Investigation on distributed scheduling with lot-streaming considering setup time based on NSGA-II in a furniture intelligent manufacturing

Abstract

Distributed flexible flowshop scheduling is getting more important in the large-scale panel furniture industry. It is vital for a higher manufacturing efficiency and economic profit. The distributed scheduling problem with lot-streaming in a flexible flow shop environment is investigated in this work. Furthermore, the actual constraints of packaging collaborative and machine setup times are considered in the proposed approach. The average order waiting time for packaging and average order delay rate is used as objectives. Non-dominated sorting method is used to handle this bi-objective optimization problem. An improved encoding method was proposed to address the large-scale orders that need to be divided into sub-lots based on genetic algorithm. The proposed approach is firstly validated by benchmark with other multi-objectives evolutionary algorithms. The results found that the proposed approach had a good convergence and diversity. Besides, the influence of the proportion of large-scale orders priority level and sub-lot size was investigated in a panel furniture manufacturing scenario. The results can be concluded that the enterprise could obtain shorter order average waiting time and delay rate when the sub-lot sizes were set as two and the order priority level was allocated in the proportion of 1:2:3:4:5.

Keywords

Distributed flexible flow shop scheduling Panel furniture manufacturing Lot-streaming Packaging collaborative Setup time

1 Introduction

The furniture industry had been developing rapidly since it turned from a traditional handicraft to mechanical automation while keep upgrading high-end equipment and employing high-level trained workers [1]. With the growing demand for customised furniture from customers, enterprises normally sorted many orders together and divided into many sub-lots for increasing production efficiency. In this period, those enterprises that did well in the collection and standardized management of information throughout the production process by means of a Manufacturing Execution System (MES) stand out in the competition [3, 4]. But they also found that an MES can not do well in the optimization of resource allocation during the production process [5]. Especially for the optimization of essential production scheduling problems [6], there is no universal solution that an MES can propose in the panel furniture industry. Therefore, it is important and promising that a further investigate into the panel furniture production scheduling problem according on the application of MES.

The main operation of panel furniture manufacturing is sawing, edge-banding, drilling and packaging [7]. At each stage, there are numbers of identical or non-identical parallel machines. Each machine can only process one workpiece at a time. Each workpiece has to be processed in the same order, but not all of them have to be processed at each stage. For example, not all workpiece needs to go through the drilling stage. Once a workpiece finished the drilling stage, it can not be directly progressed to the packaging stage until the other workpieces which belong to the same order finished drilling. In addition, there is normally more than one factory to produce a lot of orders. Insofar, a production scheduling problem in large-scale panel furniture industry can be regarded as distributed scheduling problem considering packaging collaborative in a flexible flowshop environment.

In order to handle this large-scale and complicated scheduling problem, amount of orders is normally divided into many sub-lots, which is called lot-streaming, to make the production time shorter [8]. The key is how to determine the sub-lot size form amount of orders. Besides, machine setup times normally exist in real-world production [9]. For example, a workpiece prepares to be processed at the edge-banding stage. If its previous workpiece scheduled on the same machine has a different colour or thickness, it will take time to adjust the glue unit or cutting unit to fit its demand. In summary, the panel furniture production scheduling in this case can be regarded as an integrated problem which consists of distributed flexible flowshop scheduling (DFFS) with lot-streaming considers packaging collaborative and setup time.

A DFFS problem with lot-streaming can be classified into a non-deterministic Polynomial-time hard (NP-hard) problem which closely related to applications [10], many heuristic or meta-heuristic algorithms for this problem have been studied [11-13]. For example, the factory allocation rules based on Nawaz-Enscore-Ham (NEH) heuristic were designed to arrange jobs between factories [14]. Chen designed an encoding method based on two segments of code to address the sub-lot size of lot-streaming scheduling using in the genetic algorithm (GA) [15]. Regarding the packaging collaborative scheduling problem, Guo [16] designed a decision framework based on machine learning to produce collaborative scheduling decisions in packaging products. Despite these studies into various individual factors, there is not a solution yet to combine all these individual factors for efficient panel furniture manufacturing.

