mathrm{workshop} k\\ 0 \mathrm{Else}\end{array}\right.$ is a Boolean variable; C_i is the completion time of operation i; S_i is the starting time of operation i; P_i is the processing time of operation i; $i\in \{1,2,3,\dots ,n\}$, $k\in \{1,2\}$.

The makespan of the product is the completion time of the last operation. Then,

$$C_m={max}_{i=1}^n{C}_i$$

(3)

According to the description and requirements, the constraints are as follows.

Subject to:

$$C_i-C_j+W(1-Y_{ij})\ge P_i, \forall i,j$$

(4)

$$\begin{gathered} C_i-C_j+W(1-Z_{ij})\ge P_i, \\ \forall i,j \mathrm{if} i \mathrm{and}j \mathrm{are} \mathrm{processing} \mathrm{on} \mathrm{the} \mathrm{same} \mathrm{machine} \mathrm{in} \mathrm{the} \mathrm{same} \mathrm{workshop} \end{gathered}$$

(5)

$${\displaystyle {\sum }_{k=1}^2{X}_{i,k}}=1, \forall i$$

(6)

$$S_i\ge 0, P_i>0,\forall i$$

(7)

Constraint (4) ensures that an operation can only be processed after its predecessors on the same path of processing tree has completed. W is a given large positive integer, $Y_{i,j}=\left\{\begin{array}{l}1 \mathrm{If} \mathrm{operation} j \mathrm{is} a \mathrm{predecessor} \mathrm{of} \mathrm{operation} i \mathrm{on} \mathrm{the} \mathrm{same} \mathrm{path}\\ 0 \mathrm{Else}\end{array}\right.$. Constraint (5) ensures that an operation can only be processed after other operations on the same machine have been completed. $Z_{i,j}=\left\{\begin{array}{l}1 \mathrm{If} \mathrm{operation} j \mathrm{is} \mathrm{assigned} \mathrm{earlier} \mathrm{than} i \mathrm{on} \mathrm{the} \mathrm{same} \mathrm{machine}\\ 0 \mathrm{Else}\end{array}\right.$. Constraint (6) ensures that an operation can only be processed in one workshop. Constraint (7) ensures that all the information is significant.

3. Strategy Analysis and Design

3.1. Sub-Tree Cycle Decomposition Strategy (SCDS)

The main feature of the sub-tree decomposition is to divide the nodes in a complex graph, which is dominated by the treelike structure, into several finite and related node subsets according to some rules to decompose the graph into several relatively independent parts, and then we solve them separately, and finally we synthesize them to obtain the approximate optimal solution of the original problem [19,20,21,22,23,24].

For graph $G=(V,E)$, $V$ denotes the set of all the points in the graph, and $E$ denotes the set of all the edges in the graph. One of its corresponding decomposition trees $T_G$ can be expressed as ($\left\{X_i|i\in I\right\}$,T), where $I$ is a set of consecutive integers, and its number corresponds to the number of nodes in the decomposition tree. $X_i$ is a subset of $V$, which corresponds to the ith node in the decomposition tree [25].

For the decomposition tree $T_G$=($\left\{X_i|i\in I\right\}$, T), let $u \mathrm{and} v$ be two points of $V$, and let $R_v$ and $R_u$ be the root nodes of the sub-tree corresponding to u and v on the decomposition tree $T_G$. If $(u,v)\in E$, then $R_v$ and $R_u$ must be an ancestor-descendant relationship on the decomposition tree T, that is, the constraint relation of pre-tightening operation and post-tightening operation in the digraph. If a decomposition tree of graph $G$ can be expressed as an ordered pair $T_G=(X,T)$, then the following conditions are met:

(1) $\bigcup _{i\in I}{X}_i=V$;

(2) For any data point $(u,v)\in E$, there must be a node on the sub-tree T, such that the node set $X_i$ satisfies $\left(u,v\right)$∈$X_i$;

(3) $\forall \left(u,v\right)\in E,\exists i\in I$, have $u,v\in X_i$, that is, two nodes of every edge in $G$ form a cut set.

The SCDS in this article is based on the cycle judgment of the operation attributes, such as the number of branches and the leaf nodes. The complex product process tree is decomposed into several sub-tree strings, of which the internal relations are under the priority constraints. The solution category of sub-trees is much smaller than that of the whole process tree. It can effectively avoid the long waiting time of equipment during the processing to achieve the optimization effect of reducing the overall processing time of complex products. The specific description is as follows:

Step 1: The complex product process tree is simplified into the directed graph according to the constraint relation between operations.

Step 2: Cycle decomposition:

(1) Determine whether the branch of the process tree is unique. If it is, it does not meet the conditions of complex products, therefore end and exit the procedure; if it is not, the branch root node of the process tree is taken as the cut set to decompose the process tree and establish the sub-tree operation string.

(2) Determine whether the sub-tree procedure string is a leaf node procedure. If it is, the decomposition is complete. If it is not, repeat (1).

Step 3: Decompose the complex product process tree into several sub-tree operation strings according to the determined cut set. According to the cut set, establish the sub-tree process sequence of the original complex product process tree.

3.2. Example of the SCDS

Assume that complex product A consists of 25 operations A1–A25, which are processed on four machines, M₁, M₂, M₃, and M₄, in the integrated scheduling system. The processing process tree is shown in Figure 1. The demonstration process adopting the SCDS is as follows:

Step 1: As shown in Figure 2, {A23, A24} is selected as the cut set to decompose complex products for the first time, and the operation strings, T_A1 and T_A2, are obtained as shown in Figure 3a,b.

