Dynamic Multi-Product Integrated Scheduling Algorithm with Urgent Delivery Date

Abstract

For the existing multi-product integrated scheduling algorithm considering the delivery date, the inability to fully consider the processing order of multiple products and the potential scheduling conflicts in the processing process causes problems such as equipment resource competition during the processing of products, ultimately resulting in the extension of delivery date and the failure to the deliver products on time. Aiming at this problem, a dynamic multi-product integrated scheduling algorithm with tight delivery date (ISA-TDD) is proposed. Firstly, the initial scheduling scheme of multi-product is established according to the strategy of ‘device priority + short time + layer priority’. Secondly, the product contribution value strategy is proposed to determine the scheduling order of multiple products and improve the parallel processing of products. Finally, the dynamic adjustment strategy of the game process is proposed, which solves the conflict problem of multi-product co-processing and further improves the intensity of compact scheduling of equipment. The comparative experimental results show that the algorithm achieves the optimization goal of shorter multi-product delivery date and higher overall utilization rate of equipment, and the scheduling effect is obviously better than the comparison algorithm.

1. Introduction

Intelligent manufacturing is the core technology and critical mainline for the manufacturing industry to become strong, which is not only the main direction of manufacturing power construction but also a major task to promote new industrialization. The application scale and development level of intelligent manufacturing in such fields as automobile manufacturing and electronic equipment have shown a significant leap. In recent years, countries have made every effort to promote the deep integration of digital technology and manufacturing technology, the deep integration of the digital economy and the real economy, the deep integration of information technology and industrialization, and the deep integration of artificial intelligence and manufacturing, which has greatly accelerated the process of digital transformation and intelligent transformation of enterprises. For example, in the field of production scheduling, intelligent manufacturing systems use advanced data analysis technology, intelligent scheduling algorithms, and other professional means to achieve accurate monitoring and dynamic adjustment of the production process.

Aiming to achieve high efficiency and standardization in mass production, traditional product scheduling processes products in the form of jobs, which are then sequentially processed and assembled. Scholars have carried out in-depth research on this issue and achieved abundant research results [1,2,3,4,5,6,7,8,9,10]. For example, the literature [3] constructs a mixed integer programming model for the non-waiting flow shop scheduling problem that prohibits late delivery; for the permutation flow shop scheduling problem, the literature [4] introduces an evolutionary search strategy integrating multiple agents, whose core goal is to minimize the total delay time of all tasks. The literature [5] proposed a double-layer heuristic optimization algorithm by integrating delivery date assignment and shop scheduling into an optimization model to solve the problem of integrated delivery date and scheduling.

In recent years, the complexity of the product manufacturing process has gradually increased, and enterprises have begun to pay more attention to the manufacturing of multi-variety and small-batch products. Different from traditional flow shop scheduling or job shop scheduling, the literature [11] proposes a new scheduling mode, that is, the integrated scheduling of product processing and assembly stages. Current research achievements mainly solved the general integrated scheduling problem [12,13,14], the flexible integrated scheduling problem [15,16,17], the distributed integrated scheduling problem [18,19,20], the multi-objective scheduling research [21,22,23], and the multi-shop integrated scheduling problem [24,25,26]. However, the multi-product integrated scheduling problem is different from the general integrated scheduling problem. Its main task is to arrange the order of each product reasonably, ensure that the product is processed closely on the equipment, and finally deliver the product on time. For example, Reference [27], a mixed integer programming model is established to minimize the sum of costs, and an improved grey wolf optimization algorithm is designed. However, the algorithm does not make full use of the advantage of dynamically adjusting the relative position among processes, and neighborhood structure setting and lacks the mechanism of real-time adjustment with the scheduling process. Reference [28] proposes a dynamic critical path multi-product manufacturing scheduling algorithm based on a process set. The dynamic critical path strategy and short-time scheduling strategy are adopted to better consider the two-way optimization of the tree structure in vertical and horizontal directions. However, the algorithm does not propose an effective solution to the problem of tight delivery time, which prolongs the total processing time and fails to improve the optimization effect.

