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Abstract

Research on integrated process planning and scheduling (IPPS) is of great significance to the improvement of the overall quality of machinery manufacturing system. In the actual manufacturing process, the manufacturing system is often accompanied by some unpredictable uncertain disturbance factors, for instance uncertain processing time of jobs and changes of due date, etc. These uncertain disturbance events will ultimately affect production efficiency and customer satisfaction. Consequently, this paper considers the multi-objective IPPS problem with uncertain processing time and uncertain due date. A multi-layer collaborative optimization (MLCO) method is designed for the fuzzy multi-objective IPPS (FMOIPPS) problem, including three layers. For the process planning layer, the basic genetic algorithm is used to provide various near optimal process plans for the process selection system. For the process selection layer, a multi-objective genetic algorithm (MOGA) is designed to optimize the process selection population. A sharing function method is introduced to maintain population diversity. An individual comprehensive evaluation method is introduced to evaluate non-dominated solutions. The crowded distance, fast non-dominated sorting and elite strategy based on NSGAII is adopted in the proposed MOGA. The external archive method is employed to preserve the non-dominated solutions generated during population evolution. For the scheduling layer, a MOGA with a boundary search strategy is proposed. The boundary search strategy is designed to improve the search ability of boundary solutions. Three optimization objectives are minimizing the spread of fuzzy makespan, minimizing fuzzy makespan and maximizing average customer satisfaction simultaneously. The target of scheduling layer is to make scheduling arrangements for the process information obtained by process selection layer. Through mutual cooperation among each layer, guide the overall optimization process, and finally get satisfactory solutions. Different problem examples of various scales are employed to verify feasibility and effectiveness of the MLCO method. The experimental results indicate that the MLCO method can effectively address FMOIPPS problem.

Keywords

Integrated process planning and scheduling fuzzy multi-objective optimization sharing function boundary search strategy

Introduction

Process planning and scheduling are important links in manufacturing system which are inseparable for the development of advanced manufacturing industry.^1,2 With the development of computer technology, process planning and scheduling began to develop from artificial experience to precise planning and scientific decision-making of computer-aided. In the actual production, process planning and scheduling are closely linked. But, in traditional production, process planning and scheduling are often carried out separately, resulting in mutual constraints. The main performance is that process planning often causes conflicts and unreasonable allocation of production resources. Therefore, it is difficult to reach the best scheduling plan. Meanwhile, due to the influence of many uncertain factors in the scheduling process, the process plans often needs to be modified.^3,4 In the 1980s, some scholars proposed the integrated process planning and scheduling (IPPS) problem.⁵ After decades of development, solving IPPS has accumulated abundant theoretical and practical achievements.⁶

Many results of researches show that IPPS is of great significance to resolution resource conflicts, improvement of equipment utilization ratio and production efficiency.^7,8 The research of IPPS has become an effective way to solve the problems of workshop production management. IPPS is an expansion of shop scheduling problem, so the problem solution space is larger and more complicated. Process plan flexibilities, machine flexibilities, different jobs, different operations, and various resources and operations’ constraints should be considered simultaneously for solving IPPS problems.^9,10 The IPPS problem is non-deterministic polynomial (NP)-complete problem.^11,12 With the development of new information technology, many intelligent optimization algorithms are widely used in intelligent manufacturing,^13,14 a large number of swarm intelligent algorithms are utilized in IPPS, which provide a theoretical guidance for the realization of intelligent manufacturing.

With the in-depth study of related IPPS, new problems are constantly being discovered. Workshop production is a complex process, so solving IPPS problem can not only meet a single objective, but need to consider multiple objectives. Generally, it is necessary to consider production efficiency, due date and other objectives. In the production process of the workshop, tool loading and unloading, the proficiency of personnel in the workshop, the operating status of the equipment, the transmission status of the jobs are usually uncertain conditions.^15–17 Therefore, considering the uncertain factors in the IPPS problems is more in line with the actual production status. Especially, with the requirements of on-time and the development of intelligent manufacturing technology, it is vital to fully estimate the system uncertainty. Considering multi-objective and uncertain factors is helpful to make the research closer to the actual production situation.^18,19 But it also makes the IPPS problem more complicated. Therefore, it is a significant work to design an appropriate model and an effective solution method.

At present, there are many studies on multi-objective IPPS and uncertain IPPS. The weighted method and Pareto method are adopted to solve the multi-objective problem in the existing research. The uncertainty of the system is mainly expressed by fuzzy numbers, interval numbers, and random numbers, such as fuzzy due date, fuzzy processing time, etc.²⁰ It noticed that there are few researches IPPS considering uncertainty and multi-objectives at the same time. Therefore, in this paper, an effective layered algorithm is designed for the fuzzy multi-objective IPPS (FMOIPPS) problem. Process planning and workshop scheduling are both very complex problems. The IPPS problem is extended on the basis of process planning and scheduling, which makes the problem solving more complex. The FMOIPPS problem considers the uncertain factors and multi-objective of the production process on the basis of the IPPS problem, making the solution of the problem more difficult. Therefore, a multi-objective collaborative optimization (MLCO) method is designed to solve the problem in three steps, and a better solution is finally obtained through three levels of cooperation. The MLCO method including three layers of process planning layers, process selection layers and scheduling layers is designed. The basic genetic algorithm is utilized in the process planning layer to produce the process plans for jobs. In the process selection layer, considering the processing information of all jobs, a multi-objective genetic algorithm (MOGA) is designed to select the process plan for each job. In the scheduling layer, a MOGA with a boundary search strategy is proposed to assign process information for all jobs. For the multi-objective problem, the Pareto-based method is used, and the final result is a Pareto solution set. Finally, 32 sets problem examples of different scales are employed to test feasibility and effectiveness of the MLCO method.

The remainder of this paper is organized as follows: “Literature Review” section introduce the development of the IPPS problem. “Problem description” section elaborates the FMOIPPS problem settled in this paper. “The proposed MLCO method” section presents proposed MLCO method for fuzzy multi-objective IPPS. “Experiments and discussions” section, the experiments design and result analyses are given. The conclusions and future works are reported in last section.

Literature review

IPPS was first proposed by Chryssolouris et al.⁵ After decades of development, IPPS has accumulated vast theoretical and practical values. Li et al.^4,6 summarized the research results of IPPS problem and proposed the future development direction. At present, the research on IPPS mainly focuses on model establishment and solution method.

There are three types of IPPS problem modeling, non-linear process planning, closed loop process planning, distributed process planning.^6,8 Many models for IPPS problems have been established, Jain et al.²¹ adopted a two-module integration method to select the process plan through process planning and scheduling. Li et al.²² designed an agent-based IPPS integration method, which process planning and scheduling are executed simultaneously and use optimized agents based on evolutionary algorithms to manage the interaction and communication between agents. Phanden et al.³ proposed the integration of four modules: process plan selection module, scheduling module, schedule analysis module, and plan modification module. Chu et al.²³ proposed an integrated model of uncertain bi-level programming. The upper layer solves planning problem and the lower layer solves scheduling problems.

Over the years, the algorithm for solving IPPS problem has developed rapidly. For single-objective optimization, Seker et al.²⁴ designed a hybrid algorithm by combining GA and fuzzy neural network for the IPPS problem with makespan as the optimization objective. Li et al.²⁵ proposed an active learning genetic algorithm with makepan as the objective to solve the IPPS problem. Leung et al.²⁶ used makepan as the optimization objective, and proposed an ant colony optimization algorithm based on agent system to solve the IPPS problem. Zhang et al.²⁷ designed an objective-coding genetic algorithm solution with makepan as the optimization objective and achieved well results. Sobeyko et al.²⁸ adopted a heuristic algorithm based on variable neighborhood search with the goal of total weighted tardiness. The single-objective optimization algorithm has abundant accumulation, and the algorithm mainly used makespan as the optimization objective. Recently, the use of multi-objective algorithm to solve IPPS is increasing. Shokouhi et al.²⁹ constructed a mathematical model considering multiple factors to solve the complexity of IPPS and proposed a multi-objective genetic algorithm with three objective weights, makespan, critical machine workload and machines total workload. Li et al.³⁰ designed a novel hybrid algorithm for the multi-objective IPPS problem based on Nash equilibrium in game theory. Luo et al.³¹ proposed a multi-objective genetic algorithm based on the principle of immunity and external archive for the MOIPPS problem.