This paper proposes a hybrid global production scheduling approach for large-scale panel furniture manufacturing by integrating all the practical constraints and considerations. This is implemented by proposing a simple encoding method for individuals that can address the large-scale sub-lot sizes and jobs sequence problem in non-dominated sorting genetic algorithm II (NSGA-II). Through ranking their results of average order waiting time for packaging and order delay rate by non-dominated sorting to search the optimal solutions for searching the optimal combination of algorithm parameters.

The approach has been applied to a DFFS problem with lot-streaming that considers packaging collaborative and setup times in a large-scale panel furniture manufacturer. The results showed the proposed encoding method has a good ability to handle the large-scale lot-streaming problem and a fast speed in searching the optimal solutions. The contribution of this work is threefold: 1) proposing a simple encoding method for individuals to address large-scale lot-streaming problems, 2) proposing a furniture production scheduling system with packaging collaborative and setup time which can effectively handle the contradiction between orders collaborative production and orders delay in a DFFS manufacturing environment, and 3) proposing the optimal combination of different proportions of priority level and sub-lot sizes of orders in a large-scale panel furniture manufacturing enterprise.

The rest of this paper is organised as follows. Section 2 introduces the literature review about the DFFS problem with real-world restrictions. Section 3 details the framework of a DFFS scheduling problem and propose approach to this problem. Section 4 firstly improved the performance by optimization of algorithm parameters, then validated by a large-scale instance and evaluated by a panel furniture enterprise cases. The paper is concluded in Section 5.

2 Literature review

The problem studied in this work is a DFFS problem with lot-streaming that considered packaging collaborative and setup time.

Salvador [17] was among the first who proposed the flexible flowshop scheduling problem (FFS) which presented the flow shop scheduling problem with parallel machines. In order to address the scheduling problem with parallel factories under the context of collaborative manufacturing, the DFFS problem was introduced, and it brings many actual sub-problems, such as lot-streaming and packaing collaborative. Naderi [18] proposed six mixed-integer linear program (MILP) models to address a distributed scheduling problem. Furthermore, he proposed two simple factory allocation rules and add them to the six previous models to address them. But in DFFS, different machining speed of each factory was introduced [8], and with the increase of problem scale, traditional methods mentioned above proved that they can not always produce the optimal solution when the problem becomes more complicated. Therefore, evolutionary algorithms become more and more popular in this age.

Most large-scale panel furniture manufacturers need to concern the lot-streaming and machine setup time for increasing the machining efficiency and decreasing the changeover times of machines. And the packaging is another important factor which is the key to decreasing the flow time of jobs and increasing collaborative production [19]. Chen [15] developed an improved NSGA-II to address the multi-objective scheduling with lot-streaming problems. Later, Pan [8] designed a job-factory assignment strategy based on the NEH to improve the performance of the Teaching Learning based Optimization (TLBO) algorithm in solving the distributed flowshop scheduling with lot-streaming. Regarding the scheduling with machine setup times, it has been systematically investigated [20]. The setup time mainly derived from processing auxiliary operations which occupied the production time but without profit, such as adjusting processing parameter set. It has getting more and more important when integrating setup time with scheduling problems. Tao [21] studied a distributed scheduling problem considering lot-streaming and setup time in a flowshop environment, then, he developed constructive heuristics to search the optimal solutions of makespan.

Moon [22] proposed a mathematical model with minimizing delay time under the condition of allowing a job can be progressed to a different factory between operations in the distributed scheduling environment. An encoding method based on operation priority was designed in GA, and decoding the machines allocation with the first available machine to optimize the makespan. Regarding the distributed scheduling considering packaging, it is similar to the distributed assembling scheduling problem. Jobs belonging to the same order need to be packed or assembled together after their processing. A distributed two-stage flow shop scheduling problem with an assembling machine at each stage was investigated [23]. In order to minimize both the makespan and average flow time, they converted it into single-objective optimization using the weighting method and proposed a hybrid GA strategy.

In general, although much literature has studied DFFS with different constraints, this work presents the following differences from previous studies: (1) lots of research proposed many encoding methods to handle lot-streaming scheduling problems. However, it is still desirable to propose an encoding method to solve large-scale problems; (2) most research on DFFS considered constraint of setup times, but the packaging collaborative was barely considered. It is essential to address it under the context of distributed production; (3) there were few studies focusing on the dimension of orders, such as different priority levels of orders being mixed in a production task.