Step 2: the second decomposition of operation strings T_A1 and T_A2.

The branching number of operation string T_A1 and T_A2 is 2, so the operation string T_A1 takes {A5, A12} as the cut set for the second decomposition, and the obtained operation strings T_A11 and T_A12 are shown in Figure 4a,b. The operation string T_A2 takes {A16, A22} as the cut set for the second decomposition, and the obtained operation strings T_A21 and T_A22 are shown in Figure 5a,b. After the second decomposition, the branching number of the operation string T_A21 is 1, so the third decomposition will not be carried out. Currently, the operation string T_A21 = {A13, A14, A15, A16}.

Step 3: the third decomposition of operation strings T_A11, T_A12, and T_A22.

For T_A11, the number of its branches is 2, so T_A111 and T_A112 are decomposed for the third time by taking the operation {A5} as the cut set. T_A12 takes operation {A9} as a cut set to divide T_A121 and T_A121 for the third time. After the third decomposition, T_A111 = {A1}, T_A112 = {A2, A3, A4}, backtracking T_A11={{A1}, {A2, A3, A4}, A5}, T_A121 = {A6, A7}, T_A122 = {A8}, and backtracking T_A12 = {{A6, A7}, {A8}, A9, A10, A11, A12}. On the basis of T_A11 and T_A12, backtracking T_A1 = {{{A1}, {A2, A3, A4}, A5}, {{A6, A7}, {A8}, A9, A10, A11, A12}, A23}.

After the third decomposition, T_A2 = {{A13, A14, A15, A16}, {{A17, A18}, A19, A20, A21, A22}, A24}.

Step 4: According to the determined cut sets, the process tree of complex product A is decomposed into several operation strings circularity, as follows:

T_A = {T_A1, T_A2} = {{T_A11, T_A12}, {T_A21, T_A22}} = {{T_A111, T_A112}, {T_A121, T_A122}}, {T_A21}, {T_A221, T_A222}} = {{{{A1}, {A2, A3, A4}, A5}, {{A6, A 7}, {A8}, A9, A10, A11, A12}, A23}, {{A13, A14, A15, A16}, {{A17, A18}, {A19, A20, A21, A22}}, A24}, A25}.

3.3. Operation Weight-Based Scheduling Strategy (OWSS)

Definition 5.

Layer priority.

Assuming that there are n layers in the processing tree of complex products, the priority of the root node operation is defined as 1, the priorities of all descendant node operations of the root node operation are defined as 2, and so on, until the priorities of all nodes of the nth layer are defined as n. The root node operation has the lowest priority, and the nth layer operations have the highest priorities [26].

Definition 6.

Equipment priority.

Assuming that the integrated scheduling system has m machines with equipment sequence $\{M_1,M_2,\dots ,M_m\}$, defined that the device with the largest number of operations has the highest equipment priority, the device with the second largest number of operations has the second highest equipment priority, and so on, the device with the smallest number of operations has the lowest equipment priority as 1, and the same equipment priority is allowed.

Definition 7.

Constraint degree.

Taking a certain operation as the center, the sum of all the operations directly adjacent to it that have the constraint relationship of pre-tightening operation and post-tightening operation is defined the constraint degree of this operation.

Definition 8.

Weight value.

The sum of the values of the layer priority, equipment priority, and operation constraint of an operation is defined as the weight value of this operation.

In the operation scheduling strategy, the operation weight-based scheduling strategy (OWSS) is proposed to fully consider the importance of special equipment in the integrated scheduling, that is, the influence of equipment with a large number of operations on the scheduling result. The OWSS takes all the factors, such as the layer priority of the operations, the equipment priority, and the constraint degree, as the consideration factors affecting the scheduling result. On the basis of both vertical and horizontal optimization, the operation strings are scheduled in descending order of weights to increase the intensity of continuous processing on the equipment in both workshops as far as possible and reduce the idle period of the devices. The specific description is as follows:

Step 1: According to the structural characteristics of the complex product process tree, determine the sequence of the process tree and calculate the layer priority of each operation.

Step 2: According to the attribute characteristics of the process tree of complex products, calculate the equipment priority, the constraint degree, and the weight value of each operation.

Step 3: Determine the operation string where the operation with the highest weight except the root node is located.

Step 4: Judge whether the operation string is unique. If it is, then go to Step 5. If it is not, the operation string in which the operation is located is scheduled in the descending order according to the priority of the operation layer.

Step 5: Judge whether there is an optimal scheduling time for the operation string. If it is, schedule this operation string at the optimal time. If it is not, then go to Step 6.

Step 6: Schedule the operation string, of which the operations have lower weights.

Step 7: Judge whether all the operation strings formed by the decomposition of the process tree of complex products have been scheduled. If they are, go to Step 8. If not, go to Step 4.

Step 8: Establish the scheduling sequence of operation strings, and the scheduling is finished.

Figure 6 shows the OWSS.