Due to the consideration of multi-objective to meet the actual production needs of the workshop, multi-objective algorithm has been widely used in solving the workshop scheduling problem. Li et al.³² studied the process and scheduling problems of welding workshop. Aiming at the WSSP problem in real life, a multi-objective mathematical model considering both energy efficient and production efficiency is proposed, and an effective multi-objective artificial bee colony algorithm is designed to solve the problem. Yin et al.³³ established a low-carbon scheduling model for flexible job shop scheduling problem, considering the three objectives of productivity, energy efficiency and noise reduction, and proposed a multi-objective genetic algorithm. Lu et al.³⁴ proposed a model of multi-objective dynamic welding scheduling problem and designed a multi-objective gray wolf optimization algorithm to solve the problem. Lu et al.³⁵ designed a multi-objective mathematical model considering both the maximum makespan and energy consumption for the flow shop scheduling problem, and adopted a hybrid multi-objective backtracking search algorithm. Consider multi-objective to make the production plan more in line with the requirements. So, the design of multi-objective algorithm is of great significance for the solution of practical problems.

Recently, uncertain factors have been extensive considered in various scheduling problems. Gao et al.³⁶ studied the scheduling problem of flexible workshops, established an FJSP model with uncertain processing time, the uncertainty of processing time is represented by triangular fuzzy number. Jamrus et al.³⁷ considering the uncertainty of process and equipment control process, fuzzy number is adopted to express the uncertainty of processing time, used a hybrid algorithm combining particle swarm optimization and genetic operators for flexible job shop scheduling problems. Lei³⁸ applied interval number theory to scheduling, established the scheduling model of interval number processing time, and it is proved by examples that the real makespan of each scheme is within the makespan interval indicated by the interval number. Lei³⁹ established the mathematical model of FJSP problem with fuzzy processing time, and aiming at the complexity of FJSP problem, an effective co-evolution algorithm is proposed. Hu et al.⁴⁰ proposed an uncertain job shop scheduling problem with fuzzy numbers, which represents the processing time and due date, and established a corresponding objective function to ensure that the optimal scheduling plan is obtained. According to the uncertain characteristics of the processing time in the remanufacturing workshop, Gao et al.⁴¹ use the fuzzy number to express the processing time and an effective discrete harmony search algorithm is designed to solve it. Gao et al.⁴² took the fuzzy processing time into consideration in flexible job shop scheduling, use the maximum fuzzy makespan as the objective, and an improved two-stage artificial bee colony algorithm is designed for FJSP with fuzzy processing time. Lu et al.⁴³ considering the uncertainty of processing parameters, studied the multi-objective FJSP with CPT problem and established the corresponding mathematical model. Finally, a new multi-objective discrete virus optimization algorithm is designed to solve this problem.

Gradually, uncertain factors have also been taken into account in the IPPS problem by some scholars. Zhang et al.¹⁵ adopted triangular fuzzy numbers to r express the processing time and transportation time of the machine, and solved distributed integration of fuzzy process planning and scheduling. Li et al.¹⁶ used interval numbers to represent the processing time, proposed an uncertain IPPS model, and designed a corresponding hybrid algorithm combining genetic algorithm and particle swarm algorithm. Wen et al.¹⁷ designed a multi-objective IPPS mathematical model with uncertain processing time and fuzzy due date, and designed an improved bee colony algorithm for the problem. Haddadzade et al.² used random numbers to represent the processing time, and designed a hybrid algorithm combining simulated annealing and tabu search. The research results show that taking multi-objective and uncertainty factors into account in the IPPS problem increases the difficulty of solving, but makes the problem closer to the actual situation, which is conducive to improving the quality of the scheme. It is very complicated for solving IPPS problems considering both uncertainty and multi-objectives, therefore, this paper design an effective method to solve FMOIPPS problems.

Problem description

The research of fuzzy set theory to deal with uncertain parameters of scheduling has gained wide attention in the field of scheduling. Trapezoidal fuzzy number and triangular fuzzy number are used to express the uncertainty of due date and processing time, respectively. The IPPS problem belongs to the field of shop scheduling problems, so this paper also applies fuzzy set theory to solve the FMOIPPS problem. In mathematical model, triangular fuzzy number and trapezoidal fuzzy number are used to represent fuzzy processing time and fuzzy due date respectively, and the fuzzy numbers operations are presented from Wen et al.¹⁷

The FMOIPPS problem studied in this paper can be described as follows: There is a set of jobs $J = {J_{1}, J_{2}, J_{3}, . . ., J_{n}}$ to be processed in a factory equipped with a set of machines $M = {M_{1}, M_{2}, M_{3}, . . ., M_{m}}$ . Each job has various machining features which satisfying certain precedence constraints. There are many optional operations for each feature of a job. Each process has multiple processing machines with precedence constraints among operations, however, the processing time and due date of each job are fuzzy numbers. Solving the FIPPS problem is to find a processing plan that satisfies multiple objectives at the same time based on the given resources and processing information.

The mathematical model of FMOIPPS and the fuzzy numbers operations can be found in Wen et al.¹⁷ In order to better describe the mathematical model, first define the following symbols.

n : the number of all jobs

J_i : the set of jobs.

$\tilde{C}$ : the fuzzy makespan of J_i

$ST$ : the average customer satisfaction

MS : fuzzy the spread of fuzzy makespan

C₁ : the best completion time of job

C₃ : the worst completion time of job

Fuzzy maximum makespan, fuzzy average customer satisfaction and the spread of fuzzy makespan are used as the optimization objectives for solving the FMOIPPS in this paper, the objective functions are given as follows:

f ₁: Minimizing fuzzy makespan

Min f_{1} = \tilde{C} = \max_{i \in [1, n]} \tilde{C}

(1)

f ₂: Minimizing average customer satisfaction

Max f_{2} = ST = \frac{1}{n} \sum_{i = 1}^{n} S T_{i}

(2)

f ₃: Minimizing the spread of fuzzy makespan

Min f_{3} = MS = C_{3} - C_{1}

(3)

In the above FMOIPPS problem model, the smaller the fuzzy makespan, the smaller costs and energy. The greater the average customer satisfaction, the greater the probability that jobs of on-time delivery. The evaluation method of customer satisfaction and the calculation method of average customer satisfaction were proposed by Wen et al.¹⁷ The shorter the span of the fuzzy makespan, the smaller the uncertainty of the fuzzy makespan, which is conducive to cost savings, reduced energy consumption, and improved on-time of completion time of jobs.

The following constraints also need to be taken into account.

Only one operation for each job must be processed simultaneously.

Only one operation can only be processed on each machine at a time.

Each job can only choose one process plan at a time.

Only one optional processing machine for each operation can be chosen.

The processing process is not limited by the buffer.

A simple example is given to better clarify the FMOIPPS problem in Table 1. Each job has different features with precedence constraints, some features have multiple operations companying operations sequences, each operation can be machined various machines with different fuzzy processing time, each job has different fuzzy due date. In Table 1, Job 1 includes 6 features, and the first feature F₁ must be machined before all features. The first and second features of job 1 have 2 alternative operations. The operations sequence of first candidate operations for the first feature F₁ have two operations O₁-O₂, however, the operations sequence of second candidate operations of F₁ contains only one operation O₃. There are various machines to process some operations. O₁ can be processed on two machines, which are M₁ and M₂. Moreover, different processing machines have different fuzzy processing times, and different processed jobs contain different fuzzy due date.

Table 1.

The processing information of 3 jobs processed in 10 machines.