Therefore, in order to solve the above problems, a dynamic multi-product integrated scheduling algorithm with urgent delivery time (ISA-TDD) is proposed. The main contributions are as follows:

• In the multi-product integrated scheduling, the scheduling advantages of the ‘equipment priority + short time + layer priority’ strategy are integrated, and the initial processing process set of each product is formulated based on the constraint relationship between each process, and the maximum delivery date of each product is determined;
• In multi-product integrated scheduling, the concept of product contribution value is defined to determine the priority processing order of multi-product;
• On the scheduling algorithm, a dynamic adjustment strategy for game processes is proposed. It effectively solves the problem of resource preemption in the process of multi-product processing, dynamically analyzes the relative position relationship between the processes, and realizes the dynamic adjustment of the processing sequence.

2. Problem Description and Analysis

A significant feature of the multi-product integrated scheduling system considering the delivery date is that there are various complex products in the system, and each product has its own distinct processing tree. Therefore, it is necessary to reasonably coordinate the processing sequence of each product to maximize the overall production efficiency. Specific requirements are as follows:

(1)

Each process has a unique serial number for identification and corresponds to processing equipment. The processing equipment also has a unique serial number for identification.

(2)

The necessary and sufficient condition for each process to start processing is that it has no constraints from previous order constraints.

(3)

The start time of each process of each product must be after the start time of its product, and its completion time must be within the delivery date of its product.

(4)

The end time of all processes of all products is the total production scheduling time.

For ease of description, the following concepts are defined:

Definition 1. 

Game process. The processes that occupy the same device resource at the same time.

Definition 2. 

Urgent path. The path with the highest number of processes and the longest processing time.

Definition 3. 

Urgent process. The process on the urgent path.

Definition 4. 

Maximum lead time. The total time taken for serial processing of different processes of multi-product provided that the processing equipment does not conflict.

Definition 5. 

Initial processing process set. Records the processing process set obtained after each product adopts the strategy of “equipment priority + short time + layer priority”.

Definition 6. 

Delivery shortening rate. An important indicator to measure the degree of shortening of the actual delivery time compared to the specified delivery time, the calculation formula is the shortening rate of the delivery time = (specified delivery date − actual delivery date)/specified delivery date × 100%, indicating that the greater the shortening rate of the delivery date, the smaller the actual delivery date compared to the specified delivery date.

Assume that the number of products is $X$, each product needs to go through $U_i$ different processes, the processing is completed on $M$ equipment, and the equipment resource set is $M=\left\{Mk\right\}(1\le k\le m)$. The processing start time of product i is set as $S_i$, and the maximum delivery time of the product is set as $d_i$. $T{B}_{ijk}$ and $T{E}_{ijk}$ denote the initial processing time and the completion time of the fth process of product i on equipment $M_k$, respectively. In addition, the total time required for the equipment $M_k$, to complete all relevant processes is $T{M}_k$, when $X_{ijk}=1$ and $X_{fek}=1$, the jth process of product i and the fth process of product e share the same processing equipment. Then we have:

Objective function:

$$T=min(max\{T{M}_k\})$$

(1)

Constraint condition:

$$s.t.T{E}_{ijk}\le d_i$$

(2)

$$T{B}_{ijk}\ge S_i$$

(3)

$$T{M}_k=max(T{E}_{jik})$$

(4)

$$T{E}_{jgk}\le T{B}_{ihk}$$

(5)

$$X_{ijk}=1·X_{fek}=1·(T{E}_{ijk}-T{B}_{fek})\ast X_{ijk}\ast X_{fek}\ge 0$$

(6)

Formula (1) represents the optimization objective of this paper; Formula (2) indicates that the processing must take place within the delivery period; Formula (3) indicates that the starting processing time of each process shall not be earlier than the starting processing time of the product; Formula (4) represents the latest time when each equipment completes all its processes; Formula (5) indicates that the product process needs to be processed before it is processed; Formula (6) represents the situation of equipment preemption between processes, that is, processes $X_{ijk}=1$ and $X_{fek}=1$ are game processes, and the processing of process $X_{fek}=1$ can not be started until the end of process $X_{ijk}=1$.