3 Approach

3.1 Problem description

A typical furniture manufacturer is normally mixed customer orders together firstly, and then divides them into sub-lots for production. They have distributed factories to perform the operations of sawing, edge-banding, drilling and packaging in order. There are setup times only at edge-banding stage, and the setup times varies from different colour and thickness of workpiece. And a workpiece can not be passed to another factory for processing before packaging. Machines at the same stage may not be identical. This means the processing time for a given task by different machines can be different at the same stage. The intermediate storage is regarded as infinite in this investigation. Thus, the target manufacturing environment involves factories assignment, sub-lot sizes determination and jobs sequence at each factory in a flexible flowshop with edge-banding machine setup times.

The mixing orders is normally named orders pool, denoted as O_i, i = {1, 2, ⋯ , p}. It can be divided into sub-lots, denoted as Sl_i,q, q = {1, 2, ⋯ , z_i}. There are workpieces in one sub-lot, the processing of a workpiece is called a job, denoted as J_i,q,j, j = {1, 2, ⋯ , n_q}. The cutting, edge-banding, drilling and packaging stages in order are represented as M_s, s = {1, 2, 3, 4}. Each stage can be processed by a set of machines m_s,l, l = {1, 2, ⋯ , L_s}, where l represents the l-th machine at stage s. An operation O_i,q,j,s represents the processing of job J_i,q,j at stage M_s. Each job J_i,q,j is required to carry out the same production sequence but not all stages, and it can only be processed by a single machine m_s,l at a time. A job J_i,q,j at stage M_s, s ∈ 2, 3, 4, can only be started when the upstream stage M_s-1 has been completed. In this work, P (O_i,q,j,s), S (O_i,q,j,s) and C (O_i,q,j,s) are used to denote the processing time, the starting point of the processing, and the completion point of the processing for job J_i,q,j at stage M_s, respectively. Besides, one order O_i can start packaging only if all jobs within it finished drilling operation. A technical roadmap was presented in Fig. 1.

Fig. 1

A technical roadmap of this work.

3.2 Mathematical formulation

3.2.1 Factory and job assignment strategy

In order to make a schedule of operations, the possible starting and completion times of each job J_i,q,j at stage M_s need to be determined, denoted by S′ (O_i,q,j,s) and C′ (O_i,q,j,s), respectively. Given the processing time of P (O_i,q,j,s), the following holds: C′ (O_i,q,j,s) = S′ (O_i,q,j,s) + P (O_i,q,j,s).

NSGA-II was used to generate a stochastic job sequence for the first stage, including sub-lots number and its sequence, and job numbers of one sub-lot and job sequence within it, the detailed discussion was presented in Section 3.3.1. Then, a job was always assigned to a factory with the lowest machine utilization rate. If there are more than one factory has the same machine utilization rate, then the factory with shortest makespan can be assigned. After the factory assignment, jobs sequence at each factory can be described as follows: first job comes to the next stage will be first served (FCFS). It is also assumed that a job is always be assigned to a machine that first available at that stage.

3.2.2 Estimation of makespan with setup times

The start processing time of a job J_i,q,j at stage M_s only depends on its completion time at previous stage and the first available time of machines at the next stage. Therefore, the possible completion time of the first job at first stage under the sequence generated from NSGA-II, denoted as j₁, j₂, ⋯ , j_N, $N = \sum_{i = 1}^{p} \sum_{q = 1}^{z_{i}} \sum_{j = 1}^{n_{q}} J_{i, q, j}$ : C′ (O_j₁,1) = P (O_j₁,1).

