With the development of smart manufacturing, quality has become an indispensable issue in the manufacturing process. Although there is increasing publication about inspection allocation problems, inspection allocation optimization research considering resource capability is scarce. This paper focuses on the inspection allocation problem with resource constraints in the flexible manufacturing system. Combined with the inspection resource capability model, a bi-objective model is developed to minimize the cost and balance loads of the inspection station. A modified NSGA-II algorithm with adaptive mutation operators is suggested to deal with the proposed model. Finally, a simulation experiment is conducted to test the performance of the modified algorithm and the results demonstrate that modified NSGA-II can obtain acceptable inspection solutions.
In manufacturing systems, the quality characteristics (QC) of the manufactured product may inevitably fail to meet the requirement of the design because of machines, human error, etc. Therefore, inspection is a necessary part of the product manufacturing process [
Lindsay and Bishop [
Even many scholars researched the inspection allocation problem. However, the problems that current research mainly focused on are the inspection location and inspection process planning. Many scholars regarded the inspection stations as the same inspection unit, which is an ideal processing way to some extent. In the actual condition, different inspection stations own specific inspection capabilities, such as precision, process time. Therefore, in this paper, two subproblems are taken into consideration. The first one is the inspection location problem and the second one is the inspection resource allocation problem based on the inspection capability model of Shiau [ The location of inspection in the product manufacturing cycle. Which inspection station will be used to inspect the QC?
For the type of manufacturing system, people paid more attention to the research on the IA problem in the series manufacturing system. There are few papers concerning the IA problem in the Flexible Manufacturing System (FMS). In the FMS, the product flow and inspection flow of a production line have many types of parts. The inspection stations shall inspect different types of components that contain separate product QCs, and each of the components owns a specific inspection plan. Therefore, the allocation of inspection resource becomes very complicated in a complete view.
On the other hand, plenty of scholars took the inspection cost, quality improvement, and uncertain factors into consideration in the inspection allocation problem. The inspection unit is a crucial resource, and its working load is worthy of concern. However, it seldom was mentioned by scholars. When the inspection station is put into use and each working unit gets a relatively balanced working quantity, the overall inspection efficiency will be improved. Therefore, in the paper, a multi-objective inspection allocation with resource constraints (IARC) model is built in the FMS to trade-off manufacturing cost and inspection station load. This research looks forward to reducing the planning difficulty in the inspection allocation and giving support to decision-making.
The methodology to solve the inspection planning problem varies, such as dynamic programming, integrated programming [
In this paper, the proposed IARC model belongs to a multi-object optimization problem (MOP), in which objects contract with each other. Thus, finding the optimal solutions for every object is impossible. Multi-objective heuristic algorithms are practical to solve the MOP problem; many methods were developed in recent years, such as NSGA-II, NASG-III, MOEA/D, Monarch butterfly optimization, and differential evolution algorithm, quantum-inspired differential evolution [
The following are the main contributions of this paper: Inspection-capability-based IARC is the first model which allocates inspection location and inspection station in the FMS. Besides, the IARC is a bi-objective optimization model which tries to trade-off the cost and machine load. A modified algorithm is proposed with three adaptive mutation operators for the IARC model. The simulation experiment validates the performance of the proposed algorithm.
The remainder of the paper is organized as follows. In the next Section, problem description and mathematical modeling are presented. In Section
The FMS production line includes many types of parts. Here, Ω = {1, …,
The workstation
The inspection probability based on the resource capability.
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Pursuing lower costs is a vital issue in manufacturing [
In manufacturing, each machine tool processes only one product In an inspection procedure, each inspection station inspects only one Suppose that the whole system is working correctly, and the processing flow is known
Su MC ICi, SC RC CSC
After inspecting QC
The probability of product
The probability of product
The probability of product
The probability of the product
Following the possible results of the product after each working procedure, we can calculate the inspection cost, manufacturing cost, reworking cost, discarding cost, and damage cost caused by selling the defective goods to the customer. Based on the above probabilities, we can get the related costs as follows:
The manufacturing cost of product
The inspection cost of product
The reworking cost of product
The damage caused by discarding the product
The damage caused by selling the defective goods
By integrating all the cost factors, and we can get the total cost of product
Following the distribution condition of the inspection station, the total working time of inspection station
Aiming at the optimization problem of inspection allocation under the FMS, the minimum total cost and minimum biggest inspection station load were taken as targets to build a multi-objective MIP model:
Objective function:
Formula (
The IARC model mentioned in this paper includes two fitness function formulas, (
Flow chart of Modified NSGA-II algorithm.