3.4. Balanced Workshop Allocation Strategy (BWAS)

Let $p_i$ denotes the ith operation string in the operation string sequence, $T_{p_i}$ denotes the processing time of the operation string of operation i, and $T_{p_{max}}$ denotes the operation string of the operation with the largest weight. $T_{f_{\mathit{1}.now}}$ denotes the processing time of workshop $f_{\mathit{1}}$ at the current moment, $T_{f_{\mathit{2}.now}}$ denotes the processing time of workshop $f_{\mathit{2}}$ at the current moment. $L_{∆t}$= min$\{{|T}_{f_{\mathit{1}.now}}+T_{p_i}-T_{f_{\mathit{2}.now}}$|,${|T}_{f_{\mathit{2}.now}}+T_{p_i}-T_{f_{\mathit{1}.now}}\left|\right\}$, and it denotes the current moment processing time difference between the two workshops. Then, BWAS is $min\left\{L_{∆t}\right\}$, and the detailed description is as follows:

Step 1: At the initial moment, both workshops are in the state of being processed, $T_{f_{\mathit{1}.now}}$=$T_{f_{\mathit{2}.now}}$= 0;

Step 2: initialize $L_{∆t}$ = 0;

Step 3: In the scheduling sequence of operation strings, assign $T_{p_{max}}$ to any workshop, such as workshop $f_{\mathit{1}}$, and calculate the current processing time $T_{f_{\mathit{1}.now}}$ of it.

Step 4: Assign the operation string of the operations with the second larger weight to another workshop ${ f}_{\mathit{2}}$, and calculate the current processing time ${ T}_{f_{\mathit{2}.now}}$ of it.

Step 5: Calculate $T_{p_i}$, $T_{f_{\mathit{1}.now}}$, and $T_{f_{\mathit{2}.now}}$ of each operation string that is about to be assigned to f₁ and f₂. Update the value of $L_{∆t}$.

Step 6: Assign operation string $T_{p_i}$ based on the principle of minimizing $L_{∆t}$.

Step 7: Repeat Step 5 until each operation string is allocated.

Step 8: End.

To sum up, the flowchart of the algorithm in this paper is shown in Figure 7.

4. Algorithm Complexity Analysis

Assuming that the number of operations of the complex product, the processing time of each operation, and the number of machines in both workshops are all given, then the complexity analysis of the SCDS, the OWSS, and the BWAS in the proposed algorithm is as follows:

For the SCDS, the time complexity of determining the branches is $O\left(n\right)$, and the time complexity of determining the cut sets is also $O\left(n\right)$. The time complexity of recursive calls to sub-trees is $O\left(n^2\right)$. Therefore, the time complexity of the SCDS is $O\left(n^2\right)$. For the OWSS, the time complexity of the algorithm to determine the priority of the operation layer, the equipment priority, and the constraint degree is $O\left(n\right)$. The time complexity of calculating the weights is $O\left(n\right)$. According to the weight value to establish the operation sequence, the worst situation is that an operation needs to be compared with its predecessors and the successors, so the time complexity is $O\left(n^2\right)$. Therefore, the time complexity of the OWSS is $O\left(n^2\right)$. For the BWAS, the time complexities of calculating the scheduling time of the current time of the workshop, calculating the time difference between the scheduling times of both workshops, and allocating the operation string are all $O\left(n\right)$. Therefore, the time complexity of the BWAS is $O\left(n\right)$. Based on the above analysis, the time complexity of the proposed algorithm is $O\left(n^2\right)$.

5. Scheduling Example Analysis

Assume that complex product B consists of 24 operations, as shown in Figure 8. It needs to be carried out in two workshops, and each workshop has four devices with exactly the same processing capacity.

5.1. Cyclic Decomposition of Complex Product B

Step 1: As shown in Figure 9, {B2, B9, B16, B19, B24} is the cut set to decompose the complex product B shown in Figure 8. The operation strings T_B1, T_B2, T_B3, T_B4, and T_B5 of each sub-tree obtained after the first decomposition are shown in Figure 10a–e.

After the first decomposition, the number of the branches of T_B2 with root node {B9} and T_B3 with root node {B16} are both 1, and the decomposition is finished. T_B5 with {B24} as the root node is the leaf node operation, and the decomposition is finished. At this point, T_B2 = {B9, B10, B11, B12, Bl3, B14, B15}, T_B3= {B16, B17, B18}, and T_B5 = {B24}. The number of the branches of T_B1 is 2, and the same is true for T_B4, which requires a second decomposition.

Step 2: Decompose sub-trees T_B1 and T_B4 for the second time. T_B1 is decomposed into T_B11 and T_B12 by taking {B3} as the cut set. At this time, the number of the branches of T_B11 and the number of the branches of T_B12 are 1, and the decomposition is finished. So far, T_B1 = {T_B11, T_B12} = {{B4, B5}, {B6, B7, B8}, B3, B2}.

Similarly, taking {B19} as the cut set to decompose T_B4 for the second time, T_B4={T_B41, T_B42} = {{B20, B21, B22}, {B23}, B19} is obtained.

At this point, the cyclic decomposition of complex product B is completed, and the operation strings of each sub-tree are obtained as follows: T_B = {T_B1, T_B2, T_B3, T_B4, T_B5} = {{T_B11, T_B12}, T_B2, T_B3, {T_B41, T_B42}, T_B5} = {{{B4, B5}, {B6, B7, B8}, B3, B2}, {B9, B10, B11, B12, B13, B14, B15}, {B16, B17, B18}, {{B20, B21, B22}, {B23}, B19}, {B24}}, B1}.

5.2. Process Scheduling Strategy for Complex Product B

In the integrated scheduling system for complex product B shown in Figure 8, there are five operations that need to be processed on M₁, eight operations on M₂, nine operations on M₃, and two operations on M₄. According to the definition of equipment priority, the equipment priority of M₃ is the highest, the equipment priority of M₂ is higher, the equipment priority of M₁ is lower, and M₄ has the lowest equipment priority.

Table 1. Equipment priority statistics of complex product B.

Device	Operations	Number of the Operations	Equipment Priority
M1	B5yopment the rit	5	2
M2	B3yopment the rity statistic	8	3
M3	B1yopment the rity statistics of	9	4
M4	B7yopm	2	1

lists the priorities of each device.