The positive edge-banding setup time required only when the situation that colour or thickness of consecutive jobs scheduled on the same machine at edge-banding stage occurred. Otherwise the setup time set as zero. Denoting the ST_{j_k,j_y}, x ≠ y as the edge-banding setup time of the x-th job j_x arrived at the machine which just finished processing the job j_y. Therefore, The possible starting processing time of job j₁ at stage M_s, s = 2, 3, 4 can be expressed as follows: $\begin{matrix} S^{'} (O_{j_{1}, s}) = \max (C^{'} (O_{j_{1}, s - 1}), C^{'} (O_{j_{y}, s + 1}) \\ - P (O_{j_{1}, s})) + {ST}_{j_{1}, j_{y}} \end{matrix}$ (1)

Similarity, the possible starting processing time of job j_x at the sawing stage can be divided into two situations, one is that the job j_x is the very first job scheduled at the machine, the following holds: C′ (O_{j_x,1}) = P (O_{j_x,1}). Otherwise, it can be expressed as follows: $S^{'} (O_{j_{x}, 1}) = \max (C^{'} (O_{j_{x}^{'}, 1}), C^{'} (O_{j_{y}, 2}) - P (O_{j_{1}, s})),$ (2) where $j_{x}^{'}$ represents the previous job scheduled right before the job j_x on the same machine at the same stage.

Then, the possible starting processing time of job j_x at the each stage can be obtained from the Equation 1. In addition, the possible starting processing time of job j_x at the final stage can be expressed as follows: $S^{'} (O_{j_{x}, 4}) = \max (C^{'} (O_{j_{x}, 3}), C^{'} (O_{j_{x}^{'}, 4}))$ (3)

3.3 Manufacturing Bi-objective optimisation based on NSGA-II

NSGA-II is a classic population-oriented meta-heuristic algorithm that was adopted in this work. Non-dominated sorting is used to solve the bi-objective optimization in this work. In specific, non-dominated sorting was first used to classify all the solutions to different ranks according to the fronts. A crowding distance algorithm was then used to evaluate the quality of solutions from the same rank. And the best rank will be retained.

3.3.1 Encoding and decoding

The problem addressed here involves the lot-streaming and the jobs assigned to machines. Regarding lot-streaming, it is about the determination of sub-lot size and their processing sequence. Therefore, it is necessary to propose an encoding method when addressing large-scale problems. The first step is to initialize a population $ℙ$ of individuals I_s, s ∈ {1, 2, ⋯ , n′}, as illustrated in Fig. 2.

Fig. 2

An illustration of encoding method of an individual I_i with 3 orders.

The individuals have many genes which coded in random irrational numbers with the length of total sub-lots, denoted as Sb_i,q. First of all, denoting the total job numbers in order O_i as Q_i. Then, the up-limits of sub-lots which are divided from an order hold the following rule: A sub-lot needs at least two jobs as it is. Therefore, an order at most can be divided into Q_i/2 numbers of sub-lot. In addition, the job number of the q-th sub-lot in order O_i, denoted as JQ_i,q, can be determined by the following expression: ${JQ}_{i, q} = \frac{{Sb}_{i, q}}{\sum_{q = 1}^{z_{i}} {Sb}_{i, q}} \times Q_{i}$ (4)

Another piece of information is the sub-lot sequence which can be accessed too. In specific, it can be sorted by ascending gene Sb_i,q of an individual I_s. The sub-lot size and sequence have been obtained insofar. The last step is to determine the job sequence of a sub-lot. The job sequence of a sub-lot is obtained by a series of random irrational numbers with the length of JQ_i,q. In general, the sub-lot size and sequence, as well as the job sequence within a sub-lot, can be encoded by a series of random irrational numbers. The algorithm will either terminate or go through another cycle of reproduction at this stage depending on the quality of fitness value in comparison to the best fitness value in the past iterations. If the overall improvement of the fitness value of the best individual in comparison to the average of those in the past iterations is not significantly changed, the algorithm will terminate and the best individual will be returned as the final solution for scheduling; otherwise the algorithm will return to crossover operation. In order to avoid potential non-stopping running of the algorithm, a maximum number of generations is also applied and the algorithm will terminate if the current iteration is reached regardless of the quality of the result.

3.3.2 Fitness function

There are two objective functions simultaneously optimized by non-dominated sorting in this work, including the average order waiting time for packaging and average order delay rate.