The coding scheme adopted four-layer chromosomes to express the inspection location and inspection resource assignment scheme, and it is shown in Figure
Coding scheme of modified NSGA-II.
This paper proposed three ways for initial solution generation: (1) 20% Random Generation Method (RGM), (2) 40% Minimum Load Selection Method (MLSM), and (3) 40% Earliest Stage Assignment Method (ESAM). The ESAM was proposed by Shiau [ Step 1: All QCs are traversed in order. When the QC exists, skip to step 2. When all the QC Step 2: The 0-1 variable Step 3: A random number
The multipoint crossover method is adopted in this algorithm.
Multipoint crossover method.
The most common mutation method is single point variation, which has a shortage. The generated population has a low diversity, and it further makes the solutions fall into the local optimal solution prematurely. Besides, multilayer encoding is characteristic of deep dependency between different layers of genes. If the method of single point variation is adopted, the infeasible offspring may be created, as the other layer of genes may not change with the mutated gene.
In recent years, neighborhood search strategy has been successfully applied to combinatorial problems with good performance [ Random inspection task change operator (RIC) : RIC randomly selects a QC and changes the inspection policy of the QC. If QC does not need inspection before, this QC is reassigned as an inspection procedure. Then, an available inspection station will be allocated to the QC. If the QC needs an inspection, the inspection policy of the QC is cancelled. Random equipment replacement operator (RER) : This operator can randomly select an inspection procedure and randomly reassign a piece of available equipment for this inspection procedure. Maximum load removal operator (MLR) : A total load of all devices is calculated and the inspection station
The mutation operators are designed based on human experience on the inspection allocation problem. The three mutation strategies differ from each other but do not have the same importance. In the process of population evolution, an adaptive selection strategy is proposed. For each generation of chromosome mutation, roulette is used to select the mutation operator according to the weight of the mutation operator
The computation rules are as follows:
Selection is an essential operation, which transmits excellent and adaptable parents to the next generation and maintains the quantity of the population unchanged. The individuals in the population are divided into different frontiers of inferior quality according to govern relations by using the nondominated sorting method. The elite selection strategy always retains the nondominant solution in the lower frontier to the next generation. In this case, the majority of chromosomes come from level one of the nondominant frontier of the last generation. Thus, the optimal result always entraps into the local optimum solution. The elite selection strategy is improved by adjusting the probability of the lower-level solution to the next generation. The specific steps are as follows: Step 1: The chromosomes of the parent and offspring are combined Step 2: The next generation is Step 3: The quantity of individuals that are selected to the next generation at inferior Step 4: The individual at the present level | Step 5: Selection finished.
In this Section, a simulation experiment is conducted to verify the correctness of the proposed inspection planning model and the feasibility of the modified NSGA-II algorithm. The experiment bases on a flexible production line of an aircraft manufacturer and the parameters are estimated following Shiau [
Based on the operational situation of a factory, 5 machine tools and 3 inspection tools were set in the experiment. Process capability
Parameters of the simulation experiment.
Parameter | Range |
---|---|
0.1–0.6 | |
0.001–0.06 | |
2–3 | |
20–80 | |
10–20 | |
800–2000 | |
20–80 | |
200–500 |
The experiment has 17 inspection planning tasks that deal with different types of parts. For example, the instance I6_A has 6 kinds of parts, which are randomly selected from class
The instances of the experiment.
I1_A | I1_B | NA | I2_A | I2_B | I2_AB | I4_A | I4_B | I4_AB |
---|---|---|---|---|---|---|---|---|
I6_A | I6_B | I6_AB | I8_A | I8_B | I8_AB | I10_A | I10_B | I10_AB |
To verify the effectiveness of the proposed algorithm, NSGA-II and modified NSGA-II are used to calculate the examples. Matlab R2020a was used to implement the program, which was run on the Intel (
The hyper-volume (HV) [
Normalization of the objective function values is required before calculating the hyper-volume to obtain a nondimensionalised optimization problem [
This section consists of two parts: the outcome analysis and performance analysis. The outcome analysis aims to analyse the costs and the load of inspection tasks by comparing the result of a specific inspection task leveraging by the IARC model and the NSGA-II algorithm. The objective of performance analysis is to compare the performance of NSGA-II and modified NSGA-II in the inspection planning problem by using the HV metric.