3. Algorithm Design and Analysis

We take the optimization of delivery time as the scheduling goal. Firstly, in order to establish the initial scheduling scheme of multi-product, the strategy of “equipment priority + short time + layer priority” is adopted. Secondly, the processing time and the number of urgent processes of different products are different. In order to optimize the processing sequence of multi-products, the contribution value strategy of products is proposed. Finally, in order to solve the conflict problem of multi-product co-processing and improve the intensity of compact process processing and the overall utilization rate of equipment, the core strategy of this algorithm is proposed, that is, the dynamic adjustment strategy for the game processes.

3.1. Product Contribution Strategy

The product contribution value $C_i$ is defined based on three factors: the number of urgent processes, the maximum delivery time, and the total processing time of the product.

$$C_i={\displaystyle \frac{r_i}{d_i-t_i}}$$

(7)

In Formula (7), $r_i$ represents the urgent path length of product i; $d_i$ represents the maximum delivery time of product i; $t_i$ represents the total processing time of product i. The greater the contribution value of the product, the higher the processing urgency of the product and the product must be completed close to the delivery deadline to avoid the risk of delay.

3.2. Dynamic Adjustment Strategy for Game Processes

When the same device is preempted between processes of different products, the dynamic adjustment is carried out. The strategy is as follows:

(1)

If the contribution value of product A is greater than that of product B, then schedule the urgent process of A first.

(2)

If the contribution value of product A is greater than that of product B, then schedule the process of B first when it is the only urgent process among the game processes.

The detail of the scheduling strategy of the game process is shown in Figure 1.

3.3. Algorithm Description

As shown in Figure 2, the initial scheduling scheme G0 is used as the basis for scheduling, and the first set of game processes {Pi, Qi} is selected. The schedulable processes are selected by the dynamic adjustment strategy for game processes, and the processing sequence is adjusted so as to obtain the priority scheduling scheme G1 and so on until all products are processed, and the final scheduling scheme GN is generated.

The algorithm design flow chart of the proposed algorithm is shown in Figure 3, which is described as follows:

Step 1: Input the process information of each product and construct the process tree;

Step 2: The initial process set of multiple products is determined by the strategy of “layer priority + short time + equipment priority;”

Step 3: Compare the emergency path length $r_i$ of the product and calculate the maximum delivery date $d_i$ and contribution value $C_i$ of the product, respectively;

Step 4: According to the product contribution value strategy, the product processing priority is formed;

Step 5: According to the game process dynamic adjustment strategy, select the process scheduling order;

Step 6: Determine whether there is a game operation in the initial processing operation set; if there is, then go to step 7, otherwise, put the remaining operations into the schedulable operation set and go to step 10;

Step 7: Whether the process with the largest contribution value is unique, if there is, select the process and go to step 9, otherwise, go to step 8;

Step 8: Select the more urgent process;

Step 9: Save the selected process into the schedulable process set and delete the process from the initial processing process set;

Step 10: Determine whether the initial processing procedure set is empty; if it is, go to step 8; otherwise, go to step 6;

Step 11: Output product scheduling Gantt chart;

Step 12: End and exit.

4. Algorithm Time Complexity Analysis

The time complexity of the initial processing procedure set of each product is $O(n)$ which determined by the strategy of “layer priority + short time + equipment priority”. The time complexity of calculating the product’s emergency path length is $O(n)$. The time complexity of calculating product contribution value is $O(n^2)$. The time complexity of game process dynamic adjustment strategy is $O(n^2)$. In summary, the time complexity of the proposed algorithm is $max(O(n),O(n^2))=O(n^2)$.

5. Algorithm Example Description

The algorithm in this paper is not based on specific examples but is highly universal. Compared with the scheduling results of other algorithms for complex products with tree structures, it achieves better performance. To illustrate this, complex products P and Q are randomly generated, and the scheduling examples are shown in Figure 4 and Figure 5.

5.1. Initial Process Scheduling Sequence of Multi-Product

For product P, the strategy of “equipment priority + short time + layer priority” is adopted, and the process scheduling sequence is {P1, P4, P3, P5, P2, P6, P7, P9, P8, P10, P11}, and the completion time is 190 working hours. The Gantt chart for scheduling product P is shown in Figure 6.The red box represents the process of product P and the dotted line in the following figure indicates the end of product scheduling.