The average order waiting time for packaging describes as follows: it accelerated from the first job of the order O_i finished its drilling process to the last job. Then, this order O_i can be assigned to the packaging stage. Denoting the operation of job in order O_i at stage M_s as O_i,u,s, u = {1, 2, ⋯ , N_i}. N_i means the maximum number of jobs in order O_i, it holds: $N_{i} = \sum_{q = 1}^{z_{i}} \sum_{j = 1}^{n_{q}} J_{i, q, j}$ . The aim is to minimize the average waiting time of all orders for packaging. Therefore, the first objective function can be expressed as follows: $f_{1} = \frac{\max (C^{'} (O_{i, u, 3})) - \min (C^{'} (O_{i, u, 3}))}{\sum_{i = 1}^{p} O_{i}}$ (5)

There are normally due date for each order, denoted as interval $[d_{i}^{-}, d_{i}^{+}]$ . Denoting the possible completion time at packaging stage of order O_i as C′ (O_i), it holds as follows: C′ (O_i) = max (C′ (O_i,u,4)). Denoting an index O_d as the number of orders finished between due date interval. The initial value of O_d is set as 0. Therefore, if order O_i finished processing between the due date interval, O_d can be expressed as follows: $O_{d} = O_{d} + 1, if C^{'} (O_{i}) \in [d_{i}^{-}, d_{i}^{+}]$ (6)

Therefore, the second objective function, average order delay rate, can be expressed as follows: $f_{2} = \frac{\sum_{i = 1}^{p} O_{i} - O_{d}}{\sum_{i = 1}^{p} O_{i}} \times 100 %$ (7)

4 Experimentation

The convergence and diversity of NSGA-II was evaluated by performance criteria. Second, the sensitivity analysis of proposed approach was verified by an instance from large-scale real-world panel furniture manufacturer.

4.1 Performance criteria

According to the previous literature, two following metrics are commonly adopted to evaluate the proposed NSGA-II algorithm.

Inverted Generational Distance (IGD): Given a set A as the solution of Pareto front obtained, and set Z as the objective solution set. Then, the IGD metric can be expressed as follows:

IGD (A, Z) = \frac{1}{∣ Z ∣} \sum_{i = 1}^{z} \min_{j = 1 to ∣ A ∣} d (z_{i}, a_{j}),

(8) where d (z_i, a_j) =∥ z_i - a_j ∥. The smaller the value of the IGD indicator, the better the convergence and distribution of solutions to and over the Pareto front when a set of well-distributed reference points is used.

Hypervolume (HV): It indicates the volume of the region in the target space enclosed by the non dominated solution set obtained by the algorithm and the reference point. Then, the HV metric can be expressed as follows:

HV (A) = \sum_{i = 1}^{NOb} \sum_{j = 1}^{NOp} \frac{{pt}_{i, j} - 1.2 \cdot \min_{i} F}{1.2 \cdot (\max_{i} F - \min_{i} F)},

(9) where

F

represents a set of Pareto frontier,

A \in F

, and pt ∈ X′ a point belonging to the frontier. NOb means the number of objective and NOp means the number of point in A.

\min_{i} F

and

\max_{i} F

are the best and worst values of objective i over all the frontiers in

F

. The larger the value of the HV indicator, the better the comprehensive performance of the algorithm.

4.2 Performance testing

“DTLZ3” was selected as instance to test the proposed approach, and it is used to test the ability to converge to the true Pareto optimal front [24]. Furthermore, ɛ-MOEA, MOEA/D, PESA-II and SPEA2 were selected as compared multi-objective evolutionary algorithms (MOEAs). In order to test the convergence speed of proposed approach with other MOEAs under the same parameter set, the number of objectives were set as 3. The crossover rates were all set as 1, and the mutation rates were set as 0.1. The testing results of IGD and HV were showed in Table 1.