The inspection instance I4_A was selected as an example to carry out the outcome analysis. Three methods, namely, Modified NSGA-II, NSGA-II, and all inspection strategy, were adapted to deal with this inspection task. All inspection strategy means that all the QC
The result of I4_A by using three methods.
Methods | Selected Scheme | Costs | obj1 | obj2 | HV | ||
---|---|---|---|---|---|---|---|
P1/P2/P3/P4 | MC/IC/DC/RC/CNC | ||||||
Modified NSGA-II | 8 | LI | 00011/00101/00100/01000 | 569/31/418/41/743 | 1800.7 | 65 | 0.89 |
IA | 00031/00103/00200/01000 | ||||||
NSGA-II | 2 | LI | 00011/00101/10100/01000 | 561/52/418/41/733/ | 1805.8 | 74 | 0.85 |
IA | 00011/00103/10200/02000 | ||||||
All inspection | — | LI | 11111/11111/11111/11111 | 380/351/992/30/358 | 2111.5 | 4536 | — |
IA | 31333/22231/13333/22212 |
The obj1 value is the sum of MC, IC, DC, RC, and CNC, and it means that all factors affect the result collectively. It is clear that the proportion of MC, DC, and CNC is relatively large among all factors by three methods, but the value of MC changes slightly among the three plans. Thus, it has a slight influence on obj1 as the manufacturing operations are inevitable. For the comparison of the outcome of NSGA-II and modified NSGA-II, there are more nondominated solutions by using modified NSGA-II. By comparing the outcome of all inspection strategies with heuristic algorithms, the results from the optimization algorithm are distinctly superior to the solutions from all inspection strategies. Especially, the cost of inspection and discard is visibly higher than the other two methods, because more inspection operations were set in the plan and more defective products were taken out along the process. The value of CNC in all inspection strategy is lower than others, as the inspection is strict and defective products can seldom flow into the market.
For the comparison of the two optimal algorithms, please refer to Figure
Comparison of Pareto front of three methods.
Performance analysis is conducted by dealing with the 17 instances provided in Table
The result comparison of two methods by running 10 times.
Instances | Modified NSGA-II | NSGA-II | Difference | ||||||
---|---|---|---|---|---|---|---|---|---|
Mean | var | Time | Mean | var | Time | ||||
I1_A | 0.56 | 3.1 | 4 | 5.5 | 0.59 | 1.0 | 3.1 | 5.1 | −5.40% |
I1_B | 0.73 | 5.0 | 5 | 13.3 | 0.73 | 6.4 | 3.4 | 12.3 | −0.59% |
I2_A | 0.89 | 8.2 | 5 | 10.3 | 0.89 | 7.0 | 4.3 | 9.5 | 0.04% |
I2_B | 0.99 | 1.4 | 4 | 24.1 | 0.98 | 4.7 | 2.6 | 20.5 | 0.74% |
I2_AB | 0.98 | 6.7 | 8 | 20.3 | 0.95 | 4.0 | 4.6 | 18.7 | 2.84% |
I4_A | 0.88 | 2.4 | 7 | 23.1 | 0.85 | 3.8 | 2.8 | 21.4 | 3.66% |
I4_B | 0.99 | 4.2 | 9 | 42.9 | 0.94 | 1.4 | 4.4 | 39.5 | 5.05% |
I4_AB | 0.98 | 5.2 | 8 | 35.2 | 0.93 | 2.0 | 4.4 | 32.3 | 4.83% |
I6_A | 0.98 | 2.4 | 5 | 25.7 | 0.96 | 5.4 | 3.6 | 24.0 | 1.79% |
I6_B | 0.98 | 1.4 | 7 | 56.4 | 0.95 | 2.5 | 5.4 | 49.6 | 3.57% |
I6_AB | 0.99 | 1.0 | 9 | 45.6 | 0.95 | 2.4 | 4.3 | 43.8 | 3.97% |
I8_A | 0.99 | 4.8 | 5 | 32.0 | 0.97 | 3.7 | 3.7 | 28.4 | 2.59% |
I8_B | 0.95 | 4.6 | 9 | 90.1 | 0.88 | 8.1 | 6.7 | 82.7 | 7.27% |
I8_AB | 0.97 | 1.7 | 9 | 66.2 | 0.92 | 1.1 | 5.9 | 63.3 | 5.27% |
I10_A | 0.98 | 2.1 | 6 | 61.3 | 0.91 | 4.3 | 3.8 | 58.7 | 7.65% |
I10_B | 0.96 | 9.4 | 7 | 132.5 | 0.91 | 5.2 | 6.4 | 117.3 | 6.06% |
I10_AB | 0.95 | 1.1 | 7 | 74.5 | 0.88 | 3.1 | 6.8 | 70.8 | 7.73% |
For the mean value of 10 times runs, both the Modified NSGA-II and NSGA-II can see a slight growth in mean value as the number of products involved in the experiment increased. When there is only one product in the experiment, the difference between the mean values of the two algorithms is tiny. The gap between the two results becomes large when the number of products involved in the experiment increased. The largest difference between the two algorithms is 7.73% in I10_AB and the average difference of the mean value of all instances in 10 runs is 3.36%. It means that the modified NSGA-II is excellent when dealing with complex inspection allocation task and the modified NSGA-II outperforms NSGA-II in dealing with the IARC model. The variance of the modified NSGA-II is smaller than the corresponding value of NSGA-II in most cases, which means that the solutions of the previous algorithm converge evidently than the latter one. The average number of solutions in the nondominated solutions set generated by modified NSGA-II is far more than the other one and it means that the previous one performs well in terms of variety. For the performance of algorithms in different tasks, the modified NSGA-II performs better when dealing with hybrid class AB than class A and class B. Figure
Box plot of HV metric in example I4_A.