During the integrated scheduling, the process structure of complex products can construct more complex products by means of virtual root nodes, so the processing scale and the complexity of products are not uniform. Meanwhile, due to the difference in the equipment resources, the layer priority, the equipment priority, and the constraint degrees of each operation are concrete numerical normalized before weight value calculation by the proposed algorithm. The purpose is to make no dimensional levels of data after calculation results so as not to result in the appearance of deviation. In this paper, we chose the Z-score standardized method, which is commonly used in data processing [27]. For details, see

Table 2. Statistical table of the weight of each operation of complex product B.

Operations	Equipment Priority Normalization	Layer Priority Normalization	Constraint Degree Normalization	Weights
B1	1.044	−1.676	3.577	2.945
B2	1.044	−1.093	0.097	0.048
B3	0.000	−0.510	1.257	0.747
B4	0.000	0.073	0.097	0.170
B5	−1.044	0.656	−1.063	−1.452
B6	−1.044	0.073	0.097	−0.875
B7	−2.089	0.656	0.097	−1.336
B8	1.044	1.239	−1.063	1.220
B9	0.000	−1.093	0.097	−0.996
B10	−2.089	−0.510	0.097	−2.502
B11	1.044	0.073	0.097	1.214
B12	0.000	0.656	0.097	0.752
B13	1.044	1.239	0.097	2.380
B14	0.000	1.822	0.097	1.918
B15	1.044	2.405	−1.063	2.386
B16	−1.044	−1.093	0.097	−2.041
B17	1.044	−0.510	0.097	0.631
B18	0.000	0.073	−1.063	−0.991
B19	1.044	−1.093	1.257	1.208
B20	0.000	−0.510	0.097	−0.413
B21	1.044	0.073	0.097	1.214
B22	0.000	0.656	−1.063	−0.408
B23	−1.044	−0.510	−1.063	−2.618
B24	−1.044	−1.093	−1.063	−3.201

.

The operation weight-based scheduling strategy (OWSS) is based on the subtree decomposition strategy, so the schedulable operation string can only be the one with one branch. As the root node of subtree T_B4, B19 has two branches, so it cannot be scheduled until its subtree T_B41 and T_B42 are scheduled. Similarly, B3 with two branches cannot be scheduled until operation string {B4, B5} and {B6, B7, B8} have been scheduled. Thus, the operation sequence sorted in descending order according to weights is as follows:

{B1, B15, Bl3 B14, B8, B11, B21, B19, B12, B3, B17, B4, B2, B22, B20, B6, B18, B9, B7, B5, B16, B10, B23, B24}. According to the sequence, the scheduling sequence of the operation string except the root node is established: T_B = {B15, B14, B13, B12, B11, B10, B9}; {B8, B7, B6}; {B22, B21, B20}; {B5, B4}; {B18, B17, B16}; {B3, B2}; {B23}; {B19}; {B24}.

5.3. Workshop Allocation Strategy for Complex Product B

Step 1: Choose T_B2 = {$B\mathit{15},B\mathit{14},B\mathit{13},B\mathit{12},B\mathit{11},B\mathit{10},B\mathit{9}$}, which contains operation {$B\mathit{15}$} with the highest weight value, and then, assign it to any workshop, such as workshop $f_{\mathit{1}},T_{f_{\mathit{1}.now}}$=$T_{p_{T_{B2}}}$= 20 person-hours, as shown in Figure 11. Update the scheduling sequence of operation strings: T_B = {B8, B7, B6}; {B22, B21, B20}; {B5, B4}; {B18, B17, B16}; {B3, B2}; {B23}; {B19}; {B24}.

Step 2: Choose T_B12 = {B8, B7, B6}, which contains {B8} with the higher weight, and assign it to the other workshop $f_{\mathit{2}}$, $T_{f_{\mathit{2}.now}}$=$T_{p_{T_{B12}}}$= 11 person-hours, as shown in Figure 12. Update the scheduling sequence of operation strings: T_B = {B22, B21, B20}; {B5, B4}; {B18, B17, B16}; {B3, B2}; {B23}; {B19}; {B24}.

Step 3: Calculate the processing time $T_{p_{T_{B\mathit{41}}}}$= 10 person-hours for T_B41 = {B22, B21, B20} to be allocated in the sub-tree operation string sequence, and calculate the absolute value of the difference in the processing time between the two workshops that the sequence is assigned to, respectively, that is ${|T}_{f_{\mathit{1}.now}}+T_{p_{T_{B\mathit{41}}}}{-T}_{f_{\mathit{2}.now}}|$ = $|20 +$10$- 11|$ = 19, ${|T}_{f_{\mathit{2}.now}}+T_{p_{T_{B\mathit{41}}}}{-T}_{f_{\mathit{1}.now}}|$=$|$11$+$10$- 20|$ = 1. According to $L_{∆t}=min${19,1} = 1, T_B41 is selected to be assigned to workshop $f_2$, as shown in Figure 13. Update $T_{f_{\mathit{2}.now}}$ and the scheduling sequence of operation strings: T_B = {B5, B4}; {B18, B17, B16}; {B3, B2}; {B23}; {B19}; {B24}.

Step 4: In the same way, the sub-tree operation strings, {B5, B4}, {B18, B17, B16}, and {B3, B2}, are also assigned to $f_{\mathit{2}}$. Figure 14 shows the scheduling Gantt chart for $f_{\mathit{2}}$. Update the scheduling sequence of operation strings: T_B = {B23}; {B19}; {B24}. $T_{f_{\mathit{1}.now}}$ = 20 person-hours, $T_{f_{\mathit{2}.now}}$= 16 person-hours.