For product Q, the strategy of “equipment priority + short time + layer priority” is adopted, and the process scheduling sequence is obtained as {Q2, Q1, Q3, Q5, Q4, Q6, Q7, Q8}, and the completion time is 150 working hours. The Gantt chart for scheduling product Q is shown in Figure 7, and the blue box represents the process of product Q.

5.2. Solve the Maximum Delivery Time

It can be seen from Figure 6 and Figure 7 that there is a conflict between product P and product Q during the co-processing, as shown in Figure 8.

According to the maximum delivery time strategy, on the basis of the original scheduling as shown in Figure 8, the processes of product Q are inserted into the idle time during the processing of product P according to the sequence of the initial scheduling scheme. Additionally, the processing sequence of subsequent processes is adjusted at the same time. The total scheduling hours obtained are the maximum delivery time; that is, the maximum delivery time is 285 working hours. The Gantt chart of the overall scheduling sequence of products P and Q is shown in Figure 9.

5.3. Game Process Dynamic Adjustment Strategy

According to Figure 4 and Figure 5, the urgent processes and their quantity and urgent path length of each product were calculated, and the contribution value of each product was calculated in combination with the maximum delivery time of the product. At the same time, the contribution value strategy was adopted to determine the priority order of product processing. The information table of products P and Q is shown in

Table 1. Information for products P and Q.

Product	Original Complete Time	Urgent Path Length	Max Delivery Time	Contribute Value	Schedule Order
P	190	175	285	1.84	1
Q	150	150	285	1.11	2

. According to the information in Table 1, it can be determined that the contribution value of product P is 1.84, and the contribution value of product Q is 1.11. Therefore, the processing of product P should be carried out first, followed by the processing of product Q.

Comparing the game process according to the different situations of the dynamic adjustment strategy for game processes and the game result is obtained. The specific adjustment steps are as follows:

Step 1: Compare the game process sets {P1, Q3}, {P4, Q3}: P1 and P4 are the urgent processes and the regular process, respectively, while Q3 is the regular process. Meanwhile, P1 needs to be processed before P4. Therefore, the scheduling sequence is {P1, P4, Q3}.

Step 2: Compare the game process set {P3, Q2}: P3 and Q2 are regular processes, and the scheduling sequence is {P3, Q2}.

Step 3: Compare the game process sets {P5, Q1}, {P2, Q1}: P5 and P2 are regular processes and urgent processes, respectively, while Q1 is an urgent process. Meanwhile, P5 needs to be processed before P2, and the scheduling sequence is {Q1, P5, P2}. Therefore, the scheduling time of the adjustment process Q1 is advanced from time $t=55$ to start time $t=0$, and the scheduling time of P5 and P2 is delayed to time $t=25$ and $t=65$ respectively, as shown in Figure 10.

Step 4: Compare the game process set {P6, Q6}: P6 and Q6 are both urgent processes, so the scheduling sequence is {P6, Q6}.

Step 5: Compare the game process set {P7, Q4}: P7 is the regular process, Q4 is the urgent process, so the scheduling sequence is {Q4, P7}. Therefore, the scheduling time for Q4 changes from $t=130$ advance to $t=45$, and the scheduling time for P7 is delayed to $t=75$, as shown in Figure 11.

Step 6: Compare the game process set {P9, Q6}: P9 is the regular process, and Q6 is the urgent process, so the scheduling sequence is {Q6, P9}. Therefore, Q6, which starts at time $t=190$, is scheduled at time $t=100$, and P9 is scheduled at time $t=260$, as shown in Figure 12.

Under the restriction of the immediate predecessor and successor relationship between each process, iterate repeatedly until the processing of all products is completed. The specific arrangement is shown in

Table 2. Game Process Scheduling Table.

Game Process Set	Conflict Machine	Game Result
{P1, Q3}	M2	P1
{P4, Q3}	M2	P4
{P3, Q2}	M3	P3
{P5, Q1}	M1	Q1
{P2, Q1}	M1	Q1
{P6, Q6}	M2	P6
{P7, Q4}	M3	Q4
{P9, Q6}	M2	Q6

. As shown in Figure 13, the total scheduling time of the ISA-TDD algorithm is 255 working hours.