Table 1
Orthogonal experimental design in this work

Exp. no. NSGA-II ɛ-MOEA MOEA/D PESA-II SPEA2

HV IGD HV IGD HV IGD HV IGD HV IGD

1 0.4145 0.0738 0.3905 0.0715 0.1833 0.3528 0.3778 0.0724 0.3766 0.0505

2 0.4087 0.0691 0.3936 0.0672 0.3506 0.0782 0.3751 0.0776 0.3745 0.0514

3 0.4108 0.0695 0.3819 0.0735 0.3474 0.0835 0.3856 0.0653 0.3680 0.0518

4 0.3972 0.0710 0.3943 0.0666 0.3356 0.1091 0.3865 0.0682 0.3823 0.0528

5 0.4105 0.0692 0.3483 0.0801 0.3425 0.0795 0.3851 0.0753 0.3690 0.0523

6 0.4117 0.0675 0.3837 0.0743 0.2914 0.1565 0.3709 0.0711 0.3746 0.0523

7 0.4107 0.0672 0.3653 0.0731 0.3510 0.0784 0.3820 0.0765 0.3690 0.0518

8 0.4101 0.0685 0.3804 0.0751 0.3333 0.0825 0.3519 0.0868 0.3776 0.0511

9 0.4093 0.0690 0.3722 0.0716 0.3169 0.1105 0.3835 0.0725 0.3560 0.0525

10 0.4127 0.0686 0.4014 0.0642 0.1982 0.3435 0.3966 0.0717 0.3716 0.0519

11 0.4126 0.0683 0.3629 0.0732 0.3481 0.0819 0.3807 0.0926 0.3782 0.0511

12 0.4119 0.0670 0.3891 0.0734 0.3406 0.1014 0.3924 0.0669 0.3770 0.051

13 0.4105 0.0694 0.3907 0.0753 0.3501 0.0790 0.3710 0.0723 0.3849 0.0516

14 0.4138 0.0736 0.3736 0.0760 0.3504 0.0905 0.3660 0.0763 0.3596 0.0514

15 0.4043 0.0671 0.3895 0.0714 0.1969 0.3477 0.3837 0.0702 0.3799 0.0526

16 0.4098 0.0728 0.3869 0.0764 0.3174 0.0971 0.3614 0.0804 0.3686 0.051

17 0.4066 0.0678 0.3883 0.0747 0.3458 0.0816 0.3790 0.0729 0.3757 0.0517

18 0.4091 0.0676 0.3484 0.0800 0.1898 0.3572 0.3844 0.0715 0.3725 0.0515

19 0.4107 0.0722 0.3936 0.0690 0.1890 0.3613 0.3748 0.0821 0.3651 0.0517

20 0.4110 0.0730 0.4013 0.0647 0.3006 0.1700 0.3923 0.0665 0.3762 0.0513

Average 0.4098 0.0696 0.3818 0.0726 0.2989 0.1621 0.3790 0.0745 0.3728 0.0517

Exp. no.	NSGA-II	ɛ-MOEA	MOEA/D	PESA-II	SPEA2
1	0.4145	0.0738	0.3905	0.0715	0.1833	0.3528	0.3778	0.0724	0.3766	0.0505
2	0.4087	0.0691	0.3936	0.0672	0.3506	0.0782	0.3751	0.0776	0.3745	0.0514
3	0.4108	0.0695	0.3819	0.0735	0.3474	0.0835	0.3856	0.0653	0.3680	0.0518
4	0.3972	0.0710	0.3943	0.0666	0.3356	0.1091	0.3865	0.0682	0.3823	0.0528
5	0.4105	0.0692	0.3483	0.0801	0.3425	0.0795	0.3851	0.0753	0.3690	0.0523
6	0.4117	0.0675	0.3837	0.0743	0.2914	0.1565	0.3709	0.0711	0.3746	0.0523
7	0.4107	0.0672	0.3653	0.0731	0.3510	0.0784	0.3820	0.0765	0.3690	0.0518
8	0.4101	0.0685	0.3804	0.0751	0.3333	0.0825	0.3519	0.0868	0.3776	0.0511
9	0.4093	0.0690	0.3722	0.0716	0.3169	0.1105	0.3835	0.0725	0.3560	0.0525
10	0.4127	0.0686	0.4014	0.0642	0.1982	0.3435	0.3966	0.0717	0.3716	0.0519
11	0.4126	0.0683	0.3629	0.0732	0.3481	0.0819	0.3807	0.0926	0.3782	0.0511
12	0.4119	0.0670	0.3891	0.0734	0.3406	0.1014	0.3924	0.0669	0.3770	0.051
13	0.4105	0.0694	0.3907	0.0753	0.3501	0.0790	0.3710	0.0723	0.3849	0.0516
14	0.4138	0.0736	0.3736	0.0760	0.3504	0.0905	0.3660	0.0763	0.3596	0.0514
15	0.4043	0.0671	0.3895	0.0714	0.1969	0.3477	0.3837	0.0702	0.3799	0.0526
16	0.4098	0.0728	0.3869	0.0764	0.3174	0.0971	0.3614	0.0804	0.3686	0.051
17	0.4066	0.0678	0.3883	0.0747	0.3458	0.0816	0.3790	0.0729	0.3757	0.0517
18	0.4091	0.0676	0.3484	0.0800	0.1898	0.3572	0.3844	0.0715	0.3725	0.0515
19	0.4107	0.0722	0.3936	0.0690	0.1890	0.3613	0.3748	0.0821	0.3651	0.0517
20	0.4110	0.0730	0.4013	0.0647	0.3006	0.1700	0.3923	0.0665	0.3762	0.0513
Average	0.4098	0.0696	0.3818	0.0726	0.2989	0.1621	0.3790	0.0745	0.3728	0.0517