Step 5: At this time, the scheduling sequence of operation strings is T_B = {{B23}, {B19}, {B24}}. Firstly, according to the definition of the optimal scheduling time, {B23} and {B24} as the operation string of leaf nodes can start to be scheduled at time t = 0 on M₁. However, the weight of operation string {B23} is greater than that of {B24}, so {B23} is scheduled earlier than {B24}. Secondly, according to the principle that the absolute value of the processing time difference between the two workshops is the minimum, {B23} and {B24} to be scheduled are allocated to $f_{\mathit{2}}$. Finally, the root node operation {B1} is allocated to $f_{\mathit{1}}$. The Gantt charts of complex product B processed in the two workshops are shown in Figure 15 and Figure 16. The processing times are 20 person-hours and 18 person-hours, respectively. The makespan is 20 person-hours, which is the maximum of the two completion times.

6. Comparative Analysis of Algorithms

6.1. Case Results by Five Algorithms

Taking complex product B, shown in Figure 8, as an example, the STHIS-PTCD is, respectively, compared with the algorithms in PBP, ACPM, NR, and TS in the same research field. The results show that the STHIS-PTCD has a better scheduling effect in terms of production makespan, migration times, and equipment utilization of the resource-symmetric two-workshop integrated scheduling.

The PBP takes schedulable operations as the optimization object and establishes the scheduling algorithm through the idea of balanced processing. By using this algorithm, the scheduling Gantt chart of workshop $f_{\mathit{1}}$ is shown in Figure 17, which takes 24 person-hours. The scheduling Gantt chart of workshop $f_{\mathit{2}}$ is shown in Figure 18, which takes 19 person-hours. The makespan is 24 person-hours.

The ACPM adopts the allied critical path idea of longitudinal optimization. It divides the process tree of complex products into two cases to determine the workshop allocation scheme of operations through the threshold, which is two. One is less than two, and the other is not less than two. With this algorithm, the scheduling Gantt chart of workshop $f_{\mathit{1}}$ is shown in Figure 19, which takes 23 person-hours. The scheduling Gantt chart of workshop $f_{\mathit{2}}$ is shown in Figure 20, which takes 20 person-hours. The makespan is 23 person-hours.

The NR also takes key equipment balance as the main idea. It uses the principles of neighborhood rendering and the same equipment operation in the same workshop to determine the operation allocation scheme. Inside the workshop, the operation scheduling scheme is determined according to the strategy of “dynamic critical path + short time”. With this algorithm, the scheduling Gantt chart of workshop $f_{\mathit{1}}$ is shown in Figure 21, which takes 23 person-hours. The scheduling Gantt chart of workshop $f_{\mathit{2}}$ is shown in Figure 22, which takes 19 person-hours. The makespan is 23 person-hours.

The TS “timing-selection” as the main idea of scheduling, which takes the descending order of the sum of operation paths in the sequence as the arrangement scheme, and it takes the minimum workshop completion time as the allocation scheme. With this algorithm, the scheduling Gantt chart of workshop $f_{\mathit{1}}$ is shown in Figure 23, which takes 22 person-hours. The scheduling Gantt chart of workshop $f_{\mathit{2}}$ is shown in Figure 24, which takes 22 person-hours. The makespan is 22 person-hours.

Now, the completion time and the average utilization rate of equipment in both workshops of the results obtained by the proposed algorithm are compared with the results obtained by the other four algorithms, as shown in

Table 3. Statistics of integrated scheduling results between five algorithms of two workshops in completion time and average utilization of equipment.

	Completion Time in ${\mathit{f}}_{\mathit{1}}$	Completion Time in ${\mathit{f}}_{\mathit{2}}$	Makespan	Average Utilization of Equipment
PBP	24	19	24	64.3%
ACPM	23	20	23	62.6%
NR	23	19	23	59.3%
TS	22	22	22	60.4%
STHIS-PTCD	20	18	20	67.9%

.

In the comparative analysis, we take complex product B as an example, and STHIS-PTCD has the shortest makespan, which is smaller than the other four algorithms. Additionally, the completion times are shortened by STHIS-PTCD in both $f_{\mathit{1}}$ and $f_{\mathit{2}}$f. Meanwhile, among the five algorithms, STHIS-PTCD has the highest average workshop utilization rate, which fully improves the equipment utilization rate.

6.2. Experimental Results by Five Algorithms

In order to verify the effectiveness of the STHIS-PTCD and its adaptability to complex products in symmetric two workshops, this paper randomly selects 100 instances with four operation scales, [20, 50, 100, 200], on five machines, and each scale with 25 instances. As the number of the machines is five, the average operations on each machine are [4, 10, 20, 40], which is in an ascending order. All the product instances are scheduled by the PBP, ACPM, NR, TS, and STHIS-PTCD, respectively. They are all implemented on Matlab 2019 on the same PC. The makespans of the 100 instances by five algorithms are given in Figure 25 and

Table 4. Experimental results of STHIS-PTCD and other algorithms.