Job	Features	Candidateoperations	Candidatemachines	Precedenceconstraints	Fuzzy process time	Fuzzy due date
J1	F ₁	O ₁-O₂	M₁,M₂	Before All	(7,8,9),(5,6,7)	(66,86,126,146)
			M₈		(6,7,8)
		O ₃	M₃		(11,14,16)
	F ₂	O ₄	M₈		(7,10,12)
		O ₅-O₆	M₁,M₂,M₃		(8,10,11),(5,8,12),(7,10,12)
			M₁,M₂,M₃		(9,13,15),(10,12,13),(11,14,18)
	F ₃	O ₇	M₆,M₇		(6,7,9),(5,8,9)
	F ₄	O ₈-O₉	M₁₀		(5,7,10)
		O ₁₀	M₁,M₂,M₃		(11,14,16),(8,12,18),(10,15,17)
	F ₅	O ₁₁	M₃		(10,13,16)
	F ₆	O ₁₂	M₁,M₂,M₃		(6,10,13),(8,10,12),(7,10,13)
J2	F ₁	O ₁	M₄,M₅	Before All	(5,8,10),(6,8,10)	(24,44,84,104)
	F ₂	O ₂	M₄,M₅		(15,18,20),(14,16,19)
		O ₃-O₄	M₁,M₇		(6,8,10),(7.8.9)
			M₁₀		(8,10,11)
J3	F ₁	O ₁-O₂	M₁,M₂,M₃	Before All	(5,6,8),(4,6,9),(6,8,10)	(151,171,211,231)
			M₈		(4,6,8)
	F ₂	O ₃-O₄	M₄,M₅	Before F₃	(8,10,12),(7,9,11)
			M₉		(10,13,15)
	F ₃	O ₅-O₆	M₆	Before F₄, F₅, F₆	(8,9,12)
			M₉		(11,12,15)
		O ₇-O₈	M₃		(10,13,16)
			M₉		(9,12,14)
	F ₄	O ₉	M₁,M₂,M₃	Before F₇	(12,14,16),(11,13,14),(14,16,18)
		O ₁₀-O₁₁	M₁,M₈		(7,9,12),(6,10,13)
			M₁₀		(6,8,11)
	F ₅	O ₁₂	M₁,M₂,M₃	Before F₆	(18,20,21),(15,17,22),(16,18,23)
		O ₁₃-O₁₄	M₂,M₇		(8,10,12),(7,10,13)
			M₁₀		(8,11,13)
	F ₆	O ₁₅	M₁,M₂,M₃		(10,13,15),(9,12,16),(11,12,15)
	F ₇	O ₁₆	M₁,M₂,M₃		(14,16,18),(16,18,20),(15,16,18)
		O ₁₇-O₁₈	M₂,M₇		(8,10,12),(7,11,14)
			M₁₀		(8,10,12)
	F ₇	O ₁₉	M₁, M₂, M₃		(20,22,25),(18,22,26),(20,22,24)
		O ₂₀-O₂₁	M₈,M₉		(7,9,12),(8,10,12)
			M₁₀		(12,14,16)

According to the processing information in Table 1, the processing time of the first process O₁ of the J₁ on the machine M₁ is (7, 8, 9), which means that the shortest processing time of O₁ is 7, the longest processing time is 9, and the processing time with the greatest possibility is 8. The due time of J₁ is (66, 86, 126, 146), which means that the earliest due time of the job is 66, the latest due time is 146, and the best due time is 86 to 126. When a scheduling plan is determined, the processing technology, process sequence and processing machine of each process will be determined. Through the fuzzy number summing operation, the fuzzy makespan of each job can be obtained by the operation of fuzzy number summation, and then calculate the value of the three objective functions. Get the final solution through comparison.

Proposed multi-layer collaborative optimization method for fuzzymulti-objective IPPS

Flowchart of MLCO

MLCO method is designed to address the FMOIPPS problem, including three layers. For the process planning layer, the basic genetic algorithm (GA) is applied to provide various near optimal process plans for the process selection system. For the process selection layer, a multi-objective genetic algorithm (MOGA) is designed to optimize the process selection population and select a process plan for each job, considering the overall machining process of all jobs. An individual comprehensive evaluation method is introduced to evaluate non-dominated solutions. A sharing function method is introduced to maintain population diversity. Based on the NSGAII algorithm, the basic elements of fast non-dominated sorting, elite strategy, crowed distance, and tournament selection is used in the optimization method. The external archive method is adopted to preserve the non-dominated solutions generated during population evolution. Minimizing processing time of all jobs and processing time with the maximum processing machine are selected as the optimization objectives. The goal of process selection layer is to combine optimization of the process plans of all jobs. For the scheduling layer, a MOGA with a boundary search strategy is adopted. The boundary search strategy is designed to improve the search ability of boundary solutions. Three optimization objectives are minimizing fuzzy makespan, maximizing average customer satisfaction and minimizing the spread of fuzzy makespan simultaneously. The target of scheduling layer is to make scheduling arrangements for the process information obtained by process selection layer. Through mutual cooperation among each layer, guide the overall optimization process, and finally get satisfactory solutions. The workflow of proposed MLCO method for FMOIPPS is shown in Figure 1.

Figure 1.

The workflow of MLCO method for FMOIPPS.

The main steps of MLCO method are as follows:

Step 1: Initialize process planning population.

Step 2: The optimized process plans is provided for each job through GA.

Step 3: Generate process selection population based on process plans’ information of jobs.

Step 4: Use MOGA to optimize the process selection population and select a process plan for each job, considering the overall machining process of all jobs.

Step 5: Initialize the scheduling population based on the processing information of jobs.

Step 6: Use a MOGA with a boundary search strategy to optimize scheduling population and generate the scheduling scheme.

Step 7: If the number of current generation reached, output the optimized solution. Otherwise, get on first step.

Methods for process planning layer

For the process planning layer, the basic GA is adopted to optimize the completion of each job. The process planning layer mainly provides the near optimal process plans for the process selection layer. Minimizing fuzzy processing time is selected as the optimization objectives. The workflow of GA for process planning layer is shown in Figure 2.

Figure 2.

The workflow of GA for process planning layer.

The detailed steps of population optimization in process planning layer as follows:

Step 1: Randomly initialize population for each job.

Step 2: Calculate the fitness value of each individual. Take the fuzzy processing time of job as individual fitness value.

Step 3: According to the fitness value, tournament selection is adopted to select parent individuals. The crossover operators and mutation operators proposed by Li et al.⁸ and were adopt to generate new population.

Step 4: Offspring selection: According to the comparison criteria of triangular fuzzy numbers,¹⁷ selecting individuals with small fuzzy processing time to generate new population $P_{1}$ .

Step 5: When evolutionary algebra is satisfied, go to Step 6. Otherwise, go to Step 2.

Step 6: Select near optimal process plans and input them to process selection layer.

Step 7: From Step 2 to Step 6 is repeat to complete the optimization of all jobs populations.

Encoding and decoding: The multi-dimensional encoding method and decoding method from Li et al.⁴⁴ is adopted for process planning populations. An individual consists of three chromosomes, there is a feasible encoding scheme is constructed in Figure 3. There are 3 chromosomes in total, the length of the features string in chromosome 1 and the alternative operations string in chromosome 2 is the number of the features. For the features string, the features string contains 4 features, the features sequence of first part in chromosome 1 is $F_{2} \to F_{3} \to F_{1} \to F_{4}$ . For the alternative operations string, the element 3 at the fourth position shows that the process 3 is selected as the candidate operation of the feature F₄. For the alternative machines string, the element 4 at the fifth position represents that operation O₅ chooses the candidate processed machine M₄. The length of the alternative machines string is the total number of all operations. The decoding method is presented in Li et al.⁴⁴

Figure 3.

Encoding of process planning layer.

Methods for process selection layer

The process selection is mainly responsible for providing process information for the scheduling system. The process planning population only generates process plan of a single job, but the scheduling is for all jobs. Therefore, it is important to consider the processing plan of each job base on processing information of all the jobs. The main purpose of this layer is to determine the processing information of all jobs. MOGA is designed to optimize the process selection population and select a process plan of each job based on the two indexes of the minimizing processing time of all jobs and processing time with the maximum processing machine. The MOGA algorithm adopts fast non-dominated sorting based on NSGAII, individual crowding strategy and elite retention strategy. The Pareto-based method is used to deal with multi-objective problems. Introduce the non-dominated solution obtained from the optimization process of external archive. In addition, the method of sharing function is used to reduce the individual fitness value in the solution dense area, reduce the possibility of producing offspring, and maintain the diversity of the population. A method of individual comprehensive evaluation is designed to select the branch solution to the external archive or to the new generation population. The workflow of MOGA for process selection layer is shown in Figure 4.

Figure 4.

The workflow of MOGA for process selection layer.

The detailed steps of population optimization in process selection population optimization layer are presented as follows:

Step 1: According to the information provided by process planning, initialize population $P_{2}$ randomly.

Step 2: Calculate the fitness value of individuals of P₂ population.

Step 3: Processing time of all jobs and the processing time with the maximum processing machine as evaluation indicators. The Pareto method and the triangular fuzzy number comparison method,¹⁷ non-dominated rank sorting is used to allocate dominant ranks to individuals in the population $P_{2}$ . Then update the fitness value of the individual by the sharing function.

Step 4: According to the two indexes of dominance level and population fitness value, the individual is selected as the parent from the population by using tournament selection, and the crossover and mutation operators designed by the alternative operation string are used to generate new populations.⁴⁴

Step 5: Process selection population $P_{2}$ and population $P_{2 new}$ into population $P_{2 merge}$ , and assign dominant levels to populations. Calculate the individual crowding degree, and then calculate the individual’s comprehensive score according to the individual comprehensive evaluation method.