6. Experimental Analysis

6.1. Comparison Algorithm

Taking the complex products P and Q shown in Figure 4 and Figure 5 as examples, the ISA-TDD algorithm was compared with the algorithm based on the urgency degree of delivery date (ISA-DDUD) [29] and the algorithm considering delivery date constraint (ISA-CDD) [30] in the same field.

(1)

Algorithm based on urgency degree of delivery date (ISA-DDUD). The algorithm adopts priority strategy, long-path strategy, and short-time strategy based on urgency degree to determine the scheduling order of products according to the urgency degree of product priority processing and solve the conflict problem caused by multi-product.

(2)

Algorithm considering delivery date constraint (ISA-CDD). The algorithm adopts the strategy of pre-scheduling to solve the required processing time of products, the strategy of product priority with high urgency degree, and the strategy of process insertion during idle period of equipment. It takes the urgency degree as the main basis of product scheduling sequence and uses the idle period of equipment to select the appropriate process for processing.

6.2. Comparative Test Scheduling Results and Evaluation Analysis

The ISA-DDUD and ISA-CDD algorithms are used to schedule complex products P and Q, respectively. The results are as follows:

The scheduling sequence obtained by ISA-DDUD algorithm is {Q1, Q2, Q4, Q5, Q3, Q7, Q8; P1, P5, P2, P3, P4, P6, P7, P9, P8, P10, P11}. As shown in Figure 14, the total scheduling time was 265 working hours.

The scheduling sequence obtained by the ISA-CDD algorithm is {P1, P2, P3, P5, P4, P7, P6, P9, P8, P10, P11. Q1, Q2, Q3, Q5, Q4, Q6, Q7, Q8}. As shown in Figure 15, the total scheduling time was 285 working hours.

The ISA-TDD algorithm, ISA-DDUD algorithm, and ISA-CDD algorithm are evaluated and analyzed by using four evaluation indexes: completion time, total idle time of equipment system, overall utilization of equipment, and delivery shortening rate. The specific evaluation results are shown in

Table 3. The scheduling results and evaluation of the three algorithms.

Algorithms	Completion Time	Total Idle Time of the Device	Overall Utilization of Equipment	Relative Increase Rate of the Overall Device Utilization	Delivery Shortening Rate
ISA-DDUD	265	225	70.2%	4.4%	7%
ISA-CDD	285	220	69.7%	4.9%	0%
ISA-TDD	255	175	74.6%	┄┄	10%

.

• Compared with the ISA-DDUD and the ISA-CDD algorithms, the total processing time of ISA-TDD algorithm reduced by 10 and 30 h, and the reduction rates are 3.8% and 10.5%, respectively.
• The total processing time of the ISA-TDD algorithm is 175 working hours. Compared with the ISA-DDUD and the ISA-CDD algorithms, it reduced by 22.2% and 20.5%, respectively.
• The overall device utilization rate of the ISA-TDD algorithm reached 74.6%. Compared with the ISA-DDUD and the ISA-CDD algorithms, the overall device utilization rate of the ISA-TDD algorithm increased by 4.4% and 4.9%, respectively.
• The delivery time shortening rate of ISA-TDD algorithm is 10%, which is 10% and 3% higher than that of ISA-DDUD and ISA-CDD algorithms, respectively.

6.3. Algorithm Advantage Analysis

In summary, compared with the ISA-DDUD algorithm and the ISA-CDD algorithm, the ISA-TDD algorithm has a better performance, mainly for the following reasons:

(1)

The ISA-TDD algorithm adopts the comprehensive strategy of ‘device priority + short time + layer priority,’ relying on the tree structure characteristics of the process, and solves the constraint conflict between the processes through layer-by-layer scheduling.

(2)

The ISA-TDD algorithm adopts the product contribution value strategy to determine the product processing order. It fully considered the impact of the urgent path and the number of urgent processes during the scheduling, which effectively reduces the processing time of parallel processes. However, the ISA-DDUD and the ISA-CDD algorithms do not take the impact of these two factors into account, and the equipment has unnecessary idle time during the processing, which extends the processing gap during the scheduling. It leads to a result of delays in the product completion time.