No. of Instance	PBP		ACPM		NR		TS		STHIS-PTCD
	Cm	Cs	Cm	Cs	Cm	Cs	Cm	Cs	Cm	Cs
A20_1	314	276	374	239	280	236	280	0	280	236
A20_2	343	262	424	369	224	189	209	173	209	173
A20_3	327	322	286	219	244	239	231	205	231	138
A20_4	343	311	296	120	296	232	256	256	256	156
A20_5	350	334	450	420	280	250	280	0	280	250
A20_6	275	259	284	244	273	141	245	157	245	229
A20_7	281	243	369	263	279	227	279	157	279	227
A20_8	320	304	310	159	279	263	263	246	263	211
A20_9	304	261	219	130	216	162	216	206	216	173
A20_10	335	246	428	380	305	257	263	263	263	182
A20_11	309	245	387	205	312	121	284	132	284	260
A20_12	284	263	275	207	229	208	264	264	223	202
A20_13	342	327	383	138	290	214	290	209	290	256
A20_14	376	361	354	163	348	333	348	316	348	175
A20_15	310	268	429	369	237	190	217	187	237	190
A20_16	333	287	323	139	318	180	315	303	315	269
A20_17	304	286	396	346	243	193	227	221	262	244
A20_18	252	228	241	217	248	224	228	157	248	224
A20_19	199	178	193	111	158	120	180	120	158	131
A20_20	367	336	350	319	300	269	241	210	241	210
A20_21	273	223	289	218	275	225	248	156	248	150
A20_22	219	169	272	134	253	117	210	205	210	132
A20_23	288	239	268	195	249	200	221	221	221	148
A20_24	240	206	254	186	185	158	166	163	166	158
A20_25	227	192	221	186	193	158	188	188	177	135
A50_1	618	571	984	937	533	486	506	438	506	274
A50_2	616	595	1048	1027	548	459	548	523	519	488
A50_3	626	595	885	804	536	488	452	408	538	457
A50_4	686	654	820	584	655	623	632	499	632	600
A50_5	516	471	761	641	481	436	397	389	397	286
A50_6	731	695	951	661	642	542	591	530	591	539
A50_7	610	591	987	968	492	473	505	421	480	264
A50_8	466	401	567	480	555	306	450	310	450	283
A50_9	404	386	452	312	365	347	339	271	280	268
A50_10	474	458	857	792	421	364	397	338	350	324
A50_11	694	670	973	894	586	507	539	537	542	518
A50_12	630	614	823	788	557	541	475	155	475	409
A50_13	448	407	579	478	407	306	339	316	339	324
A50_14	601	587	852	789	535	452	515	397	515	376
A50_15	641	614	858	799	601	538	584	557	507	460
A50_16	632	598	594	480	534	420	476	363	476	361
A50_17	617	572	913	757	589	433	483	463	483	368
A50_18	688	647	766	636	548	507	526	415	512	318
A50_19	603	577	946	880	565	539	494	424	494	424
A50_20	687	671	1005	857	595	579	595	519	595	458
A50_21	564	558	781	758	481	475	453	309	426	282
A50_22	644	612	666	500	577	456	521	454	521	506
A50_23	571	547	644	573	433	409	433	374	433	368
A50_24	730	720	1006	959	665	655	624	272	624	532
A50_25	563	537	784	594	430	368	464	438	410	359
A100_1	1108	1086	1602	1563	896	874	830	830	830	830
A100_2	1347	1342	1778	1614	1138	1046	1087	567	1087	995
A100_3	1341	1311	1955	1893	1187	1157	1103	549	1103	1022
A100_4	1057	1042	1881	1830	942	927	785	669	817	721
A100_5	1083	1052	1692	1506	955	924	902	833	902	716
A100_6	1003	974	1672	1589	891	839	706	491	706	670
A100_7	1150	1140	1778	1679	983	893	851	444	851	825
A100_8	987	972	1362	1147	764	697	692	677	665	645
A100_9	1274	1245	1786	1755	1041	854	992	475	992	963
A100_10	1062	1044	1415	1397	964	946	823	631	823	726
A100_11	994	980	1609	1575	747	579	696	652	681	564
A100_12	1295	1281	1822	1782	1091	1077	1024	690	1024	881
A100_13	904	843	1390	1310	838	706	727	622	649	553
A100_14	1090	1082	2088	2080	807	680	734	701	734	684
A100_15	1280	1266	1796	1680	1120	1106	1062	498	1062	1052
A100_16	1025	993	1398	1279	908	833	767	695	764	695
A100_17	1083	1063	1639	1619	1000	944	831	626	831	736
A100_18	1247	1197	1956	1822	1114	1059	978	797	978	874
A100_19	1032	1008	1778	1754	872	848	759	672	759	740
A100_20	1163	1117	1795	1593	1105	1059	962	923	962	832
A100_21	1057	1016	1484	1443	880	839	781	667	781	690
A100_22	1206	1161	2342	2231	1085	1040	981	808	981	908
A100_23	897	870	1369	1272	789	681	624	581	643	616
A100_24	1138	1097	1558	1488	977	907	827	425	827	694
A100_25	1140	1100	1699	1549	1000	884	906	790	893	741
A200_1	1995	1990	2983	2978	1661	1656	1371	907	1371	1366
A200_2	2136	2066	4062	3894	1753	1711	1530	1329	1530	1217
A200_3	2151	2127	4040	4003	1781	1744	1596	1559	1531	1528
A200_4	2401	2368	3714	3563	2033	1908	1724	1659	1697	934
A200_5	2223	2177	3810	3463	1845	1799	1534	1435	1534	1340
A200_6	2016	2011	3479	3457	1682	1644	1415	1410	1393	1369
A200_7	1640	1609	3242	3130	1376	1334	1176	1123	1113	961
A200_8	1876	1813	3205	3141	1540	1526	1240	1052	1306	1292
A200_9	2048	2000	3324	3126	1648	1450	1416	1055	1416	1339
A200_10	2277	2231	3095	2939	1885	1751	1709	637	1709	1577
A200_11	1992	1960	4115	3909	1633	1482	1280	1270	1280	1237
A200_12	2401	2372	4674	4574	2294	2191	1953	1581	2057	1894
A200_13	1749	1735	3445	3286	1512	1498	1197	1186	1197	1167
A200_14	2049	2016	3647	3614	1723	1690	1443	1286	1478	1382
A200_15	2505	2477	3708	3556	2032	1969	1891	779	1891	1828
A200_16	2284	2237	3440	3393	1817	1728	1671	1556	1671	1498
A200_17	2486	2466	3593	3437	2084	2064	1926	663	1999	1979
A200_18	2481	2396	3643	3498	1972	1935	1801	673	1812	1775
A200_19	2506	2490	3721	3705	2120	2104	1952	1888	1952	702
A200_20	1110	1066	2545	2322	1124	1080	1137	1012	1110	1071
A200_21	2072	2055	3617	3481	1645	1592	1421	1386	1421	1328
A200_22	2185	2162	3611	3532	1900	1821	1593	1522	1593	1576
A200_23	2108	2097	3421	3367	1722	1674	1530	1482	1481	1428
A200_24	1998	1983	3643	3555	1706	1691	1450	869	1545	1468
A200_25	2489	2463	3612	3586	2152	2126	1983	687	1983	1911