Step 6: The external archive 1 is updated with the non-dominated solution of the population $P_{2 merge}$ and the individual’s comprehensive scores. If the number of non-dominated solutions is larger than the size of external archive, the individual with high comprehensive score should be selected.

Step 7: Generate a new generation of population, Firstly, the non-dominated solution is liberated into a new population, and then the new population is filled according to individual crowding degree. If the number of non-dominated solutions is greater than the population $P_{2}$ , it will be selected based on the overall score.

Step 8: If the evolutionary generation is satisfied, go to Step 9. Otherwise go to Step 2.

Step 9: Randomly select an individual from the non-dominated solution of the archive 1, input the revised process plan into the scheduling system.

Encoding and decoding scheme

The encoding scheme is shown in the Figure 5. Each individual contains a chromosome. The length of the chromosome is the number of all jobs. The element 3 in the first position indicates that the job 1 selects the process plan P₃. From the process selection string, the process plans selected for the 4 jobs can be decoded as P₃→P₂→P₅→P₄. The process information is provided by the process planning layer, the processing method and processing machine of each job can be obtained. Finally, the processing time of each job can be obtained.

Figure 5.

Encoding of process selection layer.

Fitness evaluation

Considering the processing time of all jobs and processing time with the maximum processing machine, the individual fitness value is calculated as follows:

F (x) = ω_{1} \frac{t (x) - t_{min}}{t_{max} - t_{min}} + ω_{2} \frac{s (x) - s_{min}}{s_{max} - s_{min}}

(4)

In the formula (4), t(x) is the processing time of all jobs, and s(x) is the processing time with the maximum processing machine, t_max and s_max represent the maximum value found, t_min and s_min represent the minimum value found, the average value of fuzzy number is used. $ω_{1}$ and $ω_{2}$ are weight coefficients, the value of both is 0.5.

Sharing function

The sharing function is a method to determine the decline in the fitness value of an individual according to how close the individual is to other individuals within a certain distance. As the fitness value decreases, individuals in densely populated areas are less likely to generate offspring. The detailed steps of the sharing function are described as follows:

Step 1: Select individual i in the population and calculate the regional range $Z_{ij} = f_{ij} \times rc$ of each objective for individual i, $f_{ij}$ is the jth objective function value of individual i, range factor rc is equal to 0.02.

Step 2: Calculate the difference $| Δ_{i j} |$ between each objective function value of individual i and individual k.

Step 3: Compare the regional range $Z_{ij} = f_{ij} \times rc$ of each objective value and $| Δ_{i j} |$ , and use variable n to record the number where $Z_{ij}$ is greater than $| Δ_{ij} |$ .

Step 4: Update the fitness value $fv = fv' \times n \times ac$ of individual i, $fv'$ is the individual fitness value of the previous state, fitness value attenuation coefficient ac is equal to 0.98.

Step 5: Repeat Step 2 and Step 4, and complete the comparison between individual i and all individuals in the population.

Step 6: Repeat the above steps to update the fitness value of all individuals in the population.

Individual comprehensive evaluation

Multi-objective optimization is usually to obtain a solution with better overall performance. In the population, individuals with better overall performance usually carry more valuable information. A comprehensive performance evaluation method based on ranking individuals is designed in this paper. According to the ranking of individuals, the problem of dimensional non-uniformity between the objective values can be ignored, and the impact of each objective on the individual performance can be considered. At the same time, the value of a single objective excellent individual is considered, and the weight of the first ranked individual need to be increased.

Suppose a population size n is 5, and the number m of optimization objective is 3. The individual populations are represented by vectors $\vec{a}$ (6, 4, 46), and the numbers represent the function values of the three optimization objectives. According to the function value of each objective, the individuals of the population are sorted separately, and the ranking of the individual under each objective can be obtained, as shown in Table 2. Calculate the comprehensive score of each individual $C = m \times (n + 1) - (r_{1} + r_{2} + r_{3}) + Z \times W \times (n + 1)$ , where m is the number of objective, n is the population size, $r_{i}$ is ranking under ith objective of the individual, Z is the number of times the individual ranks first, the weight coefficient W is 0.5. The individual’s comprehensive scores are shown in Table 2.

Table 2.

The ranking of individual individuals under different objectives.

Individual	Ranking of objective 1	Ranking of objective 2	Ranking of objective 3	Comprehensive scores
$\vec{a}$ (6,4,46)	1	1	5	17
$\vec{b}$ (9,57,23)	2	2	1	16
$\vec{c}$ (11,75,39)	3	4	3	8
$\vec{d}$ (16,64,31)	4	3	2	9
$\vec{e}$ (51,80,45)	5	5	4	4

Comprehensive performance is used to evaluate an individual’s overall performance. A higher score indicates that the individual’s overall performance is better. Conversely, if the score is lower, the individual’s overall performance is worse.

Update external archive

In this paper, the external archive method is adopted to preserve the non-dominated solutions generated during population evolution. The solution to update the external archive can be a non-dominated population solution or other archive solutions. The external archive is updated as follows:

Step 1: Combine the non-dominated solution of the population, or the solution of other archive with the solution of the current archive to be combined into a combined solution set S.

Step 2: According to the function value of the optimization objective for the population, according to Pareto definition and comparison method of triangular fuzzy number,¹⁷ fast non-dominated sorting is used to find the non-dominated solutions in the merged solution set S.

Step 3: Empty the current archive, liberate the non-dominated in the merged solution set S into the current archive.

Methods for job shop scheduling layer

The optimization of scheduling population is the most important part of the overall optimization process. It mainly completes the arrangement of processing sequence and processing time of jobs. A MOGA with a boundary search strategy is adopted. The objectives of scheduling optimization for FMOIPPS are minimizing fuzzy makespan, shorting spread of fuzzy makespan and maximizing customer satisfaction simultaneously. The algorithm used in the scheduling layer is based on the MOGA algorithm with a boundary search strategy. Boundary search is to perform genetic operations on non-dominated solutions generated in the optimization process to enhance the search capabilities of boundary solutions. The workflow of MOGA with a boundary search strategy for scheduling layer is shown in Figure 6.

Figure 6.

The workflow of MOGA with a boundary search strategy for scheduling layer.

The detailed steps of population optimization in scheduling layer are presented as follows:

Step 1: Initialize the scheduling population $P_{3}$ according to the process information provided by the process selection layer.

Step 2: Calculate the fitness value of population $P_{3}$ .

Step 3: Fuzzy makespan, spread of fuzzy makespan and average customer satisfaction are optimization indicators. Decode processing information of each individual, and solve the three objective values of the individual. According to the Pareto method and the triangular fuzzy number comparison method,¹⁷ non-dominated rank sorting is used to allocate dominant ranks to individuals in the population $P_{3}$ . Then update the fitness value of the population $P_{3}$ by the sharing function.

Step 4: According to the two indexes of dominance rank and population fitness value, the individual is selected as the parent from the population by using the tournament method. A new population is generated by adopt crossover operation and mutation operation proposed by Li et al.⁸

Step 5: Individual selection: merge population $P_{3}$ and population $P_{3 new}$ into population $P_{3 merge}$ , and assign dominant ranks for population $P_{3 merge}$ Calculate the individual’s crowding degree, and then calculate the individual’s comprehensive score according to the individual’s comprehensive evaluation method. Update the archive 3 with the non-dominated solution in the population $P_{3 merge}$ .

Step 6: If there is a solution in population $P_{3 merge}$ that dominates the solution in external archive 2, then update the external archive 2 with the non-dominated solution in the population $P_{3 merge}$ , and then go to Step 7. Otherwise, go to Step 8.

Step 7: Perform the boundary search.

Step 7.1: Each non-dominated solution from the archive 2 is randomly crossed with other non-dominated solutions according to POX operator or a crossover operator based on job to generate offspring, and these individuals and individuals in the external archive 2 are formed into boundary population $P_{4}$ . Clear the individuals in external archive 2.

Step 7.2: Calculate the fitness value of population $P_{4}$ .

Step 7.3: According to the method of scheduling optimization in Step 3, calculate the dominance level and update he fitness value of the population $P_{4}$ .

Step 7.4: According to the method of scheduling optimization in Step 4, generate population $P_{4 new}$ .

Step 7.5: Assign dominant ranks for population $P_{3 merge}$ , calculate the individual’s crowding degree, and then calculate the individual’s comprehensive score according to the individual’s comprehensive evaluation method. Update the external archive 3 with non-dominated solutions in the population $P_{4 new}$ . If the number of non-dominated solutions is greater than the size of the archive, then update the archive according to the individual’s comprehensive scores.

Step 7.6: If the evolution times are satisfied, then go to Step 7.8. Otherwise, carry on next step.