Although the lowest average value of IGD was not obtained from NSGA-II, it still showed a feasible convergence and diversity. Besides, it is obviously that the NSGA-II obtained the highest HV value among others. The iterative performance comparison was presented in Fig. 3. It can be seen that the proposed approach based on NSGA-II has the fastest convergence speed. It can be indicated that the performance of NSGA-II is effective.

Fig. 3

The iterative performance comparison of different algorithms.

4.3 Instance test on lot-streaming in a furniture distributed flexible FlowFhop

4.3.1 Parameters in the furniture manufacturing environment

First introduced in the manufacturing environment, a furniture manufacturer has three factories as shown in Fig. 4. There are parallel machines at each stage. And jobs can not be transported to another factory before the packaging stage. This enterprise normally handles orders together and divided them into many sub-lots. This enterprise is double shift system, 10 hours in one shift, and 20 hours per day. There are five priority levels for different orders, and each level has different demands for order completion time. The order priority level and its demanding completion time can be seen in the Table 2. Apparently, level one has the highest production priority.

Fig. 4

Machine layout in this particular panel furniture factory.

Table 2

Five priority levels and its demanding completion time for different orders

Priority Level	1	2	3	4	5
Completion Time (H)	(0,20]	(20,40]	(40,60]	(60,80]	(80,100]

Setup time at the edge-banding stage was considered in this work. The value depends on the colour and thickness difference between consecutive jobs scheduled on the same edge-banding machine, and it can be seen in the Table 3. Zero in colour or thickness column means that there is no colour or thickness difference between consecutive jobs, therefore there is no setup time for them. Normally, the setup for colour and thickness can not be started together, and the colour setup time is much more than the thickness setup.

Table 3

The setup time (seconds) for colour and thickness

Colour	Colour	Thickness	Thickness	Total setup
	setup		setup	time
0	0	0	0	0
0	0	1	150	150
1	300	0	0	300
1	300	1	150	450

There are 75 orders approximately that need to be divided into sub-lots for production per day in that enterprise and 60 jobs at each order. One order can be divided into at least one sub-lot. Furthermore, the number of orders with different priority was not always the same. in order to investigate the proper distribution of priority orders, three proportions of levels can be seen in Table 4. The number in the bracket represents orders at that level. On the other hand, job processing time at each stage was based on expert knowledge and subject to discrete uniform, which followed [15,30], [90,300], [15,20] and [60,90], respectively. The algorithm parameters had the same settings with Section 4.2.

Table 4

Three order proportions with five priority levels

	Priority Level
Proportion	1	2	3	4	5
Proportion 1	1 (15)	1 (15)	1 (15)	1 (15)	1 (15)
Proportion 2	1 (5)	2 (10)	3 (15)	4 (20)	5 (25)
Proportion 3	5 (25)	4 (20)	3 (15)	2 (10)	1 (5)

4.3.2 Sensitivity analysis on the different distribution of priority orders

In order to investigate the effect of different distribution of priority orders in large-scale panel furniture manufacturing. The sub-lot size was subject to uniformly distributed [1, 5]. Table 5 showed the compared results of different proportions. The abbreviation of “NoS” means the number of Pareto solutions. At proportion 2, the lowest average orders delay rate was obtained around 13%. Proportion 2 and proportion 3 resulted in much worse than that of proportion 2. Especially in proportion 3, the average orders delay rate reached about 44%. However, the lowest average orders waiting time about 70000 seconds was resulted from proportion 3. Besides, it also had the lowest programme run time. In summary, distribution of proportion 3 could be performed well if the factory already had many delayed orders. It can greatly decrease the average orders waiting time fro packaging. But if there were not so many urgent orders, proportion 2 could be a good option for planning.