Note: The bold in table means the best-known solution for the instance. Cm: makespan Cs: shortest completion time oftwo workshops.

, and the results of the workshop with the shortest completion time are shown in Figure 26 and Table 4.

The results show that the performance of STHIS-PTCD always maintains a higher quality for each operation scale than that of the other four algorithms. Though the performance of TS is similar to it, the processing situation is more balanced than that of TS, which is reflected in the small difference between the completion times in two workshops.

In order to compare the performance of the other algorithms, we use the relative deviation index (RDI) as the measure.

R = (C_m − C_best)/C_best × 100%

(8)

where R is the RDI value of an algorithm, C_m is the makespan obtained by it, and C_best is the best makespan obtained by all the algorithms. The descriptive statistics are presented in

Table 5. Descriptive statistics based on RDI.

No. of Instance		PBP	ACPM	NR	TS	STHIS-PTCD
A20	Num	0	0	7	22	22
	Mean	25.22	34.12	7.58	1.54	1.34
	Std Dev	16.42	27.56	6.90	4.62	3.85
	Min	0.72	1.39	0.00	0.00	0.00
	Max	64.11	102.87	24.48	18.39	15.42
A50	Num	0	0	2	17	23
	Mean	26.53	72.31	11.94	3.31	0.78
	Std Dev	9.54	28.55	8.24	5.99	3.80
	Min	3.56	24.79	0.00	0.00	0.00
	Max	44.29	144.86	30.36	21.07	19.03
A100	Num	0	0	0	20	23
	Mean	32.55	102.94	13.93	0.81	0.28
	Std Dev	8.78	28.86	6.94	2.52	1.00
	Min	20.07	63.57	4.69	0.00	0.00
	Max	48.50	184.47	29.12	12.02	4.08
A200	Num	1	0	0	19	19
	Mean	38.12	135.49	15.98	0.75	0.96
	Std Dev	11.19	36.08	6.67	1.55	2.01
	Min	0.00	81.10	1.26	0.00	0.00
	Max	55.63	221.48	27.58	5.66	6.55

for each operation scale. As shown, STHIS-PTCD finds the best solutions for 22, 23, 23, and 19 out of 25 instances of each test group. Additionally, STHIS-PTCD has a better RDI value, which is the smallest for 20 operations, 50 operations, and 100 operations. Although the TS is better than it for 200 operations, this is not by much.

Furthermore, we also use Wilcoxon matched-paired signed-rank test on the results, and they are listed in

Table 6. Results of Wilcoxon matched-pair signed-rank test.

	A20		A50		A100		A200
	Z	P	Z	P	Z	P	Z	P
STHIS-PTCD & PBP	−4.373	1.23 × 10−5	−4.373	1.22 × 10−5	−4.372	1.23 × 10−5	−4.286	1.82 × 10−5
STHIS-PTCD & ACPM	−4.292	1.77 × 10−5	−4.373	1.23 × 10−5	−4.372	1.23 × 10−5	−4.373	1.23 × 10−5
STHIS-PTCD & NR	-3.132	1.74 × 10−3	−4.167	3.08 × 10−5	−4.373	1.23 × 10−5	−4.372	1.23 × 10−5
STHIS-PTCD & TS		1.00		0.105		0.578		0.506

. It shows that the differences between STHIS-PTCD and PBP, ACPM, and NR, which are significant with p < 0.05, and the difference between STHIS-PTCD and TS is not significant with p > 0.05.

Based on the above results, STHIS-PTCD performs the best.

6.3. Comparative Analysis of Scheduling Results

The main reasons why the STHIS-PTCD is the best are as follows:

(1) For the operations:

The algorithm in this paper takes “operation string” as the scheduling unit, which improves the intensity of synchronous scheduling of operations, and it effectively reduces the migration of operations between two workshops. Moreover, on the basis of ensuring sufficient parallel scheduling, the form of “string” further improves the optimization of longitudinal continuous processing of the integrated scheduling, which better makes up for the disadvantages of the existing integrated scheduling of two workshops, such as valuing horizontal relationships over vertical ones or valuing vertical relationships over horizontal ones.