Step 7.7: According to the method of scheduling optimization in Step 8, generate a new generation of population $P_{4}$ and go to Step 7.2.

Step 7.8: Combine the population $P_{4 merge}$ and the scheduling population $P_{3 merge}$ into a population $P_{merge}$ , assign a dominance level to the population, calculate the individual crowding degree, place the individual into population $P_{3 merge}$ according to the two indicators, and go to Step 9.

Step 8: update the external archive 2 with the non-dominated solution in the population $P_{3 merge}$ .

Step 9: Liberate the non-dominated population from the population $P_{3 merge}$ into a new population, and then fill the new population according to the crowding degree. If the number of non-dominated solutions is greater than the population size, select individuals to generate a new population based on comprehensive scores.

Step 10: If the evolution times are satisfied, go to last step. Otherwise, go to Step 3.

Step 11: Output the solution of external archive 3.

Encoding and decoding scheme

Operation-based encoding method⁸ is used in this paper. As seen in Figure 7, the length of the scheduling string is the number of all the operations for jobs. The element in the scheduling string represents the jobs’ serial number. The element 2 in the first position indicates the first operation of job 2 and the number 2 in the 3rd position denotes the second operation of job 2. The greedy decoding method is adopted to decode the coding scheme into active scheduling,⁸ which can get the start time, end time of each job.

Figure 7.

Encoding of scheduling optimization layer.

Fitness evaluation

Three objective functions of minimizing fuzzy makespan f₁, maximizing average customer satisfaction f₂, and minimizing the spread of fuzzy makespan f₃ are selected for scheduling layer. Considering the conflict between objectives, the individual fitness value is replaced by the individual comprehensive evaluation score.

Boundary search

The global optimal solution of many optimization problems usually exists on the boundary of the feasible region,⁴⁵ especially the non-dominated solution generated by the evolution of the population, which often carries the information closest to the optimal solution in the current population. It is very necessary to mine the information of these solutions comprehensively. So the non-dominated solutions generated during the evolution of the scheduling population are randomly combined to generate the boundary search population and perform the genetic search in this paper. The operation flow of boundary search is as follows:

Step 1: The obtained non-dominated solutions are sequentially cross-operated to generate the initial boundary population.

Step 2: Calculate individual fitness value.

Step 3: Genetic operation to generate new individuals.

Step 4: Select individuals to form a new generation of population.

Step 5: If the evolutionary algebra is satisfied, go to step 6, otherwise return to step 2.

Step 6: Merging boundary populations and scheduling populations.

Experiments and discussions

Experiments

Due to lack of FMOIPPS benchmarks, 32 sets of examples with different sizes are designed to verify the proposed MLCO method for solving the FMOIPPS problems in Table 3. The processing information of an instance with 5 jobs processed in 5 machines is shown in Table 4.

Table 3.

32 sets examples of different scale.

Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbers of machine	Numbersof job
1	5	5	9	8	8	17	12	12	25	17	17
2	5	10	10	8	12	18	12	18	26	17	26
3	5	15	11	8	16	19	12	24	27	17	34
4	5	20	12	8	20	20	12	30	28	17	42
5	7	7	13	10	10	21	15	15	29	20	20
6	7	10	14	10	15	22	15	23	30	20	30
7	7	14	15	10	20	23	15	30	31	20	40
8	7	21	16	10	25	24	15	37	32	20	50

Table 4.

The processing information of 5 jobs processed in 5 machines.

Job	Features	Candidate operations	Candidate machines	Precedence constraints	Fuzzy process Time	Fuzzy due date
J1	F ₁	O ₁	M ₁,M₂	Before All	(30,32,34),(29,32,35)	(160 230 290 310)
		O ₂-O₃	M ₃,M₄		(16,17,19),(15,18,20)
			M ₅		(14,16,17)
	F ₂	O ₄	M ₁,M₂		(22,24,26),(21,23,27)
		O ₅-O₆	M ₃,M₄		(10,12,13),(11,13,14)
			M ₅		(11,13,15)
	F ₃	O ₇	M ₁,M₂		(15,17,18),(14,16,18)
	F ₄	O ₈-O₉	M ₁		(12,14,16)
		O ₁₀	M ₂		(14,16,18)
	F ₅	O ₁₁	M ₁,M₂		(26,28,30),(27,28,29)
	F ₆	O ₁₂	M ₃,M₄		(10,12,14),(11,13,15)
J2	F ₁	O ₁-O₂	M₁,M₂	Before All	(8,10,13),(9,12,15)	(170 240 280 300)
			M₃,M₄		(10,11,13),(7,9,12)
	F ₂	O ₃	M₃,M₄		(22,25,27),(16,18,20)
	F ₃	O ₄-O₅	M₁,M₂		(22,24,26),(18,20,23)
			M₅		(9,12,15)
		O ₆-O₇	M₃,M₄		(11,13,15),(14,16,17)
			M₅		(12,15,16)
	F ₄	O ₈	M₁,M₂	Before F₅	(11,13,16),(14,16,17)
		O ₉	M₃,M₄		(12,15,17),(11,15,16)
	F ₅	O ₁₀	M₁,M₂		(20,22,24),(21,23,25)
		O ₁₁	M₅		(20,23,25)
	F ₆	O ₁₂	M₁,M₂		(14,16,18),(9,12,14)
J3	F ₁	O ₁	M₃,M₄	Before All	(22,24,26),(21,23,25)	(200 230 280 310)
	F ₂	O ₂-O₃	M₃,M₄		(14,16,18),(15,17,19)
			M₅		(15,17,18)
		O ₄	M₁,M₂		(30,32,35),(29,31,34)
	F ₃	O ₅	M₁,M₂		(20,22,24),(21,23,25)
		O ₆-O₇	M₃		(13,15,17)
			M₅		(8,10,12)
	F ₄	O ₈	M₃,M₄		(12,15,17),(13,15,17)
J4	F ₁	O ₁	M₃,M₄		(30,32,34),(29,31,33)	(180 230 270 300)
		O ₂	M₁,M₂		(29,32,34),(30,32,34)
	F ₂	O ₃	M₁,M₂	Before F₄	(24,26,28),(20,22,24)
		O ₄-O₅	M₃,M₄		(11,13,15),(9,11,13)
	F ₃	O ₆	M₅		(12,13,15)
	F ₄	O ₇	M₁,M₂		(14,,16,18),(13,16,18)
	F ₅	O ₈	M₃,M₄		(7,9,11),(8,10,12)
		O ₉-O₁₀	M₁,M₂		(18,20,23),(15,17,20)
	F ₆	O ₁₁	M₃,M₄		(9,12,14),(7,9,12)
	F ₇	O ₁₂	M₅		(9,11,13)
		O ₁₃	M₁,M₂		(30,32,33),(28,30,31)
J5	F ₁	O ₁	M₃,M₄	Before All	(23,25,26),(22,24,26)	(210 260 290 310)
	F ₂	O ₂	M₁,M₂		(15,16,17),(14,16,18)
		O ₃	M₃,M₄		(15,17,18),(13,15,17)
	F ₃	O ₄	M₁,M₂		(9,12,14),(10,12,14)
	F ₄	O ₅	M₁,M₂	Before F₅	(23,25,26),(24,26,28)
		O ₆-O₇	M₃,M₄		(11,13,15),(10,12,14)
			M₅		(12,14,16)
	F ₅	O ₈-O₉	M₃,M₄		(16,17,18),(15,17,19)
			M₅		(16,18,20)
		O ₁₀	M₁,M₂		(32,34,26),(31,34,35)
	F ₆	O ₁₁	M₃,M₄		(12,14,16),(11,15,17)

MLCO, NSGAII and NSGAII+N5 three methods are used to solve 32 sets of cases with various scales. NSGAII and NSGAII+N5 as two control groups, adopting two-stage optimization methods, namely the process planning stage and the scheduling stage. The process planning stage provides near-optimal process plans for each job, and the scheduling stage arranges the processing sequence and processing time of the jobs. GA is used in the process planning stage, and the NSGAII algorithm, NSGAII algorithm with N5 neighborhood search algorithm are employed in the scheduling stage. The performance of the algorithms is evaluated from the convergence and divergence of the algorithm and the ability to find solutions.

The operating environment is as follows: the three algorithms is programmed by Visual C++, The computer performance is intel (R)Core(TM)i5-5000 CPU@2.00GHz. The parameters in the proposed algorithm are selected after a lot of trials. The selected parameters are listed in Table 5.

Table 5.

The parameters in the proposed algorithm.