(3)

The dynamic adjustment strategy for game processes helps the ISA-TDD algorithm reduce equipment idle time and shorten product processing time. Dynamic adjustment of the processing sequence achieves more efficient resource allocation and quickly realizes the scheduling effect of the process on the equipment. For example, by the ISA-TDD algorithm, the idle time of machine $M_3$ is $t=90$ to $t=125$ and $t=155$ to $t=185$, which is smaller than that obtained by the ISA-DDUD and the ISA-CDD. For another example, by the ISA-TDD algorithm, the process on the equipment $M_3$ is completed at $t=220$, which is 20 and 65 hours earlier than that obtained by ISA-DDUD and ISA-CDD, respectively. Meanwhile, the limitation of the ISA-DDUD and the ISA-CDD algorithms is that they do not fully consider the dynamic changes of relative positions between processes, resulting in the overall utilization of the equipment failing to reach the optimal level.

6.4. Analysis of Applicability and Potential Limitations

In this paper, a dynamic multi-product integrated scheduling algorithm with tight delivery time is proposed, which is suitable for multiple tree-type complex products with tight constraints. In the aspect of process scheduling, the classical strategy of ‘equipment priority + short time + layer priority’ is integrated to reasonably allocate the production tasks of different products. In terms of optimization effect, the product contribution strategy and dynamic adjustment strategy for game processes are adopted to adjust the scheduling scheme according to the dynamic information such as equipment status and conflict process in the production process, so as to effectively avoid equipment preemption or idleness, so as to improve the overall production efficiency and reduce the production cost. In today’s highly competitive market, customers’ demand for products is becoming more and more diversified, and the delivery time of orders also varies. The algorithm in this paper can handle the multi-product comprehensive scheduling problem while taking into account the delivery time requirements of different products to ensure that each product can complete production within the specified time. For example, enterprises produce a variety of electronic products with different specifications simultaneously. The algorithm can reasonably arrange the processing sequence according to the urgency of the delivery date of each order to meet the diversified needs of customers. When the enterprise can frequently meet the customers’ delivery time requirements, the customers will be more satisfied with the enterprise’s service, thus establishing a long-term stable cooperative relationship.

In addition, the algorithm proposed in this paper also has certain limitations. When dealing with large-scale production scheduling tasks, the operating environment of equipment is complex, and there are numerous influencing factors. Special situations such as equipment failures and product rework may occur, which will affect the accuracy and practicality of the algorithm.

In summary, the algorithm proposed in this paper has high applicability in actual production and can effectively improve production efficiency and reduce costs. However, its applicability and effectiveness may vary according to different industries and production scales. Therefore, when applying this algorithm, customized adjustments can be made based on the actual situation to ensure that the algorithm exerts its maximum effectiveness.

7. Conclusions

In order to solve the conflict problem when multi-products are processed with delivery dates, a dynamic multi-product integrated scheduling algorithm (ISA-TDD) with an urgent delivery time is proposed. In the aspect of vertical optimization, the product contribution value strategy is used to determine the product processing order, which significantly reduces the time consumption of the entire process. In the aspect of horizontal optimization, the dynamic adjustment strategy for game processes not only makes the processing more compact and efficient but also improves the utilization rate of equipment. Therefore, the proposed algorithm not only ensures the timely delivery of products within the delivery date but also realizes the optimization effect of vertical and horizontal directions to the maximum extent.

However, in the actual production process, many uncertain factors need to be considered, such as broken equipment, product rework, etc., so ensuring that the algorithm can run efficiently in the actual production is a key problem. In future research, more intelligent and dynamic multi-product urgent delivery date strategies can be explored, and factors such as equipment failure and product rework can be taken into account in the algorithm. Machine learning-based algorithms can be used to predict the potential failure points of equipment and the possibility of product rework in combination with the real-time operation status monitoring data of equipment, while the production scheduling strategies can be adjusted in time. Improve the adaptability, robustness, and ability to deal with uncertainties of the algorithm, so as to achieve more efficient production scheduling.

Our future research can be further extended to the design and implementation of multi-product integrated scheduling algorithms with monitoring and forecasting functions to meet the growing customer needs and market challenges in the manufacturing industry.