PBP scheduled the operations after grouping them according to the corresponding equipment. When the operations with a constrained relationship on the longest path would not be assigned to the same workshop, not only the operation migration occurs between the two workshops, but also the lack of valuing horizontal relationships over vertical ones happens, as the scheduling of the leaf node operations is ignored. For example, {B15}, as a leaf node, cannot be scheduled in the same workshop with another operation string, {B14, B13, B12, B11, B10, B9}, which has the unique immediate predecessor and the immediate successor. It not only reduces the intensity of continuous processing, but it also adds an operation migration between the two workshops.

Although the ACPM is a classical vertical optimization idea, it neglects the effect of horizontal optimization. Thus, the parallel scheduling of operations is affected, and unnecessary operation migrations have happened. For example, {B20} and {B21} in the operation string {B20, B21, B22} have the constraint relationship of one branch. They could have been seamlessly processed in the same workshop, but they were distributed to two workshops, which increased the operation migrations and prolonged the overall completion time of complex products.

According to the NR, the operations of the operation group are equally allocated to workshops according to the descending order of parallel time. However, the equipment balancing strategy will produce unnecessary operation migration during the scheduling. The long-path strategy and the long-time strategy adopted for the key operations reduce the parallelism between operations. Moreover, it will reduce the intensity of continuous processing of operations on the same equipment no matter the relationship between the operation waiting to be scheduled whether the operation under scheduling is adjacent or with the same immediate successor. For example, in {B15, B14, Bl3, B12, B11, B10, B9}, {B15, B14, Bl3, B12} and {Bl1, B10, B9} are assigned to different workshops, which leads to the scheduling effect of splitting the longitudinal tightness of operations.

The TS first determines the arrangement scheme according to the descending order of the sum of the grouped operation paths. Then, it determines the scheme by the principle of the minimum processing time of the current partial product. Both the scheduling sequence and the allocation scheme of the algorithm take the processing time of the operations as the main consideration. It not only ignores the comprehensive control of parallel processing advantages, but it also takes insufficient consideration of the utilization of equipment resources. For example, M₄ in workshop $f_{\mathit{2}}$ is completely idle during the whole scheduling process.

(2) For the equipment:

In the PBP, ACPM and NR, the key equipment is defined by the longest duration of the equipment. In contrast, the algorithm in this paper takes the number of the operations of the equipment as the defining standard. First, the definition category of special equipment in integrated scheduling is extended. Second, the proposed algorithm sets the equipment with the most operations as the standard and embodies the factors affecting the scheduling result, such as operation constraint relation and level, in the scheduling sequence of operation strings in the way of weight value. From the perspective of optimizing equipment resources, the problem of equipment resource waste or even being idle in the integrated scheduling of two-workshop is better solved. Thus, the goal of optimizing the overall effect of the integrated scheduling of two-workshop is achieved. For example, the TS and M₄ in f₂ are always in the inactive state in the whole product processing. In the STHIS-PTCD, the average equipment utilization rate of $f_{\mathit{2}}$ reaches 83.6%, which is 8.8% higher than the PBP, 6.4% higher than the ACPM, 22.9% higher than the NR, and 32.7% higher than the TS.

(3) For the workshops:

Among the workshops, the STHIS-PTCD assigns the operation string to be scheduled according to the minimum difference in completion time between the completed workshop and the uncompleted workshop. In a workshop, the optimal scheduling time strategy is adopted so that the operation string to be scheduled can start processing on the corresponding equipment as early as possible, and the horizontal parallelism of the integrated scheduling is improved, thus the total product processing time can be shortened on the whole. For example, in $f_{\mathit{1}}$, the STHIS-PTCD makes {B23} and {B24} on M₁ of $f_{\mathit{1}}$ process from time t = 0 according to the workshop balance strategy and the optimal scheduling time strategy, which not only reduces the number of operation migrations, but also shortens the processing time. In the PBP, the problem of reducing parallel optimization and migration occurs when the scheduling of leaf nodes is ignored in a workshop. For example, {B23} and {B24} are allocated to different workshops, respectively. Similarly, in the ACPM and NR, {B23} and {B24} are also allocated to different workshops, adding one operation migration. However, in the TS, {B24} starts to be processed at t = 3, 3 h later than the proposed algorithm. Another example is that the root node {B1} is allocated to $f_{\mathit{2}}$ with relatively less processing time according to the workshop balance strategy in this paper. Therefore, on the basis of shortening the total processing time of complex product B, the average utilization rate of workshop equipment is improved.

7. Conclusions and Future Research

In this work, a symmetric two-workshop heuristic integrated scheduling algorithm based on process tree cyclic decomposition is proposed. The conclusions are drawn as follows:

(1) Aiming at the optimization problem of resource-symmetric distributed two-shop integrated scheduling, the proposed algorithm is based on the characteristics of complex product with tree-like structure, and it takes “process” and “equipment” as double optimization objects. It improved the intensity of synchronous scheduling of operations, as well as the tightness of continuous processing. The optimization effect of less scheduling time of complex products and higher average utilization rate of workshop equipment was realized. The scheduling effect is better among the other algorithms.

(2) SCDS proposed in this paper further reduces the number of transitions between operations. The proposed OWSS can fully improve the utilization rate of equipment. BWAS was proposed to realize the optimization effect of early completion of operations in two workshops.

Next, there are several research directions in the following. Firstly, we can further study the asymmetric multi-workshop distributed integrated scheduling problems. Secondly, unattended distributed integrated scheduling problems should also be considered. Thirdly, distributed integrated scheduling problems of dynamic arrival of complex product orders should also be considered.