Parameters	Process planning	Process section	Scheduling
Population size	50	50	100
Number of generations	30	50	100
Probability of mutation	0.10	0.10	0.20
Probability of crossover	0.80	0.80	0.80

Evaluation indicators of algorithm performance

Generation distance

Generation distance (GD) is used to express the convergence of the obtained solution set. The smaller the GD value, the closer the solution set obtained by the algorithm to the real Pareto optimal solution set. The GD indicator represents the distance between the known Pareto front end ( $P F_{known}$ ) and the real Pareto front end ( $P F_{ture}$ ), and its calculation formula is as follows:

GD = \frac{\sqrt{\sum_{i = 1}^{n} {d_{i}}^{2}}}{n}

(5)

In the formula (5), n is the number of solutions in $P F_{known}$ , d_i is the distance between the ith point in $P F_{known}$ and the nearest point in $P F_{ture}$ .

Inverse generation distance

Inverse generation distance (IGD) is used to simultaneously evaluate the convergence and diversity of the solution set. The smaller the IGD value, the better the algorithm performance. IGD is distance $P F_{known}$ and $P F_{ture}$ . The distance is calculated as follows:

IGD = \frac{\sqrt{\sum_{i = 1}^{m} {d_{i}}^{2}}}{m}

(6)

In the formula (6), n is the number of solutions in $P F_{ture}$ , d_i is the distance between the ith point in $P F_{known}$ and the nearest point in $P F_{ture}$ .

For each group of cases, three algorithms are used to run independently 20 times. The solutions obtained by each algorithm constitute their own $P F_{known}$ , and the un-dominated solution set obtained by combining the solutions of all algorithms constitutes $P F_{ture}$ . Due to the lack of reference points, the non-dominated solution set obtained by combining the non-dominated solutions found by the three algorithms is used as the reference point. Table 6 gives the reference points for experiment 1. The calculated results about average value of GD and IGD are shown in Table 7.

Table 6.

Reference point for experiment 1.

No.	f₁	f ₂	f ₃	No.	f₁	f ₂	f ₃
1	172 191 211	0.672595	39	22	167 183 198	0.572233	31
2	190 205 221	0.678966	31	23	172 190 205	0.645365	33
3	196 215 239	0.815735	43	24	189 205 219	0.659917	30
4	201 222 249	0.830688	48	25	188 205 220	0.743374	32
5	204 227 247	0.822495	43	26	191 208 224	0.768406	33
6	183 201 216	0.720141	33	27	199 221 240	0.807396	41
7	180 196 212	0.679272	32	28	198 219 237	0.799121	39
8	181 197 212	0.651345	31	29	200 217 236	0.795466	36
9	167 188 207	0.679068	40	30	198 217 236	0.788056	38
10	183 203 223	0.784299	40	31	189 213 229	0.805058	40
11	183 202 220	0.783083	37	32	185 201 214	0.555897	29
12	176 194 211	0.683004	35	33	182 201 219	0.726292	37
13	170 189 210	0.72971	40	34	202 227 247	0.81708	45
14	179 198 217	0.775451	38	35	175 190 207	0.639252	32
15	183 197 213	0.550062	30	36	176 195 212	0.706105	36
16	162 179 195	0.446773	33	37	180 195 211	0.642588	31
17	163 180 196	0.582375	33	38	184 211 232	0.806911	48
18	164 181 197	0.642284	33	39	170 188 206	0.645721	36
19	159 178 193	0.549509	34	40	176 195 211	0.693199	35
20	158 175 196	0.466723	38	41	184 203 218	0.725361	34
21	164 187 207	0.67371	43	42	184 204 218	0.760718	34

Table 7.

GD and IGD indicators’ average values of three algorithms.

Problems	NSGAII		NSGAII+N5		MLCO
	GD	IGD	GD	IGD	GD	IGD
5 × 5	0.68701	0.725167	0.523603	0.536922	0.412142	0.327987
5 × 10	1.22736	1.7426	1.11677	1.33619	0.225089	0
5 × 15	1.2357	1.52003	1.04367	1.05681	0.236786	0
5 × 20	0.48294	0.405397	0.495547	0.597183	0.0881965	0
7 × 7	0.312459	0.626532	0.296998	0.335882	0.292359	0.255411
7 × 10	0.80271	1.07006	0.832496	1.01293	0.271755	0.043436
7 × 14	1.49671	1.51279	1.44374	1.39721	0.397094	0
7 × 21	2.17336	2.94552	2.45204	2.38393	0.376514	0
8 × 8	0.91947	0.899956	1.14446	0.945515	0.545726	0.225922
8 × 12	1.39792	1.63744	1.37618	1.64094	0.487315	0
8 × 16	0.95386	0.850429	0.835708	0.824312	0.159151	0
8 × 20	0.22387	0.253008	0.272682	0.334704	0.0832265	0.0185052
10 × 10	0.410213	0.409253	0.320932	0.388652	0.153898	0.0557895
10 × 15	1.34111	1.47182	1.124	1.1203	0.4305	0
10 × 20	1.32736	1.63413	1.06918	1.62524	0.252991	0
10 × 25	1.17246	1.70079	1.60488	2.47439	0.379054	0
12 × 12	0.778431	0.953312	0.751103	1.00417	0.701647	1.13956
12 × 18	0.606994	0.565294	0.706244	0.508206	0.272215	0.0114631
12 × 24	0.938617	0.89353	0.802856	0.966715	0.2335	0
12 × 30	0.524848	1.21249	0.511267	1.4394	0.128325	0
15 × 15	1.12171	1.25053	1.23618	1.26972	0.469824	0.225706
15 × 23	1.26026	1.10755	0.951435	1.07226	0.189864	0
15 × 30	1.0849	1.17926	1.05238	1.73342	0.229914	0.312027
15 × 37	0.731565	0.594055	0.55088	0.687075	0.151416	0
17 × 17	1.09156	1.18831	1.11793	1.27613	0.413121	0.191345
17 × 26	1.44233	0.944394	0.985217	1.07193	0.212659	0
17 × 34	1.2975	1.95782	1.01511	1.97579	0.408189	0
17 × 42	0.634338	0.433055	0.953792	1.45164	0.264058	0.124295
20 × 20	1.98488	3.76583	1.70074	2.38991	0.681423	0
20 × 30	1.96863	2.53111	1.37009	1.8453	0.317016	0.164126
20 × 40	2.24895	2.75901	2.09016	2.36601	0	0
20 × 50	1.00646	2.25252	1.26481	2.51362	0.449539	0

The bold-faced values indicate that the values obtained by MLCO are better than the values obtained by NSGA-II and NSGA-II+N5.

According to the result in Table 7, the GD values obtained using the MLCO method are all less than the GD values obtained by NSGAII, NSGAII+N5, except for the 31 sets of experiments, the GD of other groups are less than 1 and mostly maintained at below 0.5, it shows that the solution set obtained by the MLCO method is closer to the real Pareto front end. Comparing the IGD values, in the 32 groups of experiments, when using the MLCO method, the IGD values of 31 groups of experiments were all less than the obtained IGD values by NSGAII, NSGAII+N₅, and the IGD value of 19 groups of experiments were 0. The experimental results indicated that the MLCO method can effectively solve FMOIPPS problems and demonstrate the better performance than NSGAII and NSGAII+N5.

Calculation results and discussion

1. The number and proportion of Pareto solutions.

The solution sets optimized by the three algorithms are combined, and the global non-dominated solution set is obtained according to the dominant relationship. Calculate the number and percentage of global non-dominated solutions for each algorithm. The results of Pareto solutions obtained by three algorithms are shown in Table 8.

Table 8.

The number and proportion of Pareto solutions obtained by three algorithms.