The computation complexity of the algorithms is compared, and the average time of ten runs is listed in Table
Box plot of runtime in example I4_A.
It is worth mentioning that the performance of different mutation operators is different when dealing with the IARC model. The performance is evaluated by using the average changes of weight value for mutation operators, which is shown in Table
The average changes of weight value for mutation operators.
Mutation methods | RIC | RER | MLR |
---|---|---|---|
Final weight value | 10.1 | 4.1 | 22.1 |
Through the outcome analysis, the wrong idea that all inspection strategy is always the most economically beneficial solution was corrected, but all inspection strategy is widely adopted in some factories. By using the quantitative methods, the lowest-cost solution could be found. However, the formulation of the method often considers more than a single factor in the real application. In this study, the inspection resource and the cost are both important for stakeholders. The IARC model can generate compromise solutions to make a trade-off between the cost and the load balance. Decision-makers can choose one among the multiple solutions generated by modified NSGA-II algorithms.
The experiment results show that the modified NSGA-II algorithm has a good performance in terms of diversity and convergence compared with NSGA-II in the inspection allocation problem. The modified algorithm can obtain more solutions that perform better in an objective space by using the HV metrics. It proves the effectiveness of adaptive mutation operators, which can improve the neighborhood search capability of the modified NSGA-II.
Regarding future work, more influence factors will be included in the IARC model as the objectives, such as energy efficiency, and a multi-objective optimization model will be built to find the optimum inspection solutions. Besides, a production system normally has many activities, such as production, inspection, and maintenance, and these activities need to be scheduled to improve the efficiency of the whole system. Thus, joint scheduling of production, inspection, and maintenance activities need further research. In terms of algorithm, the proposed modified algorithm could be employed in similar problems, and it is interesting to investigate the performance in other MOP problems.
The experiment result shows that the proposed IARC model has a certain guidance meaning for the realization of the intelligent, automation inspection planning systems. Although the model still has a long way to go from the practical application as it has some hypotheses, it is an interesting direction that integrates the IARC model into the IPS, which is commercially used in factories. By combining the inspection resources in the industrial field, the model can provide real-time inspection planning solutions, which can support the decision-making process.
This paper researched inspection allocation problems considering inspection resources. The main contributions are as follows: Based on the characteristics of the FMS, an IARC model is proposed to analyse the possibility and probability value of the product in the product flow by using the inspection capability model. The IARC mixed-integer programming model is built to trade-off the total cost, the load of the inspection station. To solve the multi-objective optimization model, the NSGA-II algorithm is improved. A four-layer coding scheme, the improved elite strategy, and mutation operators are integrated into NSGA-II. The neighborhood research capability of the methods is enhanced so that the solutions do not fall into the first layer of the Pareto solution prematurely. Finally, the simulation experiment was conducted. The result verified the effectiveness of the IARC model, the performance of the modified NSGA-II in terms of diversity and convergence.
The data used to support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare no conflicts of interest.
This research was supported by New Intelligent Manufacturing Model Research Program of the Ministry of Industry and Information Technology (Grant No. NISM20180123), the National Civil Aircraft Digital Manufacturing Project (Grant No. GXMJ201503A009), and Beijing digital manufacturing key laboratory project (Grant No. BDMLBH2016002).