Table 5
Different proportions of priority levels in Orders

Proportion 1 Proportion 2 Proportion 3

Solutions f ₁ f ₂ f ₁ f ₂ f ₁ f ₂

1 106862 0.32 92776 0.13 69902 0.45

2 104608 0.33 91607 0.14 71814 0.43

3 100999 0.34 70475 0.44

NoS 3 2 3

Run Time (s) 5019 5132 4616

	Proportion 1	Proportion 2	Proportion 3
1	106862	0.32	92776	0.13	69902	0.45
2	104608	0.33	91607	0.14	71814	0.43
3	100999	0.34			70475	0.44
NoS	3	2	3
Run Time (s)	5019	5132	4616

4.3.3 Sensitivity analysis on the sub-lot sizes

Sensitivity analysis on the sub-lot sizes

In order to investigate the effect of sub-lot sizes on response, the sub-lot size was set as D = {1, 2, 3, 4, 5}. The optimal proportion was adopted as proportion 2, which obtained above. The testing results was presented in Table 6. The response was greatly different when the sub-lot sizes increase. It can be confirmed that the sub-lot size had a significant impact on the average orders waiting time delay rate. Furthermore, it is obviously that the optimal solutions were obtained when sub-lot size was two. It indicates that response can be greatly improved when given a specific sub-lot size based on the distribution of proportion 2. Meanwhile, only two Pareto solutions were obtained, but the programme run time was relatively short. Therefore, the optimal solution was obtained under the condition of D = 2 and proportion 2.

Table 6
Different sub-lot sizes divided from Orders

Solutions D = 1 D = 2 D = 3 D = 4 D = 5

f ₁ f ₂ f ₁ f ₂ f ₁ f ₂ f ₁ f ₂ f ₁ f ₂

1 31577 0.07 27354 0.07 138283 0.19 95525 0.18 165976 0.22

2 27883 0.11 30952 0.06 139246 0.19 95700 0.17 160629 0.23

3 28079 0.07 128330 0.20 133496 0.23

4 32079 0.06 125599 0.21

NoS 4 2 4 2 3

Run Time (s) 5176 5128 5284 5148 5117

Solutions	D = 1		D = 2		D = 3		D = 4		D = 5
1	31577	0.07	27354	0.07	138283	0.19	95525	0.18	165976	0.22
2	27883	0.11	30952	0.06	139246	0.19	95700	0.17	160629	0.23
3	28079	0.07			128330	0.20			133496	0.23
4	32079	0.06			125599	0.21
NoS	4	2	4	2	3
Run Time (s)	5176	5128	5284	5148	5117

5 Conclusion and future work

This study explored the distributed scheduling problems with lot-streaming that consider packaging collaborative and setup times in a flexible flow shop of a panel furniture manufacturer. In specific, the possible starting processing time when considering setup time at the edge-banding stage was first investigated. Then, it proposed an encoding method that can address sub-lot sizes and sequences simultaneously. The main conclusions can be drawn as follows:

(1) The packaging collaborative and setup times were encoded in the fitness function of NSGA-II, and its ability of convergence and diversity were validated by comparing some other MOEAs;

(2) Afterwards, the effect of proportions of order priority level and sub-lot size on the order average waiting time and delay rate were investigated separately. The results showed the optimal solution can be obtained by NSGA-II when the proportion was 1:2:3:4:5. It can guide the enterprise on how to make production planning of large-scale orders with different priority levels;

(3) Into two sub-lots that one order can be divided could be the optimal solution when minimising the order average waiting time and delay rate. Based on that, the enterprise could make a global optimization for large-scale orders with the proportion (1:2:3:4:5) with two sub-lots using NSGA-II.

As for the future research on this topic, we aim at (1) studying the heuristic method for packaging collaborative; (2) considering the strategy for production interrupted by emergency orders arrival and cancelled; (3) exploring coupling relationship among distributed flexible flowshop scheduling, packaging collaborative and just-in-time production in a dynamic environment.

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