Problems	Paretosolutions	NSGAII		NSGAII+N5		MLCO
		Number	Proportion	Number	Proportion	Number	Proportion
5 × 5	43	7	16.3%	10	21.3%	26	60.5%
5 × 10	54	0	0	0	0	54	100%
5 × 15	66	0	0	0	0	66	100%
5 × 20	184	0	0	0	0	184	100%
7 × 7	96	27	28.1%	27	28.1%	42	43.8%
7 × 10	69	0	0	1	1.4%	68	98.6%
7 × 14	43	0	0	0	0	43	100%
7 × 21	41	0	0	0	0	41	100%
8 × 8	39	3	7.7%	6	15.4%	30	76.9%
8 × 12	45	0	0	0	0	45	100%
8 × 16	112	0	0	0	0	112	100%
8 × 20	330	12	3.63%	0	0	318	96.3%
10 × 10	139	5	3.6%	5	3.6%	129	92.8%
10 × 15	59	0	0	0	0	59	100%
10 × 20	63	0	0	0	0	63	100%
10 × 25	33	0	0	0	0	33	100%
12 × 12	25	8	32.0%	8	32.0%	9	36%
12 × 18	89	0	0	1	1.1%	88	98.9%
12 × 24	64	0	0	0	0	64	100%
12 × 30	117	0	0	0	0	117	100%
15 × 15	29	1	3.4%	0	0	28	96.6%
15 × 23	89	0	0	0	0	89	100%
15 × 30	67	6	9.0%	0	0	61	91.0%
15 × 37	113	0	0	0	0	113	100%
17 × 17	31	2	6.5%	0	0	29	93.5%
17 × 26	74	0	0	0	0	74	100%
17 × 34	36	0	0	0	0	36	100%
17 × 42	50	6	12%	0	0	44	88%
20 × 20	18	0	0	0	0	18	100%
20 × 30	42	1	2.4%	0	0	41	97.6%
20 × 40	44	0	0	0	0	44	100%
20 × 50	45	0	0	0	0	45	100%

The bold-faced values indicate that the values obtained by MLCO are better than the values obtained by NSGA-II and NSGA-II+N5.

According to the results in Table 8, the number of non-dominated solutions obtained by the MLCO method is more than the number of non-dominated solutions obtained by NSGAII and NSGAII+N5. In addition, as the scale of the problem increases, the number of non-dominated solutions obtained by the MLCO method becomes more and more. In the case of the same number of machines, with the increase of the number of jobs, the proportion of non-dominated solutions obtained by the MLCO method is also increasing.

2. Performance indicator of evaluating optimization results

accuracy = \frac{1}{m} \sum_{i = 1}^{m} | \frac{f_{i} - excellen t_{i}}{excellen t_{i} - worse n_{i}} |

(7)

In the formula (7), accuracy represents the optimization performance of optimization algorithm, m is the number of optimization objectives, f_i represents the number of the objectives. excellent_i and worsen_i are the best values and worst value respectively. The smaller the value of accuracy, the better the optimization result. The solution sets obtained by the three algorithms are combined, and the optimal value of each optimization objective is found from the combined solution set. The solution set is obtained from each algorithm, and the solution with the best comprehensive performance is selected as the optimal solution of the algorithm. The results are obtained according to the calculation formula of accuracy, as shown in Table 9.

Table 9.

The comparisons of results used three algorithms.

Problems	Best value			NSGAII				NSGAII+N5				MLCO
	f₁	f₂	f₃	f₁	f₂	f₃	accuracy	f₁	f₂	f₃	accuracy	f₁	f₂	f₃	accuracy
5 × 5	158 175 196	0.83068	29	180 196 212	0.67927	32	0.23443	175 190 207	0.63925	32	0.241258	164 181 197	0.64228	33	0.186269
5 × 10	234 259 282	0.74603	39	254 282 311	0.64436	57	0.184623	261 288 315	0.67022	54	0.182254	246 267 285	0.65155	39	0.116127
5 × 15	360 405 447	0.67577	60	410 446 484	0.47667	74	0.292726	378 413 446	0.50357	68	0.169397	373 407 440	0.53338	67	0.12769
5 × 20	513 569 631	0.76404	85	578 632 680	0.56961	102	0.25625	560 609 659	0.46149	99	0.29408	529 572 617	0.53504	88	0.16907
7 × 7	133 144 159	0.77399	22	155 166 180	0.45583	25	0.249411	149 159 171	0.38611	22	0.259482	136 148 160	0.21225	24	0.309035
7 × 10	165 185 205	0.74470	29	207 226 246	0.65969	39	0.219992	201 218 241	0.65324	40	0.199108	194 208 225	0.63644	31	0.166518
7 × 14	234 261 288	0.72974	41	264 284 315	0.56055	51	0.24201	271 295 324	0.62187	53	0.228658	248 270 293	0.65963	45	0.0934452
7 × 21	357 389 427	0.65537	59	391 430 469	0.47723	78	0.290136	391 424 463	0.51521	72	0.239884	359 390 425	0.61300	66	0.044966
8 × 8	163 183 211	0.71338	36	192 213 236	0.63156	44	0.239729	173 195 215	0.55820	42	0.185353	173 191 220	0.65592	47	0.103728
8 × 12	219 245 270	0.70822	43	270 299 327	0.61683	57	0.238189	263 287 313	0.50794	50	0.298415	244 268 291	0.63057	47	0.134483
8 × 16	277 309 338	0.68278	51	312 342 373	0.47765	61	0.253902	312 341 377	0.50837	65	0.233323	300 330 358	0.57656	58	0.141571
8 × 20	351 389 432	0.69706	68	387 430 469	0.36348	82	0.25107	374 411 453	0.32429	79	0.25650	369 403 440	0.33209	71	0.24148
10 × 10	173 201 233	0.74635	36	192 211 233	0.43595	41	0.243953	205 227 250	0.57657	45	0.201177	202 223 242	0.54253	40	0.207104
10 × 15	265 297 336	0.68805	48	277 305 335	0.50593	58	0.182428	286 315 343	0.52952	57	0.210212	277 300 335	0.61878	58	0.0812096
10 × 20	321 366 411	0.59370	59	357 393 432	0.47858	75	0.240217	346 384 422	0.47048	76	0.218507	338 370 400	0.50196	62	0.130202
10 × 25	381 428 480	0.72185	75	416 462 515	0.61944	99	0.179703	404 454 504	0.60545	100	0.167678	390 429 471	0.70896	81	0.0128431
12 × 12	141 161 185	0.70046	27	159 177 191	0.63746	32	0.13673	158 175 192	0.68339	34	0.085202	148 166 181	0.65109	33	0.0657602
12 × 18	187 206 229	0.70060	36	212 234 257	0.63234	45	0.182534	212 231 251	0.62015	39	0.179252	202 222 244	0.59366	42	0.162263
12 × 24	230 272 311	0.66220	44	275 299 323	0.54170	48	0.189755	259 285 310	0.43401	51	0.249385	252 274 296	0.49449	44	0.163615
12 × 30	298 348 395	0.66102	51	332 364 397	0.44801	65	0.208298	325 359 393	0.42462	68	0.21959	323 350 380	0.43407	57	0.198786
15 × 15	172 189 220	0.66876	32	204 220 236	0.57008	32	0.251727	194 214 237	0.60958	43	0.178535	181 202 220	0.64927	39	0.0603228
15 × 23	246 278 308	0.68985	45	280 308 336	0.50331	56	0.22394	270 302 329	0.51779	59	0.193411	273 297 322	0.59527	49	0.124621
15 × 30	330 370 414	0.45740	58	351 383 417	0.35794	66	0.142482	353 386 420	0.29205	67	0.226718	351 384 412	0.39712	61	0.0928578
15 × 37	403 451 498	0.68663	70	453 494 535	0.60851	82	0.179431	447 485 528	0.56197	81	0.217705	433 472 511	0.64013	78	0.0975689
17 × 17	166 184 208	0.63103	30	182 202 225	0.55906	43	0.152764	185 203 221	0.5049	36	0.206226	178 196 214	0.54798	36	0.130046
17 × 26	226 258 295	0.70097	39	272 297 320	0.62963	48	0.203587	262 289 315	0.56713	53	0.250615	252 276 295	0.58635	43	0.18106
17 × 34	303 335 368	0.70362	48	320 350 381	0.59051	61	0.159667	337 367 396	0.58434	59	0.201181	308 338 364	0.66560	56	0.045828
17 × 42	369 418 464	0.68017	65	397 432 469	0.63456	72	0.0993576	411 448 486	0.55347	75	0.246988	390 425 460	0.62710	70	0.0912583
20 × 20	204 226 245	0.52390	38	227 260 281	0.39713	54	0.250949	229 250 275	0.46886	46	0.144611	211 232 259	0.51890	48	0.0314414
20 × 30	320 340 380	0.61885	54	336 356 395	0.42616	59	0.279492	337 368 404	0.51523	67	0.188967	339 364 396	0.61848	57	0.0526256
20 × 40	418 479 541	0.63809	77	466 503 566	0.48805	100	0.274899	456 498 544	0.47917	88	0.24237	455 489 539	0.62258	84	0.0683443
20 × 50	518 601 666	0.60521	92	578 638 696	0.45926	118	0.227212	584 628 694	0.44279	110	0.240506	550 600 648	0.58021	98	0.0336005

The bold-faced values indicate that the values obtained by MLCO are better than the values obtained by NSGA-II and NSGA-II+N5.

As shown in the results of Table 9, the accuracy values of 32 sets problem examples are obtained by using the MLCO. It found that the accuracy values of 30 experiments were less than the accuracy values obtained by NSGAII and NSGAII+N5, and the accuracy values of 14 experiments were less than 0.1. It showed that the algorithm has a high performance of optimization. Table 10 is scheduling scheme for the problem 10 × 5.

Table 10.

The scheduling scheme of the problem10 × 5.

Jobs	Operations	Machines	Start time	End time	Job	Operations	Machines	Start time	End time
J1	O₁	M₂	0	29 32 35		O₃	M₄	218 239 258	231 254 275
	O₄	M₂	29 32 35	50 55 62	J6	O₆	M₂	59 67 76	77 87 99
	O₁₀	M₄	50 55 62	77 83 91		O₃	M₂	77 87 99	98 111 126
	O₇	M₁	123 132 139	138 149 157		O₂	M₁	200 216 231	209 227 244
	O₁₁	M₁	138 149 157	148 161 171	J7	O₆	M₅	0	15 17 18
	O₁₂	M₃	184 200 214	204 223 239		O₂	M₃	42 47 52	65 71 78
J2	O₁	M₁	0	8 10 13		O₁₃	M₃	65 71 78	80 89 98
	O₂	M₃	9 11 13	19 22 26		O₁₂	M₅	80 89 98	100 111 122
	O₉	M₃	80 89 98	92 104 115		O₉	M₁	112 120 126	123 132 139
	O₁₂	M₂	113 128 146	122 140 160		O₁₀	M₅	123 132 139	133 144 154
	O₆	M₃	158 170 181	169 183 196		O₁₅	M₅	133 144 154	146 159 172
	O₇	M₅	169 183 196	181 198 212		O₁	M₃	217 238 256	229 252 272
	O₃	M₄	181 198 212	197 216 232	J8	O₁	M₂	50 55 62	59 67 76
	O₁₁	M₅	197 216 232	217 239 257		O₂	M₅	59 67 79	74 83 93
J3	O₁	M₄	29 31 33	50 54 58		O₃	M₃	92 104 115	113 127 140
	O₈	M₃	113 127 140	125 142 157		O₅	M₁	184 198 211	200 216 231
	O₅	M₂	153 174 195	174 197 220		O₈	M₃	204 223 239	217 238 256
	O₄	M₂	227 247 265	256 278 299		O₉	M₅	217 239 257	231 255 275
J4	O₁	M₄	0	29 31 33		O₁₁	M₅	231 255 275	251 277 299
	O₁₁	M₁	29 31 33	59 63 66	J9	O₃	M₅	30 34 37	44 50 55
	O₈	M₂	98 111 126	113 128 146		O₈	M₅	44 50 55	54 62 70
	O₁₃	M₃	169 183 196	184 200 214		O₁	M₄	137 147 156	168 180 191
	O₃	M₂	184 200 214	204 222 238		O₄	M₂	204 222 238	227 247 265
	O₇	M₃	229 252 272	236 261 283	J10	O₁₂	M₃	0	9 11 13
	O₆	M₁	236 261 283	250 277 301		O₁₃	M₅	15 17 18	30 34 37
J5	O₁	M₃	19 22 26	42 47 52		O₃	M₁	82 88 92	112 120 126
	O₅	M₁	59 63 66	82 88 92		O₉	M₄	112 120 126	137 147 156
	O₁₁	M₄	82 88 92	93 103 109		O₂	M₃	137 147 156	158 170 181
	O₁₀	M₂	122 140 160	153 174 195		O₈	M₁	158 170 181	184 198 211
	O₄	M₁	209 227 244	218 239 258		O₁₁	M₅	184 198 212	196 213 229

The results of 32 sets of experiments show that the ability of the MLCO method to find the optimal solution is better than the other two methods. According to the data in Tables 7 and 8, it can be seen that the MLCO method has better optimization ability. The MLCO method is divided into three layers to solve the problem step by step. The process planning layer formulates near-optimal process plans for each job. The second layer considers all the processing tasks, selects the process plan for each job, and the third layer arranges processing time for all jobs. Compared with NSGAII and NSGAII+N5, MLCO method can make full use of the data information and processing information of the optimization process, and it also reduces the randomness of the optimization process. Especially with the expansion of the problem scale, reducing the randomness of the optimization is helpful to improve the ability to find the optimal solution. In addition, the MLCO method adopts a boundary search strategy in the third layer to enhance the search ability of boundary solutions. The results in Table 9 show that the accuracy value of the MLCO method is smaller, indicating that the quality of the non-dominated solutions obtained by the MLCO method is better. The individual comprehensive evaluation strategy is adopted in MLCO method to select the non-dominated solutions and improve the quality of non-dominated solutions. The experimental results show that the MLCO method proposed in this paper is effective for solving the FMOIPPS problem.

Conclusion and future works

In view of the complexity of actual production companying many uncertain factors, the diversity of demands and uncertain factors are comprehensively considered in this paper. For the multi-objective IPPS problem under uncertain conditions, the triangular fuzzy number is used to describe the uncertain processing time, and the trapezoid fuzzy number is adopted to indicate the uncertain due date. Minimize fuzzy makespan, maximize average customer satisfaction and minimize the spread of fuzzy makespan are planned as optimization objective. MLCO method is proposed to solve the FMOIPPS problem. The algorithm uses three layers of optimization methods including process planning layer, process selection layer, and scheduling layer. A sharing function strategy was adopted to maintain the population. A boundary search strategy is proposed to improve the search ability of the algorithm. An individual comprehensive evaluation strategy is proposed to selecting non-dominated solutions. The feasibility and effectiveness of the MLCO method are proved by 32 sets of examples with different scales. The results of the examples show that the algorithm has good practicability in solving uncertain multi-objective IPPS problems.

However, the actual workshop production process is often accompanied by many uncertainties, and more uncertainties should be considered in the IPPS problem. To solve the more complex and uncertain multi-objective IPPS problem and how to study the dynamic IPPS problem become the direction of further research in this paper.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research work is supported by the National Natural Science Foundation of China (grand no. 51905494 and 51775517), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (grand no. 19YJCZH185), Key Scientific and Technological Research Project in Henan Province (grand no. 202102210088).

ORCID iD

Xiaoyu Wen

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Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbers of machine	Numbersof job
1	5	5	9	8	8	17	12	12	25	17	17
2	5	10	10	8	12	18	12	18	26	17	26
3	5	15	11	8	16	19	12	24	27	17	34
4	5	20	12	8	20	20	12	30	28	17	42
5	7	7	13	10	10	21	15	15	29	20	20
6	7	10	14	10	15	22	15	23	30	20	30
7	7	14	15	10	20	23	15	30	31	20	40
8	7	21	16	10	25	24	15	37	32	20	50

Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbers of machine	Numbersof job
1	5	5	9	8	8	17	12	12	25	17	17
2	5	10	10	8	12	18	12	18	26	17	26
3	5	15	11	8	16	19	12	24	27	17	34
4	5	20	12	8	20	20	12	30	28	17	42
5	7	7	13	10	10	21	15	15	29	20	20
6	7	10	14	10	15	22	15	23	30	20	30
7	7	14	15	10	20	23	15	30	31	20	40
8	7	21	16	10	25	24	15	37	32	20	50

Multi-layer collaborative optimization method for solving fuzzy multi-objective integrated process planning and scheduling

Abstract

Keywords

Introduction

Literature review

Problem description

Proposed multi-layer collaborative optimization method for fuzzymulti-objective IPPS

Flowchart of MLCO

Methods for process planning layer

Methods for process selection layer

Encoding and decoding scheme

Fitness evaluation

Sharing function

Individual comprehensive evaluation

Update external archive

Methods for job shop scheduling layer

Encoding and decoding scheme

Fitness evaluation

Boundary search

Experiments and discussions

Experiments

Evaluation indicators of algorithm performance

Generation distance

Inverse generation distance

Calculation results and discussion

Conclusion and future works

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbersof machine	Numbersof job	Cases	Numbers of machine	Numbersof job
1	5	5	9	8	8	17	12	12	25	17	17
2	5	10	10	8	12	18	12	18	26	17	26
3	5	15	11	8	16	19	12	24	27	17	34
4	5	20	12	8	20	20	12	30	28	17	42
5	7	7	13	10	10	21	15	15	29	20	20
6	7	10	14	10	15	22	15	23	30	20	30
7	7	14	15	10	20	23	15	30	31	20	40
8	7	21	16	10	25	24	15	37	32	20	50